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Diffstat (limited to 'libtommath')
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diff --git a/libtommath/LICENSE b/libtommath/LICENSE new file mode 100644 index 0000000..5baa792 --- /dev/null +++ b/libtommath/LICENSE @@ -0,0 +1,4 @@ +LibTomMath is hereby released into the Public Domain. + +-- Tom St Denis + diff --git a/libtommath/TODO b/libtommath/TODO new file mode 100644 index 0000000..deffba1 --- /dev/null +++ b/libtommath/TODO @@ -0,0 +1,16 @@ +things for book in order of importance... + +- Fix up pseudo-code [only] for combas that are not consistent with source +- Start in chapter 3 [basics] and work up... + - re-write to prose [less abrupt] + - clean up pseudo code [spacing] + - more examples where appropriate and figures + +Goal: + - Get sync done by mid January [roughly 8-12 hours work] + - Finish ch3-6 by end of January [roughly 12-16 hours of work] + - Finish ch7-end by mid Feb [roughly 20-24 hours of work]. + +Goal isn't "first edition" but merely cleaner to read. + + diff --git a/libtommath/bn.ilg b/libtommath/bn.ilg new file mode 100644 index 0000000..3c859f0 --- /dev/null +++ b/libtommath/bn.ilg @@ -0,0 +1,6 @@ +This is makeindex, version 2.14 [02-Oct-2002] (kpathsea + Thai support). +Scanning input file bn.idx....done (79 entries accepted, 0 rejected). +Sorting entries....done (511 comparisons). +Generating output file bn.ind....done (82 lines written, 0 warnings). +Output written in bn.ind. +Transcript written in bn.ilg. diff --git a/libtommath/bn.ind b/libtommath/bn.ind new file mode 100644 index 0000000..e5f7d4a --- /dev/null +++ b/libtommath/bn.ind @@ -0,0 +1,82 @@ +\begin{theindex} + + \item mp\_add, \hyperpage{29} + \item mp\_add\_d, \hyperpage{52} + \item mp\_and, \hyperpage{29} + \item mp\_clear, \hyperpage{11} + \item mp\_clear\_multi, \hyperpage{12} + \item mp\_cmp, \hyperpage{24} + \item mp\_cmp\_d, \hyperpage{25} + \item mp\_cmp\_mag, \hyperpage{23} + \item mp\_div, \hyperpage{30} + \item mp\_div\_2, \hyperpage{26} + \item mp\_div\_2d, \hyperpage{28} + \item mp\_div\_d, \hyperpage{52} + \item mp\_dr\_reduce, \hyperpage{40} + \item mp\_dr\_setup, \hyperpage{40} + \item MP\_EQ, \hyperpage{22} + \item mp\_error\_to\_string, \hyperpage{10} + \item mp\_expt\_d, \hyperpage{43} + \item mp\_exptmod, \hyperpage{43} + \item mp\_exteuclid, \hyperpage{51} + \item mp\_gcd, \hyperpage{51} + \item mp\_get\_int, \hyperpage{20} + \item mp\_grow, \hyperpage{16} + \item MP\_GT, \hyperpage{22} + \item mp\_init, \hyperpage{11} + \item mp\_init\_copy, \hyperpage{13} + \item mp\_init\_multi, \hyperpage{12} + \item mp\_init\_set, \hyperpage{21} + \item mp\_init\_set\_int, \hyperpage{21} + \item mp\_init\_size, \hyperpage{14} + \item mp\_int, \hyperpage{10} + \item mp\_invmod, \hyperpage{52} + \item mp\_jacobi, \hyperpage{52} + \item mp\_lcm, \hyperpage{51} + \item mp\_lshd, \hyperpage{28} + \item MP\_LT, \hyperpage{22} + \item MP\_MEM, \hyperpage{9} + \item mp\_mod, \hyperpage{35} + \item mp\_mod\_d, \hyperpage{52} + \item mp\_montgomery\_calc\_normalization, \hyperpage{38} + \item mp\_montgomery\_reduce, \hyperpage{37} + \item mp\_montgomery\_setup, \hyperpage{37} + \item mp\_mul, \hyperpage{31} + \item mp\_mul\_2, \hyperpage{26} + \item mp\_mul\_2d, \hyperpage{28} + \item mp\_mul\_d, \hyperpage{52} + \item mp\_n\_root, \hyperpage{44} + \item mp\_neg, \hyperpage{29} + \item MP\_NO, \hyperpage{9} + \item MP\_OKAY, \hyperpage{9} + \item mp\_or, \hyperpage{29} + \item mp\_prime\_fermat, \hyperpage{45} + \item mp\_prime\_is\_divisible, \hyperpage{45} + \item mp\_prime\_is\_prime, \hyperpage{46} + \item mp\_prime\_miller\_rabin, \hyperpage{45} + \item mp\_prime\_next\_prime, \hyperpage{46} + \item mp\_prime\_rabin\_miller\_trials, \hyperpage{46} + \item mp\_prime\_random, \hyperpage{47} + \item mp\_prime\_random\_ex, \hyperpage{47} + \item mp\_radix\_size, \hyperpage{49} + \item mp\_read\_radix, \hyperpage{49} + \item mp\_read\_unsigned\_bin, \hyperpage{50} + \item mp\_reduce, \hyperpage{36} + \item mp\_reduce\_2k, \hyperpage{41} + \item mp\_reduce\_2k\_setup, \hyperpage{41} + \item mp\_reduce\_setup, \hyperpage{36} + \item mp\_rshd, \hyperpage{28} + \item mp\_set, \hyperpage{19} + \item mp\_set\_int, \hyperpage{20} + \item mp\_shrink, \hyperpage{15} + \item mp\_sqr, \hyperpage{33} + \item mp\_sub, \hyperpage{29} + \item mp\_sub\_d, \hyperpage{52} + \item mp\_to\_unsigned\_bin, \hyperpage{50} + \item mp\_toradix, \hyperpage{49} + \item mp\_unsigned\_bin\_size, \hyperpage{50} + \item MP\_VAL, \hyperpage{9} + \item mp\_xor, \hyperpage{29} + \item MP\_YES, \hyperpage{9} + +\end{theindex} diff --git a/libtommath/bn.pdf b/libtommath/bn.pdf Binary files differnew file mode 100644 index 0000000..9b873e1 --- /dev/null +++ b/libtommath/bn.pdf diff --git a/libtommath/bn.tex b/libtommath/bn.tex new file mode 100644 index 0000000..962d6ea --- /dev/null +++ b/libtommath/bn.tex @@ -0,0 +1,1830 @@ +\documentclass[b5paper]{book} +\usepackage{hyperref} +\usepackage{makeidx} +\usepackage{amssymb} +\usepackage{color} +\usepackage{alltt} +\usepackage{graphicx} +\usepackage{layout} +\def\union{\cup} +\def\intersect{\cap} +\def\getsrandom{\stackrel{\rm R}{\gets}} +\def\cross{\times} +\def\cat{\hspace{0.5em} \| \hspace{0.5em}} +\def\catn{$\|$} +\def\divides{\hspace{0.3em} | \hspace{0.3em}} +\def\nequiv{\not\equiv} +\def\approx{\raisebox{0.2ex}{\mbox{\small $\sim$}}} +\def\lcm{{\rm lcm}} +\def\gcd{{\rm gcd}} +\def\log{{\rm log}} +\def\ord{{\rm ord}} +\def\abs{{\mathit abs}} +\def\rep{{\mathit rep}} +\def\mod{{\mathit\ mod\ }} +\renewcommand{\pmod}[1]{\ ({\rm mod\ }{#1})} +\newcommand{\floor}[1]{\left\lfloor{#1}\right\rfloor} +\newcommand{\ceil}[1]{\left\lceil{#1}\right\rceil} +\def\Or{{\rm\ or\ }} +\def\And{{\rm\ and\ }} +\def\iff{\hspace{1em}\Longleftrightarrow\hspace{1em}} +\def\implies{\Rightarrow} +\def\undefined{{\rm ``undefined"}} +\def\Proof{\vspace{1ex}\noindent {\bf Proof:}\hspace{1em}} +\let\oldphi\phi +\def\phi{\varphi} +\def\Pr{{\rm Pr}} +\newcommand{\str}[1]{{\mathbf{#1}}} +\def\F{{\mathbb F}} +\def\N{{\mathbb N}} +\def\Z{{\mathbb Z}} +\def\R{{\mathbb R}} +\def\C{{\mathbb C}} +\def\Q{{\mathbb Q}} +\definecolor{DGray}{gray}{0.5} +\newcommand{\emailaddr}[1]{\mbox{$<${#1}$>$}} +\def\twiddle{\raisebox{0.3ex}{\mbox{\tiny $\sim$}}} +\def\gap{\vspace{0.5ex}} +\makeindex +\begin{document} +\frontmatter +\pagestyle{empty} +\title{LibTomMath User Manual \\ v0.33} +\author{Tom St Denis \\ tomstdenis@iahu.ca} +\maketitle +This text, the library and the accompanying textbook are all hereby placed in the public domain. This book has been +formatted for B5 [176x250] paper using the \LaTeX{} {\em book} macro package. + +\vspace{10cm} + +\begin{flushright}Open Source. Open Academia. Open Minds. + +\mbox{ } + +Tom St Denis, + +Ontario, Canada +\end{flushright} + +\tableofcontents +\listoffigures +\mainmatter +\pagestyle{headings} +\chapter{Introduction} +\section{What is LibTomMath?} +LibTomMath is a library of source code which provides a series of efficient and carefully written functions for manipulating +large integer numbers. It was written in portable ISO C source code so that it will build on any platform with a conforming +C compiler. + +In a nutshell the library was written from scratch with verbose comments to help instruct computer science students how +to implement ``bignum'' math. However, the resulting code has proven to be very useful. It has been used by numerous +universities, commercial and open source software developers. It has been used on a variety of platforms ranging from +Linux and Windows based x86 to ARM based Gameboys and PPC based MacOS machines. + +\section{License} +As of the v0.25 the library source code has been placed in the public domain with every new release. As of the v0.28 +release the textbook ``Implementing Multiple Precision Arithmetic'' has been placed in the public domain with every new +release as well. This textbook is meant to compliment the project by providing a more solid walkthrough of the development +algorithms used in the library. + +Since both\footnote{Note that the MPI files under mtest/ are copyrighted by Michael Fromberger. They are not required to use LibTomMath.} are in the +public domain everyone is entitled to do with them as they see fit. + +\section{Building LibTomMath} + +LibTomMath is meant to be very ``GCC friendly'' as it comes with a makefile well suited for GCC. However, the library will +also build in MSVC, Borland C out of the box. For any other ISO C compiler a makefile will have to be made by the end +developer. + +\subsection{Static Libraries} +To build as a static library for GCC issue the following +\begin{alltt} +make +\end{alltt} + +command. This will build the library and archive the object files in ``libtommath.a''. Now you link against +that and include ``tommath.h'' within your programs. Alternatively to build with MSVC issue the following +\begin{alltt} +nmake -f makefile.msvc +\end{alltt} + +This will build the library and archive the object files in ``tommath.lib''. This has been tested with MSVC +version 6.00 with service pack 5. + +\subsection{Shared Libraries} +To build as a shared library for GCC issue the following +\begin{alltt} +make -f makefile.shared +\end{alltt} +This requires the ``libtool'' package (common on most Linux/BSD systems). It will build LibTomMath as both shared +and static then install (by default) into /usr/lib as well as install the header files in /usr/include. The shared +library (resource) will be called ``libtommath.la'' while the static library called ``libtommath.a''. Generally +you use libtool to link your application against the shared object. + +There is limited support for making a ``DLL'' in windows via the ``makefile.cygwin\_dll'' makefile. It requires +Cygwin to work with since it requires the auto-export/import functionality. The resulting DLL and import library +``libtommath.dll.a'' can be used to link LibTomMath dynamically to any Windows program using Cygwin. + +\subsection{Testing} +To build the library and the test harness type + +\begin{alltt} +make test +\end{alltt} + +This will build the library, ``test'' and ``mtest/mtest''. The ``test'' program will accept test vectors and verify the +results. ``mtest/mtest'' will generate test vectors using the MPI library by Michael Fromberger\footnote{A copy of MPI +is included in the package}. Simply pipe mtest into test using + +\begin{alltt} +mtest/mtest | test +\end{alltt} + +If you do not have a ``/dev/urandom'' style RNG source you will have to write your own PRNG and simply pipe that into +mtest. For example, if your PRNG program is called ``myprng'' simply invoke + +\begin{alltt} +myprng | mtest/mtest | test +\end{alltt} + +This will output a row of numbers that are increasing. Each column is a different test (such as addition, multiplication, etc) +that is being performed. The numbers represent how many times the test was invoked. If an error is detected the program +will exit with a dump of the relevent numbers it was working with. + +\section{Build Configuration} +LibTomMath can configured at build time in three phases we shall call ``depends'', ``tweaks'' and ``trims''. +Each phase changes how the library is built and they are applied one after another respectively. + +To make the system more powerful you can tweak the build process. Classes are defined in the file +``tommath\_superclass.h''. By default, the symbol ``LTM\_ALL'' shall be defined which simply +instructs the system to build all of the functions. This is how LibTomMath used to be packaged. This will give you +access to every function LibTomMath offers. + +However, there are cases where such a build is not optional. For instance, you want to perform RSA operations. You +don't need the vast majority of the library to perform these operations. Aside from LTM\_ALL there is +another pre--defined class ``SC\_RSA\_1'' which works in conjunction with the RSA from LibTomCrypt. Additional +classes can be defined base on the need of the user. + +\subsection{Build Depends} +In the file tommath\_class.h you will see a large list of C ``defines'' followed by a series of ``ifdefs'' +which further define symbols. All of the symbols (technically they're macros $\ldots$) represent a given C source +file. For instance, BN\_MP\_ADD\_C represents the file ``bn\_mp\_add.c''. When a define has been enabled the +function in the respective file will be compiled and linked into the library. Accordingly when the define +is absent the file will not be compiled and not contribute any size to the library. + +You will also note that the header tommath\_class.h is actually recursively included (it includes itself twice). +This is to help resolve as many dependencies as possible. In the last pass the symbol LTM\_LAST will be defined. +This is useful for ``trims''. + +\subsection{Build Tweaks} +A tweak is an algorithm ``alternative''. For example, to provide tradeoffs (usually between size and space). +They can be enabled at any pass of the configuration phase. + +\begin{small} +\begin{center} +\begin{tabular}{|l|l|} +\hline \textbf{Define} & \textbf{Purpose} \\ +\hline BN\_MP\_DIV\_SMALL & Enables a slower, smaller and equally \\ + & functional mp\_div() function \\ +\hline +\end{tabular} +\end{center} +\end{small} + +\subsection{Build Trims} +A trim is a manner of removing functionality from a function that is not required. For instance, to perform +RSA cryptography you only require exponentiation with odd moduli so even moduli support can be safely removed. +Build trims are meant to be defined on the last pass of the configuration which means they are to be defined +only if LTM\_LAST has been defined. + +\subsubsection{Moduli Related} +\begin{small} +\begin{center} +\begin{tabular}{|l|l|} +\hline \textbf{Restriction} & \textbf{Undefine} \\ +\hline Exponentiation with odd moduli only & BN\_S\_MP\_EXPTMOD\_C \\ + & BN\_MP\_REDUCE\_C \\ + & BN\_MP\_REDUCE\_SETUP\_C \\ + & BN\_S\_MP\_MUL\_HIGH\_DIGS\_C \\ + & BN\_FAST\_S\_MP\_MUL\_HIGH\_DIGS\_C \\ +\hline Exponentiation with random odd moduli & (The above plus the following) \\ + & BN\_MP\_REDUCE\_2K\_C \\ + & BN\_MP\_REDUCE\_2K\_SETUP\_C \\ + & BN\_MP\_REDUCE\_IS\_2K\_C \\ + & BN\_MP\_DR\_IS\_MODULUS\_C \\ + & BN\_MP\_DR\_REDUCE\_C \\ + & BN\_MP\_DR\_SETUP\_C \\ +\hline Modular inverse odd moduli only & BN\_MP\_INVMOD\_SLOW\_C \\ +\hline Modular inverse (both, smaller/slower) & BN\_FAST\_MP\_INVMOD\_C \\ +\hline +\end{tabular} +\end{center} +\end{small} + +\subsubsection{Operand Size Related} +\begin{small} +\begin{center} +\begin{tabular}{|l|l|} +\hline \textbf{Restriction} & \textbf{Undefine} \\ +\hline Moduli $\le 2560$ bits & BN\_MP\_MONTGOMERY\_REDUCE\_C \\ + & BN\_S\_MP\_MUL\_DIGS\_C \\ + & BN\_S\_MP\_MUL\_HIGH\_DIGS\_C \\ + & BN\_S\_MP\_SQR\_C \\ +\hline Polynomial Schmolynomial & BN\_MP\_KARATSUBA\_MUL\_C \\ + & BN\_MP\_KARATSUBA\_SQR\_C \\ + & BN\_MP\_TOOM\_MUL\_C \\ + & BN\_MP\_TOOM\_SQR\_C \\ + +\hline +\end{tabular} +\end{center} +\end{small} + + +\section{Purpose of LibTomMath} +Unlike GNU MP (GMP) Library, LIP, OpenSSL or various other commercial kits (Miracl), LibTomMath was not written with +bleeding edge performance in mind. First and foremost LibTomMath was written to be entirely open. Not only is the +source code public domain (unlike various other GPL/etc licensed code), not only is the code freely downloadable but the +source code is also accessible for computer science students attempting to learn ``BigNum'' or multiple precision +arithmetic techniques. + +LibTomMath was written to be an instructive collection of source code. This is why there are many comments, only one +function per source file and often I use a ``middle-road'' approach where I don't cut corners for an extra 2\% speed +increase. + +Source code alone cannot really teach how the algorithms work which is why I also wrote a textbook that accompanies +the library (beat that!). + +So you may be thinking ``should I use LibTomMath?'' and the answer is a definite maybe. Let me tabulate what I think +are the pros and cons of LibTomMath by comparing it to the math routines from GnuPG\footnote{GnuPG v1.2.3 versus LibTomMath v0.28}. + +\newpage\begin{figure}[here] +\begin{small} +\begin{center} +\begin{tabular}{|l|c|c|l|} +\hline \textbf{Criteria} & \textbf{Pro} & \textbf{Con} & \textbf{Notes} \\ +\hline Few lines of code per file & X & & GnuPG $ = 300.9$, LibTomMath $ = 76.04$ \\ +\hline Commented function prototypes & X && GnuPG function names are cryptic. \\ +\hline Speed && X & LibTomMath is slower. \\ +\hline Totally free & X & & GPL has unfavourable restrictions.\\ +\hline Large function base & X & & GnuPG is barebones. \\ +\hline Four modular reduction algorithms & X & & Faster modular exponentiation. \\ +\hline Portable & X & & GnuPG requires configuration to build. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{LibTomMath Valuation} +\end{figure} + +It may seem odd to compare LibTomMath to GnuPG since the math in GnuPG is only a small portion of the entire application. +However, LibTomMath was written with cryptography in mind. It provides essentially all of the functions a cryptosystem +would require when working with large integers. + +So it may feel tempting to just rip the math code out of GnuPG (or GnuMP where it was taken from originally) in your +own application but I think there are reasons not to. While LibTomMath is slower than libraries such as GnuMP it is +not normally significantly slower. On x86 machines the difference is normally a factor of two when performing modular +exponentiations. + +Essentially the only time you wouldn't use LibTomMath is when blazing speed is the primary concern. + +\chapter{Getting Started with LibTomMath} +\section{Building Programs} +In order to use LibTomMath you must include ``tommath.h'' and link against the appropriate library file (typically +libtommath.a). There is no library initialization required and the entire library is thread safe. + +\section{Return Codes} +There are three possible return codes a function may return. + +\index{MP\_OKAY}\index{MP\_YES}\index{MP\_NO}\index{MP\_VAL}\index{MP\_MEM} +\begin{figure}[here!] +\begin{center} +\begin{small} +\begin{tabular}{|l|l|} +\hline \textbf{Code} & \textbf{Meaning} \\ +\hline MP\_OKAY & The function succeeded. \\ +\hline MP\_VAL & The function input was invalid. \\ +\hline MP\_MEM & Heap memory exhausted. \\ +\hline &\\ +\hline MP\_YES & Response is yes. \\ +\hline MP\_NO & Response is no. \\ +\hline +\end{tabular} +\end{small} +\end{center} +\caption{Return Codes} +\end{figure} + +The last two codes listed are not actually ``return'ed'' by a function. They are placed in an integer (the caller must +provide the address of an integer it can store to) which the caller can access. To convert one of the three return codes +to a string use the following function. + +\index{mp\_error\_to\_string} +\begin{alltt} +char *mp_error_to_string(int code); +\end{alltt} + +This will return a pointer to a string which describes the given error code. It will not work for the return codes +MP\_YES and MP\_NO. + +\section{Data Types} +The basic ``multiple precision integer'' type is known as the ``mp\_int'' within LibTomMath. This data type is used to +organize all of the data required to manipulate the integer it represents. Within LibTomMath it has been prototyped +as the following. + +\index{mp\_int} +\begin{alltt} +typedef struct \{ + int used, alloc, sign; + mp_digit *dp; +\} mp_int; +\end{alltt} + +Where ``mp\_digit'' is a data type that represents individual digits of the integer. By default, an mp\_digit is the +ISO C ``unsigned long'' data type and each digit is $28-$bits long. The mp\_digit type can be configured to suit other +platforms by defining the appropriate macros. + +All LTM functions that use the mp\_int type will expect a pointer to mp\_int structure. You must allocate memory to +hold the structure itself by yourself (whether off stack or heap it doesn't matter). The very first thing that must be +done to use an mp\_int is that it must be initialized. + +\section{Function Organization} + +The arithmetic functions of the library are all organized to have the same style prototype. That is source operands +are passed on the left and the destination is on the right. For instance, + +\begin{alltt} +mp_add(&a, &b, &c); /* c = a + b */ +mp_mul(&a, &a, &c); /* c = a * a */ +mp_div(&a, &b, &c, &d); /* c = [a/b], d = a mod b */ +\end{alltt} + +Another feature of the way the functions have been implemented is that source operands can be destination operands as well. +For instance, + +\begin{alltt} +mp_add(&a, &b, &b); /* b = a + b */ +mp_div(&a, &b, &a, &c); /* a = [a/b], c = a mod b */ +\end{alltt} + +This allows operands to be re-used which can make programming simpler. + +\section{Initialization} +\subsection{Single Initialization} +A single mp\_int can be initialized with the ``mp\_init'' function. + +\index{mp\_init} +\begin{alltt} +int mp_init (mp_int * a); +\end{alltt} + +This function expects a pointer to an mp\_int structure and will initialize the members of the structure so the mp\_int +represents the default integer which is zero. If the functions returns MP\_OKAY then the mp\_int is ready to be used +by the other LibTomMath functions. + +\begin{small} \begin{alltt} +int main(void) +\{ + mp_int number; + int result; + + if ((result = mp_init(&number)) != MP_OKAY) \{ + printf("Error initializing the number. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + /* use the number */ + + return EXIT_SUCCESS; +\} +\end{alltt} \end{small} + +\subsection{Single Free} +When you are finished with an mp\_int it is ideal to return the heap it used back to the system. The following function +provides this functionality. + +\index{mp\_clear} +\begin{alltt} +void mp_clear (mp_int * a); +\end{alltt} + +The function expects a pointer to a previously initialized mp\_int structure and frees the heap it uses. It sets the +pointer\footnote{The ``dp'' member.} within the mp\_int to \textbf{NULL} which is used to prevent double free situations. +Is is legal to call mp\_clear() twice on the same mp\_int in a row. + +\begin{small} \begin{alltt} +int main(void) +\{ + mp_int number; + int result; + + if ((result = mp_init(&number)) != MP_OKAY) \{ + printf("Error initializing the number. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + /* use the number */ + + /* We're done with it. */ + mp_clear(&number); + + return EXIT_SUCCESS; +\} +\end{alltt} \end{small} + +\subsection{Multiple Initializations} +Certain algorithms require more than one large integer. In these instances it is ideal to initialize all of the mp\_int +variables in an ``all or nothing'' fashion. That is, they are either all initialized successfully or they are all +not initialized. + +The mp\_init\_multi() function provides this functionality. + +\index{mp\_init\_multi} \index{mp\_clear\_multi} +\begin{alltt} +int mp_init_multi(mp_int *mp, ...); +\end{alltt} + +It accepts a \textbf{NULL} terminated list of pointers to mp\_int structures. It will attempt to initialize them all +at once. If the function returns MP\_OKAY then all of the mp\_int variables are ready to use, otherwise none of them +are available for use. A complementary mp\_clear\_multi() function allows multiple mp\_int variables to be free'd +from the heap at the same time. + +\begin{small} \begin{alltt} +int main(void) +\{ + mp_int num1, num2, num3; + int result; + + if ((result = mp_init_multi(&num1, + &num2, + &num3, NULL)) != MP\_OKAY) \{ + printf("Error initializing the numbers. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + /* use the numbers */ + + /* We're done with them. */ + mp_clear_multi(&num1, &num2, &num3, NULL); + + return EXIT_SUCCESS; +\} +\end{alltt} \end{small} + +\subsection{Other Initializers} +To initialized and make a copy of an mp\_int the mp\_init\_copy() function has been provided. + +\index{mp\_init\_copy} +\begin{alltt} +int mp_init_copy (mp_int * a, mp_int * b); +\end{alltt} + +This function will initialize $a$ and make it a copy of $b$ if all goes well. + +\begin{small} \begin{alltt} +int main(void) +\{ + mp_int num1, num2; + int result; + + /* initialize and do work on num1 ... */ + + /* We want a copy of num1 in num2 now */ + if ((result = mp_init_copy(&num2, &num1)) != MP_OKAY) \{ + printf("Error initializing the copy. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + /* now num2 is ready and contains a copy of num1 */ + + /* We're done with them. */ + mp_clear_multi(&num1, &num2, NULL); + + return EXIT_SUCCESS; +\} +\end{alltt} \end{small} + +Another less common initializer is mp\_init\_size() which allows the user to initialize an mp\_int with a given +default number of digits. By default, all initializers allocate \textbf{MP\_PREC} digits. This function lets +you override this behaviour. + +\index{mp\_init\_size} +\begin{alltt} +int mp_init_size (mp_int * a, int size); +\end{alltt} + +The $size$ parameter must be greater than zero. If the function succeeds the mp\_int $a$ will be initialized +to have $size$ digits (which are all initially zero). + +\begin{small} \begin{alltt} +int main(void) +\{ + mp_int number; + int result; + + /* we need a 60-digit number */ + if ((result = mp_init_size(&number, 60)) != MP_OKAY) \{ + printf("Error initializing the number. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + /* use the number */ + + return EXIT_SUCCESS; +\} +\end{alltt} \end{small} + +\section{Maintenance Functions} + +\subsection{Reducing Memory Usage} +When an mp\_int is in a state where it won't be changed again\footnote{A Diffie-Hellman modulus for instance.} excess +digits can be removed to return memory to the heap with the mp\_shrink() function. + +\index{mp\_shrink} +\begin{alltt} +int mp_shrink (mp_int * a); +\end{alltt} + +This will remove excess digits of the mp\_int $a$. If the operation fails the mp\_int should be intact without the +excess digits being removed. Note that you can use a shrunk mp\_int in further computations, however, such operations +will require heap operations which can be slow. It is not ideal to shrink mp\_int variables that you will further +modify in the system (unless you are seriously low on memory). + +\begin{small} \begin{alltt} +int main(void) +\{ + mp_int number; + int result; + + if ((result = mp_init(&number)) != MP_OKAY) \{ + printf("Error initializing the number. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + /* use the number [e.g. pre-computation] */ + + /* We're done with it for now. */ + if ((result = mp_shrink(&number)) != MP_OKAY) \{ + printf("Error shrinking the number. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + /* use it .... */ + + + /* we're done with it. */ + mp_clear(&number); + + return EXIT_SUCCESS; +\} +\end{alltt} \end{small} + +\subsection{Adding additional digits} + +Within the mp\_int structure are two parameters which control the limitations of the array of digits that represent +the integer the mp\_int is meant to equal. The \textit{used} parameter dictates how many digits are significant, that is, +contribute to the value of the mp\_int. The \textit{alloc} parameter dictates how many digits are currently available in +the array. If you need to perform an operation that requires more digits you will have to mp\_grow() the mp\_int to +your desired size. + +\index{mp\_grow} +\begin{alltt} +int mp_grow (mp_int * a, int size); +\end{alltt} + +This will grow the array of digits of $a$ to $size$. If the \textit{alloc} parameter is already bigger than +$size$ the function will not do anything. + +\begin{small} \begin{alltt} +int main(void) +\{ + mp_int number; + int result; + + if ((result = mp_init(&number)) != MP_OKAY) \{ + printf("Error initializing the number. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + /* use the number */ + + /* We need to add 20 digits to the number */ + if ((result = mp_grow(&number, number.alloc + 20)) != MP_OKAY) \{ + printf("Error growing the number. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + + /* use the number */ + + /* we're done with it. */ + mp_clear(&number); + + return EXIT_SUCCESS; +\} +\end{alltt} \end{small} + +\chapter{Basic Operations} +\section{Small Constants} +Setting mp\_ints to small constants is a relatively common operation. To accomodate these instances there are two +small constant assignment functions. The first function is used to set a single digit constant while the second sets +an ISO C style ``unsigned long'' constant. The reason for both functions is efficiency. Setting a single digit is quick but the +domain of a digit can change (it's always at least $0 \ldots 127$). + +\subsection{Single Digit} + +Setting a single digit can be accomplished with the following function. + +\index{mp\_set} +\begin{alltt} +void mp_set (mp_int * a, mp_digit b); +\end{alltt} + +This will zero the contents of $a$ and make it represent an integer equal to the value of $b$. Note that this +function has a return type of \textbf{void}. It cannot cause an error so it is safe to assume the function +succeeded. + +\begin{small} \begin{alltt} +int main(void) +\{ + mp_int number; + int result; + + if ((result = mp_init(&number)) != MP_OKAY) \{ + printf("Error initializing the number. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + /* set the number to 5 */ + mp_set(&number, 5); + + /* we're done with it. */ + mp_clear(&number); + + return EXIT_SUCCESS; +\} +\end{alltt} \end{small} + +\subsection{Long Constants} + +To set a constant that is the size of an ISO C ``unsigned long'' and larger than a single digit the following function +can be used. + +\index{mp\_set\_int} +\begin{alltt} +int mp_set_int (mp_int * a, unsigned long b); +\end{alltt} + +This will assign the value of the 32-bit variable $b$ to the mp\_int $a$. Unlike mp\_set() this function will always +accept a 32-bit input regardless of the size of a single digit. However, since the value may span several digits +this function can fail if it runs out of heap memory. + +To get the ``unsigned long'' copy of an mp\_int the following function can be used. + +\index{mp\_get\_int} +\begin{alltt} +unsigned long mp_get_int (mp_int * a); +\end{alltt} + +This will return the 32 least significant bits of the mp\_int $a$. + +\begin{small} \begin{alltt} +int main(void) +\{ + mp_int number; + int result; + + if ((result = mp_init(&number)) != MP_OKAY) \{ + printf("Error initializing the number. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + /* set the number to 654321 (note this is bigger than 127) */ + if ((result = mp_set_int(&number, 654321)) != MP_OKAY) \{ + printf("Error setting the value of the number. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + printf("number == \%lu", mp_get_int(&number)); + + /* we're done with it. */ + mp_clear(&number); + + return EXIT_SUCCESS; +\} +\end{alltt} \end{small} + +This should output the following if the program succeeds. + +\begin{alltt} +number == 654321 +\end{alltt} + +\subsection{Initialize and Setting Constants} +To both initialize and set small constants the following two functions are available. +\index{mp\_init\_set} \index{mp\_init\_set\_int} +\begin{alltt} +int mp_init_set (mp_int * a, mp_digit b); +int mp_init_set_int (mp_int * a, unsigned long b); +\end{alltt} + +Both functions work like the previous counterparts except they first mp\_init $a$ before setting the values. + +\begin{alltt} +int main(void) +\{ + mp_int number1, number2; + int result; + + /* initialize and set a single digit */ + if ((result = mp_init_set(&number1, 100)) != MP_OKAY) \{ + printf("Error setting number1: \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + /* initialize and set a long */ + if ((result = mp_init_set_int(&number2, 1023)) != MP_OKAY) \{ + printf("Error setting number2: \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + /* display */ + printf("Number1, Number2 == \%lu, \%lu", + mp_get_int(&number1), mp_get_int(&number2)); + + /* clear */ + mp_clear_multi(&number1, &number2, NULL); + + return EXIT_SUCCESS; +\} +\end{alltt} + +If this program succeeds it shall output. +\begin{alltt} +Number1, Number2 == 100, 1023 +\end{alltt} + +\section{Comparisons} + +Comparisons in LibTomMath are always performed in a ``left to right'' fashion. There are three possible return codes +for any comparison. + +\index{MP\_GT} \index{MP\_EQ} \index{MP\_LT} +\begin{figure}[here] +\begin{center} +\begin{tabular}{|c|c|} +\hline \textbf{Result Code} & \textbf{Meaning} \\ +\hline MP\_GT & $a > b$ \\ +\hline MP\_EQ & $a = b$ \\ +\hline MP\_LT & $a < b$ \\ +\hline +\end{tabular} +\end{center} +\caption{Comparison Codes for $a, b$} +\label{fig:CMP} +\end{figure} + +In figure \ref{fig:CMP} two integers $a$ and $b$ are being compared. In this case $a$ is said to be ``to the left'' of +$b$. + +\subsection{Unsigned comparison} + +An unsigned comparison considers only the digits themselves and not the associated \textit{sign} flag of the +mp\_int structures. This is analogous to an absolute comparison. The function mp\_cmp\_mag() will compare two +mp\_int variables based on their digits only. + +\index{mp\_cmp\_mag} +\begin{alltt} +int mp_cmp(mp_int * a, mp_int * b); +\end{alltt} +This will compare $a$ to $b$ placing $a$ to the left of $b$. This function cannot fail and will return one of the +three compare codes listed in figure \ref{fig:CMP}. + +\begin{small} \begin{alltt} +int main(void) +\{ + mp_int number1, number2; + int result; + + if ((result = mp_init_multi(&number1, &number2, NULL)) != MP_OKAY) \{ + printf("Error initializing the numbers. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + /* set the number1 to 5 */ + mp_set(&number1, 5); + + /* set the number2 to -6 */ + mp_set(&number2, 6); + if ((result = mp_neg(&number2, &number2)) != MP_OKAY) \{ + printf("Error negating number2. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + switch(mp_cmp_mag(&number1, &number2)) \{ + case MP_GT: printf("|number1| > |number2|"); break; + case MP_EQ: printf("|number1| = |number2|"); break; + case MP_LT: printf("|number1| < |number2|"); break; + \} + + /* we're done with it. */ + mp_clear_multi(&number1, &number2, NULL); + + return EXIT_SUCCESS; +\} +\end{alltt} \end{small} + +If this program\footnote{This function uses the mp\_neg() function which is discussed in section \ref{sec:NEG}.} completes +successfully it should print the following. + +\begin{alltt} +|number1| < |number2| +\end{alltt} + +This is because $\vert -6 \vert = 6$ and obviously $5 < 6$. + +\subsection{Signed comparison} + +To compare two mp\_int variables based on their signed value the mp\_cmp() function is provided. + +\index{mp\_cmp} +\begin{alltt} +int mp_cmp(mp_int * a, mp_int * b); +\end{alltt} + +This will compare $a$ to the left of $b$. It will first compare the signs of the two mp\_int variables. If they +differ it will return immediately based on their signs. If the signs are equal then it will compare the digits +individually. This function will return one of the compare conditions codes listed in figure \ref{fig:CMP}. + +\begin{small} \begin{alltt} +int main(void) +\{ + mp_int number1, number2; + int result; + + if ((result = mp_init_multi(&number1, &number2, NULL)) != MP_OKAY) \{ + printf("Error initializing the numbers. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + /* set the number1 to 5 */ + mp_set(&number1, 5); + + /* set the number2 to -6 */ + mp_set(&number2, 6); + if ((result = mp_neg(&number2, &number2)) != MP_OKAY) \{ + printf("Error negating number2. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + switch(mp_cmp(&number1, &number2)) \{ + case MP_GT: printf("number1 > number2"); break; + case MP_EQ: printf("number1 = number2"); break; + case MP_LT: printf("number1 < number2"); break; + \} + + /* we're done with it. */ + mp_clear_multi(&number1, &number2, NULL); + + return EXIT_SUCCESS; +\} +\end{alltt} \end{small} + +If this program\footnote{This function uses the mp\_neg() function which is discussed in section \ref{sec:NEG}.} completes +successfully it should print the following. + +\begin{alltt} +number1 > number2 +\end{alltt} + +\subsection{Single Digit} + +To compare a single digit against an mp\_int the following function has been provided. + +\index{mp\_cmp\_d} +\begin{alltt} +int mp_cmp_d(mp_int * a, mp_digit b); +\end{alltt} + +This will compare $a$ to the left of $b$ using a signed comparison. Note that it will always treat $b$ as +positive. This function is rather handy when you have to compare against small values such as $1$ (which often +comes up in cryptography). The function cannot fail and will return one of the tree compare condition codes +listed in figure \ref{fig:CMP}. + + +\begin{small} \begin{alltt} +int main(void) +\{ + mp_int number; + int result; + + if ((result = mp_init(&number)) != MP_OKAY) \{ + printf("Error initializing the number. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + /* set the number to 5 */ + mp_set(&number, 5); + + switch(mp_cmp_d(&number, 7)) \{ + case MP_GT: printf("number > 7"); break; + case MP_EQ: printf("number = 7"); break; + case MP_LT: printf("number < 7"); break; + \} + + /* we're done with it. */ + mp_clear(&number); + + return EXIT_SUCCESS; +\} +\end{alltt} \end{small} + +If this program functions properly it will print out the following. + +\begin{alltt} +number < 7 +\end{alltt} + +\section{Logical Operations} + +Logical operations are operations that can be performed either with simple shifts or boolean operators such as +AND, XOR and OR directly. These operations are very quick. + +\subsection{Multiplication by two} + +Multiplications and divisions by any power of two can be performed with quick logical shifts either left or +right depending on the operation. + +When multiplying or dividing by two a special case routine can be used which are as follows. +\index{mp\_mul\_2} \index{mp\_div\_2} +\begin{alltt} +int mp_mul_2(mp_int * a, mp_int * b); +int mp_div_2(mp_int * a, mp_int * b); +\end{alltt} + +The former will assign twice $a$ to $b$ while the latter will assign half $a$ to $b$. These functions are fast +since the shift counts and maskes are hardcoded into the routines. + +\begin{small} \begin{alltt} +int main(void) +\{ + mp_int number; + int result; + + if ((result = mp_init(&number)) != MP_OKAY) \{ + printf("Error initializing the number. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + /* set the number to 5 */ + mp_set(&number, 5); + + /* multiply by two */ + if ((result = mp\_mul\_2(&number, &number)) != MP_OKAY) \{ + printf("Error multiplying the number. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + switch(mp_cmp_d(&number, 7)) \{ + case MP_GT: printf("2*number > 7"); break; + case MP_EQ: printf("2*number = 7"); break; + case MP_LT: printf("2*number < 7"); break; + \} + + /* now divide by two */ + if ((result = mp\_div\_2(&number, &number)) != MP_OKAY) \{ + printf("Error dividing the number. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + switch(mp_cmp_d(&number, 7)) \{ + case MP_GT: printf("2*number/2 > 7"); break; + case MP_EQ: printf("2*number/2 = 7"); break; + case MP_LT: printf("2*number/2 < 7"); break; + \} + + /* we're done with it. */ + mp_clear(&number); + + return EXIT_SUCCESS; +\} +\end{alltt} \end{small} + +If this program is successful it will print out the following text. + +\begin{alltt} +2*number > 7 +2*number/2 < 7 +\end{alltt} + +Since $10 > 7$ and $5 < 7$. To multiply by a power of two the following function can be used. + +\index{mp\_mul\_2d} +\begin{alltt} +int mp_mul_2d(mp_int * a, int b, mp_int * c); +\end{alltt} + +This will multiply $a$ by $2^b$ and store the result in ``c''. If the value of $b$ is less than or equal to +zero the function will copy $a$ to ``c'' without performing any further actions. + +To divide by a power of two use the following. + +\index{mp\_div\_2d} +\begin{alltt} +int mp_div_2d (mp_int * a, int b, mp_int * c, mp_int * d); +\end{alltt} +Which will divide $a$ by $2^b$, store the quotient in ``c'' and the remainder in ``d'. If $b \le 0$ then the +function simply copies $a$ over to ``c'' and zeroes $d$. The variable $d$ may be passed as a \textbf{NULL} +value to signal that the remainder is not desired. + +\subsection{Polynomial Basis Operations} + +Strictly speaking the organization of the integers within the mp\_int structures is what is known as a +``polynomial basis''. This simply means a field element is stored by divisions of a radix. For example, if +$f(x) = \sum_{i=0}^{k} y_ix^k$ for any vector $\vec y$ then the array of digits in $\vec y$ are said to be +the polynomial basis representation of $z$ if $f(\beta) = z$ for a given radix $\beta$. + +To multiply by the polynomial $g(x) = x$ all you have todo is shift the digits of the basis left one place. The +following function provides this operation. + +\index{mp\_lshd} +\begin{alltt} +int mp_lshd (mp_int * a, int b); +\end{alltt} + +This will multiply $a$ in place by $x^b$ which is equivalent to shifting the digits left $b$ places and inserting zeroes +in the least significant digits. Similarly to divide by a power of $x$ the following function is provided. + +\index{mp\_rshd} +\begin{alltt} +void mp_rshd (mp_int * a, int b) +\end{alltt} +This will divide $a$ in place by $x^b$ and discard the remainder. This function cannot fail as it performs the operations +in place and no new digits are required to complete it. + +\subsection{AND, OR and XOR Operations} + +While AND, OR and XOR operations are not typical ``bignum functions'' they can be useful in several instances. The +three functions are prototyped as follows. + +\index{mp\_or} \index{mp\_and} \index{mp\_xor} +\begin{alltt} +int mp_or (mp_int * a, mp_int * b, mp_int * c); +int mp_and (mp_int * a, mp_int * b, mp_int * c); +int mp_xor (mp_int * a, mp_int * b, mp_int * c); +\end{alltt} + +Which compute $c = a \odot b$ where $\odot$ is one of OR, AND or XOR. + +\section{Addition and Subtraction} + +To compute an addition or subtraction the following two functions can be used. + +\index{mp\_add} \index{mp\_sub} +\begin{alltt} +int mp_add (mp_int * a, mp_int * b, mp_int * c); +int mp_sub (mp_int * a, mp_int * b, mp_int * c) +\end{alltt} + +Which perform $c = a \odot b$ where $\odot$ is one of signed addition or subtraction. The operations are fully sign +aware. + +\section{Sign Manipulation} +\subsection{Negation} +\label{sec:NEG} +Simple integer negation can be performed with the following. + +\index{mp\_neg} +\begin{alltt} +int mp_neg (mp_int * a, mp_int * b); +\end{alltt} + +Which assigns $-a$ to $b$. + +\subsection{Absolute} +Simple integer absolutes can be performed with the following. + +\index{mp\_neg} +\begin{alltt} +int mp_abs (mp_int * a, mp_int * b); +\end{alltt} + +Which assigns $\vert a \vert$ to $b$. + +\section{Integer Division and Remainder} +To perform a complete and general integer division with remainder use the following function. + +\index{mp\_div} +\begin{alltt} +int mp_div (mp_int * a, mp_int * b, mp_int * c, mp_int * d); +\end{alltt} + +This divides $a$ by $b$ and stores the quotient in $c$ and $d$. The signed quotient is computed such that +$bc + d = a$. Note that either of $c$ or $d$ can be set to \textbf{NULL} if their value is not required. If +$b$ is zero the function returns \textbf{MP\_VAL}. + + +\chapter{Multiplication and Squaring} +\section{Multiplication} +A full signed integer multiplication can be performed with the following. +\index{mp\_mul} +\begin{alltt} +int mp_mul (mp_int * a, mp_int * b, mp_int * c); +\end{alltt} +Which assigns the full signed product $ab$ to $c$. This function actually breaks into one of four cases which are +specific multiplication routines optimized for given parameters. First there are the Toom-Cook multiplications which +should only be used with very large inputs. This is followed by the Karatsuba multiplications which are for moderate +sized inputs. Then followed by the Comba and baseline multipliers. + +Fortunately for the developer you don't really need to know this unless you really want to fine tune the system. mp\_mul() +will determine on its own\footnote{Some tweaking may be required.} what routine to use automatically when it is called. + +\begin{alltt} +int main(void) +\{ + mp_int number1, number2; + int result; + + /* Initialize the numbers */ + if ((result = mp_init_multi(&number1, + &number2, NULL)) != MP_OKAY) \{ + printf("Error initializing the numbers. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + /* set the terms */ + if ((result = mp_set_int(&number, 257)) != MP_OKAY) \{ + printf("Error setting number1. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + if ((result = mp_set_int(&number2, 1023)) != MP_OKAY) \{ + printf("Error setting number2. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + /* multiply them */ + if ((result = mp_mul(&number1, &number2, + &number1)) != MP_OKAY) \{ + printf("Error multiplying terms. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + /* display */ + printf("number1 * number2 == \%lu", mp_get_int(&number1)); + + /* free terms and return */ + mp_clear_multi(&number1, &number2, NULL); + + return EXIT_SUCCESS; +\} +\end{alltt} + +If this program succeeds it shall output the following. + +\begin{alltt} +number1 * number2 == 262911 +\end{alltt} + +\section{Squaring} +Since squaring can be performed faster than multiplication it is performed it's own function instead of just using +mp\_mul(). + +\index{mp\_sqr} +\begin{alltt} +int mp_sqr (mp_int * a, mp_int * b); +\end{alltt} + +Will square $a$ and store it in $b$. Like the case of multiplication there are four different squaring +algorithms all which can be called from mp\_sqr(). It is ideal to use mp\_sqr over mp\_mul when squaring terms. + +\section{Tuning Polynomial Basis Routines} + +Both of the Toom-Cook and Karatsuba multiplication algorithms are faster than the traditional $O(n^2)$ approach that +the Comba and baseline algorithms use. At $O(n^{1.464973})$ and $O(n^{1.584962})$ running times respectfully they require +considerably less work. For example, a 10000-digit multiplication would take roughly 724,000 single precision +multiplications with Toom-Cook or 100,000,000 single precision multiplications with the standard Comba (a factor +of 138). + +So why not always use Karatsuba or Toom-Cook? The simple answer is that they have so much overhead that they're not +actually faster than Comba until you hit distinct ``cutoff'' points. For Karatsuba with the default configuration, +GCC 3.3.1 and an Athlon XP processor the cutoff point is roughly 110 digits (about 70 for the Intel P4). That is, at +110 digits Karatsuba and Comba multiplications just about break even and for 110+ digits Karatsuba is faster. + +Toom-Cook has incredible overhead and is probably only useful for very large inputs. So far no known cutoff points +exist and for the most part I just set the cutoff points very high to make sure they're not called. + +A demo program in the ``etc/'' directory of the project called ``tune.c'' can be used to find the cutoff points. This +can be built with GCC as follows + +\begin{alltt} +make XXX +\end{alltt} +Where ``XXX'' is one of the following entries from the table \ref{fig:tuning}. + +\begin{figure}[here] +\begin{center} +\begin{small} +\begin{tabular}{|l|l|} +\hline \textbf{Value of XXX} & \textbf{Meaning} \\ +\hline tune & Builds portable tuning application \\ +\hline tune86 & Builds x86 (pentium and up) program for COFF \\ +\hline tune86c & Builds x86 program for Cygwin \\ +\hline tune86l & Builds x86 program for Linux (ELF format) \\ +\hline +\end{tabular} +\end{small} +\end{center} +\caption{Build Names for Tuning Programs} +\label{fig:tuning} +\end{figure} + +When the program is running it will output a series of measurements for different cutoff points. It will first find +good Karatsuba squaring and multiplication points. Then it proceeds to find Toom-Cook points. Note that the Toom-Cook +tuning takes a very long time as the cutoff points are likely to be very high. + +\chapter{Modular Reduction} + +Modular reduction is process of taking the remainder of one quantity divided by another. Expressed +as (\ref{eqn:mod}) the modular reduction is equivalent to the remainder of $b$ divided by $c$. + +\begin{equation} +a \equiv b \mbox{ (mod }c\mbox{)} +\label{eqn:mod} +\end{equation} + +Of particular interest to cryptography are reductions where $b$ is limited to the range $0 \le b < c^2$ since particularly +fast reduction algorithms can be written for the limited range. + +Note that one of the four optimized reduction algorithms are automatically chosen in the modular exponentiation +algorithm mp\_exptmod when an appropriate modulus is detected. + +\section{Straight Division} +In order to effect an arbitrary modular reduction the following algorithm is provided. + +\index{mp\_mod} +\begin{alltt} +int mp_mod(mp_int *a, mp_int *b, mp_int *c); +\end{alltt} + +This reduces $a$ modulo $b$ and stores the result in $c$. The sign of $c$ shall agree with the sign +of $b$. This algorithm accepts an input $a$ of any range and is not limited by $0 \le a < b^2$. + +\section{Barrett Reduction} + +Barrett reduction is a generic optimized reduction algorithm that requires pre--computation to achieve +a decent speedup over straight division. First a $mu$ value must be precomputed with the following function. + +\index{mp\_reduce\_setup} +\begin{alltt} +int mp_reduce_setup(mp_int *a, mp_int *b); +\end{alltt} + +Given a modulus in $b$ this produces the required $mu$ value in $a$. For any given modulus this only has to +be computed once. Modular reduction can now be performed with the following. + +\index{mp\_reduce} +\begin{alltt} +int mp_reduce(mp_int *a, mp_int *b, mp_int *c); +\end{alltt} + +This will reduce $a$ in place modulo $b$ with the precomputed $mu$ value in $c$. $a$ must be in the range +$0 \le a < b^2$. + +\begin{alltt} +int main(void) +\{ + mp_int a, b, c, mu; + int result; + + /* initialize a,b to desired values, mp_init mu, + * c and set c to 1...we want to compute a^3 mod b + */ + + /* get mu value */ + if ((result = mp_reduce_setup(&mu, b)) != MP_OKAY) \{ + printf("Error getting mu. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + /* square a to get c = a^2 */ + if ((result = mp_sqr(&a, &c)) != MP_OKAY) \{ + printf("Error squaring. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + /* now reduce `c' modulo b */ + if ((result = mp_reduce(&c, &b, &mu)) != MP_OKAY) \{ + printf("Error reducing. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + /* multiply a to get c = a^3 */ + if ((result = mp_mul(&a, &c, &c)) != MP_OKAY) \{ + printf("Error reducing. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + /* now reduce `c' modulo b */ + if ((result = mp_reduce(&c, &b, &mu)) != MP_OKAY) \{ + printf("Error reducing. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + /* c now equals a^3 mod b */ + + return EXIT_SUCCESS; +\} +\end{alltt} + +This program will calculate $a^3 \mbox{ mod }b$ if all the functions succeed. + +\section{Montgomery Reduction} + +Montgomery is a specialized reduction algorithm for any odd moduli. Like Barrett reduction a pre--computation +step is required. This is accomplished with the following. + +\index{mp\_montgomery\_setup} +\begin{alltt} +int mp_montgomery_setup(mp_int *a, mp_digit *mp); +\end{alltt} + +For the given odd moduli $a$ the precomputation value is placed in $mp$. The reduction is computed with the +following. + +\index{mp\_montgomery\_reduce} +\begin{alltt} +int mp_montgomery_reduce(mp_int *a, mp_int *m, mp_digit mp); +\end{alltt} +This reduces $a$ in place modulo $m$ with the pre--computed value $mp$. $a$ must be in the range +$0 \le a < b^2$. + +Montgomery reduction is faster than Barrett reduction for moduli smaller than the ``comba'' limit. With the default +setup for instance, the limit is $127$ digits ($3556$--bits). Note that this function is not limited to +$127$ digits just that it falls back to a baseline algorithm after that point. + +An important observation is that this reduction does not return $a \mbox{ mod }m$ but $aR^{-1} \mbox{ mod }m$ +where $R = \beta^n$, $n$ is the n number of digits in $m$ and $\beta$ is radix used (default is $2^{28}$). + +To quickly calculate $R$ the following function was provided. + +\index{mp\_montgomery\_calc\_normalization} +\begin{alltt} +int mp_montgomery_calc_normalization(mp_int *a, mp_int *b); +\end{alltt} +Which calculates $a = R$ for the odd moduli $b$ without using multiplication or division. + +The normal modus operandi for Montgomery reductions is to normalize the integers before entering the system. For +example, to calculate $a^3 \mbox { mod }b$ using Montgomery reduction the value of $a$ can be normalized by +multiplying it by $R$. Consider the following code snippet. + +\begin{alltt} +int main(void) +\{ + mp_int a, b, c, R; + mp_digit mp; + int result; + + /* initialize a,b to desired values, + * mp_init R, c and set c to 1.... + */ + + /* get normalization */ + if ((result = mp_montgomery_calc_normalization(&R, b)) != MP_OKAY) \{ + printf("Error getting norm. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + /* get mp value */ + if ((result = mp_montgomery_setup(&c, &mp)) != MP_OKAY) \{ + printf("Error setting up montgomery. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + /* normalize `a' so now a is equal to aR */ + if ((result = mp_mulmod(&a, &R, &b, &a)) != MP_OKAY) \{ + printf("Error computing aR. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + /* square a to get c = a^2R^2 */ + if ((result = mp_sqr(&a, &c)) != MP_OKAY) \{ + printf("Error squaring. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + /* now reduce `c' back down to c = a^2R^2 * R^-1 == a^2R */ + if ((result = mp_montgomery_reduce(&c, &b, mp)) != MP_OKAY) \{ + printf("Error reducing. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + /* multiply a to get c = a^3R^2 */ + if ((result = mp_mul(&a, &c, &c)) != MP_OKAY) \{ + printf("Error reducing. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + /* now reduce `c' back down to c = a^3R^2 * R^-1 == a^3R */ + if ((result = mp_montgomery_reduce(&c, &b, mp)) != MP_OKAY) \{ + printf("Error reducing. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + /* now reduce (again) `c' back down to c = a^3R * R^-1 == a^3 */ + if ((result = mp_montgomery_reduce(&c, &b, mp)) != MP_OKAY) \{ + printf("Error reducing. \%s", + mp_error_to_string(result)); + return EXIT_FAILURE; + \} + + /* c now equals a^3 mod b */ + + return EXIT_SUCCESS; +\} +\end{alltt} + +This particular example does not look too efficient but it demonstrates the point of the algorithm. By +normalizing the inputs the reduced results are always of the form $aR$ for some variable $a$. This allows +a single final reduction to correct for the normalization and the fast reduction used within the algorithm. + +For more details consider examining the file \textit{bn\_mp\_exptmod\_fast.c}. + +\section{Restricted Dimminished Radix} + +``Dimminished Radix'' reduction refers to reduction with respect to moduli that are ameniable to simple +digit shifting and small multiplications. In this case the ``restricted'' variant refers to moduli of the +form $\beta^k - p$ for some $k \ge 0$ and $0 < p < \beta$ where $\beta$ is the radix (default to $2^{28}$). + +As in the case of Montgomery reduction there is a pre--computation phase required for a given modulus. + +\index{mp\_dr\_setup} +\begin{alltt} +void mp_dr_setup(mp_int *a, mp_digit *d); +\end{alltt} + +This computes the value required for the modulus $a$ and stores it in $d$. This function cannot fail +and does not return any error codes. After the pre--computation a reduction can be performed with the +following. + +\index{mp\_dr\_reduce} +\begin{alltt} +int mp_dr_reduce(mp_int *a, mp_int *b, mp_digit mp); +\end{alltt} + +This reduces $a$ in place modulo $b$ with the pre--computed value $mp$. $b$ must be of a restricted +dimminished radix form and $a$ must be in the range $0 \le a < b^2$. Dimminished radix reductions are +much faster than both Barrett and Montgomery reductions as they have a much lower asymtotic running time. + +Since the moduli are restricted this algorithm is not particularly useful for something like Rabin, RSA or +BBS cryptographic purposes. This reduction algorithm is useful for Diffie-Hellman and ECC where fixed +primes are acceptable. + +Note that unlike Montgomery reduction there is no normalization process. The result of this function is +equal to the correct residue. + +\section{Unrestricted Dimminshed Radix} + +Unrestricted reductions work much like the restricted counterparts except in this case the moduli is of the +form $2^k - p$ for $0 < p < \beta$. In this sense the unrestricted reductions are more flexible as they +can be applied to a wider range of numbers. + +\index{mp\_reduce\_2k\_setup} +\begin{alltt} +int mp_reduce_2k_setup(mp_int *a, mp_digit *d); +\end{alltt} + +This will compute the required $d$ value for the given moduli $a$. + +\index{mp\_reduce\_2k} +\begin{alltt} +int mp_reduce_2k(mp_int *a, mp_int *n, mp_digit d); +\end{alltt} + +This will reduce $a$ in place modulo $n$ with the pre--computed value $d$. From my experience this routine is +slower than mp\_dr\_reduce but faster for most moduli sizes than the Montgomery reduction. + +\chapter{Exponentiation} +\section{Single Digit Exponentiation} +\index{mp\_expt\_d} +\begin{alltt} +int mp_expt_d (mp_int * a, mp_digit b, mp_int * c) +\end{alltt} +This computes $c = a^b$ using a simple binary left-to-right algorithm. It is faster than repeated multiplications by +$a$ for all values of $b$ greater than three. + +\section{Modular Exponentiation} +\index{mp\_exptmod} +\begin{alltt} +int mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y) +\end{alltt} +This computes $Y \equiv G^X \mbox{ (mod }P\mbox{)}$ using a variable width sliding window algorithm. This function +will automatically detect the fastest modular reduction technique to use during the operation. For negative values of +$X$ the operation is performed as $Y \equiv (G^{-1} \mbox{ mod }P)^{\vert X \vert} \mbox{ (mod }P\mbox{)}$ provided that +$gcd(G, P) = 1$. + +This function is actually a shell around the two internal exponentiation functions. This routine will automatically +detect when Barrett, Montgomery, Restricted and Unrestricted Dimminished Radix based exponentiation can be used. Generally +moduli of the a ``restricted dimminished radix'' form lead to the fastest modular exponentiations. Followed by Montgomery +and the other two algorithms. + +\section{Root Finding} +\index{mp\_n\_root} +\begin{alltt} +int mp_n_root (mp_int * a, mp_digit b, mp_int * c) +\end{alltt} +This computes $c = a^{1/b}$ such that $c^b \le a$ and $(c+1)^b > a$. The implementation of this function is not +ideal for values of $b$ greater than three. It will work but become very slow. So unless you are working with very small +numbers (less than 1000 bits) I'd avoid $b > 3$ situations. Will return a positive root only for even roots and return +a root with the sign of the input for odd roots. For example, performing $4^{1/2}$ will return $2$ whereas $(-8)^{1/3}$ +will return $-2$. + +This algorithm uses the ``Newton Approximation'' method and will converge on the correct root fairly quickly. Since +the algorithm requires raising $a$ to the power of $b$ it is not ideal to attempt to find roots for large +values of $b$. If particularly large roots are required then a factor method could be used instead. For example, +$a^{1/16}$ is equivalent to $\left (a^{1/4} \right)^{1/4}$. + +\chapter{Prime Numbers} +\section{Trial Division} +\index{mp\_prime\_is\_divisible} +\begin{alltt} +int mp_prime_is_divisible (mp_int * a, int *result) +\end{alltt} +This will attempt to evenly divide $a$ by a list of primes\footnote{Default is the first 256 primes.} and store the +outcome in ``result''. That is if $result = 0$ then $a$ is not divisible by the primes, otherwise it is. Note that +if the function does not return \textbf{MP\_OKAY} the value in ``result'' should be considered undefined\footnote{Currently +the default is to set it to zero first.}. + +\section{Fermat Test} +\index{mp\_prime\_fermat} +\begin{alltt} +int mp_prime_fermat (mp_int * a, mp_int * b, int *result) +\end{alltt} +Performs a Fermat primality test to the base $b$. That is it computes $b^a \mbox{ mod }a$ and tests whether the value is +equal to $b$ or not. If the values are equal then $a$ is probably prime and $result$ is set to one. Otherwise $result$ +is set to zero. + +\section{Miller-Rabin Test} +\index{mp\_prime\_miller\_rabin} +\begin{alltt} +int mp_prime_miller_rabin (mp_int * a, mp_int * b, int *result) +\end{alltt} +Performs a Miller-Rabin test to the base $b$ of $a$. This test is much stronger than the Fermat test and is very hard to +fool (besides with Carmichael numbers). If $a$ passes the test (therefore is probably prime) $result$ is set to one. +Otherwise $result$ is set to zero. + +Note that is suggested that you use the Miller-Rabin test instead of the Fermat test since all of the failures of +Miller-Rabin are a subset of the failures of the Fermat test. + +\subsection{Required Number of Tests} +Generally to ensure a number is very likely to be prime you have to perform the Miller-Rabin with at least a half-dozen +or so unique bases. However, it has been proven that the probability of failure goes down as the size of the input goes up. +This is why a simple function has been provided to help out. + +\index{mp\_prime\_rabin\_miller\_trials} +\begin{alltt} +int mp_prime_rabin_miller_trials(int size) +\end{alltt} +This returns the number of trials required for a $2^{-96}$ (or lower) probability of failure for a given ``size'' expressed +in bits. This comes in handy specially since larger numbers are slower to test. For example, a 512-bit number would +require ten tests whereas a 1024-bit number would only require four tests. + +You should always still perform a trial division before a Miller-Rabin test though. + +\section{Primality Testing} +\index{mp\_prime\_is\_prime} +\begin{alltt} +int mp_prime_is_prime (mp_int * a, int t, int *result) +\end{alltt} +This will perform a trial division followed by $t$ rounds of Miller-Rabin tests on $a$ and store the result in $result$. +If $a$ passes all of the tests $result$ is set to one, otherwise it is set to zero. Note that $t$ is bounded by +$1 \le t < PRIME\_SIZE$ where $PRIME\_SIZE$ is the number of primes in the prime number table (by default this is $256$). + +\section{Next Prime} +\index{mp\_prime\_next\_prime} +\begin{alltt} +int mp_prime_next_prime(mp_int *a, int t, int bbs_style) +\end{alltt} +This finds the next prime after $a$ that passes mp\_prime\_is\_prime() with $t$ tests. Set $bbs\_style$ to one if you +want only the next prime congruent to $3 \mbox{ mod } 4$, otherwise set it to zero to find any next prime. + +\section{Random Primes} +\index{mp\_prime\_random} +\begin{alltt} +int mp_prime_random(mp_int *a, int t, int size, int bbs, + ltm_prime_callback cb, void *dat) +\end{alltt} +This will find a prime greater than $256^{size}$ which can be ``bbs\_style'' or not depending on $bbs$ and must pass +$t$ rounds of tests. The ``ltm\_prime\_callback'' is a typedef for + +\begin{alltt} +typedef int ltm_prime_callback(unsigned char *dst, int len, void *dat); +\end{alltt} + +Which is a function that must read $len$ bytes (and return the amount stored) into $dst$. The $dat$ variable is simply +copied from the original input. It can be used to pass RNG context data to the callback. The function +mp\_prime\_random() is more suitable for generating primes which must be secret (as in the case of RSA) since there +is no skew on the least significant bits. + +\textit{Note:} As of v0.30 of the LibTomMath library this function has been deprecated. It is still available +but users are encouraged to use the new mp\_prime\_random\_ex() function instead. + +\subsection{Extended Generation} +\index{mp\_prime\_random\_ex} +\begin{alltt} +int mp_prime_random_ex(mp_int *a, int t, + int size, int flags, + ltm_prime_callback cb, void *dat); +\end{alltt} +This will generate a prime in $a$ using $t$ tests of the primality testing algorithms. The variable $size$ +specifies the bit length of the prime desired. The variable $flags$ specifies one of several options available +(see fig. \ref{fig:primeopts}) which can be OR'ed together. The callback parameters are used as in +mp\_prime\_random(). + +\begin{figure}[here] +\begin{center} +\begin{small} +\begin{tabular}{|r|l|} +\hline \textbf{Flag} & \textbf{Meaning} \\ +\hline LTM\_PRIME\_BBS & Make the prime congruent to $3$ modulo $4$ \\ +\hline LTM\_PRIME\_SAFE & Make a prime $p$ such that $(p - 1)/2$ is also prime. \\ + & This option implies LTM\_PRIME\_BBS as well. \\ +\hline LTM\_PRIME\_2MSB\_OFF & Makes sure that the bit adjacent to the most significant bit \\ + & Is forced to zero. \\ +\hline LTM\_PRIME\_2MSB\_ON & Makes sure that the bit adjacent to the most significant bit \\ + & Is forced to one. \\ +\hline +\end{tabular} +\end{small} +\end{center} +\caption{Primality Generation Options} +\label{fig:primeopts} +\end{figure} + +\chapter{Input and Output} +\section{ASCII Conversions} +\subsection{To ASCII} +\index{mp\_toradix} +\begin{alltt} +int mp_toradix (mp_int * a, char *str, int radix); +\end{alltt} +This still store $a$ in ``str'' as a base-``radix'' string of ASCII chars. This function appends a NUL character +to terminate the string. Valid values of ``radix'' line in the range $[2, 64]$. To determine the size (exact) required +by the conversion before storing any data use the following function. + +\index{mp\_radix\_size} +\begin{alltt} +int mp_radix_size (mp_int * a, int radix, int *size) +\end{alltt} +This stores in ``size'' the number of characters (including space for the NUL terminator) required. Upon error this +function returns an error code and ``size'' will be zero. + +\subsection{From ASCII} +\index{mp\_read\_radix} +\begin{alltt} +int mp_read_radix (mp_int * a, char *str, int radix); +\end{alltt} +This will read the base-``radix'' NUL terminated string from ``str'' into $a$. It will stop reading when it reads a +character it does not recognize (which happens to include th NUL char... imagine that...). A single leading $-$ sign +can be used to denote a negative number. + +\section{Binary Conversions} + +Converting an mp\_int to and from binary is another keen idea. + +\index{mp\_unsigned\_bin\_size} +\begin{alltt} +int mp_unsigned_bin_size(mp_int *a); +\end{alltt} + +This will return the number of bytes (octets) required to store the unsigned copy of the integer $a$. + +\index{mp\_to\_unsigned\_bin} +\begin{alltt} +int mp_to_unsigned_bin(mp_int *a, unsigned char *b); +\end{alltt} +This will store $a$ into the buffer $b$ in big--endian format. Fortunately this is exactly what DER (or is it ASN?) +requires. It does not store the sign of the integer. + +\index{mp\_read\_unsigned\_bin} +\begin{alltt} +int mp_read_unsigned_bin(mp_int *a, unsigned char *b, int c); +\end{alltt} +This will read in an unsigned big--endian array of bytes (octets) from $b$ of length $c$ into $a$. The resulting +integer $a$ will always be positive. + +For those who acknowledge the existence of negative numbers (heretic!) there are ``signed'' versions of the +previous functions. + +\begin{alltt} +int mp_signed_bin_size(mp_int *a); +int mp_read_signed_bin(mp_int *a, unsigned char *b, int c); +int mp_to_signed_bin(mp_int *a, unsigned char *b); +\end{alltt} +They operate essentially the same as the unsigned copies except they prefix the data with zero or non--zero +byte depending on the sign. If the sign is zpos (e.g. not negative) the prefix is zero, otherwise the prefix +is non--zero. + +\chapter{Algebraic Functions} +\section{Extended Euclidean Algorithm} +\index{mp\_exteuclid} +\begin{alltt} +int mp_exteuclid(mp_int *a, mp_int *b, + mp_int *U1, mp_int *U2, mp_int *U3); +\end{alltt} + +This finds the triple U1/U2/U3 using the Extended Euclidean algorithm such that the following equation holds. + +\begin{equation} +a \cdot U1 + b \cdot U2 = U3 +\end{equation} + +Any of the U1/U2/U3 paramters can be set to \textbf{NULL} if they are not desired. + +\section{Greatest Common Divisor} +\index{mp\_gcd} +\begin{alltt} +int mp_gcd (mp_int * a, mp_int * b, mp_int * c) +\end{alltt} +This will compute the greatest common divisor of $a$ and $b$ and store it in $c$. + +\section{Least Common Multiple} +\index{mp\_lcm} +\begin{alltt} +int mp_lcm (mp_int * a, mp_int * b, mp_int * c) +\end{alltt} +This will compute the least common multiple of $a$ and $b$ and store it in $c$. + +\section{Jacobi Symbol} +\index{mp\_jacobi} +\begin{alltt} +int mp_jacobi (mp_int * a, mp_int * p, int *c) +\end{alltt} +This will compute the Jacobi symbol for $a$ with respect to $p$. If $p$ is prime this essentially computes the Legendre +symbol. The result is stored in $c$ and can take on one of three values $\lbrace -1, 0, 1 \rbrace$. If $p$ is prime +then the result will be $-1$ when $a$ is not a quadratic residue modulo $p$. The result will be $0$ if $a$ divides $p$ +and the result will be $1$ if $a$ is a quadratic residue modulo $p$. + +\section{Modular Inverse} +\index{mp\_invmod} +\begin{alltt} +int mp_invmod (mp_int * a, mp_int * b, mp_int * c) +\end{alltt} +Computes the multiplicative inverse of $a$ modulo $b$ and stores the result in $c$ such that $ac \equiv 1 \mbox{ (mod }b\mbox{)}$. + +\section{Single Digit Functions} + +For those using small numbers (\textit{snicker snicker}) there are several ``helper'' functions + +\index{mp\_add\_d} \index{mp\_sub\_d} \index{mp\_mul\_d} \index{mp\_div\_d} \index{mp\_mod\_d} +\begin{alltt} +int mp_add_d(mp_int *a, mp_digit b, mp_int *c); +int mp_sub_d(mp_int *a, mp_digit b, mp_int *c); +int mp_mul_d(mp_int *a, mp_digit b, mp_int *c); +int mp_div_d(mp_int *a, mp_digit b, mp_int *c, mp_digit *d); +int mp_mod_d(mp_int *a, mp_digit b, mp_digit *c); +\end{alltt} + +These work like the full mp\_int capable variants except the second parameter $b$ is a mp\_digit. These +functions fairly handy if you have to work with relatively small numbers since you will not have to allocate +an entire mp\_int to store a number like $1$ or $2$. + +\input{bn.ind} + +\end{document} diff --git a/libtommath/bn_error.c b/libtommath/bn_error.c new file mode 100644 index 0000000..1546784 --- /dev/null +++ b/libtommath/bn_error.c @@ -0,0 +1,43 @@ +#include <tommath.h> +#ifdef BN_ERROR_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +static const struct { + int code; + char *msg; +} msgs[] = { + { MP_OKAY, "Successful" }, + { MP_MEM, "Out of heap" }, + { MP_VAL, "Value out of range" } +}; + +/* return a char * string for a given code */ +char *mp_error_to_string(int code) +{ + int x; + + /* scan the lookup table for the given message */ + for (x = 0; x < (int)(sizeof(msgs) / sizeof(msgs[0])); x++) { + if (msgs[x].code == code) { + return msgs[x].msg; + } + } + + /* generic reply for invalid code */ + return "Invalid error code"; +} + +#endif diff --git a/libtommath/bn_fast_mp_invmod.c b/libtommath/bn_fast_mp_invmod.c new file mode 100644 index 0000000..b5b9f10 --- /dev/null +++ b/libtommath/bn_fast_mp_invmod.c @@ -0,0 +1,145 @@ +#include <tommath.h> +#ifdef BN_FAST_MP_INVMOD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* computes the modular inverse via binary extended euclidean algorithm, + * that is c = 1/a mod b + * + * Based on slow invmod except this is optimized for the case where b is + * odd as per HAC Note 14.64 on pp. 610 + */ +int +fast_mp_invmod (mp_int * a, mp_int * b, mp_int * c) +{ + mp_int x, y, u, v, B, D; + int res, neg; + + /* 2. [modified] b must be odd */ + if (mp_iseven (b) == 1) { + return MP_VAL; + } + + /* init all our temps */ + if ((res = mp_init_multi(&x, &y, &u, &v, &B, &D, NULL)) != MP_OKAY) { + return res; + } + + /* x == modulus, y == value to invert */ + if ((res = mp_copy (b, &x)) != MP_OKAY) { + goto LBL_ERR; + } + + /* we need y = |a| */ + if ((res = mp_abs (a, &y)) != MP_OKAY) { + goto LBL_ERR; + } + + /* 3. u=x, v=y, A=1, B=0, C=0,D=1 */ + if ((res = mp_copy (&x, &u)) != MP_OKAY) { + goto LBL_ERR; + } + if ((res = mp_copy (&y, &v)) != MP_OKAY) { + goto LBL_ERR; + } + mp_set (&D, 1); + +top: + /* 4. while u is even do */ + while (mp_iseven (&u) == 1) { + /* 4.1 u = u/2 */ + if ((res = mp_div_2 (&u, &u)) != MP_OKAY) { + goto LBL_ERR; + } + /* 4.2 if B is odd then */ + if (mp_isodd (&B) == 1) { + if ((res = mp_sub (&B, &x, &B)) != MP_OKAY) { + goto LBL_ERR; + } + } + /* B = B/2 */ + if ((res = mp_div_2 (&B, &B)) != MP_OKAY) { + goto LBL_ERR; + } + } + + /* 5. while v is even do */ + while (mp_iseven (&v) == 1) { + /* 5.1 v = v/2 */ + if ((res = mp_div_2 (&v, &v)) != MP_OKAY) { + goto LBL_ERR; + } + /* 5.2 if D is odd then */ + if (mp_isodd (&D) == 1) { + /* D = (D-x)/2 */ + if ((res = mp_sub (&D, &x, &D)) != MP_OKAY) { + goto LBL_ERR; + } + } + /* D = D/2 */ + if ((res = mp_div_2 (&D, &D)) != MP_OKAY) { + goto LBL_ERR; + } + } + + /* 6. if u >= v then */ + if (mp_cmp (&u, &v) != MP_LT) { + /* u = u - v, B = B - D */ + if ((res = mp_sub (&u, &v, &u)) != MP_OKAY) { + goto LBL_ERR; + } + + if ((res = mp_sub (&B, &D, &B)) != MP_OKAY) { + goto LBL_ERR; + } + } else { + /* v - v - u, D = D - B */ + if ((res = mp_sub (&v, &u, &v)) != MP_OKAY) { + goto LBL_ERR; + } + + if ((res = mp_sub (&D, &B, &D)) != MP_OKAY) { + goto LBL_ERR; + } + } + + /* if not zero goto step 4 */ + if (mp_iszero (&u) == 0) { + goto top; + } + + /* now a = C, b = D, gcd == g*v */ + + /* if v != 1 then there is no inverse */ + if (mp_cmp_d (&v, 1) != MP_EQ) { + res = MP_VAL; + goto LBL_ERR; + } + + /* b is now the inverse */ + neg = a->sign; + while (D.sign == MP_NEG) { + if ((res = mp_add (&D, b, &D)) != MP_OKAY) { + goto LBL_ERR; + } + } + mp_exch (&D, c); + c->sign = neg; + res = MP_OKAY; + +LBL_ERR:mp_clear_multi (&x, &y, &u, &v, &B, &D, NULL); + return res; +} +#endif diff --git a/libtommath/bn_fast_mp_montgomery_reduce.c b/libtommath/bn_fast_mp_montgomery_reduce.c new file mode 100644 index 0000000..7373ae6 --- /dev/null +++ b/libtommath/bn_fast_mp_montgomery_reduce.c @@ -0,0 +1,169 @@ +#include <tommath.h> +#ifdef BN_FAST_MP_MONTGOMERY_REDUCE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* computes xR**-1 == x (mod N) via Montgomery Reduction + * + * This is an optimized implementation of montgomery_reduce + * which uses the comba method to quickly calculate the columns of the + * reduction. + * + * Based on Algorithm 14.32 on pp.601 of HAC. +*/ +int +fast_mp_montgomery_reduce (mp_int * x, mp_int * n, mp_digit rho) +{ + int ix, res, olduse; + mp_word W[MP_WARRAY]; + + /* get old used count */ + olduse = x->used; + + /* grow a as required */ + if (x->alloc < n->used + 1) { + if ((res = mp_grow (x, n->used + 1)) != MP_OKAY) { + return res; + } + } + + /* first we have to get the digits of the input into + * an array of double precision words W[...] + */ + { + register mp_word *_W; + register mp_digit *tmpx; + + /* alias for the W[] array */ + _W = W; + + /* alias for the digits of x*/ + tmpx = x->dp; + + /* copy the digits of a into W[0..a->used-1] */ + for (ix = 0; ix < x->used; ix++) { + *_W++ = *tmpx++; + } + + /* zero the high words of W[a->used..m->used*2] */ + for (; ix < n->used * 2 + 1; ix++) { + *_W++ = 0; + } + } + + /* now we proceed to zero successive digits + * from the least significant upwards + */ + for (ix = 0; ix < n->used; ix++) { + /* mu = ai * m' mod b + * + * We avoid a double precision multiplication (which isn't required) + * by casting the value down to a mp_digit. Note this requires + * that W[ix-1] have the carry cleared (see after the inner loop) + */ + register mp_digit mu; + mu = (mp_digit) (((W[ix] & MP_MASK) * rho) & MP_MASK); + + /* a = a + mu * m * b**i + * + * This is computed in place and on the fly. The multiplication + * by b**i is handled by offseting which columns the results + * are added to. + * + * Note the comba method normally doesn't handle carries in the + * inner loop In this case we fix the carry from the previous + * column since the Montgomery reduction requires digits of the + * result (so far) [see above] to work. This is + * handled by fixing up one carry after the inner loop. The + * carry fixups are done in order so after these loops the + * first m->used words of W[] have the carries fixed + */ + { + register int iy; + register mp_digit *tmpn; + register mp_word *_W; + + /* alias for the digits of the modulus */ + tmpn = n->dp; + + /* Alias for the columns set by an offset of ix */ + _W = W + ix; + + /* inner loop */ + for (iy = 0; iy < n->used; iy++) { + *_W++ += ((mp_word)mu) * ((mp_word)*tmpn++); + } + } + + /* now fix carry for next digit, W[ix+1] */ + W[ix + 1] += W[ix] >> ((mp_word) DIGIT_BIT); + } + + /* now we have to propagate the carries and + * shift the words downward [all those least + * significant digits we zeroed]. + */ + { + register mp_digit *tmpx; + register mp_word *_W, *_W1; + + /* nox fix rest of carries */ + + /* alias for current word */ + _W1 = W + ix; + + /* alias for next word, where the carry goes */ + _W = W + ++ix; + + for (; ix <= n->used * 2 + 1; ix++) { + *_W++ += *_W1++ >> ((mp_word) DIGIT_BIT); + } + + /* copy out, A = A/b**n + * + * The result is A/b**n but instead of converting from an + * array of mp_word to mp_digit than calling mp_rshd + * we just copy them in the right order + */ + + /* alias for destination word */ + tmpx = x->dp; + + /* alias for shifted double precision result */ + _W = W + n->used; + + for (ix = 0; ix < n->used + 1; ix++) { + *tmpx++ = (mp_digit)(*_W++ & ((mp_word) MP_MASK)); + } + + /* zero oldused digits, if the input a was larger than + * m->used+1 we'll have to clear the digits + */ + for (; ix < olduse; ix++) { + *tmpx++ = 0; + } + } + + /* set the max used and clamp */ + x->used = n->used + 1; + mp_clamp (x); + + /* if A >= m then A = A - m */ + if (mp_cmp_mag (x, n) != MP_LT) { + return s_mp_sub (x, n, x); + } + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_fast_s_mp_mul_digs.c b/libtommath/bn_fast_s_mp_mul_digs.c new file mode 100644 index 0000000..e1ff5f3 --- /dev/null +++ b/libtommath/bn_fast_s_mp_mul_digs.c @@ -0,0 +1,106 @@ +#include <tommath.h> +#ifdef BN_FAST_S_MP_MUL_DIGS_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* Fast (comba) multiplier + * + * This is the fast column-array [comba] multiplier. It is + * designed to compute the columns of the product first + * then handle the carries afterwards. This has the effect + * of making the nested loops that compute the columns very + * simple and schedulable on super-scalar processors. + * + * This has been modified to produce a variable number of + * digits of output so if say only a half-product is required + * you don't have to compute the upper half (a feature + * required for fast Barrett reduction). + * + * Based on Algorithm 14.12 on pp.595 of HAC. + * + */ +int +fast_s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs) +{ + int olduse, res, pa, ix, iz; + mp_digit W[MP_WARRAY]; + register mp_word _W; + + /* grow the destination as required */ + if (c->alloc < digs) { + if ((res = mp_grow (c, digs)) != MP_OKAY) { + return res; + } + } + + /* number of output digits to produce */ + pa = MIN(digs, a->used + b->used); + + /* clear the carry */ + _W = 0; + for (ix = 0; ix < pa; ix++) { + int tx, ty; + int iy; + mp_digit *tmpx, *tmpy; + + /* get offsets into the two bignums */ + ty = MIN(b->used-1, ix); + tx = ix - ty; + + /* setup temp aliases */ + tmpx = a->dp + tx; + tmpy = b->dp + ty; + + /* this is the number of times the loop will iterrate, essentially its + while (tx++ < a->used && ty-- >= 0) { ... } + */ + iy = MIN(a->used-tx, ty+1); + + /* execute loop */ + for (iz = 0; iz < iy; ++iz) { + _W += ((mp_word)*tmpx++)*((mp_word)*tmpy--); + } + + /* store term */ + W[ix] = ((mp_digit)_W) & MP_MASK; + + /* make next carry */ + _W = _W >> ((mp_word)DIGIT_BIT); + } + + /* store final carry */ + W[ix] = _W; + + /* setup dest */ + olduse = c->used; + c->used = digs; + + { + register mp_digit *tmpc; + tmpc = c->dp; + for (ix = 0; ix < digs; ix++) { + /* now extract the previous digit [below the carry] */ + *tmpc++ = W[ix]; + } + + /* clear unused digits [that existed in the old copy of c] */ + for (; ix < olduse; ix++) { + *tmpc++ = 0; + } + } + mp_clamp (c); + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_fast_s_mp_mul_high_digs.c b/libtommath/bn_fast_s_mp_mul_high_digs.c new file mode 100644 index 0000000..064a9dd --- /dev/null +++ b/libtommath/bn_fast_s_mp_mul_high_digs.c @@ -0,0 +1,98 @@ +#include <tommath.h> +#ifdef BN_FAST_S_MP_MUL_HIGH_DIGS_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* this is a modified version of fast_s_mul_digs that only produces + * output digits *above* digs. See the comments for fast_s_mul_digs + * to see how it works. + * + * This is used in the Barrett reduction since for one of the multiplications + * only the higher digits were needed. This essentially halves the work. + * + * Based on Algorithm 14.12 on pp.595 of HAC. + */ +int +fast_s_mp_mul_high_digs (mp_int * a, mp_int * b, mp_int * c, int digs) +{ + int olduse, res, pa, ix, iz; + mp_digit W[MP_WARRAY]; + mp_word _W; + + /* grow the destination as required */ + pa = a->used + b->used; + if (c->alloc < pa) { + if ((res = mp_grow (c, pa)) != MP_OKAY) { + return res; + } + } + + /* number of output digits to produce */ + pa = a->used + b->used; + _W = 0; + for (ix = digs; ix < pa; ix++) { + int tx, ty, iy; + mp_digit *tmpx, *tmpy; + + /* get offsets into the two bignums */ + ty = MIN(b->used-1, ix); + tx = ix - ty; + + /* setup temp aliases */ + tmpx = a->dp + tx; + tmpy = b->dp + ty; + + /* this is the number of times the loop will iterrate, essentially its + while (tx++ < a->used && ty-- >= 0) { ... } + */ + iy = MIN(a->used-tx, ty+1); + + /* execute loop */ + for (iz = 0; iz < iy; iz++) { + _W += ((mp_word)*tmpx++)*((mp_word)*tmpy--); + } + + /* store term */ + W[ix] = ((mp_digit)_W) & MP_MASK; + + /* make next carry */ + _W = _W >> ((mp_word)DIGIT_BIT); + } + + /* store final carry */ + W[ix] = _W; + + /* setup dest */ + olduse = c->used; + c->used = pa; + + { + register mp_digit *tmpc; + + tmpc = c->dp + digs; + for (ix = digs; ix <= pa; ix++) { + /* now extract the previous digit [below the carry] */ + *tmpc++ = W[ix]; + } + + /* clear unused digits [that existed in the old copy of c] */ + for (; ix < olduse; ix++) { + *tmpc++ = 0; + } + } + mp_clamp (c); + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_fast_s_mp_sqr.c b/libtommath/bn_fast_s_mp_sqr.c new file mode 100644 index 0000000..d6014ab --- /dev/null +++ b/libtommath/bn_fast_s_mp_sqr.c @@ -0,0 +1,129 @@ +#include <tommath.h> +#ifdef BN_FAST_S_MP_SQR_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* fast squaring + * + * This is the comba method where the columns of the product + * are computed first then the carries are computed. This + * has the effect of making a very simple inner loop that + * is executed the most + * + * W2 represents the outer products and W the inner. + * + * A further optimizations is made because the inner + * products are of the form "A * B * 2". The *2 part does + * not need to be computed until the end which is good + * because 64-bit shifts are slow! + * + * Based on Algorithm 14.16 on pp.597 of HAC. + * + */ +/* the jist of squaring... + +you do like mult except the offset of the tmpx [one that starts closer to zero] +can't equal the offset of tmpy. So basically you set up iy like before then you min it with +(ty-tx) so that it never happens. You double all those you add in the inner loop + +After that loop you do the squares and add them in. + +Remove W2 and don't memset W + +*/ + +int fast_s_mp_sqr (mp_int * a, mp_int * b) +{ + int olduse, res, pa, ix, iz; + mp_digit W[MP_WARRAY], *tmpx; + mp_word W1; + + /* grow the destination as required */ + pa = a->used + a->used; + if (b->alloc < pa) { + if ((res = mp_grow (b, pa)) != MP_OKAY) { + return res; + } + } + + /* number of output digits to produce */ + W1 = 0; + for (ix = 0; ix < pa; ix++) { + int tx, ty, iy; + mp_word _W; + mp_digit *tmpy; + + /* clear counter */ + _W = 0; + + /* get offsets into the two bignums */ + ty = MIN(a->used-1, ix); + tx = ix - ty; + + /* setup temp aliases */ + tmpx = a->dp + tx; + tmpy = a->dp + ty; + + /* this is the number of times the loop will iterrate, essentially its + while (tx++ < a->used && ty-- >= 0) { ... } + */ + iy = MIN(a->used-tx, ty+1); + + /* now for squaring tx can never equal ty + * we halve the distance since they approach at a rate of 2x + * and we have to round because odd cases need to be executed + */ + iy = MIN(iy, (ty-tx+1)>>1); + + /* execute loop */ + for (iz = 0; iz < iy; iz++) { + _W += ((mp_word)*tmpx++)*((mp_word)*tmpy--); + } + + /* double the inner product and add carry */ + _W = _W + _W + W1; + + /* even columns have the square term in them */ + if ((ix&1) == 0) { + _W += ((mp_word)a->dp[ix>>1])*((mp_word)a->dp[ix>>1]); + } + + /* store it */ + W[ix] = _W; + + /* make next carry */ + W1 = _W >> ((mp_word)DIGIT_BIT); + } + + /* setup dest */ + olduse = b->used; + b->used = a->used+a->used; + + { + mp_digit *tmpb; + tmpb = b->dp; + for (ix = 0; ix < pa; ix++) { + *tmpb++ = W[ix] & MP_MASK; + } + + /* clear unused digits [that existed in the old copy of c] */ + for (; ix < olduse; ix++) { + *tmpb++ = 0; + } + } + mp_clamp (b); + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_mp_2expt.c b/libtommath/bn_mp_2expt.c new file mode 100644 index 0000000..45a6818 --- /dev/null +++ b/libtommath/bn_mp_2expt.c @@ -0,0 +1,44 @@ +#include <tommath.h> +#ifdef BN_MP_2EXPT_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* computes a = 2**b + * + * Simple algorithm which zeroes the int, grows it then just sets one bit + * as required. + */ +int +mp_2expt (mp_int * a, int b) +{ + int res; + + /* zero a as per default */ + mp_zero (a); + + /* grow a to accomodate the single bit */ + if ((res = mp_grow (a, b / DIGIT_BIT + 1)) != MP_OKAY) { + return res; + } + + /* set the used count of where the bit will go */ + a->used = b / DIGIT_BIT + 1; + + /* put the single bit in its place */ + a->dp[b / DIGIT_BIT] = ((mp_digit)1) << (b % DIGIT_BIT); + + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_mp_abs.c b/libtommath/bn_mp_abs.c new file mode 100644 index 0000000..34f810f --- /dev/null +++ b/libtommath/bn_mp_abs.c @@ -0,0 +1,39 @@ +#include <tommath.h> +#ifdef BN_MP_ABS_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* b = |a| + * + * Simple function copies the input and fixes the sign to positive + */ +int +mp_abs (mp_int * a, mp_int * b) +{ + int res; + + /* copy a to b */ + if (a != b) { + if ((res = mp_copy (a, b)) != MP_OKAY) { + return res; + } + } + + /* force the sign of b to positive */ + b->sign = MP_ZPOS; + + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_mp_add.c b/libtommath/bn_mp_add.c new file mode 100644 index 0000000..554b7f7 --- /dev/null +++ b/libtommath/bn_mp_add.c @@ -0,0 +1,49 @@ +#include <tommath.h> +#ifdef BN_MP_ADD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* high level addition (handles signs) */ +int mp_add (mp_int * a, mp_int * b, mp_int * c) +{ + int sa, sb, res; + + /* get sign of both inputs */ + sa = a->sign; + sb = b->sign; + + /* handle two cases, not four */ + if (sa == sb) { + /* both positive or both negative */ + /* add their magnitudes, copy the sign */ + c->sign = sa; + res = s_mp_add (a, b, c); + } else { + /* one positive, the other negative */ + /* subtract the one with the greater magnitude from */ + /* the one of the lesser magnitude. The result gets */ + /* the sign of the one with the greater magnitude. */ + if (mp_cmp_mag (a, b) == MP_LT) { + c->sign = sb; + res = s_mp_sub (b, a, c); + } else { + c->sign = sa; + res = s_mp_sub (a, b, c); + } + } + return res; +} + +#endif diff --git a/libtommath/bn_mp_add_d.c b/libtommath/bn_mp_add_d.c new file mode 100644 index 0000000..bdd0280 --- /dev/null +++ b/libtommath/bn_mp_add_d.c @@ -0,0 +1,105 @@ +#include <tommath.h> +#ifdef BN_MP_ADD_D_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* single digit addition */ +int +mp_add_d (mp_int * a, mp_digit b, mp_int * c) +{ + int res, ix, oldused; + mp_digit *tmpa, *tmpc, mu; + + /* grow c as required */ + if (c->alloc < a->used + 1) { + if ((res = mp_grow(c, a->used + 1)) != MP_OKAY) { + return res; + } + } + + /* if a is negative and |a| >= b, call c = |a| - b */ + if (a->sign == MP_NEG && (a->used > 1 || a->dp[0] >= b)) { + /* temporarily fix sign of a */ + a->sign = MP_ZPOS; + + /* c = |a| - b */ + res = mp_sub_d(a, b, c); + + /* fix sign */ + a->sign = c->sign = MP_NEG; + + return res; + } + + /* old number of used digits in c */ + oldused = c->used; + + /* sign always positive */ + c->sign = MP_ZPOS; + + /* source alias */ + tmpa = a->dp; + + /* destination alias */ + tmpc = c->dp; + + /* if a is positive */ + if (a->sign == MP_ZPOS) { + /* add digit, after this we're propagating + * the carry. + */ + *tmpc = *tmpa++ + b; + mu = *tmpc >> DIGIT_BIT; + *tmpc++ &= MP_MASK; + + /* now handle rest of the digits */ + for (ix = 1; ix < a->used; ix++) { + *tmpc = *tmpa++ + mu; + mu = *tmpc >> DIGIT_BIT; + *tmpc++ &= MP_MASK; + } + /* set final carry */ + ix++; + *tmpc++ = mu; + + /* setup size */ + c->used = a->used + 1; + } else { + /* a was negative and |a| < b */ + c->used = 1; + + /* the result is a single digit */ + if (a->used == 1) { + *tmpc++ = b - a->dp[0]; + } else { + *tmpc++ = b; + } + + /* setup count so the clearing of oldused + * can fall through correctly + */ + ix = 1; + } + + /* now zero to oldused */ + while (ix++ < oldused) { + *tmpc++ = 0; + } + mp_clamp(c); + + return MP_OKAY; +} + +#endif diff --git a/libtommath/bn_mp_addmod.c b/libtommath/bn_mp_addmod.c new file mode 100644 index 0000000..13eb33f --- /dev/null +++ b/libtommath/bn_mp_addmod.c @@ -0,0 +1,37 @@ +#include <tommath.h> +#ifdef BN_MP_ADDMOD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* d = a + b (mod c) */ +int +mp_addmod (mp_int * a, mp_int * b, mp_int * c, mp_int * d) +{ + int res; + mp_int t; + + if ((res = mp_init (&t)) != MP_OKAY) { + return res; + } + + if ((res = mp_add (a, b, &t)) != MP_OKAY) { + mp_clear (&t); + return res; + } + res = mp_mod (&t, c, d); + mp_clear (&t); + return res; +} +#endif diff --git a/libtommath/bn_mp_and.c b/libtommath/bn_mp_and.c new file mode 100644 index 0000000..61dc386 --- /dev/null +++ b/libtommath/bn_mp_and.c @@ -0,0 +1,53 @@ +#include <tommath.h> +#ifdef BN_MP_AND_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* AND two ints together */ +int +mp_and (mp_int * a, mp_int * b, mp_int * c) +{ + int res, ix, px; + mp_int t, *x; + + if (a->used > b->used) { + if ((res = mp_init_copy (&t, a)) != MP_OKAY) { + return res; + } + px = b->used; + x = b; + } else { + if ((res = mp_init_copy (&t, b)) != MP_OKAY) { + return res; + } + px = a->used; + x = a; + } + + for (ix = 0; ix < px; ix++) { + t.dp[ix] &= x->dp[ix]; + } + + /* zero digits above the last from the smallest mp_int */ + for (; ix < t.used; ix++) { + t.dp[ix] = 0; + } + + mp_clamp (&t); + mp_exch (c, &t); + mp_clear (&t); + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_mp_clamp.c b/libtommath/bn_mp_clamp.c new file mode 100644 index 0000000..c172611 --- /dev/null +++ b/libtommath/bn_mp_clamp.c @@ -0,0 +1,40 @@ +#include <tommath.h> +#ifdef BN_MP_CLAMP_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* trim unused digits + * + * This is used to ensure that leading zero digits are + * trimed and the leading "used" digit will be non-zero + * Typically very fast. Also fixes the sign if there + * are no more leading digits + */ +void +mp_clamp (mp_int * a) +{ + /* decrease used while the most significant digit is + * zero. + */ + while (a->used > 0 && a->dp[a->used - 1] == 0) { + --(a->used); + } + + /* reset the sign flag if used == 0 */ + if (a->used == 0) { + a->sign = MP_ZPOS; + } +} +#endif diff --git a/libtommath/bn_mp_clear.c b/libtommath/bn_mp_clear.c new file mode 100644 index 0000000..5342648 --- /dev/null +++ b/libtommath/bn_mp_clear.c @@ -0,0 +1,40 @@ +#include <tommath.h> +#ifdef BN_MP_CLEAR_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* clear one (frees) */ +void +mp_clear (mp_int * a) +{ + int i; + + /* only do anything if a hasn't been freed previously */ + if (a->dp != NULL) { + /* first zero the digits */ + for (i = 0; i < a->used; i++) { + a->dp[i] = 0; + } + + /* free ram */ + XFREE(a->dp); + + /* reset members to make debugging easier */ + a->dp = NULL; + a->alloc = a->used = 0; + a->sign = MP_ZPOS; + } +} +#endif diff --git a/libtommath/bn_mp_clear_multi.c b/libtommath/bn_mp_clear_multi.c new file mode 100644 index 0000000..24cbe73 --- /dev/null +++ b/libtommath/bn_mp_clear_multi.c @@ -0,0 +1,30 @@ +#include <tommath.h> +#ifdef BN_MP_CLEAR_MULTI_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ +#include <stdarg.h> + +void mp_clear_multi(mp_int *mp, ...) +{ + mp_int* next_mp = mp; + va_list args; + va_start(args, mp); + while (next_mp != NULL) { + mp_clear(next_mp); + next_mp = va_arg(args, mp_int*); + } + va_end(args); +} +#endif diff --git a/libtommath/bn_mp_cmp.c b/libtommath/bn_mp_cmp.c new file mode 100644 index 0000000..583b5f8 --- /dev/null +++ b/libtommath/bn_mp_cmp.c @@ -0,0 +1,39 @@ +#include <tommath.h> +#ifdef BN_MP_CMP_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* compare two ints (signed)*/ +int +mp_cmp (mp_int * a, mp_int * b) +{ + /* compare based on sign */ + if (a->sign != b->sign) { + if (a->sign == MP_NEG) { + return MP_LT; + } else { + return MP_GT; + } + } + + /* compare digits */ + if (a->sign == MP_NEG) { + /* if negative compare opposite direction */ + return mp_cmp_mag(b, a); + } else { + return mp_cmp_mag(a, b); + } +} +#endif diff --git a/libtommath/bn_mp_cmp_d.c b/libtommath/bn_mp_cmp_d.c new file mode 100644 index 0000000..882b1c9 --- /dev/null +++ b/libtommath/bn_mp_cmp_d.c @@ -0,0 +1,40 @@ +#include <tommath.h> +#ifdef BN_MP_CMP_D_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* compare a digit */ +int mp_cmp_d(mp_int * a, mp_digit b) +{ + /* compare based on sign */ + if (a->sign == MP_NEG) { + return MP_LT; + } + + /* compare based on magnitude */ + if (a->used > 1) { + return MP_GT; + } + + /* compare the only digit of a to b */ + if (a->dp[0] > b) { + return MP_GT; + } else if (a->dp[0] < b) { + return MP_LT; + } else { + return MP_EQ; + } +} +#endif diff --git a/libtommath/bn_mp_cmp_mag.c b/libtommath/bn_mp_cmp_mag.c new file mode 100644 index 0000000..a0f351c --- /dev/null +++ b/libtommath/bn_mp_cmp_mag.c @@ -0,0 +1,51 @@ +#include <tommath.h> +#ifdef BN_MP_CMP_MAG_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* compare maginitude of two ints (unsigned) */ +int mp_cmp_mag (mp_int * a, mp_int * b) +{ + int n; + mp_digit *tmpa, *tmpb; + + /* compare based on # of non-zero digits */ + if (a->used > b->used) { + return MP_GT; + } + + if (a->used < b->used) { + return MP_LT; + } + + /* alias for a */ + tmpa = a->dp + (a->used - 1); + + /* alias for b */ + tmpb = b->dp + (a->used - 1); + + /* compare based on digits */ + for (n = 0; n < a->used; ++n, --tmpa, --tmpb) { + if (*tmpa > *tmpb) { + return MP_GT; + } + + if (*tmpa < *tmpb) { + return MP_LT; + } + } + return MP_EQ; +} +#endif diff --git a/libtommath/bn_mp_cnt_lsb.c b/libtommath/bn_mp_cnt_lsb.c new file mode 100644 index 0000000..571f03f --- /dev/null +++ b/libtommath/bn_mp_cnt_lsb.c @@ -0,0 +1,49 @@ +#include <tommath.h> +#ifdef BN_MP_CNT_LSB_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +static const int lnz[16] = { + 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0 +}; + +/* Counts the number of lsbs which are zero before the first zero bit */ +int mp_cnt_lsb(mp_int *a) +{ + int x; + mp_digit q, qq; + + /* easy out */ + if (mp_iszero(a) == 1) { + return 0; + } + + /* scan lower digits until non-zero */ + for (x = 0; x < a->used && a->dp[x] == 0; x++); + q = a->dp[x]; + x *= DIGIT_BIT; + + /* now scan this digit until a 1 is found */ + if ((q & 1) == 0) { + do { + qq = q & 15; + x += lnz[qq]; + q >>= 4; + } while (qq == 0); + } + return x; +} + +#endif diff --git a/libtommath/bn_mp_copy.c b/libtommath/bn_mp_copy.c new file mode 100644 index 0000000..183ec9b --- /dev/null +++ b/libtommath/bn_mp_copy.c @@ -0,0 +1,64 @@ +#include <tommath.h> +#ifdef BN_MP_COPY_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* copy, b = a */ +int +mp_copy (mp_int * a, mp_int * b) +{ + int res, n; + + /* if dst == src do nothing */ + if (a == b) { + return MP_OKAY; + } + + /* grow dest */ + if (b->alloc < a->used) { + if ((res = mp_grow (b, a->used)) != MP_OKAY) { + return res; + } + } + + /* zero b and copy the parameters over */ + { + register mp_digit *tmpa, *tmpb; + + /* pointer aliases */ + + /* source */ + tmpa = a->dp; + + /* destination */ + tmpb = b->dp; + + /* copy all the digits */ + for (n = 0; n < a->used; n++) { + *tmpb++ = *tmpa++; + } + + /* clear high digits */ + for (; n < b->used; n++) { + *tmpb++ = 0; + } + } + + /* copy used count and sign */ + b->used = a->used; + b->sign = a->sign; + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_mp_count_bits.c b/libtommath/bn_mp_count_bits.c new file mode 100644 index 0000000..f3f85ac --- /dev/null +++ b/libtommath/bn_mp_count_bits.c @@ -0,0 +1,41 @@ +#include <tommath.h> +#ifdef BN_MP_COUNT_BITS_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* returns the number of bits in an int */ +int +mp_count_bits (mp_int * a) +{ + int r; + mp_digit q; + + /* shortcut */ + if (a->used == 0) { + return 0; + } + + /* get number of digits and add that */ + r = (a->used - 1) * DIGIT_BIT; + + /* take the last digit and count the bits in it */ + q = a->dp[a->used - 1]; + while (q > ((mp_digit) 0)) { + ++r; + q >>= ((mp_digit) 1); + } + return r; +} +#endif diff --git a/libtommath/bn_mp_div.c b/libtommath/bn_mp_div.c new file mode 100644 index 0000000..6b2b8f0 --- /dev/null +++ b/libtommath/bn_mp_div.c @@ -0,0 +1,288 @@ +#include <tommath.h> +#ifdef BN_MP_DIV_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +#ifdef BN_MP_DIV_SMALL + +/* slower bit-bang division... also smaller */ +int mp_div(mp_int * a, mp_int * b, mp_int * c, mp_int * d) +{ + mp_int ta, tb, tq, q; + int res, n, n2; + + /* is divisor zero ? */ + if (mp_iszero (b) == 1) { + return MP_VAL; + } + + /* if a < b then q=0, r = a */ + if (mp_cmp_mag (a, b) == MP_LT) { + if (d != NULL) { + res = mp_copy (a, d); + } else { + res = MP_OKAY; + } + if (c != NULL) { + mp_zero (c); + } + return res; + } + + /* init our temps */ + if ((res = mp_init_multi(&ta, &tb, &tq, &q, NULL) != MP_OKAY)) { + return res; + } + + + mp_set(&tq, 1); + n = mp_count_bits(a) - mp_count_bits(b); + if (((res = mp_abs(a, &ta)) != MP_OKAY) || + ((res = mp_abs(b, &tb)) != MP_OKAY) || + ((res = mp_mul_2d(&tb, n, &tb)) != MP_OKAY) || + ((res = mp_mul_2d(&tq, n, &tq)) != MP_OKAY)) { + goto LBL_ERR; + } + + while (n-- >= 0) { + if (mp_cmp(&tb, &ta) != MP_GT) { + if (((res = mp_sub(&ta, &tb, &ta)) != MP_OKAY) || + ((res = mp_add(&q, &tq, &q)) != MP_OKAY)) { + goto LBL_ERR; + } + } + if (((res = mp_div_2d(&tb, 1, &tb, NULL)) != MP_OKAY) || + ((res = mp_div_2d(&tq, 1, &tq, NULL)) != MP_OKAY)) { + goto LBL_ERR; + } + } + + /* now q == quotient and ta == remainder */ + n = a->sign; + n2 = (a->sign == b->sign ? MP_ZPOS : MP_NEG); + if (c != NULL) { + mp_exch(c, &q); + c->sign = (mp_iszero(c) == MP_YES) ? MP_ZPOS : n2; + } + if (d != NULL) { + mp_exch(d, &ta); + d->sign = (mp_iszero(d) == MP_YES) ? MP_ZPOS : n; + } +LBL_ERR: + mp_clear_multi(&ta, &tb, &tq, &q, NULL); + return res; +} + +#else + +/* integer signed division. + * c*b + d == a [e.g. a/b, c=quotient, d=remainder] + * HAC pp.598 Algorithm 14.20 + * + * Note that the description in HAC is horribly + * incomplete. For example, it doesn't consider + * the case where digits are removed from 'x' in + * the inner loop. It also doesn't consider the + * case that y has fewer than three digits, etc.. + * + * The overall algorithm is as described as + * 14.20 from HAC but fixed to treat these cases. +*/ +int mp_div (mp_int * a, mp_int * b, mp_int * c, mp_int * d) +{ + mp_int q, x, y, t1, t2; + int res, n, t, i, norm, neg; + + /* is divisor zero ? */ + if (mp_iszero (b) == 1) { + return MP_VAL; + } + + /* if a < b then q=0, r = a */ + if (mp_cmp_mag (a, b) == MP_LT) { + if (d != NULL) { + res = mp_copy (a, d); + } else { + res = MP_OKAY; + } + if (c != NULL) { + mp_zero (c); + } + return res; + } + + if ((res = mp_init_size (&q, a->used + 2)) != MP_OKAY) { + return res; + } + q.used = a->used + 2; + + if ((res = mp_init (&t1)) != MP_OKAY) { + goto LBL_Q; + } + + if ((res = mp_init (&t2)) != MP_OKAY) { + goto LBL_T1; + } + + if ((res = mp_init_copy (&x, a)) != MP_OKAY) { + goto LBL_T2; + } + + if ((res = mp_init_copy (&y, b)) != MP_OKAY) { + goto LBL_X; + } + + /* fix the sign */ + neg = (a->sign == b->sign) ? MP_ZPOS : MP_NEG; + x.sign = y.sign = MP_ZPOS; + + /* normalize both x and y, ensure that y >= b/2, [b == 2**DIGIT_BIT] */ + norm = mp_count_bits(&y) % DIGIT_BIT; + if (norm < (int)(DIGIT_BIT-1)) { + norm = (DIGIT_BIT-1) - norm; + if ((res = mp_mul_2d (&x, norm, &x)) != MP_OKAY) { + goto LBL_Y; + } + if ((res = mp_mul_2d (&y, norm, &y)) != MP_OKAY) { + goto LBL_Y; + } + } else { + norm = 0; + } + + /* note hac does 0 based, so if used==5 then its 0,1,2,3,4, e.g. use 4 */ + n = x.used - 1; + t = y.used - 1; + + /* while (x >= y*b**n-t) do { q[n-t] += 1; x -= y*b**{n-t} } */ + if ((res = mp_lshd (&y, n - t)) != MP_OKAY) { /* y = y*b**{n-t} */ + goto LBL_Y; + } + + while (mp_cmp (&x, &y) != MP_LT) { + ++(q.dp[n - t]); + if ((res = mp_sub (&x, &y, &x)) != MP_OKAY) { + goto LBL_Y; + } + } + + /* reset y by shifting it back down */ + mp_rshd (&y, n - t); + + /* step 3. for i from n down to (t + 1) */ + for (i = n; i >= (t + 1); i--) { + if (i > x.used) { + continue; + } + + /* step 3.1 if xi == yt then set q{i-t-1} to b-1, + * otherwise set q{i-t-1} to (xi*b + x{i-1})/yt */ + if (x.dp[i] == y.dp[t]) { + q.dp[i - t - 1] = ((((mp_digit)1) << DIGIT_BIT) - 1); + } else { + mp_word tmp; + tmp = ((mp_word) x.dp[i]) << ((mp_word) DIGIT_BIT); + tmp |= ((mp_word) x.dp[i - 1]); + tmp /= ((mp_word) y.dp[t]); + if (tmp > (mp_word) MP_MASK) + tmp = MP_MASK; + q.dp[i - t - 1] = (mp_digit) (tmp & (mp_word) (MP_MASK)); + } + + /* while (q{i-t-1} * (yt * b + y{t-1})) > + xi * b**2 + xi-1 * b + xi-2 + + do q{i-t-1} -= 1; + */ + q.dp[i - t - 1] = (q.dp[i - t - 1] + 1) & MP_MASK; + do { + q.dp[i - t - 1] = (q.dp[i - t - 1] - 1) & MP_MASK; + + /* find left hand */ + mp_zero (&t1); + t1.dp[0] = (t - 1 < 0) ? 0 : y.dp[t - 1]; + t1.dp[1] = y.dp[t]; + t1.used = 2; + if ((res = mp_mul_d (&t1, q.dp[i - t - 1], &t1)) != MP_OKAY) { + goto LBL_Y; + } + + /* find right hand */ + t2.dp[0] = (i - 2 < 0) ? 0 : x.dp[i - 2]; + t2.dp[1] = (i - 1 < 0) ? 0 : x.dp[i - 1]; + t2.dp[2] = x.dp[i]; + t2.used = 3; + } while (mp_cmp_mag(&t1, &t2) == MP_GT); + + /* step 3.3 x = x - q{i-t-1} * y * b**{i-t-1} */ + if ((res = mp_mul_d (&y, q.dp[i - t - 1], &t1)) != MP_OKAY) { + goto LBL_Y; + } + + if ((res = mp_lshd (&t1, i - t - 1)) != MP_OKAY) { + goto LBL_Y; + } + + if ((res = mp_sub (&x, &t1, &x)) != MP_OKAY) { + goto LBL_Y; + } + + /* if x < 0 then { x = x + y*b**{i-t-1}; q{i-t-1} -= 1; } */ + if (x.sign == MP_NEG) { + if ((res = mp_copy (&y, &t1)) != MP_OKAY) { + goto LBL_Y; + } + if ((res = mp_lshd (&t1, i - t - 1)) != MP_OKAY) { + goto LBL_Y; + } + if ((res = mp_add (&x, &t1, &x)) != MP_OKAY) { + goto LBL_Y; + } + + q.dp[i - t - 1] = (q.dp[i - t - 1] - 1UL) & MP_MASK; + } + } + + /* now q is the quotient and x is the remainder + * [which we have to normalize] + */ + + /* get sign before writing to c */ + x.sign = x.used == 0 ? MP_ZPOS : a->sign; + + if (c != NULL) { + mp_clamp (&q); + mp_exch (&q, c); + c->sign = neg; + } + + if (d != NULL) { + mp_div_2d (&x, norm, &x, NULL); + mp_exch (&x, d); + } + + res = MP_OKAY; + +LBL_Y:mp_clear (&y); +LBL_X:mp_clear (&x); +LBL_T2:mp_clear (&t2); +LBL_T1:mp_clear (&t1); +LBL_Q:mp_clear (&q); + return res; +} + +#endif + +#endif diff --git a/libtommath/bn_mp_div_2.c b/libtommath/bn_mp_div_2.c new file mode 100644 index 0000000..5777997 --- /dev/null +++ b/libtommath/bn_mp_div_2.c @@ -0,0 +1,64 @@ +#include <tommath.h> +#ifdef BN_MP_DIV_2_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* b = a/2 */ +int mp_div_2(mp_int * a, mp_int * b) +{ + int x, res, oldused; + + /* copy */ + if (b->alloc < a->used) { + if ((res = mp_grow (b, a->used)) != MP_OKAY) { + return res; + } + } + + oldused = b->used; + b->used = a->used; + { + register mp_digit r, rr, *tmpa, *tmpb; + + /* source alias */ + tmpa = a->dp + b->used - 1; + + /* dest alias */ + tmpb = b->dp + b->used - 1; + + /* carry */ + r = 0; + for (x = b->used - 1; x >= 0; x--) { + /* get the carry for the next iteration */ + rr = *tmpa & 1; + + /* shift the current digit, add in carry and store */ + *tmpb-- = (*tmpa-- >> 1) | (r << (DIGIT_BIT - 1)); + + /* forward carry to next iteration */ + r = rr; + } + + /* zero excess digits */ + tmpb = b->dp + b->used; + for (x = b->used; x < oldused; x++) { + *tmpb++ = 0; + } + } + b->sign = a->sign; + mp_clamp (b); + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_mp_div_2d.c b/libtommath/bn_mp_div_2d.c new file mode 100644 index 0000000..cf103f2 --- /dev/null +++ b/libtommath/bn_mp_div_2d.c @@ -0,0 +1,93 @@ +#include <tommath.h> +#ifdef BN_MP_DIV_2D_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* shift right by a certain bit count (store quotient in c, optional remainder in d) */ +int mp_div_2d (mp_int * a, int b, mp_int * c, mp_int * d) +{ + mp_digit D, r, rr; + int x, res; + mp_int t; + + + /* if the shift count is <= 0 then we do no work */ + if (b <= 0) { + res = mp_copy (a, c); + if (d != NULL) { + mp_zero (d); + } + return res; + } + + if ((res = mp_init (&t)) != MP_OKAY) { + return res; + } + + /* get the remainder */ + if (d != NULL) { + if ((res = mp_mod_2d (a, b, &t)) != MP_OKAY) { + mp_clear (&t); + return res; + } + } + + /* copy */ + if ((res = mp_copy (a, c)) != MP_OKAY) { + mp_clear (&t); + return res; + } + + /* shift by as many digits in the bit count */ + if (b >= (int)DIGIT_BIT) { + mp_rshd (c, b / DIGIT_BIT); + } + + /* shift any bit count < DIGIT_BIT */ + D = (mp_digit) (b % DIGIT_BIT); + if (D != 0) { + register mp_digit *tmpc, mask, shift; + + /* mask */ + mask = (((mp_digit)1) << D) - 1; + + /* shift for lsb */ + shift = DIGIT_BIT - D; + + /* alias */ + tmpc = c->dp + (c->used - 1); + + /* carry */ + r = 0; + for (x = c->used - 1; x >= 0; x--) { + /* get the lower bits of this word in a temp */ + rr = *tmpc & mask; + + /* shift the current word and mix in the carry bits from the previous word */ + *tmpc = (*tmpc >> D) | (r << shift); + --tmpc; + + /* set the carry to the carry bits of the current word found above */ + r = rr; + } + } + mp_clamp (c); + if (d != NULL) { + mp_exch (&t, d); + } + mp_clear (&t); + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_mp_div_3.c b/libtommath/bn_mp_div_3.c new file mode 100644 index 0000000..7cbafc1 --- /dev/null +++ b/libtommath/bn_mp_div_3.c @@ -0,0 +1,75 @@ +#include <tommath.h> +#ifdef BN_MP_DIV_3_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* divide by three (based on routine from MPI and the GMP manual) */ +int +mp_div_3 (mp_int * a, mp_int *c, mp_digit * d) +{ + mp_int q; + mp_word w, t; + mp_digit b; + int res, ix; + + /* b = 2**DIGIT_BIT / 3 */ + b = (((mp_word)1) << ((mp_word)DIGIT_BIT)) / ((mp_word)3); + + if ((res = mp_init_size(&q, a->used)) != MP_OKAY) { + return res; + } + + q.used = a->used; + q.sign = a->sign; + w = 0; + for (ix = a->used - 1; ix >= 0; ix--) { + w = (w << ((mp_word)DIGIT_BIT)) | ((mp_word)a->dp[ix]); + + if (w >= 3) { + /* multiply w by [1/3] */ + t = (w * ((mp_word)b)) >> ((mp_word)DIGIT_BIT); + + /* now subtract 3 * [w/3] from w, to get the remainder */ + w -= t+t+t; + + /* fixup the remainder as required since + * the optimization is not exact. + */ + while (w >= 3) { + t += 1; + w -= 3; + } + } else { + t = 0; + } + q.dp[ix] = (mp_digit)t; + } + + /* [optional] store the remainder */ + if (d != NULL) { + *d = (mp_digit)w; + } + + /* [optional] store the quotient */ + if (c != NULL) { + mp_clamp(&q); + mp_exch(&q, c); + } + mp_clear(&q); + + return res; +} + +#endif diff --git a/libtommath/bn_mp_div_d.c b/libtommath/bn_mp_div_d.c new file mode 100644 index 0000000..9b58aa6 --- /dev/null +++ b/libtommath/bn_mp_div_d.c @@ -0,0 +1,106 @@ +#include <tommath.h> +#ifdef BN_MP_DIV_D_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +static int s_is_power_of_two(mp_digit b, int *p) +{ + int x; + + for (x = 1; x < DIGIT_BIT; x++) { + if (b == (((mp_digit)1)<<x)) { + *p = x; + return 1; + } + } + return 0; +} + +/* single digit division (based on routine from MPI) */ +int mp_div_d (mp_int * a, mp_digit b, mp_int * c, mp_digit * d) +{ + mp_int q; + mp_word w; + mp_digit t; + int res, ix; + + /* cannot divide by zero */ + if (b == 0) { + return MP_VAL; + } + + /* quick outs */ + if (b == 1 || mp_iszero(a) == 1) { + if (d != NULL) { + *d = 0; + } + if (c != NULL) { + return mp_copy(a, c); + } + return MP_OKAY; + } + + /* power of two ? */ + if (s_is_power_of_two(b, &ix) == 1) { + if (d != NULL) { + *d = a->dp[0] & ((((mp_digit)1)<<ix) - 1); + } + if (c != NULL) { + return mp_div_2d(a, ix, c, NULL); + } + return MP_OKAY; + } + +#ifdef BN_MP_DIV_3_C + /* three? */ + if (b == 3) { + return mp_div_3(a, c, d); + } +#endif + + /* no easy answer [c'est la vie]. Just division */ + if ((res = mp_init_size(&q, a->used)) != MP_OKAY) { + return res; + } + + q.used = a->used; + q.sign = a->sign; + w = 0; + for (ix = a->used - 1; ix >= 0; ix--) { + w = (w << ((mp_word)DIGIT_BIT)) | ((mp_word)a->dp[ix]); + + if (w >= b) { + t = (mp_digit)(w / b); + w -= ((mp_word)t) * ((mp_word)b); + } else { + t = 0; + } + q.dp[ix] = (mp_digit)t; + } + + if (d != NULL) { + *d = (mp_digit)w; + } + + if (c != NULL) { + mp_clamp(&q); + mp_exch(&q, c); + } + mp_clear(&q); + + return res; +} + +#endif diff --git a/libtommath/bn_mp_dr_is_modulus.c b/libtommath/bn_mp_dr_is_modulus.c new file mode 100644 index 0000000..5ef78a3 --- /dev/null +++ b/libtommath/bn_mp_dr_is_modulus.c @@ -0,0 +1,39 @@ +#include <tommath.h> +#ifdef BN_MP_DR_IS_MODULUS_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* determines if a number is a valid DR modulus */ +int mp_dr_is_modulus(mp_int *a) +{ + int ix; + + /* must be at least two digits */ + if (a->used < 2) { + return 0; + } + + /* must be of the form b**k - a [a <= b] so all + * but the first digit must be equal to -1 (mod b). + */ + for (ix = 1; ix < a->used; ix++) { + if (a->dp[ix] != MP_MASK) { + return 0; + } + } + return 1; +} + +#endif diff --git a/libtommath/bn_mp_dr_reduce.c b/libtommath/bn_mp_dr_reduce.c new file mode 100644 index 0000000..9bb7ad7 --- /dev/null +++ b/libtommath/bn_mp_dr_reduce.c @@ -0,0 +1,90 @@ +#include <tommath.h> +#ifdef BN_MP_DR_REDUCE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* reduce "x" in place modulo "n" using the Diminished Radix algorithm. + * + * Based on algorithm from the paper + * + * "Generating Efficient Primes for Discrete Log Cryptosystems" + * Chae Hoon Lim, Pil Joong Lee, + * POSTECH Information Research Laboratories + * + * The modulus must be of a special format [see manual] + * + * Has been modified to use algorithm 7.10 from the LTM book instead + * + * Input x must be in the range 0 <= x <= (n-1)**2 + */ +int +mp_dr_reduce (mp_int * x, mp_int * n, mp_digit k) +{ + int err, i, m; + mp_word r; + mp_digit mu, *tmpx1, *tmpx2; + + /* m = digits in modulus */ + m = n->used; + + /* ensure that "x" has at least 2m digits */ + if (x->alloc < m + m) { + if ((err = mp_grow (x, m + m)) != MP_OKAY) { + return err; + } + } + +/* top of loop, this is where the code resumes if + * another reduction pass is required. + */ +top: + /* aliases for digits */ + /* alias for lower half of x */ + tmpx1 = x->dp; + + /* alias for upper half of x, or x/B**m */ + tmpx2 = x->dp + m; + + /* set carry to zero */ + mu = 0; + + /* compute (x mod B**m) + k * [x/B**m] inline and inplace */ + for (i = 0; i < m; i++) { + r = ((mp_word)*tmpx2++) * ((mp_word)k) + *tmpx1 + mu; + *tmpx1++ = (mp_digit)(r & MP_MASK); + mu = (mp_digit)(r >> ((mp_word)DIGIT_BIT)); + } + + /* set final carry */ + *tmpx1++ = mu; + + /* zero words above m */ + for (i = m + 1; i < x->used; i++) { + *tmpx1++ = 0; + } + + /* clamp, sub and return */ + mp_clamp (x); + + /* if x >= n then subtract and reduce again + * Each successive "recursion" makes the input smaller and smaller. + */ + if (mp_cmp_mag (x, n) != MP_LT) { + s_mp_sub(x, n, x); + goto top; + } + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_mp_dr_setup.c b/libtommath/bn_mp_dr_setup.c new file mode 100644 index 0000000..029d310 --- /dev/null +++ b/libtommath/bn_mp_dr_setup.c @@ -0,0 +1,28 @@ +#include <tommath.h> +#ifdef BN_MP_DR_SETUP_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* determines the setup value */ +void mp_dr_setup(mp_int *a, mp_digit *d) +{ + /* the casts are required if DIGIT_BIT is one less than + * the number of bits in a mp_digit [e.g. DIGIT_BIT==31] + */ + *d = (mp_digit)((((mp_word)1) << ((mp_word)DIGIT_BIT)) - + ((mp_word)a->dp[0])); +} + +#endif diff --git a/libtommath/bn_mp_exch.c b/libtommath/bn_mp_exch.c new file mode 100644 index 0000000..0ef485a --- /dev/null +++ b/libtommath/bn_mp_exch.c @@ -0,0 +1,30 @@ +#include <tommath.h> +#ifdef BN_MP_EXCH_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* swap the elements of two integers, for cases where you can't simply swap the + * mp_int pointers around + */ +void +mp_exch (mp_int * a, mp_int * b) +{ + mp_int t; + + t = *a; + *a = *b; + *b = t; +} +#endif diff --git a/libtommath/bn_mp_expt_d.c b/libtommath/bn_mp_expt_d.c new file mode 100644 index 0000000..fdb8bd9 --- /dev/null +++ b/libtommath/bn_mp_expt_d.c @@ -0,0 +1,53 @@ +#include <tommath.h> +#ifdef BN_MP_EXPT_D_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* calculate c = a**b using a square-multiply algorithm */ +int mp_expt_d (mp_int * a, mp_digit b, mp_int * c) +{ + int res, x; + mp_int g; + + if ((res = mp_init_copy (&g, a)) != MP_OKAY) { + return res; + } + + /* set initial result */ + mp_set (c, 1); + + for (x = 0; x < (int) DIGIT_BIT; x++) { + /* square */ + if ((res = mp_sqr (c, c)) != MP_OKAY) { + mp_clear (&g); + return res; + } + + /* if the bit is set multiply */ + if ((b & (mp_digit) (((mp_digit)1) << (DIGIT_BIT - 1))) != 0) { + if ((res = mp_mul (c, &g, c)) != MP_OKAY) { + mp_clear (&g); + return res; + } + } + + /* shift to next bit */ + b <<= 1; + } + + mp_clear (&g); + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_mp_exptmod.c b/libtommath/bn_mp_exptmod.c new file mode 100644 index 0000000..7309170 --- /dev/null +++ b/libtommath/bn_mp_exptmod.c @@ -0,0 +1,100 @@ +#include <tommath.h> +#ifdef BN_MP_EXPTMOD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + + +/* this is a shell function that calls either the normal or Montgomery + * exptmod functions. Originally the call to the montgomery code was + * embedded in the normal function but that wasted alot of stack space + * for nothing (since 99% of the time the Montgomery code would be called) + */ +int mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y) +{ + int dr; + + /* modulus P must be positive */ + if (P->sign == MP_NEG) { + return MP_VAL; + } + + /* if exponent X is negative we have to recurse */ + if (X->sign == MP_NEG) { +#ifdef BN_MP_INVMOD_C + mp_int tmpG, tmpX; + int err; + + /* first compute 1/G mod P */ + if ((err = mp_init(&tmpG)) != MP_OKAY) { + return err; + } + if ((err = mp_invmod(G, P, &tmpG)) != MP_OKAY) { + mp_clear(&tmpG); + return err; + } + + /* now get |X| */ + if ((err = mp_init(&tmpX)) != MP_OKAY) { + mp_clear(&tmpG); + return err; + } + if ((err = mp_abs(X, &tmpX)) != MP_OKAY) { + mp_clear_multi(&tmpG, &tmpX, NULL); + return err; + } + + /* and now compute (1/G)**|X| instead of G**X [X < 0] */ + err = mp_exptmod(&tmpG, &tmpX, P, Y); + mp_clear_multi(&tmpG, &tmpX, NULL); + return err; +#else + /* no invmod */ + return MP_VAL; +#endif + } + +#ifdef BN_MP_DR_IS_MODULUS_C + /* is it a DR modulus? */ + dr = mp_dr_is_modulus(P); +#else + dr = 0; +#endif + +#ifdef BN_MP_REDUCE_IS_2K_C + /* if not, is it a uDR modulus? */ + if (dr == 0) { + dr = mp_reduce_is_2k(P) << 1; + } +#endif + + /* if the modulus is odd or dr != 0 use the fast method */ +#ifdef BN_MP_EXPTMOD_FAST_C + if (mp_isodd (P) == 1 || dr != 0) { + return mp_exptmod_fast (G, X, P, Y, dr); + } else { +#endif +#ifdef BN_S_MP_EXPTMOD_C + /* otherwise use the generic Barrett reduction technique */ + return s_mp_exptmod (G, X, P, Y); +#else + /* no exptmod for evens */ + return MP_VAL; +#endif +#ifdef BN_MP_EXPTMOD_FAST_C + } +#endif +} + +#endif diff --git a/libtommath/bn_mp_exptmod_fast.c b/libtommath/bn_mp_exptmod_fast.c new file mode 100644 index 0000000..255e9d9 --- /dev/null +++ b/libtommath/bn_mp_exptmod_fast.c @@ -0,0 +1,318 @@ +#include <tommath.h> +#ifdef BN_MP_EXPTMOD_FAST_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* computes Y == G**X mod P, HAC pp.616, Algorithm 14.85 + * + * Uses a left-to-right k-ary sliding window to compute the modular exponentiation. + * The value of k changes based on the size of the exponent. + * + * Uses Montgomery or Diminished Radix reduction [whichever appropriate] + */ + +#ifdef MP_LOW_MEM + #define TAB_SIZE 32 +#else + #define TAB_SIZE 256 +#endif + +int +mp_exptmod_fast (mp_int * G, mp_int * X, mp_int * P, mp_int * Y, int redmode) +{ + mp_int M[TAB_SIZE], res; + mp_digit buf, mp; + int err, bitbuf, bitcpy, bitcnt, mode, digidx, x, y, winsize; + + /* use a pointer to the reduction algorithm. This allows us to use + * one of many reduction algorithms without modding the guts of + * the code with if statements everywhere. + */ + int (*redux)(mp_int*,mp_int*,mp_digit); + + /* find window size */ + x = mp_count_bits (X); + if (x <= 7) { + winsize = 2; + } else if (x <= 36) { + winsize = 3; + } else if (x <= 140) { + winsize = 4; + } else if (x <= 450) { + winsize = 5; + } else if (x <= 1303) { + winsize = 6; + } else if (x <= 3529) { + winsize = 7; + } else { + winsize = 8; + } + +#ifdef MP_LOW_MEM + if (winsize > 5) { + winsize = 5; + } +#endif + + /* init M array */ + /* init first cell */ + if ((err = mp_init(&M[1])) != MP_OKAY) { + return err; + } + + /* now init the second half of the array */ + for (x = 1<<(winsize-1); x < (1 << winsize); x++) { + if ((err = mp_init(&M[x])) != MP_OKAY) { + for (y = 1<<(winsize-1); y < x; y++) { + mp_clear (&M[y]); + } + mp_clear(&M[1]); + return err; + } + } + + /* determine and setup reduction code */ + if (redmode == 0) { +#ifdef BN_MP_MONTGOMERY_SETUP_C + /* now setup montgomery */ + if ((err = mp_montgomery_setup (P, &mp)) != MP_OKAY) { + goto LBL_M; + } +#else + err = MP_VAL; + goto LBL_M; +#endif + + /* automatically pick the comba one if available (saves quite a few calls/ifs) */ +#ifdef BN_FAST_MP_MONTGOMERY_REDUCE_C + if (((P->used * 2 + 1) < MP_WARRAY) && + P->used < (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) { + redux = fast_mp_montgomery_reduce; + } else +#endif + { +#ifdef BN_MP_MONTGOMERY_REDUCE_C + /* use slower baseline Montgomery method */ + redux = mp_montgomery_reduce; +#else + err = MP_VAL; + goto LBL_M; +#endif + } + } else if (redmode == 1) { +#if defined(BN_MP_DR_SETUP_C) && defined(BN_MP_DR_REDUCE_C) + /* setup DR reduction for moduli of the form B**k - b */ + mp_dr_setup(P, &mp); + redux = mp_dr_reduce; +#else + err = MP_VAL; + goto LBL_M; +#endif + } else { +#if defined(BN_MP_REDUCE_2K_SETUP_C) && defined(BN_MP_REDUCE_2K_C) + /* setup DR reduction for moduli of the form 2**k - b */ + if ((err = mp_reduce_2k_setup(P, &mp)) != MP_OKAY) { + goto LBL_M; + } + redux = mp_reduce_2k; +#else + err = MP_VAL; + goto LBL_M; +#endif + } + + /* setup result */ + if ((err = mp_init (&res)) != MP_OKAY) { + goto LBL_M; + } + + /* create M table + * + + * + * The first half of the table is not computed though accept for M[0] and M[1] + */ + + if (redmode == 0) { +#ifdef BN_MP_MONTGOMERY_CALC_NORMALIZATION_C + /* now we need R mod m */ + if ((err = mp_montgomery_calc_normalization (&res, P)) != MP_OKAY) { + goto LBL_RES; + } +#else + err = MP_VAL; + goto LBL_RES; +#endif + + /* now set M[1] to G * R mod m */ + if ((err = mp_mulmod (G, &res, P, &M[1])) != MP_OKAY) { + goto LBL_RES; + } + } else { + mp_set(&res, 1); + if ((err = mp_mod(G, P, &M[1])) != MP_OKAY) { + goto LBL_RES; + } + } + + /* compute the value at M[1<<(winsize-1)] by squaring M[1] (winsize-1) times */ + if ((err = mp_copy (&M[1], &M[1 << (winsize - 1)])) != MP_OKAY) { + goto LBL_RES; + } + + for (x = 0; x < (winsize - 1); x++) { + if ((err = mp_sqr (&M[1 << (winsize - 1)], &M[1 << (winsize - 1)])) != MP_OKAY) { + goto LBL_RES; + } + if ((err = redux (&M[1 << (winsize - 1)], P, mp)) != MP_OKAY) { + goto LBL_RES; + } + } + + /* create upper table */ + for (x = (1 << (winsize - 1)) + 1; x < (1 << winsize); x++) { + if ((err = mp_mul (&M[x - 1], &M[1], &M[x])) != MP_OKAY) { + goto LBL_RES; + } + if ((err = redux (&M[x], P, mp)) != MP_OKAY) { + goto LBL_RES; + } + } + + /* set initial mode and bit cnt */ + mode = 0; + bitcnt = 1; + buf = 0; + digidx = X->used - 1; + bitcpy = 0; + bitbuf = 0; + + for (;;) { + /* grab next digit as required */ + if (--bitcnt == 0) { + /* if digidx == -1 we are out of digits so break */ + if (digidx == -1) { + break; + } + /* read next digit and reset bitcnt */ + buf = X->dp[digidx--]; + bitcnt = (int)DIGIT_BIT; + } + + /* grab the next msb from the exponent */ + y = (mp_digit)(buf >> (DIGIT_BIT - 1)) & 1; + buf <<= (mp_digit)1; + + /* if the bit is zero and mode == 0 then we ignore it + * These represent the leading zero bits before the first 1 bit + * in the exponent. Technically this opt is not required but it + * does lower the # of trivial squaring/reductions used + */ + if (mode == 0 && y == 0) { + continue; + } + + /* if the bit is zero and mode == 1 then we square */ + if (mode == 1 && y == 0) { + if ((err = mp_sqr (&res, &res)) != MP_OKAY) { + goto LBL_RES; + } + if ((err = redux (&res, P, mp)) != MP_OKAY) { + goto LBL_RES; + } + continue; + } + + /* else we add it to the window */ + bitbuf |= (y << (winsize - ++bitcpy)); + mode = 2; + + if (bitcpy == winsize) { + /* ok window is filled so square as required and multiply */ + /* square first */ + for (x = 0; x < winsize; x++) { + if ((err = mp_sqr (&res, &res)) != MP_OKAY) { + goto LBL_RES; + } + if ((err = redux (&res, P, mp)) != MP_OKAY) { + goto LBL_RES; + } + } + + /* then multiply */ + if ((err = mp_mul (&res, &M[bitbuf], &res)) != MP_OKAY) { + goto LBL_RES; + } + if ((err = redux (&res, P, mp)) != MP_OKAY) { + goto LBL_RES; + } + + /* empty window and reset */ + bitcpy = 0; + bitbuf = 0; + mode = 1; + } + } + + /* if bits remain then square/multiply */ + if (mode == 2 && bitcpy > 0) { + /* square then multiply if the bit is set */ + for (x = 0; x < bitcpy; x++) { + if ((err = mp_sqr (&res, &res)) != MP_OKAY) { + goto LBL_RES; + } + if ((err = redux (&res, P, mp)) != MP_OKAY) { + goto LBL_RES; + } + + /* get next bit of the window */ + bitbuf <<= 1; + if ((bitbuf & (1 << winsize)) != 0) { + /* then multiply */ + if ((err = mp_mul (&res, &M[1], &res)) != MP_OKAY) { + goto LBL_RES; + } + if ((err = redux (&res, P, mp)) != MP_OKAY) { + goto LBL_RES; + } + } + } + } + + if (redmode == 0) { + /* fixup result if Montgomery reduction is used + * recall that any value in a Montgomery system is + * actually multiplied by R mod n. So we have + * to reduce one more time to cancel out the factor + * of R. + */ + if ((err = redux(&res, P, mp)) != MP_OKAY) { + goto LBL_RES; + } + } + + /* swap res with Y */ + mp_exch (&res, Y); + err = MP_OKAY; +LBL_RES:mp_clear (&res); +LBL_M: + mp_clear(&M[1]); + for (x = 1<<(winsize-1); x < (1 << winsize); x++) { + mp_clear (&M[x]); + } + return err; +} +#endif + diff --git a/libtommath/bn_mp_exteuclid.c b/libtommath/bn_mp_exteuclid.c new file mode 100644 index 0000000..545450b --- /dev/null +++ b/libtommath/bn_mp_exteuclid.c @@ -0,0 +1,71 @@ +#include <tommath.h> +#ifdef BN_MP_EXTEUCLID_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* Extended euclidean algorithm of (a, b) produces + a*u1 + b*u2 = u3 + */ +int mp_exteuclid(mp_int *a, mp_int *b, mp_int *U1, mp_int *U2, mp_int *U3) +{ + mp_int u1,u2,u3,v1,v2,v3,t1,t2,t3,q,tmp; + int err; + + if ((err = mp_init_multi(&u1, &u2, &u3, &v1, &v2, &v3, &t1, &t2, &t3, &q, &tmp, NULL)) != MP_OKAY) { + return err; + } + + /* initialize, (u1,u2,u3) = (1,0,a) */ + mp_set(&u1, 1); + if ((err = mp_copy(a, &u3)) != MP_OKAY) { goto _ERR; } + + /* initialize, (v1,v2,v3) = (0,1,b) */ + mp_set(&v2, 1); + if ((err = mp_copy(b, &v3)) != MP_OKAY) { goto _ERR; } + + /* loop while v3 != 0 */ + while (mp_iszero(&v3) == MP_NO) { + /* q = u3/v3 */ + if ((err = mp_div(&u3, &v3, &q, NULL)) != MP_OKAY) { goto _ERR; } + + /* (t1,t2,t3) = (u1,u2,u3) - (v1,v2,v3)q */ + if ((err = mp_mul(&v1, &q, &tmp)) != MP_OKAY) { goto _ERR; } + if ((err = mp_sub(&u1, &tmp, &t1)) != MP_OKAY) { goto _ERR; } + if ((err = mp_mul(&v2, &q, &tmp)) != MP_OKAY) { goto _ERR; } + if ((err = mp_sub(&u2, &tmp, &t2)) != MP_OKAY) { goto _ERR; } + if ((err = mp_mul(&v3, &q, &tmp)) != MP_OKAY) { goto _ERR; } + if ((err = mp_sub(&u3, &tmp, &t3)) != MP_OKAY) { goto _ERR; } + + /* (u1,u2,u3) = (v1,v2,v3) */ + if ((err = mp_copy(&v1, &u1)) != MP_OKAY) { goto _ERR; } + if ((err = mp_copy(&v2, &u2)) != MP_OKAY) { goto _ERR; } + if ((err = mp_copy(&v3, &u3)) != MP_OKAY) { goto _ERR; } + + /* (v1,v2,v3) = (t1,t2,t3) */ + if ((err = mp_copy(&t1, &v1)) != MP_OKAY) { goto _ERR; } + if ((err = mp_copy(&t2, &v2)) != MP_OKAY) { goto _ERR; } + if ((err = mp_copy(&t3, &v3)) != MP_OKAY) { goto _ERR; } + } + + /* copy result out */ + if (U1 != NULL) { mp_exch(U1, &u1); } + if (U2 != NULL) { mp_exch(U2, &u2); } + if (U3 != NULL) { mp_exch(U3, &u3); } + + err = MP_OKAY; +_ERR: mp_clear_multi(&u1, &u2, &u3, &v1, &v2, &v3, &t1, &t2, &t3, &q, &tmp, NULL); + return err; +} +#endif diff --git a/libtommath/bn_mp_fread.c b/libtommath/bn_mp_fread.c new file mode 100644 index 0000000..293df3f --- /dev/null +++ b/libtommath/bn_mp_fread.c @@ -0,0 +1,63 @@ +#include <tommath.h> +#ifdef BN_MP_FREAD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* read a bigint from a file stream in ASCII */ +int mp_fread(mp_int *a, int radix, FILE *stream) +{ + int err, ch, neg, y; + + /* clear a */ + mp_zero(a); + + /* if first digit is - then set negative */ + ch = fgetc(stream); + if (ch == '-') { + neg = MP_NEG; + ch = fgetc(stream); + } else { + neg = MP_ZPOS; + } + + for (;;) { + /* find y in the radix map */ + for (y = 0; y < radix; y++) { + if (mp_s_rmap[y] == ch) { + break; + } + } + if (y == radix) { + break; + } + + /* shift up and add */ + if ((err = mp_mul_d(a, radix, a)) != MP_OKAY) { + return err; + } + if ((err = mp_add_d(a, y, a)) != MP_OKAY) { + return err; + } + + ch = fgetc(stream); + } + if (mp_cmp_d(a, 0) != MP_EQ) { + a->sign = neg; + } + + return MP_OKAY; +} + +#endif diff --git a/libtommath/bn_mp_fwrite.c b/libtommath/bn_mp_fwrite.c new file mode 100644 index 0000000..8fa3129 --- /dev/null +++ b/libtommath/bn_mp_fwrite.c @@ -0,0 +1,48 @@ +#include <tommath.h> +#ifdef BN_MP_FWRITE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +int mp_fwrite(mp_int *a, int radix, FILE *stream) +{ + char *buf; + int err, len, x; + + if ((err = mp_radix_size(a, radix, &len)) != MP_OKAY) { + return err; + } + + buf = OPT_CAST(char) XMALLOC (len); + if (buf == NULL) { + return MP_MEM; + } + + if ((err = mp_toradix(a, buf, radix)) != MP_OKAY) { + XFREE (buf); + return err; + } + + for (x = 0; x < len; x++) { + if (fputc(buf[x], stream) == EOF) { + XFREE (buf); + return MP_VAL; + } + } + + XFREE (buf); + return MP_OKAY; +} + +#endif diff --git a/libtommath/bn_mp_gcd.c b/libtommath/bn_mp_gcd.c new file mode 100644 index 0000000..6265df1 --- /dev/null +++ b/libtommath/bn_mp_gcd.c @@ -0,0 +1,109 @@ +#include <tommath.h> +#ifdef BN_MP_GCD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* Greatest Common Divisor using the binary method */ +int mp_gcd (mp_int * a, mp_int * b, mp_int * c) +{ + mp_int u, v; + int k, u_lsb, v_lsb, res; + + /* either zero than gcd is the largest */ + if (mp_iszero (a) == 1 && mp_iszero (b) == 0) { + return mp_abs (b, c); + } + if (mp_iszero (a) == 0 && mp_iszero (b) == 1) { + return mp_abs (a, c); + } + + /* optimized. At this point if a == 0 then + * b must equal zero too + */ + if (mp_iszero (a) == 1) { + mp_zero(c); + return MP_OKAY; + } + + /* get copies of a and b we can modify */ + if ((res = mp_init_copy (&u, a)) != MP_OKAY) { + return res; + } + + if ((res = mp_init_copy (&v, b)) != MP_OKAY) { + goto LBL_U; + } + + /* must be positive for the remainder of the algorithm */ + u.sign = v.sign = MP_ZPOS; + + /* B1. Find the common power of two for u and v */ + u_lsb = mp_cnt_lsb(&u); + v_lsb = mp_cnt_lsb(&v); + k = MIN(u_lsb, v_lsb); + + if (k > 0) { + /* divide the power of two out */ + if ((res = mp_div_2d(&u, k, &u, NULL)) != MP_OKAY) { + goto LBL_V; + } + + if ((res = mp_div_2d(&v, k, &v, NULL)) != MP_OKAY) { + goto LBL_V; + } + } + + /* divide any remaining factors of two out */ + if (u_lsb != k) { + if ((res = mp_div_2d(&u, u_lsb - k, &u, NULL)) != MP_OKAY) { + goto LBL_V; + } + } + + if (v_lsb != k) { + if ((res = mp_div_2d(&v, v_lsb - k, &v, NULL)) != MP_OKAY) { + goto LBL_V; + } + } + + while (mp_iszero(&v) == 0) { + /* make sure v is the largest */ + if (mp_cmp_mag(&u, &v) == MP_GT) { + /* swap u and v to make sure v is >= u */ + mp_exch(&u, &v); + } + + /* subtract smallest from largest */ + if ((res = s_mp_sub(&v, &u, &v)) != MP_OKAY) { + goto LBL_V; + } + + /* Divide out all factors of two */ + if ((res = mp_div_2d(&v, mp_cnt_lsb(&v), &v, NULL)) != MP_OKAY) { + goto LBL_V; + } + } + + /* multiply by 2**k which we divided out at the beginning */ + if ((res = mp_mul_2d (&u, k, c)) != MP_OKAY) { + goto LBL_V; + } + c->sign = MP_ZPOS; + res = MP_OKAY; +LBL_V:mp_clear (&u); +LBL_U:mp_clear (&v); + return res; +} +#endif diff --git a/libtommath/bn_mp_get_int.c b/libtommath/bn_mp_get_int.c new file mode 100644 index 0000000..034467b --- /dev/null +++ b/libtommath/bn_mp_get_int.c @@ -0,0 +1,41 @@ +#include <tommath.h> +#ifdef BN_MP_GET_INT_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* get the lower 32-bits of an mp_int */ +unsigned long mp_get_int(mp_int * a) +{ + int i; + unsigned long res; + + if (a->used == 0) { + return 0; + } + + /* get number of digits of the lsb we have to read */ + i = MIN(a->used,(int)((sizeof(unsigned long)*CHAR_BIT+DIGIT_BIT-1)/DIGIT_BIT))-1; + + /* get most significant digit of result */ + res = DIGIT(a,i); + + while (--i >= 0) { + res = (res << DIGIT_BIT) | DIGIT(a,i); + } + + /* force result to 32-bits always so it is consistent on non 32-bit platforms */ + return res & 0xFFFFFFFFUL; +} +#endif diff --git a/libtommath/bn_mp_grow.c b/libtommath/bn_mp_grow.c new file mode 100644 index 0000000..12a78a8 --- /dev/null +++ b/libtommath/bn_mp_grow.c @@ -0,0 +1,53 @@ +#include <tommath.h> +#ifdef BN_MP_GROW_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* grow as required */ +int mp_grow (mp_int * a, int size) +{ + int i; + mp_digit *tmp; + + /* if the alloc size is smaller alloc more ram */ + if (a->alloc < size) { + /* ensure there are always at least MP_PREC digits extra on top */ + size += (MP_PREC * 2) - (size % MP_PREC); + + /* reallocate the array a->dp + * + * We store the return in a temporary variable + * in case the operation failed we don't want + * to overwrite the dp member of a. + */ + tmp = OPT_CAST(mp_digit) XREALLOC (a->dp, sizeof (mp_digit) * size); + if (tmp == NULL) { + /* reallocation failed but "a" is still valid [can be freed] */ + return MP_MEM; + } + + /* reallocation succeeded so set a->dp */ + a->dp = tmp; + + /* zero excess digits */ + i = a->alloc; + a->alloc = size; + for (; i < a->alloc; i++) { + a->dp[i] = 0; + } + } + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_mp_init.c b/libtommath/bn_mp_init.c new file mode 100644 index 0000000..9d70554 --- /dev/null +++ b/libtommath/bn_mp_init.c @@ -0,0 +1,42 @@ +#include <tommath.h> +#ifdef BN_MP_INIT_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* init a new mp_int */ +int mp_init (mp_int * a) +{ + int i; + + /* allocate memory required and clear it */ + a->dp = OPT_CAST(mp_digit) XMALLOC (sizeof (mp_digit) * MP_PREC); + if (a->dp == NULL) { + return MP_MEM; + } + + /* set the digits to zero */ + for (i = 0; i < MP_PREC; i++) { + a->dp[i] = 0; + } + + /* set the used to zero, allocated digits to the default precision + * and sign to positive */ + a->used = 0; + a->alloc = MP_PREC; + a->sign = MP_ZPOS; + + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_mp_init_copy.c b/libtommath/bn_mp_init_copy.c new file mode 100644 index 0000000..b1b0fa2 --- /dev/null +++ b/libtommath/bn_mp_init_copy.c @@ -0,0 +1,28 @@ +#include <tommath.h> +#ifdef BN_MP_INIT_COPY_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* creates "a" then copies b into it */ +int mp_init_copy (mp_int * a, mp_int * b) +{ + int res; + + if ((res = mp_init (a)) != MP_OKAY) { + return res; + } + return mp_copy (b, a); +} +#endif diff --git a/libtommath/bn_mp_init_multi.c b/libtommath/bn_mp_init_multi.c new file mode 100644 index 0000000..8cb123a --- /dev/null +++ b/libtommath/bn_mp_init_multi.c @@ -0,0 +1,55 @@ +#include <tommath.h> +#ifdef BN_MP_INIT_MULTI_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ +#include <stdarg.h> + +int mp_init_multi(mp_int *mp, ...) +{ + mp_err res = MP_OKAY; /* Assume ok until proven otherwise */ + int n = 0; /* Number of ok inits */ + mp_int* cur_arg = mp; + va_list args; + + va_start(args, mp); /* init args to next argument from caller */ + while (cur_arg != NULL) { + if (mp_init(cur_arg) != MP_OKAY) { + /* Oops - error! Back-track and mp_clear what we already + succeeded in init-ing, then return error. + */ + va_list clean_args; + + /* end the current list */ + va_end(args); + + /* now start cleaning up */ + cur_arg = mp; + va_start(clean_args, mp); + while (n--) { + mp_clear(cur_arg); + cur_arg = va_arg(clean_args, mp_int*); + } + va_end(clean_args); + res = MP_MEM; + break; + } + n++; + cur_arg = va_arg(args, mp_int*); + } + va_end(args); + return res; /* Assumed ok, if error flagged above. */ +} + +#endif diff --git a/libtommath/bn_mp_init_set.c b/libtommath/bn_mp_init_set.c new file mode 100644 index 0000000..0251e61 --- /dev/null +++ b/libtommath/bn_mp_init_set.c @@ -0,0 +1,28 @@ +#include <tommath.h> +#ifdef BN_MP_INIT_SET_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* initialize and set a digit */ +int mp_init_set (mp_int * a, mp_digit b) +{ + int err; + if ((err = mp_init(a)) != MP_OKAY) { + return err; + } + mp_set(a, b); + return err; +} +#endif diff --git a/libtommath/bn_mp_init_set_int.c b/libtommath/bn_mp_init_set_int.c new file mode 100644 index 0000000..f59fd19 --- /dev/null +++ b/libtommath/bn_mp_init_set_int.c @@ -0,0 +1,27 @@ +#include <tommath.h> +#ifdef BN_MP_INIT_SET_INT_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* initialize and set a digit */ +int mp_init_set_int (mp_int * a, unsigned long b) +{ + int err; + if ((err = mp_init(a)) != MP_OKAY) { + return err; + } + return mp_set_int(a, b); +} +#endif diff --git a/libtommath/bn_mp_init_size.c b/libtommath/bn_mp_init_size.c new file mode 100644 index 0000000..845ce2c --- /dev/null +++ b/libtommath/bn_mp_init_size.c @@ -0,0 +1,44 @@ +#include <tommath.h> +#ifdef BN_MP_INIT_SIZE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* init an mp_init for a given size */ +int mp_init_size (mp_int * a, int size) +{ + int x; + + /* pad size so there are always extra digits */ + size += (MP_PREC * 2) - (size % MP_PREC); + + /* alloc mem */ + a->dp = OPT_CAST(mp_digit) XMALLOC (sizeof (mp_digit) * size); + if (a->dp == NULL) { + return MP_MEM; + } + + /* set the members */ + a->used = 0; + a->alloc = size; + a->sign = MP_ZPOS; + + /* zero the digits */ + for (x = 0; x < size; x++) { + a->dp[x] = 0; + } + + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_mp_invmod.c b/libtommath/bn_mp_invmod.c new file mode 100644 index 0000000..46118ad --- /dev/null +++ b/libtommath/bn_mp_invmod.c @@ -0,0 +1,39 @@ +#include <tommath.h> +#ifdef BN_MP_INVMOD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* hac 14.61, pp608 */ +int mp_invmod (mp_int * a, mp_int * b, mp_int * c) +{ + /* b cannot be negative */ + if (b->sign == MP_NEG || mp_iszero(b) == 1) { + return MP_VAL; + } + +#ifdef BN_FAST_MP_INVMOD_C + /* if the modulus is odd we can use a faster routine instead */ + if (mp_isodd (b) == 1) { + return fast_mp_invmod (a, b, c); + } +#endif + +#ifdef BN_MP_INVMOD_SLOW_C + return mp_invmod_slow(a, b, c); +#endif + + return MP_VAL; +} +#endif diff --git a/libtommath/bn_mp_invmod_slow.c b/libtommath/bn_mp_invmod_slow.c new file mode 100644 index 0000000..c1884c0 --- /dev/null +++ b/libtommath/bn_mp_invmod_slow.c @@ -0,0 +1,171 @@ +#include <tommath.h> +#ifdef BN_MP_INVMOD_SLOW_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* hac 14.61, pp608 */ +int mp_invmod_slow (mp_int * a, mp_int * b, mp_int * c) +{ + mp_int x, y, u, v, A, B, C, D; + int res; + + /* b cannot be negative */ + if (b->sign == MP_NEG || mp_iszero(b) == 1) { + return MP_VAL; + } + + /* init temps */ + if ((res = mp_init_multi(&x, &y, &u, &v, + &A, &B, &C, &D, NULL)) != MP_OKAY) { + return res; + } + + /* x = a, y = b */ + if ((res = mp_copy (a, &x)) != MP_OKAY) { + goto LBL_ERR; + } + if ((res = mp_copy (b, &y)) != MP_OKAY) { + goto LBL_ERR; + } + + /* 2. [modified] if x,y are both even then return an error! */ + if (mp_iseven (&x) == 1 && mp_iseven (&y) == 1) { + res = MP_VAL; + goto LBL_ERR; + } + + /* 3. u=x, v=y, A=1, B=0, C=0,D=1 */ + if ((res = mp_copy (&x, &u)) != MP_OKAY) { + goto LBL_ERR; + } + if ((res = mp_copy (&y, &v)) != MP_OKAY) { + goto LBL_ERR; + } + mp_set (&A, 1); + mp_set (&D, 1); + +top: + /* 4. while u is even do */ + while (mp_iseven (&u) == 1) { + /* 4.1 u = u/2 */ + if ((res = mp_div_2 (&u, &u)) != MP_OKAY) { + goto LBL_ERR; + } + /* 4.2 if A or B is odd then */ + if (mp_isodd (&A) == 1 || mp_isodd (&B) == 1) { + /* A = (A+y)/2, B = (B-x)/2 */ + if ((res = mp_add (&A, &y, &A)) != MP_OKAY) { + goto LBL_ERR; + } + if ((res = mp_sub (&B, &x, &B)) != MP_OKAY) { + goto LBL_ERR; + } + } + /* A = A/2, B = B/2 */ + if ((res = mp_div_2 (&A, &A)) != MP_OKAY) { + goto LBL_ERR; + } + if ((res = mp_div_2 (&B, &B)) != MP_OKAY) { + goto LBL_ERR; + } + } + + /* 5. while v is even do */ + while (mp_iseven (&v) == 1) { + /* 5.1 v = v/2 */ + if ((res = mp_div_2 (&v, &v)) != MP_OKAY) { + goto LBL_ERR; + } + /* 5.2 if C or D is odd then */ + if (mp_isodd (&C) == 1 || mp_isodd (&D) == 1) { + /* C = (C+y)/2, D = (D-x)/2 */ + if ((res = mp_add (&C, &y, &C)) != MP_OKAY) { + goto LBL_ERR; + } + if ((res = mp_sub (&D, &x, &D)) != MP_OKAY) { + goto LBL_ERR; + } + } + /* C = C/2, D = D/2 */ + if ((res = mp_div_2 (&C, &C)) != MP_OKAY) { + goto LBL_ERR; + } + if ((res = mp_div_2 (&D, &D)) != MP_OKAY) { + goto LBL_ERR; + } + } + + /* 6. if u >= v then */ + if (mp_cmp (&u, &v) != MP_LT) { + /* u = u - v, A = A - C, B = B - D */ + if ((res = mp_sub (&u, &v, &u)) != MP_OKAY) { + goto LBL_ERR; + } + + if ((res = mp_sub (&A, &C, &A)) != MP_OKAY) { + goto LBL_ERR; + } + + if ((res = mp_sub (&B, &D, &B)) != MP_OKAY) { + goto LBL_ERR; + } + } else { + /* v - v - u, C = C - A, D = D - B */ + if ((res = mp_sub (&v, &u, &v)) != MP_OKAY) { + goto LBL_ERR; + } + + if ((res = mp_sub (&C, &A, &C)) != MP_OKAY) { + goto LBL_ERR; + } + + if ((res = mp_sub (&D, &B, &D)) != MP_OKAY) { + goto LBL_ERR; + } + } + + /* if not zero goto step 4 */ + if (mp_iszero (&u) == 0) + goto top; + + /* now a = C, b = D, gcd == g*v */ + + /* if v != 1 then there is no inverse */ + if (mp_cmp_d (&v, 1) != MP_EQ) { + res = MP_VAL; + goto LBL_ERR; + } + + /* if its too low */ + while (mp_cmp_d(&C, 0) == MP_LT) { + if ((res = mp_add(&C, b, &C)) != MP_OKAY) { + goto LBL_ERR; + } + } + + /* too big */ + while (mp_cmp_mag(&C, b) != MP_LT) { + if ((res = mp_sub(&C, b, &C)) != MP_OKAY) { + goto LBL_ERR; + } + } + + /* C is now the inverse */ + mp_exch (&C, c); + res = MP_OKAY; +LBL_ERR:mp_clear_multi (&x, &y, &u, &v, &A, &B, &C, &D, NULL); + return res; +} +#endif diff --git a/libtommath/bn_mp_is_square.c b/libtommath/bn_mp_is_square.c new file mode 100644 index 0000000..969d237 --- /dev/null +++ b/libtommath/bn_mp_is_square.c @@ -0,0 +1,105 @@ +#include <tommath.h> +#ifdef BN_MP_IS_SQUARE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* Check if remainders are possible squares - fast exclude non-squares */ +static const char rem_128[128] = { + 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, + 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, + 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, + 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, + 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, + 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, + 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, + 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1 +}; + +static const char rem_105[105] = { + 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, + 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, + 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, + 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, + 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, + 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, + 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1 +}; + +/* Store non-zero to ret if arg is square, and zero if not */ +int mp_is_square(mp_int *arg,int *ret) +{ + int res; + mp_digit c; + mp_int t; + unsigned long r; + + /* Default to Non-square :) */ + *ret = MP_NO; + + if (arg->sign == MP_NEG) { + return MP_VAL; + } + + /* digits used? (TSD) */ + if (arg->used == 0) { + return MP_OKAY; + } + + /* First check mod 128 (suppose that DIGIT_BIT is at least 7) */ + if (rem_128[127 & DIGIT(arg,0)] == 1) { + return MP_OKAY; + } + + /* Next check mod 105 (3*5*7) */ + if ((res = mp_mod_d(arg,105,&c)) != MP_OKAY) { + return res; + } + if (rem_105[c] == 1) { + return MP_OKAY; + } + + + if ((res = mp_init_set_int(&t,11L*13L*17L*19L*23L*29L*31L)) != MP_OKAY) { + return res; + } + if ((res = mp_mod(arg,&t,&t)) != MP_OKAY) { + goto ERR; + } + r = mp_get_int(&t); + /* Check for other prime modules, note it's not an ERROR but we must + * free "t" so the easiest way is to goto ERR. We know that res + * is already equal to MP_OKAY from the mp_mod call + */ + if ( (1L<<(r%11)) & 0x5C4L ) goto ERR; + if ( (1L<<(r%13)) & 0x9E4L ) goto ERR; + if ( (1L<<(r%17)) & 0x5CE8L ) goto ERR; + if ( (1L<<(r%19)) & 0x4F50CL ) goto ERR; + if ( (1L<<(r%23)) & 0x7ACCA0L ) goto ERR; + if ( (1L<<(r%29)) & 0xC2EDD0CL ) goto ERR; + if ( (1L<<(r%31)) & 0x6DE2B848L ) goto ERR; + + /* Final check - is sqr(sqrt(arg)) == arg ? */ + if ((res = mp_sqrt(arg,&t)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_sqr(&t,&t)) != MP_OKAY) { + goto ERR; + } + + *ret = (mp_cmp_mag(&t,arg) == MP_EQ) ? MP_YES : MP_NO; +ERR:mp_clear(&t); + return res; +} +#endif diff --git a/libtommath/bn_mp_jacobi.c b/libtommath/bn_mp_jacobi.c new file mode 100644 index 0000000..74cbbf3 --- /dev/null +++ b/libtommath/bn_mp_jacobi.c @@ -0,0 +1,101 @@ +#include <tommath.h> +#ifdef BN_MP_JACOBI_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* computes the jacobi c = (a | n) (or Legendre if n is prime) + * HAC pp. 73 Algorithm 2.149 + */ +int mp_jacobi (mp_int * a, mp_int * p, int *c) +{ + mp_int a1, p1; + int k, s, r, res; + mp_digit residue; + + /* if p <= 0 return MP_VAL */ + if (mp_cmp_d(p, 0) != MP_GT) { + return MP_VAL; + } + + /* step 1. if a == 0, return 0 */ + if (mp_iszero (a) == 1) { + *c = 0; + return MP_OKAY; + } + + /* step 2. if a == 1, return 1 */ + if (mp_cmp_d (a, 1) == MP_EQ) { + *c = 1; + return MP_OKAY; + } + + /* default */ + s = 0; + + /* step 3. write a = a1 * 2**k */ + if ((res = mp_init_copy (&a1, a)) != MP_OKAY) { + return res; + } + + if ((res = mp_init (&p1)) != MP_OKAY) { + goto LBL_A1; + } + + /* divide out larger power of two */ + k = mp_cnt_lsb(&a1); + if ((res = mp_div_2d(&a1, k, &a1, NULL)) != MP_OKAY) { + goto LBL_P1; + } + + /* step 4. if e is even set s=1 */ + if ((k & 1) == 0) { + s = 1; + } else { + /* else set s=1 if p = 1/7 (mod 8) or s=-1 if p = 3/5 (mod 8) */ + residue = p->dp[0] & 7; + + if (residue == 1 || residue == 7) { + s = 1; + } else if (residue == 3 || residue == 5) { + s = -1; + } + } + + /* step 5. if p == 3 (mod 4) *and* a1 == 3 (mod 4) then s = -s */ + if ( ((p->dp[0] & 3) == 3) && ((a1.dp[0] & 3) == 3)) { + s = -s; + } + + /* if a1 == 1 we're done */ + if (mp_cmp_d (&a1, 1) == MP_EQ) { + *c = s; + } else { + /* n1 = n mod a1 */ + if ((res = mp_mod (p, &a1, &p1)) != MP_OKAY) { + goto LBL_P1; + } + if ((res = mp_jacobi (&p1, &a1, &r)) != MP_OKAY) { + goto LBL_P1; + } + *c = s * r; + } + + /* done */ + res = MP_OKAY; +LBL_P1:mp_clear (&p1); +LBL_A1:mp_clear (&a1); + return res; +} +#endif diff --git a/libtommath/bn_mp_karatsuba_mul.c b/libtommath/bn_mp_karatsuba_mul.c new file mode 100644 index 0000000..daa78c7 --- /dev/null +++ b/libtommath/bn_mp_karatsuba_mul.c @@ -0,0 +1,163 @@ +#include <tommath.h> +#ifdef BN_MP_KARATSUBA_MUL_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* c = |a| * |b| using Karatsuba Multiplication using + * three half size multiplications + * + * Let B represent the radix [e.g. 2**DIGIT_BIT] and + * let n represent half of the number of digits in + * the min(a,b) + * + * a = a1 * B**n + a0 + * b = b1 * B**n + b0 + * + * Then, a * b => + a1b1 * B**2n + ((a1 - a0)(b1 - b0) + a0b0 + a1b1) * B + a0b0 + * + * Note that a1b1 and a0b0 are used twice and only need to be + * computed once. So in total three half size (half # of + * digit) multiplications are performed, a0b0, a1b1 and + * (a1-b1)(a0-b0) + * + * Note that a multiplication of half the digits requires + * 1/4th the number of single precision multiplications so in + * total after one call 25% of the single precision multiplications + * are saved. Note also that the call to mp_mul can end up back + * in this function if the a0, a1, b0, or b1 are above the threshold. + * This is known as divide-and-conquer and leads to the famous + * O(N**lg(3)) or O(N**1.584) work which is asymptopically lower than + * the standard O(N**2) that the baseline/comba methods use. + * Generally though the overhead of this method doesn't pay off + * until a certain size (N ~ 80) is reached. + */ +int mp_karatsuba_mul (mp_int * a, mp_int * b, mp_int * c) +{ + mp_int x0, x1, y0, y1, t1, x0y0, x1y1; + int B, err; + + /* default the return code to an error */ + err = MP_MEM; + + /* min # of digits */ + B = MIN (a->used, b->used); + + /* now divide in two */ + B = B >> 1; + + /* init copy all the temps */ + if (mp_init_size (&x0, B) != MP_OKAY) + goto ERR; + if (mp_init_size (&x1, a->used - B) != MP_OKAY) + goto X0; + if (mp_init_size (&y0, B) != MP_OKAY) + goto X1; + if (mp_init_size (&y1, b->used - B) != MP_OKAY) + goto Y0; + + /* init temps */ + if (mp_init_size (&t1, B * 2) != MP_OKAY) + goto Y1; + if (mp_init_size (&x0y0, B * 2) != MP_OKAY) + goto T1; + if (mp_init_size (&x1y1, B * 2) != MP_OKAY) + goto X0Y0; + + /* now shift the digits */ + x0.used = y0.used = B; + x1.used = a->used - B; + y1.used = b->used - B; + + { + register int x; + register mp_digit *tmpa, *tmpb, *tmpx, *tmpy; + + /* we copy the digits directly instead of using higher level functions + * since we also need to shift the digits + */ + tmpa = a->dp; + tmpb = b->dp; + + tmpx = x0.dp; + tmpy = y0.dp; + for (x = 0; x < B; x++) { + *tmpx++ = *tmpa++; + *tmpy++ = *tmpb++; + } + + tmpx = x1.dp; + for (x = B; x < a->used; x++) { + *tmpx++ = *tmpa++; + } + + tmpy = y1.dp; + for (x = B; x < b->used; x++) { + *tmpy++ = *tmpb++; + } + } + + /* only need to clamp the lower words since by definition the + * upper words x1/y1 must have a known number of digits + */ + mp_clamp (&x0); + mp_clamp (&y0); + + /* now calc the products x0y0 and x1y1 */ + /* after this x0 is no longer required, free temp [x0==t2]! */ + if (mp_mul (&x0, &y0, &x0y0) != MP_OKAY) + goto X1Y1; /* x0y0 = x0*y0 */ + if (mp_mul (&x1, &y1, &x1y1) != MP_OKAY) + goto X1Y1; /* x1y1 = x1*y1 */ + + /* now calc x1-x0 and y1-y0 */ + if (mp_sub (&x1, &x0, &t1) != MP_OKAY) + goto X1Y1; /* t1 = x1 - x0 */ + if (mp_sub (&y1, &y0, &x0) != MP_OKAY) + goto X1Y1; /* t2 = y1 - y0 */ + if (mp_mul (&t1, &x0, &t1) != MP_OKAY) + goto X1Y1; /* t1 = (x1 - x0) * (y1 - y0) */ + + /* add x0y0 */ + if (mp_add (&x0y0, &x1y1, &x0) != MP_OKAY) + goto X1Y1; /* t2 = x0y0 + x1y1 */ + if (mp_sub (&x0, &t1, &t1) != MP_OKAY) + goto X1Y1; /* t1 = x0y0 + x1y1 - (x1-x0)*(y1-y0) */ + + /* shift by B */ + if (mp_lshd (&t1, B) != MP_OKAY) + goto X1Y1; /* t1 = (x0y0 + x1y1 - (x1-x0)*(y1-y0))<<B */ + if (mp_lshd (&x1y1, B * 2) != MP_OKAY) + goto X1Y1; /* x1y1 = x1y1 << 2*B */ + + if (mp_add (&x0y0, &t1, &t1) != MP_OKAY) + goto X1Y1; /* t1 = x0y0 + t1 */ + if (mp_add (&t1, &x1y1, c) != MP_OKAY) + goto X1Y1; /* t1 = x0y0 + t1 + x1y1 */ + + /* Algorithm succeeded set the return code to MP_OKAY */ + err = MP_OKAY; + +X1Y1:mp_clear (&x1y1); +X0Y0:mp_clear (&x0y0); +T1:mp_clear (&t1); +Y1:mp_clear (&y1); +Y0:mp_clear (&y0); +X1:mp_clear (&x1); +X0:mp_clear (&x0); +ERR: + return err; +} +#endif diff --git a/libtommath/bn_mp_karatsuba_sqr.c b/libtommath/bn_mp_karatsuba_sqr.c new file mode 100644 index 0000000..315ceab --- /dev/null +++ b/libtommath/bn_mp_karatsuba_sqr.c @@ -0,0 +1,117 @@ +#include <tommath.h> +#ifdef BN_MP_KARATSUBA_SQR_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* Karatsuba squaring, computes b = a*a using three + * half size squarings + * + * See comments of karatsuba_mul for details. It + * is essentially the same algorithm but merely + * tuned to perform recursive squarings. + */ +int mp_karatsuba_sqr (mp_int * a, mp_int * b) +{ + mp_int x0, x1, t1, t2, x0x0, x1x1; + int B, err; + + err = MP_MEM; + + /* min # of digits */ + B = a->used; + + /* now divide in two */ + B = B >> 1; + + /* init copy all the temps */ + if (mp_init_size (&x0, B) != MP_OKAY) + goto ERR; + if (mp_init_size (&x1, a->used - B) != MP_OKAY) + goto X0; + + /* init temps */ + if (mp_init_size (&t1, a->used * 2) != MP_OKAY) + goto X1; + if (mp_init_size (&t2, a->used * 2) != MP_OKAY) + goto T1; + if (mp_init_size (&x0x0, B * 2) != MP_OKAY) + goto T2; + if (mp_init_size (&x1x1, (a->used - B) * 2) != MP_OKAY) + goto X0X0; + + { + register int x; + register mp_digit *dst, *src; + + src = a->dp; + + /* now shift the digits */ + dst = x0.dp; + for (x = 0; x < B; x++) { + *dst++ = *src++; + } + + dst = x1.dp; + for (x = B; x < a->used; x++) { + *dst++ = *src++; + } + } + + x0.used = B; + x1.used = a->used - B; + + mp_clamp (&x0); + + /* now calc the products x0*x0 and x1*x1 */ + if (mp_sqr (&x0, &x0x0) != MP_OKAY) + goto X1X1; /* x0x0 = x0*x0 */ + if (mp_sqr (&x1, &x1x1) != MP_OKAY) + goto X1X1; /* x1x1 = x1*x1 */ + + /* now calc (x1-x0)**2 */ + if (mp_sub (&x1, &x0, &t1) != MP_OKAY) + goto X1X1; /* t1 = x1 - x0 */ + if (mp_sqr (&t1, &t1) != MP_OKAY) + goto X1X1; /* t1 = (x1 - x0) * (x1 - x0) */ + + /* add x0y0 */ + if (s_mp_add (&x0x0, &x1x1, &t2) != MP_OKAY) + goto X1X1; /* t2 = x0x0 + x1x1 */ + if (mp_sub (&t2, &t1, &t1) != MP_OKAY) + goto X1X1; /* t1 = x0x0 + x1x1 - (x1-x0)*(x1-x0) */ + + /* shift by B */ + if (mp_lshd (&t1, B) != MP_OKAY) + goto X1X1; /* t1 = (x0x0 + x1x1 - (x1-x0)*(x1-x0))<<B */ + if (mp_lshd (&x1x1, B * 2) != MP_OKAY) + goto X1X1; /* x1x1 = x1x1 << 2*B */ + + if (mp_add (&x0x0, &t1, &t1) != MP_OKAY) + goto X1X1; /* t1 = x0x0 + t1 */ + if (mp_add (&t1, &x1x1, b) != MP_OKAY) + goto X1X1; /* t1 = x0x0 + t1 + x1x1 */ + + err = MP_OKAY; + +X1X1:mp_clear (&x1x1); +X0X0:mp_clear (&x0x0); +T2:mp_clear (&t2); +T1:mp_clear (&t1); +X1:mp_clear (&x1); +X0:mp_clear (&x0); +ERR: + return err; +} +#endif diff --git a/libtommath/bn_mp_lcm.c b/libtommath/bn_mp_lcm.c new file mode 100644 index 0000000..8e3a759 --- /dev/null +++ b/libtommath/bn_mp_lcm.c @@ -0,0 +1,56 @@ +#include <tommath.h> +#ifdef BN_MP_LCM_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* computes least common multiple as |a*b|/(a, b) */ +int mp_lcm (mp_int * a, mp_int * b, mp_int * c) +{ + int res; + mp_int t1, t2; + + + if ((res = mp_init_multi (&t1, &t2, NULL)) != MP_OKAY) { + return res; + } + + /* t1 = get the GCD of the two inputs */ + if ((res = mp_gcd (a, b, &t1)) != MP_OKAY) { + goto LBL_T; + } + + /* divide the smallest by the GCD */ + if (mp_cmp_mag(a, b) == MP_LT) { + /* store quotient in t2 such that t2 * b is the LCM */ + if ((res = mp_div(a, &t1, &t2, NULL)) != MP_OKAY) { + goto LBL_T; + } + res = mp_mul(b, &t2, c); + } else { + /* store quotient in t2 such that t2 * a is the LCM */ + if ((res = mp_div(b, &t1, &t2, NULL)) != MP_OKAY) { + goto LBL_T; + } + res = mp_mul(a, &t2, c); + } + + /* fix the sign to positive */ + c->sign = MP_ZPOS; + +LBL_T: + mp_clear_multi (&t1, &t2, NULL); + return res; +} +#endif diff --git a/libtommath/bn_mp_lshd.c b/libtommath/bn_mp_lshd.c new file mode 100644 index 0000000..398b648 --- /dev/null +++ b/libtommath/bn_mp_lshd.c @@ -0,0 +1,63 @@ +#include <tommath.h> +#ifdef BN_MP_LSHD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* shift left a certain amount of digits */ +int mp_lshd (mp_int * a, int b) +{ + int x, res; + + /* if its less than zero return */ + if (b <= 0) { + return MP_OKAY; + } + + /* grow to fit the new digits */ + if (a->alloc < a->used + b) { + if ((res = mp_grow (a, a->used + b)) != MP_OKAY) { + return res; + } + } + + { + register mp_digit *top, *bottom; + + /* increment the used by the shift amount then copy upwards */ + a->used += b; + + /* top */ + top = a->dp + a->used - 1; + + /* base */ + bottom = a->dp + a->used - 1 - b; + + /* much like mp_rshd this is implemented using a sliding window + * except the window goes the otherway around. Copying from + * the bottom to the top. see bn_mp_rshd.c for more info. + */ + for (x = a->used - 1; x >= b; x--) { + *top-- = *bottom--; + } + + /* zero the lower digits */ + top = a->dp; + for (x = 0; x < b; x++) { + *top++ = 0; + } + } + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_mp_mod.c b/libtommath/bn_mp_mod.c new file mode 100644 index 0000000..75779bb --- /dev/null +++ b/libtommath/bn_mp_mod.c @@ -0,0 +1,44 @@ +#include <tommath.h> +#ifdef BN_MP_MOD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* c = a mod b, 0 <= c < b */ +int +mp_mod (mp_int * a, mp_int * b, mp_int * c) +{ + mp_int t; + int res; + + if ((res = mp_init (&t)) != MP_OKAY) { + return res; + } + + if ((res = mp_div (a, b, NULL, &t)) != MP_OKAY) { + mp_clear (&t); + return res; + } + + if (t.sign != b->sign) { + res = mp_add (b, &t, c); + } else { + res = MP_OKAY; + mp_exch (&t, c); + } + + mp_clear (&t); + return res; +} +#endif diff --git a/libtommath/bn_mp_mod_2d.c b/libtommath/bn_mp_mod_2d.c new file mode 100644 index 0000000..589e4ba --- /dev/null +++ b/libtommath/bn_mp_mod_2d.c @@ -0,0 +1,51 @@ +#include <tommath.h> +#ifdef BN_MP_MOD_2D_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* calc a value mod 2**b */ +int +mp_mod_2d (mp_int * a, int b, mp_int * c) +{ + int x, res; + + /* if b is <= 0 then zero the int */ + if (b <= 0) { + mp_zero (c); + return MP_OKAY; + } + + /* if the modulus is larger than the value than return */ + if (b >= (int) (a->used * DIGIT_BIT)) { + res = mp_copy (a, c); + return res; + } + + /* copy */ + if ((res = mp_copy (a, c)) != MP_OKAY) { + return res; + } + + /* zero digits above the last digit of the modulus */ + for (x = (b / DIGIT_BIT) + ((b % DIGIT_BIT) == 0 ? 0 : 1); x < c->used; x++) { + c->dp[x] = 0; + } + /* clear the digit that is not completely outside/inside the modulus */ + c->dp[b / DIGIT_BIT] &= + (mp_digit) ((((mp_digit) 1) << (((mp_digit) b) % DIGIT_BIT)) - ((mp_digit) 1)); + mp_clamp (c); + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_mp_mod_d.c b/libtommath/bn_mp_mod_d.c new file mode 100644 index 0000000..8a2ad24 --- /dev/null +++ b/libtommath/bn_mp_mod_d.c @@ -0,0 +1,23 @@ +#include <tommath.h> +#ifdef BN_MP_MOD_D_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +int +mp_mod_d (mp_int * a, mp_digit b, mp_digit * c) +{ + return mp_div_d(a, b, NULL, c); +} +#endif diff --git a/libtommath/bn_mp_montgomery_calc_normalization.c b/libtommath/bn_mp_montgomery_calc_normalization.c new file mode 100644 index 0000000..0a760cf --- /dev/null +++ b/libtommath/bn_mp_montgomery_calc_normalization.c @@ -0,0 +1,56 @@ +#include <tommath.h> +#ifdef BN_MP_MONTGOMERY_CALC_NORMALIZATION_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* + * shifts with subtractions when the result is greater than b. + * + * The method is slightly modified to shift B unconditionally upto just under + * the leading bit of b. This saves alot of multiple precision shifting. + */ +int mp_montgomery_calc_normalization (mp_int * a, mp_int * b) +{ + int x, bits, res; + + /* how many bits of last digit does b use */ + bits = mp_count_bits (b) % DIGIT_BIT; + + + if (b->used > 1) { + if ((res = mp_2expt (a, (b->used - 1) * DIGIT_BIT + bits - 1)) != MP_OKAY) { + return res; + } + } else { + mp_set(a, 1); + bits = 1; + } + + + /* now compute C = A * B mod b */ + for (x = bits - 1; x < (int)DIGIT_BIT; x++) { + if ((res = mp_mul_2 (a, a)) != MP_OKAY) { + return res; + } + if (mp_cmp_mag (a, b) != MP_LT) { + if ((res = s_mp_sub (a, b, a)) != MP_OKAY) { + return res; + } + } + } + + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_mp_montgomery_reduce.c b/libtommath/bn_mp_montgomery_reduce.c new file mode 100644 index 0000000..3095fa7 --- /dev/null +++ b/libtommath/bn_mp_montgomery_reduce.c @@ -0,0 +1,114 @@ +#include <tommath.h> +#ifdef BN_MP_MONTGOMERY_REDUCE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* computes xR**-1 == x (mod N) via Montgomery Reduction */ +int +mp_montgomery_reduce (mp_int * x, mp_int * n, mp_digit rho) +{ + int ix, res, digs; + mp_digit mu; + + /* can the fast reduction [comba] method be used? + * + * Note that unlike in mul you're safely allowed *less* + * than the available columns [255 per default] since carries + * are fixed up in the inner loop. + */ + digs = n->used * 2 + 1; + if ((digs < MP_WARRAY) && + n->used < + (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) { + return fast_mp_montgomery_reduce (x, n, rho); + } + + /* grow the input as required */ + if (x->alloc < digs) { + if ((res = mp_grow (x, digs)) != MP_OKAY) { + return res; + } + } + x->used = digs; + + for (ix = 0; ix < n->used; ix++) { + /* mu = ai * rho mod b + * + * The value of rho must be precalculated via + * montgomery_setup() such that + * it equals -1/n0 mod b this allows the + * following inner loop to reduce the + * input one digit at a time + */ + mu = (mp_digit) (((mp_word)x->dp[ix]) * ((mp_word)rho) & MP_MASK); + + /* a = a + mu * m * b**i */ + { + register int iy; + register mp_digit *tmpn, *tmpx, u; + register mp_word r; + + /* alias for digits of the modulus */ + tmpn = n->dp; + + /* alias for the digits of x [the input] */ + tmpx = x->dp + ix; + + /* set the carry to zero */ + u = 0; + + /* Multiply and add in place */ + for (iy = 0; iy < n->used; iy++) { + /* compute product and sum */ + r = ((mp_word)mu) * ((mp_word)*tmpn++) + + ((mp_word) u) + ((mp_word) * tmpx); + + /* get carry */ + u = (mp_digit)(r >> ((mp_word) DIGIT_BIT)); + + /* fix digit */ + *tmpx++ = (mp_digit)(r & ((mp_word) MP_MASK)); + } + /* At this point the ix'th digit of x should be zero */ + + + /* propagate carries upwards as required*/ + while (u) { + *tmpx += u; + u = *tmpx >> DIGIT_BIT; + *tmpx++ &= MP_MASK; + } + } + } + + /* at this point the n.used'th least + * significant digits of x are all zero + * which means we can shift x to the + * right by n.used digits and the + * residue is unchanged. + */ + + /* x = x/b**n.used */ + mp_clamp(x); + mp_rshd (x, n->used); + + /* if x >= n then x = x - n */ + if (mp_cmp_mag (x, n) != MP_LT) { + return s_mp_sub (x, n, x); + } + + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_mp_montgomery_setup.c b/libtommath/bn_mp_montgomery_setup.c new file mode 100644 index 0000000..9dfc087 --- /dev/null +++ b/libtommath/bn_mp_montgomery_setup.c @@ -0,0 +1,55 @@ +#include <tommath.h> +#ifdef BN_MP_MONTGOMERY_SETUP_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* setups the montgomery reduction stuff */ +int +mp_montgomery_setup (mp_int * n, mp_digit * rho) +{ + mp_digit x, b; + +/* fast inversion mod 2**k + * + * Based on the fact that + * + * XA = 1 (mod 2**n) => (X(2-XA)) A = 1 (mod 2**2n) + * => 2*X*A - X*X*A*A = 1 + * => 2*(1) - (1) = 1 + */ + b = n->dp[0]; + + if ((b & 1) == 0) { + return MP_VAL; + } + + x = (((b + 2) & 4) << 1) + b; /* here x*a==1 mod 2**4 */ + x *= 2 - b * x; /* here x*a==1 mod 2**8 */ +#if !defined(MP_8BIT) + x *= 2 - b * x; /* here x*a==1 mod 2**16 */ +#endif +#if defined(MP_64BIT) || !(defined(MP_8BIT) || defined(MP_16BIT)) + x *= 2 - b * x; /* here x*a==1 mod 2**32 */ +#endif +#ifdef MP_64BIT + x *= 2 - b * x; /* here x*a==1 mod 2**64 */ +#endif + + /* rho = -1/m mod b */ + *rho = (((mp_word)1 << ((mp_word) DIGIT_BIT)) - x) & MP_MASK; + + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_mp_mul.c b/libtommath/bn_mp_mul.c new file mode 100644 index 0000000..f9cfa09 --- /dev/null +++ b/libtommath/bn_mp_mul.c @@ -0,0 +1,62 @@ +#include <tommath.h> +#ifdef BN_MP_MUL_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* high level multiplication (handles sign) */ +int mp_mul (mp_int * a, mp_int * b, mp_int * c) +{ + int res, neg; + neg = (a->sign == b->sign) ? MP_ZPOS : MP_NEG; + + /* use Toom-Cook? */ +#ifdef BN_MP_TOOM_MUL_C + if (MIN (a->used, b->used) >= TOOM_MUL_CUTOFF) { + res = mp_toom_mul(a, b, c); + } else +#endif +#ifdef BN_MP_KARATSUBA_MUL_C + /* use Karatsuba? */ + if (MIN (a->used, b->used) >= KARATSUBA_MUL_CUTOFF) { + res = mp_karatsuba_mul (a, b, c); + } else +#endif + { + /* can we use the fast multiplier? + * + * The fast multiplier can be used if the output will + * have less than MP_WARRAY digits and the number of + * digits won't affect carry propagation + */ + int digs = a->used + b->used + 1; + +#ifdef BN_FAST_S_MP_MUL_DIGS_C + if ((digs < MP_WARRAY) && + MIN(a->used, b->used) <= + (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) { + res = fast_s_mp_mul_digs (a, b, c, digs); + } else +#endif +#ifdef BN_S_MP_MUL_DIGS_C + res = s_mp_mul (a, b, c); /* uses s_mp_mul_digs */ +#else + res = MP_VAL; +#endif + + } + c->sign = (c->used > 0) ? neg : MP_ZPOS; + return res; +} +#endif diff --git a/libtommath/bn_mp_mul_2.c b/libtommath/bn_mp_mul_2.c new file mode 100644 index 0000000..6936681 --- /dev/null +++ b/libtommath/bn_mp_mul_2.c @@ -0,0 +1,78 @@ +#include <tommath.h> +#ifdef BN_MP_MUL_2_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* b = a*2 */ +int mp_mul_2(mp_int * a, mp_int * b) +{ + int x, res, oldused; + + /* grow to accomodate result */ + if (b->alloc < a->used + 1) { + if ((res = mp_grow (b, a->used + 1)) != MP_OKAY) { + return res; + } + } + + oldused = b->used; + b->used = a->used; + + { + register mp_digit r, rr, *tmpa, *tmpb; + + /* alias for source */ + tmpa = a->dp; + + /* alias for dest */ + tmpb = b->dp; + + /* carry */ + r = 0; + for (x = 0; x < a->used; x++) { + + /* get what will be the *next* carry bit from the + * MSB of the current digit + */ + rr = *tmpa >> ((mp_digit)(DIGIT_BIT - 1)); + + /* now shift up this digit, add in the carry [from the previous] */ + *tmpb++ = ((*tmpa++ << ((mp_digit)1)) | r) & MP_MASK; + + /* copy the carry that would be from the source + * digit into the next iteration + */ + r = rr; + } + + /* new leading digit? */ + if (r != 0) { + /* add a MSB which is always 1 at this point */ + *tmpb = 1; + ++(b->used); + } + + /* now zero any excess digits on the destination + * that we didn't write to + */ + tmpb = b->dp + b->used; + for (x = b->used; x < oldused; x++) { + *tmpb++ = 0; + } + } + b->sign = a->sign; + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_mp_mul_2d.c b/libtommath/bn_mp_mul_2d.c new file mode 100644 index 0000000..04cb8dd --- /dev/null +++ b/libtommath/bn_mp_mul_2d.c @@ -0,0 +1,81 @@ +#include <tommath.h> +#ifdef BN_MP_MUL_2D_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* shift left by a certain bit count */ +int mp_mul_2d (mp_int * a, int b, mp_int * c) +{ + mp_digit d; + int res; + + /* copy */ + if (a != c) { + if ((res = mp_copy (a, c)) != MP_OKAY) { + return res; + } + } + + if (c->alloc < (int)(c->used + b/DIGIT_BIT + 1)) { + if ((res = mp_grow (c, c->used + b / DIGIT_BIT + 1)) != MP_OKAY) { + return res; + } + } + + /* shift by as many digits in the bit count */ + if (b >= (int)DIGIT_BIT) { + if ((res = mp_lshd (c, b / DIGIT_BIT)) != MP_OKAY) { + return res; + } + } + + /* shift any bit count < DIGIT_BIT */ + d = (mp_digit) (b % DIGIT_BIT); + if (d != 0) { + register mp_digit *tmpc, shift, mask, r, rr; + register int x; + + /* bitmask for carries */ + mask = (((mp_digit)1) << d) - 1; + + /* shift for msbs */ + shift = DIGIT_BIT - d; + + /* alias */ + tmpc = c->dp; + + /* carry */ + r = 0; + for (x = 0; x < c->used; x++) { + /* get the higher bits of the current word */ + rr = (*tmpc >> shift) & mask; + + /* shift the current word and OR in the carry */ + *tmpc = ((*tmpc << d) | r) & MP_MASK; + ++tmpc; + + /* set the carry to the carry bits of the current word */ + r = rr; + } + + /* set final carry */ + if (r != 0) { + c->dp[(c->used)++] = r; + } + } + mp_clamp (c); + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_mp_mul_d.c b/libtommath/bn_mp_mul_d.c new file mode 100644 index 0000000..f936361 --- /dev/null +++ b/libtommath/bn_mp_mul_d.c @@ -0,0 +1,74 @@ +#include <tommath.h> +#ifdef BN_MP_MUL_D_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* multiply by a digit */ +int +mp_mul_d (mp_int * a, mp_digit b, mp_int * c) +{ + mp_digit u, *tmpa, *tmpc; + mp_word r; + int ix, res, olduse; + + /* make sure c is big enough to hold a*b */ + if (c->alloc < a->used + 1) { + if ((res = mp_grow (c, a->used + 1)) != MP_OKAY) { + return res; + } + } + + /* get the original destinations used count */ + olduse = c->used; + + /* set the sign */ + c->sign = a->sign; + + /* alias for a->dp [source] */ + tmpa = a->dp; + + /* alias for c->dp [dest] */ + tmpc = c->dp; + + /* zero carry */ + u = 0; + + /* compute columns */ + for (ix = 0; ix < a->used; ix++) { + /* compute product and carry sum for this term */ + r = ((mp_word) u) + ((mp_word)*tmpa++) * ((mp_word)b); + + /* mask off higher bits to get a single digit */ + *tmpc++ = (mp_digit) (r & ((mp_word) MP_MASK)); + + /* send carry into next iteration */ + u = (mp_digit) (r >> ((mp_word) DIGIT_BIT)); + } + + /* store final carry [if any] */ + *tmpc++ = u; + + /* now zero digits above the top */ + while (ix++ < olduse) { + *tmpc++ = 0; + } + + /* set used count */ + c->used = a->used + 1; + mp_clamp(c); + + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_mp_mulmod.c b/libtommath/bn_mp_mulmod.c new file mode 100644 index 0000000..d34e90a --- /dev/null +++ b/libtommath/bn_mp_mulmod.c @@ -0,0 +1,37 @@ +#include <tommath.h> +#ifdef BN_MP_MULMOD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* d = a * b (mod c) */ +int +mp_mulmod (mp_int * a, mp_int * b, mp_int * c, mp_int * d) +{ + int res; + mp_int t; + + if ((res = mp_init (&t)) != MP_OKAY) { + return res; + } + + if ((res = mp_mul (a, b, &t)) != MP_OKAY) { + mp_clear (&t); + return res; + } + res = mp_mod (&t, c, d); + mp_clear (&t); + return res; +} +#endif diff --git a/libtommath/bn_mp_n_root.c b/libtommath/bn_mp_n_root.c new file mode 100644 index 0000000..7b11aa2 --- /dev/null +++ b/libtommath/bn_mp_n_root.c @@ -0,0 +1,128 @@ +#include <tommath.h> +#ifdef BN_MP_N_ROOT_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* find the n'th root of an integer + * + * Result found such that (c)**b <= a and (c+1)**b > a + * + * This algorithm uses Newton's approximation + * x[i+1] = x[i] - f(x[i])/f'(x[i]) + * which will find the root in log(N) time where + * each step involves a fair bit. This is not meant to + * find huge roots [square and cube, etc]. + */ +int mp_n_root (mp_int * a, mp_digit b, mp_int * c) +{ + mp_int t1, t2, t3; + int res, neg; + + /* input must be positive if b is even */ + if ((b & 1) == 0 && a->sign == MP_NEG) { + return MP_VAL; + } + + if ((res = mp_init (&t1)) != MP_OKAY) { + return res; + } + + if ((res = mp_init (&t2)) != MP_OKAY) { + goto LBL_T1; + } + + if ((res = mp_init (&t3)) != MP_OKAY) { + goto LBL_T2; + } + + /* if a is negative fudge the sign but keep track */ + neg = a->sign; + a->sign = MP_ZPOS; + + /* t2 = 2 */ + mp_set (&t2, 2); + + do { + /* t1 = t2 */ + if ((res = mp_copy (&t2, &t1)) != MP_OKAY) { + goto LBL_T3; + } + + /* t2 = t1 - ((t1**b - a) / (b * t1**(b-1))) */ + + /* t3 = t1**(b-1) */ + if ((res = mp_expt_d (&t1, b - 1, &t3)) != MP_OKAY) { + goto LBL_T3; + } + + /* numerator */ + /* t2 = t1**b */ + if ((res = mp_mul (&t3, &t1, &t2)) != MP_OKAY) { + goto LBL_T3; + } + + /* t2 = t1**b - a */ + if ((res = mp_sub (&t2, a, &t2)) != MP_OKAY) { + goto LBL_T3; + } + + /* denominator */ + /* t3 = t1**(b-1) * b */ + if ((res = mp_mul_d (&t3, b, &t3)) != MP_OKAY) { + goto LBL_T3; + } + + /* t3 = (t1**b - a)/(b * t1**(b-1)) */ + if ((res = mp_div (&t2, &t3, &t3, NULL)) != MP_OKAY) { + goto LBL_T3; + } + + if ((res = mp_sub (&t1, &t3, &t2)) != MP_OKAY) { + goto LBL_T3; + } + } while (mp_cmp (&t1, &t2) != MP_EQ); + + /* result can be off by a few so check */ + for (;;) { + if ((res = mp_expt_d (&t1, b, &t2)) != MP_OKAY) { + goto LBL_T3; + } + + if (mp_cmp (&t2, a) == MP_GT) { + if ((res = mp_sub_d (&t1, 1, &t1)) != MP_OKAY) { + goto LBL_T3; + } + } else { + break; + } + } + + /* reset the sign of a first */ + a->sign = neg; + + /* set the result */ + mp_exch (&t1, c); + + /* set the sign of the result */ + c->sign = neg; + + res = MP_OKAY; + +LBL_T3:mp_clear (&t3); +LBL_T2:mp_clear (&t2); +LBL_T1:mp_clear (&t1); + return res; +} +#endif diff --git a/libtommath/bn_mp_neg.c b/libtommath/bn_mp_neg.c new file mode 100644 index 0000000..3a991db --- /dev/null +++ b/libtommath/bn_mp_neg.c @@ -0,0 +1,30 @@ +#include <tommath.h> +#ifdef BN_MP_NEG_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* b = -a */ +int mp_neg (mp_int * a, mp_int * b) +{ + int res; + if ((res = mp_copy (a, b)) != MP_OKAY) { + return res; + } + if (mp_iszero(b) != MP_YES) { + b->sign = (a->sign == MP_ZPOS) ? MP_NEG : MP_ZPOS; + } + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_mp_or.c b/libtommath/bn_mp_or.c new file mode 100644 index 0000000..dccee7e --- /dev/null +++ b/libtommath/bn_mp_or.c @@ -0,0 +1,46 @@ +#include <tommath.h> +#ifdef BN_MP_OR_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* OR two ints together */ +int mp_or (mp_int * a, mp_int * b, mp_int * c) +{ + int res, ix, px; + mp_int t, *x; + + if (a->used > b->used) { + if ((res = mp_init_copy (&t, a)) != MP_OKAY) { + return res; + } + px = b->used; + x = b; + } else { + if ((res = mp_init_copy (&t, b)) != MP_OKAY) { + return res; + } + px = a->used; + x = a; + } + + for (ix = 0; ix < px; ix++) { + t.dp[ix] |= x->dp[ix]; + } + mp_clamp (&t); + mp_exch (c, &t); + mp_clear (&t); + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_mp_prime_fermat.c b/libtommath/bn_mp_prime_fermat.c new file mode 100644 index 0000000..fd74dbe --- /dev/null +++ b/libtommath/bn_mp_prime_fermat.c @@ -0,0 +1,58 @@ +#include <tommath.h> +#ifdef BN_MP_PRIME_FERMAT_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* performs one Fermat test. + * + * If "a" were prime then b**a == b (mod a) since the order of + * the multiplicative sub-group would be phi(a) = a-1. That means + * it would be the same as b**(a mod (a-1)) == b**1 == b (mod a). + * + * Sets result to 1 if the congruence holds, or zero otherwise. + */ +int mp_prime_fermat (mp_int * a, mp_int * b, int *result) +{ + mp_int t; + int err; + + /* default to composite */ + *result = MP_NO; + + /* ensure b > 1 */ + if (mp_cmp_d(b, 1) != MP_GT) { + return MP_VAL; + } + + /* init t */ + if ((err = mp_init (&t)) != MP_OKAY) { + return err; + } + + /* compute t = b**a mod a */ + if ((err = mp_exptmod (b, a, a, &t)) != MP_OKAY) { + goto LBL_T; + } + + /* is it equal to b? */ + if (mp_cmp (&t, b) == MP_EQ) { + *result = MP_YES; + } + + err = MP_OKAY; +LBL_T:mp_clear (&t); + return err; +} +#endif diff --git a/libtommath/bn_mp_prime_is_divisible.c b/libtommath/bn_mp_prime_is_divisible.c new file mode 100644 index 0000000..f85fe7c --- /dev/null +++ b/libtommath/bn_mp_prime_is_divisible.c @@ -0,0 +1,46 @@ +#include <tommath.h> +#ifdef BN_MP_PRIME_IS_DIVISIBLE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* determines if an integers is divisible by one + * of the first PRIME_SIZE primes or not + * + * sets result to 0 if not, 1 if yes + */ +int mp_prime_is_divisible (mp_int * a, int *result) +{ + int err, ix; + mp_digit res; + + /* default to not */ + *result = MP_NO; + + for (ix = 0; ix < PRIME_SIZE; ix++) { + /* what is a mod LBL_prime_tab[ix] */ + if ((err = mp_mod_d (a, ltm_prime_tab[ix], &res)) != MP_OKAY) { + return err; + } + + /* is the residue zero? */ + if (res == 0) { + *result = MP_YES; + return MP_OKAY; + } + } + + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_mp_prime_is_prime.c b/libtommath/bn_mp_prime_is_prime.c new file mode 100644 index 0000000..188053a --- /dev/null +++ b/libtommath/bn_mp_prime_is_prime.c @@ -0,0 +1,79 @@ +#include <tommath.h> +#ifdef BN_MP_PRIME_IS_PRIME_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* performs a variable number of rounds of Miller-Rabin + * + * Probability of error after t rounds is no more than + + * + * Sets result to 1 if probably prime, 0 otherwise + */ +int mp_prime_is_prime (mp_int * a, int t, int *result) +{ + mp_int b; + int ix, err, res; + + /* default to no */ + *result = MP_NO; + + /* valid value of t? */ + if (t <= 0 || t > PRIME_SIZE) { + return MP_VAL; + } + + /* is the input equal to one of the primes in the table? */ + for (ix = 0; ix < PRIME_SIZE; ix++) { + if (mp_cmp_d(a, ltm_prime_tab[ix]) == MP_EQ) { + *result = 1; + return MP_OKAY; + } + } + + /* first perform trial division */ + if ((err = mp_prime_is_divisible (a, &res)) != MP_OKAY) { + return err; + } + + /* return if it was trivially divisible */ + if (res == MP_YES) { + return MP_OKAY; + } + + /* now perform the miller-rabin rounds */ + if ((err = mp_init (&b)) != MP_OKAY) { + return err; + } + + for (ix = 0; ix < t; ix++) { + /* set the prime */ + mp_set (&b, ltm_prime_tab[ix]); + + if ((err = mp_prime_miller_rabin (a, &b, &res)) != MP_OKAY) { + goto LBL_B; + } + + if (res == MP_NO) { + goto LBL_B; + } + } + + /* passed the test */ + *result = MP_YES; +LBL_B:mp_clear (&b); + return err; +} +#endif diff --git a/libtommath/bn_mp_prime_miller_rabin.c b/libtommath/bn_mp_prime_miller_rabin.c new file mode 100644 index 0000000..758a2c3 --- /dev/null +++ b/libtommath/bn_mp_prime_miller_rabin.c @@ -0,0 +1,99 @@ +#include <tommath.h> +#ifdef BN_MP_PRIME_MILLER_RABIN_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* Miller-Rabin test of "a" to the base of "b" as described in + * HAC pp. 139 Algorithm 4.24 + * + * Sets result to 0 if definitely composite or 1 if probably prime. + * Randomly the chance of error is no more than 1/4 and often + * very much lower. + */ +int mp_prime_miller_rabin (mp_int * a, mp_int * b, int *result) +{ + mp_int n1, y, r; + int s, j, err; + + /* default */ + *result = MP_NO; + + /* ensure b > 1 */ + if (mp_cmp_d(b, 1) != MP_GT) { + return MP_VAL; + } + + /* get n1 = a - 1 */ + if ((err = mp_init_copy (&n1, a)) != MP_OKAY) { + return err; + } + if ((err = mp_sub_d (&n1, 1, &n1)) != MP_OKAY) { + goto LBL_N1; + } + + /* set 2**s * r = n1 */ + if ((err = mp_init_copy (&r, &n1)) != MP_OKAY) { + goto LBL_N1; + } + + /* count the number of least significant bits + * which are zero + */ + s = mp_cnt_lsb(&r); + + /* now divide n - 1 by 2**s */ + if ((err = mp_div_2d (&r, s, &r, NULL)) != MP_OKAY) { + goto LBL_R; + } + + /* compute y = b**r mod a */ + if ((err = mp_init (&y)) != MP_OKAY) { + goto LBL_R; + } + if ((err = mp_exptmod (b, &r, a, &y)) != MP_OKAY) { + goto LBL_Y; + } + + /* if y != 1 and y != n1 do */ + if (mp_cmp_d (&y, 1) != MP_EQ && mp_cmp (&y, &n1) != MP_EQ) { + j = 1; + /* while j <= s-1 and y != n1 */ + while ((j <= (s - 1)) && mp_cmp (&y, &n1) != MP_EQ) { + if ((err = mp_sqrmod (&y, a, &y)) != MP_OKAY) { + goto LBL_Y; + } + + /* if y == 1 then composite */ + if (mp_cmp_d (&y, 1) == MP_EQ) { + goto LBL_Y; + } + + ++j; + } + + /* if y != n1 then composite */ + if (mp_cmp (&y, &n1) != MP_EQ) { + goto LBL_Y; + } + } + + /* probably prime now */ + *result = MP_YES; +LBL_Y:mp_clear (&y); +LBL_R:mp_clear (&r); +LBL_N1:mp_clear (&n1); + return err; +} +#endif diff --git a/libtommath/bn_mp_prime_next_prime.c b/libtommath/bn_mp_prime_next_prime.c new file mode 100644 index 0000000..24f93c4 --- /dev/null +++ b/libtommath/bn_mp_prime_next_prime.c @@ -0,0 +1,166 @@ +#include <tommath.h> +#ifdef BN_MP_PRIME_NEXT_PRIME_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* finds the next prime after the number "a" using "t" trials + * of Miller-Rabin. + * + * bbs_style = 1 means the prime must be congruent to 3 mod 4 + */ +int mp_prime_next_prime(mp_int *a, int t, int bbs_style) +{ + int err, res, x, y; + mp_digit res_tab[PRIME_SIZE], step, kstep; + mp_int b; + + /* ensure t is valid */ + if (t <= 0 || t > PRIME_SIZE) { + return MP_VAL; + } + + /* force positive */ + a->sign = MP_ZPOS; + + /* simple algo if a is less than the largest prime in the table */ + if (mp_cmp_d(a, ltm_prime_tab[PRIME_SIZE-1]) == MP_LT) { + /* find which prime it is bigger than */ + for (x = PRIME_SIZE - 2; x >= 0; x--) { + if (mp_cmp_d(a, ltm_prime_tab[x]) != MP_LT) { + if (bbs_style == 1) { + /* ok we found a prime smaller or + * equal [so the next is larger] + * + * however, the prime must be + * congruent to 3 mod 4 + */ + if ((ltm_prime_tab[x + 1] & 3) != 3) { + /* scan upwards for a prime congruent to 3 mod 4 */ + for (y = x + 1; y < PRIME_SIZE; y++) { + if ((ltm_prime_tab[y] & 3) == 3) { + mp_set(a, ltm_prime_tab[y]); + return MP_OKAY; + } + } + } + } else { + mp_set(a, ltm_prime_tab[x + 1]); + return MP_OKAY; + } + } + } + /* at this point a maybe 1 */ + if (mp_cmp_d(a, 1) == MP_EQ) { + mp_set(a, 2); + return MP_OKAY; + } + /* fall through to the sieve */ + } + + /* generate a prime congruent to 3 mod 4 or 1/3 mod 4? */ + if (bbs_style == 1) { + kstep = 4; + } else { + kstep = 2; + } + + /* at this point we will use a combination of a sieve and Miller-Rabin */ + + if (bbs_style == 1) { + /* if a mod 4 != 3 subtract the correct value to make it so */ + if ((a->dp[0] & 3) != 3) { + if ((err = mp_sub_d(a, (a->dp[0] & 3) + 1, a)) != MP_OKAY) { return err; }; + } + } else { + if (mp_iseven(a) == 1) { + /* force odd */ + if ((err = mp_sub_d(a, 1, a)) != MP_OKAY) { + return err; + } + } + } + + /* generate the restable */ + for (x = 1; x < PRIME_SIZE; x++) { + if ((err = mp_mod_d(a, ltm_prime_tab[x], res_tab + x)) != MP_OKAY) { + return err; + } + } + + /* init temp used for Miller-Rabin Testing */ + if ((err = mp_init(&b)) != MP_OKAY) { + return err; + } + + for (;;) { + /* skip to the next non-trivially divisible candidate */ + step = 0; + do { + /* y == 1 if any residue was zero [e.g. cannot be prime] */ + y = 0; + + /* increase step to next candidate */ + step += kstep; + + /* compute the new residue without using division */ + for (x = 1; x < PRIME_SIZE; x++) { + /* add the step to each residue */ + res_tab[x] += kstep; + + /* subtract the modulus [instead of using division] */ + if (res_tab[x] >= ltm_prime_tab[x]) { + res_tab[x] -= ltm_prime_tab[x]; + } + + /* set flag if zero */ + if (res_tab[x] == 0) { + y = 1; + } + } + } while (y == 1 && step < ((((mp_digit)1)<<DIGIT_BIT) - kstep)); + + /* add the step */ + if ((err = mp_add_d(a, step, a)) != MP_OKAY) { + goto LBL_ERR; + } + + /* if didn't pass sieve and step == MAX then skip test */ + if (y == 1 && step >= ((((mp_digit)1)<<DIGIT_BIT) - kstep)) { + continue; + } + + /* is this prime? */ + for (x = 0; x < t; x++) { + mp_set(&b, ltm_prime_tab[t]); + if ((err = mp_prime_miller_rabin(a, &b, &res)) != MP_OKAY) { + goto LBL_ERR; + } + if (res == MP_NO) { + break; + } + } + + if (res == MP_YES) { + break; + } + } + + err = MP_OKAY; +LBL_ERR: + mp_clear(&b); + return err; +} + +#endif diff --git a/libtommath/bn_mp_prime_rabin_miller_trials.c b/libtommath/bn_mp_prime_rabin_miller_trials.c new file mode 100644 index 0000000..d1d0867 --- /dev/null +++ b/libtommath/bn_mp_prime_rabin_miller_trials.c @@ -0,0 +1,48 @@ +#include <tommath.h> +#ifdef BN_MP_PRIME_RABIN_MILLER_TRIALS_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + + +static const struct { + int k, t; +} sizes[] = { +{ 128, 28 }, +{ 256, 16 }, +{ 384, 10 }, +{ 512, 7 }, +{ 640, 6 }, +{ 768, 5 }, +{ 896, 4 }, +{ 1024, 4 } +}; + +/* returns # of RM trials required for a given bit size */ +int mp_prime_rabin_miller_trials(int size) +{ + int x; + + for (x = 0; x < (int)(sizeof(sizes)/(sizeof(sizes[0]))); x++) { + if (sizes[x].k == size) { + return sizes[x].t; + } else if (sizes[x].k > size) { + return (x == 0) ? sizes[0].t : sizes[x - 1].t; + } + } + return sizes[x-1].t + 1; +} + + +#endif diff --git a/libtommath/bn_mp_prime_random_ex.c b/libtommath/bn_mp_prime_random_ex.c new file mode 100644 index 0000000..2010ebe --- /dev/null +++ b/libtommath/bn_mp_prime_random_ex.c @@ -0,0 +1,123 @@ +#include <tommath.h> +#ifdef BN_MP_PRIME_RANDOM_EX_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* makes a truly random prime of a given size (bits), + * + * Flags are as follows: + * + * LTM_PRIME_BBS - make prime congruent to 3 mod 4 + * LTM_PRIME_SAFE - make sure (p-1)/2 is prime as well (implies LTM_PRIME_BBS) + * LTM_PRIME_2MSB_OFF - make the 2nd highest bit zero + * LTM_PRIME_2MSB_ON - make the 2nd highest bit one + * + * You have to supply a callback which fills in a buffer with random bytes. "dat" is a parameter you can + * have passed to the callback (e.g. a state or something). This function doesn't use "dat" itself + * so it can be NULL + * + */ + +/* This is possibly the mother of all prime generation functions, muahahahahaha! */ +int mp_prime_random_ex(mp_int *a, int t, int size, int flags, ltm_prime_callback cb, void *dat) +{ + unsigned char *tmp, maskAND, maskOR_msb, maskOR_lsb; + int res, err, bsize, maskOR_msb_offset; + + /* sanity check the input */ + if (size <= 1 || t <= 0) { + return MP_VAL; + } + + /* LTM_PRIME_SAFE implies LTM_PRIME_BBS */ + if (flags & LTM_PRIME_SAFE) { + flags |= LTM_PRIME_BBS; + } + + /* calc the byte size */ + bsize = (size>>3) + ((size&7)?1:0); + + /* we need a buffer of bsize bytes */ + tmp = OPT_CAST(unsigned char) XMALLOC(bsize); + if (tmp == NULL) { + return MP_MEM; + } + + /* calc the maskAND value for the MSbyte*/ + maskAND = ((size&7) == 0) ? 0xFF : (0xFF >> (8 - (size & 7))); + + /* calc the maskOR_msb */ + maskOR_msb = 0; + maskOR_msb_offset = (size - 2) >> 3; + if (flags & LTM_PRIME_2MSB_ON) { + maskOR_msb |= 1 << ((size - 2) & 7); + } else if (flags & LTM_PRIME_2MSB_OFF) { + maskAND &= ~(1 << ((size - 2) & 7)); + } + + /* get the maskOR_lsb */ + maskOR_lsb = 0; + if (flags & LTM_PRIME_BBS) { + maskOR_lsb |= 3; + } + + do { + /* read the bytes */ + if (cb(tmp, bsize, dat) != bsize) { + err = MP_VAL; + goto error; + } + + /* work over the MSbyte */ + tmp[0] &= maskAND; + tmp[0] |= 1 << ((size - 1) & 7); + + /* mix in the maskORs */ + tmp[maskOR_msb_offset] |= maskOR_msb; + tmp[bsize-1] |= maskOR_lsb; + + /* read it in */ + if ((err = mp_read_unsigned_bin(a, tmp, bsize)) != MP_OKAY) { goto error; } + + /* is it prime? */ + if ((err = mp_prime_is_prime(a, t, &res)) != MP_OKAY) { goto error; } + if (res == MP_NO) { + continue; + } + + if (flags & LTM_PRIME_SAFE) { + /* see if (a-1)/2 is prime */ + if ((err = mp_sub_d(a, 1, a)) != MP_OKAY) { goto error; } + if ((err = mp_div_2(a, a)) != MP_OKAY) { goto error; } + + /* is it prime? */ + if ((err = mp_prime_is_prime(a, t, &res)) != MP_OKAY) { goto error; } + } + } while (res == MP_NO); + + if (flags & LTM_PRIME_SAFE) { + /* restore a to the original value */ + if ((err = mp_mul_2(a, a)) != MP_OKAY) { goto error; } + if ((err = mp_add_d(a, 1, a)) != MP_OKAY) { goto error; } + } + + err = MP_OKAY; +error: + XFREE(tmp); + return err; +} + + +#endif diff --git a/libtommath/bn_mp_radix_size.c b/libtommath/bn_mp_radix_size.c new file mode 100644 index 0000000..30b78d9 --- /dev/null +++ b/libtommath/bn_mp_radix_size.c @@ -0,0 +1,67 @@ +#include <tommath.h> +#ifdef BN_MP_RADIX_SIZE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* returns size of ASCII reprensentation */ +int mp_radix_size (mp_int * a, int radix, int *size) +{ + int res, digs; + mp_int t; + mp_digit d; + + *size = 0; + + /* special case for binary */ + if (radix == 2) { + *size = mp_count_bits (a) + (a->sign == MP_NEG ? 1 : 0) + 1; + return MP_OKAY; + } + + /* make sure the radix is in range */ + if (radix < 2 || radix > 64) { + return MP_VAL; + } + + /* init a copy of the input */ + if ((res = mp_init_copy (&t, a)) != MP_OKAY) { + return res; + } + + /* digs is the digit count */ + digs = 0; + + /* if it's negative add one for the sign */ + if (t.sign == MP_NEG) { + ++digs; + t.sign = MP_ZPOS; + } + + /* fetch out all of the digits */ + while (mp_iszero (&t) == 0) { + if ((res = mp_div_d (&t, (mp_digit) radix, &t, &d)) != MP_OKAY) { + mp_clear (&t); + return res; + } + ++digs; + } + mp_clear (&t); + + /* return digs + 1, the 1 is for the NULL byte that would be required. */ + *size = digs + 1; + return MP_OKAY; +} + +#endif diff --git a/libtommath/bn_mp_radix_smap.c b/libtommath/bn_mp_radix_smap.c new file mode 100644 index 0000000..bc7517d --- /dev/null +++ b/libtommath/bn_mp_radix_smap.c @@ -0,0 +1,20 @@ +#include <tommath.h> +#ifdef BN_MP_RADIX_SMAP_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* chars used in radix conversions */ +const char *mp_s_rmap = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz+/"; +#endif diff --git a/libtommath/bn_mp_rand.c b/libtommath/bn_mp_rand.c new file mode 100644 index 0000000..1cc47f1 --- /dev/null +++ b/libtommath/bn_mp_rand.c @@ -0,0 +1,51 @@ +#include <tommath.h> +#ifdef BN_MP_RAND_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* makes a pseudo-random int of a given size */ +int +mp_rand (mp_int * a, int digits) +{ + int res; + mp_digit d; + + mp_zero (a); + if (digits <= 0) { + return MP_OKAY; + } + + /* first place a random non-zero digit */ + do { + d = ((mp_digit) abs (rand ())); + } while (d == 0); + + if ((res = mp_add_d (a, d, a)) != MP_OKAY) { + return res; + } + + while (digits-- > 0) { + if ((res = mp_lshd (a, 1)) != MP_OKAY) { + return res; + } + + if ((res = mp_add_d (a, ((mp_digit) abs (rand ())), a)) != MP_OKAY) { + return res; + } + } + + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_mp_read_radix.c b/libtommath/bn_mp_read_radix.c new file mode 100644 index 0000000..704bd0f --- /dev/null +++ b/libtommath/bn_mp_read_radix.c @@ -0,0 +1,78 @@ +#include <tommath.h> +#ifdef BN_MP_READ_RADIX_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* read a string [ASCII] in a given radix */ +int mp_read_radix (mp_int * a, char *str, int radix) +{ + int y, res, neg; + char ch; + + /* make sure the radix is ok */ + if (radix < 2 || radix > 64) { + return MP_VAL; + } + + /* if the leading digit is a + * minus set the sign to negative. + */ + if (*str == '-') { + ++str; + neg = MP_NEG; + } else { + neg = MP_ZPOS; + } + + /* set the integer to the default of zero */ + mp_zero (a); + + /* process each digit of the string */ + while (*str) { + /* if the radix < 36 the conversion is case insensitive + * this allows numbers like 1AB and 1ab to represent the same value + * [e.g. in hex] + */ + ch = (char) ((radix < 36) ? toupper (*str) : *str); + for (y = 0; y < 64; y++) { + if (ch == mp_s_rmap[y]) { + break; + } + } + + /* if the char was found in the map + * and is less than the given radix add it + * to the number, otherwise exit the loop. + */ + if (y < radix) { + if ((res = mp_mul_d (a, (mp_digit) radix, a)) != MP_OKAY) { + return res; + } + if ((res = mp_add_d (a, (mp_digit) y, a)) != MP_OKAY) { + return res; + } + } else { + break; + } + ++str; + } + + /* set the sign only if a != 0 */ + if (mp_iszero(a) != 1) { + a->sign = neg; + } + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_mp_read_signed_bin.c b/libtommath/bn_mp_read_signed_bin.c new file mode 100644 index 0000000..814d6c1 --- /dev/null +++ b/libtommath/bn_mp_read_signed_bin.c @@ -0,0 +1,38 @@ +#include <tommath.h> +#ifdef BN_MP_READ_SIGNED_BIN_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* read signed bin, big endian, first byte is 0==positive or 1==negative */ +int +mp_read_signed_bin (mp_int * a, unsigned char *b, int c) +{ + int res; + + /* read magnitude */ + if ((res = mp_read_unsigned_bin (a, b + 1, c - 1)) != MP_OKAY) { + return res; + } + + /* first byte is 0 for positive, non-zero for negative */ + if (b[0] == 0) { + a->sign = MP_ZPOS; + } else { + a->sign = MP_NEG; + } + + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_mp_read_unsigned_bin.c b/libtommath/bn_mp_read_unsigned_bin.c new file mode 100644 index 0000000..946457d --- /dev/null +++ b/libtommath/bn_mp_read_unsigned_bin.c @@ -0,0 +1,52 @@ +#include <tommath.h> +#ifdef BN_MP_READ_UNSIGNED_BIN_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* reads a unsigned char array, assumes the msb is stored first [big endian] */ +int +mp_read_unsigned_bin (mp_int * a, unsigned char *b, int c) +{ + int res; + + /* make sure there are at least two digits */ + if (a->alloc < 2) { + if ((res = mp_grow(a, 2)) != MP_OKAY) { + return res; + } + } + + /* zero the int */ + mp_zero (a); + + /* read the bytes in */ + while (c-- > 0) { + if ((res = mp_mul_2d (a, 8, a)) != MP_OKAY) { + return res; + } + +#ifndef MP_8BIT + a->dp[0] |= *b++; + a->used += 1; +#else + a->dp[0] = (*b & MP_MASK); + a->dp[1] |= ((*b++ >> 7U) & 1); + a->used += 2; +#endif + } + mp_clamp (a); + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_mp_reduce.c b/libtommath/bn_mp_reduce.c new file mode 100644 index 0000000..cfcb55a --- /dev/null +++ b/libtommath/bn_mp_reduce.c @@ -0,0 +1,97 @@ +#include <tommath.h> +#ifdef BN_MP_REDUCE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* reduces x mod m, assumes 0 < x < m**2, mu is + * precomputed via mp_reduce_setup. + * From HAC pp.604 Algorithm 14.42 + */ +int +mp_reduce (mp_int * x, mp_int * m, mp_int * mu) +{ + mp_int q; + int res, um = m->used; + + /* q = x */ + if ((res = mp_init_copy (&q, x)) != MP_OKAY) { + return res; + } + + /* q1 = x / b**(k-1) */ + mp_rshd (&q, um - 1); + + /* according to HAC this optimization is ok */ + if (((unsigned long) um) > (((mp_digit)1) << (DIGIT_BIT - 1))) { + if ((res = mp_mul (&q, mu, &q)) != MP_OKAY) { + goto CLEANUP; + } + } else { +#ifdef BN_S_MP_MUL_HIGH_DIGS_C + if ((res = s_mp_mul_high_digs (&q, mu, &q, um - 1)) != MP_OKAY) { + goto CLEANUP; + } +#elif defined(BN_FAST_S_MP_MUL_HIGH_DIGS_C) + if ((res = fast_s_mp_mul_high_digs (&q, mu, &q, um - 1)) != MP_OKAY) { + goto CLEANUP; + } +#else + { + res = MP_VAL; + goto CLEANUP; + } +#endif + } + + /* q3 = q2 / b**(k+1) */ + mp_rshd (&q, um + 1); + + /* x = x mod b**(k+1), quick (no division) */ + if ((res = mp_mod_2d (x, DIGIT_BIT * (um + 1), x)) != MP_OKAY) { + goto CLEANUP; + } + + /* q = q * m mod b**(k+1), quick (no division) */ + if ((res = s_mp_mul_digs (&q, m, &q, um + 1)) != MP_OKAY) { + goto CLEANUP; + } + + /* x = x - q */ + if ((res = mp_sub (x, &q, x)) != MP_OKAY) { + goto CLEANUP; + } + + /* If x < 0, add b**(k+1) to it */ + if (mp_cmp_d (x, 0) == MP_LT) { + mp_set (&q, 1); + if ((res = mp_lshd (&q, um + 1)) != MP_OKAY) + goto CLEANUP; + if ((res = mp_add (x, &q, x)) != MP_OKAY) + goto CLEANUP; + } + + /* Back off if it's too big */ + while (mp_cmp (x, m) != MP_LT) { + if ((res = s_mp_sub (x, m, x)) != MP_OKAY) { + goto CLEANUP; + } + } + +CLEANUP: + mp_clear (&q); + + return res; +} +#endif diff --git a/libtommath/bn_mp_reduce_2k.c b/libtommath/bn_mp_reduce_2k.c new file mode 100644 index 0000000..a5a9c74 --- /dev/null +++ b/libtommath/bn_mp_reduce_2k.c @@ -0,0 +1,58 @@ +#include <tommath.h> +#ifdef BN_MP_REDUCE_2K_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* reduces a modulo n where n is of the form 2**p - d */ +int +mp_reduce_2k(mp_int *a, mp_int *n, mp_digit d) +{ + mp_int q; + int p, res; + + if ((res = mp_init(&q)) != MP_OKAY) { + return res; + } + + p = mp_count_bits(n); +top: + /* q = a/2**p, a = a mod 2**p */ + if ((res = mp_div_2d(a, p, &q, a)) != MP_OKAY) { + goto ERR; + } + + if (d != 1) { + /* q = q * d */ + if ((res = mp_mul_d(&q, d, &q)) != MP_OKAY) { + goto ERR; + } + } + + /* a = a + q */ + if ((res = s_mp_add(a, &q, a)) != MP_OKAY) { + goto ERR; + } + + if (mp_cmp_mag(a, n) != MP_LT) { + s_mp_sub(a, n, a); + goto top; + } + +ERR: + mp_clear(&q); + return res; +} + +#endif diff --git a/libtommath/bn_mp_reduce_2k_l.c b/libtommath/bn_mp_reduce_2k_l.c new file mode 100644 index 0000000..1d7e1f0 --- /dev/null +++ b/libtommath/bn_mp_reduce_2k_l.c @@ -0,0 +1,58 @@ +#include <tommath.h> +#ifdef BN_MP_REDUCE_2K_L_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* reduces a modulo n where n is of the form 2**p - d + This differs from reduce_2k since "d" can be larger + than a single digit. +*/ +int mp_reduce_2k_l(mp_int *a, mp_int *n, mp_int *d) +{ + mp_int q; + int p, res; + + if ((res = mp_init(&q)) != MP_OKAY) { + return res; + } + + p = mp_count_bits(n); +top: + /* q = a/2**p, a = a mod 2**p */ + if ((res = mp_div_2d(a, p, &q, a)) != MP_OKAY) { + goto ERR; + } + + /* q = q * d */ + if ((res = mp_mul(&q, d, &q)) != MP_OKAY) { + goto ERR; + } + + /* a = a + q */ + if ((res = s_mp_add(a, &q, a)) != MP_OKAY) { + goto ERR; + } + + if (mp_cmp_mag(a, n) != MP_LT) { + s_mp_sub(a, n, a); + goto top; + } + +ERR: + mp_clear(&q); + return res; +} + +#endif diff --git a/libtommath/bn_mp_reduce_2k_setup.c b/libtommath/bn_mp_reduce_2k_setup.c new file mode 100644 index 0000000..5e1fb6e --- /dev/null +++ b/libtommath/bn_mp_reduce_2k_setup.c @@ -0,0 +1,44 @@ +#include <tommath.h> +#ifdef BN_MP_REDUCE_2K_SETUP_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* determines the setup value */ +int +mp_reduce_2k_setup(mp_int *a, mp_digit *d) +{ + int res, p; + mp_int tmp; + + if ((res = mp_init(&tmp)) != MP_OKAY) { + return res; + } + + p = mp_count_bits(a); + if ((res = mp_2expt(&tmp, p)) != MP_OKAY) { + mp_clear(&tmp); + return res; + } + + if ((res = s_mp_sub(&tmp, a, &tmp)) != MP_OKAY) { + mp_clear(&tmp); + return res; + } + + *d = tmp.dp[0]; + mp_clear(&tmp); + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_mp_reduce_2k_setup_l.c b/libtommath/bn_mp_reduce_2k_setup_l.c new file mode 100644 index 0000000..810a456 --- /dev/null +++ b/libtommath/bn_mp_reduce_2k_setup_l.c @@ -0,0 +1,40 @@ +#include <tommath.h> +#ifdef BN_MP_REDUCE_2K_SETUP_L_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* determines the setup value */ +int mp_reduce_2k_setup_l(mp_int *a, mp_int *d) +{ + int res; + mp_int tmp; + + if ((res = mp_init(&tmp)) != MP_OKAY) { + return res; + } + + if ((res = mp_2expt(&tmp, mp_count_bits(a))) != MP_OKAY) { + goto ERR; + } + + if ((res = s_mp_sub(&tmp, a, d)) != MP_OKAY) { + goto ERR; + } + +ERR: + mp_clear(&tmp); + return res; +} +#endif diff --git a/libtommath/bn_mp_reduce_is_2k.c b/libtommath/bn_mp_reduce_is_2k.c new file mode 100644 index 0000000..fc81397 --- /dev/null +++ b/libtommath/bn_mp_reduce_is_2k.c @@ -0,0 +1,48 @@ +#include <tommath.h> +#ifdef BN_MP_REDUCE_IS_2K_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* determines if mp_reduce_2k can be used */ +int mp_reduce_is_2k(mp_int *a) +{ + int ix, iy, iw; + mp_digit iz; + + if (a->used == 0) { + return 0; + } else if (a->used == 1) { + return 1; + } else if (a->used > 1) { + iy = mp_count_bits(a); + iz = 1; + iw = 1; + + /* Test every bit from the second digit up, must be 1 */ + for (ix = DIGIT_BIT; ix < iy; ix++) { + if ((a->dp[iw] & iz) == 0) { + return 0; + } + iz <<= 1; + if (iz > (mp_digit)MP_MASK) { + ++iw; + iz = 1; + } + } + } + return 1; +} + +#endif diff --git a/libtommath/bn_mp_reduce_is_2k_l.c b/libtommath/bn_mp_reduce_is_2k_l.c new file mode 100644 index 0000000..ceba0ed --- /dev/null +++ b/libtommath/bn_mp_reduce_is_2k_l.c @@ -0,0 +1,40 @@ +#include <tommath.h> +#ifdef BN_MP_REDUCE_IS_2K_L_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* determines if reduce_2k_l can be used */ +int mp_reduce_is_2k_l(mp_int *a) +{ + int ix, iy; + + if (a->used == 0) { + return MP_NO; + } else if (a->used == 1) { + return MP_YES; + } else if (a->used > 1) { + /* if more than half of the digits are -1 we're sold */ + for (iy = ix = 0; ix < a->used; ix++) { + if (a->dp[ix] == MP_MASK) { + ++iy; + } + } + return (iy >= (a->used/2)) ? MP_YES : MP_NO; + + } + return MP_NO; +} + +#endif diff --git a/libtommath/bn_mp_reduce_setup.c b/libtommath/bn_mp_reduce_setup.c new file mode 100644 index 0000000..99f158a --- /dev/null +++ b/libtommath/bn_mp_reduce_setup.c @@ -0,0 +1,30 @@ +#include <tommath.h> +#ifdef BN_MP_REDUCE_SETUP_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* pre-calculate the value required for Barrett reduction + * For a given modulus "b" it calulates the value required in "a" + */ +int mp_reduce_setup (mp_int * a, mp_int * b) +{ + int res; + + if ((res = mp_2expt (a, b->used * 2 * DIGIT_BIT)) != MP_OKAY) { + return res; + } + return mp_div (a, b, a, NULL); +} +#endif diff --git a/libtommath/bn_mp_rshd.c b/libtommath/bn_mp_rshd.c new file mode 100644 index 0000000..913dda6 --- /dev/null +++ b/libtommath/bn_mp_rshd.c @@ -0,0 +1,68 @@ +#include <tommath.h> +#ifdef BN_MP_RSHD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* shift right a certain amount of digits */ +void mp_rshd (mp_int * a, int b) +{ + int x; + + /* if b <= 0 then ignore it */ + if (b <= 0) { + return; + } + + /* if b > used then simply zero it and return */ + if (a->used <= b) { + mp_zero (a); + return; + } + + { + register mp_digit *bottom, *top; + + /* shift the digits down */ + + /* bottom */ + bottom = a->dp; + + /* top [offset into digits] */ + top = a->dp + b; + + /* this is implemented as a sliding window where + * the window is b-digits long and digits from + * the top of the window are copied to the bottom + * + * e.g. + + b-2 | b-1 | b0 | b1 | b2 | ... | bb | ----> + /\ | ----> + \-------------------/ ----> + */ + for (x = 0; x < (a->used - b); x++) { + *bottom++ = *top++; + } + + /* zero the top digits */ + for (; x < a->used; x++) { + *bottom++ = 0; + } + } + + /* remove excess digits */ + a->used -= b; +} +#endif diff --git a/libtommath/bn_mp_set.c b/libtommath/bn_mp_set.c new file mode 100644 index 0000000..078fd5f --- /dev/null +++ b/libtommath/bn_mp_set.c @@ -0,0 +1,25 @@ +#include <tommath.h> +#ifdef BN_MP_SET_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* set to a digit */ +void mp_set (mp_int * a, mp_digit b) +{ + mp_zero (a); + a->dp[0] = b & MP_MASK; + a->used = (a->dp[0] != 0) ? 1 : 0; +} +#endif diff --git a/libtommath/bn_mp_set_int.c b/libtommath/bn_mp_set_int.c new file mode 100644 index 0000000..bd47136 --- /dev/null +++ b/libtommath/bn_mp_set_int.c @@ -0,0 +1,44 @@ +#include <tommath.h> +#ifdef BN_MP_SET_INT_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* set a 32-bit const */ +int mp_set_int (mp_int * a, unsigned long b) +{ + int x, res; + + mp_zero (a); + + /* set four bits at a time */ + for (x = 0; x < 8; x++) { + /* shift the number up four bits */ + if ((res = mp_mul_2d (a, 4, a)) != MP_OKAY) { + return res; + } + + /* OR in the top four bits of the source */ + a->dp[0] |= (b >> 28) & 15; + + /* shift the source up to the next four bits */ + b <<= 4; + + /* ensure that digits are not clamped off */ + a->used += 1; + } + mp_clamp (a); + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_mp_shrink.c b/libtommath/bn_mp_shrink.c new file mode 100644 index 0000000..b31f9d2 --- /dev/null +++ b/libtommath/bn_mp_shrink.c @@ -0,0 +1,31 @@ +#include <tommath.h> +#ifdef BN_MP_SHRINK_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* shrink a bignum */ +int mp_shrink (mp_int * a) +{ + mp_digit *tmp; + if (a->alloc != a->used && a->used > 0) { + if ((tmp = OPT_CAST(mp_digit) XREALLOC (a->dp, sizeof (mp_digit) * a->used)) == NULL) { + return MP_MEM; + } + a->dp = tmp; + a->alloc = a->used; + } + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_mp_signed_bin_size.c b/libtommath/bn_mp_signed_bin_size.c new file mode 100644 index 0000000..30048cb --- /dev/null +++ b/libtommath/bn_mp_signed_bin_size.c @@ -0,0 +1,23 @@ +#include <tommath.h> +#ifdef BN_MP_SIGNED_BIN_SIZE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* get the size for an signed equivalent */ +int mp_signed_bin_size (mp_int * a) +{ + return 1 + mp_unsigned_bin_size (a); +} +#endif diff --git a/libtommath/bn_mp_sqr.c b/libtommath/bn_mp_sqr.c new file mode 100644 index 0000000..b1fdb57 --- /dev/null +++ b/libtommath/bn_mp_sqr.c @@ -0,0 +1,54 @@ +#include <tommath.h> +#ifdef BN_MP_SQR_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* computes b = a*a */ +int +mp_sqr (mp_int * a, mp_int * b) +{ + int res; + +#ifdef BN_MP_TOOM_SQR_C + /* use Toom-Cook? */ + if (a->used >= TOOM_SQR_CUTOFF) { + res = mp_toom_sqr(a, b); + /* Karatsuba? */ + } else +#endif +#ifdef BN_MP_KARATSUBA_SQR_C +if (a->used >= KARATSUBA_SQR_CUTOFF) { + res = mp_karatsuba_sqr (a, b); + } else +#endif + { +#ifdef BN_FAST_S_MP_SQR_C + /* can we use the fast comba multiplier? */ + if ((a->used * 2 + 1) < MP_WARRAY && + a->used < + (1 << (sizeof(mp_word) * CHAR_BIT - 2*DIGIT_BIT - 1))) { + res = fast_s_mp_sqr (a, b); + } else +#endif +#ifdef BN_S_MP_SQR_C + res = s_mp_sqr (a, b); +#else + res = MP_VAL; +#endif + } + b->sign = MP_ZPOS; + return res; +} +#endif diff --git a/libtommath/bn_mp_sqrmod.c b/libtommath/bn_mp_sqrmod.c new file mode 100644 index 0000000..1923be4 --- /dev/null +++ b/libtommath/bn_mp_sqrmod.c @@ -0,0 +1,37 @@ +#include <tommath.h> +#ifdef BN_MP_SQRMOD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* c = a * a (mod b) */ +int +mp_sqrmod (mp_int * a, mp_int * b, mp_int * c) +{ + int res; + mp_int t; + + if ((res = mp_init (&t)) != MP_OKAY) { + return res; + } + + if ((res = mp_sqr (a, &t)) != MP_OKAY) { + mp_clear (&t); + return res; + } + res = mp_mod (&t, b, c); + mp_clear (&t); + return res; +} +#endif diff --git a/libtommath/bn_mp_sqrt.c b/libtommath/bn_mp_sqrt.c new file mode 100644 index 0000000..76cec87 --- /dev/null +++ b/libtommath/bn_mp_sqrt.c @@ -0,0 +1,77 @@ +#include <tommath.h> +#ifdef BN_MP_SQRT_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* this function is less generic than mp_n_root, simpler and faster */ +int mp_sqrt(mp_int *arg, mp_int *ret) +{ + int res; + mp_int t1,t2; + + /* must be positive */ + if (arg->sign == MP_NEG) { + return MP_VAL; + } + + /* easy out */ + if (mp_iszero(arg) == MP_YES) { + mp_zero(ret); + return MP_OKAY; + } + + if ((res = mp_init_copy(&t1, arg)) != MP_OKAY) { + return res; + } + + if ((res = mp_init(&t2)) != MP_OKAY) { + goto E2; + } + + /* First approx. (not very bad for large arg) */ + mp_rshd (&t1,t1.used/2); + + /* t1 > 0 */ + if ((res = mp_div(arg,&t1,&t2,NULL)) != MP_OKAY) { + goto E1; + } + if ((res = mp_add(&t1,&t2,&t1)) != MP_OKAY) { + goto E1; + } + if ((res = mp_div_2(&t1,&t1)) != MP_OKAY) { + goto E1; + } + /* And now t1 > sqrt(arg) */ + do { + if ((res = mp_div(arg,&t1,&t2,NULL)) != MP_OKAY) { + goto E1; + } + if ((res = mp_add(&t1,&t2,&t1)) != MP_OKAY) { + goto E1; + } + if ((res = mp_div_2(&t1,&t1)) != MP_OKAY) { + goto E1; + } + /* t1 >= sqrt(arg) >= t2 at this point */ + } while (mp_cmp_mag(&t1,&t2) == MP_GT); + + mp_exch(&t1,ret); + +E1: mp_clear(&t2); +E2: mp_clear(&t1); + return res; +} + +#endif diff --git a/libtommath/bn_mp_sub.c b/libtommath/bn_mp_sub.c new file mode 100644 index 0000000..97495f4 --- /dev/null +++ b/libtommath/bn_mp_sub.c @@ -0,0 +1,55 @@ +#include <tommath.h> +#ifdef BN_MP_SUB_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* high level subtraction (handles signs) */ +int +mp_sub (mp_int * a, mp_int * b, mp_int * c) +{ + int sa, sb, res; + + sa = a->sign; + sb = b->sign; + + if (sa != sb) { + /* subtract a negative from a positive, OR */ + /* subtract a positive from a negative. */ + /* In either case, ADD their magnitudes, */ + /* and use the sign of the first number. */ + c->sign = sa; + res = s_mp_add (a, b, c); + } else { + /* subtract a positive from a positive, OR */ + /* subtract a negative from a negative. */ + /* First, take the difference between their */ + /* magnitudes, then... */ + if (mp_cmp_mag (a, b) != MP_LT) { + /* Copy the sign from the first */ + c->sign = sa; + /* The first has a larger or equal magnitude */ + res = s_mp_sub (a, b, c); + } else { + /* The result has the *opposite* sign from */ + /* the first number. */ + c->sign = (sa == MP_ZPOS) ? MP_NEG : MP_ZPOS; + /* The second has a larger magnitude */ + res = s_mp_sub (b, a, c); + } + } + return res; +} + +#endif diff --git a/libtommath/bn_mp_sub_d.c b/libtommath/bn_mp_sub_d.c new file mode 100644 index 0000000..4923dde --- /dev/null +++ b/libtommath/bn_mp_sub_d.c @@ -0,0 +1,85 @@ +#include <tommath.h> +#ifdef BN_MP_SUB_D_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* single digit subtraction */ +int +mp_sub_d (mp_int * a, mp_digit b, mp_int * c) +{ + mp_digit *tmpa, *tmpc, mu; + int res, ix, oldused; + + /* grow c as required */ + if (c->alloc < a->used + 1) { + if ((res = mp_grow(c, a->used + 1)) != MP_OKAY) { + return res; + } + } + + /* if a is negative just do an unsigned + * addition [with fudged signs] + */ + if (a->sign == MP_NEG) { + a->sign = MP_ZPOS; + res = mp_add_d(a, b, c); + a->sign = c->sign = MP_NEG; + return res; + } + + /* setup regs */ + oldused = c->used; + tmpa = a->dp; + tmpc = c->dp; + + /* if a <= b simply fix the single digit */ + if ((a->used == 1 && a->dp[0] <= b) || a->used == 0) { + if (a->used == 1) { + *tmpc++ = b - *tmpa; + } else { + *tmpc++ = b; + } + ix = 1; + + /* negative/1digit */ + c->sign = MP_NEG; + c->used = 1; + } else { + /* positive/size */ + c->sign = MP_ZPOS; + c->used = a->used; + + /* subtract first digit */ + *tmpc = *tmpa++ - b; + mu = *tmpc >> (sizeof(mp_digit) * CHAR_BIT - 1); + *tmpc++ &= MP_MASK; + + /* handle rest of the digits */ + for (ix = 1; ix < a->used; ix++) { + *tmpc = *tmpa++ - mu; + mu = *tmpc >> (sizeof(mp_digit) * CHAR_BIT - 1); + *tmpc++ &= MP_MASK; + } + } + + /* zero excess digits */ + while (ix++ < oldused) { + *tmpc++ = 0; + } + mp_clamp(c); + return MP_OKAY; +} + +#endif diff --git a/libtommath/bn_mp_submod.c b/libtommath/bn_mp_submod.c new file mode 100644 index 0000000..b999c85 --- /dev/null +++ b/libtommath/bn_mp_submod.c @@ -0,0 +1,38 @@ +#include <tommath.h> +#ifdef BN_MP_SUBMOD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* d = a - b (mod c) */ +int +mp_submod (mp_int * a, mp_int * b, mp_int * c, mp_int * d) +{ + int res; + mp_int t; + + + if ((res = mp_init (&t)) != MP_OKAY) { + return res; + } + + if ((res = mp_sub (a, b, &t)) != MP_OKAY) { + mp_clear (&t); + return res; + } + res = mp_mod (&t, c, d); + mp_clear (&t); + return res; +} +#endif diff --git a/libtommath/bn_mp_to_signed_bin.c b/libtommath/bn_mp_to_signed_bin.c new file mode 100644 index 0000000..0e40d0f --- /dev/null +++ b/libtommath/bn_mp_to_signed_bin.c @@ -0,0 +1,30 @@ +#include <tommath.h> +#ifdef BN_MP_TO_SIGNED_BIN_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* store in signed [big endian] format */ +int +mp_to_signed_bin (mp_int * a, unsigned char *b) +{ + int res; + + if ((res = mp_to_unsigned_bin (a, b + 1)) != MP_OKAY) { + return res; + } + b[0] = (unsigned char) ((a->sign == MP_ZPOS) ? 0 : 1); + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_mp_to_signed_bin_n.c b/libtommath/bn_mp_to_signed_bin_n.c new file mode 100644 index 0000000..0f765ee --- /dev/null +++ b/libtommath/bn_mp_to_signed_bin_n.c @@ -0,0 +1,27 @@ +#include <tommath.h> +#ifdef BN_MP_TO_SIGNED_BIN_N_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* store in signed [big endian] format */ +int mp_to_signed_bin_n (mp_int * a, unsigned char *b, unsigned long *outlen) +{ + if (*outlen < (unsigned long)mp_signed_bin_size(a)) { + return MP_VAL; + } + *outlen = mp_signed_bin_size(a); + return mp_to_signed_bin(a, b); +} +#endif diff --git a/libtommath/bn_mp_to_unsigned_bin.c b/libtommath/bn_mp_to_unsigned_bin.c new file mode 100644 index 0000000..763e346 --- /dev/null +++ b/libtommath/bn_mp_to_unsigned_bin.c @@ -0,0 +1,45 @@ +#include <tommath.h> +#ifdef BN_MP_TO_UNSIGNED_BIN_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* store in unsigned [big endian] format */ +int +mp_to_unsigned_bin (mp_int * a, unsigned char *b) +{ + int x, res; + mp_int t; + + if ((res = mp_init_copy (&t, a)) != MP_OKAY) { + return res; + } + + x = 0; + while (mp_iszero (&t) == 0) { +#ifndef MP_8BIT + b[x++] = (unsigned char) (t.dp[0] & 255); +#else + b[x++] = (unsigned char) (t.dp[0] | ((t.dp[1] & 0x01) << 7)); +#endif + if ((res = mp_div_2d (&t, 8, &t, NULL)) != MP_OKAY) { + mp_clear (&t); + return res; + } + } + bn_reverse (b, x); + mp_clear (&t); + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_mp_to_unsigned_bin_n.c b/libtommath/bn_mp_to_unsigned_bin_n.c new file mode 100644 index 0000000..d0256b4 --- /dev/null +++ b/libtommath/bn_mp_to_unsigned_bin_n.c @@ -0,0 +1,27 @@ +#include <tommath.h> +#ifdef BN_MP_TO_UNSIGNED_BIN_N_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* store in unsigned [big endian] format */ +int mp_to_unsigned_bin_n (mp_int * a, unsigned char *b, unsigned long *outlen) +{ + if (*outlen < (unsigned long)mp_unsigned_bin_size(a)) { + return MP_VAL; + } + *outlen = mp_unsigned_bin_size(a); + return mp_to_unsigned_bin(a, b); +} +#endif diff --git a/libtommath/bn_mp_toom_mul.c b/libtommath/bn_mp_toom_mul.c new file mode 100644 index 0000000..2d66779 --- /dev/null +++ b/libtommath/bn_mp_toom_mul.c @@ -0,0 +1,279 @@ +#include <tommath.h> +#ifdef BN_MP_TOOM_MUL_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* multiplication using the Toom-Cook 3-way algorithm + * + * Much more complicated than Karatsuba but has a lower asymptotic running time of + * O(N**1.464). This algorithm is only particularly useful on VERY large + * inputs (we're talking 1000s of digits here...). +*/ +int mp_toom_mul(mp_int *a, mp_int *b, mp_int *c) +{ + mp_int w0, w1, w2, w3, w4, tmp1, tmp2, a0, a1, a2, b0, b1, b2; + int res, B; + + /* init temps */ + if ((res = mp_init_multi(&w0, &w1, &w2, &w3, &w4, + &a0, &a1, &a2, &b0, &b1, + &b2, &tmp1, &tmp2, NULL)) != MP_OKAY) { + return res; + } + + /* B */ + B = MIN(a->used, b->used) / 3; + + /* a = a2 * B**2 + a1 * B + a0 */ + if ((res = mp_mod_2d(a, DIGIT_BIT * B, &a0)) != MP_OKAY) { + goto ERR; + } + + if ((res = mp_copy(a, &a1)) != MP_OKAY) { + goto ERR; + } + mp_rshd(&a1, B); + mp_mod_2d(&a1, DIGIT_BIT * B, &a1); + + if ((res = mp_copy(a, &a2)) != MP_OKAY) { + goto ERR; + } + mp_rshd(&a2, B*2); + + /* b = b2 * B**2 + b1 * B + b0 */ + if ((res = mp_mod_2d(b, DIGIT_BIT * B, &b0)) != MP_OKAY) { + goto ERR; + } + + if ((res = mp_copy(b, &b1)) != MP_OKAY) { + goto ERR; + } + mp_rshd(&b1, B); + mp_mod_2d(&b1, DIGIT_BIT * B, &b1); + + if ((res = mp_copy(b, &b2)) != MP_OKAY) { + goto ERR; + } + mp_rshd(&b2, B*2); + + /* w0 = a0*b0 */ + if ((res = mp_mul(&a0, &b0, &w0)) != MP_OKAY) { + goto ERR; + } + + /* w4 = a2 * b2 */ + if ((res = mp_mul(&a2, &b2, &w4)) != MP_OKAY) { + goto ERR; + } + + /* w1 = (a2 + 2(a1 + 2a0))(b2 + 2(b1 + 2b0)) */ + if ((res = mp_mul_2(&a0, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&tmp1, &a2, &tmp1)) != MP_OKAY) { + goto ERR; + } + + if ((res = mp_mul_2(&b0, &tmp2)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&tmp2, &b2, &tmp2)) != MP_OKAY) { + goto ERR; + } + + if ((res = mp_mul(&tmp1, &tmp2, &w1)) != MP_OKAY) { + goto ERR; + } + + /* w3 = (a0 + 2(a1 + 2a2))(b0 + 2(b1 + 2b2)) */ + if ((res = mp_mul_2(&a2, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) { + goto ERR; + } + + if ((res = mp_mul_2(&b2, &tmp2)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) { + goto ERR; + } + + if ((res = mp_mul(&tmp1, &tmp2, &w3)) != MP_OKAY) { + goto ERR; + } + + + /* w2 = (a2 + a1 + a0)(b2 + b1 + b0) */ + if ((res = mp_add(&a2, &a1, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&b2, &b1, &tmp2)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_mul(&tmp1, &tmp2, &w2)) != MP_OKAY) { + goto ERR; + } + + /* now solve the matrix + + 0 0 0 0 1 + 1 2 4 8 16 + 1 1 1 1 1 + 16 8 4 2 1 + 1 0 0 0 0 + + using 12 subtractions, 4 shifts, + 2 small divisions and 1 small multiplication + */ + + /* r1 - r4 */ + if ((res = mp_sub(&w1, &w4, &w1)) != MP_OKAY) { + goto ERR; + } + /* r3 - r0 */ + if ((res = mp_sub(&w3, &w0, &w3)) != MP_OKAY) { + goto ERR; + } + /* r1/2 */ + if ((res = mp_div_2(&w1, &w1)) != MP_OKAY) { + goto ERR; + } + /* r3/2 */ + if ((res = mp_div_2(&w3, &w3)) != MP_OKAY) { + goto ERR; + } + /* r2 - r0 - r4 */ + if ((res = mp_sub(&w2, &w0, &w2)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_sub(&w2, &w4, &w2)) != MP_OKAY) { + goto ERR; + } + /* r1 - r2 */ + if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) { + goto ERR; + } + /* r3 - r2 */ + if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) { + goto ERR; + } + /* r1 - 8r0 */ + if ((res = mp_mul_2d(&w0, 3, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_sub(&w1, &tmp1, &w1)) != MP_OKAY) { + goto ERR; + } + /* r3 - 8r4 */ + if ((res = mp_mul_2d(&w4, 3, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_sub(&w3, &tmp1, &w3)) != MP_OKAY) { + goto ERR; + } + /* 3r2 - r1 - r3 */ + if ((res = mp_mul_d(&w2, 3, &w2)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_sub(&w2, &w1, &w2)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_sub(&w2, &w3, &w2)) != MP_OKAY) { + goto ERR; + } + /* r1 - r2 */ + if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) { + goto ERR; + } + /* r3 - r2 */ + if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) { + goto ERR; + } + /* r1/3 */ + if ((res = mp_div_3(&w1, &w1, NULL)) != MP_OKAY) { + goto ERR; + } + /* r3/3 */ + if ((res = mp_div_3(&w3, &w3, NULL)) != MP_OKAY) { + goto ERR; + } + + /* at this point shift W[n] by B*n */ + if ((res = mp_lshd(&w1, 1*B)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_lshd(&w2, 2*B)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_lshd(&w3, 3*B)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_lshd(&w4, 4*B)) != MP_OKAY) { + goto ERR; + } + + if ((res = mp_add(&w0, &w1, c)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&w2, &w3, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&w4, &tmp1, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&tmp1, c, c)) != MP_OKAY) { + goto ERR; + } + +ERR: + mp_clear_multi(&w0, &w1, &w2, &w3, &w4, + &a0, &a1, &a2, &b0, &b1, + &b2, &tmp1, &tmp2, NULL); + return res; +} + +#endif diff --git a/libtommath/bn_mp_toom_sqr.c b/libtommath/bn_mp_toom_sqr.c new file mode 100644 index 0000000..8c46fea --- /dev/null +++ b/libtommath/bn_mp_toom_sqr.c @@ -0,0 +1,222 @@ +#include <tommath.h> +#ifdef BN_MP_TOOM_SQR_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* squaring using Toom-Cook 3-way algorithm */ +int +mp_toom_sqr(mp_int *a, mp_int *b) +{ + mp_int w0, w1, w2, w3, w4, tmp1, a0, a1, a2; + int res, B; + + /* init temps */ + if ((res = mp_init_multi(&w0, &w1, &w2, &w3, &w4, &a0, &a1, &a2, &tmp1, NULL)) != MP_OKAY) { + return res; + } + + /* B */ + B = a->used / 3; + + /* a = a2 * B**2 + a1 * B + a0 */ + if ((res = mp_mod_2d(a, DIGIT_BIT * B, &a0)) != MP_OKAY) { + goto ERR; + } + + if ((res = mp_copy(a, &a1)) != MP_OKAY) { + goto ERR; + } + mp_rshd(&a1, B); + mp_mod_2d(&a1, DIGIT_BIT * B, &a1); + + if ((res = mp_copy(a, &a2)) != MP_OKAY) { + goto ERR; + } + mp_rshd(&a2, B*2); + + /* w0 = a0*a0 */ + if ((res = mp_sqr(&a0, &w0)) != MP_OKAY) { + goto ERR; + } + + /* w4 = a2 * a2 */ + if ((res = mp_sqr(&a2, &w4)) != MP_OKAY) { + goto ERR; + } + + /* w1 = (a2 + 2(a1 + 2a0))**2 */ + if ((res = mp_mul_2(&a0, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&tmp1, &a2, &tmp1)) != MP_OKAY) { + goto ERR; + } + + if ((res = mp_sqr(&tmp1, &w1)) != MP_OKAY) { + goto ERR; + } + + /* w3 = (a0 + 2(a1 + 2a2))**2 */ + if ((res = mp_mul_2(&a2, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) { + goto ERR; + } + + if ((res = mp_sqr(&tmp1, &w3)) != MP_OKAY) { + goto ERR; + } + + + /* w2 = (a2 + a1 + a0)**2 */ + if ((res = mp_add(&a2, &a1, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_sqr(&tmp1, &w2)) != MP_OKAY) { + goto ERR; + } + + /* now solve the matrix + + 0 0 0 0 1 + 1 2 4 8 16 + 1 1 1 1 1 + 16 8 4 2 1 + 1 0 0 0 0 + + using 12 subtractions, 4 shifts, 2 small divisions and 1 small multiplication. + */ + + /* r1 - r4 */ + if ((res = mp_sub(&w1, &w4, &w1)) != MP_OKAY) { + goto ERR; + } + /* r3 - r0 */ + if ((res = mp_sub(&w3, &w0, &w3)) != MP_OKAY) { + goto ERR; + } + /* r1/2 */ + if ((res = mp_div_2(&w1, &w1)) != MP_OKAY) { + goto ERR; + } + /* r3/2 */ + if ((res = mp_div_2(&w3, &w3)) != MP_OKAY) { + goto ERR; + } + /* r2 - r0 - r4 */ + if ((res = mp_sub(&w2, &w0, &w2)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_sub(&w2, &w4, &w2)) != MP_OKAY) { + goto ERR; + } + /* r1 - r2 */ + if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) { + goto ERR; + } + /* r3 - r2 */ + if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) { + goto ERR; + } + /* r1 - 8r0 */ + if ((res = mp_mul_2d(&w0, 3, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_sub(&w1, &tmp1, &w1)) != MP_OKAY) { + goto ERR; + } + /* r3 - 8r4 */ + if ((res = mp_mul_2d(&w4, 3, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_sub(&w3, &tmp1, &w3)) != MP_OKAY) { + goto ERR; + } + /* 3r2 - r1 - r3 */ + if ((res = mp_mul_d(&w2, 3, &w2)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_sub(&w2, &w1, &w2)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_sub(&w2, &w3, &w2)) != MP_OKAY) { + goto ERR; + } + /* r1 - r2 */ + if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) { + goto ERR; + } + /* r3 - r2 */ + if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) { + goto ERR; + } + /* r1/3 */ + if ((res = mp_div_3(&w1, &w1, NULL)) != MP_OKAY) { + goto ERR; + } + /* r3/3 */ + if ((res = mp_div_3(&w3, &w3, NULL)) != MP_OKAY) { + goto ERR; + } + + /* at this point shift W[n] by B*n */ + if ((res = mp_lshd(&w1, 1*B)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_lshd(&w2, 2*B)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_lshd(&w3, 3*B)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_lshd(&w4, 4*B)) != MP_OKAY) { + goto ERR; + } + + if ((res = mp_add(&w0, &w1, b)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&w2, &w3, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&w4, &tmp1, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&tmp1, b, b)) != MP_OKAY) { + goto ERR; + } + +ERR: + mp_clear_multi(&w0, &w1, &w2, &w3, &w4, &a0, &a1, &a2, &tmp1, NULL); + return res; +} + +#endif diff --git a/libtommath/bn_mp_toradix.c b/libtommath/bn_mp_toradix.c new file mode 100644 index 0000000..a206d5e --- /dev/null +++ b/libtommath/bn_mp_toradix.c @@ -0,0 +1,71 @@ +#include <tommath.h> +#ifdef BN_MP_TORADIX_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* stores a bignum as a ASCII string in a given radix (2..64) */ +int mp_toradix (mp_int * a, char *str, int radix) +{ + int res, digs; + mp_int t; + mp_digit d; + char *_s = str; + + /* check range of the radix */ + if (radix < 2 || radix > 64) { + return MP_VAL; + } + + /* quick out if its zero */ + if (mp_iszero(a) == 1) { + *str++ = '0'; + *str = '\0'; + return MP_OKAY; + } + + if ((res = mp_init_copy (&t, a)) != MP_OKAY) { + return res; + } + + /* if it is negative output a - */ + if (t.sign == MP_NEG) { + ++_s; + *str++ = '-'; + t.sign = MP_ZPOS; + } + + digs = 0; + while (mp_iszero (&t) == 0) { + if ((res = mp_div_d (&t, (mp_digit) radix, &t, &d)) != MP_OKAY) { + mp_clear (&t); + return res; + } + *str++ = mp_s_rmap[d]; + ++digs; + } + + /* reverse the digits of the string. In this case _s points + * to the first digit [exluding the sign] of the number] + */ + bn_reverse ((unsigned char *)_s, digs); + + /* append a NULL so the string is properly terminated */ + *str = '\0'; + + mp_clear (&t); + return MP_OKAY; +} + +#endif diff --git a/libtommath/bn_mp_toradix_n.c b/libtommath/bn_mp_toradix_n.c new file mode 100644 index 0000000..7d43558 --- /dev/null +++ b/libtommath/bn_mp_toradix_n.c @@ -0,0 +1,85 @@ +#include <tommath.h> +#ifdef BN_MP_TORADIX_N_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* stores a bignum as a ASCII string in a given radix (2..64) + * + * Stores upto maxlen-1 chars and always a NULL byte + */ +int mp_toradix_n(mp_int * a, char *str, int radix, int maxlen) +{ + int res, digs; + mp_int t; + mp_digit d; + char *_s = str; + + /* check range of the maxlen, radix */ + if (maxlen < 3 || radix < 2 || radix > 64) { + return MP_VAL; + } + + /* quick out if its zero */ + if (mp_iszero(a) == 1) { + *str++ = '0'; + *str = '\0'; + return MP_OKAY; + } + + if ((res = mp_init_copy (&t, a)) != MP_OKAY) { + return res; + } + + /* if it is negative output a - */ + if (t.sign == MP_NEG) { + /* we have to reverse our digits later... but not the - sign!! */ + ++_s; + + /* store the flag and mark the number as positive */ + *str++ = '-'; + t.sign = MP_ZPOS; + + /* subtract a char */ + --maxlen; + } + + digs = 0; + while (mp_iszero (&t) == 0) { + if ((res = mp_div_d (&t, (mp_digit) radix, &t, &d)) != MP_OKAY) { + mp_clear (&t); + return res; + } + *str++ = mp_s_rmap[d]; + ++digs; + + if (--maxlen == 1) { + /* no more room */ + break; + } + } + + /* reverse the digits of the string. In this case _s points + * to the first digit [exluding the sign] of the number] + */ + bn_reverse ((unsigned char *)_s, digs); + + /* append a NULL so the string is properly terminated */ + *str = '\0'; + + mp_clear (&t); + return MP_OKAY; +} + +#endif diff --git a/libtommath/bn_mp_unsigned_bin_size.c b/libtommath/bn_mp_unsigned_bin_size.c new file mode 100644 index 0000000..80da415 --- /dev/null +++ b/libtommath/bn_mp_unsigned_bin_size.c @@ -0,0 +1,25 @@ +#include <tommath.h> +#ifdef BN_MP_UNSIGNED_BIN_SIZE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* get the size for an unsigned equivalent */ +int +mp_unsigned_bin_size (mp_int * a) +{ + int size = mp_count_bits (a); + return (size / 8 + ((size & 7) != 0 ? 1 : 0)); +} +#endif diff --git a/libtommath/bn_mp_xor.c b/libtommath/bn_mp_xor.c new file mode 100644 index 0000000..192aacc --- /dev/null +++ b/libtommath/bn_mp_xor.c @@ -0,0 +1,47 @@ +#include <tommath.h> +#ifdef BN_MP_XOR_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* XOR two ints together */ +int +mp_xor (mp_int * a, mp_int * b, mp_int * c) +{ + int res, ix, px; + mp_int t, *x; + + if (a->used > b->used) { + if ((res = mp_init_copy (&t, a)) != MP_OKAY) { + return res; + } + px = b->used; + x = b; + } else { + if ((res = mp_init_copy (&t, b)) != MP_OKAY) { + return res; + } + px = a->used; + x = a; + } + + for (ix = 0; ix < px; ix++) { + + } + mp_clamp (&t); + mp_exch (c, &t); + mp_clear (&t); + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_mp_zero.c b/libtommath/bn_mp_zero.c new file mode 100644 index 0000000..0097598 --- /dev/null +++ b/libtommath/bn_mp_zero.c @@ -0,0 +1,26 @@ +#include <tommath.h> +#ifdef BN_MP_ZERO_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* set to zero */ +void +mp_zero (mp_int * a) +{ + a->sign = MP_ZPOS; + a->used = 0; + memset (a->dp, 0, sizeof (mp_digit) * a->alloc); +} +#endif diff --git a/libtommath/bn_prime_tab.c b/libtommath/bn_prime_tab.c new file mode 100644 index 0000000..14306c2 --- /dev/null +++ b/libtommath/bn_prime_tab.c @@ -0,0 +1,57 @@ +#include <tommath.h> +#ifdef BN_PRIME_TAB_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ +const mp_digit ltm_prime_tab[] = { + 0x0002, 0x0003, 0x0005, 0x0007, 0x000B, 0x000D, 0x0011, 0x0013, + 0x0017, 0x001D, 0x001F, 0x0025, 0x0029, 0x002B, 0x002F, 0x0035, + 0x003B, 0x003D, 0x0043, 0x0047, 0x0049, 0x004F, 0x0053, 0x0059, + 0x0061, 0x0065, 0x0067, 0x006B, 0x006D, 0x0071, 0x007F, +#ifndef MP_8BIT + 0x0083, + 0x0089, 0x008B, 0x0095, 0x0097, 0x009D, 0x00A3, 0x00A7, 0x00AD, + 0x00B3, 0x00B5, 0x00BF, 0x00C1, 0x00C5, 0x00C7, 0x00D3, 0x00DF, + 0x00E3, 0x00E5, 0x00E9, 0x00EF, 0x00F1, 0x00FB, 0x0101, 0x0107, + 0x010D, 0x010F, 0x0115, 0x0119, 0x011B, 0x0125, 0x0133, 0x0137, + + 0x0139, 0x013D, 0x014B, 0x0151, 0x015B, 0x015D, 0x0161, 0x0167, + 0x016F, 0x0175, 0x017B, 0x017F, 0x0185, 0x018D, 0x0191, 0x0199, + 0x01A3, 0x01A5, 0x01AF, 0x01B1, 0x01B7, 0x01BB, 0x01C1, 0x01C9, + 0x01CD, 0x01CF, 0x01D3, 0x01DF, 0x01E7, 0x01EB, 0x01F3, 0x01F7, + 0x01FD, 0x0209, 0x020B, 0x021D, 0x0223, 0x022D, 0x0233, 0x0239, + 0x023B, 0x0241, 0x024B, 0x0251, 0x0257, 0x0259, 0x025F, 0x0265, + 0x0269, 0x026B, 0x0277, 0x0281, 0x0283, 0x0287, 0x028D, 0x0293, + 0x0295, 0x02A1, 0x02A5, 0x02AB, 0x02B3, 0x02BD, 0x02C5, 0x02CF, + + 0x02D7, 0x02DD, 0x02E3, 0x02E7, 0x02EF, 0x02F5, 0x02F9, 0x0301, + 0x0305, 0x0313, 0x031D, 0x0329, 0x032B, 0x0335, 0x0337, 0x033B, + 0x033D, 0x0347, 0x0355, 0x0359, 0x035B, 0x035F, 0x036D, 0x0371, + 0x0373, 0x0377, 0x038B, 0x038F, 0x0397, 0x03A1, 0x03A9, 0x03AD, + 0x03B3, 0x03B9, 0x03C7, 0x03CB, 0x03D1, 0x03D7, 0x03DF, 0x03E5, + 0x03F1, 0x03F5, 0x03FB, 0x03FD, 0x0407, 0x0409, 0x040F, 0x0419, + 0x041B, 0x0425, 0x0427, 0x042D, 0x043F, 0x0443, 0x0445, 0x0449, + 0x044F, 0x0455, 0x045D, 0x0463, 0x0469, 0x047F, 0x0481, 0x048B, + + 0x0493, 0x049D, 0x04A3, 0x04A9, 0x04B1, 0x04BD, 0x04C1, 0x04C7, + 0x04CD, 0x04CF, 0x04D5, 0x04E1, 0x04EB, 0x04FD, 0x04FF, 0x0503, + 0x0509, 0x050B, 0x0511, 0x0515, 0x0517, 0x051B, 0x0527, 0x0529, + 0x052F, 0x0551, 0x0557, 0x055D, 0x0565, 0x0577, 0x0581, 0x058F, + 0x0593, 0x0595, 0x0599, 0x059F, 0x05A7, 0x05AB, 0x05AD, 0x05B3, + 0x05BF, 0x05C9, 0x05CB, 0x05CF, 0x05D1, 0x05D5, 0x05DB, 0x05E7, + 0x05F3, 0x05FB, 0x0607, 0x060D, 0x0611, 0x0617, 0x061F, 0x0623, + 0x062B, 0x062F, 0x063D, 0x0641, 0x0647, 0x0649, 0x064D, 0x0653 +#endif +}; +#endif diff --git a/libtommath/bn_reverse.c b/libtommath/bn_reverse.c new file mode 100644 index 0000000..851a6e8 --- /dev/null +++ b/libtommath/bn_reverse.c @@ -0,0 +1,35 @@ +#include <tommath.h> +#ifdef BN_REVERSE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* reverse an array, used for radix code */ +void +bn_reverse (unsigned char *s, int len) +{ + int ix, iy; + unsigned char t; + + ix = 0; + iy = len - 1; + while (ix < iy) { + t = s[ix]; + s[ix] = s[iy]; + s[iy] = t; + ++ix; + --iy; + } +} +#endif diff --git a/libtommath/bn_s_mp_add.c b/libtommath/bn_s_mp_add.c new file mode 100644 index 0000000..2b378ae --- /dev/null +++ b/libtommath/bn_s_mp_add.c @@ -0,0 +1,105 @@ +#include <tommath.h> +#ifdef BN_S_MP_ADD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* low level addition, based on HAC pp.594, Algorithm 14.7 */ +int +s_mp_add (mp_int * a, mp_int * b, mp_int * c) +{ + mp_int *x; + int olduse, res, min, max; + + /* find sizes, we let |a| <= |b| which means we have to sort + * them. "x" will point to the input with the most digits + */ + if (a->used > b->used) { + min = b->used; + max = a->used; + x = a; + } else { + min = a->used; + max = b->used; + x = b; + } + + /* init result */ + if (c->alloc < max + 1) { + if ((res = mp_grow (c, max + 1)) != MP_OKAY) { + return res; + } + } + + /* get old used digit count and set new one */ + olduse = c->used; + c->used = max + 1; + + { + register mp_digit u, *tmpa, *tmpb, *tmpc; + register int i; + + /* alias for digit pointers */ + + /* first input */ + tmpa = a->dp; + + /* second input */ + tmpb = b->dp; + + /* destination */ + tmpc = c->dp; + + /* zero the carry */ + u = 0; + for (i = 0; i < min; i++) { + /* Compute the sum at one digit, T[i] = A[i] + B[i] + U */ + *tmpc = *tmpa++ + *tmpb++ + u; + + /* U = carry bit of T[i] */ + u = *tmpc >> ((mp_digit)DIGIT_BIT); + + /* take away carry bit from T[i] */ + *tmpc++ &= MP_MASK; + } + + /* now copy higher words if any, that is in A+B + * if A or B has more digits add those in + */ + if (min != max) { + for (; i < max; i++) { + /* T[i] = X[i] + U */ + *tmpc = x->dp[i] + u; + + /* U = carry bit of T[i] */ + u = *tmpc >> ((mp_digit)DIGIT_BIT); + + /* take away carry bit from T[i] */ + *tmpc++ &= MP_MASK; + } + } + + /* add carry */ + *tmpc++ = u; + + /* clear digits above oldused */ + for (i = c->used; i < olduse; i++) { + *tmpc++ = 0; + } + } + + mp_clamp (c); + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_s_mp_exptmod.c b/libtommath/bn_s_mp_exptmod.c new file mode 100644 index 0000000..01a766f --- /dev/null +++ b/libtommath/bn_s_mp_exptmod.c @@ -0,0 +1,236 @@ +#include <tommath.h> +#ifdef BN_S_MP_EXPTMOD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +#ifdef MP_LOW_MEM + #define TAB_SIZE 32 +#else + #define TAB_SIZE 256 +#endif + +int s_mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y) +{ + mp_int M[TAB_SIZE], res, mu; + mp_digit buf; + int err, bitbuf, bitcpy, bitcnt, mode, digidx, x, y, winsize; + + /* find window size */ + x = mp_count_bits (X); + if (x <= 7) { + winsize = 2; + } else if (x <= 36) { + winsize = 3; + } else if (x <= 140) { + winsize = 4; + } else if (x <= 450) { + winsize = 5; + } else if (x <= 1303) { + winsize = 6; + } else if (x <= 3529) { + winsize = 7; + } else { + winsize = 8; + } + +#ifdef MP_LOW_MEM + if (winsize > 5) { + winsize = 5; + } +#endif + + /* init M array */ + /* init first cell */ + if ((err = mp_init(&M[1])) != MP_OKAY) { + return err; + } + + /* now init the second half of the array */ + for (x = 1<<(winsize-1); x < (1 << winsize); x++) { + if ((err = mp_init(&M[x])) != MP_OKAY) { + for (y = 1<<(winsize-1); y < x; y++) { + mp_clear (&M[y]); + } + mp_clear(&M[1]); + return err; + } + } + + /* create mu, used for Barrett reduction */ + if ((err = mp_init (&mu)) != MP_OKAY) { + goto LBL_M; + } + if ((err = mp_reduce_setup (&mu, P)) != MP_OKAY) { + goto LBL_MU; + } + + /* create M table + * + * The M table contains powers of the base, + * e.g. M[x] = G**x mod P + * + * The first half of the table is not + * computed though accept for M[0] and M[1] + */ + if ((err = mp_mod (G, P, &M[1])) != MP_OKAY) { + goto LBL_MU; + } + + /* compute the value at M[1<<(winsize-1)] by squaring + * M[1] (winsize-1) times + */ + if ((err = mp_copy (&M[1], &M[1 << (winsize - 1)])) != MP_OKAY) { + goto LBL_MU; + } + + for (x = 0; x < (winsize - 1); x++) { + if ((err = mp_sqr (&M[1 << (winsize - 1)], + &M[1 << (winsize - 1)])) != MP_OKAY) { + goto LBL_MU; + } + if ((err = mp_reduce (&M[1 << (winsize - 1)], P, &mu)) != MP_OKAY) { + goto LBL_MU; + } + } + + /* create upper table, that is M[x] = M[x-1] * M[1] (mod P) + * for x = (2**(winsize - 1) + 1) to (2**winsize - 1) + */ + for (x = (1 << (winsize - 1)) + 1; x < (1 << winsize); x++) { + if ((err = mp_mul (&M[x - 1], &M[1], &M[x])) != MP_OKAY) { + goto LBL_MU; + } + if ((err = mp_reduce (&M[x], P, &mu)) != MP_OKAY) { + goto LBL_MU; + } + } + + /* setup result */ + if ((err = mp_init (&res)) != MP_OKAY) { + goto LBL_MU; + } + mp_set (&res, 1); + + /* set initial mode and bit cnt */ + mode = 0; + bitcnt = 1; + buf = 0; + digidx = X->used - 1; + bitcpy = 0; + bitbuf = 0; + + for (;;) { + /* grab next digit as required */ + if (--bitcnt == 0) { + /* if digidx == -1 we are out of digits */ + if (digidx == -1) { + break; + } + /* read next digit and reset the bitcnt */ + buf = X->dp[digidx--]; + bitcnt = (int) DIGIT_BIT; + } + + /* grab the next msb from the exponent */ + y = (buf >> (mp_digit)(DIGIT_BIT - 1)) & 1; + buf <<= (mp_digit)1; + + /* if the bit is zero and mode == 0 then we ignore it + * These represent the leading zero bits before the first 1 bit + * in the exponent. Technically this opt is not required but it + * does lower the # of trivial squaring/reductions used + */ + if (mode == 0 && y == 0) { + continue; + } + + /* if the bit is zero and mode == 1 then we square */ + if (mode == 1 && y == 0) { + if ((err = mp_sqr (&res, &res)) != MP_OKAY) { + goto LBL_RES; + } + if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) { + goto LBL_RES; + } + continue; + } + + /* else we add it to the window */ + bitbuf |= (y << (winsize - ++bitcpy)); + mode = 2; + + if (bitcpy == winsize) { + /* ok window is filled so square as required and multiply */ + /* square first */ + for (x = 0; x < winsize; x++) { + if ((err = mp_sqr (&res, &res)) != MP_OKAY) { + goto LBL_RES; + } + if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) { + goto LBL_RES; + } + } + + /* then multiply */ + if ((err = mp_mul (&res, &M[bitbuf], &res)) != MP_OKAY) { + goto LBL_RES; + } + if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) { + goto LBL_RES; + } + + /* empty window and reset */ + bitcpy = 0; + bitbuf = 0; + mode = 1; + } + } + + /* if bits remain then square/multiply */ + if (mode == 2 && bitcpy > 0) { + /* square then multiply if the bit is set */ + for (x = 0; x < bitcpy; x++) { + if ((err = mp_sqr (&res, &res)) != MP_OKAY) { + goto LBL_RES; + } + if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) { + goto LBL_RES; + } + + bitbuf <<= 1; + if ((bitbuf & (1 << winsize)) != 0) { + /* then multiply */ + if ((err = mp_mul (&res, &M[1], &res)) != MP_OKAY) { + goto LBL_RES; + } + if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) { + goto LBL_RES; + } + } + } + } + + mp_exch (&res, Y); + err = MP_OKAY; +LBL_RES:mp_clear (&res); +LBL_MU:mp_clear (&mu); +LBL_M: + mp_clear(&M[1]); + for (x = 1<<(winsize-1); x < (1 << winsize); x++) { + mp_clear (&M[x]); + } + return err; +} +#endif diff --git a/libtommath/bn_s_mp_mul_digs.c b/libtommath/bn_s_mp_mul_digs.c new file mode 100644 index 0000000..d9f0a56 --- /dev/null +++ b/libtommath/bn_s_mp_mul_digs.c @@ -0,0 +1,87 @@ +#include <tommath.h> +#ifdef BN_S_MP_MUL_DIGS_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* multiplies |a| * |b| and only computes upto digs digits of result + * HAC pp. 595, Algorithm 14.12 Modified so you can control how + * many digits of output are created. + */ +int +s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs) +{ + mp_int t; + int res, pa, pb, ix, iy; + mp_digit u; + mp_word r; + mp_digit tmpx, *tmpt, *tmpy; + + /* can we use the fast multiplier? */ + if (((digs) < MP_WARRAY) && + MIN (a->used, b->used) < + (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) { + return fast_s_mp_mul_digs (a, b, c, digs); + } + + if ((res = mp_init_size (&t, digs)) != MP_OKAY) { + return res; + } + t.used = digs; + + /* compute the digits of the product directly */ + pa = a->used; + for (ix = 0; ix < pa; ix++) { + /* set the carry to zero */ + u = 0; + + /* limit ourselves to making digs digits of output */ + pb = MIN (b->used, digs - ix); + + /* setup some aliases */ + /* copy of the digit from a used within the nested loop */ + tmpx = a->dp[ix]; + + /* an alias for the destination shifted ix places */ + tmpt = t.dp + ix; + + /* an alias for the digits of b */ + tmpy = b->dp; + + /* compute the columns of the output and propagate the carry */ + for (iy = 0; iy < pb; iy++) { + /* compute the column as a mp_word */ + r = ((mp_word)*tmpt) + + ((mp_word)tmpx) * ((mp_word)*tmpy++) + + ((mp_word) u); + + /* the new column is the lower part of the result */ + *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK)); + + /* get the carry word from the result */ + u = (mp_digit) (r >> ((mp_word) DIGIT_BIT)); + } + /* set carry if it is placed below digs */ + if (ix + iy < digs) { + *tmpt = u; + } + } + + mp_clamp (&t); + mp_exch (&t, c); + + mp_clear (&t); + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_s_mp_mul_high_digs.c b/libtommath/bn_s_mp_mul_high_digs.c new file mode 100644 index 0000000..a060248 --- /dev/null +++ b/libtommath/bn_s_mp_mul_high_digs.c @@ -0,0 +1,77 @@ +#include <tommath.h> +#ifdef BN_S_MP_MUL_HIGH_DIGS_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* multiplies |a| * |b| and does not compute the lower digs digits + * [meant to get the higher part of the product] + */ +int +s_mp_mul_high_digs (mp_int * a, mp_int * b, mp_int * c, int digs) +{ + mp_int t; + int res, pa, pb, ix, iy; + mp_digit u; + mp_word r; + mp_digit tmpx, *tmpt, *tmpy; + + /* can we use the fast multiplier? */ +#ifdef BN_FAST_S_MP_MUL_HIGH_DIGS_C + if (((a->used + b->used + 1) < MP_WARRAY) + && MIN (a->used, b->used) < (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) { + return fast_s_mp_mul_high_digs (a, b, c, digs); + } +#endif + + if ((res = mp_init_size (&t, a->used + b->used + 1)) != MP_OKAY) { + return res; + } + t.used = a->used + b->used + 1; + + pa = a->used; + pb = b->used; + for (ix = 0; ix < pa; ix++) { + /* clear the carry */ + u = 0; + + /* left hand side of A[ix] * B[iy] */ + tmpx = a->dp[ix]; + + /* alias to the address of where the digits will be stored */ + tmpt = &(t.dp[digs]); + + /* alias for where to read the right hand side from */ + tmpy = b->dp + (digs - ix); + + for (iy = digs - ix; iy < pb; iy++) { + /* calculate the double precision result */ + r = ((mp_word)*tmpt) + + ((mp_word)tmpx) * ((mp_word)*tmpy++) + + ((mp_word) u); + + /* get the lower part */ + *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK)); + + /* carry the carry */ + u = (mp_digit) (r >> ((mp_word) DIGIT_BIT)); + } + *tmpt = u; + } + mp_clamp (&t); + mp_exch (&t, c); + mp_clear (&t); + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_s_mp_sqr.c b/libtommath/bn_s_mp_sqr.c new file mode 100644 index 0000000..4d12804 --- /dev/null +++ b/libtommath/bn_s_mp_sqr.c @@ -0,0 +1,81 @@ +#include <tommath.h> +#ifdef BN_S_MP_SQR_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* low level squaring, b = a*a, HAC pp.596-597, Algorithm 14.16 */ +int +s_mp_sqr (mp_int * a, mp_int * b) +{ + mp_int t; + int res, ix, iy, pa; + mp_word r; + mp_digit u, tmpx, *tmpt; + + pa = a->used; + if ((res = mp_init_size (&t, 2*pa + 1)) != MP_OKAY) { + return res; + } + + /* default used is maximum possible size */ + t.used = 2*pa + 1; + + for (ix = 0; ix < pa; ix++) { + /* first calculate the digit at 2*ix */ + /* calculate double precision result */ + r = ((mp_word) t.dp[2*ix]) + + ((mp_word)a->dp[ix])*((mp_word)a->dp[ix]); + + /* store lower part in result */ + t.dp[ix+ix] = (mp_digit) (r & ((mp_word) MP_MASK)); + + /* get the carry */ + u = (mp_digit)(r >> ((mp_word) DIGIT_BIT)); + + /* left hand side of A[ix] * A[iy] */ + tmpx = a->dp[ix]; + + /* alias for where to store the results */ + tmpt = t.dp + (2*ix + 1); + + for (iy = ix + 1; iy < pa; iy++) { + /* first calculate the product */ + r = ((mp_word)tmpx) * ((mp_word)a->dp[iy]); + + /* now calculate the double precision result, note we use + * addition instead of *2 since it's easier to optimize + */ + r = ((mp_word) *tmpt) + r + r + ((mp_word) u); + + /* store lower part */ + *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK)); + + /* get carry */ + u = (mp_digit)(r >> ((mp_word) DIGIT_BIT)); + } + /* propagate upwards */ + while (u != ((mp_digit) 0)) { + r = ((mp_word) *tmpt) + ((mp_word) u); + *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK)); + u = (mp_digit)(r >> ((mp_word) DIGIT_BIT)); + } + } + + mp_clamp (&t); + mp_exch (&t, b); + mp_clear (&t); + return MP_OKAY; +} +#endif diff --git a/libtommath/bn_s_mp_sub.c b/libtommath/bn_s_mp_sub.c new file mode 100644 index 0000000..5b7aef9 --- /dev/null +++ b/libtommath/bn_s_mp_sub.c @@ -0,0 +1,85 @@ +#include <tommath.h> +#ifdef BN_S_MP_SUB_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* low level subtraction (assumes |a| > |b|), HAC pp.595 Algorithm 14.9 */ +int +s_mp_sub (mp_int * a, mp_int * b, mp_int * c) +{ + int olduse, res, min, max; + + /* find sizes */ + min = b->used; + max = a->used; + + /* init result */ + if (c->alloc < max) { + if ((res = mp_grow (c, max)) != MP_OKAY) { + return res; + } + } + olduse = c->used; + c->used = max; + + { + register mp_digit u, *tmpa, *tmpb, *tmpc; + register int i; + + /* alias for digit pointers */ + tmpa = a->dp; + tmpb = b->dp; + tmpc = c->dp; + + /* set carry to zero */ + u = 0; + for (i = 0; i < min; i++) { + /* T[i] = A[i] - B[i] - U */ + *tmpc = *tmpa++ - *tmpb++ - u; + + /* U = carry bit of T[i] + * Note this saves performing an AND operation since + * if a carry does occur it will propagate all the way to the + * MSB. As a result a single shift is enough to get the carry + */ + u = *tmpc >> ((mp_digit)(CHAR_BIT * sizeof (mp_digit) - 1)); + + /* Clear carry from T[i] */ + *tmpc++ &= MP_MASK; + } + + /* now copy higher words if any, e.g. if A has more digits than B */ + for (; i < max; i++) { + /* T[i] = A[i] - U */ + *tmpc = *tmpa++ - u; + + /* U = carry bit of T[i] */ + u = *tmpc >> ((mp_digit)(CHAR_BIT * sizeof (mp_digit) - 1)); + + /* Clear carry from T[i] */ + *tmpc++ &= MP_MASK; + } + + /* clear digits above used (since we may not have grown result above) */ + for (i = c->used; i < olduse; i++) { + *tmpc++ = 0; + } + } + + mp_clamp (c); + return MP_OKAY; +} + +#endif diff --git a/libtommath/bncore.c b/libtommath/bncore.c new file mode 100644 index 0000000..cf8a15a --- /dev/null +++ b/libtommath/bncore.c @@ -0,0 +1,31 @@ +#include <tommath.h> +#ifdef BNCORE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* Known optimal configurations + + CPU /Compiler /MUL CUTOFF/SQR CUTOFF +------------------------------------------------------------- + Intel P4 Northwood /GCC v3.4.1 / 88/ 128/LTM 0.32 ;-) + +*/ + +int KARATSUBA_MUL_CUTOFF = 88, /* Min. number of digits before Karatsuba multiplication is used. */ + KARATSUBA_SQR_CUTOFF = 128, /* Min. number of digits before Karatsuba squaring is used. */ + + TOOM_MUL_CUTOFF = 350, /* no optimal values of these are known yet so set em high */ + TOOM_SQR_CUTOFF = 400; +#endif diff --git a/libtommath/booker.pl b/libtommath/booker.pl new file mode 100644 index 0000000..5c77e53 --- /dev/null +++ b/libtommath/booker.pl @@ -0,0 +1,262 @@ +#!/bin/perl +# +#Used to prepare the book "tommath.src" for LaTeX by pre-processing it into a .tex file +# +#Essentially you write the "tommath.src" as normal LaTex except where you want code snippets you put +# +#EXAM,file +# +#This preprocessor will then open "file" and insert it as a verbatim copy. +# +#Tom St Denis + +#get graphics type +if (shift =~ /PDF/) { + $graph = ""; +} else { + $graph = ".ps"; +} + +open(IN,"<tommath.src") or die "Can't open source file"; +open(OUT,">tommath.tex") or die "Can't open destination file"; + +print "Scanning for sections\n"; +$chapter = $section = $subsection = 0; +$x = 0; +while (<IN>) { + print "."; + if (!(++$x % 80)) { print "\n"; } + #update the headings + if (~($_ =~ /\*/)) { + if ($_ =~ /\\chapter{.+}/) { + ++$chapter; + $section = $subsection = 0; + } elsif ($_ =~ /\\section{.+}/) { + ++$section; + $subsection = 0; + } elsif ($_ =~ /\\subsection{.+}/) { + ++$subsection; + } + } + + if ($_ =~ m/MARK/) { + @m = split(",",$_); + chomp(@m[1]); + $index1{@m[1]} = $chapter; + $index2{@m[1]} = $section; + $index3{@m[1]} = $subsection; + } +} +close(IN); + +open(IN,"<tommath.src") or die "Can't open source file"; +$readline = $wroteline = 0; +$srcline = 0; + +while (<IN>) { + ++$readline; + ++$srcline; + + if ($_ =~ m/MARK/) { + } elsif ($_ =~ m/EXAM/ || $_ =~ m/LIST/) { + if ($_ =~ m/EXAM/) { + $skipheader = 1; + } else { + $skipheader = 0; + } + + # EXAM,file + chomp($_); + @m = split(",",$_); + open(SRC,"<$m[1]") or die "Error:$srcline:Can't open source file $m[1]"; + + print "$srcline:Inserting $m[1]:"; + + $line = 0; + $tmp = $m[1]; + $tmp =~ s/_/"\\_"/ge; + print OUT "\\vspace{+3mm}\\begin{small}\n\\hspace{-5.1mm}{\\bf File}: $tmp\n\\vspace{-3mm}\n\\begin{alltt}\n"; + $wroteline += 5; + + if ($skipheader == 1) { + # scan till next end of comment, e.g. skip license + while (<SRC>) { + $text[$line++] = $_; + last if ($_ =~ /math\.libtomcrypt\.org/); + } + <SRC>; + } + + $inline = 0; + while (<SRC>) { + $text[$line++] = $_; + ++$inline; + chomp($_); + $_ =~ s/\t/" "/ge; + $_ =~ s/{/"^{"/ge; + $_ =~ s/}/"^}"/ge; + $_ =~ s/\\/'\symbol{92}'/ge; + $_ =~ s/\^/"\\"/ge; + + printf OUT ("%03d ", $line); + for ($x = 0; $x < length($_); $x++) { + print OUT chr(vec($_, $x, 8)); + if ($x == 75) { + print OUT "\n "; + ++$wroteline; + } + } + print OUT "\n"; + ++$wroteline; + } + $totlines = $line; + print OUT "\\end{alltt}\n\\end{small}\n"; + close(SRC); + print "$inline lines\n"; + $wroteline += 2; + } elsif ($_ =~ m/@\d+,.+@/) { + # line contains [number,text] + # e.g. @14,for (ix = 0)@ + $txt = $_; + while ($txt =~ m/@\d+,.+@/) { + @m = split("@",$txt); # splits into text, one, two + @parms = split(",",$m[1]); # splits one,two into two elements + + # now search from $parms[0] down for $parms[1] + $found1 = 0; + $found2 = 0; + for ($i = $parms[0]; $i < $totlines && $found1 == 0; $i++) { + if ($text[$i] =~ m/\Q$parms[1]\E/) { + $foundline1 = $i + 1; + $found1 = 1; + } + } + + # now search backwards + for ($i = $parms[0] - 1; $i >= 0 && $found2 == 0; $i--) { + if ($text[$i] =~ m/\Q$parms[1]\E/) { + $foundline2 = $i + 1; + $found2 = 1; + } + } + + # now use the closest match or the first if tied + if ($found1 == 1 && $found2 == 0) { + $found = 1; + $foundline = $foundline1; + } elsif ($found1 == 0 && $found2 == 1) { + $found = 1; + $foundline = $foundline2; + } elsif ($found1 == 1 && $found2 == 1) { + $found = 1; + if (($foundline1 - $parms[0]) <= ($parms[0] - $foundline2)) { + $foundline = $foundline1; + } else { + $foundline = $foundline2; + } + } else { + $found = 0; + } + + # if found replace + if ($found == 1) { + $delta = $parms[0] - $foundline; + print "Found replacement tag for \"$parms[1]\" on line $srcline which refers to line $foundline (delta $delta)\n"; + $_ =~ s/@\Q$m[1]\E@/$foundline/; + } else { + print "ERROR: The tag \"$parms[1]\" on line $srcline was not found in the most recently parsed source!\n"; + } + + # remake the rest of the line + $cnt = @m; + $txt = ""; + for ($i = 2; $i < $cnt; $i++) { + $txt = $txt . $m[$i] . "@"; + } + } + print OUT $_; + ++$wroteline; + } elsif ($_ =~ /~.+~/) { + # line contains a ~text~ pair used to refer to indexing :-) + $txt = $_; + while ($txt =~ /~.+~/) { + @m = split("~", $txt); + + # word is the second position + $word = @m[1]; + $a = $index1{$word}; + $b = $index2{$word}; + $c = $index3{$word}; + + # if chapter (a) is zero it wasn't found + if ($a == 0) { + print "ERROR: the tag \"$word\" on line $srcline was not found previously marked.\n"; + } else { + # format the tag as x, x.y or x.y.z depending on the values + $str = $a; + $str = $str . ".$b" if ($b != 0); + $str = $str . ".$c" if ($c != 0); + + if ($b == 0 && $c == 0) { + # its a chapter + if ($a <= 10) { + if ($a == 1) { + $str = "chapter one"; + } elsif ($a == 2) { + $str = "chapter two"; + } elsif ($a == 3) { + $str = "chapter three"; + } elsif ($a == 4) { + $str = "chapter four"; + } elsif ($a == 5) { + $str = "chapter five"; + } elsif ($a == 6) { + $str = "chapter six"; + } elsif ($a == 7) { + $str = "chapter seven"; + } elsif ($a == 8) { + $str = "chapter eight"; + } elsif ($a == 9) { + $str = "chapter nine"; + } elsif ($a == 2) { + $str = "chapter ten"; + } + } else { + $str = "chapter " . $str; + } + } else { + $str = "section " . $str if ($b != 0 && $c == 0); + $str = "sub-section " . $str if ($b != 0 && $c != 0); + } + + #substitute + $_ =~ s/~\Q$word\E~/$str/; + + print "Found replacement tag for marker \"$word\" on line $srcline which refers to $str\n"; + } + + # remake rest of the line + $cnt = @m; + $txt = ""; + for ($i = 2; $i < $cnt; $i++) { + $txt = $txt . $m[$i] . "~"; + } + } + print OUT $_; + ++$wroteline; + } elsif ($_ =~ m/FIGU/) { + # FIGU,file,caption + chomp($_); + @m = split(",", $_); + print OUT "\\begin{center}\n\\begin{figure}[here]\n\\includegraphics{pics/$m[1]$graph}\n"; + print OUT "\\caption{$m[2]}\n\\label{pic:$m[1]}\n\\end{figure}\n\\end{center}\n"; + $wroteline += 4; + } else { + print OUT $_; + ++$wroteline; + } +} +print "Read $readline lines, wrote $wroteline lines\n"; + +close (OUT); +close (IN); diff --git a/libtommath/callgraph.txt b/libtommath/callgraph.txt new file mode 100644 index 0000000..4dc4cba --- /dev/null +++ b/libtommath/callgraph.txt @@ -0,0 +1,10193 @@ +BN_PRIME_TAB_C + + +BN_MP_SQRT_C ++--->BN_MP_N_ROOT_C +| +--->BN_MP_INIT_C +| +--->BN_MP_SET_C +| | +--->BN_MP_ZERO_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_EXPT_D_C +| | +--->BN_MP_INIT_COPY_C +| | +--->BN_MP_SQR_C +| | | +--->BN_MP_TOOM_SQR_C +| | | | +--->BN_MP_INIT_MULTI_C +| | | | | +--->BN_MP_CLEAR_C +| | | | +--->BN_MP_MOD_2D_C +| | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_MUL_2_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_DIV_2_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MUL_2D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MUL_D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_DIV_3_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_MP_CLEAR_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLEAR_MULTI_C +| | | | | +--->BN_MP_CLEAR_C +| | | +--->BN_MP_KARATSUBA_SQR_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLEAR_C +| | | +--->BN_FAST_S_MP_SQR_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_SQR_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_CLEAR_C +| | +--->BN_MP_MUL_C +| | | +--->BN_MP_TOOM_MUL_C +| | | | +--->BN_MP_INIT_MULTI_C +| | | | +--->BN_MP_MOD_2D_C +| | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_MUL_2_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_DIV_2_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MUL_2D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MUL_D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_DIV_3_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLEAR_MULTI_C +| | | +--->BN_MP_KARATSUBA_MUL_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_ZERO_C +| | | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_MUL_DIGS_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| +--->BN_MP_MUL_C +| | +--->BN_MP_TOOM_MUL_C +| | | +--->BN_MP_INIT_MULTI_C +| | | | +--->BN_MP_CLEAR_C +| | | +--->BN_MP_MOD_2D_C +| | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_MUL_2_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_SUB_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_DIV_2_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_MUL_2D_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_MUL_D_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_DIV_3_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_CLEAR_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLEAR_MULTI_C +| | | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_KARATSUBA_MUL_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_SUB_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_S_MP_MUL_DIGS_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_CLEAR_C +| +--->BN_MP_SUB_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_MUL_D_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_DIV_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_MP_ZERO_C +| | +--->BN_MP_INIT_MULTI_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_COUNT_BITS_C +| | +--->BN_MP_ABS_C +| | +--->BN_MP_MUL_2D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_2D_C +| | | +--->BN_MP_MOD_2D_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CLEAR_C +| | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_MULTI_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_INIT_COPY_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_RSHD_C +| | +--->BN_MP_RSHD_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CLEAR_C +| +--->BN_MP_CMP_C +| | +--->BN_MP_CMP_MAG_C +| +--->BN_MP_SUB_D_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_ADD_D_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_EXCH_C +| +--->BN_MP_CLEAR_C ++--->BN_MP_ZERO_C ++--->BN_MP_INIT_COPY_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C ++--->BN_MP_RSHD_C ++--->BN_MP_DIV_C +| +--->BN_MP_CMP_MAG_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_INIT_MULTI_C +| | +--->BN_MP_CLEAR_C +| +--->BN_MP_SET_C +| +--->BN_MP_COUNT_BITS_C +| +--->BN_MP_ABS_C +| +--->BN_MP_MUL_2D_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_LSHD_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CMP_C +| +--->BN_MP_SUB_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_ADD_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_DIV_2D_C +| | +--->BN_MP_MOD_2D_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CLEAR_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| +--->BN_MP_EXCH_C +| +--->BN_MP_CLEAR_MULTI_C +| | +--->BN_MP_CLEAR_C +| +--->BN_MP_INIT_SIZE_C +| +--->BN_MP_LSHD_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_MUL_D_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CLAMP_C +| +--->BN_MP_CLEAR_C ++--->BN_MP_ADD_C +| +--->BN_S_MP_ADD_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CMP_MAG_C +| +--->BN_S_MP_SUB_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C ++--->BN_MP_DIV_2_C +| +--->BN_MP_GROW_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_CMP_MAG_C ++--->BN_MP_EXCH_C ++--->BN_MP_CLEAR_C + + +BN_MP_CMP_D_C + + +BN_MP_EXCH_C + + +BN_MP_IS_SQUARE_C ++--->BN_MP_MOD_D_C +| +--->BN_MP_DIV_D_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_DIV_2D_C +| | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_INIT_C +| | | +--->BN_MP_MOD_2D_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CLEAR_C +| | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | +--->BN_MP_DIV_3_C +| | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_INIT_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_INIT_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_C ++--->BN_MP_INIT_SET_INT_C +| +--->BN_MP_INIT_C +| +--->BN_MP_SET_INT_C +| | +--->BN_MP_ZERO_C +| | +--->BN_MP_MUL_2D_C +| | | +--->BN_MP_COPY_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CLAMP_C ++--->BN_MP_MOD_C +| +--->BN_MP_INIT_C +| +--->BN_MP_DIV_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_ZERO_C +| | +--->BN_MP_INIT_MULTI_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_SET_C +| | +--->BN_MP_COUNT_BITS_C +| | +--->BN_MP_ABS_C +| | +--->BN_MP_MUL_2D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_2D_C +| | | +--->BN_MP_MOD_2D_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CLEAR_C +| | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_MULTI_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_INIT_COPY_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_RSHD_C +| | +--->BN_MP_RSHD_C +| | +--->BN_MP_MUL_D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CLEAR_C +| +--->BN_MP_CLEAR_C +| +--->BN_MP_ADD_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_EXCH_C ++--->BN_MP_GET_INT_C ++--->BN_MP_SQRT_C +| +--->BN_MP_N_ROOT_C +| | +--->BN_MP_INIT_C +| | +--->BN_MP_SET_C +| | | +--->BN_MP_ZERO_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_EXPT_D_C +| | | +--->BN_MP_INIT_COPY_C +| | | +--->BN_MP_SQR_C +| | | | +--->BN_MP_TOOM_SQR_C +| | | | | +--->BN_MP_INIT_MULTI_C +| | | | | | +--->BN_MP_CLEAR_C +| | | | | +--->BN_MP_MOD_2D_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_MUL_2_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_DIV_2_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_MUL_2D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_MUL_D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_DIV_3_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_EXCH_C +| | | | | | +--->BN_MP_CLEAR_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLEAR_MULTI_C +| | | | | | +--->BN_MP_CLEAR_C +| | | | +--->BN_MP_KARATSUBA_SQR_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLEAR_C +| | | | +--->BN_FAST_S_MP_SQR_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_S_MP_SQR_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_MP_CLEAR_C +| | | +--->BN_MP_CLEAR_C +| | | +--->BN_MP_MUL_C +| | | | +--->BN_MP_TOOM_MUL_C +| | | | | +--->BN_MP_INIT_MULTI_C +| | | | | +--->BN_MP_MOD_2D_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_MUL_2_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_DIV_2_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_MUL_2D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_MUL_D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_DIV_3_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLEAR_MULTI_C +| | | | +--->BN_MP_KARATSUBA_MUL_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | | | +--->BN_MP_ZERO_C +| | | | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_S_MP_MUL_DIGS_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | +--->BN_MP_MUL_C +| | | +--->BN_MP_TOOM_MUL_C +| | | | +--->BN_MP_INIT_MULTI_C +| | | | | +--->BN_MP_CLEAR_C +| | | | +--->BN_MP_MOD_2D_C +| | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_MUL_2_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_DIV_2_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MUL_2D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MUL_D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_DIV_3_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_MP_CLEAR_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLEAR_MULTI_C +| | | | | +--->BN_MP_CLEAR_C +| | | +--->BN_MP_KARATSUBA_MUL_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_CLEAR_C +| | | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_MUL_DIGS_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MUL_D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_INIT_MULTI_C +| | | | +--->BN_MP_CLEAR_C +| | | +--->BN_MP_COUNT_BITS_C +| | | +--->BN_MP_ABS_C +| | | +--->BN_MP_MUL_2D_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_DIV_2D_C +| | | | +--->BN_MP_MOD_2D_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CLEAR_C +| | | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_CLEAR_MULTI_C +| | | | +--->BN_MP_CLEAR_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_INIT_COPY_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_CMP_C +| | | +--->BN_MP_CMP_MAG_C +| | +--->BN_MP_SUB_D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_ADD_D_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_C +| +--->BN_MP_ZERO_C +| +--->BN_MP_INIT_COPY_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| +--->BN_MP_RSHD_C +| +--->BN_MP_DIV_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_INIT_MULTI_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_SET_C +| | +--->BN_MP_COUNT_BITS_C +| | +--->BN_MP_ABS_C +| | +--->BN_MP_MUL_2D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_2D_C +| | | +--->BN_MP_MOD_2D_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CLEAR_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_MULTI_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_MUL_D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CLEAR_C +| +--->BN_MP_ADD_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_DIV_2_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CMP_MAG_C +| +--->BN_MP_EXCH_C +| +--->BN_MP_CLEAR_C ++--->BN_MP_SQR_C +| +--->BN_MP_TOOM_SQR_C +| | +--->BN_MP_INIT_MULTI_C +| | | +--->BN_MP_INIT_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_MOD_2D_C +| | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_COPY_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_RSHD_C +| | | +--->BN_MP_ZERO_C +| | +--->BN_MP_MUL_2_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_2_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MUL_2D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MUL_D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_3_C +| | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_INIT_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_CLEAR_MULTI_C +| | | +--->BN_MP_CLEAR_C +| +--->BN_MP_KARATSUBA_SQR_C +| | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_INIT_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_ZERO_C +| | +--->BN_MP_ADD_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_MP_CLEAR_C +| +--->BN_FAST_S_MP_SQR_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_S_MP_SQR_C +| | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_INIT_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_C ++--->BN_MP_CMP_MAG_C ++--->BN_MP_CLEAR_C + + +BN_MP_NEG_C ++--->BN_MP_COPY_C +| +--->BN_MP_GROW_C + + +BN_MP_EXPTMOD_C ++--->BN_MP_INIT_C ++--->BN_MP_INVMOD_C +| +--->BN_FAST_MP_INVMOD_C +| | +--->BN_MP_INIT_MULTI_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_ABS_C +| | +--->BN_MP_SET_C +| | | +--->BN_MP_ZERO_C +| | +--->BN_MP_DIV_2_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_C +| | | +--->BN_MP_CMP_MAG_C +| | +--->BN_MP_CMP_D_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_MULTI_C +| | | +--->BN_MP_CLEAR_C +| +--->BN_MP_INVMOD_SLOW_C +| | +--->BN_MP_INIT_MULTI_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_SET_C +| | | +--->BN_MP_ZERO_C +| | +--->BN_MP_DIV_2_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_C +| | | +--->BN_MP_CMP_MAG_C +| | +--->BN_MP_CMP_D_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_MULTI_C +| | | +--->BN_MP_CLEAR_C ++--->BN_MP_CLEAR_C ++--->BN_MP_ABS_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C ++--->BN_MP_CLEAR_MULTI_C ++--->BN_MP_DR_IS_MODULUS_C ++--->BN_MP_REDUCE_IS_2K_C +| +--->BN_MP_REDUCE_2K_C +| | +--->BN_MP_COUNT_BITS_C +| | +--->BN_MP_DIV_2D_C +| | | +--->BN_MP_COPY_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_MOD_2D_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | +--->BN_MP_MUL_D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_COUNT_BITS_C ++--->BN_MP_EXPTMOD_FAST_C +| +--->BN_MP_COUNT_BITS_C +| +--->BN_MP_MONTGOMERY_SETUP_C +| +--->BN_FAST_MP_MONTGOMERY_REDUCE_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_RSHD_C +| | | +--->BN_MP_ZERO_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_S_MP_SUB_C +| +--->BN_MP_MONTGOMERY_REDUCE_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_RSHD_C +| | | +--->BN_MP_ZERO_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_S_MP_SUB_C +| +--->BN_MP_DR_SETUP_C +| +--->BN_MP_DR_REDUCE_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_S_MP_SUB_C +| +--->BN_MP_REDUCE_2K_SETUP_C +| | +--->BN_MP_2EXPT_C +| | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_GROW_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_REDUCE_2K_C +| | +--->BN_MP_DIV_2D_C +| | | +--->BN_MP_COPY_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_MOD_2D_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | +--->BN_MP_MUL_D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_MONTGOMERY_CALC_NORMALIZATION_C +| | +--->BN_MP_2EXPT_C +| | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_SET_C +| | | +--->BN_MP_ZERO_C +| | +--->BN_MP_MUL_2_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_MULMOD_C +| | +--->BN_MP_MUL_C +| | | +--->BN_MP_TOOM_MUL_C +| | | | +--->BN_MP_INIT_MULTI_C +| | | | +--->BN_MP_MOD_2D_C +| | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_COPY_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_COPY_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_MUL_2_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_DIV_2_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MUL_2D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MUL_D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_DIV_3_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_KARATSUBA_MUL_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_ZERO_C +| | | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_MUL_DIGS_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | +--->BN_MP_MOD_C +| | | +--->BN_MP_DIV_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_MP_COPY_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_INIT_MULTI_C +| | | | +--->BN_MP_SET_C +| | | | +--->BN_MP_MUL_2D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_DIV_2D_C +| | | | | +--->BN_MP_MOD_2D_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_INIT_COPY_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_MUL_D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| +--->BN_MP_SET_C +| | +--->BN_MP_ZERO_C +| +--->BN_MP_MOD_C +| | +--->BN_MP_DIV_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_MP_COPY_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_INIT_MULTI_C +| | | +--->BN_MP_MUL_2D_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_C +| | | +--->BN_MP_SUB_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_DIV_2D_C +| | | | +--->BN_MP_MOD_2D_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_INIT_COPY_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_MUL_D_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_SQR_C +| | +--->BN_MP_TOOM_SQR_C +| | | +--->BN_MP_INIT_MULTI_C +| | | +--->BN_MP_MOD_2D_C +| | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_MUL_2_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_SUB_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_DIV_2_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_MUL_2D_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_MUL_D_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_DIV_3_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_MP_KARATSUBA_SQR_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_SUB_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | +--->BN_FAST_S_MP_SQR_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_S_MP_SQR_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| +--->BN_MP_MUL_C +| | +--->BN_MP_TOOM_MUL_C +| | | +--->BN_MP_INIT_MULTI_C +| | | +--->BN_MP_MOD_2D_C +| | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_MUL_2_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_SUB_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_DIV_2_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_MUL_2D_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_MUL_D_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_DIV_3_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_MP_KARATSUBA_MUL_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_SUB_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_ZERO_C +| | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_S_MP_MUL_DIGS_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| +--->BN_MP_EXCH_C ++--->BN_S_MP_EXPTMOD_C +| +--->BN_MP_COUNT_BITS_C +| +--->BN_MP_REDUCE_SETUP_C +| | +--->BN_MP_2EXPT_C +| | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_DIV_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_MP_COPY_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_INIT_MULTI_C +| | | +--->BN_MP_SET_C +| | | +--->BN_MP_MUL_2D_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_C +| | | +--->BN_MP_SUB_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_DIV_2D_C +| | | | +--->BN_MP_MOD_2D_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_INIT_COPY_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_MUL_D_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_MOD_C +| | +--->BN_MP_DIV_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_MP_COPY_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_INIT_MULTI_C +| | | +--->BN_MP_SET_C +| | | +--->BN_MP_MUL_2D_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_C +| | | +--->BN_MP_SUB_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_DIV_2D_C +| | | | +--->BN_MP_MOD_2D_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_INIT_COPY_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_MUL_D_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_SQR_C +| | +--->BN_MP_TOOM_SQR_C +| | | +--->BN_MP_INIT_MULTI_C +| | | +--->BN_MP_MOD_2D_C +| | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_MUL_2_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_SUB_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_DIV_2_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_MUL_2D_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_MUL_D_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_DIV_3_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_MP_KARATSUBA_SQR_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_SUB_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | +--->BN_FAST_S_MP_SQR_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_S_MP_SQR_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| +--->BN_MP_REDUCE_C +| | +--->BN_MP_INIT_COPY_C +| | +--->BN_MP_RSHD_C +| | | +--->BN_MP_ZERO_C +| | +--->BN_MP_MUL_C +| | | +--->BN_MP_TOOM_MUL_C +| | | | +--->BN_MP_INIT_MULTI_C +| | | | +--->BN_MP_MOD_2D_C +| | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MUL_2_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_DIV_2_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MUL_2D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MUL_D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_DIV_3_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_KARATSUBA_MUL_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_MUL_DIGS_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | +--->BN_S_MP_MUL_HIGH_DIGS_C +| | | +--->BN_FAST_S_MP_MUL_HIGH_DIGS_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | +--->BN_FAST_S_MP_MUL_HIGH_DIGS_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MOD_2D_C +| | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_S_MP_MUL_DIGS_C +| | | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_D_C +| | +--->BN_MP_SET_C +| | | +--->BN_MP_ZERO_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_C +| | | +--->BN_MP_CMP_MAG_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_MUL_C +| | +--->BN_MP_TOOM_MUL_C +| | | +--->BN_MP_INIT_MULTI_C +| | | +--->BN_MP_MOD_2D_C +| | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_MUL_2_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_SUB_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_DIV_2_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_MUL_2D_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_MUL_D_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_DIV_3_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_MP_KARATSUBA_MUL_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_SUB_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_ZERO_C +| | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_S_MP_MUL_DIGS_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| +--->BN_MP_SET_C +| | +--->BN_MP_ZERO_C +| +--->BN_MP_EXCH_C + + +BN_MP_OR_C ++--->BN_MP_INIT_COPY_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C ++--->BN_MP_CLAMP_C ++--->BN_MP_EXCH_C ++--->BN_MP_CLEAR_C + + +BN_MP_ZERO_C + + +BN_MP_GROW_C + + +BN_MP_COUNT_BITS_C + + +BN_MP_PRIME_FERMAT_C ++--->BN_MP_CMP_D_C ++--->BN_MP_INIT_C ++--->BN_MP_EXPTMOD_C +| +--->BN_MP_INVMOD_C +| | +--->BN_FAST_MP_INVMOD_C +| | | +--->BN_MP_INIT_MULTI_C +| | | | +--->BN_MP_CLEAR_C +| | | +--->BN_MP_COPY_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_ABS_C +| | | +--->BN_MP_SET_C +| | | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_DIV_2_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_SUB_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_CLEAR_MULTI_C +| | | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_INVMOD_SLOW_C +| | | +--->BN_MP_INIT_MULTI_C +| | | | +--->BN_MP_CLEAR_C +| | | +--->BN_MP_COPY_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_SET_C +| | | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_DIV_2_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_SUB_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_CLEAR_MULTI_C +| | | | +--->BN_MP_CLEAR_C +| +--->BN_MP_CLEAR_C +| +--->BN_MP_ABS_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| +--->BN_MP_CLEAR_MULTI_C +| +--->BN_MP_DR_IS_MODULUS_C +| +--->BN_MP_REDUCE_IS_2K_C +| | +--->BN_MP_REDUCE_2K_C +| | | +--->BN_MP_COUNT_BITS_C +| | | +--->BN_MP_DIV_2D_C +| | | | +--->BN_MP_COPY_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_MOD_2D_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_MUL_D_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_COUNT_BITS_C +| +--->BN_MP_EXPTMOD_FAST_C +| | +--->BN_MP_COUNT_BITS_C +| | +--->BN_MP_MONTGOMERY_SETUP_C +| | +--->BN_FAST_MP_MONTGOMERY_REDUCE_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | +--->BN_MP_MONTGOMERY_REDUCE_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | +--->BN_MP_DR_SETUP_C +| | +--->BN_MP_DR_REDUCE_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | +--->BN_MP_REDUCE_2K_SETUP_C +| | | +--->BN_MP_2EXPT_C +| | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_REDUCE_2K_C +| | | +--->BN_MP_DIV_2D_C +| | | | +--->BN_MP_COPY_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_MOD_2D_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_MUL_D_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MONTGOMERY_CALC_NORMALIZATION_C +| | | +--->BN_MP_2EXPT_C +| | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_SET_C +| | | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_MUL_2_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MULMOD_C +| | | +--->BN_MP_MUL_C +| | | | +--->BN_MP_TOOM_MUL_C +| | | | | +--->BN_MP_INIT_MULTI_C +| | | | | +--->BN_MP_MOD_2D_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | | +--->BN_MP_COPY_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_COPY_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_MUL_2_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_DIV_2_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_MUL_2D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_MUL_D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_DIV_3_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_KARATSUBA_MUL_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | | | +--->BN_MP_ZERO_C +| | | | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_S_MP_MUL_DIGS_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_MOD_C +| | | | +--->BN_MP_DIV_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_MP_COPY_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_INIT_MULTI_C +| | | | | +--->BN_MP_SET_C +| | | | | +--->BN_MP_MUL_2D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_DIV_2D_C +| | | | | | +--->BN_MP_MOD_2D_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_INIT_COPY_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_MUL_D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | +--->BN_MP_SET_C +| | | +--->BN_MP_ZERO_C +| | +--->BN_MP_MOD_C +| | | +--->BN_MP_DIV_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_MP_COPY_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_INIT_MULTI_C +| | | | +--->BN_MP_MUL_2D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_DIV_2D_C +| | | | | +--->BN_MP_MOD_2D_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_INIT_COPY_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_MUL_D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_SQR_C +| | | +--->BN_MP_TOOM_SQR_C +| | | | +--->BN_MP_INIT_MULTI_C +| | | | +--->BN_MP_MOD_2D_C +| | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_MUL_2_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_DIV_2_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MUL_2D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MUL_D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_DIV_3_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_KARATSUBA_SQR_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | +--->BN_FAST_S_MP_SQR_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_SQR_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | +--->BN_MP_MUL_C +| | | +--->BN_MP_TOOM_MUL_C +| | | | +--->BN_MP_INIT_MULTI_C +| | | | +--->BN_MP_MOD_2D_C +| | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_MUL_2_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_DIV_2_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MUL_2D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MUL_D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_DIV_3_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_KARATSUBA_MUL_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_ZERO_C +| | | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_MUL_DIGS_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | +--->BN_MP_EXCH_C +| +--->BN_S_MP_EXPTMOD_C +| | +--->BN_MP_COUNT_BITS_C +| | +--->BN_MP_REDUCE_SETUP_C +| | | +--->BN_MP_2EXPT_C +| | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_DIV_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_MP_COPY_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_INIT_MULTI_C +| | | | +--->BN_MP_SET_C +| | | | +--->BN_MP_MUL_2D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_DIV_2D_C +| | | | | +--->BN_MP_MOD_2D_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_INIT_COPY_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_MUL_D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MOD_C +| | | +--->BN_MP_DIV_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_MP_COPY_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_INIT_MULTI_C +| | | | +--->BN_MP_SET_C +| | | | +--->BN_MP_MUL_2D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_DIV_2D_C +| | | | | +--->BN_MP_MOD_2D_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_INIT_COPY_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_MUL_D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_SQR_C +| | | +--->BN_MP_TOOM_SQR_C +| | | | +--->BN_MP_INIT_MULTI_C +| | | | +--->BN_MP_MOD_2D_C +| | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_MUL_2_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_DIV_2_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MUL_2D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MUL_D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_DIV_3_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_KARATSUBA_SQR_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | +--->BN_FAST_S_MP_SQR_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_SQR_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | +--->BN_MP_REDUCE_C +| | | +--->BN_MP_INIT_COPY_C +| | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_MUL_C +| | | | +--->BN_MP_TOOM_MUL_C +| | | | | +--->BN_MP_INIT_MULTI_C +| | | | | +--->BN_MP_MOD_2D_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_MUL_2_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_DIV_2_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_MUL_2D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_MUL_D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_DIV_3_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_KARATSUBA_MUL_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_S_MP_MUL_DIGS_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | +--->BN_S_MP_MUL_HIGH_DIGS_C +| | | | +--->BN_FAST_S_MP_MUL_HIGH_DIGS_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | +--->BN_FAST_S_MP_MUL_HIGH_DIGS_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_MOD_2D_C +| | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_MUL_DIGS_C +| | | | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_SUB_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_SET_C +| | | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MUL_C +| | | +--->BN_MP_TOOM_MUL_C +| | | | +--->BN_MP_INIT_MULTI_C +| | | | +--->BN_MP_MOD_2D_C +| | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_MUL_2_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_DIV_2_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MUL_2D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MUL_D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_DIV_3_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_KARATSUBA_MUL_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_ZERO_C +| | | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_MUL_DIGS_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | +--->BN_MP_SET_C +| | | +--->BN_MP_ZERO_C +| | +--->BN_MP_EXCH_C ++--->BN_MP_CMP_C +| +--->BN_MP_CMP_MAG_C ++--->BN_MP_CLEAR_C + + +BN_MP_SUBMOD_C ++--->BN_MP_INIT_C ++--->BN_MP_SUB_C +| +--->BN_S_MP_ADD_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CMP_MAG_C +| +--->BN_S_MP_SUB_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C ++--->BN_MP_CLEAR_C ++--->BN_MP_MOD_C +| +--->BN_MP_DIV_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_ZERO_C +| | +--->BN_MP_INIT_MULTI_C +| | +--->BN_MP_SET_C +| | +--->BN_MP_COUNT_BITS_C +| | +--->BN_MP_ABS_C +| | +--->BN_MP_MUL_2D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_2D_C +| | | +--->BN_MP_MOD_2D_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_MULTI_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_INIT_COPY_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_RSHD_C +| | +--->BN_MP_RSHD_C +| | +--->BN_MP_MUL_D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_ADD_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_EXCH_C + + +BN_MP_MOD_2D_C ++--->BN_MP_ZERO_C ++--->BN_MP_COPY_C +| +--->BN_MP_GROW_C ++--->BN_MP_CLAMP_C + + +BN_MP_TORADIX_N_C ++--->BN_MP_INIT_COPY_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C ++--->BN_MP_DIV_D_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_DIV_2D_C +| | +--->BN_MP_ZERO_C +| | +--->BN_MP_MOD_2D_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CLEAR_C +| | +--->BN_MP_RSHD_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| +--->BN_MP_DIV_3_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_C +| +--->BN_MP_INIT_SIZE_C +| +--->BN_MP_CLAMP_C +| +--->BN_MP_EXCH_C +| +--->BN_MP_CLEAR_C ++--->BN_MP_CLEAR_C + + +BN_MP_CMP_C ++--->BN_MP_CMP_MAG_C + + +BNCORE_C + + +BN_MP_TORADIX_C ++--->BN_MP_INIT_COPY_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C ++--->BN_MP_DIV_D_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_DIV_2D_C +| | +--->BN_MP_ZERO_C +| | +--->BN_MP_MOD_2D_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CLEAR_C +| | +--->BN_MP_RSHD_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| +--->BN_MP_DIV_3_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_C +| +--->BN_MP_INIT_SIZE_C +| +--->BN_MP_CLAMP_C +| +--->BN_MP_EXCH_C +| +--->BN_MP_CLEAR_C ++--->BN_MP_CLEAR_C + + +BN_MP_ADD_D_C ++--->BN_MP_GROW_C ++--->BN_MP_SUB_D_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_CLAMP_C + + +BN_MP_DIV_3_C ++--->BN_MP_INIT_SIZE_C +| +--->BN_MP_INIT_C ++--->BN_MP_CLAMP_C ++--->BN_MP_EXCH_C ++--->BN_MP_CLEAR_C + + +BN_FAST_S_MP_MUL_DIGS_C ++--->BN_MP_GROW_C ++--->BN_MP_CLAMP_C + + +BN_MP_SQRMOD_C ++--->BN_MP_INIT_C ++--->BN_MP_SQR_C +| +--->BN_MP_TOOM_SQR_C +| | +--->BN_MP_INIT_MULTI_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_MOD_2D_C +| | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_COPY_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_RSHD_C +| | | +--->BN_MP_ZERO_C +| | +--->BN_MP_MUL_2_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_2_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MUL_2D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MUL_D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_3_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_CLEAR_MULTI_C +| | | +--->BN_MP_CLEAR_C +| +--->BN_MP_KARATSUBA_SQR_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_ZERO_C +| | +--->BN_MP_ADD_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_MP_CLEAR_C +| +--->BN_FAST_S_MP_SQR_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_S_MP_SQR_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_C ++--->BN_MP_CLEAR_C ++--->BN_MP_MOD_C +| +--->BN_MP_DIV_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_ZERO_C +| | +--->BN_MP_INIT_MULTI_C +| | +--->BN_MP_SET_C +| | +--->BN_MP_COUNT_BITS_C +| | +--->BN_MP_ABS_C +| | +--->BN_MP_MUL_2D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_2D_C +| | | +--->BN_MP_MOD_2D_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_MULTI_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_INIT_COPY_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_RSHD_C +| | +--->BN_MP_RSHD_C +| | +--->BN_MP_MUL_D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_ADD_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_EXCH_C + + +BN_MP_INVMOD_C ++--->BN_FAST_MP_INVMOD_C +| +--->BN_MP_INIT_MULTI_C +| | +--->BN_MP_INIT_C +| | +--->BN_MP_CLEAR_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_ABS_C +| +--->BN_MP_SET_C +| | +--->BN_MP_ZERO_C +| +--->BN_MP_DIV_2_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_SUB_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_CMP_C +| | +--->BN_MP_CMP_MAG_C +| +--->BN_MP_CMP_D_C +| +--->BN_MP_ADD_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_EXCH_C +| +--->BN_MP_CLEAR_MULTI_C +| | +--->BN_MP_CLEAR_C ++--->BN_MP_INVMOD_SLOW_C +| +--->BN_MP_INIT_MULTI_C +| | +--->BN_MP_INIT_C +| | +--->BN_MP_CLEAR_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_SET_C +| | +--->BN_MP_ZERO_C +| +--->BN_MP_DIV_2_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_ADD_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_SUB_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_CMP_C +| | +--->BN_MP_CMP_MAG_C +| +--->BN_MP_CMP_D_C +| +--->BN_MP_CMP_MAG_C +| +--->BN_MP_EXCH_C +| +--->BN_MP_CLEAR_MULTI_C +| | +--->BN_MP_CLEAR_C + + +BN_MP_AND_C ++--->BN_MP_INIT_COPY_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C ++--->BN_MP_CLAMP_C ++--->BN_MP_EXCH_C ++--->BN_MP_CLEAR_C + + +BN_MP_MUL_D_C ++--->BN_MP_GROW_C ++--->BN_MP_CLAMP_C + + +BN_FAST_MP_INVMOD_C ++--->BN_MP_INIT_MULTI_C +| +--->BN_MP_INIT_C +| +--->BN_MP_CLEAR_C ++--->BN_MP_COPY_C +| +--->BN_MP_GROW_C ++--->BN_MP_ABS_C ++--->BN_MP_SET_C +| +--->BN_MP_ZERO_C ++--->BN_MP_DIV_2_C +| +--->BN_MP_GROW_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_SUB_C +| +--->BN_S_MP_ADD_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CMP_MAG_C +| +--->BN_S_MP_SUB_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C ++--->BN_MP_CMP_C +| +--->BN_MP_CMP_MAG_C ++--->BN_MP_CMP_D_C ++--->BN_MP_ADD_C +| +--->BN_S_MP_ADD_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CMP_MAG_C +| +--->BN_S_MP_SUB_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C ++--->BN_MP_EXCH_C ++--->BN_MP_CLEAR_MULTI_C +| +--->BN_MP_CLEAR_C + + +BN_MP_FWRITE_C ++--->BN_MP_RADIX_SIZE_C +| +--->BN_MP_COUNT_BITS_C +| +--->BN_MP_INIT_COPY_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| +--->BN_MP_DIV_D_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_DIV_2D_C +| | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_MOD_2D_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CLEAR_C +| | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | +--->BN_MP_DIV_3_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_C +| +--->BN_MP_CLEAR_C ++--->BN_MP_TORADIX_C +| +--->BN_MP_INIT_COPY_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| +--->BN_MP_DIV_D_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_DIV_2D_C +| | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_MOD_2D_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CLEAR_C +| | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | +--->BN_MP_DIV_3_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_C +| +--->BN_MP_CLEAR_C + + +BN_S_MP_SQR_C ++--->BN_MP_INIT_SIZE_C +| +--->BN_MP_INIT_C ++--->BN_MP_CLAMP_C ++--->BN_MP_EXCH_C ++--->BN_MP_CLEAR_C + + +BN_MP_N_ROOT_C ++--->BN_MP_INIT_C ++--->BN_MP_SET_C +| +--->BN_MP_ZERO_C ++--->BN_MP_COPY_C +| +--->BN_MP_GROW_C ++--->BN_MP_EXPT_D_C +| +--->BN_MP_INIT_COPY_C +| +--->BN_MP_SQR_C +| | +--->BN_MP_TOOM_SQR_C +| | | +--->BN_MP_INIT_MULTI_C +| | | | +--->BN_MP_CLEAR_C +| | | +--->BN_MP_MOD_2D_C +| | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_MUL_2_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_SUB_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_DIV_2_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_MUL_2D_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_MUL_D_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_DIV_3_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_CLEAR_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLEAR_MULTI_C +| | | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_KARATSUBA_SQR_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_SUB_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_FAST_S_MP_SQR_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_S_MP_SQR_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_CLEAR_C +| +--->BN_MP_CLEAR_C +| +--->BN_MP_MUL_C +| | +--->BN_MP_TOOM_MUL_C +| | | +--->BN_MP_INIT_MULTI_C +| | | +--->BN_MP_MOD_2D_C +| | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_MUL_2_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_SUB_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_DIV_2_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_MUL_2D_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_MUL_D_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_DIV_3_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLEAR_MULTI_C +| | +--->BN_MP_KARATSUBA_MUL_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_SUB_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_ZERO_C +| | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_S_MP_MUL_DIGS_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C ++--->BN_MP_MUL_C +| +--->BN_MP_TOOM_MUL_C +| | +--->BN_MP_INIT_MULTI_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_MOD_2D_C +| | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_RSHD_C +| | | +--->BN_MP_ZERO_C +| | +--->BN_MP_MUL_2_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_2_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MUL_2D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MUL_D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_3_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_CLEAR_MULTI_C +| | | +--->BN_MP_CLEAR_C +| +--->BN_MP_KARATSUBA_MUL_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_ZERO_C +| | +--->BN_MP_CLEAR_C +| +--->BN_FAST_S_MP_MUL_DIGS_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_S_MP_MUL_DIGS_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_C ++--->BN_MP_SUB_C +| +--->BN_S_MP_ADD_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CMP_MAG_C +| +--->BN_S_MP_SUB_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C ++--->BN_MP_MUL_D_C +| +--->BN_MP_GROW_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_DIV_C +| +--->BN_MP_CMP_MAG_C +| +--->BN_MP_ZERO_C +| +--->BN_MP_INIT_MULTI_C +| | +--->BN_MP_CLEAR_C +| +--->BN_MP_COUNT_BITS_C +| +--->BN_MP_ABS_C +| +--->BN_MP_MUL_2D_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_RSHD_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CMP_C +| +--->BN_MP_ADD_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_DIV_2D_C +| | +--->BN_MP_MOD_2D_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CLEAR_C +| | +--->BN_MP_RSHD_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| +--->BN_MP_EXCH_C +| +--->BN_MP_CLEAR_MULTI_C +| | +--->BN_MP_CLEAR_C +| +--->BN_MP_INIT_SIZE_C +| +--->BN_MP_INIT_COPY_C +| +--->BN_MP_LSHD_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_RSHD_C +| +--->BN_MP_RSHD_C +| +--->BN_MP_CLAMP_C +| +--->BN_MP_CLEAR_C ++--->BN_MP_CMP_C +| +--->BN_MP_CMP_MAG_C ++--->BN_MP_SUB_D_C +| +--->BN_MP_GROW_C +| +--->BN_MP_ADD_D_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_EXCH_C ++--->BN_MP_CLEAR_C + + +BN_MP_PRIME_RABIN_MILLER_TRIALS_C + + +BN_MP_RADIX_SIZE_C ++--->BN_MP_COUNT_BITS_C ++--->BN_MP_INIT_COPY_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C ++--->BN_MP_DIV_D_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_DIV_2D_C +| | +--->BN_MP_ZERO_C +| | +--->BN_MP_MOD_2D_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CLEAR_C +| | +--->BN_MP_RSHD_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| +--->BN_MP_DIV_3_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_C +| +--->BN_MP_INIT_SIZE_C +| +--->BN_MP_CLAMP_C +| +--->BN_MP_EXCH_C +| +--->BN_MP_CLEAR_C ++--->BN_MP_CLEAR_C + + +BN_MP_READ_SIGNED_BIN_C ++--->BN_MP_READ_UNSIGNED_BIN_C +| +--->BN_MP_GROW_C +| +--->BN_MP_ZERO_C +| +--->BN_MP_MUL_2D_C +| | +--->BN_MP_COPY_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_RSHD_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CLAMP_C + + +BN_MP_PRIME_RANDOM_EX_C ++--->BN_MP_READ_UNSIGNED_BIN_C +| +--->BN_MP_GROW_C +| +--->BN_MP_ZERO_C +| +--->BN_MP_MUL_2D_C +| | +--->BN_MP_COPY_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_RSHD_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_PRIME_IS_PRIME_C +| +--->BN_MP_CMP_D_C +| +--->BN_MP_PRIME_IS_DIVISIBLE_C +| | +--->BN_MP_MOD_D_C +| | | +--->BN_MP_DIV_D_C +| | | | +--->BN_MP_COPY_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_DIV_2D_C +| | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_INIT_C +| | | | | +--->BN_MP_MOD_2D_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CLEAR_C +| | | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_DIV_3_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_INIT_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_MP_CLEAR_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_INIT_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_CLEAR_C +| +--->BN_MP_INIT_C +| +--->BN_MP_SET_C +| | +--->BN_MP_ZERO_C +| +--->BN_MP_PRIME_MILLER_RABIN_C +| | +--->BN_MP_INIT_COPY_C +| | | +--->BN_MP_COPY_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_MP_SUB_D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_ADD_D_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CNT_LSB_C +| | +--->BN_MP_DIV_2D_C +| | | +--->BN_MP_COPY_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_MOD_2D_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CLEAR_C +| | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | +--->BN_MP_EXPTMOD_C +| | | +--->BN_MP_INVMOD_C +| | | | +--->BN_FAST_MP_INVMOD_C +| | | | | +--->BN_MP_INIT_MULTI_C +| | | | | | +--->BN_MP_CLEAR_C +| | | | | +--->BN_MP_COPY_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_ABS_C +| | | | | +--->BN_MP_DIV_2_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_MP_CLEAR_MULTI_C +| | | | | | +--->BN_MP_CLEAR_C +| | | | +--->BN_MP_INVMOD_SLOW_C +| | | | | +--->BN_MP_INIT_MULTI_C +| | | | | | +--->BN_MP_CLEAR_C +| | | | | +--->BN_MP_COPY_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_DIV_2_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_MP_CLEAR_MULTI_C +| | | | | | +--->BN_MP_CLEAR_C +| | | +--->BN_MP_CLEAR_C +| | | +--->BN_MP_ABS_C +| | | | +--->BN_MP_COPY_C +| | | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLEAR_MULTI_C +| | | +--->BN_MP_DR_IS_MODULUS_C +| | | +--->BN_MP_REDUCE_IS_2K_C +| | | | +--->BN_MP_REDUCE_2K_C +| | | | | +--->BN_MP_COUNT_BITS_C +| | | | | +--->BN_MP_MUL_D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_COUNT_BITS_C +| | | +--->BN_MP_EXPTMOD_FAST_C +| | | | +--->BN_MP_COUNT_BITS_C +| | | | +--->BN_MP_MONTGOMERY_SETUP_C +| | | | +--->BN_FAST_MP_MONTGOMERY_REDUCE_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_MONTGOMERY_REDUCE_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_DR_SETUP_C +| | | | +--->BN_MP_DR_REDUCE_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_REDUCE_2K_SETUP_C +| | | | | +--->BN_MP_2EXPT_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_REDUCE_2K_C +| | | | | +--->BN_MP_MUL_D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MONTGOMERY_CALC_NORMALIZATION_C +| | | | | +--->BN_MP_2EXPT_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_MUL_2_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MULMOD_C +| | | | | +--->BN_MP_MUL_C +| | | | | | +--->BN_MP_TOOM_MUL_C +| | | | | | | +--->BN_MP_INIT_MULTI_C +| | | | | | | +--->BN_MP_MOD_2D_C +| | | | | | | | +--->BN_MP_ZERO_C +| | | | | | | | +--->BN_MP_COPY_C +| | | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_COPY_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_RSHD_C +| | | | | | | | +--->BN_MP_ZERO_C +| | | | | | | +--->BN_MP_MUL_2_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_ADD_C +| | | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_SUB_C +| | | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_DIV_2_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_MUL_2D_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_LSHD_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_MUL_D_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_DIV_3_C +| | | | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | | +--->BN_MP_EXCH_C +| | | | | | | +--->BN_MP_LSHD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_KARATSUBA_MUL_C +| | | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_SUB_C +| | | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_ADD_C +| | | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_LSHD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_RSHD_C +| | | | | | | | | +--->BN_MP_ZERO_C +| | | | | | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_S_MP_MUL_DIGS_C +| | | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_MP_MOD_C +| | | | | | +--->BN_MP_DIV_C +| | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | +--->BN_MP_COPY_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_ZERO_C +| | | | | | | +--->BN_MP_INIT_MULTI_C +| | | | | | | +--->BN_MP_MUL_2D_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_LSHD_C +| | | | | | | | | +--->BN_MP_RSHD_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_CMP_C +| | | | | | | +--->BN_MP_SUB_C +| | | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_ADD_C +| | | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_EXCH_C +| | | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | | +--->BN_MP_LSHD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_RSHD_C +| | | | | | | +--->BN_MP_RSHD_C +| | | | | | | +--->BN_MP_MUL_D_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_ADD_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_MOD_C +| | | | | +--->BN_MP_DIV_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_MP_COPY_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | | +--->BN_MP_INIT_MULTI_C +| | | | | | +--->BN_MP_MUL_2D_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_LSHD_C +| | | | | | | | +--->BN_MP_RSHD_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_C +| | | | | | +--->BN_MP_SUB_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_ADD_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_EXCH_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_MUL_D_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_COPY_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_SQR_C +| | | | | +--->BN_MP_TOOM_SQR_C +| | | | | | +--->BN_MP_INIT_MULTI_C +| | | | | | +--->BN_MP_MOD_2D_C +| | | | | | | +--->BN_MP_ZERO_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | | | +--->BN_MP_ZERO_C +| | | | | | +--->BN_MP_MUL_2_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_ADD_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_SUB_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_DIV_2_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_MUL_2D_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_MUL_D_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_DIV_3_C +| | | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_EXCH_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_KARATSUBA_SQR_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_SUB_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_RSHD_C +| | | | | | | | +--->BN_MP_ZERO_C +| | | | | | +--->BN_MP_ADD_C +| | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_FAST_S_MP_SQR_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_S_MP_SQR_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_MUL_C +| | | | | +--->BN_MP_TOOM_MUL_C +| | | | | | +--->BN_MP_INIT_MULTI_C +| | | | | | +--->BN_MP_MOD_2D_C +| | | | | | | +--->BN_MP_ZERO_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | | | +--->BN_MP_ZERO_C +| | | | | | +--->BN_MP_MUL_2_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_ADD_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_SUB_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_DIV_2_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_MUL_2D_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_MUL_D_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_DIV_3_C +| | | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_EXCH_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_KARATSUBA_MUL_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_SUB_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_ADD_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_RSHD_C +| | | | | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_S_MP_MUL_DIGS_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_EXCH_C +| | | +--->BN_S_MP_EXPTMOD_C +| | | | +--->BN_MP_COUNT_BITS_C +| | | | +--->BN_MP_REDUCE_SETUP_C +| | | | | +--->BN_MP_2EXPT_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_DIV_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_MP_COPY_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | | +--->BN_MP_INIT_MULTI_C +| | | | | | +--->BN_MP_MUL_2D_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_LSHD_C +| | | | | | | | +--->BN_MP_RSHD_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_C +| | | | | | +--->BN_MP_SUB_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_ADD_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_EXCH_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_MUL_D_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MOD_C +| | | | | +--->BN_MP_DIV_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_MP_COPY_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | | +--->BN_MP_INIT_MULTI_C +| | | | | | +--->BN_MP_MUL_2D_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_LSHD_C +| | | | | | | | +--->BN_MP_RSHD_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_C +| | | | | | +--->BN_MP_SUB_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_ADD_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_EXCH_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_MUL_D_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_COPY_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_SQR_C +| | | | | +--->BN_MP_TOOM_SQR_C +| | | | | | +--->BN_MP_INIT_MULTI_C +| | | | | | +--->BN_MP_MOD_2D_C +| | | | | | | +--->BN_MP_ZERO_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | | | +--->BN_MP_ZERO_C +| | | | | | +--->BN_MP_MUL_2_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_ADD_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_SUB_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_DIV_2_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_MUL_2D_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_MUL_D_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_DIV_3_C +| | | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_EXCH_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_KARATSUBA_SQR_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_SUB_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_RSHD_C +| | | | | | | | +--->BN_MP_ZERO_C +| | | | | | +--->BN_MP_ADD_C +| | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_FAST_S_MP_SQR_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_S_MP_SQR_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_REDUCE_C +| | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_MUL_C +| | | | | | +--->BN_MP_TOOM_MUL_C +| | | | | | | +--->BN_MP_INIT_MULTI_C +| | | | | | | +--->BN_MP_MOD_2D_C +| | | | | | | | +--->BN_MP_ZERO_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_MUL_2_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_ADD_C +| | | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_SUB_C +| | | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_DIV_2_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_MUL_2D_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_LSHD_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_MUL_D_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_DIV_3_C +| | | | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | | +--->BN_MP_EXCH_C +| | | | | | | +--->BN_MP_LSHD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_KARATSUBA_MUL_C +| | | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_SUB_C +| | | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_ADD_C +| | | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_LSHD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_S_MP_MUL_DIGS_C +| | | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_S_MP_MUL_HIGH_DIGS_C +| | | | | | +--->BN_FAST_S_MP_MUL_HIGH_DIGS_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_FAST_S_MP_MUL_HIGH_DIGS_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_MOD_2D_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_S_MP_MUL_DIGS_C +| | | | | | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MUL_C +| | | | | +--->BN_MP_TOOM_MUL_C +| | | | | | +--->BN_MP_INIT_MULTI_C +| | | | | | +--->BN_MP_MOD_2D_C +| | | | | | | +--->BN_MP_ZERO_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | | | +--->BN_MP_ZERO_C +| | | | | | +--->BN_MP_MUL_2_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_ADD_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_SUB_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_DIV_2_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_MUL_2D_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_MUL_D_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_DIV_3_C +| | | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_EXCH_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_KARATSUBA_MUL_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_SUB_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_ADD_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_RSHD_C +| | | | | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_S_MP_MUL_DIGS_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_EXCH_C +| | +--->BN_MP_CMP_C +| | | +--->BN_MP_CMP_MAG_C +| | +--->BN_MP_SQRMOD_C +| | | +--->BN_MP_SQR_C +| | | | +--->BN_MP_TOOM_SQR_C +| | | | | +--->BN_MP_INIT_MULTI_C +| | | | | | +--->BN_MP_CLEAR_C +| | | | | +--->BN_MP_MOD_2D_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | | +--->BN_MP_COPY_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_COPY_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_MUL_2_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_DIV_2_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_MUL_2D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_MUL_D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_DIV_3_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_EXCH_C +| | | | | | +--->BN_MP_CLEAR_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLEAR_MULTI_C +| | | | | | +--->BN_MP_CLEAR_C +| | | | +--->BN_MP_KARATSUBA_SQR_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLEAR_C +| | | | +--->BN_FAST_S_MP_SQR_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_S_MP_SQR_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_MP_CLEAR_C +| | | +--->BN_MP_CLEAR_C +| | | +--->BN_MP_MOD_C +| | | | +--->BN_MP_DIV_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_MP_COPY_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_INIT_MULTI_C +| | | | | +--->BN_MP_COUNT_BITS_C +| | | | | +--->BN_MP_ABS_C +| | | | | +--->BN_MP_MUL_2D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_MP_CLEAR_MULTI_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_MUL_D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_C +| +--->BN_MP_CLEAR_C ++--->BN_MP_SUB_D_C +| +--->BN_MP_GROW_C +| +--->BN_MP_ADD_D_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_DIV_2_C +| +--->BN_MP_GROW_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_MUL_2_C +| +--->BN_MP_GROW_C ++--->BN_MP_ADD_D_C +| +--->BN_MP_GROW_C +| +--->BN_MP_CLAMP_C + + +BN_MP_KARATSUBA_SQR_C ++--->BN_MP_INIT_SIZE_C +| +--->BN_MP_INIT_C ++--->BN_MP_CLAMP_C ++--->BN_MP_SQR_C +| +--->BN_MP_TOOM_SQR_C +| | +--->BN_MP_INIT_MULTI_C +| | | +--->BN_MP_INIT_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_MOD_2D_C +| | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_COPY_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_RSHD_C +| | | +--->BN_MP_ZERO_C +| | +--->BN_MP_MUL_2_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_MP_DIV_2_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_MUL_2D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | +--->BN_MP_MUL_D_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_DIV_3_C +| | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_CLEAR_MULTI_C +| | | +--->BN_MP_CLEAR_C +| +--->BN_FAST_S_MP_SQR_C +| | +--->BN_MP_GROW_C +| +--->BN_S_MP_SQR_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_C ++--->BN_MP_SUB_C +| +--->BN_S_MP_ADD_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_CMP_MAG_C +| +--->BN_S_MP_SUB_C +| | +--->BN_MP_GROW_C ++--->BN_S_MP_ADD_C +| +--->BN_MP_GROW_C ++--->BN_MP_LSHD_C +| +--->BN_MP_GROW_C +| +--->BN_MP_RSHD_C +| | +--->BN_MP_ZERO_C ++--->BN_MP_ADD_C +| +--->BN_MP_CMP_MAG_C +| +--->BN_S_MP_SUB_C +| | +--->BN_MP_GROW_C ++--->BN_MP_CLEAR_C + + +BN_MP_INIT_COPY_C ++--->BN_MP_COPY_C +| +--->BN_MP_GROW_C + + +BN_MP_CLAMP_C + + +BN_MP_TOOM_SQR_C ++--->BN_MP_INIT_MULTI_C +| +--->BN_MP_INIT_C +| +--->BN_MP_CLEAR_C ++--->BN_MP_MOD_2D_C +| +--->BN_MP_ZERO_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_COPY_C +| +--->BN_MP_GROW_C ++--->BN_MP_RSHD_C +| +--->BN_MP_ZERO_C ++--->BN_MP_SQR_C +| +--->BN_MP_KARATSUBA_SQR_C +| | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_INIT_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_ADD_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_MP_CLEAR_C +| +--->BN_FAST_S_MP_SQR_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_S_MP_SQR_C +| | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_INIT_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_C ++--->BN_MP_MUL_2_C +| +--->BN_MP_GROW_C ++--->BN_MP_ADD_C +| +--->BN_S_MP_ADD_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CMP_MAG_C +| +--->BN_S_MP_SUB_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C ++--->BN_MP_SUB_C +| +--->BN_S_MP_ADD_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CMP_MAG_C +| +--->BN_S_MP_SUB_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C ++--->BN_MP_DIV_2_C +| +--->BN_MP_GROW_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_MUL_2D_C +| +--->BN_MP_GROW_C +| +--->BN_MP_LSHD_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_MUL_D_C +| +--->BN_MP_GROW_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_DIV_3_C +| +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_INIT_C +| +--->BN_MP_CLAMP_C +| +--->BN_MP_EXCH_C +| +--->BN_MP_CLEAR_C ++--->BN_MP_LSHD_C +| +--->BN_MP_GROW_C ++--->BN_MP_CLEAR_MULTI_C +| +--->BN_MP_CLEAR_C + + +BN_MP_MOD_C ++--->BN_MP_INIT_C ++--->BN_MP_DIV_C +| +--->BN_MP_CMP_MAG_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_ZERO_C +| +--->BN_MP_INIT_MULTI_C +| | +--->BN_MP_CLEAR_C +| +--->BN_MP_SET_C +| +--->BN_MP_COUNT_BITS_C +| +--->BN_MP_ABS_C +| +--->BN_MP_MUL_2D_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_RSHD_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CMP_C +| +--->BN_MP_SUB_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_ADD_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_DIV_2D_C +| | +--->BN_MP_MOD_2D_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CLEAR_C +| | +--->BN_MP_RSHD_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| +--->BN_MP_EXCH_C +| +--->BN_MP_CLEAR_MULTI_C +| | +--->BN_MP_CLEAR_C +| +--->BN_MP_INIT_SIZE_C +| +--->BN_MP_INIT_COPY_C +| +--->BN_MP_LSHD_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_RSHD_C +| +--->BN_MP_RSHD_C +| +--->BN_MP_MUL_D_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CLAMP_C +| +--->BN_MP_CLEAR_C ++--->BN_MP_CLEAR_C ++--->BN_MP_ADD_C +| +--->BN_S_MP_ADD_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CMP_MAG_C +| +--->BN_S_MP_SUB_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C ++--->BN_MP_EXCH_C + + +BN_MP_INIT_C + + +BN_MP_TOOM_MUL_C ++--->BN_MP_INIT_MULTI_C +| +--->BN_MP_INIT_C +| +--->BN_MP_CLEAR_C ++--->BN_MP_MOD_2D_C +| +--->BN_MP_ZERO_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_COPY_C +| +--->BN_MP_GROW_C ++--->BN_MP_RSHD_C +| +--->BN_MP_ZERO_C ++--->BN_MP_MUL_C +| +--->BN_MP_KARATSUBA_MUL_C +| | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_INIT_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_CLEAR_C +| +--->BN_FAST_S_MP_MUL_DIGS_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_S_MP_MUL_DIGS_C +| | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_INIT_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_C ++--->BN_MP_MUL_2_C +| +--->BN_MP_GROW_C ++--->BN_MP_ADD_C +| +--->BN_S_MP_ADD_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CMP_MAG_C +| +--->BN_S_MP_SUB_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C ++--->BN_MP_SUB_C +| +--->BN_S_MP_ADD_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CMP_MAG_C +| +--->BN_S_MP_SUB_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C ++--->BN_MP_DIV_2_C +| +--->BN_MP_GROW_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_MUL_2D_C +| +--->BN_MP_GROW_C +| +--->BN_MP_LSHD_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_MUL_D_C +| +--->BN_MP_GROW_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_DIV_3_C +| +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_INIT_C +| +--->BN_MP_CLAMP_C +| +--->BN_MP_EXCH_C +| +--->BN_MP_CLEAR_C ++--->BN_MP_LSHD_C +| +--->BN_MP_GROW_C ++--->BN_MP_CLEAR_MULTI_C +| +--->BN_MP_CLEAR_C + + +BN_MP_PRIME_IS_PRIME_C ++--->BN_MP_CMP_D_C ++--->BN_MP_PRIME_IS_DIVISIBLE_C +| +--->BN_MP_MOD_D_C +| | +--->BN_MP_DIV_D_C +| | | +--->BN_MP_COPY_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_DIV_2D_C +| | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_INIT_C +| | | | +--->BN_MP_MOD_2D_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CLEAR_C +| | | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_DIV_3_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_INIT_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_CLEAR_C +| | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_INIT_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_CLEAR_C ++--->BN_MP_INIT_C ++--->BN_MP_SET_C +| +--->BN_MP_ZERO_C ++--->BN_MP_PRIME_MILLER_RABIN_C +| +--->BN_MP_INIT_COPY_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| +--->BN_MP_SUB_D_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_ADD_D_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CNT_LSB_C +| +--->BN_MP_DIV_2D_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_ZERO_C +| | +--->BN_MP_MOD_2D_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CLEAR_C +| | +--->BN_MP_RSHD_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| +--->BN_MP_EXPTMOD_C +| | +--->BN_MP_INVMOD_C +| | | +--->BN_FAST_MP_INVMOD_C +| | | | +--->BN_MP_INIT_MULTI_C +| | | | | +--->BN_MP_CLEAR_C +| | | | +--->BN_MP_COPY_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ABS_C +| | | | +--->BN_MP_DIV_2_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_CLEAR_MULTI_C +| | | | | +--->BN_MP_CLEAR_C +| | | +--->BN_MP_INVMOD_SLOW_C +| | | | +--->BN_MP_INIT_MULTI_C +| | | | | +--->BN_MP_CLEAR_C +| | | | +--->BN_MP_COPY_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_DIV_2_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_CLEAR_MULTI_C +| | | | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_CLEAR_C +| | +--->BN_MP_ABS_C +| | | +--->BN_MP_COPY_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_MP_CLEAR_MULTI_C +| | +--->BN_MP_DR_IS_MODULUS_C +| | +--->BN_MP_REDUCE_IS_2K_C +| | | +--->BN_MP_REDUCE_2K_C +| | | | +--->BN_MP_COUNT_BITS_C +| | | | +--->BN_MP_MUL_D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_COUNT_BITS_C +| | +--->BN_MP_EXPTMOD_FAST_C +| | | +--->BN_MP_COUNT_BITS_C +| | | +--->BN_MP_MONTGOMERY_SETUP_C +| | | +--->BN_FAST_MP_MONTGOMERY_REDUCE_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_MONTGOMERY_REDUCE_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_DR_SETUP_C +| | | +--->BN_MP_DR_REDUCE_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_REDUCE_2K_SETUP_C +| | | | +--->BN_MP_2EXPT_C +| | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_REDUCE_2K_C +| | | | +--->BN_MP_MUL_D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_MONTGOMERY_CALC_NORMALIZATION_C +| | | | +--->BN_MP_2EXPT_C +| | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_MUL_2_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_MULMOD_C +| | | | +--->BN_MP_MUL_C +| | | | | +--->BN_MP_TOOM_MUL_C +| | | | | | +--->BN_MP_INIT_MULTI_C +| | | | | | +--->BN_MP_MOD_2D_C +| | | | | | | +--->BN_MP_ZERO_C +| | | | | | | +--->BN_MP_COPY_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_COPY_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | | | +--->BN_MP_ZERO_C +| | | | | | +--->BN_MP_MUL_2_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_ADD_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_SUB_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_DIV_2_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_MUL_2D_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_MUL_D_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_DIV_3_C +| | | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_EXCH_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_KARATSUBA_MUL_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_SUB_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_ADD_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_RSHD_C +| | | | | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_S_MP_MUL_DIGS_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_MOD_C +| | | | | +--->BN_MP_DIV_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_MP_COPY_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | | +--->BN_MP_INIT_MULTI_C +| | | | | | +--->BN_MP_MUL_2D_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_LSHD_C +| | | | | | | | +--->BN_MP_RSHD_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_C +| | | | | | +--->BN_MP_SUB_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_ADD_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_EXCH_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_MUL_D_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_MOD_C +| | | | +--->BN_MP_DIV_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_MP_COPY_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_INIT_MULTI_C +| | | | | +--->BN_MP_MUL_2D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_MUL_D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_COPY_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_SQR_C +| | | | +--->BN_MP_TOOM_SQR_C +| | | | | +--->BN_MP_INIT_MULTI_C +| | | | | +--->BN_MP_MOD_2D_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_MUL_2_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_DIV_2_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_MUL_2D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_MUL_D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_DIV_3_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_KARATSUBA_SQR_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_FAST_S_MP_SQR_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_S_MP_SQR_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_MUL_C +| | | | +--->BN_MP_TOOM_MUL_C +| | | | | +--->BN_MP_INIT_MULTI_C +| | | | | +--->BN_MP_MOD_2D_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_MUL_2_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_DIV_2_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_MUL_2D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_MUL_D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_DIV_3_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_KARATSUBA_MUL_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | | | +--->BN_MP_ZERO_C +| | | | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_S_MP_MUL_DIGS_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_EXCH_C +| | +--->BN_S_MP_EXPTMOD_C +| | | +--->BN_MP_COUNT_BITS_C +| | | +--->BN_MP_REDUCE_SETUP_C +| | | | +--->BN_MP_2EXPT_C +| | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_DIV_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_MP_COPY_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_INIT_MULTI_C +| | | | | +--->BN_MP_MUL_2D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_MUL_D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_MOD_C +| | | | +--->BN_MP_DIV_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_MP_COPY_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_INIT_MULTI_C +| | | | | +--->BN_MP_MUL_2D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_MUL_D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_COPY_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_SQR_C +| | | | +--->BN_MP_TOOM_SQR_C +| | | | | +--->BN_MP_INIT_MULTI_C +| | | | | +--->BN_MP_MOD_2D_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_MUL_2_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_DIV_2_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_MUL_2D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_MUL_D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_DIV_3_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_KARATSUBA_SQR_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_FAST_S_MP_SQR_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_S_MP_SQR_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_REDUCE_C +| | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_MUL_C +| | | | | +--->BN_MP_TOOM_MUL_C +| | | | | | +--->BN_MP_INIT_MULTI_C +| | | | | | +--->BN_MP_MOD_2D_C +| | | | | | | +--->BN_MP_ZERO_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_MUL_2_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_ADD_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_SUB_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_DIV_2_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_MUL_2D_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_MUL_D_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_DIV_3_C +| | | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_EXCH_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_KARATSUBA_MUL_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_SUB_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_ADD_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_S_MP_MUL_DIGS_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_EXCH_C +| | | | +--->BN_S_MP_MUL_HIGH_DIGS_C +| | | | | +--->BN_FAST_S_MP_MUL_HIGH_DIGS_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | +--->BN_FAST_S_MP_MUL_HIGH_DIGS_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MOD_2D_C +| | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_S_MP_MUL_DIGS_C +| | | | | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_MUL_C +| | | | +--->BN_MP_TOOM_MUL_C +| | | | | +--->BN_MP_INIT_MULTI_C +| | | | | +--->BN_MP_MOD_2D_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_MUL_2_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_DIV_2_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_MUL_2D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_MUL_D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_DIV_3_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_KARATSUBA_MUL_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | | | +--->BN_MP_ZERO_C +| | | | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_S_MP_MUL_DIGS_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_EXCH_C +| +--->BN_MP_CMP_C +| | +--->BN_MP_CMP_MAG_C +| +--->BN_MP_SQRMOD_C +| | +--->BN_MP_SQR_C +| | | +--->BN_MP_TOOM_SQR_C +| | | | +--->BN_MP_INIT_MULTI_C +| | | | | +--->BN_MP_CLEAR_C +| | | | +--->BN_MP_MOD_2D_C +| | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_COPY_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_COPY_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_MUL_2_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_DIV_2_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MUL_2D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MUL_D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_DIV_3_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_MP_CLEAR_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLEAR_MULTI_C +| | | | | +--->BN_MP_CLEAR_C +| | | +--->BN_MP_KARATSUBA_SQR_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLEAR_C +| | | +--->BN_FAST_S_MP_SQR_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_SQR_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_CLEAR_C +| | +--->BN_MP_MOD_C +| | | +--->BN_MP_DIV_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_MP_COPY_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_INIT_MULTI_C +| | | | +--->BN_MP_COUNT_BITS_C +| | | | +--->BN_MP_ABS_C +| | | | +--->BN_MP_MUL_2D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_CLEAR_MULTI_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_MUL_D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| +--->BN_MP_CLEAR_C ++--->BN_MP_CLEAR_C + + +BN_MP_COPY_C ++--->BN_MP_GROW_C + + +BN_S_MP_SUB_C ++--->BN_MP_GROW_C ++--->BN_MP_CLAMP_C + + +BN_MP_READ_UNSIGNED_BIN_C ++--->BN_MP_GROW_C ++--->BN_MP_ZERO_C ++--->BN_MP_MUL_2D_C +| +--->BN_MP_COPY_C +| +--->BN_MP_LSHD_C +| | +--->BN_MP_RSHD_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_CLAMP_C + + +BN_MP_EXPTMOD_FAST_C ++--->BN_MP_COUNT_BITS_C ++--->BN_MP_INIT_C ++--->BN_MP_CLEAR_C ++--->BN_MP_MONTGOMERY_SETUP_C ++--->BN_FAST_MP_MONTGOMERY_REDUCE_C +| +--->BN_MP_GROW_C +| +--->BN_MP_RSHD_C +| | +--->BN_MP_ZERO_C +| +--->BN_MP_CLAMP_C +| +--->BN_MP_CMP_MAG_C +| +--->BN_S_MP_SUB_C ++--->BN_MP_MONTGOMERY_REDUCE_C +| +--->BN_MP_GROW_C +| +--->BN_MP_CLAMP_C +| +--->BN_MP_RSHD_C +| | +--->BN_MP_ZERO_C +| +--->BN_MP_CMP_MAG_C +| +--->BN_S_MP_SUB_C ++--->BN_MP_DR_SETUP_C ++--->BN_MP_DR_REDUCE_C +| +--->BN_MP_GROW_C +| +--->BN_MP_CLAMP_C +| +--->BN_MP_CMP_MAG_C +| +--->BN_S_MP_SUB_C ++--->BN_MP_REDUCE_2K_SETUP_C +| +--->BN_MP_2EXPT_C +| | +--->BN_MP_ZERO_C +| | +--->BN_MP_GROW_C +| +--->BN_S_MP_SUB_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C ++--->BN_MP_REDUCE_2K_C +| +--->BN_MP_DIV_2D_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_ZERO_C +| | +--->BN_MP_MOD_2D_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_RSHD_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| +--->BN_MP_MUL_D_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_S_MP_ADD_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CMP_MAG_C +| +--->BN_S_MP_SUB_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C ++--->BN_MP_MONTGOMERY_CALC_NORMALIZATION_C +| +--->BN_MP_2EXPT_C +| | +--->BN_MP_ZERO_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_SET_C +| | +--->BN_MP_ZERO_C +| +--->BN_MP_MUL_2_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_CMP_MAG_C +| +--->BN_S_MP_SUB_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C ++--->BN_MP_MULMOD_C +| +--->BN_MP_MUL_C +| | +--->BN_MP_TOOM_MUL_C +| | | +--->BN_MP_INIT_MULTI_C +| | | +--->BN_MP_MOD_2D_C +| | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_COPY_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_COPY_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_MUL_2_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_SUB_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_DIV_2_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_MUL_2D_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_MUL_D_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_DIV_3_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLEAR_MULTI_C +| | +--->BN_MP_KARATSUBA_MUL_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_SUB_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_ZERO_C +| | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_S_MP_MUL_DIGS_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| +--->BN_MP_MOD_C +| | +--->BN_MP_DIV_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_MP_COPY_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_INIT_MULTI_C +| | | +--->BN_MP_SET_C +| | | +--->BN_MP_ABS_C +| | | +--->BN_MP_MUL_2D_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_C +| | | +--->BN_MP_SUB_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_DIV_2D_C +| | | | +--->BN_MP_MOD_2D_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_CLEAR_MULTI_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_INIT_COPY_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_MUL_D_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C ++--->BN_MP_SET_C +| +--->BN_MP_ZERO_C ++--->BN_MP_MOD_C +| +--->BN_MP_DIV_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_ZERO_C +| | +--->BN_MP_INIT_MULTI_C +| | +--->BN_MP_ABS_C +| | +--->BN_MP_MUL_2D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_2D_C +| | | +--->BN_MP_MOD_2D_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_MULTI_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_INIT_COPY_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_RSHD_C +| | +--->BN_MP_RSHD_C +| | +--->BN_MP_MUL_D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_ADD_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_EXCH_C ++--->BN_MP_COPY_C +| +--->BN_MP_GROW_C ++--->BN_MP_SQR_C +| +--->BN_MP_TOOM_SQR_C +| | +--->BN_MP_INIT_MULTI_C +| | +--->BN_MP_MOD_2D_C +| | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_RSHD_C +| | | +--->BN_MP_ZERO_C +| | +--->BN_MP_MUL_2_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_2_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MUL_2D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MUL_D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_3_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_CLEAR_MULTI_C +| +--->BN_MP_KARATSUBA_SQR_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_ZERO_C +| | +--->BN_MP_ADD_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| +--->BN_FAST_S_MP_SQR_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_S_MP_SQR_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C ++--->BN_MP_MUL_C +| +--->BN_MP_TOOM_MUL_C +| | +--->BN_MP_INIT_MULTI_C +| | +--->BN_MP_MOD_2D_C +| | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_RSHD_C +| | | +--->BN_MP_ZERO_C +| | +--->BN_MP_MUL_2_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_2_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MUL_2D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MUL_D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_3_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_CLEAR_MULTI_C +| +--->BN_MP_KARATSUBA_MUL_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_ZERO_C +| +--->BN_FAST_S_MP_MUL_DIGS_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_S_MP_MUL_DIGS_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C ++--->BN_MP_EXCH_C + + +BN_MP_TO_UNSIGNED_BIN_C ++--->BN_MP_INIT_COPY_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C ++--->BN_MP_DIV_2D_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_ZERO_C +| +--->BN_MP_MOD_2D_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CLEAR_C +| +--->BN_MP_RSHD_C +| +--->BN_MP_CLAMP_C +| +--->BN_MP_EXCH_C ++--->BN_MP_CLEAR_C + + +BN_MP_SET_INT_C ++--->BN_MP_ZERO_C ++--->BN_MP_MUL_2D_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_GROW_C +| +--->BN_MP_LSHD_C +| | +--->BN_MP_RSHD_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_CLAMP_C + + +BN_MP_MOD_D_C ++--->BN_MP_DIV_D_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_DIV_2D_C +| | +--->BN_MP_ZERO_C +| | +--->BN_MP_INIT_C +| | +--->BN_MP_MOD_2D_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CLEAR_C +| | +--->BN_MP_RSHD_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| +--->BN_MP_DIV_3_C +| | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_INIT_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_C +| +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_INIT_C +| +--->BN_MP_CLAMP_C +| +--->BN_MP_EXCH_C +| +--->BN_MP_CLEAR_C + + +BN_MP_SQR_C ++--->BN_MP_TOOM_SQR_C +| +--->BN_MP_INIT_MULTI_C +| | +--->BN_MP_INIT_C +| | +--->BN_MP_CLEAR_C +| +--->BN_MP_MOD_2D_C +| | +--->BN_MP_ZERO_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_RSHD_C +| | +--->BN_MP_ZERO_C +| +--->BN_MP_MUL_2_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_ADD_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_SUB_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_DIV_2_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_MUL_2D_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_LSHD_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_MUL_D_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_DIV_3_C +| | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_INIT_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_C +| +--->BN_MP_LSHD_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_CLEAR_MULTI_C +| | +--->BN_MP_CLEAR_C ++--->BN_MP_KARATSUBA_SQR_C +| +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_INIT_C +| +--->BN_MP_CLAMP_C +| +--->BN_MP_SUB_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| +--->BN_S_MP_ADD_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_LSHD_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_RSHD_C +| | | +--->BN_MP_ZERO_C +| +--->BN_MP_ADD_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| +--->BN_MP_CLEAR_C ++--->BN_FAST_S_MP_SQR_C +| +--->BN_MP_GROW_C +| +--->BN_MP_CLAMP_C ++--->BN_S_MP_SQR_C +| +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_INIT_C +| +--->BN_MP_CLAMP_C +| +--->BN_MP_EXCH_C +| +--->BN_MP_CLEAR_C + + +BN_MP_MULMOD_C ++--->BN_MP_INIT_C ++--->BN_MP_MUL_C +| +--->BN_MP_TOOM_MUL_C +| | +--->BN_MP_INIT_MULTI_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_MOD_2D_C +| | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_COPY_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_RSHD_C +| | | +--->BN_MP_ZERO_C +| | +--->BN_MP_MUL_2_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_2_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MUL_2D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MUL_D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_3_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_CLEAR_MULTI_C +| | | +--->BN_MP_CLEAR_C +| +--->BN_MP_KARATSUBA_MUL_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_ZERO_C +| | +--->BN_MP_CLEAR_C +| +--->BN_FAST_S_MP_MUL_DIGS_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_S_MP_MUL_DIGS_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_C ++--->BN_MP_CLEAR_C ++--->BN_MP_MOD_C +| +--->BN_MP_DIV_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_ZERO_C +| | +--->BN_MP_INIT_MULTI_C +| | +--->BN_MP_SET_C +| | +--->BN_MP_COUNT_BITS_C +| | +--->BN_MP_ABS_C +| | +--->BN_MP_MUL_2D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_2D_C +| | | +--->BN_MP_MOD_2D_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_MULTI_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_INIT_COPY_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_RSHD_C +| | +--->BN_MP_RSHD_C +| | +--->BN_MP_MUL_D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_ADD_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_EXCH_C + + +BN_MP_DIV_2D_C ++--->BN_MP_COPY_C +| +--->BN_MP_GROW_C ++--->BN_MP_ZERO_C ++--->BN_MP_INIT_C ++--->BN_MP_MOD_2D_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_CLEAR_C ++--->BN_MP_RSHD_C ++--->BN_MP_CLAMP_C ++--->BN_MP_EXCH_C + + +BN_S_MP_ADD_C ++--->BN_MP_GROW_C ++--->BN_MP_CLAMP_C + + +BN_FAST_S_MP_SQR_C ++--->BN_MP_GROW_C ++--->BN_MP_CLAMP_C + + +BN_S_MP_MUL_DIGS_C ++--->BN_FAST_S_MP_MUL_DIGS_C +| +--->BN_MP_GROW_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_INIT_SIZE_C +| +--->BN_MP_INIT_C ++--->BN_MP_CLAMP_C ++--->BN_MP_EXCH_C ++--->BN_MP_CLEAR_C + + +BN_MP_XOR_C ++--->BN_MP_INIT_COPY_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C ++--->BN_MP_CLAMP_C ++--->BN_MP_EXCH_C ++--->BN_MP_CLEAR_C + + +BN_MP_RADIX_SMAP_C + + +BN_MP_DR_IS_MODULUS_C + + +BN_MP_MONTGOMERY_CALC_NORMALIZATION_C ++--->BN_MP_COUNT_BITS_C ++--->BN_MP_2EXPT_C +| +--->BN_MP_ZERO_C +| +--->BN_MP_GROW_C ++--->BN_MP_SET_C +| +--->BN_MP_ZERO_C ++--->BN_MP_MUL_2_C +| +--->BN_MP_GROW_C ++--->BN_MP_CMP_MAG_C ++--->BN_S_MP_SUB_C +| +--->BN_MP_GROW_C +| +--->BN_MP_CLAMP_C + + +BN_MP_SUB_C ++--->BN_S_MP_ADD_C +| +--->BN_MP_GROW_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_CMP_MAG_C ++--->BN_S_MP_SUB_C +| +--->BN_MP_GROW_C +| +--->BN_MP_CLAMP_C + + +BN_MP_INIT_MULTI_C ++--->BN_MP_INIT_C ++--->BN_MP_CLEAR_C + + +BN_S_MP_MUL_HIGH_DIGS_C ++--->BN_FAST_S_MP_MUL_HIGH_DIGS_C +| +--->BN_MP_GROW_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_INIT_SIZE_C +| +--->BN_MP_INIT_C ++--->BN_MP_CLAMP_C ++--->BN_MP_EXCH_C ++--->BN_MP_CLEAR_C + + +BN_MP_PRIME_NEXT_PRIME_C ++--->BN_MP_CMP_D_C ++--->BN_MP_SET_C +| +--->BN_MP_ZERO_C ++--->BN_MP_SUB_D_C +| +--->BN_MP_GROW_C +| +--->BN_MP_ADD_D_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_MOD_D_C +| +--->BN_MP_DIV_D_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_DIV_2D_C +| | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_INIT_C +| | | +--->BN_MP_MOD_2D_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CLEAR_C +| | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | +--->BN_MP_DIV_3_C +| | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_INIT_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_INIT_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_C ++--->BN_MP_INIT_C ++--->BN_MP_ADD_D_C +| +--->BN_MP_GROW_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_PRIME_MILLER_RABIN_C +| +--->BN_MP_INIT_COPY_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| +--->BN_MP_CNT_LSB_C +| +--->BN_MP_DIV_2D_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_ZERO_C +| | +--->BN_MP_MOD_2D_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CLEAR_C +| | +--->BN_MP_RSHD_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| +--->BN_MP_EXPTMOD_C +| | +--->BN_MP_INVMOD_C +| | | +--->BN_FAST_MP_INVMOD_C +| | | | +--->BN_MP_INIT_MULTI_C +| | | | | +--->BN_MP_CLEAR_C +| | | | +--->BN_MP_COPY_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ABS_C +| | | | +--->BN_MP_DIV_2_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_CLEAR_MULTI_C +| | | | | +--->BN_MP_CLEAR_C +| | | +--->BN_MP_INVMOD_SLOW_C +| | | | +--->BN_MP_INIT_MULTI_C +| | | | | +--->BN_MP_CLEAR_C +| | | | +--->BN_MP_COPY_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_DIV_2_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_CLEAR_MULTI_C +| | | | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_CLEAR_C +| | +--->BN_MP_ABS_C +| | | +--->BN_MP_COPY_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_MP_CLEAR_MULTI_C +| | +--->BN_MP_DR_IS_MODULUS_C +| | +--->BN_MP_REDUCE_IS_2K_C +| | | +--->BN_MP_REDUCE_2K_C +| | | | +--->BN_MP_COUNT_BITS_C +| | | | +--->BN_MP_MUL_D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_COUNT_BITS_C +| | +--->BN_MP_EXPTMOD_FAST_C +| | | +--->BN_MP_COUNT_BITS_C +| | | +--->BN_MP_MONTGOMERY_SETUP_C +| | | +--->BN_FAST_MP_MONTGOMERY_REDUCE_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_MONTGOMERY_REDUCE_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_DR_SETUP_C +| | | +--->BN_MP_DR_REDUCE_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_REDUCE_2K_SETUP_C +| | | | +--->BN_MP_2EXPT_C +| | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_REDUCE_2K_C +| | | | +--->BN_MP_MUL_D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_MONTGOMERY_CALC_NORMALIZATION_C +| | | | +--->BN_MP_2EXPT_C +| | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_MUL_2_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_MULMOD_C +| | | | +--->BN_MP_MUL_C +| | | | | +--->BN_MP_TOOM_MUL_C +| | | | | | +--->BN_MP_INIT_MULTI_C +| | | | | | +--->BN_MP_MOD_2D_C +| | | | | | | +--->BN_MP_ZERO_C +| | | | | | | +--->BN_MP_COPY_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_COPY_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | | | +--->BN_MP_ZERO_C +| | | | | | +--->BN_MP_MUL_2_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_ADD_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_SUB_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_DIV_2_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_MUL_2D_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_MUL_D_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_DIV_3_C +| | | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_EXCH_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_KARATSUBA_MUL_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_SUB_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_ADD_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_RSHD_C +| | | | | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_S_MP_MUL_DIGS_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_MOD_C +| | | | | +--->BN_MP_DIV_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_MP_COPY_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | | +--->BN_MP_INIT_MULTI_C +| | | | | | +--->BN_MP_MUL_2D_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_LSHD_C +| | | | | | | | +--->BN_MP_RSHD_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_C +| | | | | | +--->BN_MP_SUB_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_ADD_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_EXCH_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_MUL_D_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_MOD_C +| | | | +--->BN_MP_DIV_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_MP_COPY_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_INIT_MULTI_C +| | | | | +--->BN_MP_MUL_2D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_MUL_D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_COPY_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_SQR_C +| | | | +--->BN_MP_TOOM_SQR_C +| | | | | +--->BN_MP_INIT_MULTI_C +| | | | | +--->BN_MP_MOD_2D_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_MUL_2_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_DIV_2_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_MUL_2D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_MUL_D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_DIV_3_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_KARATSUBA_SQR_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_FAST_S_MP_SQR_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_S_MP_SQR_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_MUL_C +| | | | +--->BN_MP_TOOM_MUL_C +| | | | | +--->BN_MP_INIT_MULTI_C +| | | | | +--->BN_MP_MOD_2D_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_MUL_2_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_DIV_2_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_MUL_2D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_MUL_D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_DIV_3_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_KARATSUBA_MUL_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | | | +--->BN_MP_ZERO_C +| | | | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_S_MP_MUL_DIGS_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_EXCH_C +| | +--->BN_S_MP_EXPTMOD_C +| | | +--->BN_MP_COUNT_BITS_C +| | | +--->BN_MP_REDUCE_SETUP_C +| | | | +--->BN_MP_2EXPT_C +| | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_DIV_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_MP_COPY_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_INIT_MULTI_C +| | | | | +--->BN_MP_MUL_2D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_MUL_D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_MOD_C +| | | | +--->BN_MP_DIV_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_MP_COPY_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_INIT_MULTI_C +| | | | | +--->BN_MP_MUL_2D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_MUL_D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_COPY_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_SQR_C +| | | | +--->BN_MP_TOOM_SQR_C +| | | | | +--->BN_MP_INIT_MULTI_C +| | | | | +--->BN_MP_MOD_2D_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_MUL_2_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_DIV_2_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_MUL_2D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_MUL_D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_DIV_3_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_KARATSUBA_SQR_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_FAST_S_MP_SQR_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_S_MP_SQR_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_REDUCE_C +| | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_MUL_C +| | | | | +--->BN_MP_TOOM_MUL_C +| | | | | | +--->BN_MP_INIT_MULTI_C +| | | | | | +--->BN_MP_MOD_2D_C +| | | | | | | +--->BN_MP_ZERO_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_MUL_2_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_ADD_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_SUB_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_DIV_2_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_MUL_2D_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_MUL_D_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_DIV_3_C +| | | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | | +--->BN_MP_EXCH_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_KARATSUBA_MUL_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_SUB_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_ADD_C +| | | | | | | +--->BN_S_MP_ADD_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | | +--->BN_S_MP_SUB_C +| | | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_S_MP_MUL_DIGS_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_EXCH_C +| | | | +--->BN_S_MP_MUL_HIGH_DIGS_C +| | | | | +--->BN_FAST_S_MP_MUL_HIGH_DIGS_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | +--->BN_FAST_S_MP_MUL_HIGH_DIGS_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MOD_2D_C +| | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_S_MP_MUL_DIGS_C +| | | | | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_MUL_C +| | | | +--->BN_MP_TOOM_MUL_C +| | | | | +--->BN_MP_INIT_MULTI_C +| | | | | +--->BN_MP_MOD_2D_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_MUL_2_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_DIV_2_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_MUL_2D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_MUL_D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_DIV_3_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_KARATSUBA_MUL_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | | | +--->BN_MP_ZERO_C +| | | | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_S_MP_MUL_DIGS_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_EXCH_C +| +--->BN_MP_CMP_C +| | +--->BN_MP_CMP_MAG_C +| +--->BN_MP_SQRMOD_C +| | +--->BN_MP_SQR_C +| | | +--->BN_MP_TOOM_SQR_C +| | | | +--->BN_MP_INIT_MULTI_C +| | | | | +--->BN_MP_CLEAR_C +| | | | +--->BN_MP_MOD_2D_C +| | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_COPY_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_COPY_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_MUL_2_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_DIV_2_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MUL_2D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MUL_D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_DIV_3_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_MP_CLEAR_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLEAR_MULTI_C +| | | | | +--->BN_MP_CLEAR_C +| | | +--->BN_MP_KARATSUBA_SQR_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLEAR_C +| | | +--->BN_FAST_S_MP_SQR_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_SQR_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_CLEAR_C +| | +--->BN_MP_MOD_C +| | | +--->BN_MP_DIV_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_MP_COPY_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_INIT_MULTI_C +| | | | +--->BN_MP_COUNT_BITS_C +| | | | +--->BN_MP_ABS_C +| | | | +--->BN_MP_MUL_2D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_CLEAR_MULTI_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_MUL_D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| +--->BN_MP_CLEAR_C ++--->BN_MP_CLEAR_C + + +BN_MP_SIGNED_BIN_SIZE_C ++--->BN_MP_UNSIGNED_BIN_SIZE_C +| +--->BN_MP_COUNT_BITS_C + + +BN_MP_INVMOD_SLOW_C ++--->BN_MP_INIT_MULTI_C +| +--->BN_MP_INIT_C +| +--->BN_MP_CLEAR_C ++--->BN_MP_COPY_C +| +--->BN_MP_GROW_C ++--->BN_MP_SET_C +| +--->BN_MP_ZERO_C ++--->BN_MP_DIV_2_C +| +--->BN_MP_GROW_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_ADD_C +| +--->BN_S_MP_ADD_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CMP_MAG_C +| +--->BN_S_MP_SUB_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C ++--->BN_MP_SUB_C +| +--->BN_S_MP_ADD_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CMP_MAG_C +| +--->BN_S_MP_SUB_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C ++--->BN_MP_CMP_C +| +--->BN_MP_CMP_MAG_C ++--->BN_MP_CMP_D_C ++--->BN_MP_CMP_MAG_C ++--->BN_MP_EXCH_C ++--->BN_MP_CLEAR_MULTI_C +| +--->BN_MP_CLEAR_C + + +BN_MP_LCM_C ++--->BN_MP_INIT_MULTI_C +| +--->BN_MP_INIT_C +| +--->BN_MP_CLEAR_C ++--->BN_MP_GCD_C +| +--->BN_MP_ABS_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| +--->BN_MP_ZERO_C +| +--->BN_MP_INIT_COPY_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| +--->BN_MP_CNT_LSB_C +| +--->BN_MP_DIV_2D_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_MOD_2D_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CLEAR_C +| | +--->BN_MP_RSHD_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| +--->BN_MP_CMP_MAG_C +| +--->BN_MP_EXCH_C +| +--->BN_S_MP_SUB_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_MUL_2D_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_RSHD_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CLEAR_C ++--->BN_MP_CMP_MAG_C ++--->BN_MP_DIV_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_ZERO_C +| +--->BN_MP_SET_C +| +--->BN_MP_COUNT_BITS_C +| +--->BN_MP_ABS_C +| +--->BN_MP_MUL_2D_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_RSHD_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CMP_C +| +--->BN_MP_SUB_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_ADD_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_DIV_2D_C +| | +--->BN_MP_INIT_C +| | +--->BN_MP_MOD_2D_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CLEAR_C +| | +--->BN_MP_RSHD_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| +--->BN_MP_EXCH_C +| +--->BN_MP_CLEAR_MULTI_C +| | +--->BN_MP_CLEAR_C +| +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_INIT_C +| +--->BN_MP_INIT_C +| +--->BN_MP_INIT_COPY_C +| +--->BN_MP_LSHD_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_RSHD_C +| +--->BN_MP_RSHD_C +| +--->BN_MP_MUL_D_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CLAMP_C +| +--->BN_MP_CLEAR_C ++--->BN_MP_MUL_C +| +--->BN_MP_TOOM_MUL_C +| | +--->BN_MP_MOD_2D_C +| | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_COPY_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_RSHD_C +| | | +--->BN_MP_ZERO_C +| | +--->BN_MP_MUL_2_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_2_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MUL_2D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MUL_D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_3_C +| | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_INIT_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_CLEAR_MULTI_C +| | | +--->BN_MP_CLEAR_C +| +--->BN_MP_KARATSUBA_MUL_C +| | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_INIT_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_ZERO_C +| | +--->BN_MP_CLEAR_C +| +--->BN_FAST_S_MP_MUL_DIGS_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_S_MP_MUL_DIGS_C +| | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_INIT_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_C ++--->BN_MP_CLEAR_MULTI_C +| +--->BN_MP_CLEAR_C + + +BN_REVERSE_C + + +BN_MP_PRIME_IS_DIVISIBLE_C ++--->BN_MP_MOD_D_C +| +--->BN_MP_DIV_D_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_DIV_2D_C +| | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_INIT_C +| | | +--->BN_MP_MOD_2D_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CLEAR_C +| | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | +--->BN_MP_DIV_3_C +| | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_INIT_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_INIT_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_C + + +BN_MP_SET_C ++--->BN_MP_ZERO_C + + +BN_MP_GCD_C ++--->BN_MP_ABS_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C ++--->BN_MP_ZERO_C ++--->BN_MP_INIT_COPY_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C ++--->BN_MP_CNT_LSB_C ++--->BN_MP_DIV_2D_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_MOD_2D_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CLEAR_C +| +--->BN_MP_RSHD_C +| +--->BN_MP_CLAMP_C +| +--->BN_MP_EXCH_C ++--->BN_MP_CMP_MAG_C ++--->BN_MP_EXCH_C ++--->BN_S_MP_SUB_C +| +--->BN_MP_GROW_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_MUL_2D_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_GROW_C +| +--->BN_MP_LSHD_C +| | +--->BN_MP_RSHD_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_CLEAR_C + + +BN_MP_READ_RADIX_C ++--->BN_MP_ZERO_C ++--->BN_MP_MUL_D_C +| +--->BN_MP_GROW_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_ADD_D_C +| +--->BN_MP_GROW_C +| +--->BN_MP_SUB_D_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CLAMP_C + + +BN_FAST_S_MP_MUL_HIGH_DIGS_C ++--->BN_MP_GROW_C ++--->BN_MP_CLAMP_C + + +BN_FAST_MP_MONTGOMERY_REDUCE_C ++--->BN_MP_GROW_C ++--->BN_MP_RSHD_C +| +--->BN_MP_ZERO_C ++--->BN_MP_CLAMP_C ++--->BN_MP_CMP_MAG_C ++--->BN_S_MP_SUB_C + + +BN_MP_DIV_D_C ++--->BN_MP_COPY_C +| +--->BN_MP_GROW_C ++--->BN_MP_DIV_2D_C +| +--->BN_MP_ZERO_C +| +--->BN_MP_INIT_C +| +--->BN_MP_MOD_2D_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CLEAR_C +| +--->BN_MP_RSHD_C +| +--->BN_MP_CLAMP_C +| +--->BN_MP_EXCH_C ++--->BN_MP_DIV_3_C +| +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_INIT_C +| +--->BN_MP_CLAMP_C +| +--->BN_MP_EXCH_C +| +--->BN_MP_CLEAR_C ++--->BN_MP_INIT_SIZE_C +| +--->BN_MP_INIT_C ++--->BN_MP_CLAMP_C ++--->BN_MP_EXCH_C ++--->BN_MP_CLEAR_C + + +BN_MP_REDUCE_2K_SETUP_C ++--->BN_MP_INIT_C ++--->BN_MP_COUNT_BITS_C ++--->BN_MP_2EXPT_C +| +--->BN_MP_ZERO_C +| +--->BN_MP_GROW_C ++--->BN_MP_CLEAR_C ++--->BN_S_MP_SUB_C +| +--->BN_MP_GROW_C +| +--->BN_MP_CLAMP_C + + +BN_MP_INIT_SET_C ++--->BN_MP_INIT_C ++--->BN_MP_SET_C +| +--->BN_MP_ZERO_C + + +BN_MP_REDUCE_2K_C ++--->BN_MP_INIT_C ++--->BN_MP_COUNT_BITS_C ++--->BN_MP_DIV_2D_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_ZERO_C +| +--->BN_MP_MOD_2D_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CLEAR_C +| +--->BN_MP_RSHD_C +| +--->BN_MP_CLAMP_C +| +--->BN_MP_EXCH_C ++--->BN_MP_MUL_D_C +| +--->BN_MP_GROW_C +| +--->BN_MP_CLAMP_C ++--->BN_S_MP_ADD_C +| +--->BN_MP_GROW_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_CMP_MAG_C ++--->BN_S_MP_SUB_C +| +--->BN_MP_GROW_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_CLEAR_C + + +BN_ERROR_C + + +BN_MP_EXPT_D_C ++--->BN_MP_INIT_COPY_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C ++--->BN_MP_SET_C +| +--->BN_MP_ZERO_C ++--->BN_MP_SQR_C +| +--->BN_MP_TOOM_SQR_C +| | +--->BN_MP_INIT_MULTI_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_MOD_2D_C +| | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_COPY_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_RSHD_C +| | | +--->BN_MP_ZERO_C +| | +--->BN_MP_MUL_2_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_2_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MUL_2D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MUL_D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_3_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_CLEAR_MULTI_C +| | | +--->BN_MP_CLEAR_C +| +--->BN_MP_KARATSUBA_SQR_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_ZERO_C +| | +--->BN_MP_ADD_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_MP_CLEAR_C +| +--->BN_FAST_S_MP_SQR_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_S_MP_SQR_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_C ++--->BN_MP_CLEAR_C ++--->BN_MP_MUL_C +| +--->BN_MP_TOOM_MUL_C +| | +--->BN_MP_INIT_MULTI_C +| | +--->BN_MP_MOD_2D_C +| | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_COPY_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_RSHD_C +| | | +--->BN_MP_ZERO_C +| | +--->BN_MP_MUL_2_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_2_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MUL_2D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MUL_D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_3_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_CLEAR_MULTI_C +| +--->BN_MP_KARATSUBA_MUL_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_ZERO_C +| +--->BN_FAST_S_MP_MUL_DIGS_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_S_MP_MUL_DIGS_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C + + +BN_S_MP_EXPTMOD_C ++--->BN_MP_COUNT_BITS_C ++--->BN_MP_INIT_C ++--->BN_MP_CLEAR_C ++--->BN_MP_REDUCE_SETUP_C +| +--->BN_MP_2EXPT_C +| | +--->BN_MP_ZERO_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_DIV_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_ZERO_C +| | +--->BN_MP_INIT_MULTI_C +| | +--->BN_MP_SET_C +| | +--->BN_MP_ABS_C +| | +--->BN_MP_MUL_2D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_2D_C +| | | +--->BN_MP_MOD_2D_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_MULTI_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_INIT_COPY_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_RSHD_C +| | +--->BN_MP_RSHD_C +| | +--->BN_MP_MUL_D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CLAMP_C ++--->BN_MP_MOD_C +| +--->BN_MP_DIV_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_ZERO_C +| | +--->BN_MP_INIT_MULTI_C +| | +--->BN_MP_SET_C +| | +--->BN_MP_ABS_C +| | +--->BN_MP_MUL_2D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_2D_C +| | | +--->BN_MP_MOD_2D_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_MULTI_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_INIT_COPY_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_RSHD_C +| | +--->BN_MP_RSHD_C +| | +--->BN_MP_MUL_D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_ADD_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_EXCH_C ++--->BN_MP_COPY_C +| +--->BN_MP_GROW_C ++--->BN_MP_SQR_C +| +--->BN_MP_TOOM_SQR_C +| | +--->BN_MP_INIT_MULTI_C +| | +--->BN_MP_MOD_2D_C +| | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_RSHD_C +| | | +--->BN_MP_ZERO_C +| | +--->BN_MP_MUL_2_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_2_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MUL_2D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MUL_D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_3_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_CLEAR_MULTI_C +| +--->BN_MP_KARATSUBA_SQR_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_ZERO_C +| | +--->BN_MP_ADD_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| +--->BN_FAST_S_MP_SQR_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_S_MP_SQR_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C ++--->BN_MP_REDUCE_C +| +--->BN_MP_INIT_COPY_C +| +--->BN_MP_RSHD_C +| | +--->BN_MP_ZERO_C +| +--->BN_MP_MUL_C +| | +--->BN_MP_TOOM_MUL_C +| | | +--->BN_MP_INIT_MULTI_C +| | | +--->BN_MP_MOD_2D_C +| | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_MUL_2_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_SUB_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_DIV_2_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_MUL_2D_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_MUL_D_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_DIV_3_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLEAR_MULTI_C +| | +--->BN_MP_KARATSUBA_MUL_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_SUB_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_S_MP_MUL_DIGS_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| +--->BN_S_MP_MUL_HIGH_DIGS_C +| | +--->BN_FAST_S_MP_MUL_HIGH_DIGS_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| +--->BN_FAST_S_MP_MUL_HIGH_DIGS_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_MOD_2D_C +| | +--->BN_MP_ZERO_C +| | +--->BN_MP_CLAMP_C +| +--->BN_S_MP_MUL_DIGS_C +| | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| +--->BN_MP_SUB_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_CMP_D_C +| +--->BN_MP_SET_C +| | +--->BN_MP_ZERO_C +| +--->BN_MP_LSHD_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_ADD_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_CMP_C +| | +--->BN_MP_CMP_MAG_C +| +--->BN_S_MP_SUB_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C ++--->BN_MP_MUL_C +| +--->BN_MP_TOOM_MUL_C +| | +--->BN_MP_INIT_MULTI_C +| | +--->BN_MP_MOD_2D_C +| | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_RSHD_C +| | | +--->BN_MP_ZERO_C +| | +--->BN_MP_MUL_2_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_2_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MUL_2D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MUL_D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_3_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_CLEAR_MULTI_C +| +--->BN_MP_KARATSUBA_MUL_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_ZERO_C +| +--->BN_FAST_S_MP_MUL_DIGS_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_S_MP_MUL_DIGS_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C ++--->BN_MP_SET_C +| +--->BN_MP_ZERO_C ++--->BN_MP_EXCH_C + + +BN_MP_ABS_C ++--->BN_MP_COPY_C +| +--->BN_MP_GROW_C + + +BN_MP_INIT_SET_INT_C ++--->BN_MP_INIT_C ++--->BN_MP_SET_INT_C +| +--->BN_MP_ZERO_C +| +--->BN_MP_MUL_2D_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_RSHD_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CLAMP_C + + +BN_MP_SUB_D_C ++--->BN_MP_GROW_C ++--->BN_MP_ADD_D_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_CLAMP_C + + +BN_MP_TO_SIGNED_BIN_C ++--->BN_MP_TO_UNSIGNED_BIN_C +| +--->BN_MP_INIT_COPY_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| +--->BN_MP_DIV_2D_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_ZERO_C +| | +--->BN_MP_MOD_2D_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CLEAR_C +| | +--->BN_MP_RSHD_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| +--->BN_MP_CLEAR_C + + +BN_MP_DIV_2_C ++--->BN_MP_GROW_C ++--->BN_MP_CLAMP_C + + +BN_MP_REDUCE_IS_2K_C ++--->BN_MP_REDUCE_2K_C +| +--->BN_MP_INIT_C +| +--->BN_MP_COUNT_BITS_C +| +--->BN_MP_DIV_2D_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_ZERO_C +| | +--->BN_MP_MOD_2D_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CLEAR_C +| | +--->BN_MP_RSHD_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| +--->BN_MP_MUL_D_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_S_MP_ADD_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CMP_MAG_C +| +--->BN_S_MP_SUB_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CLEAR_C ++--->BN_MP_COUNT_BITS_C + + +BN_MP_INIT_SIZE_C ++--->BN_MP_INIT_C + + +BN_MP_DIV_C ++--->BN_MP_CMP_MAG_C ++--->BN_MP_COPY_C +| +--->BN_MP_GROW_C ++--->BN_MP_ZERO_C ++--->BN_MP_INIT_MULTI_C +| +--->BN_MP_INIT_C +| +--->BN_MP_CLEAR_C ++--->BN_MP_SET_C ++--->BN_MP_COUNT_BITS_C ++--->BN_MP_ABS_C ++--->BN_MP_MUL_2D_C +| +--->BN_MP_GROW_C +| +--->BN_MP_LSHD_C +| | +--->BN_MP_RSHD_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_CMP_C ++--->BN_MP_SUB_C +| +--->BN_S_MP_ADD_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_S_MP_SUB_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C ++--->BN_MP_ADD_C +| +--->BN_S_MP_ADD_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_S_MP_SUB_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C ++--->BN_MP_DIV_2D_C +| +--->BN_MP_INIT_C +| +--->BN_MP_MOD_2D_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CLEAR_C +| +--->BN_MP_RSHD_C +| +--->BN_MP_CLAMP_C +| +--->BN_MP_EXCH_C ++--->BN_MP_EXCH_C ++--->BN_MP_CLEAR_MULTI_C +| +--->BN_MP_CLEAR_C ++--->BN_MP_INIT_SIZE_C +| +--->BN_MP_INIT_C ++--->BN_MP_INIT_C ++--->BN_MP_INIT_COPY_C ++--->BN_MP_LSHD_C +| +--->BN_MP_GROW_C +| +--->BN_MP_RSHD_C ++--->BN_MP_RSHD_C ++--->BN_MP_MUL_D_C +| +--->BN_MP_GROW_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_CLAMP_C ++--->BN_MP_CLEAR_C + + +BN_MP_CLEAR_C + + +BN_MP_MONTGOMERY_REDUCE_C ++--->BN_FAST_MP_MONTGOMERY_REDUCE_C +| +--->BN_MP_GROW_C +| +--->BN_MP_RSHD_C +| | +--->BN_MP_ZERO_C +| +--->BN_MP_CLAMP_C +| +--->BN_MP_CMP_MAG_C +| +--->BN_S_MP_SUB_C ++--->BN_MP_GROW_C ++--->BN_MP_CLAMP_C ++--->BN_MP_RSHD_C +| +--->BN_MP_ZERO_C ++--->BN_MP_CMP_MAG_C ++--->BN_S_MP_SUB_C + + +BN_MP_MUL_2_C ++--->BN_MP_GROW_C + + +BN_MP_UNSIGNED_BIN_SIZE_C ++--->BN_MP_COUNT_BITS_C + + +BN_MP_ADDMOD_C ++--->BN_MP_INIT_C ++--->BN_MP_ADD_C +| +--->BN_S_MP_ADD_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CMP_MAG_C +| +--->BN_S_MP_SUB_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C ++--->BN_MP_CLEAR_C ++--->BN_MP_MOD_C +| +--->BN_MP_DIV_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_ZERO_C +| | +--->BN_MP_INIT_MULTI_C +| | +--->BN_MP_SET_C +| | +--->BN_MP_COUNT_BITS_C +| | +--->BN_MP_ABS_C +| | +--->BN_MP_MUL_2D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_2D_C +| | | +--->BN_MP_MOD_2D_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_MULTI_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_INIT_COPY_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_RSHD_C +| | +--->BN_MP_RSHD_C +| | +--->BN_MP_MUL_D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_EXCH_C + + +BN_MP_ADD_C ++--->BN_S_MP_ADD_C +| +--->BN_MP_GROW_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_CMP_MAG_C ++--->BN_S_MP_SUB_C +| +--->BN_MP_GROW_C +| +--->BN_MP_CLAMP_C + + +BN_MP_RAND_C ++--->BN_MP_ZERO_C ++--->BN_MP_ADD_D_C +| +--->BN_MP_GROW_C +| +--->BN_MP_SUB_D_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_LSHD_C +| +--->BN_MP_GROW_C +| +--->BN_MP_RSHD_C + + +BN_MP_CNT_LSB_C + + +BN_MP_2EXPT_C ++--->BN_MP_ZERO_C ++--->BN_MP_GROW_C + + +BN_MP_RSHD_C ++--->BN_MP_ZERO_C + + +BN_MP_SHRINK_C + + +BN_MP_REDUCE_C ++--->BN_MP_REDUCE_SETUP_C +| +--->BN_MP_2EXPT_C +| | +--->BN_MP_ZERO_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_DIV_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_ZERO_C +| | +--->BN_MP_INIT_MULTI_C +| | | +--->BN_MP_INIT_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_SET_C +| | +--->BN_MP_COUNT_BITS_C +| | +--->BN_MP_ABS_C +| | +--->BN_MP_MUL_2D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_2D_C +| | | +--->BN_MP_INIT_C +| | | +--->BN_MP_MOD_2D_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CLEAR_C +| | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_MULTI_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_INIT_C +| | +--->BN_MP_INIT_C +| | +--->BN_MP_INIT_COPY_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_RSHD_C +| | +--->BN_MP_RSHD_C +| | +--->BN_MP_MUL_D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CLEAR_C ++--->BN_MP_INIT_COPY_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C ++--->BN_MP_RSHD_C +| +--->BN_MP_ZERO_C ++--->BN_MP_MUL_C +| +--->BN_MP_TOOM_MUL_C +| | +--->BN_MP_INIT_MULTI_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_MOD_2D_C +| | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_COPY_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_MUL_2_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_2_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MUL_2D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MUL_D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_3_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_CLEAR_MULTI_C +| | | +--->BN_MP_CLEAR_C +| +--->BN_MP_KARATSUBA_MUL_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_CLEAR_C +| +--->BN_FAST_S_MP_MUL_DIGS_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_S_MP_MUL_DIGS_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_C ++--->BN_S_MP_MUL_HIGH_DIGS_C +| +--->BN_FAST_S_MP_MUL_HIGH_DIGS_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_INIT_SIZE_C +| +--->BN_MP_CLAMP_C +| +--->BN_MP_EXCH_C +| +--->BN_MP_CLEAR_C ++--->BN_FAST_S_MP_MUL_HIGH_DIGS_C +| +--->BN_MP_GROW_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_MOD_2D_C +| +--->BN_MP_ZERO_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_CLAMP_C ++--->BN_S_MP_MUL_DIGS_C +| +--->BN_FAST_S_MP_MUL_DIGS_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_INIT_SIZE_C +| +--->BN_MP_CLAMP_C +| +--->BN_MP_EXCH_C +| +--->BN_MP_CLEAR_C ++--->BN_MP_SUB_C +| +--->BN_S_MP_ADD_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CMP_MAG_C +| +--->BN_S_MP_SUB_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C ++--->BN_MP_CMP_D_C ++--->BN_MP_SET_C +| +--->BN_MP_ZERO_C ++--->BN_MP_LSHD_C +| +--->BN_MP_GROW_C ++--->BN_MP_ADD_C +| +--->BN_S_MP_ADD_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CMP_MAG_C +| +--->BN_S_MP_SUB_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C ++--->BN_MP_CMP_C +| +--->BN_MP_CMP_MAG_C ++--->BN_S_MP_SUB_C +| +--->BN_MP_GROW_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_CLEAR_C + + +BN_MP_MUL_2D_C ++--->BN_MP_COPY_C +| +--->BN_MP_GROW_C ++--->BN_MP_GROW_C ++--->BN_MP_LSHD_C +| +--->BN_MP_RSHD_C +| | +--->BN_MP_ZERO_C ++--->BN_MP_CLAMP_C + + +BN_MP_GET_INT_C + + +BN_MP_JACOBI_C ++--->BN_MP_CMP_D_C ++--->BN_MP_INIT_COPY_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C ++--->BN_MP_CNT_LSB_C ++--->BN_MP_DIV_2D_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_ZERO_C +| +--->BN_MP_MOD_2D_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CLEAR_C +| +--->BN_MP_RSHD_C +| +--->BN_MP_CLAMP_C +| +--->BN_MP_EXCH_C ++--->BN_MP_MOD_C +| +--->BN_MP_DIV_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_ZERO_C +| | +--->BN_MP_INIT_MULTI_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_SET_C +| | +--->BN_MP_COUNT_BITS_C +| | +--->BN_MP_ABS_C +| | +--->BN_MP_MUL_2D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_MULTI_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_RSHD_C +| | +--->BN_MP_RSHD_C +| | +--->BN_MP_MUL_D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CLEAR_C +| +--->BN_MP_CLEAR_C +| +--->BN_MP_ADD_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_EXCH_C ++--->BN_MP_CLEAR_C + + +BN_MP_CLEAR_MULTI_C ++--->BN_MP_CLEAR_C + + +BN_MP_MUL_C ++--->BN_MP_TOOM_MUL_C +| +--->BN_MP_INIT_MULTI_C +| | +--->BN_MP_INIT_C +| | +--->BN_MP_CLEAR_C +| +--->BN_MP_MOD_2D_C +| | +--->BN_MP_ZERO_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_RSHD_C +| | +--->BN_MP_ZERO_C +| +--->BN_MP_MUL_2_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_ADD_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_SUB_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_DIV_2_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_MUL_2D_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_LSHD_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_MUL_D_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_DIV_3_C +| | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_INIT_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_C +| +--->BN_MP_LSHD_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_CLEAR_MULTI_C +| | +--->BN_MP_CLEAR_C ++--->BN_MP_KARATSUBA_MUL_C +| +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_INIT_C +| +--->BN_MP_CLAMP_C +| +--->BN_MP_SUB_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| +--->BN_MP_ADD_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_CMP_MAG_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| +--->BN_MP_LSHD_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_RSHD_C +| | | +--->BN_MP_ZERO_C +| +--->BN_MP_CLEAR_C ++--->BN_FAST_S_MP_MUL_DIGS_C +| +--->BN_MP_GROW_C +| +--->BN_MP_CLAMP_C ++--->BN_S_MP_MUL_DIGS_C +| +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_INIT_C +| +--->BN_MP_CLAMP_C +| +--->BN_MP_EXCH_C +| +--->BN_MP_CLEAR_C + + +BN_MP_EXTEUCLID_C ++--->BN_MP_INIT_MULTI_C +| +--->BN_MP_INIT_C +| +--->BN_MP_CLEAR_C ++--->BN_MP_SET_C +| +--->BN_MP_ZERO_C ++--->BN_MP_COPY_C +| +--->BN_MP_GROW_C ++--->BN_MP_DIV_C +| +--->BN_MP_CMP_MAG_C +| +--->BN_MP_ZERO_C +| +--->BN_MP_COUNT_BITS_C +| +--->BN_MP_ABS_C +| +--->BN_MP_MUL_2D_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_RSHD_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CMP_C +| +--->BN_MP_SUB_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_ADD_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_DIV_2D_C +| | +--->BN_MP_INIT_C +| | +--->BN_MP_MOD_2D_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CLEAR_C +| | +--->BN_MP_RSHD_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| +--->BN_MP_EXCH_C +| +--->BN_MP_CLEAR_MULTI_C +| | +--->BN_MP_CLEAR_C +| +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_INIT_C +| +--->BN_MP_INIT_C +| +--->BN_MP_INIT_COPY_C +| +--->BN_MP_LSHD_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_RSHD_C +| +--->BN_MP_RSHD_C +| +--->BN_MP_MUL_D_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CLAMP_C +| +--->BN_MP_CLEAR_C ++--->BN_MP_MUL_C +| +--->BN_MP_TOOM_MUL_C +| | +--->BN_MP_MOD_2D_C +| | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_RSHD_C +| | | +--->BN_MP_ZERO_C +| | +--->BN_MP_MUL_2_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_2_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MUL_2D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MUL_D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_3_C +| | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_INIT_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_CLEAR_MULTI_C +| | | +--->BN_MP_CLEAR_C +| +--->BN_MP_KARATSUBA_MUL_C +| | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_INIT_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_ZERO_C +| | +--->BN_MP_CLEAR_C +| +--->BN_FAST_S_MP_MUL_DIGS_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_S_MP_MUL_DIGS_C +| | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_INIT_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_C ++--->BN_MP_SUB_C +| +--->BN_S_MP_ADD_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CMP_MAG_C +| +--->BN_S_MP_SUB_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C ++--->BN_MP_EXCH_C ++--->BN_MP_CLEAR_MULTI_C +| +--->BN_MP_CLEAR_C + + +BN_MP_DR_REDUCE_C ++--->BN_MP_GROW_C ++--->BN_MP_CLAMP_C ++--->BN_MP_CMP_MAG_C ++--->BN_S_MP_SUB_C + + +BN_MP_FREAD_C ++--->BN_MP_ZERO_C ++--->BN_MP_MUL_D_C +| +--->BN_MP_GROW_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_ADD_D_C +| +--->BN_MP_GROW_C +| +--->BN_MP_SUB_D_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_CMP_D_C + + +BN_MP_REDUCE_SETUP_C ++--->BN_MP_2EXPT_C +| +--->BN_MP_ZERO_C +| +--->BN_MP_GROW_C ++--->BN_MP_DIV_C +| +--->BN_MP_CMP_MAG_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_ZERO_C +| +--->BN_MP_INIT_MULTI_C +| | +--->BN_MP_INIT_C +| | +--->BN_MP_CLEAR_C +| +--->BN_MP_SET_C +| +--->BN_MP_COUNT_BITS_C +| +--->BN_MP_ABS_C +| +--->BN_MP_MUL_2D_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_RSHD_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CMP_C +| +--->BN_MP_SUB_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_ADD_C +| | +--->BN_S_MP_ADD_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_S_MP_SUB_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| +--->BN_MP_DIV_2D_C +| | +--->BN_MP_INIT_C +| | +--->BN_MP_MOD_2D_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_CLEAR_C +| | +--->BN_MP_RSHD_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| +--->BN_MP_EXCH_C +| +--->BN_MP_CLEAR_MULTI_C +| | +--->BN_MP_CLEAR_C +| +--->BN_MP_INIT_SIZE_C +| | +--->BN_MP_INIT_C +| +--->BN_MP_INIT_C +| +--->BN_MP_INIT_COPY_C +| +--->BN_MP_LSHD_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_RSHD_C +| +--->BN_MP_RSHD_C +| +--->BN_MP_MUL_D_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CLAMP_C +| +--->BN_MP_CLEAR_C + + +BN_MP_MONTGOMERY_SETUP_C + + +BN_MP_KARATSUBA_MUL_C ++--->BN_MP_MUL_C +| +--->BN_MP_TOOM_MUL_C +| | +--->BN_MP_INIT_MULTI_C +| | | +--->BN_MP_INIT_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_MOD_2D_C +| | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_COPY_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_RSHD_C +| | | +--->BN_MP_ZERO_C +| | +--->BN_MP_MUL_2_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_SUB_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_2_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MUL_2D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MUL_D_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_DIV_3_C +| | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_INIT_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_LSHD_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_CLEAR_MULTI_C +| | | +--->BN_MP_CLEAR_C +| +--->BN_FAST_S_MP_MUL_DIGS_C +| | +--->BN_MP_GROW_C +| | +--->BN_MP_CLAMP_C +| +--->BN_S_MP_MUL_DIGS_C +| | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_INIT_C +| | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C +| | +--->BN_MP_CLEAR_C ++--->BN_MP_INIT_SIZE_C +| +--->BN_MP_INIT_C ++--->BN_MP_CLAMP_C ++--->BN_MP_SUB_C +| +--->BN_S_MP_ADD_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_CMP_MAG_C +| +--->BN_S_MP_SUB_C +| | +--->BN_MP_GROW_C ++--->BN_MP_ADD_C +| +--->BN_S_MP_ADD_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_CMP_MAG_C +| +--->BN_S_MP_SUB_C +| | +--->BN_MP_GROW_C ++--->BN_MP_LSHD_C +| +--->BN_MP_GROW_C +| +--->BN_MP_RSHD_C +| | +--->BN_MP_ZERO_C ++--->BN_MP_CLEAR_C + + +BN_MP_LSHD_C ++--->BN_MP_GROW_C ++--->BN_MP_RSHD_C +| +--->BN_MP_ZERO_C + + +BN_MP_PRIME_MILLER_RABIN_C ++--->BN_MP_CMP_D_C ++--->BN_MP_INIT_COPY_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C ++--->BN_MP_SUB_D_C +| +--->BN_MP_GROW_C +| +--->BN_MP_ADD_D_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CLAMP_C ++--->BN_MP_CNT_LSB_C ++--->BN_MP_DIV_2D_C +| +--->BN_MP_COPY_C +| | +--->BN_MP_GROW_C +| +--->BN_MP_ZERO_C +| +--->BN_MP_MOD_2D_C +| | +--->BN_MP_CLAMP_C +| +--->BN_MP_CLEAR_C +| +--->BN_MP_RSHD_C +| +--->BN_MP_CLAMP_C +| +--->BN_MP_EXCH_C ++--->BN_MP_EXPTMOD_C +| +--->BN_MP_INVMOD_C +| | +--->BN_FAST_MP_INVMOD_C +| | | +--->BN_MP_INIT_MULTI_C +| | | | +--->BN_MP_CLEAR_C +| | | +--->BN_MP_COPY_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_ABS_C +| | | +--->BN_MP_SET_C +| | | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_DIV_2_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_SUB_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_CLEAR_MULTI_C +| | | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_INVMOD_SLOW_C +| | | +--->BN_MP_INIT_MULTI_C +| | | | +--->BN_MP_CLEAR_C +| | | +--->BN_MP_COPY_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_SET_C +| | | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_DIV_2_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_SUB_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_CLEAR_MULTI_C +| | | | +--->BN_MP_CLEAR_C +| +--->BN_MP_CLEAR_C +| +--->BN_MP_ABS_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| +--->BN_MP_CLEAR_MULTI_C +| +--->BN_MP_DR_IS_MODULUS_C +| +--->BN_MP_REDUCE_IS_2K_C +| | +--->BN_MP_REDUCE_2K_C +| | | +--->BN_MP_COUNT_BITS_C +| | | +--->BN_MP_MUL_D_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_COUNT_BITS_C +| +--->BN_MP_EXPTMOD_FAST_C +| | +--->BN_MP_COUNT_BITS_C +| | +--->BN_MP_MONTGOMERY_SETUP_C +| | +--->BN_FAST_MP_MONTGOMERY_REDUCE_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | +--->BN_MP_MONTGOMERY_REDUCE_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | +--->BN_MP_DR_SETUP_C +| | +--->BN_MP_DR_REDUCE_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | +--->BN_MP_REDUCE_2K_SETUP_C +| | | +--->BN_MP_2EXPT_C +| | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_REDUCE_2K_C +| | | +--->BN_MP_MUL_D_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MONTGOMERY_CALC_NORMALIZATION_C +| | | +--->BN_MP_2EXPT_C +| | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_SET_C +| | | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_MUL_2_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MULMOD_C +| | | +--->BN_MP_MUL_C +| | | | +--->BN_MP_TOOM_MUL_C +| | | | | +--->BN_MP_INIT_MULTI_C +| | | | | +--->BN_MP_MOD_2D_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | | +--->BN_MP_COPY_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_COPY_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_MUL_2_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_DIV_2_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_MUL_2D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_MUL_D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_DIV_3_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_KARATSUBA_MUL_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | | | +--->BN_MP_ZERO_C +| | | | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_S_MP_MUL_DIGS_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_MOD_C +| | | | +--->BN_MP_DIV_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_MP_COPY_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_INIT_MULTI_C +| | | | | +--->BN_MP_SET_C +| | | | | +--->BN_MP_MUL_2D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_MUL_D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | +--->BN_MP_SET_C +| | | +--->BN_MP_ZERO_C +| | +--->BN_MP_MOD_C +| | | +--->BN_MP_DIV_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_MP_COPY_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_INIT_MULTI_C +| | | | +--->BN_MP_MUL_2D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_MUL_D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_SQR_C +| | | +--->BN_MP_TOOM_SQR_C +| | | | +--->BN_MP_INIT_MULTI_C +| | | | +--->BN_MP_MOD_2D_C +| | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_MUL_2_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_DIV_2_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MUL_2D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MUL_D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_DIV_3_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_KARATSUBA_SQR_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | +--->BN_FAST_S_MP_SQR_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_SQR_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | +--->BN_MP_MUL_C +| | | +--->BN_MP_TOOM_MUL_C +| | | | +--->BN_MP_INIT_MULTI_C +| | | | +--->BN_MP_MOD_2D_C +| | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_MUL_2_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_DIV_2_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MUL_2D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MUL_D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_DIV_3_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_KARATSUBA_MUL_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_ZERO_C +| | | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_MUL_DIGS_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | +--->BN_MP_EXCH_C +| +--->BN_S_MP_EXPTMOD_C +| | +--->BN_MP_COUNT_BITS_C +| | +--->BN_MP_REDUCE_SETUP_C +| | | +--->BN_MP_2EXPT_C +| | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_DIV_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_MP_COPY_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_INIT_MULTI_C +| | | | +--->BN_MP_SET_C +| | | | +--->BN_MP_MUL_2D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_MUL_D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MOD_C +| | | +--->BN_MP_DIV_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_MP_COPY_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_INIT_MULTI_C +| | | | +--->BN_MP_SET_C +| | | | +--->BN_MP_MUL_2D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_MUL_D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | +--->BN_MP_COPY_C +| | | +--->BN_MP_GROW_C +| | +--->BN_MP_SQR_C +| | | +--->BN_MP_TOOM_SQR_C +| | | | +--->BN_MP_INIT_MULTI_C +| | | | +--->BN_MP_MOD_2D_C +| | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_MUL_2_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_DIV_2_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MUL_2D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MUL_D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_DIV_3_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_KARATSUBA_SQR_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | +--->BN_FAST_S_MP_SQR_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_SQR_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | +--->BN_MP_REDUCE_C +| | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_MUL_C +| | | | +--->BN_MP_TOOM_MUL_C +| | | | | +--->BN_MP_INIT_MULTI_C +| | | | | +--->BN_MP_MOD_2D_C +| | | | | | +--->BN_MP_ZERO_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_MUL_2_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_DIV_2_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_MUL_2D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_MUL_D_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_DIV_3_C +| | | | | | +--->BN_MP_INIT_SIZE_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | | +--->BN_MP_EXCH_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_KARATSUBA_MUL_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_SUB_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_ADD_C +| | | | | | +--->BN_S_MP_ADD_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CMP_MAG_C +| | | | | | +--->BN_S_MP_SUB_C +| | | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_S_MP_MUL_DIGS_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | +--->BN_S_MP_MUL_HIGH_DIGS_C +| | | | +--->BN_FAST_S_MP_MUL_HIGH_DIGS_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | +--->BN_FAST_S_MP_MUL_HIGH_DIGS_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_MOD_2D_C +| | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_MUL_DIGS_C +| | | | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_SUB_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_SET_C +| | | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_MUL_C +| | | +--->BN_MP_TOOM_MUL_C +| | | | +--->BN_MP_INIT_MULTI_C +| | | | +--->BN_MP_MOD_2D_C +| | | | | +--->BN_MP_ZERO_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_MUL_2_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_DIV_2_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MUL_2D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_MUL_D_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_DIV_3_C +| | | | | +--->BN_MP_INIT_SIZE_C +| | | | | +--->BN_MP_CLAMP_C +| | | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_KARATSUBA_MUL_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_SUB_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_ADD_C +| | | | | +--->BN_S_MP_ADD_C +| | | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CMP_MAG_C +| | | | | +--->BN_S_MP_SUB_C +| | | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_RSHD_C +| | | | | | +--->BN_MP_ZERO_C +| | | +--->BN_FAST_S_MP_MUL_DIGS_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_S_MP_MUL_DIGS_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | +--->BN_MP_SET_C +| | | +--->BN_MP_ZERO_C +| | +--->BN_MP_EXCH_C ++--->BN_MP_CMP_C +| +--->BN_MP_CMP_MAG_C ++--->BN_MP_SQRMOD_C +| +--->BN_MP_SQR_C +| | +--->BN_MP_TOOM_SQR_C +| | | +--->BN_MP_INIT_MULTI_C +| | | | +--->BN_MP_CLEAR_C +| | | +--->BN_MP_MOD_2D_C +| | | | +--->BN_MP_ZERO_C +| | | | +--->BN_MP_COPY_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_COPY_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_MUL_2_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_SUB_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_DIV_2_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_MUL_2D_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_MUL_D_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_DIV_3_C +| | | | +--->BN_MP_INIT_SIZE_C +| | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_MP_EXCH_C +| | | | +--->BN_MP_CLEAR_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLEAR_MULTI_C +| | | | +--->BN_MP_CLEAR_C +| | +--->BN_MP_KARATSUBA_SQR_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_SUB_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_RSHD_C +| | | | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_MP_CMP_MAG_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLEAR_C +| | +--->BN_FAST_S_MP_SQR_C +| | | +--->BN_MP_GROW_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_S_MP_SQR_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_CLEAR_C +| +--->BN_MP_CLEAR_C +| +--->BN_MP_MOD_C +| | +--->BN_MP_DIV_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_MP_COPY_C +| | | | +--->BN_MP_GROW_C +| | | +--->BN_MP_ZERO_C +| | | +--->BN_MP_INIT_MULTI_C +| | | +--->BN_MP_SET_C +| | | +--->BN_MP_COUNT_BITS_C +| | | +--->BN_MP_ABS_C +| | | +--->BN_MP_MUL_2D_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_LSHD_C +| | | | | +--->BN_MP_RSHD_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_SUB_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_ADD_C +| | | | +--->BN_S_MP_ADD_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | | +--->BN_S_MP_SUB_C +| | | | | +--->BN_MP_GROW_C +| | | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_EXCH_C +| | | +--->BN_MP_CLEAR_MULTI_C +| | | +--->BN_MP_INIT_SIZE_C +| | | +--->BN_MP_LSHD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_RSHD_C +| | | +--->BN_MP_MUL_D_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_ADD_C +| | | +--->BN_S_MP_ADD_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | | +--->BN_MP_CMP_MAG_C +| | | +--->BN_S_MP_SUB_C +| | | | +--->BN_MP_GROW_C +| | | | +--->BN_MP_CLAMP_C +| | +--->BN_MP_EXCH_C ++--->BN_MP_CLEAR_C + + +BN_MP_DR_SETUP_C + + +BN_MP_CMP_MAG_C + + diff --git a/libtommath/changes.txt b/libtommath/changes.txt new file mode 100644 index 0000000..0d1ec2e --- /dev/null +++ b/libtommath/changes.txt @@ -0,0 +1,337 @@ +December 23rd, 2004 +v0.33 -- Fixed "small" variant for mp_div() which would munge with negative dividends... + -- Fixed bug in mp_prime_random_ex() which would set the most significant byte to zero when + no special flags were set + -- Fixed overflow [minor] bug in fast_s_mp_sqr() + -- Made the makefiles easier to configure the group/user that ltm will install as + -- Fixed "final carry" bug in comba multipliers. (Volkan Ceylan) + -- Matt Johnston pointed out a missing semi-colon in mp_exptmod + +October 29th, 2004 +v0.32 -- Added "makefile.shared" for shared object support + -- Added more to the build options/configs in the manual + -- Started the Depends framework, wrote dep.pl to scan deps and + produce "callgraph.txt" ;-) + -- Wrote SC_RSA_1 which will enable close to the minimum required to perform + RSA on 32-bit [or 64-bit] platforms with LibTomCrypt + -- Merged in the small/slower mp_div replacement. You can now toggle which + you want to use as your mp_div() at build time. Saves roughly 8KB or so. + -- Renamed a few files and changed some comments to make depends system work better. + (No changes to function names) + -- Merged in new Combas that perform 2 reads per inner loop instead of the older + 3reads/2writes per inner loop of the old code. Really though if you want speed + learn to use TomsFastMath ;-) + +August 9th, 2004 +v0.31 -- "profiled" builds now :-) new timings for Intel Northwoods + -- Added "pretty" build target + -- Update mp_init() to actually assign 0's instead of relying on calloc() + -- "Wolfgang Ehrhardt" <Wolfgang.Ehrhardt@munich.netsurf.de> found a bug in mp_mul() where if + you multiply a negative by zero you get negative zero as the result. Oops. + -- J Harper from PeerSec let me toy with his AMD64 and I got 60-bit digits working properly + [this also means that I fixed a bug where if sizeof(int) < sizeof(mp_digit) it would bug] + +April 11th, 2004 +v0.30 -- Added "mp_toradix_n" which stores upto "n-1" least significant digits of an mp_int + -- Johan Lindh sent a patch so MSVC wouldn't whine about redefining malloc [in weird dll modes] + -- Henrik Goldman spotted a missing OPT_CAST in mp_fwrite() + -- Tuned tommath.h so that when MP_LOW_MEM is defined MP_PREC shall be reduced. + [I also allow MP_PREC to be externally defined now] + -- Sped up mp_cnt_lsb() by using a 4x4 table [e.g. 4x speedup] + -- Added mp_prime_random_ex() which is a more versatile prime generator accurate to + exact bit lengths (unlike the deprecated but still available mp_prime_random() which + is only accurate to byte lengths). See the new LTM_PRIME_* flags ;-) + -- Alex Polushin contributed an optimized mp_sqrt() as well as mp_get_int() and mp_is_square(). + I've cleaned them all up to be a little more consistent [along with one bug fix] for this release. + -- Added mp_init_set and mp_init_set_int to initialize and set small constants with one function + call. + -- Removed /etclib directory [um LibTomPoly deprecates this]. + -- Fixed mp_mod() so the sign of the result agrees with the sign of the modulus. + ++ N.B. My semester is almost up so expect updates to the textbook to be posted to the libtomcrypt.org + website. + +Jan 25th, 2004 +v0.29 ++ Note: "Henrik" from the v0.28 changelog refers to Henrik Goldman ;-) + -- Added fix to mp_shrink to prevent a realloc when used == 0 [e.g. realloc zero bytes???] + -- Made the mp_prime_rabin_miller_trials() function internal table smaller and also + set the minimum number of tests to two (sounds a bit safer). + -- Added a mp_exteuclid() which computes the extended euclidean algorithm. + -- Fixed a memory leak in s_mp_exptmod() [called when Barrett reduction is to be used] which would arise + if a multiplication or subsequent reduction failed [would not free the temp result]. + -- Made an API change to mp_radix_size(). It now returns an error code and stores the required size + through an "int star" passed to it. + +Dec 24th, 2003 +v0.28 -- Henrik Goldman suggested I add casts to the montomgery code [stores into mu...] so compilers wouldn't + spew [erroneous] diagnostics... fixed. + -- Henrik Goldman also spotted two typos. One in mp_radix_size() and another in mp_toradix(). + -- Added fix to mp_shrink() to avoid a memory leak. + -- Added mp_prime_random() which requires a callback to make truly random primes of a given nature + (idea from chat with Niels Ferguson at Crypto'03) + -- Picked up a second wind. I'm filled with Gooo. Mission Gooo! + -- Removed divisions from mp_reduce_is_2k() + -- Sped up mp_div_d() [general case] to use only one division per digit instead of two. + -- Added the heap macros from LTC to LTM. Now you can easily [by editing four lines of tommath.h] + change the name of the heap functions used in LTM [also compatible with LTC via MPI mode] + -- Added bn_prime_rabin_miller_trials() which gives the number of Rabin-Miller trials to achieve + a failure rate of less than 2^-96 + -- fixed bug in fast_mp_invmod(). The initial testing logic was wrong. An invalid input is not when + "a" and "b" are even it's when "b" is even [the algo is for odd moduli only]. + -- Started a new manual [finally]. It is incomplete and will be finished as time goes on. I had to stop + adding full demos around half way in chapter three so I could at least get a good portion of the + manual done. If you really need help using the library you can always email me! + -- My Textbook is now included as part of the package [all Public Domain] + +Sept 19th, 2003 +v0.27 -- Removed changes.txt~ which was made by accident since "kate" decided it was + a good time to re-enable backups... [kde is fun!] + -- In mp_grow() "a->dp" is not overwritten by realloc call [re: memory leak] + Now if mp_grow() fails the mp_int is still valid and can be cleared via + mp_clear() to reclaim the memory. + -- Henrik Goldman found a buffer overflow bug in mp_add_d(). Fixed. + -- Cleaned up mp_mul_d() to be much easier to read and follow. + +Aug 29th, 2003 +v0.26 -- Fixed typo that caused warning with GCC 3.2 + -- Martin Marcel noticed a bug in mp_neg() that allowed negative zeroes. + Also, Martin is the fellow who noted the bugs in mp_gcd() of 0.24/0.25. + -- Martin Marcel noticed an optimization [and slight bug] in mp_lcm(). + -- Added fix to mp_read_unsigned_bin to prevent a buffer overflow. + -- Beefed up the comments in the baseline multipliers [and montgomery] + -- Added "mont" demo to the makefile.msvc in etc/ + -- Optimized sign compares in mp_cmp from 4 to 2 cases. + +Aug 4th, 2003 +v0.25 -- Fix to mp_gcd again... oops (0,-a) == (-a, 0) == a + -- Fix to mp_clear which didn't reset the sign [Greg Rose] + -- Added mp_error_to_string() to convert return codes to strings. [Greg Rose] + -- Optimized fast_mp_invmod() to do the test for invalid inputs [both even] + first so temps don't have to be initialized if it's going to fail. + -- Optimized mp_gcd() by removing mp_div_2d calls for when one of the inputs + is odd. + -- Tons of new comments, some indentation fixups, etc. + -- mp_jacobi() returns MP_VAL if the modulus is less than or equal to zero. + -- fixed two typos in the header of each file :-) + -- LibTomMath is officially Public Domain [see LICENSE] + +July 15th, 2003 +v0.24 -- Optimized mp_add_d and mp_sub_d to not allocate temporary variables + -- Fixed mp_gcd() so the gcd of 0,0 is 0. Allows the gcd operation to be chained + e.g. (0,0,a) == a [instead of 1] + -- Should be one of the last release for a while. Working on LibTomMath book now. + -- optimized the pprime demo [/etc/pprime.c] to first make a huge table of single + digit primes then it reads them randomly instead of randomly choosing/testing single + digit primes. + +July 12th, 2003 +v0.23 -- Optimized mp_prime_next_prime() to not use mp_mod [via is_divisible()] in each + iteration. Instead now a smaller table is kept of the residues which can be updated + without division. + -- Fixed a bug in next_prime() where an input of zero would be treated as odd and + have two added to it [to move to the next odd]. + -- fixed a bug in prime_fermat() and prime_miller_rabin() which allowed the base + to be negative, zero or one. Normally the test is only valid if the base is + greater than one. + -- changed the next_prime() prototype to accept a new parameter "bbs_style" which + will find the next prime congruent to 3 mod 4. The default [bbs_style==0] will + make primes which are either congruent to 1 or 3 mod 4. + -- fixed mp_read_unsigned_bin() so that it doesn't include both code for + the case DIGIT_BIT < 8 and >= 8 + -- optimized div_d() to easy out on division by 1 [or if a == 0] and use + logical shifts if the divisor is a power of two. + -- the default DIGIT_BIT type was not int for non-default builds. Fixed. + +July 2nd, 2003 +v0.22 -- Fixed up mp_invmod so the result is properly in range now [was always congruent to the inverse...] + -- Fixed up s_mp_exptmod and mp_exptmod_fast so the lower half of the pre-computed table isn't allocated + which makes the algorithm use half as much ram. + -- Fixed the install script not to make the book :-) [which isn't included anyways] + -- added mp_cnt_lsb() which counts how many of the lsbs are zero + -- optimized mp_gcd() to use the new mp_cnt_lsb() to replace multiple divisions by two by a single division. + -- applied similar optimization to mp_prime_miller_rabin(). + -- Fixed a bug in both mp_invmod() and fast_mp_invmod() which tested for odd + via "mp_iseven() == 0" which is not valid [since zero is not even either]. + +June 19th, 2003 +v0.21 -- Fixed bug in mp_mul_d which would not handle sign correctly [would not always forward it] + -- Removed the #line lines from gen.pl [was in violation of ISO C] + +June 8th, 2003 +v0.20 -- Removed the book from the package. Added the TDCAL license document. + -- This release is officially pure-bred TDCAL again [last officially TDCAL based release was v0.16] + +June 6th, 2003 +v0.19 -- Fixed a bug in mp_montgomery_reduce() which was introduced when I tweaked mp_rshd() in the previous release. + Essentially the digits were not trimmed before the compare which cause a subtraction to occur all the time. + -- Fixed up etc/tune.c a bit to stop testing new cutoffs after 16 failures [to find more optimal points]. + Brute force ho! + + +May 29th, 2003 +v0.18 -- Fixed a bug in s_mp_sqr which would handle carries properly just not very elegantly. + (e.g. correct result, just bad looking code) + -- Fixed bug in mp_sqr which still had a 512 constant instead of MP_WARRAY + -- Added Toom-Cook multipliers [needs tuning!] + -- Added efficient divide by 3 algorithm mp_div_3 + -- Re-wrote mp_div_d to be faster than calling mp_div + -- Added in a donated BCC makefile and a single page LTM poster (ahalhabsi@sbcglobal.net) + -- Added mp_reduce_2k which reduces an input modulo n = 2**p - k for any single digit k + -- Made the exptmod system be aware of the 2k reduction algorithms. + -- Rewrote mp_dr_reduce to be smaller, simpler and easier to understand. + +May 17th, 2003 +v0.17 -- Benjamin Goldberg submitted optimized mp_add and mp_sub routines. A new gen.pl as well + as several smaller suggestions. Thanks! + -- removed call to mp_cmp in inner loop of mp_div and put mp_cmp_mag in its place :-) + -- Fixed bug in mp_exptmod that would cause it to fail for odd moduli when DIGIT_BIT != 28 + -- mp_exptmod now also returns errors if the modulus is negative and will handle negative exponents + -- mp_prime_is_prime will now return true if the input is one of the primes in the prime table + -- Damian M Gryski (dgryski@uwaterloo.ca) found a index out of bounds error in the + mp_fast_s_mp_mul_high_digs function which didn't come up before. (fixed) + -- Refactored the DR reduction code so there is only one function per file. + -- Fixed bug in the mp_mul() which would erroneously avoid the faster multiplier [comba] when it was + allowed. The bug would not cause the incorrect value to be produced just less efficient (fixed) + -- Fixed similar bug in the Montgomery reduction code. + -- Added tons of (mp_digit) casts so the 7/15/28/31 bit digit code will work flawlessly out of the box. + Also added limited support for 64-bit machines with a 60-bit digit. Both thanks to Tom Wu (tom@arcot.com) + -- Added new comments here and there, cleaned up some code [style stuff] + -- Fixed a lingering typo in mp_exptmod* that would set bitcnt to zero then one. Very silly stuff :-) + -- Fixed up mp_exptmod_fast so it would set "redux" to the comba Montgomery reduction if allowed. This + saves quite a few calls and if statements. + -- Added etc/mont.c a test of the Montgomery reduction [assuming all else works :-| ] + -- Fixed up etc/tune.c to use a wider test range [more appropriate] also added a x86 based addition which + uses RDTSC for high precision timing. + -- Updated demo/demo.c to remove MPI stuff [won't work anyways], made the tests run for 2 seconds each so its + not so insanely slow. Also made the output space delimited [and fixed up various errors] + -- Added logs directory, logs/graph.dem which will use gnuplot to make a series of PNG files + that go with the pre-made index.html. You have to build [via make timing] and run ltmtest first in the + root of the package. + -- Fixed a bug in mp_sub and mp_add where "-a - -a" or "-a + a" would produce -0 as the result [obviously invalid]. + -- Fixed a bug in mp_rshd. If the count == a.used it should zero/return [instead of shifting] + -- Fixed a "off-by-one" bug in mp_mul2d. The initial size check on alloc would be off by one if the residue + shifting caused a carry. + -- Fixed a bug where s_mp_mul_digs() would not call the Comba based routine if allowed. This made Barrett reduction + slower than it had to be. + +Mar 29th, 2003 +v0.16 -- Sped up mp_div by making normalization one shift call + -- Sped up mp_mul_2d/mp_div_2d by aliasing pointers :-) + -- Cleaned up mp_gcd to use the macros for odd/even detection + -- Added comments here and there, mostly there but occasionally here too. + +Mar 22nd, 2003 +v0.15 -- Added series of prime testing routines to lib + -- Fixed up etc/tune.c + -- Added DR reduction algorithm + -- Beefed up the manual more. + -- Fixed up demo/demo.c so it doesn't have so many warnings and it does the full series of + tests + -- Added "pre-gen" directory which will hold a "gen.pl"'ed copy of the entire lib [done at + zipup time so its always the latest] + -- Added conditional casts for C++ users [boo!] + +Mar 15th, 2003 +v0.14 -- Tons of manual updates + -- cleaned up the directory + -- added MSVC makefiles + -- source changes [that I don't recall] + -- Fixed up the lshd/rshd code to use pointer aliasing + -- Fixed up the mul_2d and div_2d to not call rshd/lshd unless needed + -- Fixed up etc/tune.c a tad + -- fixed up demo/demo.c to output comma-delimited results of timing + also fixed up timing demo to use a finer granularity for various functions + -- fixed up demo/demo.c testing to pause during testing so my Duron won't catch on fire + [stays around 31-35C during testing :-)] + +Feb 13th, 2003 +v0.13 -- tons of minor speed-ups in low level add, sub, mul_2 and div_2 which propagate + to other functions like mp_invmod, mp_div, etc... + -- Sped up mp_exptmod_fast by using new code to find R mod m [e.g. B^n mod m] + -- minor fixes + +Jan 17th, 2003 +v0.12 -- re-wrote the majority of the makefile so its more portable and will + install via "make install" on most *nix platforms + -- Re-packaged all the source as seperate files. Means the library a single + file packagage any more. Instead of just adding "bn.c" you have to add + libtommath.a + -- Renamed "bn.h" to "tommath.h" + -- Changes to the manual to reflect all of this + -- Used GNU Indent to clean up the source + +Jan 15th, 2003 +v0.11 -- More subtle fixes + -- Moved to gentoo linux [hurrah!] so made *nix specific fixes to the make process + -- Sped up the montgomery reduction code quite a bit + -- fixed up demo so when building timing for the x86 it assumes ELF format now + +Jan 9th, 2003 +v0.10 -- Pekka Riikonen suggested fixes to the radix conversion code. + -- Added baseline montgomery and comba montgomery reductions, sped up exptmods + [to a point, see bn.h for MONTGOMERY_EXPT_CUTOFF] + +Jan 6th, 2003 +v0.09 -- Updated the manual to reflect recent changes. :-) + -- Added Jacobi function (mp_jacobi) to supplement the number theory side of the lib + -- Added a Mersenne prime finder demo in ./etc/mersenne.c + +Jan 2nd, 2003 +v0.08 -- Sped up the multipliers by moving the inner loop variables into a smaller scope + -- Corrected a bunch of small "warnings" + -- Added more comments + -- Made "mtest" be able to use /dev/random, /dev/urandom or stdin for RNG data + -- Corrected some bugs where error messages were potentially ignored + -- add etc/pprime.c program which makes numbers which are provably prime. + +Jan 1st, 2003 +v0.07 -- Removed alot of heap operations from core functions to speed them up + -- Added a root finding function [and mp_sqrt macro like from MPI] + -- Added more to manual + +Dec 31st, 2002 +v0.06 -- Sped up the s_mp_add, s_mp_sub which inturn sped up mp_invmod, mp_exptmod, etc... + -- Cleaned up the header a bit more + +Dec 30th, 2002 +v0.05 -- Builds with MSVC out of the box + -- Fixed a bug in mp_invmod w.r.t. even moduli + -- Made mp_toradix and mp_read_radix use char instead of unsigned char arrays + -- Fixed up exptmod to use fewer multiplications + -- Fixed up mp_init_size to use only one heap operation + -- Note there is a slight "off-by-one" bug in the library somewhere + without the padding (see the source for comment) the library + crashes in libtomcrypt. Anyways a reasonable workaround is to pad the + numbers which will always correct it since as the numbers grow the padding + will still be beyond the end of the number + -- Added more to the manual + +Dec 29th, 2002 +v0.04 -- Fixed a memory leak in mp_to_unsigned_bin + -- optimized invmod code + -- Fixed bug in mp_div + -- use exchange instead of copy for results + -- added a bit more to the manual + +Dec 27th, 2002 +v0.03 -- Sped up s_mp_mul_high_digs by not computing the carries of the lower digits + -- Fixed a bug where mp_set_int wouldn't zero the value first and set the used member. + -- fixed a bug in s_mp_mul_high_digs where the limit placed on the result digits was not calculated properly + -- fixed bugs in add/sub/mul/sqr_mod functions where if the modulus and dest were the same it wouldn't work + -- fixed a bug in mp_mod and mp_mod_d concerning negative inputs + -- mp_mul_d didn't preserve sign + -- Many many many many fixes + -- Works in LibTomCrypt now :-) + -- Added iterations to the timing demos... more accurate. + -- Tom needs a job. + +Dec 26th, 2002 +v0.02 -- Fixed a few "slips" in the manual. This is "LibTomMath" afterall :-) + -- Added mp_cmp_mag, mp_neg, mp_abs and mp_radix_size that were missing. + -- Sped up the fast [comba] multipliers more [yahoo!] + +Dec 25th,2002 +v0.01 -- Initial release. Gimme a break. + -- Todo list, + add details to manual [e.g. algorithms] + more comments in code + example programs diff --git a/libtommath/demo/demo.c b/libtommath/demo/demo.c new file mode 100644 index 0000000..62615cd --- /dev/null +++ b/libtommath/demo/demo.c @@ -0,0 +1,515 @@ +#include <time.h> + +#ifdef IOWNANATHLON +#include <unistd.h> +#define SLEEP sleep(4) +#else +#define SLEEP +#endif + +#include "tommath.h" + +void ndraw(mp_int *a, char *name) +{ + char buf[16000]; + printf("%s: ", name); + mp_toradix(a, buf, 10); + printf("%s\n", buf); +} + +static void draw(mp_int *a) +{ + ndraw(a, ""); +} + + +unsigned long lfsr = 0xAAAAAAAAUL; + +int lbit(void) +{ + if (lfsr & 0x80000000UL) { + lfsr = ((lfsr << 1) ^ 0x8000001BUL) & 0xFFFFFFFFUL; + return 1; + } else { + lfsr <<= 1; + return 0; + } +} + +int myrng(unsigned char *dst, int len, void *dat) +{ + int x; + for (x = 0; x < len; x++) dst[x] = rand() & 0xFF; + return len; +} + + + + char cmd[4096], buf[4096]; +int main(void) +{ + mp_int a, b, c, d, e, f; + unsigned long expt_n, add_n, sub_n, mul_n, div_n, sqr_n, mul2d_n, div2d_n, gcd_n, lcm_n, inv_n, + div2_n, mul2_n, add_d_n, sub_d_n, t; + unsigned rr; + int i, n, err, cnt, ix, old_kara_m, old_kara_s; + + + mp_init(&a); + mp_init(&b); + mp_init(&c); + mp_init(&d); + mp_init(&e); + mp_init(&f); + + srand(time(NULL)); + +#if 0 + // test mp_get_int + printf("Testing: mp_get_int\n"); + for(i=0;i<1000;++i) { + t = ((unsigned long)rand()*rand()+1)&0xFFFFFFFF; + mp_set_int(&a,t); + if (t!=mp_get_int(&a)) { + printf("mp_get_int() bad result!\n"); + return 1; + } + } + mp_set_int(&a,0); + if (mp_get_int(&a)!=0) + { printf("mp_get_int() bad result!\n"); + return 1; + } + mp_set_int(&a,0xffffffff); + if (mp_get_int(&a)!=0xffffffff) + { printf("mp_get_int() bad result!\n"); + return 1; + } + + // test mp_sqrt + printf("Testing: mp_sqrt\n"); + for (i=0;i<1000;++i) { + printf("%6d\r", i); fflush(stdout); + n = (rand()&15)+1; + mp_rand(&a,n); + if (mp_sqrt(&a,&b) != MP_OKAY) + { printf("mp_sqrt() error!\n"); + return 1; + } + mp_n_root(&a,2,&a); + if (mp_cmp_mag(&b,&a) != MP_EQ) + { printf("mp_sqrt() bad result!\n"); + return 1; + } + } + + printf("\nTesting: mp_is_square\n"); + for (i=0;i<1000;++i) { + printf("%6d\r", i); fflush(stdout); + + /* test mp_is_square false negatives */ + n = (rand()&7)+1; + mp_rand(&a,n); + mp_sqr(&a,&a); + if (mp_is_square(&a,&n)!=MP_OKAY) { + printf("fn:mp_is_square() error!\n"); + return 1; + } + if (n==0) { + printf("fn:mp_is_square() bad result!\n"); + return 1; + } + + /* test for false positives */ + mp_add_d(&a, 1, &a); + if (mp_is_square(&a,&n)!=MP_OKAY) { + printf("fp:mp_is_square() error!\n"); + return 1; + } + if (n==1) { + printf("fp:mp_is_square() bad result!\n"); + return 1; + } + + } + printf("\n\n"); + + /* test for size */ + for (ix = 10; ix < 256; ix++) { + printf("Testing (not safe-prime): %9d bits \r", ix); fflush(stdout); + err = mp_prime_random_ex(&a, 8, ix, (rand()&1)?LTM_PRIME_2MSB_OFF:LTM_PRIME_2MSB_ON, myrng, NULL); + if (err != MP_OKAY) { + printf("failed with err code %d\n", err); + return EXIT_FAILURE; + } + if (mp_count_bits(&a) != ix) { + printf("Prime is %d not %d bits!!!\n", mp_count_bits(&a), ix); + return EXIT_FAILURE; + } + } + + for (ix = 16; ix < 256; ix++) { + printf("Testing ( safe-prime): %9d bits \r", ix); fflush(stdout); + err = mp_prime_random_ex(&a, 8, ix, ((rand()&1)?LTM_PRIME_2MSB_OFF:LTM_PRIME_2MSB_ON)|LTM_PRIME_SAFE, myrng, NULL); + if (err != MP_OKAY) { + printf("failed with err code %d\n", err); + return EXIT_FAILURE; + } + if (mp_count_bits(&a) != ix) { + printf("Prime is %d not %d bits!!!\n", mp_count_bits(&a), ix); + return EXIT_FAILURE; + } + /* let's see if it's really a safe prime */ + mp_sub_d(&a, 1, &a); + mp_div_2(&a, &a); + mp_prime_is_prime(&a, 8, &cnt); + if (cnt != MP_YES) { + printf("sub is not prime!\n"); + return EXIT_FAILURE; + } + } + + printf("\n\n"); + + mp_read_radix(&a, "123456", 10); + mp_toradix_n(&a, buf, 10, 3); + printf("a == %s\n", buf); + mp_toradix_n(&a, buf, 10, 4); + printf("a == %s\n", buf); + mp_toradix_n(&a, buf, 10, 30); + printf("a == %s\n", buf); + + +#if 0 + for (;;) { + fgets(buf, sizeof(buf), stdin); + mp_read_radix(&a, buf, 10); + mp_prime_next_prime(&a, 5, 1); + mp_toradix(&a, buf, 10); + printf("%s, %lu\n", buf, a.dp[0] & 3); + } +#endif + + /* test mp_cnt_lsb */ + printf("testing mp_cnt_lsb...\n"); + mp_set(&a, 1); + for (ix = 0; ix < 1024; ix++) { + if (mp_cnt_lsb(&a) != ix) { + printf("Failed at %d, %d\n", ix, mp_cnt_lsb(&a)); + return 0; + } + mp_mul_2(&a, &a); + } + +/* test mp_reduce_2k */ + printf("Testing mp_reduce_2k...\n"); + for (cnt = 3; cnt <= 128; ++cnt) { + mp_digit tmp; + mp_2expt(&a, cnt); + mp_sub_d(&a, 2, &a); /* a = 2**cnt - 2 */ + + + printf("\nTesting %4d bits", cnt); + printf("(%d)", mp_reduce_is_2k(&a)); + mp_reduce_2k_setup(&a, &tmp); + printf("(%d)", tmp); + for (ix = 0; ix < 1000; ix++) { + if (!(ix & 127)) {printf("."); fflush(stdout); } + mp_rand(&b, (cnt/DIGIT_BIT + 1) * 2); + mp_copy(&c, &b); + mp_mod(&c, &a, &c); + mp_reduce_2k(&b, &a, 1); + if (mp_cmp(&c, &b)) { + printf("FAILED\n"); + exit(0); + } + } + } + +/* test mp_div_3 */ + printf("Testing mp_div_3...\n"); + mp_set(&d, 3); + for (cnt = 0; cnt < 10000; ) { + mp_digit r1, r2; + + if (!(++cnt & 127)) printf("%9d\r", cnt); + mp_rand(&a, abs(rand()) % 128 + 1); + mp_div(&a, &d, &b, &e); + mp_div_3(&a, &c, &r2); + + if (mp_cmp(&b, &c) || mp_cmp_d(&e, r2)) { + printf("\n\nmp_div_3 => Failure\n"); + } + } + printf("\n\nPassed div_3 testing\n"); + +/* test the DR reduction */ + printf("testing mp_dr_reduce...\n"); + for (cnt = 2; cnt < 32; cnt++) { + printf("%d digit modulus\n", cnt); + mp_grow(&a, cnt); + mp_zero(&a); + for (ix = 1; ix < cnt; ix++) { + a.dp[ix] = MP_MASK; + } + a.used = cnt; + a.dp[0] = 3; + + mp_rand(&b, cnt - 1); + mp_copy(&b, &c); + + rr = 0; + do { + if (!(rr & 127)) { printf("%9lu\r", rr); fflush(stdout); } + mp_sqr(&b, &b); mp_add_d(&b, 1, &b); + mp_copy(&b, &c); + + mp_mod(&b, &a, &b); + mp_dr_reduce(&c, &a, (((mp_digit)1)<<DIGIT_BIT)-a.dp[0]); + + if (mp_cmp(&b, &c) != MP_EQ) { + printf("Failed on trial %lu\n", rr); exit(-1); + + } + } while (++rr < 500); + printf("Passed DR test for %d digits\n", cnt); + } + +#endif + + div2_n = mul2_n = inv_n = expt_n = lcm_n = gcd_n = add_n = + sub_n = mul_n = div_n = sqr_n = mul2d_n = div2d_n = cnt = add_d_n = sub_d_n= 0; + + /* force KARA and TOOM to enable despite cutoffs */ + KARATSUBA_SQR_CUTOFF = KARATSUBA_MUL_CUTOFF = 110; + TOOM_SQR_CUTOFF = TOOM_MUL_CUTOFF = 150; + + for (;;) { + /* randomly clear and re-init one variable, this has the affect of triming the alloc space */ + switch (abs(rand()) % 7) { + case 0: mp_clear(&a); mp_init(&a); break; + case 1: mp_clear(&b); mp_init(&b); break; + case 2: mp_clear(&c); mp_init(&c); break; + case 3: mp_clear(&d); mp_init(&d); break; + case 4: mp_clear(&e); mp_init(&e); break; + case 5: mp_clear(&f); mp_init(&f); break; + case 6: break; /* don't clear any */ + } + + + printf("%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu ", add_n, sub_n, mul_n, div_n, sqr_n, mul2d_n, div2d_n, gcd_n, lcm_n, expt_n, inv_n, div2_n, mul2_n, add_d_n, sub_d_n); + fgets(cmd, 4095, stdin); + cmd[strlen(cmd)-1] = 0; + printf("%s ]\r",cmd); fflush(stdout); + if (!strcmp(cmd, "mul2d")) { ++mul2d_n; + fgets(buf, 4095, stdin); mp_read_radix(&a, buf, 64); + fgets(buf, 4095, stdin); sscanf(buf, "%d", &rr); + fgets(buf, 4095, stdin); mp_read_radix(&b, buf, 64); + + mp_mul_2d(&a, rr, &a); + a.sign = b.sign; + if (mp_cmp(&a, &b) != MP_EQ) { + printf("mul2d failed, rr == %d\n",rr); + draw(&a); + draw(&b); + return 0; + } + } else if (!strcmp(cmd, "div2d")) { ++div2d_n; + fgets(buf, 4095, stdin); mp_read_radix(&a, buf, 64); + fgets(buf, 4095, stdin); sscanf(buf, "%d", &rr); + fgets(buf, 4095, stdin); mp_read_radix(&b, buf, 64); + + mp_div_2d(&a, rr, &a, &e); + a.sign = b.sign; + if (a.used == b.used && a.used == 0) { a.sign = b.sign = MP_ZPOS; } + if (mp_cmp(&a, &b) != MP_EQ) { + printf("div2d failed, rr == %d\n",rr); + draw(&a); + draw(&b); + return 0; + } + } else if (!strcmp(cmd, "add")) { ++add_n; + fgets(buf, 4095, stdin); mp_read_radix(&a, buf, 64); + fgets(buf, 4095, stdin); mp_read_radix(&b, buf, 64); + fgets(buf, 4095, stdin); mp_read_radix(&c, buf, 64); + mp_copy(&a, &d); + mp_add(&d, &b, &d); + if (mp_cmp(&c, &d) != MP_EQ) { + printf("add %lu failure!\n", add_n); +draw(&a);draw(&b);draw(&c);draw(&d); + return 0; + } + + /* test the sign/unsigned storage functions */ + + rr = mp_signed_bin_size(&c); + mp_to_signed_bin(&c, (unsigned char *)cmd); + memset(cmd+rr, rand()&255, sizeof(cmd)-rr); + mp_read_signed_bin(&d, (unsigned char *)cmd, rr); + if (mp_cmp(&c, &d) != MP_EQ) { + printf("mp_signed_bin failure!\n"); + draw(&c); + draw(&d); + return 0; + } + + + rr = mp_unsigned_bin_size(&c); + mp_to_unsigned_bin(&c, (unsigned char *)cmd); + memset(cmd+rr, rand()&255, sizeof(cmd)-rr); + mp_read_unsigned_bin(&d, (unsigned char *)cmd, rr); + if (mp_cmp_mag(&c, &d) != MP_EQ) { + printf("mp_unsigned_bin failure!\n"); + draw(&c); + draw(&d); + return 0; + } + + } else if (!strcmp(cmd, "sub")) { ++sub_n; + fgets(buf, 4095, stdin); mp_read_radix(&a, buf, 64); + fgets(buf, 4095, stdin); mp_read_radix(&b, buf, 64); + fgets(buf, 4095, stdin); mp_read_radix(&c, buf, 64); + mp_copy(&a, &d); + mp_sub(&d, &b, &d); + if (mp_cmp(&c, &d) != MP_EQ) { + printf("sub %lu failure!\n", sub_n); +draw(&a);draw(&b);draw(&c);draw(&d); + return 0; + } + } else if (!strcmp(cmd, "mul")) { ++mul_n; + fgets(buf, 4095, stdin); mp_read_radix(&a, buf, 64); + fgets(buf, 4095, stdin); mp_read_radix(&b, buf, 64); + fgets(buf, 4095, stdin); mp_read_radix(&c, buf, 64); + mp_copy(&a, &d); + mp_mul(&d, &b, &d); + if (mp_cmp(&c, &d) != MP_EQ) { + printf("mul %lu failure!\n", mul_n); +draw(&a);draw(&b);draw(&c);draw(&d); + return 0; + } + } else if (!strcmp(cmd, "div")) { ++div_n; + fgets(buf, 4095, stdin); mp_read_radix(&a, buf, 64); + fgets(buf, 4095, stdin); mp_read_radix(&b, buf, 64); + fgets(buf, 4095, stdin); mp_read_radix(&c, buf, 64); + fgets(buf, 4095, stdin); mp_read_radix(&d, buf, 64); + + mp_div(&a, &b, &e, &f); + if (mp_cmp(&c, &e) != MP_EQ || mp_cmp(&d, &f) != MP_EQ) { + printf("div %lu %d, %d, failure!\n", div_n, mp_cmp(&c, &e), mp_cmp(&d, &f)); +draw(&a);draw(&b);draw(&c);draw(&d); draw(&e); draw(&f); + return 0; + } + + } else if (!strcmp(cmd, "sqr")) { ++sqr_n; + fgets(buf, 4095, stdin); mp_read_radix(&a, buf, 64); + fgets(buf, 4095, stdin); mp_read_radix(&b, buf, 64); + mp_copy(&a, &c); + mp_sqr(&c, &c); + if (mp_cmp(&b, &c) != MP_EQ) { + printf("sqr %lu failure!\n", sqr_n); +draw(&a);draw(&b);draw(&c); + return 0; + } + } else if (!strcmp(cmd, "gcd")) { ++gcd_n; + fgets(buf, 4095, stdin); mp_read_radix(&a, buf, 64); + fgets(buf, 4095, stdin); mp_read_radix(&b, buf, 64); + fgets(buf, 4095, stdin); mp_read_radix(&c, buf, 64); + mp_copy(&a, &d); + mp_gcd(&d, &b, &d); + d.sign = c.sign; + if (mp_cmp(&c, &d) != MP_EQ) { + printf("gcd %lu failure!\n", gcd_n); +draw(&a);draw(&b);draw(&c);draw(&d); + return 0; + } + } else if (!strcmp(cmd, "lcm")) { ++lcm_n; + fgets(buf, 4095, stdin); mp_read_radix(&a, buf, 64); + fgets(buf, 4095, stdin); mp_read_radix(&b, buf, 64); + fgets(buf, 4095, stdin); mp_read_radix(&c, buf, 64); + mp_copy(&a, &d); + mp_lcm(&d, &b, &d); + d.sign = c.sign; + if (mp_cmp(&c, &d) != MP_EQ) { + printf("lcm %lu failure!\n", lcm_n); + draw(&a);draw(&b);draw(&c);draw(&d); + return 0; + } + } else if (!strcmp(cmd, "expt")) { ++expt_n; + fgets(buf, 4095, stdin); mp_read_radix(&a, buf, 64); + fgets(buf, 4095, stdin); mp_read_radix(&b, buf, 64); + fgets(buf, 4095, stdin); mp_read_radix(&c, buf, 64); + fgets(buf, 4095, stdin); mp_read_radix(&d, buf, 64); + mp_copy(&a, &e); + mp_exptmod(&e, &b, &c, &e); + if (mp_cmp(&d, &e) != MP_EQ) { + printf("expt %lu failure!\n", expt_n); + draw(&a);draw(&b);draw(&c);draw(&d); draw(&e); + return 0; + } + } else if (!strcmp(cmd, "invmod")) { ++inv_n; + fgets(buf, 4095, stdin); mp_read_radix(&a, buf, 64); + fgets(buf, 4095, stdin); mp_read_radix(&b, buf, 64); + fgets(buf, 4095, stdin); mp_read_radix(&c, buf, 64); + mp_invmod(&a, &b, &d); + mp_mulmod(&d,&a,&b,&e); + if (mp_cmp_d(&e, 1) != MP_EQ) { + printf("inv [wrong value from MPI?!] failure\n"); + draw(&a);draw(&b);draw(&c);draw(&d); + mp_gcd(&a, &b, &e); + draw(&e); + return 0; + } + + } else if (!strcmp(cmd, "div2")) { ++div2_n; + fgets(buf, 4095, stdin); mp_read_radix(&a, buf, 64); + fgets(buf, 4095, stdin); mp_read_radix(&b, buf, 64); + mp_div_2(&a, &c); + if (mp_cmp(&c, &b) != MP_EQ) { + printf("div_2 %lu failure\n", div2_n); + draw(&a); + draw(&b); + draw(&c); + return 0; + } + } else if (!strcmp(cmd, "mul2")) { ++mul2_n; + fgets(buf, 4095, stdin); mp_read_radix(&a, buf, 64); + fgets(buf, 4095, stdin); mp_read_radix(&b, buf, 64); + mp_mul_2(&a, &c); + if (mp_cmp(&c, &b) != MP_EQ) { + printf("mul_2 %lu failure\n", mul2_n); + draw(&a); + draw(&b); + draw(&c); + return 0; + } + } else if (!strcmp(cmd, "add_d")) { ++add_d_n; + fgets(buf, 4095, stdin); mp_read_radix(&a, buf, 64); + fgets(buf, 4095, stdin); sscanf(buf, "%d", &ix); + fgets(buf, 4095, stdin); mp_read_radix(&b, buf, 64); + mp_add_d(&a, ix, &c); + if (mp_cmp(&b, &c) != MP_EQ) { + printf("add_d %lu failure\n", add_d_n); + draw(&a); + draw(&b); + draw(&c); + printf("d == %d\n", ix); + return 0; + } + } else if (!strcmp(cmd, "sub_d")) { ++sub_d_n; + fgets(buf, 4095, stdin); mp_read_radix(&a, buf, 64); + fgets(buf, 4095, stdin); sscanf(buf, "%d", &ix); + fgets(buf, 4095, stdin); mp_read_radix(&b, buf, 64); + mp_sub_d(&a, ix, &c); + if (mp_cmp(&b, &c) != MP_EQ) { + printf("sub_d %lu failure\n", sub_d_n); + draw(&a); + draw(&b); + draw(&c); + printf("d == %d\n", ix); + return 0; + } + } + } + return 0; +} + diff --git a/libtommath/demo/timing.c b/libtommath/demo/timing.c new file mode 100644 index 0000000..7b27d53 --- /dev/null +++ b/libtommath/demo/timing.c @@ -0,0 +1,287 @@ +#include <tommath.h> +#include <time.h> + +ulong64 _tt; + +#ifdef IOWNANATHLON +#include <unistd.h> +#define SLEEP sleep(4) +#else +#define SLEEP +#endif + + +void ndraw(mp_int *a, char *name) +{ + char buf[4096]; + printf("%s: ", name); + mp_toradix(a, buf, 64); + printf("%s\n", buf); +} + +static void draw(mp_int *a) +{ + ndraw(a, ""); +} + + +unsigned long lfsr = 0xAAAAAAAAUL; + +int lbit(void) +{ + if (lfsr & 0x80000000UL) { + lfsr = ((lfsr << 1) ^ 0x8000001BUL) & 0xFFFFFFFFUL; + return 1; + } else { + lfsr <<= 1; + return 0; + } +} + +/* RDTSC from Scott Duplichan */ +static ulong64 TIMFUNC (void) + { + #if defined __GNUC__ + #if defined(__i386__) || defined(__x86_64__) + unsigned long long a; + __asm__ __volatile__ ("rdtsc\nmovl %%eax,%0\nmovl %%edx,4+%0\n"::"m"(a):"%eax","%edx"); + return a; + #else /* gcc-IA64 version */ + unsigned long result; + __asm__ __volatile__("mov %0=ar.itc" : "=r"(result) :: "memory"); + while (__builtin_expect ((int) result == -1, 0)) + __asm__ __volatile__("mov %0=ar.itc" : "=r"(result) :: "memory"); + return result; + #endif + + // Microsoft and Intel Windows compilers + #elif defined _M_IX86 + __asm rdtsc + #elif defined _M_AMD64 + return __rdtsc (); + #elif defined _M_IA64 + #if defined __INTEL_COMPILER + #include <ia64intrin.h> + #endif + return __getReg (3116); + #else + #error need rdtsc function for this build + #endif + } + +#define DO(x) x; x; +//#define DO4(x) DO2(x); DO2(x); +//#define DO8(x) DO4(x); DO4(x); +//#define DO(x) DO8(x); DO8(x); + +int main(void) +{ + ulong64 tt, gg, CLK_PER_SEC; + FILE *log, *logb, *logc; + mp_int a, b, c, d, e, f; + int n, cnt, ix, old_kara_m, old_kara_s; + unsigned rr; + + mp_init(&a); + mp_init(&b); + mp_init(&c); + mp_init(&d); + mp_init(&e); + mp_init(&f); + + srand(time(NULL)); + + + /* temp. turn off TOOM */ + TOOM_MUL_CUTOFF = TOOM_SQR_CUTOFF = 100000; + + CLK_PER_SEC = TIMFUNC(); + sleep(1); + CLK_PER_SEC = TIMFUNC() - CLK_PER_SEC; + + printf("CLK_PER_SEC == %llu\n", CLK_PER_SEC); + + log = fopen("logs/add.log", "w"); + for (cnt = 8; cnt <= 128; cnt += 8) { + SLEEP; + mp_rand(&a, cnt); + mp_rand(&b, cnt); + rr = 0; + tt = -1; + do { + gg = TIMFUNC(); + DO(mp_add(&a,&b,&c)); + gg = (TIMFUNC() - gg)>>1; + if (tt > gg) tt = gg; + } while (++rr < 100000); + printf("Adding\t\t%4d-bit => %9llu/sec, %9llu cycles\n", mp_count_bits(&a), CLK_PER_SEC/tt, tt); + fprintf(log, "%d %9llu\n", cnt*DIGIT_BIT, tt); fflush(log); + } + fclose(log); + + log = fopen("logs/sub.log", "w"); + for (cnt = 8; cnt <= 128; cnt += 8) { + SLEEP; + mp_rand(&a, cnt); + mp_rand(&b, cnt); + rr = 0; + tt = -1; + do { + gg = TIMFUNC(); + DO(mp_sub(&a,&b,&c)); + gg = (TIMFUNC() - gg)>>1; + if (tt > gg) tt = gg; + } while (++rr < 100000); + + printf("Subtracting\t\t%4d-bit => %9llu/sec, %9llu cycles\n", mp_count_bits(&a), CLK_PER_SEC/tt, tt); + fprintf(log, "%d %9llu\n", cnt*DIGIT_BIT, tt); fflush(log); + } + fclose(log); + + /* do mult/square twice, first without karatsuba and second with */ + old_kara_m = KARATSUBA_MUL_CUTOFF; + old_kara_s = KARATSUBA_SQR_CUTOFF; + for (ix = 0; ix < 1; ix++) { + printf("With%s Karatsuba\n", (ix==0)?"out":""); + + KARATSUBA_MUL_CUTOFF = (ix==0)?9999:old_kara_m; + KARATSUBA_SQR_CUTOFF = (ix==0)?9999:old_kara_s; + + log = fopen((ix==0)?"logs/mult.log":"logs/mult_kara.log", "w"); + for (cnt = 4; cnt <= 288; cnt += 2) { + SLEEP; + mp_rand(&a, cnt); + mp_rand(&b, cnt); + rr = 0; + tt = -1; + do { + gg = TIMFUNC(); + DO(mp_mul(&a, &b, &c)); + gg = (TIMFUNC() - gg)>>1; + if (tt > gg) tt = gg; + } while (++rr < 100); + printf("Multiplying\t%4d-bit => %9llu/sec, %9llu cycles\n", mp_count_bits(&a), CLK_PER_SEC/tt, tt); + fprintf(log, "%d %9llu\n", mp_count_bits(&a), tt); fflush(log); + } + fclose(log); + + log = fopen((ix==0)?"logs/sqr.log":"logs/sqr_kara.log", "w"); + for (cnt = 4; cnt <= 288; cnt += 2) { + SLEEP; + mp_rand(&a, cnt); + rr = 0; + tt = -1; + do { + gg = TIMFUNC(); + DO(mp_sqr(&a, &b)); + gg = (TIMFUNC() - gg)>>1; + if (tt > gg) tt = gg; + } while (++rr < 100); + printf("Squaring\t%4d-bit => %9llu/sec, %9llu cycles\n", mp_count_bits(&a), CLK_PER_SEC/tt, tt); + fprintf(log, "%d %9llu\n", mp_count_bits(&a), tt); fflush(log); + } + fclose(log); + + } + + { + char *primes[] = { + /* 2K moduli mersenne primes */ + "6864797660130609714981900799081393217269435300143305409394463459185543183397656052122559640661454554977296311391480858037121987999716643812574028291115057151", + "531137992816767098689588206552468627329593117727031923199444138200403559860852242739162502265229285668889329486246501015346579337652707239409519978766587351943831270835393219031728127", + "10407932194664399081925240327364085538615262247266704805319112350403608059673360298012239441732324184842421613954281007791383566248323464908139906605677320762924129509389220345773183349661583550472959420547689811211693677147548478866962501384438260291732348885311160828538416585028255604666224831890918801847068222203140521026698435488732958028878050869736186900714720710555703168729087", + "1475979915214180235084898622737381736312066145333169775147771216478570297878078949377407337049389289382748507531496480477281264838760259191814463365330269540496961201113430156902396093989090226259326935025281409614983499388222831448598601834318536230923772641390209490231836446899608210795482963763094236630945410832793769905399982457186322944729636418890623372171723742105636440368218459649632948538696905872650486914434637457507280441823676813517852099348660847172579408422316678097670224011990280170474894487426924742108823536808485072502240519452587542875349976558572670229633962575212637477897785501552646522609988869914013540483809865681250419497686697771007", + "259117086013202627776246767922441530941818887553125427303974923161874019266586362086201209516800483406550695241733194177441689509238807017410377709597512042313066624082916353517952311186154862265604547691127595848775610568757931191017711408826252153849035830401185072116424747461823031471398340229288074545677907941037288235820705892351068433882986888616658650280927692080339605869308790500409503709875902119018371991620994002568935113136548829739112656797303241986517250116412703509705427773477972349821676443446668383119322540099648994051790241624056519054483690809616061625743042361721863339415852426431208737266591962061753535748892894599629195183082621860853400937932839420261866586142503251450773096274235376822938649407127700846077124211823080804139298087057504713825264571448379371125032081826126566649084251699453951887789613650248405739378594599444335231188280123660406262468609212150349937584782292237144339628858485938215738821232393687046160677362909315071", + "190797007524439073807468042969529173669356994749940177394741882673528979787005053706368049835514900244303495954950709725762186311224148828811920216904542206960744666169364221195289538436845390250168663932838805192055137154390912666527533007309292687539092257043362517857366624699975402375462954490293259233303137330643531556539739921926201438606439020075174723029056838272505051571967594608350063404495977660656269020823960825567012344189908927956646011998057988548630107637380993519826582389781888135705408653045219655801758081251164080554609057468028203308718724654081055323215860189611391296030471108443146745671967766308925858547271507311563765171008318248647110097614890313562856541784154881743146033909602737947385055355960331855614540900081456378659068370317267696980001187750995491090350108417050917991562167972281070161305972518044872048331306383715094854938415738549894606070722584737978176686422134354526989443028353644037187375385397838259511833166416134323695660367676897722287918773420968982326089026150031515424165462111337527431154890666327374921446276833564519776797633875503548665093914556482031482248883127023777039667707976559857333357013727342079099064400455741830654320379350833236245819348824064783585692924881021978332974949906122664421376034687815350484991", + + /* DR moduli */ + "14059105607947488696282932836518693308967803494693489478439861164411992439598399594747002144074658928593502845729752797260025831423419686528151609940203368612079", + "101745825697019260773923519755878567461315282017759829107608914364075275235254395622580447400994175578963163918967182013639660669771108475957692810857098847138903161308502419410142185759152435680068435915159402496058513611411688900243039", + "736335108039604595805923406147184530889923370574768772191969612422073040099331944991573923112581267542507986451953227192970402893063850485730703075899286013451337291468249027691733891486704001513279827771740183629161065194874727962517148100775228363421083691764065477590823919364012917984605619526140821797602431", + "38564998830736521417281865696453025806593491967131023221754800625044118265468851210705360385717536794615180260494208076605798671660719333199513807806252394423283413430106003596332513246682903994829528690198205120921557533726473585751382193953592127439965050261476810842071573684505878854588706623484573925925903505747545471088867712185004135201289273405614415899438276535626346098904241020877974002916168099951885406379295536200413493190419727789712076165162175783", + "542189391331696172661670440619180536749994166415993334151601745392193484590296600979602378676624808129613777993466242203025054573692562689251250471628358318743978285860720148446448885701001277560572526947619392551574490839286458454994488665744991822837769918095117129546414124448777033941223565831420390846864429504774477949153794689948747680362212954278693335653935890352619041936727463717926744868338358149568368643403037768649616778526013610493696186055899318268339432671541328195724261329606699831016666359440874843103020666106568222401047720269951530296879490444224546654729111504346660859907296364097126834834235287147", + "1487259134814709264092032648525971038895865645148901180585340454985524155135260217788758027400478312256339496385275012465661575576202252063145698732079880294664220579764848767704076761853197216563262660046602703973050798218246170835962005598561669706844469447435461092542265792444947706769615695252256130901271870341005768912974433684521436211263358097522726462083917939091760026658925757076733484173202927141441492573799914240222628795405623953109131594523623353044898339481494120112723445689647986475279242446083151413667587008191682564376412347964146113898565886683139407005941383669325997475076910488086663256335689181157957571445067490187939553165903773554290260531009121879044170766615232300936675369451260747671432073394867530820527479172464106442450727640226503746586340279816318821395210726268291535648506190714616083163403189943334431056876038286530365757187367147446004855912033137386225053275419626102417236133948503", + "1095121115716677802856811290392395128588168592409109494900178008967955253005183831872715423151551999734857184538199864469605657805519106717529655044054833197687459782636297255219742994736751541815269727940751860670268774903340296040006114013971309257028332849679096824800250742691718610670812374272414086863715763724622797509437062518082383056050144624962776302147890521249477060215148275163688301275847155316042279405557632639366066847442861422164832655874655824221577849928863023018366835675399949740429332468186340518172487073360822220449055340582568461568645259954873303616953776393853174845132081121976327462740354930744487429617202585015510744298530101547706821590188733515880733527449780963163909830077616357506845523215289297624086914545378511082534229620116563260168494523906566709418166011112754529766183554579321224940951177394088465596712620076240067370589036924024728375076210477267488679008016579588696191194060127319035195370137160936882402244399699172017835144537488486396906144217720028992863941288217185353914991583400421682751000603596655790990815525126154394344641336397793791497068253936771017031980867706707490224041075826337383538651825493679503771934836094655802776331664261631740148281763487765852746577808019633679", + + /* generic unrestricted moduli */ + "17933601194860113372237070562165128350027320072176844226673287945873370751245439587792371960615073855669274087805055507977323024886880985062002853331424203", + "2893527720709661239493896562339544088620375736490408468011883030469939904368086092336458298221245707898933583190713188177399401852627749210994595974791782790253946539043962213027074922559572312141181787434278708783207966459019479487", + "347743159439876626079252796797422223177535447388206607607181663903045907591201940478223621722118173270898487582987137708656414344685816179420855160986340457973820182883508387588163122354089264395604796675278966117567294812714812796820596564876450716066283126720010859041484786529056457896367683122960411136319", + "47266428956356393164697365098120418976400602706072312735924071745438532218237979333351774907308168340693326687317443721193266215155735814510792148768576498491199122744351399489453533553203833318691678263241941706256996197460424029012419012634671862283532342656309677173602509498417976091509154360039893165037637034737020327399910409885798185771003505320583967737293415979917317338985837385734747478364242020380416892056650841470869294527543597349250299539682430605173321029026555546832473048600327036845781970289288898317888427517364945316709081173840186150794397479045034008257793436817683392375274635794835245695887", + "436463808505957768574894870394349739623346440601945961161254440072143298152040105676491048248110146278752857839930515766167441407021501229924721335644557342265864606569000117714935185566842453630868849121480179691838399545644365571106757731317371758557990781880691336695584799313313687287468894148823761785582982549586183756806449017542622267874275103877481475534991201849912222670102069951687572917937634467778042874315463238062009202992087620963771759666448266532858079402669920025224220613419441069718482837399612644978839925207109870840278194042158748845445131729137117098529028886770063736487420613144045836803985635654192482395882603511950547826439092832800532152534003936926017612446606135655146445620623395788978726744728503058670046885876251527122350275750995227", + "11424167473351836398078306042624362277956429440521137061889702611766348760692206243140413411077394583180726863277012016602279290144126785129569474909173584789822341986742719230331946072730319555984484911716797058875905400999504305877245849119687509023232790273637466821052576859232452982061831009770786031785669030271542286603956118755585683996118896215213488875253101894663403069677745948305893849505434201763745232895780711972432011344857521691017896316861403206449421332243658855453435784006517202894181640562433575390821384210960117518650374602256601091379644034244332285065935413233557998331562749140202965844219336298970011513882564935538704289446968322281451907487362046511461221329799897350993370560697505809686438782036235372137015731304779072430260986460269894522159103008260495503005267165927542949439526272736586626709581721032189532726389643625590680105784844246152702670169304203783072275089194754889511973916207", + "1214855636816562637502584060163403830270705000634713483015101384881871978446801224798536155406895823305035467591632531067547890948695117172076954220727075688048751022421198712032848890056357845974246560748347918630050853933697792254955890439720297560693579400297062396904306270145886830719309296352765295712183040773146419022875165382778007040109957609739589875590885701126197906063620133954893216612678838507540777138437797705602453719559017633986486649523611975865005712371194067612263330335590526176087004421363598470302731349138773205901447704682181517904064735636518462452242791676541725292378925568296858010151852326316777511935037531017413910506921922450666933202278489024521263798482237150056835746454842662048692127173834433089016107854491097456725016327709663199738238442164843147132789153725513257167915555162094970853584447993125488607696008169807374736711297007473812256272245489405898470297178738029484459690836250560495461579533254473316340608217876781986188705928270735695752830825527963838355419762516246028680280988020401914551825487349990306976304093109384451438813251211051597392127491464898797406789175453067960072008590614886532333015881171367104445044718144312416815712216611576221546455968770801413440778423979", + NULL + }; + log = fopen("logs/expt.log", "w"); + logb = fopen("logs/expt_dr.log", "w"); + logc = fopen("logs/expt_2k.log", "w"); + for (n = 0; primes[n]; n++) { + SLEEP; + mp_read_radix(&a, primes[n], 10); + mp_zero(&b); + for (rr = 0; rr < (unsigned)mp_count_bits(&a); rr++) { + mp_mul_2(&b, &b); + b.dp[0] |= lbit(); + b.used += 1; + } + mp_sub_d(&a, 1, &c); + mp_mod(&b, &c, &b); + mp_set(&c, 3); + rr = 0; + tt = -1; + do { + gg = TIMFUNC(); + DO(mp_exptmod(&c, &b, &a, &d)); + gg = (TIMFUNC() - gg)>>1; + if (tt > gg) tt = gg; + } while (++rr < 10); + mp_sub_d(&a, 1, &e); + mp_sub(&e, &b, &b); + mp_exptmod(&c, &b, &a, &e); /* c^(p-1-b) mod a */ + mp_mulmod(&e, &d, &a, &d); /* c^b * c^(p-1-b) == c^p-1 == 1 */ + if (mp_cmp_d(&d, 1)) { + printf("Different (%d)!!!\n", mp_count_bits(&a)); + draw(&d); + exit(0); + } + printf("Exponentiating\t%4d-bit => %9llu/sec, %9llu cycles\n", mp_count_bits(&a), CLK_PER_SEC/tt, tt); + fprintf((n < 6) ? logc : (n < 13) ? logb : log, "%d %9llu\n", mp_count_bits(&a), tt); + } + } + fclose(log); + fclose(logb); + fclose(logc); + + log = fopen("logs/invmod.log", "w"); + for (cnt = 4; cnt <= 128; cnt += 4) { + SLEEP; + mp_rand(&a, cnt); + mp_rand(&b, cnt); + + do { + mp_add_d(&b, 1, &b); + mp_gcd(&a, &b, &c); + } while (mp_cmp_d(&c, 1) != MP_EQ); + + rr = 0; + tt = -1; + do { + gg = TIMFUNC(); + DO(mp_invmod(&b, &a, &c)); + gg = (TIMFUNC() - gg)>>1; + if (tt > gg) tt = gg; + } while (++rr < 1000); + mp_mulmod(&b, &c, &a, &d); + if (mp_cmp_d(&d, 1) != MP_EQ) { + printf("Failed to invert\n"); + return 0; + } + printf("Inverting mod\t%4d-bit => %9llu/sec, %9llu cycles\n", mp_count_bits(&a), CLK_PER_SEC/tt, tt); + fprintf(log, "%d %9llu\n", cnt*DIGIT_BIT, tt); + } + fclose(log); + + return 0; +} + diff --git a/libtommath/dep.pl b/libtommath/dep.pl new file mode 100644 index 0000000..22266e3 --- /dev/null +++ b/libtommath/dep.pl @@ -0,0 +1,121 @@ +#!/usr/bin/perl +# +# Walk through source, add labels and make classes +# +#use strict; + +my %deplist; + +#open class file and write preamble +open(CLASS, ">tommath_class.h") or die "Couldn't open tommath_class.h for writing\n"; +print CLASS "#if !(defined(LTM1) && defined(LTM2) && defined(LTM3))\n#if defined(LTM2)\n#define LTM3\n#endif\n#if defined(LTM1)\n#define LTM2\n#endif\n#define LTM1\n\n#if defined(LTM_ALL)\n"; + +foreach my $filename (glob "bn*.c") { + my $define = $filename; + + # convert filename to upper case so we can use it as a define + $define =~ tr/[a-z]/[A-Z]/; + $define =~ tr/\./_/; + print CLASS "#define $define\n"; + + # now copy text and apply #ifdef as required + my $apply = 0; + open(SRC, "<$filename"); + open(OUT, ">tmp"); + + # first line will be the #ifdef + my $line = <SRC>; + if ($line =~ /include/) { + print OUT $line; + } else { + print OUT "#include <tommath.h>\n#ifdef $define\n$line"; + $apply = 1; + } + while (<SRC>) { + if (!($_ =~ /tommath\.h/)) { + print OUT $_; + } + } + if ($apply == 1) { + print OUT "#endif\n"; + } + close SRC; + close OUT; + + unlink($filename); + rename("tmp", $filename); +} +print CLASS "#endif\n\n"; + +# now do classes + +foreach my $filename (glob "bn*.c") { + open(SRC, "<$filename") or die "Can't open source file!\n"; + + # convert filename to upper case so we can use it as a define + $filename =~ tr/[a-z]/[A-Z]/; + $filename =~ tr/\./_/; + + print CLASS "#if defined($filename)\n"; + my $list = $filename; + + # scan for mp_* and make classes + while (<SRC>) { + my $line = $_; + while ($line =~ m/(fast_)*(s_)*mp\_[a-z_0-9]*/) { + $line = $'; + # now $& is the match, we want to skip over LTM keywords like + # mp_int, mp_word, mp_digit + if (!($& eq "mp_digit") && !($& eq "mp_word") && !($& eq "mp_int")) { + my $a = $&; + $a =~ tr/[a-z]/[A-Z]/; + $a = "BN_" . $a . "_C"; + if (!($list =~ /$a/)) { + print CLASS " #define $a\n"; + } + $list = $list . "," . $a; + } + } + } + @deplist{$filename} = $list; + + print CLASS "#endif\n\n"; + close SRC; +} + +print CLASS "#ifdef LTM3\n#define LTM_LAST\n#endif\n#include <tommath_superclass.h>\n#include <tommath_class.h>\n#else\n#define LTM_LAST\n#endif\n"; +close CLASS; + +#now let's make a cool call graph... + +open(OUT,">callgraph.txt"); +$indent = 0; +foreach (keys %deplist) { + $list = ""; + draw_func(@deplist{$_}); + print OUT "\n\n"; +} +close(OUT); + +sub draw_func() +{ + my @funcs = split(",", $_[0]); + if ($list =~ /@funcs[0]/) { + return; + } else { + $list = $list . @funcs[0]; + } + if ($indent == 0) { } + elsif ($indent >= 1) { print OUT "| " x ($indent - 1) . "+--->"; } + print OUT @funcs[0] . "\n"; + shift @funcs; + my $temp = $list; + foreach my $i (@funcs) { + ++$indent; + draw_func(@deplist{$i}); + --$indent; + } + $list = $temp; +} + + diff --git a/libtommath/etc/2kprime.1 b/libtommath/etc/2kprime.1 new file mode 100644 index 0000000..c41ded1 --- /dev/null +++ b/libtommath/etc/2kprime.1 @@ -0,0 +1,2 @@ +256-bits (k = 36113) = 115792089237316195423570985008687907853269984665640564039457584007913129603823 +512-bits (k = 38117) = 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006045979 diff --git a/libtommath/etc/2kprime.c b/libtommath/etc/2kprime.c new file mode 100644 index 0000000..d48b83e --- /dev/null +++ b/libtommath/etc/2kprime.c @@ -0,0 +1,80 @@ +/* Makes safe primes of a 2k nature */ +#include <tommath.h> +#include <time.h> + +int sizes[] = {256, 512, 768, 1024, 1536, 2048, 3072, 4096}; + +int main(void) +{ + char buf[2000]; + int x, y; + mp_int q, p; + FILE *out; + clock_t t1; + mp_digit z; + + mp_init_multi(&q, &p, NULL); + + out = fopen("2kprime.1", "w"); + for (x = 0; x < (int)(sizeof(sizes) / sizeof(sizes[0])); x++) { + top: + mp_2expt(&q, sizes[x]); + mp_add_d(&q, 3, &q); + z = -3; + + t1 = clock(); + for(;;) { + mp_sub_d(&q, 4, &q); + z += 4; + + if (z > MP_MASK) { + printf("No primes of size %d found\n", sizes[x]); + break; + } + + if (clock() - t1 > CLOCKS_PER_SEC) { + printf("."); fflush(stdout); +// sleep((clock() - t1 + CLOCKS_PER_SEC/2)/CLOCKS_PER_SEC); + t1 = clock(); + } + + /* quick test on q */ + mp_prime_is_prime(&q, 1, &y); + if (y == 0) { + continue; + } + + /* find (q-1)/2 */ + mp_sub_d(&q, 1, &p); + mp_div_2(&p, &p); + mp_prime_is_prime(&p, 3, &y); + if (y == 0) { + continue; + } + + /* test on q */ + mp_prime_is_prime(&q, 3, &y); + if (y == 0) { + continue; + } + + break; + } + + if (y == 0) { + ++sizes[x]; + goto top; + } + + mp_toradix(&q, buf, 10); + printf("\n\n%d-bits (k = %lu) = %s\n", sizes[x], z, buf); + fprintf(out, "%d-bits (k = %lu) = %s\n", sizes[x], z, buf); fflush(out); + } + + return 0; +} + + + + + diff --git a/libtommath/etc/drprime.c b/libtommath/etc/drprime.c new file mode 100644 index 0000000..0ab8ea6 --- /dev/null +++ b/libtommath/etc/drprime.c @@ -0,0 +1,60 @@ +/* Makes safe primes of a DR nature */ +#include <tommath.h> + +int sizes[] = { 1+256/DIGIT_BIT, 1+512/DIGIT_BIT, 1+768/DIGIT_BIT, 1+1024/DIGIT_BIT, 1+2048/DIGIT_BIT, 1+4096/DIGIT_BIT }; +int main(void) +{ + int res, x, y; + char buf[4096]; + FILE *out; + mp_int a, b; + + mp_init(&a); + mp_init(&b); + + out = fopen("drprimes.txt", "w"); + for (x = 0; x < (int)(sizeof(sizes)/sizeof(sizes[0])); x++) { + top: + printf("Seeking a %d-bit safe prime\n", sizes[x] * DIGIT_BIT); + mp_grow(&a, sizes[x]); + mp_zero(&a); + for (y = 1; y < sizes[x]; y++) { + a.dp[y] = MP_MASK; + } + + /* make a DR modulus */ + a.dp[0] = -1; + a.used = sizes[x]; + + /* now loop */ + res = 0; + for (;;) { + a.dp[0] += 4; + if (a.dp[0] >= MP_MASK) break; + mp_prime_is_prime(&a, 1, &res); + if (res == 0) continue; + printf("."); fflush(stdout); + mp_sub_d(&a, 1, &b); + mp_div_2(&b, &b); + mp_prime_is_prime(&b, 3, &res); + if (res == 0) continue; + mp_prime_is_prime(&a, 3, &res); + if (res == 1) break; + } + + if (res != 1) { + printf("Error not DR modulus\n"); sizes[x] += 1; goto top; + } else { + mp_toradix(&a, buf, 10); + printf("\n\np == %s\n\n", buf); + fprintf(out, "%d-bit prime:\np == %s\n\n", mp_count_bits(&a), buf); fflush(out); + } + } + fclose(out); + + mp_clear(&a); + mp_clear(&b); + + return 0; +} + diff --git a/libtommath/etc/drprimes.28 b/libtommath/etc/drprimes.28 new file mode 100644 index 0000000..9d438ad --- /dev/null +++ b/libtommath/etc/drprimes.28 @@ -0,0 +1,25 @@ +DR safe primes for 28-bit digits. + +224-bit prime: +p == 26959946667150639794667015087019630673637144422540572481103341844143 + +532-bit prime: +p == 14059105607947488696282932836518693308967803494693489478439861164411992439598399594747002144074658928593502845729752797260025831423419686528151609940203368691747 + +784-bit prime: +p == 101745825697019260773923519755878567461315282017759829107608914364075275235254395622580447400994175578963163918967182013639660669771108475957692810857098847138903161308502419410142185759152435680068435915159402496058513611411688900243039 + +1036-bit prime: +p == 736335108039604595805923406147184530889923370574768772191969612422073040099331944991573923112581267542507986451953227192970402893063850485730703075899286013451337291468249027691733891486704001513279827771740183629161065194874727962517148100775228363421083691764065477590823919364012917984605619526140821798437127 + +1540-bit prime: +p == 38564998830736521417281865696453025806593491967131023221754800625044118265468851210705360385717536794615180260494208076605798671660719333199513807806252394423283413430106003596332513246682903994829528690198205120921557533726473585751382193953592127439965050261476810842071573684505878854588706623484573925925903505747545471088867712185004135201289273405614415899438276535626346098904241020877974002916168099951885406379295536200413493190419727789712076165162175783 + +2072-bit prime: +p == 542189391331696172661670440619180536749994166415993334151601745392193484590296600979602378676624808129613777993466242203025054573692562689251250471628358318743978285860720148446448885701001277560572526947619392551574490839286458454994488665744991822837769918095117129546414124448777033941223565831420390846864429504774477949153794689948747680362212954278693335653935890352619041936727463717926744868338358149568368643403037768649616778526013610493696186055899318268339432671541328195724261329606699831016666359440874843103020666106568222401047720269951530296879490444224546654729111504346660859907296364097126834834235287147 + +3080-bit prime: +p == 1487259134814709264092032648525971038895865645148901180585340454985524155135260217788758027400478312256339496385275012465661575576202252063145698732079880294664220579764848767704076761853197216563262660046602703973050798218246170835962005598561669706844469447435461092542265792444947706769615695252256130901271870341005768912974433684521436211263358097522726462083917939091760026658925757076733484173202927141441492573799914240222628795405623953109131594523623353044898339481494120112723445689647986475279242446083151413667587008191682564376412347964146113898565886683139407005941383669325997475076910488086663256335689181157957571445067490187939553165903773554290260531009121879044170766615232300936675369451260747671432073394867530820527479172464106442450727640226503746586340279816318821395210726268291535648506190714616083163403189943334431056876038286530365757187367147446004855912033137386225053275419626102417236133948503 + +4116-bit prime: +p == 1095121115716677802856811290392395128588168592409109494900178008967955253005183831872715423151551999734857184538199864469605657805519106717529655044054833197687459782636297255219742994736751541815269727940751860670268774903340296040006114013971309257028332849679096824800250742691718610670812374272414086863715763724622797509437062518082383056050144624962776302147890521249477060215148275163688301275847155316042279405557632639366066847442861422164832655874655824221577849928863023018366835675399949740429332468186340518172487073360822220449055340582568461568645259954873303616953776393853174845132081121976327462740354930744487429617202585015510744298530101547706821590188733515880733527449780963163909830077616357506845523215289297624086914545378511082534229620116563260168494523906566709418166011112754529766183554579321224940951177394088465596712620076240067370589036924024728375076210477267488679008016579588696191194060127319035195370137160936882402244399699172017835144537488486396906144217720028992863941288217185353914991583400421682751000603596655790990815525126154394344641336397793791497068253936771017031980867706707490224041075826337383538651825493679503771934836094655802776331664261631740148281763487765852746577808019633679 diff --git a/libtommath/etc/drprimes.txt b/libtommath/etc/drprimes.txt new file mode 100644 index 0000000..2c887ea --- /dev/null +++ b/libtommath/etc/drprimes.txt @@ -0,0 +1,6 @@ +280-bit prime: +p == 1942668892225729070919461906823518906642406839052139521251812409738904285204940164839 + +532-bit prime: +p == 14059105607947488696282932836518693308967803494693489478439861164411992439598399594747002144074658928593502845729752797260025831423419686528151609940203368691747 + diff --git a/libtommath/etc/makefile b/libtommath/etc/makefile new file mode 100644 index 0000000..99154d8 --- /dev/null +++ b/libtommath/etc/makefile @@ -0,0 +1,50 @@ +CFLAGS += -Wall -W -Wshadow -O3 -fomit-frame-pointer -funroll-loops -I../ + +# default lib name (requires install with root) +# LIBNAME=-ltommath + +# libname when you can't install the lib with install +LIBNAME=../libtommath.a + +#provable primes +pprime: pprime.o + $(CC) pprime.o $(LIBNAME) -o pprime + +# portable [well requires clock()] tuning app +tune: tune.o + $(CC) tune.o $(LIBNAME) -o tune + +# same app but using RDTSC for higher precision [requires 80586+], coff based gcc installs [e.g. ming, cygwin, djgpp] +tune86: tune.c + nasm -f coff timer.asm + $(CC) -DX86_TIMER $(CFLAGS) tune.c timer.o $(LIBNAME) -o tune86 + +# for cygwin +tune86c: tune.c + nasm -f gnuwin32 timer.asm + $(CC) -DX86_TIMER $(CFLAGS) tune.c timer.o $(LIBNAME) -o tune86 + +#make tune86 for linux or any ELF format +tune86l: tune.c + nasm -f elf -DUSE_ELF timer.asm + $(CC) -DX86_TIMER $(CFLAGS) tune.c timer.o $(LIBNAME) -o tune86l + +# spits out mersenne primes +mersenne: mersenne.o + $(CC) mersenne.o $(LIBNAME) -o mersenne + +# fines DR safe primes for the given config +drprime: drprime.o + $(CC) drprime.o $(LIBNAME) -o drprime + +# fines 2k safe primes for the given config +2kprime: 2kprime.o + $(CC) 2kprime.o $(LIBNAME) -o 2kprime + +mont: mont.o + $(CC) mont.o $(LIBNAME) -o mont + + +clean: + rm -f *.log *.o *.obj *.exe pprime tune mersenne drprime tune86 tune86l mont 2kprime pprime.dat \ + *.da *.dyn *.dpi *~ diff --git a/libtommath/etc/makefile.icc b/libtommath/etc/makefile.icc new file mode 100644 index 0000000..0a50728 --- /dev/null +++ b/libtommath/etc/makefile.icc @@ -0,0 +1,67 @@ +CC = icc + +CFLAGS += -I../ + +# optimize for SPEED +# +# -mcpu= can be pentium, pentiumpro (covers PII through PIII) or pentium4 +# -ax? specifies make code specifically for ? but compatible with IA-32 +# -x? specifies compile solely for ? [not specifically IA-32 compatible] +# +# where ? is +# K - PIII +# W - first P4 [Williamette] +# N - P4 Northwood +# P - P4 Prescott +# B - Blend of P4 and PM [mobile] +# +# Default to just generic max opts +CFLAGS += -O3 -xN -ip + +# default lib name (requires install with root) +# LIBNAME=-ltommath + +# libname when you can't install the lib with install +LIBNAME=../libtommath.a + +#provable primes +pprime: pprime.o + $(CC) pprime.o $(LIBNAME) -o pprime + +# portable [well requires clock()] tuning app +tune: tune.o + $(CC) tune.o $(LIBNAME) -o tune + +# same app but using RDTSC for higher precision [requires 80586+], coff based gcc installs [e.g. ming, cygwin, djgpp] +tune86: tune.c + nasm -f coff timer.asm + $(CC) -DX86_TIMER $(CFLAGS) tune.c timer.o $(LIBNAME) -o tune86 + +# for cygwin +tune86c: tune.c + nasm -f gnuwin32 timer.asm + $(CC) -DX86_TIMER $(CFLAGS) tune.c timer.o $(LIBNAME) -o tune86 + +#make tune86 for linux or any ELF format +tune86l: tune.c + nasm -f elf -DUSE_ELF timer.asm + $(CC) -DX86_TIMER $(CFLAGS) tune.c timer.o $(LIBNAME) -o tune86l + +# spits out mersenne primes +mersenne: mersenne.o + $(CC) mersenne.o $(LIBNAME) -o mersenne + +# fines DR safe primes for the given config +drprime: drprime.o + $(CC) drprime.o $(LIBNAME) -o drprime + +# fines 2k safe primes for the given config +2kprime: 2kprime.o + $(CC) 2kprime.o $(LIBNAME) -o 2kprime + +mont: mont.o + $(CC) mont.o $(LIBNAME) -o mont + + +clean: + rm -f *.log *.o *.obj *.exe pprime tune mersenne drprime tune86 tune86l mont 2kprime pprime.dat *.il diff --git a/libtommath/etc/makefile.msvc b/libtommath/etc/makefile.msvc new file mode 100644 index 0000000..2833372 --- /dev/null +++ b/libtommath/etc/makefile.msvc @@ -0,0 +1,23 @@ +#MSVC Makefile +# +#Tom St Denis + +CFLAGS = /I../ /Ox /DWIN32 /W3 + +pprime: pprime.obj + cl pprime.obj ../tommath.lib + +mersenne: mersenne.obj + cl mersenne.obj ../tommath.lib + +tune: tune.obj + cl tune.obj ../tommath.lib + +mont: mont.obj + cl mont.obj ../tommath.lib + +drprime: drprime.obj + cl drprime.obj ../tommath.lib + +2kprime: 2kprime.obj + cl 2kprime.obj ../tommath.lib diff --git a/libtommath/etc/mersenne.c b/libtommath/etc/mersenne.c new file mode 100644 index 0000000..1cd5b50 --- /dev/null +++ b/libtommath/etc/mersenne.c @@ -0,0 +1,140 @@ +/* Finds Mersenne primes using the Lucas-Lehmer test + * + * Tom St Denis, tomstdenis@iahu.ca + */ +#include <time.h> +#include <tommath.h> + +int +is_mersenne (long s, int *pp) +{ + mp_int n, u; + int res, k; + + *pp = 0; + + if ((res = mp_init (&n)) != MP_OKAY) { + return res; + } + + if ((res = mp_init (&u)) != MP_OKAY) { + goto LBL_N; + } + + /* n = 2^s - 1 */ + if ((res = mp_2expt(&n, s)) != MP_OKAY) { + goto LBL_MU; + } + if ((res = mp_sub_d (&n, 1, &n)) != MP_OKAY) { + goto LBL_MU; + } + + /* set u=4 */ + mp_set (&u, 4); + + /* for k=1 to s-2 do */ + for (k = 1; k <= s - 2; k++) { + /* u = u^2 - 2 mod n */ + if ((res = mp_sqr (&u, &u)) != MP_OKAY) { + goto LBL_MU; + } + if ((res = mp_sub_d (&u, 2, &u)) != MP_OKAY) { + goto LBL_MU; + } + + /* make sure u is positive */ + while (u.sign == MP_NEG) { + if ((res = mp_add (&u, &n, &u)) != MP_OKAY) { + goto LBL_MU; + } + } + + /* reduce */ + if ((res = mp_reduce_2k (&u, &n, 1)) != MP_OKAY) { + goto LBL_MU; + } + } + + /* if u == 0 then its prime */ + if (mp_iszero (&u) == 1) { + mp_prime_is_prime(&n, 8, pp); + if (*pp != 1) printf("FAILURE\n"); + } + + res = MP_OKAY; +LBL_MU:mp_clear (&u); +LBL_N:mp_clear (&n); + return res; +} + +/* square root of a long < 65536 */ +long +i_sqrt (long x) +{ + long x1, x2; + + x2 = 16; + do { + x1 = x2; + x2 = x1 - ((x1 * x1) - x) / (2 * x1); + } while (x1 != x2); + + if (x1 * x1 > x) { + --x1; + } + + return x1; +} + +/* is the long prime by brute force */ +int +isprime (long k) +{ + long y, z; + + y = i_sqrt (k); + for (z = 2; z <= y; z++) { + if ((k % z) == 0) + return 0; + } + return 1; +} + + +int +main (void) +{ + int pp; + long k; + clock_t tt; + + k = 3; + + for (;;) { + /* start time */ + tt = clock (); + + /* test if 2^k - 1 is prime */ + if (is_mersenne (k, &pp) != MP_OKAY) { + printf ("Whoa error\n"); + return -1; + } + + if (pp == 1) { + /* count time */ + tt = clock () - tt; + + /* display if prime */ + printf ("2^%-5ld - 1 is prime, test took %ld ticks\n", k, tt); + } + + /* goto next odd exponent */ + k += 2; + + /* but make sure its prime */ + while (isprime (k) == 0) { + k += 2; + } + } + return 0; +} diff --git a/libtommath/etc/mont.c b/libtommath/etc/mont.c new file mode 100644 index 0000000..dbf1735 --- /dev/null +++ b/libtommath/etc/mont.c @@ -0,0 +1,46 @@ +/* tests the montgomery routines */ +#include <tommath.h> + +int main(void) +{ + mp_int modulus, R, p, pp; + mp_digit mp; + long x, y; + + srand(time(NULL)); + mp_init_multi(&modulus, &R, &p, &pp, NULL); + + /* loop through various sizes */ + for (x = 4; x < 256; x++) { + printf("DIGITS == %3ld...", x); fflush(stdout); + + /* make up the odd modulus */ + mp_rand(&modulus, x); + modulus.dp[0] |= 1; + + /* now find the R value */ + mp_montgomery_calc_normalization(&R, &modulus); + mp_montgomery_setup(&modulus, &mp); + + /* now run through a bunch tests */ + for (y = 0; y < 1000; y++) { + mp_rand(&p, x/2); /* p = random */ + mp_mul(&p, &R, &pp); /* pp = R * p */ + mp_montgomery_reduce(&pp, &modulus, mp); + + /* should be equal to p */ + if (mp_cmp(&pp, &p) != MP_EQ) { + printf("FAILURE!\n"); + exit(-1); + } + } + printf("PASSED\n"); + } + + return 0; +} + + + + + diff --git a/libtommath/etc/pprime.c b/libtommath/etc/pprime.c new file mode 100644 index 0000000..26e0d84 --- /dev/null +++ b/libtommath/etc/pprime.c @@ -0,0 +1,396 @@ +/* Generates provable primes + * + * See http://iahu.ca:8080/papers/pp.pdf for more info. + * + * Tom St Denis, tomstdenis@iahu.ca, http://tom.iahu.ca + */ +#include <time.h> +#include "tommath.h" + +int n_prime; +FILE *primes; + +/* fast square root */ +static mp_digit +i_sqrt (mp_word x) +{ + mp_word x1, x2; + + x2 = x; + do { + x1 = x2; + x2 = x1 - ((x1 * x1) - x) / (2 * x1); + } while (x1 != x2); + + if (x1 * x1 > x) { + --x1; + } + + return x1; +} + + +/* generates a prime digit */ +static void gen_prime (void) +{ + mp_digit r, x, y, next; + FILE *out; + + out = fopen("pprime.dat", "wb"); + + /* write first set of primes */ + r = 3; fwrite(&r, 1, sizeof(mp_digit), out); + r = 5; fwrite(&r, 1, sizeof(mp_digit), out); + r = 7; fwrite(&r, 1, sizeof(mp_digit), out); + r = 11; fwrite(&r, 1, sizeof(mp_digit), out); + r = 13; fwrite(&r, 1, sizeof(mp_digit), out); + r = 17; fwrite(&r, 1, sizeof(mp_digit), out); + r = 19; fwrite(&r, 1, sizeof(mp_digit), out); + r = 23; fwrite(&r, 1, sizeof(mp_digit), out); + r = 29; fwrite(&r, 1, sizeof(mp_digit), out); + r = 31; fwrite(&r, 1, sizeof(mp_digit), out); + + /* get square root, since if 'r' is composite its factors must be < than this */ + y = i_sqrt (r); + next = (y + 1) * (y + 1); + + for (;;) { + do { + r += 2; /* next candidate */ + r &= MP_MASK; + if (r < 31) break; + + /* update sqrt ? */ + if (next <= r) { + ++y; + next = (y + 1) * (y + 1); + } + + /* loop if divisible by 3,5,7,11,13,17,19,23,29 */ + if ((r % 3) == 0) { + x = 0; + continue; + } + if ((r % 5) == 0) { + x = 0; + continue; + } + if ((r % 7) == 0) { + x = 0; + continue; + } + if ((r % 11) == 0) { + x = 0; + continue; + } + if ((r % 13) == 0) { + x = 0; + continue; + } + if ((r % 17) == 0) { + x = 0; + continue; + } + if ((r % 19) == 0) { + x = 0; + continue; + } + if ((r % 23) == 0) { + x = 0; + continue; + } + if ((r % 29) == 0) { + x = 0; + continue; + } + + /* now check if r is divisible by x + k={1,7,11,13,17,19,23,29} */ + for (x = 30; x <= y; x += 30) { + if ((r % (x + 1)) == 0) { + x = 0; + break; + } + if ((r % (x + 7)) == 0) { + x = 0; + break; + } + if ((r % (x + 11)) == 0) { + x = 0; + break; + } + if ((r % (x + 13)) == 0) { + x = 0; + break; + } + if ((r % (x + 17)) == 0) { + x = 0; + break; + } + if ((r % (x + 19)) == 0) { + x = 0; + break; + } + if ((r % (x + 23)) == 0) { + x = 0; + break; + } + if ((r % (x + 29)) == 0) { + x = 0; + break; + } + } + } while (x == 0); + if (r > 31) { fwrite(&r, 1, sizeof(mp_digit), out); printf("%9d\r", r); fflush(stdout); } + if (r < 31) break; + } + + fclose(out); +} + +void load_tab(void) +{ + primes = fopen("pprime.dat", "rb"); + if (primes == NULL) { + gen_prime(); + primes = fopen("pprime.dat", "rb"); + } + fseek(primes, 0, SEEK_END); + n_prime = ftell(primes) / sizeof(mp_digit); +} + +mp_digit prime_digit(void) +{ + int n; + mp_digit d; + + n = abs(rand()) % n_prime; + fseek(primes, n * sizeof(mp_digit), SEEK_SET); + fread(&d, 1, sizeof(mp_digit), primes); + return d; +} + + +/* makes a prime of at least k bits */ +int +pprime (int k, int li, mp_int * p, mp_int * q) +{ + mp_int a, b, c, n, x, y, z, v; + int res, ii; + static const mp_digit bases[] = { 2, 3, 5, 7, 11, 13, 17, 19 }; + + /* single digit ? */ + if (k <= (int) DIGIT_BIT) { + mp_set (p, prime_digit ()); + return MP_OKAY; + } + + if ((res = mp_init (&c)) != MP_OKAY) { + return res; + } + + if ((res = mp_init (&v)) != MP_OKAY) { + goto LBL_C; + } + + /* product of first 50 primes */ + if ((res = + mp_read_radix (&v, + "19078266889580195013601891820992757757219839668357012055907516904309700014933909014729740190", + 10)) != MP_OKAY) { + goto LBL_V; + } + + if ((res = mp_init (&a)) != MP_OKAY) { + goto LBL_V; + } + + /* set the prime */ + mp_set (&a, prime_digit ()); + + if ((res = mp_init (&b)) != MP_OKAY) { + goto LBL_A; + } + + if ((res = mp_init (&n)) != MP_OKAY) { + goto LBL_B; + } + + if ((res = mp_init (&x)) != MP_OKAY) { + goto LBL_N; + } + + if ((res = mp_init (&y)) != MP_OKAY) { + goto LBL_X; + } + + if ((res = mp_init (&z)) != MP_OKAY) { + goto LBL_Y; + } + + /* now loop making the single digit */ + while (mp_count_bits (&a) < k) { + fprintf (stderr, "prime has %4d bits left\r", k - mp_count_bits (&a)); + fflush (stderr); + top: + mp_set (&b, prime_digit ()); + + /* now compute z = a * b * 2 */ + if ((res = mp_mul (&a, &b, &z)) != MP_OKAY) { /* z = a * b */ + goto LBL_Z; + } + + if ((res = mp_copy (&z, &c)) != MP_OKAY) { /* c = a * b */ + goto LBL_Z; + } + + if ((res = mp_mul_2 (&z, &z)) != MP_OKAY) { /* z = 2 * a * b */ + goto LBL_Z; + } + + /* n = z + 1 */ + if ((res = mp_add_d (&z, 1, &n)) != MP_OKAY) { /* n = z + 1 */ + goto LBL_Z; + } + + /* check (n, v) == 1 */ + if ((res = mp_gcd (&n, &v, &y)) != MP_OKAY) { /* y = (n, v) */ + goto LBL_Z; + } + + if (mp_cmp_d (&y, 1) != MP_EQ) + goto top; + + /* now try base x=bases[ii] */ + for (ii = 0; ii < li; ii++) { + mp_set (&x, bases[ii]); + + /* compute x^a mod n */ + if ((res = mp_exptmod (&x, &a, &n, &y)) != MP_OKAY) { /* y = x^a mod n */ + goto LBL_Z; + } + + /* if y == 1 loop */ + if (mp_cmp_d (&y, 1) == MP_EQ) + continue; + + /* now x^2a mod n */ + if ((res = mp_sqrmod (&y, &n, &y)) != MP_OKAY) { /* y = x^2a mod n */ + goto LBL_Z; + } + + if (mp_cmp_d (&y, 1) == MP_EQ) + continue; + + /* compute x^b mod n */ + if ((res = mp_exptmod (&x, &b, &n, &y)) != MP_OKAY) { /* y = x^b mod n */ + goto LBL_Z; + } + + /* if y == 1 loop */ + if (mp_cmp_d (&y, 1) == MP_EQ) + continue; + + /* now x^2b mod n */ + if ((res = mp_sqrmod (&y, &n, &y)) != MP_OKAY) { /* y = x^2b mod n */ + goto LBL_Z; + } + + if (mp_cmp_d (&y, 1) == MP_EQ) + continue; + + /* compute x^c mod n == x^ab mod n */ + if ((res = mp_exptmod (&x, &c, &n, &y)) != MP_OKAY) { /* y = x^ab mod n */ + goto LBL_Z; + } + + /* if y == 1 loop */ + if (mp_cmp_d (&y, 1) == MP_EQ) + continue; + + /* now compute (x^c mod n)^2 */ + if ((res = mp_sqrmod (&y, &n, &y)) != MP_OKAY) { /* y = x^2ab mod n */ + goto LBL_Z; + } + + /* y should be 1 */ + if (mp_cmp_d (&y, 1) != MP_EQ) + continue; + break; + } + + /* no bases worked? */ + if (ii == li) + goto top; + +{ + char buf[4096]; + + mp_toradix(&n, buf, 10); + printf("Certificate of primality for:\n%s\n\n", buf); + mp_toradix(&a, buf, 10); + printf("A == \n%s\n\n", buf); + mp_toradix(&b, buf, 10); + printf("B == \n%s\n\nG == %d\n", buf, bases[ii]); + printf("----------------------------------------------------------------\n"); +} + + /* a = n */ + mp_copy (&n, &a); + } + + /* get q to be the order of the large prime subgroup */ + mp_sub_d (&n, 1, q); + mp_div_2 (q, q); + mp_div (q, &b, q, NULL); + + mp_exch (&n, p); + + res = MP_OKAY; +LBL_Z:mp_clear (&z); +LBL_Y:mp_clear (&y); +LBL_X:mp_clear (&x); +LBL_N:mp_clear (&n); +LBL_B:mp_clear (&b); +LBL_A:mp_clear (&a); +LBL_V:mp_clear (&v); +LBL_C:mp_clear (&c); + return res; +} + + +int +main (void) +{ + mp_int p, q; + char buf[4096]; + int k, li; + clock_t t1; + + srand (time (NULL)); + load_tab(); + + printf ("Enter # of bits: \n"); + fgets (buf, sizeof (buf), stdin); + sscanf (buf, "%d", &k); + + printf ("Enter number of bases to try (1 to 8):\n"); + fgets (buf, sizeof (buf), stdin); + sscanf (buf, "%d", &li); + + + mp_init (&p); + mp_init (&q); + + t1 = clock (); + pprime (k, li, &p, &q); + t1 = clock () - t1; + + printf ("\n\nTook %ld ticks, %d bits\n", t1, mp_count_bits (&p)); + + mp_toradix (&p, buf, 10); + printf ("P == %s\n", buf); + mp_toradix (&q, buf, 10); + printf ("Q == %s\n", buf); + + return 0; +} diff --git a/libtommath/etc/prime.1024 b/libtommath/etc/prime.1024 new file mode 100644 index 0000000..5636e2d --- /dev/null +++ b/libtommath/etc/prime.1024 @@ -0,0 +1,414 @@ +Enter # of bits: +Enter number of bases to try (1 to 8): +Certificate of primality for: +36360080703173363 + +A == +89963569 + +B == +202082249 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +4851595597739856136987139 + +A == +36360080703173363 + +B == +66715963 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +19550639734462621430325731591027 + +A == +4851595597739856136987139 + +B == +2014867 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +10409036141344317165691858509923818734539 + +A == +19550639734462621430325731591027 + +B == +266207047 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +1049829549988285012736475602118094726647504414203 + +A == +10409036141344317165691858509923818734539 + +B == +50428759 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +77194737385528288387712399596835459931920358844586615003 + +A == +1049829549988285012736475602118094726647504414203 + +B == +36765367 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +35663756695365208574443215955488689578374232732893628896541201763 + +A == +77194737385528288387712399596835459931920358844586615003 + +B == +230998627 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +16711831463502165169495622246023119698415848120292671294127567620396469803 + +A == +35663756695365208574443215955488689578374232732893628896541201763 + +B == +234297127 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +6163534781560285962890718925972249753147470953579266394395432475622345597103528739 + +A == +16711831463502165169495622246023119698415848120292671294127567620396469803 + +B == +184406323 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +814258256205243497704094951432575867360065658372158511036259934640748088306764553488803787 + +A == +6163534781560285962890718925972249753147470953579266394395432475622345597103528739 + +B == +66054487 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +176469695533271657902814176811660357049007467856432383037590673407330246967781451723764079581998187 + +A == +814258256205243497704094951432575867360065658372158511036259934640748088306764553488803787 + +B == +108362239 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +44924492859445516541759485198544012102424796403707253610035148063863073596051272171194806669756971406400419 + +A == +176469695533271657902814176811660357049007467856432383037590673407330246967781451723764079581998187 + +B == +127286707 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +20600996927219343383225424320134474929609459588323857796871086845924186191561749519858600696159932468024710985371059 + +A == +44924492859445516541759485198544012102424796403707253610035148063863073596051272171194806669756971406400419 + +B == +229284691 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +6295696427695493110141186605837397185848992307978456138112526915330347715236378041486547994708748840844217371233735072572979 + +A == +20600996927219343383225424320134474929609459588323857796871086845924186191561749519858600696159932468024710985371059 + +B == +152800771 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +3104984078042317488749073016454213579257792635142218294052134804187631661145261015102617582090263808696699966840735333252107678792123 + +A == +6295696427695493110141186605837397185848992307978456138112526915330347715236378041486547994708748840844217371233735072572979 + +B == +246595759 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +26405175827665701256325699315126705508919255051121452292124404943796947287968603975320562847910946802396632302209435206627913466015741799499 + +A == +3104984078042317488749073016454213579257792635142218294052134804187631661145261015102617582090263808696699966840735333252107678792123 + +B == +4252063 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +11122146237908413610034600609460545703591095894418599759742741406628055069007082998134905595800236452010905900391505454890446585211975124558601770163 + +A == +26405175827665701256325699315126705508919255051121452292124404943796947287968603975320562847910946802396632302209435206627913466015741799499 + +B == +210605419 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +1649861642047798890580354082088712649911849362201343649289384923147797960364736011515757482030049342943790127685185806092659832129486307035500638595572396187 + +A == +11122146237908413610034600609460545703591095894418599759742741406628055069007082998134905595800236452010905900391505454890446585211975124558601770163 + +B == +74170111 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +857983367126266717607389719637086684134462613006415859877666235955788392464081914127715967940968197765042399904117392707518175220864852816390004264107201177394565363 + +A == +1649861642047798890580354082088712649911849362201343649289384923147797960364736011515757482030049342943790127685185806092659832129486307035500638595572396187 + +B == +260016763 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +175995909353623703257072120479340610010337144085688850745292031336724691277374210929188442230237711063783727092685448718515661641054886101716698390145283196296702450566161283 + +A == +857983367126266717607389719637086684134462613006415859877666235955788392464081914127715967940968197765042399904117392707518175220864852816390004264107201177394565363 + +B == +102563707 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: 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for: +6355194692790533601105154341731997464407930009404822926832136060319955058388106456084549316415200519472481147942263916585428906582726749131479465958107142228236909665306781538860053107680830113869123 + +A == +13156468011529105025061495011938518171328604045212410096476697450506055664012861932372156505805788068791146986282263016790631108386790291275939575123375304599622623328517354163964228279867403 + +B == +241523587 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +3157116676535430302794438027544146642863331358530722860333745617571010460905857862561870488000265751138954271040017454405707755458702044884023184574412221802502351503929935224995314581932097706874819348858083 + +A == +6355194692790533601105154341731997464407930009404822926832136060319955058388106456084549316415200519472481147942263916585428906582726749131479465958107142228236909665306781538860053107680830113869123 + +B == +248388667 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +390533129219992506725320633489467713907837370444962163378727819939092929448752905310115311180032249230394348337568973177802874166228132778126338883671958897238722734394783244237133367055422297736215754829839364158067 + +A == +3157116676535430302794438027544146642863331358530722860333745617571010460905857862561870488000265751138954271040017454405707755458702044884023184574412221802502351503929935224995314581932097706874819348858083 + +B == +61849651 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +48583654555070224891047847050732516652910250240135992225139515777200432486685999462997073444468380434359929499498804723793106565291183220444221080449740542884172281158126259373095216435009661050109711341419005972852770440739 + +A == 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primality for: +168459393231883505975876919268398655632763956627405508859662408056221544310200546265681845397346956580604208064328814319465940958080244889692368602591598503944015835190587740756859842792554282496742843600573336023639256008687581291233481455395123454655488735304365627 + +A == +392847529056126766528615419937165193421166694172790666626558750047057558168124866940509180171236517681470100877687445134633784815352076138790217228749332398026714192707447855731679485746120589851992221508292976900578299504461333767437280988393026452846013683 + +B == +214408111 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +14865774288636941404884923981945833072113667565310054952177860608355263252462409554658728941191929400198053290113492910272458441655458514080123870132092365833472436407455910185221474386718838138135065780840839893113912689594815485706154461164071775481134379794909690501684643 + +A == +168459393231883505975876919268398655632763956627405508859662408056221544310200546265681845397346956580604208064328814319465940958080244889692368602591598503944015835190587740756859842792554282496742843600573336023639256008687581291233481455395123454655488735304365627 + +B == +44122723 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +1213301773203241614897109856134894783021668292000023984098824423682568173639394290886185366993108292039068940333907505157813934962357206131450244004178619265868614859794316361031904412926604138893775068853175215502104744339658944443630407632290152772487455298652998368296998719996019 + +A == +14865774288636941404884923981945833072113667565310054952177860608355263252462409554658728941191929400198053290113492910272458441655458514080123870132092365833472436407455910185221474386718838138135065780840839893113912689594815485706154461164071775481134379794909690501684643 + +B == +40808563 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +186935245989515158127969129347464851990429060640910951266513740972248428651109062997368144722015290092846666943896556191257222521203647606911446635194198213436423080005867489516421559330500722264446765608763224572386410155413161172707802334865729654109050873820610813855041667633843601286843 + +A == +1213301773203241614897109856134894783021668292000023984098824423682568173639394290886185366993108292039068940333907505157813934962357206131450244004178619265868614859794316361031904412926604138893775068853175215502104744339658944443630407632290152772487455298652998368296998719996019 + +B == +77035759 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +83142661079751490510739960019112406284111408348732592580459037404394946037094409915127399165633756159385609671956087845517678367844901424617866988187132480585966721962585586730693443536100138246516868613250009028187662080828012497191775172228832247706080044971423654632146928165751885302331924491683 + +A == +186935245989515158127969129347464851990429060640910951266513740972248428651109062997368144722015290092846666943896556191257222521203647606911446635194198213436423080005867489516421559330500722264446765608763224572386410155413161172707802334865729654109050873820610813855041667633843601286843 + +B == +222383587 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +3892354773803809855317742245039794448230625839512638747643814927766738642436392673485997449586432241626440927010641564064764336402368634186618250134234189066179771240232458249806850838490410473462391401438160528157981942499581634732706904411807195259620779379274017704050790865030808501633772117217899534443 + +A == +83142661079751490510739960019112406284111408348732592580459037404394946037094409915127399165633756159385609671956087845517678367844901424617866988187132480585966721962585586730693443536100138246516868613250009028187662080828012497191775172228832247706080044971423654632146928165751885302331924491683 + +B == +23407687 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +1663606652988091811284014366560171522582683318514519379924950390627250155440313691226744227787921928894551755219495501365555370027257568506349958010457682898612082048959464465369892842603765280317696116552850664773291371490339084156052244256635115997453399761029567033971998617303988376172539172702246575225837054723 + +A == +3892354773803809855317742245039794448230625839512638747643814927766738642436392673485997449586432241626440927010641564064764336402368634186618250134234189066179771240232458249806850838490410473462391401438160528157981942499581634732706904411807195259620779379274017704050790865030808501633772117217899534443 + +B == +213701827 + +G == 2 +---------------------------------------------------------------- + + +Took 33057 ticks, 1048 bits +P == 1663606652988091811284014366560171522582683318514519379924950390627250155440313691226744227787921928894551755219495501365555370027257568506349958010457682898612082048959464465369892842603765280317696116552850664773291371490339084156052244256635115997453399761029567033971998617303988376172539172702246575225837054723 +Q == 3892354773803809855317742245039794448230625839512638747643814927766738642436392673485997449586432241626440927010641564064764336402368634186618250134234189066179771240232458249806850838490410473462391401438160528157981942499581634732706904411807195259620779379274017704050790865030808501633772117217899534443 diff --git a/libtommath/etc/prime.512 b/libtommath/etc/prime.512 new file mode 100644 index 0000000..cb6ec30 --- /dev/null +++ b/libtommath/etc/prime.512 @@ -0,0 +1,205 @@ +Enter # of bits: +Enter number of bases to try (1 to 8): +Certificate of primality for: +85933926807634727 + +A == +253758023 + +B == +169322581 + +G == 5 +---------------------------------------------------------------- +Certificate of primality for: +23930198825086241462113799 + +A == +85933926807634727 + +B == +139236037 + +G == 11 +---------------------------------------------------------------- +Certificate of primality for: +6401844647261612602378676572510019 + +A == +23930198825086241462113799 + +B == +133760791 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +269731366027728777712034888684015329354259 + +A == +6401844647261612602378676572510019 + +B == +21066691 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +37942338209025571690075025099189467992329684223707 + +A == +269731366027728777712034888684015329354259 + +B == +70333567 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +15306904714258982484473490774101705363308327436988160248323 + +A == +37942338209025571690075025099189467992329684223707 + +B == +201712723 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +1616744757018513392810355191503853040357155275733333124624513530099 + +A == +15306904714258982484473490774101705363308327436988160248323 + +B == +52810963 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +464222094814208047161771036072622485188658077940154689939306386289983787983 + +A == +1616744757018513392810355191503853040357155275733333124624513530099 + +B == +143566909 + +G == 5 +---------------------------------------------------------------- +Certificate of primality for: +187429931674053784626487560729643601208757374994177258429930699354770049369025096447 + +A == +464222094814208047161771036072622485188658077940154689939306386289983787983 + +B == +201875281 + +G == 5 +---------------------------------------------------------------- +Certificate of primality for: +100579220846502621074093727119851331775052664444339632682598589456666938521976625305832917563 + +A == +187429931674053784626487560729643601208757374994177258429930699354770049369025096447 + +B == +268311523 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +1173616081309758475197022137833792133815753368965945885089720153370737965497134878651384030219765163 + +A == +100579220846502621074093727119851331775052664444339632682598589456666938521976625305832917563 + +B == +5834287 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +191456913489905913185935197655672585713573070349044195411728114905691721186574907738081340754373032735283623 + +A == +1173616081309758475197022137833792133815753368965945885089720153370737965497134878651384030219765163 + +B == +81567097 + +G == 5 +---------------------------------------------------------------- +Certificate of primality for: +57856530489201750164178576399448868489243874083056587683743345599898489554401618943240901541005080049321706789987519 + +A == +191456913489905913185935197655672585713573070349044195411728114905691721186574907738081340754373032735283623 + +B == +151095433 + +G == 7 +---------------------------------------------------------------- +Certificate of primality for: +13790529750452576698109671710773784949185621244122040804792403407272729038377767162233653248852099545134831722512085881814803 + +A == +57856530489201750164178576399448868489243874083056587683743345599898489554401618943240901541005080049321706789987519 + +B == +119178679 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +7075985989000817742677547821106534174334812111605018857703825637170140040509067704269696198231266351631132464035671858077052876058979 + +A == +13790529750452576698109671710773784949185621244122040804792403407272729038377767162233653248852099545134831722512085881814803 + +B == +256552363 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +1227273006232588072907488910282307435921226646895131225407452056677899411162892829564455154080310937471747140942360789623819327234258162420463 + +A == +7075985989000817742677547821106534174334812111605018857703825637170140040509067704269696198231266351631132464035671858077052876058979 + +B == +86720989 + +G == 5 +---------------------------------------------------------------- +Certificate of primality for: +446764896913554613686067036908702877942872355053329937790398156069936255759889884246832779737114032666318220500106499161852193765380831330106375235763 + +A == +1227273006232588072907488910282307435921226646895131225407452056677899411162892829564455154080310937471747140942360789623819327234258162420463 + +B == +182015287 + +G == 2 +---------------------------------------------------------------- +Certificate of primality for: +5290203010849586596974953717018896543907195901082056939587768479377028575911127944611236020459652034082251335583308070846379514569838984811187823420951275243 + +A == +446764896913554613686067036908702877942872355053329937790398156069936255759889884246832779737114032666318220500106499161852193765380831330106375235763 + +B == +5920567 + +G == 2 +---------------------------------------------------------------- + + +Took 3454 ticks, 521 bits +P == 5290203010849586596974953717018896543907195901082056939587768479377028575911127944611236020459652034082251335583308070846379514569838984811187823420951275243 +Q == 446764896913554613686067036908702877942872355053329937790398156069936255759889884246832779737114032666318220500106499161852193765380831330106375235763 diff --git a/libtommath/etc/timer.asm b/libtommath/etc/timer.asm new file mode 100644 index 0000000..35890d9 --- /dev/null +++ b/libtommath/etc/timer.asm @@ -0,0 +1,37 @@ +; x86 timer in NASM
+;
+; Tom St Denis, tomstdenis@iahu.ca
+[bits 32]
+[section .data]
+time dd 0, 0
+
+[section .text]
+
+%ifdef USE_ELF
+[global t_start]
+t_start:
+%else
+[global _t_start]
+_t_start:
+%endif
+ push edx
+ push eax
+ rdtsc
+ mov [time+0],edx
+ mov [time+4],eax
+ pop eax
+ pop edx
+ ret
+
+%ifdef USE_ELF
+[global t_read]
+t_read:
+%else
+[global _t_read]
+_t_read:
+%endif
+ rdtsc
+ sub eax,[time+4]
+ sbb edx,[time+0]
+ ret
+
\ No newline at end of file diff --git a/libtommath/etc/tune.c b/libtommath/etc/tune.c new file mode 100644 index 0000000..14aace2 --- /dev/null +++ b/libtommath/etc/tune.c @@ -0,0 +1,107 @@ +/* Tune the Karatsuba parameters + * + * Tom St Denis, tomstdenis@iahu.ca + */ +#include <tommath.h> +#include <time.h> + +/* how many times todo each size mult. Depends on your computer. For slow computers + * this can be low like 5 or 10. For fast [re: Athlon] should be 25 - 50 or so + */ +#define TIMES (1UL<<14UL) + + +#ifndef X86_TIMER + +/* generic ISO C timer */ +ulong64 LBL_T; +void t_start(void) { LBL_T = clock(); } +ulong64 t_read(void) { return clock() - LBL_T; } + +#else +extern void t_start(void); +extern ulong64 t_read(void); +#endif + +ulong64 time_mult(int size, int s) +{ + unsigned long x; + mp_int a, b, c; + ulong64 t1; + + mp_init (&a); + mp_init (&b); + mp_init (&c); + + mp_rand (&a, size); + mp_rand (&b, size); + + if (s == 1) { + KARATSUBA_MUL_CUTOFF = size; + } else { + KARATSUBA_MUL_CUTOFF = 100000; + } + + t_start(); + for (x = 0; x < TIMES; x++) { + mp_mul(&a,&b,&c); + } + t1 = t_read(); + mp_clear (&a); + mp_clear (&b); + mp_clear (&c); + return t1; +} + +ulong64 time_sqr(int size, int s) +{ + unsigned long x; + mp_int a, b; + ulong64 t1; + + mp_init (&a); + mp_init (&b); + + mp_rand (&a, size); + + if (s == 1) { + KARATSUBA_SQR_CUTOFF = size; + } else { + KARATSUBA_SQR_CUTOFF = 100000; + } + + t_start(); + for (x = 0; x < TIMES; x++) { + mp_sqr(&a,&b); + } + t1 = t_read(); + mp_clear (&a); + mp_clear (&b); + return t1; +} + +int +main (void) +{ + ulong64 t1, t2; + int x, y; + + for (x = 8; ; x += 2) { + t1 = time_mult(x, 0); + t2 = time_mult(x, 1); + printf("%d: %9llu %9llu, %9llu\n", x, t1, t2, t2 - t1); + if (t2 < t1) break; + } + y = x; + + for (x = 8; ; x += 2) { + t1 = time_sqr(x, 0); + t2 = time_sqr(x, 1); + printf("%d: %9llu %9llu, %9llu\n", x, t1, t2, t2 - t1); + if (t2 < t1) break; + } + printf("KARATSUBA_MUL_CUTOFF = %d\n", y); + printf("KARATSUBA_SQR_CUTOFF = %d\n", x); + + return 0; +} diff --git a/libtommath/gen.pl b/libtommath/gen.pl new file mode 100644 index 0000000..7236591 --- /dev/null +++ b/libtommath/gen.pl @@ -0,0 +1,17 @@ +#!/usr/bin/perl -w +# +# Generates a "single file" you can use to quickly +# add the whole source without any makefile troubles +# +use strict; + +open( OUT, ">mpi.c" ) or die "Couldn't open mpi.c for writing: $!"; +foreach my $filename (glob "bn*.c") { + open( SRC, "<$filename" ) or die "Couldn't open $filename for reading: $!"; + print OUT "/* Start: $filename */\n"; + print OUT while <SRC>; + print OUT "\n/* End: $filename */\n\n"; + close SRC or die "Error closing $filename after reading: $!"; +} +print OUT "\n/* EOF */\n"; +close OUT or die "Error closing mpi.c after writing: $!";
\ No newline at end of file diff --git a/libtommath/logs/README b/libtommath/logs/README new file mode 100644 index 0000000..ea20c81 --- /dev/null +++ b/libtommath/logs/README @@ -0,0 +1,13 @@ +To use the pretty graphs you have to first build/run the ltmtest from the root directory of the package.
+Todo this type
+
+make timing ; ltmtest
+
+in the root. It will run for a while [about ten minutes on most PCs] and produce a series of .log files in logs/.
+
+After doing that run "gnuplot graphs.dem" to make the PNGs. If you managed todo that all so far just open index.html to view
+them all :-)
+
+Have fun
+
+Tom
\ No newline at end of file diff --git a/libtommath/logs/add.log b/libtommath/logs/add.log new file mode 100644 index 0000000..fa11039 --- /dev/null +++ b/libtommath/logs/add.log @@ -0,0 +1,16 @@ +480 88 +960 113 +1440 138 +1920 163 +2400 202 +2880 226 +3360 251 +3840 272 +4320 296 +4800 320 +5280 344 +5760 368 +6240 392 +6720 416 +7200 440 +7680 464 diff --git a/libtommath/logs/addsub.png b/libtommath/logs/addsub.png Binary files differnew file mode 100644 index 0000000..a5679ac --- /dev/null +++ b/libtommath/logs/addsub.png diff --git a/libtommath/logs/expt.log b/libtommath/logs/expt.log new file mode 100644 index 0000000..e65e927 --- /dev/null +++ b/libtommath/logs/expt.log @@ -0,0 +1,7 @@ +513 1499509 +769 3682671 +1025 8098887 +2049 49332743 +2561 89647783 +3073 149440713 +4097 326135364 diff --git a/libtommath/logs/expt.png b/libtommath/logs/expt.png Binary files differnew file mode 100644 index 0000000..9ee8bb7 --- /dev/null +++ b/libtommath/logs/expt.png diff --git a/libtommath/logs/expt_2k.log b/libtommath/logs/expt_2k.log new file mode 100644 index 0000000..d106280 --- /dev/null +++ b/libtommath/logs/expt_2k.log @@ -0,0 +1,6 @@ +521 1423346 +607 1841305 +1279 8375656 +2203 34104708 +3217 83830729 +4253 167916804 diff --git a/libtommath/logs/expt_2kl.log b/libtommath/logs/expt_2kl.log new file mode 100644 index 0000000..b2eb8c2 --- /dev/null +++ b/libtommath/logs/expt_2kl.log @@ -0,0 +1,4 @@ +1024 6954080 +2048 35993987 +4096 176068521 +521 1683720 diff --git a/libtommath/logs/expt_dr.log b/libtommath/logs/expt_dr.log new file mode 100644 index 0000000..6cfc874 --- /dev/null +++ b/libtommath/logs/expt_dr.log @@ -0,0 +1,7 @@ +532 1803110 +784 3607375 +1036 6089790 +1540 14739797 +2072 33251589 +3080 82794331 +4116 165212734 diff --git a/libtommath/logs/graphs.dem b/libtommath/logs/graphs.dem new file mode 100644 index 0000000..dfaf613 --- /dev/null +++ b/libtommath/logs/graphs.dem @@ -0,0 +1,17 @@ +set terminal png +set size 1.75 +set ylabel "Cycles per Operation" +set xlabel "Operand size (bits)" + +set output "addsub.png" +plot 'add.log' smooth bezier title "Addition", 'sub.log' smooth bezier title "Subtraction" + +set output "mult.png" +plot 'sqr.log' smooth bezier title "Squaring (without Karatsuba)", 'sqr_kara.log' smooth bezier title "Squaring (Karatsuba)", 'mult.log' smooth bezier title "Multiplication (without Karatsuba)", 'mult_kara.log' smooth bezier title "Multiplication (Karatsuba)" + +set output "expt.png" +plot 'expt.log' smooth bezier title "Exptmod (Montgomery)", 'expt_dr.log' smooth bezier title "Exptmod (Dimminished Radix)", 'expt_2k.log' smooth bezier title "Exptmod (2k Reduction)" + +set output "invmod.png" +plot 'invmod.log' smooth bezier title "Modular Inverse" + diff --git a/libtommath/logs/index.html b/libtommath/logs/index.html new file mode 100644 index 0000000..19fe403 --- /dev/null +++ b/libtommath/logs/index.html @@ -0,0 +1,24 @@ +<html> +<head> +<title>LibTomMath Log Plots</title> +</head> +<body> + +<h1>Addition and Subtraction</h1> +<center><img src=addsub.png></center> +<hr> + +<h1>Multipliers</h1> +<center><img src=mult.png></center> +<hr> + +<h1>Exptmod</h1> +<center><img src=expt.png></center> +<hr> + +<h1>Modular Inverse</h1> +<center><img src=invmod.png></center> +<hr> + +</body> +</html>
\ No newline at end of file diff --git a/libtommath/logs/invmod.log b/libtommath/logs/invmod.log new file mode 100644 index 0000000..e69de29 --- /dev/null +++ b/libtommath/logs/invmod.log diff --git a/libtommath/logs/invmod.png b/libtommath/logs/invmod.png Binary files differnew file mode 100644 index 0000000..0a8a4ad --- /dev/null +++ b/libtommath/logs/invmod.png diff --git a/libtommath/logs/mult.log b/libtommath/logs/mult.log new file mode 100644 index 0000000..864de46 --- /dev/null +++ b/libtommath/logs/mult.log @@ -0,0 +1,143 @@ +271 580 +390 861 +511 1177 +630 1598 +749 2115 +871 2670 +991 3276 +1111 3987 +1231 4722 +1351 5474 +1471 6281 +1589 7126 +1710 8114 +1831 8988 +1946 10038 +2071 10995 +2188 12286 +2310 13152 +2430 14480 +2549 15521 +2671 17171 +2790 18081 +2911 19754 +3031 20809 +3150 22849 +3269 23757 +3391 25772 +3508 26832 +3631 29304 +3750 30149 +3865 32581 +3988 33644 +4111 36565 +4231 37309 +4351 40152 +4471 41188 +4590 44658 +4710 45256 +4827 48538 +4951 49490 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367526 +14189 365330 +14311 381551 +14429 374149 +14549 392203 +14670 389764 +14791 406761 +14910 398652 +15026 417718 +15150 414733 +15269 432759 +15390 1037071 +15511 1053454 +15631 1069198 +15748 1086164 +15871 1112820 +15991 1129676 +16111 1145924 +16230 1163016 +16345 1179911 +16471 1197048 +16586 1214352 +16711 1232095 +16829 1249338 +16947 1266987 +17071 1284181 +17188 1302521 +17311 1320539 diff --git a/libtommath/logs/mult.png b/libtommath/logs/mult.png Binary files differnew file mode 100644 index 0000000..4f7a4ee --- /dev/null +++ b/libtommath/logs/mult.png diff --git a/libtommath/logs/mult_kara.log b/libtommath/logs/mult_kara.log new file mode 100644 index 0000000..086feaf --- /dev/null +++ b/libtommath/logs/mult_kara.log @@ -0,0 +1,33 @@ +924 16686 +1146 25334 +1371 35304 +1591 47122 +1820 61500 +2044 75254 +2266 91732 +2492 111656 +2716 129428 +2937 147508 +3164 167758 +3388 188248 +3612 210826 +3836 233814 +4059 256898 +4284 280210 +4508 310372 +4731 333902 +4955 376502 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927989 +15870 940790 +15991 954948 +16110 969483 +16231 984544 +16350 997837 +16470 1012445 +16590 1027834 +16710 1043032 +16831 1056394 +16951 1071408 +17069 1097263 +17191 1113364 +17306 1123650 diff --git a/libtommath/logs/sqr_kara.log b/libtommath/logs/sqr_kara.log new file mode 100644 index 0000000..cafe458 --- /dev/null +++ b/libtommath/logs/sqr_kara.log @@ -0,0 +1,33 @@ +922 11272 +1148 16004 +1370 21958 +1596 28684 +1817 37832 +2044 46386 +2262 56218 +2492 66388 +2716 77478 +2940 89380 +3163 103680 +3385 116274 +3612 135334 +3836 151332 +4057 164938 +4284 183178 +4508 198864 +4731 215222 +4954 231986 +5180 251660 +5404 269414 +5626 288454 +5850 307806 +6076 329458 +6299 347726 +6523 369864 +6748 387832 +6971 413010 +7194 453310 +7415 476936 +7643 497118 +7867 521394 +8091 540224 diff --git a/libtommath/logs/sub.log b/libtommath/logs/sub.log new file mode 100644 index 0000000..a42d91e --- /dev/null +++ b/libtommath/logs/sub.log @@ -0,0 +1,16 @@ +480 87 +960 114 +1440 139 +1920 159 +2400 204 +2880 228 +3360 250 +3840 273 +4320 300 +4800 321 +5280 348 +5760 370 +6240 393 +6720 420 +7200 444 +7680 466 diff --git a/libtommath/makefile b/libtommath/makefile new file mode 100644 index 0000000..164a0ab --- /dev/null +++ b/libtommath/makefile @@ -0,0 +1,157 @@ +#Makefile for GCC +# +#Tom St Denis + +#version of library +VERSION=0.33 + +CFLAGS += -I./ -Wall -W -Wshadow -Wsign-compare + +#for speed +CFLAGS += -O3 -funroll-all-loops + +#for size +#CFLAGS += -Os + +#x86 optimizations [should be valid for any GCC install though] +CFLAGS += -fomit-frame-pointer + +#debug +#CFLAGS += -g3 + +#install as this user +USER=root +GROUP=root + +default: libtommath.a + +#default files to install +LIBNAME=libtommath.a +HEADERS=tommath.h tommath_class.h tommath_superclass.h + +#LIBPATH-The directory for libtommath to be installed to. +#INCPATH-The directory to install the header files for libtommath. +#DATAPATH-The directory to install the pdf docs. +DESTDIR= +LIBPATH=/usr/lib +INCPATH=/usr/include +DATAPATH=/usr/share/doc/libtommath/pdf + +OBJECTS=bncore.o bn_mp_init.o bn_mp_clear.o bn_mp_exch.o bn_mp_grow.o bn_mp_shrink.o \ +bn_mp_clamp.o bn_mp_zero.o bn_mp_set.o bn_mp_set_int.o bn_mp_init_size.o bn_mp_copy.o \ +bn_mp_init_copy.o bn_mp_abs.o bn_mp_neg.o bn_mp_cmp_mag.o bn_mp_cmp.o bn_mp_cmp_d.o \ +bn_mp_rshd.o bn_mp_lshd.o bn_mp_mod_2d.o bn_mp_div_2d.o bn_mp_mul_2d.o bn_mp_div_2.o \ +bn_mp_mul_2.o bn_s_mp_add.o bn_s_mp_sub.o bn_fast_s_mp_mul_digs.o bn_s_mp_mul_digs.o \ +bn_fast_s_mp_mul_high_digs.o bn_s_mp_mul_high_digs.o bn_fast_s_mp_sqr.o bn_s_mp_sqr.o \ +bn_mp_add.o bn_mp_sub.o bn_mp_karatsuba_mul.o bn_mp_mul.o bn_mp_karatsuba_sqr.o \ +bn_mp_sqr.o bn_mp_div.o bn_mp_mod.o bn_mp_add_d.o bn_mp_sub_d.o bn_mp_mul_d.o \ +bn_mp_div_d.o bn_mp_mod_d.o bn_mp_expt_d.o bn_mp_addmod.o bn_mp_submod.o \ +bn_mp_mulmod.o bn_mp_sqrmod.o bn_mp_gcd.o bn_mp_lcm.o bn_fast_mp_invmod.o bn_mp_invmod.o \ +bn_mp_reduce.o bn_mp_montgomery_setup.o bn_fast_mp_montgomery_reduce.o bn_mp_montgomery_reduce.o \ +bn_mp_exptmod_fast.o bn_mp_exptmod.o bn_mp_2expt.o bn_mp_n_root.o bn_mp_jacobi.o bn_reverse.o \ +bn_mp_count_bits.o bn_mp_read_unsigned_bin.o bn_mp_read_signed_bin.o bn_mp_to_unsigned_bin.o \ +bn_mp_to_signed_bin.o bn_mp_unsigned_bin_size.o bn_mp_signed_bin_size.o \ +bn_mp_xor.o bn_mp_and.o bn_mp_or.o bn_mp_rand.o bn_mp_montgomery_calc_normalization.o \ +bn_mp_prime_is_divisible.o bn_prime_tab.o bn_mp_prime_fermat.o bn_mp_prime_miller_rabin.o \ +bn_mp_prime_is_prime.o bn_mp_prime_next_prime.o bn_mp_dr_reduce.o \ +bn_mp_dr_is_modulus.o bn_mp_dr_setup.o bn_mp_reduce_setup.o \ +bn_mp_toom_mul.o bn_mp_toom_sqr.o bn_mp_div_3.o bn_s_mp_exptmod.o \ +bn_mp_reduce_2k.o bn_mp_reduce_is_2k.o bn_mp_reduce_2k_setup.o \ +bn_mp_radix_smap.o bn_mp_read_radix.o bn_mp_toradix.o bn_mp_radix_size.o \ +bn_mp_fread.o bn_mp_fwrite.o bn_mp_cnt_lsb.o bn_error.o \ +bn_mp_init_multi.o bn_mp_clear_multi.o bn_mp_exteuclid.o bn_mp_toradix_n.o \ +bn_mp_prime_random_ex.o bn_mp_get_int.o bn_mp_sqrt.o bn_mp_is_square.o bn_mp_init_set.o \ +bn_mp_init_set_int.o bn_mp_invmod_slow.o bn_mp_prime_rabin_miller_trials.o + +libtommath.a: $(OBJECTS) + $(AR) $(ARFLAGS) libtommath.a $(OBJECTS) + ranlib libtommath.a + +#make a profiled library (takes a while!!!) +# +# This will build the library with profile generation +# then run the test demo and rebuild the library. +# +# So far I've seen improvements in the MP math +profiled: + make CFLAGS="$(CFLAGS) -fprofile-arcs -DTESTING" timing + ./ltmtest + rm -f *.a *.o ltmtest + make CFLAGS="$(CFLAGS) -fbranch-probabilities" + +#make a single object profiled library +profiled_single: + perl gen.pl + $(CC) $(CFLAGS) -fprofile-arcs -DTESTING -c mpi.c -o mpi.o + $(CC) $(CFLAGS) -DTESTING -DTIMER demo/timing.c mpi.o -o ltmtest + ./ltmtest + rm -f *.o ltmtest + $(CC) $(CFLAGS) -fbranch-probabilities -DTESTING -c mpi.c -o mpi.o + $(AR) $(ARFLAGS) libtommath.a mpi.o + ranlib libtommath.a + +install: libtommath.a + install -d -g $(GROUP) -o $(USER) $(DESTDIR)$(LIBPATH) + install -d -g $(GROUP) -o $(USER) $(DESTDIR)$(INCPATH) + install -g $(GROUP) -o $(USER) $(LIBNAME) $(DESTDIR)$(LIBPATH) + install -g $(GROUP) -o $(USER) $(HEADERS) $(DESTDIR)$(INCPATH) + +test: libtommath.a demo/demo.o + $(CC) $(CFLAGS) demo/demo.o libtommath.a -o test + +mtest: test + cd mtest ; $(CC) $(CFLAGS) mtest.c -o mtest + +timing: libtommath.a + $(CC) $(CFLAGS) -DTIMER demo/timing.c libtommath.a -o ltmtest + +# makes the LTM book DVI file, requires tetex, perl and makeindex [part of tetex I think] +docdvi: tommath.src + cd pics ; make + echo "hello" > tommath.ind + perl booker.pl + latex tommath > /dev/null + latex tommath > /dev/null + makeindex tommath + latex tommath > /dev/null + +# poster, makes the single page PDF poster +poster: poster.tex + pdflatex poster + rm -f poster.aux poster.log + +# makes the LTM book PDF file, requires tetex, cleans up the LaTeX temp files +docs: docdvi + dvipdf tommath + rm -f tommath.log tommath.aux tommath.dvi tommath.idx tommath.toc tommath.lof tommath.ind tommath.ilg + cd pics ; make clean + +#LTM user manual +mandvi: bn.tex + echo "hello" > bn.ind + latex bn > /dev/null + latex bn > /dev/null + makeindex bn + latex bn > /dev/null + +#LTM user manual [pdf] +manual: mandvi + pdflatex bn >/dev/null + rm -f bn.aux bn.dvi bn.log bn.idx bn.lof bn.out bn.toc + +pretty: + perl pretty.build + +clean: + rm -f *.bat *.pdf *.o *.a *.obj *.lib *.exe *.dll etclib/*.o demo/demo.o test ltmtest mpitest mtest/mtest mtest/mtest.exe \ + *.idx *.toc *.log *.aux *.dvi *.lof *.ind *.ilg *.ps *.log *.s mpi.c *.da *.dyn *.dpi tommath.tex `find -type f | grep [~] | xargs` *.lo *.la + rm -rf .libs + cd etc ; make clean + cd pics ; make clean + +zipup: clean manual poster docs + perl gen.pl ; mv mpi.c pre_gen/ ; \ + cd .. ; rm -rf ltm* libtommath-$(VERSION) ; mkdir libtommath-$(VERSION) ; \ + cp -R ./libtommath/* ./libtommath-$(VERSION)/ ; \ + tar -c libtommath-$(VERSION)/* | bzip2 -9vvc > ltm-$(VERSION).tar.bz2 ; \ + zip -9 -r ltm-$(VERSION).zip libtommath-$(VERSION)/* diff --git a/libtommath/makefile.bcc b/libtommath/makefile.bcc new file mode 100644 index 0000000..775e9ff --- /dev/null +++ b/libtommath/makefile.bcc @@ -0,0 +1,42 @@ +# +# Borland C++Builder Makefile (makefile.bcc) +# + + +LIB = tlib +CC = bcc32 +CFLAGS = -c -O2 -I. + +OBJECTS=bncore.obj bn_mp_init.obj bn_mp_clear.obj bn_mp_exch.obj bn_mp_grow.obj bn_mp_shrink.obj \ +bn_mp_clamp.obj bn_mp_zero.obj bn_mp_set.obj bn_mp_set_int.obj bn_mp_init_size.obj bn_mp_copy.obj \ +bn_mp_init_copy.obj bn_mp_abs.obj bn_mp_neg.obj bn_mp_cmp_mag.obj bn_mp_cmp.obj bn_mp_cmp_d.obj \ +bn_mp_rshd.obj bn_mp_lshd.obj bn_mp_mod_2d.obj bn_mp_div_2d.obj bn_mp_mul_2d.obj bn_mp_div_2.obj \ +bn_mp_mul_2.obj bn_s_mp_add.obj bn_s_mp_sub.obj bn_fast_s_mp_mul_digs.obj bn_s_mp_mul_digs.obj \ +bn_fast_s_mp_mul_high_digs.obj bn_s_mp_mul_high_digs.obj bn_fast_s_mp_sqr.obj bn_s_mp_sqr.obj \ +bn_mp_add.obj bn_mp_sub.obj bn_mp_karatsuba_mul.obj bn_mp_mul.obj bn_mp_karatsuba_sqr.obj \ +bn_mp_sqr.obj bn_mp_div.obj bn_mp_mod.obj bn_mp_add_d.obj bn_mp_sub_d.obj bn_mp_mul_d.obj \ +bn_mp_div_d.obj bn_mp_mod_d.obj bn_mp_expt_d.obj bn_mp_addmod.obj bn_mp_submod.obj \ +bn_mp_mulmod.obj bn_mp_sqrmod.obj bn_mp_gcd.obj bn_mp_lcm.obj bn_fast_mp_invmod.obj bn_mp_invmod.obj \ +bn_mp_reduce.obj bn_mp_montgomery_setup.obj bn_fast_mp_montgomery_reduce.obj bn_mp_montgomery_reduce.obj \ +bn_mp_exptmod_fast.obj bn_mp_exptmod.obj bn_mp_2expt.obj bn_mp_n_root.obj bn_mp_jacobi.obj bn_reverse.obj \ +bn_mp_count_bits.obj bn_mp_read_unsigned_bin.obj bn_mp_read_signed_bin.obj bn_mp_to_unsigned_bin.obj \ +bn_mp_to_signed_bin.obj bn_mp_unsigned_bin_size.obj bn_mp_signed_bin_size.obj \ +bn_mp_xor.obj bn_mp_and.obj bn_mp_or.obj bn_mp_rand.obj bn_mp_montgomery_calc_normalization.obj \ +bn_mp_prime_is_divisible.obj bn_prime_tab.obj bn_mp_prime_fermat.obj bn_mp_prime_miller_rabin.obj \ +bn_mp_prime_is_prime.obj bn_mp_prime_next_prime.obj bn_mp_dr_reduce.obj \ +bn_mp_dr_is_modulus.obj bn_mp_dr_setup.obj bn_mp_reduce_setup.obj \ +bn_mp_toom_mul.obj bn_mp_toom_sqr.obj bn_mp_div_3.obj bn_s_mp_exptmod.obj \ +bn_mp_reduce_2k.obj bn_mp_reduce_is_2k.obj bn_mp_reduce_2k_setup.obj \ +bn_mp_radix_smap.obj bn_mp_read_radix.obj bn_mp_toradix.obj bn_mp_radix_size.obj \ +bn_mp_fread.obj bn_mp_fwrite.obj bn_mp_cnt_lsb.obj bn_error.obj \ +bn_mp_init_multi.obj bn_mp_clear_multi.obj bn_mp_exteuclid.obj bn_mp_toradix_n.obj \ +bn_mp_prime_random_ex.obj bn_mp_get_int.obj bn_mp_sqrt.obj bn_mp_is_square.obj \ +bn_mp_init_set.obj bn_mp_init_set_int.obj bn_mp_invmod_slow.obj bn_mp_prime_rabin_miller_trials.obj + +TARGET = libtommath.lib + +$(TARGET): $(OBJECTS) + +.c.objbjbjbj: + $(CC) $(CFLAGS) $< + $(LIB) $(TARGET) -+$@ diff --git a/libtommath/makefile.cygwin_dll b/libtommath/makefile.cygwin_dll new file mode 100644 index 0000000..c90e5d9 --- /dev/null +++ b/libtommath/makefile.cygwin_dll @@ -0,0 +1,49 @@ +#Makefile for Cygwin-GCC +# +#This makefile will build a Windows DLL [doesn't require cygwin to run] in the file +#libtommath.dll. The import library is in libtommath.dll.a. Remember to add +#"-Wl,--enable-auto-import" to your client build to avoid the auto-import warnings +# +#Tom St Denis +CFLAGS += -I./ -Wall -W -Wshadow -O3 -funroll-loops -mno-cygwin + +#x86 optimizations [should be valid for any GCC install though] +CFLAGS += -fomit-frame-pointer + +default: windll + +OBJECTS=bncore.o bn_mp_init.o bn_mp_clear.o bn_mp_exch.o bn_mp_grow.o bn_mp_shrink.o \ +bn_mp_clamp.o bn_mp_zero.o bn_mp_set.o bn_mp_set_int.o bn_mp_init_size.o bn_mp_copy.o \ +bn_mp_init_copy.o bn_mp_abs.o bn_mp_neg.o bn_mp_cmp_mag.o bn_mp_cmp.o bn_mp_cmp_d.o \ +bn_mp_rshd.o bn_mp_lshd.o bn_mp_mod_2d.o bn_mp_div_2d.o bn_mp_mul_2d.o bn_mp_div_2.o \ +bn_mp_mul_2.o bn_s_mp_add.o bn_s_mp_sub.o bn_fast_s_mp_mul_digs.o bn_s_mp_mul_digs.o \ +bn_fast_s_mp_mul_high_digs.o bn_s_mp_mul_high_digs.o bn_fast_s_mp_sqr.o bn_s_mp_sqr.o \ +bn_mp_add.o bn_mp_sub.o bn_mp_karatsuba_mul.o bn_mp_mul.o bn_mp_karatsuba_sqr.o \ +bn_mp_sqr.o bn_mp_div.o bn_mp_mod.o bn_mp_add_d.o bn_mp_sub_d.o bn_mp_mul_d.o \ +bn_mp_div_d.o bn_mp_mod_d.o bn_mp_expt_d.o bn_mp_addmod.o bn_mp_submod.o \ +bn_mp_mulmod.o bn_mp_sqrmod.o bn_mp_gcd.o bn_mp_lcm.o bn_fast_mp_invmod.o bn_mp_invmod.o \ +bn_mp_reduce.o bn_mp_montgomery_setup.o bn_fast_mp_montgomery_reduce.o bn_mp_montgomery_reduce.o \ +bn_mp_exptmod_fast.o bn_mp_exptmod.o bn_mp_2expt.o bn_mp_n_root.o bn_mp_jacobi.o bn_reverse.o \ +bn_mp_count_bits.o bn_mp_read_unsigned_bin.o bn_mp_read_signed_bin.o bn_mp_to_unsigned_bin.o \ +bn_mp_to_signed_bin.o bn_mp_unsigned_bin_size.o bn_mp_signed_bin_size.o \ +bn_mp_xor.o bn_mp_and.o bn_mp_or.o bn_mp_rand.o bn_mp_montgomery_calc_normalization.o \ +bn_mp_prime_is_divisible.o bn_prime_tab.o bn_mp_prime_fermat.o bn_mp_prime_miller_rabin.o \ +bn_mp_prime_is_prime.o bn_mp_prime_next_prime.o bn_mp_dr_reduce.o \ +bn_mp_dr_is_modulus.o bn_mp_dr_setup.o bn_mp_reduce_setup.o \ +bn_mp_toom_mul.o bn_mp_toom_sqr.o bn_mp_div_3.o bn_s_mp_exptmod.o \ +bn_mp_reduce_2k.o bn_mp_reduce_is_2k.o bn_mp_reduce_2k_setup.o \ +bn_mp_radix_smap.o bn_mp_read_radix.o bn_mp_toradix.o bn_mp_radix_size.o \ +bn_mp_fread.o bn_mp_fwrite.o bn_mp_cnt_lsb.o bn_error.o \ +bn_mp_init_multi.o bn_mp_clear_multi.o bn_mp_exteuclid.o bn_mp_toradix_n.o \ +bn_mp_prime_random_ex.o bn_mp_get_int.o bn_mp_sqrt.o bn_mp_is_square.o bn_mp_init_set.o \ +bn_mp_init_set_int.o bn_mp_invmod_slow.o bn_mp_prime_rabin_miller_trials.o + +# make a Windows DLL via Cygwin +windll: $(OBJECTS) + gcc -mno-cygwin -mdll -o libtommath.dll -Wl,--out-implib=libtommath.dll.a -Wl,--export-all-symbols *.o + ranlib libtommath.dll.a + +# build the test program using the windows DLL +test: $(OBJECTS) windll + gcc $(CFLAGS) demo/demo.c libtommath.dll.a -Wl,--enable-auto-import -o test -s + cd mtest ; $(CC) -O3 -fomit-frame-pointer -funroll-loops mtest.c -o mtest -s diff --git a/libtommath/makefile.icc b/libtommath/makefile.icc new file mode 100644 index 0000000..3775b20 --- /dev/null +++ b/libtommath/makefile.icc @@ -0,0 +1,114 @@ +#Makefile for ICC +# +#Tom St Denis +CC=icc + +CFLAGS += -I./ + +# optimize for SPEED +# +# -mcpu= can be pentium, pentiumpro (covers PII through PIII) or pentium4 +# -ax? specifies make code specifically for ? but compatible with IA-32 +# -x? specifies compile solely for ? [not specifically IA-32 compatible] +# +# where ? is +# K - PIII +# W - first P4 [Williamette] +# N - P4 Northwood +# P - P4 Prescott +# B - Blend of P4 and PM [mobile] +# +# Default to just generic max opts +CFLAGS += -O3 -xN + +#install as this user +USER=root +GROUP=root + +default: libtommath.a + +#default files to install +LIBNAME=libtommath.a +HEADERS=tommath.h + +#LIBPATH-The directory for libtomcrypt to be installed to. +#INCPATH-The directory to install the header files for libtommath. +#DATAPATH-The directory to install the pdf docs. +DESTDIR= +LIBPATH=/usr/lib +INCPATH=/usr/include +DATAPATH=/usr/share/doc/libtommath/pdf + +OBJECTS=bncore.o bn_mp_init.o bn_mp_clear.o bn_mp_exch.o bn_mp_grow.o bn_mp_shrink.o \ +bn_mp_clamp.o bn_mp_zero.o bn_mp_set.o bn_mp_set_int.o bn_mp_init_size.o bn_mp_copy.o \ +bn_mp_init_copy.o bn_mp_abs.o bn_mp_neg.o bn_mp_cmp_mag.o bn_mp_cmp.o bn_mp_cmp_d.o \ +bn_mp_rshd.o bn_mp_lshd.o bn_mp_mod_2d.o bn_mp_div_2d.o bn_mp_mul_2d.o bn_mp_div_2.o \ +bn_mp_mul_2.o bn_s_mp_add.o bn_s_mp_sub.o bn_fast_s_mp_mul_digs.o bn_s_mp_mul_digs.o \ +bn_fast_s_mp_mul_high_digs.o bn_s_mp_mul_high_digs.o bn_fast_s_mp_sqr.o bn_s_mp_sqr.o \ +bn_mp_add.o bn_mp_sub.o bn_mp_karatsuba_mul.o bn_mp_mul.o bn_mp_karatsuba_sqr.o \ +bn_mp_sqr.o bn_mp_div.o bn_mp_mod.o bn_mp_add_d.o bn_mp_sub_d.o bn_mp_mul_d.o \ +bn_mp_div_d.o bn_mp_mod_d.o bn_mp_expt_d.o bn_mp_addmod.o bn_mp_submod.o \ +bn_mp_mulmod.o bn_mp_sqrmod.o bn_mp_gcd.o bn_mp_lcm.o bn_fast_mp_invmod.o bn_mp_invmod.o \ +bn_mp_reduce.o bn_mp_montgomery_setup.o bn_fast_mp_montgomery_reduce.o bn_mp_montgomery_reduce.o \ +bn_mp_exptmod_fast.o bn_mp_exptmod.o bn_mp_2expt.o bn_mp_n_root.o bn_mp_jacobi.o bn_reverse.o \ +bn_mp_count_bits.o bn_mp_read_unsigned_bin.o bn_mp_read_signed_bin.o bn_mp_to_unsigned_bin.o \ +bn_mp_to_signed_bin.o bn_mp_unsigned_bin_size.o bn_mp_signed_bin_size.o \ +bn_mp_xor.o bn_mp_and.o bn_mp_or.o bn_mp_rand.o bn_mp_montgomery_calc_normalization.o \ +bn_mp_prime_is_divisible.o bn_prime_tab.o bn_mp_prime_fermat.o bn_mp_prime_miller_rabin.o \ +bn_mp_prime_is_prime.o bn_mp_prime_next_prime.o bn_mp_dr_reduce.o \ +bn_mp_dr_is_modulus.o bn_mp_dr_setup.o bn_mp_reduce_setup.o \ +bn_mp_toom_mul.o bn_mp_toom_sqr.o bn_mp_div_3.o bn_s_mp_exptmod.o \ +bn_mp_reduce_2k.o bn_mp_reduce_is_2k.o bn_mp_reduce_2k_setup.o \ +bn_mp_radix_smap.o bn_mp_read_radix.o bn_mp_toradix.o bn_mp_radix_size.o \ +bn_mp_fread.o bn_mp_fwrite.o bn_mp_cnt_lsb.o bn_error.o \ +bn_mp_init_multi.o bn_mp_clear_multi.o bn_mp_exteuclid.o bn_mp_toradix_n.o \ +bn_mp_prime_random_ex.o bn_mp_get_int.o bn_mp_sqrt.o bn_mp_is_square.o bn_mp_init_set.o \ +bn_mp_init_set_int.o bn_mp_invmod_slow.o bn_mp_prime_rabin_miller_trials.o + +libtommath.a: $(OBJECTS) + $(AR) $(ARFLAGS) libtommath.a $(OBJECTS) + ranlib libtommath.a + +#make a profiled library (takes a while!!!) +# +# This will build the library with profile generation +# then run the test demo and rebuild the library. +# +# So far I've seen improvements in the MP math +profiled: + make -f makefile.icc CFLAGS="$(CFLAGS) -prof_gen -DTESTING" timing + ./ltmtest + rm -f *.a *.o ltmtest + make -f makefile.icc CFLAGS="$(CFLAGS) -prof_use" + +#make a single object profiled library +profiled_single: + perl gen.pl + $(CC) $(CFLAGS) -prof_gen -DTESTING -c mpi.c -o mpi.o + $(CC) $(CFLAGS) -DTESTING -DTIMER demo/demo.c mpi.o -o ltmtest + ./ltmtest + rm -f *.o ltmtest + $(CC) $(CFLAGS) -prof_use -ip -DTESTING -c mpi.c -o mpi.o + $(AR) $(ARFLAGS) libtommath.a mpi.o + ranlib libtommath.a + +install: libtommath.a + install -d -g $(GROUP) -o $(USER) $(DESTDIR)$(LIBPATH) + install -d -g $(GROUP) -o $(USER) $(DESTDIR)$(INCPATH) + install -g $(GROUP) -o $(USER) $(LIBNAME) $(DESTDIR)$(LIBPATH) + install -g $(GROUP) -o $(USER) $(HEADERS) $(DESTDIR)$(INCPATH) + +test: libtommath.a demo/demo.o + $(CC) demo/demo.o libtommath.a -o test + +mtest: test + cd mtest ; $(CC) $(CFLAGS) mtest.c -o mtest + +timing: libtommath.a + $(CC) $(CFLAGS) -DTIMER demo/timing.c libtommath.a -o ltmtest + +clean: + rm -f *.bat *.pdf *.o *.a *.obj *.lib *.exe *.dll etclib/*.o demo/demo.o test ltmtest mpitest mtest/mtest mtest/mtest.exe \ + *.idx *.toc *.log *.aux *.dvi *.lof *.ind *.ilg *.ps *.log *.s mpi.c *.il etc/*.il *.dyn + cd etc ; make clean + cd pics ; make clean diff --git a/libtommath/makefile.msvc b/libtommath/makefile.msvc new file mode 100644 index 0000000..cf59943 --- /dev/null +++ b/libtommath/makefile.msvc @@ -0,0 +1,36 @@ +#MSVC Makefile +# +#Tom St Denis + +CFLAGS = /I. /Ox /DWIN32 /W4 + +default: library + +OBJECTS=bncore.obj bn_mp_init.obj bn_mp_clear.obj bn_mp_exch.obj bn_mp_grow.obj bn_mp_shrink.obj \ +bn_mp_clamp.obj bn_mp_zero.obj bn_mp_set.obj bn_mp_set_int.obj bn_mp_init_size.obj bn_mp_copy.obj \ +bn_mp_init_copy.obj bn_mp_abs.obj bn_mp_neg.obj bn_mp_cmp_mag.obj bn_mp_cmp.obj bn_mp_cmp_d.obj \ +bn_mp_rshd.obj bn_mp_lshd.obj bn_mp_mod_2d.obj bn_mp_div_2d.obj bn_mp_mul_2d.obj bn_mp_div_2.obj \ +bn_mp_mul_2.obj bn_s_mp_add.obj bn_s_mp_sub.obj bn_fast_s_mp_mul_digs.obj bn_s_mp_mul_digs.obj \ +bn_fast_s_mp_mul_high_digs.obj bn_s_mp_mul_high_digs.obj bn_fast_s_mp_sqr.obj bn_s_mp_sqr.obj \ +bn_mp_add.obj bn_mp_sub.obj bn_mp_karatsuba_mul.obj bn_mp_mul.obj bn_mp_karatsuba_sqr.obj \ +bn_mp_sqr.obj bn_mp_div.obj bn_mp_mod.obj bn_mp_add_d.obj bn_mp_sub_d.obj bn_mp_mul_d.obj \ +bn_mp_div_d.obj bn_mp_mod_d.obj bn_mp_expt_d.obj bn_mp_addmod.obj bn_mp_submod.obj \ +bn_mp_mulmod.obj bn_mp_sqrmod.obj bn_mp_gcd.obj bn_mp_lcm.obj bn_fast_mp_invmod.obj bn_mp_invmod.obj \ +bn_mp_reduce.obj bn_mp_montgomery_setup.obj bn_fast_mp_montgomery_reduce.obj bn_mp_montgomery_reduce.obj \ +bn_mp_exptmod_fast.obj bn_mp_exptmod.obj bn_mp_2expt.obj bn_mp_n_root.obj bn_mp_jacobi.obj bn_reverse.obj \ +bn_mp_count_bits.obj bn_mp_read_unsigned_bin.obj bn_mp_read_signed_bin.obj bn_mp_to_unsigned_bin.obj \ +bn_mp_to_signed_bin.obj bn_mp_unsigned_bin_size.obj bn_mp_signed_bin_size.obj \ +bn_mp_xor.obj bn_mp_and.obj bn_mp_or.obj bn_mp_rand.obj bn_mp_montgomery_calc_normalization.obj \ +bn_mp_prime_is_divisible.obj bn_prime_tab.obj bn_mp_prime_fermat.obj bn_mp_prime_miller_rabin.obj \ +bn_mp_prime_is_prime.obj bn_mp_prime_next_prime.obj bn_mp_dr_reduce.obj \ +bn_mp_dr_is_modulus.obj bn_mp_dr_setup.obj bn_mp_reduce_setup.obj \ +bn_mp_toom_mul.obj bn_mp_toom_sqr.obj bn_mp_div_3.obj bn_s_mp_exptmod.obj \ +bn_mp_reduce_2k.obj bn_mp_reduce_is_2k.obj bn_mp_reduce_2k_setup.obj \ +bn_mp_radix_smap.obj bn_mp_read_radix.obj bn_mp_toradix.obj bn_mp_radix_size.obj \ +bn_mp_fread.obj bn_mp_fwrite.obj bn_mp_cnt_lsb.obj bn_error.obj \ +bn_mp_init_multi.obj bn_mp_clear_multi.obj bn_mp_exteuclid.obj bn_mp_toradix_n.obj \ +bn_mp_prime_random_ex.obj bn_mp_get_int.obj bn_mp_sqrt.obj bn_mp_is_square.obj \ +bn_mp_init_set.obj bn_mp_init_set_int.obj bn_mp_invmod_slow.obj bn_mp_prime_rabin_miller_trials.obj + +library: $(OBJECTS) + lib /out:tommath.lib $(OBJECTS) diff --git a/libtommath/makefile.shared b/libtommath/makefile.shared new file mode 100644 index 0000000..86a3786 --- /dev/null +++ b/libtommath/makefile.shared @@ -0,0 +1,77 @@ +#Makefile for GCC +# +#Tom St Denis +VERSION=0:33 + +CC = libtool --mode=compile gcc +CFLAGS += -I./ -Wall -W -Wshadow -Wsign-compare + +#for speed +CFLAGS += -O3 -funroll-loops + +#for size +#CFLAGS += -Os + +#x86 optimizations [should be valid for any GCC install though] +CFLAGS += -fomit-frame-pointer + +#install as this user +USER=root +GROUP=root + +default: libtommath.la + +#default files to install +LIBNAME=libtommath.la +HEADERS=tommath.h tommath_class.h tommath_superclass.h + +#LIBPATH-The directory for libtommath to be installed to. +#INCPATH-The directory to install the header files for libtommath. +#DATAPATH-The directory to install the pdf docs. +DESTDIR= +LIBPATH=/usr/lib +INCPATH=/usr/include +DATAPATH=/usr/share/doc/libtommath/pdf + +OBJECTS=bncore.o bn_mp_init.o bn_mp_clear.o bn_mp_exch.o bn_mp_grow.o bn_mp_shrink.o \ +bn_mp_clamp.o bn_mp_zero.o bn_mp_set.o bn_mp_set_int.o bn_mp_init_size.o bn_mp_copy.o \ +bn_mp_init_copy.o bn_mp_abs.o bn_mp_neg.o bn_mp_cmp_mag.o bn_mp_cmp.o bn_mp_cmp_d.o \ +bn_mp_rshd.o bn_mp_lshd.o bn_mp_mod_2d.o bn_mp_div_2d.o bn_mp_mul_2d.o bn_mp_div_2.o \ +bn_mp_mul_2.o bn_s_mp_add.o bn_s_mp_sub.o bn_fast_s_mp_mul_digs.o bn_s_mp_mul_digs.o \ +bn_fast_s_mp_mul_high_digs.o bn_s_mp_mul_high_digs.o bn_fast_s_mp_sqr.o bn_s_mp_sqr.o \ +bn_mp_add.o bn_mp_sub.o bn_mp_karatsuba_mul.o bn_mp_mul.o bn_mp_karatsuba_sqr.o \ +bn_mp_sqr.o bn_mp_div.o bn_mp_mod.o bn_mp_add_d.o bn_mp_sub_d.o bn_mp_mul_d.o \ +bn_mp_div_d.o bn_mp_mod_d.o bn_mp_expt_d.o bn_mp_addmod.o bn_mp_submod.o \ +bn_mp_mulmod.o bn_mp_sqrmod.o bn_mp_gcd.o bn_mp_lcm.o bn_fast_mp_invmod.o bn_mp_invmod.o \ +bn_mp_reduce.o bn_mp_montgomery_setup.o bn_fast_mp_montgomery_reduce.o bn_mp_montgomery_reduce.o \ +bn_mp_exptmod_fast.o bn_mp_exptmod.o bn_mp_2expt.o bn_mp_n_root.o bn_mp_jacobi.o bn_reverse.o \ +bn_mp_count_bits.o bn_mp_read_unsigned_bin.o bn_mp_read_signed_bin.o bn_mp_to_unsigned_bin.o \ +bn_mp_to_signed_bin.o bn_mp_unsigned_bin_size.o bn_mp_signed_bin_size.o \ +bn_mp_xor.o bn_mp_and.o bn_mp_or.o bn_mp_rand.o bn_mp_montgomery_calc_normalization.o \ +bn_mp_prime_is_divisible.o bn_prime_tab.o bn_mp_prime_fermat.o bn_mp_prime_miller_rabin.o \ +bn_mp_prime_is_prime.o bn_mp_prime_next_prime.o bn_mp_dr_reduce.o \ +bn_mp_dr_is_modulus.o bn_mp_dr_setup.o bn_mp_reduce_setup.o \ +bn_mp_toom_mul.o bn_mp_toom_sqr.o bn_mp_div_3.o bn_s_mp_exptmod.o \ +bn_mp_reduce_2k.o bn_mp_reduce_is_2k.o bn_mp_reduce_2k_setup.o \ +bn_mp_radix_smap.o bn_mp_read_radix.o bn_mp_toradix.o bn_mp_radix_size.o \ +bn_mp_fread.o bn_mp_fwrite.o bn_mp_cnt_lsb.o bn_error.o \ +bn_mp_init_multi.o bn_mp_clear_multi.o bn_mp_exteuclid.o bn_mp_toradix_n.o \ +bn_mp_prime_random_ex.o bn_mp_get_int.o bn_mp_sqrt.o bn_mp_is_square.o bn_mp_init_set.o \ +bn_mp_init_set_int.o bn_mp_invmod_slow.o bn_mp_prime_rabin_miller_trials.o + +libtommath.la: $(OBJECTS) + libtool --mode=link gcc *.lo -o libtommath.la -rpath $(LIBPATH) -version-info $(VERSION) + libtool --mode=link gcc *.o -o libtommath.a + libtool --mode=install install -c libtommath.la $(LIBPATH)/libtommath.la + install -d -g $(GROUP) -o $(USER) $(DESTDIR)$(INCPATH) + install -g $(GROUP) -o $(USER) $(HEADERS) $(DESTDIR)$(INCPATH) + +test: libtommath.a demo/demo.o + gcc $(CFLAGS) -c demo/demo.c -o demo/demo.o + libtool --mode=link gcc -o test demo/demo.o libtommath.la + +mtest: test + cd mtest ; gcc $(CFLAGS) mtest.c -o mtest -s + +timing: libtommath.la + gcc $(CFLAGS) -DTIMER demo/timing.c libtommath.a -o ltmtest -s diff --git a/libtommath/mess.sh b/libtommath/mess.sh new file mode 100644 index 0000000..bf639ce --- /dev/null +++ b/libtommath/mess.sh @@ -0,0 +1,4 @@ +#!/bin/bash +if cvs log $1 >/dev/null 2>/dev/null; then exit 0; else echo "$1 shouldn't be here" ; exit 1; fi + + diff --git a/libtommath/mtest/logtab.h b/libtommath/mtest/logtab.h new file mode 100644 index 0000000..68462bd --- /dev/null +++ b/libtommath/mtest/logtab.h @@ -0,0 +1,20 @@ +const float s_logv_2[] = { + 0.000000000, 0.000000000, 1.000000000, 0.630929754, /* 0 1 2 3 */ + 0.500000000, 0.430676558, 0.386852807, 0.356207187, /* 4 5 6 7 */ + 0.333333333, 0.315464877, 0.301029996, 0.289064826, /* 8 9 10 11 */ + 0.278942946, 0.270238154, 0.262649535, 0.255958025, /* 12 13 14 15 */ + 0.250000000, 0.244650542, 0.239812467, 0.235408913, /* 16 17 18 19 */ + 0.231378213, 0.227670249, 0.224243824, 0.221064729, /* 20 21 22 23 */ + 0.218104292, 0.215338279, 0.212746054, 0.210309918, /* 24 25 26 27 */ + 0.208014598, 0.205846832, 0.203795047, 0.201849087, /* 28 29 30 31 */ + 0.200000000, 0.198239863, 0.196561632, 0.194959022, /* 32 33 34 35 */ + 0.193426404, 0.191958720, 0.190551412, 0.189200360, /* 36 37 38 39 */ + 0.187901825, 0.186652411, 0.185449023, 0.184288833, /* 40 41 42 43 */ + 0.183169251, 0.182087900, 0.181042597, 0.180031327, /* 44 45 46 47 */ + 0.179052232, 0.178103594, 0.177183820, 0.176291434, /* 48 49 50 51 */ + 0.175425064, 0.174583430, 0.173765343, 0.172969690, /* 52 53 54 55 */ + 0.172195434, 0.171441601, 0.170707280, 0.169991616, /* 56 57 58 59 */ + 0.169293808, 0.168613099, 0.167948779, 0.167300179, /* 60 61 62 63 */ + 0.166666667 +}; + diff --git a/libtommath/mtest/mpi-config.h b/libtommath/mtest/mpi-config.h new file mode 100644 index 0000000..dcdbd35 --- /dev/null +++ b/libtommath/mtest/mpi-config.h @@ -0,0 +1,86 @@ +/* Default configuration for MPI library */ +/* $Id: mpi-config.h,v 1.1.1.1 2005/01/19 22:41:29 kennykb Exp $ */ + +#ifndef MPI_CONFIG_H_ +#define MPI_CONFIG_H_ + +/* + For boolean options, + 0 = no + 1 = yes + + Other options are documented individually. + + */ + +#ifndef MP_IOFUNC +#define MP_IOFUNC 0 /* include mp_print() ? */ +#endif + +#ifndef MP_MODARITH +#define MP_MODARITH 1 /* include modular arithmetic ? */ +#endif + +#ifndef MP_NUMTH +#define MP_NUMTH 1 /* include number theoretic functions? */ +#endif + +#ifndef MP_LOGTAB +#define MP_LOGTAB 1 /* use table of logs instead of log()? */ +#endif + +#ifndef MP_MEMSET +#define MP_MEMSET 1 /* use memset() to zero buffers? */ +#endif + +#ifndef MP_MEMCPY +#define MP_MEMCPY 1 /* use memcpy() to copy buffers? */ +#endif + +#ifndef MP_CRYPTO +#define MP_CRYPTO 1 /* erase memory on free? */ +#endif + +#ifndef MP_ARGCHK +/* + 0 = no parameter checks + 1 = runtime checks, continue execution and return an error to caller + 2 = assertions; dump core on parameter errors + */ +#define MP_ARGCHK 2 /* how to check input arguments */ +#endif + +#ifndef MP_DEBUG +#define MP_DEBUG 0 /* print diagnostic output? */ +#endif + +#ifndef MP_DEFPREC +#define MP_DEFPREC 64 /* default precision, in digits */ +#endif + +#ifndef MP_MACRO +#define MP_MACRO 1 /* use macros for frequent calls? */ +#endif + +#ifndef MP_SQUARE +#define MP_SQUARE 1 /* use separate squaring code? */ +#endif + +#ifndef MP_PTAB_SIZE +/* + When building mpprime.c, we build in a table of small prime + values to use for primality testing. The more you include, + the more space they take up. See primes.c for the possible + values (currently 16, 32, 64, 128, 256, and 6542) + */ +#define MP_PTAB_SIZE 128 /* how many built-in primes? */ +#endif + +#ifndef MP_COMPAT_MACROS +#define MP_COMPAT_MACROS 1 /* define compatibility macros? */ +#endif + +#endif /* ifndef MPI_CONFIG_H_ */ + + +/* crc==3287762869, version==2, Sat Feb 02 06:43:53 2002 */ diff --git a/libtommath/mtest/mpi-types.h b/libtommath/mtest/mpi-types.h new file mode 100644 index 0000000..e097188 --- /dev/null +++ b/libtommath/mtest/mpi-types.h @@ -0,0 +1,16 @@ +/* Type definitions generated by 'types.pl' */ +typedef char mp_sign; +typedef unsigned short mp_digit; /* 2 byte type */ +typedef unsigned int mp_word; /* 4 byte type */ +typedef unsigned int mp_size; +typedef int mp_err; + +#define MP_DIGIT_BIT (CHAR_BIT*sizeof(mp_digit)) +#define MP_DIGIT_MAX USHRT_MAX +#define MP_WORD_BIT (CHAR_BIT*sizeof(mp_word)) +#define MP_WORD_MAX UINT_MAX + +#define MP_DIGIT_SIZE 2 +#define DIGIT_FMT "%04X" +#define RADIX (MP_DIGIT_MAX+1) + diff --git a/libtommath/mtest/mpi.c b/libtommath/mtest/mpi.c new file mode 100644 index 0000000..0517602 --- /dev/null +++ b/libtommath/mtest/mpi.c @@ -0,0 +1,3981 @@ +/* + mpi.c + + by Michael J. Fromberger <sting@linguist.dartmouth.edu> + Copyright (C) 1998 Michael J. Fromberger, All Rights Reserved + + Arbitrary precision integer arithmetic library + + $Id: mpi.c,v 1.1.1.1 2005/01/19 22:41:29 kennykb Exp $ + */ + +#include "mpi.h" +#include <stdlib.h> +#include <string.h> +#include <ctype.h> + +#if MP_DEBUG +#include <stdio.h> + +#define DIAG(T,V) {fprintf(stderr,T);mp_print(V,stderr);fputc('\n',stderr);} +#else +#define DIAG(T,V) +#endif + +/* + If MP_LOGTAB is not defined, use the math library to compute the + logarithms on the fly. Otherwise, use the static table below. + Pick which works best for your system. + */ +#if MP_LOGTAB + +/* {{{ s_logv_2[] - log table for 2 in various bases */ + +/* + A table of the logs of 2 for various bases (the 0 and 1 entries of + this table are meaningless and should not be referenced). + + This table is used to compute output lengths for the mp_toradix() + function. Since a number n in radix r takes up about log_r(n) + digits, we estimate the output size by taking the least integer + greater than log_r(n), where: + + log_r(n) = log_2(n) * log_r(2) + + This table, therefore, is a table of log_r(2) for 2 <= r <= 36, + which are the output bases supported. + */ + +#include "logtab.h" + +/* }}} */ +#define LOG_V_2(R) s_logv_2[(R)] + +#else + +#include <math.h> +#define LOG_V_2(R) (log(2.0)/log(R)) + +#endif + +/* Default precision for newly created mp_int's */ +static unsigned int s_mp_defprec = MP_DEFPREC; + +/* {{{ Digit arithmetic macros */ + +/* + When adding and multiplying digits, the results can be larger than + can be contained in an mp_digit. Thus, an mp_word is used. These + macros mask off the upper and lower digits of the mp_word (the + mp_word may be more than 2 mp_digits wide, but we only concern + ourselves with the low-order 2 mp_digits) + + If your mp_word DOES have more than 2 mp_digits, you need to + uncomment the first line, and comment out the second. + */ + +/* #define CARRYOUT(W) (((W)>>DIGIT_BIT)&MP_DIGIT_MAX) */ +#define CARRYOUT(W) ((W)>>DIGIT_BIT) +#define ACCUM(W) ((W)&MP_DIGIT_MAX) + +/* }}} */ + +/* {{{ Comparison constants */ + +#define MP_LT -1 +#define MP_EQ 0 +#define MP_GT 1 + +/* }}} */ + +/* {{{ Constant strings */ + +/* Constant strings returned by mp_strerror() */ +static const char *mp_err_string[] = { + "unknown result code", /* say what? */ + "boolean true", /* MP_OKAY, MP_YES */ + "boolean false", /* MP_NO */ + "out of memory", /* MP_MEM */ + "argument out of range", /* MP_RANGE */ + "invalid input parameter", /* MP_BADARG */ + "result is undefined" /* MP_UNDEF */ +}; + +/* Value to digit maps for radix conversion */ + +/* s_dmap_1 - standard digits and letters */ +static const char *s_dmap_1 = + "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz+/"; + +#if 0 +/* s_dmap_2 - base64 ordering for digits */ +static const char *s_dmap_2 = + "ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789+/"; +#endif + +/* }}} */ + +/* {{{ Static function declarations */ + +/* + If MP_MACRO is false, these will be defined as actual functions; + otherwise, suitable macro definitions will be used. This works + around the fact that ANSI C89 doesn't support an 'inline' keyword + (although I hear C9x will ... about bloody time). At present, the + macro definitions are identical to the function bodies, but they'll + expand in place, instead of generating a function call. + + I chose these particular functions to be made into macros because + some profiling showed they are called a lot on a typical workload, + and yet they are primarily housekeeping. + */ +#if MP_MACRO == 0 + void s_mp_setz(mp_digit *dp, mp_size count); /* zero digits */ + void s_mp_copy(mp_digit *sp, mp_digit *dp, mp_size count); /* copy */ + void *s_mp_alloc(size_t nb, size_t ni); /* general allocator */ + void s_mp_free(void *ptr); /* general free function */ +#else + + /* Even if these are defined as macros, we need to respect the settings + of the MP_MEMSET and MP_MEMCPY configuration options... + */ + #if MP_MEMSET == 0 + #define s_mp_setz(dp, count) \ + {int ix;for(ix=0;ix<(count);ix++)(dp)[ix]=0;} + #else + #define s_mp_setz(dp, count) memset(dp, 0, (count) * sizeof(mp_digit)) + #endif /* MP_MEMSET */ + + #if MP_MEMCPY == 0 + #define s_mp_copy(sp, dp, count) \ + {int ix;for(ix=0;ix<(count);ix++)(dp)[ix]=(sp)[ix];} + #else + #define s_mp_copy(sp, dp, count) memcpy(dp, sp, (count) * sizeof(mp_digit)) + #endif /* MP_MEMCPY */ + + #define s_mp_alloc(nb, ni) calloc(nb, ni) + #define s_mp_free(ptr) {if(ptr) free(ptr);} +#endif /* MP_MACRO */ + +mp_err s_mp_grow(mp_int *mp, mp_size min); /* increase allocated size */ +mp_err s_mp_pad(mp_int *mp, mp_size min); /* left pad with zeroes */ + +void s_mp_clamp(mp_int *mp); /* clip leading zeroes */ + +void s_mp_exch(mp_int *a, mp_int *b); /* swap a and b in place */ + +mp_err s_mp_lshd(mp_int *mp, mp_size p); /* left-shift by p digits */ +void s_mp_rshd(mp_int *mp, mp_size p); /* right-shift by p digits */ +void s_mp_div_2d(mp_int *mp, mp_digit d); /* divide by 2^d in place */ +void s_mp_mod_2d(mp_int *mp, mp_digit d); /* modulo 2^d in place */ +mp_err s_mp_mul_2d(mp_int *mp, mp_digit d); /* multiply by 2^d in place*/ +void s_mp_div_2(mp_int *mp); /* divide by 2 in place */ +mp_err s_mp_mul_2(mp_int *mp); /* multiply by 2 in place */ +mp_digit s_mp_norm(mp_int *a, mp_int *b); /* normalize for division */ +mp_err s_mp_add_d(mp_int *mp, mp_digit d); /* unsigned digit addition */ +mp_err s_mp_sub_d(mp_int *mp, mp_digit d); /* unsigned digit subtract */ +mp_err s_mp_mul_d(mp_int *mp, mp_digit d); /* unsigned digit multiply */ +mp_err s_mp_div_d(mp_int *mp, mp_digit d, mp_digit *r); + /* unsigned digit divide */ +mp_err s_mp_reduce(mp_int *x, mp_int *m, mp_int *mu); + /* Barrett reduction */ +mp_err s_mp_add(mp_int *a, mp_int *b); /* magnitude addition */ +mp_err s_mp_sub(mp_int *a, mp_int *b); /* magnitude subtract */ +mp_err s_mp_mul(mp_int *a, mp_int *b); /* magnitude multiply */ +#if 0 +void s_mp_kmul(mp_digit *a, mp_digit *b, mp_digit *out, mp_size len); + /* multiply buffers in place */ +#endif +#if MP_SQUARE +mp_err s_mp_sqr(mp_int *a); /* magnitude square */ +#else +#define s_mp_sqr(a) s_mp_mul(a, a) +#endif +mp_err s_mp_div(mp_int *a, mp_int *b); /* magnitude divide */ +mp_err s_mp_2expt(mp_int *a, mp_digit k); /* a = 2^k */ +int s_mp_cmp(mp_int *a, mp_int *b); /* magnitude comparison */ +int s_mp_cmp_d(mp_int *a, mp_digit d); /* magnitude digit compare */ +int s_mp_ispow2(mp_int *v); /* is v a power of 2? */ +int s_mp_ispow2d(mp_digit d); /* is d a power of 2? */ + +int s_mp_tovalue(char ch, int r); /* convert ch to value */ +char s_mp_todigit(int val, int r, int low); /* convert val to digit */ +int s_mp_outlen(int bits, int r); /* output length in bytes */ + +/* }}} */ + +/* {{{ Default precision manipulation */ + +unsigned int mp_get_prec(void) +{ + return s_mp_defprec; + +} /* end mp_get_prec() */ + +void mp_set_prec(unsigned int prec) +{ + if(prec == 0) + s_mp_defprec = MP_DEFPREC; + else + s_mp_defprec = prec; + +} /* end mp_set_prec() */ + +/* }}} */ + +/*------------------------------------------------------------------------*/ +/* {{{ mp_init(mp) */ + +/* + mp_init(mp) + + Initialize a new zero-valued mp_int. Returns MP_OKAY if successful, + MP_MEM if memory could not be allocated for the structure. + */ + +mp_err mp_init(mp_int *mp) +{ + return mp_init_size(mp, s_mp_defprec); + +} /* end mp_init() */ + +/* }}} */ + +/* {{{ mp_init_array(mp[], count) */ + +mp_err mp_init_array(mp_int mp[], int count) +{ + mp_err res; + int pos; + + ARGCHK(mp !=NULL && count > 0, MP_BADARG); + + for(pos = 0; pos < count; ++pos) { + if((res = mp_init(&mp[pos])) != MP_OKAY) + goto CLEANUP; + } + + return MP_OKAY; + + CLEANUP: + while(--pos >= 0) + mp_clear(&mp[pos]); + + return res; + +} /* end mp_init_array() */ + +/* }}} */ + +/* {{{ mp_init_size(mp, prec) */ + +/* + mp_init_size(mp, prec) + + Initialize a new zero-valued mp_int with at least the given + precision; returns MP_OKAY if successful, or MP_MEM if memory could + not be allocated for the structure. + */ + +mp_err mp_init_size(mp_int *mp, mp_size prec) +{ + ARGCHK(mp != NULL && prec > 0, MP_BADARG); + + if((DIGITS(mp) = s_mp_alloc(prec, sizeof(mp_digit))) == NULL) + return MP_MEM; + + SIGN(mp) = MP_ZPOS; + USED(mp) = 1; + ALLOC(mp) = prec; + + return MP_OKAY; + +} /* end mp_init_size() */ + +/* }}} */ + +/* {{{ mp_init_copy(mp, from) */ + +/* + mp_init_copy(mp, from) + + Initialize mp as an exact copy of from. Returns MP_OKAY if + successful, MP_MEM if memory could not be allocated for the new + structure. + */ + +mp_err mp_init_copy(mp_int *mp, mp_int *from) +{ + ARGCHK(mp != NULL && from != NULL, MP_BADARG); + + if(mp == from) + return MP_OKAY; + + if((DIGITS(mp) = s_mp_alloc(USED(from), sizeof(mp_digit))) == NULL) + return MP_MEM; + + s_mp_copy(DIGITS(from), DIGITS(mp), USED(from)); + USED(mp) = USED(from); + ALLOC(mp) = USED(from); + SIGN(mp) = SIGN(from); + + return MP_OKAY; + +} /* end mp_init_copy() */ + +/* }}} */ + +/* {{{ mp_copy(from, to) */ + +/* + mp_copy(from, to) + + Copies the mp_int 'from' to the mp_int 'to'. It is presumed that + 'to' has already been initialized (if not, use mp_init_copy() + instead). If 'from' and 'to' are identical, nothing happens. + */ + +mp_err mp_copy(mp_int *from, mp_int *to) +{ + ARGCHK(from != NULL && to != NULL, MP_BADARG); + + if(from == to) + return MP_OKAY; + + { /* copy */ + mp_digit *tmp; + + /* + If the allocated buffer in 'to' already has enough space to hold + all the used digits of 'from', we'll re-use it to avoid hitting + the memory allocater more than necessary; otherwise, we'd have + to grow anyway, so we just allocate a hunk and make the copy as + usual + */ + if(ALLOC(to) >= USED(from)) { + s_mp_setz(DIGITS(to) + USED(from), ALLOC(to) - USED(from)); + s_mp_copy(DIGITS(from), DIGITS(to), USED(from)); + + } else { + if((tmp = s_mp_alloc(USED(from), sizeof(mp_digit))) == NULL) + return MP_MEM; + + s_mp_copy(DIGITS(from), tmp, USED(from)); + + if(DIGITS(to) != NULL) { +#if MP_CRYPTO + s_mp_setz(DIGITS(to), ALLOC(to)); +#endif + s_mp_free(DIGITS(to)); + } + + DIGITS(to) = tmp; + ALLOC(to) = USED(from); + } + + /* Copy the precision and sign from the original */ + USED(to) = USED(from); + SIGN(to) = SIGN(from); + } /* end copy */ + + return MP_OKAY; + +} /* end mp_copy() */ + +/* }}} */ + +/* {{{ mp_exch(mp1, mp2) */ + +/* + mp_exch(mp1, mp2) + + Exchange mp1 and mp2 without allocating any intermediate memory + (well, unless you count the stack space needed for this call and the + locals it creates...). This cannot fail. + */ + +void mp_exch(mp_int *mp1, mp_int *mp2) +{ +#if MP_ARGCHK == 2 + assert(mp1 != NULL && mp2 != NULL); +#else + if(mp1 == NULL || mp2 == NULL) + return; +#endif + + s_mp_exch(mp1, mp2); + +} /* end mp_exch() */ + +/* }}} */ + +/* {{{ mp_clear(mp) */ + +/* + mp_clear(mp) + + Release the storage used by an mp_int, and void its fields so that + if someone calls mp_clear() again for the same int later, we won't + get tollchocked. + */ + +void mp_clear(mp_int *mp) +{ + if(mp == NULL) + return; + + if(DIGITS(mp) != NULL) { +#if MP_CRYPTO + s_mp_setz(DIGITS(mp), ALLOC(mp)); +#endif + s_mp_free(DIGITS(mp)); + DIGITS(mp) = NULL; + } + + USED(mp) = 0; + ALLOC(mp) = 0; + +} /* end mp_clear() */ + +/* }}} */ + +/* {{{ mp_clear_array(mp[], count) */ + +void mp_clear_array(mp_int mp[], int count) +{ + ARGCHK(mp != NULL && count > 0, MP_BADARG); + + while(--count >= 0) + mp_clear(&mp[count]); + +} /* end mp_clear_array() */ + +/* }}} */ + +/* {{{ mp_zero(mp) */ + +/* + mp_zero(mp) + + Set mp to zero. Does not change the allocated size of the structure, + and therefore cannot fail (except on a bad argument, which we ignore) + */ +void mp_zero(mp_int *mp) +{ + if(mp == NULL) + return; + + s_mp_setz(DIGITS(mp), ALLOC(mp)); + USED(mp) = 1; + SIGN(mp) = MP_ZPOS; + +} /* end mp_zero() */ + +/* }}} */ + +/* {{{ mp_set(mp, d) */ + +void mp_set(mp_int *mp, mp_digit d) +{ + if(mp == NULL) + return; + + mp_zero(mp); + DIGIT(mp, 0) = d; + +} /* end mp_set() */ + +/* }}} */ + +/* {{{ mp_set_int(mp, z) */ + +mp_err mp_set_int(mp_int *mp, long z) +{ + int ix; + unsigned long v = abs(z); + mp_err res; + + ARGCHK(mp != NULL, MP_BADARG); + + mp_zero(mp); + if(z == 0) + return MP_OKAY; /* shortcut for zero */ + + for(ix = sizeof(long) - 1; ix >= 0; ix--) { + + if((res = s_mp_mul_2d(mp, CHAR_BIT)) != MP_OKAY) + return res; + + res = s_mp_add_d(mp, + (mp_digit)((v >> (ix * CHAR_BIT)) & UCHAR_MAX)); + if(res != MP_OKAY) + return res; + + } + + if(z < 0) + SIGN(mp) = MP_NEG; + + return MP_OKAY; + +} /* end mp_set_int() */ + +/* }}} */ + +/*------------------------------------------------------------------------*/ +/* {{{ Digit arithmetic */ + +/* {{{ mp_add_d(a, d, b) */ + +/* + mp_add_d(a, d, b) + + Compute the sum b = a + d, for a single digit d. Respects the sign of + its primary addend (single digits are unsigned anyway). + */ + +mp_err mp_add_d(mp_int *a, mp_digit d, mp_int *b) +{ + mp_err res = MP_OKAY; + + ARGCHK(a != NULL && b != NULL, MP_BADARG); + + if((res = mp_copy(a, b)) != MP_OKAY) + return res; + + if(SIGN(b) == MP_ZPOS) { + res = s_mp_add_d(b, d); + } else if(s_mp_cmp_d(b, d) >= 0) { + res = s_mp_sub_d(b, d); + } else { + SIGN(b) = MP_ZPOS; + + DIGIT(b, 0) = d - DIGIT(b, 0); + } + + return res; + +} /* end mp_add_d() */ + +/* }}} */ + +/* {{{ mp_sub_d(a, d, b) */ + +/* + mp_sub_d(a, d, b) + + Compute the difference b = a - d, for a single digit d. Respects the + sign of its subtrahend (single digits are unsigned anyway). + */ + +mp_err mp_sub_d(mp_int *a, mp_digit d, mp_int *b) +{ + mp_err res; + + ARGCHK(a != NULL && b != NULL, MP_BADARG); + + if((res = mp_copy(a, b)) != MP_OKAY) + return res; + + if(SIGN(b) == MP_NEG) { + if((res = s_mp_add_d(b, d)) != MP_OKAY) + return res; + + } else if(s_mp_cmp_d(b, d) >= 0) { + if((res = s_mp_sub_d(b, d)) != MP_OKAY) + return res; + + } else { + mp_neg(b, b); + + DIGIT(b, 0) = d - DIGIT(b, 0); + SIGN(b) = MP_NEG; + } + + if(s_mp_cmp_d(b, 0) == 0) + SIGN(b) = MP_ZPOS; + + return MP_OKAY; + +} /* end mp_sub_d() */ + +/* }}} */ + +/* {{{ mp_mul_d(a, d, b) */ + +/* + mp_mul_d(a, d, b) + + Compute the product b = a * d, for a single digit d. Respects the sign + of its multiplicand (single digits are unsigned anyway) + */ + +mp_err mp_mul_d(mp_int *a, mp_digit d, mp_int *b) +{ + mp_err res; + + ARGCHK(a != NULL && b != NULL, MP_BADARG); + + if(d == 0) { + mp_zero(b); + return MP_OKAY; + } + + if((res = mp_copy(a, b)) != MP_OKAY) + return res; + + res = s_mp_mul_d(b, d); + + return res; + +} /* end mp_mul_d() */ + +/* }}} */ + +/* {{{ mp_mul_2(a, c) */ + +mp_err mp_mul_2(mp_int *a, mp_int *c) +{ + mp_err res; + + ARGCHK(a != NULL && c != NULL, MP_BADARG); + + if((res = mp_copy(a, c)) != MP_OKAY) + return res; + + return s_mp_mul_2(c); + +} /* end mp_mul_2() */ + +/* }}} */ + +/* {{{ mp_div_d(a, d, q, r) */ + +/* + mp_div_d(a, d, q, r) + + Compute the quotient q = a / d and remainder r = a mod d, for a + single digit d. Respects the sign of its divisor (single digits are + unsigned anyway). + */ + +mp_err mp_div_d(mp_int *a, mp_digit d, mp_int *q, mp_digit *r) +{ + mp_err res; + mp_digit rem; + int pow; + + ARGCHK(a != NULL, MP_BADARG); + + if(d == 0) + return MP_RANGE; + + /* Shortcut for powers of two ... */ + if((pow = s_mp_ispow2d(d)) >= 0) { + mp_digit mask; + + mask = (1 << pow) - 1; + rem = DIGIT(a, 0) & mask; + + if(q) { + mp_copy(a, q); + s_mp_div_2d(q, pow); + } + + if(r) + *r = rem; + + return MP_OKAY; + } + + /* + If the quotient is actually going to be returned, we'll try to + avoid hitting the memory allocator by copying the dividend into it + and doing the division there. This can't be any _worse_ than + always copying, and will sometimes be better (since it won't make + another copy) + + If it's not going to be returned, we need to allocate a temporary + to hold the quotient, which will just be discarded. + */ + if(q) { + if((res = mp_copy(a, q)) != MP_OKAY) + return res; + + res = s_mp_div_d(q, d, &rem); + if(s_mp_cmp_d(q, 0) == MP_EQ) + SIGN(q) = MP_ZPOS; + + } else { + mp_int qp; + + if((res = mp_init_copy(&qp, a)) != MP_OKAY) + return res; + + res = s_mp_div_d(&qp, d, &rem); + if(s_mp_cmp_d(&qp, 0) == 0) + SIGN(&qp) = MP_ZPOS; + + mp_clear(&qp); + } + + if(r) + *r = rem; + + return res; + +} /* end mp_div_d() */ + +/* }}} */ + +/* {{{ mp_div_2(a, c) */ + +/* + mp_div_2(a, c) + + Compute c = a / 2, disregarding the remainder. + */ + +mp_err mp_div_2(mp_int *a, mp_int *c) +{ + mp_err res; + + ARGCHK(a != NULL && c != NULL, MP_BADARG); + + if((res = mp_copy(a, c)) != MP_OKAY) + return res; + + s_mp_div_2(c); + + return MP_OKAY; + +} /* end mp_div_2() */ + +/* }}} */ + +/* {{{ mp_expt_d(a, d, b) */ + +mp_err mp_expt_d(mp_int *a, mp_digit d, mp_int *c) +{ + mp_int s, x; + mp_err res; + + ARGCHK(a != NULL && c != NULL, MP_BADARG); + + if((res = mp_init(&s)) != MP_OKAY) + return res; + if((res = mp_init_copy(&x, a)) != MP_OKAY) + goto X; + + DIGIT(&s, 0) = 1; + + while(d != 0) { + if(d & 1) { + if((res = s_mp_mul(&s, &x)) != MP_OKAY) + goto CLEANUP; + } + + d >>= 1; + + if((res = s_mp_sqr(&x)) != MP_OKAY) + goto CLEANUP; + } + + s_mp_exch(&s, c); + +CLEANUP: + mp_clear(&x); +X: + mp_clear(&s); + + return res; + +} /* end mp_expt_d() */ + +/* }}} */ + +/* }}} */ + +/*------------------------------------------------------------------------*/ +/* {{{ Full arithmetic */ + +/* {{{ mp_abs(a, b) */ + +/* + mp_abs(a, b) + + Compute b = |a|. 'a' and 'b' may be identical. + */ + +mp_err mp_abs(mp_int *a, mp_int *b) +{ + mp_err res; + + ARGCHK(a != NULL && b != NULL, MP_BADARG); + + if((res = mp_copy(a, b)) != MP_OKAY) + return res; + + SIGN(b) = MP_ZPOS; + + return MP_OKAY; + +} /* end mp_abs() */ + +/* }}} */ + +/* {{{ mp_neg(a, b) */ + +/* + mp_neg(a, b) + + Compute b = -a. 'a' and 'b' may be identical. + */ + +mp_err mp_neg(mp_int *a, mp_int *b) +{ + mp_err res; + + ARGCHK(a != NULL && b != NULL, MP_BADARG); + + if((res = mp_copy(a, b)) != MP_OKAY) + return res; + + if(s_mp_cmp_d(b, 0) == MP_EQ) + SIGN(b) = MP_ZPOS; + else + SIGN(b) = (SIGN(b) == MP_NEG) ? MP_ZPOS : MP_NEG; + + return MP_OKAY; + +} /* end mp_neg() */ + +/* }}} */ + +/* {{{ mp_add(a, b, c) */ + +/* + mp_add(a, b, c) + + Compute c = a + b. All parameters may be identical. + */ + +mp_err mp_add(mp_int *a, mp_int *b, mp_int *c) +{ + mp_err res; + int cmp; + + ARGCHK(a != NULL && b != NULL && c != NULL, MP_BADARG); + + if(SIGN(a) == SIGN(b)) { /* same sign: add values, keep sign */ + + /* Commutativity of addition lets us do this in either order, + so we avoid having to use a temporary even if the result + is supposed to replace the output + */ + if(c == b) { + if((res = s_mp_add(c, a)) != MP_OKAY) + return res; + } else { + if(c != a && (res = mp_copy(a, c)) != MP_OKAY) + return res; + + if((res = s_mp_add(c, b)) != MP_OKAY) + return res; + } + + } else if((cmp = s_mp_cmp(a, b)) > 0) { /* different sign: a > b */ + + /* If the output is going to be clobbered, we will use a temporary + variable; otherwise, we'll do it without touching the memory + allocator at all, if possible + */ + if(c == b) { + mp_int tmp; + + if((res = mp_init_copy(&tmp, a)) != MP_OKAY) + return res; + if((res = s_mp_sub(&tmp, b)) != MP_OKAY) { + mp_clear(&tmp); + return res; + } + + s_mp_exch(&tmp, c); + mp_clear(&tmp); + + } else { + + if(c != a && (res = mp_copy(a, c)) != MP_OKAY) + return res; + if((res = s_mp_sub(c, b)) != MP_OKAY) + return res; + + } + + } else if(cmp == 0) { /* different sign, a == b */ + + mp_zero(c); + return MP_OKAY; + + } else { /* different sign: a < b */ + + /* See above... */ + if(c == a) { + mp_int tmp; + + if((res = mp_init_copy(&tmp, b)) != MP_OKAY) + return res; + if((res = s_mp_sub(&tmp, a)) != MP_OKAY) { + mp_clear(&tmp); + return res; + } + + s_mp_exch(&tmp, c); + mp_clear(&tmp); + + } else { + + if(c != b && (res = mp_copy(b, c)) != MP_OKAY) + return res; + if((res = s_mp_sub(c, a)) != MP_OKAY) + return res; + + } + } + + if(USED(c) == 1 && DIGIT(c, 0) == 0) + SIGN(c) = MP_ZPOS; + + return MP_OKAY; + +} /* end mp_add() */ + +/* }}} */ + +/* {{{ mp_sub(a, b, c) */ + +/* + mp_sub(a, b, c) + + Compute c = a - b. All parameters may be identical. + */ + +mp_err mp_sub(mp_int *a, mp_int *b, mp_int *c) +{ + mp_err res; + int cmp; + + ARGCHK(a != NULL && b != NULL && c != NULL, MP_BADARG); + + if(SIGN(a) != SIGN(b)) { + if(c == a) { + if((res = s_mp_add(c, b)) != MP_OKAY) + return res; + } else { + if(c != b && ((res = mp_copy(b, c)) != MP_OKAY)) + return res; + if((res = s_mp_add(c, a)) != MP_OKAY) + return res; + SIGN(c) = SIGN(a); + } + + } else if((cmp = s_mp_cmp(a, b)) > 0) { /* Same sign, a > b */ + if(c == b) { + mp_int tmp; + + if((res = mp_init_copy(&tmp, a)) != MP_OKAY) + return res; + if((res = s_mp_sub(&tmp, b)) != MP_OKAY) { + mp_clear(&tmp); + return res; + } + s_mp_exch(&tmp, c); + mp_clear(&tmp); + + } else { + if(c != a && ((res = mp_copy(a, c)) != MP_OKAY)) + return res; + + if((res = s_mp_sub(c, b)) != MP_OKAY) + return res; + } + + } else if(cmp == 0) { /* Same sign, equal magnitude */ + mp_zero(c); + return MP_OKAY; + + } else { /* Same sign, b > a */ + if(c == a) { + mp_int tmp; + + if((res = mp_init_copy(&tmp, b)) != MP_OKAY) + return res; + + if((res = s_mp_sub(&tmp, a)) != MP_OKAY) { + mp_clear(&tmp); + return res; + } + s_mp_exch(&tmp, c); + mp_clear(&tmp); + + } else { + if(c != b && ((res = mp_copy(b, c)) != MP_OKAY)) + return res; + + if((res = s_mp_sub(c, a)) != MP_OKAY) + return res; + } + + SIGN(c) = !SIGN(b); + } + + if(USED(c) == 1 && DIGIT(c, 0) == 0) + SIGN(c) = MP_ZPOS; + + return MP_OKAY; + +} /* end mp_sub() */ + +/* }}} */ + +/* {{{ mp_mul(a, b, c) */ + +/* + mp_mul(a, b, c) + + Compute c = a * b. All parameters may be identical. + */ + +mp_err mp_mul(mp_int *a, mp_int *b, mp_int *c) +{ + mp_err res; + mp_sign sgn; + + ARGCHK(a != NULL && b != NULL && c != NULL, MP_BADARG); + + sgn = (SIGN(a) == SIGN(b)) ? MP_ZPOS : MP_NEG; + + if(c == b) { + if((res = s_mp_mul(c, a)) != MP_OKAY) + return res; + + } else { + if((res = mp_copy(a, c)) != MP_OKAY) + return res; + + if((res = s_mp_mul(c, b)) != MP_OKAY) + return res; + } + + if(sgn == MP_ZPOS || s_mp_cmp_d(c, 0) == MP_EQ) + SIGN(c) = MP_ZPOS; + else + SIGN(c) = sgn; + + return MP_OKAY; + +} /* end mp_mul() */ + +/* }}} */ + +/* {{{ mp_mul_2d(a, d, c) */ + +/* + mp_mul_2d(a, d, c) + + Compute c = a * 2^d. a may be the same as c. + */ + +mp_err mp_mul_2d(mp_int *a, mp_digit d, mp_int *c) +{ + mp_err res; + + ARGCHK(a != NULL && c != NULL, MP_BADARG); + + if((res = mp_copy(a, c)) != MP_OKAY) + return res; + + if(d == 0) + return MP_OKAY; + + return s_mp_mul_2d(c, d); + +} /* end mp_mul() */ + +/* }}} */ + +/* {{{ mp_sqr(a, b) */ + +#if MP_SQUARE +mp_err mp_sqr(mp_int *a, mp_int *b) +{ + mp_err res; + + ARGCHK(a != NULL && b != NULL, MP_BADARG); + + if((res = mp_copy(a, b)) != MP_OKAY) + return res; + + if((res = s_mp_sqr(b)) != MP_OKAY) + return res; + + SIGN(b) = MP_ZPOS; + + return MP_OKAY; + +} /* end mp_sqr() */ +#endif + +/* }}} */ + +/* {{{ mp_div(a, b, q, r) */ + +/* + mp_div(a, b, q, r) + + Compute q = a / b and r = a mod b. Input parameters may be re-used + as output parameters. If q or r is NULL, that portion of the + computation will be discarded (although it will still be computed) + + Pay no attention to the hacker behind the curtain. + */ + +mp_err mp_div(mp_int *a, mp_int *b, mp_int *q, mp_int *r) +{ + mp_err res; + mp_int qtmp, rtmp; + int cmp; + + ARGCHK(a != NULL && b != NULL, MP_BADARG); + + if(mp_cmp_z(b) == MP_EQ) + return MP_RANGE; + + /* If a <= b, we can compute the solution without division, and + avoid any memory allocation + */ + if((cmp = s_mp_cmp(a, b)) < 0) { + if(r) { + if((res = mp_copy(a, r)) != MP_OKAY) + return res; + } + + if(q) + mp_zero(q); + + return MP_OKAY; + + } else if(cmp == 0) { + + /* Set quotient to 1, with appropriate sign */ + if(q) { + int qneg = (SIGN(a) != SIGN(b)); + + mp_set(q, 1); + if(qneg) + SIGN(q) = MP_NEG; + } + + if(r) + mp_zero(r); + + return MP_OKAY; + } + + /* If we get here, it means we actually have to do some division */ + + /* Set up some temporaries... */ + if((res = mp_init_copy(&qtmp, a)) != MP_OKAY) + return res; + if((res = mp_init_copy(&rtmp, b)) != MP_OKAY) + goto CLEANUP; + + if((res = s_mp_div(&qtmp, &rtmp)) != MP_OKAY) + goto CLEANUP; + + /* Compute the signs for the output */ + SIGN(&rtmp) = SIGN(a); /* Sr = Sa */ + if(SIGN(a) == SIGN(b)) + SIGN(&qtmp) = MP_ZPOS; /* Sq = MP_ZPOS if Sa = Sb */ + else + SIGN(&qtmp) = MP_NEG; /* Sq = MP_NEG if Sa != Sb */ + + if(s_mp_cmp_d(&qtmp, 0) == MP_EQ) + SIGN(&qtmp) = MP_ZPOS; + if(s_mp_cmp_d(&rtmp, 0) == MP_EQ) + SIGN(&rtmp) = MP_ZPOS; + + /* Copy output, if it is needed */ + if(q) + s_mp_exch(&qtmp, q); + + if(r) + s_mp_exch(&rtmp, r); + +CLEANUP: + mp_clear(&rtmp); + mp_clear(&qtmp); + + return res; + +} /* end mp_div() */ + +/* }}} */ + +/* {{{ mp_div_2d(a, d, q, r) */ + +mp_err mp_div_2d(mp_int *a, mp_digit d, mp_int *q, mp_int *r) +{ + mp_err res; + + ARGCHK(a != NULL, MP_BADARG); + + if(q) { + if((res = mp_copy(a, q)) != MP_OKAY) + return res; + + s_mp_div_2d(q, d); + } + + if(r) { + if((res = mp_copy(a, r)) != MP_OKAY) + return res; + + s_mp_mod_2d(r, d); + } + + return MP_OKAY; + +} /* end mp_div_2d() */ + +/* }}} */ + +/* {{{ mp_expt(a, b, c) */ + +/* + mp_expt(a, b, c) + + Compute c = a ** b, that is, raise a to the b power. Uses a + standard iterative square-and-multiply technique. + */ + +mp_err mp_expt(mp_int *a, mp_int *b, mp_int *c) +{ + mp_int s, x; + mp_err res; + mp_digit d; + int dig, bit; + + ARGCHK(a != NULL && b != NULL && c != NULL, MP_BADARG); + + if(mp_cmp_z(b) < 0) + return MP_RANGE; + + if((res = mp_init(&s)) != MP_OKAY) + return res; + + mp_set(&s, 1); + + if((res = mp_init_copy(&x, a)) != MP_OKAY) + goto X; + + /* Loop over low-order digits in ascending order */ + for(dig = 0; dig < (USED(b) - 1); dig++) { + d = DIGIT(b, dig); + + /* Loop over bits of each non-maximal digit */ + for(bit = 0; bit < DIGIT_BIT; bit++) { + if(d & 1) { + if((res = s_mp_mul(&s, &x)) != MP_OKAY) + goto CLEANUP; + } + + d >>= 1; + + if((res = s_mp_sqr(&x)) != MP_OKAY) + goto CLEANUP; + } + } + + /* Consider now the last digit... */ + d = DIGIT(b, dig); + + while(d) { + if(d & 1) { + if((res = s_mp_mul(&s, &x)) != MP_OKAY) + goto CLEANUP; + } + + d >>= 1; + + if((res = s_mp_sqr(&x)) != MP_OKAY) + goto CLEANUP; + } + + if(mp_iseven(b)) + SIGN(&s) = SIGN(a); + + res = mp_copy(&s, c); + +CLEANUP: + mp_clear(&x); +X: + mp_clear(&s); + + return res; + +} /* end mp_expt() */ + +/* }}} */ + +/* {{{ mp_2expt(a, k) */ + +/* Compute a = 2^k */ + +mp_err mp_2expt(mp_int *a, mp_digit k) +{ + ARGCHK(a != NULL, MP_BADARG); + + return s_mp_2expt(a, k); + +} /* end mp_2expt() */ + +/* }}} */ + +/* {{{ mp_mod(a, m, c) */ + +/* + mp_mod(a, m, c) + + Compute c = a (mod m). Result will always be 0 <= c < m. + */ + +mp_err mp_mod(mp_int *a, mp_int *m, mp_int *c) +{ + mp_err res; + int mag; + + ARGCHK(a != NULL && m != NULL && c != NULL, MP_BADARG); + + if(SIGN(m) == MP_NEG) + return MP_RANGE; + + /* + If |a| > m, we need to divide to get the remainder and take the + absolute value. + + If |a| < m, we don't need to do any division, just copy and adjust + the sign (if a is negative). + + If |a| == m, we can simply set the result to zero. + + This order is intended to minimize the average path length of the + comparison chain on common workloads -- the most frequent cases are + that |a| != m, so we do those first. + */ + if((mag = s_mp_cmp(a, m)) > 0) { + if((res = mp_div(a, m, NULL, c)) != MP_OKAY) + return res; + + if(SIGN(c) == MP_NEG) { + if((res = mp_add(c, m, c)) != MP_OKAY) + return res; + } + + } else if(mag < 0) { + if((res = mp_copy(a, c)) != MP_OKAY) + return res; + + if(mp_cmp_z(a) < 0) { + if((res = mp_add(c, m, c)) != MP_OKAY) + return res; + + } + + } else { + mp_zero(c); + + } + + return MP_OKAY; + +} /* end mp_mod() */ + +/* }}} */ + +/* {{{ mp_mod_d(a, d, c) */ + +/* + mp_mod_d(a, d, c) + + Compute c = a (mod d). Result will always be 0 <= c < d + */ +mp_err mp_mod_d(mp_int *a, mp_digit d, mp_digit *c) +{ + mp_err res; + mp_digit rem; + + ARGCHK(a != NULL && c != NULL, MP_BADARG); + + if(s_mp_cmp_d(a, d) > 0) { + if((res = mp_div_d(a, d, NULL, &rem)) != MP_OKAY) + return res; + + } else { + if(SIGN(a) == MP_NEG) + rem = d - DIGIT(a, 0); + else + rem = DIGIT(a, 0); + } + + if(c) + *c = rem; + + return MP_OKAY; + +} /* end mp_mod_d() */ + +/* }}} */ + +/* {{{ mp_sqrt(a, b) */ + +/* + mp_sqrt(a, b) + + Compute the integer square root of a, and store the result in b. + Uses an integer-arithmetic version of Newton's iterative linear + approximation technique to determine this value; the result has the + following two properties: + + b^2 <= a + (b+1)^2 >= a + + It is a range error to pass a negative value. + */ +mp_err mp_sqrt(mp_int *a, mp_int *b) +{ + mp_int x, t; + mp_err res; + + ARGCHK(a != NULL && b != NULL, MP_BADARG); + + /* Cannot take square root of a negative value */ + if(SIGN(a) == MP_NEG) + return MP_RANGE; + + /* Special cases for zero and one, trivial */ + if(mp_cmp_d(a, 0) == MP_EQ || mp_cmp_d(a, 1) == MP_EQ) + return mp_copy(a, b); + + /* Initialize the temporaries we'll use below */ + if((res = mp_init_size(&t, USED(a))) != MP_OKAY) + return res; + + /* Compute an initial guess for the iteration as a itself */ + if((res = mp_init_copy(&x, a)) != MP_OKAY) + goto X; + +s_mp_rshd(&x, (USED(&x)/2)+1); +mp_add_d(&x, 1, &x); + + for(;;) { + /* t = (x * x) - a */ + mp_copy(&x, &t); /* can't fail, t is big enough for original x */ + if((res = mp_sqr(&t, &t)) != MP_OKAY || + (res = mp_sub(&t, a, &t)) != MP_OKAY) + goto CLEANUP; + + /* t = t / 2x */ + s_mp_mul_2(&x); + if((res = mp_div(&t, &x, &t, NULL)) != MP_OKAY) + goto CLEANUP; + s_mp_div_2(&x); + + /* Terminate the loop, if the quotient is zero */ + if(mp_cmp_z(&t) == MP_EQ) + break; + + /* x = x - t */ + if((res = mp_sub(&x, &t, &x)) != MP_OKAY) + goto CLEANUP; + + } + + /* Copy result to output parameter */ + mp_sub_d(&x, 1, &x); + s_mp_exch(&x, b); + + CLEANUP: + mp_clear(&x); + X: + mp_clear(&t); + + return res; + +} /* end mp_sqrt() */ + +/* }}} */ + +/* }}} */ + +/*------------------------------------------------------------------------*/ +/* {{{ Modular arithmetic */ + +#if MP_MODARITH +/* {{{ mp_addmod(a, b, m, c) */ + +/* + mp_addmod(a, b, m, c) + + Compute c = (a + b) mod m + */ + +mp_err mp_addmod(mp_int *a, mp_int *b, mp_int *m, mp_int *c) +{ + mp_err res; + + ARGCHK(a != NULL && b != NULL && m != NULL && c != NULL, MP_BADARG); + + if((res = mp_add(a, b, c)) != MP_OKAY) + return res; + if((res = mp_mod(c, m, c)) != MP_OKAY) + return res; + + return MP_OKAY; + +} + +/* }}} */ + +/* {{{ mp_submod(a, b, m, c) */ + +/* + mp_submod(a, b, m, c) + + Compute c = (a - b) mod m + */ + +mp_err mp_submod(mp_int *a, mp_int *b, mp_int *m, mp_int *c) +{ + mp_err res; + + ARGCHK(a != NULL && b != NULL && m != NULL && c != NULL, MP_BADARG); + + if((res = mp_sub(a, b, c)) != MP_OKAY) + return res; + if((res = mp_mod(c, m, c)) != MP_OKAY) + return res; + + return MP_OKAY; + +} + +/* }}} */ + +/* {{{ mp_mulmod(a, b, m, c) */ + +/* + mp_mulmod(a, b, m, c) + + Compute c = (a * b) mod m + */ + +mp_err mp_mulmod(mp_int *a, mp_int *b, mp_int *m, mp_int *c) +{ + mp_err res; + + ARGCHK(a != NULL && b != NULL && m != NULL && c != NULL, MP_BADARG); + + if((res = mp_mul(a, b, c)) != MP_OKAY) + return res; + if((res = mp_mod(c, m, c)) != MP_OKAY) + return res; + + return MP_OKAY; + +} + +/* }}} */ + +/* {{{ mp_sqrmod(a, m, c) */ + +#if MP_SQUARE +mp_err mp_sqrmod(mp_int *a, mp_int *m, mp_int *c) +{ + mp_err res; + + ARGCHK(a != NULL && m != NULL && c != NULL, MP_BADARG); + + if((res = mp_sqr(a, c)) != MP_OKAY) + return res; + if((res = mp_mod(c, m, c)) != MP_OKAY) + return res; + + return MP_OKAY; + +} /* end mp_sqrmod() */ +#endif + +/* }}} */ + +/* {{{ mp_exptmod(a, b, m, c) */ + +/* + mp_exptmod(a, b, m, c) + + Compute c = (a ** b) mod m. Uses a standard square-and-multiply + method with modular reductions at each step. (This is basically the + same code as mp_expt(), except for the addition of the reductions) + + The modular reductions are done using Barrett's algorithm (see + s_mp_reduce() below for details) + */ + +mp_err mp_exptmod(mp_int *a, mp_int *b, mp_int *m, mp_int *c) +{ + mp_int s, x, mu; + mp_err res; + mp_digit d, *db = DIGITS(b); + mp_size ub = USED(b); + int dig, bit; + + ARGCHK(a != NULL && b != NULL && c != NULL, MP_BADARG); + + if(mp_cmp_z(b) < 0 || mp_cmp_z(m) <= 0) + return MP_RANGE; + + if((res = mp_init(&s)) != MP_OKAY) + return res; + if((res = mp_init_copy(&x, a)) != MP_OKAY) + goto X; + if((res = mp_mod(&x, m, &x)) != MP_OKAY || + (res = mp_init(&mu)) != MP_OKAY) + goto MU; + + mp_set(&s, 1); + + /* mu = b^2k / m */ + s_mp_add_d(&mu, 1); + s_mp_lshd(&mu, 2 * USED(m)); + if((res = mp_div(&mu, m, &mu, NULL)) != MP_OKAY) + goto CLEANUP; + + /* Loop over digits of b in ascending order, except highest order */ + for(dig = 0; dig < (ub - 1); dig++) { + d = *db++; + + /* Loop over the bits of the lower-order digits */ + for(bit = 0; bit < DIGIT_BIT; bit++) { + if(d & 1) { + if((res = s_mp_mul(&s, &x)) != MP_OKAY) + goto CLEANUP; + if((res = s_mp_reduce(&s, m, &mu)) != MP_OKAY) + goto CLEANUP; + } + + d >>= 1; + + if((res = s_mp_sqr(&x)) != MP_OKAY) + goto CLEANUP; + if((res = s_mp_reduce(&x, m, &mu)) != MP_OKAY) + goto CLEANUP; + } + } + + /* Now do the last digit... */ + d = *db; + + while(d) { + if(d & 1) { + if((res = s_mp_mul(&s, &x)) != MP_OKAY) + goto CLEANUP; + if((res = s_mp_reduce(&s, m, &mu)) != MP_OKAY) + goto CLEANUP; + } + + d >>= 1; + + if((res = s_mp_sqr(&x)) != MP_OKAY) + goto CLEANUP; + if((res = s_mp_reduce(&x, m, &mu)) != MP_OKAY) + goto CLEANUP; + } + + s_mp_exch(&s, c); + + CLEANUP: + mp_clear(&mu); + MU: + mp_clear(&x); + X: + mp_clear(&s); + + return res; + +} /* end mp_exptmod() */ + +/* }}} */ + +/* {{{ mp_exptmod_d(a, d, m, c) */ + +mp_err mp_exptmod_d(mp_int *a, mp_digit d, mp_int *m, mp_int *c) +{ + mp_int s, x; + mp_err res; + + ARGCHK(a != NULL && c != NULL, MP_BADARG); + + if((res = mp_init(&s)) != MP_OKAY) + return res; + if((res = mp_init_copy(&x, a)) != MP_OKAY) + goto X; + + mp_set(&s, 1); + + while(d != 0) { + if(d & 1) { + if((res = s_mp_mul(&s, &x)) != MP_OKAY || + (res = mp_mod(&s, m, &s)) != MP_OKAY) + goto CLEANUP; + } + + d /= 2; + + if((res = s_mp_sqr(&x)) != MP_OKAY || + (res = mp_mod(&x, m, &x)) != MP_OKAY) + goto CLEANUP; + } + + s_mp_exch(&s, c); + +CLEANUP: + mp_clear(&x); +X: + mp_clear(&s); + + return res; + +} /* end mp_exptmod_d() */ + +/* }}} */ +#endif /* if MP_MODARITH */ + +/* }}} */ + +/*------------------------------------------------------------------------*/ +/* {{{ Comparison functions */ + +/* {{{ mp_cmp_z(a) */ + +/* + mp_cmp_z(a) + + Compare a <=> 0. Returns <0 if a<0, 0 if a=0, >0 if a>0. + */ + +int mp_cmp_z(mp_int *a) +{ + if(SIGN(a) == MP_NEG) + return MP_LT; + else if(USED(a) == 1 && DIGIT(a, 0) == 0) + return MP_EQ; + else + return MP_GT; + +} /* end mp_cmp_z() */ + +/* }}} */ + +/* {{{ mp_cmp_d(a, d) */ + +/* + mp_cmp_d(a, d) + + Compare a <=> d. Returns <0 if a<d, 0 if a=d, >0 if a>d + */ + +int mp_cmp_d(mp_int *a, mp_digit d) +{ + ARGCHK(a != NULL, MP_EQ); + + if(SIGN(a) == MP_NEG) + return MP_LT; + + return s_mp_cmp_d(a, d); + +} /* end mp_cmp_d() */ + +/* }}} */ + +/* {{{ mp_cmp(a, b) */ + +int mp_cmp(mp_int *a, mp_int *b) +{ + ARGCHK(a != NULL && b != NULL, MP_EQ); + + if(SIGN(a) == SIGN(b)) { + int mag; + + if((mag = s_mp_cmp(a, b)) == MP_EQ) + return MP_EQ; + + if(SIGN(a) == MP_ZPOS) + return mag; + else + return -mag; + + } else if(SIGN(a) == MP_ZPOS) { + return MP_GT; + } else { + return MP_LT; + } + +} /* end mp_cmp() */ + +/* }}} */ + +/* {{{ mp_cmp_mag(a, b) */ + +/* + mp_cmp_mag(a, b) + + Compares |a| <=> |b|, and returns an appropriate comparison result + */ + +int mp_cmp_mag(mp_int *a, mp_int *b) +{ + ARGCHK(a != NULL && b != NULL, MP_EQ); + + return s_mp_cmp(a, b); + +} /* end mp_cmp_mag() */ + +/* }}} */ + +/* {{{ mp_cmp_int(a, z) */ + +/* + This just converts z to an mp_int, and uses the existing comparison + routines. This is sort of inefficient, but it's not clear to me how + frequently this wil get used anyway. For small positive constants, + you can always use mp_cmp_d(), and for zero, there is mp_cmp_z(). + */ +int mp_cmp_int(mp_int *a, long z) +{ + mp_int tmp; + int out; + + ARGCHK(a != NULL, MP_EQ); + + mp_init(&tmp); mp_set_int(&tmp, z); + out = mp_cmp(a, &tmp); + mp_clear(&tmp); + + return out; + +} /* end mp_cmp_int() */ + +/* }}} */ + +/* {{{ mp_isodd(a) */ + +/* + mp_isodd(a) + + Returns a true (non-zero) value if a is odd, false (zero) otherwise. + */ +int mp_isodd(mp_int *a) +{ + ARGCHK(a != NULL, 0); + + return (DIGIT(a, 0) & 1); + +} /* end mp_isodd() */ + +/* }}} */ + +/* {{{ mp_iseven(a) */ + +int mp_iseven(mp_int *a) +{ + return !mp_isodd(a); + +} /* end mp_iseven() */ + +/* }}} */ + +/* }}} */ + +/*------------------------------------------------------------------------*/ +/* {{{ Number theoretic functions */ + +#if MP_NUMTH +/* {{{ mp_gcd(a, b, c) */ + +/* + Like the old mp_gcd() function, except computes the GCD using the + binary algorithm due to Josef Stein in 1961 (via Knuth). + */ +mp_err mp_gcd(mp_int *a, mp_int *b, mp_int *c) +{ + mp_err res; + mp_int u, v, t; + mp_size k = 0; + + ARGCHK(a != NULL && b != NULL && c != NULL, MP_BADARG); + + if(mp_cmp_z(a) == MP_EQ && mp_cmp_z(b) == MP_EQ) + return MP_RANGE; + if(mp_cmp_z(a) == MP_EQ) { + return mp_copy(b, c); + } else if(mp_cmp_z(b) == MP_EQ) { + return mp_copy(a, c); + } + + if((res = mp_init(&t)) != MP_OKAY) + return res; + if((res = mp_init_copy(&u, a)) != MP_OKAY) + goto U; + if((res = mp_init_copy(&v, b)) != MP_OKAY) + goto V; + + SIGN(&u) = MP_ZPOS; + SIGN(&v) = MP_ZPOS; + + /* Divide out common factors of 2 until at least 1 of a, b is even */ + while(mp_iseven(&u) && mp_iseven(&v)) { + s_mp_div_2(&u); + s_mp_div_2(&v); + ++k; + } + + /* Initialize t */ + if(mp_isodd(&u)) { + if((res = mp_copy(&v, &t)) != MP_OKAY) + goto CLEANUP; + + /* t = -v */ + if(SIGN(&v) == MP_ZPOS) + SIGN(&t) = MP_NEG; + else + SIGN(&t) = MP_ZPOS; + + } else { + if((res = mp_copy(&u, &t)) != MP_OKAY) + goto CLEANUP; + + } + + for(;;) { + while(mp_iseven(&t)) { + s_mp_div_2(&t); + } + + if(mp_cmp_z(&t) == MP_GT) { + if((res = mp_copy(&t, &u)) != MP_OKAY) + goto CLEANUP; + + } else { + if((res = mp_copy(&t, &v)) != MP_OKAY) + goto CLEANUP; + + /* v = -t */ + if(SIGN(&t) == MP_ZPOS) + SIGN(&v) = MP_NEG; + else + SIGN(&v) = MP_ZPOS; + } + + if((res = mp_sub(&u, &v, &t)) != MP_OKAY) + goto CLEANUP; + + if(s_mp_cmp_d(&t, 0) == MP_EQ) + break; + } + + s_mp_2expt(&v, k); /* v = 2^k */ + res = mp_mul(&u, &v, c); /* c = u * v */ + + CLEANUP: + mp_clear(&v); + V: + mp_clear(&u); + U: + mp_clear(&t); + + return res; + +} /* end mp_bgcd() */ + +/* }}} */ + +/* {{{ mp_lcm(a, b, c) */ + +/* We compute the least common multiple using the rule: + + ab = [a, b](a, b) + + ... by computing the product, and dividing out the gcd. + */ + +mp_err mp_lcm(mp_int *a, mp_int *b, mp_int *c) +{ + mp_int gcd, prod; + mp_err res; + + ARGCHK(a != NULL && b != NULL && c != NULL, MP_BADARG); + + /* Set up temporaries */ + if((res = mp_init(&gcd)) != MP_OKAY) + return res; + if((res = mp_init(&prod)) != MP_OKAY) + goto GCD; + + if((res = mp_mul(a, b, &prod)) != MP_OKAY) + goto CLEANUP; + if((res = mp_gcd(a, b, &gcd)) != MP_OKAY) + goto CLEANUP; + + res = mp_div(&prod, &gcd, c, NULL); + + CLEANUP: + mp_clear(&prod); + GCD: + mp_clear(&gcd); + + return res; + +} /* end mp_lcm() */ + +/* }}} */ + +/* {{{ mp_xgcd(a, b, g, x, y) */ + +/* + mp_xgcd(a, b, g, x, y) + + Compute g = (a, b) and values x and y satisfying Bezout's identity + (that is, ax + by = g). This uses the extended binary GCD algorithm + based on the Stein algorithm used for mp_gcd() + */ + +mp_err mp_xgcd(mp_int *a, mp_int *b, mp_int *g, mp_int *x, mp_int *y) +{ + mp_int gx, xc, yc, u, v, A, B, C, D; + mp_int *clean[9]; + mp_err res; + int last = -1; + + if(mp_cmp_z(b) == 0) + return MP_RANGE; + + /* Initialize all these variables we need */ + if((res = mp_init(&u)) != MP_OKAY) goto CLEANUP; + clean[++last] = &u; + if((res = mp_init(&v)) != MP_OKAY) goto CLEANUP; + clean[++last] = &v; + if((res = mp_init(&gx)) != MP_OKAY) goto CLEANUP; + clean[++last] = &gx; + if((res = mp_init(&A)) != MP_OKAY) goto CLEANUP; + clean[++last] = &A; + if((res = mp_init(&B)) != MP_OKAY) goto CLEANUP; + clean[++last] = &B; + if((res = mp_init(&C)) != MP_OKAY) goto CLEANUP; + clean[++last] = &C; + if((res = mp_init(&D)) != MP_OKAY) goto CLEANUP; + clean[++last] = &D; + if((res = mp_init_copy(&xc, a)) != MP_OKAY) goto CLEANUP; + clean[++last] = &xc; + mp_abs(&xc, &xc); + if((res = mp_init_copy(&yc, b)) != MP_OKAY) goto CLEANUP; + clean[++last] = &yc; + mp_abs(&yc, &yc); + + mp_set(&gx, 1); + + /* Divide by two until at least one of them is even */ + while(mp_iseven(&xc) && mp_iseven(&yc)) { + s_mp_div_2(&xc); + s_mp_div_2(&yc); + if((res = s_mp_mul_2(&gx)) != MP_OKAY) + goto CLEANUP; + } + + mp_copy(&xc, &u); + mp_copy(&yc, &v); + mp_set(&A, 1); mp_set(&D, 1); + + /* Loop through binary GCD algorithm */ + for(;;) { + while(mp_iseven(&u)) { + s_mp_div_2(&u); + + if(mp_iseven(&A) && mp_iseven(&B)) { + s_mp_div_2(&A); s_mp_div_2(&B); + } else { + if((res = mp_add(&A, &yc, &A)) != MP_OKAY) goto CLEANUP; + s_mp_div_2(&A); + if((res = mp_sub(&B, &xc, &B)) != MP_OKAY) goto CLEANUP; + s_mp_div_2(&B); + } + } + + while(mp_iseven(&v)) { + s_mp_div_2(&v); + + if(mp_iseven(&C) && mp_iseven(&D)) { + s_mp_div_2(&C); s_mp_div_2(&D); + } else { + if((res = mp_add(&C, &yc, &C)) != MP_OKAY) goto CLEANUP; + s_mp_div_2(&C); + if((res = mp_sub(&D, &xc, &D)) != MP_OKAY) goto CLEANUP; + s_mp_div_2(&D); + } + } + + if(mp_cmp(&u, &v) >= 0) { + if((res = mp_sub(&u, &v, &u)) != MP_OKAY) goto CLEANUP; + if((res = mp_sub(&A, &C, &A)) != MP_OKAY) goto CLEANUP; + if((res = mp_sub(&B, &D, &B)) != MP_OKAY) goto CLEANUP; + + } else { + if((res = mp_sub(&v, &u, &v)) != MP_OKAY) goto CLEANUP; + if((res = mp_sub(&C, &A, &C)) != MP_OKAY) goto CLEANUP; + if((res = mp_sub(&D, &B, &D)) != MP_OKAY) goto CLEANUP; + + } + + /* If we're done, copy results to output */ + if(mp_cmp_z(&u) == 0) { + if(x) + if((res = mp_copy(&C, x)) != MP_OKAY) goto CLEANUP; + + if(y) + if((res = mp_copy(&D, y)) != MP_OKAY) goto CLEANUP; + + if(g) + if((res = mp_mul(&gx, &v, g)) != MP_OKAY) goto CLEANUP; + + break; + } + } + + CLEANUP: + while(last >= 0) + mp_clear(clean[last--]); + + return res; + +} /* end mp_xgcd() */ + +/* }}} */ + +/* {{{ mp_invmod(a, m, c) */ + +/* + mp_invmod(a, m, c) + + Compute c = a^-1 (mod m), if there is an inverse for a (mod m). + This is equivalent to the question of whether (a, m) = 1. If not, + MP_UNDEF is returned, and there is no inverse. + */ + +mp_err mp_invmod(mp_int *a, mp_int *m, mp_int *c) +{ + mp_int g, x; + mp_err res; + + ARGCHK(a && m && c, MP_BADARG); + + if(mp_cmp_z(a) == 0 || mp_cmp_z(m) == 0) + return MP_RANGE; + + if((res = mp_init(&g)) != MP_OKAY) + return res; + if((res = mp_init(&x)) != MP_OKAY) + goto X; + + if((res = mp_xgcd(a, m, &g, &x, NULL)) != MP_OKAY) + goto CLEANUP; + + if(mp_cmp_d(&g, 1) != MP_EQ) { + res = MP_UNDEF; + goto CLEANUP; + } + + res = mp_mod(&x, m, c); + SIGN(c) = SIGN(a); + +CLEANUP: + mp_clear(&x); +X: + mp_clear(&g); + + return res; + +} /* end mp_invmod() */ + +/* }}} */ +#endif /* if MP_NUMTH */ + +/* }}} */ + +/*------------------------------------------------------------------------*/ +/* {{{ mp_print(mp, ofp) */ + +#if MP_IOFUNC +/* + mp_print(mp, ofp) + + Print a textual representation of the given mp_int on the output + stream 'ofp'. Output is generated using the internal radix. + */ + +void mp_print(mp_int *mp, FILE *ofp) +{ + int ix; + + if(mp == NULL || ofp == NULL) + return; + + fputc((SIGN(mp) == MP_NEG) ? '-' : '+', ofp); + + for(ix = USED(mp) - 1; ix >= 0; ix--) { + fprintf(ofp, DIGIT_FMT, DIGIT(mp, ix)); + } + +} /* end mp_print() */ + +#endif /* if MP_IOFUNC */ + +/* }}} */ + +/*------------------------------------------------------------------------*/ +/* {{{ More I/O Functions */ + +/* {{{ mp_read_signed_bin(mp, str, len) */ + +/* + mp_read_signed_bin(mp, str, len) + + Read in a raw value (base 256) into the given mp_int + */ + +mp_err mp_read_signed_bin(mp_int *mp, unsigned char *str, int len) +{ + mp_err res; + + ARGCHK(mp != NULL && str != NULL && len > 0, MP_BADARG); + + if((res = mp_read_unsigned_bin(mp, str + 1, len - 1)) == MP_OKAY) { + /* Get sign from first byte */ + if(str[0]) + SIGN(mp) = MP_NEG; + else + SIGN(mp) = MP_ZPOS; + } + + return res; + +} /* end mp_read_signed_bin() */ + +/* }}} */ + +/* {{{ mp_signed_bin_size(mp) */ + +int mp_signed_bin_size(mp_int *mp) +{ + ARGCHK(mp != NULL, 0); + + return mp_unsigned_bin_size(mp) + 1; + +} /* end mp_signed_bin_size() */ + +/* }}} */ + +/* {{{ mp_to_signed_bin(mp, str) */ + +mp_err mp_to_signed_bin(mp_int *mp, unsigned char *str) +{ + ARGCHK(mp != NULL && str != NULL, MP_BADARG); + + /* Caller responsible for allocating enough memory (use mp_raw_size(mp)) */ + str[0] = (char)SIGN(mp); + + return mp_to_unsigned_bin(mp, str + 1); + +} /* end mp_to_signed_bin() */ + +/* }}} */ + +/* {{{ mp_read_unsigned_bin(mp, str, len) */ + +/* + mp_read_unsigned_bin(mp, str, len) + + Read in an unsigned value (base 256) into the given mp_int + */ + +mp_err mp_read_unsigned_bin(mp_int *mp, unsigned char *str, int len) +{ + int ix; + mp_err res; + + ARGCHK(mp != NULL && str != NULL && len > 0, MP_BADARG); + + mp_zero(mp); + + for(ix = 0; ix < len; ix++) { + if((res = s_mp_mul_2d(mp, CHAR_BIT)) != MP_OKAY) + return res; + + if((res = mp_add_d(mp, str[ix], mp)) != MP_OKAY) + return res; + } + + return MP_OKAY; + +} /* end mp_read_unsigned_bin() */ + +/* }}} */ + +/* {{{ mp_unsigned_bin_size(mp) */ + +int mp_unsigned_bin_size(mp_int *mp) +{ + mp_digit topdig; + int count; + + ARGCHK(mp != NULL, 0); + + /* Special case for the value zero */ + if(USED(mp) == 1 && DIGIT(mp, 0) == 0) + return 1; + + count = (USED(mp) - 1) * sizeof(mp_digit); + topdig = DIGIT(mp, USED(mp) - 1); + + while(topdig != 0) { + ++count; + topdig >>= CHAR_BIT; + } + + return count; + +} /* end mp_unsigned_bin_size() */ + +/* }}} */ + +/* {{{ mp_to_unsigned_bin(mp, str) */ + +mp_err mp_to_unsigned_bin(mp_int *mp, unsigned char *str) +{ + mp_digit *dp, *end, d; + unsigned char *spos; + + ARGCHK(mp != NULL && str != NULL, MP_BADARG); + + dp = DIGITS(mp); + end = dp + USED(mp) - 1; + spos = str; + + /* Special case for zero, quick test */ + if(dp == end && *dp == 0) { + *str = '\0'; + return MP_OKAY; + } + + /* Generate digits in reverse order */ + while(dp < end) { + int ix; + + d = *dp; + for(ix = 0; ix < sizeof(mp_digit); ++ix) { + *spos = d & UCHAR_MAX; + d >>= CHAR_BIT; + ++spos; + } + + ++dp; + } + + /* Now handle last digit specially, high order zeroes are not written */ + d = *end; + while(d != 0) { + *spos = d & UCHAR_MAX; + d >>= CHAR_BIT; + ++spos; + } + + /* Reverse everything to get digits in the correct order */ + while(--spos > str) { + unsigned char t = *str; + *str = *spos; + *spos = t; + + ++str; + } + + return MP_OKAY; + +} /* end mp_to_unsigned_bin() */ + +/* }}} */ + +/* {{{ mp_count_bits(mp) */ + +int mp_count_bits(mp_int *mp) +{ + int len; + mp_digit d; + + ARGCHK(mp != NULL, MP_BADARG); + + len = DIGIT_BIT * (USED(mp) - 1); + d = DIGIT(mp, USED(mp) - 1); + + while(d != 0) { + ++len; + d >>= 1; + } + + return len; + +} /* end mp_count_bits() */ + +/* }}} */ + +/* {{{ mp_read_radix(mp, str, radix) */ + +/* + mp_read_radix(mp, str, radix) + + Read an integer from the given string, and set mp to the resulting + value. The input is presumed to be in base 10. Leading non-digit + characters are ignored, and the function reads until a non-digit + character or the end of the string. + */ + +mp_err mp_read_radix(mp_int *mp, unsigned char *str, int radix) +{ + int ix = 0, val = 0; + mp_err res; + mp_sign sig = MP_ZPOS; + + ARGCHK(mp != NULL && str != NULL && radix >= 2 && radix <= MAX_RADIX, + MP_BADARG); + + mp_zero(mp); + + /* Skip leading non-digit characters until a digit or '-' or '+' */ + while(str[ix] && + (s_mp_tovalue(str[ix], radix) < 0) && + str[ix] != '-' && + str[ix] != '+') { + ++ix; + } + + if(str[ix] == '-') { + sig = MP_NEG; + ++ix; + } else if(str[ix] == '+') { + sig = MP_ZPOS; /* this is the default anyway... */ + ++ix; + } + + while((val = s_mp_tovalue(str[ix], radix)) >= 0) { + if((res = s_mp_mul_d(mp, radix)) != MP_OKAY) + return res; + if((res = s_mp_add_d(mp, val)) != MP_OKAY) + return res; + ++ix; + } + + if(s_mp_cmp_d(mp, 0) == MP_EQ) + SIGN(mp) = MP_ZPOS; + else + SIGN(mp) = sig; + + return MP_OKAY; + +} /* end mp_read_radix() */ + +/* }}} */ + +/* {{{ mp_radix_size(mp, radix) */ + +int mp_radix_size(mp_int *mp, int radix) +{ + int len; + ARGCHK(mp != NULL, 0); + + len = s_mp_outlen(mp_count_bits(mp), radix) + 1; /* for NUL terminator */ + + if(mp_cmp_z(mp) < 0) + ++len; /* for sign */ + + return len; + +} /* end mp_radix_size() */ + +/* }}} */ + +/* {{{ mp_value_radix_size(num, qty, radix) */ + +/* num = number of digits + qty = number of bits per digit + radix = target base + + Return the number of digits in the specified radix that would be + needed to express 'num' digits of 'qty' bits each. + */ +int mp_value_radix_size(int num, int qty, int radix) +{ + ARGCHK(num >= 0 && qty > 0 && radix >= 2 && radix <= MAX_RADIX, 0); + + return s_mp_outlen(num * qty, radix); + +} /* end mp_value_radix_size() */ + +/* }}} */ + +/* {{{ mp_toradix(mp, str, radix) */ + +mp_err mp_toradix(mp_int *mp, unsigned char *str, int radix) +{ + int ix, pos = 0; + + ARGCHK(mp != NULL && str != NULL, MP_BADARG); + ARGCHK(radix > 1 && radix <= MAX_RADIX, MP_RANGE); + + if(mp_cmp_z(mp) == MP_EQ) { + str[0] = '0'; + str[1] = '\0'; + } else { + mp_err res; + mp_int tmp; + mp_sign sgn; + mp_digit rem, rdx = (mp_digit)radix; + char ch; + + if((res = mp_init_copy(&tmp, mp)) != MP_OKAY) + return res; + + /* Save sign for later, and take absolute value */ + sgn = SIGN(&tmp); SIGN(&tmp) = MP_ZPOS; + + /* Generate output digits in reverse order */ + while(mp_cmp_z(&tmp) != 0) { + if((res = s_mp_div_d(&tmp, rdx, &rem)) != MP_OKAY) { + mp_clear(&tmp); + return res; + } + + /* Generate digits, use capital letters */ + ch = s_mp_todigit(rem, radix, 0); + + str[pos++] = ch; + } + + /* Add - sign if original value was negative */ + if(sgn == MP_NEG) + str[pos++] = '-'; + + /* Add trailing NUL to end the string */ + str[pos--] = '\0'; + + /* Reverse the digits and sign indicator */ + ix = 0; + while(ix < pos) { + char tmp = str[ix]; + + str[ix] = str[pos]; + str[pos] = tmp; + ++ix; + --pos; + } + + mp_clear(&tmp); + } + + return MP_OKAY; + +} /* end mp_toradix() */ + +/* }}} */ + +/* {{{ mp_char2value(ch, r) */ + +int mp_char2value(char ch, int r) +{ + return s_mp_tovalue(ch, r); + +} /* end mp_tovalue() */ + +/* }}} */ + +/* }}} */ + +/* {{{ mp_strerror(ec) */ + +/* + mp_strerror(ec) + + Return a string describing the meaning of error code 'ec'. The + string returned is allocated in static memory, so the caller should + not attempt to modify or free the memory associated with this + string. + */ +const char *mp_strerror(mp_err ec) +{ + int aec = (ec < 0) ? -ec : ec; + + /* Code values are negative, so the senses of these comparisons + are accurate */ + if(ec < MP_LAST_CODE || ec > MP_OKAY) { + return mp_err_string[0]; /* unknown error code */ + } else { + return mp_err_string[aec + 1]; + } + +} /* end mp_strerror() */ + +/* }}} */ + +/*========================================================================*/ +/*------------------------------------------------------------------------*/ +/* Static function definitions (internal use only) */ + +/* {{{ Memory management */ + +/* {{{ s_mp_grow(mp, min) */ + +/* Make sure there are at least 'min' digits allocated to mp */ +mp_err s_mp_grow(mp_int *mp, mp_size min) +{ + if(min > ALLOC(mp)) { + mp_digit *tmp; + + /* Set min to next nearest default precision block size */ + min = ((min + (s_mp_defprec - 1)) / s_mp_defprec) * s_mp_defprec; + + if((tmp = s_mp_alloc(min, sizeof(mp_digit))) == NULL) + return MP_MEM; + + s_mp_copy(DIGITS(mp), tmp, USED(mp)); + +#if MP_CRYPTO + s_mp_setz(DIGITS(mp), ALLOC(mp)); +#endif + s_mp_free(DIGITS(mp)); + DIGITS(mp) = tmp; + ALLOC(mp) = min; + } + + return MP_OKAY; + +} /* end s_mp_grow() */ + +/* }}} */ + +/* {{{ s_mp_pad(mp, min) */ + +/* Make sure the used size of mp is at least 'min', growing if needed */ +mp_err s_mp_pad(mp_int *mp, mp_size min) +{ + if(min > USED(mp)) { + mp_err res; + + /* Make sure there is room to increase precision */ + if(min > ALLOC(mp) && (res = s_mp_grow(mp, min)) != MP_OKAY) + return res; + + /* Increase precision; should already be 0-filled */ + USED(mp) = min; + } + + return MP_OKAY; + +} /* end s_mp_pad() */ + +/* }}} */ + +/* {{{ s_mp_setz(dp, count) */ + +#if MP_MACRO == 0 +/* Set 'count' digits pointed to by dp to be zeroes */ +void s_mp_setz(mp_digit *dp, mp_size count) +{ +#if MP_MEMSET == 0 + int ix; + + for(ix = 0; ix < count; ix++) + dp[ix] = 0; +#else + memset(dp, 0, count * sizeof(mp_digit)); +#endif + +} /* end s_mp_setz() */ +#endif + +/* }}} */ + +/* {{{ s_mp_copy(sp, dp, count) */ + +#if MP_MACRO == 0 +/* Copy 'count' digits from sp to dp */ +void s_mp_copy(mp_digit *sp, mp_digit *dp, mp_size count) +{ +#if MP_MEMCPY == 0 + int ix; + + for(ix = 0; ix < count; ix++) + dp[ix] = sp[ix]; +#else + memcpy(dp, sp, count * sizeof(mp_digit)); +#endif + +} /* end s_mp_copy() */ +#endif + +/* }}} */ + +/* {{{ s_mp_alloc(nb, ni) */ + +#if MP_MACRO == 0 +/* Allocate ni records of nb bytes each, and return a pointer to that */ +void *s_mp_alloc(size_t nb, size_t ni) +{ + return calloc(nb, ni); + +} /* end s_mp_alloc() */ +#endif + +/* }}} */ + +/* {{{ s_mp_free(ptr) */ + +#if MP_MACRO == 0 +/* Free the memory pointed to by ptr */ +void s_mp_free(void *ptr) +{ + if(ptr) + free(ptr); + +} /* end s_mp_free() */ +#endif + +/* }}} */ + +/* {{{ s_mp_clamp(mp) */ + +/* Remove leading zeroes from the given value */ +void s_mp_clamp(mp_int *mp) +{ + mp_size du = USED(mp); + mp_digit *zp = DIGITS(mp) + du - 1; + + while(du > 1 && !*zp--) + --du; + + USED(mp) = du; + +} /* end s_mp_clamp() */ + + +/* }}} */ + +/* {{{ s_mp_exch(a, b) */ + +/* Exchange the data for a and b; (b, a) = (a, b) */ +void s_mp_exch(mp_int *a, mp_int *b) +{ + mp_int tmp; + + tmp = *a; + *a = *b; + *b = tmp; + +} /* end s_mp_exch() */ + +/* }}} */ + +/* }}} */ + +/* {{{ Arithmetic helpers */ + +/* {{{ s_mp_lshd(mp, p) */ + +/* + Shift mp leftward by p digits, growing if needed, and zero-filling + the in-shifted digits at the right end. This is a convenient + alternative to multiplication by powers of the radix + */ + +mp_err s_mp_lshd(mp_int *mp, mp_size p) +{ + mp_err res; + mp_size pos; + mp_digit *dp; + int ix; + + if(p == 0) + return MP_OKAY; + + if((res = s_mp_pad(mp, USED(mp) + p)) != MP_OKAY) + return res; + + pos = USED(mp) - 1; + dp = DIGITS(mp); + + /* Shift all the significant figures over as needed */ + for(ix = pos - p; ix >= 0; ix--) + dp[ix + p] = dp[ix]; + + /* Fill the bottom digits with zeroes */ + for(ix = 0; ix < p; ix++) + dp[ix] = 0; + + return MP_OKAY; + +} /* end s_mp_lshd() */ + +/* }}} */ + +/* {{{ s_mp_rshd(mp, p) */ + +/* + Shift mp rightward by p digits. Maintains the invariant that + digits above the precision are all zero. Digits shifted off the + end are lost. Cannot fail. + */ + +void s_mp_rshd(mp_int *mp, mp_size p) +{ + mp_size ix; + mp_digit *dp; + + if(p == 0) + return; + + /* Shortcut when all digits are to be shifted off */ + if(p >= USED(mp)) { + s_mp_setz(DIGITS(mp), ALLOC(mp)); + USED(mp) = 1; + SIGN(mp) = MP_ZPOS; + return; + } + + /* Shift all the significant figures over as needed */ + dp = DIGITS(mp); + for(ix = p; ix < USED(mp); ix++) + dp[ix - p] = dp[ix]; + + /* Fill the top digits with zeroes */ + ix -= p; + while(ix < USED(mp)) + dp[ix++] = 0; + + /* Strip off any leading zeroes */ + s_mp_clamp(mp); + +} /* end s_mp_rshd() */ + +/* }}} */ + +/* {{{ s_mp_div_2(mp) */ + +/* Divide by two -- take advantage of radix properties to do it fast */ +void s_mp_div_2(mp_int *mp) +{ + s_mp_div_2d(mp, 1); + +} /* end s_mp_div_2() */ + +/* }}} */ + +/* {{{ s_mp_mul_2(mp) */ + +mp_err s_mp_mul_2(mp_int *mp) +{ + int ix; + mp_digit kin = 0, kout, *dp = DIGITS(mp); + mp_err res; + + /* Shift digits leftward by 1 bit */ + for(ix = 0; ix < USED(mp); ix++) { + kout = (dp[ix] >> (DIGIT_BIT - 1)) & 1; + dp[ix] = (dp[ix] << 1) | kin; + + kin = kout; + } + + /* Deal with rollover from last digit */ + if(kin) { + if(ix >= ALLOC(mp)) { + if((res = s_mp_grow(mp, ALLOC(mp) + 1)) != MP_OKAY) + return res; + dp = DIGITS(mp); + } + + dp[ix] = kin; + USED(mp) += 1; + } + + return MP_OKAY; + +} /* end s_mp_mul_2() */ + +/* }}} */ + +/* {{{ s_mp_mod_2d(mp, d) */ + +/* + Remainder the integer by 2^d, where d is a number of bits. This + amounts to a bitwise AND of the value, and does not require the full + division code + */ +void s_mp_mod_2d(mp_int *mp, mp_digit d) +{ + unsigned int ndig = (d / DIGIT_BIT), nbit = (d % DIGIT_BIT); + unsigned int ix; + mp_digit dmask, *dp = DIGITS(mp); + + if(ndig >= USED(mp)) + return; + + /* Flush all the bits above 2^d in its digit */ + dmask = (1 << nbit) - 1; + dp[ndig] &= dmask; + + /* Flush all digits above the one with 2^d in it */ + for(ix = ndig + 1; ix < USED(mp); ix++) + dp[ix] = 0; + + s_mp_clamp(mp); + +} /* end s_mp_mod_2d() */ + +/* }}} */ + +/* {{{ s_mp_mul_2d(mp, d) */ + +/* + Multiply by the integer 2^d, where d is a number of bits. This + amounts to a bitwise shift of the value, and does not require the + full multiplication code. + */ +mp_err s_mp_mul_2d(mp_int *mp, mp_digit d) +{ + mp_err res; + mp_digit save, next, mask, *dp; + mp_size used; + int ix; + + if((res = s_mp_lshd(mp, d / DIGIT_BIT)) != MP_OKAY) + return res; + + dp = DIGITS(mp); used = USED(mp); + d %= DIGIT_BIT; + + mask = (1 << d) - 1; + + /* If the shift requires another digit, make sure we've got one to + work with */ + if((dp[used - 1] >> (DIGIT_BIT - d)) & mask) { + if((res = s_mp_grow(mp, used + 1)) != MP_OKAY) + return res; + dp = DIGITS(mp); + } + + /* Do the shifting... */ + save = 0; + for(ix = 0; ix < used; ix++) { + next = (dp[ix] >> (DIGIT_BIT - d)) & mask; + dp[ix] = (dp[ix] << d) | save; + save = next; + } + + /* If, at this point, we have a nonzero carryout into the next + digit, we'll increase the size by one digit, and store it... + */ + if(save) { + dp[used] = save; + USED(mp) += 1; + } + + s_mp_clamp(mp); + return MP_OKAY; + +} /* end s_mp_mul_2d() */ + +/* }}} */ + +/* {{{ s_mp_div_2d(mp, d) */ + +/* + Divide the integer by 2^d, where d is a number of bits. This + amounts to a bitwise shift of the value, and does not require the + full division code (used in Barrett reduction, see below) + */ +void s_mp_div_2d(mp_int *mp, mp_digit d) +{ + int ix; + mp_digit save, next, mask, *dp = DIGITS(mp); + + s_mp_rshd(mp, d / DIGIT_BIT); + d %= DIGIT_BIT; + + mask = (1 << d) - 1; + + save = 0; + for(ix = USED(mp) - 1; ix >= 0; ix--) { + next = dp[ix] & mask; + dp[ix] = (dp[ix] >> d) | (save << (DIGIT_BIT - d)); + save = next; + } + + s_mp_clamp(mp); + +} /* end s_mp_div_2d() */ + +/* }}} */ + +/* {{{ s_mp_norm(a, b) */ + +/* + s_mp_norm(a, b) + + Normalize a and b for division, where b is the divisor. In order + that we might make good guesses for quotient digits, we want the + leading digit of b to be at least half the radix, which we + accomplish by multiplying a and b by a constant. This constant is + returned (so that it can be divided back out of the remainder at the + end of the division process). + + We multiply by the smallest power of 2 that gives us a leading digit + at least half the radix. By choosing a power of 2, we simplify the + multiplication and division steps to simple shifts. + */ +mp_digit s_mp_norm(mp_int *a, mp_int *b) +{ + mp_digit t, d = 0; + + t = DIGIT(b, USED(b) - 1); + while(t < (RADIX / 2)) { + t <<= 1; + ++d; + } + + if(d != 0) { + s_mp_mul_2d(a, d); + s_mp_mul_2d(b, d); + } + + return d; + +} /* end s_mp_norm() */ + +/* }}} */ + +/* }}} */ + +/* {{{ Primitive digit arithmetic */ + +/* {{{ s_mp_add_d(mp, d) */ + +/* Add d to |mp| in place */ +mp_err s_mp_add_d(mp_int *mp, mp_digit d) /* unsigned digit addition */ +{ + mp_word w, k = 0; + mp_size ix = 1, used = USED(mp); + mp_digit *dp = DIGITS(mp); + + w = dp[0] + d; + dp[0] = ACCUM(w); + k = CARRYOUT(w); + + while(ix < used && k) { + w = dp[ix] + k; + dp[ix] = ACCUM(w); + k = CARRYOUT(w); + ++ix; + } + + if(k != 0) { + mp_err res; + + if((res = s_mp_pad(mp, USED(mp) + 1)) != MP_OKAY) + return res; + + DIGIT(mp, ix) = k; + } + + return MP_OKAY; + +} /* end s_mp_add_d() */ + +/* }}} */ + +/* {{{ s_mp_sub_d(mp, d) */ + +/* Subtract d from |mp| in place, assumes |mp| > d */ +mp_err s_mp_sub_d(mp_int *mp, mp_digit d) /* unsigned digit subtract */ +{ + mp_word w, b = 0; + mp_size ix = 1, used = USED(mp); + mp_digit *dp = DIGITS(mp); + + /* Compute initial subtraction */ + w = (RADIX + dp[0]) - d; + b = CARRYOUT(w) ? 0 : 1; + dp[0] = ACCUM(w); + + /* Propagate borrows leftward */ + while(b && ix < used) { + w = (RADIX + dp[ix]) - b; + b = CARRYOUT(w) ? 0 : 1; + dp[ix] = ACCUM(w); + ++ix; + } + + /* Remove leading zeroes */ + s_mp_clamp(mp); + + /* If we have a borrow out, it's a violation of the input invariant */ + if(b) + return MP_RANGE; + else + return MP_OKAY; + +} /* end s_mp_sub_d() */ + +/* }}} */ + +/* {{{ s_mp_mul_d(a, d) */ + +/* Compute a = a * d, single digit multiplication */ +mp_err s_mp_mul_d(mp_int *a, mp_digit d) +{ + mp_word w, k = 0; + mp_size ix, max; + mp_err res; + mp_digit *dp = DIGITS(a); + + /* + Single-digit multiplication will increase the precision of the + output by at most one digit. However, we can detect when this + will happen -- if the high-order digit of a, times d, gives a + two-digit result, then the precision of the result will increase; + otherwise it won't. We use this fact to avoid calling s_mp_pad() + unless absolutely necessary. + */ + max = USED(a); + w = dp[max - 1] * d; + if(CARRYOUT(w) != 0) { + if((res = s_mp_pad(a, max + 1)) != MP_OKAY) + return res; + dp = DIGITS(a); + } + + for(ix = 0; ix < max; ix++) { + w = (dp[ix] * d) + k; + dp[ix] = ACCUM(w); + k = CARRYOUT(w); + } + + /* If there is a precision increase, take care of it here; the above + test guarantees we have enough storage to do this safely. + */ + if(k) { + dp[max] = k; + USED(a) = max + 1; + } + + s_mp_clamp(a); + + return MP_OKAY; + +} /* end s_mp_mul_d() */ + +/* }}} */ + +/* {{{ s_mp_div_d(mp, d, r) */ + +/* + s_mp_div_d(mp, d, r) + + Compute the quotient mp = mp / d and remainder r = mp mod d, for a + single digit d. If r is null, the remainder will be discarded. + */ + +mp_err s_mp_div_d(mp_int *mp, mp_digit d, mp_digit *r) +{ + mp_word w = 0, t; + mp_int quot; + mp_err res; + mp_digit *dp = DIGITS(mp), *qp; + int ix; + + if(d == 0) + return MP_RANGE; + + /* Make room for the quotient */ + if((res = mp_init_size(", USED(mp))) != MP_OKAY) + return res; + + USED(") = USED(mp); /* so clamping will work below */ + qp = DIGITS("); + + /* Divide without subtraction */ + for(ix = USED(mp) - 1; ix >= 0; ix--) { + w = (w << DIGIT_BIT) | dp[ix]; + + if(w >= d) { + t = w / d; + w = w % d; + } else { + t = 0; + } + + qp[ix] = t; + } + + /* Deliver the remainder, if desired */ + if(r) + *r = w; + + s_mp_clamp("); + mp_exch(", mp); + mp_clear("); + + return MP_OKAY; + +} /* end s_mp_div_d() */ + +/* }}} */ + +/* }}} */ + +/* {{{ Primitive full arithmetic */ + +/* {{{ s_mp_add(a, b) */ + +/* Compute a = |a| + |b| */ +mp_err s_mp_add(mp_int *a, mp_int *b) /* magnitude addition */ +{ + mp_word w = 0; + mp_digit *pa, *pb; + mp_size ix, used = USED(b); + mp_err res; + + /* Make sure a has enough precision for the output value */ + if((used > USED(a)) && (res = s_mp_pad(a, used)) != MP_OKAY) + return res; + + /* + Add up all digits up to the precision of b. If b had initially + the same precision as a, or greater, we took care of it by the + padding step above, so there is no problem. If b had initially + less precision, we'll have to make sure the carry out is duly + propagated upward among the higher-order digits of the sum. + */ + pa = DIGITS(a); + pb = DIGITS(b); + for(ix = 0; ix < used; ++ix) { + w += *pa + *pb++; + *pa++ = ACCUM(w); + w = CARRYOUT(w); + } + + /* If we run out of 'b' digits before we're actually done, make + sure the carries get propagated upward... + */ + used = USED(a); + while(w && ix < used) { + w += *pa; + *pa++ = ACCUM(w); + w = CARRYOUT(w); + ++ix; + } + + /* If there's an overall carry out, increase precision and include + it. We could have done this initially, but why touch the memory + allocator unless we're sure we have to? + */ + if(w) { + if((res = s_mp_pad(a, used + 1)) != MP_OKAY) + return res; + + DIGIT(a, ix) = w; /* pa may not be valid after s_mp_pad() call */ + } + + return MP_OKAY; + +} /* end s_mp_add() */ + +/* }}} */ + +/* {{{ s_mp_sub(a, b) */ + +/* Compute a = |a| - |b|, assumes |a| >= |b| */ +mp_err s_mp_sub(mp_int *a, mp_int *b) /* magnitude subtract */ +{ + mp_word w = 0; + mp_digit *pa, *pb; + mp_size ix, used = USED(b); + + /* + Subtract and propagate borrow. Up to the precision of b, this + accounts for the digits of b; after that, we just make sure the + carries get to the right place. This saves having to pad b out to + the precision of a just to make the loops work right... + */ + pa = DIGITS(a); + pb = DIGITS(b); + + for(ix = 0; ix < used; ++ix) { + w = (RADIX + *pa) - w - *pb++; + *pa++ = ACCUM(w); + w = CARRYOUT(w) ? 0 : 1; + } + + used = USED(a); + while(ix < used) { + w = RADIX + *pa - w; + *pa++ = ACCUM(w); + w = CARRYOUT(w) ? 0 : 1; + ++ix; + } + + /* Clobber any leading zeroes we created */ + s_mp_clamp(a); + + /* + If there was a borrow out, then |b| > |a| in violation + of our input invariant. We've already done the work, + but we'll at least complain about it... + */ + if(w) + return MP_RANGE; + else + return MP_OKAY; + +} /* end s_mp_sub() */ + +/* }}} */ + +mp_err s_mp_reduce(mp_int *x, mp_int *m, mp_int *mu) +{ + mp_int q; + mp_err res; + mp_size um = USED(m); + + if((res = mp_init_copy(&q, x)) != MP_OKAY) + return res; + + s_mp_rshd(&q, um - 1); /* q1 = x / b^(k-1) */ + s_mp_mul(&q, mu); /* q2 = q1 * mu */ + s_mp_rshd(&q, um + 1); /* q3 = q2 / b^(k+1) */ + + /* x = x mod b^(k+1), quick (no division) */ + s_mp_mod_2d(x, (mp_digit)(DIGIT_BIT * (um + 1))); + + /* q = q * m mod b^(k+1), quick (no division), uses the short multiplier */ +#ifndef SHRT_MUL + s_mp_mul(&q, m); + s_mp_mod_2d(&q, (mp_digit)(DIGIT_BIT * (um + 1))); +#else + s_mp_mul_dig(&q, m, um + 1); +#endif + + /* x = x - q */ + if((res = mp_sub(x, &q, x)) != MP_OKAY) + goto CLEANUP; + + /* If x < 0, add b^(k+1) to it */ + if(mp_cmp_z(x) < 0) { + mp_set(&q, 1); + if((res = s_mp_lshd(&q, um + 1)) != MP_OKAY) + goto CLEANUP; + if((res = mp_add(x, &q, x)) != MP_OKAY) + goto CLEANUP; + } + + /* Back off if it's too big */ + while(mp_cmp(x, m) >= 0) { + if((res = s_mp_sub(x, m)) != MP_OKAY) + break; + } + + CLEANUP: + mp_clear(&q); + + return res; + +} /* end s_mp_reduce() */ + + + +/* {{{ s_mp_mul(a, b) */ + +/* Compute a = |a| * |b| */ +mp_err s_mp_mul(mp_int *a, mp_int *b) +{ + mp_word w, k = 0; + mp_int tmp; + mp_err res; + mp_size ix, jx, ua = USED(a), ub = USED(b); + mp_digit *pa, *pb, *pt, *pbt; + + if((res = mp_init_size(&tmp, ua + ub)) != MP_OKAY) + return res; + + /* This has the effect of left-padding with zeroes... */ + USED(&tmp) = ua + ub; + + /* We're going to need the base value each iteration */ + pbt = DIGITS(&tmp); + + /* Outer loop: Digits of b */ + + pb = DIGITS(b); + for(ix = 0; ix < ub; ++ix, ++pb) { + if(*pb == 0) + continue; + + /* Inner product: Digits of a */ + pa = DIGITS(a); + for(jx = 0; jx < ua; ++jx, ++pa) { + pt = pbt + ix + jx; + w = *pb * *pa + k + *pt; + *pt = ACCUM(w); + k = CARRYOUT(w); + } + + pbt[ix + jx] = k; + k = 0; + } + + s_mp_clamp(&tmp); + s_mp_exch(&tmp, a); + + mp_clear(&tmp); + + return MP_OKAY; + +} /* end s_mp_mul() */ + +/* }}} */ + +/* {{{ s_mp_kmul(a, b, out, len) */ + +#if 0 +void s_mp_kmul(mp_digit *a, mp_digit *b, mp_digit *out, mp_size len) +{ + mp_word w, k = 0; + mp_size ix, jx; + mp_digit *pa, *pt; + + for(ix = 0; ix < len; ++ix, ++b) { + if(*b == 0) + continue; + + pa = a; + for(jx = 0; jx < len; ++jx, ++pa) { + pt = out + ix + jx; + w = *b * *pa + k + *pt; + *pt = ACCUM(w); + k = CARRYOUT(w); + } + + out[ix + jx] = k; + k = 0; + } + +} /* end s_mp_kmul() */ +#endif + +/* }}} */ + +/* {{{ s_mp_sqr(a) */ + +/* + Computes the square of a, in place. This can be done more + efficiently than a general multiplication, because many of the + computation steps are redundant when squaring. The inner product + step is a bit more complicated, but we save a fair number of + iterations of the multiplication loop. + */ +#if MP_SQUARE +mp_err s_mp_sqr(mp_int *a) +{ + mp_word w, k = 0; + mp_int tmp; + mp_err res; + mp_size ix, jx, kx, used = USED(a); + mp_digit *pa1, *pa2, *pt, *pbt; + + if((res = mp_init_size(&tmp, 2 * used)) != MP_OKAY) + return res; + + /* Left-pad with zeroes */ + USED(&tmp) = 2 * used; + + /* We need the base value each time through the loop */ + pbt = DIGITS(&tmp); + + pa1 = DIGITS(a); + for(ix = 0; ix < used; ++ix, ++pa1) { + if(*pa1 == 0) + continue; + + w = DIGIT(&tmp, ix + ix) + (*pa1 * *pa1); + + pbt[ix + ix] = ACCUM(w); + k = CARRYOUT(w); + + /* + The inner product is computed as: + + (C, S) = t[i,j] + 2 a[i] a[j] + C + + This can overflow what can be represented in an mp_word, and + since C arithmetic does not provide any way to check for + overflow, we have to check explicitly for overflow conditions + before they happen. + */ + for(jx = ix + 1, pa2 = DIGITS(a) + jx; jx < used; ++jx, ++pa2) { + mp_word u = 0, v; + + /* Store this in a temporary to avoid indirections later */ + pt = pbt + ix + jx; + + /* Compute the multiplicative step */ + w = *pa1 * *pa2; + + /* If w is more than half MP_WORD_MAX, the doubling will + overflow, and we need to record a carry out into the next + word */ + u = (w >> (MP_WORD_BIT - 1)) & 1; + + /* Double what we've got, overflow will be ignored as defined + for C arithmetic (we've already noted if it is to occur) + */ + w *= 2; + + /* Compute the additive step */ + v = *pt + k; + + /* If we do not already have an overflow carry, check to see + if the addition will cause one, and set the carry out if so + */ + u |= ((MP_WORD_MAX - v) < w); + + /* Add in the rest, again ignoring overflow */ + w += v; + + /* Set the i,j digit of the output */ + *pt = ACCUM(w); + + /* Save carry information for the next iteration of the loop. + This is why k must be an mp_word, instead of an mp_digit */ + k = CARRYOUT(w) | (u << DIGIT_BIT); + + } /* for(jx ...) */ + + /* Set the last digit in the cycle and reset the carry */ + k = DIGIT(&tmp, ix + jx) + k; + pbt[ix + jx] = ACCUM(k); + k = CARRYOUT(k); + + /* If we are carrying out, propagate the carry to the next digit + in the output. This may cascade, so we have to be somewhat + circumspect -- but we will have enough precision in the output + that we won't overflow + */ + kx = 1; + while(k) { + k = pbt[ix + jx + kx] + 1; + pbt[ix + jx + kx] = ACCUM(k); + k = CARRYOUT(k); + ++kx; + } + } /* for(ix ...) */ + + s_mp_clamp(&tmp); + s_mp_exch(&tmp, a); + + mp_clear(&tmp); + + return MP_OKAY; + +} /* end s_mp_sqr() */ +#endif + +/* }}} */ + +/* {{{ s_mp_div(a, b) */ + +/* + s_mp_div(a, b) + + Compute a = a / b and b = a mod b. Assumes b > a. + */ + +mp_err s_mp_div(mp_int *a, mp_int *b) +{ + mp_int quot, rem, t; + mp_word q; + mp_err res; + mp_digit d; + int ix; + + if(mp_cmp_z(b) == 0) + return MP_RANGE; + + /* Shortcut if b is power of two */ + if((ix = s_mp_ispow2(b)) >= 0) { + mp_copy(a, b); /* need this for remainder */ + s_mp_div_2d(a, (mp_digit)ix); + s_mp_mod_2d(b, (mp_digit)ix); + + return MP_OKAY; + } + + /* Allocate space to store the quotient */ + if((res = mp_init_size(", USED(a))) != MP_OKAY) + return res; + + /* A working temporary for division */ + if((res = mp_init_size(&t, USED(a))) != MP_OKAY) + goto T; + + /* Allocate space for the remainder */ + if((res = mp_init_size(&rem, USED(a))) != MP_OKAY) + goto REM; + + /* Normalize to optimize guessing */ + d = s_mp_norm(a, b); + + /* Perform the division itself...woo! */ + ix = USED(a) - 1; + + while(ix >= 0) { + /* Find a partial substring of a which is at least b */ + while(s_mp_cmp(&rem, b) < 0 && ix >= 0) { + if((res = s_mp_lshd(&rem, 1)) != MP_OKAY) + goto CLEANUP; + + if((res = s_mp_lshd(", 1)) != MP_OKAY) + goto CLEANUP; + + DIGIT(&rem, 0) = DIGIT(a, ix); + s_mp_clamp(&rem); + --ix; + } + + /* If we didn't find one, we're finished dividing */ + if(s_mp_cmp(&rem, b) < 0) + break; + + /* Compute a guess for the next quotient digit */ + q = DIGIT(&rem, USED(&rem) - 1); + if(q <= DIGIT(b, USED(b) - 1) && USED(&rem) > 1) + q = (q << DIGIT_BIT) | DIGIT(&rem, USED(&rem) - 2); + + q /= DIGIT(b, USED(b) - 1); + + /* The guess can be as much as RADIX + 1 */ + if(q >= RADIX) + q = RADIX - 1; + + /* See what that multiplies out to */ + mp_copy(b, &t); + if((res = s_mp_mul_d(&t, q)) != MP_OKAY) + goto CLEANUP; + + /* + If it's too big, back it off. We should not have to do this + more than once, or, in rare cases, twice. Knuth describes a + method by which this could be reduced to a maximum of once, but + I didn't implement that here. + */ + while(s_mp_cmp(&t, &rem) > 0) { + --q; + s_mp_sub(&t, b); + } + + /* At this point, q should be the right next digit */ + if((res = s_mp_sub(&rem, &t)) != MP_OKAY) + goto CLEANUP; + + /* + Include the digit in the quotient. We allocated enough memory + for any quotient we could ever possibly get, so we should not + have to check for failures here + */ + DIGIT(", 0) = q; + } + + /* Denormalize remainder */ + if(d != 0) + s_mp_div_2d(&rem, d); + + s_mp_clamp("); + s_mp_clamp(&rem); + + /* Copy quotient back to output */ + s_mp_exch(", a); + + /* Copy remainder back to output */ + s_mp_exch(&rem, b); + +CLEANUP: + mp_clear(&rem); +REM: + mp_clear(&t); +T: + mp_clear("); + + return res; + +} /* end s_mp_div() */ + +/* }}} */ + +/* {{{ s_mp_2expt(a, k) */ + +mp_err s_mp_2expt(mp_int *a, mp_digit k) +{ + mp_err res; + mp_size dig, bit; + + dig = k / DIGIT_BIT; + bit = k % DIGIT_BIT; + + mp_zero(a); + if((res = s_mp_pad(a, dig + 1)) != MP_OKAY) + return res; + + DIGIT(a, dig) |= (1 << bit); + + return MP_OKAY; + +} /* end s_mp_2expt() */ + +/* }}} */ + + +/* }}} */ + +/* }}} */ + +/* {{{ Primitive comparisons */ + +/* {{{ s_mp_cmp(a, b) */ + +/* Compare |a| <=> |b|, return 0 if equal, <0 if a<b, >0 if a>b */ +int s_mp_cmp(mp_int *a, mp_int *b) +{ + mp_size ua = USED(a), ub = USED(b); + + if(ua > ub) + return MP_GT; + else if(ua < ub) + return MP_LT; + else { + int ix = ua - 1; + mp_digit *ap = DIGITS(a) + ix, *bp = DIGITS(b) + ix; + + while(ix >= 0) { + if(*ap > *bp) + return MP_GT; + else if(*ap < *bp) + return MP_LT; + + --ap; --bp; --ix; + } + + return MP_EQ; + } + +} /* end s_mp_cmp() */ + +/* }}} */ + +/* {{{ s_mp_cmp_d(a, d) */ + +/* Compare |a| <=> d, return 0 if equal, <0 if a<d, >0 if a>d */ +int s_mp_cmp_d(mp_int *a, mp_digit d) +{ + mp_size ua = USED(a); + mp_digit *ap = DIGITS(a); + + if(ua > 1) + return MP_GT; + + if(*ap < d) + return MP_LT; + else if(*ap > d) + return MP_GT; + else + return MP_EQ; + +} /* end s_mp_cmp_d() */ + +/* }}} */ + +/* {{{ s_mp_ispow2(v) */ + +/* + Returns -1 if the value is not a power of two; otherwise, it returns + k such that v = 2^k, i.e. lg(v). + */ +int s_mp_ispow2(mp_int *v) +{ + mp_digit d, *dp; + mp_size uv = USED(v); + int extra = 0, ix; + + d = DIGIT(v, uv - 1); /* most significant digit of v */ + + while(d && ((d & 1) == 0)) { + d >>= 1; + ++extra; + } + + if(d == 1) { + ix = uv - 2; + dp = DIGITS(v) + ix; + + while(ix >= 0) { + if(*dp) + return -1; /* not a power of two */ + + --dp; --ix; + } + + return ((uv - 1) * DIGIT_BIT) + extra; + } + + return -1; + +} /* end s_mp_ispow2() */ + +/* }}} */ + +/* {{{ s_mp_ispow2d(d) */ + +int s_mp_ispow2d(mp_digit d) +{ + int pow = 0; + + while((d & 1) == 0) { + ++pow; d >>= 1; + } + + if(d == 1) + return pow; + + return -1; + +} /* end s_mp_ispow2d() */ + +/* }}} */ + +/* }}} */ + +/* {{{ Primitive I/O helpers */ + +/* {{{ s_mp_tovalue(ch, r) */ + +/* + Convert the given character to its digit value, in the given radix. + If the given character is not understood in the given radix, -1 is + returned. Otherwise the digit's numeric value is returned. + + The results will be odd if you use a radix < 2 or > 62, you are + expected to know what you're up to. + */ +int s_mp_tovalue(char ch, int r) +{ + int val, xch; + + if(r > 36) + xch = ch; + else + xch = toupper(ch); + + if(isdigit(xch)) + val = xch - '0'; + else if(isupper(xch)) + val = xch - 'A' + 10; + else if(islower(xch)) + val = xch - 'a' + 36; + else if(xch == '+') + val = 62; + else if(xch == '/') + val = 63; + else + return -1; + + if(val < 0 || val >= r) + return -1; + + return val; + +} /* end s_mp_tovalue() */ + +/* }}} */ + +/* {{{ s_mp_todigit(val, r, low) */ + +/* + Convert val to a radix-r digit, if possible. If val is out of range + for r, returns zero. Otherwise, returns an ASCII character denoting + the value in the given radix. + + The results may be odd if you use a radix < 2 or > 64, you are + expected to know what you're doing. + */ + +char s_mp_todigit(int val, int r, int low) +{ + char ch; + + if(val < 0 || val >= r) + return 0; + + ch = s_dmap_1[val]; + + if(r <= 36 && low) + ch = tolower(ch); + + return ch; + +} /* end s_mp_todigit() */ + +/* }}} */ + +/* {{{ s_mp_outlen(bits, radix) */ + +/* + Return an estimate for how long a string is needed to hold a radix + r representation of a number with 'bits' significant bits. + + Does not include space for a sign or a NUL terminator. + */ +int s_mp_outlen(int bits, int r) +{ + return (int)((double)bits * LOG_V_2(r)); + +} /* end s_mp_outlen() */ + +/* }}} */ + +/* }}} */ + +/*------------------------------------------------------------------------*/ +/* HERE THERE BE DRAGONS */ +/* crc==4242132123, version==2, Sat Feb 02 06:43:52 2002 */ diff --git a/libtommath/mtest/mpi.h b/libtommath/mtest/mpi.h new file mode 100644 index 0000000..b7a8cb5 --- /dev/null +++ b/libtommath/mtest/mpi.h @@ -0,0 +1,227 @@ +/* + mpi.h + + by Michael J. Fromberger <sting@linguist.dartmouth.edu> + Copyright (C) 1998 Michael J. Fromberger, All Rights Reserved + + Arbitrary precision integer arithmetic library + + $Id: mpi.h,v 1.1.1.1 2005/01/19 22:41:29 kennykb Exp $ + */ + +#ifndef _H_MPI_ +#define _H_MPI_ + +#include "mpi-config.h" + +#define MP_LT -1 +#define MP_EQ 0 +#define MP_GT 1 + +#if MP_DEBUG +#undef MP_IOFUNC +#define MP_IOFUNC 1 +#endif + +#if MP_IOFUNC +#include <stdio.h> +#include <ctype.h> +#endif + +#include <limits.h> + +#define MP_NEG 1 +#define MP_ZPOS 0 + +/* Included for compatibility... */ +#define NEG MP_NEG +#define ZPOS MP_ZPOS + +#define MP_OKAY 0 /* no error, all is well */ +#define MP_YES 0 /* yes (boolean result) */ +#define MP_NO -1 /* no (boolean result) */ +#define MP_MEM -2 /* out of memory */ +#define MP_RANGE -3 /* argument out of range */ +#define MP_BADARG -4 /* invalid parameter */ +#define MP_UNDEF -5 /* answer is undefined */ +#define MP_LAST_CODE MP_UNDEF + +#include "mpi-types.h" + +/* Included for compatibility... */ +#define DIGIT_BIT MP_DIGIT_BIT +#define DIGIT_MAX MP_DIGIT_MAX + +/* Macros for accessing the mp_int internals */ +#define SIGN(MP) ((MP)->sign) +#define USED(MP) ((MP)->used) +#define ALLOC(MP) ((MP)->alloc) +#define DIGITS(MP) ((MP)->dp) +#define DIGIT(MP,N) (MP)->dp[(N)] + +#if MP_ARGCHK == 1 +#define ARGCHK(X,Y) {if(!(X)){return (Y);}} +#elif MP_ARGCHK == 2 +#include <assert.h> +#define ARGCHK(X,Y) assert(X) +#else +#define ARGCHK(X,Y) /* */ +#endif + +/* This defines the maximum I/O base (minimum is 2) */ +#define MAX_RADIX 64 + +typedef struct { + mp_sign sign; /* sign of this quantity */ + mp_size alloc; /* how many digits allocated */ + mp_size used; /* how many digits used */ + mp_digit *dp; /* the digits themselves */ +} mp_int; + +/*------------------------------------------------------------------------*/ +/* Default precision */ + +unsigned int mp_get_prec(void); +void mp_set_prec(unsigned int prec); + +/*------------------------------------------------------------------------*/ +/* Memory management */ + +mp_err mp_init(mp_int *mp); +mp_err mp_init_array(mp_int mp[], int count); +mp_err mp_init_size(mp_int *mp, mp_size prec); +mp_err mp_init_copy(mp_int *mp, mp_int *from); +mp_err mp_copy(mp_int *from, mp_int *to); +void mp_exch(mp_int *mp1, mp_int *mp2); +void mp_clear(mp_int *mp); +void mp_clear_array(mp_int mp[], int count); +void mp_zero(mp_int *mp); +void mp_set(mp_int *mp, mp_digit d); +mp_err mp_set_int(mp_int *mp, long z); +mp_err mp_shrink(mp_int *a); + + +/*------------------------------------------------------------------------*/ +/* Single digit arithmetic */ + +mp_err mp_add_d(mp_int *a, mp_digit d, mp_int *b); +mp_err mp_sub_d(mp_int *a, mp_digit d, mp_int *b); +mp_err mp_mul_d(mp_int *a, mp_digit d, mp_int *b); +mp_err mp_mul_2(mp_int *a, mp_int *c); +mp_err mp_div_d(mp_int *a, mp_digit d, mp_int *q, mp_digit *r); +mp_err mp_div_2(mp_int *a, mp_int *c); +mp_err mp_expt_d(mp_int *a, mp_digit d, mp_int *c); + +/*------------------------------------------------------------------------*/ +/* Sign manipulations */ + +mp_err mp_abs(mp_int *a, mp_int *b); +mp_err mp_neg(mp_int *a, mp_int *b); + +/*------------------------------------------------------------------------*/ +/* Full arithmetic */ + +mp_err mp_add(mp_int *a, mp_int *b, mp_int *c); +mp_err mp_sub(mp_int *a, mp_int *b, mp_int *c); +mp_err mp_mul(mp_int *a, mp_int *b, mp_int *c); +mp_err mp_mul_2d(mp_int *a, mp_digit d, mp_int *c); +#if MP_SQUARE +mp_err mp_sqr(mp_int *a, mp_int *b); +#else +#define mp_sqr(a, b) mp_mul(a, a, b) +#endif +mp_err mp_div(mp_int *a, mp_int *b, mp_int *q, mp_int *r); +mp_err mp_div_2d(mp_int *a, mp_digit d, mp_int *q, mp_int *r); +mp_err mp_expt(mp_int *a, mp_int *b, mp_int *c); +mp_err mp_2expt(mp_int *a, mp_digit k); +mp_err mp_sqrt(mp_int *a, mp_int *b); + +/*------------------------------------------------------------------------*/ +/* Modular arithmetic */ + +#if MP_MODARITH +mp_err mp_mod(mp_int *a, mp_int *m, mp_int *c); +mp_err mp_mod_d(mp_int *a, mp_digit d, mp_digit *c); +mp_err mp_addmod(mp_int *a, mp_int *b, mp_int *m, mp_int *c); +mp_err mp_submod(mp_int *a, mp_int *b, mp_int *m, mp_int *c); +mp_err mp_mulmod(mp_int *a, mp_int *b, mp_int *m, mp_int *c); +#if MP_SQUARE +mp_err mp_sqrmod(mp_int *a, mp_int *m, mp_int *c); +#else +#define mp_sqrmod(a, m, c) mp_mulmod(a, a, m, c) +#endif +mp_err mp_exptmod(mp_int *a, mp_int *b, mp_int *m, mp_int *c); +mp_err mp_exptmod_d(mp_int *a, mp_digit d, mp_int *m, mp_int *c); +#endif /* MP_MODARITH */ + +/*------------------------------------------------------------------------*/ +/* Comparisons */ + +int mp_cmp_z(mp_int *a); +int mp_cmp_d(mp_int *a, mp_digit d); +int mp_cmp(mp_int *a, mp_int *b); +int mp_cmp_mag(mp_int *a, mp_int *b); +int mp_cmp_int(mp_int *a, long z); +int mp_isodd(mp_int *a); +int mp_iseven(mp_int *a); + +/*------------------------------------------------------------------------*/ +/* Number theoretic */ + +#if MP_NUMTH +mp_err mp_gcd(mp_int *a, mp_int *b, mp_int *c); +mp_err mp_lcm(mp_int *a, mp_int *b, mp_int *c); +mp_err mp_xgcd(mp_int *a, mp_int *b, mp_int *g, mp_int *x, mp_int *y); +mp_err mp_invmod(mp_int *a, mp_int *m, mp_int *c); +#endif /* end MP_NUMTH */ + +/*------------------------------------------------------------------------*/ +/* Input and output */ + +#if MP_IOFUNC +void mp_print(mp_int *mp, FILE *ofp); +#endif /* end MP_IOFUNC */ + +/*------------------------------------------------------------------------*/ +/* Base conversion */ + +#define BITS 1 +#define BYTES CHAR_BIT + +mp_err mp_read_signed_bin(mp_int *mp, unsigned char *str, int len); +int mp_signed_bin_size(mp_int *mp); +mp_err mp_to_signed_bin(mp_int *mp, unsigned char *str); + +mp_err mp_read_unsigned_bin(mp_int *mp, unsigned char *str, int len); +int mp_unsigned_bin_size(mp_int *mp); +mp_err mp_to_unsigned_bin(mp_int *mp, unsigned char *str); + +int mp_count_bits(mp_int *mp); + +#if MP_COMPAT_MACROS +#define mp_read_raw(mp, str, len) mp_read_signed_bin((mp), (str), (len)) +#define mp_raw_size(mp) mp_signed_bin_size(mp) +#define mp_toraw(mp, str) mp_to_signed_bin((mp), (str)) +#define mp_read_mag(mp, str, len) mp_read_unsigned_bin((mp), (str), (len)) +#define mp_mag_size(mp) mp_unsigned_bin_size(mp) +#define mp_tomag(mp, str) mp_to_unsigned_bin((mp), (str)) +#endif + +mp_err mp_read_radix(mp_int *mp, unsigned char *str, int radix); +int mp_radix_size(mp_int *mp, int radix); +int mp_value_radix_size(int num, int qty, int radix); +mp_err mp_toradix(mp_int *mp, unsigned char *str, int radix); + +int mp_char2value(char ch, int r); + +#define mp_tobinary(M, S) mp_toradix((M), (S), 2) +#define mp_tooctal(M, S) mp_toradix((M), (S), 8) +#define mp_todecimal(M, S) mp_toradix((M), (S), 10) +#define mp_tohex(M, S) mp_toradix((M), (S), 16) + +/*------------------------------------------------------------------------*/ +/* Error strings */ + +const char *mp_strerror(mp_err ec); + +#endif /* end _H_MPI_ */ diff --git a/libtommath/mtest/mtest.c b/libtommath/mtest/mtest.c new file mode 100644 index 0000000..d46f456 --- /dev/null +++ b/libtommath/mtest/mtest.c @@ -0,0 +1,304 @@ +/* makes a bignum test harness with NUM tests per operation + * + * the output is made in the following format [one parameter per line] + +operation +operand1 +operand2 +[... operandN] +result1 +result2 +[... resultN] + +So for example "a * b mod n" would be + +mulmod +a +b +n +a*b mod n + +e.g. if a=3, b=4 n=11 then + +mulmod +3 +4 +11 +1 + + */ + +#ifdef MP_8BIT +#define THE_MASK 127 +#else +#define THE_MASK 32767 +#endif + +#include <stdio.h> +#include <stdlib.h> +#include <time.h> +#include "mpi.c" + +FILE *rng; + +void rand_num(mp_int *a) +{ + int n, size; + unsigned char buf[2048]; + + size = 1 + ((fgetc(rng)<<8) + fgetc(rng)) % 101; + buf[0] = (fgetc(rng)&1)?1:0; + fread(buf+1, 1, size, rng); + while (buf[1] == 0) buf[1] = fgetc(rng); + mp_read_raw(a, buf, 1+size); +} + +void rand_num2(mp_int *a) +{ + int n, size; + unsigned char buf[2048]; + + size = 10 + ((fgetc(rng)<<8) + fgetc(rng)) % 101; + buf[0] = (fgetc(rng)&1)?1:0; + fread(buf+1, 1, size, rng); + while (buf[1] == 0) buf[1] = fgetc(rng); + mp_read_raw(a, buf, 1+size); +} + +#define mp_to64(a, b) mp_toradix(a, b, 64) + +int main(void) +{ + int n, tmp; + mp_int a, b, c, d, e; + clock_t t1; + char buf[4096]; + + mp_init(&a); + mp_init(&b); + mp_init(&c); + mp_init(&d); + mp_init(&e); + + + /* initial (2^n - 1)^2 testing, makes sure the comba multiplier works [it has the new carry code] */ +/* + mp_set(&a, 1); + for (n = 1; n < 8192; n++) { + mp_mul(&a, &a, &c); + printf("mul\n"); + mp_to64(&a, buf); + printf("%s\n%s\n", buf, buf); + mp_to64(&c, buf); + printf("%s\n", buf); + + mp_add_d(&a, 1, &a); + mp_mul_2(&a, &a); + mp_sub_d(&a, 1, &a); + } +*/ + + rng = fopen("/dev/urandom", "rb"); + if (rng == NULL) { + rng = fopen("/dev/random", "rb"); + if (rng == NULL) { + fprintf(stderr, "\nWarning: stdin used as random source\n\n"); + rng = stdin; + } + } + + t1 = clock(); + for (;;) { +#if 0 + if (clock() - t1 > CLOCKS_PER_SEC) { + sleep(2); + t1 = clock(); + } +#endif + n = fgetc(rng) % 15; + + if (n == 0) { + /* add tests */ + rand_num(&a); + rand_num(&b); + mp_add(&a, &b, &c); + printf("add\n"); + mp_to64(&a, buf); + printf("%s\n", buf); + mp_to64(&b, buf); + printf("%s\n", buf); + mp_to64(&c, buf); + printf("%s\n", buf); + } else if (n == 1) { + /* sub tests */ + rand_num(&a); + rand_num(&b); + mp_sub(&a, &b, &c); + printf("sub\n"); + mp_to64(&a, buf); + printf("%s\n", buf); + mp_to64(&b, buf); + printf("%s\n", buf); + mp_to64(&c, buf); + printf("%s\n", buf); + } else if (n == 2) { + /* mul tests */ + rand_num(&a); + rand_num(&b); + mp_mul(&a, &b, &c); + printf("mul\n"); + mp_to64(&a, buf); + printf("%s\n", buf); + mp_to64(&b, buf); + printf("%s\n", buf); + mp_to64(&c, buf); + printf("%s\n", buf); + } else if (n == 3) { + /* div tests */ + rand_num(&a); + rand_num(&b); + mp_div(&a, &b, &c, &d); + printf("div\n"); + mp_to64(&a, buf); + printf("%s\n", buf); + mp_to64(&b, buf); + printf("%s\n", buf); + mp_to64(&c, buf); + printf("%s\n", buf); + mp_to64(&d, buf); + printf("%s\n", buf); + } else if (n == 4) { + /* sqr tests */ + rand_num(&a); + mp_sqr(&a, &b); + printf("sqr\n"); + mp_to64(&a, buf); + printf("%s\n", buf); + mp_to64(&b, buf); + printf("%s\n", buf); + } else if (n == 5) { + /* mul_2d test */ + rand_num(&a); + mp_copy(&a, &b); + n = fgetc(rng) & 63; + mp_mul_2d(&b, n, &b); + mp_to64(&a, buf); + printf("mul2d\n"); + printf("%s\n", buf); + printf("%d\n", n); + mp_to64(&b, buf); + printf("%s\n", buf); + } else if (n == 6) { + /* div_2d test */ + rand_num(&a); + mp_copy(&a, &b); + n = fgetc(rng) & 63; + mp_div_2d(&b, n, &b, NULL); + mp_to64(&a, buf); + printf("div2d\n"); + printf("%s\n", buf); + printf("%d\n", n); + mp_to64(&b, buf); + printf("%s\n", buf); + } else if (n == 7) { + /* gcd test */ + rand_num(&a); + rand_num(&b); + a.sign = MP_ZPOS; + b.sign = MP_ZPOS; + mp_gcd(&a, &b, &c); + printf("gcd\n"); + mp_to64(&a, buf); + printf("%s\n", buf); + mp_to64(&b, buf); + printf("%s\n", buf); + mp_to64(&c, buf); + printf("%s\n", buf); + } else if (n == 8) { + /* lcm test */ + rand_num(&a); + rand_num(&b); + a.sign = MP_ZPOS; + b.sign = MP_ZPOS; + mp_lcm(&a, &b, &c); + printf("lcm\n"); + mp_to64(&a, buf); + printf("%s\n", buf); + mp_to64(&b, buf); + printf("%s\n", buf); + mp_to64(&c, buf); + printf("%s\n", buf); + } else if (n == 9) { + /* exptmod test */ + rand_num2(&a); + rand_num2(&b); + rand_num2(&c); +// if (c.dp[0]&1) mp_add_d(&c, 1, &c); + a.sign = b.sign = c.sign = 0; + mp_exptmod(&a, &b, &c, &d); + printf("expt\n"); + mp_to64(&a, buf); + printf("%s\n", buf); + mp_to64(&b, buf); + printf("%s\n", buf); + mp_to64(&c, buf); + printf("%s\n", buf); + mp_to64(&d, buf); + printf("%s\n", buf); + } else if (n == 10) { + /* invmod test */ + rand_num2(&a); + rand_num2(&b); + b.sign = MP_ZPOS; + a.sign = MP_ZPOS; + mp_gcd(&a, &b, &c); + if (mp_cmp_d(&c, 1) != 0) continue; + if (mp_cmp_d(&b, 1) == 0) continue; + mp_invmod(&a, &b, &c); + printf("invmod\n"); + mp_to64(&a, buf); + printf("%s\n", buf); + mp_to64(&b, buf); + printf("%s\n", buf); + mp_to64(&c, buf); + printf("%s\n", buf); + } else if (n == 11) { + rand_num(&a); + mp_mul_2(&a, &a); + mp_div_2(&a, &b); + printf("div2\n"); + mp_to64(&a, buf); + printf("%s\n", buf); + mp_to64(&b, buf); + printf("%s\n", buf); + } else if (n == 12) { + rand_num2(&a); + mp_mul_2(&a, &b); + printf("mul2\n"); + mp_to64(&a, buf); + printf("%s\n", buf); + mp_to64(&b, buf); + printf("%s\n", buf); + } else if (n == 13) { + rand_num2(&a); + tmp = abs(rand()) & THE_MASK; + mp_add_d(&a, tmp, &b); + printf("add_d\n"); + mp_to64(&a, buf); + printf("%s\n%d\n", buf, tmp); + mp_to64(&b, buf); + printf("%s\n", buf); + } else if (n == 14) { + rand_num2(&a); + tmp = abs(rand()) & THE_MASK; + mp_sub_d(&a, tmp, &b); + printf("sub_d\n"); + mp_to64(&a, buf); + printf("%s\n%d\n", buf, tmp); + mp_to64(&b, buf); + printf("%s\n", buf); + } + } + fclose(rng); + return 0; +} diff --git a/libtommath/pics/design_process.sxd b/libtommath/pics/design_process.sxd Binary files differnew file mode 100644 index 0000000..7414dbb --- /dev/null +++ b/libtommath/pics/design_process.sxd diff --git a/libtommath/pics/design_process.tif b/libtommath/pics/design_process.tif Binary files differnew file mode 100644 index 0000000..4a0c012 --- /dev/null +++ b/libtommath/pics/design_process.tif diff --git a/libtommath/pics/expt_state.sxd b/libtommath/pics/expt_state.sxd Binary files differnew file mode 100644 index 0000000..6518404 --- /dev/null +++ b/libtommath/pics/expt_state.sxd diff --git a/libtommath/pics/expt_state.tif b/libtommath/pics/expt_state.tif Binary files differnew file mode 100644 index 0000000..cb06e8e --- /dev/null +++ b/libtommath/pics/expt_state.tif diff --git a/libtommath/pics/makefile b/libtommath/pics/makefile new file mode 100644 index 0000000..3ecb02f --- /dev/null +++ b/libtommath/pics/makefile @@ -0,0 +1,35 @@ +# makes the images... yeah + +default: pses + +design_process.ps: design_process.tif + tiff2ps -s -e design_process.tif > design_process.ps + +sliding_window.ps: sliding_window.tif + tiff2ps -s -e sliding_window.tif > sliding_window.ps + +expt_state.ps: expt_state.tif + tiff2ps -s -e expt_state.tif > expt_state.ps + +primality.ps: primality.tif + tiff2ps -s -e primality.tif > primality.ps + +design_process.pdf: design_process.ps + epstopdf design_process.ps + +sliding_window.pdf: sliding_window.ps + epstopdf sliding_window.ps + +expt_state.pdf: expt_state.ps + epstopdf expt_state.ps + +primality.pdf: primality.ps + epstopdf primality.ps + + +pses: sliding_window.ps expt_state.ps primality.ps design_process.ps +pdfes: sliding_window.pdf expt_state.pdf primality.pdf design_process.pdf + +clean: + rm -rf *.ps *.pdf .xvpics +
\ No newline at end of file diff --git a/libtommath/pics/primality.tif b/libtommath/pics/primality.tif Binary files differnew file mode 100644 index 0000000..76d6be3 --- /dev/null +++ b/libtommath/pics/primality.tif diff --git a/libtommath/pics/radix.sxd b/libtommath/pics/radix.sxd Binary files differnew file mode 100644 index 0000000..b9eb9a0 --- /dev/null +++ b/libtommath/pics/radix.sxd diff --git a/libtommath/pics/sliding_window.sxd b/libtommath/pics/sliding_window.sxd Binary files differnew file mode 100644 index 0000000..91e7c0d --- /dev/null +++ b/libtommath/pics/sliding_window.sxd diff --git a/libtommath/pics/sliding_window.tif b/libtommath/pics/sliding_window.tif Binary files differnew file mode 100644 index 0000000..bb4cb96 --- /dev/null +++ b/libtommath/pics/sliding_window.tif diff --git a/libtommath/poster.out b/libtommath/poster.out new file mode 100644 index 0000000..e69de29 --- /dev/null +++ b/libtommath/poster.out diff --git a/libtommath/poster.pdf b/libtommath/poster.pdf Binary files differnew file mode 100644 index 0000000..e0b4f84 --- /dev/null +++ b/libtommath/poster.pdf diff --git a/libtommath/poster.tex b/libtommath/poster.tex new file mode 100644 index 0000000..e7388f4 --- /dev/null +++ b/libtommath/poster.tex @@ -0,0 +1,35 @@ +\documentclass[landscape,11pt]{article} +\usepackage{amsmath, amssymb} +\usepackage{hyperref} +\begin{document} +\hspace*{-3in} +\begin{tabular}{llllll} +$c = a + b$ & {\tt mp\_add(\&a, \&b, \&c)} & $b = 2a$ & {\tt mp\_mul\_2(\&a, \&b)} & \\ +$c = a - b$ & {\tt mp\_sub(\&a, \&b, \&c)} & $b = a/2$ & {\tt mp\_div\_2(\&a, \&b)} & \\ +$c = ab $ & {\tt mp\_mul(\&a, \&b, \&c)} & $c = 2^ba$ & {\tt mp\_mul\_2d(\&a, b, \&c)} \\ +$b = a^2 $ & {\tt mp\_sqr(\&a, \&b)} & $c = a/2^b, d = a \mod 2^b$ & {\tt mp\_div\_2d(\&a, b, \&c, \&d)} \\ +$c = \lfloor a/b \rfloor, d = a \mod b$ & {\tt mp\_div(\&a, \&b, \&c, \&d)} & $c = a \mod 2^b $ & {\tt mp\_mod\_2d(\&a, b, \&c)} \\ + && \\ +$a = b $ & {\tt mp\_set\_int(\&a, b)} & $c = a \vee b$ & {\tt mp\_or(\&a, \&b, \&c)} \\ +$b = a $ & {\tt mp\_copy(\&a, \&b)} & $c = a \wedge b$ & {\tt mp\_and(\&a, \&b, \&c)} \\ + && $c = a \oplus b$ & {\tt mp\_xor(\&a, \&b, \&c)} \\ + & \\ +$b = -a $ & {\tt mp\_neg(\&a, \&b)} & $d = a + b \mod c$ & {\tt mp\_addmod(\&a, \&b, \&c, \&d)} \\ +$b = |a| $ & {\tt mp\_abs(\&a, \&b)} & $d = a - b \mod c$ & {\tt mp\_submod(\&a, \&b, \&c, \&d)} \\ + && $d = ab \mod c$ & {\tt mp\_mulmod(\&a, \&b, \&c, \&d)} \\ +Compare $a$ and $b$ & {\tt mp\_cmp(\&a, \&b)} & $c = a^2 \mod b$ & {\tt mp\_sqrmod(\&a, \&b, \&c)} \\ +Is Zero? & {\tt mp\_iszero(\&a)} & $c = a^{-1} \mod b$ & {\tt mp\_invmod(\&a, \&b, \&c)} \\ +Is Even? & {\tt mp\_iseven(\&a)} & $d = a^b \mod c$ & {\tt mp\_exptmod(\&a, \&b, \&c, \&d)} \\ +Is Odd ? & {\tt mp\_isodd(\&a)} \\ +&\\ +$\vert \vert a \vert \vert$ & {\tt mp\_unsigned\_bin\_size(\&a)} & $res$ = 1 if $a$ prime to $t$ rounds? & {\tt mp\_prime\_is\_prime(\&a, t, \&res)} \\ +$buf \leftarrow a$ & {\tt mp\_to\_unsigned\_bin(\&a, buf)} & Next prime after $a$ to $t$ rounds. & {\tt mp\_prime\_next\_prime(\&a, t, bbs\_style)} \\ +$a \leftarrow buf[0..len-1]$ & {\tt mp\_read\_unsigned\_bin(\&a, buf, len)} \\ +&\\ +$b = \sqrt{a}$ & {\tt mp\_sqrt(\&a, \&b)} & $c = \mbox{gcd}(a, b)$ & {\tt mp\_gcd(\&a, \&b, \&c)} \\ +$c = a^{1/b}$ & {\tt mp\_n\_root(\&a, b, \&c)} & $c = \mbox{lcm}(a, b)$ & {\tt mp\_lcm(\&a, \&b, \&c)} \\ +&\\ +Greater Than & MP\_GT & Equal To & MP\_EQ \\ +Less Than & MP\_LT & Bits per digit & DIGIT\_BIT \\ +\end{tabular} +\end{document} diff --git a/libtommath/pre_gen/mpi.c b/libtommath/pre_gen/mpi.c new file mode 100644 index 0000000..7d832e7 --- /dev/null +++ b/libtommath/pre_gen/mpi.c @@ -0,0 +1,8819 @@ +/* Start: bn_error.c */ +#include <tommath.h> +#ifdef BN_ERROR_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +static const struct { + int code; + char *msg; +} msgs[] = { + { MP_OKAY, "Successful" }, + { MP_MEM, "Out of heap" }, + { MP_VAL, "Value out of range" } +}; + +/* return a char * string for a given code */ +char *mp_error_to_string(int code) +{ + int x; + + /* scan the lookup table for the given message */ + for (x = 0; x < (int)(sizeof(msgs) / sizeof(msgs[0])); x++) { + if (msgs[x].code == code) { + return msgs[x].msg; + } + } + + /* generic reply for invalid code */ + return "Invalid error code"; +} + +#endif + +/* End: bn_error.c */ + +/* Start: bn_fast_mp_invmod.c */ +#include <tommath.h> +#ifdef BN_FAST_MP_INVMOD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* computes the modular inverse via binary extended euclidean algorithm, + * that is c = 1/a mod b + * + * Based on slow invmod except this is optimized for the case where b is + * odd as per HAC Note 14.64 on pp. 610 + */ +int +fast_mp_invmod (mp_int * a, mp_int * b, mp_int * c) +{ + mp_int x, y, u, v, B, D; + int res, neg; + + /* 2. [modified] b must be odd */ + if (mp_iseven (b) == 1) { + return MP_VAL; + } + + /* init all our temps */ + if ((res = mp_init_multi(&x, &y, &u, &v, &B, &D, NULL)) != MP_OKAY) { + return res; + } + + /* x == modulus, y == value to invert */ + if ((res = mp_copy (b, &x)) != MP_OKAY) { + goto LBL_ERR; + } + + /* we need y = |a| */ + if ((res = mp_abs (a, &y)) != MP_OKAY) { + goto LBL_ERR; + } + + /* 3. u=x, v=y, A=1, B=0, C=0,D=1 */ + if ((res = mp_copy (&x, &u)) != MP_OKAY) { + goto LBL_ERR; + } + if ((res = mp_copy (&y, &v)) != MP_OKAY) { + goto LBL_ERR; + } + mp_set (&D, 1); + +top: + /* 4. while u is even do */ + while (mp_iseven (&u) == 1) { + /* 4.1 u = u/2 */ + if ((res = mp_div_2 (&u, &u)) != MP_OKAY) { + goto LBL_ERR; + } + /* 4.2 if B is odd then */ + if (mp_isodd (&B) == 1) { + if ((res = mp_sub (&B, &x, &B)) != MP_OKAY) { + goto LBL_ERR; + } + } + /* B = B/2 */ + if ((res = mp_div_2 (&B, &B)) != MP_OKAY) { + goto LBL_ERR; + } + } + + /* 5. while v is even do */ + while (mp_iseven (&v) == 1) { + /* 5.1 v = v/2 */ + if ((res = mp_div_2 (&v, &v)) != MP_OKAY) { + goto LBL_ERR; + } + /* 5.2 if D is odd then */ + if (mp_isodd (&D) == 1) { + /* D = (D-x)/2 */ + if ((res = mp_sub (&D, &x, &D)) != MP_OKAY) { + goto LBL_ERR; + } + } + /* D = D/2 */ + if ((res = mp_div_2 (&D, &D)) != MP_OKAY) { + goto LBL_ERR; + } + } + + /* 6. if u >= v then */ + if (mp_cmp (&u, &v) != MP_LT) { + /* u = u - v, B = B - D */ + if ((res = mp_sub (&u, &v, &u)) != MP_OKAY) { + goto LBL_ERR; + } + + if ((res = mp_sub (&B, &D, &B)) != MP_OKAY) { + goto LBL_ERR; + } + } else { + /* v - v - u, D = D - B */ + if ((res = mp_sub (&v, &u, &v)) != MP_OKAY) { + goto LBL_ERR; + } + + if ((res = mp_sub (&D, &B, &D)) != MP_OKAY) { + goto LBL_ERR; + } + } + + /* if not zero goto step 4 */ + if (mp_iszero (&u) == 0) { + goto top; + } + + /* now a = C, b = D, gcd == g*v */ + + /* if v != 1 then there is no inverse */ + if (mp_cmp_d (&v, 1) != MP_EQ) { + res = MP_VAL; + goto LBL_ERR; + } + + /* b is now the inverse */ + neg = a->sign; + while (D.sign == MP_NEG) { + if ((res = mp_add (&D, b, &D)) != MP_OKAY) { + goto LBL_ERR; + } + } + mp_exch (&D, c); + c->sign = neg; + res = MP_OKAY; + +LBL_ERR:mp_clear_multi (&x, &y, &u, &v, &B, &D, NULL); + return res; +} +#endif + +/* End: bn_fast_mp_invmod.c */ + +/* Start: bn_fast_mp_montgomery_reduce.c */ +#include <tommath.h> +#ifdef BN_FAST_MP_MONTGOMERY_REDUCE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* computes xR**-1 == x (mod N) via Montgomery Reduction + * + * This is an optimized implementation of montgomery_reduce + * which uses the comba method to quickly calculate the columns of the + * reduction. + * + * Based on Algorithm 14.32 on pp.601 of HAC. +*/ +int +fast_mp_montgomery_reduce (mp_int * x, mp_int * n, mp_digit rho) +{ + int ix, res, olduse; + mp_word W[MP_WARRAY]; + + /* get old used count */ + olduse = x->used; + + /* grow a as required */ + if (x->alloc < n->used + 1) { + if ((res = mp_grow (x, n->used + 1)) != MP_OKAY) { + return res; + } + } + + /* first we have to get the digits of the input into + * an array of double precision words W[...] + */ + { + register mp_word *_W; + register mp_digit *tmpx; + + /* alias for the W[] array */ + _W = W; + + /* alias for the digits of x*/ + tmpx = x->dp; + + /* copy the digits of a into W[0..a->used-1] */ + for (ix = 0; ix < x->used; ix++) { + *_W++ = *tmpx++; + } + + /* zero the high words of W[a->used..m->used*2] */ + for (; ix < n->used * 2 + 1; ix++) { + *_W++ = 0; + } + } + + /* now we proceed to zero successive digits + * from the least significant upwards + */ + for (ix = 0; ix < n->used; ix++) { + /* mu = ai * m' mod b + * + * We avoid a double precision multiplication (which isn't required) + * by casting the value down to a mp_digit. Note this requires + * that W[ix-1] have the carry cleared (see after the inner loop) + */ + register mp_digit mu; + mu = (mp_digit) (((W[ix] & MP_MASK) * rho) & MP_MASK); + + /* a = a + mu * m * b**i + * + * This is computed in place and on the fly. The multiplication + * by b**i is handled by offseting which columns the results + * are added to. + * + * Note the comba method normally doesn't handle carries in the + * inner loop In this case we fix the carry from the previous + * column since the Montgomery reduction requires digits of the + * result (so far) [see above] to work. This is + * handled by fixing up one carry after the inner loop. The + * carry fixups are done in order so after these loops the + * first m->used words of W[] have the carries fixed + */ + { + register int iy; + register mp_digit *tmpn; + register mp_word *_W; + + /* alias for the digits of the modulus */ + tmpn = n->dp; + + /* Alias for the columns set by an offset of ix */ + _W = W + ix; + + /* inner loop */ + for (iy = 0; iy < n->used; iy++) { + *_W++ += ((mp_word)mu) * ((mp_word)*tmpn++); + } + } + + /* now fix carry for next digit, W[ix+1] */ + W[ix + 1] += W[ix] >> ((mp_word) DIGIT_BIT); + } + + /* now we have to propagate the carries and + * shift the words downward [all those least + * significant digits we zeroed]. + */ + { + register mp_digit *tmpx; + register mp_word *_W, *_W1; + + /* nox fix rest of carries */ + + /* alias for current word */ + _W1 = W + ix; + + /* alias for next word, where the carry goes */ + _W = W + ++ix; + + for (; ix <= n->used * 2 + 1; ix++) { + *_W++ += *_W1++ >> ((mp_word) DIGIT_BIT); + } + + /* copy out, A = A/b**n + * + * The result is A/b**n but instead of converting from an + * array of mp_word to mp_digit than calling mp_rshd + * we just copy them in the right order + */ + + /* alias for destination word */ + tmpx = x->dp; + + /* alias for shifted double precision result */ + _W = W + n->used; + + for (ix = 0; ix < n->used + 1; ix++) { + *tmpx++ = (mp_digit)(*_W++ & ((mp_word) MP_MASK)); + } + + /* zero oldused digits, if the input a was larger than + * m->used+1 we'll have to clear the digits + */ + for (; ix < olduse; ix++) { + *tmpx++ = 0; + } + } + + /* set the max used and clamp */ + x->used = n->used + 1; + mp_clamp (x); + + /* if A >= m then A = A - m */ + if (mp_cmp_mag (x, n) != MP_LT) { + return s_mp_sub (x, n, x); + } + return MP_OKAY; +} +#endif + +/* End: bn_fast_mp_montgomery_reduce.c */ + +/* Start: bn_fast_s_mp_mul_digs.c */ +#include <tommath.h> +#ifdef BN_FAST_S_MP_MUL_DIGS_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* Fast (comba) multiplier + * + * This is the fast column-array [comba] multiplier. It is + * designed to compute the columns of the product first + * then handle the carries afterwards. This has the effect + * of making the nested loops that compute the columns very + * simple and schedulable on super-scalar processors. + * + * This has been modified to produce a variable number of + * digits of output so if say only a half-product is required + * you don't have to compute the upper half (a feature + * required for fast Barrett reduction). + * + * Based on Algorithm 14.12 on pp.595 of HAC. + * + */ +int +fast_s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs) +{ + int olduse, res, pa, ix, iz; + mp_digit W[MP_WARRAY]; + register mp_word _W; + + /* grow the destination as required */ + if (c->alloc < digs) { + if ((res = mp_grow (c, digs)) != MP_OKAY) { + return res; + } + } + + /* number of output digits to produce */ + pa = MIN(digs, a->used + b->used); + + /* clear the carry */ + _W = 0; + for (ix = 0; ix < pa; ix++) { + int tx, ty; + int iy; + mp_digit *tmpx, *tmpy; + + /* get offsets into the two bignums */ + ty = MIN(b->used-1, ix); + tx = ix - ty; + + /* setup temp aliases */ + tmpx = a->dp + tx; + tmpy = b->dp + ty; + + /* this is the number of times the loop will iterrate, essentially its + while (tx++ < a->used && ty-- >= 0) { ... } + */ + iy = MIN(a->used-tx, ty+1); + + /* execute loop */ + for (iz = 0; iz < iy; ++iz) { + _W += ((mp_word)*tmpx++)*((mp_word)*tmpy--); + } + + /* store term */ + W[ix] = ((mp_digit)_W) & MP_MASK; + + /* make next carry */ + _W = _W >> ((mp_word)DIGIT_BIT); + } + + /* store final carry */ + W[ix] = _W; + + /* setup dest */ + olduse = c->used; + c->used = digs; + + { + register mp_digit *tmpc; + tmpc = c->dp; + for (ix = 0; ix < digs; ix++) { + /* now extract the previous digit [below the carry] */ + *tmpc++ = W[ix]; + } + + /* clear unused digits [that existed in the old copy of c] */ + for (; ix < olduse; ix++) { + *tmpc++ = 0; + } + } + mp_clamp (c); + return MP_OKAY; +} +#endif + +/* End: bn_fast_s_mp_mul_digs.c */ + +/* Start: bn_fast_s_mp_mul_high_digs.c */ +#include <tommath.h> +#ifdef BN_FAST_S_MP_MUL_HIGH_DIGS_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* this is a modified version of fast_s_mul_digs that only produces + * output digits *above* digs. See the comments for fast_s_mul_digs + * to see how it works. + * + * This is used in the Barrett reduction since for one of the multiplications + * only the higher digits were needed. This essentially halves the work. + * + * Based on Algorithm 14.12 on pp.595 of HAC. + */ +int +fast_s_mp_mul_high_digs (mp_int * a, mp_int * b, mp_int * c, int digs) +{ + int olduse, res, pa, ix, iz; + mp_digit W[MP_WARRAY]; + mp_word _W; + + /* grow the destination as required */ + pa = a->used + b->used; + if (c->alloc < pa) { + if ((res = mp_grow (c, pa)) != MP_OKAY) { + return res; + } + } + + /* number of output digits to produce */ + pa = a->used + b->used; + _W = 0; + for (ix = digs; ix < pa; ix++) { + int tx, ty, iy; + mp_digit *tmpx, *tmpy; + + /* get offsets into the two bignums */ + ty = MIN(b->used-1, ix); + tx = ix - ty; + + /* setup temp aliases */ + tmpx = a->dp + tx; + tmpy = b->dp + ty; + + /* this is the number of times the loop will iterrate, essentially its + while (tx++ < a->used && ty-- >= 0) { ... } + */ + iy = MIN(a->used-tx, ty+1); + + /* execute loop */ + for (iz = 0; iz < iy; iz++) { + _W += ((mp_word)*tmpx++)*((mp_word)*tmpy--); + } + + /* store term */ + W[ix] = ((mp_digit)_W) & MP_MASK; + + /* make next carry */ + _W = _W >> ((mp_word)DIGIT_BIT); + } + + /* store final carry */ + W[ix] = _W; + + /* setup dest */ + olduse = c->used; + c->used = pa; + + { + register mp_digit *tmpc; + + tmpc = c->dp + digs; + for (ix = digs; ix <= pa; ix++) { + /* now extract the previous digit [below the carry] */ + *tmpc++ = W[ix]; + } + + /* clear unused digits [that existed in the old copy of c] */ + for (; ix < olduse; ix++) { + *tmpc++ = 0; + } + } + mp_clamp (c); + return MP_OKAY; +} +#endif + +/* End: bn_fast_s_mp_mul_high_digs.c */ + +/* Start: bn_fast_s_mp_sqr.c */ +#include <tommath.h> +#ifdef BN_FAST_S_MP_SQR_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* fast squaring + * + * This is the comba method where the columns of the product + * are computed first then the carries are computed. This + * has the effect of making a very simple inner loop that + * is executed the most + * + * W2 represents the outer products and W the inner. + * + * A further optimizations is made because the inner + * products are of the form "A * B * 2". The *2 part does + * not need to be computed until the end which is good + * because 64-bit shifts are slow! + * + * Based on Algorithm 14.16 on pp.597 of HAC. + * + */ +/* the jist of squaring... + +you do like mult except the offset of the tmpx [one that starts closer to zero] +can't equal the offset of tmpy. So basically you set up iy like before then you min it with +(ty-tx) so that it never happens. You double all those you add in the inner loop + +After that loop you do the squares and add them in. + +Remove W2 and don't memset W + +*/ + +int fast_s_mp_sqr (mp_int * a, mp_int * b) +{ + int olduse, res, pa, ix, iz; + mp_digit W[MP_WARRAY], *tmpx; + mp_word W1; + + /* grow the destination as required */ + pa = a->used + a->used; + if (b->alloc < pa) { + if ((res = mp_grow (b, pa)) != MP_OKAY) { + return res; + } + } + + /* number of output digits to produce */ + W1 = 0; + for (ix = 0; ix < pa; ix++) { + int tx, ty, iy; + mp_word _W; + mp_digit *tmpy; + + /* clear counter */ + _W = 0; + + /* get offsets into the two bignums */ + ty = MIN(a->used-1, ix); + tx = ix - ty; + + /* setup temp aliases */ + tmpx = a->dp + tx; + tmpy = a->dp + ty; + + /* this is the number of times the loop will iterrate, essentially its + while (tx++ < a->used && ty-- >= 0) { ... } + */ + iy = MIN(a->used-tx, ty+1); + + /* now for squaring tx can never equal ty + * we halve the distance since they approach at a rate of 2x + * and we have to round because odd cases need to be executed + */ + iy = MIN(iy, (ty-tx+1)>>1); + + /* execute loop */ + for (iz = 0; iz < iy; iz++) { + _W += ((mp_word)*tmpx++)*((mp_word)*tmpy--); + } + + /* double the inner product and add carry */ + _W = _W + _W + W1; + + /* even columns have the square term in them */ + if ((ix&1) == 0) { + _W += ((mp_word)a->dp[ix>>1])*((mp_word)a->dp[ix>>1]); + } + + /* store it */ + W[ix] = _W; + + /* make next carry */ + W1 = _W >> ((mp_word)DIGIT_BIT); + } + + /* setup dest */ + olduse = b->used; + b->used = a->used+a->used; + + { + mp_digit *tmpb; + tmpb = b->dp; + for (ix = 0; ix < pa; ix++) { + *tmpb++ = W[ix] & MP_MASK; + } + + /* clear unused digits [that existed in the old copy of c] */ + for (; ix < olduse; ix++) { + *tmpb++ = 0; + } + } + mp_clamp (b); + return MP_OKAY; +} +#endif + +/* End: bn_fast_s_mp_sqr.c */ + +/* Start: bn_mp_2expt.c */ +#include <tommath.h> +#ifdef BN_MP_2EXPT_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* computes a = 2**b + * + * Simple algorithm which zeroes the int, grows it then just sets one bit + * as required. + */ +int +mp_2expt (mp_int * a, int b) +{ + int res; + + /* zero a as per default */ + mp_zero (a); + + /* grow a to accomodate the single bit */ + if ((res = mp_grow (a, b / DIGIT_BIT + 1)) != MP_OKAY) { + return res; + } + + /* set the used count of where the bit will go */ + a->used = b / DIGIT_BIT + 1; + + /* put the single bit in its place */ + a->dp[b / DIGIT_BIT] = ((mp_digit)1) << (b % DIGIT_BIT); + + return MP_OKAY; +} +#endif + +/* End: bn_mp_2expt.c */ + +/* Start: bn_mp_abs.c */ +#include <tommath.h> +#ifdef BN_MP_ABS_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* b = |a| + * + * Simple function copies the input and fixes the sign to positive + */ +int +mp_abs (mp_int * a, mp_int * b) +{ + int res; + + /* copy a to b */ + if (a != b) { + if ((res = mp_copy (a, b)) != MP_OKAY) { + return res; + } + } + + /* force the sign of b to positive */ + b->sign = MP_ZPOS; + + return MP_OKAY; +} +#endif + +/* End: bn_mp_abs.c */ + +/* Start: bn_mp_add.c */ +#include <tommath.h> +#ifdef BN_MP_ADD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* high level addition (handles signs) */ +int mp_add (mp_int * a, mp_int * b, mp_int * c) +{ + int sa, sb, res; + + /* get sign of both inputs */ + sa = a->sign; + sb = b->sign; + + /* handle two cases, not four */ + if (sa == sb) { + /* both positive or both negative */ + /* add their magnitudes, copy the sign */ + c->sign = sa; + res = s_mp_add (a, b, c); + } else { + /* one positive, the other negative */ + /* subtract the one with the greater magnitude from */ + /* the one of the lesser magnitude. The result gets */ + /* the sign of the one with the greater magnitude. */ + if (mp_cmp_mag (a, b) == MP_LT) { + c->sign = sb; + res = s_mp_sub (b, a, c); + } else { + c->sign = sa; + res = s_mp_sub (a, b, c); + } + } + return res; +} + +#endif + +/* End: bn_mp_add.c */ + +/* Start: bn_mp_add_d.c */ +#include <tommath.h> +#ifdef BN_MP_ADD_D_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* single digit addition */ +int +mp_add_d (mp_int * a, mp_digit b, mp_int * c) +{ + int res, ix, oldused; + mp_digit *tmpa, *tmpc, mu; + + /* grow c as required */ + if (c->alloc < a->used + 1) { + if ((res = mp_grow(c, a->used + 1)) != MP_OKAY) { + return res; + } + } + + /* if a is negative and |a| >= b, call c = |a| - b */ + if (a->sign == MP_NEG && (a->used > 1 || a->dp[0] >= b)) { + /* temporarily fix sign of a */ + a->sign = MP_ZPOS; + + /* c = |a| - b */ + res = mp_sub_d(a, b, c); + + /* fix sign */ + a->sign = c->sign = MP_NEG; + + return res; + } + + /* old number of used digits in c */ + oldused = c->used; + + /* sign always positive */ + c->sign = MP_ZPOS; + + /* source alias */ + tmpa = a->dp; + + /* destination alias */ + tmpc = c->dp; + + /* if a is positive */ + if (a->sign == MP_ZPOS) { + /* add digit, after this we're propagating + * the carry. + */ + *tmpc = *tmpa++ + b; + mu = *tmpc >> DIGIT_BIT; + *tmpc++ &= MP_MASK; + + /* now handle rest of the digits */ + for (ix = 1; ix < a->used; ix++) { + *tmpc = *tmpa++ + mu; + mu = *tmpc >> DIGIT_BIT; + *tmpc++ &= MP_MASK; + } + /* set final carry */ + ix++; + *tmpc++ = mu; + + /* setup size */ + c->used = a->used + 1; + } else { + /* a was negative and |a| < b */ + c->used = 1; + + /* the result is a single digit */ + if (a->used == 1) { + *tmpc++ = b - a->dp[0]; + } else { + *tmpc++ = b; + } + + /* setup count so the clearing of oldused + * can fall through correctly + */ + ix = 1; + } + + /* now zero to oldused */ + while (ix++ < oldused) { + *tmpc++ = 0; + } + mp_clamp(c); + + return MP_OKAY; +} + +#endif + +/* End: bn_mp_add_d.c */ + +/* Start: bn_mp_addmod.c */ +#include <tommath.h> +#ifdef BN_MP_ADDMOD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* d = a + b (mod c) */ +int +mp_addmod (mp_int * a, mp_int * b, mp_int * c, mp_int * d) +{ + int res; + mp_int t; + + if ((res = mp_init (&t)) != MP_OKAY) { + return res; + } + + if ((res = mp_add (a, b, &t)) != MP_OKAY) { + mp_clear (&t); + return res; + } + res = mp_mod (&t, c, d); + mp_clear (&t); + return res; +} +#endif + +/* End: bn_mp_addmod.c */ + +/* Start: bn_mp_and.c */ +#include <tommath.h> +#ifdef BN_MP_AND_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* AND two ints together */ +int +mp_and (mp_int * a, mp_int * b, mp_int * c) +{ + int res, ix, px; + mp_int t, *x; + + if (a->used > b->used) { + if ((res = mp_init_copy (&t, a)) != MP_OKAY) { + return res; + } + px = b->used; + x = b; + } else { + if ((res = mp_init_copy (&t, b)) != MP_OKAY) { + return res; + } + px = a->used; + x = a; + } + + for (ix = 0; ix < px; ix++) { + t.dp[ix] &= x->dp[ix]; + } + + /* zero digits above the last from the smallest mp_int */ + for (; ix < t.used; ix++) { + t.dp[ix] = 0; + } + + mp_clamp (&t); + mp_exch (c, &t); + mp_clear (&t); + return MP_OKAY; +} +#endif + +/* End: bn_mp_and.c */ + +/* Start: bn_mp_clamp.c */ +#include <tommath.h> +#ifdef BN_MP_CLAMP_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* trim unused digits + * + * This is used to ensure that leading zero digits are + * trimed and the leading "used" digit will be non-zero + * Typically very fast. Also fixes the sign if there + * are no more leading digits + */ +void +mp_clamp (mp_int * a) +{ + /* decrease used while the most significant digit is + * zero. + */ + while (a->used > 0 && a->dp[a->used - 1] == 0) { + --(a->used); + } + + /* reset the sign flag if used == 0 */ + if (a->used == 0) { + a->sign = MP_ZPOS; + } +} +#endif + +/* End: bn_mp_clamp.c */ + +/* Start: bn_mp_clear.c */ +#include <tommath.h> +#ifdef BN_MP_CLEAR_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* clear one (frees) */ +void +mp_clear (mp_int * a) +{ + int i; + + /* only do anything if a hasn't been freed previously */ + if (a->dp != NULL) { + /* first zero the digits */ + for (i = 0; i < a->used; i++) { + a->dp[i] = 0; + } + + /* free ram */ + XFREE(a->dp); + + /* reset members to make debugging easier */ + a->dp = NULL; + a->alloc = a->used = 0; + a->sign = MP_ZPOS; + } +} +#endif + +/* End: bn_mp_clear.c */ + +/* Start: bn_mp_clear_multi.c */ +#include <tommath.h> +#ifdef BN_MP_CLEAR_MULTI_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ +#include <stdarg.h> + +void mp_clear_multi(mp_int *mp, ...) +{ + mp_int* next_mp = mp; + va_list args; + va_start(args, mp); + while (next_mp != NULL) { + mp_clear(next_mp); + next_mp = va_arg(args, mp_int*); + } + va_end(args); +} +#endif + +/* End: bn_mp_clear_multi.c */ + +/* Start: bn_mp_cmp.c */ +#include <tommath.h> +#ifdef BN_MP_CMP_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* compare two ints (signed)*/ +int +mp_cmp (mp_int * a, mp_int * b) +{ + /* compare based on sign */ + if (a->sign != b->sign) { + if (a->sign == MP_NEG) { + return MP_LT; + } else { + return MP_GT; + } + } + + /* compare digits */ + if (a->sign == MP_NEG) { + /* if negative compare opposite direction */ + return mp_cmp_mag(b, a); + } else { + return mp_cmp_mag(a, b); + } +} +#endif + +/* End: bn_mp_cmp.c */ + +/* Start: bn_mp_cmp_d.c */ +#include <tommath.h> +#ifdef BN_MP_CMP_D_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* compare a digit */ +int mp_cmp_d(mp_int * a, mp_digit b) +{ + /* compare based on sign */ + if (a->sign == MP_NEG) { + return MP_LT; + } + + /* compare based on magnitude */ + if (a->used > 1) { + return MP_GT; + } + + /* compare the only digit of a to b */ + if (a->dp[0] > b) { + return MP_GT; + } else if (a->dp[0] < b) { + return MP_LT; + } else { + return MP_EQ; + } +} +#endif + +/* End: bn_mp_cmp_d.c */ + +/* Start: bn_mp_cmp_mag.c */ +#include <tommath.h> +#ifdef BN_MP_CMP_MAG_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* compare maginitude of two ints (unsigned) */ +int mp_cmp_mag (mp_int * a, mp_int * b) +{ + int n; + mp_digit *tmpa, *tmpb; + + /* compare based on # of non-zero digits */ + if (a->used > b->used) { + return MP_GT; + } + + if (a->used < b->used) { + return MP_LT; + } + + /* alias for a */ + tmpa = a->dp + (a->used - 1); + + /* alias for b */ + tmpb = b->dp + (a->used - 1); + + /* compare based on digits */ + for (n = 0; n < a->used; ++n, --tmpa, --tmpb) { + if (*tmpa > *tmpb) { + return MP_GT; + } + + if (*tmpa < *tmpb) { + return MP_LT; + } + } + return MP_EQ; +} +#endif + +/* End: bn_mp_cmp_mag.c */ + +/* Start: bn_mp_cnt_lsb.c */ +#include <tommath.h> +#ifdef BN_MP_CNT_LSB_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +static const int lnz[16] = { + 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0 +}; + +/* Counts the number of lsbs which are zero before the first zero bit */ +int mp_cnt_lsb(mp_int *a) +{ + int x; + mp_digit q, qq; + + /* easy out */ + if (mp_iszero(a) == 1) { + return 0; + } + + /* scan lower digits until non-zero */ + for (x = 0; x < a->used && a->dp[x] == 0; x++); + q = a->dp[x]; + x *= DIGIT_BIT; + + /* now scan this digit until a 1 is found */ + if ((q & 1) == 0) { + do { + qq = q & 15; + x += lnz[qq]; + q >>= 4; + } while (qq == 0); + } + return x; +} + +#endif + +/* End: bn_mp_cnt_lsb.c */ + +/* Start: bn_mp_copy.c */ +#include <tommath.h> +#ifdef BN_MP_COPY_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* copy, b = a */ +int +mp_copy (mp_int * a, mp_int * b) +{ + int res, n; + + /* if dst == src do nothing */ + if (a == b) { + return MP_OKAY; + } + + /* grow dest */ + if (b->alloc < a->used) { + if ((res = mp_grow (b, a->used)) != MP_OKAY) { + return res; + } + } + + /* zero b and copy the parameters over */ + { + register mp_digit *tmpa, *tmpb; + + /* pointer aliases */ + + /* source */ + tmpa = a->dp; + + /* destination */ + tmpb = b->dp; + + /* copy all the digits */ + for (n = 0; n < a->used; n++) { + *tmpb++ = *tmpa++; + } + + /* clear high digits */ + for (; n < b->used; n++) { + *tmpb++ = 0; + } + } + + /* copy used count and sign */ + b->used = a->used; + b->sign = a->sign; + return MP_OKAY; +} +#endif + +/* End: bn_mp_copy.c */ + +/* Start: bn_mp_count_bits.c */ +#include <tommath.h> +#ifdef BN_MP_COUNT_BITS_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* returns the number of bits in an int */ +int +mp_count_bits (mp_int * a) +{ + int r; + mp_digit q; + + /* shortcut */ + if (a->used == 0) { + return 0; + } + + /* get number of digits and add that */ + r = (a->used - 1) * DIGIT_BIT; + + /* take the last digit and count the bits in it */ + q = a->dp[a->used - 1]; + while (q > ((mp_digit) 0)) { + ++r; + q >>= ((mp_digit) 1); + } + return r; +} +#endif + +/* End: bn_mp_count_bits.c */ + +/* Start: bn_mp_div.c */ +#include <tommath.h> +#ifdef BN_MP_DIV_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +#ifdef BN_MP_DIV_SMALL + +/* slower bit-bang division... also smaller */ +int mp_div(mp_int * a, mp_int * b, mp_int * c, mp_int * d) +{ + mp_int ta, tb, tq, q; + int res, n, n2; + + /* is divisor zero ? */ + if (mp_iszero (b) == 1) { + return MP_VAL; + } + + /* if a < b then q=0, r = a */ + if (mp_cmp_mag (a, b) == MP_LT) { + if (d != NULL) { + res = mp_copy (a, d); + } else { + res = MP_OKAY; + } + if (c != NULL) { + mp_zero (c); + } + return res; + } + + /* init our temps */ + if ((res = mp_init_multi(&ta, &tb, &tq, &q, NULL) != MP_OKAY)) { + return res; + } + + + mp_set(&tq, 1); + n = mp_count_bits(a) - mp_count_bits(b); + if (((res = mp_abs(a, &ta)) != MP_OKAY) || + ((res = mp_abs(b, &tb)) != MP_OKAY) || + ((res = mp_mul_2d(&tb, n, &tb)) != MP_OKAY) || + ((res = mp_mul_2d(&tq, n, &tq)) != MP_OKAY)) { + goto LBL_ERR; + } + + while (n-- >= 0) { + if (mp_cmp(&tb, &ta) != MP_GT) { + if (((res = mp_sub(&ta, &tb, &ta)) != MP_OKAY) || + ((res = mp_add(&q, &tq, &q)) != MP_OKAY)) { + goto LBL_ERR; + } + } + if (((res = mp_div_2d(&tb, 1, &tb, NULL)) != MP_OKAY) || + ((res = mp_div_2d(&tq, 1, &tq, NULL)) != MP_OKAY)) { + goto LBL_ERR; + } + } + + /* now q == quotient and ta == remainder */ + n = a->sign; + n2 = (a->sign == b->sign ? MP_ZPOS : MP_NEG); + if (c != NULL) { + mp_exch(c, &q); + c->sign = (mp_iszero(c) == MP_YES) ? MP_ZPOS : n2; + } + if (d != NULL) { + mp_exch(d, &ta); + d->sign = (mp_iszero(d) == MP_YES) ? MP_ZPOS : n; + } +LBL_ERR: + mp_clear_multi(&ta, &tb, &tq, &q, NULL); + return res; +} + +#else + +/* integer signed division. + * c*b + d == a [e.g. a/b, c=quotient, d=remainder] + * HAC pp.598 Algorithm 14.20 + * + * Note that the description in HAC is horribly + * incomplete. For example, it doesn't consider + * the case where digits are removed from 'x' in + * the inner loop. It also doesn't consider the + * case that y has fewer than three digits, etc.. + * + * The overall algorithm is as described as + * 14.20 from HAC but fixed to treat these cases. +*/ +int mp_div (mp_int * a, mp_int * b, mp_int * c, mp_int * d) +{ + mp_int q, x, y, t1, t2; + int res, n, t, i, norm, neg; + + /* is divisor zero ? */ + if (mp_iszero (b) == 1) { + return MP_VAL; + } + + /* if a < b then q=0, r = a */ + if (mp_cmp_mag (a, b) == MP_LT) { + if (d != NULL) { + res = mp_copy (a, d); + } else { + res = MP_OKAY; + } + if (c != NULL) { + mp_zero (c); + } + return res; + } + + if ((res = mp_init_size (&q, a->used + 2)) != MP_OKAY) { + return res; + } + q.used = a->used + 2; + + if ((res = mp_init (&t1)) != MP_OKAY) { + goto LBL_Q; + } + + if ((res = mp_init (&t2)) != MP_OKAY) { + goto LBL_T1; + } + + if ((res = mp_init_copy (&x, a)) != MP_OKAY) { + goto LBL_T2; + } + + if ((res = mp_init_copy (&y, b)) != MP_OKAY) { + goto LBL_X; + } + + /* fix the sign */ + neg = (a->sign == b->sign) ? MP_ZPOS : MP_NEG; + x.sign = y.sign = MP_ZPOS; + + /* normalize both x and y, ensure that y >= b/2, [b == 2**DIGIT_BIT] */ + norm = mp_count_bits(&y) % DIGIT_BIT; + if (norm < (int)(DIGIT_BIT-1)) { + norm = (DIGIT_BIT-1) - norm; + if ((res = mp_mul_2d (&x, norm, &x)) != MP_OKAY) { + goto LBL_Y; + } + if ((res = mp_mul_2d (&y, norm, &y)) != MP_OKAY) { + goto LBL_Y; + } + } else { + norm = 0; + } + + /* note hac does 0 based, so if used==5 then its 0,1,2,3,4, e.g. use 4 */ + n = x.used - 1; + t = y.used - 1; + + /* while (x >= y*b**n-t) do { q[n-t] += 1; x -= y*b**{n-t} } */ + if ((res = mp_lshd (&y, n - t)) != MP_OKAY) { /* y = y*b**{n-t} */ + goto LBL_Y; + } + + while (mp_cmp (&x, &y) != MP_LT) { + ++(q.dp[n - t]); + if ((res = mp_sub (&x, &y, &x)) != MP_OKAY) { + goto LBL_Y; + } + } + + /* reset y by shifting it back down */ + mp_rshd (&y, n - t); + + /* step 3. for i from n down to (t + 1) */ + for (i = n; i >= (t + 1); i--) { + if (i > x.used) { + continue; + } + + /* step 3.1 if xi == yt then set q{i-t-1} to b-1, + * otherwise set q{i-t-1} to (xi*b + x{i-1})/yt */ + if (x.dp[i] == y.dp[t]) { + q.dp[i - t - 1] = ((((mp_digit)1) << DIGIT_BIT) - 1); + } else { + mp_word tmp; + tmp = ((mp_word) x.dp[i]) << ((mp_word) DIGIT_BIT); + tmp |= ((mp_word) x.dp[i - 1]); + tmp /= ((mp_word) y.dp[t]); + if (tmp > (mp_word) MP_MASK) + tmp = MP_MASK; + q.dp[i - t - 1] = (mp_digit) (tmp & (mp_word) (MP_MASK)); + } + + /* while (q{i-t-1} * (yt * b + y{t-1})) > + xi * b**2 + xi-1 * b + xi-2 + + do q{i-t-1} -= 1; + */ + q.dp[i - t - 1] = (q.dp[i - t - 1] + 1) & MP_MASK; + do { + q.dp[i - t - 1] = (q.dp[i - t - 1] - 1) & MP_MASK; + + /* find left hand */ + mp_zero (&t1); + t1.dp[0] = (t - 1 < 0) ? 0 : y.dp[t - 1]; + t1.dp[1] = y.dp[t]; + t1.used = 2; + if ((res = mp_mul_d (&t1, q.dp[i - t - 1], &t1)) != MP_OKAY) { + goto LBL_Y; + } + + /* find right hand */ + t2.dp[0] = (i - 2 < 0) ? 0 : x.dp[i - 2]; + t2.dp[1] = (i - 1 < 0) ? 0 : x.dp[i - 1]; + t2.dp[2] = x.dp[i]; + t2.used = 3; + } while (mp_cmp_mag(&t1, &t2) == MP_GT); + + /* step 3.3 x = x - q{i-t-1} * y * b**{i-t-1} */ + if ((res = mp_mul_d (&y, q.dp[i - t - 1], &t1)) != MP_OKAY) { + goto LBL_Y; + } + + if ((res = mp_lshd (&t1, i - t - 1)) != MP_OKAY) { + goto LBL_Y; + } + + if ((res = mp_sub (&x, &t1, &x)) != MP_OKAY) { + goto LBL_Y; + } + + /* if x < 0 then { x = x + y*b**{i-t-1}; q{i-t-1} -= 1; } */ + if (x.sign == MP_NEG) { + if ((res = mp_copy (&y, &t1)) != MP_OKAY) { + goto LBL_Y; + } + if ((res = mp_lshd (&t1, i - t - 1)) != MP_OKAY) { + goto LBL_Y; + } + if ((res = mp_add (&x, &t1, &x)) != MP_OKAY) { + goto LBL_Y; + } + + q.dp[i - t - 1] = (q.dp[i - t - 1] - 1UL) & MP_MASK; + } + } + + /* now q is the quotient and x is the remainder + * [which we have to normalize] + */ + + /* get sign before writing to c */ + x.sign = x.used == 0 ? MP_ZPOS : a->sign; + + if (c != NULL) { + mp_clamp (&q); + mp_exch (&q, c); + c->sign = neg; + } + + if (d != NULL) { + mp_div_2d (&x, norm, &x, NULL); + mp_exch (&x, d); + } + + res = MP_OKAY; + +LBL_Y:mp_clear (&y); +LBL_X:mp_clear (&x); +LBL_T2:mp_clear (&t2); +LBL_T1:mp_clear (&t1); +LBL_Q:mp_clear (&q); + return res; +} + +#endif + +#endif + +/* End: bn_mp_div.c */ + +/* Start: bn_mp_div_2.c */ +#include <tommath.h> +#ifdef BN_MP_DIV_2_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* b = a/2 */ +int mp_div_2(mp_int * a, mp_int * b) +{ + int x, res, oldused; + + /* copy */ + if (b->alloc < a->used) { + if ((res = mp_grow (b, a->used)) != MP_OKAY) { + return res; + } + } + + oldused = b->used; + b->used = a->used; + { + register mp_digit r, rr, *tmpa, *tmpb; + + /* source alias */ + tmpa = a->dp + b->used - 1; + + /* dest alias */ + tmpb = b->dp + b->used - 1; + + /* carry */ + r = 0; + for (x = b->used - 1; x >= 0; x--) { + /* get the carry for the next iteration */ + rr = *tmpa & 1; + + /* shift the current digit, add in carry and store */ + *tmpb-- = (*tmpa-- >> 1) | (r << (DIGIT_BIT - 1)); + + /* forward carry to next iteration */ + r = rr; + } + + /* zero excess digits */ + tmpb = b->dp + b->used; + for (x = b->used; x < oldused; x++) { + *tmpb++ = 0; + } + } + b->sign = a->sign; + mp_clamp (b); + return MP_OKAY; +} +#endif + +/* End: bn_mp_div_2.c */ + +/* Start: bn_mp_div_2d.c */ +#include <tommath.h> +#ifdef BN_MP_DIV_2D_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* shift right by a certain bit count (store quotient in c, optional remainder in d) */ +int mp_div_2d (mp_int * a, int b, mp_int * c, mp_int * d) +{ + mp_digit D, r, rr; + int x, res; + mp_int t; + + + /* if the shift count is <= 0 then we do no work */ + if (b <= 0) { + res = mp_copy (a, c); + if (d != NULL) { + mp_zero (d); + } + return res; + } + + if ((res = mp_init (&t)) != MP_OKAY) { + return res; + } + + /* get the remainder */ + if (d != NULL) { + if ((res = mp_mod_2d (a, b, &t)) != MP_OKAY) { + mp_clear (&t); + return res; + } + } + + /* copy */ + if ((res = mp_copy (a, c)) != MP_OKAY) { + mp_clear (&t); + return res; + } + + /* shift by as many digits in the bit count */ + if (b >= (int)DIGIT_BIT) { + mp_rshd (c, b / DIGIT_BIT); + } + + /* shift any bit count < DIGIT_BIT */ + D = (mp_digit) (b % DIGIT_BIT); + if (D != 0) { + register mp_digit *tmpc, mask, shift; + + /* mask */ + mask = (((mp_digit)1) << D) - 1; + + /* shift for lsb */ + shift = DIGIT_BIT - D; + + /* alias */ + tmpc = c->dp + (c->used - 1); + + /* carry */ + r = 0; + for (x = c->used - 1; x >= 0; x--) { + /* get the lower bits of this word in a temp */ + rr = *tmpc & mask; + + /* shift the current word and mix in the carry bits from the previous word */ + *tmpc = (*tmpc >> D) | (r << shift); + --tmpc; + + /* set the carry to the carry bits of the current word found above */ + r = rr; + } + } + mp_clamp (c); + if (d != NULL) { + mp_exch (&t, d); + } + mp_clear (&t); + return MP_OKAY; +} +#endif + +/* End: bn_mp_div_2d.c */ + +/* Start: bn_mp_div_3.c */ +#include <tommath.h> +#ifdef BN_MP_DIV_3_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* divide by three (based on routine from MPI and the GMP manual) */ +int +mp_div_3 (mp_int * a, mp_int *c, mp_digit * d) +{ + mp_int q; + mp_word w, t; + mp_digit b; + int res, ix; + + /* b = 2**DIGIT_BIT / 3 */ + b = (((mp_word)1) << ((mp_word)DIGIT_BIT)) / ((mp_word)3); + + if ((res = mp_init_size(&q, a->used)) != MP_OKAY) { + return res; + } + + q.used = a->used; + q.sign = a->sign; + w = 0; + for (ix = a->used - 1; ix >= 0; ix--) { + w = (w << ((mp_word)DIGIT_BIT)) | ((mp_word)a->dp[ix]); + + if (w >= 3) { + /* multiply w by [1/3] */ + t = (w * ((mp_word)b)) >> ((mp_word)DIGIT_BIT); + + /* now subtract 3 * [w/3] from w, to get the remainder */ + w -= t+t+t; + + /* fixup the remainder as required since + * the optimization is not exact. + */ + while (w >= 3) { + t += 1; + w -= 3; + } + } else { + t = 0; + } + q.dp[ix] = (mp_digit)t; + } + + /* [optional] store the remainder */ + if (d != NULL) { + *d = (mp_digit)w; + } + + /* [optional] store the quotient */ + if (c != NULL) { + mp_clamp(&q); + mp_exch(&q, c); + } + mp_clear(&q); + + return res; +} + +#endif + +/* End: bn_mp_div_3.c */ + +/* Start: bn_mp_div_d.c */ +#include <tommath.h> +#ifdef BN_MP_DIV_D_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +static int s_is_power_of_two(mp_digit b, int *p) +{ + int x; + + for (x = 1; x < DIGIT_BIT; x++) { + if (b == (((mp_digit)1)<<x)) { + *p = x; + return 1; + } + } + return 0; +} + +/* single digit division (based on routine from MPI) */ +int mp_div_d (mp_int * a, mp_digit b, mp_int * c, mp_digit * d) +{ + mp_int q; + mp_word w; + mp_digit t; + int res, ix; + + /* cannot divide by zero */ + if (b == 0) { + return MP_VAL; + } + + /* quick outs */ + if (b == 1 || mp_iszero(a) == 1) { + if (d != NULL) { + *d = 0; + } + if (c != NULL) { + return mp_copy(a, c); + } + return MP_OKAY; + } + + /* power of two ? */ + if (s_is_power_of_two(b, &ix) == 1) { + if (d != NULL) { + *d = a->dp[0] & ((((mp_digit)1)<<ix) - 1); + } + if (c != NULL) { + return mp_div_2d(a, ix, c, NULL); + } + return MP_OKAY; + } + +#ifdef BN_MP_DIV_3_C + /* three? */ + if (b == 3) { + return mp_div_3(a, c, d); + } +#endif + + /* no easy answer [c'est la vie]. Just division */ + if ((res = mp_init_size(&q, a->used)) != MP_OKAY) { + return res; + } + + q.used = a->used; + q.sign = a->sign; + w = 0; + for (ix = a->used - 1; ix >= 0; ix--) { + w = (w << ((mp_word)DIGIT_BIT)) | ((mp_word)a->dp[ix]); + + if (w >= b) { + t = (mp_digit)(w / b); + w -= ((mp_word)t) * ((mp_word)b); + } else { + t = 0; + } + q.dp[ix] = (mp_digit)t; + } + + if (d != NULL) { + *d = (mp_digit)w; + } + + if (c != NULL) { + mp_clamp(&q); + mp_exch(&q, c); + } + mp_clear(&q); + + return res; +} + +#endif + +/* End: bn_mp_div_d.c */ + +/* Start: bn_mp_dr_is_modulus.c */ +#include <tommath.h> +#ifdef BN_MP_DR_IS_MODULUS_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* determines if a number is a valid DR modulus */ +int mp_dr_is_modulus(mp_int *a) +{ + int ix; + + /* must be at least two digits */ + if (a->used < 2) { + return 0; + } + + /* must be of the form b**k - a [a <= b] so all + * but the first digit must be equal to -1 (mod b). + */ + for (ix = 1; ix < a->used; ix++) { + if (a->dp[ix] != MP_MASK) { + return 0; + } + } + return 1; +} + +#endif + +/* End: bn_mp_dr_is_modulus.c */ + +/* Start: bn_mp_dr_reduce.c */ +#include <tommath.h> +#ifdef BN_MP_DR_REDUCE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* reduce "x" in place modulo "n" using the Diminished Radix algorithm. + * + * Based on algorithm from the paper + * + * "Generating Efficient Primes for Discrete Log Cryptosystems" + * Chae Hoon Lim, Pil Joong Lee, + * POSTECH Information Research Laboratories + * + * The modulus must be of a special format [see manual] + * + * Has been modified to use algorithm 7.10 from the LTM book instead + * + * Input x must be in the range 0 <= x <= (n-1)**2 + */ +int +mp_dr_reduce (mp_int * x, mp_int * n, mp_digit k) +{ + int err, i, m; + mp_word r; + mp_digit mu, *tmpx1, *tmpx2; + + /* m = digits in modulus */ + m = n->used; + + /* ensure that "x" has at least 2m digits */ + if (x->alloc < m + m) { + if ((err = mp_grow (x, m + m)) != MP_OKAY) { + return err; + } + } + +/* top of loop, this is where the code resumes if + * another reduction pass is required. + */ +top: + /* aliases for digits */ + /* alias for lower half of x */ + tmpx1 = x->dp; + + /* alias for upper half of x, or x/B**m */ + tmpx2 = x->dp + m; + + /* set carry to zero */ + mu = 0; + + /* compute (x mod B**m) + k * [x/B**m] inline and inplace */ + for (i = 0; i < m; i++) { + r = ((mp_word)*tmpx2++) * ((mp_word)k) + *tmpx1 + mu; + *tmpx1++ = (mp_digit)(r & MP_MASK); + mu = (mp_digit)(r >> ((mp_word)DIGIT_BIT)); + } + + /* set final carry */ + *tmpx1++ = mu; + + /* zero words above m */ + for (i = m + 1; i < x->used; i++) { + *tmpx1++ = 0; + } + + /* clamp, sub and return */ + mp_clamp (x); + + /* if x >= n then subtract and reduce again + * Each successive "recursion" makes the input smaller and smaller. + */ + if (mp_cmp_mag (x, n) != MP_LT) { + s_mp_sub(x, n, x); + goto top; + } + return MP_OKAY; +} +#endif + +/* End: bn_mp_dr_reduce.c */ + +/* Start: bn_mp_dr_setup.c */ +#include <tommath.h> +#ifdef BN_MP_DR_SETUP_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* determines the setup value */ +void mp_dr_setup(mp_int *a, mp_digit *d) +{ + /* the casts are required if DIGIT_BIT is one less than + * the number of bits in a mp_digit [e.g. DIGIT_BIT==31] + */ + *d = (mp_digit)((((mp_word)1) << ((mp_word)DIGIT_BIT)) - + ((mp_word)a->dp[0])); +} + +#endif + +/* End: bn_mp_dr_setup.c */ + +/* Start: bn_mp_exch.c */ +#include <tommath.h> +#ifdef BN_MP_EXCH_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* swap the elements of two integers, for cases where you can't simply swap the + * mp_int pointers around + */ +void +mp_exch (mp_int * a, mp_int * b) +{ + mp_int t; + + t = *a; + *a = *b; + *b = t; +} +#endif + +/* End: bn_mp_exch.c */ + +/* Start: bn_mp_expt_d.c */ +#include <tommath.h> +#ifdef BN_MP_EXPT_D_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* calculate c = a**b using a square-multiply algorithm */ +int mp_expt_d (mp_int * a, mp_digit b, mp_int * c) +{ + int res, x; + mp_int g; + + if ((res = mp_init_copy (&g, a)) != MP_OKAY) { + return res; + } + + /* set initial result */ + mp_set (c, 1); + + for (x = 0; x < (int) DIGIT_BIT; x++) { + /* square */ + if ((res = mp_sqr (c, c)) != MP_OKAY) { + mp_clear (&g); + return res; + } + + /* if the bit is set multiply */ + if ((b & (mp_digit) (((mp_digit)1) << (DIGIT_BIT - 1))) != 0) { + if ((res = mp_mul (c, &g, c)) != MP_OKAY) { + mp_clear (&g); + return res; + } + } + + /* shift to next bit */ + b <<= 1; + } + + mp_clear (&g); + return MP_OKAY; +} +#endif + +/* End: bn_mp_expt_d.c */ + +/* Start: bn_mp_exptmod.c */ +#include <tommath.h> +#ifdef BN_MP_EXPTMOD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + + +/* this is a shell function that calls either the normal or Montgomery + * exptmod functions. Originally the call to the montgomery code was + * embedded in the normal function but that wasted alot of stack space + * for nothing (since 99% of the time the Montgomery code would be called) + */ +int mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y) +{ + int dr; + + /* modulus P must be positive */ + if (P->sign == MP_NEG) { + return MP_VAL; + } + + /* if exponent X is negative we have to recurse */ + if (X->sign == MP_NEG) { +#ifdef BN_MP_INVMOD_C + mp_int tmpG, tmpX; + int err; + + /* first compute 1/G mod P */ + if ((err = mp_init(&tmpG)) != MP_OKAY) { + return err; + } + if ((err = mp_invmod(G, P, &tmpG)) != MP_OKAY) { + mp_clear(&tmpG); + return err; + } + + /* now get |X| */ + if ((err = mp_init(&tmpX)) != MP_OKAY) { + mp_clear(&tmpG); + return err; + } + if ((err = mp_abs(X, &tmpX)) != MP_OKAY) { + mp_clear_multi(&tmpG, &tmpX, NULL); + return err; + } + + /* and now compute (1/G)**|X| instead of G**X [X < 0] */ + err = mp_exptmod(&tmpG, &tmpX, P, Y); + mp_clear_multi(&tmpG, &tmpX, NULL); + return err; +#else + /* no invmod */ + return MP_VAL; +#endif + } + +#ifdef BN_MP_DR_IS_MODULUS_C + /* is it a DR modulus? */ + dr = mp_dr_is_modulus(P); +#else + dr = 0; +#endif + +#ifdef BN_MP_REDUCE_IS_2K_C + /* if not, is it a uDR modulus? */ + if (dr == 0) { + dr = mp_reduce_is_2k(P) << 1; + } +#endif + + /* if the modulus is odd or dr != 0 use the fast method */ +#ifdef BN_MP_EXPTMOD_FAST_C + if (mp_isodd (P) == 1 || dr != 0) { + return mp_exptmod_fast (G, X, P, Y, dr); + } else { +#endif +#ifdef BN_S_MP_EXPTMOD_C + /* otherwise use the generic Barrett reduction technique */ + return s_mp_exptmod (G, X, P, Y); +#else + /* no exptmod for evens */ + return MP_VAL; +#endif +#ifdef BN_MP_EXPTMOD_FAST_C + } +#endif +} + +#endif + +/* End: bn_mp_exptmod.c */ + +/* Start: bn_mp_exptmod_fast.c */ +#include <tommath.h> +#ifdef BN_MP_EXPTMOD_FAST_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* computes Y == G**X mod P, HAC pp.616, Algorithm 14.85 + * + * Uses a left-to-right k-ary sliding window to compute the modular exponentiation. + * The value of k changes based on the size of the exponent. + * + * Uses Montgomery or Diminished Radix reduction [whichever appropriate] + */ + +#ifdef MP_LOW_MEM + #define TAB_SIZE 32 +#else + #define TAB_SIZE 256 +#endif + +int +mp_exptmod_fast (mp_int * G, mp_int * X, mp_int * P, mp_int * Y, int redmode) +{ + mp_int M[TAB_SIZE], res; + mp_digit buf, mp; + int err, bitbuf, bitcpy, bitcnt, mode, digidx, x, y, winsize; + + /* use a pointer to the reduction algorithm. This allows us to use + * one of many reduction algorithms without modding the guts of + * the code with if statements everywhere. + */ + int (*redux)(mp_int*,mp_int*,mp_digit); + + /* find window size */ + x = mp_count_bits (X); + if (x <= 7) { + winsize = 2; + } else if (x <= 36) { + winsize = 3; + } else if (x <= 140) { + winsize = 4; + } else if (x <= 450) { + winsize = 5; + } else if (x <= 1303) { + winsize = 6; + } else if (x <= 3529) { + winsize = 7; + } else { + winsize = 8; + } + +#ifdef MP_LOW_MEM + if (winsize > 5) { + winsize = 5; + } +#endif + + /* init M array */ + /* init first cell */ + if ((err = mp_init(&M[1])) != MP_OKAY) { + return err; + } + + /* now init the second half of the array */ + for (x = 1<<(winsize-1); x < (1 << winsize); x++) { + if ((err = mp_init(&M[x])) != MP_OKAY) { + for (y = 1<<(winsize-1); y < x; y++) { + mp_clear (&M[y]); + } + mp_clear(&M[1]); + return err; + } + } + + /* determine and setup reduction code */ + if (redmode == 0) { +#ifdef BN_MP_MONTGOMERY_SETUP_C + /* now setup montgomery */ + if ((err = mp_montgomery_setup (P, &mp)) != MP_OKAY) { + goto LBL_M; + } +#else + err = MP_VAL; + goto LBL_M; +#endif + + /* automatically pick the comba one if available (saves quite a few calls/ifs) */ +#ifdef BN_FAST_MP_MONTGOMERY_REDUCE_C + if (((P->used * 2 + 1) < MP_WARRAY) && + P->used < (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) { + redux = fast_mp_montgomery_reduce; + } else +#endif + { +#ifdef BN_MP_MONTGOMERY_REDUCE_C + /* use slower baseline Montgomery method */ + redux = mp_montgomery_reduce; +#else + err = MP_VAL; + goto LBL_M; +#endif + } + } else if (redmode == 1) { +#if defined(BN_MP_DR_SETUP_C) && defined(BN_MP_DR_REDUCE_C) + /* setup DR reduction for moduli of the form B**k - b */ + mp_dr_setup(P, &mp); + redux = mp_dr_reduce; +#else + err = MP_VAL; + goto LBL_M; +#endif + } else { +#if defined(BN_MP_REDUCE_2K_SETUP_C) && defined(BN_MP_REDUCE_2K_C) + /* setup DR reduction for moduli of the form 2**k - b */ + if ((err = mp_reduce_2k_setup(P, &mp)) != MP_OKAY) { + goto LBL_M; + } + redux = mp_reduce_2k; +#else + err = MP_VAL; + goto LBL_M; +#endif + } + + /* setup result */ + if ((err = mp_init (&res)) != MP_OKAY) { + goto LBL_M; + } + + /* create M table + * + + * + * The first half of the table is not computed though accept for M[0] and M[1] + */ + + if (redmode == 0) { +#ifdef BN_MP_MONTGOMERY_CALC_NORMALIZATION_C + /* now we need R mod m */ + if ((err = mp_montgomery_calc_normalization (&res, P)) != MP_OKAY) { + goto LBL_RES; + } +#else + err = MP_VAL; + goto LBL_RES; +#endif + + /* now set M[1] to G * R mod m */ + if ((err = mp_mulmod (G, &res, P, &M[1])) != MP_OKAY) { + goto LBL_RES; + } + } else { + mp_set(&res, 1); + if ((err = mp_mod(G, P, &M[1])) != MP_OKAY) { + goto LBL_RES; + } + } + + /* compute the value at M[1<<(winsize-1)] by squaring M[1] (winsize-1) times */ + if ((err = mp_copy (&M[1], &M[1 << (winsize - 1)])) != MP_OKAY) { + goto LBL_RES; + } + + for (x = 0; x < (winsize - 1); x++) { + if ((err = mp_sqr (&M[1 << (winsize - 1)], &M[1 << (winsize - 1)])) != MP_OKAY) { + goto LBL_RES; + } + if ((err = redux (&M[1 << (winsize - 1)], P, mp)) != MP_OKAY) { + goto LBL_RES; + } + } + + /* create upper table */ + for (x = (1 << (winsize - 1)) + 1; x < (1 << winsize); x++) { + if ((err = mp_mul (&M[x - 1], &M[1], &M[x])) != MP_OKAY) { + goto LBL_RES; + } + if ((err = redux (&M[x], P, mp)) != MP_OKAY) { + goto LBL_RES; + } + } + + /* set initial mode and bit cnt */ + mode = 0; + bitcnt = 1; + buf = 0; + digidx = X->used - 1; + bitcpy = 0; + bitbuf = 0; + + for (;;) { + /* grab next digit as required */ + if (--bitcnt == 0) { + /* if digidx == -1 we are out of digits so break */ + if (digidx == -1) { + break; + } + /* read next digit and reset bitcnt */ + buf = X->dp[digidx--]; + bitcnt = (int)DIGIT_BIT; + } + + /* grab the next msb from the exponent */ + y = (mp_digit)(buf >> (DIGIT_BIT - 1)) & 1; + buf <<= (mp_digit)1; + + /* if the bit is zero and mode == 0 then we ignore it + * These represent the leading zero bits before the first 1 bit + * in the exponent. Technically this opt is not required but it + * does lower the # of trivial squaring/reductions used + */ + if (mode == 0 && y == 0) { + continue; + } + + /* if the bit is zero and mode == 1 then we square */ + if (mode == 1 && y == 0) { + if ((err = mp_sqr (&res, &res)) != MP_OKAY) { + goto LBL_RES; + } + if ((err = redux (&res, P, mp)) != MP_OKAY) { + goto LBL_RES; + } + continue; + } + + /* else we add it to the window */ + bitbuf |= (y << (winsize - ++bitcpy)); + mode = 2; + + if (bitcpy == winsize) { + /* ok window is filled so square as required and multiply */ + /* square first */ + for (x = 0; x < winsize; x++) { + if ((err = mp_sqr (&res, &res)) != MP_OKAY) { + goto LBL_RES; + } + if ((err = redux (&res, P, mp)) != MP_OKAY) { + goto LBL_RES; + } + } + + /* then multiply */ + if ((err = mp_mul (&res, &M[bitbuf], &res)) != MP_OKAY) { + goto LBL_RES; + } + if ((err = redux (&res, P, mp)) != MP_OKAY) { + goto LBL_RES; + } + + /* empty window and reset */ + bitcpy = 0; + bitbuf = 0; + mode = 1; + } + } + + /* if bits remain then square/multiply */ + if (mode == 2 && bitcpy > 0) { + /* square then multiply if the bit is set */ + for (x = 0; x < bitcpy; x++) { + if ((err = mp_sqr (&res, &res)) != MP_OKAY) { + goto LBL_RES; + } + if ((err = redux (&res, P, mp)) != MP_OKAY) { + goto LBL_RES; + } + + /* get next bit of the window */ + bitbuf <<= 1; + if ((bitbuf & (1 << winsize)) != 0) { + /* then multiply */ + if ((err = mp_mul (&res, &M[1], &res)) != MP_OKAY) { + goto LBL_RES; + } + if ((err = redux (&res, P, mp)) != MP_OKAY) { + goto LBL_RES; + } + } + } + } + + if (redmode == 0) { + /* fixup result if Montgomery reduction is used + * recall that any value in a Montgomery system is + * actually multiplied by R mod n. So we have + * to reduce one more time to cancel out the factor + * of R. + */ + if ((err = redux(&res, P, mp)) != MP_OKAY) { + goto LBL_RES; + } + } + + /* swap res with Y */ + mp_exch (&res, Y); + err = MP_OKAY; +LBL_RES:mp_clear (&res); +LBL_M: + mp_clear(&M[1]); + for (x = 1<<(winsize-1); x < (1 << winsize); x++) { + mp_clear (&M[x]); + } + return err; +} +#endif + + +/* End: bn_mp_exptmod_fast.c */ + +/* Start: bn_mp_exteuclid.c */ +#include <tommath.h> +#ifdef BN_MP_EXTEUCLID_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* Extended euclidean algorithm of (a, b) produces + a*u1 + b*u2 = u3 + */ +int mp_exteuclid(mp_int *a, mp_int *b, mp_int *U1, mp_int *U2, mp_int *U3) +{ + mp_int u1,u2,u3,v1,v2,v3,t1,t2,t3,q,tmp; + int err; + + if ((err = mp_init_multi(&u1, &u2, &u3, &v1, &v2, &v3, &t1, &t2, &t3, &q, &tmp, NULL)) != MP_OKAY) { + return err; + } + + /* initialize, (u1,u2,u3) = (1,0,a) */ + mp_set(&u1, 1); + if ((err = mp_copy(a, &u3)) != MP_OKAY) { goto _ERR; } + + /* initialize, (v1,v2,v3) = (0,1,b) */ + mp_set(&v2, 1); + if ((err = mp_copy(b, &v3)) != MP_OKAY) { goto _ERR; } + + /* loop while v3 != 0 */ + while (mp_iszero(&v3) == MP_NO) { + /* q = u3/v3 */ + if ((err = mp_div(&u3, &v3, &q, NULL)) != MP_OKAY) { goto _ERR; } + + /* (t1,t2,t3) = (u1,u2,u3) - (v1,v2,v3)q */ + if ((err = mp_mul(&v1, &q, &tmp)) != MP_OKAY) { goto _ERR; } + if ((err = mp_sub(&u1, &tmp, &t1)) != MP_OKAY) { goto _ERR; } + if ((err = mp_mul(&v2, &q, &tmp)) != MP_OKAY) { goto _ERR; } + if ((err = mp_sub(&u2, &tmp, &t2)) != MP_OKAY) { goto _ERR; } + if ((err = mp_mul(&v3, &q, &tmp)) != MP_OKAY) { goto _ERR; } + if ((err = mp_sub(&u3, &tmp, &t3)) != MP_OKAY) { goto _ERR; } + + /* (u1,u2,u3) = (v1,v2,v3) */ + if ((err = mp_copy(&v1, &u1)) != MP_OKAY) { goto _ERR; } + if ((err = mp_copy(&v2, &u2)) != MP_OKAY) { goto _ERR; } + if ((err = mp_copy(&v3, &u3)) != MP_OKAY) { goto _ERR; } + + /* (v1,v2,v3) = (t1,t2,t3) */ + if ((err = mp_copy(&t1, &v1)) != MP_OKAY) { goto _ERR; } + if ((err = mp_copy(&t2, &v2)) != MP_OKAY) { goto _ERR; } + if ((err = mp_copy(&t3, &v3)) != MP_OKAY) { goto _ERR; } + } + + /* copy result out */ + if (U1 != NULL) { mp_exch(U1, &u1); } + if (U2 != NULL) { mp_exch(U2, &u2); } + if (U3 != NULL) { mp_exch(U3, &u3); } + + err = MP_OKAY; +_ERR: mp_clear_multi(&u1, &u2, &u3, &v1, &v2, &v3, &t1, &t2, &t3, &q, &tmp, NULL); + return err; +} +#endif + +/* End: bn_mp_exteuclid.c */ + +/* Start: bn_mp_fread.c */ +#include <tommath.h> +#ifdef BN_MP_FREAD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* read a bigint from a file stream in ASCII */ +int mp_fread(mp_int *a, int radix, FILE *stream) +{ + int err, ch, neg, y; + + /* clear a */ + mp_zero(a); + + /* if first digit is - then set negative */ + ch = fgetc(stream); + if (ch == '-') { + neg = MP_NEG; + ch = fgetc(stream); + } else { + neg = MP_ZPOS; + } + + for (;;) { + /* find y in the radix map */ + for (y = 0; y < radix; y++) { + if (mp_s_rmap[y] == ch) { + break; + } + } + if (y == radix) { + break; + } + + /* shift up and add */ + if ((err = mp_mul_d(a, radix, a)) != MP_OKAY) { + return err; + } + if ((err = mp_add_d(a, y, a)) != MP_OKAY) { + return err; + } + + ch = fgetc(stream); + } + if (mp_cmp_d(a, 0) != MP_EQ) { + a->sign = neg; + } + + return MP_OKAY; +} + +#endif + +/* End: bn_mp_fread.c */ + +/* Start: bn_mp_fwrite.c */ +#include <tommath.h> +#ifdef BN_MP_FWRITE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +int mp_fwrite(mp_int *a, int radix, FILE *stream) +{ + char *buf; + int err, len, x; + + if ((err = mp_radix_size(a, radix, &len)) != MP_OKAY) { + return err; + } + + buf = OPT_CAST(char) XMALLOC (len); + if (buf == NULL) { + return MP_MEM; + } + + if ((err = mp_toradix(a, buf, radix)) != MP_OKAY) { + XFREE (buf); + return err; + } + + for (x = 0; x < len; x++) { + if (fputc(buf[x], stream) == EOF) { + XFREE (buf); + return MP_VAL; + } + } + + XFREE (buf); + return MP_OKAY; +} + +#endif + +/* End: bn_mp_fwrite.c */ + +/* Start: bn_mp_gcd.c */ +#include <tommath.h> +#ifdef BN_MP_GCD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* Greatest Common Divisor using the binary method */ +int mp_gcd (mp_int * a, mp_int * b, mp_int * c) +{ + mp_int u, v; + int k, u_lsb, v_lsb, res; + + /* either zero than gcd is the largest */ + if (mp_iszero (a) == 1 && mp_iszero (b) == 0) { + return mp_abs (b, c); + } + if (mp_iszero (a) == 0 && mp_iszero (b) == 1) { + return mp_abs (a, c); + } + + /* optimized. At this point if a == 0 then + * b must equal zero too + */ + if (mp_iszero (a) == 1) { + mp_zero(c); + return MP_OKAY; + } + + /* get copies of a and b we can modify */ + if ((res = mp_init_copy (&u, a)) != MP_OKAY) { + return res; + } + + if ((res = mp_init_copy (&v, b)) != MP_OKAY) { + goto LBL_U; + } + + /* must be positive for the remainder of the algorithm */ + u.sign = v.sign = MP_ZPOS; + + /* B1. Find the common power of two for u and v */ + u_lsb = mp_cnt_lsb(&u); + v_lsb = mp_cnt_lsb(&v); + k = MIN(u_lsb, v_lsb); + + if (k > 0) { + /* divide the power of two out */ + if ((res = mp_div_2d(&u, k, &u, NULL)) != MP_OKAY) { + goto LBL_V; + } + + if ((res = mp_div_2d(&v, k, &v, NULL)) != MP_OKAY) { + goto LBL_V; + } + } + + /* divide any remaining factors of two out */ + if (u_lsb != k) { + if ((res = mp_div_2d(&u, u_lsb - k, &u, NULL)) != MP_OKAY) { + goto LBL_V; + } + } + + if (v_lsb != k) { + if ((res = mp_div_2d(&v, v_lsb - k, &v, NULL)) != MP_OKAY) { + goto LBL_V; + } + } + + while (mp_iszero(&v) == 0) { + /* make sure v is the largest */ + if (mp_cmp_mag(&u, &v) == MP_GT) { + /* swap u and v to make sure v is >= u */ + mp_exch(&u, &v); + } + + /* subtract smallest from largest */ + if ((res = s_mp_sub(&v, &u, &v)) != MP_OKAY) { + goto LBL_V; + } + + /* Divide out all factors of two */ + if ((res = mp_div_2d(&v, mp_cnt_lsb(&v), &v, NULL)) != MP_OKAY) { + goto LBL_V; + } + } + + /* multiply by 2**k which we divided out at the beginning */ + if ((res = mp_mul_2d (&u, k, c)) != MP_OKAY) { + goto LBL_V; + } + c->sign = MP_ZPOS; + res = MP_OKAY; +LBL_V:mp_clear (&u); +LBL_U:mp_clear (&v); + return res; +} +#endif + +/* End: bn_mp_gcd.c */ + +/* Start: bn_mp_get_int.c */ +#include <tommath.h> +#ifdef BN_MP_GET_INT_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* get the lower 32-bits of an mp_int */ +unsigned long mp_get_int(mp_int * a) +{ + int i; + unsigned long res; + + if (a->used == 0) { + return 0; + } + + /* get number of digits of the lsb we have to read */ + i = MIN(a->used,(int)((sizeof(unsigned long)*CHAR_BIT+DIGIT_BIT-1)/DIGIT_BIT))-1; + + /* get most significant digit of result */ + res = DIGIT(a,i); + + while (--i >= 0) { + res = (res << DIGIT_BIT) | DIGIT(a,i); + } + + /* force result to 32-bits always so it is consistent on non 32-bit platforms */ + return res & 0xFFFFFFFFUL; +} +#endif + +/* End: bn_mp_get_int.c */ + +/* Start: bn_mp_grow.c */ +#include <tommath.h> +#ifdef BN_MP_GROW_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* grow as required */ +int mp_grow (mp_int * a, int size) +{ + int i; + mp_digit *tmp; + + /* if the alloc size is smaller alloc more ram */ + if (a->alloc < size) { + /* ensure there are always at least MP_PREC digits extra on top */ + size += (MP_PREC * 2) - (size % MP_PREC); + + /* reallocate the array a->dp + * + * We store the return in a temporary variable + * in case the operation failed we don't want + * to overwrite the dp member of a. + */ + tmp = OPT_CAST(mp_digit) XREALLOC (a->dp, sizeof (mp_digit) * size); + if (tmp == NULL) { + /* reallocation failed but "a" is still valid [can be freed] */ + return MP_MEM; + } + + /* reallocation succeeded so set a->dp */ + a->dp = tmp; + + /* zero excess digits */ + i = a->alloc; + a->alloc = size; + for (; i < a->alloc; i++) { + a->dp[i] = 0; + } + } + return MP_OKAY; +} +#endif + +/* End: bn_mp_grow.c */ + +/* Start: bn_mp_init.c */ +#include <tommath.h> +#ifdef BN_MP_INIT_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* init a new mp_int */ +int mp_init (mp_int * a) +{ + int i; + + /* allocate memory required and clear it */ + a->dp = OPT_CAST(mp_digit) XMALLOC (sizeof (mp_digit) * MP_PREC); + if (a->dp == NULL) { + return MP_MEM; + } + + /* set the digits to zero */ + for (i = 0; i < MP_PREC; i++) { + a->dp[i] = 0; + } + + /* set the used to zero, allocated digits to the default precision + * and sign to positive */ + a->used = 0; + a->alloc = MP_PREC; + a->sign = MP_ZPOS; + + return MP_OKAY; +} +#endif + +/* End: bn_mp_init.c */ + +/* Start: bn_mp_init_copy.c */ +#include <tommath.h> +#ifdef BN_MP_INIT_COPY_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* creates "a" then copies b into it */ +int mp_init_copy (mp_int * a, mp_int * b) +{ + int res; + + if ((res = mp_init (a)) != MP_OKAY) { + return res; + } + return mp_copy (b, a); +} +#endif + +/* End: bn_mp_init_copy.c */ + +/* Start: bn_mp_init_multi.c */ +#include <tommath.h> +#ifdef BN_MP_INIT_MULTI_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ +#include <stdarg.h> + +int mp_init_multi(mp_int *mp, ...) +{ + mp_err res = MP_OKAY; /* Assume ok until proven otherwise */ + int n = 0; /* Number of ok inits */ + mp_int* cur_arg = mp; + va_list args; + + va_start(args, mp); /* init args to next argument from caller */ + while (cur_arg != NULL) { + if (mp_init(cur_arg) != MP_OKAY) { + /* Oops - error! Back-track and mp_clear what we already + succeeded in init-ing, then return error. + */ + va_list clean_args; + + /* end the current list */ + va_end(args); + + /* now start cleaning up */ + cur_arg = mp; + va_start(clean_args, mp); + while (n--) { + mp_clear(cur_arg); + cur_arg = va_arg(clean_args, mp_int*); + } + va_end(clean_args); + res = MP_MEM; + break; + } + n++; + cur_arg = va_arg(args, mp_int*); + } + va_end(args); + return res; /* Assumed ok, if error flagged above. */ +} + +#endif + +/* End: bn_mp_init_multi.c */ + +/* Start: bn_mp_init_set.c */ +#include <tommath.h> +#ifdef BN_MP_INIT_SET_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* initialize and set a digit */ +int mp_init_set (mp_int * a, mp_digit b) +{ + int err; + if ((err = mp_init(a)) != MP_OKAY) { + return err; + } + mp_set(a, b); + return err; +} +#endif + +/* End: bn_mp_init_set.c */ + +/* Start: bn_mp_init_set_int.c */ +#include <tommath.h> +#ifdef BN_MP_INIT_SET_INT_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* initialize and set a digit */ +int mp_init_set_int (mp_int * a, unsigned long b) +{ + int err; + if ((err = mp_init(a)) != MP_OKAY) { + return err; + } + return mp_set_int(a, b); +} +#endif + +/* End: bn_mp_init_set_int.c */ + +/* Start: bn_mp_init_size.c */ +#include <tommath.h> +#ifdef BN_MP_INIT_SIZE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* init an mp_init for a given size */ +int mp_init_size (mp_int * a, int size) +{ + int x; + + /* pad size so there are always extra digits */ + size += (MP_PREC * 2) - (size % MP_PREC); + + /* alloc mem */ + a->dp = OPT_CAST(mp_digit) XMALLOC (sizeof (mp_digit) * size); + if (a->dp == NULL) { + return MP_MEM; + } + + /* set the members */ + a->used = 0; + a->alloc = size; + a->sign = MP_ZPOS; + + /* zero the digits */ + for (x = 0; x < size; x++) { + a->dp[x] = 0; + } + + return MP_OKAY; +} +#endif + +/* End: bn_mp_init_size.c */ + +/* Start: bn_mp_invmod.c */ +#include <tommath.h> +#ifdef BN_MP_INVMOD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* hac 14.61, pp608 */ +int mp_invmod (mp_int * a, mp_int * b, mp_int * c) +{ + /* b cannot be negative */ + if (b->sign == MP_NEG || mp_iszero(b) == 1) { + return MP_VAL; + } + +#ifdef BN_FAST_MP_INVMOD_C + /* if the modulus is odd we can use a faster routine instead */ + if (mp_isodd (b) == 1) { + return fast_mp_invmod (a, b, c); + } +#endif + +#ifdef BN_MP_INVMOD_SLOW_C + return mp_invmod_slow(a, b, c); +#endif + + return MP_VAL; +} +#endif + +/* End: bn_mp_invmod.c */ + +/* Start: bn_mp_invmod_slow.c */ +#include <tommath.h> +#ifdef BN_MP_INVMOD_SLOW_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* hac 14.61, pp608 */ +int mp_invmod_slow (mp_int * a, mp_int * b, mp_int * c) +{ + mp_int x, y, u, v, A, B, C, D; + int res; + + /* b cannot be negative */ + if (b->sign == MP_NEG || mp_iszero(b) == 1) { + return MP_VAL; + } + + /* init temps */ + if ((res = mp_init_multi(&x, &y, &u, &v, + &A, &B, &C, &D, NULL)) != MP_OKAY) { + return res; + } + + /* x = a, y = b */ + if ((res = mp_copy (a, &x)) != MP_OKAY) { + goto LBL_ERR; + } + if ((res = mp_copy (b, &y)) != MP_OKAY) { + goto LBL_ERR; + } + + /* 2. [modified] if x,y are both even then return an error! */ + if (mp_iseven (&x) == 1 && mp_iseven (&y) == 1) { + res = MP_VAL; + goto LBL_ERR; + } + + /* 3. u=x, v=y, A=1, B=0, C=0,D=1 */ + if ((res = mp_copy (&x, &u)) != MP_OKAY) { + goto LBL_ERR; + } + if ((res = mp_copy (&y, &v)) != MP_OKAY) { + goto LBL_ERR; + } + mp_set (&A, 1); + mp_set (&D, 1); + +top: + /* 4. while u is even do */ + while (mp_iseven (&u) == 1) { + /* 4.1 u = u/2 */ + if ((res = mp_div_2 (&u, &u)) != MP_OKAY) { + goto LBL_ERR; + } + /* 4.2 if A or B is odd then */ + if (mp_isodd (&A) == 1 || mp_isodd (&B) == 1) { + /* A = (A+y)/2, B = (B-x)/2 */ + if ((res = mp_add (&A, &y, &A)) != MP_OKAY) { + goto LBL_ERR; + } + if ((res = mp_sub (&B, &x, &B)) != MP_OKAY) { + goto LBL_ERR; + } + } + /* A = A/2, B = B/2 */ + if ((res = mp_div_2 (&A, &A)) != MP_OKAY) { + goto LBL_ERR; + } + if ((res = mp_div_2 (&B, &B)) != MP_OKAY) { + goto LBL_ERR; + } + } + + /* 5. while v is even do */ + while (mp_iseven (&v) == 1) { + /* 5.1 v = v/2 */ + if ((res = mp_div_2 (&v, &v)) != MP_OKAY) { + goto LBL_ERR; + } + /* 5.2 if C or D is odd then */ + if (mp_isodd (&C) == 1 || mp_isodd (&D) == 1) { + /* C = (C+y)/2, D = (D-x)/2 */ + if ((res = mp_add (&C, &y, &C)) != MP_OKAY) { + goto LBL_ERR; + } + if ((res = mp_sub (&D, &x, &D)) != MP_OKAY) { + goto LBL_ERR; + } + } + /* C = C/2, D = D/2 */ + if ((res = mp_div_2 (&C, &C)) != MP_OKAY) { + goto LBL_ERR; + } + if ((res = mp_div_2 (&D, &D)) != MP_OKAY) { + goto LBL_ERR; + } + } + + /* 6. if u >= v then */ + if (mp_cmp (&u, &v) != MP_LT) { + /* u = u - v, A = A - C, B = B - D */ + if ((res = mp_sub (&u, &v, &u)) != MP_OKAY) { + goto LBL_ERR; + } + + if ((res = mp_sub (&A, &C, &A)) != MP_OKAY) { + goto LBL_ERR; + } + + if ((res = mp_sub (&B, &D, &B)) != MP_OKAY) { + goto LBL_ERR; + } + } else { + /* v - v - u, C = C - A, D = D - B */ + if ((res = mp_sub (&v, &u, &v)) != MP_OKAY) { + goto LBL_ERR; + } + + if ((res = mp_sub (&C, &A, &C)) != MP_OKAY) { + goto LBL_ERR; + } + + if ((res = mp_sub (&D, &B, &D)) != MP_OKAY) { + goto LBL_ERR; + } + } + + /* if not zero goto step 4 */ + if (mp_iszero (&u) == 0) + goto top; + + /* now a = C, b = D, gcd == g*v */ + + /* if v != 1 then there is no inverse */ + if (mp_cmp_d (&v, 1) != MP_EQ) { + res = MP_VAL; + goto LBL_ERR; + } + + /* if its too low */ + while (mp_cmp_d(&C, 0) == MP_LT) { + if ((res = mp_add(&C, b, &C)) != MP_OKAY) { + goto LBL_ERR; + } + } + + /* too big */ + while (mp_cmp_mag(&C, b) != MP_LT) { + if ((res = mp_sub(&C, b, &C)) != MP_OKAY) { + goto LBL_ERR; + } + } + + /* C is now the inverse */ + mp_exch (&C, c); + res = MP_OKAY; +LBL_ERR:mp_clear_multi (&x, &y, &u, &v, &A, &B, &C, &D, NULL); + return res; +} +#endif + +/* End: bn_mp_invmod_slow.c */ + +/* Start: bn_mp_is_square.c */ +#include <tommath.h> +#ifdef BN_MP_IS_SQUARE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* Check if remainders are possible squares - fast exclude non-squares */ +static const char rem_128[128] = { + 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, + 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, + 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, + 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, + 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, + 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, + 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, + 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1 +}; + +static const char rem_105[105] = { + 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, + 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, + 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, + 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, + 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, + 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, + 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1 +}; + +/* Store non-zero to ret if arg is square, and zero if not */ +int mp_is_square(mp_int *arg,int *ret) +{ + int res; + mp_digit c; + mp_int t; + unsigned long r; + + /* Default to Non-square :) */ + *ret = MP_NO; + + if (arg->sign == MP_NEG) { + return MP_VAL; + } + + /* digits used? (TSD) */ + if (arg->used == 0) { + return MP_OKAY; + } + + /* First check mod 128 (suppose that DIGIT_BIT is at least 7) */ + if (rem_128[127 & DIGIT(arg,0)] == 1) { + return MP_OKAY; + } + + /* Next check mod 105 (3*5*7) */ + if ((res = mp_mod_d(arg,105,&c)) != MP_OKAY) { + return res; + } + if (rem_105[c] == 1) { + return MP_OKAY; + } + + + if ((res = mp_init_set_int(&t,11L*13L*17L*19L*23L*29L*31L)) != MP_OKAY) { + return res; + } + if ((res = mp_mod(arg,&t,&t)) != MP_OKAY) { + goto ERR; + } + r = mp_get_int(&t); + /* Check for other prime modules, note it's not an ERROR but we must + * free "t" so the easiest way is to goto ERR. We know that res + * is already equal to MP_OKAY from the mp_mod call + */ + if ( (1L<<(r%11)) & 0x5C4L ) goto ERR; + if ( (1L<<(r%13)) & 0x9E4L ) goto ERR; + if ( (1L<<(r%17)) & 0x5CE8L ) goto ERR; + if ( (1L<<(r%19)) & 0x4F50CL ) goto ERR; + if ( (1L<<(r%23)) & 0x7ACCA0L ) goto ERR; + if ( (1L<<(r%29)) & 0xC2EDD0CL ) goto ERR; + if ( (1L<<(r%31)) & 0x6DE2B848L ) goto ERR; + + /* Final check - is sqr(sqrt(arg)) == arg ? */ + if ((res = mp_sqrt(arg,&t)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_sqr(&t,&t)) != MP_OKAY) { + goto ERR; + } + + *ret = (mp_cmp_mag(&t,arg) == MP_EQ) ? MP_YES : MP_NO; +ERR:mp_clear(&t); + return res; +} +#endif + +/* End: bn_mp_is_square.c */ + +/* Start: bn_mp_jacobi.c */ +#include <tommath.h> +#ifdef BN_MP_JACOBI_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* computes the jacobi c = (a | n) (or Legendre if n is prime) + * HAC pp. 73 Algorithm 2.149 + */ +int mp_jacobi (mp_int * a, mp_int * p, int *c) +{ + mp_int a1, p1; + int k, s, r, res; + mp_digit residue; + + /* if p <= 0 return MP_VAL */ + if (mp_cmp_d(p, 0) != MP_GT) { + return MP_VAL; + } + + /* step 1. if a == 0, return 0 */ + if (mp_iszero (a) == 1) { + *c = 0; + return MP_OKAY; + } + + /* step 2. if a == 1, return 1 */ + if (mp_cmp_d (a, 1) == MP_EQ) { + *c = 1; + return MP_OKAY; + } + + /* default */ + s = 0; + + /* step 3. write a = a1 * 2**k */ + if ((res = mp_init_copy (&a1, a)) != MP_OKAY) { + return res; + } + + if ((res = mp_init (&p1)) != MP_OKAY) { + goto LBL_A1; + } + + /* divide out larger power of two */ + k = mp_cnt_lsb(&a1); + if ((res = mp_div_2d(&a1, k, &a1, NULL)) != MP_OKAY) { + goto LBL_P1; + } + + /* step 4. if e is even set s=1 */ + if ((k & 1) == 0) { + s = 1; + } else { + /* else set s=1 if p = 1/7 (mod 8) or s=-1 if p = 3/5 (mod 8) */ + residue = p->dp[0] & 7; + + if (residue == 1 || residue == 7) { + s = 1; + } else if (residue == 3 || residue == 5) { + s = -1; + } + } + + /* step 5. if p == 3 (mod 4) *and* a1 == 3 (mod 4) then s = -s */ + if ( ((p->dp[0] & 3) == 3) && ((a1.dp[0] & 3) == 3)) { + s = -s; + } + + /* if a1 == 1 we're done */ + if (mp_cmp_d (&a1, 1) == MP_EQ) { + *c = s; + } else { + /* n1 = n mod a1 */ + if ((res = mp_mod (p, &a1, &p1)) != MP_OKAY) { + goto LBL_P1; + } + if ((res = mp_jacobi (&p1, &a1, &r)) != MP_OKAY) { + goto LBL_P1; + } + *c = s * r; + } + + /* done */ + res = MP_OKAY; +LBL_P1:mp_clear (&p1); +LBL_A1:mp_clear (&a1); + return res; +} +#endif + +/* End: bn_mp_jacobi.c */ + +/* Start: bn_mp_karatsuba_mul.c */ +#include <tommath.h> +#ifdef BN_MP_KARATSUBA_MUL_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* c = |a| * |b| using Karatsuba Multiplication using + * three half size multiplications + * + * Let B represent the radix [e.g. 2**DIGIT_BIT] and + * let n represent half of the number of digits in + * the min(a,b) + * + * a = a1 * B**n + a0 + * b = b1 * B**n + b0 + * + * Then, a * b => + a1b1 * B**2n + ((a1 - a0)(b1 - b0) + a0b0 + a1b1) * B + a0b0 + * + * Note that a1b1 and a0b0 are used twice and only need to be + * computed once. So in total three half size (half # of + * digit) multiplications are performed, a0b0, a1b1 and + * (a1-b1)(a0-b0) + * + * Note that a multiplication of half the digits requires + * 1/4th the number of single precision multiplications so in + * total after one call 25% of the single precision multiplications + * are saved. Note also that the call to mp_mul can end up back + * in this function if the a0, a1, b0, or b1 are above the threshold. + * This is known as divide-and-conquer and leads to the famous + * O(N**lg(3)) or O(N**1.584) work which is asymptopically lower than + * the standard O(N**2) that the baseline/comba methods use. + * Generally though the overhead of this method doesn't pay off + * until a certain size (N ~ 80) is reached. + */ +int mp_karatsuba_mul (mp_int * a, mp_int * b, mp_int * c) +{ + mp_int x0, x1, y0, y1, t1, x0y0, x1y1; + int B, err; + + /* default the return code to an error */ + err = MP_MEM; + + /* min # of digits */ + B = MIN (a->used, b->used); + + /* now divide in two */ + B = B >> 1; + + /* init copy all the temps */ + if (mp_init_size (&x0, B) != MP_OKAY) + goto ERR; + if (mp_init_size (&x1, a->used - B) != MP_OKAY) + goto X0; + if (mp_init_size (&y0, B) != MP_OKAY) + goto X1; + if (mp_init_size (&y1, b->used - B) != MP_OKAY) + goto Y0; + + /* init temps */ + if (mp_init_size (&t1, B * 2) != MP_OKAY) + goto Y1; + if (mp_init_size (&x0y0, B * 2) != MP_OKAY) + goto T1; + if (mp_init_size (&x1y1, B * 2) != MP_OKAY) + goto X0Y0; + + /* now shift the digits */ + x0.used = y0.used = B; + x1.used = a->used - B; + y1.used = b->used - B; + + { + register int x; + register mp_digit *tmpa, *tmpb, *tmpx, *tmpy; + + /* we copy the digits directly instead of using higher level functions + * since we also need to shift the digits + */ + tmpa = a->dp; + tmpb = b->dp; + + tmpx = x0.dp; + tmpy = y0.dp; + for (x = 0; x < B; x++) { + *tmpx++ = *tmpa++; + *tmpy++ = *tmpb++; + } + + tmpx = x1.dp; + for (x = B; x < a->used; x++) { + *tmpx++ = *tmpa++; + } + + tmpy = y1.dp; + for (x = B; x < b->used; x++) { + *tmpy++ = *tmpb++; + } + } + + /* only need to clamp the lower words since by definition the + * upper words x1/y1 must have a known number of digits + */ + mp_clamp (&x0); + mp_clamp (&y0); + + /* now calc the products x0y0 and x1y1 */ + /* after this x0 is no longer required, free temp [x0==t2]! */ + if (mp_mul (&x0, &y0, &x0y0) != MP_OKAY) + goto X1Y1; /* x0y0 = x0*y0 */ + if (mp_mul (&x1, &y1, &x1y1) != MP_OKAY) + goto X1Y1; /* x1y1 = x1*y1 */ + + /* now calc x1-x0 and y1-y0 */ + if (mp_sub (&x1, &x0, &t1) != MP_OKAY) + goto X1Y1; /* t1 = x1 - x0 */ + if (mp_sub (&y1, &y0, &x0) != MP_OKAY) + goto X1Y1; /* t2 = y1 - y0 */ + if (mp_mul (&t1, &x0, &t1) != MP_OKAY) + goto X1Y1; /* t1 = (x1 - x0) * (y1 - y0) */ + + /* add x0y0 */ + if (mp_add (&x0y0, &x1y1, &x0) != MP_OKAY) + goto X1Y1; /* t2 = x0y0 + x1y1 */ + if (mp_sub (&x0, &t1, &t1) != MP_OKAY) + goto X1Y1; /* t1 = x0y0 + x1y1 - (x1-x0)*(y1-y0) */ + + /* shift by B */ + if (mp_lshd (&t1, B) != MP_OKAY) + goto X1Y1; /* t1 = (x0y0 + x1y1 - (x1-x0)*(y1-y0))<<B */ + if (mp_lshd (&x1y1, B * 2) != MP_OKAY) + goto X1Y1; /* x1y1 = x1y1 << 2*B */ + + if (mp_add (&x0y0, &t1, &t1) != MP_OKAY) + goto X1Y1; /* t1 = x0y0 + t1 */ + if (mp_add (&t1, &x1y1, c) != MP_OKAY) + goto X1Y1; /* t1 = x0y0 + t1 + x1y1 */ + + /* Algorithm succeeded set the return code to MP_OKAY */ + err = MP_OKAY; + +X1Y1:mp_clear (&x1y1); +X0Y0:mp_clear (&x0y0); +T1:mp_clear (&t1); +Y1:mp_clear (&y1); +Y0:mp_clear (&y0); +X1:mp_clear (&x1); +X0:mp_clear (&x0); +ERR: + return err; +} +#endif + +/* End: bn_mp_karatsuba_mul.c */ + +/* Start: bn_mp_karatsuba_sqr.c */ +#include <tommath.h> +#ifdef BN_MP_KARATSUBA_SQR_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* Karatsuba squaring, computes b = a*a using three + * half size squarings + * + * See comments of karatsuba_mul for details. It + * is essentially the same algorithm but merely + * tuned to perform recursive squarings. + */ +int mp_karatsuba_sqr (mp_int * a, mp_int * b) +{ + mp_int x0, x1, t1, t2, x0x0, x1x1; + int B, err; + + err = MP_MEM; + + /* min # of digits */ + B = a->used; + + /* now divide in two */ + B = B >> 1; + + /* init copy all the temps */ + if (mp_init_size (&x0, B) != MP_OKAY) + goto ERR; + if (mp_init_size (&x1, a->used - B) != MP_OKAY) + goto X0; + + /* init temps */ + if (mp_init_size (&t1, a->used * 2) != MP_OKAY) + goto X1; + if (mp_init_size (&t2, a->used * 2) != MP_OKAY) + goto T1; + if (mp_init_size (&x0x0, B * 2) != MP_OKAY) + goto T2; + if (mp_init_size (&x1x1, (a->used - B) * 2) != MP_OKAY) + goto X0X0; + + { + register int x; + register mp_digit *dst, *src; + + src = a->dp; + + /* now shift the digits */ + dst = x0.dp; + for (x = 0; x < B; x++) { + *dst++ = *src++; + } + + dst = x1.dp; + for (x = B; x < a->used; x++) { + *dst++ = *src++; + } + } + + x0.used = B; + x1.used = a->used - B; + + mp_clamp (&x0); + + /* now calc the products x0*x0 and x1*x1 */ + if (mp_sqr (&x0, &x0x0) != MP_OKAY) + goto X1X1; /* x0x0 = x0*x0 */ + if (mp_sqr (&x1, &x1x1) != MP_OKAY) + goto X1X1; /* x1x1 = x1*x1 */ + + /* now calc (x1-x0)**2 */ + if (mp_sub (&x1, &x0, &t1) != MP_OKAY) + goto X1X1; /* t1 = x1 - x0 */ + if (mp_sqr (&t1, &t1) != MP_OKAY) + goto X1X1; /* t1 = (x1 - x0) * (x1 - x0) */ + + /* add x0y0 */ + if (s_mp_add (&x0x0, &x1x1, &t2) != MP_OKAY) + goto X1X1; /* t2 = x0x0 + x1x1 */ + if (mp_sub (&t2, &t1, &t1) != MP_OKAY) + goto X1X1; /* t1 = x0x0 + x1x1 - (x1-x0)*(x1-x0) */ + + /* shift by B */ + if (mp_lshd (&t1, B) != MP_OKAY) + goto X1X1; /* t1 = (x0x0 + x1x1 - (x1-x0)*(x1-x0))<<B */ + if (mp_lshd (&x1x1, B * 2) != MP_OKAY) + goto X1X1; /* x1x1 = x1x1 << 2*B */ + + if (mp_add (&x0x0, &t1, &t1) != MP_OKAY) + goto X1X1; /* t1 = x0x0 + t1 */ + if (mp_add (&t1, &x1x1, b) != MP_OKAY) + goto X1X1; /* t1 = x0x0 + t1 + x1x1 */ + + err = MP_OKAY; + +X1X1:mp_clear (&x1x1); +X0X0:mp_clear (&x0x0); +T2:mp_clear (&t2); +T1:mp_clear (&t1); +X1:mp_clear (&x1); +X0:mp_clear (&x0); +ERR: + return err; +} +#endif + +/* End: bn_mp_karatsuba_sqr.c */ + +/* Start: bn_mp_lcm.c */ +#include <tommath.h> +#ifdef BN_MP_LCM_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* computes least common multiple as |a*b|/(a, b) */ +int mp_lcm (mp_int * a, mp_int * b, mp_int * c) +{ + int res; + mp_int t1, t2; + + + if ((res = mp_init_multi (&t1, &t2, NULL)) != MP_OKAY) { + return res; + } + + /* t1 = get the GCD of the two inputs */ + if ((res = mp_gcd (a, b, &t1)) != MP_OKAY) { + goto LBL_T; + } + + /* divide the smallest by the GCD */ + if (mp_cmp_mag(a, b) == MP_LT) { + /* store quotient in t2 such that t2 * b is the LCM */ + if ((res = mp_div(a, &t1, &t2, NULL)) != MP_OKAY) { + goto LBL_T; + } + res = mp_mul(b, &t2, c); + } else { + /* store quotient in t2 such that t2 * a is the LCM */ + if ((res = mp_div(b, &t1, &t2, NULL)) != MP_OKAY) { + goto LBL_T; + } + res = mp_mul(a, &t2, c); + } + + /* fix the sign to positive */ + c->sign = MP_ZPOS; + +LBL_T: + mp_clear_multi (&t1, &t2, NULL); + return res; +} +#endif + +/* End: bn_mp_lcm.c */ + +/* Start: bn_mp_lshd.c */ +#include <tommath.h> +#ifdef BN_MP_LSHD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* shift left a certain amount of digits */ +int mp_lshd (mp_int * a, int b) +{ + int x, res; + + /* if its less than zero return */ + if (b <= 0) { + return MP_OKAY; + } + + /* grow to fit the new digits */ + if (a->alloc < a->used + b) { + if ((res = mp_grow (a, a->used + b)) != MP_OKAY) { + return res; + } + } + + { + register mp_digit *top, *bottom; + + /* increment the used by the shift amount then copy upwards */ + a->used += b; + + /* top */ + top = a->dp + a->used - 1; + + /* base */ + bottom = a->dp + a->used - 1 - b; + + /* much like mp_rshd this is implemented using a sliding window + * except the window goes the otherway around. Copying from + * the bottom to the top. see bn_mp_rshd.c for more info. + */ + for (x = a->used - 1; x >= b; x--) { + *top-- = *bottom--; + } + + /* zero the lower digits */ + top = a->dp; + for (x = 0; x < b; x++) { + *top++ = 0; + } + } + return MP_OKAY; +} +#endif + +/* End: bn_mp_lshd.c */ + +/* Start: bn_mp_mod.c */ +#include <tommath.h> +#ifdef BN_MP_MOD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* c = a mod b, 0 <= c < b */ +int +mp_mod (mp_int * a, mp_int * b, mp_int * c) +{ + mp_int t; + int res; + + if ((res = mp_init (&t)) != MP_OKAY) { + return res; + } + + if ((res = mp_div (a, b, NULL, &t)) != MP_OKAY) { + mp_clear (&t); + return res; + } + + if (t.sign != b->sign) { + res = mp_add (b, &t, c); + } else { + res = MP_OKAY; + mp_exch (&t, c); + } + + mp_clear (&t); + return res; +} +#endif + +/* End: bn_mp_mod.c */ + +/* Start: bn_mp_mod_2d.c */ +#include <tommath.h> +#ifdef BN_MP_MOD_2D_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* calc a value mod 2**b */ +int +mp_mod_2d (mp_int * a, int b, mp_int * c) +{ + int x, res; + + /* if b is <= 0 then zero the int */ + if (b <= 0) { + mp_zero (c); + return MP_OKAY; + } + + /* if the modulus is larger than the value than return */ + if (b >= (int) (a->used * DIGIT_BIT)) { + res = mp_copy (a, c); + return res; + } + + /* copy */ + if ((res = mp_copy (a, c)) != MP_OKAY) { + return res; + } + + /* zero digits above the last digit of the modulus */ + for (x = (b / DIGIT_BIT) + ((b % DIGIT_BIT) == 0 ? 0 : 1); x < c->used; x++) { + c->dp[x] = 0; + } + /* clear the digit that is not completely outside/inside the modulus */ + c->dp[b / DIGIT_BIT] &= + (mp_digit) ((((mp_digit) 1) << (((mp_digit) b) % DIGIT_BIT)) - ((mp_digit) 1)); + mp_clamp (c); + return MP_OKAY; +} +#endif + +/* End: bn_mp_mod_2d.c */ + +/* Start: bn_mp_mod_d.c */ +#include <tommath.h> +#ifdef BN_MP_MOD_D_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +int +mp_mod_d (mp_int * a, mp_digit b, mp_digit * c) +{ + return mp_div_d(a, b, NULL, c); +} +#endif + +/* End: bn_mp_mod_d.c */ + +/* Start: bn_mp_montgomery_calc_normalization.c */ +#include <tommath.h> +#ifdef BN_MP_MONTGOMERY_CALC_NORMALIZATION_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* + * shifts with subtractions when the result is greater than b. + * + * The method is slightly modified to shift B unconditionally upto just under + * the leading bit of b. This saves alot of multiple precision shifting. + */ +int mp_montgomery_calc_normalization (mp_int * a, mp_int * b) +{ + int x, bits, res; + + /* how many bits of last digit does b use */ + bits = mp_count_bits (b) % DIGIT_BIT; + + + if (b->used > 1) { + if ((res = mp_2expt (a, (b->used - 1) * DIGIT_BIT + bits - 1)) != MP_OKAY) { + return res; + } + } else { + mp_set(a, 1); + bits = 1; + } + + + /* now compute C = A * B mod b */ + for (x = bits - 1; x < (int)DIGIT_BIT; x++) { + if ((res = mp_mul_2 (a, a)) != MP_OKAY) { + return res; + } + if (mp_cmp_mag (a, b) != MP_LT) { + if ((res = s_mp_sub (a, b, a)) != MP_OKAY) { + return res; + } + } + } + + return MP_OKAY; +} +#endif + +/* End: bn_mp_montgomery_calc_normalization.c */ + +/* Start: bn_mp_montgomery_reduce.c */ +#include <tommath.h> +#ifdef BN_MP_MONTGOMERY_REDUCE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* computes xR**-1 == x (mod N) via Montgomery Reduction */ +int +mp_montgomery_reduce (mp_int * x, mp_int * n, mp_digit rho) +{ + int ix, res, digs; + mp_digit mu; + + /* can the fast reduction [comba] method be used? + * + * Note that unlike in mul you're safely allowed *less* + * than the available columns [255 per default] since carries + * are fixed up in the inner loop. + */ + digs = n->used * 2 + 1; + if ((digs < MP_WARRAY) && + n->used < + (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) { + return fast_mp_montgomery_reduce (x, n, rho); + } + + /* grow the input as required */ + if (x->alloc < digs) { + if ((res = mp_grow (x, digs)) != MP_OKAY) { + return res; + } + } + x->used = digs; + + for (ix = 0; ix < n->used; ix++) { + /* mu = ai * rho mod b + * + * The value of rho must be precalculated via + * montgomery_setup() such that + * it equals -1/n0 mod b this allows the + * following inner loop to reduce the + * input one digit at a time + */ + mu = (mp_digit) (((mp_word)x->dp[ix]) * ((mp_word)rho) & MP_MASK); + + /* a = a + mu * m * b**i */ + { + register int iy; + register mp_digit *tmpn, *tmpx, u; + register mp_word r; + + /* alias for digits of the modulus */ + tmpn = n->dp; + + /* alias for the digits of x [the input] */ + tmpx = x->dp + ix; + + /* set the carry to zero */ + u = 0; + + /* Multiply and add in place */ + for (iy = 0; iy < n->used; iy++) { + /* compute product and sum */ + r = ((mp_word)mu) * ((mp_word)*tmpn++) + + ((mp_word) u) + ((mp_word) * tmpx); + + /* get carry */ + u = (mp_digit)(r >> ((mp_word) DIGIT_BIT)); + + /* fix digit */ + *tmpx++ = (mp_digit)(r & ((mp_word) MP_MASK)); + } + /* At this point the ix'th digit of x should be zero */ + + + /* propagate carries upwards as required*/ + while (u) { + *tmpx += u; + u = *tmpx >> DIGIT_BIT; + *tmpx++ &= MP_MASK; + } + } + } + + /* at this point the n.used'th least + * significant digits of x are all zero + * which means we can shift x to the + * right by n.used digits and the + * residue is unchanged. + */ + + /* x = x/b**n.used */ + mp_clamp(x); + mp_rshd (x, n->used); + + /* if x >= n then x = x - n */ + if (mp_cmp_mag (x, n) != MP_LT) { + return s_mp_sub (x, n, x); + } + + return MP_OKAY; +} +#endif + +/* End: bn_mp_montgomery_reduce.c */ + +/* Start: bn_mp_montgomery_setup.c */ +#include <tommath.h> +#ifdef BN_MP_MONTGOMERY_SETUP_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* setups the montgomery reduction stuff */ +int +mp_montgomery_setup (mp_int * n, mp_digit * rho) +{ + mp_digit x, b; + +/* fast inversion mod 2**k + * + * Based on the fact that + * + * XA = 1 (mod 2**n) => (X(2-XA)) A = 1 (mod 2**2n) + * => 2*X*A - X*X*A*A = 1 + * => 2*(1) - (1) = 1 + */ + b = n->dp[0]; + + if ((b & 1) == 0) { + return MP_VAL; + } + + x = (((b + 2) & 4) << 1) + b; /* here x*a==1 mod 2**4 */ + x *= 2 - b * x; /* here x*a==1 mod 2**8 */ +#if !defined(MP_8BIT) + x *= 2 - b * x; /* here x*a==1 mod 2**16 */ +#endif +#if defined(MP_64BIT) || !(defined(MP_8BIT) || defined(MP_16BIT)) + x *= 2 - b * x; /* here x*a==1 mod 2**32 */ +#endif +#ifdef MP_64BIT + x *= 2 - b * x; /* here x*a==1 mod 2**64 */ +#endif + + /* rho = -1/m mod b */ + *rho = (((mp_word)1 << ((mp_word) DIGIT_BIT)) - x) & MP_MASK; + + return MP_OKAY; +} +#endif + +/* End: bn_mp_montgomery_setup.c */ + +/* Start: bn_mp_mul.c */ +#include <tommath.h> +#ifdef BN_MP_MUL_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* high level multiplication (handles sign) */ +int mp_mul (mp_int * a, mp_int * b, mp_int * c) +{ + int res, neg; + neg = (a->sign == b->sign) ? MP_ZPOS : MP_NEG; + + /* use Toom-Cook? */ +#ifdef BN_MP_TOOM_MUL_C + if (MIN (a->used, b->used) >= TOOM_MUL_CUTOFF) { + res = mp_toom_mul(a, b, c); + } else +#endif +#ifdef BN_MP_KARATSUBA_MUL_C + /* use Karatsuba? */ + if (MIN (a->used, b->used) >= KARATSUBA_MUL_CUTOFF) { + res = mp_karatsuba_mul (a, b, c); + } else +#endif + { + /* can we use the fast multiplier? + * + * The fast multiplier can be used if the output will + * have less than MP_WARRAY digits and the number of + * digits won't affect carry propagation + */ + int digs = a->used + b->used + 1; + +#ifdef BN_FAST_S_MP_MUL_DIGS_C + if ((digs < MP_WARRAY) && + MIN(a->used, b->used) <= + (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) { + res = fast_s_mp_mul_digs (a, b, c, digs); + } else +#endif +#ifdef BN_S_MP_MUL_DIGS_C + res = s_mp_mul (a, b, c); /* uses s_mp_mul_digs */ +#else + res = MP_VAL; +#endif + + } + c->sign = (c->used > 0) ? neg : MP_ZPOS; + return res; +} +#endif + +/* End: bn_mp_mul.c */ + +/* Start: bn_mp_mul_2.c */ +#include <tommath.h> +#ifdef BN_MP_MUL_2_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* b = a*2 */ +int mp_mul_2(mp_int * a, mp_int * b) +{ + int x, res, oldused; + + /* grow to accomodate result */ + if (b->alloc < a->used + 1) { + if ((res = mp_grow (b, a->used + 1)) != MP_OKAY) { + return res; + } + } + + oldused = b->used; + b->used = a->used; + + { + register mp_digit r, rr, *tmpa, *tmpb; + + /* alias for source */ + tmpa = a->dp; + + /* alias for dest */ + tmpb = b->dp; + + /* carry */ + r = 0; + for (x = 0; x < a->used; x++) { + + /* get what will be the *next* carry bit from the + * MSB of the current digit + */ + rr = *tmpa >> ((mp_digit)(DIGIT_BIT - 1)); + + /* now shift up this digit, add in the carry [from the previous] */ + *tmpb++ = ((*tmpa++ << ((mp_digit)1)) | r) & MP_MASK; + + /* copy the carry that would be from the source + * digit into the next iteration + */ + r = rr; + } + + /* new leading digit? */ + if (r != 0) { + /* add a MSB which is always 1 at this point */ + *tmpb = 1; + ++(b->used); + } + + /* now zero any excess digits on the destination + * that we didn't write to + */ + tmpb = b->dp + b->used; + for (x = b->used; x < oldused; x++) { + *tmpb++ = 0; + } + } + b->sign = a->sign; + return MP_OKAY; +} +#endif + +/* End: bn_mp_mul_2.c */ + +/* Start: bn_mp_mul_2d.c */ +#include <tommath.h> +#ifdef BN_MP_MUL_2D_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* shift left by a certain bit count */ +int mp_mul_2d (mp_int * a, int b, mp_int * c) +{ + mp_digit d; + int res; + + /* copy */ + if (a != c) { + if ((res = mp_copy (a, c)) != MP_OKAY) { + return res; + } + } + + if (c->alloc < (int)(c->used + b/DIGIT_BIT + 1)) { + if ((res = mp_grow (c, c->used + b / DIGIT_BIT + 1)) != MP_OKAY) { + return res; + } + } + + /* shift by as many digits in the bit count */ + if (b >= (int)DIGIT_BIT) { + if ((res = mp_lshd (c, b / DIGIT_BIT)) != MP_OKAY) { + return res; + } + } + + /* shift any bit count < DIGIT_BIT */ + d = (mp_digit) (b % DIGIT_BIT); + if (d != 0) { + register mp_digit *tmpc, shift, mask, r, rr; + register int x; + + /* bitmask for carries */ + mask = (((mp_digit)1) << d) - 1; + + /* shift for msbs */ + shift = DIGIT_BIT - d; + + /* alias */ + tmpc = c->dp; + + /* carry */ + r = 0; + for (x = 0; x < c->used; x++) { + /* get the higher bits of the current word */ + rr = (*tmpc >> shift) & mask; + + /* shift the current word and OR in the carry */ + *tmpc = ((*tmpc << d) | r) & MP_MASK; + ++tmpc; + + /* set the carry to the carry bits of the current word */ + r = rr; + } + + /* set final carry */ + if (r != 0) { + c->dp[(c->used)++] = r; + } + } + mp_clamp (c); + return MP_OKAY; +} +#endif + +/* End: bn_mp_mul_2d.c */ + +/* Start: bn_mp_mul_d.c */ +#include <tommath.h> +#ifdef BN_MP_MUL_D_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* multiply by a digit */ +int +mp_mul_d (mp_int * a, mp_digit b, mp_int * c) +{ + mp_digit u, *tmpa, *tmpc; + mp_word r; + int ix, res, olduse; + + /* make sure c is big enough to hold a*b */ + if (c->alloc < a->used + 1) { + if ((res = mp_grow (c, a->used + 1)) != MP_OKAY) { + return res; + } + } + + /* get the original destinations used count */ + olduse = c->used; + + /* set the sign */ + c->sign = a->sign; + + /* alias for a->dp [source] */ + tmpa = a->dp; + + /* alias for c->dp [dest] */ + tmpc = c->dp; + + /* zero carry */ + u = 0; + + /* compute columns */ + for (ix = 0; ix < a->used; ix++) { + /* compute product and carry sum for this term */ + r = ((mp_word) u) + ((mp_word)*tmpa++) * ((mp_word)b); + + /* mask off higher bits to get a single digit */ + *tmpc++ = (mp_digit) (r & ((mp_word) MP_MASK)); + + /* send carry into next iteration */ + u = (mp_digit) (r >> ((mp_word) DIGIT_BIT)); + } + + /* store final carry [if any] */ + *tmpc++ = u; + + /* now zero digits above the top */ + while (ix++ < olduse) { + *tmpc++ = 0; + } + + /* set used count */ + c->used = a->used + 1; + mp_clamp(c); + + return MP_OKAY; +} +#endif + +/* End: bn_mp_mul_d.c */ + +/* Start: bn_mp_mulmod.c */ +#include <tommath.h> +#ifdef BN_MP_MULMOD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* d = a * b (mod c) */ +int +mp_mulmod (mp_int * a, mp_int * b, mp_int * c, mp_int * d) +{ + int res; + mp_int t; + + if ((res = mp_init (&t)) != MP_OKAY) { + return res; + } + + if ((res = mp_mul (a, b, &t)) != MP_OKAY) { + mp_clear (&t); + return res; + } + res = mp_mod (&t, c, d); + mp_clear (&t); + return res; +} +#endif + +/* End: bn_mp_mulmod.c */ + +/* Start: bn_mp_n_root.c */ +#include <tommath.h> +#ifdef BN_MP_N_ROOT_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* find the n'th root of an integer + * + * Result found such that (c)**b <= a and (c+1)**b > a + * + * This algorithm uses Newton's approximation + * x[i+1] = x[i] - f(x[i])/f'(x[i]) + * which will find the root in log(N) time where + * each step involves a fair bit. This is not meant to + * find huge roots [square and cube, etc]. + */ +int mp_n_root (mp_int * a, mp_digit b, mp_int * c) +{ + mp_int t1, t2, t3; + int res, neg; + + /* input must be positive if b is even */ + if ((b & 1) == 0 && a->sign == MP_NEG) { + return MP_VAL; + } + + if ((res = mp_init (&t1)) != MP_OKAY) { + return res; + } + + if ((res = mp_init (&t2)) != MP_OKAY) { + goto LBL_T1; + } + + if ((res = mp_init (&t3)) != MP_OKAY) { + goto LBL_T2; + } + + /* if a is negative fudge the sign but keep track */ + neg = a->sign; + a->sign = MP_ZPOS; + + /* t2 = 2 */ + mp_set (&t2, 2); + + do { + /* t1 = t2 */ + if ((res = mp_copy (&t2, &t1)) != MP_OKAY) { + goto LBL_T3; + } + + /* t2 = t1 - ((t1**b - a) / (b * t1**(b-1))) */ + + /* t3 = t1**(b-1) */ + if ((res = mp_expt_d (&t1, b - 1, &t3)) != MP_OKAY) { + goto LBL_T3; + } + + /* numerator */ + /* t2 = t1**b */ + if ((res = mp_mul (&t3, &t1, &t2)) != MP_OKAY) { + goto LBL_T3; + } + + /* t2 = t1**b - a */ + if ((res = mp_sub (&t2, a, &t2)) != MP_OKAY) { + goto LBL_T3; + } + + /* denominator */ + /* t3 = t1**(b-1) * b */ + if ((res = mp_mul_d (&t3, b, &t3)) != MP_OKAY) { + goto LBL_T3; + } + + /* t3 = (t1**b - a)/(b * t1**(b-1)) */ + if ((res = mp_div (&t2, &t3, &t3, NULL)) != MP_OKAY) { + goto LBL_T3; + } + + if ((res = mp_sub (&t1, &t3, &t2)) != MP_OKAY) { + goto LBL_T3; + } + } while (mp_cmp (&t1, &t2) != MP_EQ); + + /* result can be off by a few so check */ + for (;;) { + if ((res = mp_expt_d (&t1, b, &t2)) != MP_OKAY) { + goto LBL_T3; + } + + if (mp_cmp (&t2, a) == MP_GT) { + if ((res = mp_sub_d (&t1, 1, &t1)) != MP_OKAY) { + goto LBL_T3; + } + } else { + break; + } + } + + /* reset the sign of a first */ + a->sign = neg; + + /* set the result */ + mp_exch (&t1, c); + + /* set the sign of the result */ + c->sign = neg; + + res = MP_OKAY; + +LBL_T3:mp_clear (&t3); +LBL_T2:mp_clear (&t2); +LBL_T1:mp_clear (&t1); + return res; +} +#endif + +/* End: bn_mp_n_root.c */ + +/* Start: bn_mp_neg.c */ +#include <tommath.h> +#ifdef BN_MP_NEG_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* b = -a */ +int mp_neg (mp_int * a, mp_int * b) +{ + int res; + if ((res = mp_copy (a, b)) != MP_OKAY) { + return res; + } + if (mp_iszero(b) != MP_YES) { + b->sign = (a->sign == MP_ZPOS) ? MP_NEG : MP_ZPOS; + } + return MP_OKAY; +} +#endif + +/* End: bn_mp_neg.c */ + +/* Start: bn_mp_or.c */ +#include <tommath.h> +#ifdef BN_MP_OR_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* OR two ints together */ +int mp_or (mp_int * a, mp_int * b, mp_int * c) +{ + int res, ix, px; + mp_int t, *x; + + if (a->used > b->used) { + if ((res = mp_init_copy (&t, a)) != MP_OKAY) { + return res; + } + px = b->used; + x = b; + } else { + if ((res = mp_init_copy (&t, b)) != MP_OKAY) { + return res; + } + px = a->used; + x = a; + } + + for (ix = 0; ix < px; ix++) { + t.dp[ix] |= x->dp[ix]; + } + mp_clamp (&t); + mp_exch (c, &t); + mp_clear (&t); + return MP_OKAY; +} +#endif + +/* End: bn_mp_or.c */ + +/* Start: bn_mp_prime_fermat.c */ +#include <tommath.h> +#ifdef BN_MP_PRIME_FERMAT_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* performs one Fermat test. + * + * If "a" were prime then b**a == b (mod a) since the order of + * the multiplicative sub-group would be phi(a) = a-1. That means + * it would be the same as b**(a mod (a-1)) == b**1 == b (mod a). + * + * Sets result to 1 if the congruence holds, or zero otherwise. + */ +int mp_prime_fermat (mp_int * a, mp_int * b, int *result) +{ + mp_int t; + int err; + + /* default to composite */ + *result = MP_NO; + + /* ensure b > 1 */ + if (mp_cmp_d(b, 1) != MP_GT) { + return MP_VAL; + } + + /* init t */ + if ((err = mp_init (&t)) != MP_OKAY) { + return err; + } + + /* compute t = b**a mod a */ + if ((err = mp_exptmod (b, a, a, &t)) != MP_OKAY) { + goto LBL_T; + } + + /* is it equal to b? */ + if (mp_cmp (&t, b) == MP_EQ) { + *result = MP_YES; + } + + err = MP_OKAY; +LBL_T:mp_clear (&t); + return err; +} +#endif + +/* End: bn_mp_prime_fermat.c */ + +/* Start: bn_mp_prime_is_divisible.c */ +#include <tommath.h> +#ifdef BN_MP_PRIME_IS_DIVISIBLE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* determines if an integers is divisible by one + * of the first PRIME_SIZE primes or not + * + * sets result to 0 if not, 1 if yes + */ +int mp_prime_is_divisible (mp_int * a, int *result) +{ + int err, ix; + mp_digit res; + + /* default to not */ + *result = MP_NO; + + for (ix = 0; ix < PRIME_SIZE; ix++) { + /* what is a mod LBL_prime_tab[ix] */ + if ((err = mp_mod_d (a, ltm_prime_tab[ix], &res)) != MP_OKAY) { + return err; + } + + /* is the residue zero? */ + if (res == 0) { + *result = MP_YES; + return MP_OKAY; + } + } + + return MP_OKAY; +} +#endif + +/* End: bn_mp_prime_is_divisible.c */ + +/* Start: bn_mp_prime_is_prime.c */ +#include <tommath.h> +#ifdef BN_MP_PRIME_IS_PRIME_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* performs a variable number of rounds of Miller-Rabin + * + * Probability of error after t rounds is no more than + + * + * Sets result to 1 if probably prime, 0 otherwise + */ +int mp_prime_is_prime (mp_int * a, int t, int *result) +{ + mp_int b; + int ix, err, res; + + /* default to no */ + *result = MP_NO; + + /* valid value of t? */ + if (t <= 0 || t > PRIME_SIZE) { + return MP_VAL; + } + + /* is the input equal to one of the primes in the table? */ + for (ix = 0; ix < PRIME_SIZE; ix++) { + if (mp_cmp_d(a, ltm_prime_tab[ix]) == MP_EQ) { + *result = 1; + return MP_OKAY; + } + } + + /* first perform trial division */ + if ((err = mp_prime_is_divisible (a, &res)) != MP_OKAY) { + return err; + } + + /* return if it was trivially divisible */ + if (res == MP_YES) { + return MP_OKAY; + } + + /* now perform the miller-rabin rounds */ + if ((err = mp_init (&b)) != MP_OKAY) { + return err; + } + + for (ix = 0; ix < t; ix++) { + /* set the prime */ + mp_set (&b, ltm_prime_tab[ix]); + + if ((err = mp_prime_miller_rabin (a, &b, &res)) != MP_OKAY) { + goto LBL_B; + } + + if (res == MP_NO) { + goto LBL_B; + } + } + + /* passed the test */ + *result = MP_YES; +LBL_B:mp_clear (&b); + return err; +} +#endif + +/* End: bn_mp_prime_is_prime.c */ + +/* Start: bn_mp_prime_miller_rabin.c */ +#include <tommath.h> +#ifdef BN_MP_PRIME_MILLER_RABIN_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* Miller-Rabin test of "a" to the base of "b" as described in + * HAC pp. 139 Algorithm 4.24 + * + * Sets result to 0 if definitely composite or 1 if probably prime. + * Randomly the chance of error is no more than 1/4 and often + * very much lower. + */ +int mp_prime_miller_rabin (mp_int * a, mp_int * b, int *result) +{ + mp_int n1, y, r; + int s, j, err; + + /* default */ + *result = MP_NO; + + /* ensure b > 1 */ + if (mp_cmp_d(b, 1) != MP_GT) { + return MP_VAL; + } + + /* get n1 = a - 1 */ + if ((err = mp_init_copy (&n1, a)) != MP_OKAY) { + return err; + } + if ((err = mp_sub_d (&n1, 1, &n1)) != MP_OKAY) { + goto LBL_N1; + } + + /* set 2**s * r = n1 */ + if ((err = mp_init_copy (&r, &n1)) != MP_OKAY) { + goto LBL_N1; + } + + /* count the number of least significant bits + * which are zero + */ + s = mp_cnt_lsb(&r); + + /* now divide n - 1 by 2**s */ + if ((err = mp_div_2d (&r, s, &r, NULL)) != MP_OKAY) { + goto LBL_R; + } + + /* compute y = b**r mod a */ + if ((err = mp_init (&y)) != MP_OKAY) { + goto LBL_R; + } + if ((err = mp_exptmod (b, &r, a, &y)) != MP_OKAY) { + goto LBL_Y; + } + + /* if y != 1 and y != n1 do */ + if (mp_cmp_d (&y, 1) != MP_EQ && mp_cmp (&y, &n1) != MP_EQ) { + j = 1; + /* while j <= s-1 and y != n1 */ + while ((j <= (s - 1)) && mp_cmp (&y, &n1) != MP_EQ) { + if ((err = mp_sqrmod (&y, a, &y)) != MP_OKAY) { + goto LBL_Y; + } + + /* if y == 1 then composite */ + if (mp_cmp_d (&y, 1) == MP_EQ) { + goto LBL_Y; + } + + ++j; + } + + /* if y != n1 then composite */ + if (mp_cmp (&y, &n1) != MP_EQ) { + goto LBL_Y; + } + } + + /* probably prime now */ + *result = MP_YES; +LBL_Y:mp_clear (&y); +LBL_R:mp_clear (&r); +LBL_N1:mp_clear (&n1); + return err; +} +#endif + +/* End: bn_mp_prime_miller_rabin.c */ + +/* Start: bn_mp_prime_next_prime.c */ +#include <tommath.h> +#ifdef BN_MP_PRIME_NEXT_PRIME_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* finds the next prime after the number "a" using "t" trials + * of Miller-Rabin. + * + * bbs_style = 1 means the prime must be congruent to 3 mod 4 + */ +int mp_prime_next_prime(mp_int *a, int t, int bbs_style) +{ + int err, res, x, y; + mp_digit res_tab[PRIME_SIZE], step, kstep; + mp_int b; + + /* ensure t is valid */ + if (t <= 0 || t > PRIME_SIZE) { + return MP_VAL; + } + + /* force positive */ + a->sign = MP_ZPOS; + + /* simple algo if a is less than the largest prime in the table */ + if (mp_cmp_d(a, ltm_prime_tab[PRIME_SIZE-1]) == MP_LT) { + /* find which prime it is bigger than */ + for (x = PRIME_SIZE - 2; x >= 0; x--) { + if (mp_cmp_d(a, ltm_prime_tab[x]) != MP_LT) { + if (bbs_style == 1) { + /* ok we found a prime smaller or + * equal [so the next is larger] + * + * however, the prime must be + * congruent to 3 mod 4 + */ + if ((ltm_prime_tab[x + 1] & 3) != 3) { + /* scan upwards for a prime congruent to 3 mod 4 */ + for (y = x + 1; y < PRIME_SIZE; y++) { + if ((ltm_prime_tab[y] & 3) == 3) { + mp_set(a, ltm_prime_tab[y]); + return MP_OKAY; + } + } + } + } else { + mp_set(a, ltm_prime_tab[x + 1]); + return MP_OKAY; + } + } + } + /* at this point a maybe 1 */ + if (mp_cmp_d(a, 1) == MP_EQ) { + mp_set(a, 2); + return MP_OKAY; + } + /* fall through to the sieve */ + } + + /* generate a prime congruent to 3 mod 4 or 1/3 mod 4? */ + if (bbs_style == 1) { + kstep = 4; + } else { + kstep = 2; + } + + /* at this point we will use a combination of a sieve and Miller-Rabin */ + + if (bbs_style == 1) { + /* if a mod 4 != 3 subtract the correct value to make it so */ + if ((a->dp[0] & 3) != 3) { + if ((err = mp_sub_d(a, (a->dp[0] & 3) + 1, a)) != MP_OKAY) { return err; }; + } + } else { + if (mp_iseven(a) == 1) { + /* force odd */ + if ((err = mp_sub_d(a, 1, a)) != MP_OKAY) { + return err; + } + } + } + + /* generate the restable */ + for (x = 1; x < PRIME_SIZE; x++) { + if ((err = mp_mod_d(a, ltm_prime_tab[x], res_tab + x)) != MP_OKAY) { + return err; + } + } + + /* init temp used for Miller-Rabin Testing */ + if ((err = mp_init(&b)) != MP_OKAY) { + return err; + } + + for (;;) { + /* skip to the next non-trivially divisible candidate */ + step = 0; + do { + /* y == 1 if any residue was zero [e.g. cannot be prime] */ + y = 0; + + /* increase step to next candidate */ + step += kstep; + + /* compute the new residue without using division */ + for (x = 1; x < PRIME_SIZE; x++) { + /* add the step to each residue */ + res_tab[x] += kstep; + + /* subtract the modulus [instead of using division] */ + if (res_tab[x] >= ltm_prime_tab[x]) { + res_tab[x] -= ltm_prime_tab[x]; + } + + /* set flag if zero */ + if (res_tab[x] == 0) { + y = 1; + } + } + } while (y == 1 && step < ((((mp_digit)1)<<DIGIT_BIT) - kstep)); + + /* add the step */ + if ((err = mp_add_d(a, step, a)) != MP_OKAY) { + goto LBL_ERR; + } + + /* if didn't pass sieve and step == MAX then skip test */ + if (y == 1 && step >= ((((mp_digit)1)<<DIGIT_BIT) - kstep)) { + continue; + } + + /* is this prime? */ + for (x = 0; x < t; x++) { + mp_set(&b, ltm_prime_tab[t]); + if ((err = mp_prime_miller_rabin(a, &b, &res)) != MP_OKAY) { + goto LBL_ERR; + } + if (res == MP_NO) { + break; + } + } + + if (res == MP_YES) { + break; + } + } + + err = MP_OKAY; +LBL_ERR: + mp_clear(&b); + return err; +} + +#endif + +/* End: bn_mp_prime_next_prime.c */ + +/* Start: bn_mp_prime_rabin_miller_trials.c */ +#include <tommath.h> +#ifdef BN_MP_PRIME_RABIN_MILLER_TRIALS_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + + +static const struct { + int k, t; +} sizes[] = { +{ 128, 28 }, +{ 256, 16 }, +{ 384, 10 }, +{ 512, 7 }, +{ 640, 6 }, +{ 768, 5 }, +{ 896, 4 }, +{ 1024, 4 } +}; + +/* returns # of RM trials required for a given bit size */ +int mp_prime_rabin_miller_trials(int size) +{ + int x; + + for (x = 0; x < (int)(sizeof(sizes)/(sizeof(sizes[0]))); x++) { + if (sizes[x].k == size) { + return sizes[x].t; + } else if (sizes[x].k > size) { + return (x == 0) ? sizes[0].t : sizes[x - 1].t; + } + } + return sizes[x-1].t + 1; +} + + +#endif + +/* End: bn_mp_prime_rabin_miller_trials.c */ + +/* Start: bn_mp_prime_random_ex.c */ +#include <tommath.h> +#ifdef BN_MP_PRIME_RANDOM_EX_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* makes a truly random prime of a given size (bits), + * + * Flags are as follows: + * + * LTM_PRIME_BBS - make prime congruent to 3 mod 4 + * LTM_PRIME_SAFE - make sure (p-1)/2 is prime as well (implies LTM_PRIME_BBS) + * LTM_PRIME_2MSB_OFF - make the 2nd highest bit zero + * LTM_PRIME_2MSB_ON - make the 2nd highest bit one + * + * You have to supply a callback which fills in a buffer with random bytes. "dat" is a parameter you can + * have passed to the callback (e.g. a state or something). This function doesn't use "dat" itself + * so it can be NULL + * + */ + +/* This is possibly the mother of all prime generation functions, muahahahahaha! */ +int mp_prime_random_ex(mp_int *a, int t, int size, int flags, ltm_prime_callback cb, void *dat) +{ + unsigned char *tmp, maskAND, maskOR_msb, maskOR_lsb; + int res, err, bsize, maskOR_msb_offset; + + /* sanity check the input */ + if (size <= 1 || t <= 0) { + return MP_VAL; + } + + /* LTM_PRIME_SAFE implies LTM_PRIME_BBS */ + if (flags & LTM_PRIME_SAFE) { + flags |= LTM_PRIME_BBS; + } + + /* calc the byte size */ + bsize = (size>>3) + ((size&7)?1:0); + + /* we need a buffer of bsize bytes */ + tmp = OPT_CAST(unsigned char) XMALLOC(bsize); + if (tmp == NULL) { + return MP_MEM; + } + + /* calc the maskAND value for the MSbyte*/ + maskAND = ((size&7) == 0) ? 0xFF : (0xFF >> (8 - (size & 7))); + + /* calc the maskOR_msb */ + maskOR_msb = 0; + maskOR_msb_offset = (size - 2) >> 3; + if (flags & LTM_PRIME_2MSB_ON) { + maskOR_msb |= 1 << ((size - 2) & 7); + } else if (flags & LTM_PRIME_2MSB_OFF) { + maskAND &= ~(1 << ((size - 2) & 7)); + } + + /* get the maskOR_lsb */ + maskOR_lsb = 0; + if (flags & LTM_PRIME_BBS) { + maskOR_lsb |= 3; + } + + do { + /* read the bytes */ + if (cb(tmp, bsize, dat) != bsize) { + err = MP_VAL; + goto error; + } + + /* work over the MSbyte */ + tmp[0] &= maskAND; + tmp[0] |= 1 << ((size - 1) & 7); + + /* mix in the maskORs */ + tmp[maskOR_msb_offset] |= maskOR_msb; + tmp[bsize-1] |= maskOR_lsb; + + /* read it in */ + if ((err = mp_read_unsigned_bin(a, tmp, bsize)) != MP_OKAY) { goto error; } + + /* is it prime? */ + if ((err = mp_prime_is_prime(a, t, &res)) != MP_OKAY) { goto error; } + if (res == MP_NO) { + continue; + } + + if (flags & LTM_PRIME_SAFE) { + /* see if (a-1)/2 is prime */ + if ((err = mp_sub_d(a, 1, a)) != MP_OKAY) { goto error; } + if ((err = mp_div_2(a, a)) != MP_OKAY) { goto error; } + + /* is it prime? */ + if ((err = mp_prime_is_prime(a, t, &res)) != MP_OKAY) { goto error; } + } + } while (res == MP_NO); + + if (flags & LTM_PRIME_SAFE) { + /* restore a to the original value */ + if ((err = mp_mul_2(a, a)) != MP_OKAY) { goto error; } + if ((err = mp_add_d(a, 1, a)) != MP_OKAY) { goto error; } + } + + err = MP_OKAY; +error: + XFREE(tmp); + return err; +} + + +#endif + +/* End: bn_mp_prime_random_ex.c */ + +/* Start: bn_mp_radix_size.c */ +#include <tommath.h> +#ifdef BN_MP_RADIX_SIZE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* returns size of ASCII reprensentation */ +int mp_radix_size (mp_int * a, int radix, int *size) +{ + int res, digs; + mp_int t; + mp_digit d; + + *size = 0; + + /* special case for binary */ + if (radix == 2) { + *size = mp_count_bits (a) + (a->sign == MP_NEG ? 1 : 0) + 1; + return MP_OKAY; + } + + /* make sure the radix is in range */ + if (radix < 2 || radix > 64) { + return MP_VAL; + } + + /* init a copy of the input */ + if ((res = mp_init_copy (&t, a)) != MP_OKAY) { + return res; + } + + /* digs is the digit count */ + digs = 0; + + /* if it's negative add one for the sign */ + if (t.sign == MP_NEG) { + ++digs; + t.sign = MP_ZPOS; + } + + /* fetch out all of the digits */ + while (mp_iszero (&t) == 0) { + if ((res = mp_div_d (&t, (mp_digit) radix, &t, &d)) != MP_OKAY) { + mp_clear (&t); + return res; + } + ++digs; + } + mp_clear (&t); + + /* return digs + 1, the 1 is for the NULL byte that would be required. */ + *size = digs + 1; + return MP_OKAY; +} + +#endif + +/* End: bn_mp_radix_size.c */ + +/* Start: bn_mp_radix_smap.c */ +#include <tommath.h> +#ifdef BN_MP_RADIX_SMAP_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* chars used in radix conversions */ +const char *mp_s_rmap = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz+/"; +#endif + +/* End: bn_mp_radix_smap.c */ + +/* Start: bn_mp_rand.c */ +#include <tommath.h> +#ifdef BN_MP_RAND_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* makes a pseudo-random int of a given size */ +int +mp_rand (mp_int * a, int digits) +{ + int res; + mp_digit d; + + mp_zero (a); + if (digits <= 0) { + return MP_OKAY; + } + + /* first place a random non-zero digit */ + do { + d = ((mp_digit) abs (rand ())); + } while (d == 0); + + if ((res = mp_add_d (a, d, a)) != MP_OKAY) { + return res; + } + + while (digits-- > 0) { + if ((res = mp_lshd (a, 1)) != MP_OKAY) { + return res; + } + + if ((res = mp_add_d (a, ((mp_digit) abs (rand ())), a)) != MP_OKAY) { + return res; + } + } + + return MP_OKAY; +} +#endif + +/* End: bn_mp_rand.c */ + +/* Start: bn_mp_read_radix.c */ +#include <tommath.h> +#ifdef BN_MP_READ_RADIX_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* read a string [ASCII] in a given radix */ +int mp_read_radix (mp_int * a, char *str, int radix) +{ + int y, res, neg; + char ch; + + /* make sure the radix is ok */ + if (radix < 2 || radix > 64) { + return MP_VAL; + } + + /* if the leading digit is a + * minus set the sign to negative. + */ + if (*str == '-') { + ++str; + neg = MP_NEG; + } else { + neg = MP_ZPOS; + } + + /* set the integer to the default of zero */ + mp_zero (a); + + /* process each digit of the string */ + while (*str) { + /* if the radix < 36 the conversion is case insensitive + * this allows numbers like 1AB and 1ab to represent the same value + * [e.g. in hex] + */ + ch = (char) ((radix < 36) ? toupper (*str) : *str); + for (y = 0; y < 64; y++) { + if (ch == mp_s_rmap[y]) { + break; + } + } + + /* if the char was found in the map + * and is less than the given radix add it + * to the number, otherwise exit the loop. + */ + if (y < radix) { + if ((res = mp_mul_d (a, (mp_digit) radix, a)) != MP_OKAY) { + return res; + } + if ((res = mp_add_d (a, (mp_digit) y, a)) != MP_OKAY) { + return res; + } + } else { + break; + } + ++str; + } + + /* set the sign only if a != 0 */ + if (mp_iszero(a) != 1) { + a->sign = neg; + } + return MP_OKAY; +} +#endif + +/* End: bn_mp_read_radix.c */ + +/* Start: bn_mp_read_signed_bin.c */ +#include <tommath.h> +#ifdef BN_MP_READ_SIGNED_BIN_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* read signed bin, big endian, first byte is 0==positive or 1==negative */ +int +mp_read_signed_bin (mp_int * a, unsigned char *b, int c) +{ + int res; + + /* read magnitude */ + if ((res = mp_read_unsigned_bin (a, b + 1, c - 1)) != MP_OKAY) { + return res; + } + + /* first byte is 0 for positive, non-zero for negative */ + if (b[0] == 0) { + a->sign = MP_ZPOS; + } else { + a->sign = MP_NEG; + } + + return MP_OKAY; +} +#endif + +/* End: bn_mp_read_signed_bin.c */ + +/* Start: bn_mp_read_unsigned_bin.c */ +#include <tommath.h> +#ifdef BN_MP_READ_UNSIGNED_BIN_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* reads a unsigned char array, assumes the msb is stored first [big endian] */ +int +mp_read_unsigned_bin (mp_int * a, unsigned char *b, int c) +{ + int res; + + /* make sure there are at least two digits */ + if (a->alloc < 2) { + if ((res = mp_grow(a, 2)) != MP_OKAY) { + return res; + } + } + + /* zero the int */ + mp_zero (a); + + /* read the bytes in */ + while (c-- > 0) { + if ((res = mp_mul_2d (a, 8, a)) != MP_OKAY) { + return res; + } + +#ifndef MP_8BIT + a->dp[0] |= *b++; + a->used += 1; +#else + a->dp[0] = (*b & MP_MASK); + a->dp[1] |= ((*b++ >> 7U) & 1); + a->used += 2; +#endif + } + mp_clamp (a); + return MP_OKAY; +} +#endif + +/* End: bn_mp_read_unsigned_bin.c */ + +/* Start: bn_mp_reduce.c */ +#include <tommath.h> +#ifdef BN_MP_REDUCE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* reduces x mod m, assumes 0 < x < m**2, mu is + * precomputed via mp_reduce_setup. + * From HAC pp.604 Algorithm 14.42 + */ +int +mp_reduce (mp_int * x, mp_int * m, mp_int * mu) +{ + mp_int q; + int res, um = m->used; + + /* q = x */ + if ((res = mp_init_copy (&q, x)) != MP_OKAY) { + return res; + } + + /* q1 = x / b**(k-1) */ + mp_rshd (&q, um - 1); + + /* according to HAC this optimization is ok */ + if (((unsigned long) um) > (((mp_digit)1) << (DIGIT_BIT - 1))) { + if ((res = mp_mul (&q, mu, &q)) != MP_OKAY) { + goto CLEANUP; + } + } else { +#ifdef BN_S_MP_MUL_HIGH_DIGS_C + if ((res = s_mp_mul_high_digs (&q, mu, &q, um - 1)) != MP_OKAY) { + goto CLEANUP; + } +#elif defined(BN_FAST_S_MP_MUL_HIGH_DIGS_C) + if ((res = fast_s_mp_mul_high_digs (&q, mu, &q, um - 1)) != MP_OKAY) { + goto CLEANUP; + } +#else + { + res = MP_VAL; + goto CLEANUP; + } +#endif + } + + /* q3 = q2 / b**(k+1) */ + mp_rshd (&q, um + 1); + + /* x = x mod b**(k+1), quick (no division) */ + if ((res = mp_mod_2d (x, DIGIT_BIT * (um + 1), x)) != MP_OKAY) { + goto CLEANUP; + } + + /* q = q * m mod b**(k+1), quick (no division) */ + if ((res = s_mp_mul_digs (&q, m, &q, um + 1)) != MP_OKAY) { + goto CLEANUP; + } + + /* x = x - q */ + if ((res = mp_sub (x, &q, x)) != MP_OKAY) { + goto CLEANUP; + } + + /* If x < 0, add b**(k+1) to it */ + if (mp_cmp_d (x, 0) == MP_LT) { + mp_set (&q, 1); + if ((res = mp_lshd (&q, um + 1)) != MP_OKAY) + goto CLEANUP; + if ((res = mp_add (x, &q, x)) != MP_OKAY) + goto CLEANUP; + } + + /* Back off if it's too big */ + while (mp_cmp (x, m) != MP_LT) { + if ((res = s_mp_sub (x, m, x)) != MP_OKAY) { + goto CLEANUP; + } + } + +CLEANUP: + mp_clear (&q); + + return res; +} +#endif + +/* End: bn_mp_reduce.c */ + +/* Start: bn_mp_reduce_2k.c */ +#include <tommath.h> +#ifdef BN_MP_REDUCE_2K_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* reduces a modulo n where n is of the form 2**p - d */ +int +mp_reduce_2k(mp_int *a, mp_int *n, mp_digit d) +{ + mp_int q; + int p, res; + + if ((res = mp_init(&q)) != MP_OKAY) { + return res; + } + + p = mp_count_bits(n); +top: + /* q = a/2**p, a = a mod 2**p */ + if ((res = mp_div_2d(a, p, &q, a)) != MP_OKAY) { + goto ERR; + } + + if (d != 1) { + /* q = q * d */ + if ((res = mp_mul_d(&q, d, &q)) != MP_OKAY) { + goto ERR; + } + } + + /* a = a + q */ + if ((res = s_mp_add(a, &q, a)) != MP_OKAY) { + goto ERR; + } + + if (mp_cmp_mag(a, n) != MP_LT) { + s_mp_sub(a, n, a); + goto top; + } + +ERR: + mp_clear(&q); + return res; +} + +#endif + +/* End: bn_mp_reduce_2k.c */ + +/* Start: bn_mp_reduce_2k_setup.c */ +#include <tommath.h> +#ifdef BN_MP_REDUCE_2K_SETUP_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* determines the setup value */ +int +mp_reduce_2k_setup(mp_int *a, mp_digit *d) +{ + int res, p; + mp_int tmp; + + if ((res = mp_init(&tmp)) != MP_OKAY) { + return res; + } + + p = mp_count_bits(a); + if ((res = mp_2expt(&tmp, p)) != MP_OKAY) { + mp_clear(&tmp); + return res; + } + + if ((res = s_mp_sub(&tmp, a, &tmp)) != MP_OKAY) { + mp_clear(&tmp); + return res; + } + + *d = tmp.dp[0]; + mp_clear(&tmp); + return MP_OKAY; +} +#endif + +/* End: bn_mp_reduce_2k_setup.c */ + +/* Start: bn_mp_reduce_is_2k.c */ +#include <tommath.h> +#ifdef BN_MP_REDUCE_IS_2K_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* determines if mp_reduce_2k can be used */ +int mp_reduce_is_2k(mp_int *a) +{ + int ix, iy, iw; + mp_digit iz; + + if (a->used == 0) { + return 0; + } else if (a->used == 1) { + return 1; + } else if (a->used > 1) { + iy = mp_count_bits(a); + iz = 1; + iw = 1; + + /* Test every bit from the second digit up, must be 1 */ + for (ix = DIGIT_BIT; ix < iy; ix++) { + if ((a->dp[iw] & iz) == 0) { + return 0; + } + iz <<= 1; + if (iz > (mp_digit)MP_MASK) { + ++iw; + iz = 1; + } + } + } + return 1; +} + +#endif + +/* End: bn_mp_reduce_is_2k.c */ + +/* Start: bn_mp_reduce_setup.c */ +#include <tommath.h> +#ifdef BN_MP_REDUCE_SETUP_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* pre-calculate the value required for Barrett reduction + * For a given modulus "b" it calulates the value required in "a" + */ +int mp_reduce_setup (mp_int * a, mp_int * b) +{ + int res; + + if ((res = mp_2expt (a, b->used * 2 * DIGIT_BIT)) != MP_OKAY) { + return res; + } + return mp_div (a, b, a, NULL); +} +#endif + +/* End: bn_mp_reduce_setup.c */ + +/* Start: bn_mp_rshd.c */ +#include <tommath.h> +#ifdef BN_MP_RSHD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* shift right a certain amount of digits */ +void mp_rshd (mp_int * a, int b) +{ + int x; + + /* if b <= 0 then ignore it */ + if (b <= 0) { + return; + } + + /* if b > used then simply zero it and return */ + if (a->used <= b) { + mp_zero (a); + return; + } + + { + register mp_digit *bottom, *top; + + /* shift the digits down */ + + /* bottom */ + bottom = a->dp; + + /* top [offset into digits] */ + top = a->dp + b; + + /* this is implemented as a sliding window where + * the window is b-digits long and digits from + * the top of the window are copied to the bottom + * + * e.g. + + b-2 | b-1 | b0 | b1 | b2 | ... | bb | ----> + /\ | ----> + \-------------------/ ----> + */ + for (x = 0; x < (a->used - b); x++) { + *bottom++ = *top++; + } + + /* zero the top digits */ + for (; x < a->used; x++) { + *bottom++ = 0; + } + } + + /* remove excess digits */ + a->used -= b; +} +#endif + +/* End: bn_mp_rshd.c */ + +/* Start: bn_mp_set.c */ +#include <tommath.h> +#ifdef BN_MP_SET_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* set to a digit */ +void mp_set (mp_int * a, mp_digit b) +{ + mp_zero (a); + a->dp[0] = b & MP_MASK; + a->used = (a->dp[0] != 0) ? 1 : 0; +} +#endif + +/* End: bn_mp_set.c */ + +/* Start: bn_mp_set_int.c */ +#include <tommath.h> +#ifdef BN_MP_SET_INT_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* set a 32-bit const */ +int mp_set_int (mp_int * a, unsigned long b) +{ + int x, res; + + mp_zero (a); + + /* set four bits at a time */ + for (x = 0; x < 8; x++) { + /* shift the number up four bits */ + if ((res = mp_mul_2d (a, 4, a)) != MP_OKAY) { + return res; + } + + /* OR in the top four bits of the source */ + a->dp[0] |= (b >> 28) & 15; + + /* shift the source up to the next four bits */ + b <<= 4; + + /* ensure that digits are not clamped off */ + a->used += 1; + } + mp_clamp (a); + return MP_OKAY; +} +#endif + +/* End: bn_mp_set_int.c */ + +/* Start: bn_mp_shrink.c */ +#include <tommath.h> +#ifdef BN_MP_SHRINK_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* shrink a bignum */ +int mp_shrink (mp_int * a) +{ + mp_digit *tmp; + if (a->alloc != a->used && a->used > 0) { + if ((tmp = OPT_CAST(mp_digit) XREALLOC (a->dp, sizeof (mp_digit) * a->used)) == NULL) { + return MP_MEM; + } + a->dp = tmp; + a->alloc = a->used; + } + return MP_OKAY; +} +#endif + +/* End: bn_mp_shrink.c */ + +/* Start: bn_mp_signed_bin_size.c */ +#include <tommath.h> +#ifdef BN_MP_SIGNED_BIN_SIZE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* get the size for an signed equivalent */ +int mp_signed_bin_size (mp_int * a) +{ + return 1 + mp_unsigned_bin_size (a); +} +#endif + +/* End: bn_mp_signed_bin_size.c */ + +/* Start: bn_mp_sqr.c */ +#include <tommath.h> +#ifdef BN_MP_SQR_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* computes b = a*a */ +int +mp_sqr (mp_int * a, mp_int * b) +{ + int res; + +#ifdef BN_MP_TOOM_SQR_C + /* use Toom-Cook? */ + if (a->used >= TOOM_SQR_CUTOFF) { + res = mp_toom_sqr(a, b); + /* Karatsuba? */ + } else +#endif +#ifdef BN_MP_KARATSUBA_SQR_C +if (a->used >= KARATSUBA_SQR_CUTOFF) { + res = mp_karatsuba_sqr (a, b); + } else +#endif + { +#ifdef BN_FAST_S_MP_SQR_C + /* can we use the fast comba multiplier? */ + if ((a->used * 2 + 1) < MP_WARRAY && + a->used < + (1 << (sizeof(mp_word) * CHAR_BIT - 2*DIGIT_BIT - 1))) { + res = fast_s_mp_sqr (a, b); + } else +#endif +#ifdef BN_S_MP_SQR_C + res = s_mp_sqr (a, b); +#else + res = MP_VAL; +#endif + } + b->sign = MP_ZPOS; + return res; +} +#endif + +/* End: bn_mp_sqr.c */ + +/* Start: bn_mp_sqrmod.c */ +#include <tommath.h> +#ifdef BN_MP_SQRMOD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* c = a * a (mod b) */ +int +mp_sqrmod (mp_int * a, mp_int * b, mp_int * c) +{ + int res; + mp_int t; + + if ((res = mp_init (&t)) != MP_OKAY) { + return res; + } + + if ((res = mp_sqr (a, &t)) != MP_OKAY) { + mp_clear (&t); + return res; + } + res = mp_mod (&t, b, c); + mp_clear (&t); + return res; +} +#endif + +/* End: bn_mp_sqrmod.c */ + +/* Start: bn_mp_sqrt.c */ +#include <tommath.h> +#ifdef BN_MP_SQRT_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* this function is less generic than mp_n_root, simpler and faster */ +int mp_sqrt(mp_int *arg, mp_int *ret) +{ + int res; + mp_int t1,t2; + + /* must be positive */ + if (arg->sign == MP_NEG) { + return MP_VAL; + } + + /* easy out */ + if (mp_iszero(arg) == MP_YES) { + mp_zero(ret); + return MP_OKAY; + } + + if ((res = mp_init_copy(&t1, arg)) != MP_OKAY) { + return res; + } + + if ((res = mp_init(&t2)) != MP_OKAY) { + goto E2; + } + + /* First approx. (not very bad for large arg) */ + mp_rshd (&t1,t1.used/2); + + /* t1 > 0 */ + if ((res = mp_div(arg,&t1,&t2,NULL)) != MP_OKAY) { + goto E1; + } + if ((res = mp_add(&t1,&t2,&t1)) != MP_OKAY) { + goto E1; + } + if ((res = mp_div_2(&t1,&t1)) != MP_OKAY) { + goto E1; + } + /* And now t1 > sqrt(arg) */ + do { + if ((res = mp_div(arg,&t1,&t2,NULL)) != MP_OKAY) { + goto E1; + } + if ((res = mp_add(&t1,&t2,&t1)) != MP_OKAY) { + goto E1; + } + if ((res = mp_div_2(&t1,&t1)) != MP_OKAY) { + goto E1; + } + /* t1 >= sqrt(arg) >= t2 at this point */ + } while (mp_cmp_mag(&t1,&t2) == MP_GT); + + mp_exch(&t1,ret); + +E1: mp_clear(&t2); +E2: mp_clear(&t1); + return res; +} + +#endif + +/* End: bn_mp_sqrt.c */ + +/* Start: bn_mp_sub.c */ +#include <tommath.h> +#ifdef BN_MP_SUB_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* high level subtraction (handles signs) */ +int +mp_sub (mp_int * a, mp_int * b, mp_int * c) +{ + int sa, sb, res; + + sa = a->sign; + sb = b->sign; + + if (sa != sb) { + /* subtract a negative from a positive, OR */ + /* subtract a positive from a negative. */ + /* In either case, ADD their magnitudes, */ + /* and use the sign of the first number. */ + c->sign = sa; + res = s_mp_add (a, b, c); + } else { + /* subtract a positive from a positive, OR */ + /* subtract a negative from a negative. */ + /* First, take the difference between their */ + /* magnitudes, then... */ + if (mp_cmp_mag (a, b) != MP_LT) { + /* Copy the sign from the first */ + c->sign = sa; + /* The first has a larger or equal magnitude */ + res = s_mp_sub (a, b, c); + } else { + /* The result has the *opposite* sign from */ + /* the first number. */ + c->sign = (sa == MP_ZPOS) ? MP_NEG : MP_ZPOS; + /* The second has a larger magnitude */ + res = s_mp_sub (b, a, c); + } + } + return res; +} + +#endif + +/* End: bn_mp_sub.c */ + +/* Start: bn_mp_sub_d.c */ +#include <tommath.h> +#ifdef BN_MP_SUB_D_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* single digit subtraction */ +int +mp_sub_d (mp_int * a, mp_digit b, mp_int * c) +{ + mp_digit *tmpa, *tmpc, mu; + int res, ix, oldused; + + /* grow c as required */ + if (c->alloc < a->used + 1) { + if ((res = mp_grow(c, a->used + 1)) != MP_OKAY) { + return res; + } + } + + /* if a is negative just do an unsigned + * addition [with fudged signs] + */ + if (a->sign == MP_NEG) { + a->sign = MP_ZPOS; + res = mp_add_d(a, b, c); + a->sign = c->sign = MP_NEG; + return res; + } + + /* setup regs */ + oldused = c->used; + tmpa = a->dp; + tmpc = c->dp; + + /* if a <= b simply fix the single digit */ + if ((a->used == 1 && a->dp[0] <= b) || a->used == 0) { + if (a->used == 1) { + *tmpc++ = b - *tmpa; + } else { + *tmpc++ = b; + } + ix = 1; + + /* negative/1digit */ + c->sign = MP_NEG; + c->used = 1; + } else { + /* positive/size */ + c->sign = MP_ZPOS; + c->used = a->used; + + /* subtract first digit */ + *tmpc = *tmpa++ - b; + mu = *tmpc >> (sizeof(mp_digit) * CHAR_BIT - 1); + *tmpc++ &= MP_MASK; + + /* handle rest of the digits */ + for (ix = 1; ix < a->used; ix++) { + *tmpc = *tmpa++ - mu; + mu = *tmpc >> (sizeof(mp_digit) * CHAR_BIT - 1); + *tmpc++ &= MP_MASK; + } + } + + /* zero excess digits */ + while (ix++ < oldused) { + *tmpc++ = 0; + } + mp_clamp(c); + return MP_OKAY; +} + +#endif + +/* End: bn_mp_sub_d.c */ + +/* Start: bn_mp_submod.c */ +#include <tommath.h> +#ifdef BN_MP_SUBMOD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* d = a - b (mod c) */ +int +mp_submod (mp_int * a, mp_int * b, mp_int * c, mp_int * d) +{ + int res; + mp_int t; + + + if ((res = mp_init (&t)) != MP_OKAY) { + return res; + } + + if ((res = mp_sub (a, b, &t)) != MP_OKAY) { + mp_clear (&t); + return res; + } + res = mp_mod (&t, c, d); + mp_clear (&t); + return res; +} +#endif + +/* End: bn_mp_submod.c */ + +/* Start: bn_mp_to_signed_bin.c */ +#include <tommath.h> +#ifdef BN_MP_TO_SIGNED_BIN_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* store in signed [big endian] format */ +int +mp_to_signed_bin (mp_int * a, unsigned char *b) +{ + int res; + + if ((res = mp_to_unsigned_bin (a, b + 1)) != MP_OKAY) { + return res; + } + b[0] = (unsigned char) ((a->sign == MP_ZPOS) ? 0 : 1); + return MP_OKAY; +} +#endif + +/* End: bn_mp_to_signed_bin.c */ + +/* Start: bn_mp_to_unsigned_bin.c */ +#include <tommath.h> +#ifdef BN_MP_TO_UNSIGNED_BIN_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* store in unsigned [big endian] format */ +int +mp_to_unsigned_bin (mp_int * a, unsigned char *b) +{ + int x, res; + mp_int t; + + if ((res = mp_init_copy (&t, a)) != MP_OKAY) { + return res; + } + + x = 0; + while (mp_iszero (&t) == 0) { +#ifndef MP_8BIT + b[x++] = (unsigned char) (t.dp[0] & 255); +#else + b[x++] = (unsigned char) (t.dp[0] | ((t.dp[1] & 0x01) << 7)); +#endif + if ((res = mp_div_2d (&t, 8, &t, NULL)) != MP_OKAY) { + mp_clear (&t); + return res; + } + } + bn_reverse (b, x); + mp_clear (&t); + return MP_OKAY; +} +#endif + +/* End: bn_mp_to_unsigned_bin.c */ + +/* Start: bn_mp_toom_mul.c */ +#include <tommath.h> +#ifdef BN_MP_TOOM_MUL_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* multiplication using the Toom-Cook 3-way algorithm + * + * Much more complicated than Karatsuba but has a lower asymptotic running time of + * O(N**1.464). This algorithm is only particularly useful on VERY large + * inputs (we're talking 1000s of digits here...). +*/ +int mp_toom_mul(mp_int *a, mp_int *b, mp_int *c) +{ + mp_int w0, w1, w2, w3, w4, tmp1, tmp2, a0, a1, a2, b0, b1, b2; + int res, B; + + /* init temps */ + if ((res = mp_init_multi(&w0, &w1, &w2, &w3, &w4, + &a0, &a1, &a2, &b0, &b1, + &b2, &tmp1, &tmp2, NULL)) != MP_OKAY) { + return res; + } + + /* B */ + B = MIN(a->used, b->used) / 3; + + /* a = a2 * B**2 + a1 * B + a0 */ + if ((res = mp_mod_2d(a, DIGIT_BIT * B, &a0)) != MP_OKAY) { + goto ERR; + } + + if ((res = mp_copy(a, &a1)) != MP_OKAY) { + goto ERR; + } + mp_rshd(&a1, B); + mp_mod_2d(&a1, DIGIT_BIT * B, &a1); + + if ((res = mp_copy(a, &a2)) != MP_OKAY) { + goto ERR; + } + mp_rshd(&a2, B*2); + + /* b = b2 * B**2 + b1 * B + b0 */ + if ((res = mp_mod_2d(b, DIGIT_BIT * B, &b0)) != MP_OKAY) { + goto ERR; + } + + if ((res = mp_copy(b, &b1)) != MP_OKAY) { + goto ERR; + } + mp_rshd(&b1, B); + mp_mod_2d(&b1, DIGIT_BIT * B, &b1); + + if ((res = mp_copy(b, &b2)) != MP_OKAY) { + goto ERR; + } + mp_rshd(&b2, B*2); + + /* w0 = a0*b0 */ + if ((res = mp_mul(&a0, &b0, &w0)) != MP_OKAY) { + goto ERR; + } + + /* w4 = a2 * b2 */ + if ((res = mp_mul(&a2, &b2, &w4)) != MP_OKAY) { + goto ERR; + } + + /* w1 = (a2 + 2(a1 + 2a0))(b2 + 2(b1 + 2b0)) */ + if ((res = mp_mul_2(&a0, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&tmp1, &a2, &tmp1)) != MP_OKAY) { + goto ERR; + } + + if ((res = mp_mul_2(&b0, &tmp2)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&tmp2, &b2, &tmp2)) != MP_OKAY) { + goto ERR; + } + + if ((res = mp_mul(&tmp1, &tmp2, &w1)) != MP_OKAY) { + goto ERR; + } + + /* w3 = (a0 + 2(a1 + 2a2))(b0 + 2(b1 + 2b2)) */ + if ((res = mp_mul_2(&a2, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) { + goto ERR; + } + + if ((res = mp_mul_2(&b2, &tmp2)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) { + goto ERR; + } + + if ((res = mp_mul(&tmp1, &tmp2, &w3)) != MP_OKAY) { + goto ERR; + } + + + /* w2 = (a2 + a1 + a0)(b2 + b1 + b0) */ + if ((res = mp_add(&a2, &a1, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&b2, &b1, &tmp2)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_mul(&tmp1, &tmp2, &w2)) != MP_OKAY) { + goto ERR; + } + + /* now solve the matrix + + 0 0 0 0 1 + 1 2 4 8 16 + 1 1 1 1 1 + 16 8 4 2 1 + 1 0 0 0 0 + + using 12 subtractions, 4 shifts, + 2 small divisions and 1 small multiplication + */ + + /* r1 - r4 */ + if ((res = mp_sub(&w1, &w4, &w1)) != MP_OKAY) { + goto ERR; + } + /* r3 - r0 */ + if ((res = mp_sub(&w3, &w0, &w3)) != MP_OKAY) { + goto ERR; + } + /* r1/2 */ + if ((res = mp_div_2(&w1, &w1)) != MP_OKAY) { + goto ERR; + } + /* r3/2 */ + if ((res = mp_div_2(&w3, &w3)) != MP_OKAY) { + goto ERR; + } + /* r2 - r0 - r4 */ + if ((res = mp_sub(&w2, &w0, &w2)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_sub(&w2, &w4, &w2)) != MP_OKAY) { + goto ERR; + } + /* r1 - r2 */ + if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) { + goto ERR; + } + /* r3 - r2 */ + if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) { + goto ERR; + } + /* r1 - 8r0 */ + if ((res = mp_mul_2d(&w0, 3, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_sub(&w1, &tmp1, &w1)) != MP_OKAY) { + goto ERR; + } + /* r3 - 8r4 */ + if ((res = mp_mul_2d(&w4, 3, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_sub(&w3, &tmp1, &w3)) != MP_OKAY) { + goto ERR; + } + /* 3r2 - r1 - r3 */ + if ((res = mp_mul_d(&w2, 3, &w2)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_sub(&w2, &w1, &w2)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_sub(&w2, &w3, &w2)) != MP_OKAY) { + goto ERR; + } + /* r1 - r2 */ + if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) { + goto ERR; + } + /* r3 - r2 */ + if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) { + goto ERR; + } + /* r1/3 */ + if ((res = mp_div_3(&w1, &w1, NULL)) != MP_OKAY) { + goto ERR; + } + /* r3/3 */ + if ((res = mp_div_3(&w3, &w3, NULL)) != MP_OKAY) { + goto ERR; + } + + /* at this point shift W[n] by B*n */ + if ((res = mp_lshd(&w1, 1*B)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_lshd(&w2, 2*B)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_lshd(&w3, 3*B)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_lshd(&w4, 4*B)) != MP_OKAY) { + goto ERR; + } + + if ((res = mp_add(&w0, &w1, c)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&w2, &w3, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&w4, &tmp1, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&tmp1, c, c)) != MP_OKAY) { + goto ERR; + } + +ERR: + mp_clear_multi(&w0, &w1, &w2, &w3, &w4, + &a0, &a1, &a2, &b0, &b1, + &b2, &tmp1, &tmp2, NULL); + return res; +} + +#endif + +/* End: bn_mp_toom_mul.c */ + +/* Start: bn_mp_toom_sqr.c */ +#include <tommath.h> +#ifdef BN_MP_TOOM_SQR_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* squaring using Toom-Cook 3-way algorithm */ +int +mp_toom_sqr(mp_int *a, mp_int *b) +{ + mp_int w0, w1, w2, w3, w4, tmp1, a0, a1, a2; + int res, B; + + /* init temps */ + if ((res = mp_init_multi(&w0, &w1, &w2, &w3, &w4, &a0, &a1, &a2, &tmp1, NULL)) != MP_OKAY) { + return res; + } + + /* B */ + B = a->used / 3; + + /* a = a2 * B**2 + a1 * B + a0 */ + if ((res = mp_mod_2d(a, DIGIT_BIT * B, &a0)) != MP_OKAY) { + goto ERR; + } + + if ((res = mp_copy(a, &a1)) != MP_OKAY) { + goto ERR; + } + mp_rshd(&a1, B); + mp_mod_2d(&a1, DIGIT_BIT * B, &a1); + + if ((res = mp_copy(a, &a2)) != MP_OKAY) { + goto ERR; + } + mp_rshd(&a2, B*2); + + /* w0 = a0*a0 */ + if ((res = mp_sqr(&a0, &w0)) != MP_OKAY) { + goto ERR; + } + + /* w4 = a2 * a2 */ + if ((res = mp_sqr(&a2, &w4)) != MP_OKAY) { + goto ERR; + } + + /* w1 = (a2 + 2(a1 + 2a0))**2 */ + if ((res = mp_mul_2(&a0, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&tmp1, &a2, &tmp1)) != MP_OKAY) { + goto ERR; + } + + if ((res = mp_sqr(&tmp1, &w1)) != MP_OKAY) { + goto ERR; + } + + /* w3 = (a0 + 2(a1 + 2a2))**2 */ + if ((res = mp_mul_2(&a2, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) { + goto ERR; + } + + if ((res = mp_sqr(&tmp1, &w3)) != MP_OKAY) { + goto ERR; + } + + + /* w2 = (a2 + a1 + a0)**2 */ + if ((res = mp_add(&a2, &a1, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_sqr(&tmp1, &w2)) != MP_OKAY) { + goto ERR; + } + + /* now solve the matrix + + 0 0 0 0 1 + 1 2 4 8 16 + 1 1 1 1 1 + 16 8 4 2 1 + 1 0 0 0 0 + + using 12 subtractions, 4 shifts, 2 small divisions and 1 small multiplication. + */ + + /* r1 - r4 */ + if ((res = mp_sub(&w1, &w4, &w1)) != MP_OKAY) { + goto ERR; + } + /* r3 - r0 */ + if ((res = mp_sub(&w3, &w0, &w3)) != MP_OKAY) { + goto ERR; + } + /* r1/2 */ + if ((res = mp_div_2(&w1, &w1)) != MP_OKAY) { + goto ERR; + } + /* r3/2 */ + if ((res = mp_div_2(&w3, &w3)) != MP_OKAY) { + goto ERR; + } + /* r2 - r0 - r4 */ + if ((res = mp_sub(&w2, &w0, &w2)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_sub(&w2, &w4, &w2)) != MP_OKAY) { + goto ERR; + } + /* r1 - r2 */ + if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) { + goto ERR; + } + /* r3 - r2 */ + if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) { + goto ERR; + } + /* r1 - 8r0 */ + if ((res = mp_mul_2d(&w0, 3, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_sub(&w1, &tmp1, &w1)) != MP_OKAY) { + goto ERR; + } + /* r3 - 8r4 */ + if ((res = mp_mul_2d(&w4, 3, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_sub(&w3, &tmp1, &w3)) != MP_OKAY) { + goto ERR; + } + /* 3r2 - r1 - r3 */ + if ((res = mp_mul_d(&w2, 3, &w2)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_sub(&w2, &w1, &w2)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_sub(&w2, &w3, &w2)) != MP_OKAY) { + goto ERR; + } + /* r1 - r2 */ + if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) { + goto ERR; + } + /* r3 - r2 */ + if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) { + goto ERR; + } + /* r1/3 */ + if ((res = mp_div_3(&w1, &w1, NULL)) != MP_OKAY) { + goto ERR; + } + /* r3/3 */ + if ((res = mp_div_3(&w3, &w3, NULL)) != MP_OKAY) { + goto ERR; + } + + /* at this point shift W[n] by B*n */ + if ((res = mp_lshd(&w1, 1*B)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_lshd(&w2, 2*B)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_lshd(&w3, 3*B)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_lshd(&w4, 4*B)) != MP_OKAY) { + goto ERR; + } + + if ((res = mp_add(&w0, &w1, b)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&w2, &w3, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&w4, &tmp1, &tmp1)) != MP_OKAY) { + goto ERR; + } + if ((res = mp_add(&tmp1, b, b)) != MP_OKAY) { + goto ERR; + } + +ERR: + mp_clear_multi(&w0, &w1, &w2, &w3, &w4, &a0, &a1, &a2, &tmp1, NULL); + return res; +} + +#endif + +/* End: bn_mp_toom_sqr.c */ + +/* Start: bn_mp_toradix.c */ +#include <tommath.h> +#ifdef BN_MP_TORADIX_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* stores a bignum as a ASCII string in a given radix (2..64) */ +int mp_toradix (mp_int * a, char *str, int radix) +{ + int res, digs; + mp_int t; + mp_digit d; + char *_s = str; + + /* check range of the radix */ + if (radix < 2 || radix > 64) { + return MP_VAL; + } + + /* quick out if its zero */ + if (mp_iszero(a) == 1) { + *str++ = '0'; + *str = '\0'; + return MP_OKAY; + } + + if ((res = mp_init_copy (&t, a)) != MP_OKAY) { + return res; + } + + /* if it is negative output a - */ + if (t.sign == MP_NEG) { + ++_s; + *str++ = '-'; + t.sign = MP_ZPOS; + } + + digs = 0; + while (mp_iszero (&t) == 0) { + if ((res = mp_div_d (&t, (mp_digit) radix, &t, &d)) != MP_OKAY) { + mp_clear (&t); + return res; + } + *str++ = mp_s_rmap[d]; + ++digs; + } + + /* reverse the digits of the string. In this case _s points + * to the first digit [exluding the sign] of the number] + */ + bn_reverse ((unsigned char *)_s, digs); + + /* append a NULL so the string is properly terminated */ + *str = '\0'; + + mp_clear (&t); + return MP_OKAY; +} + +#endif + +/* End: bn_mp_toradix.c */ + +/* Start: bn_mp_toradix_n.c */ +#include <tommath.h> +#ifdef BN_MP_TORADIX_N_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* stores a bignum as a ASCII string in a given radix (2..64) + * + * Stores upto maxlen-1 chars and always a NULL byte + */ +int mp_toradix_n(mp_int * a, char *str, int radix, int maxlen) +{ + int res, digs; + mp_int t; + mp_digit d; + char *_s = str; + + /* check range of the maxlen, radix */ + if (maxlen < 3 || radix < 2 || radix > 64) { + return MP_VAL; + } + + /* quick out if its zero */ + if (mp_iszero(a) == 1) { + *str++ = '0'; + *str = '\0'; + return MP_OKAY; + } + + if ((res = mp_init_copy (&t, a)) != MP_OKAY) { + return res; + } + + /* if it is negative output a - */ + if (t.sign == MP_NEG) { + /* we have to reverse our digits later... but not the - sign!! */ + ++_s; + + /* store the flag and mark the number as positive */ + *str++ = '-'; + t.sign = MP_ZPOS; + + /* subtract a char */ + --maxlen; + } + + digs = 0; + while (mp_iszero (&t) == 0) { + if ((res = mp_div_d (&t, (mp_digit) radix, &t, &d)) != MP_OKAY) { + mp_clear (&t); + return res; + } + *str++ = mp_s_rmap[d]; + ++digs; + + if (--maxlen == 1) { + /* no more room */ + break; + } + } + + /* reverse the digits of the string. In this case _s points + * to the first digit [exluding the sign] of the number] + */ + bn_reverse ((unsigned char *)_s, digs); + + /* append a NULL so the string is properly terminated */ + *str = '\0'; + + mp_clear (&t); + return MP_OKAY; +} + +#endif + +/* End: bn_mp_toradix_n.c */ + +/* Start: bn_mp_unsigned_bin_size.c */ +#include <tommath.h> +#ifdef BN_MP_UNSIGNED_BIN_SIZE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* get the size for an unsigned equivalent */ +int +mp_unsigned_bin_size (mp_int * a) +{ + int size = mp_count_bits (a); + return (size / 8 + ((size & 7) != 0 ? 1 : 0)); +} +#endif + +/* End: bn_mp_unsigned_bin_size.c */ + +/* Start: bn_mp_xor.c */ +#include <tommath.h> +#ifdef BN_MP_XOR_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* XOR two ints together */ +int +mp_xor (mp_int * a, mp_int * b, mp_int * c) +{ + int res, ix, px; + mp_int t, *x; + + if (a->used > b->used) { + if ((res = mp_init_copy (&t, a)) != MP_OKAY) { + return res; + } + px = b->used; + x = b; + } else { + if ((res = mp_init_copy (&t, b)) != MP_OKAY) { + return res; + } + px = a->used; + x = a; + } + + for (ix = 0; ix < px; ix++) { + + } + mp_clamp (&t); + mp_exch (c, &t); + mp_clear (&t); + return MP_OKAY; +} +#endif + +/* End: bn_mp_xor.c */ + +/* Start: bn_mp_zero.c */ +#include <tommath.h> +#ifdef BN_MP_ZERO_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* set to zero */ +void +mp_zero (mp_int * a) +{ + a->sign = MP_ZPOS; + a->used = 0; + memset (a->dp, 0, sizeof (mp_digit) * a->alloc); +} +#endif + +/* End: bn_mp_zero.c */ + +/* Start: bn_prime_tab.c */ +#include <tommath.h> +#ifdef BN_PRIME_TAB_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ +const mp_digit ltm_prime_tab[] = { + 0x0002, 0x0003, 0x0005, 0x0007, 0x000B, 0x000D, 0x0011, 0x0013, + 0x0017, 0x001D, 0x001F, 0x0025, 0x0029, 0x002B, 0x002F, 0x0035, + 0x003B, 0x003D, 0x0043, 0x0047, 0x0049, 0x004F, 0x0053, 0x0059, + 0x0061, 0x0065, 0x0067, 0x006B, 0x006D, 0x0071, 0x007F, +#ifndef MP_8BIT + 0x0083, + 0x0089, 0x008B, 0x0095, 0x0097, 0x009D, 0x00A3, 0x00A7, 0x00AD, + 0x00B3, 0x00B5, 0x00BF, 0x00C1, 0x00C5, 0x00C7, 0x00D3, 0x00DF, + 0x00E3, 0x00E5, 0x00E9, 0x00EF, 0x00F1, 0x00FB, 0x0101, 0x0107, + 0x010D, 0x010F, 0x0115, 0x0119, 0x011B, 0x0125, 0x0133, 0x0137, + + 0x0139, 0x013D, 0x014B, 0x0151, 0x015B, 0x015D, 0x0161, 0x0167, + 0x016F, 0x0175, 0x017B, 0x017F, 0x0185, 0x018D, 0x0191, 0x0199, + 0x01A3, 0x01A5, 0x01AF, 0x01B1, 0x01B7, 0x01BB, 0x01C1, 0x01C9, + 0x01CD, 0x01CF, 0x01D3, 0x01DF, 0x01E7, 0x01EB, 0x01F3, 0x01F7, + 0x01FD, 0x0209, 0x020B, 0x021D, 0x0223, 0x022D, 0x0233, 0x0239, + 0x023B, 0x0241, 0x024B, 0x0251, 0x0257, 0x0259, 0x025F, 0x0265, + 0x0269, 0x026B, 0x0277, 0x0281, 0x0283, 0x0287, 0x028D, 0x0293, + 0x0295, 0x02A1, 0x02A5, 0x02AB, 0x02B3, 0x02BD, 0x02C5, 0x02CF, + + 0x02D7, 0x02DD, 0x02E3, 0x02E7, 0x02EF, 0x02F5, 0x02F9, 0x0301, + 0x0305, 0x0313, 0x031D, 0x0329, 0x032B, 0x0335, 0x0337, 0x033B, + 0x033D, 0x0347, 0x0355, 0x0359, 0x035B, 0x035F, 0x036D, 0x0371, + 0x0373, 0x0377, 0x038B, 0x038F, 0x0397, 0x03A1, 0x03A9, 0x03AD, + 0x03B3, 0x03B9, 0x03C7, 0x03CB, 0x03D1, 0x03D7, 0x03DF, 0x03E5, + 0x03F1, 0x03F5, 0x03FB, 0x03FD, 0x0407, 0x0409, 0x040F, 0x0419, + 0x041B, 0x0425, 0x0427, 0x042D, 0x043F, 0x0443, 0x0445, 0x0449, + 0x044F, 0x0455, 0x045D, 0x0463, 0x0469, 0x047F, 0x0481, 0x048B, + + 0x0493, 0x049D, 0x04A3, 0x04A9, 0x04B1, 0x04BD, 0x04C1, 0x04C7, + 0x04CD, 0x04CF, 0x04D5, 0x04E1, 0x04EB, 0x04FD, 0x04FF, 0x0503, + 0x0509, 0x050B, 0x0511, 0x0515, 0x0517, 0x051B, 0x0527, 0x0529, + 0x052F, 0x0551, 0x0557, 0x055D, 0x0565, 0x0577, 0x0581, 0x058F, + 0x0593, 0x0595, 0x0599, 0x059F, 0x05A7, 0x05AB, 0x05AD, 0x05B3, + 0x05BF, 0x05C9, 0x05CB, 0x05CF, 0x05D1, 0x05D5, 0x05DB, 0x05E7, + 0x05F3, 0x05FB, 0x0607, 0x060D, 0x0611, 0x0617, 0x061F, 0x0623, + 0x062B, 0x062F, 0x063D, 0x0641, 0x0647, 0x0649, 0x064D, 0x0653 +#endif +}; +#endif + +/* End: bn_prime_tab.c */ + +/* Start: bn_reverse.c */ +#include <tommath.h> +#ifdef BN_REVERSE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* reverse an array, used for radix code */ +void +bn_reverse (unsigned char *s, int len) +{ + int ix, iy; + unsigned char t; + + ix = 0; + iy = len - 1; + while (ix < iy) { + t = s[ix]; + s[ix] = s[iy]; + s[iy] = t; + ++ix; + --iy; + } +} +#endif + +/* End: bn_reverse.c */ + +/* Start: bn_s_mp_add.c */ +#include <tommath.h> +#ifdef BN_S_MP_ADD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* low level addition, based on HAC pp.594, Algorithm 14.7 */ +int +s_mp_add (mp_int * a, mp_int * b, mp_int * c) +{ + mp_int *x; + int olduse, res, min, max; + + /* find sizes, we let |a| <= |b| which means we have to sort + * them. "x" will point to the input with the most digits + */ + if (a->used > b->used) { + min = b->used; + max = a->used; + x = a; + } else { + min = a->used; + max = b->used; + x = b; + } + + /* init result */ + if (c->alloc < max + 1) { + if ((res = mp_grow (c, max + 1)) != MP_OKAY) { + return res; + } + } + + /* get old used digit count and set new one */ + olduse = c->used; + c->used = max + 1; + + { + register mp_digit u, *tmpa, *tmpb, *tmpc; + register int i; + + /* alias for digit pointers */ + + /* first input */ + tmpa = a->dp; + + /* second input */ + tmpb = b->dp; + + /* destination */ + tmpc = c->dp; + + /* zero the carry */ + u = 0; + for (i = 0; i < min; i++) { + /* Compute the sum at one digit, T[i] = A[i] + B[i] + U */ + *tmpc = *tmpa++ + *tmpb++ + u; + + /* U = carry bit of T[i] */ + u = *tmpc >> ((mp_digit)DIGIT_BIT); + + /* take away carry bit from T[i] */ + *tmpc++ &= MP_MASK; + } + + /* now copy higher words if any, that is in A+B + * if A or B has more digits add those in + */ + if (min != max) { + for (; i < max; i++) { + /* T[i] = X[i] + U */ + *tmpc = x->dp[i] + u; + + /* U = carry bit of T[i] */ + u = *tmpc >> ((mp_digit)DIGIT_BIT); + + /* take away carry bit from T[i] */ + *tmpc++ &= MP_MASK; + } + } + + /* add carry */ + *tmpc++ = u; + + /* clear digits above oldused */ + for (i = c->used; i < olduse; i++) { + *tmpc++ = 0; + } + } + + mp_clamp (c); + return MP_OKAY; +} +#endif + +/* End: bn_s_mp_add.c */ + +/* Start: bn_s_mp_exptmod.c */ +#include <tommath.h> +#ifdef BN_S_MP_EXPTMOD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +#ifdef MP_LOW_MEM + #define TAB_SIZE 32 +#else + #define TAB_SIZE 256 +#endif + +int s_mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y) +{ + mp_int M[TAB_SIZE], res, mu; + mp_digit buf; + int err, bitbuf, bitcpy, bitcnt, mode, digidx, x, y, winsize; + + /* find window size */ + x = mp_count_bits (X); + if (x <= 7) { + winsize = 2; + } else if (x <= 36) { + winsize = 3; + } else if (x <= 140) { + winsize = 4; + } else if (x <= 450) { + winsize = 5; + } else if (x <= 1303) { + winsize = 6; + } else if (x <= 3529) { + winsize = 7; + } else { + winsize = 8; + } + +#ifdef MP_LOW_MEM + if (winsize > 5) { + winsize = 5; + } +#endif + + /* init M array */ + /* init first cell */ + if ((err = mp_init(&M[1])) != MP_OKAY) { + return err; + } + + /* now init the second half of the array */ + for (x = 1<<(winsize-1); x < (1 << winsize); x++) { + if ((err = mp_init(&M[x])) != MP_OKAY) { + for (y = 1<<(winsize-1); y < x; y++) { + mp_clear (&M[y]); + } + mp_clear(&M[1]); + return err; + } + } + + /* create mu, used for Barrett reduction */ + if ((err = mp_init (&mu)) != MP_OKAY) { + goto LBL_M; + } + if ((err = mp_reduce_setup (&mu, P)) != MP_OKAY) { + goto LBL_MU; + } + + /* create M table + * + * The M table contains powers of the base, + * e.g. M[x] = G**x mod P + * + * The first half of the table is not + * computed though accept for M[0] and M[1] + */ + if ((err = mp_mod (G, P, &M[1])) != MP_OKAY) { + goto LBL_MU; + } + + /* compute the value at M[1<<(winsize-1)] by squaring + * M[1] (winsize-1) times + */ + if ((err = mp_copy (&M[1], &M[1 << (winsize - 1)])) != MP_OKAY) { + goto LBL_MU; + } + + for (x = 0; x < (winsize - 1); x++) { + if ((err = mp_sqr (&M[1 << (winsize - 1)], + &M[1 << (winsize - 1)])) != MP_OKAY) { + goto LBL_MU; + } + if ((err = mp_reduce (&M[1 << (winsize - 1)], P, &mu)) != MP_OKAY) { + goto LBL_MU; + } + } + + /* create upper table, that is M[x] = M[x-1] * M[1] (mod P) + * for x = (2**(winsize - 1) + 1) to (2**winsize - 1) + */ + for (x = (1 << (winsize - 1)) + 1; x < (1 << winsize); x++) { + if ((err = mp_mul (&M[x - 1], &M[1], &M[x])) != MP_OKAY) { + goto LBL_MU; + } + if ((err = mp_reduce (&M[x], P, &mu)) != MP_OKAY) { + goto LBL_MU; + } + } + + /* setup result */ + if ((err = mp_init (&res)) != MP_OKAY) { + goto LBL_MU; + } + mp_set (&res, 1); + + /* set initial mode and bit cnt */ + mode = 0; + bitcnt = 1; + buf = 0; + digidx = X->used - 1; + bitcpy = 0; + bitbuf = 0; + + for (;;) { + /* grab next digit as required */ + if (--bitcnt == 0) { + /* if digidx == -1 we are out of digits */ + if (digidx == -1) { + break; + } + /* read next digit and reset the bitcnt */ + buf = X->dp[digidx--]; + bitcnt = (int) DIGIT_BIT; + } + + /* grab the next msb from the exponent */ + y = (buf >> (mp_digit)(DIGIT_BIT - 1)) & 1; + buf <<= (mp_digit)1; + + /* if the bit is zero and mode == 0 then we ignore it + * These represent the leading zero bits before the first 1 bit + * in the exponent. Technically this opt is not required but it + * does lower the # of trivial squaring/reductions used + */ + if (mode == 0 && y == 0) { + continue; + } + + /* if the bit is zero and mode == 1 then we square */ + if (mode == 1 && y == 0) { + if ((err = mp_sqr (&res, &res)) != MP_OKAY) { + goto LBL_RES; + } + if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) { + goto LBL_RES; + } + continue; + } + + /* else we add it to the window */ + bitbuf |= (y << (winsize - ++bitcpy)); + mode = 2; + + if (bitcpy == winsize) { + /* ok window is filled so square as required and multiply */ + /* square first */ + for (x = 0; x < winsize; x++) { + if ((err = mp_sqr (&res, &res)) != MP_OKAY) { + goto LBL_RES; + } + if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) { + goto LBL_RES; + } + } + + /* then multiply */ + if ((err = mp_mul (&res, &M[bitbuf], &res)) != MP_OKAY) { + goto LBL_RES; + } + if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) { + goto LBL_RES; + } + + /* empty window and reset */ + bitcpy = 0; + bitbuf = 0; + mode = 1; + } + } + + /* if bits remain then square/multiply */ + if (mode == 2 && bitcpy > 0) { + /* square then multiply if the bit is set */ + for (x = 0; x < bitcpy; x++) { + if ((err = mp_sqr (&res, &res)) != MP_OKAY) { + goto LBL_RES; + } + if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) { + goto LBL_RES; + } + + bitbuf <<= 1; + if ((bitbuf & (1 << winsize)) != 0) { + /* then multiply */ + if ((err = mp_mul (&res, &M[1], &res)) != MP_OKAY) { + goto LBL_RES; + } + if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) { + goto LBL_RES; + } + } + } + } + + mp_exch (&res, Y); + err = MP_OKAY; +LBL_RES:mp_clear (&res); +LBL_MU:mp_clear (&mu); +LBL_M: + mp_clear(&M[1]); + for (x = 1<<(winsize-1); x < (1 << winsize); x++) { + mp_clear (&M[x]); + } + return err; +} +#endif + +/* End: bn_s_mp_exptmod.c */ + +/* Start: bn_s_mp_mul_digs.c */ +#include <tommath.h> +#ifdef BN_S_MP_MUL_DIGS_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* multiplies |a| * |b| and only computes upto digs digits of result + * HAC pp. 595, Algorithm 14.12 Modified so you can control how + * many digits of output are created. + */ +int +s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs) +{ + mp_int t; + int res, pa, pb, ix, iy; + mp_digit u; + mp_word r; + mp_digit tmpx, *tmpt, *tmpy; + + /* can we use the fast multiplier? */ + if (((digs) < MP_WARRAY) && + MIN (a->used, b->used) < + (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) { + return fast_s_mp_mul_digs (a, b, c, digs); + } + + if ((res = mp_init_size (&t, digs)) != MP_OKAY) { + return res; + } + t.used = digs; + + /* compute the digits of the product directly */ + pa = a->used; + for (ix = 0; ix < pa; ix++) { + /* set the carry to zero */ + u = 0; + + /* limit ourselves to making digs digits of output */ + pb = MIN (b->used, digs - ix); + + /* setup some aliases */ + /* copy of the digit from a used within the nested loop */ + tmpx = a->dp[ix]; + + /* an alias for the destination shifted ix places */ + tmpt = t.dp + ix; + + /* an alias for the digits of b */ + tmpy = b->dp; + + /* compute the columns of the output and propagate the carry */ + for (iy = 0; iy < pb; iy++) { + /* compute the column as a mp_word */ + r = ((mp_word)*tmpt) + + ((mp_word)tmpx) * ((mp_word)*tmpy++) + + ((mp_word) u); + + /* the new column is the lower part of the result */ + *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK)); + + /* get the carry word from the result */ + u = (mp_digit) (r >> ((mp_word) DIGIT_BIT)); + } + /* set carry if it is placed below digs */ + if (ix + iy < digs) { + *tmpt = u; + } + } + + mp_clamp (&t); + mp_exch (&t, c); + + mp_clear (&t); + return MP_OKAY; +} +#endif + +/* End: bn_s_mp_mul_digs.c */ + +/* Start: bn_s_mp_mul_high_digs.c */ +#include <tommath.h> +#ifdef BN_S_MP_MUL_HIGH_DIGS_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* multiplies |a| * |b| and does not compute the lower digs digits + * [meant to get the higher part of the product] + */ +int +s_mp_mul_high_digs (mp_int * a, mp_int * b, mp_int * c, int digs) +{ + mp_int t; + int res, pa, pb, ix, iy; + mp_digit u; + mp_word r; + mp_digit tmpx, *tmpt, *tmpy; + + /* can we use the fast multiplier? */ +#ifdef BN_FAST_S_MP_MUL_HIGH_DIGS_C + if (((a->used + b->used + 1) < MP_WARRAY) + && MIN (a->used, b->used) < (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) { + return fast_s_mp_mul_high_digs (a, b, c, digs); + } +#endif + + if ((res = mp_init_size (&t, a->used + b->used + 1)) != MP_OKAY) { + return res; + } + t.used = a->used + b->used + 1; + + pa = a->used; + pb = b->used; + for (ix = 0; ix < pa; ix++) { + /* clear the carry */ + u = 0; + + /* left hand side of A[ix] * B[iy] */ + tmpx = a->dp[ix]; + + /* alias to the address of where the digits will be stored */ + tmpt = &(t.dp[digs]); + + /* alias for where to read the right hand side from */ + tmpy = b->dp + (digs - ix); + + for (iy = digs - ix; iy < pb; iy++) { + /* calculate the double precision result */ + r = ((mp_word)*tmpt) + + ((mp_word)tmpx) * ((mp_word)*tmpy++) + + ((mp_word) u); + + /* get the lower part */ + *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK)); + + /* carry the carry */ + u = (mp_digit) (r >> ((mp_word) DIGIT_BIT)); + } + *tmpt = u; + } + mp_clamp (&t); + mp_exch (&t, c); + mp_clear (&t); + return MP_OKAY; +} +#endif + +/* End: bn_s_mp_mul_high_digs.c */ + +/* Start: bn_s_mp_sqr.c */ +#include <tommath.h> +#ifdef BN_S_MP_SQR_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* low level squaring, b = a*a, HAC pp.596-597, Algorithm 14.16 */ +int +s_mp_sqr (mp_int * a, mp_int * b) +{ + mp_int t; + int res, ix, iy, pa; + mp_word r; + mp_digit u, tmpx, *tmpt; + + pa = a->used; + if ((res = mp_init_size (&t, 2*pa + 1)) != MP_OKAY) { + return res; + } + + /* default used is maximum possible size */ + t.used = 2*pa + 1; + + for (ix = 0; ix < pa; ix++) { + /* first calculate the digit at 2*ix */ + /* calculate double precision result */ + r = ((mp_word) t.dp[2*ix]) + + ((mp_word)a->dp[ix])*((mp_word)a->dp[ix]); + + /* store lower part in result */ + t.dp[ix+ix] = (mp_digit) (r & ((mp_word) MP_MASK)); + + /* get the carry */ + u = (mp_digit)(r >> ((mp_word) DIGIT_BIT)); + + /* left hand side of A[ix] * A[iy] */ + tmpx = a->dp[ix]; + + /* alias for where to store the results */ + tmpt = t.dp + (2*ix + 1); + + for (iy = ix + 1; iy < pa; iy++) { + /* first calculate the product */ + r = ((mp_word)tmpx) * ((mp_word)a->dp[iy]); + + /* now calculate the double precision result, note we use + * addition instead of *2 since it's easier to optimize + */ + r = ((mp_word) *tmpt) + r + r + ((mp_word) u); + + /* store lower part */ + *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK)); + + /* get carry */ + u = (mp_digit)(r >> ((mp_word) DIGIT_BIT)); + } + /* propagate upwards */ + while (u != ((mp_digit) 0)) { + r = ((mp_word) *tmpt) + ((mp_word) u); + *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK)); + u = (mp_digit)(r >> ((mp_word) DIGIT_BIT)); + } + } + + mp_clamp (&t); + mp_exch (&t, b); + mp_clear (&t); + return MP_OKAY; +} +#endif + +/* End: bn_s_mp_sqr.c */ + +/* Start: bn_s_mp_sub.c */ +#include <tommath.h> +#ifdef BN_S_MP_SUB_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* low level subtraction (assumes |a| > |b|), HAC pp.595 Algorithm 14.9 */ +int +s_mp_sub (mp_int * a, mp_int * b, mp_int * c) +{ + int olduse, res, min, max; + + /* find sizes */ + min = b->used; + max = a->used; + + /* init result */ + if (c->alloc < max) { + if ((res = mp_grow (c, max)) != MP_OKAY) { + return res; + } + } + olduse = c->used; + c->used = max; + + { + register mp_digit u, *tmpa, *tmpb, *tmpc; + register int i; + + /* alias for digit pointers */ + tmpa = a->dp; + tmpb = b->dp; + tmpc = c->dp; + + /* set carry to zero */ + u = 0; + for (i = 0; i < min; i++) { + /* T[i] = A[i] - B[i] - U */ + *tmpc = *tmpa++ - *tmpb++ - u; + + /* U = carry bit of T[i] + * Note this saves performing an AND operation since + * if a carry does occur it will propagate all the way to the + * MSB. As a result a single shift is enough to get the carry + */ + u = *tmpc >> ((mp_digit)(CHAR_BIT * sizeof (mp_digit) - 1)); + + /* Clear carry from T[i] */ + *tmpc++ &= MP_MASK; + } + + /* now copy higher words if any, e.g. if A has more digits than B */ + for (; i < max; i++) { + /* T[i] = A[i] - U */ + *tmpc = *tmpa++ - u; + + /* U = carry bit of T[i] */ + u = *tmpc >> ((mp_digit)(CHAR_BIT * sizeof (mp_digit) - 1)); + + /* Clear carry from T[i] */ + *tmpc++ &= MP_MASK; + } + + /* clear digits above used (since we may not have grown result above) */ + for (i = c->used; i < olduse; i++) { + *tmpc++ = 0; + } + } + + mp_clamp (c); + return MP_OKAY; +} + +#endif + +/* End: bn_s_mp_sub.c */ + +/* Start: bncore.c */ +#include <tommath.h> +#ifdef BNCORE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ + +/* Known optimal configurations + + CPU /Compiler /MUL CUTOFF/SQR CUTOFF +------------------------------------------------------------- + Intel P4 Northwood /GCC v3.4.1 / 88/ 128/LTM 0.32 ;-) + +*/ + +int KARATSUBA_MUL_CUTOFF = 88, /* Min. number of digits before Karatsuba multiplication is used. */ + KARATSUBA_SQR_CUTOFF = 128, /* Min. number of digits before Karatsuba squaring is used. */ + + TOOM_MUL_CUTOFF = 350, /* no optimal values of these are known yet so set em high */ + TOOM_SQR_CUTOFF = 400; +#endif + +/* End: bncore.c */ + + +/* EOF */ diff --git a/libtommath/pretty.build b/libtommath/pretty.build new file mode 100644 index 0000000..a708b8a --- /dev/null +++ b/libtommath/pretty.build @@ -0,0 +1,66 @@ +#!/bin/perl -w +# +# Cute little builder for perl +# Total waste of development time... +# +# This will build all the object files and then the archive .a file +# requires GCC, GNU make and a sense of humour. +# +# Tom St Denis +use strict; + +my $count = 0; +my $starttime = time; +my $rate = 0; +print "Scanning for source files...\n"; +foreach my $filename (glob "*.c") { + ++$count; +} +print "Source files to build: $count\nBuilding...\n"; +my $i = 0; +my $lines = 0; +my $filesbuilt = 0; +foreach my $filename (glob "*.c") { + printf("Building %3.2f%%, ", (++$i/$count)*100.0); + if ($i % 4 == 0) { print "/, "; } + if ($i % 4 == 1) { print "-, "; } + if ($i % 4 == 2) { print "\\, "; } + if ($i % 4 == 3) { print "|, "; } + if ($rate > 0) { + my $tleft = ($count - $i) / $rate; + my $tsec = $tleft%60; + my $tmin = ($tleft/60)%60; + my $thour = ($tleft/3600)%60; + printf("%2d:%02d:%02d left, ", $thour, $tmin, $tsec); + } + my $cnt = ($i/$count)*30.0; + my $x = 0; + print "["; + for (; $x < $cnt; $x++) { print "#"; } + for (; $x < 30; $x++) { print " "; } + print "]\r"; + my $tmp = $filename; + $tmp =~ s/\.c/".o"/ge; + if (open(SRC, "<$tmp")) { + close SRC; + } else { + !system("make $tmp > /dev/null 2>/dev/null") or die "\nERROR: Failed to make $tmp!!!\n"; + open( SRC, "<$filename" ) or die "Couldn't open $filename for reading: $!"; + ++$lines while (<SRC>); + close SRC or die "Error closing $filename after reading: $!"; + ++$filesbuilt; + } + + # update timer + if (time != $starttime) { + my $delay = time - $starttime; + $rate = $i/$delay; + } +} + +# finish building the library +printf("\nFinished building source (%d seconds, %3.2f files per second).\n", time - $starttime, $rate); +print "Compiled approximately $filesbuilt files and $lines lines of code.\n"; +print "Doing final make (building archive...)\n"; +!system("make > /dev/null 2>/dev/null") or die "\nERROR: Failed to perform last make command!!!\n"; +print "done.\n";
\ No newline at end of file diff --git a/libtommath/tombc/grammar.txt b/libtommath/tombc/grammar.txt new file mode 100644 index 0000000..a780e75 --- /dev/null +++ b/libtommath/tombc/grammar.txt @@ -0,0 +1,35 @@ +program := program statement | statement | empty +statement := { statement } | + identifier = numexpression; | + identifier[numexpression] = numexpression; | + function(expressionlist); | + for (identifer = numexpression; numexpression; identifier = numexpression) { statement } | + while (numexpression) { statement } | + if (numexpresion) { statement } elif | + break; | + continue; + +elif := else statement | empty +function := abs | countbits | exptmod | jacobi | print | isprime | nextprime | issquare | readinteger | exit +expressionlist := expressionlist, expression | expression + +// LR(1) !!!? +expression := string | numexpression +numexpression := cmpexpr && cmpexpr | cmpexpr \|\| cmpexpr | cmpexpr +cmpexpr := boolexpr < boolexpr | boolexpr > boolexpr | boolexpr == boolexpr | + boolexpr <= boolexpr | boolexpr >= boolexpr | boolexpr +boolexpr := shiftexpr & shiftexpr | shiftexpr ^ shiftexpr | shiftexpr \| shiftexpr | shiftexpr +shiftexpr := addsubexpr << addsubexpr | addsubexpr >> addsubexpr | addsubexpr +addsubexpr := mulexpr + mulexpr | mulexpr - mulexpr | mulexpr +mulexpr := expr * expr | expr / expr | expr % expr | expr +expr := -nexpr | nexpr +nexpr := integer | identifier | ( numexpression ) | identifier[numexpression] + +identifier := identifer digits | identifier alpha | alpha +alpha := a ... z | A ... Z +integer := hexnumber | digits +hexnumber := 0xhexdigits +hexdigits := hexdigits hexdigit | hexdigit +hexdigit := 0 ... 9 | a ... f | A ... F +digits := digits digit | digit +digit := 0 ... 9 diff --git a/libtommath/tommath.h b/libtommath/tommath.h new file mode 100644 index 0000000..7cc92c2 --- /dev/null +++ b/libtommath/tommath.h @@ -0,0 +1,567 @@ +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org + */ +#ifndef BN_H_ +#define BN_H_ + +#include <stdio.h> +#include <string.h> +#include <stdlib.h> +#include <ctype.h> +#include <limits.h> + +#include <tommath_class.h> + +#undef MIN +#define MIN(x,y) ((x)<(y)?(x):(y)) +#undef MAX +#define MAX(x,y) ((x)>(y)?(x):(y)) + +#ifdef __cplusplus +extern "C" { + +/* C++ compilers don't like assigning void * to mp_digit * */ +#define OPT_CAST(x) (x *) + +#else + +/* C on the other hand doesn't care */ +#define OPT_CAST(x) + +#endif + + +/* detect 64-bit mode if possible */ +#if defined(__x86_64__) + #if !(defined(MP_64BIT) && defined(MP_16BIT) && defined(MP_8BIT)) + #define MP_64BIT + #endif +#endif + +/* some default configurations. + * + * A "mp_digit" must be able to hold DIGIT_BIT + 1 bits + * A "mp_word" must be able to hold 2*DIGIT_BIT + 1 bits + * + * At the very least a mp_digit must be able to hold 7 bits + * [any size beyond that is ok provided it doesn't overflow the data type] + */ +#ifdef MP_8BIT + typedef unsigned char mp_digit; + typedef unsigned short mp_word; +#elif defined(MP_16BIT) + typedef unsigned short mp_digit; + typedef unsigned long mp_word; +#elif defined(MP_64BIT) + /* for GCC only on supported platforms */ +#ifndef CRYPT + typedef unsigned long long ulong64; + typedef signed long long long64; +#endif + + typedef unsigned long mp_digit; + typedef unsigned long mp_word __attribute__ ((mode(TI))); + + #define DIGIT_BIT 60 +#else + /* this is the default case, 28-bit digits */ + + /* this is to make porting into LibTomCrypt easier :-) */ +#ifndef CRYPT + #if defined(_MSC_VER) || defined(__BORLANDC__) + typedef unsigned __int64 ulong64; + typedef signed __int64 long64; + #else + typedef unsigned long long ulong64; + typedef signed long long long64; + #endif +#endif + + typedef unsigned long mp_digit; + typedef ulong64 mp_word; + +#ifdef MP_31BIT + /* this is an extension that uses 31-bit digits */ + #define DIGIT_BIT 31 +#else + /* default case is 28-bit digits, defines MP_28BIT as a handy macro to test */ + #define DIGIT_BIT 28 + #define MP_28BIT +#endif +#endif + +/* define heap macros */ +#ifndef CRYPT + /* default to libc stuff */ + #ifndef XMALLOC + #define XMALLOC malloc + #define XFREE free + #define XREALLOC realloc + #define XCALLOC calloc + #else + /* prototypes for our heap functions */ + extern void *XMALLOC(size_t n); + extern void *REALLOC(void *p, size_t n); + extern void *XCALLOC(size_t n, size_t s); + extern void XFREE(void *p); + #endif +#endif + + +/* otherwise the bits per digit is calculated automatically from the size of a mp_digit */ +#ifndef DIGIT_BIT + #define DIGIT_BIT ((int)((CHAR_BIT * sizeof(mp_digit) - 1))) /* bits per digit */ +#endif + +#define MP_DIGIT_BIT DIGIT_BIT +#define MP_MASK ((((mp_digit)1)<<((mp_digit)DIGIT_BIT))-((mp_digit)1)) +#define MP_DIGIT_MAX MP_MASK + +/* equalities */ +#define MP_LT -1 /* less than */ +#define MP_EQ 0 /* equal to */ +#define MP_GT 1 /* greater than */ + +#define MP_ZPOS 0 /* positive integer */ +#define MP_NEG 1 /* negative */ + +#define MP_OKAY 0 /* ok result */ +#define MP_MEM -2 /* out of mem */ +#define MP_VAL -3 /* invalid input */ +#define MP_RANGE MP_VAL + +#define MP_YES 1 /* yes response */ +#define MP_NO 0 /* no response */ + +/* Primality generation flags */ +#define LTM_PRIME_BBS 0x0001 /* BBS style prime */ +#define LTM_PRIME_SAFE 0x0002 /* Safe prime (p-1)/2 == prime */ +#define LTM_PRIME_2MSB_OFF 0x0004 /* force 2nd MSB to 0 */ +#define LTM_PRIME_2MSB_ON 0x0008 /* force 2nd MSB to 1 */ + +typedef int mp_err; + +/* you'll have to tune these... */ +extern int KARATSUBA_MUL_CUTOFF, + KARATSUBA_SQR_CUTOFF, + TOOM_MUL_CUTOFF, + TOOM_SQR_CUTOFF; + +/* define this to use lower memory usage routines (exptmods mostly) */ +/* #define MP_LOW_MEM */ + +/* default precision */ +#ifndef MP_PREC + #ifndef MP_LOW_MEM + #define MP_PREC 64 /* default digits of precision */ + #else + #define MP_PREC 8 /* default digits of precision */ + #endif +#endif + +/* size of comba arrays, should be at least 2 * 2**(BITS_PER_WORD - BITS_PER_DIGIT*2) */ +#define MP_WARRAY (1 << (sizeof(mp_word) * CHAR_BIT - 2 * DIGIT_BIT + 1)) + +/* the infamous mp_int structure */ +typedef struct { + int used, alloc, sign; + mp_digit *dp; +} mp_int; + +/* callback for mp_prime_random, should fill dst with random bytes and return how many read [upto len] */ +typedef int ltm_prime_callback(unsigned char *dst, int len, void *dat); + + +#define USED(m) ((m)->used) +#define DIGIT(m,k) ((m)->dp[(k)]) +#define SIGN(m) ((m)->sign) + +/* error code to char* string */ +char *mp_error_to_string(int code); + +/* ---> init and deinit bignum functions <--- */ +/* init a bignum */ +int mp_init(mp_int *a); + +/* free a bignum */ +void mp_clear(mp_int *a); + +/* init a null terminated series of arguments */ +int mp_init_multi(mp_int *mp, ...); + +/* clear a null terminated series of arguments */ +void mp_clear_multi(mp_int *mp, ...); + +/* exchange two ints */ +void mp_exch(mp_int *a, mp_int *b); + +/* shrink ram required for a bignum */ +int mp_shrink(mp_int *a); + +/* grow an int to a given size */ +int mp_grow(mp_int *a, int size); + +/* init to a given number of digits */ +int mp_init_size(mp_int *a, int size); + +/* ---> Basic Manipulations <--- */ +#define mp_iszero(a) (((a)->used == 0) ? MP_YES : MP_NO) +#define mp_iseven(a) (((a)->used > 0 && (((a)->dp[0] & 1) == 0)) ? MP_YES : MP_NO) +#define mp_isodd(a) (((a)->used > 0 && (((a)->dp[0] & 1) == 1)) ? MP_YES : MP_NO) + +/* set to zero */ +void mp_zero(mp_int *a); + +/* set to a digit */ +void mp_set(mp_int *a, mp_digit b); + +/* set a 32-bit const */ +int mp_set_int(mp_int *a, unsigned long b); + +/* get a 32-bit value */ +unsigned long mp_get_int(mp_int * a); + +/* initialize and set a digit */ +int mp_init_set (mp_int * a, mp_digit b); + +/* initialize and set 32-bit value */ +int mp_init_set_int (mp_int * a, unsigned long b); + +/* copy, b = a */ +int mp_copy(mp_int *a, mp_int *b); + +/* inits and copies, a = b */ +int mp_init_copy(mp_int *a, mp_int *b); + +/* trim unused digits */ +void mp_clamp(mp_int *a); + +/* ---> digit manipulation <--- */ + +/* right shift by "b" digits */ +void mp_rshd(mp_int *a, int b); + +/* left shift by "b" digits */ +int mp_lshd(mp_int *a, int b); + +/* c = a / 2**b */ +int mp_div_2d(mp_int *a, int b, mp_int *c, mp_int *d); + +/* b = a/2 */ +int mp_div_2(mp_int *a, mp_int *b); + +/* c = a * 2**b */ +int mp_mul_2d(mp_int *a, int b, mp_int *c); + +/* b = a*2 */ +int mp_mul_2(mp_int *a, mp_int *b); + +/* c = a mod 2**d */ +int mp_mod_2d(mp_int *a, int b, mp_int *c); + +/* computes a = 2**b */ +int mp_2expt(mp_int *a, int b); + +/* Counts the number of lsbs which are zero before the first zero bit */ +int mp_cnt_lsb(mp_int *a); + +/* I Love Earth! */ + +/* makes a pseudo-random int of a given size */ +int mp_rand(mp_int *a, int digits); + +/* ---> binary operations <--- */ +/* c = a XOR b */ +int mp_xor(mp_int *a, mp_int *b, mp_int *c); + +/* c = a OR b */ +int mp_or(mp_int *a, mp_int *b, mp_int *c); + +/* c = a AND b */ +int mp_and(mp_int *a, mp_int *b, mp_int *c); + +/* ---> Basic arithmetic <--- */ + +/* b = -a */ +int mp_neg(mp_int *a, mp_int *b); + +/* b = |a| */ +int mp_abs(mp_int *a, mp_int *b); + +/* compare a to b */ +int mp_cmp(mp_int *a, mp_int *b); + +/* compare |a| to |b| */ +int mp_cmp_mag(mp_int *a, mp_int *b); + +/* c = a + b */ +int mp_add(mp_int *a, mp_int *b, mp_int *c); + +/* c = a - b */ +int mp_sub(mp_int *a, mp_int *b, mp_int *c); + +/* c = a * b */ +int mp_mul(mp_int *a, mp_int *b, mp_int *c); + +/* b = a*a */ +int mp_sqr(mp_int *a, mp_int *b); + +/* a/b => cb + d == a */ +int mp_div(mp_int *a, mp_int *b, mp_int *c, mp_int *d); + +/* c = a mod b, 0 <= c < b */ +int mp_mod(mp_int *a, mp_int *b, mp_int *c); + +/* ---> single digit functions <--- */ + +/* compare against a single digit */ +int mp_cmp_d(mp_int *a, mp_digit b); + +/* c = a + b */ +int mp_add_d(mp_int *a, mp_digit b, mp_int *c); + +/* c = a - b */ +int mp_sub_d(mp_int *a, mp_digit b, mp_int *c); + +/* c = a * b */ +int mp_mul_d(mp_int *a, mp_digit b, mp_int *c); + +/* a/b => cb + d == a */ +int mp_div_d(mp_int *a, mp_digit b, mp_int *c, mp_digit *d); + +/* a/3 => 3c + d == a */ +int mp_div_3(mp_int *a, mp_int *c, mp_digit *d); + +/* c = a**b */ +int mp_expt_d(mp_int *a, mp_digit b, mp_int *c); + +/* c = a mod b, 0 <= c < b */ +int mp_mod_d(mp_int *a, mp_digit b, mp_digit *c); + +/* ---> number theory <--- */ + +/* d = a + b (mod c) */ +int mp_addmod(mp_int *a, mp_int *b, mp_int *c, mp_int *d); + +/* d = a - b (mod c) */ +int mp_submod(mp_int *a, mp_int *b, mp_int *c, mp_int *d); + +/* d = a * b (mod c) */ +int mp_mulmod(mp_int *a, mp_int *b, mp_int *c, mp_int *d); + +/* c = a * a (mod b) */ +int mp_sqrmod(mp_int *a, mp_int *b, mp_int *c); + +/* c = 1/a (mod b) */ +int mp_invmod(mp_int *a, mp_int *b, mp_int *c); + +/* c = (a, b) */ +int mp_gcd(mp_int *a, mp_int *b, mp_int *c); + +/* produces value such that U1*a + U2*b = U3 */ +int mp_exteuclid(mp_int *a, mp_int *b, mp_int *U1, mp_int *U2, mp_int *U3); + +/* c = [a, b] or (a*b)/(a, b) */ +int mp_lcm(mp_int *a, mp_int *b, mp_int *c); + +/* finds one of the b'th root of a, such that |c|**b <= |a| + * + * returns error if a < 0 and b is even + */ +int mp_n_root(mp_int *a, mp_digit b, mp_int *c); + +/* special sqrt algo */ +int mp_sqrt(mp_int *arg, mp_int *ret); + +/* is number a square? */ +int mp_is_square(mp_int *arg, int *ret); + +/* computes the jacobi c = (a | n) (or Legendre if b is prime) */ +int mp_jacobi(mp_int *a, mp_int *n, int *c); + +/* used to setup the Barrett reduction for a given modulus b */ +int mp_reduce_setup(mp_int *a, mp_int *b); + +/* Barrett Reduction, computes a (mod b) with a precomputed value c + * + * Assumes that 0 < a <= b*b, note if 0 > a > -(b*b) then you can merely + * compute the reduction as -1 * mp_reduce(mp_abs(a)) [pseudo code]. + */ +int mp_reduce(mp_int *a, mp_int *b, mp_int *c); + +/* setups the montgomery reduction */ +int mp_montgomery_setup(mp_int *a, mp_digit *mp); + +/* computes a = B**n mod b without division or multiplication useful for + * normalizing numbers in a Montgomery system. + */ +int mp_montgomery_calc_normalization(mp_int *a, mp_int *b); + +/* computes x/R == x (mod N) via Montgomery Reduction */ +int mp_montgomery_reduce(mp_int *a, mp_int *m, mp_digit mp); + +/* returns 1 if a is a valid DR modulus */ +int mp_dr_is_modulus(mp_int *a); + +/* sets the value of "d" required for mp_dr_reduce */ +void mp_dr_setup(mp_int *a, mp_digit *d); + +/* reduces a modulo b using the Diminished Radix method */ +int mp_dr_reduce(mp_int *a, mp_int *b, mp_digit mp); + +/* returns true if a can be reduced with mp_reduce_2k */ +int mp_reduce_is_2k(mp_int *a); + +/* determines k value for 2k reduction */ +int mp_reduce_2k_setup(mp_int *a, mp_digit *d); + +/* reduces a modulo b where b is of the form 2**p - k [0 <= a] */ +int mp_reduce_2k(mp_int *a, mp_int *n, mp_digit d); + +/* d = a**b (mod c) */ +int mp_exptmod(mp_int *a, mp_int *b, mp_int *c, mp_int *d); + +/* ---> Primes <--- */ + +/* number of primes */ +#ifdef MP_8BIT + #define PRIME_SIZE 31 +#else + #define PRIME_SIZE 256 +#endif + +/* table of first PRIME_SIZE primes */ +extern const mp_digit ltm_prime_tab[]; + +/* result=1 if a is divisible by one of the first PRIME_SIZE primes */ +int mp_prime_is_divisible(mp_int *a, int *result); + +/* performs one Fermat test of "a" using base "b". + * Sets result to 0 if composite or 1 if probable prime + */ +int mp_prime_fermat(mp_int *a, mp_int *b, int *result); + +/* performs one Miller-Rabin test of "a" using base "b". + * Sets result to 0 if composite or 1 if probable prime + */ +int mp_prime_miller_rabin(mp_int *a, mp_int *b, int *result); + +/* This gives [for a given bit size] the number of trials required + * such that Miller-Rabin gives a prob of failure lower than 2^-96 + */ +int mp_prime_rabin_miller_trials(int size); + +/* performs t rounds of Miller-Rabin on "a" using the first + * t prime bases. Also performs an initial sieve of trial + * division. Determines if "a" is prime with probability + * of error no more than (1/4)**t. + * + * Sets result to 1 if probably prime, 0 otherwise + */ +int mp_prime_is_prime(mp_int *a, int t, int *result); + +/* finds the next prime after the number "a" using "t" trials + * of Miller-Rabin. + * + * bbs_style = 1 means the prime must be congruent to 3 mod 4 + */ +int mp_prime_next_prime(mp_int *a, int t, int bbs_style); + +/* makes a truly random prime of a given size (bytes), + * call with bbs = 1 if you want it to be congruent to 3 mod 4 + * + * You have to supply a callback which fills in a buffer with random bytes. "dat" is a parameter you can + * have passed to the callback (e.g. a state or something). This function doesn't use "dat" itself + * so it can be NULL + * + * The prime generated will be larger than 2^(8*size). + */ +#define mp_prime_random(a, t, size, bbs, cb, dat) mp_prime_random_ex(a, t, ((size) * 8) + 1, (bbs==1)?LTM_PRIME_BBS:0, cb, dat) + +/* makes a truly random prime of a given size (bits), + * + * Flags are as follows: + * + * LTM_PRIME_BBS - make prime congruent to 3 mod 4 + * LTM_PRIME_SAFE - make sure (p-1)/2 is prime as well (implies LTM_PRIME_BBS) + * LTM_PRIME_2MSB_OFF - make the 2nd highest bit zero + * LTM_PRIME_2MSB_ON - make the 2nd highest bit one + * + * You have to supply a callback which fills in a buffer with random bytes. "dat" is a parameter you can + * have passed to the callback (e.g. a state or something). This function doesn't use "dat" itself + * so it can be NULL + * + */ +int mp_prime_random_ex(mp_int *a, int t, int size, int flags, ltm_prime_callback cb, void *dat); + +/* ---> radix conversion <--- */ +int mp_count_bits(mp_int *a); + +int mp_unsigned_bin_size(mp_int *a); +int mp_read_unsigned_bin(mp_int *a, unsigned char *b, int c); +int mp_to_unsigned_bin(mp_int *a, unsigned char *b); + +int mp_signed_bin_size(mp_int *a); +int mp_read_signed_bin(mp_int *a, unsigned char *b, int c); +int mp_to_signed_bin(mp_int *a, unsigned char *b); + +int mp_read_radix(mp_int *a, char *str, int radix); +int mp_toradix(mp_int *a, char *str, int radix); +int mp_toradix_n(mp_int * a, char *str, int radix, int maxlen); +int mp_radix_size(mp_int *a, int radix, int *size); + +int mp_fread(mp_int *a, int radix, FILE *stream); +int mp_fwrite(mp_int *a, int radix, FILE *stream); + +#define mp_read_raw(mp, str, len) mp_read_signed_bin((mp), (str), (len)) +#define mp_raw_size(mp) mp_signed_bin_size(mp) +#define mp_toraw(mp, str) mp_to_signed_bin((mp), (str)) +#define mp_read_mag(mp, str, len) mp_read_unsigned_bin((mp), (str), (len)) +#define mp_mag_size(mp) mp_unsigned_bin_size(mp) +#define mp_tomag(mp, str) mp_to_unsigned_bin((mp), (str)) + +#define mp_tobinary(M, S) mp_toradix((M), (S), 2) +#define mp_tooctal(M, S) mp_toradix((M), (S), 8) +#define mp_todecimal(M, S) mp_toradix((M), (S), 10) +#define mp_tohex(M, S) mp_toradix((M), (S), 16) + +/* lowlevel functions, do not call! */ +int s_mp_add(mp_int *a, mp_int *b, mp_int *c); +int s_mp_sub(mp_int *a, mp_int *b, mp_int *c); +#define s_mp_mul(a, b, c) s_mp_mul_digs(a, b, c, (a)->used + (b)->used + 1) +int fast_s_mp_mul_digs(mp_int *a, mp_int *b, mp_int *c, int digs); +int s_mp_mul_digs(mp_int *a, mp_int *b, mp_int *c, int digs); +int fast_s_mp_mul_high_digs(mp_int *a, mp_int *b, mp_int *c, int digs); +int s_mp_mul_high_digs(mp_int *a, mp_int *b, mp_int *c, int digs); +int fast_s_mp_sqr(mp_int *a, mp_int *b); +int s_mp_sqr(mp_int *a, mp_int *b); +int mp_karatsuba_mul(mp_int *a, mp_int *b, mp_int *c); +int mp_toom_mul(mp_int *a, mp_int *b, mp_int *c); +int mp_karatsuba_sqr(mp_int *a, mp_int *b); +int mp_toom_sqr(mp_int *a, mp_int *b); +int fast_mp_invmod(mp_int *a, mp_int *b, mp_int *c); +int mp_invmod_slow (mp_int * a, mp_int * b, mp_int * c); +int fast_mp_montgomery_reduce(mp_int *a, mp_int *m, mp_digit mp); +int mp_exptmod_fast(mp_int *G, mp_int *X, mp_int *P, mp_int *Y, int mode); +int s_mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y); +void bn_reverse(unsigned char *s, int len); + +extern const char *mp_s_rmap; + +#ifdef __cplusplus + } +#endif + +#endif + diff --git a/libtommath/tommath.out b/libtommath/tommath.out new file mode 100644 index 0000000..9f62617 --- /dev/null +++ b/libtommath/tommath.out @@ -0,0 +1,139 @@ +\BOOKMARK [0][-]{chapter.1}{Introduction}{} +\BOOKMARK [1][-]{section.1.1}{Multiple Precision Arithmetic}{chapter.1} +\BOOKMARK [2][-]{subsection.1.1.1}{What is Multiple Precision Arithmetic?}{section.1.1} +\BOOKMARK [2][-]{subsection.1.1.2}{The Need for Multiple Precision Arithmetic}{section.1.1} +\BOOKMARK [2][-]{subsection.1.1.3}{Benefits of Multiple Precision Arithmetic}{section.1.1} +\BOOKMARK [1][-]{section.1.2}{Purpose of This Text}{chapter.1} +\BOOKMARK [1][-]{section.1.3}{Discussion and Notation}{chapter.1} +\BOOKMARK [2][-]{subsection.1.3.1}{Notation}{section.1.3} +\BOOKMARK [2][-]{subsection.1.3.2}{Precision Notation}{section.1.3} +\BOOKMARK [2][-]{subsection.1.3.3}{Algorithm Inputs and Outputs}{section.1.3} +\BOOKMARK [2][-]{subsection.1.3.4}{Mathematical Expressions}{section.1.3} +\BOOKMARK [2][-]{subsection.1.3.5}{Work Effort}{section.1.3} +\BOOKMARK [1][-]{section.1.4}{Exercises}{chapter.1} +\BOOKMARK [1][-]{section.1.5}{Introduction to LibTomMath}{chapter.1} +\BOOKMARK [2][-]{subsection.1.5.1}{What is LibTomMath?}{section.1.5} +\BOOKMARK [2][-]{subsection.1.5.2}{Goals of LibTomMath}{section.1.5} +\BOOKMARK [1][-]{section.1.6}{Choice of LibTomMath}{chapter.1} +\BOOKMARK [2][-]{subsection.1.6.1}{Code Base}{section.1.6} +\BOOKMARK [2][-]{subsection.1.6.2}{API Simplicity}{section.1.6} +\BOOKMARK [2][-]{subsection.1.6.3}{Optimizations}{section.1.6} +\BOOKMARK [2][-]{subsection.1.6.4}{Portability and Stability}{section.1.6} +\BOOKMARK [2][-]{subsection.1.6.5}{Choice}{section.1.6} +\BOOKMARK [0][-]{chapter.2}{Getting Started}{} +\BOOKMARK [1][-]{section.2.1}{Library Basics}{chapter.2} +\BOOKMARK [1][-]{section.2.2}{What is a Multiple Precision Integer?}{chapter.2} +\BOOKMARK [2][-]{subsection.2.2.1}{The mp\137int Structure}{section.2.2} +\BOOKMARK [1][-]{section.2.3}{Argument Passing}{chapter.2} +\BOOKMARK [1][-]{section.2.4}{Return Values}{chapter.2} +\BOOKMARK [1][-]{section.2.5}{Initialization and Clearing}{chapter.2} +\BOOKMARK [2][-]{subsection.2.5.1}{Initializing an mp\137int}{section.2.5} +\BOOKMARK [2][-]{subsection.2.5.2}{Clearing an mp\137int}{section.2.5} +\BOOKMARK [1][-]{section.2.6}{Maintenance Algorithms}{chapter.2} +\BOOKMARK [2][-]{subsection.2.6.1}{Augmenting an mp\137int's Precision}{section.2.6} +\BOOKMARK [2][-]{subsection.2.6.2}{Initializing Variable Precision mp\137ints}{section.2.6} +\BOOKMARK [2][-]{subsection.2.6.3}{Multiple Integer Initializations and Clearings}{section.2.6} +\BOOKMARK [2][-]{subsection.2.6.4}{Clamping Excess Digits}{section.2.6} +\BOOKMARK [0][-]{chapter.3}{Basic Operations}{} +\BOOKMARK [1][-]{section.3.1}{Introduction}{chapter.3} +\BOOKMARK [1][-]{section.3.2}{Assigning Values to mp\137int Structures}{chapter.3} +\BOOKMARK [2][-]{subsection.3.2.1}{Copying an mp\137int}{section.3.2} +\BOOKMARK [2][-]{subsection.3.2.2}{Creating a Clone}{section.3.2} +\BOOKMARK [1][-]{section.3.3}{Zeroing an Integer}{chapter.3} +\BOOKMARK [1][-]{section.3.4}{Sign Manipulation}{chapter.3} +\BOOKMARK [2][-]{subsection.3.4.1}{Absolute Value}{section.3.4} +\BOOKMARK [2][-]{subsection.3.4.2}{Integer Negation}{section.3.4} +\BOOKMARK [1][-]{section.3.5}{Small Constants}{chapter.3} +\BOOKMARK [2][-]{subsection.3.5.1}{Setting Small Constants}{section.3.5} +\BOOKMARK [2][-]{subsection.3.5.2}{Setting Large Constants}{section.3.5} +\BOOKMARK [1][-]{section.3.6}{Comparisons}{chapter.3} +\BOOKMARK [2][-]{subsection.3.6.1}{Unsigned Comparisions}{section.3.6} +\BOOKMARK [2][-]{subsection.3.6.2}{Signed Comparisons}{section.3.6} +\BOOKMARK [0][-]{chapter.4}{Basic Arithmetic}{} +\BOOKMARK [1][-]{section.4.1}{Introduction}{chapter.4} +\BOOKMARK [1][-]{section.4.2}{Addition and Subtraction}{chapter.4} +\BOOKMARK [2][-]{subsection.4.2.1}{Low Level Addition}{section.4.2} +\BOOKMARK [2][-]{subsection.4.2.2}{Low Level Subtraction}{section.4.2} +\BOOKMARK [2][-]{subsection.4.2.3}{High Level Addition}{section.4.2} +\BOOKMARK [2][-]{subsection.4.2.4}{High Level Subtraction}{section.4.2} +\BOOKMARK [1][-]{section.4.3}{Bit and Digit Shifting}{chapter.4} +\BOOKMARK [2][-]{subsection.4.3.1}{Multiplication by Two}{section.4.3} +\BOOKMARK [2][-]{subsection.4.3.2}{Division by Two}{section.4.3} +\BOOKMARK [1][-]{section.4.4}{Polynomial Basis Operations}{chapter.4} +\BOOKMARK [2][-]{subsection.4.4.1}{Multiplication by x}{section.4.4} +\BOOKMARK [2][-]{subsection.4.4.2}{Division by x}{section.4.4} +\BOOKMARK [1][-]{section.4.5}{Powers of Two}{chapter.4} +\BOOKMARK [2][-]{subsection.4.5.1}{Multiplication by Power of Two}{section.4.5} +\BOOKMARK [2][-]{subsection.4.5.2}{Division by Power of Two}{section.4.5} +\BOOKMARK [2][-]{subsection.4.5.3}{Remainder of Division by Power of Two}{section.4.5} +\BOOKMARK [0][-]{chapter.5}{Multiplication and Squaring}{} +\BOOKMARK [1][-]{section.5.1}{The Multipliers}{chapter.5} +\BOOKMARK [1][-]{section.5.2}{Multiplication}{chapter.5} +\BOOKMARK [2][-]{subsection.5.2.1}{The Baseline Multiplication}{section.5.2} +\BOOKMARK [2][-]{subsection.5.2.2}{Faster Multiplication by the ``Comba'' Method}{section.5.2} +\BOOKMARK [2][-]{subsection.5.2.3}{Polynomial Basis Multiplication}{section.5.2} +\BOOKMARK [2][-]{subsection.5.2.4}{Karatsuba Multiplication}{section.5.2} +\BOOKMARK [2][-]{subsection.5.2.5}{Toom-Cook 3-Way Multiplication}{section.5.2} +\BOOKMARK [2][-]{subsection.5.2.6}{Signed Multiplication}{section.5.2} +\BOOKMARK [1][-]{section.5.3}{Squaring}{chapter.5} +\BOOKMARK [2][-]{subsection.5.3.1}{The Baseline Squaring Algorithm}{section.5.3} +\BOOKMARK [2][-]{subsection.5.3.2}{Faster Squaring by the ``Comba'' Method}{section.5.3} +\BOOKMARK [2][-]{subsection.5.3.3}{Polynomial Basis Squaring}{section.5.3} +\BOOKMARK [2][-]{subsection.5.3.4}{Karatsuba Squaring}{section.5.3} +\BOOKMARK [2][-]{subsection.5.3.5}{Toom-Cook Squaring}{section.5.3} +\BOOKMARK [2][-]{subsection.5.3.6}{High Level Squaring}{section.5.3} +\BOOKMARK [0][-]{chapter.6}{Modular Reduction}{} +\BOOKMARK [1][-]{section.6.1}{Basics of Modular Reduction}{chapter.6} +\BOOKMARK [1][-]{section.6.2}{The Barrett Reduction}{chapter.6} +\BOOKMARK [2][-]{subsection.6.2.1}{Fixed Point Arithmetic}{section.6.2} +\BOOKMARK [2][-]{subsection.6.2.2}{Choosing a Radix Point}{section.6.2} +\BOOKMARK [2][-]{subsection.6.2.3}{Trimming the Quotient}{section.6.2} +\BOOKMARK [2][-]{subsection.6.2.4}{Trimming the Residue}{section.6.2} +\BOOKMARK [2][-]{subsection.6.2.5}{The Barrett Algorithm}{section.6.2} +\BOOKMARK [2][-]{subsection.6.2.6}{The Barrett Setup Algorithm}{section.6.2} +\BOOKMARK [1][-]{section.6.3}{The Montgomery Reduction}{chapter.6} +\BOOKMARK [2][-]{subsection.6.3.1}{Digit Based Montgomery Reduction}{section.6.3} +\BOOKMARK [2][-]{subsection.6.3.2}{Baseline Montgomery Reduction}{section.6.3} +\BOOKMARK [2][-]{subsection.6.3.3}{Faster ``Comba'' Montgomery Reduction}{section.6.3} +\BOOKMARK [2][-]{subsection.6.3.4}{Montgomery Setup}{section.6.3} +\BOOKMARK [1][-]{section.6.4}{The Diminished Radix Algorithm}{chapter.6} +\BOOKMARK [2][-]{subsection.6.4.1}{Choice of Moduli}{section.6.4} +\BOOKMARK [2][-]{subsection.6.4.2}{Choice of k}{section.6.4} +\BOOKMARK [2][-]{subsection.6.4.3}{Restricted Diminished Radix Reduction}{section.6.4} +\BOOKMARK [2][-]{subsection.6.4.4}{Unrestricted Diminished Radix Reduction}{section.6.4} +\BOOKMARK [1][-]{section.6.5}{Algorithm Comparison}{chapter.6} +\BOOKMARK [0][-]{chapter.7}{Exponentiation}{} +\BOOKMARK [1][-]{section.7.1}{Exponentiation Basics}{chapter.7} +\BOOKMARK [2][-]{subsection.7.1.1}{Single Digit Exponentiation}{section.7.1} +\BOOKMARK [1][-]{section.7.2}{k-ary Exponentiation}{chapter.7} +\BOOKMARK [2][-]{subsection.7.2.1}{Optimal Values of k}{section.7.2} +\BOOKMARK [2][-]{subsection.7.2.2}{Sliding-Window Exponentiation}{section.7.2} +\BOOKMARK [1][-]{section.7.3}{Modular Exponentiation}{chapter.7} +\BOOKMARK [2][-]{subsection.7.3.1}{Barrett Modular Exponentiation}{section.7.3} +\BOOKMARK [1][-]{section.7.4}{Quick Power of Two}{chapter.7} +\BOOKMARK [0][-]{chapter.8}{Higher Level Algorithms}{} +\BOOKMARK [1][-]{section.8.1}{Integer Division with Remainder}{chapter.8} +\BOOKMARK [2][-]{subsection.8.1.1}{Quotient Estimation}{section.8.1} +\BOOKMARK [2][-]{subsection.8.1.2}{Normalized Integers}{section.8.1} +\BOOKMARK [2][-]{subsection.8.1.3}{Radix- Division with Remainder}{section.8.1} +\BOOKMARK [1][-]{section.8.2}{Single Digit Helpers}{chapter.8} +\BOOKMARK [2][-]{subsection.8.2.1}{Single Digit Addition and Subtraction}{section.8.2} +\BOOKMARK [2][-]{subsection.8.2.2}{Single Digit Multiplication}{section.8.2} +\BOOKMARK [2][-]{subsection.8.2.3}{Single Digit Division}{section.8.2} +\BOOKMARK [2][-]{subsection.8.2.4}{Single Digit Root Extraction}{section.8.2} +\BOOKMARK [1][-]{section.8.3}{Random Number Generation}{chapter.8} +\BOOKMARK [1][-]{section.8.4}{Formatted Representations}{chapter.8} +\BOOKMARK [2][-]{subsection.8.4.1}{Reading Radix-n Input}{section.8.4} +\BOOKMARK [2][-]{subsection.8.4.2}{Generating Radix-n Output}{section.8.4} +\BOOKMARK [0][-]{chapter.9}{Number Theoretic Algorithms}{} +\BOOKMARK [1][-]{section.9.1}{Greatest Common Divisor}{chapter.9} +\BOOKMARK [2][-]{subsection.9.1.1}{Complete Greatest Common Divisor}{section.9.1} +\BOOKMARK [1][-]{section.9.2}{Least Common Multiple}{chapter.9} +\BOOKMARK [1][-]{section.9.3}{Jacobi Symbol Computation}{chapter.9} +\BOOKMARK [2][-]{subsection.9.3.1}{Jacobi Symbol}{section.9.3} +\BOOKMARK [1][-]{section.9.4}{Modular Inverse}{chapter.9} +\BOOKMARK [2][-]{subsection.9.4.1}{General Case}{section.9.4} +\BOOKMARK [1][-]{section.9.5}{Primality Tests}{chapter.9} +\BOOKMARK [2][-]{subsection.9.5.1}{Trial Division}{section.9.5} +\BOOKMARK [2][-]{subsection.9.5.2}{The Fermat Test}{section.9.5} +\BOOKMARK [2][-]{subsection.9.5.3}{The Miller-Rabin Test}{section.9.5} diff --git a/libtommath/tommath.pdf b/libtommath/tommath.pdf Binary files differnew file mode 100644 index 0000000..88e2dc7 --- /dev/null +++ b/libtommath/tommath.pdf diff --git a/libtommath/tommath.src b/libtommath/tommath.src new file mode 100644 index 0000000..6ee842d --- /dev/null +++ b/libtommath/tommath.src @@ -0,0 +1,6314 @@ +\documentclass[b5paper]{book} +\usepackage{hyperref} +\usepackage{makeidx} +\usepackage{amssymb} +\usepackage{color} +\usepackage{alltt} +\usepackage{graphicx} +\usepackage{layout} +\def\union{\cup} +\def\intersect{\cap} +\def\getsrandom{\stackrel{\rm R}{\gets}} +\def\cross{\times} +\def\cat{\hspace{0.5em} \| \hspace{0.5em}} +\def\catn{$\|$} +\def\divides{\hspace{0.3em} | \hspace{0.3em}} +\def\nequiv{\not\equiv} +\def\approx{\raisebox{0.2ex}{\mbox{\small $\sim$}}} +\def\lcm{{\rm lcm}} +\def\gcd{{\rm gcd}} +\def\log{{\rm log}} +\def\ord{{\rm ord}} +\def\abs{{\mathit abs}} +\def\rep{{\mathit rep}} +\def\mod{{\mathit\ mod\ }} +\renewcommand{\pmod}[1]{\ ({\rm mod\ }{#1})} +\newcommand{\floor}[1]{\left\lfloor{#1}\right\rfloor} +\newcommand{\ceil}[1]{\left\lceil{#1}\right\rceil} +\def\Or{{\rm\ or\ }} +\def\And{{\rm\ and\ }} +\def\iff{\hspace{1em}\Longleftrightarrow\hspace{1em}} +\def\implies{\Rightarrow} +\def\undefined{{\rm ``undefined"}} +\def\Proof{\vspace{1ex}\noindent {\bf Proof:}\hspace{1em}} +\let\oldphi\phi +\def\phi{\varphi} +\def\Pr{{\rm Pr}} +\newcommand{\str}[1]{{\mathbf{#1}}} +\def\F{{\mathbb F}} +\def\N{{\mathbb N}} +\def\Z{{\mathbb Z}} +\def\R{{\mathbb R}} +\def\C{{\mathbb C}} +\def\Q{{\mathbb Q}} +\definecolor{DGray}{gray}{0.5} +\newcommand{\emailaddr}[1]{\mbox{$<${#1}$>$}} +\def\twiddle{\raisebox{0.3ex}{\mbox{\tiny $\sim$}}} +\def\gap{\vspace{0.5ex}} +\makeindex +\begin{document} +\frontmatter +\pagestyle{empty} +\title{Implementing Multiple Precision Arithmetic \\ ~ \\ Draft Edition } +\author{\mbox{ +%\begin{small} +\begin{tabular}{c} +Tom St Denis \\ +Algonquin College \\ +\\ +Mads Rasmussen \\ +Open Communications Security \\ +\\ +Greg Rose \\ +QUALCOMM Australia \\ +\end{tabular} +%\end{small} +} +} +\maketitle +This text has been placed in the public domain. This text corresponds to the v0.30 release of the +LibTomMath project. + +\begin{alltt} +Tom St Denis +111 Banning Rd +Ottawa, Ontario +K2L 1C3 +Canada + +Phone: 1-613-836-3160 +Email: tomstdenis@iahu.ca +\end{alltt} + +This text is formatted to the international B5 paper size of 176mm wide by 250mm tall using the \LaTeX{} +{\em book} macro package and the Perl {\em booker} package. + +\tableofcontents +\listoffigures +\chapter*{Prefaces to the Draft Edition} +I started this text in April 2003 to complement my LibTomMath library. That is, explain how to implement the functions +contained in LibTomMath. The goal is to have a textbook that any Computer Science student can use when implementing their +own multiple precision arithmetic. The plan I wanted to follow was flesh out all the +ideas and concepts I had floating around in my head and then work on it afterwards refining a little bit at a time. Chance +would have it that I ended up with my summer off from Algonquin College and I was given four months solid to work on the +text. + +Choosing to not waste any time I dove right into the project even before my spring semester was finished. I wrote a bit +off and on at first. The moment my exams were finished I jumped into long 12 to 16 hour days. The result after only +a couple of months was a ten chapter, three hundred page draft that I quickly had distributed to anyone who wanted +to read it. I had Jean-Luc Cooke print copies for me and I brought them to Crypto'03 in Santa Barbara. So far I have +managed to grab a certain level of attention having people from around the world ask me for copies of the text was certain +rewarding. + +Now we are past December 2003. By this time I had pictured that I would have at least finished my second draft of the text. +Currently I am far off from this goal. I've done partial re-writes of chapters one, two and three but they are not even +finished yet. I haven't given up on the project, only had some setbacks. First O'Reilly declined to publish the text then +Addison-Wesley and Greg is tried another which I don't know the name of. However, at this point I want to focus my energy +onto finishing the book not securing a contract. + +So why am I writing this text? It seems like a lot of work right? Most certainly it is a lot of work writing a textbook. +Even the simplest introductory material has to be lined with references and figures. A lot of the text has to be re-written +from point form to prose form to ensure an easier read. Why am I doing all this work for free then? Simple. My philosophy +is quite simply ``Open Source. Open Academia. Open Minds'' which means that to achieve a goal of open minds, that is, +people willing to accept new ideas and explore the unknown you have to make available material they can access freely +without hinderance. + +I've been writing free software since I was about sixteen but only recently have I hit upon software that people have come +to depend upon. I started LibTomCrypt in December 2001 and now several major companies use it as integral portions of their +software. Several educational institutions use it as a matter of course and many freelance developers use it as +part of their projects. To further my contributions I started the LibTomMath project in December 2002 aimed at providing +multiple precision arithmetic routines that students could learn from. That is write routines that are not only easy +to understand and follow but provide quite impressive performance considering they are all in standard portable ISO C. + +The second leg of my philosophy is ``Open Academia'' which is where this textbook comes in. In the end, when all is +said and done the text will be useable by educational institutions as a reference on multiple precision arithmetic. + +At this time I feel I should share a little information about myself. The most common question I was asked at +Crypto'03, perhaps just out of professional courtesy, was which school I either taught at or attended. The unfortunate +truth is that I neither teach at or attend a school of academic reputation. I'm currently at Algonquin College which +is what I'd like to call ``somewhat academic but mostly vocational'' college. In otherwords, job training. + +I'm a 21 year old computer science student mostly self-taught in the areas I am aware of (which includes a half-dozen +computer science fields, a few fields of mathematics and some English). I look forward to teaching someday but I am +still far off from that goal. + +Now it would be improper for me to not introduce the rest of the texts co-authors. While they are only contributing +corrections and editorial feedback their support has been tremendously helpful in presenting the concepts laid out +in the text so far. Greg has always been there for me. He has tracked my LibTom projects since their inception and even +sent cheques to help pay tuition from time to time. His background has provided a wonderful source to bounce ideas off +of and improve the quality of my writing. Mads is another fellow who has just ``been there''. I don't even recall what +his interest in the LibTom projects is but I'm definitely glad he has been around. His ability to catch logical errors +in my written English have saved me on several occasions to say the least. + +What to expect next? Well this is still a rough draft. I've only had the chance to update a few chapters. However, I've +been getting the feeling that people are starting to use my text and I owe them some updated material. My current tenative +plan is to edit one chapter every two weeks starting January 4th. It seems insane but my lower course load at college +should provide ample time. By Crypto'04 I plan to have a 2nd draft of the text polished and ready to hand out to as many +people who will take it. + +\begin{flushright} Tom St Denis \end{flushright} + +\newpage +I found the opportunity to work with Tom appealing for several reasons, not only could I broaden my own horizons, but also +contribute to educate others facing the problem of having to handle big number mathematical calculations. + +This book is Tom's child and he has been caring and fostering the project ever since the beginning with a clear mind of +how he wanted the project to turn out. I have helped by proofreading the text and we have had several discussions about +the layout and language used. + +I hold a masters degree in cryptography from the University of Southern Denmark and have always been interested in the +practical aspects of cryptography. + +Having worked in the security consultancy business for several years in S\~{a}o Paulo, Brazil, I have been in touch with a +great deal of work in which multiple precision mathematics was needed. Understanding the possibilities for speeding up +multiple precision calculations is often very important since we deal with outdated machine architecture where modular +reductions, for example, become painfully slow. + +This text is for people who stop and wonder when first examining algorithms such as RSA for the first time and asks +themselves, ``You tell me this is only secure for large numbers, fine; but how do you implement these numbers?'' + +\begin{flushright} +Mads Rasmussen + +S\~{a}o Paulo - SP + +Brazil +\end{flushright} + +\newpage +It's all because I broke my leg. That just happened to be at about the same time that Tom asked for someone to review the section of the book about +Karatsuba multiplication. I was laid up, alone and immobile, and thought ``Why not?'' I vaguely knew what Karatsuba multiplication was, but not +really, so I thought I could help, learn, and stop myself from watching daytime cable TV, all at once. + +At the time of writing this, I've still not met Tom or Mads in meatspace. I've been following Tom's progress since his first splash on the +sci.crypt Usenet news group. I watched him go from a clueless newbie, to the cryptographic equivalent of a reformed smoker, to a real +contributor to the field, over a period of about two years. I've been impressed with his obvious intelligence, and astounded by his productivity. +Of course, he's young enough to be my own child, so he doesn't have my problems with staying awake. + +When I reviewed that single section of the book, in its very earliest form, I was very pleasantly surprised. So I decided to collaborate more fully, +and at least review all of it, and perhaps write some bits too. There's still a long way to go with it, and I have watched a number of close +friends go through the mill of publication, so I think that the way to go is longer than Tom thinks it is. Nevertheless, it's a good effort, +and I'm pleased to be involved with it. + +\begin{flushright} +Greg Rose, Sydney, Australia, June 2003. +\end{flushright} + +\mainmatter +\pagestyle{headings} +\chapter{Introduction} +\section{Multiple Precision Arithmetic} + +\subsection{What is Multiple Precision Arithmetic?} +When we think of long-hand arithmetic such as addition or multiplication we rarely consider the fact that we instinctively +raise or lower the precision of the numbers we are dealing with. For example, in decimal we almost immediate can +reason that $7$ times $6$ is $42$. However, $42$ has two digits of precision as opposed to one digit we started with. +Further multiplications of say $3$ result in a larger precision result $126$. In these few examples we have multiple +precisions for the numbers we are working with. Despite the various levels of precision a single subset\footnote{With the occasional optimization.} + of algorithms can be designed to accomodate them. + +By way of comparison a fixed or single precision operation would lose precision on various operations. For example, in +the decimal system with fixed precision $6 \cdot 7 = 2$. + +Essentially at the heart of computer based multiple precision arithmetic are the same long-hand algorithms taught in +schools to manually add, subtract, multiply and divide. + +\subsection{The Need for Multiple Precision Arithmetic} +The most prevalent need for multiple precision arithmetic, often referred to as ``bignum'' math, is within the implementation +of public-key cryptography algorithms. Algorithms such as RSA \cite{RSAREF} and Diffie-Hellman \cite{DHREF} require +integers of significant magnitude to resist known cryptanalytic attacks. For example, at the time of this writing a +typical RSA modulus would be at least greater than $10^{309}$. However, modern programming languages such as ISO C \cite{ISOC} and +Java \cite{JAVA} only provide instrinsic support for integers which are relatively small and single precision. + +\begin{figure}[!here] +\begin{center} +\begin{tabular}{|r|c|} +\hline \textbf{Data Type} & \textbf{Range} \\ +\hline char & $-128 \ldots 127$ \\ +\hline short & $-32768 \ldots 32767$ \\ +\hline long & $-2147483648 \ldots 2147483647$ \\ +\hline long long & $-9223372036854775808 \ldots 9223372036854775807$ \\ +\hline +\end{tabular} +\end{center} +\caption{Typical Data Types for the C Programming Language} +\label{fig:ISOC} +\end{figure} + +The largest data type guaranteed to be provided by the ISO C programming +language\footnote{As per the ISO C standard. However, each compiler vendor is allowed to augment the precision as they +see fit.} can only represent values up to $10^{19}$ as shown in figure \ref{fig:ISOC}. On its own the C language is +insufficient to accomodate the magnitude required for the problem at hand. An RSA modulus of magnitude $10^{19}$ could be +trivially factored\footnote{A Pollard-Rho factoring would take only $2^{16}$ time.} on the average desktop computer, +rendering any protocol based on the algorithm insecure. Multiple precision algorithms solve this very problem by +extending the range of representable integers while using single precision data types. + +Most advancements in fast multiple precision arithmetic stem from the need for faster and more efficient cryptographic +primitives. Faster modular reduction and exponentiation algorithms such as Barrett's algorithm, which have appeared in +various cryptographic journals, can render algorithms such as RSA and Diffie-Hellman more efficient. In fact, several +major companies such as RSA Security, Certicom and Entrust have built entire product lines on the implementation and +deployment of efficient algorithms. + +However, cryptography is not the only field of study that can benefit from fast multiple precision integer routines. +Another auxiliary use of multiple precision integers is high precision floating point data types. +The basic IEEE \cite{IEEE} standard floating point type is made up of an integer mantissa $q$, an exponent $e$ and a sign bit $s$. +Numbers are given in the form $n = q \cdot b^e \cdot -1^s$ where $b = 2$ is the most common base for IEEE. Since IEEE +floating point is meant to be implemented in hardware the precision of the mantissa is often fairly small +(\textit{23, 48 and 64 bits}). The mantissa is merely an integer and a multiple precision integer could be used to create +a mantissa of much larger precision than hardware alone can efficiently support. This approach could be useful where +scientific applications must minimize the total output error over long calculations. + +Yet another use for large integers is within arithmetic on polynomials of large characteristic (i.e. $GF(p)[x]$ for large $p$). +In fact the library discussed within this text has already been used to form a polynomial basis library\footnote{See \url{http://poly.libtomcrypt.org} for more details.}. + +\subsection{Benefits of Multiple Precision Arithmetic} +\index{precision} +The benefit of multiple precision representations over single or fixed precision representations is that +no precision is lost while representing the result of an operation which requires excess precision. For example, +the product of two $n$-bit integers requires at least $2n$ bits of precision to be represented faithfully. A multiple +precision algorithm would augment the precision of the destination to accomodate the result while a single precision system +would truncate excess bits to maintain a fixed level of precision. + +It is possible to implement algorithms which require large integers with fixed precision algorithms. For example, elliptic +curve cryptography (\textit{ECC}) is often implemented on smartcards by fixing the precision of the integers to the maximum +size the system will ever need. Such an approach can lead to vastly simpler algorithms which can accomodate the +integers required even if the host platform cannot natively accomodate them\footnote{For example, the average smartcard +processor has an 8 bit accumulator.}. However, as efficient as such an approach may be, the resulting source code is not +normally very flexible. It cannot, at runtime, accomodate inputs of higher magnitude than the designer anticipated. + +Multiple precision algorithms have the most overhead of any style of arithmetic. For the the most part the +overhead can be kept to a minimum with careful planning, but overall, it is not well suited for most memory starved +platforms. However, multiple precision algorithms do offer the most flexibility in terms of the magnitude of the +inputs. That is, the same algorithms based on multiple precision integers can accomodate any reasonable size input +without the designer's explicit forethought. This leads to lower cost of ownership for the code as it only has to +be written and tested once. + +\section{Purpose of This Text} +The purpose of this text is to instruct the reader regarding how to implement efficient multiple precision algorithms. +That is to not only explain a limited subset of the core theory behind the algorithms but also the various ``house keeping'' +elements that are neglected by authors of other texts on the subject. Several well reknowned texts \cite{TAOCPV2,HAC} +give considerably detailed explanations of the theoretical aspects of algorithms and often very little information +regarding the practical implementation aspects. + +In most cases how an algorithm is explained and how it is actually implemented are two very different concepts. For +example, the Handbook of Applied Cryptography (\textit{HAC}), algorithm 14.7 on page 594, gives a relatively simple +algorithm for performing multiple precision integer addition. However, the description lacks any discussion concerning +the fact that the two integer inputs may be of differing magnitudes. As a result the implementation is not as simple +as the text would lead people to believe. Similarly the division routine (\textit{algorithm 14.20, pp. 598}) does not +discuss how to handle sign or handle the dividend's decreasing magnitude in the main loop (\textit{step \#3}). + +Both texts also do not discuss several key optimal algorithms required such as ``Comba'' and Karatsuba multipliers +and fast modular inversion, which we consider practical oversights. These optimal algorithms are vital to achieve +any form of useful performance in non-trivial applications. + +To solve this problem the focus of this text is on the practical aspects of implementing a multiple precision integer +package. As a case study the ``LibTomMath''\footnote{Available at \url{http://math.libtomcrypt.org}} package is used +to demonstrate algorithms with real implementations\footnote{In the ISO C programming language.} that have been field +tested and work very well. The LibTomMath library is freely available on the Internet for all uses and this text +discusses a very large portion of the inner workings of the library. + +The algorithms that are presented will always include at least one ``pseudo-code'' description followed +by the actual C source code that implements the algorithm. The pseudo-code can be used to implement the same +algorithm in other programming languages as the reader sees fit. + +This text shall also serve as a walkthrough of the creation of multiple precision algorithms from scratch. Showing +the reader how the algorithms fit together as well as where to start on various taskings. + +\section{Discussion and Notation} +\subsection{Notation} +A multiple precision integer of $n$-digits shall be denoted as $x = (x_{n-1}, \ldots, x_1, x_0)_{ \beta }$ and represent +the integer $x \equiv \sum_{i=0}^{n-1} x_i\beta^i$. The elements of the array $x$ are said to be the radix $\beta$ digits +of the integer. For example, $x = (1,2,3)_{10}$ would represent the integer +$1\cdot 10^2 + 2\cdot10^1 + 3\cdot10^0 = 123$. + +\index{mp\_int} +The term ``mp\_int'' shall refer to a composite structure which contains the digits of the integer it represents, as well +as auxilary data required to manipulate the data. These additional members are discussed further in section +\ref{sec:MPINT}. For the purposes of this text a ``multiple precision integer'' and an ``mp\_int'' are assumed to be +synonymous. When an algorithm is specified to accept an mp\_int variable it is assumed the various auxliary data members +are present as well. An expression of the type \textit{variablename.item} implies that it should evaluate to the +member named ``item'' of the variable. For example, a string of characters may have a member ``length'' which would +evaluate to the number of characters in the string. If the string $a$ equals ``hello'' then it follows that +$a.length = 5$. + +For certain discussions more generic algorithms are presented to help the reader understand the final algorithm used +to solve a given problem. When an algorithm is described as accepting an integer input it is assumed the input is +a plain integer with no additional multiple-precision members. That is, algorithms that use integers as opposed to +mp\_ints as inputs do not concern themselves with the housekeeping operations required such as memory management. These +algorithms will be used to establish the relevant theory which will subsequently be used to describe a multiple +precision algorithm to solve the same problem. + +\subsection{Precision Notation} +The variable $\beta$ represents the radix of a single digit of a multiple precision integer and +must be of the form $q^p$ for $q, p \in \Z^+$. A single precision variable must be able to represent integers in +the range $0 \le x < q \beta$ while a double precision variable must be able to represent integers in the range +$0 \le x < q \beta^2$. The extra radix-$q$ factor allows additions and subtractions to proceed without truncation of the +carry. Since all modern computers are binary, it is assumed that $q$ is two. + +\index{mp\_digit} \index{mp\_word} +Within the source code that will be presented for each algorithm, the data type \textbf{mp\_digit} will represent +a single precision integer type, while, the data type \textbf{mp\_word} will represent a double precision integer type. In +several algorithms (notably the Comba routines) temporary results will be stored in arrays of double precision mp\_words. +For the purposes of this text $x_j$ will refer to the $j$'th digit of a single precision array and $\hat x_j$ will refer to +the $j$'th digit of a double precision array. Whenever an expression is to be assigned to a double precision +variable it is assumed that all single precision variables are promoted to double precision during the evaluation. +Expressions that are assigned to a single precision variable are truncated to fit within the precision of a single +precision data type. + +For example, if $\beta = 10^2$ a single precision data type may represent a value in the +range $0 \le x < 10^3$, while a double precision data type may represent a value in the range $0 \le x < 10^5$. Let +$a = 23$ and $b = 49$ represent two single precision variables. The single precision product shall be written +as $c \leftarrow a \cdot b$ while the double precision product shall be written as $\hat c \leftarrow a \cdot b$. +In this particular case, $\hat c = 1127$ and $c = 127$. The most significant digit of the product would not fit +in a single precision data type and as a result $c \ne \hat c$. + +\subsection{Algorithm Inputs and Outputs} +Within the algorithm descriptions all variables are assumed to be scalars of either single or double precision +as indicated. The only exception to this rule is when variables have been indicated to be of type mp\_int. This +distinction is important as scalars are often used as array indicies and various other counters. + +\subsection{Mathematical Expressions} +The $\lfloor \mbox{ } \rfloor$ brackets imply an expression truncated to an integer not greater than the expression +itself. For example, $\lfloor 5.7 \rfloor = 5$. Similarly the $\lceil \mbox{ } \rceil$ brackets imply an expression +rounded to an integer not less than the expression itself. For example, $\lceil 5.1 \rceil = 6$. Typically when +the $/$ division symbol is used the intention is to perform an integer division with truncation. For example, +$5/2 = 2$ which will often be written as $\lfloor 5/2 \rfloor = 2$ for clarity. When an expression is written as a +fraction a real value division is implied, for example ${5 \over 2} = 2.5$. + +The norm of a multiple precision integer, for example $\vert \vert x \vert \vert$, will be used to represent the number of digits in the representation +of the integer. For example, $\vert \vert 123 \vert \vert = 3$ and $\vert \vert 79452 \vert \vert = 5$. + +\subsection{Work Effort} +\index{big-Oh} +To measure the efficiency of the specified algorithms, a modified big-Oh notation is used. In this system all +single precision operations are considered to have the same cost\footnote{Except where explicitly noted.}. +That is a single precision addition, multiplication and division are assumed to take the same time to +complete. While this is generally not true in practice, it will simplify the discussions considerably. + +Some algorithms have slight advantages over others which is why some constants will not be removed in +the notation. For example, a normal baseline multiplication (section \ref{sec:basemult}) requires $O(n^2)$ work while a +baseline squaring (section \ref{sec:basesquare}) requires $O({{n^2 + n}\over 2})$ work. In standard big-Oh notation these +would both be said to be equivalent to $O(n^2)$. However, +in the context of the this text this is not the case as the magnitude of the inputs will typically be rather small. As a +result small constant factors in the work effort will make an observable difference in algorithm efficiency. + +All of the algorithms presented in this text have a polynomial time work level. That is, of the form +$O(n^k)$ for $n, k \in \Z^{+}$. This will help make useful comparisons in terms of the speed of the algorithms and how +various optimizations will help pay off in the long run. + +\section{Exercises} +Within the more advanced chapters a section will be set aside to give the reader some challenging exercises related to +the discussion at hand. These exercises are not designed to be prize winning problems, but instead to be thought +provoking. Wherever possible the problems are forward minded, stating problems that will be answered in subsequent +chapters. The reader is encouraged to finish the exercises as they appear to get a better understanding of the +subject material. + +That being said, the problems are designed to affirm knowledge of a particular subject matter. Students in particular +are encouraged to verify they can answer the problems correctly before moving on. + +Similar to the exercises of \cite[pp. ix]{TAOCPV2} these exercises are given a scoring system based on the difficulty of +the problem. However, unlike \cite{TAOCPV2} the problems do not get nearly as hard. The scoring of these +exercises ranges from one (the easiest) to five (the hardest). The following table sumarizes the +scoring system used. + +\begin{figure}[here] +\begin{center} +\begin{small} +\begin{tabular}{|c|l|} +\hline $\left [ 1 \right ]$ & An easy problem that should only take the reader a manner of \\ + & minutes to solve. Usually does not involve much computer time \\ + & to solve. \\ +\hline $\left [ 2 \right ]$ & An easy problem that involves a marginal amount of computer \\ + & time usage. Usually requires a program to be written to \\ + & solve the problem. \\ +\hline $\left [ 3 \right ]$ & A moderately hard problem that requires a non-trivial amount \\ + & of work. Usually involves trivial research and development of \\ + & new theory from the perspective of a student. \\ +\hline $\left [ 4 \right ]$ & A moderately hard problem that involves a non-trivial amount \\ + & of work and research, the solution to which will demonstrate \\ + & a higher mastery of the subject matter. \\ +\hline $\left [ 5 \right ]$ & A hard problem that involves concepts that are difficult for a \\ + & novice to solve. Solutions to these problems will demonstrate a \\ + & complete mastery of the given subject. \\ +\hline +\end{tabular} +\end{small} +\end{center} +\caption{Exercise Scoring System} +\end{figure} + +Problems at the first level are meant to be simple questions that the reader can answer quickly without programming a solution or +devising new theory. These problems are quick tests to see if the material is understood. Problems at the second level +are also designed to be easy but will require a program or algorithm to be implemented to arrive at the answer. These +two levels are essentially entry level questions. + +Problems at the third level are meant to be a bit more difficult than the first two levels. The answer is often +fairly obvious but arriving at an exacting solution requires some thought and skill. These problems will almost always +involve devising a new algorithm or implementing a variation of another algorithm previously presented. Readers who can +answer these questions will feel comfortable with the concepts behind the topic at hand. + +Problems at the fourth level are meant to be similar to those of the level three questions except they will require +additional research to be completed. The reader will most likely not know the answer right away, nor will the text provide +the exact details of the answer until a subsequent chapter. + +Problems at the fifth level are meant to be the hardest +problems relative to all the other problems in the chapter. People who can correctly answer fifth level problems have a +mastery of the subject matter at hand. + +Often problems will be tied together. The purpose of this is to start a chain of thought that will be discussed in future chapters. The reader +is encouraged to answer the follow-up problems and try to draw the relevance of problems. + +\section{Introduction to LibTomMath} + +\subsection{What is LibTomMath?} +LibTomMath is a free and open source multiple precision integer library written entirely in portable ISO C. By portable it +is meant that the library does not contain any code that is computer platform dependent or otherwise problematic to use on +any given platform. + +The library has been successfully tested under numerous operating systems including Unix\footnote{All of these +trademarks belong to their respective rightful owners.}, MacOS, Windows, Linux, PalmOS and on standalone hardware such +as the Gameboy Advance. The library is designed to contain enough functionality to be able to develop applications such +as public key cryptosystems and still maintain a relatively small footprint. + +\subsection{Goals of LibTomMath} + +Libraries which obtain the most efficiency are rarely written in a high level programming language such as C. However, +even though this library is written entirely in ISO C, considerable care has been taken to optimize the algorithm implementations within the +library. Specifically the code has been written to work well with the GNU C Compiler (\textit{GCC}) on both x86 and ARM +processors. Wherever possible, highly efficient algorithms, such as Karatsuba multiplication, sliding window +exponentiation and Montgomery reduction have been provided to make the library more efficient. + +Even with the nearly optimal and specialized algorithms that have been included the Application Programing Interface +(\textit{API}) has been kept as simple as possible. Often generic place holder routines will make use of specialized +algorithms automatically without the developer's specific attention. One such example is the generic multiplication +algorithm \textbf{mp\_mul()} which will automatically use Toom--Cook, Karatsuba, Comba or baseline multiplication +based on the magnitude of the inputs and the configuration of the library. + +Making LibTomMath as efficient as possible is not the only goal of the LibTomMath project. Ideally the library should +be source compatible with another popular library which makes it more attractive for developers to use. In this case the +MPI library was used as a API template for all the basic functions. MPI was chosen because it is another library that fits +in the same niche as LibTomMath. Even though LibTomMath uses MPI as the template for the function names and argument +passing conventions, it has been written from scratch by Tom St Denis. + +The project is also meant to act as a learning tool for students, the logic being that no easy-to-follow ``bignum'' +library exists which can be used to teach computer science students how to perform fast and reliable multiple precision +integer arithmetic. To this end the source code has been given quite a few comments and algorithm discussion points. + +\section{Choice of LibTomMath} +LibTomMath was chosen as the case study of this text not only because the author of both projects is one and the same but +for more worthy reasons. Other libraries such as GMP \cite{GMP}, MPI \cite{MPI}, LIP \cite{LIP} and OpenSSL +\cite{OPENSSL} have multiple precision integer arithmetic routines but would not be ideal for this text for +reasons that will be explained in the following sub-sections. + +\subsection{Code Base} +The LibTomMath code base is all portable ISO C source code. This means that there are no platform dependent conditional +segments of code littered throughout the source. This clean and uncluttered approach to the library means that a +developer can more readily discern the true intent of a given section of source code without trying to keep track of +what conditional code will be used. + +The code base of LibTomMath is well organized. Each function is in its own separate source code file +which allows the reader to find a given function very quickly. On average there are $76$ lines of code per source +file which makes the source very easily to follow. By comparison MPI and LIP are single file projects making code tracing +very hard. GMP has many conditional code segments which also hinder tracing. + +When compiled with GCC for the x86 processor and optimized for speed the entire library is approximately $100$KiB\footnote{The notation ``KiB'' means $2^{10}$ octets, similarly ``MiB'' means $2^{20}$ octets.} + which is fairly small compared to GMP (over $250$KiB). LibTomMath is slightly larger than MPI (which compiles to about +$50$KiB) but LibTomMath is also much faster and more complete than MPI. + +\subsection{API Simplicity} +LibTomMath is designed after the MPI library and shares the API design. Quite often programs that use MPI will build +with LibTomMath without change. The function names correlate directly to the action they perform. Almost all of the +functions share the same parameter passing convention. The learning curve is fairly shallow with the API provided +which is an extremely valuable benefit for the student and developer alike. + +The LIP library is an example of a library with an API that is awkward to work with. LIP uses function names that are often ``compressed'' to +illegible short hand. LibTomMath does not share this characteristic. + +The GMP library also does not return error codes. Instead it uses a POSIX.1 \cite{POSIX1} signal system where errors +are signaled to the host application. This happens to be the fastest approach but definitely not the most versatile. In +effect a math error (i.e. invalid input, heap error, etc) can cause a program to stop functioning which is definitely +undersireable in many situations. + +\subsection{Optimizations} +While LibTomMath is certainly not the fastest library (GMP often beats LibTomMath by a factor of two) it does +feature a set of optimal algorithms for tasks such as modular reduction, exponentiation, multiplication and squaring. GMP +and LIP also feature such optimizations while MPI only uses baseline algorithms with no optimizations. GMP lacks a few +of the additional modular reduction optimizations that LibTomMath features\footnote{At the time of this writing GMP +only had Barrett and Montgomery modular reduction algorithms.}. + +LibTomMath is almost always an order of magnitude faster than the MPI library at computationally expensive tasks such as modular +exponentiation. In the grand scheme of ``bignum'' libraries LibTomMath is faster than the average library and usually +slower than the best libraries such as GMP and OpenSSL by only a small factor. + +\subsection{Portability and Stability} +LibTomMath will build ``out of the box'' on any platform equipped with a modern version of the GNU C Compiler +(\textit{GCC}). This means that without changes the library will build without configuration or setting up any +variables. LIP and MPI will build ``out of the box'' as well but have numerous known bugs. Most notably the author of +MPI has recently stopped working on his library and LIP has long since been discontinued. + +GMP requires a configuration script to run and will not build out of the box. GMP and LibTomMath are still in active +development and are very stable across a variety of platforms. + +\subsection{Choice} +LibTomMath is a relatively compact, well documented, highly optimized and portable library which seems only natural for +the case study of this text. Various source files from the LibTomMath project will be included within the text. However, +the reader is encouraged to download their own copy of the library to actually be able to work with the library. + +\chapter{Getting Started} +\section{Library Basics} +The trick to writing any useful library of source code is to build a solid foundation and work outwards from it. First, +a problem along with allowable solution parameters should be identified and analyzed. In this particular case the +inability to accomodate multiple precision integers is the problem. Futhermore, the solution must be written +as portable source code that is reasonably efficient across several different computer platforms. + +After a foundation is formed the remainder of the library can be designed and implemented in a hierarchical fashion. +That is, to implement the lowest level dependencies first and work towards the most abstract functions last. For example, +before implementing a modular exponentiation algorithm one would implement a modular reduction algorithm. +By building outwards from a base foundation instead of using a parallel design methodology the resulting project is +highly modular. Being highly modular is a desirable property of any project as it often means the resulting product +has a small footprint and updates are easy to perform. + +Usually when I start a project I will begin with the header files. I define the data types I think I will need and +prototype the initial functions that are not dependent on other functions (within the library). After I +implement these base functions I prototype more dependent functions and implement them. The process repeats until +I implement all of the functions I require. For example, in the case of LibTomMath I implemented functions such as +mp\_init() well before I implemented mp\_mul() and even further before I implemented mp\_exptmod(). As an example as to +why this design works note that the Karatsuba and Toom-Cook multipliers were written \textit{after} the +dependent function mp\_exptmod() was written. Adding the new multiplication algorithms did not require changes to the +mp\_exptmod() function itself and lowered the total cost of ownership (\textit{so to speak}) and of development +for new algorithms. This methodology allows new algorithms to be tested in a complete framework with relative ease. + +FIGU,design_process,Design Flow of the First Few Original LibTomMath Functions. + +Only after the majority of the functions were in place did I pursue a less hierarchical approach to auditing and optimizing +the source code. For example, one day I may audit the multipliers and the next day the polynomial basis functions. + +It only makes sense to begin the text with the preliminary data types and support algorithms required as well. +This chapter discusses the core algorithms of the library which are the dependents for every other algorithm. + +\section{What is a Multiple Precision Integer?} +Recall that most programming languages, in particular ISO C \cite{ISOC}, only have fixed precision data types that on their own cannot +be used to represent values larger than their precision will allow. The purpose of multiple precision algorithms is +to use fixed precision data types to create and manipulate multiple precision integers which may represent values +that are very large. + +As a well known analogy, school children are taught how to form numbers larger than nine by prepending more radix ten digits. In the decimal system +the largest single digit value is $9$. However, by concatenating digits together larger numbers may be represented. Newly prepended digits +(\textit{to the left}) are said to be in a different power of ten column. That is, the number $123$ can be described as having a $1$ in the hundreds +column, $2$ in the tens column and $3$ in the ones column. Or more formally $123 = 1 \cdot 10^2 + 2 \cdot 10^1 + 3 \cdot 10^0$. Computer based +multiple precision arithmetic is essentially the same concept. Larger integers are represented by adjoining fixed +precision computer words with the exception that a different radix is used. + +What most people probably do not think about explicitly are the various other attributes that describe a multiple precision +integer. For example, the integer $154_{10}$ has two immediately obvious properties. First, the integer is positive, +that is the sign of this particular integer is positive as opposed to negative. Second, the integer has three digits in +its representation. There is an additional property that the integer posesses that does not concern pencil-and-paper +arithmetic. The third property is how many digits placeholders are available to hold the integer. + +The human analogy of this third property is ensuring there is enough space on the paper to write the integer. For example, +if one starts writing a large number too far to the right on a piece of paper they will have to erase it and move left. +Similarly, computer algorithms must maintain strict control over memory usage to ensure that the digits of an integer +will not exceed the allowed boundaries. These three properties make up what is known as a multiple precision +integer or mp\_int for short. + +\subsection{The mp\_int Structure} +\label{sec:MPINT} +The mp\_int structure is the ISO C based manifestation of what represents a multiple precision integer. The ISO C standard does not provide for +any such data type but it does provide for making composite data types known as structures. The following is the structure definition +used within LibTomMath. + +\index{mp\_int} +\begin{figure}[here] +\begin{center} +\begin{small} +%\begin{verbatim} +\begin{tabular}{|l|} +\hline +typedef struct \{ \\ +\hspace{3mm}int used, alloc, sign;\\ +\hspace{3mm}mp\_digit *dp;\\ +\} \textbf{mp\_int}; \\ +\hline +\end{tabular} +%\end{verbatim} +\end{small} +\caption{The mp\_int Structure} +\label{fig:mpint} +\end{center} +\end{figure} + +The mp\_int structure (fig. \ref{fig:mpint}) can be broken down as follows. + +\begin{enumerate} +\item The \textbf{used} parameter denotes how many digits of the array \textbf{dp} contain the digits used to represent +a given integer. The \textbf{used} count must be positive (or zero) and may not exceed the \textbf{alloc} count. + +\item The \textbf{alloc} parameter denotes how +many digits are available in the array to use by functions before it has to increase in size. When the \textbf{used} count +of a result would exceed the \textbf{alloc} count all of the algorithms will automatically increase the size of the +array to accommodate the precision of the result. + +\item The pointer \textbf{dp} points to a dynamically allocated array of digits that represent the given multiple +precision integer. It is padded with $(\textbf{alloc} - \textbf{used})$ zero digits. The array is maintained in a least +significant digit order. As a pencil and paper analogy the array is organized such that the right most digits are stored +first starting at the location indexed by zero\footnote{In C all arrays begin at zero.} in the array. For example, +if \textbf{dp} contains $\lbrace a, b, c, \ldots \rbrace$ where \textbf{dp}$_0 = a$, \textbf{dp}$_1 = b$, \textbf{dp}$_2 = c$, $\ldots$ then +it would represent the integer $a + b\beta + c\beta^2 + \ldots$ + +\index{MP\_ZPOS} \index{MP\_NEG} +\item The \textbf{sign} parameter denotes the sign as either zero/positive (\textbf{MP\_ZPOS}) or negative (\textbf{MP\_NEG}). +\end{enumerate} + +\subsubsection{Valid mp\_int Structures} +Several rules are placed on the state of an mp\_int structure and are assumed to be followed for reasons of efficiency. +The only exceptions are when the structure is passed to initialization functions such as mp\_init() and mp\_init\_copy(). + +\begin{enumerate} +\item The value of \textbf{alloc} may not be less than one. That is \textbf{dp} always points to a previously allocated +array of digits. +\item The value of \textbf{used} may not exceed \textbf{alloc} and must be greater than or equal to zero. +\item The value of \textbf{used} implies the digit at index $(used - 1)$ of the \textbf{dp} array is non-zero. That is, +leading zero digits in the most significant positions must be trimmed. + \begin{enumerate} + \item Digits in the \textbf{dp} array at and above the \textbf{used} location must be zero. + \end{enumerate} +\item The value of \textbf{sign} must be \textbf{MP\_ZPOS} if \textbf{used} is zero; +this represents the mp\_int value of zero. +\end{enumerate} + +\section{Argument Passing} +A convention of argument passing must be adopted early on in the development of any library. Making the function +prototypes consistent will help eliminate many headaches in the future as the library grows to significant complexity. +In LibTomMath the multiple precision integer functions accept parameters from left to right as pointers to mp\_int +structures. That means that the source (input) operands are placed on the left and the destination (output) on the right. +Consider the following examples. + +\begin{verbatim} + mp_mul(&a, &b, &c); /* c = a * b */ + mp_add(&a, &b, &a); /* a = a + b */ + mp_sqr(&a, &b); /* b = a * a */ +\end{verbatim} + +The left to right order is a fairly natural way to implement the functions since it lets the developer read aloud the +functions and make sense of them. For example, the first function would read ``multiply a and b and store in c''. + +Certain libraries (\textit{LIP by Lenstra for instance}) accept parameters the other way around, to mimic the order +of assignment expressions. That is, the destination (output) is on the left and arguments (inputs) are on the right. In +truth, it is entirely a matter of preference. In the case of LibTomMath the convention from the MPI library has been +adopted. + +Another very useful design consideration, provided for in LibTomMath, is whether to allow argument sources to also be a +destination. For example, the second example (\textit{mp\_add}) adds $a$ to $b$ and stores in $a$. This is an important +feature to implement since it allows the calling functions to cut down on the number of variables it must maintain. +However, to implement this feature specific care has to be given to ensure the destination is not modified before the +source is fully read. + +\section{Return Values} +A well implemented application, no matter what its purpose, should trap as many runtime errors as possible and return them +to the caller. By catching runtime errors a library can be guaranteed to prevent undefined behaviour. However, the end +developer can still manage to cause a library to crash. For example, by passing an invalid pointer an application may +fault by dereferencing memory not owned by the application. + +In the case of LibTomMath the only errors that are checked for are related to inappropriate inputs (division by zero for +instance) and memory allocation errors. It will not check that the mp\_int passed to any function is valid nor +will it check pointers for validity. Any function that can cause a runtime error will return an error code as an +\textbf{int} data type with one of the following values (fig \ref{fig:errcodes}). + +\index{MP\_OKAY} \index{MP\_VAL} \index{MP\_MEM} +\begin{figure}[here] +\begin{center} +\begin{tabular}{|l|l|} +\hline \textbf{Value} & \textbf{Meaning} \\ +\hline \textbf{MP\_OKAY} & The function was successful \\ +\hline \textbf{MP\_VAL} & One of the input value(s) was invalid \\ +\hline \textbf{MP\_MEM} & The function ran out of heap memory \\ +\hline +\end{tabular} +\end{center} +\caption{LibTomMath Error Codes} +\label{fig:errcodes} +\end{figure} + +When an error is detected within a function it should free any memory it allocated, often during the initialization of +temporary mp\_ints, and return as soon as possible. The goal is to leave the system in the same state it was when the +function was called. Error checking with this style of API is fairly simple. + +\begin{verbatim} + int err; + if ((err = mp_add(&a, &b, &c)) != MP_OKAY) { + printf("Error: %s\n", mp_error_to_string(err)); + exit(EXIT_FAILURE); + } +\end{verbatim} + +The GMP \cite{GMP} library uses C style \textit{signals} to flag errors which is of questionable use. Not all errors are fatal +and it was not deemed ideal by the author of LibTomMath to force developers to have signal handlers for such cases. + +\section{Initialization and Clearing} +The logical starting point when actually writing multiple precision integer functions is the initialization and +clearing of the mp\_int structures. These two algorithms will be used by the majority of the higher level algorithms. + +Given the basic mp\_int structure an initialization routine must first allocate memory to hold the digits of +the integer. Often it is optimal to allocate a sufficiently large pre-set number of digits even though +the initial integer will represent zero. If only a single digit were allocated quite a few subsequent re-allocations +would occur when operations are performed on the integers. There is a tradeoff between how many default digits to allocate +and how many re-allocations are tolerable. Obviously allocating an excessive amount of digits initially will waste +memory and become unmanageable. + +If the memory for the digits has been successfully allocated then the rest of the members of the structure must +be initialized. Since the initial state of an mp\_int is to represent the zero integer, the allocated digits must be set +to zero. The \textbf{used} count set to zero and \textbf{sign} set to \textbf{MP\_ZPOS}. + +\subsection{Initializing an mp\_int} +An mp\_int is said to be initialized if it is set to a valid, preferably default, state such that all of the members of the +structure are set to valid values. The mp\_init algorithm will perform such an action. + +\index{mp\_init} +\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_init}. \\ +\textbf{Input}. An mp\_int $a$ \\ +\textbf{Output}. Allocate memory and initialize $a$ to a known valid mp\_int state. \\ +\hline \\ +1. Allocate memory for \textbf{MP\_PREC} digits. \\ +2. If the allocation failed return(\textit{MP\_MEM}) \\ +3. for $n$ from $0$ to $MP\_PREC - 1$ do \\ +\hspace{3mm}3.1 $a_n \leftarrow 0$\\ +4. $a.sign \leftarrow MP\_ZPOS$\\ +5. $a.used \leftarrow 0$\\ +6. $a.alloc \leftarrow MP\_PREC$\\ +7. Return(\textit{MP\_OKAY})\\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_init} +\end{figure} + +\textbf{Algorithm mp\_init.} +The purpose of this function is to initialize an mp\_int structure so that the rest of the library can properly +manipulte it. It is assumed that the input may not have had any of its members previously initialized which is certainly +a valid assumption if the input resides on the stack. + +Before any of the members such as \textbf{sign}, \textbf{used} or \textbf{alloc} are initialized the memory for +the digits is allocated. If this fails the function returns before setting any of the other members. The \textbf{MP\_PREC} +name represents a constant\footnote{Defined in the ``tommath.h'' header file within LibTomMath.} +used to dictate the minimum precision of newly initialized mp\_int integers. Ideally, it is at least equal to the smallest +precision number you'll be working with. + +Allocating a block of digits at first instead of a single digit has the benefit of lowering the number of usually slow +heap operations later functions will have to perform in the future. If \textbf{MP\_PREC} is set correctly the slack +memory and the number of heap operations will be trivial. + +Once the allocation has been made the digits have to be set to zero as well as the \textbf{used}, \textbf{sign} and +\textbf{alloc} members initialized. This ensures that the mp\_int will always represent the default state of zero regardless +of the original condition of the input. + +\textbf{Remark.} +This function introduces the idiosyncrasy that all iterative loops, commonly initiated with the ``for'' keyword, iterate incrementally +when the ``to'' keyword is placed between two expressions. For example, ``for $a$ from $b$ to $c$ do'' means that +a subsequent expression (or body of expressions) are to be evaluated upto $c - b$ times so long as $b \le c$. In each +iteration the variable $a$ is substituted for a new integer that lies inclusively between $b$ and $c$. If $b > c$ occured +the loop would not iterate. By contrast if the ``downto'' keyword were used in place of ``to'' the loop would iterate +decrementally. + +EXAM,bn_mp_init.c + +One immediate observation of this initializtion function is that it does not return a pointer to a mp\_int structure. It +is assumed that the caller has already allocated memory for the mp\_int structure, typically on the application stack. The +call to mp\_init() is used only to initialize the members of the structure to a known default state. + +Here we see (line @23,XMALLOC@) the memory allocation is performed first. This allows us to exit cleanly and quickly +if there is an error. If the allocation fails the routine will return \textbf{MP\_MEM} to the caller to indicate there +was a memory error. The function XMALLOC is what actually allocates the memory. Technically XMALLOC is not a function +but a macro defined in ``tommath.h``. By default, XMALLOC will evaluate to malloc() which is the C library's built--in +memory allocation routine. + +In order to assure the mp\_int is in a known state the digits must be set to zero. On most platforms this could have been +accomplished by using calloc() instead of malloc(). However, to correctly initialize a integer type to a given value in a +portable fashion you have to actually assign the value. The for loop (line @28,for@) performs this required +operation. + +After the memory has been successfully initialized the remainder of the members are initialized +(lines @29,used@ through @31,sign@) to their respective default states. At this point the algorithm has succeeded and +a success code is returned to the calling function. If this function returns \textbf{MP\_OKAY} it is safe to assume the +mp\_int structure has been properly initialized and is safe to use with other functions within the library. + +\subsection{Clearing an mp\_int} +When an mp\_int is no longer required by the application, the memory that has been allocated for its digits must be +returned to the application's memory pool with the mp\_clear algorithm. + +\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_clear}. \\ +\textbf{Input}. An mp\_int $a$ \\ +\textbf{Output}. The memory for $a$ shall be deallocated. \\ +\hline \\ +1. If $a$ has been previously freed then return(\textit{MP\_OKAY}). \\ +2. for $n$ from 0 to $a.used - 1$ do \\ +\hspace{3mm}2.1 $a_n \leftarrow 0$ \\ +3. Free the memory allocated for the digits of $a$. \\ +4. $a.used \leftarrow 0$ \\ +5. $a.alloc \leftarrow 0$ \\ +6. $a.sign \leftarrow MP\_ZPOS$ \\ +7. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_clear} +\end{figure} + +\textbf{Algorithm mp\_clear.} +This algorithm accomplishes two goals. First, it clears the digits and the other mp\_int members. This ensures that +if a developer accidentally re-uses a cleared structure it is less likely to cause problems. The second goal +is to free the allocated memory. + +The logic behind the algorithm is extended by marking cleared mp\_int structures so that subsequent calls to this +algorithm will not try to free the memory multiple times. Cleared mp\_ints are detectable by having a pre-defined invalid +digit pointer \textbf{dp} setting. + +Once an mp\_int has been cleared the mp\_int structure is no longer in a valid state for any other algorithm +with the exception of algorithms mp\_init, mp\_init\_copy, mp\_init\_size and mp\_clear. + +EXAM,bn_mp_clear.c + +The algorithm only operates on the mp\_int if it hasn't been previously cleared. The if statement (line @23,a->dp != NULL@) +checks to see if the \textbf{dp} member is not \textbf{NULL}. If the mp\_int is a valid mp\_int then \textbf{dp} cannot be +\textbf{NULL} in which case the if statement will evaluate to true. + +The digits of the mp\_int are cleared by the for loop (line @25,for@) which assigns a zero to every digit. Similar to mp\_init() +the digits are assigned zero instead of using block memory operations (such as memset()) since this is more portable. + +The digits are deallocated off the heap via the XFREE macro. Similar to XMALLOC the XFREE macro actually evaluates to +a standard C library function. In this case the free() function. Since free() only deallocates the memory the pointer +still has to be reset to \textbf{NULL} manually (line @33,NULL@). + +Now that the digits have been cleared and deallocated the other members are set to their final values (lines @34,= 0@ and @35,ZPOS@). + +\section{Maintenance Algorithms} + +The previous sections describes how to initialize and clear an mp\_int structure. To further support operations +that are to be performed on mp\_int structures (such as addition and multiplication) the dependent algorithms must be +able to augment the precision of an mp\_int and +initialize mp\_ints with differing initial conditions. + +These algorithms complete the set of low level algorithms required to work with mp\_int structures in the higher level +algorithms such as addition, multiplication and modular exponentiation. + +\subsection{Augmenting an mp\_int's Precision} +When storing a value in an mp\_int structure, a sufficient number of digits must be available to accomodate the entire +result of an operation without loss of precision. Quite often the size of the array given by the \textbf{alloc} member +is large enough to simply increase the \textbf{used} digit count. However, when the size of the array is too small it +must be re-sized appropriately to accomodate the result. The mp\_grow algorithm will provide this functionality. + +\newpage\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_grow}. \\ +\textbf{Input}. An mp\_int $a$ and an integer $b$. \\ +\textbf{Output}. $a$ is expanded to accomodate $b$ digits. \\ +\hline \\ +1. if $a.alloc \ge b$ then return(\textit{MP\_OKAY}) \\ +2. $u \leftarrow b\mbox{ (mod }MP\_PREC\mbox{)}$ \\ +3. $v \leftarrow b + 2 \cdot MP\_PREC - u$ \\ +4. Re-allocate the array of digits $a$ to size $v$ \\ +5. If the allocation failed then return(\textit{MP\_MEM}). \\ +6. for n from a.alloc to $v - 1$ do \\ +\hspace{+3mm}6.1 $a_n \leftarrow 0$ \\ +7. $a.alloc \leftarrow v$ \\ +8. Return(\textit{MP\_OKAY}) \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_grow} +\end{figure} + +\textbf{Algorithm mp\_grow.} +It is ideal to prevent re-allocations from being performed if they are not required (step one). This is useful to +prevent mp\_ints from growing excessively in code that erroneously calls mp\_grow. + +The requested digit count is padded up to next multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} (steps two and three). +This helps prevent many trivial reallocations that would grow an mp\_int by trivially small values. + +It is assumed that the reallocation (step four) leaves the lower $a.alloc$ digits of the mp\_int intact. This is much +akin to how the \textit{realloc} function from the standard C library works. Since the newly allocated digits are +assumed to contain undefined values they are initially set to zero. + +EXAM,bn_mp_grow.c + +A quick optimization is to first determine if a memory re-allocation is required at all. The if statement (line @23,if@) checks +if the \textbf{alloc} member of the mp\_int is smaller than the requested digit count. If the count is not larger than \textbf{alloc} +the function skips the re-allocation part thus saving time. + +When a re-allocation is performed it is turned into an optimal request to save time in the future. The requested digit count is +padded upwards to 2nd multiple of \textbf{MP\_PREC} larger than \textbf{alloc} (line @25, size@). The XREALLOC function is used +to re-allocate the memory. As per the other functions XREALLOC is actually a macro which evaluates to realloc by default. The realloc +function leaves the base of the allocation intact which means the first \textbf{alloc} digits of the mp\_int are the same as before +the re-allocation. All that is left is to clear the newly allocated digits and return. + +Note that the re-allocation result is actually stored in a temporary pointer $tmp$. This is to allow this function to return +an error with a valid pointer. Earlier releases of the library stored the result of XREALLOC into the mp\_int $a$. That would +result in a memory leak if XREALLOC ever failed. + +\subsection{Initializing Variable Precision mp\_ints} +Occasionally the number of digits required will be known in advance of an initialization, based on, for example, the size +of input mp\_ints to a given algorithm. The purpose of algorithm mp\_init\_size is similar to mp\_init except that it +will allocate \textit{at least} a specified number of digits. + +\begin{figure}[here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_init\_size}. \\ +\textbf{Input}. An mp\_int $a$ and the requested number of digits $b$. \\ +\textbf{Output}. $a$ is initialized to hold at least $b$ digits. \\ +\hline \\ +1. $u \leftarrow b \mbox{ (mod }MP\_PREC\mbox{)}$ \\ +2. $v \leftarrow b + 2 \cdot MP\_PREC - u$ \\ +3. Allocate $v$ digits. \\ +4. for $n$ from $0$ to $v - 1$ do \\ +\hspace{3mm}4.1 $a_n \leftarrow 0$ \\ +5. $a.sign \leftarrow MP\_ZPOS$\\ +6. $a.used \leftarrow 0$\\ +7. $a.alloc \leftarrow v$\\ +8. Return(\textit{MP\_OKAY})\\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_init\_size} +\end{figure} + +\textbf{Algorithm mp\_init\_size.} +This algorithm will initialize an mp\_int structure $a$ like algorithm mp\_init with the exception that the number of +digits allocated can be controlled by the second input argument $b$. The input size is padded upwards so it is a +multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} digits. This padding is used to prevent trivial +allocations from becoming a bottleneck in the rest of the algorithms. + +Like algorithm mp\_init, the mp\_int structure is initialized to a default state representing the integer zero. This +particular algorithm is useful if it is known ahead of time the approximate size of the input. If the approximation is +correct no further memory re-allocations are required to work with the mp\_int. + +EXAM,bn_mp_init_size.c + +The number of digits $b$ requested is padded (line @22,MP_PREC@) by first augmenting it to the next multiple of +\textbf{MP\_PREC} and then adding \textbf{MP\_PREC} to the result. If the memory can be successfully allocated the +mp\_int is placed in a default state representing the integer zero. Otherwise, the error code \textbf{MP\_MEM} will be +returned (line @27,return@). + +The digits are allocated and set to zero at the same time with the calloc() function (line @25,XCALLOC@). The +\textbf{used} count is set to zero, the \textbf{alloc} count set to the padded digit count and the \textbf{sign} flag set +to \textbf{MP\_ZPOS} to achieve a default valid mp\_int state (lines @29,used@, @30,alloc@ and @31,sign@). If the function +returns succesfully then it is correct to assume that the mp\_int structure is in a valid state for the remainder of the +functions to work with. + +\subsection{Multiple Integer Initializations and Clearings} +Occasionally a function will require a series of mp\_int data types to be made available simultaneously. +The purpose of algorithm mp\_init\_multi is to initialize a variable length array of mp\_int structures in a single +statement. It is essentially a shortcut to multiple initializations. + +\newpage\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_init\_multi}. \\ +\textbf{Input}. Variable length array $V_k$ of mp\_int variables of length $k$. \\ +\textbf{Output}. The array is initialized such that each mp\_int of $V_k$ is ready to use. \\ +\hline \\ +1. for $n$ from 0 to $k - 1$ do \\ +\hspace{+3mm}1.1. Initialize the mp\_int $V_n$ (\textit{mp\_init}) \\ +\hspace{+3mm}1.2. If initialization failed then do \\ +\hspace{+6mm}1.2.1. for $j$ from $0$ to $n$ do \\ +\hspace{+9mm}1.2.1.1. Free the mp\_int $V_j$ (\textit{mp\_clear}) \\ +\hspace{+6mm}1.2.2. Return(\textit{MP\_MEM}) \\ +2. Return(\textit{MP\_OKAY}) \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_init\_multi} +\end{figure} + +\textbf{Algorithm mp\_init\_multi.} +The algorithm will initialize the array of mp\_int variables one at a time. If a runtime error has been detected +(\textit{step 1.2}) all of the previously initialized variables are cleared. The goal is an ``all or nothing'' +initialization which allows for quick recovery from runtime errors. + +EXAM,bn_mp_init_multi.c + +This function intializes a variable length list of mp\_int structure pointers. However, instead of having the mp\_int +structures in an actual C array they are simply passed as arguments to the function. This function makes use of the +``...'' argument syntax of the C programming language. The list is terminated with a final \textbf{NULL} argument +appended on the right. + +The function uses the ``stdarg.h'' \textit{va} functions to step portably through the arguments to the function. A count +$n$ of succesfully initialized mp\_int structures is maintained (line @47,n++@) such that if a failure does occur, +the algorithm can backtrack and free the previously initialized structures (lines @27,if@ to @46,}@). + + +\subsection{Clamping Excess Digits} +When a function anticipates a result will be $n$ digits it is simpler to assume this is true within the body of +the function instead of checking during the computation. For example, a multiplication of a $i$ digit number by a +$j$ digit produces a result of at most $i + j$ digits. It is entirely possible that the result is $i + j - 1$ +though, with no final carry into the last position. However, suppose the destination had to be first expanded +(\textit{via mp\_grow}) to accomodate $i + j - 1$ digits than further expanded to accomodate the final carry. +That would be a considerable waste of time since heap operations are relatively slow. + +The ideal solution is to always assume the result is $i + j$ and fix up the \textbf{used} count after the function +terminates. This way a single heap operation (\textit{at most}) is required. However, if the result was not checked +there would be an excess high order zero digit. + +For example, suppose the product of two integers was $x_n = (0x_{n-1}x_{n-2}...x_0)_{\beta}$. The leading zero digit +will not contribute to the precision of the result. In fact, through subsequent operations more leading zero digits would +accumulate to the point the size of the integer would be prohibitive. As a result even though the precision is very +low the representation is excessively large. + +The mp\_clamp algorithm is designed to solve this very problem. It will trim high-order zeros by decrementing the +\textbf{used} count until a non-zero most significant digit is found. Also in this system, zero is considered to be a +positive number which means that if the \textbf{used} count is decremented to zero, the sign must be set to +\textbf{MP\_ZPOS}. + +\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_clamp}. \\ +\textbf{Input}. An mp\_int $a$ \\ +\textbf{Output}. Any excess leading zero digits of $a$ are removed \\ +\hline \\ +1. while $a.used > 0$ and $a_{a.used - 1} = 0$ do \\ +\hspace{+3mm}1.1 $a.used \leftarrow a.used - 1$ \\ +2. if $a.used = 0$ then do \\ +\hspace{+3mm}2.1 $a.sign \leftarrow MP\_ZPOS$ \\ +\hline \\ +\end{tabular} +\end{center} +\caption{Algorithm mp\_clamp} +\end{figure} + +\textbf{Algorithm mp\_clamp.} +As can be expected this algorithm is very simple. The loop on step one is expected to iterate only once or twice at +the most. For example, this will happen in cases where there is not a carry to fill the last position. Step two fixes the sign for +when all of the digits are zero to ensure that the mp\_int is valid at all times. + +EXAM,bn_mp_clamp.c + +Note on line @27,while@ how to test for the \textbf{used} count is made on the left of the \&\& operator. In the C programming +language the terms to \&\& are evaluated left to right with a boolean short-circuit if any condition fails. This is +important since if the \textbf{used} is zero the test on the right would fetch below the array. That is obviously +undesirable. The parenthesis on line @28,a->used@ is used to make sure the \textbf{used} count is decremented and not +the pointer ``a''. + +\section*{Exercises} +\begin{tabular}{cl} +$\left [ 1 \right ]$ & Discuss the relevance of the \textbf{used} member of the mp\_int structure. \\ + & \\ +$\left [ 1 \right ]$ & Discuss the consequences of not using padding when performing allocations. \\ + & \\ +$\left [ 2 \right ]$ & Estimate an ideal value for \textbf{MP\_PREC} when performing 1024-bit RSA \\ + & encryption when $\beta = 2^{28}$. \\ + & \\ +$\left [ 1 \right ]$ & Discuss the relevance of the algorithm mp\_clamp. What does it prevent? \\ + & \\ +$\left [ 1 \right ]$ & Give an example of when the algorithm mp\_init\_copy might be useful. \\ + & \\ +\end{tabular} + + +%%% +% CHAPTER FOUR +%%% + +\chapter{Basic Operations} + +\section{Introduction} +In the previous chapter a series of low level algorithms were established that dealt with initializing and maintaining +mp\_int structures. This chapter will discuss another set of seemingly non-algebraic algorithms which will form the low +level basis of the entire library. While these algorithm are relatively trivial it is important to understand how they +work before proceeding since these algorithms will be used almost intrinsically in the following chapters. + +The algorithms in this chapter deal primarily with more ``programmer'' related tasks such as creating copies of +mp\_int structures, assigning small values to mp\_int structures and comparisons of the values mp\_int structures +represent. + +\section{Assigning Values to mp\_int Structures} +\subsection{Copying an mp\_int} +Assigning the value that a given mp\_int structure represents to another mp\_int structure shall be known as making +a copy for the purposes of this text. The copy of the mp\_int will be a separate entity that represents the same +value as the mp\_int it was copied from. The mp\_copy algorithm provides this functionality. + +\newpage\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_copy}. \\ +\textbf{Input}. An mp\_int $a$ and $b$. \\ +\textbf{Output}. Store a copy of $a$ in $b$. \\ +\hline \\ +1. If $b.alloc < a.used$ then grow $b$ to $a.used$ digits. (\textit{mp\_grow}) \\ +2. for $n$ from 0 to $a.used - 1$ do \\ +\hspace{3mm}2.1 $b_{n} \leftarrow a_{n}$ \\ +3. for $n$ from $a.used$ to $b.used - 1$ do \\ +\hspace{3mm}3.1 $b_{n} \leftarrow 0$ \\ +4. $b.used \leftarrow a.used$ \\ +5. $b.sign \leftarrow a.sign$ \\ +6. return(\textit{MP\_OKAY}) \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_copy} +\end{figure} + +\textbf{Algorithm mp\_copy.} +This algorithm copies the mp\_int $a$ such that upon succesful termination of the algorithm the mp\_int $b$ will +represent the same integer as the mp\_int $a$. The mp\_int $b$ shall be a complete and distinct copy of the +mp\_int $a$ meaing that the mp\_int $a$ can be modified and it shall not affect the value of the mp\_int $b$. + +If $b$ does not have enough room for the digits of $a$ it must first have its precision augmented via the mp\_grow +algorithm. The digits of $a$ are copied over the digits of $b$ and any excess digits of $b$ are set to zero (step two +and three). The \textbf{used} and \textbf{sign} members of $a$ are finally copied over the respective members of +$b$. + +\textbf{Remark.} This algorithm also introduces a new idiosyncrasy that will be used throughout the rest of the +text. The error return codes of other algorithms are not explicitly checked in the pseudo-code presented. For example, in +step one of the mp\_copy algorithm the return of mp\_grow is not explicitly checked to ensure it succeeded. Text space is +limited so it is assumed that if a algorithm fails it will clear all temporarily allocated mp\_ints and return +the error code itself. However, the C code presented will demonstrate all of the error handling logic required to +implement the pseudo-code. + +EXAM,bn_mp_copy.c + +Occasionally a dependent algorithm may copy an mp\_int effectively into itself such as when the input and output +mp\_int structures passed to a function are one and the same. For this case it is optimal to return immediately without +copying digits (line @24,a == b@). + +The mp\_int $b$ must have enough digits to accomodate the used digits of the mp\_int $a$. If $b.alloc$ is less than +$a.used$ the algorithm mp\_grow is used to augment the precision of $b$ (lines @29,alloc@ to @33,}@). In order to +simplify the inner loop that copies the digits from $a$ to $b$, two aliases $tmpa$ and $tmpb$ point directly at the digits +of the mp\_ints $a$ and $b$ respectively. These aliases (lines @42,tmpa@ and @45,tmpb@) allow the compiler to access the digits without first dereferencing the +mp\_int pointers and then subsequently the pointer to the digits. + +After the aliases are established the digits from $a$ are copied into $b$ (lines @48,for@ to @50,}@) and then the excess +digits of $b$ are set to zero (lines @53,for@ to @55,}@). Both ``for'' loops make use of the pointer aliases and in +fact the alias for $b$ is carried through into the second ``for'' loop to clear the excess digits. This optimization +allows the alias to stay in a machine register fairly easy between the two loops. + +\textbf{Remarks.} The use of pointer aliases is an implementation methodology first introduced in this function that will +be used considerably in other functions. Technically, a pointer alias is simply a short hand alias used to lower the +number of pointer dereferencing operations required to access data. For example, a for loop may resemble + +\begin{alltt} +for (x = 0; x < 100; x++) \{ + a->num[4]->dp[x] = 0; +\} +\end{alltt} + +This could be re-written using aliases as + +\begin{alltt} +mp_digit *tmpa; +a = a->num[4]->dp; +for (x = 0; x < 100; x++) \{ + *a++ = 0; +\} +\end{alltt} + +In this case an alias is used to access the +array of digits within an mp\_int structure directly. It may seem that a pointer alias is strictly not required +as a compiler may optimize out the redundant pointer operations. However, there are two dominant reasons to use aliases. + +The first reason is that most compilers will not effectively optimize pointer arithmetic. For example, some optimizations +may work for the Microsoft Visual C++ compiler (MSVC) and not for the GNU C Compiler (GCC). Also some optimizations may +work for GCC and not MSVC. As such it is ideal to find a common ground for as many compilers as possible. Pointer +aliases optimize the code considerably before the compiler even reads the source code which means the end compiled code +stands a better chance of being faster. + +The second reason is that pointer aliases often can make an algorithm simpler to read. Consider the first ``for'' +loop of the function mp\_copy() re-written to not use pointer aliases. + +\begin{alltt} + /* copy all the digits */ + for (n = 0; n < a->used; n++) \{ + b->dp[n] = a->dp[n]; + \} +\end{alltt} + +Whether this code is harder to read depends strongly on the individual. However, it is quantifiably slightly more +complicated as there are four variables within the statement instead of just two. + +\subsubsection{Nested Statements} +Another commonly used technique in the source routines is that certain sections of code are nested. This is used in +particular with the pointer aliases to highlight code phases. For example, a Comba multiplier (discussed in chapter six) +will typically have three different phases. First the temporaries are initialized, then the columns calculated and +finally the carries are propagated. In this example the middle column production phase will typically be nested as it +uses temporary variables and aliases the most. + +The nesting also simplies the source code as variables that are nested are only valid for their scope. As a result +the various temporary variables required do not propagate into other sections of code. + + +\subsection{Creating a Clone} +Another common operation is to make a local temporary copy of an mp\_int argument. To initialize an mp\_int +and then copy another existing mp\_int into the newly intialized mp\_int will be known as creating a clone. This is +useful within functions that need to modify an argument but do not wish to actually modify the original copy. The +mp\_init\_copy algorithm has been designed to help perform this task. + +\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_init\_copy}. \\ +\textbf{Input}. An mp\_int $a$ and $b$\\ +\textbf{Output}. $a$ is initialized to be a copy of $b$. \\ +\hline \\ +1. Init $a$. (\textit{mp\_init}) \\ +2. Copy $b$ to $a$. (\textit{mp\_copy}) \\ +3. Return the status of the copy operation. \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_init\_copy} +\end{figure} + +\textbf{Algorithm mp\_init\_copy.} +This algorithm will initialize an mp\_int variable and copy another previously initialized mp\_int variable into it. As +such this algorithm will perform two operations in one step. + +EXAM,bn_mp_init_copy.c + +This will initialize \textbf{a} and make it a verbatim copy of the contents of \textbf{b}. Note that +\textbf{a} will have its own memory allocated which means that \textbf{b} may be cleared after the call +and \textbf{a} will be left intact. + +\section{Zeroing an Integer} +Reseting an mp\_int to the default state is a common step in many algorithms. The mp\_zero algorithm will be the algorithm used to +perform this task. + +\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_zero}. \\ +\textbf{Input}. An mp\_int $a$ \\ +\textbf{Output}. Zero the contents of $a$ \\ +\hline \\ +1. $a.used \leftarrow 0$ \\ +2. $a.sign \leftarrow$ MP\_ZPOS \\ +3. for $n$ from 0 to $a.alloc - 1$ do \\ +\hspace{3mm}3.1 $a_n \leftarrow 0$ \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_zero} +\end{figure} + +\textbf{Algorithm mp\_zero.} +This algorithm simply resets a mp\_int to the default state. + +EXAM,bn_mp_zero.c + +After the function is completed, all of the digits are zeroed, the \textbf{used} count is zeroed and the +\textbf{sign} variable is set to \textbf{MP\_ZPOS}. + +\section{Sign Manipulation} +\subsection{Absolute Value} +With the mp\_int representation of an integer, calculating the absolute value is trivial. The mp\_abs algorithm will compute +the absolute value of an mp\_int. + +\newpage\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_abs}. \\ +\textbf{Input}. An mp\_int $a$ \\ +\textbf{Output}. Computes $b = \vert a \vert$ \\ +\hline \\ +1. Copy $a$ to $b$. (\textit{mp\_copy}) \\ +2. If the copy failed return(\textit{MP\_MEM}). \\ +3. $b.sign \leftarrow MP\_ZPOS$ \\ +4. Return(\textit{MP\_OKAY}) \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_abs} +\end{figure} + +\textbf{Algorithm mp\_abs.} +This algorithm computes the absolute of an mp\_int input. First it copies $a$ over $b$. This is an example of an +algorithm where the check in mp\_copy that determines if the source and destination are equal proves useful. This allows, +for instance, the developer to pass the same mp\_int as the source and destination to this function without addition +logic to handle it. + +EXAM,bn_mp_abs.c + +\subsection{Integer Negation} +With the mp\_int representation of an integer, calculating the negation is also trivial. The mp\_neg algorithm will compute +the negative of an mp\_int input. + +\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_neg}. \\ +\textbf{Input}. An mp\_int $a$ \\ +\textbf{Output}. Computes $b = -a$ \\ +\hline \\ +1. Copy $a$ to $b$. (\textit{mp\_copy}) \\ +2. If the copy failed return(\textit{MP\_MEM}). \\ +3. If $a.used = 0$ then return(\textit{MP\_OKAY}). \\ +4. If $a.sign = MP\_ZPOS$ then do \\ +\hspace{3mm}4.1 $b.sign = MP\_NEG$. \\ +5. else do \\ +\hspace{3mm}5.1 $b.sign = MP\_ZPOS$. \\ +6. Return(\textit{MP\_OKAY}) \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_neg} +\end{figure} + +\textbf{Algorithm mp\_neg.} +This algorithm computes the negation of an input. First it copies $a$ over $b$. If $a$ has no used digits then +the algorithm returns immediately. Otherwise it flips the sign flag and stores the result in $b$. Note that if +$a$ had no digits then it must be positive by definition. Had step three been omitted then the algorithm would return +zero as negative. + +EXAM,bn_mp_neg.c + +\section{Small Constants} +\subsection{Setting Small Constants} +Often a mp\_int must be set to a relatively small value such as $1$ or $2$. For these cases the mp\_set algorithm is useful. + +\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_set}. \\ +\textbf{Input}. An mp\_int $a$ and a digit $b$ \\ +\textbf{Output}. Make $a$ equivalent to $b$ \\ +\hline \\ +1. Zero $a$ (\textit{mp\_zero}). \\ +2. $a_0 \leftarrow b \mbox{ (mod }\beta\mbox{)}$ \\ +3. $a.used \leftarrow \left \lbrace \begin{array}{ll} + 1 & \mbox{if }a_0 > 0 \\ + 0 & \mbox{if }a_0 = 0 + \end{array} \right .$ \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_set} +\end{figure} + +\textbf{Algorithm mp\_set.} +This algorithm sets a mp\_int to a small single digit value. Step number 1 ensures that the integer is reset to the default state. The +single digit is set (\textit{modulo $\beta$}) and the \textbf{used} count is adjusted accordingly. + +EXAM,bn_mp_set.c + +Line @21,mp_zero@ calls mp\_zero() to clear the mp\_int and reset the sign. Line @22,MP_MASK@ copies the digit +into the least significant location. Note the usage of a new constant \textbf{MP\_MASK}. This constant is used to quickly +reduce an integer modulo $\beta$. Since $\beta$ is of the form $2^k$ for any suitable $k$ it suffices to perform a binary AND with +$MP\_MASK = 2^k - 1$ to perform the reduction. Finally line @23,a->used@ will set the \textbf{used} member with respect to the +digit actually set. This function will always make the integer positive. + +One important limitation of this function is that it will only set one digit. The size of a digit is not fixed, meaning source that uses +this function should take that into account. Only trivially small constants can be set using this function. + +\subsection{Setting Large Constants} +To overcome the limitations of the mp\_set algorithm the mp\_set\_int algorithm is ideal. It accepts a ``long'' +data type as input and will always treat it as a 32-bit integer. + +\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_set\_int}. \\ +\textbf{Input}. An mp\_int $a$ and a ``long'' integer $b$ \\ +\textbf{Output}. Make $a$ equivalent to $b$ \\ +\hline \\ +1. Zero $a$ (\textit{mp\_zero}) \\ +2. for $n$ from 0 to 7 do \\ +\hspace{3mm}2.1 $a \leftarrow a \cdot 16$ (\textit{mp\_mul2d}) \\ +\hspace{3mm}2.2 $u \leftarrow \lfloor b / 2^{4(7 - n)} \rfloor \mbox{ (mod }16\mbox{)}$\\ +\hspace{3mm}2.3 $a_0 \leftarrow a_0 + u$ \\ +\hspace{3mm}2.4 $a.used \leftarrow a.used + 1$ \\ +3. Clamp excess used digits (\textit{mp\_clamp}) \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_set\_int} +\end{figure} + +\textbf{Algorithm mp\_set\_int.} +The algorithm performs eight iterations of a simple loop where in each iteration four bits from the source are added to the +mp\_int. Step 2.1 will multiply the current result by sixteen making room for four more bits in the less significant positions. In step 2.2 the +next four bits from the source are extracted and are added to the mp\_int. The \textbf{used} digit count is +incremented to reflect the addition. The \textbf{used} digit counter is incremented since if any of the leading digits were zero the mp\_int would have +zero digits used and the newly added four bits would be ignored. + +Excess zero digits are trimmed in steps 2.1 and 3 by using higher level algorithms mp\_mul2d and mp\_clamp. + +EXAM,bn_mp_set_int.c + +This function sets four bits of the number at a time to handle all practical \textbf{DIGIT\_BIT} sizes. The weird +addition on line @38,a->used@ ensures that the newly added in bits are added to the number of digits. While it may not +seem obvious as to why the digit counter does not grow exceedingly large it is because of the shift on line @27,mp_mul_2d@ +as well as the call to mp\_clamp() on line @40,mp_clamp@. Both functions will clamp excess leading digits which keeps +the number of used digits low. + +\section{Comparisons} +\subsection{Unsigned Comparisions} +Comparing a multiple precision integer is performed with the exact same algorithm used to compare two decimal numbers. For example, +to compare $1,234$ to $1,264$ the digits are extracted by their positions. That is we compare $1 \cdot 10^3 + 2 \cdot 10^2 + 3 \cdot 10^1 + 4 \cdot 10^0$ +to $1 \cdot 10^3 + 2 \cdot 10^2 + 6 \cdot 10^1 + 4 \cdot 10^0$ by comparing single digits at a time starting with the highest magnitude +positions. If any leading digit of one integer is greater than a digit in the same position of another integer then obviously it must be greater. + +The first comparision routine that will be developed is the unsigned magnitude compare which will perform a comparison based on the digits of two +mp\_int variables alone. It will ignore the sign of the two inputs. Such a function is useful when an absolute comparison is required or if the +signs are known to agree in advance. + +To facilitate working with the results of the comparison functions three constants are required. + +\begin{figure}[here] +\begin{center} +\begin{tabular}{|r|l|} +\hline \textbf{Constant} & \textbf{Meaning} \\ +\hline \textbf{MP\_GT} & Greater Than \\ +\hline \textbf{MP\_EQ} & Equal To \\ +\hline \textbf{MP\_LT} & Less Than \\ +\hline +\end{tabular} +\end{center} +\caption{Comparison Return Codes} +\end{figure} + +\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_cmp\_mag}. \\ +\textbf{Input}. Two mp\_ints $a$ and $b$. \\ +\textbf{Output}. Unsigned comparison results ($a$ to the left of $b$). \\ +\hline \\ +1. If $a.used > b.used$ then return(\textit{MP\_GT}) \\ +2. If $a.used < b.used$ then return(\textit{MP\_LT}) \\ +3. for n from $a.used - 1$ to 0 do \\ +\hspace{+3mm}3.1 if $a_n > b_n$ then return(\textit{MP\_GT}) \\ +\hspace{+3mm}3.2 if $a_n < b_n$ then return(\textit{MP\_LT}) \\ +4. Return(\textit{MP\_EQ}) \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_cmp\_mag} +\end{figure} + +\textbf{Algorithm mp\_cmp\_mag.} +By saying ``$a$ to the left of $b$'' it is meant that the comparison is with respect to $a$, that is if $a$ is greater than $b$ it will return +\textbf{MP\_GT} and similar with respect to when $a = b$ and $a < b$. The first two steps compare the number of digits used in both $a$ and $b$. +Obviously if the digit counts differ there would be an imaginary zero digit in the smaller number where the leading digit of the larger number is. +If both have the same number of digits than the actual digits themselves must be compared starting at the leading digit. + +By step three both inputs must have the same number of digits so its safe to start from either $a.used - 1$ or $b.used - 1$ and count down to +the zero'th digit. If after all of the digits have been compared, no difference is found, the algorithm returns \textbf{MP\_EQ}. + +EXAM,bn_mp_cmp_mag.c + +The two if statements on lines @24,if@ and @28,if@ compare the number of digits in the two inputs. These two are performed before all of the digits +are compared since it is a very cheap test to perform and can potentially save considerable time. The implementation given is also not valid +without those two statements. $b.alloc$ may be smaller than $a.used$, meaning that undefined values will be read from $b$ past the end of the +array of digits. + +\subsection{Signed Comparisons} +Comparing with sign considerations is also fairly critical in several routines (\textit{division for example}). Based on an unsigned magnitude +comparison a trivial signed comparison algorithm can be written. + +\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_cmp}. \\ +\textbf{Input}. Two mp\_ints $a$ and $b$ \\ +\textbf{Output}. Signed Comparison Results ($a$ to the left of $b$) \\ +\hline \\ +1. if $a.sign = MP\_NEG$ and $b.sign = MP\_ZPOS$ then return(\textit{MP\_LT}) \\ +2. if $a.sign = MP\_ZPOS$ and $b.sign = MP\_NEG$ then return(\textit{MP\_GT}) \\ +3. if $a.sign = MP\_NEG$ then \\ +\hspace{+3mm}3.1 Return the unsigned comparison of $b$ and $a$ (\textit{mp\_cmp\_mag}) \\ +4 Otherwise \\ +\hspace{+3mm}4.1 Return the unsigned comparison of $a$ and $b$ \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_cmp} +\end{figure} + +\textbf{Algorithm mp\_cmp.} +The first two steps compare the signs of the two inputs. If the signs do not agree then it can return right away with the appropriate +comparison code. When the signs are equal the digits of the inputs must be compared to determine the correct result. In step +three the unsigned comparision flips the order of the arguments since they are both negative. For instance, if $-a > -b$ then +$\vert a \vert < \vert b \vert$. Step number four will compare the two when they are both positive. + +EXAM,bn_mp_cmp.c + +The two if statements on lines @22,if@ and @26,if@ perform the initial sign comparison. If the signs are not the equal then which ever +has the positive sign is larger. At line @30,if@, the inputs are compared based on magnitudes. If the signs were both negative then +the unsigned comparison is performed in the opposite direction (\textit{line @31,mp_cmp_mag@}). Otherwise, the signs are assumed to +be both positive and a forward direction unsigned comparison is performed. + +\section*{Exercises} +\begin{tabular}{cl} +$\left [ 2 \right ]$ & Modify algorithm mp\_set\_int to accept as input a variable length array of bits. \\ + & \\ +$\left [ 3 \right ]$ & Give the probability that algorithm mp\_cmp\_mag will have to compare $k$ digits \\ + & of two random digits (of equal magnitude) before a difference is found. \\ + & \\ +$\left [ 1 \right ]$ & Suggest a simple method to speed up the implementation of mp\_cmp\_mag based \\ + & on the observations made in the previous problem. \\ + & +\end{tabular} + +\chapter{Basic Arithmetic} +\section{Introduction} +At this point algorithms for initialization, clearing, zeroing, copying, comparing and setting small constants have been +established. The next logical set of algorithms to develop are addition, subtraction and digit shifting algorithms. These +algorithms make use of the lower level algorithms and are the cruicial building block for the multiplication algorithms. It is very important +that these algorithms are highly optimized. On their own they are simple $O(n)$ algorithms but they can be called from higher level algorithms +which easily places them at $O(n^2)$ or even $O(n^3)$ work levels. + +MARK,SHIFTS +All of the algorithms within this chapter make use of the logical bit shift operations denoted by $<<$ and $>>$ for left and right +logical shifts respectively. A logical shift is analogous to sliding the decimal point of radix-10 representations. For example, the real +number $0.9345$ is equivalent to $93.45\%$ which is found by sliding the the decimal two places to the right (\textit{multiplying by $\beta^2 = 10^2$}). +Algebraically a binary logical shift is equivalent to a division or multiplication by a power of two. +For example, $a << k = a \cdot 2^k$ while $a >> k = \lfloor a/2^k \rfloor$. + +One significant difference between a logical shift and the way decimals are shifted is that digits below the zero'th position are removed +from the number. For example, consider $1101_2 >> 1$ using decimal notation this would produce $110.1_2$. However, with a logical shift the +result is $110_2$. + +\section{Addition and Subtraction} +In common twos complement fixed precision arithmetic negative numbers are easily represented by subtraction from the modulus. For example, with 32-bit integers +$a - b\mbox{ (mod }2^{32}\mbox{)}$ is the same as $a + (2^{32} - b) \mbox{ (mod }2^{32}\mbox{)}$ since $2^{32} \equiv 0 \mbox{ (mod }2^{32}\mbox{)}$. +As a result subtraction can be performed with a trivial series of logical operations and an addition. + +However, in multiple precision arithmetic negative numbers are not represented in the same way. Instead a sign flag is used to keep track of the +sign of the integer. As a result signed addition and subtraction are actually implemented as conditional usage of lower level addition or +subtraction algorithms with the sign fixed up appropriately. + +The lower level algorithms will add or subtract integers without regard to the sign flag. That is they will add or subtract the magnitude of +the integers respectively. + +\subsection{Low Level Addition} +An unsigned addition of multiple precision integers is performed with the same long-hand algorithm used to add decimal numbers. That is to add the +trailing digits first and propagate the resulting carry upwards. Since this is a lower level algorithm the name will have a ``s\_'' prefix. +Historically that convention stems from the MPI library where ``s\_'' stood for static functions that were hidden from the developer entirely. + +\newpage +\begin{figure}[!here] +\begin{center} +\begin{small} +\begin{tabular}{l} +\hline Algorithm \textbf{s\_mp\_add}. \\ +\textbf{Input}. Two mp\_ints $a$ and $b$ \\ +\textbf{Output}. The unsigned addition $c = \vert a \vert + \vert b \vert$. \\ +\hline \\ +1. if $a.used > b.used$ then \\ +\hspace{+3mm}1.1 $min \leftarrow b.used$ \\ +\hspace{+3mm}1.2 $max \leftarrow a.used$ \\ +\hspace{+3mm}1.3 $x \leftarrow a$ \\ +2. else \\ +\hspace{+3mm}2.1 $min \leftarrow a.used$ \\ +\hspace{+3mm}2.2 $max \leftarrow b.used$ \\ +\hspace{+3mm}2.3 $x \leftarrow b$ \\ +3. If $c.alloc < max + 1$ then grow $c$ to hold at least $max + 1$ digits (\textit{mp\_grow}) \\ +4. $oldused \leftarrow c.used$ \\ +5. $c.used \leftarrow max + 1$ \\ +6. $u \leftarrow 0$ \\ +7. for $n$ from $0$ to $min - 1$ do \\ +\hspace{+3mm}7.1 $c_n \leftarrow a_n + b_n + u$ \\ +\hspace{+3mm}7.2 $u \leftarrow c_n >> lg(\beta)$ \\ +\hspace{+3mm}7.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ +8. if $min \ne max$ then do \\ +\hspace{+3mm}8.1 for $n$ from $min$ to $max - 1$ do \\ +\hspace{+6mm}8.1.1 $c_n \leftarrow x_n + u$ \\ +\hspace{+6mm}8.1.2 $u \leftarrow c_n >> lg(\beta)$ \\ +\hspace{+6mm}8.1.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ +9. $c_{max} \leftarrow u$ \\ +10. if $olduse > max$ then \\ +\hspace{+3mm}10.1 for $n$ from $max + 1$ to $oldused - 1$ do \\ +\hspace{+6mm}10.1.1 $c_n \leftarrow 0$ \\ +11. Clamp excess digits in $c$. (\textit{mp\_clamp}) \\ +12. Return(\textit{MP\_OKAY}) \\ +\hline +\end{tabular} +\end{small} +\end{center} +\caption{Algorithm s\_mp\_add} +\end{figure} + +\textbf{Algorithm s\_mp\_add.} +This algorithm is loosely based on algorithm 14.7 of HAC \cite[pp. 594]{HAC} but has been extended to allow the inputs to have different magnitudes. +Coincidentally the description of algorithm A in Knuth \cite[pp. 266]{TAOCPV2} shares the same deficiency as the algorithm from \cite{HAC}. Even the +MIX pseudo machine code presented by Knuth \cite[pp. 266-267]{TAOCPV2} is incapable of handling inputs which are of different magnitudes. + +The first thing that has to be accomplished is to sort out which of the two inputs is the largest. The addition logic +will simply add all of the smallest input to the largest input and store that first part of the result in the +destination. Then it will apply a simpler addition loop to excess digits of the larger input. + +The first two steps will handle sorting the inputs such that $min$ and $max$ hold the digit counts of the two +inputs. The variable $x$ will be an mp\_int alias for the largest input or the second input $b$ if they have the +same number of digits. After the inputs are sorted the destination $c$ is grown as required to accomodate the sum +of the two inputs. The original \textbf{used} count of $c$ is copied and set to the new used count. + +At this point the first addition loop will go through as many digit positions that both inputs have. The carry +variable $\mu$ is set to zero outside the loop. Inside the loop an ``addition'' step requires three statements to produce +one digit of the summand. First +two digits from $a$ and $b$ are added together along with the carry $\mu$. The carry of this step is extracted and stored +in $\mu$ and finally the digit of the result $c_n$ is truncated within the range $0 \le c_n < \beta$. + +Now all of the digit positions that both inputs have in common have been exhausted. If $min \ne max$ then $x$ is an alias +for one of the inputs that has more digits. A simplified addition loop is then used to essentially copy the remaining digits +and the carry to the destination. + +The final carry is stored in $c_{max}$ and digits above $max$ upto $oldused$ are zeroed which completes the addition. + + +EXAM,bn_s_mp_add.c + +Lines @27,if@ to @35,}@ perform the initial sorting of the inputs and determine the $min$ and $max$ variables. Note that $x$ is a pointer to a +mp\_int assigned to the largest input, in effect it is a local alias. Lines @37,init@ to @42,}@ ensure that the destination is grown to +accomodate the result of the addition. + +Similar to the implementation of mp\_copy this function uses the braced code and local aliases coding style. The three aliases that are on +lines @56,tmpa@, @59,tmpb@ and @62,tmpc@ represent the two inputs and destination variables respectively. These aliases are used to ensure the +compiler does not have to dereference $a$, $b$ or $c$ (respectively) to access the digits of the respective mp\_int. + +The initial carry $u$ is cleared on line @65,u = 0@, note that $u$ is of type mp\_digit which ensures type compatibility within the +implementation. The initial addition loop begins on line @66,for@ and ends on line @75,}@. Similarly the conditional addition loop +begins on line @81,for@ and ends on line @90,}@. The addition is finished with the final carry being stored in $tmpc$ on line @94,tmpc++@. +Note the ``++'' operator on the same line. After line @94,tmpc++@ $tmpc$ will point to the $c.used$'th digit of the mp\_int $c$. This is useful +for the next loop on lines @97,for@ to @99,}@ which set any old upper digits to zero. + +\subsection{Low Level Subtraction} +The low level unsigned subtraction algorithm is very similar to the low level unsigned addition algorithm. The principle difference is that the +unsigned subtraction algorithm requires the result to be positive. That is when computing $a - b$ the condition $\vert a \vert \ge \vert b\vert$ must +be met for this algorithm to function properly. Keep in mind this low level algorithm is not meant to be used in higher level algorithms directly. +This algorithm as will be shown can be used to create functional signed addition and subtraction algorithms. + +MARK,GAMMA + +For this algorithm a new variable is required to make the description simpler. Recall from section 1.3.1 that a mp\_digit must be able to represent +the range $0 \le x < 2\beta$ for the algorithms to work correctly. However, it is allowable that a mp\_digit represent a larger range of values. For +this algorithm we will assume that the variable $\gamma$ represents the number of bits available in a +mp\_digit (\textit{this implies $2^{\gamma} > \beta$}). + +For example, the default for LibTomMath is to use a ``unsigned long'' for the mp\_digit ``type'' while $\beta = 2^{28}$. In ISO C an ``unsigned long'' +data type must be able to represent $0 \le x < 2^{32}$ meaning that in this case $\gamma = 32$. + +\newpage\begin{figure}[!here] +\begin{center} +\begin{small} +\begin{tabular}{l} +\hline Algorithm \textbf{s\_mp\_sub}. \\ +\textbf{Input}. Two mp\_ints $a$ and $b$ ($\vert a \vert \ge \vert b \vert$) \\ +\textbf{Output}. The unsigned subtraction $c = \vert a \vert - \vert b \vert$. \\ +\hline \\ +1. $min \leftarrow b.used$ \\ +2. $max \leftarrow a.used$ \\ +3. If $c.alloc < max$ then grow $c$ to hold at least $max$ digits. (\textit{mp\_grow}) \\ +4. $oldused \leftarrow c.used$ \\ +5. $c.used \leftarrow max$ \\ +6. $u \leftarrow 0$ \\ +7. for $n$ from $0$ to $min - 1$ do \\ +\hspace{3mm}7.1 $c_n \leftarrow a_n - b_n - u$ \\ +\hspace{3mm}7.2 $u \leftarrow c_n >> (\gamma - 1)$ \\ +\hspace{3mm}7.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ +8. if $min < max$ then do \\ +\hspace{3mm}8.1 for $n$ from $min$ to $max - 1$ do \\ +\hspace{6mm}8.1.1 $c_n \leftarrow a_n - u$ \\ +\hspace{6mm}8.1.2 $u \leftarrow c_n >> (\gamma - 1)$ \\ +\hspace{6mm}8.1.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ +9. if $oldused > max$ then do \\ +\hspace{3mm}9.1 for $n$ from $max$ to $oldused - 1$ do \\ +\hspace{6mm}9.1.1 $c_n \leftarrow 0$ \\ +10. Clamp excess digits of $c$. (\textit{mp\_clamp}). \\ +11. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{small} +\end{center} +\caption{Algorithm s\_mp\_sub} +\end{figure} + +\textbf{Algorithm s\_mp\_sub.} +This algorithm performs the unsigned subtraction of two mp\_int variables under the restriction that the result must be positive. That is when +passing variables $a$ and $b$ the condition that $\vert a \vert \ge \vert b \vert$ must be met for the algorithm to function correctly. This +algorithm is loosely based on algorithm 14.9 \cite[pp. 595]{HAC} and is similar to algorithm S in \cite[pp. 267]{TAOCPV2} as well. As was the case +of the algorithm s\_mp\_add both other references lack discussion concerning various practical details such as when the inputs differ in magnitude. + +The initial sorting of the inputs is trivial in this algorithm since $a$ is guaranteed to have at least the same magnitude of $b$. Steps 1 and 2 +set the $min$ and $max$ variables. Unlike the addition routine there is guaranteed to be no carry which means that the final result can be at +most $max$ digits in length as opposed to $max + 1$. Similar to the addition algorithm the \textbf{used} count of $c$ is copied locally and +set to the maximal count for the operation. + +The subtraction loop that begins on step seven is essentially the same as the addition loop of algorithm s\_mp\_add except single precision +subtraction is used instead. Note the use of the $\gamma$ variable to extract the carry (\textit{also known as the borrow}) within the subtraction +loops. Under the assumption that two's complement single precision arithmetic is used this will successfully extract the desired carry. + +For example, consider subtracting $0101_2$ from $0100_2$ where $\gamma = 4$ and $\beta = 2$. The least significant bit will force a carry upwards to +the third bit which will be set to zero after the borrow. After the very first bit has been subtracted $4 - 1 \equiv 0011_2$ will remain, When the +third bit of $0101_2$ is subtracted from the result it will cause another carry. In this case though the carry will be forced to propagate all the +way to the most significant bit. + +Recall that $\beta < 2^{\gamma}$. This means that if a carry does occur just before the $lg(\beta)$'th bit it will propagate all the way to the most +significant bit. Thus, the high order bits of the mp\_digit that are not part of the actual digit will either be all zero, or all one. All that +is needed is a single zero or one bit for the carry. Therefore a single logical shift right by $\gamma - 1$ positions is sufficient to extract the +carry. This method of carry extraction may seem awkward but the reason for it becomes apparent when the implementation is discussed. + +If $b$ has a smaller magnitude than $a$ then step 9 will force the carry and copy operation to propagate through the larger input $a$ into $c$. Step +10 will ensure that any leading digits of $c$ above the $max$'th position are zeroed. + +EXAM,bn_s_mp_sub.c + +Line @24,min@ and @25,max@ perform the initial hardcoded sorting of the inputs. In reality the $min$ and $max$ variables are only aliases and are only +used to make the source code easier to read. Again the pointer alias optimization is used within this algorithm. Lines @42,tmpa@, @43,tmpb@ and @44,tmpc@ initialize the aliases for +$a$, $b$ and $c$ respectively. + +The first subtraction loop occurs on lines @47,u = 0@ through @61,}@. The theory behind the subtraction loop is exactly the same as that for +the addition loop. As remarked earlier there is an implementation reason for using the ``awkward'' method of extracting the carry +(\textit{see line @57, >>@}). The traditional method for extracting the carry would be to shift by $lg(\beta)$ positions and logically AND +the least significant bit. The AND operation is required because all of the bits above the $\lg(\beta)$'th bit will be set to one after a carry +occurs from subtraction. This carry extraction requires two relatively cheap operations to extract the carry. The other method is to simply +shift the most significant bit to the least significant bit thus extracting the carry with a single cheap operation. This optimization only works on +twos compliment machines which is a safe assumption to make. + +If $a$ has a larger magnitude than $b$ an additional loop (\textit{see lines @64,for@ through @73,}@}) is required to propagate the carry through +$a$ and copy the result to $c$. + +\subsection{High Level Addition} +Now that both lower level addition and subtraction algorithms have been established an effective high level signed addition algorithm can be +established. This high level addition algorithm will be what other algorithms and developers will use to perform addition of mp\_int data +types. + +Recall from section 5.2 that an mp\_int represents an integer with an unsigned mantissa (\textit{the array of digits}) and a \textbf{sign} +flag. A high level addition is actually performed as a series of eight separate cases which can be optimized down to three unique cases. + +\begin{figure}[!here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_add}. \\ +\textbf{Input}. Two mp\_ints $a$ and $b$ \\ +\textbf{Output}. The signed addition $c = a + b$. \\ +\hline \\ +1. if $a.sign = b.sign$ then do \\ +\hspace{3mm}1.1 $c.sign \leftarrow a.sign$ \\ +\hspace{3mm}1.2 $c \leftarrow \vert a \vert + \vert b \vert$ (\textit{s\_mp\_add})\\ +2. else do \\ +\hspace{3mm}2.1 if $\vert a \vert < \vert b \vert$ then do (\textit{mp\_cmp\_mag}) \\ +\hspace{6mm}2.1.1 $c.sign \leftarrow b.sign$ \\ +\hspace{6mm}2.1.2 $c \leftarrow \vert b \vert - \vert a \vert$ (\textit{s\_mp\_sub}) \\ +\hspace{3mm}2.2 else do \\ +\hspace{6mm}2.2.1 $c.sign \leftarrow a.sign$ \\ +\hspace{6mm}2.2.2 $c \leftarrow \vert a \vert - \vert b \vert$ \\ +3. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_add} +\end{figure} + +\textbf{Algorithm mp\_add.} +This algorithm performs the signed addition of two mp\_int variables. There is no reference algorithm to draw upon from +either \cite{TAOCPV2} or \cite{HAC} since they both only provide unsigned operations. The algorithm is fairly +straightforward but restricted since subtraction can only produce positive results. + +\begin{figure}[here] +\begin{small} +\begin{center} +\begin{tabular}{|c|c|c|c|c|} +\hline \textbf{Sign of $a$} & \textbf{Sign of $b$} & \textbf{$\vert a \vert > \vert b \vert $} & \textbf{Unsigned Operation} & \textbf{Result Sign Flag} \\ +\hline $+$ & $+$ & Yes & $c = a + b$ & $a.sign$ \\ +\hline $+$ & $+$ & No & $c = a + b$ & $a.sign$ \\ +\hline $-$ & $-$ & Yes & $c = a + b$ & $a.sign$ \\ +\hline $-$ & $-$ & No & $c = a + b$ & $a.sign$ \\ +\hline &&&&\\ + +\hline $+$ & $-$ & No & $c = b - a$ & $b.sign$ \\ +\hline $-$ & $+$ & No & $c = b - a$ & $b.sign$ \\ + +\hline &&&&\\ + +\hline $+$ & $-$ & Yes & $c = a - b$ & $a.sign$ \\ +\hline $-$ & $+$ & Yes & $c = a - b$ & $a.sign$ \\ + +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Addition Guide Chart} +\label{fig:AddChart} +\end{figure} + +Figure~\ref{fig:AddChart} lists all of the eight possible input combinations and is sorted to show that only three +specific cases need to be handled. The return code of the unsigned operations at step 1.2, 2.1.2 and 2.2.2 are +forwarded to step three to check for errors. This simplifies the description of the algorithm considerably and best +follows how the implementation actually was achieved. + +Also note how the \textbf{sign} is set before the unsigned addition or subtraction is performed. Recall from the descriptions of algorithms +s\_mp\_add and s\_mp\_sub that the mp\_clamp function is used at the end to trim excess digits. The mp\_clamp algorithm will set the \textbf{sign} +to \textbf{MP\_ZPOS} when the \textbf{used} digit count reaches zero. + +For example, consider performing $-a + a$ with algorithm mp\_add. By the description of the algorithm the sign is set to \textbf{MP\_NEG} which would +produce a result of $-0$. However, since the sign is set first then the unsigned addition is performed the subsequent usage of algorithm mp\_clamp +within algorithm s\_mp\_add will force $-0$ to become $0$. + +EXAM,bn_mp_add.c + +The source code follows the algorithm fairly closely. The most notable new source code addition is the usage of the $res$ integer variable which +is used to pass result of the unsigned operations forward. Unlike in the algorithm, the variable $res$ is merely returned as is without +explicitly checking it and returning the constant \textbf{MP\_OKAY}. The observation is this algorithm will succeed or fail only if the lower +level functions do so. Returning their return code is sufficient. + +\subsection{High Level Subtraction} +The high level signed subtraction algorithm is essentially the same as the high level signed addition algorithm. + +\newpage\begin{figure}[!here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_sub}. \\ +\textbf{Input}. Two mp\_ints $a$ and $b$ \\ +\textbf{Output}. The signed subtraction $c = a - b$. \\ +\hline \\ +1. if $a.sign \ne b.sign$ then do \\ +\hspace{3mm}1.1 $c.sign \leftarrow a.sign$ \\ +\hspace{3mm}1.2 $c \leftarrow \vert a \vert + \vert b \vert$ (\textit{s\_mp\_add}) \\ +2. else do \\ +\hspace{3mm}2.1 if $\vert a \vert \ge \vert b \vert$ then do (\textit{mp\_cmp\_mag}) \\ +\hspace{6mm}2.1.1 $c.sign \leftarrow a.sign$ \\ +\hspace{6mm}2.1.2 $c \leftarrow \vert a \vert - \vert b \vert$ (\textit{s\_mp\_sub}) \\ +\hspace{3mm}2.2 else do \\ +\hspace{6mm}2.2.1 $c.sign \leftarrow \left \lbrace \begin{array}{ll} + MP\_ZPOS & \mbox{if }a.sign = MP\_NEG \\ + MP\_NEG & \mbox{otherwise} \\ + \end{array} \right .$ \\ +\hspace{6mm}2.2.2 $c \leftarrow \vert b \vert - \vert a \vert$ \\ +3. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_sub} +\end{figure} + +\textbf{Algorithm mp\_sub.} +This algorithm performs the signed subtraction of two inputs. Similar to algorithm mp\_add there is no reference in either \cite{TAOCPV2} or +\cite{HAC}. Also this algorithm is restricted by algorithm s\_mp\_sub. Chart \ref{fig:SubChart} lists the eight possible inputs and +the operations required. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{|c|c|c|c|c|} +\hline \textbf{Sign of $a$} & \textbf{Sign of $b$} & \textbf{$\vert a \vert \ge \vert b \vert $} & \textbf{Unsigned Operation} & \textbf{Result Sign Flag} \\ +\hline $+$ & $-$ & Yes & $c = a + b$ & $a.sign$ \\ +\hline $+$ & $-$ & No & $c = a + b$ & $a.sign$ \\ +\hline $-$ & $+$ & Yes & $c = a + b$ & $a.sign$ \\ +\hline $-$ & $+$ & No & $c = a + b$ & $a.sign$ \\ +\hline &&&& \\ +\hline $+$ & $+$ & Yes & $c = a - b$ & $a.sign$ \\ +\hline $-$ & $-$ & Yes & $c = a - b$ & $a.sign$ \\ +\hline &&&& \\ +\hline $+$ & $+$ & No & $c = b - a$ & $\mbox{opposite of }a.sign$ \\ +\hline $-$ & $-$ & No & $c = b - a$ & $\mbox{opposite of }a.sign$ \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Subtraction Guide Chart} +\label{fig:SubChart} +\end{figure} + +Similar to the case of algorithm mp\_add the \textbf{sign} is set first before the unsigned addition or subtraction. That is to prevent the +algorithm from producing $-a - -a = -0$ as a result. + +EXAM,bn_mp_sub.c + +Much like the implementation of algorithm mp\_add the variable $res$ is used to catch the return code of the unsigned addition or subtraction operations +and forward it to the end of the function. On line @38, != MP_LT@ the ``not equal to'' \textbf{MP\_LT} expression is used to emulate a +``greater than or equal to'' comparison. + +\section{Bit and Digit Shifting} +MARK,POLY +It is quite common to think of a multiple precision integer as a polynomial in $x$, that is $y = f(\beta)$ where $f(x) = \sum_{i=0}^{n-1} a_i x^i$. +This notation arises within discussion of Montgomery and Diminished Radix Reduction as well as Karatsuba multiplication and squaring. + +In order to facilitate operations on polynomials in $x$ as above a series of simple ``digit'' algorithms have to be established. That is to shift +the digits left or right as well to shift individual bits of the digits left and right. It is important to note that not all ``shift'' operations +are on radix-$\beta$ digits. + +\subsection{Multiplication by Two} + +In a binary system where the radix is a power of two multiplication by two not only arises often in other algorithms it is a fairly efficient +operation to perform. A single precision logical shift left is sufficient to multiply a single digit by two. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_mul\_2}. \\ +\textbf{Input}. One mp\_int $a$ \\ +\textbf{Output}. $b = 2a$. \\ +\hline \\ +1. If $b.alloc < a.used + 1$ then grow $b$ to hold $a.used + 1$ digits. (\textit{mp\_grow}) \\ +2. $oldused \leftarrow b.used$ \\ +3. $b.used \leftarrow a.used$ \\ +4. $r \leftarrow 0$ \\ +5. for $n$ from 0 to $a.used - 1$ do \\ +\hspace{3mm}5.1 $rr \leftarrow a_n >> (lg(\beta) - 1)$ \\ +\hspace{3mm}5.2 $b_n \leftarrow (a_n << 1) + r \mbox{ (mod }\beta\mbox{)}$ \\ +\hspace{3mm}5.3 $r \leftarrow rr$ \\ +6. If $r \ne 0$ then do \\ +\hspace{3mm}6.1 $b_{n + 1} \leftarrow r$ \\ +\hspace{3mm}6.2 $b.used \leftarrow b.used + 1$ \\ +7. If $b.used < oldused - 1$ then do \\ +\hspace{3mm}7.1 for $n$ from $b.used$ to $oldused - 1$ do \\ +\hspace{6mm}7.1.1 $b_n \leftarrow 0$ \\ +8. $b.sign \leftarrow a.sign$ \\ +9. Return(\textit{MP\_OKAY}).\\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_mul\_2} +\end{figure} + +\textbf{Algorithm mp\_mul\_2.} +This algorithm will quickly multiply a mp\_int by two provided $\beta$ is a power of two. Neither \cite{TAOCPV2} nor \cite{HAC} describe such +an algorithm despite the fact it arises often in other algorithms. The algorithm is setup much like the lower level algorithm s\_mp\_add since +it is for all intents and purposes equivalent to the operation $b = \vert a \vert + \vert a \vert$. + +Step 1 and 2 grow the input as required to accomodate the maximum number of \textbf{used} digits in the result. The initial \textbf{used} count +is set to $a.used$ at step 4. Only if there is a final carry will the \textbf{used} count require adjustment. + +Step 6 is an optimization implementation of the addition loop for this specific case. That is since the two values being added together +are the same there is no need to perform two reads from the digits of $a$. Step 6.1 performs a single precision shift on the current digit $a_n$ to +obtain what will be the carry for the next iteration. Step 6.2 calculates the $n$'th digit of the result as single precision shift of $a_n$ plus +the previous carry. Recall from ~SHIFTS~ that $a_n << 1$ is equivalent to $a_n \cdot 2$. An iteration of the addition loop is finished with +forwarding the carry to the next iteration. + +Step 7 takes care of any final carry by setting the $a.used$'th digit of the result to the carry and augmenting the \textbf{used} count of $b$. +Step 8 clears any leading digits of $b$ in case it originally had a larger magnitude than $a$. + +EXAM,bn_mp_mul_2.c + +This implementation is essentially an optimized implementation of s\_mp\_add for the case of doubling an input. The only noteworthy difference +is the use of the logical shift operator on line @52,<<@ to perform a single precision doubling. + +\subsection{Division by Two} +A division by two can just as easily be accomplished with a logical shift right as multiplication by two can be with a logical shift left. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_div\_2}. \\ +\textbf{Input}. One mp\_int $a$ \\ +\textbf{Output}. $b = a/2$. \\ +\hline \\ +1. If $b.alloc < a.used$ then grow $b$ to hold $a.used$ digits. (\textit{mp\_grow}) \\ +2. If the reallocation failed return(\textit{MP\_MEM}). \\ +3. $oldused \leftarrow b.used$ \\ +4. $b.used \leftarrow a.used$ \\ +5. $r \leftarrow 0$ \\ +6. for $n$ from $b.used - 1$ to $0$ do \\ +\hspace{3mm}6.1 $rr \leftarrow a_n \mbox{ (mod }2\mbox{)}$\\ +\hspace{3mm}6.2 $b_n \leftarrow (a_n >> 1) + (r << (lg(\beta) - 1)) \mbox{ (mod }\beta\mbox{)}$ \\ +\hspace{3mm}6.3 $r \leftarrow rr$ \\ +7. If $b.used < oldused - 1$ then do \\ +\hspace{3mm}7.1 for $n$ from $b.used$ to $oldused - 1$ do \\ +\hspace{6mm}7.1.1 $b_n \leftarrow 0$ \\ +8. $b.sign \leftarrow a.sign$ \\ +9. Clamp excess digits of $b$. (\textit{mp\_clamp}) \\ +10. Return(\textit{MP\_OKAY}).\\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_div\_2} +\end{figure} + +\textbf{Algorithm mp\_div\_2.} +This algorithm will divide an mp\_int by two using logical shifts to the right. Like mp\_mul\_2 it uses a modified low level addition +core as the basis of the algorithm. Unlike mp\_mul\_2 the shift operations work from the leading digit to the trailing digit. The algorithm +could be written to work from the trailing digit to the leading digit however, it would have to stop one short of $a.used - 1$ digits to prevent +reading past the end of the array of digits. + +Essentially the loop at step 6 is similar to that of mp\_mul\_2 except the logical shifts go in the opposite direction and the carry is at the +least significant bit not the most significant bit. + +EXAM,bn_mp_div_2.c + +\section{Polynomial Basis Operations} +Recall from ~POLY~ that any integer can be represented as a polynomial in $x$ as $y = f(\beta)$. Such a representation is also known as +the polynomial basis \cite[pp. 48]{ROSE}. Given such a notation a multiplication or division by $x$ amounts to shifting whole digits a single +place. The need for such operations arises in several other higher level algorithms such as Barrett and Montgomery reduction, integer +division and Karatsuba multiplication. + +Converting from an array of digits to polynomial basis is very simple. Consider the integer $y \equiv (a_2, a_1, a_0)_{\beta}$ and recall that +$y = \sum_{i=0}^{2} a_i \beta^i$. Simply replace $\beta$ with $x$ and the expression is in polynomial basis. For example, $f(x) = 8x + 9$ is the +polynomial basis representation for $89$ using radix ten. That is, $f(10) = 8(10) + 9 = 89$. + +\subsection{Multiplication by $x$} + +Given a polynomial in $x$ such as $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0$ multiplying by $x$ amounts to shifting the coefficients up one +degree. In this case $f(x) \cdot x = a_n x^{n+1} + a_{n-1} x^n + ... + a_0 x$. From a scalar basis point of view multiplying by $x$ is equivalent to +multiplying by the integer $\beta$. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_lshd}. \\ +\textbf{Input}. One mp\_int $a$ and an integer $b$ \\ +\textbf{Output}. $a \leftarrow a \cdot \beta^b$ (equivalent to multiplication by $x^b$). \\ +\hline \\ +1. If $b \le 0$ then return(\textit{MP\_OKAY}). \\ +2. If $a.alloc < a.used + b$ then grow $a$ to at least $a.used + b$ digits. (\textit{mp\_grow}). \\ +3. If the reallocation failed return(\textit{MP\_MEM}). \\ +4. $a.used \leftarrow a.used + b$ \\ +5. $i \leftarrow a.used - 1$ \\ +6. $j \leftarrow a.used - 1 - b$ \\ +7. for $n$ from $a.used - 1$ to $b$ do \\ +\hspace{3mm}7.1 $a_{i} \leftarrow a_{j}$ \\ +\hspace{3mm}7.2 $i \leftarrow i - 1$ \\ +\hspace{3mm}7.3 $j \leftarrow j - 1$ \\ +8. for $n$ from 0 to $b - 1$ do \\ +\hspace{3mm}8.1 $a_n \leftarrow 0$ \\ +9. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_lshd} +\end{figure} + +\textbf{Algorithm mp\_lshd.} +This algorithm multiplies an mp\_int by the $b$'th power of $x$. This is equivalent to multiplying by $\beta^b$. The algorithm differs +from the other algorithms presented so far as it performs the operation in place instead storing the result in a separate location. The +motivation behind this change is due to the way this function is typically used. Algorithms such as mp\_add store the result in an optionally +different third mp\_int because the original inputs are often still required. Algorithm mp\_lshd (\textit{and similarly algorithm mp\_rshd}) is +typically used on values where the original value is no longer required. The algorithm will return success immediately if +$b \le 0$ since the rest of algorithm is only valid when $b > 0$. + +First the destination $a$ is grown as required to accomodate the result. The counters $i$ and $j$ are used to form a \textit{sliding window} over +the digits of $a$ of length $b$. The head of the sliding window is at $i$ (\textit{the leading digit}) and the tail at $j$ (\textit{the trailing digit}). +The loop on step 7 copies the digit from the tail to the head. In each iteration the window is moved down one digit. The last loop on +step 8 sets the lower $b$ digits to zero. + +\newpage +FIGU,sliding_window,Sliding Window Movement + +EXAM,bn_mp_lshd.c + +The if statement on line @24,if@ ensures that the $b$ variable is greater than zero. The \textbf{used} count is incremented by $b$ before +the copy loop begins. This elminates the need for an additional variable in the for loop. The variable $top$ on line @42,top@ is an alias +for the leading digit while $bottom$ on line @45,bottom@ is an alias for the trailing edge. The aliases form a window of exactly $b$ digits +over the input. + +\subsection{Division by $x$} + +Division by powers of $x$ is easily achieved by shifting the digits right and removing any that will end up to the right of the zero'th digit. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_rshd}. \\ +\textbf{Input}. One mp\_int $a$ and an integer $b$ \\ +\textbf{Output}. $a \leftarrow a / \beta^b$ (Divide by $x^b$). \\ +\hline \\ +1. If $b \le 0$ then return. \\ +2. If $a.used \le b$ then do \\ +\hspace{3mm}2.1 Zero $a$. (\textit{mp\_zero}). \\ +\hspace{3mm}2.2 Return. \\ +3. $i \leftarrow 0$ \\ +4. $j \leftarrow b$ \\ +5. for $n$ from 0 to $a.used - b - 1$ do \\ +\hspace{3mm}5.1 $a_i \leftarrow a_j$ \\ +\hspace{3mm}5.2 $i \leftarrow i + 1$ \\ +\hspace{3mm}5.3 $j \leftarrow j + 1$ \\ +6. for $n$ from $a.used - b$ to $a.used - 1$ do \\ +\hspace{3mm}6.1 $a_n \leftarrow 0$ \\ +7. $a.used \leftarrow a.used - b$ \\ +8. Return. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_rshd} +\end{figure} + +\textbf{Algorithm mp\_rshd.} +This algorithm divides the input in place by the $b$'th power of $x$. It is analogous to dividing by a $\beta^b$ but much quicker since +it does not require single precision division. This algorithm does not actually return an error code as it cannot fail. + +If the input $b$ is less than one the algorithm quickly returns without performing any work. If the \textbf{used} count is less than or equal +to the shift count $b$ then it will simply zero the input and return. + +After the trivial cases of inputs have been handled the sliding window is setup. Much like the case of algorithm mp\_lshd a sliding window that +is $b$ digits wide is used to copy the digits. Unlike mp\_lshd the window slides in the opposite direction from the trailing to the leading digit. +Also the digits are copied from the leading to the trailing edge. + +Once the window copy is complete the upper digits must be zeroed and the \textbf{used} count decremented. + +EXAM,bn_mp_rshd.c + +The only noteworthy element of this routine is the lack of a return type. + +-- Will update later to give it a return type...Tom + +\section{Powers of Two} + +Now that algorithms for moving single bits as well as whole digits exist algorithms for moving the ``in between'' distances are required. For +example, to quickly multiply by $2^k$ for any $k$ without using a full multiplier algorithm would prove useful. Instead of performing single +shifts $k$ times to achieve a multiplication by $2^{\pm k}$ a mixture of whole digit shifting and partial digit shifting is employed. + +\subsection{Multiplication by Power of Two} + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_mul\_2d}. \\ +\textbf{Input}. One mp\_int $a$ and an integer $b$ \\ +\textbf{Output}. $c \leftarrow a \cdot 2^b$. \\ +\hline \\ +1. $c \leftarrow a$. (\textit{mp\_copy}) \\ +2. If $c.alloc < c.used + \lfloor b / lg(\beta) \rfloor + 2$ then grow $c$ accordingly. \\ +3. If the reallocation failed return(\textit{MP\_MEM}). \\ +4. If $b \ge lg(\beta)$ then \\ +\hspace{3mm}4.1 $c \leftarrow c \cdot \beta^{\lfloor b / lg(\beta) \rfloor}$ (\textit{mp\_lshd}). \\ +\hspace{3mm}4.2 If step 4.1 failed return(\textit{MP\_MEM}). \\ +5. $d \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\ +6. If $d \ne 0$ then do \\ +\hspace{3mm}6.1 $mask \leftarrow 2^d$ \\ +\hspace{3mm}6.2 $r \leftarrow 0$ \\ +\hspace{3mm}6.3 for $n$ from $0$ to $c.used - 1$ do \\ +\hspace{6mm}6.3.1 $rr \leftarrow c_n >> (lg(\beta) - d) \mbox{ (mod }mask\mbox{)}$ \\ +\hspace{6mm}6.3.2 $c_n \leftarrow (c_n << d) + r \mbox{ (mod }\beta\mbox{)}$ \\ +\hspace{6mm}6.3.3 $r \leftarrow rr$ \\ +\hspace{3mm}6.4 If $r > 0$ then do \\ +\hspace{6mm}6.4.1 $c_{c.used} \leftarrow r$ \\ +\hspace{6mm}6.4.2 $c.used \leftarrow c.used + 1$ \\ +7. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_mul\_2d} +\end{figure} + +\textbf{Algorithm mp\_mul\_2d.} +This algorithm multiplies $a$ by $2^b$ and stores the result in $c$. The algorithm uses algorithm mp\_lshd and a derivative of algorithm mp\_mul\_2 to +quickly compute the product. + +First the algorithm will multiply $a$ by $x^{\lfloor b / lg(\beta) \rfloor}$ which will ensure that the remainder multiplicand is less than +$\beta$. For example, if $b = 37$ and $\beta = 2^{28}$ then this step will multiply by $x$ leaving a multiplication by $2^{37 - 28} = 2^{9}$ +left. + +After the digits have been shifted appropriately at most $lg(\beta) - 1$ shifts are left to perform. Step 5 calculates the number of remaining shifts +required. If it is non-zero a modified shift loop is used to calculate the remaining product. +Essentially the loop is a generic version of algorith mp\_mul2 designed to handle any shift count in the range $1 \le x < lg(\beta)$. The $mask$ +variable is used to extract the upper $d$ bits to form the carry for the next iteration. + +This algorithm is loosely measured as a $O(2n)$ algorithm which means that if the input is $n$-digits that it takes $2n$ ``time'' to +complete. It is possible to optimize this algorithm down to a $O(n)$ algorithm at a cost of making the algorithm slightly harder to follow. + +EXAM,bn_mp_mul_2d.c + +Notes to be revised when code is updated. -- Tom + +\subsection{Division by Power of Two} + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_div\_2d}. \\ +\textbf{Input}. One mp\_int $a$ and an integer $b$ \\ +\textbf{Output}. $c \leftarrow \lfloor a / 2^b \rfloor, d \leftarrow a \mbox{ (mod }2^b\mbox{)}$. \\ +\hline \\ +1. If $b \le 0$ then do \\ +\hspace{3mm}1.1 $c \leftarrow a$ (\textit{mp\_copy}) \\ +\hspace{3mm}1.2 $d \leftarrow 0$ (\textit{mp\_zero}) \\ +\hspace{3mm}1.3 Return(\textit{MP\_OKAY}). \\ +2. $c \leftarrow a$ \\ +3. $d \leftarrow a \mbox{ (mod }2^b\mbox{)}$ (\textit{mp\_mod\_2d}) \\ +4. If $b \ge lg(\beta)$ then do \\ +\hspace{3mm}4.1 $c \leftarrow \lfloor c/\beta^{\lfloor b/lg(\beta) \rfloor} \rfloor$ (\textit{mp\_rshd}). \\ +5. $k \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\ +6. If $k \ne 0$ then do \\ +\hspace{3mm}6.1 $mask \leftarrow 2^k$ \\ +\hspace{3mm}6.2 $r \leftarrow 0$ \\ +\hspace{3mm}6.3 for $n$ from $c.used - 1$ to $0$ do \\ +\hspace{6mm}6.3.1 $rr \leftarrow c_n \mbox{ (mod }mask\mbox{)}$ \\ +\hspace{6mm}6.3.2 $c_n \leftarrow (c_n >> k) + (r << (lg(\beta) - k))$ \\ +\hspace{6mm}6.3.3 $r \leftarrow rr$ \\ +7. Clamp excess digits of $c$. (\textit{mp\_clamp}) \\ +8. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_div\_2d} +\end{figure} + +\textbf{Algorithm mp\_div\_2d.} +This algorithm will divide an input $a$ by $2^b$ and produce the quotient and remainder. The algorithm is designed much like algorithm +mp\_mul\_2d by first using whole digit shifts then single precision shifts. This algorithm will also produce the remainder of the division +by using algorithm mp\_mod\_2d. + +EXAM,bn_mp_div_2d.c + +The implementation of algorithm mp\_div\_2d is slightly different than the algorithm specifies. The remainder $d$ may be optionally +ignored by passing \textbf{NULL} as the pointer to the mp\_int variable. The temporary mp\_int variable $t$ is used to hold the +result of the remainder operation until the end. This allows $d$ and $a$ to represent the same mp\_int without modifying $a$ before +the quotient is obtained. + +The remainder of the source code is essentially the same as the source code for mp\_mul\_2d. (-- Fix this paragraph up later, Tom). + +\subsection{Remainder of Division by Power of Two} + +The last algorithm in the series of polynomial basis power of two algorithms is calculating the remainder of division by $2^b$. This +algorithm benefits from the fact that in twos complement arithmetic $a \mbox{ (mod }2^b\mbox{)}$ is the same as $a$ AND $2^b - 1$. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_mod\_2d}. \\ +\textbf{Input}. One mp\_int $a$ and an integer $b$ \\ +\textbf{Output}. $c \leftarrow a \mbox{ (mod }2^b\mbox{)}$. \\ +\hline \\ +1. If $b \le 0$ then do \\ +\hspace{3mm}1.1 $c \leftarrow 0$ (\textit{mp\_zero}) \\ +\hspace{3mm}1.2 Return(\textit{MP\_OKAY}). \\ +2. If $b > a.used \cdot lg(\beta)$ then do \\ +\hspace{3mm}2.1 $c \leftarrow a$ (\textit{mp\_copy}) \\ +\hspace{3mm}2.2 Return the result of step 2.1. \\ +3. $c \leftarrow a$ \\ +4. If step 3 failed return(\textit{MP\_MEM}). \\ +5. for $n$ from $\lceil b / lg(\beta) \rceil$ to $c.used$ do \\ +\hspace{3mm}5.1 $c_n \leftarrow 0$ \\ +6. $k \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\ +7. $c_{\lfloor b / lg(\beta) \rfloor} \leftarrow c_{\lfloor b / lg(\beta) \rfloor} \mbox{ (mod }2^{k}\mbox{)}$. \\ +8. Clamp excess digits of $c$. (\textit{mp\_clamp}) \\ +9. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_mod\_2d} +\end{figure} + +\textbf{Algorithm mp\_mod\_2d.} +This algorithm will quickly calculate the value of $a \mbox{ (mod }2^b\mbox{)}$. First if $b$ is less than or equal to zero the +result is set to zero. If $b$ is greater than the number of bits in $a$ then it simply copies $a$ to $c$ and returns. Otherwise, $a$ +is copied to $b$, leading digits are removed and the remaining leading digit is trimed to the exact bit count. + +EXAM,bn_mp_mod_2d.c + +-- Add comments later, Tom. + +\section*{Exercises} +\begin{tabular}{cl} +$\left [ 3 \right ] $ & Devise an algorithm that performs $a \cdot 2^b$ for generic values of $b$ \\ + & in $O(n)$ time. \\ + &\\ +$\left [ 3 \right ] $ & Devise an efficient algorithm to multiply by small low hamming \\ + & weight values such as $3$, $5$ and $9$. Extend it to handle all values \\ + & upto $64$ with a hamming weight less than three. \\ + &\\ +$\left [ 2 \right ] $ & Modify the preceding algorithm to handle values of the form \\ + & $2^k - 1$ as well. \\ + &\\ +$\left [ 3 \right ] $ & Using only algorithms mp\_mul\_2, mp\_div\_2 and mp\_add create an \\ + & algorithm to multiply two integers in roughly $O(2n^2)$ time for \\ + & any $n$-bit input. Note that the time of addition is ignored in the \\ + & calculation. \\ + & \\ +$\left [ 5 \right ] $ & Improve the previous algorithm to have a working time of at most \\ + & $O \left (2^{(k-1)}n + \left ({2n^2 \over k} \right ) \right )$ for an appropriate choice of $k$. Again ignore \\ + & the cost of addition. \\ + & \\ +$\left [ 2 \right ] $ & Devise a chart to find optimal values of $k$ for the previous problem \\ + & for $n = 64 \ldots 1024$ in steps of $64$. \\ + & \\ +$\left [ 2 \right ] $ & Using only algorithms mp\_abs and mp\_sub devise another method for \\ + & calculating the result of a signed comparison. \\ + & +\end{tabular} + +\chapter{Multiplication and Squaring} +\section{The Multipliers} +For most number theoretic problems including certain public key cryptographic algorithms, the ``multipliers'' form the most important subset of +algorithms of any multiple precision integer package. The set of multiplier algorithms include integer multiplication, squaring and modular reduction +where in each of the algorithms single precision multiplication is the dominant operation performed. This chapter will discuss integer multiplication +and squaring, leaving modular reductions for the subsequent chapter. + +The importance of the multiplier algorithms is for the most part driven by the fact that certain popular public key algorithms are based on modular +exponentiation, that is computing $d \equiv a^b \mbox{ (mod }c\mbox{)}$ for some arbitrary choice of $a$, $b$, $c$ and $d$. During a modular +exponentiation the majority\footnote{Roughly speaking a modular exponentiation will spend about 40\% of the time performing modular reductions, +35\% of the time performing squaring and 25\% of the time performing multiplications.} of the processor time is spent performing single precision +multiplications. + +For centuries general purpose multiplication has required a lengthly $O(n^2)$ process, whereby each digit of one multiplicand has to be multiplied +against every digit of the other multiplicand. Traditional long-hand multiplication is based on this process; while the techniques can differ the +overall algorithm used is essentially the same. Only ``recently'' have faster algorithms been studied. First Karatsuba multiplication was discovered in +1962. This algorithm can multiply two numbers with considerably fewer single precision multiplications when compared to the long-hand approach. +This technique led to the discovery of polynomial basis algorithms (\textit{good reference?}) and subquently Fourier Transform based solutions. + +\section{Multiplication} +\subsection{The Baseline Multiplication} +\label{sec:basemult} +\index{baseline multiplication} +Computing the product of two integers in software can be achieved using a trivial adaptation of the standard $O(n^2)$ long-hand multiplication +algorithm that school children are taught. The algorithm is considered an $O(n^2)$ algorithm since for two $n$-digit inputs $n^2$ single precision +multiplications are required. More specifically for a $m$ and $n$ digit input $m \cdot n$ single precision multiplications are required. To +simplify most discussions, it will be assumed that the inputs have comparable number of digits. + +The ``baseline multiplication'' algorithm is designed to act as the ``catch-all'' algorithm, only to be used when the faster algorithms cannot be +used. This algorithm does not use any particularly interesting optimizations and should ideally be avoided if possible. One important +facet of this algorithm, is that it has been modified to only produce a certain amount of output digits as resolution. The importance of this +modification will become evident during the discussion of Barrett modular reduction. Recall that for a $n$ and $m$ digit input the product +will be at most $n + m$ digits. Therefore, this algorithm can be reduced to a full multiplier by having it produce $n + m$ digits of the product. + +Recall from ~GAMMA~ the definition of $\gamma$ as the number of bits in the type \textbf{mp\_digit}. We shall now extend the variable set to +include $\alpha$ which shall represent the number of bits in the type \textbf{mp\_word}. This implies that $2^{\alpha} > 2 \cdot \beta^2$. The +constant $\delta = 2^{\alpha - 2lg(\beta)}$ will represent the maximal weight of any column in a product (\textit{see ~COMBA~ for more information}). + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{s\_mp\_mul\_digs}. \\ +\textbf{Input}. mp\_int $a$, mp\_int $b$ and an integer $digs$ \\ +\textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert \mbox{ (mod }\beta^{digs}\mbox{)}$. \\ +\hline \\ +1. If min$(a.used, b.used) < \delta$ then do \\ +\hspace{3mm}1.1 Calculate $c = \vert a \vert \cdot \vert b \vert$ by the Comba method (\textit{see algorithm~\ref{fig:COMBAMULT}}). \\ +\hspace{3mm}1.2 Return the result of step 1.1 \\ +\\ +Allocate and initialize a temporary mp\_int. \\ +2. Init $t$ to be of size $digs$ \\ +3. If step 2 failed return(\textit{MP\_MEM}). \\ +4. $t.used \leftarrow digs$ \\ +\\ +Compute the product. \\ +5. for $ix$ from $0$ to $a.used - 1$ do \\ +\hspace{3mm}5.1 $u \leftarrow 0$ \\ +\hspace{3mm}5.2 $pb \leftarrow \mbox{min}(b.used, digs - ix)$ \\ +\hspace{3mm}5.3 If $pb < 1$ then goto step 6. \\ +\hspace{3mm}5.4 for $iy$ from $0$ to $pb - 1$ do \\ +\hspace{6mm}5.4.1 $\hat r \leftarrow t_{iy + ix} + a_{ix} \cdot b_{iy} + u$ \\ +\hspace{6mm}5.4.2 $t_{iy + ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ +\hspace{6mm}5.4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ +\hspace{3mm}5.5 if $ix + pb < digs$ then do \\ +\hspace{6mm}5.5.1 $t_{ix + pb} \leftarrow u$ \\ +6. Clamp excess digits of $t$. \\ +7. Swap $c$ with $t$ \\ +8. Clear $t$ \\ +9. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm s\_mp\_mul\_digs} +\end{figure} + +\textbf{Algorithm s\_mp\_mul\_digs.} +This algorithm computes the unsigned product of two inputs $a$ and $b$, limited to an output precision of $digs$ digits. While it may seem +a bit awkward to modify the function from its simple $O(n^2)$ description, the usefulness of partial multipliers will arise in a subsequent +algorithm. The algorithm is loosely based on algorithm 14.12 from \cite[pp. 595]{HAC} and is similar to Algorithm M of Knuth \cite[pp. 268]{TAOCPV2}. +Algorithm s\_mp\_mul\_digs differs from these cited references since it can produce a variable output precision regardless of the precision of the +inputs. + +The first thing this algorithm checks for is whether a Comba multiplier can be used instead. If the minimum digit count of either +input is less than $\delta$, then the Comba method may be used instead. After the Comba method is ruled out, the baseline algorithm begins. A +temporary mp\_int variable $t$ is used to hold the intermediate result of the product. This allows the algorithm to be used to +compute products when either $a = c$ or $b = c$ without overwriting the inputs. + +All of step 5 is the infamous $O(n^2)$ multiplication loop slightly modified to only produce upto $digs$ digits of output. The $pb$ variable +is given the count of digits to read from $b$ inside the nested loop. If $pb \le 1$ then no more output digits can be produced and the algorithm +will exit the loop. The best way to think of the loops are as a series of $pb \times 1$ multiplications. That is, in each pass of the +innermost loop $a_{ix}$ is multiplied against $b$ and the result is added (\textit{with an appropriate shift}) to $t$. + +For example, consider multiplying $576$ by $241$. That is equivalent to computing $10^0(1)(576) + 10^1(4)(576) + 10^2(2)(576)$ which is best +visualized in the following table. + +\begin{figure}[here] +\begin{center} +\begin{tabular}{|c|c|c|c|c|c|l|} +\hline && & 5 & 7 & 6 & \\ +\hline $\times$&& & 2 & 4 & 1 & \\ +\hline &&&&&&\\ + && & 5 & 7 & 6 & $10^0(1)(576)$ \\ + &2 & 3 & 6 & 1 & 6 & $10^1(4)(576) + 10^0(1)(576)$ \\ + 1 & 3 & 8 & 8 & 1 & 6 & $10^2(2)(576) + 10^1(4)(576) + 10^0(1)(576)$ \\ +\hline +\end{tabular} +\end{center} +\caption{Long-Hand Multiplication Diagram} +\end{figure} + +Each row of the product is added to the result after being shifted to the left (\textit{multiplied by a power of the radix}) by the appropriate +count. That is in pass $ix$ of the inner loop the product is added starting at the $ix$'th digit of the reult. + +Step 5.4.1 introduces the hat symbol (\textit{e.g. $\hat r$}) which represents a double precision variable. The multiplication on that step +is assumed to be a double wide output single precision multiplication. That is, two single precision variables are multiplied to produce a +double precision result. The step is somewhat optimized from a long-hand multiplication algorithm because the carry from the addition in step +5.4.1 is propagated through the nested loop. If the carry was not propagated immediately it would overflow the single precision digit +$t_{ix+iy}$ and the result would be lost. + +At step 5.5 the nested loop is finished and any carry that was left over should be forwarded. The carry does not have to be added to the $ix+pb$'th +digit since that digit is assumed to be zero at this point. However, if $ix + pb \ge digs$ the carry is not set as it would make the result +exceed the precision requested. + +EXAM,bn_s_mp_mul_digs.c + +Lines @31,if@ to @35,}@ determine if the Comba method can be used first. The conditions for using the Comba routine are that min$(a.used, b.used) < \delta$ and +the number of digits of output is less than \textbf{MP\_WARRAY}. This new constant is used to control +the stack usage in the Comba routines. By default it is set to $\delta$ but can be reduced when memory is at a premium. + +Of particular importance is the calculation of the $ix+iy$'th column on lines @64,mp_word@, @65,mp_word@ and @66,mp_word@. Note how all of the +variables are cast to the type \textbf{mp\_word}, which is also the type of variable $\hat r$. That is to ensure that double precision operations +are used instead of single precision. The multiplication on line @65,) * (@ makes use of a specific GCC optimizer behaviour. On the outset it looks like +the compiler will have to use a double precision multiplication to produce the result required. Such an operation would be horribly slow on most +processors and drag this to a crawl. However, GCC is smart enough to realize that double wide output single precision multipliers can be used. For +example, the instruction ``MUL'' on the x86 processor can multiply two 32-bit values and produce a 64-bit result. + +\subsection{Faster Multiplication by the ``Comba'' Method} +MARK,COMBA + +One of the huge drawbacks of the ``baseline'' algorithms is that at the $O(n^2)$ level the carry must be computed and propagated upwards. This +makes the nested loop very sequential and hard to unroll and implement in parallel. The ``Comba'' \cite{COMBA} method is named after little known +(\textit{in cryptographic venues}) Paul G. Comba who described a method of implementing fast multipliers that do not require nested +carry fixup operations. As an interesting aside it seems that Paul Barrett describes a similar technique in +his 1986 paper \cite{BARRETT} written five years before. + +At the heart of the Comba technique is once again the long-hand algorithm. Except in this case a slight twist is placed on how +the columns of the result are produced. In the standard long-hand algorithm rows of products are produced then added together to form the +final result. In the baseline algorithm the columns are added together after each iteration to get the result instantaneously. + +In the Comba algorithm the columns of the result are produced entirely independently of each other. That is at the $O(n^2)$ level a +simple multiplication and addition step is performed. The carries of the columns are propagated after the nested loop to reduce the amount +of work requiored. Succintly the first step of the algorithm is to compute the product vector $\vec x$ as follows. + +\begin{equation} +\vec x_n = \sum_{i+j = n} a_ib_j, \forall n \in \lbrace 0, 1, 2, \ldots, i + j \rbrace +\end{equation} + +Where $\vec x_n$ is the $n'th$ column of the output vector. Consider the following example which computes the vector $\vec x$ for the multiplication +of $576$ and $241$. + +\newpage\begin{figure}[here] +\begin{small} +\begin{center} +\begin{tabular}{|c|c|c|c|c|c|} + \hline & & 5 & 7 & 6 & First Input\\ + \hline $\times$ & & 2 & 4 & 1 & Second Input\\ +\hline & & $1 \cdot 5 = 5$ & $1 \cdot 7 = 7$ & $1 \cdot 6 = 6$ & First pass \\ + & $4 \cdot 5 = 20$ & $4 \cdot 7+5=33$ & $4 \cdot 6+7=31$ & 6 & Second pass \\ + $2 \cdot 5 = 10$ & $2 \cdot 7 + 20 = 34$ & $2 \cdot 6+33=45$ & 31 & 6 & Third pass \\ +\hline 10 & 34 & 45 & 31 & 6 & Final Result \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Comba Multiplication Diagram} +\end{figure} + +At this point the vector $x = \left < 10, 34, 45, 31, 6 \right >$ is the result of the first step of the Comba multipler. +Now the columns must be fixed by propagating the carry upwards. The resultant vector will have one extra dimension over the input vector which is +congruent to adding a leading zero digit. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{Comba Fixup}. \\ +\textbf{Input}. Vector $\vec x$ of dimension $k$ \\ +\textbf{Output}. Vector $\vec x$ such that the carries have been propagated. \\ +\hline \\ +1. for $n$ from $0$ to $k - 1$ do \\ +\hspace{3mm}1.1 $\vec x_{n+1} \leftarrow \vec x_{n+1} + \lfloor \vec x_{n}/\beta \rfloor$ \\ +\hspace{3mm}1.2 $\vec x_{n} \leftarrow \vec x_{n} \mbox{ (mod }\beta\mbox{)}$ \\ +2. Return($\vec x$). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm Comba Fixup} +\end{figure} + +With that algorithm and $k = 5$ and $\beta = 10$ the following vector is produced $\vec x= \left < 1, 3, 8, 8, 1, 6 \right >$. In this case +$241 \cdot 576$ is in fact $138816$ and the procedure succeeded. If the algorithm is correct and as will be demonstrated shortly more +efficient than the baseline algorithm why not simply always use this algorithm? + +\subsubsection{Column Weight.} +At the nested $O(n^2)$ level the Comba method adds the product of two single precision variables to each column of the output +independently. A serious obstacle is if the carry is lost, due to lack of precision before the algorithm has a chance to fix +the carries. For example, in the multiplication of two three-digit numbers the third column of output will be the sum of +three single precision multiplications. If the precision of the accumulator for the output digits is less then $3 \cdot (\beta - 1)^2$ then +an overflow can occur and the carry information will be lost. For any $m$ and $n$ digit inputs the maximum weight of any column is +min$(m, n)$ which is fairly obvious. + +The maximum number of terms in any column of a product is known as the ``column weight'' and strictly governs when the algorithm can be used. Recall +from earlier that a double precision type has $\alpha$ bits of resolution and a single precision digit has $lg(\beta)$ bits of precision. Given these +two quantities we must not violate the following + +\begin{equation} +k \cdot \left (\beta - 1 \right )^2 < 2^{\alpha} +\end{equation} + +Which reduces to + +\begin{equation} +k \cdot \left ( \beta^2 - 2\beta + 1 \right ) < 2^{\alpha} +\end{equation} + +Let $\rho = lg(\beta)$ represent the number of bits in a single precision digit. By further re-arrangement of the equation the final solution is +found. + +\begin{equation} +k < {{2^{\alpha}} \over {\left (2^{2\rho} - 2^{\rho + 1} + 1 \right )}} +\end{equation} + +The defaults for LibTomMath are $\beta = 2^{28}$ and $\alpha = 2^{64}$ which means that $k$ is bounded by $k < 257$. In this configuration +the smaller input may not have more than $256$ digits if the Comba method is to be used. This is quite satisfactory for most applications since +$256$ digits would allow for numbers in the range of $0 \le x < 2^{7168}$ which, is much larger than most public key cryptographic algorithms require. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{fast\_s\_mp\_mul\_digs}. \\ +\textbf{Input}. mp\_int $a$, mp\_int $b$ and an integer $digs$ \\ +\textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert \mbox{ (mod }\beta^{digs}\mbox{)}$. \\ +\hline \\ +Place an array of \textbf{MP\_WARRAY} double precision digits named $\hat W$ on the stack. \\ +1. If $c.alloc < digs$ then grow $c$ to $digs$ digits. (\textit{mp\_grow}) \\ +2. If step 1 failed return(\textit{MP\_MEM}).\\ +\\ +Zero the temporary array $\hat W$. \\ +3. for $n$ from $0$ to $digs - 1$ do \\ +\hspace{3mm}3.1 $\hat W_n \leftarrow 0$ \\ +\\ +Compute the columns. \\ +4. for $ix$ from $0$ to $a.used - 1$ do \\ +\hspace{3mm}4.1 $pb \leftarrow \mbox{min}(b.used, digs - ix)$ \\ +\hspace{3mm}4.2 If $pb < 1$ then goto step 5. \\ +\hspace{3mm}4.3 for $iy$ from $0$ to $pb - 1$ do \\ +\hspace{6mm}4.3.1 $\hat W_{ix+iy} \leftarrow \hat W_{ix+iy} + a_{ix}b_{iy}$ \\ +\\ +Propagate the carries upwards. \\ +5. $oldused \leftarrow c.used$ \\ +6. $c.used \leftarrow digs$ \\ +7. If $digs > 1$ then do \\ +\hspace{3mm}7.1. for $ix$ from $1$ to $digs - 1$ do \\ +\hspace{6mm}7.1.1 $\hat W_{ix} \leftarrow \hat W_{ix} + \lfloor \hat W_{ix-1} / \beta \rfloor$ \\ +\hspace{6mm}7.1.2 $c_{ix - 1} \leftarrow \hat W_{ix - 1} \mbox{ (mod }\beta\mbox{)}$ \\ +8. else do \\ +\hspace{3mm}8.1 $ix \leftarrow 0$ \\ +9. $c_{ix} \leftarrow \hat W_{ix} \mbox{ (mod }\beta\mbox{)}$ \\ +\\ +Zero excess digits. \\ +10. If $digs < oldused$ then do \\ +\hspace{3mm}10.1 for $n$ from $digs$ to $oldused - 1$ do \\ +\hspace{6mm}10.1.1 $c_n \leftarrow 0$ \\ +11. Clamp excessive digits of $c$. (\textit{mp\_clamp}) \\ +12. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm fast\_s\_mp\_mul\_digs} +\label{fig:COMBAMULT} +\end{figure} + +\textbf{Algorithm fast\_s\_mp\_mul\_digs.} +This algorithm performs the unsigned multiplication of $a$ and $b$ using the Comba method limited to $digs$ digits of precision. The algorithm +essentially peforms the same calculation as algorithm s\_mp\_mul\_digs, just much faster. + +The array $\hat W$ is meant to be on the stack when the algorithm is used. The size of the array does not change which is ideal. Note also that +unlike algorithm s\_mp\_mul\_digs no temporary mp\_int is required since the result is calculated directly in $\hat W$. + +The $O(n^2)$ loop on step four is where the Comba method's advantages begin to show through in comparison to the baseline algorithm. The lack of +a carry variable or propagation in this loop allows the loop to be performed with only single precision multiplication and additions. Now that each +iteration of the inner loop can be performed independent of the others the inner loop can be performed with a high level of parallelism. + +To measure the benefits of the Comba method over the baseline method consider the number of operations that are required. If the +cost in terms of time of a multiply and addition is $p$ and the cost of a carry propagation is $q$ then a baseline multiplication would require +$O \left ((p + q)n^2 \right )$ time to multiply two $n$-digit numbers. The Comba method requires only $O(pn^2 + qn)$ time, however in practice, +the speed increase is actually much more. With $O(n)$ space the algorithm can be reduced to $O(pn + qn)$ time by implementing the $n$ multiply +and addition operations in the nested loop in parallel. + +EXAM,bn_fast_s_mp_mul_digs.c + +The memset on line @47,memset@ clears the initial $\hat W$ array to zero in a single step. Like the slower baseline multiplication +implementation a series of aliases (\textit{lines @67, tmpx@, @70, tmpy@ and @75,_W@}) are used to simplify the inner $O(n^2)$ loop. +In this case a new alias $\_\hat W$ has been added which refers to the double precision columns offset by $ix$ in each pass. + +The inner loop on lines @83,for@, @84,mp_word@ and @85,}@ is where the algorithm will spend the majority of the time, which is why it has been +stripped to the bones of any extra baggage\footnote{Hence the pointer aliases.}. On x86 processors the multiplication and additions amount to at the +very least five instructions (\textit{two loads, two additions, one multiply}) while on the ARMv4 processors they amount to only three +(\textit{one load, one store, one multiply-add}). For both of the x86 and ARMv4 processors the GCC compiler performs a good job at unrolling the loop +and scheduling the instructions so there are very few dependency stalls. + +In theory the difference between the baseline and comba algorithms is a mere $O(qn)$ time difference. However, in the $O(n^2)$ nested loop of the +baseline method there are dependency stalls as the algorithm must wait for the multiplier to finish before propagating the carry to the next +digit. As a result fewer of the often multiple execution units\footnote{The AMD Athlon has three execution units and the Intel P4 has four.} can +be simultaneously used. + +\subsection{Polynomial Basis Multiplication} +To break the $O(n^2)$ barrier in multiplication requires a completely different look at integer multiplication. In the following algorithms +the use of polynomial basis representation for two integers $a$ and $b$ as $f(x) = \sum_{i=0}^{n} a_i x^i$ and +$g(x) = \sum_{i=0}^{n} b_i x^i$ respectively, is required. In this system both $f(x)$ and $g(x)$ have $n + 1$ terms and are of the $n$'th degree. + +The product $a \cdot b \equiv f(x)g(x)$ is the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$. The coefficients $w_i$ will +directly yield the desired product when $\beta$ is substituted for $x$. The direct solution to solve for the $2n + 1$ coefficients +requires $O(n^2)$ time and would in practice be slower than the Comba technique. + +However, numerical analysis theory indicates that only $2n + 1$ distinct points in $W(x)$ are required to determine the values of the $2n + 1$ unknown +coefficients. This means by finding $\zeta_y = W(y)$ for $2n + 1$ small values of $y$ the coefficients of $W(x)$ can be found with +Gaussian elimination. This technique is also occasionally refered to as the \textit{interpolation technique} (\textit{references please...}) since in +effect an interpolation based on $2n + 1$ points will yield a polynomial equivalent to $W(x)$. + +The coefficients of the polynomial $W(x)$ are unknown which makes finding $W(y)$ for any value of $y$ impossible. However, since +$W(x) = f(x)g(x)$ the equivalent $\zeta_y = f(y) g(y)$ can be used in its place. The benefit of this technique stems from the +fact that $f(y)$ and $g(y)$ are much smaller than either $a$ or $b$ respectively. As a result finding the $2n + 1$ relations required +by multiplying $f(y)g(y)$ involves multiplying integers that are much smaller than either of the inputs. + +When picking points to gather relations there are always three obvious points to choose, $y = 0, 1$ and $ \infty$. The $\zeta_0$ term +is simply the product $W(0) = w_0 = a_0 \cdot b_0$. The $\zeta_1$ term is the product +$W(1) = \left (\sum_{i = 0}^{n} a_i \right ) \left (\sum_{i = 0}^{n} b_i \right )$. The third point $\zeta_{\infty}$ is less obvious but rather +simple to explain. The $2n + 1$'th coefficient of $W(x)$ is numerically equivalent to the most significant column in an integer multiplication. +The point at $\infty$ is used symbolically to represent the most significant column, that is $W(\infty) = w_{2n} = a_nb_n$. Note that the +points at $y = 0$ and $\infty$ yield the coefficients $w_0$ and $w_{2n}$ directly. + +If more points are required they should be of small values and powers of two such as $2^q$ and the related \textit{mirror points} +$\left (2^q \right )^{2n} \cdot \zeta_{2^{-q}}$ for small values of $q$. The term ``mirror point'' stems from the fact that +$\left (2^q \right )^{2n} \cdot \zeta_{2^{-q}}$ can be calculated in the exact opposite fashion as $\zeta_{2^q}$. For +example, when $n = 2$ and $q = 1$ then following two equations are equivalent to the point $\zeta_{2}$ and its mirror. + +\begin{eqnarray} +\zeta_{2} = f(2)g(2) = (4a_2 + 2a_1 + a_0)(4b_2 + 2b_1 + b_0) \nonumber \\ +16 \cdot \zeta_{1 \over 2} = 4f({1\over 2}) \cdot 4g({1 \over 2}) = (a_2 + 2a_1 + 4a_0)(b_2 + 2b_1 + 4b_0) +\end{eqnarray} + +Using such points will allow the values of $f(y)$ and $g(y)$ to be independently calculated using only left shifts. For example, when $n = 2$ the +polynomial $f(2^q)$ is equal to $2^q((2^qa_2) + a_1) + a_0$. This technique of polynomial representation is known as Horner's method. + +As a general rule of the algorithm when the inputs are split into $n$ parts each there are $2n - 1$ multiplications. Each multiplication is of +multiplicands that have $n$ times fewer digits than the inputs. The asymptotic running time of this algorithm is +$O \left ( k^{lg_n(2n - 1)} \right )$ for $k$ digit inputs (\textit{assuming they have the same number of digits}). Figure~\ref{fig:exponent} +summarizes the exponents for various values of $n$. + +\begin{figure} +\begin{center} +\begin{tabular}{|c|c|c|} +\hline \textbf{Split into $n$ Parts} & \textbf{Exponent} & \textbf{Notes}\\ +\hline $2$ & $1.584962501$ & This is Karatsuba Multiplication. \\ +\hline $3$ & $1.464973520$ & This is Toom-Cook Multiplication. \\ +\hline $4$ & $1.403677461$ &\\ +\hline $5$ & $1.365212389$ &\\ +\hline $10$ & $1.278753601$ &\\ +\hline $100$ & $1.149426538$ &\\ +\hline $1000$ & $1.100270931$ &\\ +\hline $10000$ & $1.075252070$ &\\ +\hline +\end{tabular} +\end{center} +\caption{Asymptotic Running Time of Polynomial Basis Multiplication} +\label{fig:exponent} +\end{figure} + +At first it may seem like a good idea to choose $n = 1000$ since the exponent is approximately $1.1$. However, the overhead +of solving for the 2001 terms of $W(x)$ will certainly consume any savings the algorithm could offer for all but exceedingly large +numbers. + +\subsubsection{Cutoff Point} +The polynomial basis multiplication algorithms all require fewer single precision multiplications than a straight Comba approach. However, +the algorithms incur an overhead (\textit{at the $O(n)$ work level}) since they require a system of equations to be solved. This makes the +polynomial basis approach more costly to use with small inputs. + +Let $m$ represent the number of digits in the multiplicands (\textit{assume both multiplicands have the same number of digits}). There exists a +point $y$ such that when $m < y$ the polynomial basis algorithms are more costly than Comba, when $m = y$ they are roughly the same cost and +when $m > y$ the Comba methods are slower than the polynomial basis algorithms. + +The exact location of $y$ depends on several key architectural elements of the computer platform in question. + +\begin{enumerate} +\item The ratio of clock cycles for single precision multiplication versus other simpler operations such as addition, shifting, etc. For example +on the AMD Athlon the ratio is roughly $17 : 1$ while on the Intel P4 it is $29 : 1$. The higher the ratio in favour of multiplication the lower +the cutoff point $y$ will be. + +\item The complexity of the linear system of equations (\textit{for the coefficients of $W(x)$}) is. Generally speaking as the number of splits +grows the complexity grows substantially. Ideally solving the system will only involve addition, subtraction and shifting of integers. This +directly reflects on the ratio previous mentioned. + +\item To a lesser extent memory bandwidth and function call overheads. Provided the values are in the processor cache this is less of an +influence over the cutoff point. + +\end{enumerate} + +A clean cutoff point separation occurs when a point $y$ is found such that all of the cutoff point conditions are met. For example, if the point +is too low then there will be values of $m$ such that $m > y$ and the Comba method is still faster. Finding the cutoff points is fairly simple when +a high resolution timer is available. + +\subsection{Karatsuba Multiplication} +Karatsuba \cite{KARA} multiplication when originally proposed in 1962 was among the first set of algorithms to break the $O(n^2)$ barrier for +general purpose multiplication. Given two polynomial basis representations $f(x) = ax + b$ and $g(x) = cx + d$, Karatsuba proved with +light algebra \cite{KARAP} that the following polynomial is equivalent to multiplication of the two integers the polynomials represent. + +\begin{equation} +f(x) \cdot g(x) = acx^2 + ((a - b)(c - d) - (ac + bd))x + bd +\end{equation} + +Using the observation that $ac$ and $bd$ could be re-used only three half sized multiplications would be required to produce the product. Applying +this algorithm recursively, the work factor becomes $O(n^{lg(3)})$ which is substantially better than the work factor $O(n^2)$ of the Comba technique. It turns +out what Karatsuba did not know or at least did not publish was that this is simply polynomial basis multiplication with the points +$\zeta_0$, $\zeta_{\infty}$ and $-\zeta_{-1}$. Consider the resultant system of equations. + +\begin{center} +\begin{tabular}{rcrcrcrc} +$\zeta_{0}$ & $=$ & & & & & $w_0$ \\ +$-\zeta_{-1}$ & $=$ & $-w_2$ & $+$ & $w_1$ & $-$ & $w_0$ \\ +$\zeta_{\infty}$ & $=$ & $w_2$ & & & & \\ +\end{tabular} +\end{center} + +By adding the first and last equation to the equation in the middle the term $w_1$ can be isolated and all three coefficients solved for. The simplicity +of this system of equations has made Karatsuba fairly popular. In fact the cutoff point is often fairly low\footnote{With LibTomMath 0.18 it is 70 and 109 digits for the Intel P4 and AMD Athlon respectively.} +making it an ideal algorithm to speed up certain public key cryptosystems such as RSA and Diffie-Hellman. It is worth noting that the point +$\zeta_1$ could be substituted for $-\zeta_{-1}$. In this case the first and third row are subtracted instead of added to the second row. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_karatsuba\_mul}. \\ +\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ +\textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert$ \\ +\hline \\ +1. Init the following mp\_int variables: $x0$, $x1$, $y0$, $y1$, $t1$, $x0y0$, $x1y1$.\\ +2. If step 2 failed then return(\textit{MP\_MEM}). \\ +\\ +Split the input. e.g. $a = x1 \cdot \beta^B + x0$ \\ +3. $B \leftarrow \mbox{min}(a.used, b.used)/2$ \\ +4. $x0 \leftarrow a \mbox{ (mod }\beta^B\mbox{)}$ (\textit{mp\_mod\_2d}) \\ +5. $y0 \leftarrow b \mbox{ (mod }\beta^B\mbox{)}$ \\ +6. $x1 \leftarrow \lfloor a / \beta^B \rfloor$ (\textit{mp\_rshd}) \\ +7. $y1 \leftarrow \lfloor b / \beta^B \rfloor$ \\ +\\ +Calculate the three products. \\ +8. $x0y0 \leftarrow x0 \cdot y0$ (\textit{mp\_mul}) \\ +9. $x1y1 \leftarrow x1 \cdot y1$ \\ +10. $t1 \leftarrow x1 - x0$ (\textit{mp\_sub}) \\ +11. $x0 \leftarrow y1 - y0$ \\ +12. $t1 \leftarrow t1 \cdot x0$ \\ +\\ +Calculate the middle term. \\ +13. $x0 \leftarrow x0y0 + x1y1$ \\ +14. $t1 \leftarrow x0 - t1$ \\ +\\ +Calculate the final product. \\ +15. $t1 \leftarrow t1 \cdot \beta^B$ (\textit{mp\_lshd}) \\ +16. $x1y1 \leftarrow x1y1 \cdot \beta^{2B}$ \\ +17. $t1 \leftarrow x0y0 + t1$ \\ +18. $c \leftarrow t1 + x1y1$ \\ +19. Clear all of the temporary variables. \\ +20. Return(\textit{MP\_OKAY}).\\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_karatsuba\_mul} +\end{figure} + +\textbf{Algorithm mp\_karatsuba\_mul.} +This algorithm computes the unsigned product of two inputs using the Karatsuba multiplication algorithm. It is loosely based on the description +from Knuth \cite[pp. 294-295]{TAOCPV2}. + +\index{radix point} +In order to split the two inputs into their respective halves, a suitable \textit{radix point} must be chosen. The radix point chosen must +be used for both of the inputs meaning that it must be smaller than the smallest input. Step 3 chooses the radix point $B$ as half of the +smallest input \textbf{used} count. After the radix point is chosen the inputs are split into lower and upper halves. Step 4 and 5 +compute the lower halves. Step 6 and 7 computer the upper halves. + +After the halves have been computed the three intermediate half-size products must be computed. Step 8 and 9 compute the trivial products +$x0 \cdot y0$ and $x1 \cdot y1$. The mp\_int $x0$ is used as a temporary variable after $x1 - x0$ has been computed. By using $x0$ instead +of an additional temporary variable, the algorithm can avoid an addition memory allocation operation. + +The remaining steps 13 through 18 compute the Karatsuba polynomial through a variety of digit shifting and addition operations. + +EXAM,bn_mp_karatsuba_mul.c + +The new coding element in this routine, not seen in previous routines, is the usage of goto statements. The conventional +wisdom is that goto statements should be avoided. This is generally true, however when every single function call can fail, it makes sense +to handle error recovery with a single piece of code. Lines @61,if@ to @75,if@ handle initializing all of the temporary variables +required. Note how each of the if statements goes to a different label in case of failure. This allows the routine to correctly free only +the temporaries that have been successfully allocated so far. + +The temporary variables are all initialized using the mp\_init\_size routine since they are expected to be large. This saves the +additional reallocation that would have been necessary. Also $x0$, $x1$, $y0$ and $y1$ have to be able to hold at least their respective +number of digits for the next section of code. + +The first algebraic portion of the algorithm is to split the two inputs into their halves. However, instead of using mp\_mod\_2d and mp\_rshd +to extract the halves, the respective code has been placed inline within the body of the function. To initialize the halves, the \textbf{used} and +\textbf{sign} members are copied first. The first for loop on line @98,for@ copies the lower halves. Since they are both the same magnitude it +is simpler to calculate both lower halves in a single loop. The for loop on lines @104,for@ and @109,for@ calculate the upper halves $x1$ and +$y1$ respectively. + +By inlining the calculation of the halves, the Karatsuba multiplier has a slightly lower overhead and can be used for smaller magnitude inputs. + +When line @152,err@ is reached, the algorithm has completed succesfully. The ``error status'' variable $err$ is set to \textbf{MP\_OKAY} so that +the same code that handles errors can be used to clear the temporary variables and return. + +\subsection{Toom-Cook $3$-Way Multiplication} +Toom-Cook $3$-Way \cite{TOOM} multiplication is essentially the polynomial basis algorithm for $n = 2$ except that the points are +chosen such that $\zeta$ is easy to compute and the resulting system of equations easy to reduce. Here, the points $\zeta_{0}$, +$16 \cdot \zeta_{1 \over 2}$, $\zeta_1$, $\zeta_2$ and $\zeta_{\infty}$ make up the five required points to solve for the coefficients +of the $W(x)$. + +With the five relations that Toom-Cook specifies, the following system of equations is formed. + +\begin{center} +\begin{tabular}{rcrcrcrcrcr} +$\zeta_0$ & $=$ & $0w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $1w_0$ \\ +$16 \cdot \zeta_{1 \over 2}$ & $=$ & $1w_4$ & $+$ & $2w_3$ & $+$ & $4w_2$ & $+$ & $8w_1$ & $+$ & $16w_0$ \\ +$\zeta_1$ & $=$ & $1w_4$ & $+$ & $1w_3$ & $+$ & $1w_2$ & $+$ & $1w_1$ & $+$ & $1w_0$ \\ +$\zeta_2$ & $=$ & $16w_4$ & $+$ & $8w_3$ & $+$ & $4w_2$ & $+$ & $2w_1$ & $+$ & $1w_0$ \\ +$\zeta_{\infty}$ & $=$ & $1w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $0w_0$ \\ +\end{tabular} +\end{center} + +A trivial solution to this matrix requires $12$ subtractions, two multiplications by a small power of two, two divisions by a small power +of two, two divisions by three and one multiplication by three. All of these $19$ sub-operations require less than quadratic time, meaning that +the algorithm can be faster than a baseline multiplication. However, the greater complexity of this algorithm places the cutoff point +(\textbf{TOOM\_MUL\_CUTOFF}) where Toom-Cook becomes more efficient much higher than the Karatsuba cutoff point. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_toom\_mul}. \\ +\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ +\textbf{Output}. $c \leftarrow a \cdot b $ \\ +\hline \\ +Split $a$ and $b$ into three pieces. E.g. $a = a_2 \beta^{2k} + a_1 \beta^{k} + a_0$ \\ +1. $k \leftarrow \lfloor \mbox{min}(a.used, b.used) / 3 \rfloor$ \\ +2. $a_0 \leftarrow a \mbox{ (mod }\beta^{k}\mbox{)}$ \\ +3. $a_1 \leftarrow \lfloor a / \beta^k \rfloor$, $a_1 \leftarrow a_1 \mbox{ (mod }\beta^{k}\mbox{)}$ \\ +4. $a_2 \leftarrow \lfloor a / \beta^{2k} \rfloor$, $a_2 \leftarrow a_2 \mbox{ (mod }\beta^{k}\mbox{)}$ \\ +5. $b_0 \leftarrow a \mbox{ (mod }\beta^{k}\mbox{)}$ \\ +6. $b_1 \leftarrow \lfloor a / \beta^k \rfloor$, $b_1 \leftarrow b_1 \mbox{ (mod }\beta^{k}\mbox{)}$ \\ +7. $b_2 \leftarrow \lfloor a / \beta^{2k} \rfloor$, $b_2 \leftarrow b_2 \mbox{ (mod }\beta^{k}\mbox{)}$ \\ +\\ +Find the five equations for $w_0, w_1, ..., w_4$. \\ +8. $w_0 \leftarrow a_0 \cdot b_0$ \\ +9. $w_4 \leftarrow a_2 \cdot b_2$ \\ +10. $tmp_1 \leftarrow 2 \cdot a_0$, $tmp_1 \leftarrow a_1 + tmp_1$, $tmp_1 \leftarrow 2 \cdot tmp_1$, $tmp_1 \leftarrow tmp_1 + a_2$ \\ +11. $tmp_2 \leftarrow 2 \cdot b_0$, $tmp_2 \leftarrow b_1 + tmp_2$, $tmp_2 \leftarrow 2 \cdot tmp_2$, $tmp_2 \leftarrow tmp_2 + b_2$ \\ +12. $w_1 \leftarrow tmp_1 \cdot tmp_2$ \\ +13. $tmp_1 \leftarrow 2 \cdot a_2$, $tmp_1 \leftarrow a_1 + tmp_1$, $tmp_1 \leftarrow 2 \cdot tmp_1$, $tmp_1 \leftarrow tmp_1 + a_0$ \\ +14. $tmp_2 \leftarrow 2 \cdot b_2$, $tmp_2 \leftarrow b_1 + tmp_2$, $tmp_2 \leftarrow 2 \cdot tmp_2$, $tmp_2 \leftarrow tmp_2 + b_0$ \\ +15. $w_3 \leftarrow tmp_1 \cdot tmp_2$ \\ +16. $tmp_1 \leftarrow a_0 + a_1$, $tmp_1 \leftarrow tmp_1 + a_2$, $tmp_2 \leftarrow b_0 + b_1$, $tmp_2 \leftarrow tmp_2 + b_2$ \\ +17. $w_2 \leftarrow tmp_1 \cdot tmp_2$ \\ +\\ +Continued on the next page.\\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_toom\_mul} +\end{figure} + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_toom\_mul} (continued). \\ +\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ +\textbf{Output}. $c \leftarrow a \cdot b $ \\ +\hline \\ +Now solve the system of equations. \\ +18. $w_1 \leftarrow w_4 - w_1$, $w_3 \leftarrow w_3 - w_0$ \\ +19. $w_1 \leftarrow \lfloor w_1 / 2 \rfloor$, $w_3 \leftarrow \lfloor w_3 / 2 \rfloor$ \\ +20. $w_2 \leftarrow w_2 - w_0$, $w_2 \leftarrow w_2 - w_4$ \\ +21. $w_1 \leftarrow w_1 - w_2$, $w_3 \leftarrow w_3 - w_2$ \\ +22. $tmp_1 \leftarrow 8 \cdot w_0$, $w_1 \leftarrow w_1 - tmp_1$, $tmp_1 \leftarrow 8 \cdot w_4$, $w_3 \leftarrow w_3 - tmp_1$ \\ +23. $w_2 \leftarrow 3 \cdot w_2$, $w_2 \leftarrow w_2 - w_1$, $w_2 \leftarrow w_2 - w_3$ \\ +24. $w_1 \leftarrow w_1 - w_2$, $w_3 \leftarrow w_3 - w_2$ \\ +25. $w_1 \leftarrow \lfloor w_1 / 3 \rfloor, w_3 \leftarrow \lfloor w_3 / 3 \rfloor$ \\ +\\ +Now substitute $\beta^k$ for $x$ by shifting $w_0, w_1, ..., w_4$. \\ +26. for $n$ from $1$ to $4$ do \\ +\hspace{3mm}26.1 $w_n \leftarrow w_n \cdot \beta^{nk}$ \\ +27. $c \leftarrow w_0 + w_1$, $c \leftarrow c + w_2$, $c \leftarrow c + w_3$, $c \leftarrow c + w_4$ \\ +28. Return(\textit{MP\_OKAY}) \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_toom\_mul (continued)} +\end{figure} + +\textbf{Algorithm mp\_toom\_mul.} +This algorithm computes the product of two mp\_int variables $a$ and $b$ using the Toom-Cook approach. Compared to the Karatsuba multiplication, this +algorithm has a lower asymptotic running time of approximately $O(n^{1.464})$ but at an obvious cost in overhead. In this +description, several statements have been compounded to save space. The intention is that the statements are executed from left to right across +any given step. + +The two inputs $a$ and $b$ are first split into three $k$-digit integers $a_0, a_1, a_2$ and $b_0, b_1, b_2$ respectively. From these smaller +integers the coefficients of the polynomial basis representations $f(x)$ and $g(x)$ are known and can be used to find the relations required. + +The first two relations $w_0$ and $w_4$ are the points $\zeta_{0}$ and $\zeta_{\infty}$ respectively. The relation $w_1, w_2$ and $w_3$ correspond +to the points $16 \cdot \zeta_{1 \over 2}, \zeta_{2}$ and $\zeta_{1}$ respectively. These are found using logical shifts to independently find +$f(y)$ and $g(y)$ which significantly speeds up the algorithm. + +After the five relations $w_0, w_1, \ldots, w_4$ have been computed, the system they represent must be solved in order for the unknown coefficients +$w_1, w_2$ and $w_3$ to be isolated. The steps 18 through 25 perform the system reduction required as previously described. Each step of +the reduction represents the comparable matrix operation that would be performed had this been performed by pencil. For example, step 18 indicates +that row $1$ must be subtracted from row $4$ and simultaneously row $0$ subtracted from row $3$. + +Once the coeffients have been isolated, the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$ is known. By substituting $\beta^{k}$ for $x$, the integer +result $a \cdot b$ is produced. + +EXAM,bn_mp_toom_mul.c + +-- Comments to be added during editing phase. + +\subsection{Signed Multiplication} +Now that algorithms to handle multiplications of every useful dimensions have been developed, a rather simple finishing touch is required. So far all +of the multiplication algorithms have been unsigned multiplications which leaves only a signed multiplication algorithm to be established. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_mul}. \\ +\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ +\textbf{Output}. $c \leftarrow a \cdot b$ \\ +\hline \\ +1. If $a.sign = b.sign$ then \\ +\hspace{3mm}1.1 $sign = MP\_ZPOS$ \\ +2. else \\ +\hspace{3mm}2.1 $sign = MP\_ZNEG$ \\ +3. If min$(a.used, b.used) \ge TOOM\_MUL\_CUTOFF$ then \\ +\hspace{3mm}3.1 $c \leftarrow a \cdot b$ using algorithm mp\_toom\_mul \\ +4. else if min$(a.used, b.used) \ge KARATSUBA\_MUL\_CUTOFF$ then \\ +\hspace{3mm}4.1 $c \leftarrow a \cdot b$ using algorithm mp\_karatsuba\_mul \\ +5. else \\ +\hspace{3mm}5.1 $digs \leftarrow a.used + b.used + 1$ \\ +\hspace{3mm}5.2 If $digs < MP\_ARRAY$ and min$(a.used, b.used) \le \delta$ then \\ +\hspace{6mm}5.2.1 $c \leftarrow a \cdot b \mbox{ (mod }\beta^{digs}\mbox{)}$ using algorithm fast\_s\_mp\_mul\_digs. \\ +\hspace{3mm}5.3 else \\ +\hspace{6mm}5.3.1 $c \leftarrow a \cdot b \mbox{ (mod }\beta^{digs}\mbox{)}$ using algorithm s\_mp\_mul\_digs. \\ +6. $c.sign \leftarrow sign$ \\ +7. Return the result of the unsigned multiplication performed. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_mul} +\end{figure} + +\textbf{Algorithm mp\_mul.} +This algorithm performs the signed multiplication of two inputs. It will make use of any of the three unsigned multiplication algorithms +available when the input is of appropriate size. The \textbf{sign} of the result is not set until the end of the algorithm since algorithm +s\_mp\_mul\_digs will clear it. + +EXAM,bn_mp_mul.c + +The implementation is rather simplistic and is not particularly noteworthy. Line @22,?@ computes the sign of the result using the ``?'' +operator from the C programming language. Line @37,<<@ computes $\delta$ using the fact that $1 << k$ is equal to $2^k$. + +\section{Squaring} +\label{sec:basesquare} + +Squaring is a special case of multiplication where both multiplicands are equal. At first it may seem like there is no significant optimization +available but in fact there is. Consider the multiplication of $576$ against $241$. In total there will be nine single precision multiplications +performed which are $1\cdot 6$, $1 \cdot 7$, $1 \cdot 5$, $4 \cdot 6$, $4 \cdot 7$, $4 \cdot 5$, $2 \cdot 6$, $2 \cdot 7$ and $2 \cdot 5$. Now consider +the multiplication of $123$ against $123$. The nine products are $3 \cdot 3$, $3 \cdot 2$, $3 \cdot 1$, $2 \cdot 3$, $2 \cdot 2$, $2 \cdot 1$, +$1 \cdot 3$, $1 \cdot 2$ and $1 \cdot 1$. On closer inspection some of the products are equivalent. For example, $3 \cdot 2 = 2 \cdot 3$ +and $3 \cdot 1 = 1 \cdot 3$. + +For any $n$-digit input, there are ${{\left (n^2 + n \right)}\over 2}$ possible unique single precision multiplications required compared to the $n^2$ +required for multiplication. The following diagram gives an example of the operations required. + +\begin{figure}[here] +\begin{center} +\begin{tabular}{ccccc|c} +&&1&2&3&\\ +$\times$ &&1&2&3&\\ +\hline && $3 \cdot 1$ & $3 \cdot 2$ & $3 \cdot 3$ & Row 0\\ + & $2 \cdot 1$ & $2 \cdot 2$ & $2 \cdot 3$ && Row 1 \\ + $1 \cdot 1$ & $1 \cdot 2$ & $1 \cdot 3$ &&& Row 2 \\ +\end{tabular} +\end{center} +\caption{Squaring Optimization Diagram} +\end{figure} + +MARK,SQUARE +Starting from zero and numbering the columns from right to left a very simple pattern becomes obvious. For the purposes of this discussion let $x$ +represent the number being squared. The first observation is that in row $k$ the $2k$'th column of the product has a $\left (x_k \right)^2$ term in it. + +The second observation is that every column $j$ in row $k$ where $j \ne 2k$ is part of a double product. Every non-square term of a column will +appear twice hence the name ``double product''. Every odd column is made up entirely of double products. In fact every column is made up of double +products and at most one square (\textit{see the exercise section}). + +The third and final observation is that for row $k$ the first unique non-square term, that is, one that hasn't already appeared in an earlier row, +occurs at column $2k + 1$. For example, on row $1$ of the previous squaring, column one is part of the double product with column one from row zero. +Column two of row one is a square and column three is the first unique column. + +\subsection{The Baseline Squaring Algorithm} +The baseline squaring algorithm is meant to be a catch-all squaring algorithm. It will handle any of the input sizes that the faster routines +will not handle. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{s\_mp\_sqr}. \\ +\textbf{Input}. mp\_int $a$ \\ +\textbf{Output}. $b \leftarrow a^2$ \\ +\hline \\ +1. Init a temporary mp\_int of at least $2 \cdot a.used +1$ digits. (\textit{mp\_init\_size}) \\ +2. If step 1 failed return(\textit{MP\_MEM}) \\ +3. $t.used \leftarrow 2 \cdot a.used + 1$ \\ +4. For $ix$ from 0 to $a.used - 1$ do \\ +\hspace{3mm}Calculate the square. \\ +\hspace{3mm}4.1 $\hat r \leftarrow t_{2ix} + \left (a_{ix} \right )^2$ \\ +\hspace{3mm}4.2 $t_{2ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ +\hspace{3mm}Calculate the double products after the square. \\ +\hspace{3mm}4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ +\hspace{3mm}4.4 For $iy$ from $ix + 1$ to $a.used - 1$ do \\ +\hspace{6mm}4.4.1 $\hat r \leftarrow 2 \cdot a_{ix}a_{iy} + t_{ix + iy} + u$ \\ +\hspace{6mm}4.4.2 $t_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ +\hspace{6mm}4.4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ +\hspace{3mm}Set the last carry. \\ +\hspace{3mm}4.5 While $u > 0$ do \\ +\hspace{6mm}4.5.1 $iy \leftarrow iy + 1$ \\ +\hspace{6mm}4.5.2 $\hat r \leftarrow t_{ix + iy} + u$ \\ +\hspace{6mm}4.5.3 $t_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ +\hspace{6mm}4.5.4 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ +5. Clamp excess digits of $t$. (\textit{mp\_clamp}) \\ +6. Exchange $b$ and $t$. \\ +7. Clear $t$ (\textit{mp\_clear}) \\ +8. Return(\textit{MP\_OKAY}) \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm s\_mp\_sqr} +\end{figure} + +\textbf{Algorithm s\_mp\_sqr.} +This algorithm computes the square of an input using the three observations on squaring. It is based fairly faithfully on algorithm 14.16 of HAC +\cite[pp.596-597]{HAC}. Similar to algorithm s\_mp\_mul\_digs, a temporary mp\_int is allocated to hold the result of the squaring. This allows the +destination mp\_int to be the same as the source mp\_int. + +The outer loop of this algorithm begins on step 4. It is best to think of the outer loop as walking down the rows of the partial results, while +the inner loop computes the columns of the partial result. Step 4.1 and 4.2 compute the square term for each row, and step 4.3 and 4.4 propagate +the carry and compute the double products. + +The requirement that a mp\_word be able to represent the range $0 \le x < 2 \beta^2$ arises from this +very algorithm. The product $a_{ix}a_{iy}$ will lie in the range $0 \le x \le \beta^2 - 2\beta + 1$ which is obviously less than $\beta^2$ meaning that +when it is multiplied by two, it can be properly represented by a mp\_word. + +Similar to algorithm s\_mp\_mul\_digs, after every pass of the inner loop, the destination is correctly set to the sum of all of the partial +results calculated so far. This involves expensive carry propagation which will be eliminated in the next algorithm. + +EXAM,bn_s_mp_sqr.c + +Inside the outer loop (\textit{see line @32,for@}) the square term is calculated on line @35,r =@. Line @42,>>@ extracts the carry from the square +term. Aliases for $a_{ix}$ and $t_{ix+iy}$ are initialized on lines @45,tmpx@ and @48,tmpt@ respectively. The doubling is performed using two +additions (\textit{see line @57,r + r@}) since it is usually faster than shifting,if not at least as fast. + +\subsection{Faster Squaring by the ``Comba'' Method} +A major drawback to the baseline method is the requirement for single precision shifting inside the $O(n^2)$ nested loop. Squaring has an additional +drawback that it must double the product inside the inner loop as well. As for multiplication, the Comba technique can be used to eliminate these +performance hazards. + +The first obvious solution is to make an array of mp\_words which will hold all of the columns. This will indeed eliminate all of the carry +propagation operations from the inner loop. However, the inner product must still be doubled $O(n^2)$ times. The solution stems from the simple fact +that $2a + 2b + 2c = 2(a + b + c)$. That is the sum of all of the double products is equal to double the sum of all the products. For example, +$ab + ba + ac + ca = 2ab + 2ac = 2(ab + ac)$. + +However, we cannot simply double all of the columns, since the squares appear only once per row. The most practical solution is to have two mp\_word +arrays. One array will hold the squares and the other array will hold the double products. With both arrays the doubling and carry propagation can be +moved to a $O(n)$ work level outside the $O(n^2)$ level. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{fast\_s\_mp\_sqr}. \\ +\textbf{Input}. mp\_int $a$ \\ +\textbf{Output}. $b \leftarrow a^2$ \\ +\hline \\ +Place two arrays of \textbf{MP\_WARRAY} mp\_words named $\hat W$ and $\hat {X}$ on the stack. \\ +1. If $b.alloc < 2a.used + 1$ then grow $b$ to $2a.used + 1$ digits. (\textit{mp\_grow}). \\ +2. If step 1 failed return(\textit{MP\_MEM}). \\ +3. for $ix$ from $0$ to $2a.used + 1$ do \\ +\hspace{3mm}3.1 $\hat W_{ix} \leftarrow 0$ \\ +\hspace{3mm}3.2 $\hat {X}_{ix} \leftarrow 0$ \\ +4. for $ix$ from $0$ to $a.used - 1$ do \\ +\hspace{3mm}Compute the square.\\ +\hspace{3mm}4.1 $\hat {X}_{ix+ix} \leftarrow \left ( a_{ix} \right )^2$ \\ +\\ +\hspace{3mm}Compute the double products.\\ +\hspace{3mm}4.2 for $iy$ from $ix + 1$ to $a.used - 1$ do \\ +\hspace{6mm}4.2.1 $\hat W_{ix+iy} \leftarrow \hat W_{ix+iy} + a_{ix}a_{iy}$ \\ +5. $oldused \leftarrow b.used$ \\ +6. $b.used \leftarrow 2a.used + 1$ \\ +\\ +Double the products and propagate the carries simultaneously. \\ +7. $\hat W_0 \leftarrow 2 \hat W_0 + \hat {X}_0$ \\ +8. for $ix$ from $1$ to $2a.used$ do \\ +\hspace{3mm}8.1 $\hat W_{ix} \leftarrow 2 \hat W_{ix} + \hat {X}_{ix}$ \\ +\hspace{3mm}8.2 $\hat W_{ix} \leftarrow \hat W_{ix} + \lfloor \hat W_{ix - 1} / \beta \rfloor$ \\ +\hspace{3mm}8.3 $b_{ix-1} \leftarrow W_{ix-1} \mbox{ (mod }\beta\mbox{)}$ \\ +9. $b_{2a.used} \leftarrow \hat W_{2a.used} \mbox{ (mod }\beta\mbox{)}$ \\ +10. if $2a.used + 1 < oldused$ then do \\ +\hspace{3mm}10.1 for $ix$ from $2a.used + 1$ to $oldused$ do \\ +\hspace{6mm}10.1.1 $b_{ix} \leftarrow 0$ \\ +11. Clamp excess digits from $b$. (\textit{mp\_clamp}) \\ +12. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm fast\_s\_mp\_sqr} +\end{figure} + +\textbf{Algorithm fast\_s\_mp\_sqr.} +This algorithm computes the square of an input using the Comba technique. It is designed to be a replacement for algorithm s\_mp\_sqr when +the number of input digits is less than \textbf{MP\_WARRAY} and less than $\delta \over 2$. + +This routine requires two arrays of mp\_words to be placed on the stack. The first array $\hat W$ will hold the double products and the second +array $\hat X$ will hold the squares. Though only at most $MP\_WARRAY \over 2$ words of $\hat X$ are used, it has proven faster on most +processors to simply make it a full size array. + +The loop on step 3 will zero the two arrays to prepare them for the squaring step. Step 4.1 computes the squares of the product. Note how +it simply assigns the value into the $\hat X$ array. The nested loop on step 4.2 computes the doubles of the products. This loop +computes the sum of the products for each column. They are not doubled until later. + +After the squaring loop, the products stored in $\hat W$ musted be doubled and the carries propagated forwards. It makes sense to do both +operations at the same time. The expression $\hat W_{ix} \leftarrow 2 \hat W_{ix} + \hat {X}_{ix}$ computes the sum of the double product and the +squares in place. + +EXAM,bn_fast_s_mp_sqr.c + +-- Write something deep and insightful later, Tom. + +\subsection{Polynomial Basis Squaring} +The same algorithm that performs optimal polynomial basis multiplication can be used to perform polynomial basis squaring. The minor exception +is that $\zeta_y = f(y)g(y)$ is actually equivalent to $\zeta_y = f(y)^2$ since $f(y) = g(y)$. Instead of performing $2n + 1$ +multiplications to find the $\zeta$ relations, squaring operations are performed instead. + +\subsection{Karatsuba Squaring} +Let $f(x) = ax + b$ represent the polynomial basis representation of a number to square. +Let $h(x) = \left ( f(x) \right )^2$ represent the square of the polynomial. The Karatsuba equation can be modified to square a +number with the following equation. + +\begin{equation} +h(x) = a^2x^2 + \left (a^2 + b^2 - (a - b)^2 \right )x + b^2 +\end{equation} + +Upon closer inspection this equation only requires the calculation of three half-sized squares: $a^2$, $b^2$ and $(a - b)^2$. As in +Karatsuba multiplication, this algorithm can be applied recursively on the input and will achieve an asymptotic running time of +$O \left ( n^{lg(3)} \right )$. + +If the asymptotic times of Karatsuba squaring and multiplication are the same, why not simply use the multiplication algorithm +instead? The answer to this arises from the cutoff point for squaring. As in multiplication there exists a cutoff point, at which the +time required for a Comba based squaring and a Karatsuba based squaring meet. Due to the overhead inherent in the Karatsuba method, the cutoff +point is fairly high. For example, on an AMD Athlon XP processor with $\beta = 2^{28}$, the cutoff point is around 127 digits. + +Consider squaring a 200 digit number with this technique. It will be split into two 100 digit halves which are subsequently squared. +The 100 digit halves will not be squared using Karatsuba, but instead using the faster Comba based squaring algorithm. If Karatsuba multiplication +were used instead, the 100 digit numbers would be squared with a slower Comba based multiplication. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_karatsuba\_sqr}. \\ +\textbf{Input}. mp\_int $a$ \\ +\textbf{Output}. $b \leftarrow a^2$ \\ +\hline \\ +1. Initialize the following temporary mp\_ints: $x0$, $x1$, $t1$, $t2$, $x0x0$ and $x1x1$. \\ +2. If any of the initializations on step 1 failed return(\textit{MP\_MEM}). \\ +\\ +Split the input. e.g. $a = x1\beta^B + x0$ \\ +3. $B \leftarrow \lfloor a.used / 2 \rfloor$ \\ +4. $x0 \leftarrow a \mbox{ (mod }\beta^B\mbox{)}$ (\textit{mp\_mod\_2d}) \\ +5. $x1 \leftarrow \lfloor a / \beta^B \rfloor$ (\textit{mp\_lshd}) \\ +\\ +Calculate the three squares. \\ +6. $x0x0 \leftarrow x0^2$ (\textit{mp\_sqr}) \\ +7. $x1x1 \leftarrow x1^2$ \\ +8. $t1 \leftarrow x1 - x0$ (\textit{mp\_sub}) \\ +9. $t1 \leftarrow t1^2$ \\ +\\ +Compute the middle term. \\ +10. $t2 \leftarrow x0x0 + x1x1$ (\textit{s\_mp\_add}) \\ +11. $t1 \leftarrow t2 - t1$ \\ +\\ +Compute final product. \\ +12. $t1 \leftarrow t1\beta^B$ (\textit{mp\_lshd}) \\ +13. $x1x1 \leftarrow x1x1\beta^{2B}$ \\ +14. $t1 \leftarrow t1 + x0x0$ \\ +15. $b \leftarrow t1 + x1x1$ \\ +16. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_karatsuba\_sqr} +\end{figure} + +\textbf{Algorithm mp\_karatsuba\_sqr.} +This algorithm computes the square of an input $a$ using the Karatsuba technique. This algorithm is very similar to the Karatsuba based +multiplication algorithm with the exception that the three half-size multiplications have been replaced with three half-size squarings. + +The radix point for squaring is simply placed exactly in the middle of the digits when the input has an odd number of digits, otherwise it is +placed just below the middle. Step 3, 4 and 5 compute the two halves required using $B$ +as the radix point. The first two squares in steps 6 and 7 are rather straightforward while the last square is of a more compact form. + +By expanding $\left (x1 - x0 \right )^2$, the $x1^2$ and $x0^2$ terms in the middle disappear, that is $x1^2 + x0^2 - (x1 - x0)^2 = 2 \cdot x0 \cdot x1$. +Now if $5n$ single precision additions and a squaring of $n$-digits is faster than multiplying two $n$-digit numbers and doubling then +this method is faster. Assuming no further recursions occur, the difference can be estimated with the following inequality. + +Let $p$ represent the cost of a single precision addition and $q$ the cost of a single precision multiplication both in terms of time\footnote{Or +machine clock cycles.}. + +\begin{equation} +5pn +{{q(n^2 + n)} \over 2} \le pn + qn^2 +\end{equation} + +For example, on an AMD Athlon XP processor $p = {1 \over 3}$ and $q = 6$. This implies that the following inequality should hold. +\begin{center} +\begin{tabular}{rcl} +${5n \over 3} + 3n^2 + 3n$ & $<$ & ${n \over 3} + 6n^2$ \\ +${5 \over 3} + 3n + 3$ & $<$ & ${1 \over 3} + 6n$ \\ +${13 \over 9}$ & $<$ & $n$ \\ +\end{tabular} +\end{center} + +This results in a cutoff point around $n = 2$. As a consequence it is actually faster to compute the middle term the ``long way'' on processors +where multiplication is substantially slower\footnote{On the Athlon there is a 1:17 ratio between clock cycles for addition and multiplication. On +the Intel P4 processor this ratio is 1:29 making this method even more beneficial. The only common exception is the ARMv4 processor which has a +ratio of 1:7. } than simpler operations such as addition. + +EXAM,bn_mp_karatsuba_sqr.c + +This implementation is largely based on the implementation of algorithm mp\_karatsuba\_mul. It uses the same inline style to copy and +shift the input into the two halves. The loop from line @54,{@ to line @70,}@ has been modified since only one input exists. The \textbf{used} +count of both $x0$ and $x1$ is fixed up and $x0$ is clamped before the calculations begin. At this point $x1$ and $x0$ are valid equivalents +to the respective halves as if mp\_rshd and mp\_mod\_2d had been used. + +By inlining the copy and shift operations the cutoff point for Karatsuba multiplication can be lowered. On the Athlon the cutoff point +is exactly at the point where Comba squaring can no longer be used (\textit{128 digits}). On slower processors such as the Intel P4 +it is actually below the Comba limit (\textit{at 110 digits}). + +This routine uses the same error trap coding style as mp\_karatsuba\_sqr. As the temporary variables are initialized errors are redirected to +the error trap higher up. If the algorithm completes without error the error code is set to \textbf{MP\_OKAY} and mp\_clears are executed normally. + +\textit{Last paragraph sucks. re-write! -- Tom} + +\subsection{Toom-Cook Squaring} +The Toom-Cook squaring algorithm mp\_toom\_sqr is heavily based on the algorithm mp\_toom\_mul with the exception that squarings are used +instead of multiplication to find the five relations.. The reader is encouraged to read the description of the latter algorithm and try to +derive their own Toom-Cook squaring algorithm. + +\subsection{High Level Squaring} +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_sqr}. \\ +\textbf{Input}. mp\_int $a$ \\ +\textbf{Output}. $b \leftarrow a^2$ \\ +\hline \\ +1. If $a.used \ge TOOM\_SQR\_CUTOFF$ then \\ +\hspace{3mm}1.1 $b \leftarrow a^2$ using algorithm mp\_toom\_sqr \\ +2. else if $a.used \ge KARATSUBA\_SQR\_CUTOFF$ then \\ +\hspace{3mm}2.1 $b \leftarrow a^2$ using algorithm mp\_karatsuba\_sqr \\ +3. else \\ +\hspace{3mm}3.1 $digs \leftarrow a.used + b.used + 1$ \\ +\hspace{3mm}3.2 If $digs < MP\_ARRAY$ and $a.used \le \delta$ then \\ +\hspace{6mm}3.2.1 $b \leftarrow a^2$ using algorithm fast\_s\_mp\_sqr. \\ +\hspace{3mm}3.3 else \\ +\hspace{6mm}3.3.1 $b \leftarrow a^2$ using algorithm s\_mp\_sqr. \\ +4. $b.sign \leftarrow MP\_ZPOS$ \\ +5. Return the result of the unsigned squaring performed. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_sqr} +\end{figure} + +\textbf{Algorithm mp\_sqr.} +This algorithm computes the square of the input using one of four different algorithms. If the input is very large and has at least +\textbf{TOOM\_SQR\_CUTOFF} or \textbf{KARATSUBA\_SQR\_CUTOFF} digits then either the Toom-Cook or the Karatsuba Squaring algorithm is used. If +neither of the polynomial basis algorithms should be used then either the Comba or baseline algorithm is used. + +EXAM,bn_mp_sqr.c + +\section*{Exercises} +\begin{tabular}{cl} +$\left [ 3 \right ] $ & Devise an efficient algorithm for selection of the radix point to handle inputs \\ + & that have different number of digits in Karatsuba multiplication. \\ + & \\ +$\left [ 3 \right ] $ & In ~SQUARE~ the fact that every column of a squaring is made up \\ + & of double products and at most one square is stated. Prove this statement. \\ + & \\ +$\left [ 2 \right ] $ & In the Comba squaring algorithm half of the $\hat X$ variables are not used. \\ + & Revise algorithm fast\_s\_mp\_sqr to shrink the $\hat X$ array. \\ + & \\ +$\left [ 3 \right ] $ & Prove the equation for Karatsuba squaring. \\ + & \\ +$\left [ 1 \right ] $ & Prove that Karatsuba squaring requires $O \left (n^{lg(3)} \right )$ time. \\ + & \\ +$\left [ 2 \right ] $ & Determine the minimal ratio between addition and multiplication clock cycles \\ + & required for equation $6.7$ to be true. \\ + & \\ +\end{tabular} + +\chapter{Modular Reduction} +MARK,REDUCTION +\section{Basics of Modular Reduction} +\index{modular residue} +Modular reduction is an operation that arises quite often within public key cryptography algorithms and various number theoretic algorithms, +such as factoring. Modular reduction algorithms are the third class of algorithms of the ``multipliers'' set. A number $a$ is said to be \textit{reduced} +modulo another number $b$ by finding the remainder of the division $a/b$. Full integer division with remainder is a topic to be covered +in~\ref{sec:division}. + +Modular reduction is equivalent to solving for $r$ in the following equation. $a = bq + r$ where $q = \lfloor a/b \rfloor$. The result +$r$ is said to be ``congruent to $a$ modulo $b$'' which is also written as $r \equiv a \mbox{ (mod }b\mbox{)}$. In other vernacular $r$ is known as the +``modular residue'' which leads to ``quadratic residue''\footnote{That's fancy talk for $b \equiv a^2 \mbox{ (mod }p\mbox{)}$.} and +other forms of residues. + +Modular reductions are normally used to create either finite groups, rings or fields. The most common usage for performance driven modular reductions +is in modular exponentiation algorithms. That is to compute $d = a^b \mbox{ (mod }c\mbox{)}$ as fast as possible. This operation is used in the +RSA and Diffie-Hellman public key algorithms, for example. Modular multiplication and squaring also appears as a fundamental operation in +Elliptic Curve cryptographic algorithms. As will be discussed in the subsequent chapter there exist fast algorithms for computing modular +exponentiations without having to perform (\textit{in this example}) $b - 1$ multiplications. These algorithms will produce partial results in the +range $0 \le x < c^2$ which can be taken advantage of to create several efficient algorithms. They have also been used to create redundancy check +algorithms known as CRCs, error correction codes such as Reed-Solomon and solve a variety of number theoeretic problems. + +\section{The Barrett Reduction} +The Barrett reduction algorithm \cite{BARRETT} was inspired by fast division algorithms which multiply by the reciprocal to emulate +division. Barretts observation was that the residue $c$ of $a$ modulo $b$ is equal to + +\begin{equation} +c = a - b \cdot \lfloor a/b \rfloor +\end{equation} + +Since algorithms such as modular exponentiation would be using the same modulus extensively, typical DSP\footnote{It is worth noting that Barrett's paper +targeted the DSP56K processor.} intuition would indicate the next step would be to replace $a/b$ by a multiplication by the reciprocal. However, +DSP intuition on its own will not work as these numbers are considerably larger than the precision of common DSP floating point data types. +It would take another common optimization to optimize the algorithm. + +\subsection{Fixed Point Arithmetic} +The trick used to optimize the above equation is based on a technique of emulating floating point data types with fixed precision integers. Fixed +point arithmetic would become very popular as it greatly optimize the ``3d-shooter'' genre of games in the mid 1990s when floating point units were +fairly slow if not unavailable. The idea behind fixed point arithmetic is to take a normal $k$-bit integer data type and break it into $p$-bit +integer and a $q$-bit fraction part (\textit{where $p+q = k$}). + +In this system a $k$-bit integer $n$ would actually represent $n/2^q$. For example, with $q = 4$ the integer $n = 37$ would actually represent the +value $2.3125$. To multiply two fixed point numbers the integers are multiplied using traditional arithmetic and subsequently normalized by +moving the implied decimal point back to where it should be. For example, with $q = 4$ to multiply the integers $9$ and $5$ they must be converted +to fixed point first by multiplying by $2^q$. Let $a = 9(2^q)$ represent the fixed point representation of $9$ and $b = 5(2^q)$ represent the +fixed point representation of $5$. The product $ab$ is equal to $45(2^{2q})$ which when normalized by dividing by $2^q$ produces $45(2^q)$. + +This technique became popular since a normal integer multiplication and logical shift right are the only required operations to perform a multiplication +of two fixed point numbers. Using fixed point arithmetic, division can be easily approximated by multiplying by the reciprocal. If $2^q$ is +equivalent to one than $2^q/b$ is equivalent to the fixed point approximation of $1/b$ using real arithmetic. Using this fact dividing an integer +$a$ by another integer $b$ can be achieved with the following expression. + +\begin{equation} +\lfloor a / b \rfloor \mbox{ }\approx\mbox{ } \lfloor (a \cdot \lfloor 2^q / b \rfloor)/2^q \rfloor +\end{equation} + +The precision of the division is proportional to the value of $q$. If the divisor $b$ is used frequently as is the case with +modular exponentiation pre-computing $2^q/b$ will allow a division to be performed with a multiplication and a right shift. Both operations +are considerably faster than division on most processors. + +Consider dividing $19$ by $5$. The correct result is $\lfloor 19/5 \rfloor = 3$. With $q = 3$ the reciprocal is $\lfloor 2^q/5 \rfloor = 1$ which +leads to a product of $19$ which when divided by $2^q$ produces $2$. However, with $q = 4$ the reciprocal is $\lfloor 2^q/5 \rfloor = 3$ and +the result of the emulated division is $\lfloor 3 \cdot 19 / 2^q \rfloor = 3$ which is correct. The value of $2^q$ must be close to or ideally +larger than the dividend. In effect if $a$ is the dividend then $q$ should allow $0 \le \lfloor a/2^q \rfloor \le 1$ in order for this approach +to work correctly. Plugging this form of divison into the original equation the following modular residue equation arises. + +\begin{equation} +c = a - b \cdot \lfloor (a \cdot \lfloor 2^q / b \rfloor)/2^q \rfloor +\end{equation} + +Using the notation from \cite{BARRETT} the value of $\lfloor 2^q / b \rfloor$ will be represented by the $\mu$ symbol. Using the $\mu$ +variable also helps re-inforce the idea that it is meant to be computed once and re-used. + +\begin{equation} +c = a - b \cdot \lfloor (a \cdot \mu)/2^q \rfloor +\end{equation} + +Provided that $2^q \ge a$ this algorithm will produce a quotient that is either exactly correct or off by a value of one. In the context of Barrett +reduction the value of $a$ is bound by $0 \le a \le (b - 1)^2$ meaning that $2^q \ge b^2$ is sufficient to ensure the reciprocal will have enough +precision. + +Let $n$ represent the number of digits in $b$. This algorithm requires approximately $2n^2$ single precision multiplications to produce the quotient and +another $n^2$ single precision multiplications to find the residue. In total $3n^2$ single precision multiplications are required to +reduce the number. + +For example, if $b = 1179677$ and $q = 41$ ($2^q > b^2$), then the reciprocal $\mu$ is equal to $\lfloor 2^q / b \rfloor = 1864089$. Consider reducing +$a = 180388626447$ modulo $b$ using the above reduction equation. The quotient using the new formula is $\lfloor (a \cdot \mu) / 2^q \rfloor = 152913$. +By subtracting $152913b$ from $a$ the correct residue $a \equiv 677346 \mbox{ (mod }b\mbox{)}$ is found. + +\subsection{Choosing a Radix Point} +Using the fixed point representation a modular reduction can be performed with $3n^2$ single precision multiplications. If that were the best +that could be achieved a full division\footnote{A division requires approximately $O(2cn^2)$ single precision multiplications for a small value of $c$. +See~\ref{sec:division} for further details.} might as well be used in its place. The key to optimizing the reduction is to reduce the precision of +the initial multiplication that finds the quotient. + +Let $a$ represent the number of which the residue is sought. Let $b$ represent the modulus used to find the residue. Let $m$ represent +the number of digits in $b$. For the purposes of this discussion we will assume that the number of digits in $a$ is $2m$, which is generally true if +two $m$-digit numbers have been multiplied. Dividing $a$ by $b$ is the same as dividing a $2m$ digit integer by a $m$ digit integer. Digits below the +$m - 1$'th digit of $a$ will contribute at most a value of $1$ to the quotient because $\beta^k < b$ for any $0 \le k \le m - 1$. Another way to +express this is by re-writing $a$ as two parts. If $a' \equiv a \mbox{ (mod }b^m\mbox{)}$ and $a'' = a - a'$ then +${a \over b} \equiv {{a' + a''} \over b}$ which is equivalent to ${a' \over b} + {a'' \over b}$. Since $a'$ is bound to be less than $b$ the quotient +is bound by $0 \le {a' \over b} < 1$. + +Since the digits of $a'$ do not contribute much to the quotient the observation is that they might as well be zero. However, if the digits +``might as well be zero'' they might as well not be there in the first place. Let $q_0 = \lfloor a/\beta^{m-1} \rfloor$ represent the input +with the irrelevant digits trimmed. Now the modular reduction is trimmed to the almost equivalent equation + +\begin{equation} +c = a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor +\end{equation} + +Note that the original divisor $2^q$ has been replaced with $\beta^{m+1}$ where in this case $q$ is a multiple of $lg(\beta)$. Also note that the +exponent on the divisor when added to the amount $q_0$ was shifted by equals $2m$. If the optimization had not been performed the divisor +would have the exponent $2m$ so in the end the exponents do ``add up''. Using the above equation the quotient +$\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ can be off from the true quotient by at most two. The original fixed point quotient can be off +by as much as one (\textit{provided the radix point is chosen suitably}) and now that the lower irrelevent digits have been trimmed the quotient +can be off by an additional value of one for a total of at most two. This implies that +$0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$. By first subtracting $b$ times the quotient and then conditionally subtracting +$b$ once or twice the residue is found. + +The quotient is now found using $(m + 1)(m) = m^2 + m$ single precision multiplications and the residue with an additional $m^2$ single +precision multiplications, ignoring the subtractions required. In total $2m^2 + m$ single precision multiplications are required to find the residue. +This is considerably faster than the original attempt. + +For example, let $\beta = 10$ represent the radix of the digits. Let $b = 9999$ represent the modulus which implies $m = 4$. Let $a = 99929878$ +represent the value of which the residue is desired. In this case $q = 8$ since $10^7 < 9999^2$ meaning that $\mu = \lfloor \beta^{q}/b \rfloor = 10001$. +With the new observation the multiplicand for the quotient is equal to $q_0 = \lfloor a / \beta^{m - 1} \rfloor = 99929$. The quotient is then +$\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor = 9993$. Subtracting $9993b$ from $a$ and the correct residue $a \equiv 9871 \mbox{ (mod }b\mbox{)}$ +is found. + +\subsection{Trimming the Quotient} +So far the reduction algorithm has been optimized from $3m^2$ single precision multiplications down to $2m^2 + m$ single precision multiplications. As +it stands now the algorithm is already fairly fast compared to a full integer division algorithm. However, there is still room for +optimization. + +After the first multiplication inside the quotient ($q_0 \cdot \mu$) the value is shifted right by $m + 1$ places effectively nullifying the lower +half of the product. It would be nice to be able to remove those digits from the product to effectively cut down the number of single precision +multiplications. If the number of digits in the modulus $m$ is far less than $\beta$ a full product is not required for the algorithm to work properly. +In fact the lower $m - 2$ digits will not affect the upper half of the product at all and do not need to be computed. + +The value of $\mu$ is a $m$-digit number and $q_0$ is a $m + 1$ digit number. Using a full multiplier $(m + 1)(m) = m^2 + m$ single precision +multiplications would be required. Using a multiplier that will only produce digits at and above the $m - 1$'th digit reduces the number +of single precision multiplications to ${m^2 + m} \over 2$ single precision multiplications. + +\subsection{Trimming the Residue} +After the quotient has been calculated it is used to reduce the input. As previously noted the algorithm is not exact and it can be off by a small +multiple of the modulus, that is $0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$. If $b$ is $m$ digits than the +result of reduction equation is a value of at most $m + 1$ digits (\textit{provided $3 < \beta$}) implying that the upper $m - 1$ digits are +implicitly zero. + +The next optimization arises from this very fact. Instead of computing $b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ using a full +$O(m^2)$ multiplication algorithm only the lower $m+1$ digits of the product have to be computed. Similarly the value of $a$ can +be reduced modulo $\beta^{m+1}$ before the multiple of $b$ is subtracted which simplifes the subtraction as well. A multiplication that produces +only the lower $m+1$ digits requires ${m^2 + 3m - 2} \over 2$ single precision multiplications. + +With both optimizations in place the algorithm is the algorithm Barrett proposed. It requires $m^2 + 2m - 1$ single precision multiplications which +is considerably faster than the straightforward $3m^2$ method. + +\subsection{The Barrett Algorithm} +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_reduce}. \\ +\textbf{Input}. mp\_int $a$, mp\_int $b$ and $\mu = \lfloor \beta^{2m}/b \rfloor, m = \lceil lg_{\beta}(b) \rceil, (0 \le a < b^2, b > 1)$ \\ +\textbf{Output}. $a \mbox{ (mod }b\mbox{)}$ \\ +\hline \\ +Let $m$ represent the number of digits in $b$. \\ +1. Make a copy of $a$ and store it in $q$. (\textit{mp\_init\_copy}) \\ +2. $q \leftarrow \lfloor q / \beta^{m - 1} \rfloor$ (\textit{mp\_rshd}) \\ +\\ +Produce the quotient. \\ +3. $q \leftarrow q \cdot \mu$ (\textit{note: only produce digits at or above $m-1$}) \\ +4. $q \leftarrow \lfloor q / \beta^{m + 1} \rfloor$ \\ +\\ +Subtract the multiple of modulus from the input. \\ +5. $a \leftarrow a \mbox{ (mod }\beta^{m+1}\mbox{)}$ (\textit{mp\_mod\_2d}) \\ +6. $q \leftarrow q \cdot b \mbox{ (mod }\beta^{m+1}\mbox{)}$ (\textit{s\_mp\_mul\_digs}) \\ +7. $a \leftarrow a - q$ (\textit{mp\_sub}) \\ +\\ +Add $\beta^{m+1}$ if a carry occured. \\ +8. If $a < 0$ then (\textit{mp\_cmp\_d}) \\ +\hspace{3mm}8.1 $q \leftarrow 1$ (\textit{mp\_set}) \\ +\hspace{3mm}8.2 $q \leftarrow q \cdot \beta^{m+1}$ (\textit{mp\_lshd}) \\ +\hspace{3mm}8.3 $a \leftarrow a + q$ \\ +\\ +Now subtract the modulus if the residue is too large (e.g. quotient too small). \\ +9. While $a \ge b$ do (\textit{mp\_cmp}) \\ +\hspace{3mm}9.1 $c \leftarrow a - b$ \\ +10. Clear $q$. \\ +11. Return(\textit{MP\_OKAY}) \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_reduce} +\end{figure} + +\textbf{Algorithm mp\_reduce.} +This algorithm will reduce the input $a$ modulo $b$ in place using the Barrett algorithm. It is loosely based on algorithm 14.42 of HAC +\cite[pp. 602]{HAC} which is based on the paper from Paul Barrett \cite{BARRETT}. The algorithm has several restrictions and assumptions which must +be adhered to for the algorithm to work. + +First the modulus $b$ is assumed to be positive and greater than one. If the modulus were less than or equal to one than subtracting +a multiple of it would either accomplish nothing or actually enlarge the input. The input $a$ must be in the range $0 \le a < b^2$ in order +for the quotient to have enough precision. If $a$ is the product of two numbers that were already reduced modulo $b$, this will not be a problem. +Technically the algorithm will still work if $a \ge b^2$ but it will take much longer to finish. The value of $\mu$ is passed as an argument to this +algorithm and is assumed to be calculated and stored before the algorithm is used. + +Recall that the multiplication for the quotient on step 3 must only produce digits at or above the $m-1$'th position. An algorithm called +$s\_mp\_mul\_high\_digs$ which has not been presented is used to accomplish this task. The algorithm is based on $s\_mp\_mul\_digs$ except that +instead of stopping at a given level of precision it starts at a given level of precision. This optimal algorithm can only be used if the number +of digits in $b$ is very much smaller than $\beta$. + +While it is known that +$a \ge b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ only the lower $m+1$ digits are being used to compute the residue, so an implied +``borrow'' from the higher digits might leave a negative result. After the multiple of the modulus has been subtracted from $a$ the residue must be +fixed up in case it is negative. The invariant $\beta^{m+1}$ must be added to the residue to make it positive again. + +The while loop at step 9 will subtract $b$ until the residue is less than $b$. If the algorithm is performed correctly this step is +performed at most twice, and on average once. However, if $a \ge b^2$ than it will iterate substantially more times than it should. + +EXAM,bn_mp_reduce.c + +The first multiplication that determines the quotient can be performed by only producing the digits from $m - 1$ and up. This essentially halves +the number of single precision multiplications required. However, the optimization is only safe if $\beta$ is much larger than the number of digits +in the modulus. In the source code this is evaluated on lines @36,if@ to @44,}@ where algorithm s\_mp\_mul\_high\_digs is used when it is +safe to do so. + +\subsection{The Barrett Setup Algorithm} +In order to use algorithm mp\_reduce the value of $\mu$ must be calculated in advance. Ideally this value should be computed once and stored for +future use so that the Barrett algorithm can be used without delay. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_reduce\_setup}. \\ +\textbf{Input}. mp\_int $a$ ($a > 1$) \\ +\textbf{Output}. $\mu \leftarrow \lfloor \beta^{2m}/a \rfloor$ \\ +\hline \\ +1. $\mu \leftarrow 2^{2 \cdot lg(\beta) \cdot m}$ (\textit{mp\_2expt}) \\ +2. $\mu \leftarrow \lfloor \mu / b \rfloor$ (\textit{mp\_div}) \\ +3. Return(\textit{MP\_OKAY}) \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_reduce\_setup} +\end{figure} + +\textbf{Algorithm mp\_reduce\_setup.} +This algorithm computes the reciprocal $\mu$ required for Barrett reduction. First $\beta^{2m}$ is calculated as $2^{2 \cdot lg(\beta) \cdot m}$ which +is equivalent and much faster. The final value is computed by taking the integer quotient of $\lfloor \mu / b \rfloor$. + +EXAM,bn_mp_reduce_setup.c + +This simple routine calculates the reciprocal $\mu$ required by Barrett reduction. Note the extended usage of algorithm mp\_div where the variable +which would received the remainder is passed as NULL. As will be discussed in~\ref{sec:division} the division routine allows both the quotient and the +remainder to be passed as NULL meaning to ignore the value. + +\section{The Montgomery Reduction} +Montgomery reduction\footnote{Thanks to Niels Ferguson for his insightful explanation of the algorithm.} \cite{MONT} is by far the most interesting +form of reduction in common use. It computes a modular residue which is not actually equal to the residue of the input yet instead equal to a +residue times a constant. However, as perplexing as this may sound the algorithm is relatively simple and very efficient. + +Throughout this entire section the variable $n$ will represent the modulus used to form the residue. As will be discussed shortly the value of +$n$ must be odd. The variable $x$ will represent the quantity of which the residue is sought. Similar to the Barrett algorithm the input +is restricted to $0 \le x < n^2$. To begin the description some simple number theory facts must be established. + +\textbf{Fact 1.} Adding $n$ to $x$ does not change the residue since in effect it adds one to the quotient $\lfloor x / n \rfloor$. Another way +to explain this is that $n$ is (\textit{or multiples of $n$ are}) congruent to zero modulo $n$. Adding zero will not change the value of the residue. + +\textbf{Fact 2.} If $x$ is even then performing a division by two in $\Z$ is congruent to $x \cdot 2^{-1} \mbox{ (mod }n\mbox{)}$. Actually +this is an application of the fact that if $x$ is evenly divisible by any $k \in \Z$ then division in $\Z$ will be congruent to +multiplication by $k^{-1}$ modulo $n$. + +From these two simple facts the following simple algorithm can be derived. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{Montgomery Reduction}. \\ +\textbf{Input}. Integer $x$, $n$ and $k$ \\ +\textbf{Output}. $2^{-k}x \mbox{ (mod }n\mbox{)}$ \\ +\hline \\ +1. for $t$ from $1$ to $k$ do \\ +\hspace{3mm}1.1 If $x$ is odd then \\ +\hspace{6mm}1.1.1 $x \leftarrow x + n$ \\ +\hspace{3mm}1.2 $x \leftarrow x/2$ \\ +2. Return $x$. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm Montgomery Reduction} +\end{figure} + +The algorithm reduces the input one bit at a time using the two congruencies stated previously. Inside the loop $n$, which is odd, is +added to $x$ if $x$ is odd. This forces $x$ to be even which allows the division by two in $\Z$ to be congruent to a modular division by two. Since +$x$ is assumed to be initially much larger than $n$ the addition of $n$ will contribute an insignificant magnitude to $x$. Let $r$ represent the +final result of the Montgomery algorithm. If $k > lg(n)$ and $0 \le x < n^2$ then the final result is limited to +$0 \le r < \lfloor x/2^k \rfloor + n$. As a result at most a single subtraction is required to get the residue desired. + +\begin{figure}[here] +\begin{small} +\begin{center} +\begin{tabular}{|c|l|} +\hline \textbf{Step number ($t$)} & \textbf{Result ($x$)} \\ +\hline $1$ & $x + n = 5812$, $x/2 = 2906$ \\ +\hline $2$ & $x/2 = 1453$ \\ +\hline $3$ & $x + n = 1710$, $x/2 = 855$ \\ +\hline $4$ & $x + n = 1112$, $x/2 = 556$ \\ +\hline $5$ & $x/2 = 278$ \\ +\hline $6$ & $x/2 = 139$ \\ +\hline $7$ & $x + n = 396$, $x/2 = 198$ \\ +\hline $8$ & $x/2 = 99$ \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Example of Montgomery Reduction (I)} +\label{fig:MONT1} +\end{figure} + +Consider the example in figure~\ref{fig:MONT1} which reduces $x = 5555$ modulo $n = 257$ when $k = 8$. The result of the algorithm $r = 99$ is +congruent to the value of $2^{-8} \cdot 5555 \mbox{ (mod }257\mbox{)}$. When $r$ is multiplied by $2^8$ modulo $257$ the correct residue +$r \equiv 158$ is produced. + +Let $k = \lfloor lg(n) \rfloor + 1$ represent the number of bits in $n$. The current algorithm requires $2k^2$ single precision shifts +and $k^2$ single precision additions. At this rate the algorithm is most certainly slower than Barrett reduction and not terribly useful. +Fortunately there exists an alternative representation of the algorithm. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{Montgomery Reduction} (modified I). \\ +\textbf{Input}. Integer $x$, $n$ and $k$ \\ +\textbf{Output}. $2^{-k}x \mbox{ (mod }n\mbox{)}$ \\ +\hline \\ +1. for $t$ from $0$ to $k - 1$ do \\ +\hspace{3mm}1.1 If the $t$'th bit of $x$ is one then \\ +\hspace{6mm}1.1.1 $x \leftarrow x + 2^tn$ \\ +2. Return $x/2^k$. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm Montgomery Reduction (modified I)} +\end{figure} + +This algorithm is equivalent since $2^tn$ is a multiple of $n$ and the lower $k$ bits of $x$ are zero by step 2. The number of single +precision shifts has now been reduced from $2k^2$ to $k^2 + k$ which is only a small improvement. + +\begin{figure}[here] +\begin{small} +\begin{center} +\begin{tabular}{|c|l|r|} +\hline \textbf{Step number ($t$)} & \textbf{Result ($x$)} & \textbf{Result ($x$) in Binary} \\ +\hline -- & $5555$ & $1010110110011$ \\ +\hline $1$ & $x + 2^{0}n = 5812$ & $1011010110100$ \\ +\hline $2$ & $5812$ & $1011010110100$ \\ +\hline $3$ & $x + 2^{2}n = 6840$ & $1101010111000$ \\ +\hline $4$ & $x + 2^{3}n = 8896$ & $10001011000000$ \\ +\hline $5$ & $8896$ & $10001011000000$ \\ +\hline $6$ & $8896$ & $10001011000000$ \\ +\hline $7$ & $x + 2^{6}n = 25344$ & $110001100000000$ \\ +\hline $8$ & $25344$ & $110001100000000$ \\ +\hline -- & $x/2^k = 99$ & \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Example of Montgomery Reduction (II)} +\label{fig:MONT2} +\end{figure} + +Figure~\ref{fig:MONT2} demonstrates the modified algorithm reducing $x = 5555$ modulo $n = 257$ with $k = 8$. +With this algorithm a single shift right at the end is the only right shift required to reduce the input instead of $k$ right shifts inside the +loop. Note that for the iterations $t = 2, 5, 6$ and $8$ where the result $x$ is not changed. In those iterations the $t$'th bit of $x$ is +zero and the appropriate multiple of $n$ does not need to be added to force the $t$'th bit of the result to zero. + +\subsection{Digit Based Montgomery Reduction} +Instead of computing the reduction on a bit-by-bit basis it is actually much faster to compute it on digit-by-digit basis. Consider the +previous algorithm re-written to compute the Montgomery reduction in this new fashion. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{Montgomery Reduction} (modified II). \\ +\textbf{Input}. Integer $x$, $n$ and $k$ \\ +\textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\ +\hline \\ +1. for $t$ from $0$ to $k - 1$ do \\ +\hspace{3mm}1.1 $x \leftarrow x + \mu n \beta^t$ \\ +2. Return $x/\beta^k$. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm Montgomery Reduction (modified II)} +\end{figure} + +The value $\mu n \beta^t$ is a multiple of the modulus $n$ meaning that it will not change the residue. If the first digit of +the value $\mu n \beta^t$ equals the negative (modulo $\beta$) of the $t$'th digit of $x$ then the addition will result in a zero digit. This +problem breaks down to solving the following congruency. + +\begin{center} +\begin{tabular}{rcl} +$x_t + \mu n_0$ & $\equiv$ & $0 \mbox{ (mod }\beta\mbox{)}$ \\ +$\mu n_0$ & $\equiv$ & $-x_t \mbox{ (mod }\beta\mbox{)}$ \\ +$\mu$ & $\equiv$ & $-x_t/n_0 \mbox{ (mod }\beta\mbox{)}$ \\ +\end{tabular} +\end{center} + +In each iteration of the loop on step 1 a new value of $\mu$ must be calculated. The value of $-1/n_0 \mbox{ (mod }\beta\mbox{)}$ is used +extensively in this algorithm and should be precomputed. Let $\rho$ represent the negative of the modular inverse of $n_0$ modulo $\beta$. + +For example, let $\beta = 10$ represent the radix. Let $n = 17$ represent the modulus which implies $k = 2$ and $\rho \equiv 7$. Let $x = 33$ +represent the value to reduce. + +\newpage\begin{figure} +\begin{center} +\begin{tabular}{|c|c|c|} +\hline \textbf{Step ($t$)} & \textbf{Value of $x$} & \textbf{Value of $\mu$} \\ +\hline -- & $33$ & --\\ +\hline $0$ & $33 + \mu n = 50$ & $1$ \\ +\hline $1$ & $50 + \mu n \beta = 900$ & $5$ \\ +\hline +\end{tabular} +\end{center} +\caption{Example of Montgomery Reduction} +\end{figure} + +The final result $900$ is then divided by $\beta^k$ to produce the final result $9$. The first observation is that $9 \nequiv x \mbox{ (mod }n\mbox{)}$ +which implies the result is not the modular residue of $x$ modulo $n$. However, recall that the residue is actually multiplied by $\beta^{-k}$ in +the algorithm. To get the true residue the value must be multiplied by $\beta^k$. In this case $\beta^k \equiv 15 \mbox{ (mod }n\mbox{)}$ and +the correct residue is $9 \cdot 15 \equiv 16 \mbox{ (mod }n\mbox{)}$. + +\subsection{Baseline Montgomery Reduction} +The baseline Montgomery reduction algorithm will produce the residue for any size input. It is designed to be a catch-all algororithm for +Montgomery reductions. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_montgomery\_reduce}. \\ +\textbf{Input}. mp\_int $x$, mp\_int $n$ and a digit $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\ +\hspace{11.5mm}($0 \le x < n^2, n > 1, (n, \beta) = 1, \beta^k > n$) \\ +\textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\ +\hline \\ +1. $digs \leftarrow 2n.used + 1$ \\ +2. If $digs < MP\_ARRAY$ and $m.used < \delta$ then \\ +\hspace{3mm}2.1 Use algorithm fast\_mp\_montgomery\_reduce instead. \\ +\\ +Setup $x$ for the reduction. \\ +3. If $x.alloc < digs$ then grow $x$ to $digs$ digits. \\ +4. $x.used \leftarrow digs$ \\ +\\ +Eliminate the lower $k$ digits. \\ +5. For $ix$ from $0$ to $k - 1$ do \\ +\hspace{3mm}5.1 $\mu \leftarrow x_{ix} \cdot \rho \mbox{ (mod }\beta\mbox{)}$ \\ +\hspace{3mm}5.2 $u \leftarrow 0$ \\ +\hspace{3mm}5.3 For $iy$ from $0$ to $k - 1$ do \\ +\hspace{6mm}5.3.1 $\hat r \leftarrow \mu n_{iy} + x_{ix + iy} + u$ \\ +\hspace{6mm}5.3.2 $x_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ +\hspace{6mm}5.3.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ +\hspace{3mm}5.4 While $u > 0$ do \\ +\hspace{6mm}5.4.1 $iy \leftarrow iy + 1$ \\ +\hspace{6mm}5.4.2 $x_{ix + iy} \leftarrow x_{ix + iy} + u$ \\ +\hspace{6mm}5.4.3 $u \leftarrow \lfloor x_{ix+iy} / \beta \rfloor$ \\ +\hspace{6mm}5.4.4 $x_{ix + iy} \leftarrow x_{ix+iy} \mbox{ (mod }\beta\mbox{)}$ \\ +\\ +Divide by $\beta^k$ and fix up as required. \\ +6. $x \leftarrow \lfloor x / \beta^k \rfloor$ \\ +7. If $x \ge n$ then \\ +\hspace{3mm}7.1 $x \leftarrow x - n$ \\ +8. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_montgomery\_reduce} +\end{figure} + +\textbf{Algorithm mp\_montgomery\_reduce.} +This algorithm reduces the input $x$ modulo $n$ in place using the Montgomery reduction algorithm. The algorithm is loosely based +on algorithm 14.32 of \cite[pp.601]{HAC} except it merges the multiplication of $\mu n \beta^t$ with the addition in the inner loop. The +restrictions on this algorithm are fairly easy to adapt to. First $0 \le x < n^2$ bounds the input to numbers in the same range as +for the Barrett algorithm. Additionally if $n > 1$ and $n$ is odd there will exist a modular inverse $\rho$. $\rho$ must be calculated in +advance of this algorithm. Finally the variable $k$ is fixed and a pseudonym for $n.used$. + +Step 2 decides whether a faster Montgomery algorithm can be used. It is based on the Comba technique meaning that there are limits on +the size of the input. This algorithm is discussed in ~COMBARED~. + +Step 5 is the main reduction loop of the algorithm. The value of $\mu$ is calculated once per iteration in the outer loop. The inner loop +calculates $x + \mu n \beta^{ix}$ by multiplying $\mu n$ and adding the result to $x$ shifted by $ix$ digits. Both the addition and +multiplication are performed in the same loop to save time and memory. Step 5.4 will handle any additional carries that escape the inner loop. + +Using a quick inspection this algorithm requires $n$ single precision multiplications for the outer loop and $n^2$ single precision multiplications +in the inner loop. In total $n^2 + n$ single precision multiplications which compares favourably to Barrett at $n^2 + 2n - 1$ single precision +multiplications. + +EXAM,bn_mp_montgomery_reduce.c + +This is the baseline implementation of the Montgomery reduction algorithm. Lines @30,digs@ to @35,}@ determine if the Comba based +routine can be used instead. Line @47,mu@ computes the value of $\mu$ for that particular iteration of the outer loop. + +The multiplication $\mu n \beta^{ix}$ is performed in one step in the inner loop. The alias $tmpx$ refers to the $ix$'th digit of $x$ and +the alias $tmpn$ refers to the modulus $n$. + +\subsection{Faster ``Comba'' Montgomery Reduction} +MARK,COMBARED + +The Montgomery reduction requires fewer single precision multiplications than a Barrett reduction, however it is much slower due to the serial +nature of the inner loop. The Barrett reduction algorithm requires two slightly modified multipliers which can be implemented with the Comba +technique. The Montgomery reduction algorithm cannot directly use the Comba technique to any significant advantage since the inner loop calculates +a $k \times 1$ product $k$ times. + +The biggest obstacle is that at the $ix$'th iteration of the outer loop the value of $x_{ix}$ is required to calculate $\mu$. This means the +carries from $0$ to $ix - 1$ must have been propagated upwards to form a valid $ix$'th digit. The solution as it turns out is very simple. +Perform a Comba like multiplier and inside the outer loop just after the inner loop fix up the $ix + 1$'th digit by forwarding the carry. + +With this change in place the Montgomery reduction algorithm can be performed with a Comba style multiplication loop which substantially increases +the speed of the algorithm. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{fast\_mp\_montgomery\_reduce}. \\ +\textbf{Input}. mp\_int $x$, mp\_int $n$ and a digit $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\ +\hspace{11.5mm}($0 \le x < n^2, n > 1, (n, \beta) = 1, \beta^k > n$) \\ +\textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\ +\hline \\ +Place an array of \textbf{MP\_WARRAY} mp\_word variables called $\hat W$ on the stack. \\ +1. if $x.alloc < n.used + 1$ then grow $x$ to $n.used + 1$ digits. \\ +Copy the digits of $x$ into the array $\hat W$ \\ +2. For $ix$ from $0$ to $x.used - 1$ do \\ +\hspace{3mm}2.1 $\hat W_{ix} \leftarrow x_{ix}$ \\ +3. For $ix$ from $x.used$ to $2n.used - 1$ do \\ +\hspace{3mm}3.1 $\hat W_{ix} \leftarrow 0$ \\ +Elimiate the lower $k$ digits. \\ +4. for $ix$ from $0$ to $n.used - 1$ do \\ +\hspace{3mm}4.1 $\mu \leftarrow \hat W_{ix} \cdot \rho \mbox{ (mod }\beta\mbox{)}$ \\ +\hspace{3mm}4.2 For $iy$ from $0$ to $n.used - 1$ do \\ +\hspace{6mm}4.2.1 $\hat W_{iy + ix} \leftarrow \hat W_{iy + ix} + \mu \cdot n_{iy}$ \\ +\hspace{3mm}4.3 $\hat W_{ix + 1} \leftarrow \hat W_{ix + 1} + \lfloor \hat W_{ix} / \beta \rfloor$ \\ +Propagate carries upwards. \\ +5. for $ix$ from $n.used$ to $2n.used + 1$ do \\ +\hspace{3mm}5.1 $\hat W_{ix + 1} \leftarrow \hat W_{ix + 1} + \lfloor \hat W_{ix} / \beta \rfloor$ \\ +Shift right and reduce modulo $\beta$ simultaneously. \\ +6. for $ix$ from $0$ to $n.used + 1$ do \\ +\hspace{3mm}6.1 $x_{ix} \leftarrow \hat W_{ix + n.used} \mbox{ (mod }\beta\mbox{)}$ \\ +Zero excess digits and fixup $x$. \\ +7. if $x.used > n.used + 1$ then do \\ +\hspace{3mm}7.1 for $ix$ from $n.used + 1$ to $x.used - 1$ do \\ +\hspace{6mm}7.1.1 $x_{ix} \leftarrow 0$ \\ +8. $x.used \leftarrow n.used + 1$ \\ +9. Clamp excessive digits of $x$. \\ +10. If $x \ge n$ then \\ +\hspace{3mm}10.1 $x \leftarrow x - n$ \\ +11. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm fast\_mp\_montgomery\_reduce} +\end{figure} + +\textbf{Algorithm fast\_mp\_montgomery\_reduce.} +This algorithm will compute the Montgomery reduction of $x$ modulo $n$ using the Comba technique. It is on most computer platforms significantly +faster than algorithm mp\_montgomery\_reduce and algorithm mp\_reduce (\textit{Barrett reduction}). The algorithm has the same restrictions +on the input as the baseline reduction algorithm. An additional two restrictions are imposed on this algorithm. The number of digits $k$ in the +the modulus $n$ must not violate $MP\_WARRAY > 2k +1$ and $n < \delta$. When $\beta = 2^{28}$ this algorithm can be used to reduce modulo +a modulus of at most $3,556$ bits in length. + +As in the other Comba reduction algorithms there is a $\hat W$ array which stores the columns of the product. It is initially filled with the +contents of $x$ with the excess digits zeroed. The reduction loop is very similar the to the baseline loop at heart. The multiplication on step +4.1 can be single precision only since $ab \mbox{ (mod }\beta\mbox{)} \equiv (a \mbox{ mod }\beta)(b \mbox{ mod }\beta)$. Some multipliers such +as those on the ARM processors take a variable length time to complete depending on the number of bytes of result it must produce. By performing +a single precision multiplication instead half the amount of time is spent. + +Also note that digit $\hat W_{ix}$ must have the carry from the $ix - 1$'th digit propagated upwards in order for this to work. That is what step +4.3 will do. In effect over the $n.used$ iterations of the outer loop the $n.used$'th lower columns all have the their carries propagated forwards. Note +how the upper bits of those same words are not reduced modulo $\beta$. This is because those values will be discarded shortly and there is no +point. + +Step 5 will propagate the remainder of the carries upwards. On step 6 the columns are reduced modulo $\beta$ and shifted simultaneously as they are +stored in the destination $x$. + +EXAM,bn_fast_mp_montgomery_reduce.c + +The $\hat W$ array is first filled with digits of $x$ on line @49,for@ then the rest of the digits are zeroed on line @54,for@. Both loops share +the same alias variables to make the code easier to read. + +The value of $\mu$ is calculated in an interesting fashion. First the value $\hat W_{ix}$ is reduced modulo $\beta$ and cast to a mp\_digit. This +forces the compiler to use a single precision multiplication and prevents any concerns about loss of precision. Line @101,>>@ fixes the carry +for the next iteration of the loop by propagating the carry from $\hat W_{ix}$ to $\hat W_{ix+1}$. + +The for loop on line @113,for@ propagates the rest of the carries upwards through the columns. The for loop on line @126,for@ reduces the columns +modulo $\beta$ and shifts them $k$ places at the same time. The alias $\_ \hat W$ actually refers to the array $\hat W$ starting at the $n.used$'th +digit, that is $\_ \hat W_{t} = \hat W_{n.used + t}$. + +\subsection{Montgomery Setup} +To calculate the variable $\rho$ a relatively simple algorithm will be required. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_montgomery\_setup}. \\ +\textbf{Input}. mp\_int $n$ ($n > 1$ and $(n, 2) = 1$) \\ +\textbf{Output}. $\rho \equiv -1/n_0 \mbox{ (mod }\beta\mbox{)}$ \\ +\hline \\ +1. $b \leftarrow n_0$ \\ +2. If $b$ is even return(\textit{MP\_VAL}) \\ +3. $x \leftarrow ((b + 2) \mbox{ AND } 4) << 1) + b$ \\ +4. for $k$ from 0 to $\lceil lg(lg(\beta)) \rceil - 2$ do \\ +\hspace{3mm}4.1 $x \leftarrow x \cdot (2 - bx)$ \\ +5. $\rho \leftarrow \beta - x \mbox{ (mod }\beta\mbox{)}$ \\ +6. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_montgomery\_setup} +\end{figure} + +\textbf{Algorithm mp\_montgomery\_setup.} +This algorithm will calculate the value of $\rho$ required within the Montgomery reduction algorithms. It uses a very interesting trick +to calculate $1/n_0$ when $\beta$ is a power of two. + +EXAM,bn_mp_montgomery_setup.c + +This source code computes the value of $\rho$ required to perform Montgomery reduction. It has been modified to avoid performing excess +multiplications when $\beta$ is not the default 28-bits. + +\section{The Diminished Radix Algorithm} +The Diminished Radix method of modular reduction \cite{DRMET} is a fairly clever technique which can be more efficient than either the Barrett +or Montgomery methods for certain forms of moduli. The technique is based on the following simple congruence. + +\begin{equation} +(x \mbox{ mod } n) + k \lfloor x / n \rfloor \equiv x \mbox{ (mod }(n - k)\mbox{)} +\end{equation} + +This observation was used in the MMB \cite{MMB} block cipher to create a diffusion primitive. It used the fact that if $n = 2^{31}$ and $k=1$ that +then a x86 multiplier could produce the 62-bit product and use the ``shrd'' instruction to perform a double-precision right shift. The proof +of the above equation is very simple. First write $x$ in the product form. + +\begin{equation} +x = qn + r +\end{equation} + +Now reduce both sides modulo $(n - k)$. + +\begin{equation} +x \equiv qk + r \mbox{ (mod }(n-k)\mbox{)} +\end{equation} + +The variable $n$ reduces modulo $n - k$ to $k$. By putting $q = \lfloor x/n \rfloor$ and $r = x \mbox{ mod } n$ +into the equation the original congruence is reproduced, thus concluding the proof. The following algorithm is based on this observation. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{Diminished Radix Reduction}. \\ +\textbf{Input}. Integer $x$, $n$, $k$ \\ +\textbf{Output}. $x \mbox{ mod } (n - k)$ \\ +\hline \\ +1. $q \leftarrow \lfloor x / n \rfloor$ \\ +2. $q \leftarrow k \cdot q$ \\ +3. $x \leftarrow x \mbox{ (mod }n\mbox{)}$ \\ +4. $x \leftarrow x + q$ \\ +5. If $x \ge (n - k)$ then \\ +\hspace{3mm}5.1 $x \leftarrow x - (n - k)$ \\ +\hspace{3mm}5.2 Goto step 1. \\ +6. Return $x$ \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm Diminished Radix Reduction} +\label{fig:DR} +\end{figure} + +This algorithm will reduce $x$ modulo $n - k$ and return the residue. If $0 \le x < (n - k)^2$ then the algorithm will loop almost always +once or twice and occasionally three times. For simplicity sake the value of $x$ is bounded by the following simple polynomial. + +\begin{equation} +0 \le x < n^2 + k^2 - 2nk +\end{equation} + +The true bound is $0 \le x < (n - k - 1)^2$ but this has quite a few more terms. The value of $q$ after step 1 is bounded by the following. + +\begin{equation} +q < n - 2k - k^2/n +\end{equation} + +Since $k^2$ is going to be considerably smaller than $n$ that term will always be zero. The value of $x$ after step 3 is bounded trivially as +$0 \le x < n$. By step four the sum $x + q$ is bounded by + +\begin{equation} +0 \le q + x < (k + 1)n - 2k^2 - 1 +\end{equation} + +With a second pass $q$ will be loosely bounded by $0 \le q < k^2$ after step 2 while $x$ will still be loosely bounded by $0 \le x < n$ after step 3. After the second pass it is highly unlike that the +sum in step 4 will exceed $n - k$. In practice fewer than three passes of the algorithm are required to reduce virtually every input in the +range $0 \le x < (n - k - 1)^2$. + +\begin{figure} +\begin{small} +\begin{center} +\begin{tabular}{|l|} +\hline +$x = 123456789, n = 256, k = 3$ \\ +\hline $q \leftarrow \lfloor x/n \rfloor = 482253$ \\ +$q \leftarrow q*k = 1446759$ \\ +$x \leftarrow x \mbox{ mod } n = 21$ \\ +$x \leftarrow x + q = 1446780$ \\ +$x \leftarrow x - (n - k) = 1446527$ \\ +\hline +$q \leftarrow \lfloor x/n \rfloor = 5650$ \\ +$q \leftarrow q*k = 16950$ \\ +$x \leftarrow x \mbox{ mod } n = 127$ \\ +$x \leftarrow x + q = 17077$ \\ +$x \leftarrow x - (n - k) = 16824$ \\ +\hline +$q \leftarrow \lfloor x/n \rfloor = 65$ \\ +$q \leftarrow q*k = 195$ \\ +$x \leftarrow x \mbox{ mod } n = 184$ \\ +$x \leftarrow x + q = 379$ \\ +$x \leftarrow x - (n - k) = 126$ \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Example Diminished Radix Reduction} +\label{fig:EXDR} +\end{figure} + +Figure~\ref{fig:EXDR} demonstrates the reduction of $x = 123456789$ modulo $n - k = 253$ when $n = 256$ and $k = 3$. Note that even while $x$ +is considerably larger than $(n - k - 1)^2 = 63504$ the algorithm still converges on the modular residue exceedingly fast. In this case only +three passes were required to find the residue $x \equiv 126$. + + +\subsection{Choice of Moduli} +On the surface this algorithm looks like a very expensive algorithm. It requires a couple of subtractions followed by multiplication and other +modular reductions. The usefulness of this algorithm becomes exceedingly clear when an appropriate modulus is chosen. + +Division in general is a very expensive operation to perform. The one exception is when the division is by a power of the radix of representation used. +Division by ten for example is simple for pencil and paper mathematics since it amounts to shifting the decimal place to the right. Similarly division +by two (\textit{or powers of two}) is very simple for binary computers to perform. It would therefore seem logical to choose $n$ of the form $2^p$ +which would imply that $\lfloor x / n \rfloor$ is a simple shift of $x$ right $p$ bits. + +However, there is one operation related to division of power of twos that is even faster than this. If $n = \beta^p$ then the division may be +performed by moving whole digits to the right $p$ places. In practice division by $\beta^p$ is much faster than division by $2^p$ for any $p$. +Also with the choice of $n = \beta^p$ reducing $x$ modulo $n$ merely requires zeroing the digits above the $p-1$'th digit of $x$. + +Throughout the next section the term ``restricted modulus'' will refer to a modulus of the form $\beta^p - k$ whereas the term ``unrestricted +modulus'' will refer to a modulus of the form $2^p - k$. The word ``restricted'' in this case refers to the fact that it is based on the +$2^p$ logic except $p$ must be a multiple of $lg(\beta)$. + +\subsection{Choice of $k$} +Now that division and reduction (\textit{step 1 and 3 of figure~\ref{fig:DR}}) have been optimized to simple digit operations the multiplication by $k$ +in step 2 is the most expensive operation. Fortunately the choice of $k$ is not terribly limited. For all intents and purposes it might +as well be a single digit. The smaller the value of $k$ is the faster the algorithm will be. + +\subsection{Restricted Diminished Radix Reduction} +The restricted Diminished Radix algorithm can quickly reduce an input modulo a modulus of the form $n = \beta^p - k$. This algorithm can reduce +an input $x$ within the range $0 \le x < n^2$ using only a couple passes of the algorithm demonstrated in figure~\ref{fig:DR}. The implementation +of this algorithm has been optimized to avoid additional overhead associated with a division by $\beta^p$, the multiplication by $k$ or the addition +of $x$ and $q$. The resulting algorithm is very efficient and can lead to substantial improvements over Barrett and Montgomery reduction when modular +exponentiations are performed. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_dr\_reduce}. \\ +\textbf{Input}. mp\_int $x$, $n$ and a mp\_digit $k = \beta - n_0$ \\ +\hspace{11.5mm}($0 \le x < n^2$, $n > 1$, $0 < k < \beta$) \\ +\textbf{Output}. $x \mbox{ mod } n$ \\ +\hline \\ +1. $m \leftarrow n.used$ \\ +2. If $x.alloc < 2m$ then grow $x$ to $2m$ digits. \\ +3. $\mu \leftarrow 0$ \\ +4. for $i$ from $0$ to $m - 1$ do \\ +\hspace{3mm}4.1 $\hat r \leftarrow k \cdot x_{m+i} + x_{i} + \mu$ \\ +\hspace{3mm}4.2 $x_{i} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ +\hspace{3mm}4.3 $\mu \leftarrow \lfloor \hat r / \beta \rfloor$ \\ +5. $x_{m} \leftarrow \mu$ \\ +6. for $i$ from $m + 1$ to $x.used - 1$ do \\ +\hspace{3mm}6.1 $x_{i} \leftarrow 0$ \\ +7. Clamp excess digits of $x$. \\ +8. If $x \ge n$ then \\ +\hspace{3mm}8.1 $x \leftarrow x - n$ \\ +\hspace{3mm}8.2 Goto step 3. \\ +9. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_dr\_reduce} +\end{figure} + +\textbf{Algorithm mp\_dr\_reduce.} +This algorithm will perform the Dimished Radix reduction of $x$ modulo $n$. It has similar restrictions to that of the Barrett reduction +with the addition that $n$ must be of the form $n = \beta^m - k$ where $0 < k <\beta$. + +This algorithm essentially implements the pseudo-code in figure~\ref{fig:DR} except with a slight optimization. The division by $\beta^m$, multiplication by $k$ +and addition of $x \mbox{ mod }\beta^m$ are all performed simultaneously inside the loop on step 4. The division by $\beta^m$ is emulated by accessing +the term at the $m+i$'th position which is subsequently multiplied by $k$ and added to the term at the $i$'th position. After the loop the $m$'th +digit is set to the carry and the upper digits are zeroed. Steps 5 and 6 emulate the reduction modulo $\beta^m$ that should have happend to +$x$ before the addition of the multiple of the upper half. + +At step 8 if $x$ is still larger than $n$ another pass of the algorithm is required. First $n$ is subtracted from $x$ and then the algorithm resumes +at step 3. + +EXAM,bn_mp_dr_reduce.c + +The first step is to grow $x$ as required to $2m$ digits since the reduction is performed in place on $x$. The label on line @49,top:@ is where +the algorithm will resume if further reduction passes are required. In theory it could be placed at the top of the function however, the size of +the modulus and question of whether $x$ is large enough are invariant after the first pass meaning that it would be a waste of time. + +The aliases $tmpx1$ and $tmpx2$ refer to the digits of $x$ where the latter is offset by $m$ digits. By reading digits from $x$ offset by $m$ digits +a division by $\beta^m$ can be simulated virtually for free. The loop on line @61,for@ performs the bulk of the work (\textit{corresponds to step 4 of algorithm 7.11}) +in this algorithm. + +By line @68,mu@ the pointer $tmpx1$ points to the $m$'th digit of $x$ which is where the final carry will be placed. Similarly by line @71,for@ the +same pointer will point to the $m+1$'th digit where the zeroes will be placed. + +Since the algorithm is only valid if both $x$ and $n$ are greater than zero an unsigned comparison suffices to determine if another pass is required. +With the same logic at line @82,sub@ the value of $x$ is known to be greater than or equal to $n$ meaning that an unsigned subtraction can be used +as well. Since the destination of the subtraction is the larger of the inputs the call to algorithm s\_mp\_sub cannot fail and the return code +does not need to be checked. + +\subsubsection{Setup} +To setup the restricted Diminished Radix algorithm the value $k = \beta - n_0$ is required. This algorithm is not really complicated but provided for +completeness. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_dr\_setup}. \\ +\textbf{Input}. mp\_int $n$ \\ +\textbf{Output}. $k = \beta - n_0$ \\ +\hline \\ +1. $k \leftarrow \beta - n_0$ \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_dr\_setup} +\end{figure} + +EXAM,bn_mp_dr_setup.c + +\subsubsection{Modulus Detection} +Another algorithm which will be useful is the ability to detect a restricted Diminished Radix modulus. An integer is said to be +of restricted Diminished Radix form if all of the digits are equal to $\beta - 1$ except the trailing digit which may be any value. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_dr\_is\_modulus}. \\ +\textbf{Input}. mp\_int $n$ \\ +\textbf{Output}. $1$ if $n$ is in D.R form, $0$ otherwise \\ +\hline +1. If $n.used < 2$ then return($0$). \\ +2. for $ix$ from $1$ to $n.used - 1$ do \\ +\hspace{3mm}2.1 If $n_{ix} \ne \beta - 1$ return($0$). \\ +3. Return($1$). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_dr\_is\_modulus} +\end{figure} + +\textbf{Algorithm mp\_dr\_is\_modulus.} +This algorithm determines if a value is in Diminished Radix form. Step 1 rejects obvious cases where fewer than two digits are +in the mp\_int. Step 2 tests all but the first digit to see if they are equal to $\beta - 1$. If the algorithm manages to get to +step 3 then $n$ must be of Diminished Radix form. + +EXAM,bn_mp_dr_is_modulus.c + +\subsection{Unrestricted Diminished Radix Reduction} +The unrestricted Diminished Radix algorithm allows modular reductions to be performed when the modulus is of the form $2^p - k$. This algorithm +is a straightforward adaptation of algorithm~\ref{fig:DR}. + +In general the restricted Diminished Radix reduction algorithm is much faster since it has considerably lower overhead. However, this new +algorithm is much faster than either Montgomery or Barrett reduction when the moduli are of the appropriate form. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_reduce\_2k}. \\ +\textbf{Input}. mp\_int $a$ and $n$. mp\_digit $k$ \\ +\hspace{11.5mm}($a \ge 0$, $n > 1$, $0 < k < \beta$, $n + k$ is a power of two) \\ +\textbf{Output}. $a \mbox{ (mod }n\mbox{)}$ \\ +\hline +1. $p \leftarrow \lceil lg(n) \rceil$ (\textit{mp\_count\_bits}) \\ +2. While $a \ge n$ do \\ +\hspace{3mm}2.1 $q \leftarrow \lfloor a / 2^p \rfloor$ (\textit{mp\_div\_2d}) \\ +\hspace{3mm}2.2 $a \leftarrow a \mbox{ (mod }2^p\mbox{)}$ (\textit{mp\_mod\_2d}) \\ +\hspace{3mm}2.3 $q \leftarrow q \cdot k$ (\textit{mp\_mul\_d}) \\ +\hspace{3mm}2.4 $a \leftarrow a - q$ (\textit{s\_mp\_sub}) \\ +\hspace{3mm}2.5 If $a \ge n$ then do \\ +\hspace{6mm}2.5.1 $a \leftarrow a - n$ \\ +3. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_reduce\_2k} +\end{figure} + +\textbf{Algorithm mp\_reduce\_2k.} +This algorithm quickly reduces an input $a$ modulo an unrestricted Diminished Radix modulus $n$. Division by $2^p$ is emulated with a right +shift which makes the algorithm fairly inexpensive to use. + +EXAM,bn_mp_reduce_2k.c + +The algorithm mp\_count\_bits calculates the number of bits in an mp\_int which is used to find the initial value of $p$. The call to mp\_div\_2d +on line @31,mp_div_2d@ calculates both the quotient $q$ and the remainder $a$ required. By doing both in a single function call the code size +is kept fairly small. The multiplication by $k$ is only performed if $k > 1$. This allows reductions modulo $2^p - 1$ to be performed without +any multiplications. + +The unsigned s\_mp\_add, mp\_cmp\_mag and s\_mp\_sub are used in place of their full sign counterparts since the inputs are only valid if they are +positive. By using the unsigned versions the overhead is kept to a minimum. + +\subsubsection{Unrestricted Setup} +To setup this reduction algorithm the value of $k = 2^p - n$ is required. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_reduce\_2k\_setup}. \\ +\textbf{Input}. mp\_int $n$ \\ +\textbf{Output}. $k = 2^p - n$ \\ +\hline +1. $p \leftarrow \lceil lg(n) \rceil$ (\textit{mp\_count\_bits}) \\ +2. $x \leftarrow 2^p$ (\textit{mp\_2expt}) \\ +3. $x \leftarrow x - n$ (\textit{mp\_sub}) \\ +4. $k \leftarrow x_0$ \\ +5. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_reduce\_2k\_setup} +\end{figure} + +\textbf{Algorithm mp\_reduce\_2k\_setup.} +This algorithm computes the value of $k$ required for the algorithm mp\_reduce\_2k. By making a temporary variable $x$ equal to $2^p$ a subtraction +is sufficient to solve for $k$. Alternatively if $n$ has more than one digit the value of $k$ is simply $\beta - n_0$. + +EXAM,bn_mp_reduce_2k_setup.c + +\subsubsection{Unrestricted Detection} +An integer $n$ is a valid unrestricted Diminished Radix modulus if either of the following are true. + +\begin{enumerate} +\item The number has only one digit. +\item The number has more than one digit and every bit from the $\beta$'th to the most significant is one. +\end{enumerate} + +If either condition is true than there is a power of two $2^p$ such that $0 < 2^p - n < \beta$. If the input is only +one digit than it will always be of the correct form. Otherwise all of the bits above the first digit must be one. This arises from the fact +that there will be value of $k$ that when added to the modulus causes a carry in the first digit which propagates all the way to the most +significant bit. The resulting sum will be a power of two. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_reduce\_is\_2k}. \\ +\textbf{Input}. mp\_int $n$ \\ +\textbf{Output}. $1$ if of proper form, $0$ otherwise \\ +\hline +1. If $n.used = 0$ then return($0$). \\ +2. If $n.used = 1$ then return($1$). \\ +3. $p \leftarrow \lceil lg(n) \rceil$ (\textit{mp\_count\_bits}) \\ +4. for $x$ from $lg(\beta)$ to $p$ do \\ +\hspace{3mm}4.1 If the ($x \mbox{ mod }lg(\beta)$)'th bit of the $\lfloor x / lg(\beta) \rfloor$ of $n$ is zero then return($0$). \\ +5. Return($1$). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_reduce\_is\_2k} +\end{figure} + +\textbf{Algorithm mp\_reduce\_is\_2k.} +This algorithm quickly determines if a modulus is of the form required for algorithm mp\_reduce\_2k to function properly. + +EXAM,bn_mp_reduce_is_2k.c + + + +\section{Algorithm Comparison} +So far three very different algorithms for modular reduction have been discussed. Each of the algorithms have their own strengths and weaknesses +that makes having such a selection very useful. The following table sumarizes the three algorithms along with comparisons of work factors. Since +all three algorithms have the restriction that $0 \le x < n^2$ and $n > 1$ those limitations are not included in the table. + +\begin{center} +\begin{small} +\begin{tabular}{|c|c|c|c|c|c|} +\hline \textbf{Method} & \textbf{Work Required} & \textbf{Limitations} & \textbf{$m = 8$} & \textbf{$m = 32$} & \textbf{$m = 64$} \\ +\hline Barrett & $m^2 + 2m - 1$ & None & $79$ & $1087$ & $4223$ \\ +\hline Montgomery & $m^2 + m$ & $n$ must be odd & $72$ & $1056$ & $4160$ \\ +\hline D.R. & $2m$ & $n = \beta^m - k$ & $16$ & $64$ & $128$ \\ +\hline +\end{tabular} +\end{small} +\end{center} + +In theory Montgomery and Barrett reductions would require roughly the same amount of time to complete. However, in practice since Montgomery +reduction can be written as a single function with the Comba technique it is much faster. Barrett reduction suffers from the overhead of +calling the half precision multipliers, addition and division by $\beta$ algorithms. + +For almost every cryptographic algorithm Montgomery reduction is the algorithm of choice. The one set of algorithms where Diminished Radix reduction truly +shines are based on the discrete logarithm problem such as Diffie-Hellman \cite{DH} and ElGamal \cite{ELGAMAL}. In these algorithms +primes of the form $\beta^m - k$ can be found and shared amongst users. These primes will allow the Diminished Radix algorithm to be used in +modular exponentiation to greatly speed up the operation. + + + +\section*{Exercises} +\begin{tabular}{cl} +$\left [ 3 \right ]$ & Prove that the ``trick'' in algorithm mp\_montgomery\_setup actually \\ + & calculates the correct value of $\rho$. \\ + & \\ +$\left [ 2 \right ]$ & Devise an algorithm to reduce modulo $n + k$ for small $k$ quickly. \\ + & \\ +$\left [ 4 \right ]$ & Prove that the pseudo-code algorithm ``Diminished Radix Reduction'' \\ + & (\textit{figure~\ref{fig:DR}}) terminates. Also prove the probability that it will \\ + & terminate within $1 \le k \le 10$ iterations. \\ + & \\ +\end{tabular} + + +\chapter{Exponentiation} +Exponentiation is the operation of raising one variable to the power of another, for example, $a^b$. A variant of exponentiation, computed +in a finite field or ring, is called modular exponentiation. This latter style of operation is typically used in public key +cryptosystems such as RSA and Diffie-Hellman. The ability to quickly compute modular exponentiations is of great benefit to any +such cryptosystem and many methods have been sought to speed it up. + +\section{Exponentiation Basics} +A trivial algorithm would simply multiply $a$ against itself $b - 1$ times to compute the exponentiation desired. However, as $b$ grows in size +the number of multiplications becomes prohibitive. Imagine what would happen if $b$ $\approx$ $2^{1024}$ as is the case when computing an RSA signature +with a $1024$-bit key. Such a calculation could never be completed as it would take simply far too long. + +Fortunately there is a very simple algorithm based on the laws of exponents. Recall that $lg_a(a^b) = b$ and that $lg_a(a^ba^c) = b + c$ which +are two trivial relationships between the base and the exponent. Let $b_i$ represent the $i$'th bit of $b$ starting from the least +significant bit. If $b$ is a $k$-bit integer than the following equation is true. + +\begin{equation} +a^b = \prod_{i=0}^{k-1} a^{2^i \cdot b_i} +\end{equation} + +By taking the base $a$ logarithm of both sides of the equation the following equation is the result. + +\begin{equation} +b = \sum_{i=0}^{k-1}2^i \cdot b_i +\end{equation} + +The term $a^{2^i}$ can be found from the $i - 1$'th term by squaring the term since $\left ( a^{2^i} \right )^2$ is equal to +$a^{2^{i+1}}$. This observation forms the basis of essentially all fast exponentiation algorithms. It requires $k$ squarings and on average +$k \over 2$ multiplications to compute the result. This is indeed quite an improvement over simply multiplying by $a$ a total of $b-1$ times. + +While this current method is a considerable speed up there are further improvements to be made. For example, the $a^{2^i}$ term does not need to +be computed in an auxilary variable. Consider the following equivalent algorithm. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{Left to Right Exponentiation}. \\ +\textbf{Input}. Integer $a$, $b$ and $k$ \\ +\textbf{Output}. $c = a^b$ \\ +\hline \\ +1. $c \leftarrow 1$ \\ +2. for $i$ from $k - 1$ to $0$ do \\ +\hspace{3mm}2.1 $c \leftarrow c^2$ \\ +\hspace{3mm}2.2 $c \leftarrow c \cdot a^{b_i}$ \\ +3. Return $c$. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Left to Right Exponentiation} +\label{fig:LTOR} +\end{figure} + +This algorithm starts from the most significant bit and works towards the least significant bit. When the $i$'th bit of $b$ is set $a$ is +multiplied against the current product. In each iteration the product is squared which doubles the exponent of the individual terms of the +product. + +For example, let $b = 101100_2 \equiv 44_{10}$. The following chart demonstrates the actions of the algorithm. + +\newpage\begin{figure} +\begin{center} +\begin{tabular}{|c|c|} +\hline \textbf{Value of $i$} & \textbf{Value of $c$} \\ +\hline - & $1$ \\ +\hline $5$ & $a$ \\ +\hline $4$ & $a^2$ \\ +\hline $3$ & $a^4 \cdot a$ \\ +\hline $2$ & $a^8 \cdot a^2 \cdot a$ \\ +\hline $1$ & $a^{16} \cdot a^4 \cdot a^2$ \\ +\hline $0$ & $a^{32} \cdot a^8 \cdot a^4$ \\ +\hline +\end{tabular} +\end{center} +\caption{Example of Left to Right Exponentiation} +\end{figure} + +When the product $a^{32} \cdot a^8 \cdot a^4$ is simplified it is equal $a^{44}$ which is the desired exponentiation. This particular algorithm is +called ``Left to Right'' because it reads the exponent in that order. All of the exponentiation algorithms that will be presented are of this nature. + +\subsection{Single Digit Exponentiation} +The first algorithm in the series of exponentiation algorithms will be an unbounded algorithm where the exponent is a single digit. It is intended +to be used when a small power of an input is required (\textit{e.g. $a^5$}). It is faster than simply multiplying $b - 1$ times for all values of +$b$ that are greater than three. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_expt\_d}. \\ +\textbf{Input}. mp\_int $a$ and mp\_digit $b$ \\ +\textbf{Output}. $c = a^b$ \\ +\hline \\ +1. $g \leftarrow a$ (\textit{mp\_init\_copy}) \\ +2. $c \leftarrow 1$ (\textit{mp\_set}) \\ +3. for $x$ from 1 to $lg(\beta)$ do \\ +\hspace{3mm}3.1 $c \leftarrow c^2$ (\textit{mp\_sqr}) \\ +\hspace{3mm}3.2 If $b$ AND $2^{lg(\beta) - 1} \ne 0$ then \\ +\hspace{6mm}3.2.1 $c \leftarrow c \cdot g$ (\textit{mp\_mul}) \\ +\hspace{3mm}3.3 $b \leftarrow b << 1$ \\ +4. Clear $g$. \\ +5. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_expt\_d} +\end{figure} + +\textbf{Algorithm mp\_expt\_d.} +This algorithm computes the value of $a$ raised to the power of a single digit $b$. It uses the left to right exponentiation algorithm to +quickly compute the exponentiation. It is loosely based on algorithm 14.79 of HAC \cite[pp. 615]{HAC} with the difference that the +exponent is a fixed width. + +A copy of $a$ is made first to allow destination variable $c$ be the same as the source variable $a$. The result is set to the initial value of +$1$ in the subsequent step. + +Inside the loop the exponent is read from the most significant bit first down to the least significant bit. First $c$ is invariably squared +on step 3.1. In the following step if the most significant bit of $b$ is one the copy of $a$ is multiplied against $c$. The value +of $b$ is shifted left one bit to make the next bit down from the most signficant bit the new most significant bit. In effect each +iteration of the loop moves the bits of the exponent $b$ upwards to the most significant location. + +EXAM,bn_mp_expt_d.c + +Line @29,mp_set@ sets the initial value of the result to $1$. Next the loop on line @31,for@ steps through each bit of the exponent starting from +the most significant down towards the least significant. The invariant squaring operation placed on line @333,mp_sqr@ is performed first. After +the squaring the result $c$ is multiplied by the base $g$ if and only if the most significant bit of the exponent is set. The shift on line +@47,<<@ moves all of the bits of the exponent upwards towards the most significant location. + +\section{$k$-ary Exponentiation} +When calculating an exponentiation the most time consuming bottleneck is the multiplications which are in general a small factor +slower than squaring. Recall from the previous algorithm that $b_{i}$ refers to the $i$'th bit of the exponent $b$. Suppose instead it referred to +the $i$'th $k$-bit digit of the exponent of $b$. For $k = 1$ the definitions are synonymous and for $k > 1$ algorithm~\ref{fig:KARY} +computes the same exponentiation. A group of $k$ bits from the exponent is called a \textit{window}. That is it is a small window on only a +portion of the entire exponent. Consider the following modification to the basic left to right exponentiation algorithm. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{$k$-ary Exponentiation}. \\ +\textbf{Input}. Integer $a$, $b$, $k$ and $t$ \\ +\textbf{Output}. $c = a^b$ \\ +\hline \\ +1. $c \leftarrow 1$ \\ +2. for $i$ from $t - 1$ to $0$ do \\ +\hspace{3mm}2.1 $c \leftarrow c^{2^k} $ \\ +\hspace{3mm}2.2 Extract the $i$'th $k$-bit word from $b$ and store it in $g$. \\ +\hspace{3mm}2.3 $c \leftarrow c \cdot a^g$ \\ +3. Return $c$. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{$k$-ary Exponentiation} +\label{fig:KARY} +\end{figure} + +The squaring on step 2.1 can be calculated by squaring the value $c$ successively $k$ times. If the values of $a^g$ for $0 < g < 2^k$ have been +precomputed this algorithm requires only $t$ multiplications and $tk$ squarings. The table can be generated with $2^{k - 1} - 1$ squarings and +$2^{k - 1} + 1$ multiplications. This algorithm assumes that the number of bits in the exponent is evenly divisible by $k$. +However, when it is not the remaining $0 < x \le k - 1$ bits can be handled with algorithm~\ref{fig:LTOR}. + +Suppose $k = 4$ and $t = 100$. This modified algorithm will require $109$ multiplications and $408$ squarings to compute the exponentiation. The +original algorithm would on average have required $200$ multiplications and $400$ squrings to compute the same value. The total number of squarings +has increased slightly but the number of multiplications has nearly halved. + +\subsection{Optimal Values of $k$} +An optimal value of $k$ will minimize $2^{k} + \lceil n / k \rceil + n - 1$ for a fixed number of bits in the exponent $n$. The simplest +approach is to brute force search amongst the values $k = 2, 3, \ldots, 8$ for the lowest result. Table~\ref{fig:OPTK} lists optimal values of $k$ +for various exponent sizes and compares the number of multiplication and squarings required against algorithm~\ref{fig:LTOR}. + +\begin{figure}[here] +\begin{center} +\begin{small} +\begin{tabular}{|c|c|c|c|c|c|} +\hline \textbf{Exponent (bits)} & \textbf{Optimal $k$} & \textbf{Work at $k$} & \textbf{Work with ~\ref{fig:LTOR}} \\ +\hline $16$ & $2$ & $27$ & $24$ \\ +\hline $32$ & $3$ & $49$ & $48$ \\ +\hline $64$ & $3$ & $92$ & $96$ \\ +\hline $128$ & $4$ & $175$ & $192$ \\ +\hline $256$ & $4$ & $335$ & $384$ \\ +\hline $512$ & $5$ & $645$ & $768$ \\ +\hline $1024$ & $6$ & $1257$ & $1536$ \\ +\hline $2048$ & $6$ & $2452$ & $3072$ \\ +\hline $4096$ & $7$ & $4808$ & $6144$ \\ +\hline +\end{tabular} +\end{small} +\end{center} +\caption{Optimal Values of $k$ for $k$-ary Exponentiation} +\label{fig:OPTK} +\end{figure} + +\subsection{Sliding-Window Exponentiation} +A simple modification to the previous algorithm is only generate the upper half of the table in the range $2^{k-1} \le g < 2^k$. Essentially +this is a table for all values of $g$ where the most significant bit of $g$ is a one. However, in order for this to be allowed in the +algorithm values of $g$ in the range $0 \le g < 2^{k-1}$ must be avoided. + +Table~\ref{fig:OPTK2} lists optimal values of $k$ for various exponent sizes and compares the work required against algorithm~\ref{fig:KARY}. + +\begin{figure}[here] +\begin{center} +\begin{small} +\begin{tabular}{|c|c|c|c|c|c|} +\hline \textbf{Exponent (bits)} & \textbf{Optimal $k$} & \textbf{Work at $k$} & \textbf{Work with ~\ref{fig:KARY}} \\ +\hline $16$ & $3$ & $24$ & $27$ \\ +\hline $32$ & $3$ & $45$ & $49$ \\ +\hline $64$ & $4$ & $87$ & $92$ \\ +\hline $128$ & $4$ & $167$ & $175$ \\ +\hline $256$ & $5$ & $322$ & $335$ \\ +\hline $512$ & $6$ & $628$ & $645$ \\ +\hline $1024$ & $6$ & $1225$ & $1257$ \\ +\hline $2048$ & $7$ & $2403$ & $2452$ \\ +\hline $4096$ & $8$ & $4735$ & $4808$ \\ +\hline +\end{tabular} +\end{small} +\end{center} +\caption{Optimal Values of $k$ for Sliding Window Exponentiation} +\label{fig:OPTK2} +\end{figure} + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{Sliding Window $k$-ary Exponentiation}. \\ +\textbf{Input}. Integer $a$, $b$, $k$ and $t$ \\ +\textbf{Output}. $c = a^b$ \\ +\hline \\ +1. $c \leftarrow 1$ \\ +2. for $i$ from $t - 1$ to $0$ do \\ +\hspace{3mm}2.1 If the $i$'th bit of $b$ is a zero then \\ +\hspace{6mm}2.1.1 $c \leftarrow c^2$ \\ +\hspace{3mm}2.2 else do \\ +\hspace{6mm}2.2.1 $c \leftarrow c^{2^k}$ \\ +\hspace{6mm}2.2.2 Extract the $k$ bits from $(b_{i}b_{i-1}\ldots b_{i-(k-1)})$ and store it in $g$. \\ +\hspace{6mm}2.2.3 $c \leftarrow c \cdot a^g$ \\ +\hspace{6mm}2.2.4 $i \leftarrow i - k$ \\ +3. Return $c$. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Sliding Window $k$-ary Exponentiation} +\end{figure} + +Similar to the previous algorithm this algorithm must have a special handler when fewer than $k$ bits are left in the exponent. While this +algorithm requires the same number of squarings it can potentially have fewer multiplications. The pre-computed table $a^g$ is also half +the size as the previous table. + +Consider the exponent $b = 111101011001000_2 \equiv 31432_{10}$ with $k = 3$ using both algorithms. The first algorithm will divide the exponent up as +the following five $3$-bit words $b \equiv \left ( 111, 101, 011, 001, 000 \right )_{2}$. The second algorithm will break the +exponent as $b \equiv \left ( 111, 101, 0, 110, 0, 100, 0 \right )_{2}$. The single digit $0$ in the second representation are where +a single squaring took place instead of a squaring and multiplication. In total the first method requires $10$ multiplications and $18$ +squarings. The second method requires $8$ multiplications and $18$ squarings. + +In general the sliding window method is never slower than the generic $k$-ary method and often it is slightly faster. + +\section{Modular Exponentiation} + +Modular exponentiation is essentially computing the power of a base within a finite field or ring. For example, computing +$d \equiv a^b \mbox{ (mod }c\mbox{)}$ is a modular exponentiation. Instead of first computing $a^b$ and then reducing it +modulo $c$ the intermediate result is reduced modulo $c$ after every squaring or multiplication operation. + +This guarantees that any intermediate result is bounded by $0 \le d \le c^2 - 2c + 1$ and can be reduced modulo $c$ quickly using +one of the algorithms presented in ~REDUCTION~. + +Before the actual modular exponentiation algorithm can be written a wrapper algorithm must be written first. This algorithm +will allow the exponent $b$ to be negative which is computed as $c \equiv \left (1 / a \right )^{\vert b \vert} \mbox{(mod }d\mbox{)}$. The +value of $(1/a) \mbox{ mod }c$ is computed using the modular inverse (\textit{see \ref{sec;modinv}}). If no inverse exists the algorithm +terminates with an error. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_exptmod}. \\ +\textbf{Input}. mp\_int $a$, $b$ and $c$ \\ +\textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\ +\hline \\ +1. If $c.sign = MP\_NEG$ return(\textit{MP\_VAL}). \\ +2. If $b.sign = MP\_NEG$ then \\ +\hspace{3mm}2.1 $g' \leftarrow g^{-1} \mbox{ (mod }c\mbox{)}$ \\ +\hspace{3mm}2.2 $x' \leftarrow \vert x \vert$ \\ +\hspace{3mm}2.3 Compute $d \equiv g'^{x'} \mbox{ (mod }c\mbox{)}$ via recursion. \\ +3. if $p$ is odd \textbf{OR} $p$ is a D.R. modulus then \\ +\hspace{3mm}3.1 Compute $y \equiv g^{x} \mbox{ (mod }p\mbox{)}$ via algorithm mp\_exptmod\_fast. \\ +4. else \\ +\hspace{3mm}4.1 Compute $y \equiv g^{x} \mbox{ (mod }p\mbox{)}$ via algorithm s\_mp\_exptmod. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_exptmod} +\end{figure} + +\textbf{Algorithm mp\_exptmod.} +The first algorithm which actually performs modular exponentiation is algorithm s\_mp\_exptmod. It is a sliding window $k$-ary algorithm +which uses Barrett reduction to reduce the product modulo $p$. The second algorithm mp\_exptmod\_fast performs the same operation +except it uses either Montgomery or Diminished Radix reduction. The two latter reduction algorithms are clumped in the same exponentiation +algorithm since their arguments are essentially the same (\textit{two mp\_ints and one mp\_digit}). + +EXAM,bn_mp_exptmod.c + +In order to keep the algorithms in a known state the first step on line @29,if@ is to reject any negative modulus as input. If the exponent is +negative the algorithm tries to perform a modular exponentiation with the modular inverse of the base $G$. The temporary variable $tmpG$ is assigned +the modular inverse of $G$ and $tmpX$ is assigned the absolute value of $X$. The algorithm will recuse with these new values with a positive +exponent. + +If the exponent is positive the algorithm resumes the exponentiation. Line @63,dr_@ determines if the modulus is of the restricted Diminished Radix +form. If it is not line @65,reduce@ attempts to determine if it is of a unrestricted Diminished Radix form. The integer $dr$ will take on one +of three values. + +\begin{enumerate} +\item $dr = 0$ means that the modulus is not of either restricted or unrestricted Diminished Radix form. +\item $dr = 1$ means that the modulus is of restricted Diminished Radix form. +\item $dr = 2$ means that the modulus is of unrestricted Diminished Radix form. +\end{enumerate} + +Line @69,if@ determines if the fast modular exponentiation algorithm can be used. It is allowed if $dr \ne 0$ or if the modulus is odd. Otherwise, +the slower s\_mp\_exptmod algorithm is used which uses Barrett reduction. + +\subsection{Barrett Modular Exponentiation} + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{s\_mp\_exptmod}. \\ +\textbf{Input}. mp\_int $a$, $b$ and $c$ \\ +\textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\ +\hline \\ +1. $k \leftarrow lg(x)$ \\ +2. $winsize \leftarrow \left \lbrace \begin{array}{ll} + 2 & \mbox{if }k \le 7 \\ + 3 & \mbox{if }7 < k \le 36 \\ + 4 & \mbox{if }36 < k \le 140 \\ + 5 & \mbox{if }140 < k \le 450 \\ + 6 & \mbox{if }450 < k \le 1303 \\ + 7 & \mbox{if }1303 < k \le 3529 \\ + 8 & \mbox{if }3529 < k \\ + \end{array} \right .$ \\ +3. Initialize $2^{winsize}$ mp\_ints in an array named $M$ and one mp\_int named $\mu$ \\ +4. Calculate the $\mu$ required for Barrett Reduction (\textit{mp\_reduce\_setup}). \\ +5. $M_1 \leftarrow g \mbox{ (mod }p\mbox{)}$ \\ +\\ +Setup the table of small powers of $g$. First find $g^{2^{winsize}}$ and then all multiples of it. \\ +6. $k \leftarrow 2^{winsize - 1}$ \\ +7. $M_{k} \leftarrow M_1$ \\ +8. for $ix$ from 0 to $winsize - 2$ do \\ +\hspace{3mm}8.1 $M_k \leftarrow \left ( M_k \right )^2$ (\textit{mp\_sqr}) \\ +\hspace{3mm}8.2 $M_k \leftarrow M_k \mbox{ (mod }p\mbox{)}$ (\textit{mp\_reduce}) \\ +9. for $ix$ from $2^{winsize - 1} + 1$ to $2^{winsize} - 1$ do \\ +\hspace{3mm}9.1 $M_{ix} \leftarrow M_{ix - 1} \cdot M_{1}$ (\textit{mp\_mul}) \\ +\hspace{3mm}9.2 $M_{ix} \leftarrow M_{ix} \mbox{ (mod }p\mbox{)}$ (\textit{mp\_reduce}) \\ +10. $res \leftarrow 1$ \\ +\\ +Start Sliding Window. \\ +11. $mode \leftarrow 0, bitcnt \leftarrow 1, buf \leftarrow 0, digidx \leftarrow x.used - 1, bitcpy \leftarrow 0, bitbuf \leftarrow 0$ \\ +12. Loop \\ +\hspace{3mm}12.1 $bitcnt \leftarrow bitcnt - 1$ \\ +\hspace{3mm}12.2 If $bitcnt = 0$ then do \\ +\hspace{6mm}12.2.1 If $digidx = -1$ goto step 13. \\ +\hspace{6mm}12.2.2 $buf \leftarrow x_{digidx}$ \\ +\hspace{6mm}12.2.3 $digidx \leftarrow digidx - 1$ \\ +\hspace{6mm}12.2.4 $bitcnt \leftarrow lg(\beta)$ \\ +Continued on next page. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm s\_mp\_exptmod} +\end{figure} + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{s\_mp\_exptmod} (\textit{continued}). \\ +\textbf{Input}. mp\_int $a$, $b$ and $c$ \\ +\textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\ +\hline \\ +\hspace{3mm}12.3 $y \leftarrow (buf >> (lg(\beta) - 1))$ AND $1$ \\ +\hspace{3mm}12.4 $buf \leftarrow buf << 1$ \\ +\hspace{3mm}12.5 if $mode = 0$ and $y = 0$ then goto step 12. \\ +\hspace{3mm}12.6 if $mode = 1$ and $y = 0$ then do \\ +\hspace{6mm}12.6.1 $res \leftarrow res^2$ \\ +\hspace{6mm}12.6.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ +\hspace{6mm}12.6.3 Goto step 12. \\ +\hspace{3mm}12.7 $bitcpy \leftarrow bitcpy + 1$ \\ +\hspace{3mm}12.8 $bitbuf \leftarrow bitbuf + (y << (winsize - bitcpy))$ \\ +\hspace{3mm}12.9 $mode \leftarrow 2$ \\ +\hspace{3mm}12.10 If $bitcpy = winsize$ then do \\ +\hspace{6mm}Window is full so perform the squarings and single multiplication. \\ +\hspace{6mm}12.10.1 for $ix$ from $0$ to $winsize -1$ do \\ +\hspace{9mm}12.10.1.1 $res \leftarrow res^2$ \\ +\hspace{9mm}12.10.1.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ +\hspace{6mm}12.10.2 $res \leftarrow res \cdot M_{bitbuf}$ \\ +\hspace{6mm}12.10.3 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ +\hspace{6mm}Reset the window. \\ +\hspace{6mm}12.10.4 $bitcpy \leftarrow 0, bitbuf \leftarrow 0, mode \leftarrow 1$ \\ +\\ +No more windows left. Check for residual bits of exponent. \\ +13. If $mode = 2$ and $bitcpy > 0$ then do \\ +\hspace{3mm}13.1 for $ix$ form $0$ to $bitcpy - 1$ do \\ +\hspace{6mm}13.1.1 $res \leftarrow res^2$ \\ +\hspace{6mm}13.1.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ +\hspace{6mm}13.1.3 $bitbuf \leftarrow bitbuf << 1$ \\ +\hspace{6mm}13.1.4 If $bitbuf$ AND $2^{winsize} \ne 0$ then do \\ +\hspace{9mm}13.1.4.1 $res \leftarrow res \cdot M_{1}$ \\ +\hspace{9mm}13.1.4.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ +14. $y \leftarrow res$ \\ +15. Clear $res$, $mu$ and the $M$ array. \\ +16. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm s\_mp\_exptmod (continued)} +\end{figure} + +\textbf{Algorithm s\_mp\_exptmod.} +This algorithm computes the $x$'th power of $g$ modulo $p$ and stores the result in $y$. It takes advantage of the Barrett reduction +algorithm to keep the product small throughout the algorithm. + +The first two steps determine the optimal window size based on the number of bits in the exponent. The larger the exponent the +larger the window size becomes. After a window size $winsize$ has been chosen an array of $2^{winsize}$ mp\_int variables is allocated. This +table will hold the values of $g^x \mbox{ (mod }p\mbox{)}$ for $2^{winsize - 1} \le x < 2^{winsize}$. + +After the table is allocated the first power of $g$ is found. Since $g \ge p$ is allowed it must be first reduced modulo $p$ to make +the rest of the algorithm more efficient. The first element of the table at $2^{winsize - 1}$ is found by squaring $M_1$ successively $winsize - 2$ +times. The rest of the table elements are found by multiplying the previous element by $M_1$ modulo $p$. + +Now that the table is available the sliding window may begin. The following list describes the functions of all the variables in the window. +\begin{enumerate} +\item The variable $mode$ dictates how the bits of the exponent are interpreted. +\begin{enumerate} + \item When $mode = 0$ the bits are ignored since no non-zero bit of the exponent has been seen yet. For example, if the exponent were simply + $1$ then there would be $lg(\beta) - 1$ zero bits before the first non-zero bit. In this case bits are ignored until a non-zero bit is found. + \item When $mode = 1$ a non-zero bit has been seen before and a new $winsize$-bit window has not been formed yet. In this mode leading $0$ bits + are read and a single squaring is performed. If a non-zero bit is read a new window is created. + \item When $mode = 2$ the algorithm is in the middle of forming a window and new bits are appended to the window from the most significant bit + downwards. +\end{enumerate} +\item The variable $bitcnt$ indicates how many bits are left in the current digit of the exponent left to be read. When it reaches zero a new digit + is fetched from the exponent. +\item The variable $buf$ holds the currently read digit of the exponent. +\item The variable $digidx$ is an index into the exponents digits. It starts at the leading digit $x.used - 1$ and moves towards the trailing digit. +\item The variable $bitcpy$ indicates how many bits are in the currently formed window. When it reaches $winsize$ the window is flushed and + the appropriate operations performed. +\item The variable $bitbuf$ holds the current bits of the window being formed. +\end{enumerate} + +All of step 12 is the window processing loop. It will iterate while there are digits available form the exponent to read. The first step +inside this loop is to extract a new digit if no more bits are available in the current digit. If there are no bits left a new digit is +read and if there are no digits left than the loop terminates. + +After a digit is made available step 12.3 will extract the most significant bit of the current digit and move all other bits in the digit +upwards. In effect the digit is read from most significant bit to least significant bit and since the digits are read from leading to +trailing edges the entire exponent is read from most significant bit to least significant bit. + +At step 12.5 if the $mode$ and currently extracted bit $y$ are both zero the bit is ignored and the next bit is read. This prevents the +algorithm from having to perform trivial squaring and reduction operations before the first non-zero bit is read. Step 12.6 and 12.7-10 handle +the two cases of $mode = 1$ and $mode = 2$ respectively. + +FIGU,expt_state,Sliding Window State Diagram + +By step 13 there are no more digits left in the exponent. However, there may be partial bits in the window left. If $mode = 2$ then +a Left-to-Right algorithm is used to process the remaining few bits. + +EXAM,bn_s_mp_exptmod.c + +Lines @26,if@ through @40,}@ determine the optimal window size based on the length of the exponent in bits. The window divisions are sorted +from smallest to greatest so that in each \textbf{if} statement only one condition must be tested. For example, by the \textbf{if} statement +on line @32,if@ the value of $x$ is already known to be greater than $140$. + +The conditional piece of code beginning on line @42,ifdef@ allows the window size to be restricted to five bits. This logic is used to ensure +the table of precomputed powers of $G$ remains relatively small. + +The for loop on line @49,for@ initializes the $M$ array while lines @59,mp_init@ and @62,mp_reduce@ compute the value of $\mu$ required for +Barrett reduction. + +-- More later. + +\section{Quick Power of Two} +Calculating $b = 2^a$ can be performed much quicker than with any of the previous algorithms. Recall that a logical shift left $m << k$ is +equivalent to $m \cdot 2^k$. By this logic when $m = 1$ a quick power of two can be achieved. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_2expt}. \\ +\textbf{Input}. integer $b$ \\ +\textbf{Output}. $a \leftarrow 2^b$ \\ +\hline \\ +1. $a \leftarrow 0$ \\ +2. If $a.alloc < \lfloor b / lg(\beta) \rfloor + 1$ then grow $a$ appropriately. \\ +3. $a.used \leftarrow \lfloor b / lg(\beta) \rfloor + 1$ \\ +4. $a_{\lfloor b / lg(\beta) \rfloor} \leftarrow 1 << (b \mbox{ mod } lg(\beta))$ \\ +5. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_2expt} +\end{figure} + +\textbf{Algorithm mp\_2expt.} + +EXAM,bn_mp_2expt.c + +\chapter{Higher Level Algorithms} + +This chapter discusses the various higher level algorithms that are required to complete a well rounded multiple precision integer package. These +routines are less performance oriented than the algorithms of chapters five, six and seven but are no less important. + +The first section describes a method of integer division with remainder that is universally well known. It provides the signed division logic +for the package. The subsequent section discusses a set of algorithms which allow a single digit to be the 2nd operand for a variety of operations. +These algorithms serve mostly to simplify other algorithms where small constants are required. The last two sections discuss how to manipulate +various representations of integers. For example, converting from an mp\_int to a string of character. + +\section{Integer Division with Remainder} +\label{sec:division} + +Integer division aside from modular exponentiation is the most intensive algorithm to compute. Like addition, subtraction and multiplication +the basis of this algorithm is the long-hand division algorithm taught to school children. Throughout this discussion several common variables +will be used. Let $x$ represent the divisor and $y$ represent the dividend. Let $q$ represent the integer quotient $\lfloor y / x \rfloor$ and +let $r$ represent the remainder $r = y - x \lfloor y / x \rfloor$. The following simple algorithm will be used to start the discussion. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{Radix-$\beta$ Integer Division}. \\ +\textbf{Input}. integer $x$ and $y$ \\ +\textbf{Output}. $q = \lfloor y/x\rfloor, r = y - xq$ \\ +\hline \\ +1. $q \leftarrow 0$ \\ +2. $n \leftarrow \vert \vert y \vert \vert - \vert \vert x \vert \vert$ \\ +3. for $t$ from $n$ down to $0$ do \\ +\hspace{3mm}3.1 Maximize $k$ such that $kx\beta^t$ is less than or equal to $y$ and $(k + 1)x\beta^t$ is greater. \\ +\hspace{3mm}3.2 $q \leftarrow q + k\beta^t$ \\ +\hspace{3mm}3.3 $y \leftarrow y - kx\beta^t$ \\ +4. $r \leftarrow y$ \\ +5. Return($q, r$) \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm Radix-$\beta$ Integer Division} +\label{fig:raddiv} +\end{figure} + +As children we are taught this very simple algorithm for the case of $\beta = 10$. Almost instinctively several optimizations are taught for which +their reason of existing are never explained. For this example let $y = 5471$ represent the dividend and $x = 23$ represent the divisor. + +To find the first digit of the quotient the value of $k$ must be maximized such that $kx\beta^t$ is less than or equal to $y$ and +simultaneously $(k + 1)x\beta^t$ is greater than $y$. Implicitly $k$ is the maximum value the $t$'th digit of the quotient may have. The habitual method +used to find the maximum is to ``eyeball'' the two numbers, typically only the leading digits and quickly estimate a quotient. By only using leading +digits a much simpler division may be used to form an educated guess at what the value must be. In this case $k = \lfloor 54/23\rfloor = 2$ quickly +arises as a possible solution. Indeed $2x\beta^2 = 4600$ is less than $y = 5471$ and simultaneously $(k + 1)x\beta^2 = 6900$ is larger than $y$. +As a result $k\beta^2$ is added to the quotient which now equals $q = 200$ and $4600$ is subtracted from $y$ to give a remainder of $y = 841$. + +Again this process is repeated to produce the quotient digit $k = 3$ which makes the quotient $q = 200 + 3\beta = 230$ and the remainder +$y = 841 - 3x\beta = 181$. Finally the last iteration of the loop produces $k = 7$ which leads to the quotient $q = 230 + 7 = 237$ and the +remainder $y = 181 - 7x = 20$. The final quotient and remainder found are $q = 237$ and $r = y = 20$ which are indeed correct since +$237 \cdot 23 + 20 = 5471$ is true. + +\subsection{Quotient Estimation} +\label{sec:divest} +As alluded to earlier the quotient digit $k$ can be estimated from only the leading digits of both the divisor and dividend. When $p$ leading +digits are used from both the divisor and dividend to form an estimation the accuracy of the estimation rises as $p$ grows. Technically +speaking the estimation is based on assuming the lower $\vert \vert y \vert \vert - p$ and $\vert \vert x \vert \vert - p$ lower digits of the +dividend and divisor are zero. + +The value of the estimation may off by a few values in either direction and in general is fairly correct. A simplification \cite[pp. 271]{TAOCPV2} +of the estimation technique is to use $t + 1$ digits of the dividend and $t$ digits of the divisor, in particularly when $t = 1$. The estimate +using this technique is never too small. For the following proof let $t = \vert \vert y \vert \vert - 1$ and $s = \vert \vert x \vert \vert - 1$ +represent the most significant digits of the dividend and divisor respectively. + +\textbf{Proof.}\textit{ The quotient $\hat k = \lfloor (y_t\beta + y_{t-1}) / x_s \rfloor$ is greater than or equal to +$k = \lfloor y / (x \cdot \beta^{\vert \vert y \vert \vert - \vert \vert x \vert \vert - 1}) \rfloor$. } +The first obvious case is when $\hat k = \beta - 1$ in which case the proof is concluded since the real quotient cannot be larger. For all other +cases $\hat k = \lfloor (y_t\beta + y_{t-1}) / x_s \rfloor$ and $\hat k x_s \ge y_t\beta + y_{t-1} - x_s + 1$. The latter portion of the inequalility +$-x_s + 1$ arises from the fact that a truncated integer division will give the same quotient for at most $x_s - 1$ values. Next a series of +inequalities will prove the hypothesis. + +\begin{equation} +y - \hat k x \le y - \hat k x_s\beta^s +\end{equation} + +This is trivially true since $x \ge x_s\beta^s$. Next we replace $\hat kx_s\beta^s$ by the previous inequality for $\hat kx_s$. + +\begin{equation} +y - \hat k x \le y_t\beta^t + \ldots + y_0 - (y_t\beta^t + y_{t-1}\beta^{t-1} - x_s\beta^t + \beta^s) +\end{equation} + +By simplifying the previous inequality the following inequality is formed. + +\begin{equation} +y - \hat k x \le y_{t-2}\beta^{t-2} + \ldots + y_0 + x_s\beta^s - \beta^s +\end{equation} + +Subsequently, + +\begin{equation} +y_{t-2}\beta^{t-2} + \ldots + y_0 + x_s\beta^s - \beta^s < x_s\beta^s \le x +\end{equation} + +Which proves that $y - \hat kx \le x$ and by consequence $\hat k \ge k$ which concludes the proof. \textbf{QED} + + +\subsection{Normalized Integers} +For the purposes of division a normalized input is when the divisors leading digit $x_n$ is greater than or equal to $\beta / 2$. By multiplying both +$x$ and $y$ by $j = \lfloor (\beta / 2) / x_n \rfloor$ the quotient remains unchanged and the remainder is simply $j$ times the original +remainder. The purpose of normalization is to ensure the leading digit of the divisor is sufficiently large such that the estimated quotient will +lie in the domain of a single digit. Consider the maximum dividend $(\beta - 1) \cdot \beta + (\beta - 1)$ and the minimum divisor $\beta / 2$. + +\begin{equation} +{{\beta^2 - 1} \over { \beta / 2}} \le 2\beta - {2 \over \beta} +\end{equation} + +At most the quotient approaches $2\beta$, however, in practice this will not occur since that would imply the previous quotient digit was too small. + +\subsection{Radix-$\beta$ Division with Remainder} +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_div}. \\ +\textbf{Input}. mp\_int $a, b$ \\ +\textbf{Output}. $c = \lfloor a/b \rfloor$, $d = a - bc$ \\ +\hline \\ +1. If $b = 0$ return(\textit{MP\_VAL}). \\ +2. If $\vert a \vert < \vert b \vert$ then do \\ +\hspace{3mm}2.1 $d \leftarrow a$ \\ +\hspace{3mm}2.2 $c \leftarrow 0$ \\ +\hspace{3mm}2.3 Return(\textit{MP\_OKAY}). \\ +\\ +Setup the quotient to receive the digits. \\ +3. Grow $q$ to $a.used + 2$ digits. \\ +4. $q \leftarrow 0$ \\ +5. $x \leftarrow \vert a \vert , y \leftarrow \vert b \vert$ \\ +6. $sign \leftarrow \left \lbrace \begin{array}{ll} + MP\_ZPOS & \mbox{if }a.sign = b.sign \\ + MP\_NEG & \mbox{otherwise} \\ + \end{array} \right .$ \\ +\\ +Normalize the inputs such that the leading digit of $y$ is greater than or equal to $\beta / 2$. \\ +7. $norm \leftarrow (lg(\beta) - 1) - (\lceil lg(y) \rceil \mbox{ (mod }lg(\beta)\mbox{)})$ \\ +8. $x \leftarrow x \cdot 2^{norm}, y \leftarrow y \cdot 2^{norm}$ \\ +\\ +Find the leading digit of the quotient. \\ +9. $n \leftarrow x.used - 1, t \leftarrow y.used - 1$ \\ +10. $y \leftarrow y \cdot \beta^{n - t}$ \\ +11. While ($x \ge y$) do \\ +\hspace{3mm}11.1 $q_{n - t} \leftarrow q_{n - t} + 1$ \\ +\hspace{3mm}11.2 $x \leftarrow x - y$ \\ +12. $y \leftarrow \lfloor y / \beta^{n-t} \rfloor$ \\ +\\ +Continued on the next page. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_div} +\end{figure} + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_div} (continued). \\ +\textbf{Input}. mp\_int $a, b$ \\ +\textbf{Output}. $c = \lfloor a/b \rfloor$, $d = a - bc$ \\ +\hline \\ +Now find the remainder fo the digits. \\ +13. for $i$ from $n$ down to $(t + 1)$ do \\ +\hspace{3mm}13.1 If $i > x.used$ then jump to the next iteration of this loop. \\ +\hspace{3mm}13.2 If $x_{i} = y_{t}$ then \\ +\hspace{6mm}13.2.1 $q_{i - t - 1} \leftarrow \beta - 1$ \\ +\hspace{3mm}13.3 else \\ +\hspace{6mm}13.3.1 $\hat r \leftarrow x_{i} \cdot \beta + x_{i - 1}$ \\ +\hspace{6mm}13.3.2 $\hat r \leftarrow \lfloor \hat r / y_{t} \rfloor$ \\ +\hspace{6mm}13.3.3 $q_{i - t - 1} \leftarrow \hat r$ \\ +\hspace{3mm}13.4 $q_{i - t - 1} \leftarrow q_{i - t - 1} + 1$ \\ +\\ +Fixup quotient estimation. \\ +\hspace{3mm}13.5 Loop \\ +\hspace{6mm}13.5.1 $q_{i - t - 1} \leftarrow q_{i - t - 1} - 1$ \\ +\hspace{6mm}13.5.2 t$1 \leftarrow 0$ \\ +\hspace{6mm}13.5.3 t$1_0 \leftarrow y_{t - 1}, $ t$1_1 \leftarrow y_t,$ t$1.used \leftarrow 2$ \\ +\hspace{6mm}13.5.4 $t1 \leftarrow t1 \cdot q_{i - t - 1}$ \\ +\hspace{6mm}13.5.5 t$2_0 \leftarrow x_{i - 2}, $ t$2_1 \leftarrow x_{i - 1}, $ t$2_2 \leftarrow x_i, $ t$2.used \leftarrow 3$ \\ +\hspace{6mm}13.5.6 If $\vert t1 \vert > \vert t2 \vert$ then goto step 13.5. \\ +\hspace{3mm}13.6 t$1 \leftarrow y \cdot q_{i - t - 1}$ \\ +\hspace{3mm}13.7 t$1 \leftarrow $ t$1 \cdot \beta^{i - t - 1}$ \\ +\hspace{3mm}13.8 $x \leftarrow x - $ t$1$ \\ +\hspace{3mm}13.9 If $x.sign = MP\_NEG$ then \\ +\hspace{6mm}13.10 t$1 \leftarrow y$ \\ +\hspace{6mm}13.11 t$1 \leftarrow $ t$1 \cdot \beta^{i - t - 1}$ \\ +\hspace{6mm}13.12 $x \leftarrow x + $ t$1$ \\ +\hspace{6mm}13.13 $q_{i - t - 1} \leftarrow q_{i - t - 1} - 1$ \\ +\\ +Finalize the result. \\ +14. Clamp excess digits of $q$ \\ +15. $c \leftarrow q, c.sign \leftarrow sign$ \\ +16. $x.sign \leftarrow a.sign$ \\ +17. $d \leftarrow \lfloor x / 2^{norm} \rfloor$ \\ +18. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_div (continued)} +\end{figure} +\textbf{Algorithm mp\_div.} +This algorithm will calculate quotient and remainder from an integer division given a dividend and divisor. The algorithm is a signed +division and will produce a fully qualified quotient and remainder. + +First the divisor $b$ must be non-zero which is enforced in step one. If the divisor is larger than the dividend than the quotient is implicitly +zero and the remainder is the dividend. + +After the first two trivial cases of inputs are handled the variable $q$ is setup to receive the digits of the quotient. Two unsigned copies of the +divisor $y$ and dividend $x$ are made as well. The core of the division algorithm is an unsigned division and will only work if the values are +positive. Now the two values $x$ and $y$ must be normalized such that the leading digit of $y$ is greater than or equal to $\beta / 2$. +This is performed by shifting both to the left by enough bits to get the desired normalization. + +At this point the division algorithm can begin producing digits of the quotient. Recall that maximum value of the estimation used is +$2\beta - {2 \over \beta}$ which means that a digit of the quotient must be first produced by another means. In this case $y$ is shifted +to the left (\textit{step ten}) so that it has the same number of digits as $x$. The loop on step eleven will subtract multiples of the +shifted copy of $y$ until $x$ is smaller. Since the leading digit of $y$ is greater than or equal to $\beta/2$ this loop will iterate at most two +times to produce the desired leading digit of the quotient. + +Now the remainder of the digits can be produced. The equation $\hat q = \lfloor {{x_i \beta + x_{i-1}}\over y_t} \rfloor$ is used to fairly +accurately approximate the true quotient digit. The estimation can in theory produce an estimation as high as $2\beta - {2 \over \beta}$ but by +induction the upper quotient digit is correct (\textit{as established on step eleven}) and the estimate must be less than $\beta$. + +Recall from section~\ref{sec:divest} that the estimation is never too low but may be too high. The next step of the estimation process is +to refine the estimation. The loop on step 13.5 uses $x_i\beta^2 + x_{i-1}\beta + x_{i-2}$ and $q_{i - t - 1}(y_t\beta + y_{t-1})$ as a higher +order approximation to adjust the quotient digit. + +After both phases of estimation the quotient digit may still be off by a value of one\footnote{This is similar to the error introduced +by optimizing Barrett reduction.}. Steps 13.6 and 13.7 subtract the multiple of the divisor from the dividend (\textit{Similar to step 3.3 of +algorithm~\ref{fig:raddiv}} and then subsequently add a multiple of the divisor if the quotient was too large. + +Now that the quotient has been determine finializing the result is a matter of clamping the quotient, fixing the sizes and de-normalizing the +remainder. An important aspect of this algorithm seemingly overlooked in other descriptions such as that of Algorithm 14.20 HAC \cite[pp. 598]{HAC} +is that when the estimations are being made (\textit{inside the loop on step 13.5}) that the digits $y_{t-1}$, $x_{i-2}$ and $x_{i-1}$ may lie +outside their respective boundaries. For example, if $t = 0$ or $i \le 1$ then the digits would be undefined. In those cases the digits should +respectively be replaced with a zero. + +EXAM,bn_mp_div.c + +The implementation of this algorithm differs slightly from the pseudo code presented previously. In this algorithm either of the quotient $c$ or +remainder $d$ may be passed as a \textbf{NULL} pointer which indicates their value is not desired. For example, the C code to call the division +algorithm with only the quotient is + +\begin{verbatim} +mp_div(&a, &b, &c, NULL); /* c = [a/b] */ +\end{verbatim} + +Lines @37,if@ and @42,if@ handle the two trivial cases of inputs which are division by zero and dividend smaller than the divisor +respectively. After the two trivial cases all of the temporary variables are initialized. Line @76,neg@ determines the sign of +the quotient and line @77,sign@ ensures that both $x$ and $y$ are positive. + +The number of bits in the leading digit is calculated on line @80,norm@. Implictly an mp\_int with $r$ digits will require $lg(\beta)(r-1) + k$ bits +of precision which when reduced modulo $lg(\beta)$ produces the value of $k$. In this case $k$ is the number of bits in the leading digit which is +exactly what is required. For the algorithm to operate $k$ must equal $lg(\beta) - 1$ and when it does not the inputs must be normalized by shifting +them to the left by $lg(\beta) - 1 - k$ bits. + +Throughout the variables $n$ and $t$ will represent the highest digit of $x$ and $y$ respectively. These are first used to produce the +leading digit of the quotient. The loop beginning on line @113,for@ will produce the remainder of the quotient digits. + +The conditional ``continue'' on line @114,if@ is used to prevent the algorithm from reading past the leading edge of $x$ which can occur when the +algorithm eliminates multiple non-zero digits in a single iteration. This ensures that $x_i$ is always non-zero since by definition the digits +above the $i$'th position $x$ must be zero in order for the quotient to be precise\footnote{Precise as far as integer division is concerned.}. + +Lines @142,t1@, @143,t1@ and @150,t2@ through @152,t2@ manually construct the high accuracy estimations by setting the digits of the two mp\_int +variables directly. + +\section{Single Digit Helpers} + +This section briefly describes a series of single digit helper algorithms which come in handy when working with small constants. All of +the helper functions assume the single digit input is positive and will treat them as such. + +\subsection{Single Digit Addition and Subtraction} + +Both addition and subtraction are performed by ``cheating'' and using mp\_set followed by the higher level addition or subtraction +algorithms. As a result these algorithms are subtantially simpler with a slight cost in performance. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_add\_d}. \\ +\textbf{Input}. mp\_int $a$ and a mp\_digit $b$ \\ +\textbf{Output}. $c = a + b$ \\ +\hline \\ +1. $t \leftarrow b$ (\textit{mp\_set}) \\ +2. $c \leftarrow a + t$ \\ +3. Return(\textit{MP\_OKAY}) \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_add\_d} +\end{figure} + +\textbf{Algorithm mp\_add\_d.} +This algorithm initiates a temporary mp\_int with the value of the single digit and uses algorithm mp\_add to add the two values together. + +EXAM,bn_mp_add_d.c + +Clever use of the letter 't'. + +\subsubsection{Subtraction} +The single digit subtraction algorithm mp\_sub\_d is essentially the same except it uses mp\_sub to subtract the digit from the mp\_int. + +\subsection{Single Digit Multiplication} +Single digit multiplication arises enough in division and radix conversion that it ought to be implement as a special case of the baseline +multiplication algorithm. Essentially this algorithm is a modified version of algorithm s\_mp\_mul\_digs where one of the multiplicands +only has one digit. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_mul\_d}. \\ +\textbf{Input}. mp\_int $a$ and a mp\_digit $b$ \\ +\textbf{Output}. $c = ab$ \\ +\hline \\ +1. $pa \leftarrow a.used$ \\ +2. Grow $c$ to at least $pa + 1$ digits. \\ +3. $oldused \leftarrow c.used$ \\ +4. $c.used \leftarrow pa + 1$ \\ +5. $c.sign \leftarrow a.sign$ \\ +6. $\mu \leftarrow 0$ \\ +7. for $ix$ from $0$ to $pa - 1$ do \\ +\hspace{3mm}7.1 $\hat r \leftarrow \mu + a_{ix}b$ \\ +\hspace{3mm}7.2 $c_{ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ +\hspace{3mm}7.3 $\mu \leftarrow \lfloor \hat r / \beta \rfloor$ \\ +8. $c_{pa} \leftarrow \mu$ \\ +9. for $ix$ from $pa + 1$ to $oldused$ do \\ +\hspace{3mm}9.1 $c_{ix} \leftarrow 0$ \\ +10. Clamp excess digits of $c$. \\ +11. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_mul\_d} +\end{figure} +\textbf{Algorithm mp\_mul\_d.} +This algorithm quickly multiplies an mp\_int by a small single digit value. It is specially tailored to the job and has a minimal of overhead. +Unlike the full multiplication algorithms this algorithm does not require any significnat temporary storage or memory allocations. + +EXAM,bn_mp_mul_d.c + +In this implementation the destination $c$ may point to the same mp\_int as the source $a$ since the result is written after the digit is +read from the source. This function uses pointer aliases $tmpa$ and $tmpc$ for the digits of $a$ and $c$ respectively. + +\subsection{Single Digit Division} +Like the single digit multiplication algorithm, single digit division is also a fairly common algorithm used in radix conversion. Since the +divisor is only a single digit a specialized variant of the division algorithm can be used to compute the quotient. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_div\_d}. \\ +\textbf{Input}. mp\_int $a$ and a mp\_digit $b$ \\ +\textbf{Output}. $c = \lfloor a / b \rfloor, d = a - cb$ \\ +\hline \\ +1. If $b = 0$ then return(\textit{MP\_VAL}).\\ +2. If $b = 3$ then use algorithm mp\_div\_3 instead. \\ +3. Init $q$ to $a.used$ digits. \\ +4. $q.used \leftarrow a.used$ \\ +5. $q.sign \leftarrow a.sign$ \\ +6. $\hat w \leftarrow 0$ \\ +7. for $ix$ from $a.used - 1$ down to $0$ do \\ +\hspace{3mm}7.1 $\hat w \leftarrow \hat w \beta + a_{ix}$ \\ +\hspace{3mm}7.2 If $\hat w \ge b$ then \\ +\hspace{6mm}7.2.1 $t \leftarrow \lfloor \hat w / b \rfloor$ \\ +\hspace{6mm}7.2.2 $\hat w \leftarrow \hat w \mbox{ (mod }b\mbox{)}$ \\ +\hspace{3mm}7.3 else\\ +\hspace{6mm}7.3.1 $t \leftarrow 0$ \\ +\hspace{3mm}7.4 $q_{ix} \leftarrow t$ \\ +8. $d \leftarrow \hat w$ \\ +9. Clamp excess digits of $q$. \\ +10. $c \leftarrow q$ \\ +11. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_div\_d} +\end{figure} +\textbf{Algorithm mp\_div\_d.} +This algorithm divides the mp\_int $a$ by the single mp\_digit $b$ using an optimized approach. Essentially in every iteration of the +algorithm another digit of the dividend is reduced and another digit of quotient produced. Provided $b < \beta$ the value of $\hat w$ +after step 7.1 will be limited such that $0 \le \lfloor \hat w / b \rfloor < \beta$. + +If the divisor $b$ is equal to three a variant of this algorithm is used which is called mp\_div\_3. It replaces the division by three with +a multiplication by $\lfloor \beta / 3 \rfloor$ and the appropriate shift and residual fixup. In essence it is much like the Barrett reduction +from chapter seven. + +EXAM,bn_mp_div_d.c + +Like the implementation of algorithm mp\_div this algorithm allows either of the quotient or remainder to be passed as a \textbf{NULL} pointer to +indicate the respective value is not required. This allows a trivial single digit modular reduction algorithm, mp\_mod\_d to be created. + +The division and remainder on lines @44,/@ and @45,%@ can be replaced often by a single division on most processors. For example, the 32-bit x86 based +processors can divide a 64-bit quantity by a 32-bit quantity and produce the quotient and remainder simultaneously. Unfortunately the GCC +compiler does not recognize that optimization and will actually produce two function calls to find the quotient and remainder respectively. + +\subsection{Single Digit Root Extraction} + +Finding the $n$'th root of an integer is fairly easy as far as numerical analysis is concerned. Algorithms such as the Newton-Raphson approximation +(\ref{eqn:newton}) series will converge very quickly to a root for any continuous function $f(x)$. + +\begin{equation} +x_{i+1} = x_i - {f(x_i) \over f'(x_i)} +\label{eqn:newton} +\end{equation} + +In this case the $n$'th root is desired and $f(x) = x^n - a$ where $a$ is the integer of which the root is desired. The derivative of $f(x)$ is +simply $f'(x) = nx^{n - 1}$. Of particular importance is that this algorithm will be used over the integers not over the a more continuous domain +such as the real numbers. As a result the root found can be above the true root by few and must be manually adjusted. Ideally at the end of the +algorithm the $n$'th root $b$ of an integer $a$ is desired such that $b^n \le a$. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_n\_root}. \\ +\textbf{Input}. mp\_int $a$ and a mp\_digit $b$ \\ +\textbf{Output}. $c^b \le a$ \\ +\hline \\ +1. If $b$ is even and $a.sign = MP\_NEG$ return(\textit{MP\_VAL}). \\ +2. $sign \leftarrow a.sign$ \\ +3. $a.sign \leftarrow MP\_ZPOS$ \\ +4. t$2 \leftarrow 2$ \\ +5. Loop \\ +\hspace{3mm}5.1 t$1 \leftarrow $ t$2$ \\ +\hspace{3mm}5.2 t$3 \leftarrow $ t$1^{b - 1}$ \\ +\hspace{3mm}5.3 t$2 \leftarrow $ t$3 $ $\cdot$ t$1$ \\ +\hspace{3mm}5.4 t$2 \leftarrow $ t$2 - a$ \\ +\hspace{3mm}5.5 t$3 \leftarrow $ t$3 \cdot b$ \\ +\hspace{3mm}5.6 t$3 \leftarrow \lfloor $t$2 / $t$3 \rfloor$ \\ +\hspace{3mm}5.7 t$2 \leftarrow $ t$1 - $ t$3$ \\ +\hspace{3mm}5.8 If t$1 \ne $ t$2$ then goto step 5. \\ +6. Loop \\ +\hspace{3mm}6.1 t$2 \leftarrow $ t$1^b$ \\ +\hspace{3mm}6.2 If t$2 > a$ then \\ +\hspace{6mm}6.2.1 t$1 \leftarrow $ t$1 - 1$ \\ +\hspace{6mm}6.2.2 Goto step 6. \\ +7. $a.sign \leftarrow sign$ \\ +8. $c \leftarrow $ t$1$ \\ +9. $c.sign \leftarrow sign$ \\ +10. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_n\_root} +\end{figure} +\textbf{Algorithm mp\_n\_root.} +This algorithm finds the integer $n$'th root of an input using the Newton-Raphson approach. It is partially optimized based on the observation +that the numerator of ${f(x) \over f'(x)}$ can be derived from a partial denominator. That is at first the denominator is calculated by finding +$x^{b - 1}$. This value can then be multiplied by $x$ and have $a$ subtracted from it to find the numerator. This saves a total of $b - 1$ +multiplications by t$1$ inside the loop. + +The initial value of the approximation is t$2 = 2$ which allows the algorithm to start with very small values and quickly converge on the +root. Ideally this algorithm is meant to find the $n$'th root of an input where $n$ is bounded by $2 \le n \le 5$. + +EXAM,bn_mp_n_root.c + +\section{Random Number Generation} + +Random numbers come up in a variety of activities from public key cryptography to simple simulations and various randomized algorithms. Pollard-Rho +factoring for example, can make use of random values as starting points to find factors of a composite integer. In this case the algorithm presented +is solely for simulations and not intended for cryptographic use. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_rand}. \\ +\textbf{Input}. An integer $b$ \\ +\textbf{Output}. A pseudo-random number of $b$ digits \\ +\hline \\ +1. $a \leftarrow 0$ \\ +2. If $b \le 0$ return(\textit{MP\_OKAY}) \\ +3. Pick a non-zero random digit $d$. \\ +4. $a \leftarrow a + d$ \\ +5. for $ix$ from 1 to $d - 1$ do \\ +\hspace{3mm}5.1 $a \leftarrow a \cdot \beta$ \\ +\hspace{3mm}5.2 Pick a random digit $d$. \\ +\hspace{3mm}5.3 $a \leftarrow a + d$ \\ +6. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_rand} +\end{figure} +\textbf{Algorithm mp\_rand.} +This algorithm produces a pseudo-random integer of $b$ digits. By ensuring that the first digit is non-zero the algorithm also guarantees that the +final result has at least $b$ digits. It relies heavily on a third-part random number generator which should ideally generate uniformly all of +the integers from $0$ to $\beta - 1$. + +EXAM,bn_mp_rand.c + +\section{Formatted Representations} +The ability to emit a radix-$n$ textual representation of an integer is useful for interacting with human parties. For example, the ability to +be given a string of characters such as ``114585'' and turn it into the radix-$\beta$ equivalent would make it easier to enter numbers +into a program. + +\subsection{Reading Radix-n Input} +For the purposes of this text we will assume that a simple lower ASCII map (\ref{fig:ASC}) is used for the values of from $0$ to $63$ to +printable characters. For example, when the character ``N'' is read it represents the integer $23$. The first $16$ characters of the +map are for the common representations up to hexadecimal. After that they match the ``base64'' encoding scheme which are suitable chosen +such that they are printable. While outputting as base64 may not be too helpful for human operators it does allow communication via non binary +mediums. + +\newpage\begin{figure}[here] +\begin{center} +\begin{tabular}{cc|cc|cc|cc} +\hline \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} \\ +\hline +0 & 0 & 1 & 1 & 2 & 2 & 3 & 3 \\ +4 & 4 & 5 & 5 & 6 & 6 & 7 & 7 \\ +8 & 8 & 9 & 9 & 10 & A & 11 & B \\ +12 & C & 13 & D & 14 & E & 15 & F \\ +16 & G & 17 & H & 18 & I & 19 & J \\ +20 & K & 21 & L & 22 & M & 23 & N \\ +24 & O & 25 & P & 26 & Q & 27 & R \\ +28 & S & 29 & T & 30 & U & 31 & V \\ +32 & W & 33 & X & 34 & Y & 35 & Z \\ +36 & a & 37 & b & 38 & c & 39 & d \\ +40 & e & 41 & f & 42 & g & 43 & h \\ +44 & i & 45 & j & 46 & k & 47 & l \\ +48 & m & 49 & n & 50 & o & 51 & p \\ +52 & q & 53 & r & 54 & s & 55 & t \\ +56 & u & 57 & v & 58 & w & 59 & x \\ +60 & y & 61 & z & 62 & $+$ & 63 & $/$ \\ +\hline +\end{tabular} +\end{center} +\caption{Lower ASCII Map} +\label{fig:ASC} +\end{figure} + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_read\_radix}. \\ +\textbf{Input}. A string $str$ of length $sn$ and radix $r$. \\ +\textbf{Output}. The radix-$\beta$ equivalent mp\_int. \\ +\hline \\ +1. If $r < 2$ or $r > 64$ return(\textit{MP\_VAL}). \\ +2. $ix \leftarrow 0$ \\ +3. If $str_0 =$ ``-'' then do \\ +\hspace{3mm}3.1 $ix \leftarrow ix + 1$ \\ +\hspace{3mm}3.2 $sign \leftarrow MP\_NEG$ \\ +4. else \\ +\hspace{3mm}4.1 $sign \leftarrow MP\_ZPOS$ \\ +5. $a \leftarrow 0$ \\ +6. for $iy$ from $ix$ to $sn - 1$ do \\ +\hspace{3mm}6.1 Let $y$ denote the position in the map of $str_{iy}$. \\ +\hspace{3mm}6.2 If $str_{iy}$ is not in the map or $y \ge r$ then goto step 7. \\ +\hspace{3mm}6.3 $a \leftarrow a \cdot r$ \\ +\hspace{3mm}6.4 $a \leftarrow a + y$ \\ +7. If $a \ne 0$ then $a.sign \leftarrow sign$ \\ +8. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_read\_radix} +\end{figure} +\textbf{Algorithm mp\_read\_radix.} +This algorithm will read an ASCII string and produce the radix-$\beta$ mp\_int representation of the same integer. A minus symbol ``-'' may precede the +string to indicate the value is negative, otherwise it is assumed to be positive. The algorithm will read up to $sn$ characters from the input +and will stop when it reads a character it cannot map the algorithm stops reading characters from the string. This allows numbers to be embedded +as part of larger input without any significant problem. + +EXAM,bn_mp_read_radix.c + +\subsection{Generating Radix-$n$ Output} +Generating radix-$n$ output is fairly trivial with a division and remainder algorithm. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_toradix}. \\ +\textbf{Input}. A mp\_int $a$ and an integer $r$\\ +\textbf{Output}. The radix-$r$ representation of $a$ \\ +\hline \\ +1. If $r < 2$ or $r > 64$ return(\textit{MP\_VAL}). \\ +2. If $a = 0$ then $str = $ ``$0$'' and return(\textit{MP\_OKAY}). \\ +3. $t \leftarrow a$ \\ +4. $str \leftarrow$ ``'' \\ +5. if $t.sign = MP\_NEG$ then \\ +\hspace{3mm}5.1 $str \leftarrow str + $ ``-'' \\ +\hspace{3mm}5.2 $t.sign = MP\_ZPOS$ \\ +6. While ($t \ne 0$) do \\ +\hspace{3mm}6.1 $d \leftarrow t \mbox{ (mod }r\mbox{)}$ \\ +\hspace{3mm}6.2 $t \leftarrow \lfloor t / r \rfloor$ \\ +\hspace{3mm}6.3 Look up $d$ in the map and store the equivalent character in $y$. \\ +\hspace{3mm}6.4 $str \leftarrow str + y$ \\ +7. If $str_0 = $``$-$'' then \\ +\hspace{3mm}7.1 Reverse the digits $str_1, str_2, \ldots str_n$. \\ +8. Otherwise \\ +\hspace{3mm}8.1 Reverse the digits $str_0, str_1, \ldots str_n$. \\ +9. Return(\textit{MP\_OKAY}).\\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_toradix} +\end{figure} +\textbf{Algorithm mp\_toradix.} +This algorithm computes the radix-$r$ representation of an mp\_int $a$. The ``digits'' of the representation are extracted by reducing +successive powers of $\lfloor a / r^k \rfloor$ the input modulo $r$ until $r^k > a$. Note that instead of actually dividing by $r^k$ in +each iteration the quotient $\lfloor a / r \rfloor$ is saved for the next iteration. As a result a series of trivial $n \times 1$ divisions +are required instead of a series of $n \times k$ divisions. One design flaw of this approach is that the digits are produced in the reverse order +(see~\ref{fig:mpradix}). To remedy this flaw the digits must be swapped or simply ``reversed''. + +\begin{figure} +\begin{center} +\begin{tabular}{|c|c|c|} +\hline \textbf{Value of $a$} & \textbf{Value of $d$} & \textbf{Value of $str$} \\ +\hline $1234$ & -- & -- \\ +\hline $123$ & $4$ & ``4'' \\ +\hline $12$ & $3$ & ``43'' \\ +\hline $1$ & $2$ & ``432'' \\ +\hline $0$ & $1$ & ``4321'' \\ +\hline +\end{tabular} +\end{center} +\caption{Example of Algorithm mp\_toradix.} +\label{fig:mpradix} +\end{figure} + +EXAM,bn_mp_toradix.c + +\chapter{Number Theoretic Algorithms} +This chapter discusses several fundamental number theoretic algorithms such as the greatest common divisor, least common multiple and Jacobi +symbol computation. These algorithms arise as essential components in several key cryptographic algorithms such as the RSA public key algorithm and +various Sieve based factoring algorithms. + +\section{Greatest Common Divisor} +The greatest common divisor of two integers $a$ and $b$, often denoted as $(a, b)$ is the largest integer $k$ that is a proper divisor of +both $a$ and $b$. That is, $k$ is the largest integer such that $0 \equiv a \mbox{ (mod }k\mbox{)}$ and $0 \equiv b \mbox{ (mod }k\mbox{)}$ occur +simultaneously. + +The most common approach (cite) is to reduce one input modulo another. That is if $a$ and $b$ are divisible by some integer $k$ and if $qa + r = b$ then +$r$ is also divisible by $k$. The reduction pattern follows $\left < a , b \right > \rightarrow \left < b, a \mbox{ mod } b \right >$. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{Greatest Common Divisor (I)}. \\ +\textbf{Input}. Two positive integers $a$ and $b$ greater than zero. \\ +\textbf{Output}. The greatest common divisor $(a, b)$. \\ +\hline \\ +1. While ($b > 0$) do \\ +\hspace{3mm}1.1 $r \leftarrow a \mbox{ (mod }b\mbox{)}$ \\ +\hspace{3mm}1.2 $a \leftarrow b$ \\ +\hspace{3mm}1.3 $b \leftarrow r$ \\ +2. Return($a$). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm Greatest Common Divisor (I)} +\label{fig:gcd1} +\end{figure} + +This algorithm will quickly converge on the greatest common divisor since the residue $r$ tends diminish rapidly. However, divisions are +relatively expensive operations to perform and should ideally be avoided. There is another approach based on a similar relationship of +greatest common divisors. The faster approach is based on the observation that if $k$ divides both $a$ and $b$ it will also divide $a - b$. +In particular, we would like $a - b$ to decrease in magnitude which implies that $b \ge a$. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{Greatest Common Divisor (II)}. \\ +\textbf{Input}. Two positive integers $a$ and $b$ greater than zero. \\ +\textbf{Output}. The greatest common divisor $(a, b)$. \\ +\hline \\ +1. While ($b > 0$) do \\ +\hspace{3mm}1.1 Swap $a$ and $b$ such that $a$ is the smallest of the two. \\ +\hspace{3mm}1.2 $b \leftarrow b - a$ \\ +2. Return($a$). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm Greatest Common Divisor (II)} +\label{fig:gcd2} +\end{figure} + +\textbf{Proof} \textit{Algorithm~\ref{fig:gcd2} will return the greatest common divisor of $a$ and $b$.} +The algorithm in figure~\ref{fig:gcd2} will eventually terminate since $b \ge a$ the subtraction in step 1.2 will be a value less than $b$. In other +words in every iteration that tuple $\left < a, b \right >$ decrease in magnitude until eventually $a = b$. Since both $a$ and $b$ are always +divisible by the greatest common divisor (\textit{until the last iteration}) and in the last iteration of the algorithm $b = 0$, therefore, in the +second to last iteration of the algorithm $b = a$ and clearly $(a, a) = a$ which concludes the proof. \textbf{QED}. + +As a matter of practicality algorithm \ref{fig:gcd1} decreases far too slowly to be useful. Specially if $b$ is much larger than $a$ such that +$b - a$ is still very much larger than $a$. A simple addition to the algorithm is to divide $b - a$ by a power of some integer $p$ which does +not divide the greatest common divisor but will divide $b - a$. In this case ${b - a} \over p$ is also an integer and still divisible by +the greatest common divisor. + +However, instead of factoring $b - a$ to find a suitable value of $p$ the powers of $p$ can be removed from $a$ and $b$ that are in common first. +Then inside the loop whenever $b - a$ is divisible by some power of $p$ it can be safely removed. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{Greatest Common Divisor (III)}. \\ +\textbf{Input}. Two positive integers $a$ and $b$ greater than zero. \\ +\textbf{Output}. The greatest common divisor $(a, b)$. \\ +\hline \\ +1. $k \leftarrow 0$ \\ +2. While $a$ and $b$ are both divisible by $p$ do \\ +\hspace{3mm}2.1 $a \leftarrow \lfloor a / p \rfloor$ \\ +\hspace{3mm}2.2 $b \leftarrow \lfloor b / p \rfloor$ \\ +\hspace{3mm}2.3 $k \leftarrow k + 1$ \\ +3. While $a$ is divisible by $p$ do \\ +\hspace{3mm}3.1 $a \leftarrow \lfloor a / p \rfloor$ \\ +4. While $b$ is divisible by $p$ do \\ +\hspace{3mm}4.1 $b \leftarrow \lfloor b / p \rfloor$ \\ +5. While ($b > 0$) do \\ +\hspace{3mm}5.1 Swap $a$ and $b$ such that $a$ is the smallest of the two. \\ +\hspace{3mm}5.2 $b \leftarrow b - a$ \\ +\hspace{3mm}5.3 While $b$ is divisible by $p$ do \\ +\hspace{6mm}5.3.1 $b \leftarrow \lfloor b / p \rfloor$ \\ +6. Return($a \cdot p^k$). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm Greatest Common Divisor (III)} +\label{fig:gcd3} +\end{figure} + +This algorithm is based on the first except it removes powers of $p$ first and inside the main loop to ensure the tuple $\left < a, b \right >$ +decreases more rapidly. The first loop on step two removes powers of $p$ that are in common. A count, $k$, is kept which will present a common +divisor of $p^k$. After step two the remaining common divisor of $a$ and $b$ cannot be divisible by $p$. This means that $p$ can be safely +divided out of the difference $b - a$ so long as the division leaves no remainder. + +In particular the value of $p$ should be chosen such that the division on step 5.3.1 occur often. It also helps that division by $p$ be easy +to compute. The ideal choice of $p$ is two since division by two amounts to a right logical shift. Another important observation is that by +step five both $a$ and $b$ are odd. Therefore, the diffrence $b - a$ must be even which means that each iteration removes one bit from the +largest of the pair. + +\subsection{Complete Greatest Common Divisor} +The algorithms presented so far cannot handle inputs which are zero or negative. The following algorithm can handle all input cases properly +and will produce the greatest common divisor. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_gcd}. \\ +\textbf{Input}. mp\_int $a$ and $b$ \\ +\textbf{Output}. The greatest common divisor $c = (a, b)$. \\ +\hline \\ +1. If $a = 0$ and $b \ne 0$ then \\ +\hspace{3mm}1.1 $c \leftarrow b$ \\ +\hspace{3mm}1.2 Return(\textit{MP\_OKAY}). \\ +2. If $a \ne 0$ and $b = 0$ then \\ +\hspace{3mm}2.1 $c \leftarrow a$ \\ +\hspace{3mm}2.2 Return(\textit{MP\_OKAY}). \\ +3. If $a = b = 0$ then \\ +\hspace{3mm}3.1 $c \leftarrow 1$ \\ +\hspace{3mm}3.2 Return(\textit{MP\_OKAY}). \\ +4. $u \leftarrow \vert a \vert, v \leftarrow \vert b \vert$ \\ +5. $k \leftarrow 0$ \\ +6. While $u.used > 0$ and $v.used > 0$ and $u_0 \equiv v_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ +\hspace{3mm}6.1 $k \leftarrow k + 1$ \\ +\hspace{3mm}6.2 $u \leftarrow \lfloor u / 2 \rfloor$ \\ +\hspace{3mm}6.3 $v \leftarrow \lfloor v / 2 \rfloor$ \\ +7. While $u.used > 0$ and $u_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ +\hspace{3mm}7.1 $u \leftarrow \lfloor u / 2 \rfloor$ \\ +8. While $v.used > 0$ and $v_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ +\hspace{3mm}8.1 $v \leftarrow \lfloor v / 2 \rfloor$ \\ +9. While $v.used > 0$ \\ +\hspace{3mm}9.1 If $\vert u \vert > \vert v \vert$ then \\ +\hspace{6mm}9.1.1 Swap $u$ and $v$. \\ +\hspace{3mm}9.2 $v \leftarrow \vert v \vert - \vert u \vert$ \\ +\hspace{3mm}9.3 While $v.used > 0$ and $v_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ +\hspace{6mm}9.3.1 $v \leftarrow \lfloor v / 2 \rfloor$ \\ +10. $c \leftarrow u \cdot 2^k$ \\ +11. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_gcd} +\end{figure} +\textbf{Algorithm mp\_gcd.} +This algorithm will produce the greatest common divisor of two mp\_ints $a$ and $b$. The algorithm was originally based on Algorithm B of +Knuth \cite[pp. 338]{TAOCPV2} but has been modified to be simpler to explain. In theory it achieves the same asymptotic working time as +Algorithm B and in practice this appears to be true. + +The first three steps handle the cases where either one of or both inputs are zero. If either input is zero the greatest common divisor is the +largest input or zero if they are both zero. If the inputs are not trivial than $u$ and $v$ are assigned the absolute values of +$a$ and $b$ respectively and the algorithm will proceed to reduce the pair. + +Step six will divide out any common factors of two and keep track of the count in the variable $k$. After this step two is no longer a +factor of the remaining greatest common divisor between $u$ and $v$ and can be safely evenly divided out of either whenever they are even. Step +seven and eight ensure that the $u$ and $v$ respectively have no more factors of two. At most only one of the while loops will iterate since +they cannot both be even. + +By step nine both of $u$ and $v$ are odd which is required for the inner logic. First the pair are swapped such that $v$ is equal to +or greater than $u$. This ensures that the subtraction on step 9.2 will always produce a positive and even result. Step 9.3 removes any +factors of two from the difference $u$ to ensure that in the next iteration of the loop both are once again odd. + +After $v = 0$ occurs the variable $u$ has the greatest common divisor of the pair $\left < u, v \right >$ just after step six. The result +must be adjusted by multiplying by the common factors of two ($2^k$) removed earlier. + +EXAM,bn_mp_gcd.c + +This function makes use of the macros mp\_iszero and mp\_iseven. The former evaluates to $1$ if the input mp\_int is equivalent to the +integer zero otherwise it evaluates to $0$. The latter evaluates to $1$ if the input mp\_int represents a non-zero even integer otherwise +it evaluates to $0$. Note that just because mp\_iseven may evaluate to $0$ does not mean the input is odd, it could also be zero. The three +trivial cases of inputs are handled on lines @25,zero@ through @34,}@. After those lines the inputs are assumed to be non-zero. + +Lines @36,if@ and @40,if@ make local copies $u$ and $v$ of the inputs $a$ and $b$ respectively. At this point the common factors of two +must be divided out of the two inputs. The while loop on line @49,while@ iterates so long as both are even. The local integer $k$ is used to +keep track of how many factors of $2$ are pulled out of both values. It is assumed that the number of factors will not exceed the maximum +value of a C ``int'' data type\footnote{Strictly speaking no array in C may have more than entries than are accessible by an ``int'' so this is not +a limitation.}. + +At this point there are no more common factors of two in the two values. The while loops on lines @60,while@ and @65,while@ remove any independent +factors of two such that both $u$ and $v$ are guaranteed to be an odd integer before hitting the main body of the algorithm. The while loop +on line @71, while@ performs the reduction of the pair until $v$ is equal to zero. The unsigned comparison and subtraction algorithms are used in +place of the full signed routines since both values are guaranteed to be positive and the result of the subtraction is guaranteed to be non-negative. + +\section{Least Common Multiple} +The least common multiple of a pair of integers is their product divided by their greatest common divisor. For two integers $a$ and $b$ the +least common multiple is normally denoted as $[ a, b ]$ and numerically equivalent to ${ab} \over {(a, b)}$. For example, if $a = 2 \cdot 2 \cdot 3 = 12$ +and $b = 2 \cdot 3 \cdot 3 \cdot 7 = 126$ the least common multiple is ${126 \over {(12, 126)}} = {126 \over 6} = 21$. + +The least common multiple arises often in coding theory as well as number theory. If two functions have periods of $a$ and $b$ respectively they will +collide, that is be in synchronous states, after only $[ a, b ]$ iterations. This is why, for example, random number generators based on +Linear Feedback Shift Registers (LFSR) tend to use registers with periods which are co-prime (\textit{e.g. the greatest common divisor is one.}). +Similarly in number theory if a composite $n$ has two prime factors $p$ and $q$ then maximal order of any unit of $\Z/n\Z$ will be $[ p - 1, q - 1] $. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_lcm}. \\ +\textbf{Input}. mp\_int $a$ and $b$ \\ +\textbf{Output}. The least common multiple $c = [a, b]$. \\ +\hline \\ +1. $c \leftarrow (a, b)$ \\ +2. $t \leftarrow a \cdot b$ \\ +3. $c \leftarrow \lfloor t / c \rfloor$ \\ +4. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_lcm} +\end{figure} +\textbf{Algorithm mp\_lcm.} +This algorithm computes the least common multiple of two mp\_int inputs $a$ and $b$. It computes the least common multiple directly by +dividing the product of the two inputs by their greatest common divisor. + +EXAM,bn_mp_lcm.c + +\section{Jacobi Symbol Computation} +To explain the Jacobi Symbol we shall first discuss the Legendre function\footnote{Arrg. What is the name of this?} off which the Jacobi symbol is +defined. The Legendre function computes whether or not an integer $a$ is a quadratic residue modulo an odd prime $p$. Numerically it is +equivalent to equation \ref{eqn:legendre}. + +\begin{equation} +a^{(p-1)/2} \equiv \begin{array}{rl} + -1 & \mbox{if }a\mbox{ is a quadratic non-residue.} \\ + 0 & \mbox{if }a\mbox{ divides }p\mbox{.} \\ + 1 & \mbox{if }a\mbox{ is a quadratic residue}. + \end{array} \mbox{ (mod }p\mbox{)} +\label{eqn:legendre} +\end{equation} + +\textbf{Proof.} \textit{Equation \ref{eqn:legendre} correctly identifies the residue status of an integer $a$ modulo a prime $p$.} +An integer $a$ is a quadratic residue if the following equation has a solution. + +\begin{equation} +x^2 \equiv a \mbox{ (mod }p\mbox{)} +\label{eqn:root} +\end{equation} + +Consider the following equation. + +\begin{equation} +0 \equiv x^{p-1} - 1 \equiv \left \lbrace \left (x^2 \right )^{(p-1)/2} - a^{(p-1)/2} \right \rbrace + \left ( a^{(p-1)/2} - 1 \right ) \mbox{ (mod }p\mbox{)} +\label{eqn:rooti} +\end{equation} + +Whether equation \ref{eqn:root} has a solution or not equation \ref{eqn:rooti} is always true. If $a^{(p-1)/2} - 1 \equiv 0 \mbox{ (mod }p\mbox{)}$ +then the quantity in the braces must be zero. By reduction, + +\begin{eqnarray} +\left (x^2 \right )^{(p-1)/2} - a^{(p-1)/2} \equiv 0 \nonumber \\ +\left (x^2 \right )^{(p-1)/2} \equiv a^{(p-1)/2} \nonumber \\ +x^2 \equiv a \mbox{ (mod }p\mbox{)} +\end{eqnarray} + +As a result there must be a solution to the quadratic equation and in turn $a$ must be a quadratic residue. If $a$ does not divide $p$ and $a$ +is not a quadratic residue then the only other value $a^{(p-1)/2}$ may be congruent to is $-1$ since +\begin{equation} +0 \equiv a^{p - 1} - 1 \equiv (a^{(p-1)/2} + 1)(a^{(p-1)/2} - 1) \mbox{ (mod }p\mbox{)} +\end{equation} +One of the terms on the right hand side must be zero. \textbf{QED} + +\subsection{Jacobi Symbol} +The Jacobi symbol is a generalization of the Legendre function for any odd non prime moduli $p$ greater than 2. If $p = \prod_{i=0}^n p_i$ then +the Jacobi symbol $\left ( { a \over p } \right )$ is equal to the following equation. + +\begin{equation} +\left ( { a \over p } \right ) = \left ( { a \over p_0} \right ) \left ( { a \over p_1} \right ) \ldots \left ( { a \over p_n} \right ) +\end{equation} + +By inspection if $p$ is prime the Jacobi symbol is equivalent to the Legendre function. The following facts\footnote{See HAC \cite[pp. 72-74]{HAC} for +further details.} will be used to derive an efficient Jacobi symbol algorithm. Where $p$ is an odd integer greater than two and $a, b \in \Z$ the +following are true. + +\begin{enumerate} +\item $\left ( { a \over p} \right )$ equals $-1$, $0$ or $1$. +\item $\left ( { ab \over p} \right ) = \left ( { a \over p} \right )\left ( { b \over p} \right )$. +\item If $a \equiv b$ then $\left ( { a \over p} \right ) = \left ( { b \over p} \right )$. +\item $\left ( { 2 \over p} \right )$ equals $1$ if $p \equiv 1$ or $7 \mbox{ (mod }8\mbox{)}$. Otherwise, it equals $-1$. +\item $\left ( { a \over p} \right ) \equiv \left ( { p \over a} \right ) \cdot (-1)^{(p-1)(a-1)/4}$. More specifically +$\left ( { a \over p} \right ) = \left ( { p \over a} \right )$ if $p \equiv a \equiv 1 \mbox{ (mod }4\mbox{)}$. +\end{enumerate} + +Using these facts if $a = 2^k \cdot a'$ then + +\begin{eqnarray} +\left ( { a \over p } \right ) = \left ( {{2^k} \over p } \right ) \left ( {a' \over p} \right ) \nonumber \\ + = \left ( {2 \over p } \right )^k \left ( {a' \over p} \right ) +\label{eqn:jacobi} +\end{eqnarray} + +By fact five, + +\begin{equation} +\left ( { a \over p } \right ) = \left ( { p \over a } \right ) \cdot (-1)^{(p-1)(a-1)/4} +\end{equation} + +Subsequently by fact three since $p \equiv (p \mbox{ mod }a) \mbox{ (mod }a\mbox{)}$ then + +\begin{equation} +\left ( { a \over p } \right ) = \left ( { {p \mbox{ mod } a} \over a } \right ) \cdot (-1)^{(p-1)(a-1)/4} +\end{equation} + +By putting both observations into equation \ref{eqn:jacobi} the following simplified equation is formed. + +\begin{equation} +\left ( { a \over p } \right ) = \left ( {2 \over p } \right )^k \left ( {{p\mbox{ mod }a'} \over a'} \right ) \cdot (-1)^{(p-1)(a'-1)/4} +\end{equation} + +The value of $\left ( {{p \mbox{ mod }a'} \over a'} \right )$ can be found by using the same equation recursively. The value of +$\left ( {2 \over p } \right )^k$ equals $1$ if $k$ is even otherwise it equals $\left ( {2 \over p } \right )$. Using this approach the +factors of $p$ do not have to be known. Furthermore, if $(a, p) = 1$ then the algorithm will terminate when the recursion requests the +Jacobi symbol computation of $\left ( {1 \over a'} \right )$ which is simply $1$. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_jacobi}. \\ +\textbf{Input}. mp\_int $a$ and $p$, $a \ge 0$, $p \ge 3$, $p \equiv 1 \mbox{ (mod }2\mbox{)}$ \\ +\textbf{Output}. The Jacobi symbol $c = \left ( {a \over p } \right )$. \\ +\hline \\ +1. If $a = 0$ then \\ +\hspace{3mm}1.1 $c \leftarrow 0$ \\ +\hspace{3mm}1.2 Return(\textit{MP\_OKAY}). \\ +2. If $a = 1$ then \\ +\hspace{3mm}2.1 $c \leftarrow 1$ \\ +\hspace{3mm}2.2 Return(\textit{MP\_OKAY}). \\ +3. $a' \leftarrow a$ \\ +4. $k \leftarrow 0$ \\ +5. While $a'.used > 0$ and $a'_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ +\hspace{3mm}5.1 $k \leftarrow k + 1$ \\ +\hspace{3mm}5.2 $a' \leftarrow \lfloor a' / 2 \rfloor$ \\ +6. If $k \equiv 0 \mbox{ (mod }2\mbox{)}$ then \\ +\hspace{3mm}6.1 $s \leftarrow 1$ \\ +7. else \\ +\hspace{3mm}7.1 $r \leftarrow p_0 \mbox{ (mod }8\mbox{)}$ \\ +\hspace{3mm}7.2 If $r = 1$ or $r = 7$ then \\ +\hspace{6mm}7.2.1 $s \leftarrow 1$ \\ +\hspace{3mm}7.3 else \\ +\hspace{6mm}7.3.1 $s \leftarrow -1$ \\ +8. If $p_0 \equiv a'_0 \equiv 3 \mbox{ (mod }4\mbox{)}$ then \\ +\hspace{3mm}8.1 $s \leftarrow -s$ \\ +9. If $a' \ne 1$ then \\ +\hspace{3mm}9.1 $p' \leftarrow p \mbox{ (mod }a'\mbox{)}$ \\ +\hspace{3mm}9.2 $s \leftarrow s \cdot \mbox{mp\_jacobi}(p', a')$ \\ +10. $c \leftarrow s$ \\ +11. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_jacobi} +\end{figure} +\textbf{Algorithm mp\_jacobi.} +This algorithm computes the Jacobi symbol for an arbitrary positive integer $a$ with respect to an odd integer $p$ greater than three. The algorithm +is based on algorithm 2.149 of HAC \cite[pp. 73]{HAC}. + +Step numbers one and two handle the trivial cases of $a = 0$ and $a = 1$ respectively. Step five determines the number of two factors in the +input $a$. If $k$ is even than the term $\left ( { 2 \over p } \right )^k$ must always evaluate to one. If $k$ is odd than the term evaluates to one +if $p_0$ is congruent to one or seven modulo eight, otherwise it evaluates to $-1$. After the the $\left ( { 2 \over p } \right )^k$ term is handled +the $(-1)^{(p-1)(a'-1)/4}$ is computed and multiplied against the current product $s$. The latter term evaluates to one if both $p$ and $a'$ +are congruent to one modulo four, otherwise it evaluates to negative one. + +By step nine if $a'$ does not equal one a recursion is required. Step 9.1 computes $p' \equiv p \mbox{ (mod }a'\mbox{)}$ and will recurse to compute +$\left ( {p' \over a'} \right )$ which is multiplied against the current Jacobi product. + +EXAM,bn_mp_jacobi.c + +As a matter of practicality the variable $a'$ as per the pseudo-code is reprensented by the variable $a1$ since the $'$ symbol is not valid for a C +variable name character. + +The two simple cases of $a = 0$ and $a = 1$ are handled at the very beginning to simplify the algorithm. If the input is non-trivial the algorithm +has to proceed compute the Jacobi. The variable $s$ is used to hold the current Jacobi product. Note that $s$ is merely a C ``int'' data type since +the values it may obtain are merely $-1$, $0$ and $1$. + +After a local copy of $a$ is made all of the factors of two are divided out and the total stored in $k$. Technically only the least significant +bit of $k$ is required, however, it makes the algorithm simpler to follow to perform an addition. In practice an exclusive-or and addition have the same +processor requirements and neither is faster than the other. + +Line @59, if@ through @70, }@ determines the value of $\left ( { 2 \over p } \right )^k$. If the least significant bit of $k$ is zero than +$k$ is even and the value is one. Otherwise, the value of $s$ depends on which residue class $p$ belongs to modulo eight. The value of +$(-1)^{(p-1)(a'-1)/4}$ is compute and multiplied against $s$ on lines @73, if@ through @75, }@. + +Finally, if $a1$ does not equal one the algorithm must recurse and compute $\left ( {p' \over a'} \right )$. + +\textit{-- Comment about default $s$ and such...} + +\section{Modular Inverse} +\label{sec:modinv} +The modular inverse of a number actually refers to the modular multiplicative inverse. Essentially for any integer $a$ such that $(a, p) = 1$ there +exist another integer $b$ such that $ab \equiv 1 \mbox{ (mod }p\mbox{)}$. The integer $b$ is called the multiplicative inverse of $a$ which is +denoted as $b = a^{-1}$. Technically speaking modular inversion is a well defined operation for any finite ring or field not just for rings and +fields of integers. However, the former will be the matter of discussion. + +The simplest approach is to compute the algebraic inverse of the input. That is to compute $b \equiv a^{\Phi(p) - 1}$. If $\Phi(p)$ is the +order of the multiplicative subgroup modulo $p$ then $b$ must be the multiplicative inverse of $a$. The proof of which is trivial. + +\begin{equation} +ab \equiv a \left (a^{\Phi(p) - 1} \right ) \equiv a^{\Phi(p)} \equiv a^0 \equiv 1 \mbox{ (mod }p\mbox{)} +\end{equation} + +However, as simple as this approach may be it has two serious flaws. It requires that the value of $\Phi(p)$ be known which if $p$ is composite +requires all of the prime factors. This approach also is very slow as the size of $p$ grows. + +A simpler approach is based on the observation that solving for the multiplicative inverse is equivalent to solving the linear +Diophantine\footnote{See LeVeque \cite[pp. 40-43]{LeVeque} for more information.} equation. + +\begin{equation} +ab + pq = 1 +\end{equation} + +Where $a$, $b$, $p$ and $q$ are all integers. If such a pair of integers $ \left < b, q \right >$ exist than $b$ is the multiplicative inverse of +$a$ modulo $p$. The extended Euclidean algorithm (Knuth \cite[pp. 342]{TAOCPV2}) can be used to solve such equations provided $(a, p) = 1$. +However, instead of using that algorithm directly a variant known as the binary Extended Euclidean algorithm will be used in its place. The +binary approach is very similar to the binary greatest common divisor algorithm except it will produce a full solution to the Diophantine +equation. + +\subsection{General Case} +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_invmod}. \\ +\textbf{Input}. mp\_int $a$ and $b$, $(a, b) = 1$, $p \ge 2$, $0 < a < p$. \\ +\textbf{Output}. The modular inverse $c \equiv a^{-1} \mbox{ (mod }b\mbox{)}$. \\ +\hline \\ +1. If $b \le 0$ then return(\textit{MP\_VAL}). \\ +2. If $b_0 \equiv 1 \mbox{ (mod }2\mbox{)}$ then use algorithm fast\_mp\_invmod. \\ +3. $x \leftarrow \vert a \vert, y \leftarrow b$ \\ +4. If $x_0 \equiv y_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ then return(\textit{MP\_VAL}). \\ +5. $B \leftarrow 0, C \leftarrow 0, A \leftarrow 1, D \leftarrow 1$ \\ +6. While $u.used > 0$ and $u_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ +\hspace{3mm}6.1 $u \leftarrow \lfloor u / 2 \rfloor$ \\ +\hspace{3mm}6.2 If ($A.used > 0$ and $A_0 \equiv 1 \mbox{ (mod }2\mbox{)}$) or ($B.used > 0$ and $B_0 \equiv 1 \mbox{ (mod }2\mbox{)}$) then \\ +\hspace{6mm}6.2.1 $A \leftarrow A + y$ \\ +\hspace{6mm}6.2.2 $B \leftarrow B - x$ \\ +\hspace{3mm}6.3 $A \leftarrow \lfloor A / 2 \rfloor$ \\ +\hspace{3mm}6.4 $B \leftarrow \lfloor B / 2 \rfloor$ \\ +7. While $v.used > 0$ and $v_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ +\hspace{3mm}7.1 $v \leftarrow \lfloor v / 2 \rfloor$ \\ +\hspace{3mm}7.2 If ($C.used > 0$ and $C_0 \equiv 1 \mbox{ (mod }2\mbox{)}$) or ($D.used > 0$ and $D_0 \equiv 1 \mbox{ (mod }2\mbox{)}$) then \\ +\hspace{6mm}7.2.1 $C \leftarrow C + y$ \\ +\hspace{6mm}7.2.2 $D \leftarrow D - x$ \\ +\hspace{3mm}7.3 $C \leftarrow \lfloor C / 2 \rfloor$ \\ +\hspace{3mm}7.4 $D \leftarrow \lfloor D / 2 \rfloor$ \\ +8. If $u \ge v$ then \\ +\hspace{3mm}8.1 $u \leftarrow u - v$ \\ +\hspace{3mm}8.2 $A \leftarrow A - C$ \\ +\hspace{3mm}8.3 $B \leftarrow B - D$ \\ +9. else \\ +\hspace{3mm}9.1 $v \leftarrow v - u$ \\ +\hspace{3mm}9.2 $C \leftarrow C - A$ \\ +\hspace{3mm}9.3 $D \leftarrow D - B$ \\ +10. If $u \ne 0$ goto step 6. \\ +11. If $v \ne 1$ return(\textit{MP\_VAL}). \\ +12. While $C \le 0$ do \\ +\hspace{3mm}12.1 $C \leftarrow C + b$ \\ +13. While $C \ge b$ do \\ +\hspace{3mm}13.1 $C \leftarrow C - b$ \\ +14. $c \leftarrow C$ \\ +15. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\end{figure} +\textbf{Algorithm mp\_invmod.} +This algorithm computes the modular multiplicative inverse of an integer $a$ modulo an integer $b$. This algorithm is a variation of the +extended binary Euclidean algorithm from HAC \cite[pp. 608]{HAC}. It has been modified to only compute the modular inverse and not a complete +Diophantine solution. + +If $b \le 0$ than the modulus is invalid and MP\_VAL is returned. Similarly if both $a$ and $b$ are even then there cannot be a multiplicative +inverse for $a$ and the error is reported. + +The astute reader will observe that steps seven through nine are very similar to the binary greatest common divisor algorithm mp\_gcd. In this case +the other variables to the Diophantine equation are solved. The algorithm terminates when $u = 0$ in which case the solution is + +\begin{equation} +Ca + Db = v +\end{equation} + +If $v$, the greatest common divisor of $a$ and $b$ is not equal to one then the algorithm will report an error as no inverse exists. Otherwise, $C$ +is the modular inverse of $a$. The actual value of $C$ is congruent to, but not necessarily equal to, the ideal modular inverse which should lie +within $1 \le a^{-1} < b$. Step numbers twelve and thirteen adjust the inverse until it is in range. If the original input $a$ is within $0 < a < p$ +then only a couple of additions or subtractions will be required to adjust the inverse. + +EXAM,bn_mp_invmod.c + +\subsubsection{Odd Moduli} + +When the modulus $b$ is odd the variables $A$ and $C$ are fixed and are not required to compute the inverse. In particular by attempting to solve +the Diophantine $Cb + Da = 1$ only $B$ and $D$ are required to find the inverse of $a$. + +The algorithm fast\_mp\_invmod is a direct adaptation of algorithm mp\_invmod with all all steps involving either $A$ or $C$ removed. This +optimization will halve the time required to compute the modular inverse. + +\section{Primality Tests} + +A non-zero integer $a$ is said to be prime if it is not divisible by any other integer excluding one and itself. For example, $a = 7$ is prime +since the integers $2 \ldots 6$ do not evenly divide $a$. By contrast, $a = 6$ is not prime since $a = 6 = 2 \cdot 3$. + +Prime numbers arise in cryptography considerably as they allow finite fields to be formed. The ability to determine whether an integer is prime or +not quickly has been a viable subject in cryptography and number theory for considerable time. The algorithms that will be presented are all +probablistic algorithms in that when they report an integer is composite it must be composite. However, when the algorithms report an integer is +prime the algorithm may be incorrect. + +As will be discussed it is possible to limit the probability of error so well that for practical purposes the probablity of error might as +well be zero. For the purposes of these discussions let $n$ represent the candidate integer of which the primality is in question. + +\subsection{Trial Division} + +Trial division means to attempt to evenly divide a candidate integer by small prime integers. If the candidate can be evenly divided it obviously +cannot be prime. By dividing by all primes $1 < p \le \sqrt{n}$ this test can actually prove whether an integer is prime. However, such a test +would require a prohibitive amount of time as $n$ grows. + +Instead of dividing by every prime, a smaller, more mangeable set of primes may be used instead. By performing trial division with only a subset +of the primes less than $\sqrt{n} + 1$ the algorithm cannot prove if a candidate is prime. However, often it can prove a candidate is not prime. + +The benefit of this test is that trial division by small values is fairly efficient. Specially compared to the other algorithms that will be +discussed shortly. The probability that this approach correctly identifies a composite candidate when tested with all primes upto $q$ is given by +$1 - {1.12 \over ln(q)}$. The graph (\ref{pic:primality}, will be added later) demonstrates the probability of success for the range +$3 \le q \le 100$. + +At approximately $q = 30$ the gain of performing further tests diminishes fairly quickly. At $q = 90$ further testing is generally not going to +be of any practical use. In the case of LibTomMath the default limit $q = 256$ was chosen since it is not too high and will eliminate +approximately $80\%$ of all candidate integers. The constant \textbf{PRIME\_SIZE} is equal to the number of primes in the test base. The +array \_\_prime\_tab is an array of the first \textbf{PRIME\_SIZE} prime numbers. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_prime\_is\_divisible}. \\ +\textbf{Input}. mp\_int $a$ \\ +\textbf{Output}. $c = 1$ if $n$ is divisible by a small prime, otherwise $c = 0$. \\ +\hline \\ +1. for $ix$ from $0$ to $PRIME\_SIZE$ do \\ +\hspace{3mm}1.1 $d \leftarrow n \mbox{ (mod }\_\_prime\_tab_{ix}\mbox{)}$ \\ +\hspace{3mm}1.2 If $d = 0$ then \\ +\hspace{6mm}1.2.1 $c \leftarrow 1$ \\ +\hspace{6mm}1.2.2 Return(\textit{MP\_OKAY}). \\ +2. $c \leftarrow 0$ \\ +3. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_prime\_is\_divisible} +\end{figure} +\textbf{Algorithm mp\_prime\_is\_divisible.} +This algorithm attempts to determine if a candidate integer $n$ is composite by performing trial divisions. + +EXAM,bn_mp_prime_is_divisible.c + +The algorithm defaults to a return of $0$ in case an error occurs. The values in the prime table are all specified to be in the range of a +mp\_digit. The table \_\_prime\_tab is defined in the following file. + +EXAM,bn_prime_tab.c + +Note that there are two possible tables. When an mp\_digit is 7-bits long only the primes upto $127$ may be included, otherwise the primes +upto $1619$ are used. Note that the value of \textbf{PRIME\_SIZE} is a constant dependent on the size of a mp\_digit. + +\subsection{The Fermat Test} +The Fermat test is probably one the oldest tests to have a non-trivial probability of success. It is based on the fact that if $n$ is in +fact prime then $a^{n} \equiv a \mbox{ (mod }n\mbox{)}$ for all $0 < a < n$. The reason being that if $n$ is prime than the order of +the multiplicative sub group is $n - 1$. Any base $a$ must have an order which divides $n - 1$ and as such $a^n$ is equivalent to +$a^1 = a$. + +If $n$ is composite then any given base $a$ does not have to have a period which divides $n - 1$. In which case +it is possible that $a^n \nequiv a \mbox{ (mod }n\mbox{)}$. However, this test is not absolute as it is possible that the order +of a base will divide $n - 1$ which would then be reported as prime. Such a base yields what is known as a Fermat pseudo-prime. Several +integers known as Carmichael numbers will be a pseudo-prime to all valid bases. Fortunately such numbers are extremely rare as $n$ grows +in size. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_prime\_fermat}. \\ +\textbf{Input}. mp\_int $a$ and $b$, $a \ge 2$, $0 < b < a$. \\ +\textbf{Output}. $c = 1$ if $b^a \equiv b \mbox{ (mod }a\mbox{)}$, otherwise $c = 0$. \\ +\hline \\ +1. $t \leftarrow b^a \mbox{ (mod }a\mbox{)}$ \\ +2. If $t = b$ then \\ +\hspace{3mm}2.1 $c = 1$ \\ +3. else \\ +\hspace{3mm}3.1 $c = 0$ \\ +4. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_prime\_fermat} +\end{figure} +\textbf{Algorithm mp\_prime\_fermat.} +This algorithm determines whether an mp\_int $a$ is a Fermat prime to the base $b$ or not. It uses a single modular exponentiation to +determine the result. + +EXAM,bn_mp_prime_fermat.c + +\subsection{The Miller-Rabin Test} +The Miller-Rabin (citation) test is another primality test which has tighter error bounds than the Fermat test specifically with sequentially chosen +candidate integers. The algorithm is based on the observation that if $n - 1 = 2^kr$ and if $b^r \nequiv \pm 1$ then after upto $k - 1$ squarings the +value must be equal to $-1$. The squarings are stopped as soon as $-1$ is observed. If the value of $1$ is observed first it means that +some value not congruent to $\pm 1$ when squared equals one which cannot occur if $n$ is prime. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_prime\_miller\_rabin}. \\ +\textbf{Input}. mp\_int $a$ and $b$, $a \ge 2$, $0 < b < a$. \\ +\textbf{Output}. $c = 1$ if $a$ is a Miller-Rabin prime to the base $a$, otherwise $c = 0$. \\ +\hline +1. $a' \leftarrow a - 1$ \\ +2. $r \leftarrow n1$ \\ +3. $c \leftarrow 0, s \leftarrow 0$ \\ +4. While $r.used > 0$ and $r_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ +\hspace{3mm}4.1 $s \leftarrow s + 1$ \\ +\hspace{3mm}4.2 $r \leftarrow \lfloor r / 2 \rfloor$ \\ +5. $y \leftarrow b^r \mbox{ (mod }a\mbox{)}$ \\ +6. If $y \nequiv \pm 1$ then \\ +\hspace{3mm}6.1 $j \leftarrow 1$ \\ +\hspace{3mm}6.2 While $j \le (s - 1)$ and $y \nequiv a'$ \\ +\hspace{6mm}6.2.1 $y \leftarrow y^2 \mbox{ (mod }a\mbox{)}$ \\ +\hspace{6mm}6.2.2 If $y = 1$ then goto step 8. \\ +\hspace{6mm}6.2.3 $j \leftarrow j + 1$ \\ +\hspace{3mm}6.3 If $y \nequiv a'$ goto step 8. \\ +7. $c \leftarrow 1$\\ +8. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_prime\_miller\_rabin} +\end{figure} +\textbf{Algorithm mp\_prime\_miller\_rabin.} +This algorithm performs one trial round of the Miller-Rabin algorithm to the base $b$. It will set $c = 1$ if the algorithm cannot determine +if $b$ is composite or $c = 0$ if $b$ is provably composite. The values of $s$ and $r$ are computed such that $a' = a - 1 = 2^sr$. + +If the value $y \equiv b^r$ is congruent to $\pm 1$ then the algorithm cannot prove if $a$ is composite or not. Otherwise, the algorithm will +square $y$ upto $s - 1$ times stopping only when $y \equiv -1$. If $y^2 \equiv 1$ and $y \nequiv \pm 1$ then the algorithm can report that $a$ +is provably composite. If the algorithm performs $s - 1$ squarings and $y \nequiv -1$ then $a$ is provably composite. If $a$ is not provably +composite then it is \textit{probably} prime. + +EXAM,bn_mp_prime_miller_rabin.c + + + + +\backmatter +\appendix +\begin{thebibliography}{ABCDEF} +\bibitem[1]{TAOCPV2} +Donald Knuth, \textit{The Art of Computer Programming}, Third Edition, Volume Two, Seminumerical Algorithms, Addison-Wesley, 1998 + +\bibitem[2]{HAC} +A. Menezes, P. van Oorschot, S. Vanstone, \textit{Handbook of Applied Cryptography}, CRC Press, 1996 + +\bibitem[3]{ROSE} +Michael Rosing, \textit{Implementing Elliptic Curve Cryptography}, Manning Publications, 1999 + +\bibitem[4]{COMBA} +Paul G. Comba, \textit{Exponentiation Cryptosystems on the IBM PC}. IBM Systems Journal 29(4): 526-538 (1990) + +\bibitem[5]{KARA} +A. Karatsuba, Doklay Akad. Nauk SSSR 145 (1962), pp.293-294 + +\bibitem[6]{KARAP} +Andre Weimerskirch and Christof Paar, \textit{Generalizations of the Karatsuba Algorithm for Polynomial Multiplication}, Submitted to Design, Codes and Cryptography, March 2002 + +\bibitem[7]{BARRETT} +Paul Barrett, \textit{Implementing the Rivest Shamir and Adleman Public Key Encryption Algorithm on a Standard Digital Signal Processor}, Advances in Cryptology, Crypto '86, Springer-Verlag. + +\bibitem[8]{MONT} +P.L.Montgomery. \textit{Modular multiplication without trial division}. Mathematics of Computation, 44(170):519-521, April 1985. + +\bibitem[9]{DRMET} +Chae Hoon Lim and Pil Joong Lee, \textit{Generating Efficient Primes for Discrete Log Cryptosystems}, POSTECH Information Research Laboratories + +\bibitem[10]{MMB} +J. Daemen and R. Govaerts and J. Vandewalle, \textit{Block ciphers based on Modular Arithmetic}, State and {P}rogress in the {R}esearch of {C}ryptography, 1993, pp. 80-89 + +\bibitem[11]{RSAREF} +R.L. Rivest, A. Shamir, L. Adleman, \textit{A Method for Obtaining Digital Signatures and Public-Key Cryptosystems} + +\bibitem[12]{DHREF} +Whitfield Diffie, Martin E. Hellman, \textit{New Directions in Cryptography}, IEEE Transactions on Information Theory, 1976 + +\bibitem[13]{IEEE} +IEEE Standard for Binary Floating-Point Arithmetic (ANSI/IEEE Std 754-1985) + +\bibitem[14]{GMP} +GNU Multiple Precision (GMP), \url{http://www.swox.com/gmp/} + +\bibitem[15]{MPI} +Multiple Precision Integer Library (MPI), Michael Fromberger, \url{http://thayer.dartmouth.edu/~sting/mpi/} + +\bibitem[16]{OPENSSL} +OpenSSL Cryptographic Toolkit, \url{http://openssl.org} + +\bibitem[17]{LIP} +Large Integer Package, \url{http://home.hetnet.nl/~ecstr/LIP.zip} + +\bibitem[18]{ISOC} +JTC1/SC22/WG14, ISO/IEC 9899:1999, ``A draft rationale for the C99 standard.'' + +\bibitem[19]{JAVA} +The Sun Java Website, \url{http://java.sun.com/} + +\end{thebibliography} + +\input{tommath.ind} + +\end{document} diff --git a/libtommath/tommath.tex b/libtommath/tommath.tex new file mode 100644 index 0000000..9c4dc82 --- /dev/null +++ b/libtommath/tommath.tex @@ -0,0 +1,10771 @@ +\documentclass[b5paper]{book} +\usepackage{hyperref} +\usepackage{makeidx} +\usepackage{amssymb} +\usepackage{color} +\usepackage{alltt} +\usepackage{graphicx} +\usepackage{layout} +\def\union{\cup} +\def\intersect{\cap} +\def\getsrandom{\stackrel{\rm R}{\gets}} +\def\cross{\times} +\def\cat{\hspace{0.5em} \| \hspace{0.5em}} +\def\catn{$\|$} +\def\divides{\hspace{0.3em} | \hspace{0.3em}} +\def\nequiv{\not\equiv} +\def\approx{\raisebox{0.2ex}{\mbox{\small $\sim$}}} +\def\lcm{{\rm lcm}} +\def\gcd{{\rm gcd}} +\def\log{{\rm log}} +\def\ord{{\rm ord}} +\def\abs{{\mathit abs}} +\def\rep{{\mathit rep}} +\def\mod{{\mathit\ mod\ }} +\renewcommand{\pmod}[1]{\ ({\rm mod\ }{#1})} +\newcommand{\floor}[1]{\left\lfloor{#1}\right\rfloor} +\newcommand{\ceil}[1]{\left\lceil{#1}\right\rceil} +\def\Or{{\rm\ or\ }} +\def\And{{\rm\ and\ }} +\def\iff{\hspace{1em}\Longleftrightarrow\hspace{1em}} +\def\implies{\Rightarrow} +\def\undefined{{\rm ``undefined"}} +\def\Proof{\vspace{1ex}\noindent {\bf Proof:}\hspace{1em}} +\let\oldphi\phi +\def\phi{\varphi} +\def\Pr{{\rm Pr}} +\newcommand{\str}[1]{{\mathbf{#1}}} +\def\F{{\mathbb F}} +\def\N{{\mathbb N}} +\def\Z{{\mathbb Z}} +\def\R{{\mathbb R}} +\def\C{{\mathbb C}} +\def\Q{{\mathbb Q}} +\definecolor{DGray}{gray}{0.5} +\newcommand{\emailaddr}[1]{\mbox{$<${#1}$>$}} +\def\twiddle{\raisebox{0.3ex}{\mbox{\tiny $\sim$}}} +\def\gap{\vspace{0.5ex}} +\makeindex +\begin{document} +\frontmatter +\pagestyle{empty} +\title{Implementing Multiple Precision Arithmetic \\ ~ \\ Draft Edition } +\author{\mbox{ +%\begin{small} +\begin{tabular}{c} +Tom St Denis \\ +Algonquin College \\ +\\ +Mads Rasmussen \\ +Open Communications Security \\ +\\ +Greg Rose \\ +QUALCOMM Australia \\ +\end{tabular} +%\end{small} +} +} +\maketitle +This text has been placed in the public domain. This text corresponds to the v0.30 release of the +LibTomMath project. + +\begin{alltt} +Tom St Denis +111 Banning Rd +Ottawa, Ontario +K2L 1C3 +Canada + +Phone: 1-613-836-3160 +Email: tomstdenis@iahu.ca +\end{alltt} + +This text is formatted to the international B5 paper size of 176mm wide by 250mm tall using the \LaTeX{} +{\em book} macro package and the Perl {\em booker} package. + +\tableofcontents +\listoffigures +\chapter*{Prefaces to the Draft Edition} +I started this text in April 2003 to complement my LibTomMath library. That is, explain how to implement the functions +contained in LibTomMath. The goal is to have a textbook that any Computer Science student can use when implementing their +own multiple precision arithmetic. The plan I wanted to follow was flesh out all the +ideas and concepts I had floating around in my head and then work on it afterwards refining a little bit at a time. Chance +would have it that I ended up with my summer off from Algonquin College and I was given four months solid to work on the +text. + +Choosing to not waste any time I dove right into the project even before my spring semester was finished. I wrote a bit +off and on at first. The moment my exams were finished I jumped into long 12 to 16 hour days. The result after only +a couple of months was a ten chapter, three hundred page draft that I quickly had distributed to anyone who wanted +to read it. I had Jean-Luc Cooke print copies for me and I brought them to Crypto'03 in Santa Barbara. So far I have +managed to grab a certain level of attention having people from around the world ask me for copies of the text was certain +rewarding. + +Now we are past December 2003. By this time I had pictured that I would have at least finished my second draft of the text. +Currently I am far off from this goal. I've done partial re-writes of chapters one, two and three but they are not even +finished yet. I haven't given up on the project, only had some setbacks. First O'Reilly declined to publish the text then +Addison-Wesley and Greg is tried another which I don't know the name of. However, at this point I want to focus my energy +onto finishing the book not securing a contract. + +So why am I writing this text? It seems like a lot of work right? Most certainly it is a lot of work writing a textbook. +Even the simplest introductory material has to be lined with references and figures. A lot of the text has to be re-written +from point form to prose form to ensure an easier read. Why am I doing all this work for free then? Simple. My philosophy +is quite simply ``Open Source. Open Academia. Open Minds'' which means that to achieve a goal of open minds, that is, +people willing to accept new ideas and explore the unknown you have to make available material they can access freely +without hinderance. + +I've been writing free software since I was about sixteen but only recently have I hit upon software that people have come +to depend upon. I started LibTomCrypt in December 2001 and now several major companies use it as integral portions of their +software. Several educational institutions use it as a matter of course and many freelance developers use it as +part of their projects. To further my contributions I started the LibTomMath project in December 2002 aimed at providing +multiple precision arithmetic routines that students could learn from. That is write routines that are not only easy +to understand and follow but provide quite impressive performance considering they are all in standard portable ISO C. + +The second leg of my philosophy is ``Open Academia'' which is where this textbook comes in. In the end, when all is +said and done the text will be useable by educational institutions as a reference on multiple precision arithmetic. + +At this time I feel I should share a little information about myself. The most common question I was asked at +Crypto'03, perhaps just out of professional courtesy, was which school I either taught at or attended. The unfortunate +truth is that I neither teach at or attend a school of academic reputation. I'm currently at Algonquin College which +is what I'd like to call ``somewhat academic but mostly vocational'' college. In otherwords, job training. + +I'm a 21 year old computer science student mostly self-taught in the areas I am aware of (which includes a half-dozen +computer science fields, a few fields of mathematics and some English). I look forward to teaching someday but I am +still far off from that goal. + +Now it would be improper for me to not introduce the rest of the texts co-authors. While they are only contributing +corrections and editorial feedback their support has been tremendously helpful in presenting the concepts laid out +in the text so far. Greg has always been there for me. He has tracked my LibTom projects since their inception and even +sent cheques to help pay tuition from time to time. His background has provided a wonderful source to bounce ideas off +of and improve the quality of my writing. Mads is another fellow who has just ``been there''. I don't even recall what +his interest in the LibTom projects is but I'm definitely glad he has been around. His ability to catch logical errors +in my written English have saved me on several occasions to say the least. + +What to expect next? Well this is still a rough draft. I've only had the chance to update a few chapters. However, I've +been getting the feeling that people are starting to use my text and I owe them some updated material. My current tenative +plan is to edit one chapter every two weeks starting January 4th. It seems insane but my lower course load at college +should provide ample time. By Crypto'04 I plan to have a 2nd draft of the text polished and ready to hand out to as many +people who will take it. + +\begin{flushright} Tom St Denis \end{flushright} + +\newpage +I found the opportunity to work with Tom appealing for several reasons, not only could I broaden my own horizons, but also +contribute to educate others facing the problem of having to handle big number mathematical calculations. + +This book is Tom's child and he has been caring and fostering the project ever since the beginning with a clear mind of +how he wanted the project to turn out. I have helped by proofreading the text and we have had several discussions about +the layout and language used. + +I hold a masters degree in cryptography from the University of Southern Denmark and have always been interested in the +practical aspects of cryptography. + +Having worked in the security consultancy business for several years in S\~{a}o Paulo, Brazil, I have been in touch with a +great deal of work in which multiple precision mathematics was needed. Understanding the possibilities for speeding up +multiple precision calculations is often very important since we deal with outdated machine architecture where modular +reductions, for example, become painfully slow. + +This text is for people who stop and wonder when first examining algorithms such as RSA for the first time and asks +themselves, ``You tell me this is only secure for large numbers, fine; but how do you implement these numbers?'' + +\begin{flushright} +Mads Rasmussen + +S\~{a}o Paulo - SP + +Brazil +\end{flushright} + +\newpage +It's all because I broke my leg. That just happened to be at about the same time that Tom asked for someone to review the section of the book about +Karatsuba multiplication. I was laid up, alone and immobile, and thought ``Why not?'' I vaguely knew what Karatsuba multiplication was, but not +really, so I thought I could help, learn, and stop myself from watching daytime cable TV, all at once. + +At the time of writing this, I've still not met Tom or Mads in meatspace. I've been following Tom's progress since his first splash on the +sci.crypt Usenet news group. I watched him go from a clueless newbie, to the cryptographic equivalent of a reformed smoker, to a real +contributor to the field, over a period of about two years. I've been impressed with his obvious intelligence, and astounded by his productivity. +Of course, he's young enough to be my own child, so he doesn't have my problems with staying awake. + +When I reviewed that single section of the book, in its very earliest form, I was very pleasantly surprised. So I decided to collaborate more fully, +and at least review all of it, and perhaps write some bits too. There's still a long way to go with it, and I have watched a number of close +friends go through the mill of publication, so I think that the way to go is longer than Tom thinks it is. Nevertheless, it's a good effort, +and I'm pleased to be involved with it. + +\begin{flushright} +Greg Rose, Sydney, Australia, June 2003. +\end{flushright} + +\mainmatter +\pagestyle{headings} +\chapter{Introduction} +\section{Multiple Precision Arithmetic} + +\subsection{What is Multiple Precision Arithmetic?} +When we think of long-hand arithmetic such as addition or multiplication we rarely consider the fact that we instinctively +raise or lower the precision of the numbers we are dealing with. For example, in decimal we almost immediate can +reason that $7$ times $6$ is $42$. However, $42$ has two digits of precision as opposed to one digit we started with. +Further multiplications of say $3$ result in a larger precision result $126$. In these few examples we have multiple +precisions for the numbers we are working with. Despite the various levels of precision a single subset\footnote{With the occasional optimization.} + of algorithms can be designed to accomodate them. + +By way of comparison a fixed or single precision operation would lose precision on various operations. For example, in +the decimal system with fixed precision $6 \cdot 7 = 2$. + +Essentially at the heart of computer based multiple precision arithmetic are the same long-hand algorithms taught in +schools to manually add, subtract, multiply and divide. + +\subsection{The Need for Multiple Precision Arithmetic} +The most prevalent need for multiple precision arithmetic, often referred to as ``bignum'' math, is within the implementation +of public-key cryptography algorithms. Algorithms such as RSA \cite{RSAREF} and Diffie-Hellman \cite{DHREF} require +integers of significant magnitude to resist known cryptanalytic attacks. For example, at the time of this writing a +typical RSA modulus would be at least greater than $10^{309}$. However, modern programming languages such as ISO C \cite{ISOC} and +Java \cite{JAVA} only provide instrinsic support for integers which are relatively small and single precision. + +\begin{figure}[!here] +\begin{center} +\begin{tabular}{|r|c|} +\hline \textbf{Data Type} & \textbf{Range} \\ +\hline char & $-128 \ldots 127$ \\ +\hline short & $-32768 \ldots 32767$ \\ +\hline long & $-2147483648 \ldots 2147483647$ \\ +\hline long long & $-9223372036854775808 \ldots 9223372036854775807$ \\ +\hline +\end{tabular} +\end{center} +\caption{Typical Data Types for the C Programming Language} +\label{fig:ISOC} +\end{figure} + +The largest data type guaranteed to be provided by the ISO C programming +language\footnote{As per the ISO C standard. However, each compiler vendor is allowed to augment the precision as they +see fit.} can only represent values up to $10^{19}$ as shown in figure \ref{fig:ISOC}. On its own the C language is +insufficient to accomodate the magnitude required for the problem at hand. An RSA modulus of magnitude $10^{19}$ could be +trivially factored\footnote{A Pollard-Rho factoring would take only $2^{16}$ time.} on the average desktop computer, +rendering any protocol based on the algorithm insecure. Multiple precision algorithms solve this very problem by +extending the range of representable integers while using single precision data types. + +Most advancements in fast multiple precision arithmetic stem from the need for faster and more efficient cryptographic +primitives. Faster modular reduction and exponentiation algorithms such as Barrett's algorithm, which have appeared in +various cryptographic journals, can render algorithms such as RSA and Diffie-Hellman more efficient. In fact, several +major companies such as RSA Security, Certicom and Entrust have built entire product lines on the implementation and +deployment of efficient algorithms. + +However, cryptography is not the only field of study that can benefit from fast multiple precision integer routines. +Another auxiliary use of multiple precision integers is high precision floating point data types. +The basic IEEE \cite{IEEE} standard floating point type is made up of an integer mantissa $q$, an exponent $e$ and a sign bit $s$. +Numbers are given in the form $n = q \cdot b^e \cdot -1^s$ where $b = 2$ is the most common base for IEEE. Since IEEE +floating point is meant to be implemented in hardware the precision of the mantissa is often fairly small +(\textit{23, 48 and 64 bits}). The mantissa is merely an integer and a multiple precision integer could be used to create +a mantissa of much larger precision than hardware alone can efficiently support. This approach could be useful where +scientific applications must minimize the total output error over long calculations. + +Yet another use for large integers is within arithmetic on polynomials of large characteristic (i.e. $GF(p)[x]$ for large $p$). +In fact the library discussed within this text has already been used to form a polynomial basis library\footnote{See \url{http://poly.libtomcrypt.org} for more details.}. + +\subsection{Benefits of Multiple Precision Arithmetic} +\index{precision} +The benefit of multiple precision representations over single or fixed precision representations is that +no precision is lost while representing the result of an operation which requires excess precision. For example, +the product of two $n$-bit integers requires at least $2n$ bits of precision to be represented faithfully. A multiple +precision algorithm would augment the precision of the destination to accomodate the result while a single precision system +would truncate excess bits to maintain a fixed level of precision. + +It is possible to implement algorithms which require large integers with fixed precision algorithms. For example, elliptic +curve cryptography (\textit{ECC}) is often implemented on smartcards by fixing the precision of the integers to the maximum +size the system will ever need. Such an approach can lead to vastly simpler algorithms which can accomodate the +integers required even if the host platform cannot natively accomodate them\footnote{For example, the average smartcard +processor has an 8 bit accumulator.}. However, as efficient as such an approach may be, the resulting source code is not +normally very flexible. It cannot, at runtime, accomodate inputs of higher magnitude than the designer anticipated. + +Multiple precision algorithms have the most overhead of any style of arithmetic. For the the most part the +overhead can be kept to a minimum with careful planning, but overall, it is not well suited for most memory starved +platforms. However, multiple precision algorithms do offer the most flexibility in terms of the magnitude of the +inputs. That is, the same algorithms based on multiple precision integers can accomodate any reasonable size input +without the designer's explicit forethought. This leads to lower cost of ownership for the code as it only has to +be written and tested once. + +\section{Purpose of This Text} +The purpose of this text is to instruct the reader regarding how to implement efficient multiple precision algorithms. +That is to not only explain a limited subset of the core theory behind the algorithms but also the various ``house keeping'' +elements that are neglected by authors of other texts on the subject. Several well reknowned texts \cite{TAOCPV2,HAC} +give considerably detailed explanations of the theoretical aspects of algorithms and often very little information +regarding the practical implementation aspects. + +In most cases how an algorithm is explained and how it is actually implemented are two very different concepts. For +example, the Handbook of Applied Cryptography (\textit{HAC}), algorithm 14.7 on page 594, gives a relatively simple +algorithm for performing multiple precision integer addition. However, the description lacks any discussion concerning +the fact that the two integer inputs may be of differing magnitudes. As a result the implementation is not as simple +as the text would lead people to believe. Similarly the division routine (\textit{algorithm 14.20, pp. 598}) does not +discuss how to handle sign or handle the dividend's decreasing magnitude in the main loop (\textit{step \#3}). + +Both texts also do not discuss several key optimal algorithms required such as ``Comba'' and Karatsuba multipliers +and fast modular inversion, which we consider practical oversights. These optimal algorithms are vital to achieve +any form of useful performance in non-trivial applications. + +To solve this problem the focus of this text is on the practical aspects of implementing a multiple precision integer +package. As a case study the ``LibTomMath''\footnote{Available at \url{http://math.libtomcrypt.org}} package is used +to demonstrate algorithms with real implementations\footnote{In the ISO C programming language.} that have been field +tested and work very well. The LibTomMath library is freely available on the Internet for all uses and this text +discusses a very large portion of the inner workings of the library. + +The algorithms that are presented will always include at least one ``pseudo-code'' description followed +by the actual C source code that implements the algorithm. The pseudo-code can be used to implement the same +algorithm in other programming languages as the reader sees fit. + +This text shall also serve as a walkthrough of the creation of multiple precision algorithms from scratch. Showing +the reader how the algorithms fit together as well as where to start on various taskings. + +\section{Discussion and Notation} +\subsection{Notation} +A multiple precision integer of $n$-digits shall be denoted as $x = (x_{n-1}, \ldots, x_1, x_0)_{ \beta }$ and represent +the integer $x \equiv \sum_{i=0}^{n-1} x_i\beta^i$. The elements of the array $x$ are said to be the radix $\beta$ digits +of the integer. For example, $x = (1,2,3)_{10}$ would represent the integer +$1\cdot 10^2 + 2\cdot10^1 + 3\cdot10^0 = 123$. + +\index{mp\_int} +The term ``mp\_int'' shall refer to a composite structure which contains the digits of the integer it represents, as well +as auxilary data required to manipulate the data. These additional members are discussed further in section +\ref{sec:MPINT}. For the purposes of this text a ``multiple precision integer'' and an ``mp\_int'' are assumed to be +synonymous. When an algorithm is specified to accept an mp\_int variable it is assumed the various auxliary data members +are present as well. An expression of the type \textit{variablename.item} implies that it should evaluate to the +member named ``item'' of the variable. For example, a string of characters may have a member ``length'' which would +evaluate to the number of characters in the string. If the string $a$ equals ``hello'' then it follows that +$a.length = 5$. + +For certain discussions more generic algorithms are presented to help the reader understand the final algorithm used +to solve a given problem. When an algorithm is described as accepting an integer input it is assumed the input is +a plain integer with no additional multiple-precision members. That is, algorithms that use integers as opposed to +mp\_ints as inputs do not concern themselves with the housekeeping operations required such as memory management. These +algorithms will be used to establish the relevant theory which will subsequently be used to describe a multiple +precision algorithm to solve the same problem. + +\subsection{Precision Notation} +The variable $\beta$ represents the radix of a single digit of a multiple precision integer and +must be of the form $q^p$ for $q, p \in \Z^+$. A single precision variable must be able to represent integers in +the range $0 \le x < q \beta$ while a double precision variable must be able to represent integers in the range +$0 \le x < q \beta^2$. The extra radix-$q$ factor allows additions and subtractions to proceed without truncation of the +carry. Since all modern computers are binary, it is assumed that $q$ is two. + +\index{mp\_digit} \index{mp\_word} +Within the source code that will be presented for each algorithm, the data type \textbf{mp\_digit} will represent +a single precision integer type, while, the data type \textbf{mp\_word} will represent a double precision integer type. In +several algorithms (notably the Comba routines) temporary results will be stored in arrays of double precision mp\_words. +For the purposes of this text $x_j$ will refer to the $j$'th digit of a single precision array and $\hat x_j$ will refer to +the $j$'th digit of a double precision array. Whenever an expression is to be assigned to a double precision +variable it is assumed that all single precision variables are promoted to double precision during the evaluation. +Expressions that are assigned to a single precision variable are truncated to fit within the precision of a single +precision data type. + +For example, if $\beta = 10^2$ a single precision data type may represent a value in the +range $0 \le x < 10^3$, while a double precision data type may represent a value in the range $0 \le x < 10^5$. Let +$a = 23$ and $b = 49$ represent two single precision variables. The single precision product shall be written +as $c \leftarrow a \cdot b$ while the double precision product shall be written as $\hat c \leftarrow a \cdot b$. +In this particular case, $\hat c = 1127$ and $c = 127$. The most significant digit of the product would not fit +in a single precision data type and as a result $c \ne \hat c$. + +\subsection{Algorithm Inputs and Outputs} +Within the algorithm descriptions all variables are assumed to be scalars of either single or double precision +as indicated. The only exception to this rule is when variables have been indicated to be of type mp\_int. This +distinction is important as scalars are often used as array indicies and various other counters. + +\subsection{Mathematical Expressions} +The $\lfloor \mbox{ } \rfloor$ brackets imply an expression truncated to an integer not greater than the expression +itself. For example, $\lfloor 5.7 \rfloor = 5$. Similarly the $\lceil \mbox{ } \rceil$ brackets imply an expression +rounded to an integer not less than the expression itself. For example, $\lceil 5.1 \rceil = 6$. Typically when +the $/$ division symbol is used the intention is to perform an integer division with truncation. For example, +$5/2 = 2$ which will often be written as $\lfloor 5/2 \rfloor = 2$ for clarity. When an expression is written as a +fraction a real value division is implied, for example ${5 \over 2} = 2.5$. + +The norm of a multiple precision integer, for example $\vert \vert x \vert \vert$, will be used to represent the number of digits in the representation +of the integer. For example, $\vert \vert 123 \vert \vert = 3$ and $\vert \vert 79452 \vert \vert = 5$. + +\subsection{Work Effort} +\index{big-Oh} +To measure the efficiency of the specified algorithms, a modified big-Oh notation is used. In this system all +single precision operations are considered to have the same cost\footnote{Except where explicitly noted.}. +That is a single precision addition, multiplication and division are assumed to take the same time to +complete. While this is generally not true in practice, it will simplify the discussions considerably. + +Some algorithms have slight advantages over others which is why some constants will not be removed in +the notation. For example, a normal baseline multiplication (section \ref{sec:basemult}) requires $O(n^2)$ work while a +baseline squaring (section \ref{sec:basesquare}) requires $O({{n^2 + n}\over 2})$ work. In standard big-Oh notation these +would both be said to be equivalent to $O(n^2)$. However, +in the context of the this text this is not the case as the magnitude of the inputs will typically be rather small. As a +result small constant factors in the work effort will make an observable difference in algorithm efficiency. + +All of the algorithms presented in this text have a polynomial time work level. That is, of the form +$O(n^k)$ for $n, k \in \Z^{+}$. This will help make useful comparisons in terms of the speed of the algorithms and how +various optimizations will help pay off in the long run. + +\section{Exercises} +Within the more advanced chapters a section will be set aside to give the reader some challenging exercises related to +the discussion at hand. These exercises are not designed to be prize winning problems, but instead to be thought +provoking. Wherever possible the problems are forward minded, stating problems that will be answered in subsequent +chapters. The reader is encouraged to finish the exercises as they appear to get a better understanding of the +subject material. + +That being said, the problems are designed to affirm knowledge of a particular subject matter. Students in particular +are encouraged to verify they can answer the problems correctly before moving on. + +Similar to the exercises of \cite[pp. ix]{TAOCPV2} these exercises are given a scoring system based on the difficulty of +the problem. However, unlike \cite{TAOCPV2} the problems do not get nearly as hard. The scoring of these +exercises ranges from one (the easiest) to five (the hardest). The following table sumarizes the +scoring system used. + +\begin{figure}[here] +\begin{center} +\begin{small} +\begin{tabular}{|c|l|} +\hline $\left [ 1 \right ]$ & An easy problem that should only take the reader a manner of \\ + & minutes to solve. Usually does not involve much computer time \\ + & to solve. \\ +\hline $\left [ 2 \right ]$ & An easy problem that involves a marginal amount of computer \\ + & time usage. Usually requires a program to be written to \\ + & solve the problem. \\ +\hline $\left [ 3 \right ]$ & A moderately hard problem that requires a non-trivial amount \\ + & of work. Usually involves trivial research and development of \\ + & new theory from the perspective of a student. \\ +\hline $\left [ 4 \right ]$ & A moderately hard problem that involves a non-trivial amount \\ + & of work and research, the solution to which will demonstrate \\ + & a higher mastery of the subject matter. \\ +\hline $\left [ 5 \right ]$ & A hard problem that involves concepts that are difficult for a \\ + & novice to solve. Solutions to these problems will demonstrate a \\ + & complete mastery of the given subject. \\ +\hline +\end{tabular} +\end{small} +\end{center} +\caption{Exercise Scoring System} +\end{figure} + +Problems at the first level are meant to be simple questions that the reader can answer quickly without programming a solution or +devising new theory. These problems are quick tests to see if the material is understood. Problems at the second level +are also designed to be easy but will require a program or algorithm to be implemented to arrive at the answer. These +two levels are essentially entry level questions. + +Problems at the third level are meant to be a bit more difficult than the first two levels. The answer is often +fairly obvious but arriving at an exacting solution requires some thought and skill. These problems will almost always +involve devising a new algorithm or implementing a variation of another algorithm previously presented. Readers who can +answer these questions will feel comfortable with the concepts behind the topic at hand. + +Problems at the fourth level are meant to be similar to those of the level three questions except they will require +additional research to be completed. The reader will most likely not know the answer right away, nor will the text provide +the exact details of the answer until a subsequent chapter. + +Problems at the fifth level are meant to be the hardest +problems relative to all the other problems in the chapter. People who can correctly answer fifth level problems have a +mastery of the subject matter at hand. + +Often problems will be tied together. The purpose of this is to start a chain of thought that will be discussed in future chapters. The reader +is encouraged to answer the follow-up problems and try to draw the relevance of problems. + +\section{Introduction to LibTomMath} + +\subsection{What is LibTomMath?} +LibTomMath is a free and open source multiple precision integer library written entirely in portable ISO C. By portable it +is meant that the library does not contain any code that is computer platform dependent or otherwise problematic to use on +any given platform. + +The library has been successfully tested under numerous operating systems including Unix\footnote{All of these +trademarks belong to their respective rightful owners.}, MacOS, Windows, Linux, PalmOS and on standalone hardware such +as the Gameboy Advance. The library is designed to contain enough functionality to be able to develop applications such +as public key cryptosystems and still maintain a relatively small footprint. + +\subsection{Goals of LibTomMath} + +Libraries which obtain the most efficiency are rarely written in a high level programming language such as C. However, +even though this library is written entirely in ISO C, considerable care has been taken to optimize the algorithm implementations within the +library. Specifically the code has been written to work well with the GNU C Compiler (\textit{GCC}) on both x86 and ARM +processors. Wherever possible, highly efficient algorithms, such as Karatsuba multiplication, sliding window +exponentiation and Montgomery reduction have been provided to make the library more efficient. + +Even with the nearly optimal and specialized algorithms that have been included the Application Programing Interface +(\textit{API}) has been kept as simple as possible. Often generic place holder routines will make use of specialized +algorithms automatically without the developer's specific attention. One such example is the generic multiplication +algorithm \textbf{mp\_mul()} which will automatically use Toom--Cook, Karatsuba, Comba or baseline multiplication +based on the magnitude of the inputs and the configuration of the library. + +Making LibTomMath as efficient as possible is not the only goal of the LibTomMath project. Ideally the library should +be source compatible with another popular library which makes it more attractive for developers to use. In this case the +MPI library was used as a API template for all the basic functions. MPI was chosen because it is another library that fits +in the same niche as LibTomMath. Even though LibTomMath uses MPI as the template for the function names and argument +passing conventions, it has been written from scratch by Tom St Denis. + +The project is also meant to act as a learning tool for students, the logic being that no easy-to-follow ``bignum'' +library exists which can be used to teach computer science students how to perform fast and reliable multiple precision +integer arithmetic. To this end the source code has been given quite a few comments and algorithm discussion points. + +\section{Choice of LibTomMath} +LibTomMath was chosen as the case study of this text not only because the author of both projects is one and the same but +for more worthy reasons. Other libraries such as GMP \cite{GMP}, MPI \cite{MPI}, LIP \cite{LIP} and OpenSSL +\cite{OPENSSL} have multiple precision integer arithmetic routines but would not be ideal for this text for +reasons that will be explained in the following sub-sections. + +\subsection{Code Base} +The LibTomMath code base is all portable ISO C source code. This means that there are no platform dependent conditional +segments of code littered throughout the source. This clean and uncluttered approach to the library means that a +developer can more readily discern the true intent of a given section of source code without trying to keep track of +what conditional code will be used. + +The code base of LibTomMath is well organized. Each function is in its own separate source code file +which allows the reader to find a given function very quickly. On average there are $76$ lines of code per source +file which makes the source very easily to follow. By comparison MPI and LIP are single file projects making code tracing +very hard. GMP has many conditional code segments which also hinder tracing. + +When compiled with GCC for the x86 processor and optimized for speed the entire library is approximately $100$KiB\footnote{The notation ``KiB'' means $2^{10}$ octets, similarly ``MiB'' means $2^{20}$ octets.} + which is fairly small compared to GMP (over $250$KiB). LibTomMath is slightly larger than MPI (which compiles to about +$50$KiB) but LibTomMath is also much faster and more complete than MPI. + +\subsection{API Simplicity} +LibTomMath is designed after the MPI library and shares the API design. Quite often programs that use MPI will build +with LibTomMath without change. The function names correlate directly to the action they perform. Almost all of the +functions share the same parameter passing convention. The learning curve is fairly shallow with the API provided +which is an extremely valuable benefit for the student and developer alike. + +The LIP library is an example of a library with an API that is awkward to work with. LIP uses function names that are often ``compressed'' to +illegible short hand. LibTomMath does not share this characteristic. + +The GMP library also does not return error codes. Instead it uses a POSIX.1 \cite{POSIX1} signal system where errors +are signaled to the host application. This happens to be the fastest approach but definitely not the most versatile. In +effect a math error (i.e. invalid input, heap error, etc) can cause a program to stop functioning which is definitely +undersireable in many situations. + +\subsection{Optimizations} +While LibTomMath is certainly not the fastest library (GMP often beats LibTomMath by a factor of two) it does +feature a set of optimal algorithms for tasks such as modular reduction, exponentiation, multiplication and squaring. GMP +and LIP also feature such optimizations while MPI only uses baseline algorithms with no optimizations. GMP lacks a few +of the additional modular reduction optimizations that LibTomMath features\footnote{At the time of this writing GMP +only had Barrett and Montgomery modular reduction algorithms.}. + +LibTomMath is almost always an order of magnitude faster than the MPI library at computationally expensive tasks such as modular +exponentiation. In the grand scheme of ``bignum'' libraries LibTomMath is faster than the average library and usually +slower than the best libraries such as GMP and OpenSSL by only a small factor. + +\subsection{Portability and Stability} +LibTomMath will build ``out of the box'' on any platform equipped with a modern version of the GNU C Compiler +(\textit{GCC}). This means that without changes the library will build without configuration or setting up any +variables. LIP and MPI will build ``out of the box'' as well but have numerous known bugs. Most notably the author of +MPI has recently stopped working on his library and LIP has long since been discontinued. + +GMP requires a configuration script to run and will not build out of the box. GMP and LibTomMath are still in active +development and are very stable across a variety of platforms. + +\subsection{Choice} +LibTomMath is a relatively compact, well documented, highly optimized and portable library which seems only natural for +the case study of this text. Various source files from the LibTomMath project will be included within the text. However, +the reader is encouraged to download their own copy of the library to actually be able to work with the library. + +\chapter{Getting Started} +\section{Library Basics} +The trick to writing any useful library of source code is to build a solid foundation and work outwards from it. First, +a problem along with allowable solution parameters should be identified and analyzed. In this particular case the +inability to accomodate multiple precision integers is the problem. Futhermore, the solution must be written +as portable source code that is reasonably efficient across several different computer platforms. + +After a foundation is formed the remainder of the library can be designed and implemented in a hierarchical fashion. +That is, to implement the lowest level dependencies first and work towards the most abstract functions last. For example, +before implementing a modular exponentiation algorithm one would implement a modular reduction algorithm. +By building outwards from a base foundation instead of using a parallel design methodology the resulting project is +highly modular. Being highly modular is a desirable property of any project as it often means the resulting product +has a small footprint and updates are easy to perform. + +Usually when I start a project I will begin with the header files. I define the data types I think I will need and +prototype the initial functions that are not dependent on other functions (within the library). After I +implement these base functions I prototype more dependent functions and implement them. The process repeats until +I implement all of the functions I require. For example, in the case of LibTomMath I implemented functions such as +mp\_init() well before I implemented mp\_mul() and even further before I implemented mp\_exptmod(). As an example as to +why this design works note that the Karatsuba and Toom-Cook multipliers were written \textit{after} the +dependent function mp\_exptmod() was written. Adding the new multiplication algorithms did not require changes to the +mp\_exptmod() function itself and lowered the total cost of ownership (\textit{so to speak}) and of development +for new algorithms. This methodology allows new algorithms to be tested in a complete framework with relative ease. + +\begin{center} +\begin{figure}[here] +\includegraphics{pics/design_process.ps} +\caption{Design Flow of the First Few Original LibTomMath Functions.} +\label{pic:design_process} +\end{figure} +\end{center} + +Only after the majority of the functions were in place did I pursue a less hierarchical approach to auditing and optimizing +the source code. For example, one day I may audit the multipliers and the next day the polynomial basis functions. + +It only makes sense to begin the text with the preliminary data types and support algorithms required as well. +This chapter discusses the core algorithms of the library which are the dependents for every other algorithm. + +\section{What is a Multiple Precision Integer?} +Recall that most programming languages, in particular ISO C \cite{ISOC}, only have fixed precision data types that on their own cannot +be used to represent values larger than their precision will allow. The purpose of multiple precision algorithms is +to use fixed precision data types to create and manipulate multiple precision integers which may represent values +that are very large. + +As a well known analogy, school children are taught how to form numbers larger than nine by prepending more radix ten digits. In the decimal system +the largest single digit value is $9$. However, by concatenating digits together larger numbers may be represented. Newly prepended digits +(\textit{to the left}) are said to be in a different power of ten column. That is, the number $123$ can be described as having a $1$ in the hundreds +column, $2$ in the tens column and $3$ in the ones column. Or more formally $123 = 1 \cdot 10^2 + 2 \cdot 10^1 + 3 \cdot 10^0$. Computer based +multiple precision arithmetic is essentially the same concept. Larger integers are represented by adjoining fixed +precision computer words with the exception that a different radix is used. + +What most people probably do not think about explicitly are the various other attributes that describe a multiple precision +integer. For example, the integer $154_{10}$ has two immediately obvious properties. First, the integer is positive, +that is the sign of this particular integer is positive as opposed to negative. Second, the integer has three digits in +its representation. There is an additional property that the integer posesses that does not concern pencil-and-paper +arithmetic. The third property is how many digits placeholders are available to hold the integer. + +The human analogy of this third property is ensuring there is enough space on the paper to write the integer. For example, +if one starts writing a large number too far to the right on a piece of paper they will have to erase it and move left. +Similarly, computer algorithms must maintain strict control over memory usage to ensure that the digits of an integer +will not exceed the allowed boundaries. These three properties make up what is known as a multiple precision +integer or mp\_int for short. + +\subsection{The mp\_int Structure} +\label{sec:MPINT} +The mp\_int structure is the ISO C based manifestation of what represents a multiple precision integer. The ISO C standard does not provide for +any such data type but it does provide for making composite data types known as structures. The following is the structure definition +used within LibTomMath. + +\index{mp\_int} +\begin{figure}[here] +\begin{center} +\begin{small} +%\begin{verbatim} +\begin{tabular}{|l|} +\hline +typedef struct \{ \\ +\hspace{3mm}int used, alloc, sign;\\ +\hspace{3mm}mp\_digit *dp;\\ +\} \textbf{mp\_int}; \\ +\hline +\end{tabular} +%\end{verbatim} +\end{small} +\caption{The mp\_int Structure} +\label{fig:mpint} +\end{center} +\end{figure} + +The mp\_int structure (fig. \ref{fig:mpint}) can be broken down as follows. + +\begin{enumerate} +\item The \textbf{used} parameter denotes how many digits of the array \textbf{dp} contain the digits used to represent +a given integer. The \textbf{used} count must be positive (or zero) and may not exceed the \textbf{alloc} count. + +\item The \textbf{alloc} parameter denotes how +many digits are available in the array to use by functions before it has to increase in size. When the \textbf{used} count +of a result would exceed the \textbf{alloc} count all of the algorithms will automatically increase the size of the +array to accommodate the precision of the result. + +\item The pointer \textbf{dp} points to a dynamically allocated array of digits that represent the given multiple +precision integer. It is padded with $(\textbf{alloc} - \textbf{used})$ zero digits. The array is maintained in a least +significant digit order. As a pencil and paper analogy the array is organized such that the right most digits are stored +first starting at the location indexed by zero\footnote{In C all arrays begin at zero.} in the array. For example, +if \textbf{dp} contains $\lbrace a, b, c, \ldots \rbrace$ where \textbf{dp}$_0 = a$, \textbf{dp}$_1 = b$, \textbf{dp}$_2 = c$, $\ldots$ then +it would represent the integer $a + b\beta + c\beta^2 + \ldots$ + +\index{MP\_ZPOS} \index{MP\_NEG} +\item The \textbf{sign} parameter denotes the sign as either zero/positive (\textbf{MP\_ZPOS}) or negative (\textbf{MP\_NEG}). +\end{enumerate} + +\subsubsection{Valid mp\_int Structures} +Several rules are placed on the state of an mp\_int structure and are assumed to be followed for reasons of efficiency. +The only exceptions are when the structure is passed to initialization functions such as mp\_init() and mp\_init\_copy(). + +\begin{enumerate} +\item The value of \textbf{alloc} may not be less than one. That is \textbf{dp} always points to a previously allocated +array of digits. +\item The value of \textbf{used} may not exceed \textbf{alloc} and must be greater than or equal to zero. +\item The value of \textbf{used} implies the digit at index $(used - 1)$ of the \textbf{dp} array is non-zero. That is, +leading zero digits in the most significant positions must be trimmed. + \begin{enumerate} + \item Digits in the \textbf{dp} array at and above the \textbf{used} location must be zero. + \end{enumerate} +\item The value of \textbf{sign} must be \textbf{MP\_ZPOS} if \textbf{used} is zero; +this represents the mp\_int value of zero. +\end{enumerate} + +\section{Argument Passing} +A convention of argument passing must be adopted early on in the development of any library. Making the function +prototypes consistent will help eliminate many headaches in the future as the library grows to significant complexity. +In LibTomMath the multiple precision integer functions accept parameters from left to right as pointers to mp\_int +structures. That means that the source (input) operands are placed on the left and the destination (output) on the right. +Consider the following examples. + +\begin{verbatim} + mp_mul(&a, &b, &c); /* c = a * b */ + mp_add(&a, &b, &a); /* a = a + b */ + mp_sqr(&a, &b); /* b = a * a */ +\end{verbatim} + +The left to right order is a fairly natural way to implement the functions since it lets the developer read aloud the +functions and make sense of them. For example, the first function would read ``multiply a and b and store in c''. + +Certain libraries (\textit{LIP by Lenstra for instance}) accept parameters the other way around, to mimic the order +of assignment expressions. That is, the destination (output) is on the left and arguments (inputs) are on the right. In +truth, it is entirely a matter of preference. In the case of LibTomMath the convention from the MPI library has been +adopted. + +Another very useful design consideration, provided for in LibTomMath, is whether to allow argument sources to also be a +destination. For example, the second example (\textit{mp\_add}) adds $a$ to $b$ and stores in $a$. This is an important +feature to implement since it allows the calling functions to cut down on the number of variables it must maintain. +However, to implement this feature specific care has to be given to ensure the destination is not modified before the +source is fully read. + +\section{Return Values} +A well implemented application, no matter what its purpose, should trap as many runtime errors as possible and return them +to the caller. By catching runtime errors a library can be guaranteed to prevent undefined behaviour. However, the end +developer can still manage to cause a library to crash. For example, by passing an invalid pointer an application may +fault by dereferencing memory not owned by the application. + +In the case of LibTomMath the only errors that are checked for are related to inappropriate inputs (division by zero for +instance) and memory allocation errors. It will not check that the mp\_int passed to any function is valid nor +will it check pointers for validity. Any function that can cause a runtime error will return an error code as an +\textbf{int} data type with one of the following values (fig \ref{fig:errcodes}). + +\index{MP\_OKAY} \index{MP\_VAL} \index{MP\_MEM} +\begin{figure}[here] +\begin{center} +\begin{tabular}{|l|l|} +\hline \textbf{Value} & \textbf{Meaning} \\ +\hline \textbf{MP\_OKAY} & The function was successful \\ +\hline \textbf{MP\_VAL} & One of the input value(s) was invalid \\ +\hline \textbf{MP\_MEM} & The function ran out of heap memory \\ +\hline +\end{tabular} +\end{center} +\caption{LibTomMath Error Codes} +\label{fig:errcodes} +\end{figure} + +When an error is detected within a function it should free any memory it allocated, often during the initialization of +temporary mp\_ints, and return as soon as possible. The goal is to leave the system in the same state it was when the +function was called. Error checking with this style of API is fairly simple. + +\begin{verbatim} + int err; + if ((err = mp_add(&a, &b, &c)) != MP_OKAY) { + printf("Error: %s\n", mp_error_to_string(err)); + exit(EXIT_FAILURE); + } +\end{verbatim} + +The GMP \cite{GMP} library uses C style \textit{signals} to flag errors which is of questionable use. Not all errors are fatal +and it was not deemed ideal by the author of LibTomMath to force developers to have signal handlers for such cases. + +\section{Initialization and Clearing} +The logical starting point when actually writing multiple precision integer functions is the initialization and +clearing of the mp\_int structures. These two algorithms will be used by the majority of the higher level algorithms. + +Given the basic mp\_int structure an initialization routine must first allocate memory to hold the digits of +the integer. Often it is optimal to allocate a sufficiently large pre-set number of digits even though +the initial integer will represent zero. If only a single digit were allocated quite a few subsequent re-allocations +would occur when operations are performed on the integers. There is a tradeoff between how many default digits to allocate +and how many re-allocations are tolerable. Obviously allocating an excessive amount of digits initially will waste +memory and become unmanageable. + +If the memory for the digits has been successfully allocated then the rest of the members of the structure must +be initialized. Since the initial state of an mp\_int is to represent the zero integer, the allocated digits must be set +to zero. The \textbf{used} count set to zero and \textbf{sign} set to \textbf{MP\_ZPOS}. + +\subsection{Initializing an mp\_int} +An mp\_int is said to be initialized if it is set to a valid, preferably default, state such that all of the members of the +structure are set to valid values. The mp\_init algorithm will perform such an action. + +\index{mp\_init} +\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_init}. \\ +\textbf{Input}. An mp\_int $a$ \\ +\textbf{Output}. Allocate memory and initialize $a$ to a known valid mp\_int state. \\ +\hline \\ +1. Allocate memory for \textbf{MP\_PREC} digits. \\ +2. If the allocation failed return(\textit{MP\_MEM}) \\ +3. for $n$ from $0$ to $MP\_PREC - 1$ do \\ +\hspace{3mm}3.1 $a_n \leftarrow 0$\\ +4. $a.sign \leftarrow MP\_ZPOS$\\ +5. $a.used \leftarrow 0$\\ +6. $a.alloc \leftarrow MP\_PREC$\\ +7. Return(\textit{MP\_OKAY})\\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_init} +\end{figure} + +\textbf{Algorithm mp\_init.} +The purpose of this function is to initialize an mp\_int structure so that the rest of the library can properly +manipulte it. It is assumed that the input may not have had any of its members previously initialized which is certainly +a valid assumption if the input resides on the stack. + +Before any of the members such as \textbf{sign}, \textbf{used} or \textbf{alloc} are initialized the memory for +the digits is allocated. If this fails the function returns before setting any of the other members. The \textbf{MP\_PREC} +name represents a constant\footnote{Defined in the ``tommath.h'' header file within LibTomMath.} +used to dictate the minimum precision of newly initialized mp\_int integers. Ideally, it is at least equal to the smallest +precision number you'll be working with. + +Allocating a block of digits at first instead of a single digit has the benefit of lowering the number of usually slow +heap operations later functions will have to perform in the future. If \textbf{MP\_PREC} is set correctly the slack +memory and the number of heap operations will be trivial. + +Once the allocation has been made the digits have to be set to zero as well as the \textbf{used}, \textbf{sign} and +\textbf{alloc} members initialized. This ensures that the mp\_int will always represent the default state of zero regardless +of the original condition of the input. + +\textbf{Remark.} +This function introduces the idiosyncrasy that all iterative loops, commonly initiated with the ``for'' keyword, iterate incrementally +when the ``to'' keyword is placed between two expressions. For example, ``for $a$ from $b$ to $c$ do'' means that +a subsequent expression (or body of expressions) are to be evaluated upto $c - b$ times so long as $b \le c$. In each +iteration the variable $a$ is substituted for a new integer that lies inclusively between $b$ and $c$. If $b > c$ occured +the loop would not iterate. By contrast if the ``downto'' keyword were used in place of ``to'' the loop would iterate +decrementally. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_init.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* init a new mp_int */ +018 int mp_init (mp_int * a) +019 \{ +020 int i; +021 +022 /* allocate memory required and clear it */ +023 a->dp = OPT_CAST(mp_digit) XMALLOC (sizeof (mp_digit) * MP_PREC); +024 if (a->dp == NULL) \{ +025 return MP_MEM; +026 \} +027 +028 /* set the digits to zero */ +029 for (i = 0; i < MP_PREC; i++) \{ +030 a->dp[i] = 0; +031 \} +032 +033 /* set the used to zero, allocated digits to the default precision +034 * and sign to positive */ +035 a->used = 0; +036 a->alloc = MP_PREC; +037 a->sign = MP_ZPOS; +038 +039 return MP_OKAY; +040 \} +041 #endif +\end{alltt} +\end{small} + +One immediate observation of this initializtion function is that it does not return a pointer to a mp\_int structure. It +is assumed that the caller has already allocated memory for the mp\_int structure, typically on the application stack. The +call to mp\_init() is used only to initialize the members of the structure to a known default state. + +Here we see (line 23) the memory allocation is performed first. This allows us to exit cleanly and quickly +if there is an error. If the allocation fails the routine will return \textbf{MP\_MEM} to the caller to indicate there +was a memory error. The function XMALLOC is what actually allocates the memory. Technically XMALLOC is not a function +but a macro defined in ``tommath.h``. By default, XMALLOC will evaluate to malloc() which is the C library's built--in +memory allocation routine. + +In order to assure the mp\_int is in a known state the digits must be set to zero. On most platforms this could have been +accomplished by using calloc() instead of malloc(). However, to correctly initialize a integer type to a given value in a +portable fashion you have to actually assign the value. The for loop (line 29) performs this required +operation. + +After the memory has been successfully initialized the remainder of the members are initialized +(lines 33 through 34) to their respective default states. At this point the algorithm has succeeded and +a success code is returned to the calling function. If this function returns \textbf{MP\_OKAY} it is safe to assume the +mp\_int structure has been properly initialized and is safe to use with other functions within the library. + +\subsection{Clearing an mp\_int} +When an mp\_int is no longer required by the application, the memory that has been allocated for its digits must be +returned to the application's memory pool with the mp\_clear algorithm. + +\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_clear}. \\ +\textbf{Input}. An mp\_int $a$ \\ +\textbf{Output}. The memory for $a$ shall be deallocated. \\ +\hline \\ +1. If $a$ has been previously freed then return(\textit{MP\_OKAY}). \\ +2. for $n$ from 0 to $a.used - 1$ do \\ +\hspace{3mm}2.1 $a_n \leftarrow 0$ \\ +3. Free the memory allocated for the digits of $a$. \\ +4. $a.used \leftarrow 0$ \\ +5. $a.alloc \leftarrow 0$ \\ +6. $a.sign \leftarrow MP\_ZPOS$ \\ +7. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_clear} +\end{figure} + +\textbf{Algorithm mp\_clear.} +This algorithm accomplishes two goals. First, it clears the digits and the other mp\_int members. This ensures that +if a developer accidentally re-uses a cleared structure it is less likely to cause problems. The second goal +is to free the allocated memory. + +The logic behind the algorithm is extended by marking cleared mp\_int structures so that subsequent calls to this +algorithm will not try to free the memory multiple times. Cleared mp\_ints are detectable by having a pre-defined invalid +digit pointer \textbf{dp} setting. + +Once an mp\_int has been cleared the mp\_int structure is no longer in a valid state for any other algorithm +with the exception of algorithms mp\_init, mp\_init\_copy, mp\_init\_size and mp\_clear. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_clear.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* clear one (frees) */ +018 void +019 mp_clear (mp_int * a) +020 \{ +021 int i; +022 +023 /* only do anything if a hasn't been freed previously */ +024 if (a->dp != NULL) \{ +025 /* first zero the digits */ +026 for (i = 0; i < a->used; i++) \{ +027 a->dp[i] = 0; +028 \} +029 +030 /* free ram */ +031 XFREE(a->dp); +032 +033 /* reset members to make debugging easier */ +034 a->dp = NULL; +035 a->alloc = a->used = 0; +036 a->sign = MP_ZPOS; +037 \} +038 \} +039 #endif +\end{alltt} +\end{small} + +The algorithm only operates on the mp\_int if it hasn't been previously cleared. The if statement (line 24) +checks to see if the \textbf{dp} member is not \textbf{NULL}. If the mp\_int is a valid mp\_int then \textbf{dp} cannot be +\textbf{NULL} in which case the if statement will evaluate to true. + +The digits of the mp\_int are cleared by the for loop (line 26) which assigns a zero to every digit. Similar to mp\_init() +the digits are assigned zero instead of using block memory operations (such as memset()) since this is more portable. + +The digits are deallocated off the heap via the XFREE macro. Similar to XMALLOC the XFREE macro actually evaluates to +a standard C library function. In this case the free() function. Since free() only deallocates the memory the pointer +still has to be reset to \textbf{NULL} manually (line 34). + +Now that the digits have been cleared and deallocated the other members are set to their final values (lines 35 and 36). + +\section{Maintenance Algorithms} + +The previous sections describes how to initialize and clear an mp\_int structure. To further support operations +that are to be performed on mp\_int structures (such as addition and multiplication) the dependent algorithms must be +able to augment the precision of an mp\_int and +initialize mp\_ints with differing initial conditions. + +These algorithms complete the set of low level algorithms required to work with mp\_int structures in the higher level +algorithms such as addition, multiplication and modular exponentiation. + +\subsection{Augmenting an mp\_int's Precision} +When storing a value in an mp\_int structure, a sufficient number of digits must be available to accomodate the entire +result of an operation without loss of precision. Quite often the size of the array given by the \textbf{alloc} member +is large enough to simply increase the \textbf{used} digit count. However, when the size of the array is too small it +must be re-sized appropriately to accomodate the result. The mp\_grow algorithm will provide this functionality. + +\newpage\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_grow}. \\ +\textbf{Input}. An mp\_int $a$ and an integer $b$. \\ +\textbf{Output}. $a$ is expanded to accomodate $b$ digits. \\ +\hline \\ +1. if $a.alloc \ge b$ then return(\textit{MP\_OKAY}) \\ +2. $u \leftarrow b\mbox{ (mod }MP\_PREC\mbox{)}$ \\ +3. $v \leftarrow b + 2 \cdot MP\_PREC - u$ \\ +4. Re-allocate the array of digits $a$ to size $v$ \\ +5. If the allocation failed then return(\textit{MP\_MEM}). \\ +6. for n from a.alloc to $v - 1$ do \\ +\hspace{+3mm}6.1 $a_n \leftarrow 0$ \\ +7. $a.alloc \leftarrow v$ \\ +8. Return(\textit{MP\_OKAY}) \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_grow} +\end{figure} + +\textbf{Algorithm mp\_grow.} +It is ideal to prevent re-allocations from being performed if they are not required (step one). This is useful to +prevent mp\_ints from growing excessively in code that erroneously calls mp\_grow. + +The requested digit count is padded up to next multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} (steps two and three). +This helps prevent many trivial reallocations that would grow an mp\_int by trivially small values. + +It is assumed that the reallocation (step four) leaves the lower $a.alloc$ digits of the mp\_int intact. This is much +akin to how the \textit{realloc} function from the standard C library works. Since the newly allocated digits are +assumed to contain undefined values they are initially set to zero. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_grow.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* grow as required */ +018 int mp_grow (mp_int * a, int size) +019 \{ +020 int i; +021 mp_digit *tmp; +022 +023 /* if the alloc size is smaller alloc more ram */ +024 if (a->alloc < size) \{ +025 /* ensure there are always at least MP_PREC digits extra on top */ +026 size += (MP_PREC * 2) - (size % MP_PREC); +027 +028 /* reallocate the array a->dp +029 * +030 * We store the return in a temporary variable +031 * in case the operation failed we don't want +032 * to overwrite the dp member of a. +033 */ +034 tmp = OPT_CAST(mp_digit) XREALLOC (a->dp, sizeof (mp_digit) * size); +035 if (tmp == NULL) \{ +036 /* reallocation failed but "a" is still valid [can be freed] */ +037 return MP_MEM; +038 \} +039 +040 /* reallocation succeeded so set a->dp */ +041 a->dp = tmp; +042 +043 /* zero excess digits */ +044 i = a->alloc; +045 a->alloc = size; +046 for (; i < a->alloc; i++) \{ +047 a->dp[i] = 0; +048 \} +049 \} +050 return MP_OKAY; +051 \} +052 #endif +\end{alltt} +\end{small} + +A quick optimization is to first determine if a memory re-allocation is required at all. The if statement (line 23) checks +if the \textbf{alloc} member of the mp\_int is smaller than the requested digit count. If the count is not larger than \textbf{alloc} +the function skips the re-allocation part thus saving time. + +When a re-allocation is performed it is turned into an optimal request to save time in the future. The requested digit count is +padded upwards to 2nd multiple of \textbf{MP\_PREC} larger than \textbf{alloc} (line 26). The XREALLOC function is used +to re-allocate the memory. As per the other functions XREALLOC is actually a macro which evaluates to realloc by default. The realloc +function leaves the base of the allocation intact which means the first \textbf{alloc} digits of the mp\_int are the same as before +the re-allocation. All that is left is to clear the newly allocated digits and return. + +Note that the re-allocation result is actually stored in a temporary pointer $tmp$. This is to allow this function to return +an error with a valid pointer. Earlier releases of the library stored the result of XREALLOC into the mp\_int $a$. That would +result in a memory leak if XREALLOC ever failed. + +\subsection{Initializing Variable Precision mp\_ints} +Occasionally the number of digits required will be known in advance of an initialization, based on, for example, the size +of input mp\_ints to a given algorithm. The purpose of algorithm mp\_init\_size is similar to mp\_init except that it +will allocate \textit{at least} a specified number of digits. + +\begin{figure}[here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_init\_size}. \\ +\textbf{Input}. An mp\_int $a$ and the requested number of digits $b$. \\ +\textbf{Output}. $a$ is initialized to hold at least $b$ digits. \\ +\hline \\ +1. $u \leftarrow b \mbox{ (mod }MP\_PREC\mbox{)}$ \\ +2. $v \leftarrow b + 2 \cdot MP\_PREC - u$ \\ +3. Allocate $v$ digits. \\ +4. for $n$ from $0$ to $v - 1$ do \\ +\hspace{3mm}4.1 $a_n \leftarrow 0$ \\ +5. $a.sign \leftarrow MP\_ZPOS$\\ +6. $a.used \leftarrow 0$\\ +7. $a.alloc \leftarrow v$\\ +8. Return(\textit{MP\_OKAY})\\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_init\_size} +\end{figure} + +\textbf{Algorithm mp\_init\_size.} +This algorithm will initialize an mp\_int structure $a$ like algorithm mp\_init with the exception that the number of +digits allocated can be controlled by the second input argument $b$. The input size is padded upwards so it is a +multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} digits. This padding is used to prevent trivial +allocations from becoming a bottleneck in the rest of the algorithms. + +Like algorithm mp\_init, the mp\_int structure is initialized to a default state representing the integer zero. This +particular algorithm is useful if it is known ahead of time the approximate size of the input. If the approximation is +correct no further memory re-allocations are required to work with the mp\_int. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_init\_size.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* init an mp_init for a given size */ +018 int mp_init_size (mp_int * a, int size) +019 \{ +020 int x; +021 +022 /* pad size so there are always extra digits */ +023 size += (MP_PREC * 2) - (size % MP_PREC); +024 +025 /* alloc mem */ +026 a->dp = OPT_CAST(mp_digit) XMALLOC (sizeof (mp_digit) * size); +027 if (a->dp == NULL) \{ +028 return MP_MEM; +029 \} +030 +031 /* set the members */ +032 a->used = 0; +033 a->alloc = size; +034 a->sign = MP_ZPOS; +035 +036 /* zero the digits */ +037 for (x = 0; x < size; x++) \{ +038 a->dp[x] = 0; +039 \} +040 +041 return MP_OKAY; +042 \} +043 #endif +\end{alltt} +\end{small} + +The number of digits $b$ requested is padded (line 23) by first augmenting it to the next multiple of +\textbf{MP\_PREC} and then adding \textbf{MP\_PREC} to the result. If the memory can be successfully allocated the +mp\_int is placed in a default state representing the integer zero. Otherwise, the error code \textbf{MP\_MEM} will be +returned (line 28). + +The digits are allocated and set to zero at the same time with the calloc() function (line @25,XCALLOC@). The +\textbf{used} count is set to zero, the \textbf{alloc} count set to the padded digit count and the \textbf{sign} flag set +to \textbf{MP\_ZPOS} to achieve a default valid mp\_int state (lines 32, 33 and 34). If the function +returns succesfully then it is correct to assume that the mp\_int structure is in a valid state for the remainder of the +functions to work with. + +\subsection{Multiple Integer Initializations and Clearings} +Occasionally a function will require a series of mp\_int data types to be made available simultaneously. +The purpose of algorithm mp\_init\_multi is to initialize a variable length array of mp\_int structures in a single +statement. It is essentially a shortcut to multiple initializations. + +\newpage\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_init\_multi}. \\ +\textbf{Input}. Variable length array $V_k$ of mp\_int variables of length $k$. \\ +\textbf{Output}. The array is initialized such that each mp\_int of $V_k$ is ready to use. \\ +\hline \\ +1. for $n$ from 0 to $k - 1$ do \\ +\hspace{+3mm}1.1. Initialize the mp\_int $V_n$ (\textit{mp\_init}) \\ +\hspace{+3mm}1.2. If initialization failed then do \\ +\hspace{+6mm}1.2.1. for $j$ from $0$ to $n$ do \\ +\hspace{+9mm}1.2.1.1. Free the mp\_int $V_j$ (\textit{mp\_clear}) \\ +\hspace{+6mm}1.2.2. Return(\textit{MP\_MEM}) \\ +2. Return(\textit{MP\_OKAY}) \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_init\_multi} +\end{figure} + +\textbf{Algorithm mp\_init\_multi.} +The algorithm will initialize the array of mp\_int variables one at a time. If a runtime error has been detected +(\textit{step 1.2}) all of the previously initialized variables are cleared. The goal is an ``all or nothing'' +initialization which allows for quick recovery from runtime errors. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_init\_multi.c +\vspace{-3mm} +\begin{alltt} +016 #include <stdarg.h> +017 +018 int mp_init_multi(mp_int *mp, ...) +019 \{ +020 mp_err res = MP_OKAY; /* Assume ok until proven otherwise */ +021 int n = 0; /* Number of ok inits */ +022 mp_int* cur_arg = mp; +023 va_list args; +024 +025 va_start(args, mp); /* init args to next argument from caller */ +026 while (cur_arg != NULL) \{ +027 if (mp_init(cur_arg) != MP_OKAY) \{ +028 /* Oops - error! Back-track and mp_clear what we already +029 succeeded in init-ing, then return error. +030 */ +031 va_list clean_args; +032 +033 /* end the current list */ +034 va_end(args); +035 +036 /* now start cleaning up */ +037 cur_arg = mp; +038 va_start(clean_args, mp); +039 while (n--) \{ +040 mp_clear(cur_arg); +041 cur_arg = va_arg(clean_args, mp_int*); +042 \} +043 va_end(clean_args); +044 res = MP_MEM; +045 break; +046 \} +047 n++; +048 cur_arg = va_arg(args, mp_int*); +049 \} +050 va_end(args); +051 return res; /* Assumed ok, if error flagged above. */ +052 \} +053 +054 #endif +\end{alltt} +\end{small} + +This function intializes a variable length list of mp\_int structure pointers. However, instead of having the mp\_int +structures in an actual C array they are simply passed as arguments to the function. This function makes use of the +``...'' argument syntax of the C programming language. The list is terminated with a final \textbf{NULL} argument +appended on the right. + +The function uses the ``stdarg.h'' \textit{va} functions to step portably through the arguments to the function. A count +$n$ of succesfully initialized mp\_int structures is maintained (line 47) such that if a failure does occur, +the algorithm can backtrack and free the previously initialized structures (lines 27 to 46). + + +\subsection{Clamping Excess Digits} +When a function anticipates a result will be $n$ digits it is simpler to assume this is true within the body of +the function instead of checking during the computation. For example, a multiplication of a $i$ digit number by a +$j$ digit produces a result of at most $i + j$ digits. It is entirely possible that the result is $i + j - 1$ +though, with no final carry into the last position. However, suppose the destination had to be first expanded +(\textit{via mp\_grow}) to accomodate $i + j - 1$ digits than further expanded to accomodate the final carry. +That would be a considerable waste of time since heap operations are relatively slow. + +The ideal solution is to always assume the result is $i + j$ and fix up the \textbf{used} count after the function +terminates. This way a single heap operation (\textit{at most}) is required. However, if the result was not checked +there would be an excess high order zero digit. + +For example, suppose the product of two integers was $x_n = (0x_{n-1}x_{n-2}...x_0)_{\beta}$. The leading zero digit +will not contribute to the precision of the result. In fact, through subsequent operations more leading zero digits would +accumulate to the point the size of the integer would be prohibitive. As a result even though the precision is very +low the representation is excessively large. + +The mp\_clamp algorithm is designed to solve this very problem. It will trim high-order zeros by decrementing the +\textbf{used} count until a non-zero most significant digit is found. Also in this system, zero is considered to be a +positive number which means that if the \textbf{used} count is decremented to zero, the sign must be set to +\textbf{MP\_ZPOS}. + +\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_clamp}. \\ +\textbf{Input}. An mp\_int $a$ \\ +\textbf{Output}. Any excess leading zero digits of $a$ are removed \\ +\hline \\ +1. while $a.used > 0$ and $a_{a.used - 1} = 0$ do \\ +\hspace{+3mm}1.1 $a.used \leftarrow a.used - 1$ \\ +2. if $a.used = 0$ then do \\ +\hspace{+3mm}2.1 $a.sign \leftarrow MP\_ZPOS$ \\ +\hline \\ +\end{tabular} +\end{center} +\caption{Algorithm mp\_clamp} +\end{figure} + +\textbf{Algorithm mp\_clamp.} +As can be expected this algorithm is very simple. The loop on step one is expected to iterate only once or twice at +the most. For example, this will happen in cases where there is not a carry to fill the last position. Step two fixes the sign for +when all of the digits are zero to ensure that the mp\_int is valid at all times. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_clamp.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* trim unused digits +018 * +019 * This is used to ensure that leading zero digits are +020 * trimed and the leading "used" digit will be non-zero +021 * Typically very fast. Also fixes the sign if there +022 * are no more leading digits +023 */ +024 void +025 mp_clamp (mp_int * a) +026 \{ +027 /* decrease used while the most significant digit is +028 * zero. +029 */ +030 while (a->used > 0 && a->dp[a->used - 1] == 0) \{ +031 --(a->used); +032 \} +033 +034 /* reset the sign flag if used == 0 */ +035 if (a->used == 0) \{ +036 a->sign = MP_ZPOS; +037 \} +038 \} +039 #endif +\end{alltt} +\end{small} + +Note on line 27 how to test for the \textbf{used} count is made on the left of the \&\& operator. In the C programming +language the terms to \&\& are evaluated left to right with a boolean short-circuit if any condition fails. This is +important since if the \textbf{used} is zero the test on the right would fetch below the array. That is obviously +undesirable. The parenthesis on line 30 is used to make sure the \textbf{used} count is decremented and not +the pointer ``a''. + +\section*{Exercises} +\begin{tabular}{cl} +$\left [ 1 \right ]$ & Discuss the relevance of the \textbf{used} member of the mp\_int structure. \\ + & \\ +$\left [ 1 \right ]$ & Discuss the consequences of not using padding when performing allocations. \\ + & \\ +$\left [ 2 \right ]$ & Estimate an ideal value for \textbf{MP\_PREC} when performing 1024-bit RSA \\ + & encryption when $\beta = 2^{28}$. \\ + & \\ +$\left [ 1 \right ]$ & Discuss the relevance of the algorithm mp\_clamp. What does it prevent? \\ + & \\ +$\left [ 1 \right ]$ & Give an example of when the algorithm mp\_init\_copy might be useful. \\ + & \\ +\end{tabular} + + +%%% +% CHAPTER FOUR +%%% + +\chapter{Basic Operations} + +\section{Introduction} +In the previous chapter a series of low level algorithms were established that dealt with initializing and maintaining +mp\_int structures. This chapter will discuss another set of seemingly non-algebraic algorithms which will form the low +level basis of the entire library. While these algorithm are relatively trivial it is important to understand how they +work before proceeding since these algorithms will be used almost intrinsically in the following chapters. + +The algorithms in this chapter deal primarily with more ``programmer'' related tasks such as creating copies of +mp\_int structures, assigning small values to mp\_int structures and comparisons of the values mp\_int structures +represent. + +\section{Assigning Values to mp\_int Structures} +\subsection{Copying an mp\_int} +Assigning the value that a given mp\_int structure represents to another mp\_int structure shall be known as making +a copy for the purposes of this text. The copy of the mp\_int will be a separate entity that represents the same +value as the mp\_int it was copied from. The mp\_copy algorithm provides this functionality. + +\newpage\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_copy}. \\ +\textbf{Input}. An mp\_int $a$ and $b$. \\ +\textbf{Output}. Store a copy of $a$ in $b$. \\ +\hline \\ +1. If $b.alloc < a.used$ then grow $b$ to $a.used$ digits. (\textit{mp\_grow}) \\ +2. for $n$ from 0 to $a.used - 1$ do \\ +\hspace{3mm}2.1 $b_{n} \leftarrow a_{n}$ \\ +3. for $n$ from $a.used$ to $b.used - 1$ do \\ +\hspace{3mm}3.1 $b_{n} \leftarrow 0$ \\ +4. $b.used \leftarrow a.used$ \\ +5. $b.sign \leftarrow a.sign$ \\ +6. return(\textit{MP\_OKAY}) \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_copy} +\end{figure} + +\textbf{Algorithm mp\_copy.} +This algorithm copies the mp\_int $a$ such that upon succesful termination of the algorithm the mp\_int $b$ will +represent the same integer as the mp\_int $a$. The mp\_int $b$ shall be a complete and distinct copy of the +mp\_int $a$ meaing that the mp\_int $a$ can be modified and it shall not affect the value of the mp\_int $b$. + +If $b$ does not have enough room for the digits of $a$ it must first have its precision augmented via the mp\_grow +algorithm. The digits of $a$ are copied over the digits of $b$ and any excess digits of $b$ are set to zero (step two +and three). The \textbf{used} and \textbf{sign} members of $a$ are finally copied over the respective members of +$b$. + +\textbf{Remark.} This algorithm also introduces a new idiosyncrasy that will be used throughout the rest of the +text. The error return codes of other algorithms are not explicitly checked in the pseudo-code presented. For example, in +step one of the mp\_copy algorithm the return of mp\_grow is not explicitly checked to ensure it succeeded. Text space is +limited so it is assumed that if a algorithm fails it will clear all temporarily allocated mp\_ints and return +the error code itself. However, the C code presented will demonstrate all of the error handling logic required to +implement the pseudo-code. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_copy.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* copy, b = a */ +018 int +019 mp_copy (mp_int * a, mp_int * b) +020 \{ +021 int res, n; +022 +023 /* if dst == src do nothing */ +024 if (a == b) \{ +025 return MP_OKAY; +026 \} +027 +028 /* grow dest */ +029 if (b->alloc < a->used) \{ +030 if ((res = mp_grow (b, a->used)) != MP_OKAY) \{ +031 return res; +032 \} +033 \} +034 +035 /* zero b and copy the parameters over */ +036 \{ +037 register mp_digit *tmpa, *tmpb; +038 +039 /* pointer aliases */ +040 +041 /* source */ +042 tmpa = a->dp; +043 +044 /* destination */ +045 tmpb = b->dp; +046 +047 /* copy all the digits */ +048 for (n = 0; n < a->used; n++) \{ +049 *tmpb++ = *tmpa++; +050 \} +051 +052 /* clear high digits */ +053 for (; n < b->used; n++) \{ +054 *tmpb++ = 0; +055 \} +056 \} +057 +058 /* copy used count and sign */ +059 b->used = a->used; +060 b->sign = a->sign; +061 return MP_OKAY; +062 \} +063 #endif +\end{alltt} +\end{small} + +Occasionally a dependent algorithm may copy an mp\_int effectively into itself such as when the input and output +mp\_int structures passed to a function are one and the same. For this case it is optimal to return immediately without +copying digits (line 24). + +The mp\_int $b$ must have enough digits to accomodate the used digits of the mp\_int $a$. If $b.alloc$ is less than +$a.used$ the algorithm mp\_grow is used to augment the precision of $b$ (lines 29 to 33). In order to +simplify the inner loop that copies the digits from $a$ to $b$, two aliases $tmpa$ and $tmpb$ point directly at the digits +of the mp\_ints $a$ and $b$ respectively. These aliases (lines 42 and 45) allow the compiler to access the digits without first dereferencing the +mp\_int pointers and then subsequently the pointer to the digits. + +After the aliases are established the digits from $a$ are copied into $b$ (lines 48 to 50) and then the excess +digits of $b$ are set to zero (lines 53 to 55). Both ``for'' loops make use of the pointer aliases and in +fact the alias for $b$ is carried through into the second ``for'' loop to clear the excess digits. This optimization +allows the alias to stay in a machine register fairly easy between the two loops. + +\textbf{Remarks.} The use of pointer aliases is an implementation methodology first introduced in this function that will +be used considerably in other functions. Technically, a pointer alias is simply a short hand alias used to lower the +number of pointer dereferencing operations required to access data. For example, a for loop may resemble + +\begin{alltt} +for (x = 0; x < 100; x++) \{ + a->num[4]->dp[x] = 0; +\} +\end{alltt} + +This could be re-written using aliases as + +\begin{alltt} +mp_digit *tmpa; +a = a->num[4]->dp; +for (x = 0; x < 100; x++) \{ + *a++ = 0; +\} +\end{alltt} + +In this case an alias is used to access the +array of digits within an mp\_int structure directly. It may seem that a pointer alias is strictly not required +as a compiler may optimize out the redundant pointer operations. However, there are two dominant reasons to use aliases. + +The first reason is that most compilers will not effectively optimize pointer arithmetic. For example, some optimizations +may work for the Microsoft Visual C++ compiler (MSVC) and not for the GNU C Compiler (GCC). Also some optimizations may +work for GCC and not MSVC. As such it is ideal to find a common ground for as many compilers as possible. Pointer +aliases optimize the code considerably before the compiler even reads the source code which means the end compiled code +stands a better chance of being faster. + +The second reason is that pointer aliases often can make an algorithm simpler to read. Consider the first ``for'' +loop of the function mp\_copy() re-written to not use pointer aliases. + +\begin{alltt} + /* copy all the digits */ + for (n = 0; n < a->used; n++) \{ + b->dp[n] = a->dp[n]; + \} +\end{alltt} + +Whether this code is harder to read depends strongly on the individual. However, it is quantifiably slightly more +complicated as there are four variables within the statement instead of just two. + +\subsubsection{Nested Statements} +Another commonly used technique in the source routines is that certain sections of code are nested. This is used in +particular with the pointer aliases to highlight code phases. For example, a Comba multiplier (discussed in chapter six) +will typically have three different phases. First the temporaries are initialized, then the columns calculated and +finally the carries are propagated. In this example the middle column production phase will typically be nested as it +uses temporary variables and aliases the most. + +The nesting also simplies the source code as variables that are nested are only valid for their scope. As a result +the various temporary variables required do not propagate into other sections of code. + + +\subsection{Creating a Clone} +Another common operation is to make a local temporary copy of an mp\_int argument. To initialize an mp\_int +and then copy another existing mp\_int into the newly intialized mp\_int will be known as creating a clone. This is +useful within functions that need to modify an argument but do not wish to actually modify the original copy. The +mp\_init\_copy algorithm has been designed to help perform this task. + +\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_init\_copy}. \\ +\textbf{Input}. An mp\_int $a$ and $b$\\ +\textbf{Output}. $a$ is initialized to be a copy of $b$. \\ +\hline \\ +1. Init $a$. (\textit{mp\_init}) \\ +2. Copy $b$ to $a$. (\textit{mp\_copy}) \\ +3. Return the status of the copy operation. \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_init\_copy} +\end{figure} + +\textbf{Algorithm mp\_init\_copy.} +This algorithm will initialize an mp\_int variable and copy another previously initialized mp\_int variable into it. As +such this algorithm will perform two operations in one step. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_init\_copy.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* creates "a" then copies b into it */ +018 int mp_init_copy (mp_int * a, mp_int * b) +019 \{ +020 int res; +021 +022 if ((res = mp_init (a)) != MP_OKAY) \{ +023 return res; +024 \} +025 return mp_copy (b, a); +026 \} +027 #endif +\end{alltt} +\end{small} + +This will initialize \textbf{a} and make it a verbatim copy of the contents of \textbf{b}. Note that +\textbf{a} will have its own memory allocated which means that \textbf{b} may be cleared after the call +and \textbf{a} will be left intact. + +\section{Zeroing an Integer} +Reseting an mp\_int to the default state is a common step in many algorithms. The mp\_zero algorithm will be the algorithm used to +perform this task. + +\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_zero}. \\ +\textbf{Input}. An mp\_int $a$ \\ +\textbf{Output}. Zero the contents of $a$ \\ +\hline \\ +1. $a.used \leftarrow 0$ \\ +2. $a.sign \leftarrow$ MP\_ZPOS \\ +3. for $n$ from 0 to $a.alloc - 1$ do \\ +\hspace{3mm}3.1 $a_n \leftarrow 0$ \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_zero} +\end{figure} + +\textbf{Algorithm mp\_zero.} +This algorithm simply resets a mp\_int to the default state. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_zero.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* set to zero */ +018 void +019 mp_zero (mp_int * a) +020 \{ +021 a->sign = MP_ZPOS; +022 a->used = 0; +023 memset (a->dp, 0, sizeof (mp_digit) * a->alloc); +024 \} +025 #endif +\end{alltt} +\end{small} + +After the function is completed, all of the digits are zeroed, the \textbf{used} count is zeroed and the +\textbf{sign} variable is set to \textbf{MP\_ZPOS}. + +\section{Sign Manipulation} +\subsection{Absolute Value} +With the mp\_int representation of an integer, calculating the absolute value is trivial. The mp\_abs algorithm will compute +the absolute value of an mp\_int. + +\newpage\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_abs}. \\ +\textbf{Input}. An mp\_int $a$ \\ +\textbf{Output}. Computes $b = \vert a \vert$ \\ +\hline \\ +1. Copy $a$ to $b$. (\textit{mp\_copy}) \\ +2. If the copy failed return(\textit{MP\_MEM}). \\ +3. $b.sign \leftarrow MP\_ZPOS$ \\ +4. Return(\textit{MP\_OKAY}) \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_abs} +\end{figure} + +\textbf{Algorithm mp\_abs.} +This algorithm computes the absolute of an mp\_int input. First it copies $a$ over $b$. This is an example of an +algorithm where the check in mp\_copy that determines if the source and destination are equal proves useful. This allows, +for instance, the developer to pass the same mp\_int as the source and destination to this function without addition +logic to handle it. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_abs.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* b = |a| +018 * +019 * Simple function copies the input and fixes the sign to positive +020 */ +021 int +022 mp_abs (mp_int * a, mp_int * b) +023 \{ +024 int res; +025 +026 /* copy a to b */ +027 if (a != b) \{ +028 if ((res = mp_copy (a, b)) != MP_OKAY) \{ +029 return res; +030 \} +031 \} +032 +033 /* force the sign of b to positive */ +034 b->sign = MP_ZPOS; +035 +036 return MP_OKAY; +037 \} +038 #endif +\end{alltt} +\end{small} + +\subsection{Integer Negation} +With the mp\_int representation of an integer, calculating the negation is also trivial. The mp\_neg algorithm will compute +the negative of an mp\_int input. + +\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_neg}. \\ +\textbf{Input}. An mp\_int $a$ \\ +\textbf{Output}. Computes $b = -a$ \\ +\hline \\ +1. Copy $a$ to $b$. (\textit{mp\_copy}) \\ +2. If the copy failed return(\textit{MP\_MEM}). \\ +3. If $a.used = 0$ then return(\textit{MP\_OKAY}). \\ +4. If $a.sign = MP\_ZPOS$ then do \\ +\hspace{3mm}4.1 $b.sign = MP\_NEG$. \\ +5. else do \\ +\hspace{3mm}5.1 $b.sign = MP\_ZPOS$. \\ +6. Return(\textit{MP\_OKAY}) \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_neg} +\end{figure} + +\textbf{Algorithm mp\_neg.} +This algorithm computes the negation of an input. First it copies $a$ over $b$. If $a$ has no used digits then +the algorithm returns immediately. Otherwise it flips the sign flag and stores the result in $b$. Note that if +$a$ had no digits then it must be positive by definition. Had step three been omitted then the algorithm would return +zero as negative. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_neg.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* b = -a */ +018 int mp_neg (mp_int * a, mp_int * b) +019 \{ +020 int res; +021 if ((res = mp_copy (a, b)) != MP_OKAY) \{ +022 return res; +023 \} +024 if (mp_iszero(b) != MP_YES) \{ +025 b->sign = (a->sign == MP_ZPOS) ? MP_NEG : MP_ZPOS; +026 \} +027 return MP_OKAY; +028 \} +029 #endif +\end{alltt} +\end{small} + +\section{Small Constants} +\subsection{Setting Small Constants} +Often a mp\_int must be set to a relatively small value such as $1$ or $2$. For these cases the mp\_set algorithm is useful. + +\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_set}. \\ +\textbf{Input}. An mp\_int $a$ and a digit $b$ \\ +\textbf{Output}. Make $a$ equivalent to $b$ \\ +\hline \\ +1. Zero $a$ (\textit{mp\_zero}). \\ +2. $a_0 \leftarrow b \mbox{ (mod }\beta\mbox{)}$ \\ +3. $a.used \leftarrow \left \lbrace \begin{array}{ll} + 1 & \mbox{if }a_0 > 0 \\ + 0 & \mbox{if }a_0 = 0 + \end{array} \right .$ \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_set} +\end{figure} + +\textbf{Algorithm mp\_set.} +This algorithm sets a mp\_int to a small single digit value. Step number 1 ensures that the integer is reset to the default state. The +single digit is set (\textit{modulo $\beta$}) and the \textbf{used} count is adjusted accordingly. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_set.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* set to a digit */ +018 void mp_set (mp_int * a, mp_digit b) +019 \{ +020 mp_zero (a); +021 a->dp[0] = b & MP_MASK; +022 a->used = (a->dp[0] != 0) ? 1 : 0; +023 \} +024 #endif +\end{alltt} +\end{small} + +Line 20 calls mp\_zero() to clear the mp\_int and reset the sign. Line 21 copies the digit +into the least significant location. Note the usage of a new constant \textbf{MP\_MASK}. This constant is used to quickly +reduce an integer modulo $\beta$. Since $\beta$ is of the form $2^k$ for any suitable $k$ it suffices to perform a binary AND with +$MP\_MASK = 2^k - 1$ to perform the reduction. Finally line 22 will set the \textbf{used} member with respect to the +digit actually set. This function will always make the integer positive. + +One important limitation of this function is that it will only set one digit. The size of a digit is not fixed, meaning source that uses +this function should take that into account. Only trivially small constants can be set using this function. + +\subsection{Setting Large Constants} +To overcome the limitations of the mp\_set algorithm the mp\_set\_int algorithm is ideal. It accepts a ``long'' +data type as input and will always treat it as a 32-bit integer. + +\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_set\_int}. \\ +\textbf{Input}. An mp\_int $a$ and a ``long'' integer $b$ \\ +\textbf{Output}. Make $a$ equivalent to $b$ \\ +\hline \\ +1. Zero $a$ (\textit{mp\_zero}) \\ +2. for $n$ from 0 to 7 do \\ +\hspace{3mm}2.1 $a \leftarrow a \cdot 16$ (\textit{mp\_mul2d}) \\ +\hspace{3mm}2.2 $u \leftarrow \lfloor b / 2^{4(7 - n)} \rfloor \mbox{ (mod }16\mbox{)}$\\ +\hspace{3mm}2.3 $a_0 \leftarrow a_0 + u$ \\ +\hspace{3mm}2.4 $a.used \leftarrow a.used + 1$ \\ +3. Clamp excess used digits (\textit{mp\_clamp}) \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_set\_int} +\end{figure} + +\textbf{Algorithm mp\_set\_int.} +The algorithm performs eight iterations of a simple loop where in each iteration four bits from the source are added to the +mp\_int. Step 2.1 will multiply the current result by sixteen making room for four more bits in the less significant positions. In step 2.2 the +next four bits from the source are extracted and are added to the mp\_int. The \textbf{used} digit count is +incremented to reflect the addition. The \textbf{used} digit counter is incremented since if any of the leading digits were zero the mp\_int would have +zero digits used and the newly added four bits would be ignored. + +Excess zero digits are trimmed in steps 2.1 and 3 by using higher level algorithms mp\_mul2d and mp\_clamp. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_set\_int.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* set a 32-bit const */ +018 int mp_set_int (mp_int * a, unsigned long b) +019 \{ +020 int x, res; +021 +022 mp_zero (a); +023 +024 /* set four bits at a time */ +025 for (x = 0; x < 8; x++) \{ +026 /* shift the number up four bits */ +027 if ((res = mp_mul_2d (a, 4, a)) != MP_OKAY) \{ +028 return res; +029 \} +030 +031 /* OR in the top four bits of the source */ +032 a->dp[0] |= (b >> 28) & 15; +033 +034 /* shift the source up to the next four bits */ +035 b <<= 4; +036 +037 /* ensure that digits are not clamped off */ +038 a->used += 1; +039 \} +040 mp_clamp (a); +041 return MP_OKAY; +042 \} +043 #endif +\end{alltt} +\end{small} + +This function sets four bits of the number at a time to handle all practical \textbf{DIGIT\_BIT} sizes. The weird +addition on line 38 ensures that the newly added in bits are added to the number of digits. While it may not +seem obvious as to why the digit counter does not grow exceedingly large it is because of the shift on line 27 +as well as the call to mp\_clamp() on line 40. Both functions will clamp excess leading digits which keeps +the number of used digits low. + +\section{Comparisons} +\subsection{Unsigned Comparisions} +Comparing a multiple precision integer is performed with the exact same algorithm used to compare two decimal numbers. For example, +to compare $1,234$ to $1,264$ the digits are extracted by their positions. That is we compare $1 \cdot 10^3 + 2 \cdot 10^2 + 3 \cdot 10^1 + 4 \cdot 10^0$ +to $1 \cdot 10^3 + 2 \cdot 10^2 + 6 \cdot 10^1 + 4 \cdot 10^0$ by comparing single digits at a time starting with the highest magnitude +positions. If any leading digit of one integer is greater than a digit in the same position of another integer then obviously it must be greater. + +The first comparision routine that will be developed is the unsigned magnitude compare which will perform a comparison based on the digits of two +mp\_int variables alone. It will ignore the sign of the two inputs. Such a function is useful when an absolute comparison is required or if the +signs are known to agree in advance. + +To facilitate working with the results of the comparison functions three constants are required. + +\begin{figure}[here] +\begin{center} +\begin{tabular}{|r|l|} +\hline \textbf{Constant} & \textbf{Meaning} \\ +\hline \textbf{MP\_GT} & Greater Than \\ +\hline \textbf{MP\_EQ} & Equal To \\ +\hline \textbf{MP\_LT} & Less Than \\ +\hline +\end{tabular} +\end{center} +\caption{Comparison Return Codes} +\end{figure} + +\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_cmp\_mag}. \\ +\textbf{Input}. Two mp\_ints $a$ and $b$. \\ +\textbf{Output}. Unsigned comparison results ($a$ to the left of $b$). \\ +\hline \\ +1. If $a.used > b.used$ then return(\textit{MP\_GT}) \\ +2. If $a.used < b.used$ then return(\textit{MP\_LT}) \\ +3. for n from $a.used - 1$ to 0 do \\ +\hspace{+3mm}3.1 if $a_n > b_n$ then return(\textit{MP\_GT}) \\ +\hspace{+3mm}3.2 if $a_n < b_n$ then return(\textit{MP\_LT}) \\ +4. Return(\textit{MP\_EQ}) \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_cmp\_mag} +\end{figure} + +\textbf{Algorithm mp\_cmp\_mag.} +By saying ``$a$ to the left of $b$'' it is meant that the comparison is with respect to $a$, that is if $a$ is greater than $b$ it will return +\textbf{MP\_GT} and similar with respect to when $a = b$ and $a < b$. The first two steps compare the number of digits used in both $a$ and $b$. +Obviously if the digit counts differ there would be an imaginary zero digit in the smaller number where the leading digit of the larger number is. +If both have the same number of digits than the actual digits themselves must be compared starting at the leading digit. + +By step three both inputs must have the same number of digits so its safe to start from either $a.used - 1$ or $b.used - 1$ and count down to +the zero'th digit. If after all of the digits have been compared, no difference is found, the algorithm returns \textbf{MP\_EQ}. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_cmp\_mag.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* compare maginitude of two ints (unsigned) */ +018 int mp_cmp_mag (mp_int * a, mp_int * b) +019 \{ +020 int n; +021 mp_digit *tmpa, *tmpb; +022 +023 /* compare based on # of non-zero digits */ +024 if (a->used > b->used) \{ +025 return MP_GT; +026 \} +027 +028 if (a->used < b->used) \{ +029 return MP_LT; +030 \} +031 +032 /* alias for a */ +033 tmpa = a->dp + (a->used - 1); +034 +035 /* alias for b */ +036 tmpb = b->dp + (a->used - 1); +037 +038 /* compare based on digits */ +039 for (n = 0; n < a->used; ++n, --tmpa, --tmpb) \{ +040 if (*tmpa > *tmpb) \{ +041 return MP_GT; +042 \} +043 +044 if (*tmpa < *tmpb) \{ +045 return MP_LT; +046 \} +047 \} +048 return MP_EQ; +049 \} +050 #endif +\end{alltt} +\end{small} + +The two if statements on lines 24 and 28 compare the number of digits in the two inputs. These two are performed before all of the digits +are compared since it is a very cheap test to perform and can potentially save considerable time. The implementation given is also not valid +without those two statements. $b.alloc$ may be smaller than $a.used$, meaning that undefined values will be read from $b$ past the end of the +array of digits. + +\subsection{Signed Comparisons} +Comparing with sign considerations is also fairly critical in several routines (\textit{division for example}). Based on an unsigned magnitude +comparison a trivial signed comparison algorithm can be written. + +\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_cmp}. \\ +\textbf{Input}. Two mp\_ints $a$ and $b$ \\ +\textbf{Output}. Signed Comparison Results ($a$ to the left of $b$) \\ +\hline \\ +1. if $a.sign = MP\_NEG$ and $b.sign = MP\_ZPOS$ then return(\textit{MP\_LT}) \\ +2. if $a.sign = MP\_ZPOS$ and $b.sign = MP\_NEG$ then return(\textit{MP\_GT}) \\ +3. if $a.sign = MP\_NEG$ then \\ +\hspace{+3mm}3.1 Return the unsigned comparison of $b$ and $a$ (\textit{mp\_cmp\_mag}) \\ +4 Otherwise \\ +\hspace{+3mm}4.1 Return the unsigned comparison of $a$ and $b$ \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_cmp} +\end{figure} + +\textbf{Algorithm mp\_cmp.} +The first two steps compare the signs of the two inputs. If the signs do not agree then it can return right away with the appropriate +comparison code. When the signs are equal the digits of the inputs must be compared to determine the correct result. In step +three the unsigned comparision flips the order of the arguments since they are both negative. For instance, if $-a > -b$ then +$\vert a \vert < \vert b \vert$. Step number four will compare the two when they are both positive. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_cmp.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* compare two ints (signed)*/ +018 int +019 mp_cmp (mp_int * a, mp_int * b) +020 \{ +021 /* compare based on sign */ +022 if (a->sign != b->sign) \{ +023 if (a->sign == MP_NEG) \{ +024 return MP_LT; +025 \} else \{ +026 return MP_GT; +027 \} +028 \} +029 +030 /* compare digits */ +031 if (a->sign == MP_NEG) \{ +032 /* if negative compare opposite direction */ +033 return mp_cmp_mag(b, a); +034 \} else \{ +035 return mp_cmp_mag(a, b); +036 \} +037 \} +038 #endif +\end{alltt} +\end{small} + +The two if statements on lines 22 and 23 perform the initial sign comparison. If the signs are not the equal then which ever +has the positive sign is larger. At line 31, the inputs are compared based on magnitudes. If the signs were both negative then +the unsigned comparison is performed in the opposite direction (\textit{line 33}). Otherwise, the signs are assumed to +be both positive and a forward direction unsigned comparison is performed. + +\section*{Exercises} +\begin{tabular}{cl} +$\left [ 2 \right ]$ & Modify algorithm mp\_set\_int to accept as input a variable length array of bits. \\ + & \\ +$\left [ 3 \right ]$ & Give the probability that algorithm mp\_cmp\_mag will have to compare $k$ digits \\ + & of two random digits (of equal magnitude) before a difference is found. \\ + & \\ +$\left [ 1 \right ]$ & Suggest a simple method to speed up the implementation of mp\_cmp\_mag based \\ + & on the observations made in the previous problem. \\ + & +\end{tabular} + +\chapter{Basic Arithmetic} +\section{Introduction} +At this point algorithms for initialization, clearing, zeroing, copying, comparing and setting small constants have been +established. The next logical set of algorithms to develop are addition, subtraction and digit shifting algorithms. These +algorithms make use of the lower level algorithms and are the cruicial building block for the multiplication algorithms. It is very important +that these algorithms are highly optimized. On their own they are simple $O(n)$ algorithms but they can be called from higher level algorithms +which easily places them at $O(n^2)$ or even $O(n^3)$ work levels. + +All of the algorithms within this chapter make use of the logical bit shift operations denoted by $<<$ and $>>$ for left and right +logical shifts respectively. A logical shift is analogous to sliding the decimal point of radix-10 representations. For example, the real +number $0.9345$ is equivalent to $93.45\%$ which is found by sliding the the decimal two places to the right (\textit{multiplying by $\beta^2 = 10^2$}). +Algebraically a binary logical shift is equivalent to a division or multiplication by a power of two. +For example, $a << k = a \cdot 2^k$ while $a >> k = \lfloor a/2^k \rfloor$. + +One significant difference between a logical shift and the way decimals are shifted is that digits below the zero'th position are removed +from the number. For example, consider $1101_2 >> 1$ using decimal notation this would produce $110.1_2$. However, with a logical shift the +result is $110_2$. + +\section{Addition and Subtraction} +In common twos complement fixed precision arithmetic negative numbers are easily represented by subtraction from the modulus. For example, with 32-bit integers +$a - b\mbox{ (mod }2^{32}\mbox{)}$ is the same as $a + (2^{32} - b) \mbox{ (mod }2^{32}\mbox{)}$ since $2^{32} \equiv 0 \mbox{ (mod }2^{32}\mbox{)}$. +As a result subtraction can be performed with a trivial series of logical operations and an addition. + +However, in multiple precision arithmetic negative numbers are not represented in the same way. Instead a sign flag is used to keep track of the +sign of the integer. As a result signed addition and subtraction are actually implemented as conditional usage of lower level addition or +subtraction algorithms with the sign fixed up appropriately. + +The lower level algorithms will add or subtract integers without regard to the sign flag. That is they will add or subtract the magnitude of +the integers respectively. + +\subsection{Low Level Addition} +An unsigned addition of multiple precision integers is performed with the same long-hand algorithm used to add decimal numbers. That is to add the +trailing digits first and propagate the resulting carry upwards. Since this is a lower level algorithm the name will have a ``s\_'' prefix. +Historically that convention stems from the MPI library where ``s\_'' stood for static functions that were hidden from the developer entirely. + +\newpage +\begin{figure}[!here] +\begin{center} +\begin{small} +\begin{tabular}{l} +\hline Algorithm \textbf{s\_mp\_add}. \\ +\textbf{Input}. Two mp\_ints $a$ and $b$ \\ +\textbf{Output}. The unsigned addition $c = \vert a \vert + \vert b \vert$. \\ +\hline \\ +1. if $a.used > b.used$ then \\ +\hspace{+3mm}1.1 $min \leftarrow b.used$ \\ +\hspace{+3mm}1.2 $max \leftarrow a.used$ \\ +\hspace{+3mm}1.3 $x \leftarrow a$ \\ +2. else \\ +\hspace{+3mm}2.1 $min \leftarrow a.used$ \\ +\hspace{+3mm}2.2 $max \leftarrow b.used$ \\ +\hspace{+3mm}2.3 $x \leftarrow b$ \\ +3. If $c.alloc < max + 1$ then grow $c$ to hold at least $max + 1$ digits (\textit{mp\_grow}) \\ +4. $oldused \leftarrow c.used$ \\ +5. $c.used \leftarrow max + 1$ \\ +6. $u \leftarrow 0$ \\ +7. for $n$ from $0$ to $min - 1$ do \\ +\hspace{+3mm}7.1 $c_n \leftarrow a_n + b_n + u$ \\ +\hspace{+3mm}7.2 $u \leftarrow c_n >> lg(\beta)$ \\ +\hspace{+3mm}7.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ +8. if $min \ne max$ then do \\ +\hspace{+3mm}8.1 for $n$ from $min$ to $max - 1$ do \\ +\hspace{+6mm}8.1.1 $c_n \leftarrow x_n + u$ \\ +\hspace{+6mm}8.1.2 $u \leftarrow c_n >> lg(\beta)$ \\ +\hspace{+6mm}8.1.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ +9. $c_{max} \leftarrow u$ \\ +10. if $olduse > max$ then \\ +\hspace{+3mm}10.1 for $n$ from $max + 1$ to $oldused - 1$ do \\ +\hspace{+6mm}10.1.1 $c_n \leftarrow 0$ \\ +11. Clamp excess digits in $c$. (\textit{mp\_clamp}) \\ +12. Return(\textit{MP\_OKAY}) \\ +\hline +\end{tabular} +\end{small} +\end{center} +\caption{Algorithm s\_mp\_add} +\end{figure} + +\textbf{Algorithm s\_mp\_add.} +This algorithm is loosely based on algorithm 14.7 of HAC \cite[pp. 594]{HAC} but has been extended to allow the inputs to have different magnitudes. +Coincidentally the description of algorithm A in Knuth \cite[pp. 266]{TAOCPV2} shares the same deficiency as the algorithm from \cite{HAC}. Even the +MIX pseudo machine code presented by Knuth \cite[pp. 266-267]{TAOCPV2} is incapable of handling inputs which are of different magnitudes. + +The first thing that has to be accomplished is to sort out which of the two inputs is the largest. The addition logic +will simply add all of the smallest input to the largest input and store that first part of the result in the +destination. Then it will apply a simpler addition loop to excess digits of the larger input. + +The first two steps will handle sorting the inputs such that $min$ and $max$ hold the digit counts of the two +inputs. The variable $x$ will be an mp\_int alias for the largest input or the second input $b$ if they have the +same number of digits. After the inputs are sorted the destination $c$ is grown as required to accomodate the sum +of the two inputs. The original \textbf{used} count of $c$ is copied and set to the new used count. + +At this point the first addition loop will go through as many digit positions that both inputs have. The carry +variable $\mu$ is set to zero outside the loop. Inside the loop an ``addition'' step requires three statements to produce +one digit of the summand. First +two digits from $a$ and $b$ are added together along with the carry $\mu$. The carry of this step is extracted and stored +in $\mu$ and finally the digit of the result $c_n$ is truncated within the range $0 \le c_n < \beta$. + +Now all of the digit positions that both inputs have in common have been exhausted. If $min \ne max$ then $x$ is an alias +for one of the inputs that has more digits. A simplified addition loop is then used to essentially copy the remaining digits +and the carry to the destination. + +The final carry is stored in $c_{max}$ and digits above $max$ upto $oldused$ are zeroed which completes the addition. + + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_add.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* low level addition, based on HAC pp.594, Algorithm 14.7 */ +018 int +019 s_mp_add (mp_int * a, mp_int * b, mp_int * c) +020 \{ +021 mp_int *x; +022 int olduse, res, min, max; +023 +024 /* find sizes, we let |a| <= |b| which means we have to sort +025 * them. "x" will point to the input with the most digits +026 */ +027 if (a->used > b->used) \{ +028 min = b->used; +029 max = a->used; +030 x = a; +031 \} else \{ +032 min = a->used; +033 max = b->used; +034 x = b; +035 \} +036 +037 /* init result */ +038 if (c->alloc < max + 1) \{ +039 if ((res = mp_grow (c, max + 1)) != MP_OKAY) \{ +040 return res; +041 \} +042 \} +043 +044 /* get old used digit count and set new one */ +045 olduse = c->used; +046 c->used = max + 1; +047 +048 \{ +049 register mp_digit u, *tmpa, *tmpb, *tmpc; +050 register int i; +051 +052 /* alias for digit pointers */ +053 +054 /* first input */ +055 tmpa = a->dp; +056 +057 /* second input */ +058 tmpb = b->dp; +059 +060 /* destination */ +061 tmpc = c->dp; +062 +063 /* zero the carry */ +064 u = 0; +065 for (i = 0; i < min; i++) \{ +066 /* Compute the sum at one digit, T[i] = A[i] + B[i] + U */ +067 *tmpc = *tmpa++ + *tmpb++ + u; +068 +069 /* U = carry bit of T[i] */ +070 u = *tmpc >> ((mp_digit)DIGIT_BIT); +071 +072 /* take away carry bit from T[i] */ +073 *tmpc++ &= MP_MASK; +074 \} +075 +076 /* now copy higher words if any, that is in A+B +077 * if A or B has more digits add those in +078 */ +079 if (min != max) \{ +080 for (; i < max; i++) \{ +081 /* T[i] = X[i] + U */ +082 *tmpc = x->dp[i] + u; +083 +084 /* U = carry bit of T[i] */ +085 u = *tmpc >> ((mp_digit)DIGIT_BIT); +086 +087 /* take away carry bit from T[i] */ +088 *tmpc++ &= MP_MASK; +089 \} +090 \} +091 +092 /* add carry */ +093 *tmpc++ = u; +094 +095 /* clear digits above oldused */ +096 for (i = c->used; i < olduse; i++) \{ +097 *tmpc++ = 0; +098 \} +099 \} +100 +101 mp_clamp (c); +102 return MP_OKAY; +103 \} +104 #endif +\end{alltt} +\end{small} + +Lines 27 to 35 perform the initial sorting of the inputs and determine the $min$ and $max$ variables. Note that $x$ is a pointer to a +mp\_int assigned to the largest input, in effect it is a local alias. Lines 37 to 42 ensure that the destination is grown to +accomodate the result of the addition. + +Similar to the implementation of mp\_copy this function uses the braced code and local aliases coding style. The three aliases that are on +lines 55, 58 and 61 represent the two inputs and destination variables respectively. These aliases are used to ensure the +compiler does not have to dereference $a$, $b$ or $c$ (respectively) to access the digits of the respective mp\_int. + +The initial carry $u$ is cleared on line 64, note that $u$ is of type mp\_digit which ensures type compatibility within the +implementation. The initial addition loop begins on line 65 and ends on line 74. Similarly the conditional addition loop +begins on line 80 and ends on line 90. The addition is finished with the final carry being stored in $tmpc$ on line 93. +Note the ``++'' operator on the same line. After line 93 $tmpc$ will point to the $c.used$'th digit of the mp\_int $c$. This is useful +for the next loop on lines 96 to 99 which set any old upper digits to zero. + +\subsection{Low Level Subtraction} +The low level unsigned subtraction algorithm is very similar to the low level unsigned addition algorithm. The principle difference is that the +unsigned subtraction algorithm requires the result to be positive. That is when computing $a - b$ the condition $\vert a \vert \ge \vert b\vert$ must +be met for this algorithm to function properly. Keep in mind this low level algorithm is not meant to be used in higher level algorithms directly. +This algorithm as will be shown can be used to create functional signed addition and subtraction algorithms. + + +For this algorithm a new variable is required to make the description simpler. Recall from section 1.3.1 that a mp\_digit must be able to represent +the range $0 \le x < 2\beta$ for the algorithms to work correctly. However, it is allowable that a mp\_digit represent a larger range of values. For +this algorithm we will assume that the variable $\gamma$ represents the number of bits available in a +mp\_digit (\textit{this implies $2^{\gamma} > \beta$}). + +For example, the default for LibTomMath is to use a ``unsigned long'' for the mp\_digit ``type'' while $\beta = 2^{28}$. In ISO C an ``unsigned long'' +data type must be able to represent $0 \le x < 2^{32}$ meaning that in this case $\gamma = 32$. + +\newpage\begin{figure}[!here] +\begin{center} +\begin{small} +\begin{tabular}{l} +\hline Algorithm \textbf{s\_mp\_sub}. \\ +\textbf{Input}. Two mp\_ints $a$ and $b$ ($\vert a \vert \ge \vert b \vert$) \\ +\textbf{Output}. The unsigned subtraction $c = \vert a \vert - \vert b \vert$. \\ +\hline \\ +1. $min \leftarrow b.used$ \\ +2. $max \leftarrow a.used$ \\ +3. If $c.alloc < max$ then grow $c$ to hold at least $max$ digits. (\textit{mp\_grow}) \\ +4. $oldused \leftarrow c.used$ \\ +5. $c.used \leftarrow max$ \\ +6. $u \leftarrow 0$ \\ +7. for $n$ from $0$ to $min - 1$ do \\ +\hspace{3mm}7.1 $c_n \leftarrow a_n - b_n - u$ \\ +\hspace{3mm}7.2 $u \leftarrow c_n >> (\gamma - 1)$ \\ +\hspace{3mm}7.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ +8. if $min < max$ then do \\ +\hspace{3mm}8.1 for $n$ from $min$ to $max - 1$ do \\ +\hspace{6mm}8.1.1 $c_n \leftarrow a_n - u$ \\ +\hspace{6mm}8.1.2 $u \leftarrow c_n >> (\gamma - 1)$ \\ +\hspace{6mm}8.1.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ +9. if $oldused > max$ then do \\ +\hspace{3mm}9.1 for $n$ from $max$ to $oldused - 1$ do \\ +\hspace{6mm}9.1.1 $c_n \leftarrow 0$ \\ +10. Clamp excess digits of $c$. (\textit{mp\_clamp}). \\ +11. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{small} +\end{center} +\caption{Algorithm s\_mp\_sub} +\end{figure} + +\textbf{Algorithm s\_mp\_sub.} +This algorithm performs the unsigned subtraction of two mp\_int variables under the restriction that the result must be positive. That is when +passing variables $a$ and $b$ the condition that $\vert a \vert \ge \vert b \vert$ must be met for the algorithm to function correctly. This +algorithm is loosely based on algorithm 14.9 \cite[pp. 595]{HAC} and is similar to algorithm S in \cite[pp. 267]{TAOCPV2} as well. As was the case +of the algorithm s\_mp\_add both other references lack discussion concerning various practical details such as when the inputs differ in magnitude. + +The initial sorting of the inputs is trivial in this algorithm since $a$ is guaranteed to have at least the same magnitude of $b$. Steps 1 and 2 +set the $min$ and $max$ variables. Unlike the addition routine there is guaranteed to be no carry which means that the final result can be at +most $max$ digits in length as opposed to $max + 1$. Similar to the addition algorithm the \textbf{used} count of $c$ is copied locally and +set to the maximal count for the operation. + +The subtraction loop that begins on step seven is essentially the same as the addition loop of algorithm s\_mp\_add except single precision +subtraction is used instead. Note the use of the $\gamma$ variable to extract the carry (\textit{also known as the borrow}) within the subtraction +loops. Under the assumption that two's complement single precision arithmetic is used this will successfully extract the desired carry. + +For example, consider subtracting $0101_2$ from $0100_2$ where $\gamma = 4$ and $\beta = 2$. The least significant bit will force a carry upwards to +the third bit which will be set to zero after the borrow. After the very first bit has been subtracted $4 - 1 \equiv 0011_2$ will remain, When the +third bit of $0101_2$ is subtracted from the result it will cause another carry. In this case though the carry will be forced to propagate all the +way to the most significant bit. + +Recall that $\beta < 2^{\gamma}$. This means that if a carry does occur just before the $lg(\beta)$'th bit it will propagate all the way to the most +significant bit. Thus, the high order bits of the mp\_digit that are not part of the actual digit will either be all zero, or all one. All that +is needed is a single zero or one bit for the carry. Therefore a single logical shift right by $\gamma - 1$ positions is sufficient to extract the +carry. This method of carry extraction may seem awkward but the reason for it becomes apparent when the implementation is discussed. + +If $b$ has a smaller magnitude than $a$ then step 9 will force the carry and copy operation to propagate through the larger input $a$ into $c$. Step +10 will ensure that any leading digits of $c$ above the $max$'th position are zeroed. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_sub.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* low level subtraction (assumes |a| > |b|), HAC pp.595 Algorithm 14.9 */ +018 int +019 s_mp_sub (mp_int * a, mp_int * b, mp_int * c) +020 \{ +021 int olduse, res, min, max; +022 +023 /* find sizes */ +024 min = b->used; +025 max = a->used; +026 +027 /* init result */ +028 if (c->alloc < max) \{ +029 if ((res = mp_grow (c, max)) != MP_OKAY) \{ +030 return res; +031 \} +032 \} +033 olduse = c->used; +034 c->used = max; +035 +036 \{ +037 register mp_digit u, *tmpa, *tmpb, *tmpc; +038 register int i; +039 +040 /* alias for digit pointers */ +041 tmpa = a->dp; +042 tmpb = b->dp; +043 tmpc = c->dp; +044 +045 /* set carry to zero */ +046 u = 0; +047 for (i = 0; i < min; i++) \{ +048 /* T[i] = A[i] - B[i] - U */ +049 *tmpc = *tmpa++ - *tmpb++ - u; +050 +051 /* U = carry bit of T[i] +052 * Note this saves performing an AND operation since +053 * if a carry does occur it will propagate all the way to the +054 * MSB. As a result a single shift is enough to get the carry +055 */ +056 u = *tmpc >> ((mp_digit)(CHAR_BIT * sizeof (mp_digit) - 1)); +057 +058 /* Clear carry from T[i] */ +059 *tmpc++ &= MP_MASK; +060 \} +061 +062 /* now copy higher words if any, e.g. if A has more digits than B */ +063 for (; i < max; i++) \{ +064 /* T[i] = A[i] - U */ +065 *tmpc = *tmpa++ - u; +066 +067 /* U = carry bit of T[i] */ +068 u = *tmpc >> ((mp_digit)(CHAR_BIT * sizeof (mp_digit) - 1)); +069 +070 /* Clear carry from T[i] */ +071 *tmpc++ &= MP_MASK; +072 \} +073 +074 /* clear digits above used (since we may not have grown result above) */ + +075 for (i = c->used; i < olduse; i++) \{ +076 *tmpc++ = 0; +077 \} +078 \} +079 +080 mp_clamp (c); +081 return MP_OKAY; +082 \} +083 +084 #endif +\end{alltt} +\end{small} + +Line 24 and 25 perform the initial hardcoded sorting of the inputs. In reality the $min$ and $max$ variables are only aliases and are only +used to make the source code easier to read. Again the pointer alias optimization is used within this algorithm. Lines 41, 42 and 43 initialize the aliases for +$a$, $b$ and $c$ respectively. + +The first subtraction loop occurs on lines 46 through 60. The theory behind the subtraction loop is exactly the same as that for +the addition loop. As remarked earlier there is an implementation reason for using the ``awkward'' method of extracting the carry +(\textit{see line 56}). The traditional method for extracting the carry would be to shift by $lg(\beta)$ positions and logically AND +the least significant bit. The AND operation is required because all of the bits above the $\lg(\beta)$'th bit will be set to one after a carry +occurs from subtraction. This carry extraction requires two relatively cheap operations to extract the carry. The other method is to simply +shift the most significant bit to the least significant bit thus extracting the carry with a single cheap operation. This optimization only works on +twos compliment machines which is a safe assumption to make. + +If $a$ has a larger magnitude than $b$ an additional loop (\textit{see lines 63 through 72}) is required to propagate the carry through +$a$ and copy the result to $c$. + +\subsection{High Level Addition} +Now that both lower level addition and subtraction algorithms have been established an effective high level signed addition algorithm can be +established. This high level addition algorithm will be what other algorithms and developers will use to perform addition of mp\_int data +types. + +Recall from section 5.2 that an mp\_int represents an integer with an unsigned mantissa (\textit{the array of digits}) and a \textbf{sign} +flag. A high level addition is actually performed as a series of eight separate cases which can be optimized down to three unique cases. + +\begin{figure}[!here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_add}. \\ +\textbf{Input}. Two mp\_ints $a$ and $b$ \\ +\textbf{Output}. The signed addition $c = a + b$. \\ +\hline \\ +1. if $a.sign = b.sign$ then do \\ +\hspace{3mm}1.1 $c.sign \leftarrow a.sign$ \\ +\hspace{3mm}1.2 $c \leftarrow \vert a \vert + \vert b \vert$ (\textit{s\_mp\_add})\\ +2. else do \\ +\hspace{3mm}2.1 if $\vert a \vert < \vert b \vert$ then do (\textit{mp\_cmp\_mag}) \\ +\hspace{6mm}2.1.1 $c.sign \leftarrow b.sign$ \\ +\hspace{6mm}2.1.2 $c \leftarrow \vert b \vert - \vert a \vert$ (\textit{s\_mp\_sub}) \\ +\hspace{3mm}2.2 else do \\ +\hspace{6mm}2.2.1 $c.sign \leftarrow a.sign$ \\ +\hspace{6mm}2.2.2 $c \leftarrow \vert a \vert - \vert b \vert$ \\ +3. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_add} +\end{figure} + +\textbf{Algorithm mp\_add.} +This algorithm performs the signed addition of two mp\_int variables. There is no reference algorithm to draw upon from +either \cite{TAOCPV2} or \cite{HAC} since they both only provide unsigned operations. The algorithm is fairly +straightforward but restricted since subtraction can only produce positive results. + +\begin{figure}[here] +\begin{small} +\begin{center} +\begin{tabular}{|c|c|c|c|c|} +\hline \textbf{Sign of $a$} & \textbf{Sign of $b$} & \textbf{$\vert a \vert > \vert b \vert $} & \textbf{Unsigned Operation} & \textbf{Result Sign Flag} \\ +\hline $+$ & $+$ & Yes & $c = a + b$ & $a.sign$ \\ +\hline $+$ & $+$ & No & $c = a + b$ & $a.sign$ \\ +\hline $-$ & $-$ & Yes & $c = a + b$ & $a.sign$ \\ +\hline $-$ & $-$ & No & $c = a + b$ & $a.sign$ \\ +\hline &&&&\\ + +\hline $+$ & $-$ & No & $c = b - a$ & $b.sign$ \\ +\hline $-$ & $+$ & No & $c = b - a$ & $b.sign$ \\ + +\hline &&&&\\ + +\hline $+$ & $-$ & Yes & $c = a - b$ & $a.sign$ \\ +\hline $-$ & $+$ & Yes & $c = a - b$ & $a.sign$ \\ + +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Addition Guide Chart} +\label{fig:AddChart} +\end{figure} + +Figure~\ref{fig:AddChart} lists all of the eight possible input combinations and is sorted to show that only three +specific cases need to be handled. The return code of the unsigned operations at step 1.2, 2.1.2 and 2.2.2 are +forwarded to step three to check for errors. This simplifies the description of the algorithm considerably and best +follows how the implementation actually was achieved. + +Also note how the \textbf{sign} is set before the unsigned addition or subtraction is performed. Recall from the descriptions of algorithms +s\_mp\_add and s\_mp\_sub that the mp\_clamp function is used at the end to trim excess digits. The mp\_clamp algorithm will set the \textbf{sign} +to \textbf{MP\_ZPOS} when the \textbf{used} digit count reaches zero. + +For example, consider performing $-a + a$ with algorithm mp\_add. By the description of the algorithm the sign is set to \textbf{MP\_NEG} which would +produce a result of $-0$. However, since the sign is set first then the unsigned addition is performed the subsequent usage of algorithm mp\_clamp +within algorithm s\_mp\_add will force $-0$ to become $0$. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_add.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* high level addition (handles signs) */ +018 int mp_add (mp_int * a, mp_int * b, mp_int * c) +019 \{ +020 int sa, sb, res; +021 +022 /* get sign of both inputs */ +023 sa = a->sign; +024 sb = b->sign; +025 +026 /* handle two cases, not four */ +027 if (sa == sb) \{ +028 /* both positive or both negative */ +029 /* add their magnitudes, copy the sign */ +030 c->sign = sa; +031 res = s_mp_add (a, b, c); +032 \} else \{ +033 /* one positive, the other negative */ +034 /* subtract the one with the greater magnitude from */ +035 /* the one of the lesser magnitude. The result gets */ +036 /* the sign of the one with the greater magnitude. */ +037 if (mp_cmp_mag (a, b) == MP_LT) \{ +038 c->sign = sb; +039 res = s_mp_sub (b, a, c); +040 \} else \{ +041 c->sign = sa; +042 res = s_mp_sub (a, b, c); +043 \} +044 \} +045 return res; +046 \} +047 +048 #endif +\end{alltt} +\end{small} + +The source code follows the algorithm fairly closely. The most notable new source code addition is the usage of the $res$ integer variable which +is used to pass result of the unsigned operations forward. Unlike in the algorithm, the variable $res$ is merely returned as is without +explicitly checking it and returning the constant \textbf{MP\_OKAY}. The observation is this algorithm will succeed or fail only if the lower +level functions do so. Returning their return code is sufficient. + +\subsection{High Level Subtraction} +The high level signed subtraction algorithm is essentially the same as the high level signed addition algorithm. + +\newpage\begin{figure}[!here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_sub}. \\ +\textbf{Input}. Two mp\_ints $a$ and $b$ \\ +\textbf{Output}. The signed subtraction $c = a - b$. \\ +\hline \\ +1. if $a.sign \ne b.sign$ then do \\ +\hspace{3mm}1.1 $c.sign \leftarrow a.sign$ \\ +\hspace{3mm}1.2 $c \leftarrow \vert a \vert + \vert b \vert$ (\textit{s\_mp\_add}) \\ +2. else do \\ +\hspace{3mm}2.1 if $\vert a \vert \ge \vert b \vert$ then do (\textit{mp\_cmp\_mag}) \\ +\hspace{6mm}2.1.1 $c.sign \leftarrow a.sign$ \\ +\hspace{6mm}2.1.2 $c \leftarrow \vert a \vert - \vert b \vert$ (\textit{s\_mp\_sub}) \\ +\hspace{3mm}2.2 else do \\ +\hspace{6mm}2.2.1 $c.sign \leftarrow \left \lbrace \begin{array}{ll} + MP\_ZPOS & \mbox{if }a.sign = MP\_NEG \\ + MP\_NEG & \mbox{otherwise} \\ + \end{array} \right .$ \\ +\hspace{6mm}2.2.2 $c \leftarrow \vert b \vert - \vert a \vert$ \\ +3. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_sub} +\end{figure} + +\textbf{Algorithm mp\_sub.} +This algorithm performs the signed subtraction of two inputs. Similar to algorithm mp\_add there is no reference in either \cite{TAOCPV2} or +\cite{HAC}. Also this algorithm is restricted by algorithm s\_mp\_sub. Chart \ref{fig:SubChart} lists the eight possible inputs and +the operations required. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{|c|c|c|c|c|} +\hline \textbf{Sign of $a$} & \textbf{Sign of $b$} & \textbf{$\vert a \vert \ge \vert b \vert $} & \textbf{Unsigned Operation} & \textbf{Result Sign Flag} \\ +\hline $+$ & $-$ & Yes & $c = a + b$ & $a.sign$ \\ +\hline $+$ & $-$ & No & $c = a + b$ & $a.sign$ \\ +\hline $-$ & $+$ & Yes & $c = a + b$ & $a.sign$ \\ +\hline $-$ & $+$ & No & $c = a + b$ & $a.sign$ \\ +\hline &&&& \\ +\hline $+$ & $+$ & Yes & $c = a - b$ & $a.sign$ \\ +\hline $-$ & $-$ & Yes & $c = a - b$ & $a.sign$ \\ +\hline &&&& \\ +\hline $+$ & $+$ & No & $c = b - a$ & $\mbox{opposite of }a.sign$ \\ +\hline $-$ & $-$ & No & $c = b - a$ & $\mbox{opposite of }a.sign$ \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Subtraction Guide Chart} +\label{fig:SubChart} +\end{figure} + +Similar to the case of algorithm mp\_add the \textbf{sign} is set first before the unsigned addition or subtraction. That is to prevent the +algorithm from producing $-a - -a = -0$ as a result. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_sub.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* high level subtraction (handles signs) */ +018 int +019 mp_sub (mp_int * a, mp_int * b, mp_int * c) +020 \{ +021 int sa, sb, res; +022 +023 sa = a->sign; +024 sb = b->sign; +025 +026 if (sa != sb) \{ +027 /* subtract a negative from a positive, OR */ +028 /* subtract a positive from a negative. */ +029 /* In either case, ADD their magnitudes, */ +030 /* and use the sign of the first number. */ +031 c->sign = sa; +032 res = s_mp_add (a, b, c); +033 \} else \{ +034 /* subtract a positive from a positive, OR */ +035 /* subtract a negative from a negative. */ +036 /* First, take the difference between their */ +037 /* magnitudes, then... */ +038 if (mp_cmp_mag (a, b) != MP_LT) \{ +039 /* Copy the sign from the first */ +040 c->sign = sa; +041 /* The first has a larger or equal magnitude */ +042 res = s_mp_sub (a, b, c); +043 \} else \{ +044 /* The result has the *opposite* sign from */ +045 /* the first number. */ +046 c->sign = (sa == MP_ZPOS) ? MP_NEG : MP_ZPOS; +047 /* The second has a larger magnitude */ +048 res = s_mp_sub (b, a, c); +049 \} +050 \} +051 return res; +052 \} +053 +054 #endif +\end{alltt} +\end{small} + +Much like the implementation of algorithm mp\_add the variable $res$ is used to catch the return code of the unsigned addition or subtraction operations +and forward it to the end of the function. On line 38 the ``not equal to'' \textbf{MP\_LT} expression is used to emulate a +``greater than or equal to'' comparison. + +\section{Bit and Digit Shifting} +It is quite common to think of a multiple precision integer as a polynomial in $x$, that is $y = f(\beta)$ where $f(x) = \sum_{i=0}^{n-1} a_i x^i$. +This notation arises within discussion of Montgomery and Diminished Radix Reduction as well as Karatsuba multiplication and squaring. + +In order to facilitate operations on polynomials in $x$ as above a series of simple ``digit'' algorithms have to be established. That is to shift +the digits left or right as well to shift individual bits of the digits left and right. It is important to note that not all ``shift'' operations +are on radix-$\beta$ digits. + +\subsection{Multiplication by Two} + +In a binary system where the radix is a power of two multiplication by two not only arises often in other algorithms it is a fairly efficient +operation to perform. A single precision logical shift left is sufficient to multiply a single digit by two. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_mul\_2}. \\ +\textbf{Input}. One mp\_int $a$ \\ +\textbf{Output}. $b = 2a$. \\ +\hline \\ +1. If $b.alloc < a.used + 1$ then grow $b$ to hold $a.used + 1$ digits. (\textit{mp\_grow}) \\ +2. $oldused \leftarrow b.used$ \\ +3. $b.used \leftarrow a.used$ \\ +4. $r \leftarrow 0$ \\ +5. for $n$ from 0 to $a.used - 1$ do \\ +\hspace{3mm}5.1 $rr \leftarrow a_n >> (lg(\beta) - 1)$ \\ +\hspace{3mm}5.2 $b_n \leftarrow (a_n << 1) + r \mbox{ (mod }\beta\mbox{)}$ \\ +\hspace{3mm}5.3 $r \leftarrow rr$ \\ +6. If $r \ne 0$ then do \\ +\hspace{3mm}6.1 $b_{n + 1} \leftarrow r$ \\ +\hspace{3mm}6.2 $b.used \leftarrow b.used + 1$ \\ +7. If $b.used < oldused - 1$ then do \\ +\hspace{3mm}7.1 for $n$ from $b.used$ to $oldused - 1$ do \\ +\hspace{6mm}7.1.1 $b_n \leftarrow 0$ \\ +8. $b.sign \leftarrow a.sign$ \\ +9. Return(\textit{MP\_OKAY}).\\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_mul\_2} +\end{figure} + +\textbf{Algorithm mp\_mul\_2.} +This algorithm will quickly multiply a mp\_int by two provided $\beta$ is a power of two. Neither \cite{TAOCPV2} nor \cite{HAC} describe such +an algorithm despite the fact it arises often in other algorithms. The algorithm is setup much like the lower level algorithm s\_mp\_add since +it is for all intents and purposes equivalent to the operation $b = \vert a \vert + \vert a \vert$. + +Step 1 and 2 grow the input as required to accomodate the maximum number of \textbf{used} digits in the result. The initial \textbf{used} count +is set to $a.used$ at step 4. Only if there is a final carry will the \textbf{used} count require adjustment. + +Step 6 is an optimization implementation of the addition loop for this specific case. That is since the two values being added together +are the same there is no need to perform two reads from the digits of $a$. Step 6.1 performs a single precision shift on the current digit $a_n$ to +obtain what will be the carry for the next iteration. Step 6.2 calculates the $n$'th digit of the result as single precision shift of $a_n$ plus +the previous carry. Recall from section 4.1 that $a_n << 1$ is equivalent to $a_n \cdot 2$. An iteration of the addition loop is finished with +forwarding the carry to the next iteration. + +Step 7 takes care of any final carry by setting the $a.used$'th digit of the result to the carry and augmenting the \textbf{used} count of $b$. +Step 8 clears any leading digits of $b$ in case it originally had a larger magnitude than $a$. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_mul\_2.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* b = a*2 */ +018 int mp_mul_2(mp_int * a, mp_int * b) +019 \{ +020 int x, res, oldused; +021 +022 /* grow to accomodate result */ +023 if (b->alloc < a->used + 1) \{ +024 if ((res = mp_grow (b, a->used + 1)) != MP_OKAY) \{ +025 return res; +026 \} +027 \} +028 +029 oldused = b->used; +030 b->used = a->used; +031 +032 \{ +033 register mp_digit r, rr, *tmpa, *tmpb; +034 +035 /* alias for source */ +036 tmpa = a->dp; +037 +038 /* alias for dest */ +039 tmpb = b->dp; +040 +041 /* carry */ +042 r = 0; +043 for (x = 0; x < a->used; x++) \{ +044 +045 /* get what will be the *next* carry bit from the +046 * MSB of the current digit +047 */ +048 rr = *tmpa >> ((mp_digit)(DIGIT_BIT - 1)); +049 +050 /* now shift up this digit, add in the carry [from the previous] */ +051 *tmpb++ = ((*tmpa++ << ((mp_digit)1)) | r) & MP_MASK; +052 +053 /* copy the carry that would be from the source +054 * digit into the next iteration +055 */ +056 r = rr; +057 \} +058 +059 /* new leading digit? */ +060 if (r != 0) \{ +061 /* add a MSB which is always 1 at this point */ +062 *tmpb = 1; +063 ++(b->used); +064 \} +065 +066 /* now zero any excess digits on the destination +067 * that we didn't write to +068 */ +069 tmpb = b->dp + b->used; +070 for (x = b->used; x < oldused; x++) \{ +071 *tmpb++ = 0; +072 \} +073 \} +074 b->sign = a->sign; +075 return MP_OKAY; +076 \} +077 #endif +\end{alltt} +\end{small} + +This implementation is essentially an optimized implementation of s\_mp\_add for the case of doubling an input. The only noteworthy difference +is the use of the logical shift operator on line 51 to perform a single precision doubling. + +\subsection{Division by Two} +A division by two can just as easily be accomplished with a logical shift right as multiplication by two can be with a logical shift left. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_div\_2}. \\ +\textbf{Input}. One mp\_int $a$ \\ +\textbf{Output}. $b = a/2$. \\ +\hline \\ +1. If $b.alloc < a.used$ then grow $b$ to hold $a.used$ digits. (\textit{mp\_grow}) \\ +2. If the reallocation failed return(\textit{MP\_MEM}). \\ +3. $oldused \leftarrow b.used$ \\ +4. $b.used \leftarrow a.used$ \\ +5. $r \leftarrow 0$ \\ +6. for $n$ from $b.used - 1$ to $0$ do \\ +\hspace{3mm}6.1 $rr \leftarrow a_n \mbox{ (mod }2\mbox{)}$\\ +\hspace{3mm}6.2 $b_n \leftarrow (a_n >> 1) + (r << (lg(\beta) - 1)) \mbox{ (mod }\beta\mbox{)}$ \\ +\hspace{3mm}6.3 $r \leftarrow rr$ \\ +7. If $b.used < oldused - 1$ then do \\ +\hspace{3mm}7.1 for $n$ from $b.used$ to $oldused - 1$ do \\ +\hspace{6mm}7.1.1 $b_n \leftarrow 0$ \\ +8. $b.sign \leftarrow a.sign$ \\ +9. Clamp excess digits of $b$. (\textit{mp\_clamp}) \\ +10. Return(\textit{MP\_OKAY}).\\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_div\_2} +\end{figure} + +\textbf{Algorithm mp\_div\_2.} +This algorithm will divide an mp\_int by two using logical shifts to the right. Like mp\_mul\_2 it uses a modified low level addition +core as the basis of the algorithm. Unlike mp\_mul\_2 the shift operations work from the leading digit to the trailing digit. The algorithm +could be written to work from the trailing digit to the leading digit however, it would have to stop one short of $a.used - 1$ digits to prevent +reading past the end of the array of digits. + +Essentially the loop at step 6 is similar to that of mp\_mul\_2 except the logical shifts go in the opposite direction and the carry is at the +least significant bit not the most significant bit. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_div\_2.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* b = a/2 */ +018 int mp_div_2(mp_int * a, mp_int * b) +019 \{ +020 int x, res, oldused; +021 +022 /* copy */ +023 if (b->alloc < a->used) \{ +024 if ((res = mp_grow (b, a->used)) != MP_OKAY) \{ +025 return res; +026 \} +027 \} +028 +029 oldused = b->used; +030 b->used = a->used; +031 \{ +032 register mp_digit r, rr, *tmpa, *tmpb; +033 +034 /* source alias */ +035 tmpa = a->dp + b->used - 1; +036 +037 /* dest alias */ +038 tmpb = b->dp + b->used - 1; +039 +040 /* carry */ +041 r = 0; +042 for (x = b->used - 1; x >= 0; x--) \{ +043 /* get the carry for the next iteration */ +044 rr = *tmpa & 1; +045 +046 /* shift the current digit, add in carry and store */ +047 *tmpb-- = (*tmpa-- >> 1) | (r << (DIGIT_BIT - 1)); +048 +049 /* forward carry to next iteration */ +050 r = rr; +051 \} +052 +053 /* zero excess digits */ +054 tmpb = b->dp + b->used; +055 for (x = b->used; x < oldused; x++) \{ +056 *tmpb++ = 0; +057 \} +058 \} +059 b->sign = a->sign; +060 mp_clamp (b); +061 return MP_OKAY; +062 \} +063 #endif +\end{alltt} +\end{small} + +\section{Polynomial Basis Operations} +Recall from section 4.3 that any integer can be represented as a polynomial in $x$ as $y = f(\beta)$. Such a representation is also known as +the polynomial basis \cite[pp. 48]{ROSE}. Given such a notation a multiplication or division by $x$ amounts to shifting whole digits a single +place. The need for such operations arises in several other higher level algorithms such as Barrett and Montgomery reduction, integer +division and Karatsuba multiplication. + +Converting from an array of digits to polynomial basis is very simple. Consider the integer $y \equiv (a_2, a_1, a_0)_{\beta}$ and recall that +$y = \sum_{i=0}^{2} a_i \beta^i$. Simply replace $\beta$ with $x$ and the expression is in polynomial basis. For example, $f(x) = 8x + 9$ is the +polynomial basis representation for $89$ using radix ten. That is, $f(10) = 8(10) + 9 = 89$. + +\subsection{Multiplication by $x$} + +Given a polynomial in $x$ such as $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0$ multiplying by $x$ amounts to shifting the coefficients up one +degree. In this case $f(x) \cdot x = a_n x^{n+1} + a_{n-1} x^n + ... + a_0 x$. From a scalar basis point of view multiplying by $x$ is equivalent to +multiplying by the integer $\beta$. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_lshd}. \\ +\textbf{Input}. One mp\_int $a$ and an integer $b$ \\ +\textbf{Output}. $a \leftarrow a \cdot \beta^b$ (equivalent to multiplication by $x^b$). \\ +\hline \\ +1. If $b \le 0$ then return(\textit{MP\_OKAY}). \\ +2. If $a.alloc < a.used + b$ then grow $a$ to at least $a.used + b$ digits. (\textit{mp\_grow}). \\ +3. If the reallocation failed return(\textit{MP\_MEM}). \\ +4. $a.used \leftarrow a.used + b$ \\ +5. $i \leftarrow a.used - 1$ \\ +6. $j \leftarrow a.used - 1 - b$ \\ +7. for $n$ from $a.used - 1$ to $b$ do \\ +\hspace{3mm}7.1 $a_{i} \leftarrow a_{j}$ \\ +\hspace{3mm}7.2 $i \leftarrow i - 1$ \\ +\hspace{3mm}7.3 $j \leftarrow j - 1$ \\ +8. for $n$ from 0 to $b - 1$ do \\ +\hspace{3mm}8.1 $a_n \leftarrow 0$ \\ +9. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_lshd} +\end{figure} + +\textbf{Algorithm mp\_lshd.} +This algorithm multiplies an mp\_int by the $b$'th power of $x$. This is equivalent to multiplying by $\beta^b$. The algorithm differs +from the other algorithms presented so far as it performs the operation in place instead storing the result in a separate location. The +motivation behind this change is due to the way this function is typically used. Algorithms such as mp\_add store the result in an optionally +different third mp\_int because the original inputs are often still required. Algorithm mp\_lshd (\textit{and similarly algorithm mp\_rshd}) is +typically used on values where the original value is no longer required. The algorithm will return success immediately if +$b \le 0$ since the rest of algorithm is only valid when $b > 0$. + +First the destination $a$ is grown as required to accomodate the result. The counters $i$ and $j$ are used to form a \textit{sliding window} over +the digits of $a$ of length $b$. The head of the sliding window is at $i$ (\textit{the leading digit}) and the tail at $j$ (\textit{the trailing digit}). +The loop on step 7 copies the digit from the tail to the head. In each iteration the window is moved down one digit. The last loop on +step 8 sets the lower $b$ digits to zero. + +\newpage +\begin{center} +\begin{figure}[here] +\includegraphics{pics/sliding_window.ps} +\caption{Sliding Window Movement} +\label{pic:sliding_window} +\end{figure} +\end{center} + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_lshd.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* shift left a certain amount of digits */ +018 int mp_lshd (mp_int * a, int b) +019 \{ +020 int x, res; +021 +022 /* if its less than zero return */ +023 if (b <= 0) \{ +024 return MP_OKAY; +025 \} +026 +027 /* grow to fit the new digits */ +028 if (a->alloc < a->used + b) \{ +029 if ((res = mp_grow (a, a->used + b)) != MP_OKAY) \{ +030 return res; +031 \} +032 \} +033 +034 \{ +035 register mp_digit *top, *bottom; +036 +037 /* increment the used by the shift amount then copy upwards */ +038 a->used += b; +039 +040 /* top */ +041 top = a->dp + a->used - 1; +042 +043 /* base */ +044 bottom = a->dp + a->used - 1 - b; +045 +046 /* much like mp_rshd this is implemented using a sliding window +047 * except the window goes the otherway around. Copying from +048 * the bottom to the top. see bn_mp_rshd.c for more info. +049 */ +050 for (x = a->used - 1; x >= b; x--) \{ +051 *top-- = *bottom--; +052 \} +053 +054 /* zero the lower digits */ +055 top = a->dp; +056 for (x = 0; x < b; x++) \{ +057 *top++ = 0; +058 \} +059 \} +060 return MP_OKAY; +061 \} +062 #endif +\end{alltt} +\end{small} + +The if statement on line 23 ensures that the $b$ variable is greater than zero. The \textbf{used} count is incremented by $b$ before +the copy loop begins. This elminates the need for an additional variable in the for loop. The variable $top$ on line 41 is an alias +for the leading digit while $bottom$ on line 44 is an alias for the trailing edge. The aliases form a window of exactly $b$ digits +over the input. + +\subsection{Division by $x$} + +Division by powers of $x$ is easily achieved by shifting the digits right and removing any that will end up to the right of the zero'th digit. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_rshd}. \\ +\textbf{Input}. One mp\_int $a$ and an integer $b$ \\ +\textbf{Output}. $a \leftarrow a / \beta^b$ (Divide by $x^b$). \\ +\hline \\ +1. If $b \le 0$ then return. \\ +2. If $a.used \le b$ then do \\ +\hspace{3mm}2.1 Zero $a$. (\textit{mp\_zero}). \\ +\hspace{3mm}2.2 Return. \\ +3. $i \leftarrow 0$ \\ +4. $j \leftarrow b$ \\ +5. for $n$ from 0 to $a.used - b - 1$ do \\ +\hspace{3mm}5.1 $a_i \leftarrow a_j$ \\ +\hspace{3mm}5.2 $i \leftarrow i + 1$ \\ +\hspace{3mm}5.3 $j \leftarrow j + 1$ \\ +6. for $n$ from $a.used - b$ to $a.used - 1$ do \\ +\hspace{3mm}6.1 $a_n \leftarrow 0$ \\ +7. $a.used \leftarrow a.used - b$ \\ +8. Return. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_rshd} +\end{figure} + +\textbf{Algorithm mp\_rshd.} +This algorithm divides the input in place by the $b$'th power of $x$. It is analogous to dividing by a $\beta^b$ but much quicker since +it does not require single precision division. This algorithm does not actually return an error code as it cannot fail. + +If the input $b$ is less than one the algorithm quickly returns without performing any work. If the \textbf{used} count is less than or equal +to the shift count $b$ then it will simply zero the input and return. + +After the trivial cases of inputs have been handled the sliding window is setup. Much like the case of algorithm mp\_lshd a sliding window that +is $b$ digits wide is used to copy the digits. Unlike mp\_lshd the window slides in the opposite direction from the trailing to the leading digit. +Also the digits are copied from the leading to the trailing edge. + +Once the window copy is complete the upper digits must be zeroed and the \textbf{used} count decremented. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_rshd.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* shift right a certain amount of digits */ +018 void mp_rshd (mp_int * a, int b) +019 \{ +020 int x; +021 +022 /* if b <= 0 then ignore it */ +023 if (b <= 0) \{ +024 return; +025 \} +026 +027 /* if b > used then simply zero it and return */ +028 if (a->used <= b) \{ +029 mp_zero (a); +030 return; +031 \} +032 +033 \{ +034 register mp_digit *bottom, *top; +035 +036 /* shift the digits down */ +037 +038 /* bottom */ +039 bottom = a->dp; +040 +041 /* top [offset into digits] */ +042 top = a->dp + b; +043 +044 /* this is implemented as a sliding window where +045 * the window is b-digits long and digits from +046 * the top of the window are copied to the bottom +047 * +048 * e.g. +049 +050 b-2 | b-1 | b0 | b1 | b2 | ... | bb | ----> +051 /\symbol{92} | ----> +052 \symbol{92}-------------------/ ----> +053 */ +054 for (x = 0; x < (a->used - b); x++) \{ +055 *bottom++ = *top++; +056 \} +057 +058 /* zero the top digits */ +059 for (; x < a->used; x++) \{ +060 *bottom++ = 0; +061 \} +062 \} +063 +064 /* remove excess digits */ +065 a->used -= b; +066 \} +067 #endif +\end{alltt} +\end{small} + +The only noteworthy element of this routine is the lack of a return type. + +-- Will update later to give it a return type...Tom + +\section{Powers of Two} + +Now that algorithms for moving single bits as well as whole digits exist algorithms for moving the ``in between'' distances are required. For +example, to quickly multiply by $2^k$ for any $k$ without using a full multiplier algorithm would prove useful. Instead of performing single +shifts $k$ times to achieve a multiplication by $2^{\pm k}$ a mixture of whole digit shifting and partial digit shifting is employed. + +\subsection{Multiplication by Power of Two} + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_mul\_2d}. \\ +\textbf{Input}. One mp\_int $a$ and an integer $b$ \\ +\textbf{Output}. $c \leftarrow a \cdot 2^b$. \\ +\hline \\ +1. $c \leftarrow a$. (\textit{mp\_copy}) \\ +2. If $c.alloc < c.used + \lfloor b / lg(\beta) \rfloor + 2$ then grow $c$ accordingly. \\ +3. If the reallocation failed return(\textit{MP\_MEM}). \\ +4. If $b \ge lg(\beta)$ then \\ +\hspace{3mm}4.1 $c \leftarrow c \cdot \beta^{\lfloor b / lg(\beta) \rfloor}$ (\textit{mp\_lshd}). \\ +\hspace{3mm}4.2 If step 4.1 failed return(\textit{MP\_MEM}). \\ +5. $d \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\ +6. If $d \ne 0$ then do \\ +\hspace{3mm}6.1 $mask \leftarrow 2^d$ \\ +\hspace{3mm}6.2 $r \leftarrow 0$ \\ +\hspace{3mm}6.3 for $n$ from $0$ to $c.used - 1$ do \\ +\hspace{6mm}6.3.1 $rr \leftarrow c_n >> (lg(\beta) - d) \mbox{ (mod }mask\mbox{)}$ \\ +\hspace{6mm}6.3.2 $c_n \leftarrow (c_n << d) + r \mbox{ (mod }\beta\mbox{)}$ \\ +\hspace{6mm}6.3.3 $r \leftarrow rr$ \\ +\hspace{3mm}6.4 If $r > 0$ then do \\ +\hspace{6mm}6.4.1 $c_{c.used} \leftarrow r$ \\ +\hspace{6mm}6.4.2 $c.used \leftarrow c.used + 1$ \\ +7. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_mul\_2d} +\end{figure} + +\textbf{Algorithm mp\_mul\_2d.} +This algorithm multiplies $a$ by $2^b$ and stores the result in $c$. The algorithm uses algorithm mp\_lshd and a derivative of algorithm mp\_mul\_2 to +quickly compute the product. + +First the algorithm will multiply $a$ by $x^{\lfloor b / lg(\beta) \rfloor}$ which will ensure that the remainder multiplicand is less than +$\beta$. For example, if $b = 37$ and $\beta = 2^{28}$ then this step will multiply by $x$ leaving a multiplication by $2^{37 - 28} = 2^{9}$ +left. + +After the digits have been shifted appropriately at most $lg(\beta) - 1$ shifts are left to perform. Step 5 calculates the number of remaining shifts +required. If it is non-zero a modified shift loop is used to calculate the remaining product. +Essentially the loop is a generic version of algorith mp\_mul2 designed to handle any shift count in the range $1 \le x < lg(\beta)$. The $mask$ +variable is used to extract the upper $d$ bits to form the carry for the next iteration. + +This algorithm is loosely measured as a $O(2n)$ algorithm which means that if the input is $n$-digits that it takes $2n$ ``time'' to +complete. It is possible to optimize this algorithm down to a $O(n)$ algorithm at a cost of making the algorithm slightly harder to follow. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_mul\_2d.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* shift left by a certain bit count */ +018 int mp_mul_2d (mp_int * a, int b, mp_int * c) +019 \{ +020 mp_digit d; +021 int res; +022 +023 /* copy */ +024 if (a != c) \{ +025 if ((res = mp_copy (a, c)) != MP_OKAY) \{ +026 return res; +027 \} +028 \} +029 +030 if (c->alloc < (int)(c->used + b/DIGIT_BIT + 1)) \{ +031 if ((res = mp_grow (c, c->used + b / DIGIT_BIT + 1)) != MP_OKAY) \{ +032 return res; +033 \} +034 \} +035 +036 /* shift by as many digits in the bit count */ +037 if (b >= (int)DIGIT_BIT) \{ +038 if ((res = mp_lshd (c, b / DIGIT_BIT)) != MP_OKAY) \{ +039 return res; +040 \} +041 \} +042 +043 /* shift any bit count < DIGIT_BIT */ +044 d = (mp_digit) (b % DIGIT_BIT); +045 if (d != 0) \{ +046 register mp_digit *tmpc, shift, mask, r, rr; +047 register int x; +048 +049 /* bitmask for carries */ +050 mask = (((mp_digit)1) << d) - 1; +051 +052 /* shift for msbs */ +053 shift = DIGIT_BIT - d; +054 +055 /* alias */ +056 tmpc = c->dp; +057 +058 /* carry */ +059 r = 0; +060 for (x = 0; x < c->used; x++) \{ +061 /* get the higher bits of the current word */ +062 rr = (*tmpc >> shift) & mask; +063 +064 /* shift the current word and OR in the carry */ +065 *tmpc = ((*tmpc << d) | r) & MP_MASK; +066 ++tmpc; +067 +068 /* set the carry to the carry bits of the current word */ +069 r = rr; +070 \} +071 +072 /* set final carry */ +073 if (r != 0) \{ +074 c->dp[(c->used)++] = r; +075 \} +076 \} +077 mp_clamp (c); +078 return MP_OKAY; +079 \} +080 #endif +\end{alltt} +\end{small} + +Notes to be revised when code is updated. -- Tom + +\subsection{Division by Power of Two} + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_div\_2d}. \\ +\textbf{Input}. One mp\_int $a$ and an integer $b$ \\ +\textbf{Output}. $c \leftarrow \lfloor a / 2^b \rfloor, d \leftarrow a \mbox{ (mod }2^b\mbox{)}$. \\ +\hline \\ +1. If $b \le 0$ then do \\ +\hspace{3mm}1.1 $c \leftarrow a$ (\textit{mp\_copy}) \\ +\hspace{3mm}1.2 $d \leftarrow 0$ (\textit{mp\_zero}) \\ +\hspace{3mm}1.3 Return(\textit{MP\_OKAY}). \\ +2. $c \leftarrow a$ \\ +3. $d \leftarrow a \mbox{ (mod }2^b\mbox{)}$ (\textit{mp\_mod\_2d}) \\ +4. If $b \ge lg(\beta)$ then do \\ +\hspace{3mm}4.1 $c \leftarrow \lfloor c/\beta^{\lfloor b/lg(\beta) \rfloor} \rfloor$ (\textit{mp\_rshd}). \\ +5. $k \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\ +6. If $k \ne 0$ then do \\ +\hspace{3mm}6.1 $mask \leftarrow 2^k$ \\ +\hspace{3mm}6.2 $r \leftarrow 0$ \\ +\hspace{3mm}6.3 for $n$ from $c.used - 1$ to $0$ do \\ +\hspace{6mm}6.3.1 $rr \leftarrow c_n \mbox{ (mod }mask\mbox{)}$ \\ +\hspace{6mm}6.3.2 $c_n \leftarrow (c_n >> k) + (r << (lg(\beta) - k))$ \\ +\hspace{6mm}6.3.3 $r \leftarrow rr$ \\ +7. Clamp excess digits of $c$. (\textit{mp\_clamp}) \\ +8. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_div\_2d} +\end{figure} + +\textbf{Algorithm mp\_div\_2d.} +This algorithm will divide an input $a$ by $2^b$ and produce the quotient and remainder. The algorithm is designed much like algorithm +mp\_mul\_2d by first using whole digit shifts then single precision shifts. This algorithm will also produce the remainder of the division +by using algorithm mp\_mod\_2d. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_div\_2d.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* shift right by a certain bit count (store quotient in c, optional remaind + er in d) */ +018 int mp_div_2d (mp_int * a, int b, mp_int * c, mp_int * d) +019 \{ +020 mp_digit D, r, rr; +021 int x, res; +022 mp_int t; +023 +024 +025 /* if the shift count is <= 0 then we do no work */ +026 if (b <= 0) \{ +027 res = mp_copy (a, c); +028 if (d != NULL) \{ +029 mp_zero (d); +030 \} +031 return res; +032 \} +033 +034 if ((res = mp_init (&t)) != MP_OKAY) \{ +035 return res; +036 \} +037 +038 /* get the remainder */ +039 if (d != NULL) \{ +040 if ((res = mp_mod_2d (a, b, &t)) != MP_OKAY) \{ +041 mp_clear (&t); +042 return res; +043 \} +044 \} +045 +046 /* copy */ +047 if ((res = mp_copy (a, c)) != MP_OKAY) \{ +048 mp_clear (&t); +049 return res; +050 \} +051 +052 /* shift by as many digits in the bit count */ +053 if (b >= (int)DIGIT_BIT) \{ +054 mp_rshd (c, b / DIGIT_BIT); +055 \} +056 +057 /* shift any bit count < DIGIT_BIT */ +058 D = (mp_digit) (b % DIGIT_BIT); +059 if (D != 0) \{ +060 register mp_digit *tmpc, mask, shift; +061 +062 /* mask */ +063 mask = (((mp_digit)1) << D) - 1; +064 +065 /* shift for lsb */ +066 shift = DIGIT_BIT - D; +067 +068 /* alias */ +069 tmpc = c->dp + (c->used - 1); +070 +071 /* carry */ +072 r = 0; +073 for (x = c->used - 1; x >= 0; x--) \{ +074 /* get the lower bits of this word in a temp */ +075 rr = *tmpc & mask; +076 +077 /* shift the current word and mix in the carry bits from the previous + word */ +078 *tmpc = (*tmpc >> D) | (r << shift); +079 --tmpc; +080 +081 /* set the carry to the carry bits of the current word found above */ +082 r = rr; +083 \} +084 \} +085 mp_clamp (c); +086 if (d != NULL) \{ +087 mp_exch (&t, d); +088 \} +089 mp_clear (&t); +090 return MP_OKAY; +091 \} +092 #endif +\end{alltt} +\end{small} + +The implementation of algorithm mp\_div\_2d is slightly different than the algorithm specifies. The remainder $d$ may be optionally +ignored by passing \textbf{NULL} as the pointer to the mp\_int variable. The temporary mp\_int variable $t$ is used to hold the +result of the remainder operation until the end. This allows $d$ and $a$ to represent the same mp\_int without modifying $a$ before +the quotient is obtained. + +The remainder of the source code is essentially the same as the source code for mp\_mul\_2d. (-- Fix this paragraph up later, Tom). + +\subsection{Remainder of Division by Power of Two} + +The last algorithm in the series of polynomial basis power of two algorithms is calculating the remainder of division by $2^b$. This +algorithm benefits from the fact that in twos complement arithmetic $a \mbox{ (mod }2^b\mbox{)}$ is the same as $a$ AND $2^b - 1$. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_mod\_2d}. \\ +\textbf{Input}. One mp\_int $a$ and an integer $b$ \\ +\textbf{Output}. $c \leftarrow a \mbox{ (mod }2^b\mbox{)}$. \\ +\hline \\ +1. If $b \le 0$ then do \\ +\hspace{3mm}1.1 $c \leftarrow 0$ (\textit{mp\_zero}) \\ +\hspace{3mm}1.2 Return(\textit{MP\_OKAY}). \\ +2. If $b > a.used \cdot lg(\beta)$ then do \\ +\hspace{3mm}2.1 $c \leftarrow a$ (\textit{mp\_copy}) \\ +\hspace{3mm}2.2 Return the result of step 2.1. \\ +3. $c \leftarrow a$ \\ +4. If step 3 failed return(\textit{MP\_MEM}). \\ +5. for $n$ from $\lceil b / lg(\beta) \rceil$ to $c.used$ do \\ +\hspace{3mm}5.1 $c_n \leftarrow 0$ \\ +6. $k \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\ +7. $c_{\lfloor b / lg(\beta) \rfloor} \leftarrow c_{\lfloor b / lg(\beta) \rfloor} \mbox{ (mod }2^{k}\mbox{)}$. \\ +8. Clamp excess digits of $c$. (\textit{mp\_clamp}) \\ +9. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_mod\_2d} +\end{figure} + +\textbf{Algorithm mp\_mod\_2d.} +This algorithm will quickly calculate the value of $a \mbox{ (mod }2^b\mbox{)}$. First if $b$ is less than or equal to zero the +result is set to zero. If $b$ is greater than the number of bits in $a$ then it simply copies $a$ to $c$ and returns. Otherwise, $a$ +is copied to $b$, leading digits are removed and the remaining leading digit is trimed to the exact bit count. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_mod\_2d.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* calc a value mod 2**b */ +018 int +019 mp_mod_2d (mp_int * a, int b, mp_int * c) +020 \{ +021 int x, res; +022 +023 /* if b is <= 0 then zero the int */ +024 if (b <= 0) \{ +025 mp_zero (c); +026 return MP_OKAY; +027 \} +028 +029 /* if the modulus is larger than the value than return */ +030 if (b >= (int) (a->used * DIGIT_BIT)) \{ +031 res = mp_copy (a, c); +032 return res; +033 \} +034 +035 /* copy */ +036 if ((res = mp_copy (a, c)) != MP_OKAY) \{ +037 return res; +038 \} +039 +040 /* zero digits above the last digit of the modulus */ +041 for (x = (b / DIGIT_BIT) + ((b % DIGIT_BIT) == 0 ? 0 : 1); x < c->used; x+ + +) \{ +042 c->dp[x] = 0; +043 \} +044 /* clear the digit that is not completely outside/inside the modulus */ +045 c->dp[b / DIGIT_BIT] &= +046 (mp_digit) ((((mp_digit) 1) << (((mp_digit) b) % DIGIT_BIT)) - ((mp_digi + t) 1)); +047 mp_clamp (c); +048 return MP_OKAY; +049 \} +050 #endif +\end{alltt} +\end{small} + +-- Add comments later, Tom. + +\section*{Exercises} +\begin{tabular}{cl} +$\left [ 3 \right ] $ & Devise an algorithm that performs $a \cdot 2^b$ for generic values of $b$ \\ + & in $O(n)$ time. \\ + &\\ +$\left [ 3 \right ] $ & Devise an efficient algorithm to multiply by small low hamming \\ + & weight values such as $3$, $5$ and $9$. Extend it to handle all values \\ + & upto $64$ with a hamming weight less than three. \\ + &\\ +$\left [ 2 \right ] $ & Modify the preceding algorithm to handle values of the form \\ + & $2^k - 1$ as well. \\ + &\\ +$\left [ 3 \right ] $ & Using only algorithms mp\_mul\_2, mp\_div\_2 and mp\_add create an \\ + & algorithm to multiply two integers in roughly $O(2n^2)$ time for \\ + & any $n$-bit input. Note that the time of addition is ignored in the \\ + & calculation. \\ + & \\ +$\left [ 5 \right ] $ & Improve the previous algorithm to have a working time of at most \\ + & $O \left (2^{(k-1)}n + \left ({2n^2 \over k} \right ) \right )$ for an appropriate choice of $k$. Again ignore \\ + & the cost of addition. \\ + & \\ +$\left [ 2 \right ] $ & Devise a chart to find optimal values of $k$ for the previous problem \\ + & for $n = 64 \ldots 1024$ in steps of $64$. \\ + & \\ +$\left [ 2 \right ] $ & Using only algorithms mp\_abs and mp\_sub devise another method for \\ + & calculating the result of a signed comparison. \\ + & +\end{tabular} + +\chapter{Multiplication and Squaring} +\section{The Multipliers} +For most number theoretic problems including certain public key cryptographic algorithms, the ``multipliers'' form the most important subset of +algorithms of any multiple precision integer package. The set of multiplier algorithms include integer multiplication, squaring and modular reduction +where in each of the algorithms single precision multiplication is the dominant operation performed. This chapter will discuss integer multiplication +and squaring, leaving modular reductions for the subsequent chapter. + +The importance of the multiplier algorithms is for the most part driven by the fact that certain popular public key algorithms are based on modular +exponentiation, that is computing $d \equiv a^b \mbox{ (mod }c\mbox{)}$ for some arbitrary choice of $a$, $b$, $c$ and $d$. During a modular +exponentiation the majority\footnote{Roughly speaking a modular exponentiation will spend about 40\% of the time performing modular reductions, +35\% of the time performing squaring and 25\% of the time performing multiplications.} of the processor time is spent performing single precision +multiplications. + +For centuries general purpose multiplication has required a lengthly $O(n^2)$ process, whereby each digit of one multiplicand has to be multiplied +against every digit of the other multiplicand. Traditional long-hand multiplication is based on this process; while the techniques can differ the +overall algorithm used is essentially the same. Only ``recently'' have faster algorithms been studied. First Karatsuba multiplication was discovered in +1962. This algorithm can multiply two numbers with considerably fewer single precision multiplications when compared to the long-hand approach. +This technique led to the discovery of polynomial basis algorithms (\textit{good reference?}) and subquently Fourier Transform based solutions. + +\section{Multiplication} +\subsection{The Baseline Multiplication} +\label{sec:basemult} +\index{baseline multiplication} +Computing the product of two integers in software can be achieved using a trivial adaptation of the standard $O(n^2)$ long-hand multiplication +algorithm that school children are taught. The algorithm is considered an $O(n^2)$ algorithm since for two $n$-digit inputs $n^2$ single precision +multiplications are required. More specifically for a $m$ and $n$ digit input $m \cdot n$ single precision multiplications are required. To +simplify most discussions, it will be assumed that the inputs have comparable number of digits. + +The ``baseline multiplication'' algorithm is designed to act as the ``catch-all'' algorithm, only to be used when the faster algorithms cannot be +used. This algorithm does not use any particularly interesting optimizations and should ideally be avoided if possible. One important +facet of this algorithm, is that it has been modified to only produce a certain amount of output digits as resolution. The importance of this +modification will become evident during the discussion of Barrett modular reduction. Recall that for a $n$ and $m$ digit input the product +will be at most $n + m$ digits. Therefore, this algorithm can be reduced to a full multiplier by having it produce $n + m$ digits of the product. + +Recall from sub-section 4.2.2 the definition of $\gamma$ as the number of bits in the type \textbf{mp\_digit}. We shall now extend the variable set to +include $\alpha$ which shall represent the number of bits in the type \textbf{mp\_word}. This implies that $2^{\alpha} > 2 \cdot \beta^2$. The +constant $\delta = 2^{\alpha - 2lg(\beta)}$ will represent the maximal weight of any column in a product (\textit{see sub-section 5.2.2 for more information}). + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{s\_mp\_mul\_digs}. \\ +\textbf{Input}. mp\_int $a$, mp\_int $b$ and an integer $digs$ \\ +\textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert \mbox{ (mod }\beta^{digs}\mbox{)}$. \\ +\hline \\ +1. If min$(a.used, b.used) < \delta$ then do \\ +\hspace{3mm}1.1 Calculate $c = \vert a \vert \cdot \vert b \vert$ by the Comba method (\textit{see algorithm~\ref{fig:COMBAMULT}}). \\ +\hspace{3mm}1.2 Return the result of step 1.1 \\ +\\ +Allocate and initialize a temporary mp\_int. \\ +2. Init $t$ to be of size $digs$ \\ +3. If step 2 failed return(\textit{MP\_MEM}). \\ +4. $t.used \leftarrow digs$ \\ +\\ +Compute the product. \\ +5. for $ix$ from $0$ to $a.used - 1$ do \\ +\hspace{3mm}5.1 $u \leftarrow 0$ \\ +\hspace{3mm}5.2 $pb \leftarrow \mbox{min}(b.used, digs - ix)$ \\ +\hspace{3mm}5.3 If $pb < 1$ then goto step 6. \\ +\hspace{3mm}5.4 for $iy$ from $0$ to $pb - 1$ do \\ +\hspace{6mm}5.4.1 $\hat r \leftarrow t_{iy + ix} + a_{ix} \cdot b_{iy} + u$ \\ +\hspace{6mm}5.4.2 $t_{iy + ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ +\hspace{6mm}5.4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ +\hspace{3mm}5.5 if $ix + pb < digs$ then do \\ +\hspace{6mm}5.5.1 $t_{ix + pb} \leftarrow u$ \\ +6. Clamp excess digits of $t$. \\ +7. Swap $c$ with $t$ \\ +8. Clear $t$ \\ +9. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm s\_mp\_mul\_digs} +\end{figure} + +\textbf{Algorithm s\_mp\_mul\_digs.} +This algorithm computes the unsigned product of two inputs $a$ and $b$, limited to an output precision of $digs$ digits. While it may seem +a bit awkward to modify the function from its simple $O(n^2)$ description, the usefulness of partial multipliers will arise in a subsequent +algorithm. The algorithm is loosely based on algorithm 14.12 from \cite[pp. 595]{HAC} and is similar to Algorithm M of Knuth \cite[pp. 268]{TAOCPV2}. +Algorithm s\_mp\_mul\_digs differs from these cited references since it can produce a variable output precision regardless of the precision of the +inputs. + +The first thing this algorithm checks for is whether a Comba multiplier can be used instead. If the minimum digit count of either +input is less than $\delta$, then the Comba method may be used instead. After the Comba method is ruled out, the baseline algorithm begins. A +temporary mp\_int variable $t$ is used to hold the intermediate result of the product. This allows the algorithm to be used to +compute products when either $a = c$ or $b = c$ without overwriting the inputs. + +All of step 5 is the infamous $O(n^2)$ multiplication loop slightly modified to only produce upto $digs$ digits of output. The $pb$ variable +is given the count of digits to read from $b$ inside the nested loop. If $pb \le 1$ then no more output digits can be produced and the algorithm +will exit the loop. The best way to think of the loops are as a series of $pb \times 1$ multiplications. That is, in each pass of the +innermost loop $a_{ix}$ is multiplied against $b$ and the result is added (\textit{with an appropriate shift}) to $t$. + +For example, consider multiplying $576$ by $241$. That is equivalent to computing $10^0(1)(576) + 10^1(4)(576) + 10^2(2)(576)$ which is best +visualized in the following table. + +\begin{figure}[here] +\begin{center} +\begin{tabular}{|c|c|c|c|c|c|l|} +\hline && & 5 & 7 & 6 & \\ +\hline $\times$&& & 2 & 4 & 1 & \\ +\hline &&&&&&\\ + && & 5 & 7 & 6 & $10^0(1)(576)$ \\ + &2 & 3 & 6 & 1 & 6 & $10^1(4)(576) + 10^0(1)(576)$ \\ + 1 & 3 & 8 & 8 & 1 & 6 & $10^2(2)(576) + 10^1(4)(576) + 10^0(1)(576)$ \\ +\hline +\end{tabular} +\end{center} +\caption{Long-Hand Multiplication Diagram} +\end{figure} + +Each row of the product is added to the result after being shifted to the left (\textit{multiplied by a power of the radix}) by the appropriate +count. That is in pass $ix$ of the inner loop the product is added starting at the $ix$'th digit of the reult. + +Step 5.4.1 introduces the hat symbol (\textit{e.g. $\hat r$}) which represents a double precision variable. The multiplication on that step +is assumed to be a double wide output single precision multiplication. That is, two single precision variables are multiplied to produce a +double precision result. The step is somewhat optimized from a long-hand multiplication algorithm because the carry from the addition in step +5.4.1 is propagated through the nested loop. If the carry was not propagated immediately it would overflow the single precision digit +$t_{ix+iy}$ and the result would be lost. + +At step 5.5 the nested loop is finished and any carry that was left over should be forwarded. The carry does not have to be added to the $ix+pb$'th +digit since that digit is assumed to be zero at this point. However, if $ix + pb \ge digs$ the carry is not set as it would make the result +exceed the precision requested. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_mul\_digs.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* multiplies |a| * |b| and only computes upto digs digits of result +018 * HAC pp. 595, Algorithm 14.12 Modified so you can control how +019 * many digits of output are created. +020 */ +021 int +022 s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs) +023 \{ +024 mp_int t; +025 int res, pa, pb, ix, iy; +026 mp_digit u; +027 mp_word r; +028 mp_digit tmpx, *tmpt, *tmpy; +029 +030 /* can we use the fast multiplier? */ +031 if (((digs) < MP_WARRAY) && +032 MIN (a->used, b->used) < +033 (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) \{ +034 return fast_s_mp_mul_digs (a, b, c, digs); +035 \} +036 +037 if ((res = mp_init_size (&t, digs)) != MP_OKAY) \{ +038 return res; +039 \} +040 t.used = digs; +041 +042 /* compute the digits of the product directly */ +043 pa = a->used; +044 for (ix = 0; ix < pa; ix++) \{ +045 /* set the carry to zero */ +046 u = 0; +047 +048 /* limit ourselves to making digs digits of output */ +049 pb = MIN (b->used, digs - ix); +050 +051 /* setup some aliases */ +052 /* copy of the digit from a used within the nested loop */ +053 tmpx = a->dp[ix]; +054 +055 /* an alias for the destination shifted ix places */ +056 tmpt = t.dp + ix; +057 +058 /* an alias for the digits of b */ +059 tmpy = b->dp; +060 +061 /* compute the columns of the output and propagate the carry */ +062 for (iy = 0; iy < pb; iy++) \{ +063 /* compute the column as a mp_word */ +064 r = ((mp_word)*tmpt) + +065 ((mp_word)tmpx) * ((mp_word)*tmpy++) + +066 ((mp_word) u); +067 +068 /* the new column is the lower part of the result */ +069 *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK)); +070 +071 /* get the carry word from the result */ +072 u = (mp_digit) (r >> ((mp_word) DIGIT_BIT)); +073 \} +074 /* set carry if it is placed below digs */ +075 if (ix + iy < digs) \{ +076 *tmpt = u; +077 \} +078 \} +079 +080 mp_clamp (&t); +081 mp_exch (&t, c); +082 +083 mp_clear (&t); +084 return MP_OKAY; +085 \} +086 #endif +\end{alltt} +\end{small} + +Lines 31 to 35 determine if the Comba method can be used first. The conditions for using the Comba routine are that min$(a.used, b.used) < \delta$ and +the number of digits of output is less than \textbf{MP\_WARRAY}. This new constant is used to control +the stack usage in the Comba routines. By default it is set to $\delta$ but can be reduced when memory is at a premium. + +Of particular importance is the calculation of the $ix+iy$'th column on lines 64, 65 and 66. Note how all of the +variables are cast to the type \textbf{mp\_word}, which is also the type of variable $\hat r$. That is to ensure that double precision operations +are used instead of single precision. The multiplication on line 65 makes use of a specific GCC optimizer behaviour. On the outset it looks like +the compiler will have to use a double precision multiplication to produce the result required. Such an operation would be horribly slow on most +processors and drag this to a crawl. However, GCC is smart enough to realize that double wide output single precision multipliers can be used. For +example, the instruction ``MUL'' on the x86 processor can multiply two 32-bit values and produce a 64-bit result. + +\subsection{Faster Multiplication by the ``Comba'' Method} + +One of the huge drawbacks of the ``baseline'' algorithms is that at the $O(n^2)$ level the carry must be computed and propagated upwards. This +makes the nested loop very sequential and hard to unroll and implement in parallel. The ``Comba'' \cite{COMBA} method is named after little known +(\textit{in cryptographic venues}) Paul G. Comba who described a method of implementing fast multipliers that do not require nested +carry fixup operations. As an interesting aside it seems that Paul Barrett describes a similar technique in +his 1986 paper \cite{BARRETT} written five years before. + +At the heart of the Comba technique is once again the long-hand algorithm. Except in this case a slight twist is placed on how +the columns of the result are produced. In the standard long-hand algorithm rows of products are produced then added together to form the +final result. In the baseline algorithm the columns are added together after each iteration to get the result instantaneously. + +In the Comba algorithm the columns of the result are produced entirely independently of each other. That is at the $O(n^2)$ level a +simple multiplication and addition step is performed. The carries of the columns are propagated after the nested loop to reduce the amount +of work requiored. Succintly the first step of the algorithm is to compute the product vector $\vec x$ as follows. + +\begin{equation} +\vec x_n = \sum_{i+j = n} a_ib_j, \forall n \in \lbrace 0, 1, 2, \ldots, i + j \rbrace +\end{equation} + +Where $\vec x_n$ is the $n'th$ column of the output vector. Consider the following example which computes the vector $\vec x$ for the multiplication +of $576$ and $241$. + +\newpage\begin{figure}[here] +\begin{small} +\begin{center} +\begin{tabular}{|c|c|c|c|c|c|} + \hline & & 5 & 7 & 6 & First Input\\ + \hline $\times$ & & 2 & 4 & 1 & Second Input\\ +\hline & & $1 \cdot 5 = 5$ & $1 \cdot 7 = 7$ & $1 \cdot 6 = 6$ & First pass \\ + & $4 \cdot 5 = 20$ & $4 \cdot 7+5=33$ & $4 \cdot 6+7=31$ & 6 & Second pass \\ + $2 \cdot 5 = 10$ & $2 \cdot 7 + 20 = 34$ & $2 \cdot 6+33=45$ & 31 & 6 & Third pass \\ +\hline 10 & 34 & 45 & 31 & 6 & Final Result \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Comba Multiplication Diagram} +\end{figure} + +At this point the vector $x = \left < 10, 34, 45, 31, 6 \right >$ is the result of the first step of the Comba multipler. +Now the columns must be fixed by propagating the carry upwards. The resultant vector will have one extra dimension over the input vector which is +congruent to adding a leading zero digit. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{Comba Fixup}. \\ +\textbf{Input}. Vector $\vec x$ of dimension $k$ \\ +\textbf{Output}. Vector $\vec x$ such that the carries have been propagated. \\ +\hline \\ +1. for $n$ from $0$ to $k - 1$ do \\ +\hspace{3mm}1.1 $\vec x_{n+1} \leftarrow \vec x_{n+1} + \lfloor \vec x_{n}/\beta \rfloor$ \\ +\hspace{3mm}1.2 $\vec x_{n} \leftarrow \vec x_{n} \mbox{ (mod }\beta\mbox{)}$ \\ +2. Return($\vec x$). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm Comba Fixup} +\end{figure} + +With that algorithm and $k = 5$ and $\beta = 10$ the following vector is produced $\vec x= \left < 1, 3, 8, 8, 1, 6 \right >$. In this case +$241 \cdot 576$ is in fact $138816$ and the procedure succeeded. If the algorithm is correct and as will be demonstrated shortly more +efficient than the baseline algorithm why not simply always use this algorithm? + +\subsubsection{Column Weight.} +At the nested $O(n^2)$ level the Comba method adds the product of two single precision variables to each column of the output +independently. A serious obstacle is if the carry is lost, due to lack of precision before the algorithm has a chance to fix +the carries. For example, in the multiplication of two three-digit numbers the third column of output will be the sum of +three single precision multiplications. If the precision of the accumulator for the output digits is less then $3 \cdot (\beta - 1)^2$ then +an overflow can occur and the carry information will be lost. For any $m$ and $n$ digit inputs the maximum weight of any column is +min$(m, n)$ which is fairly obvious. + +The maximum number of terms in any column of a product is known as the ``column weight'' and strictly governs when the algorithm can be used. Recall +from earlier that a double precision type has $\alpha$ bits of resolution and a single precision digit has $lg(\beta)$ bits of precision. Given these +two quantities we must not violate the following + +\begin{equation} +k \cdot \left (\beta - 1 \right )^2 < 2^{\alpha} +\end{equation} + +Which reduces to + +\begin{equation} +k \cdot \left ( \beta^2 - 2\beta + 1 \right ) < 2^{\alpha} +\end{equation} + +Let $\rho = lg(\beta)$ represent the number of bits in a single precision digit. By further re-arrangement of the equation the final solution is +found. + +\begin{equation} +k < {{2^{\alpha}} \over {\left (2^{2\rho} - 2^{\rho + 1} + 1 \right )}} +\end{equation} + +The defaults for LibTomMath are $\beta = 2^{28}$ and $\alpha = 2^{64}$ which means that $k$ is bounded by $k < 257$. In this configuration +the smaller input may not have more than $256$ digits if the Comba method is to be used. This is quite satisfactory for most applications since +$256$ digits would allow for numbers in the range of $0 \le x < 2^{7168}$ which, is much larger than most public key cryptographic algorithms require. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{fast\_s\_mp\_mul\_digs}. \\ +\textbf{Input}. mp\_int $a$, mp\_int $b$ and an integer $digs$ \\ +\textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert \mbox{ (mod }\beta^{digs}\mbox{)}$. \\ +\hline \\ +Place an array of \textbf{MP\_WARRAY} double precision digits named $\hat W$ on the stack. \\ +1. If $c.alloc < digs$ then grow $c$ to $digs$ digits. (\textit{mp\_grow}) \\ +2. If step 1 failed return(\textit{MP\_MEM}).\\ +\\ +Zero the temporary array $\hat W$. \\ +3. for $n$ from $0$ to $digs - 1$ do \\ +\hspace{3mm}3.1 $\hat W_n \leftarrow 0$ \\ +\\ +Compute the columns. \\ +4. for $ix$ from $0$ to $a.used - 1$ do \\ +\hspace{3mm}4.1 $pb \leftarrow \mbox{min}(b.used, digs - ix)$ \\ +\hspace{3mm}4.2 If $pb < 1$ then goto step 5. \\ +\hspace{3mm}4.3 for $iy$ from $0$ to $pb - 1$ do \\ +\hspace{6mm}4.3.1 $\hat W_{ix+iy} \leftarrow \hat W_{ix+iy} + a_{ix}b_{iy}$ \\ +\\ +Propagate the carries upwards. \\ +5. $oldused \leftarrow c.used$ \\ +6. $c.used \leftarrow digs$ \\ +7. If $digs > 1$ then do \\ +\hspace{3mm}7.1. for $ix$ from $1$ to $digs - 1$ do \\ +\hspace{6mm}7.1.1 $\hat W_{ix} \leftarrow \hat W_{ix} + \lfloor \hat W_{ix-1} / \beta \rfloor$ \\ +\hspace{6mm}7.1.2 $c_{ix - 1} \leftarrow \hat W_{ix - 1} \mbox{ (mod }\beta\mbox{)}$ \\ +8. else do \\ +\hspace{3mm}8.1 $ix \leftarrow 0$ \\ +9. $c_{ix} \leftarrow \hat W_{ix} \mbox{ (mod }\beta\mbox{)}$ \\ +\\ +Zero excess digits. \\ +10. If $digs < oldused$ then do \\ +\hspace{3mm}10.1 for $n$ from $digs$ to $oldused - 1$ do \\ +\hspace{6mm}10.1.1 $c_n \leftarrow 0$ \\ +11. Clamp excessive digits of $c$. (\textit{mp\_clamp}) \\ +12. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm fast\_s\_mp\_mul\_digs} +\label{fig:COMBAMULT} +\end{figure} + +\textbf{Algorithm fast\_s\_mp\_mul\_digs.} +This algorithm performs the unsigned multiplication of $a$ and $b$ using the Comba method limited to $digs$ digits of precision. The algorithm +essentially peforms the same calculation as algorithm s\_mp\_mul\_digs, just much faster. + +The array $\hat W$ is meant to be on the stack when the algorithm is used. The size of the array does not change which is ideal. Note also that +unlike algorithm s\_mp\_mul\_digs no temporary mp\_int is required since the result is calculated directly in $\hat W$. + +The $O(n^2)$ loop on step four is where the Comba method's advantages begin to show through in comparison to the baseline algorithm. The lack of +a carry variable or propagation in this loop allows the loop to be performed with only single precision multiplication and additions. Now that each +iteration of the inner loop can be performed independent of the others the inner loop can be performed with a high level of parallelism. + +To measure the benefits of the Comba method over the baseline method consider the number of operations that are required. If the +cost in terms of time of a multiply and addition is $p$ and the cost of a carry propagation is $q$ then a baseline multiplication would require +$O \left ((p + q)n^2 \right )$ time to multiply two $n$-digit numbers. The Comba method requires only $O(pn^2 + qn)$ time, however in practice, +the speed increase is actually much more. With $O(n)$ space the algorithm can be reduced to $O(pn + qn)$ time by implementing the $n$ multiply +and addition operations in the nested loop in parallel. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_fast\_s\_mp\_mul\_digs.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* Fast (comba) multiplier +018 * +019 * This is the fast column-array [comba] multiplier. It is +020 * designed to compute the columns of the product first +021 * then handle the carries afterwards. This has the effect +022 * of making the nested loops that compute the columns very +023 * simple and schedulable on super-scalar processors. +024 * +025 * This has been modified to produce a variable number of +026 * digits of output so if say only a half-product is required +027 * you don't have to compute the upper half (a feature +028 * required for fast Barrett reduction). +029 * +030 * Based on Algorithm 14.12 on pp.595 of HAC. +031 * +032 */ +033 int +034 fast_s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs) +035 \{ +036 int olduse, res, pa, ix, iz; +037 mp_digit W[MP_WARRAY]; +038 register mp_word _W; +039 +040 /* grow the destination as required */ +041 if (c->alloc < digs) \{ +042 if ((res = mp_grow (c, digs)) != MP_OKAY) \{ +043 return res; +044 \} +045 \} +046 +047 /* number of output digits to produce */ +048 pa = MIN(digs, a->used + b->used); +049 +050 /* clear the carry */ +051 _W = 0; +052 for (ix = 0; ix < pa; ix++) \{ +053 int tx, ty; +054 int iy; +055 mp_digit *tmpx, *tmpy; +056 +057 /* get offsets into the two bignums */ +058 ty = MIN(b->used-1, ix); +059 tx = ix - ty; +060 +061 /* setup temp aliases */ +062 tmpx = a->dp + tx; +063 tmpy = b->dp + ty; +064 +065 /* this is the number of times the loop will iterrate, essentially its + +066 while (tx++ < a->used && ty-- >= 0) \{ ... \} +067 */ +068 iy = MIN(a->used-tx, ty+1); +069 +070 /* execute loop */ +071 for (iz = 0; iz < iy; ++iz) \{ +072 _W += ((mp_word)*tmpx++)*((mp_word)*tmpy--); +073 \} +074 +075 /* store term */ +076 W[ix] = ((mp_digit)_W) & MP_MASK; +077 +078 /* make next carry */ +079 _W = _W >> ((mp_word)DIGIT_BIT); +080 \} +081 +082 /* store final carry */ +083 W[ix] = _W; +084 +085 /* setup dest */ +086 olduse = c->used; +087 c->used = digs; +088 +089 \{ +090 register mp_digit *tmpc; +091 tmpc = c->dp; +092 for (ix = 0; ix < digs; ix++) \{ +093 /* now extract the previous digit [below the carry] */ +094 *tmpc++ = W[ix]; +095 \} +096 +097 /* clear unused digits [that existed in the old copy of c] */ +098 for (; ix < olduse; ix++) \{ +099 *tmpc++ = 0; +100 \} +101 \} +102 mp_clamp (c); +103 return MP_OKAY; +104 \} +105 #endif +\end{alltt} +\end{small} + +The memset on line @47,memset@ clears the initial $\hat W$ array to zero in a single step. Like the slower baseline multiplication +implementation a series of aliases (\textit{lines 62, 63 and 76}) are used to simplify the inner $O(n^2)$ loop. +In this case a new alias $\_\hat W$ has been added which refers to the double precision columns offset by $ix$ in each pass. + +The inner loop on lines 92, 79 and 80 is where the algorithm will spend the majority of the time, which is why it has been +stripped to the bones of any extra baggage\footnote{Hence the pointer aliases.}. On x86 processors the multiplication and additions amount to at the +very least five instructions (\textit{two loads, two additions, one multiply}) while on the ARMv4 processors they amount to only three +(\textit{one load, one store, one multiply-add}). For both of the x86 and ARMv4 processors the GCC compiler performs a good job at unrolling the loop +and scheduling the instructions so there are very few dependency stalls. + +In theory the difference between the baseline and comba algorithms is a mere $O(qn)$ time difference. However, in the $O(n^2)$ nested loop of the +baseline method there are dependency stalls as the algorithm must wait for the multiplier to finish before propagating the carry to the next +digit. As a result fewer of the often multiple execution units\footnote{The AMD Athlon has three execution units and the Intel P4 has four.} can +be simultaneously used. + +\subsection{Polynomial Basis Multiplication} +To break the $O(n^2)$ barrier in multiplication requires a completely different look at integer multiplication. In the following algorithms +the use of polynomial basis representation for two integers $a$ and $b$ as $f(x) = \sum_{i=0}^{n} a_i x^i$ and +$g(x) = \sum_{i=0}^{n} b_i x^i$ respectively, is required. In this system both $f(x)$ and $g(x)$ have $n + 1$ terms and are of the $n$'th degree. + +The product $a \cdot b \equiv f(x)g(x)$ is the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$. The coefficients $w_i$ will +directly yield the desired product when $\beta$ is substituted for $x$. The direct solution to solve for the $2n + 1$ coefficients +requires $O(n^2)$ time and would in practice be slower than the Comba technique. + +However, numerical analysis theory indicates that only $2n + 1$ distinct points in $W(x)$ are required to determine the values of the $2n + 1$ unknown +coefficients. This means by finding $\zeta_y = W(y)$ for $2n + 1$ small values of $y$ the coefficients of $W(x)$ can be found with +Gaussian elimination. This technique is also occasionally refered to as the \textit{interpolation technique} (\textit{references please...}) since in +effect an interpolation based on $2n + 1$ points will yield a polynomial equivalent to $W(x)$. + +The coefficients of the polynomial $W(x)$ are unknown which makes finding $W(y)$ for any value of $y$ impossible. However, since +$W(x) = f(x)g(x)$ the equivalent $\zeta_y = f(y) g(y)$ can be used in its place. The benefit of this technique stems from the +fact that $f(y)$ and $g(y)$ are much smaller than either $a$ or $b$ respectively. As a result finding the $2n + 1$ relations required +by multiplying $f(y)g(y)$ involves multiplying integers that are much smaller than either of the inputs. + +When picking points to gather relations there are always three obvious points to choose, $y = 0, 1$ and $ \infty$. The $\zeta_0$ term +is simply the product $W(0) = w_0 = a_0 \cdot b_0$. The $\zeta_1$ term is the product +$W(1) = \left (\sum_{i = 0}^{n} a_i \right ) \left (\sum_{i = 0}^{n} b_i \right )$. The third point $\zeta_{\infty}$ is less obvious but rather +simple to explain. The $2n + 1$'th coefficient of $W(x)$ is numerically equivalent to the most significant column in an integer multiplication. +The point at $\infty$ is used symbolically to represent the most significant column, that is $W(\infty) = w_{2n} = a_nb_n$. Note that the +points at $y = 0$ and $\infty$ yield the coefficients $w_0$ and $w_{2n}$ directly. + +If more points are required they should be of small values and powers of two such as $2^q$ and the related \textit{mirror points} +$\left (2^q \right )^{2n} \cdot \zeta_{2^{-q}}$ for small values of $q$. The term ``mirror point'' stems from the fact that +$\left (2^q \right )^{2n} \cdot \zeta_{2^{-q}}$ can be calculated in the exact opposite fashion as $\zeta_{2^q}$. For +example, when $n = 2$ and $q = 1$ then following two equations are equivalent to the point $\zeta_{2}$ and its mirror. + +\begin{eqnarray} +\zeta_{2} = f(2)g(2) = (4a_2 + 2a_1 + a_0)(4b_2 + 2b_1 + b_0) \nonumber \\ +16 \cdot \zeta_{1 \over 2} = 4f({1\over 2}) \cdot 4g({1 \over 2}) = (a_2 + 2a_1 + 4a_0)(b_2 + 2b_1 + 4b_0) +\end{eqnarray} + +Using such points will allow the values of $f(y)$ and $g(y)$ to be independently calculated using only left shifts. For example, when $n = 2$ the +polynomial $f(2^q)$ is equal to $2^q((2^qa_2) + a_1) + a_0$. This technique of polynomial representation is known as Horner's method. + +As a general rule of the algorithm when the inputs are split into $n$ parts each there are $2n - 1$ multiplications. Each multiplication is of +multiplicands that have $n$ times fewer digits than the inputs. The asymptotic running time of this algorithm is +$O \left ( k^{lg_n(2n - 1)} \right )$ for $k$ digit inputs (\textit{assuming they have the same number of digits}). Figure~\ref{fig:exponent} +summarizes the exponents for various values of $n$. + +\begin{figure} +\begin{center} +\begin{tabular}{|c|c|c|} +\hline \textbf{Split into $n$ Parts} & \textbf{Exponent} & \textbf{Notes}\\ +\hline $2$ & $1.584962501$ & This is Karatsuba Multiplication. \\ +\hline $3$ & $1.464973520$ & This is Toom-Cook Multiplication. \\ +\hline $4$ & $1.403677461$ &\\ +\hline $5$ & $1.365212389$ &\\ +\hline $10$ & $1.278753601$ &\\ +\hline $100$ & $1.149426538$ &\\ +\hline $1000$ & $1.100270931$ &\\ +\hline $10000$ & $1.075252070$ &\\ +\hline +\end{tabular} +\end{center} +\caption{Asymptotic Running Time of Polynomial Basis Multiplication} +\label{fig:exponent} +\end{figure} + +At first it may seem like a good idea to choose $n = 1000$ since the exponent is approximately $1.1$. However, the overhead +of solving for the 2001 terms of $W(x)$ will certainly consume any savings the algorithm could offer for all but exceedingly large +numbers. + +\subsubsection{Cutoff Point} +The polynomial basis multiplication algorithms all require fewer single precision multiplications than a straight Comba approach. However, +the algorithms incur an overhead (\textit{at the $O(n)$ work level}) since they require a system of equations to be solved. This makes the +polynomial basis approach more costly to use with small inputs. + +Let $m$ represent the number of digits in the multiplicands (\textit{assume both multiplicands have the same number of digits}). There exists a +point $y$ such that when $m < y$ the polynomial basis algorithms are more costly than Comba, when $m = y$ they are roughly the same cost and +when $m > y$ the Comba methods are slower than the polynomial basis algorithms. + +The exact location of $y$ depends on several key architectural elements of the computer platform in question. + +\begin{enumerate} +\item The ratio of clock cycles for single precision multiplication versus other simpler operations such as addition, shifting, etc. For example +on the AMD Athlon the ratio is roughly $17 : 1$ while on the Intel P4 it is $29 : 1$. The higher the ratio in favour of multiplication the lower +the cutoff point $y$ will be. + +\item The complexity of the linear system of equations (\textit{for the coefficients of $W(x)$}) is. Generally speaking as the number of splits +grows the complexity grows substantially. Ideally solving the system will only involve addition, subtraction and shifting of integers. This +directly reflects on the ratio previous mentioned. + +\item To a lesser extent memory bandwidth and function call overheads. Provided the values are in the processor cache this is less of an +influence over the cutoff point. + +\end{enumerate} + +A clean cutoff point separation occurs when a point $y$ is found such that all of the cutoff point conditions are met. For example, if the point +is too low then there will be values of $m$ such that $m > y$ and the Comba method is still faster. Finding the cutoff points is fairly simple when +a high resolution timer is available. + +\subsection{Karatsuba Multiplication} +Karatsuba \cite{KARA} multiplication when originally proposed in 1962 was among the first set of algorithms to break the $O(n^2)$ barrier for +general purpose multiplication. Given two polynomial basis representations $f(x) = ax + b$ and $g(x) = cx + d$, Karatsuba proved with +light algebra \cite{KARAP} that the following polynomial is equivalent to multiplication of the two integers the polynomials represent. + +\begin{equation} +f(x) \cdot g(x) = acx^2 + ((a - b)(c - d) - (ac + bd))x + bd +\end{equation} + +Using the observation that $ac$ and $bd$ could be re-used only three half sized multiplications would be required to produce the product. Applying +this algorithm recursively, the work factor becomes $O(n^{lg(3)})$ which is substantially better than the work factor $O(n^2)$ of the Comba technique. It turns +out what Karatsuba did not know or at least did not publish was that this is simply polynomial basis multiplication with the points +$\zeta_0$, $\zeta_{\infty}$ and $-\zeta_{-1}$. Consider the resultant system of equations. + +\begin{center} +\begin{tabular}{rcrcrcrc} +$\zeta_{0}$ & $=$ & & & & & $w_0$ \\ +$-\zeta_{-1}$ & $=$ & $-w_2$ & $+$ & $w_1$ & $-$ & $w_0$ \\ +$\zeta_{\infty}$ & $=$ & $w_2$ & & & & \\ +\end{tabular} +\end{center} + +By adding the first and last equation to the equation in the middle the term $w_1$ can be isolated and all three coefficients solved for. The simplicity +of this system of equations has made Karatsuba fairly popular. In fact the cutoff point is often fairly low\footnote{With LibTomMath 0.18 it is 70 and 109 digits for the Intel P4 and AMD Athlon respectively.} +making it an ideal algorithm to speed up certain public key cryptosystems such as RSA and Diffie-Hellman. It is worth noting that the point +$\zeta_1$ could be substituted for $-\zeta_{-1}$. In this case the first and third row are subtracted instead of added to the second row. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_karatsuba\_mul}. \\ +\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ +\textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert$ \\ +\hline \\ +1. Init the following mp\_int variables: $x0$, $x1$, $y0$, $y1$, $t1$, $x0y0$, $x1y1$.\\ +2. If step 2 failed then return(\textit{MP\_MEM}). \\ +\\ +Split the input. e.g. $a = x1 \cdot \beta^B + x0$ \\ +3. $B \leftarrow \mbox{min}(a.used, b.used)/2$ \\ +4. $x0 \leftarrow a \mbox{ (mod }\beta^B\mbox{)}$ (\textit{mp\_mod\_2d}) \\ +5. $y0 \leftarrow b \mbox{ (mod }\beta^B\mbox{)}$ \\ +6. $x1 \leftarrow \lfloor a / \beta^B \rfloor$ (\textit{mp\_rshd}) \\ +7. $y1 \leftarrow \lfloor b / \beta^B \rfloor$ \\ +\\ +Calculate the three products. \\ +8. $x0y0 \leftarrow x0 \cdot y0$ (\textit{mp\_mul}) \\ +9. $x1y1 \leftarrow x1 \cdot y1$ \\ +10. $t1 \leftarrow x1 - x0$ (\textit{mp\_sub}) \\ +11. $x0 \leftarrow y1 - y0$ \\ +12. $t1 \leftarrow t1 \cdot x0$ \\ +\\ +Calculate the middle term. \\ +13. $x0 \leftarrow x0y0 + x1y1$ \\ +14. $t1 \leftarrow x0 - t1$ \\ +\\ +Calculate the final product. \\ +15. $t1 \leftarrow t1 \cdot \beta^B$ (\textit{mp\_lshd}) \\ +16. $x1y1 \leftarrow x1y1 \cdot \beta^{2B}$ \\ +17. $t1 \leftarrow x0y0 + t1$ \\ +18. $c \leftarrow t1 + x1y1$ \\ +19. Clear all of the temporary variables. \\ +20. Return(\textit{MP\_OKAY}).\\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_karatsuba\_mul} +\end{figure} + +\textbf{Algorithm mp\_karatsuba\_mul.} +This algorithm computes the unsigned product of two inputs using the Karatsuba multiplication algorithm. It is loosely based on the description +from Knuth \cite[pp. 294-295]{TAOCPV2}. + +\index{radix point} +In order to split the two inputs into their respective halves, a suitable \textit{radix point} must be chosen. The radix point chosen must +be used for both of the inputs meaning that it must be smaller than the smallest input. Step 3 chooses the radix point $B$ as half of the +smallest input \textbf{used} count. After the radix point is chosen the inputs are split into lower and upper halves. Step 4 and 5 +compute the lower halves. Step 6 and 7 computer the upper halves. + +After the halves have been computed the three intermediate half-size products must be computed. Step 8 and 9 compute the trivial products +$x0 \cdot y0$ and $x1 \cdot y1$. The mp\_int $x0$ is used as a temporary variable after $x1 - x0$ has been computed. By using $x0$ instead +of an additional temporary variable, the algorithm can avoid an addition memory allocation operation. + +The remaining steps 13 through 18 compute the Karatsuba polynomial through a variety of digit shifting and addition operations. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_karatsuba\_mul.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* c = |a| * |b| using Karatsuba Multiplication using +018 * three half size multiplications +019 * +020 * Let B represent the radix [e.g. 2**DIGIT_BIT] and +021 * let n represent half of the number of digits in +022 * the min(a,b) +023 * +024 * a = a1 * B**n + a0 +025 * b = b1 * B**n + b0 +026 * +027 * Then, a * b => +028 a1b1 * B**2n + ((a1 - a0)(b1 - b0) + a0b0 + a1b1) * B + a0b0 +029 * +030 * Note that a1b1 and a0b0 are used twice and only need to be +031 * computed once. So in total three half size (half # of +032 * digit) multiplications are performed, a0b0, a1b1 and +033 * (a1-b1)(a0-b0) +034 * +035 * Note that a multiplication of half the digits requires +036 * 1/4th the number of single precision multiplications so in +037 * total after one call 25% of the single precision multiplications +038 * are saved. Note also that the call to mp_mul can end up back +039 * in this function if the a0, a1, b0, or b1 are above the threshold. +040 * This is known as divide-and-conquer and leads to the famous +041 * O(N**lg(3)) or O(N**1.584) work which is asymptopically lower than +042 * the standard O(N**2) that the baseline/comba methods use. +043 * Generally though the overhead of this method doesn't pay off +044 * until a certain size (N ~ 80) is reached. +045 */ +046 int mp_karatsuba_mul (mp_int * a, mp_int * b, mp_int * c) +047 \{ +048 mp_int x0, x1, y0, y1, t1, x0y0, x1y1; +049 int B, err; +050 +051 /* default the return code to an error */ +052 err = MP_MEM; +053 +054 /* min # of digits */ +055 B = MIN (a->used, b->used); +056 +057 /* now divide in two */ +058 B = B >> 1; +059 +060 /* init copy all the temps */ +061 if (mp_init_size (&x0, B) != MP_OKAY) +062 goto ERR; +063 if (mp_init_size (&x1, a->used - B) != MP_OKAY) +064 goto X0; +065 if (mp_init_size (&y0, B) != MP_OKAY) +066 goto X1; +067 if (mp_init_size (&y1, b->used - B) != MP_OKAY) +068 goto Y0; +069 +070 /* init temps */ +071 if (mp_init_size (&t1, B * 2) != MP_OKAY) +072 goto Y1; +073 if (mp_init_size (&x0y0, B * 2) != MP_OKAY) +074 goto T1; +075 if (mp_init_size (&x1y1, B * 2) != MP_OKAY) +076 goto X0Y0; +077 +078 /* now shift the digits */ +079 x0.used = y0.used = B; +080 x1.used = a->used - B; +081 y1.used = b->used - B; +082 +083 \{ +084 register int x; +085 register mp_digit *tmpa, *tmpb, *tmpx, *tmpy; +086 +087 /* we copy the digits directly instead of using higher level functions +088 * since we also need to shift the digits +089 */ +090 tmpa = a->dp; +091 tmpb = b->dp; +092 +093 tmpx = x0.dp; +094 tmpy = y0.dp; +095 for (x = 0; x < B; x++) \{ +096 *tmpx++ = *tmpa++; +097 *tmpy++ = *tmpb++; +098 \} +099 +100 tmpx = x1.dp; +101 for (x = B; x < a->used; x++) \{ +102 *tmpx++ = *tmpa++; +103 \} +104 +105 tmpy = y1.dp; +106 for (x = B; x < b->used; x++) \{ +107 *tmpy++ = *tmpb++; +108 \} +109 \} +110 +111 /* only need to clamp the lower words since by definition the +112 * upper words x1/y1 must have a known number of digits +113 */ +114 mp_clamp (&x0); +115 mp_clamp (&y0); +116 +117 /* now calc the products x0y0 and x1y1 */ +118 /* after this x0 is no longer required, free temp [x0==t2]! */ +119 if (mp_mul (&x0, &y0, &x0y0) != MP_OKAY) +120 goto X1Y1; /* x0y0 = x0*y0 */ +121 if (mp_mul (&x1, &y1, &x1y1) != MP_OKAY) +122 goto X1Y1; /* x1y1 = x1*y1 */ +123 +124 /* now calc x1-x0 and y1-y0 */ +125 if (mp_sub (&x1, &x0, &t1) != MP_OKAY) +126 goto X1Y1; /* t1 = x1 - x0 */ +127 if (mp_sub (&y1, &y0, &x0) != MP_OKAY) +128 goto X1Y1; /* t2 = y1 - y0 */ +129 if (mp_mul (&t1, &x0, &t1) != MP_OKAY) +130 goto X1Y1; /* t1 = (x1 - x0) * (y1 - y0) */ +131 +132 /* add x0y0 */ +133 if (mp_add (&x0y0, &x1y1, &x0) != MP_OKAY) +134 goto X1Y1; /* t2 = x0y0 + x1y1 */ +135 if (mp_sub (&x0, &t1, &t1) != MP_OKAY) +136 goto X1Y1; /* t1 = x0y0 + x1y1 - (x1-x0)*(y1-y0) */ +137 +138 /* shift by B */ +139 if (mp_lshd (&t1, B) != MP_OKAY) +140 goto X1Y1; /* t1 = (x0y0 + x1y1 - (x1-x0)*(y1-y0))<<B */ +141 if (mp_lshd (&x1y1, B * 2) != MP_OKAY) +142 goto X1Y1; /* x1y1 = x1y1 << 2*B */ +143 +144 if (mp_add (&x0y0, &t1, &t1) != MP_OKAY) +145 goto X1Y1; /* t1 = x0y0 + t1 */ +146 if (mp_add (&t1, &x1y1, c) != MP_OKAY) +147 goto X1Y1; /* t1 = x0y0 + t1 + x1y1 */ +148 +149 /* Algorithm succeeded set the return code to MP_OKAY */ +150 err = MP_OKAY; +151 +152 X1Y1:mp_clear (&x1y1); +153 X0Y0:mp_clear (&x0y0); +154 T1:mp_clear (&t1); +155 Y1:mp_clear (&y1); +156 Y0:mp_clear (&y0); +157 X1:mp_clear (&x1); +158 X0:mp_clear (&x0); +159 ERR: +160 return err; +161 \} +162 #endif +\end{alltt} +\end{small} + +The new coding element in this routine, not seen in previous routines, is the usage of goto statements. The conventional +wisdom is that goto statements should be avoided. This is generally true, however when every single function call can fail, it makes sense +to handle error recovery with a single piece of code. Lines 61 to 75 handle initializing all of the temporary variables +required. Note how each of the if statements goes to a different label in case of failure. This allows the routine to correctly free only +the temporaries that have been successfully allocated so far. + +The temporary variables are all initialized using the mp\_init\_size routine since they are expected to be large. This saves the +additional reallocation that would have been necessary. Also $x0$, $x1$, $y0$ and $y1$ have to be able to hold at least their respective +number of digits for the next section of code. + +The first algebraic portion of the algorithm is to split the two inputs into their halves. However, instead of using mp\_mod\_2d and mp\_rshd +to extract the halves, the respective code has been placed inline within the body of the function. To initialize the halves, the \textbf{used} and +\textbf{sign} members are copied first. The first for loop on line 101 copies the lower halves. Since they are both the same magnitude it +is simpler to calculate both lower halves in a single loop. The for loop on lines 106 and 106 calculate the upper halves $x1$ and +$y1$ respectively. + +By inlining the calculation of the halves, the Karatsuba multiplier has a slightly lower overhead and can be used for smaller magnitude inputs. + +When line 150 is reached, the algorithm has completed succesfully. The ``error status'' variable $err$ is set to \textbf{MP\_OKAY} so that +the same code that handles errors can be used to clear the temporary variables and return. + +\subsection{Toom-Cook $3$-Way Multiplication} +Toom-Cook $3$-Way \cite{TOOM} multiplication is essentially the polynomial basis algorithm for $n = 2$ except that the points are +chosen such that $\zeta$ is easy to compute and the resulting system of equations easy to reduce. Here, the points $\zeta_{0}$, +$16 \cdot \zeta_{1 \over 2}$, $\zeta_1$, $\zeta_2$ and $\zeta_{\infty}$ make up the five required points to solve for the coefficients +of the $W(x)$. + +With the five relations that Toom-Cook specifies, the following system of equations is formed. + +\begin{center} +\begin{tabular}{rcrcrcrcrcr} +$\zeta_0$ & $=$ & $0w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $1w_0$ \\ +$16 \cdot \zeta_{1 \over 2}$ & $=$ & $1w_4$ & $+$ & $2w_3$ & $+$ & $4w_2$ & $+$ & $8w_1$ & $+$ & $16w_0$ \\ +$\zeta_1$ & $=$ & $1w_4$ & $+$ & $1w_3$ & $+$ & $1w_2$ & $+$ & $1w_1$ & $+$ & $1w_0$ \\ +$\zeta_2$ & $=$ & $16w_4$ & $+$ & $8w_3$ & $+$ & $4w_2$ & $+$ & $2w_1$ & $+$ & $1w_0$ \\ +$\zeta_{\infty}$ & $=$ & $1w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $0w_0$ \\ +\end{tabular} +\end{center} + +A trivial solution to this matrix requires $12$ subtractions, two multiplications by a small power of two, two divisions by a small power +of two, two divisions by three and one multiplication by three. All of these $19$ sub-operations require less than quadratic time, meaning that +the algorithm can be faster than a baseline multiplication. However, the greater complexity of this algorithm places the cutoff point +(\textbf{TOOM\_MUL\_CUTOFF}) where Toom-Cook becomes more efficient much higher than the Karatsuba cutoff point. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_toom\_mul}. \\ +\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ +\textbf{Output}. $c \leftarrow a \cdot b $ \\ +\hline \\ +Split $a$ and $b$ into three pieces. E.g. $a = a_2 \beta^{2k} + a_1 \beta^{k} + a_0$ \\ +1. $k \leftarrow \lfloor \mbox{min}(a.used, b.used) / 3 \rfloor$ \\ +2. $a_0 \leftarrow a \mbox{ (mod }\beta^{k}\mbox{)}$ \\ +3. $a_1 \leftarrow \lfloor a / \beta^k \rfloor$, $a_1 \leftarrow a_1 \mbox{ (mod }\beta^{k}\mbox{)}$ \\ +4. $a_2 \leftarrow \lfloor a / \beta^{2k} \rfloor$, $a_2 \leftarrow a_2 \mbox{ (mod }\beta^{k}\mbox{)}$ \\ +5. $b_0 \leftarrow a \mbox{ (mod }\beta^{k}\mbox{)}$ \\ +6. $b_1 \leftarrow \lfloor a / \beta^k \rfloor$, $b_1 \leftarrow b_1 \mbox{ (mod }\beta^{k}\mbox{)}$ \\ +7. $b_2 \leftarrow \lfloor a / \beta^{2k} \rfloor$, $b_2 \leftarrow b_2 \mbox{ (mod }\beta^{k}\mbox{)}$ \\ +\\ +Find the five equations for $w_0, w_1, ..., w_4$. \\ +8. $w_0 \leftarrow a_0 \cdot b_0$ \\ +9. $w_4 \leftarrow a_2 \cdot b_2$ \\ +10. $tmp_1 \leftarrow 2 \cdot a_0$, $tmp_1 \leftarrow a_1 + tmp_1$, $tmp_1 \leftarrow 2 \cdot tmp_1$, $tmp_1 \leftarrow tmp_1 + a_2$ \\ +11. $tmp_2 \leftarrow 2 \cdot b_0$, $tmp_2 \leftarrow b_1 + tmp_2$, $tmp_2 \leftarrow 2 \cdot tmp_2$, $tmp_2 \leftarrow tmp_2 + b_2$ \\ +12. $w_1 \leftarrow tmp_1 \cdot tmp_2$ \\ +13. $tmp_1 \leftarrow 2 \cdot a_2$, $tmp_1 \leftarrow a_1 + tmp_1$, $tmp_1 \leftarrow 2 \cdot tmp_1$, $tmp_1 \leftarrow tmp_1 + a_0$ \\ +14. $tmp_2 \leftarrow 2 \cdot b_2$, $tmp_2 \leftarrow b_1 + tmp_2$, $tmp_2 \leftarrow 2 \cdot tmp_2$, $tmp_2 \leftarrow tmp_2 + b_0$ \\ +15. $w_3 \leftarrow tmp_1 \cdot tmp_2$ \\ +16. $tmp_1 \leftarrow a_0 + a_1$, $tmp_1 \leftarrow tmp_1 + a_2$, $tmp_2 \leftarrow b_0 + b_1$, $tmp_2 \leftarrow tmp_2 + b_2$ \\ +17. $w_2 \leftarrow tmp_1 \cdot tmp_2$ \\ +\\ +Continued on the next page.\\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_toom\_mul} +\end{figure} + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_toom\_mul} (continued). \\ +\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ +\textbf{Output}. $c \leftarrow a \cdot b $ \\ +\hline \\ +Now solve the system of equations. \\ +18. $w_1 \leftarrow w_4 - w_1$, $w_3 \leftarrow w_3 - w_0$ \\ +19. $w_1 \leftarrow \lfloor w_1 / 2 \rfloor$, $w_3 \leftarrow \lfloor w_3 / 2 \rfloor$ \\ +20. $w_2 \leftarrow w_2 - w_0$, $w_2 \leftarrow w_2 - w_4$ \\ +21. $w_1 \leftarrow w_1 - w_2$, $w_3 \leftarrow w_3 - w_2$ \\ +22. $tmp_1 \leftarrow 8 \cdot w_0$, $w_1 \leftarrow w_1 - tmp_1$, $tmp_1 \leftarrow 8 \cdot w_4$, $w_3 \leftarrow w_3 - tmp_1$ \\ +23. $w_2 \leftarrow 3 \cdot w_2$, $w_2 \leftarrow w_2 - w_1$, $w_2 \leftarrow w_2 - w_3$ \\ +24. $w_1 \leftarrow w_1 - w_2$, $w_3 \leftarrow w_3 - w_2$ \\ +25. $w_1 \leftarrow \lfloor w_1 / 3 \rfloor, w_3 \leftarrow \lfloor w_3 / 3 \rfloor$ \\ +\\ +Now substitute $\beta^k$ for $x$ by shifting $w_0, w_1, ..., w_4$. \\ +26. for $n$ from $1$ to $4$ do \\ +\hspace{3mm}26.1 $w_n \leftarrow w_n \cdot \beta^{nk}$ \\ +27. $c \leftarrow w_0 + w_1$, $c \leftarrow c + w_2$, $c \leftarrow c + w_3$, $c \leftarrow c + w_4$ \\ +28. Return(\textit{MP\_OKAY}) \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_toom\_mul (continued)} +\end{figure} + +\textbf{Algorithm mp\_toom\_mul.} +This algorithm computes the product of two mp\_int variables $a$ and $b$ using the Toom-Cook approach. Compared to the Karatsuba multiplication, this +algorithm has a lower asymptotic running time of approximately $O(n^{1.464})$ but at an obvious cost in overhead. In this +description, several statements have been compounded to save space. The intention is that the statements are executed from left to right across +any given step. + +The two inputs $a$ and $b$ are first split into three $k$-digit integers $a_0, a_1, a_2$ and $b_0, b_1, b_2$ respectively. From these smaller +integers the coefficients of the polynomial basis representations $f(x)$ and $g(x)$ are known and can be used to find the relations required. + +The first two relations $w_0$ and $w_4$ are the points $\zeta_{0}$ and $\zeta_{\infty}$ respectively. The relation $w_1, w_2$ and $w_3$ correspond +to the points $16 \cdot \zeta_{1 \over 2}, \zeta_{2}$ and $\zeta_{1}$ respectively. These are found using logical shifts to independently find +$f(y)$ and $g(y)$ which significantly speeds up the algorithm. + +After the five relations $w_0, w_1, \ldots, w_4$ have been computed, the system they represent must be solved in order for the unknown coefficients +$w_1, w_2$ and $w_3$ to be isolated. The steps 18 through 25 perform the system reduction required as previously described. Each step of +the reduction represents the comparable matrix operation that would be performed had this been performed by pencil. For example, step 18 indicates +that row $1$ must be subtracted from row $4$ and simultaneously row $0$ subtracted from row $3$. + +Once the coeffients have been isolated, the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$ is known. By substituting $\beta^{k}$ for $x$, the integer +result $a \cdot b$ is produced. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_toom\_mul.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* multiplication using the Toom-Cook 3-way algorithm +018 * +019 * Much more complicated than Karatsuba but has a lower asymptotic running t + ime of +020 * O(N**1.464). This algorithm is only particularly useful on VERY large +021 * inputs (we're talking 1000s of digits here...). +022 */ +023 int mp_toom_mul(mp_int *a, mp_int *b, mp_int *c) +024 \{ +025 mp_int w0, w1, w2, w3, w4, tmp1, tmp2, a0, a1, a2, b0, b1, b2; +026 int res, B; +027 +028 /* init temps */ +029 if ((res = mp_init_multi(&w0, &w1, &w2, &w3, &w4, +030 &a0, &a1, &a2, &b0, &b1, +031 &b2, &tmp1, &tmp2, NULL)) != MP_OKAY) \{ +032 return res; +033 \} +034 +035 /* B */ +036 B = MIN(a->used, b->used) / 3; +037 +038 /* a = a2 * B**2 + a1 * B + a0 */ +039 if ((res = mp_mod_2d(a, DIGIT_BIT * B, &a0)) != MP_OKAY) \{ +040 goto ERR; +041 \} +042 +043 if ((res = mp_copy(a, &a1)) != MP_OKAY) \{ +044 goto ERR; +045 \} +046 mp_rshd(&a1, B); +047 mp_mod_2d(&a1, DIGIT_BIT * B, &a1); +048 +049 if ((res = mp_copy(a, &a2)) != MP_OKAY) \{ +050 goto ERR; +051 \} +052 mp_rshd(&a2, B*2); +053 +054 /* b = b2 * B**2 + b1 * B + b0 */ +055 if ((res = mp_mod_2d(b, DIGIT_BIT * B, &b0)) != MP_OKAY) \{ +056 goto ERR; +057 \} +058 +059 if ((res = mp_copy(b, &b1)) != MP_OKAY) \{ +060 goto ERR; +061 \} +062 mp_rshd(&b1, B); +063 mp_mod_2d(&b1, DIGIT_BIT * B, &b1); +064 +065 if ((res = mp_copy(b, &b2)) != MP_OKAY) \{ +066 goto ERR; +067 \} +068 mp_rshd(&b2, B*2); +069 +070 /* w0 = a0*b0 */ +071 if ((res = mp_mul(&a0, &b0, &w0)) != MP_OKAY) \{ +072 goto ERR; +073 \} +074 +075 /* w4 = a2 * b2 */ +076 if ((res = mp_mul(&a2, &b2, &w4)) != MP_OKAY) \{ +077 goto ERR; +078 \} +079 +080 /* w1 = (a2 + 2(a1 + 2a0))(b2 + 2(b1 + 2b0)) */ +081 if ((res = mp_mul_2(&a0, &tmp1)) != MP_OKAY) \{ +082 goto ERR; +083 \} +084 if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) \{ +085 goto ERR; +086 \} +087 if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) \{ +088 goto ERR; +089 \} +090 if ((res = mp_add(&tmp1, &a2, &tmp1)) != MP_OKAY) \{ +091 goto ERR; +092 \} +093 +094 if ((res = mp_mul_2(&b0, &tmp2)) != MP_OKAY) \{ +095 goto ERR; +096 \} +097 if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) \{ +098 goto ERR; +099 \} +100 if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) \{ +101 goto ERR; +102 \} +103 if ((res = mp_add(&tmp2, &b2, &tmp2)) != MP_OKAY) \{ +104 goto ERR; +105 \} +106 +107 if ((res = mp_mul(&tmp1, &tmp2, &w1)) != MP_OKAY) \{ +108 goto ERR; +109 \} +110 +111 /* w3 = (a0 + 2(a1 + 2a2))(b0 + 2(b1 + 2b2)) */ +112 if ((res = mp_mul_2(&a2, &tmp1)) != MP_OKAY) \{ +113 goto ERR; +114 \} +115 if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) \{ +116 goto ERR; +117 \} +118 if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) \{ +119 goto ERR; +120 \} +121 if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) \{ +122 goto ERR; +123 \} +124 +125 if ((res = mp_mul_2(&b2, &tmp2)) != MP_OKAY) \{ +126 goto ERR; +127 \} +128 if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) \{ +129 goto ERR; +130 \} +131 if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) \{ +132 goto ERR; +133 \} +134 if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) \{ +135 goto ERR; +136 \} +137 +138 if ((res = mp_mul(&tmp1, &tmp2, &w3)) != MP_OKAY) \{ +139 goto ERR; +140 \} +141 +142 +143 /* w2 = (a2 + a1 + a0)(b2 + b1 + b0) */ +144 if ((res = mp_add(&a2, &a1, &tmp1)) != MP_OKAY) \{ +145 goto ERR; +146 \} +147 if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) \{ +148 goto ERR; +149 \} +150 if ((res = mp_add(&b2, &b1, &tmp2)) != MP_OKAY) \{ +151 goto ERR; +152 \} +153 if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) \{ +154 goto ERR; +155 \} +156 if ((res = mp_mul(&tmp1, &tmp2, &w2)) != MP_OKAY) \{ +157 goto ERR; +158 \} +159 +160 /* now solve the matrix +161 +162 0 0 0 0 1 +163 1 2 4 8 16 +164 1 1 1 1 1 +165 16 8 4 2 1 +166 1 0 0 0 0 +167 +168 using 12 subtractions, 4 shifts, +169 2 small divisions and 1 small multiplication +170 */ +171 +172 /* r1 - r4 */ +173 if ((res = mp_sub(&w1, &w4, &w1)) != MP_OKAY) \{ +174 goto ERR; +175 \} +176 /* r3 - r0 */ +177 if ((res = mp_sub(&w3, &w0, &w3)) != MP_OKAY) \{ +178 goto ERR; +179 \} +180 /* r1/2 */ +181 if ((res = mp_div_2(&w1, &w1)) != MP_OKAY) \{ +182 goto ERR; +183 \} +184 /* r3/2 */ +185 if ((res = mp_div_2(&w3, &w3)) != MP_OKAY) \{ +186 goto ERR; +187 \} +188 /* r2 - r0 - r4 */ +189 if ((res = mp_sub(&w2, &w0, &w2)) != MP_OKAY) \{ +190 goto ERR; +191 \} +192 if ((res = mp_sub(&w2, &w4, &w2)) != MP_OKAY) \{ +193 goto ERR; +194 \} +195 /* r1 - r2 */ +196 if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) \{ +197 goto ERR; +198 \} +199 /* r3 - r2 */ +200 if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) \{ +201 goto ERR; +202 \} +203 /* r1 - 8r0 */ +204 if ((res = mp_mul_2d(&w0, 3, &tmp1)) != MP_OKAY) \{ +205 goto ERR; +206 \} +207 if ((res = mp_sub(&w1, &tmp1, &w1)) != MP_OKAY) \{ +208 goto ERR; +209 \} +210 /* r3 - 8r4 */ +211 if ((res = mp_mul_2d(&w4, 3, &tmp1)) != MP_OKAY) \{ +212 goto ERR; +213 \} +214 if ((res = mp_sub(&w3, &tmp1, &w3)) != MP_OKAY) \{ +215 goto ERR; +216 \} +217 /* 3r2 - r1 - r3 */ +218 if ((res = mp_mul_d(&w2, 3, &w2)) != MP_OKAY) \{ +219 goto ERR; +220 \} +221 if ((res = mp_sub(&w2, &w1, &w2)) != MP_OKAY) \{ +222 goto ERR; +223 \} +224 if ((res = mp_sub(&w2, &w3, &w2)) != MP_OKAY) \{ +225 goto ERR; +226 \} +227 /* r1 - r2 */ +228 if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) \{ +229 goto ERR; +230 \} +231 /* r3 - r2 */ +232 if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) \{ +233 goto ERR; +234 \} +235 /* r1/3 */ +236 if ((res = mp_div_3(&w1, &w1, NULL)) != MP_OKAY) \{ +237 goto ERR; +238 \} +239 /* r3/3 */ +240 if ((res = mp_div_3(&w3, &w3, NULL)) != MP_OKAY) \{ +241 goto ERR; +242 \} +243 +244 /* at this point shift W[n] by B*n */ +245 if ((res = mp_lshd(&w1, 1*B)) != MP_OKAY) \{ +246 goto ERR; +247 \} +248 if ((res = mp_lshd(&w2, 2*B)) != MP_OKAY) \{ +249 goto ERR; +250 \} +251 if ((res = mp_lshd(&w3, 3*B)) != MP_OKAY) \{ +252 goto ERR; +253 \} +254 if ((res = mp_lshd(&w4, 4*B)) != MP_OKAY) \{ +255 goto ERR; +256 \} +257 +258 if ((res = mp_add(&w0, &w1, c)) != MP_OKAY) \{ +259 goto ERR; +260 \} +261 if ((res = mp_add(&w2, &w3, &tmp1)) != MP_OKAY) \{ +262 goto ERR; +263 \} +264 if ((res = mp_add(&w4, &tmp1, &tmp1)) != MP_OKAY) \{ +265 goto ERR; +266 \} +267 if ((res = mp_add(&tmp1, c, c)) != MP_OKAY) \{ +268 goto ERR; +269 \} +270 +271 ERR: +272 mp_clear_multi(&w0, &w1, &w2, &w3, &w4, +273 &a0, &a1, &a2, &b0, &b1, +274 &b2, &tmp1, &tmp2, NULL); +275 return res; +276 \} +277 +278 #endif +\end{alltt} +\end{small} + +-- Comments to be added during editing phase. + +\subsection{Signed Multiplication} +Now that algorithms to handle multiplications of every useful dimensions have been developed, a rather simple finishing touch is required. So far all +of the multiplication algorithms have been unsigned multiplications which leaves only a signed multiplication algorithm to be established. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_mul}. \\ +\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ +\textbf{Output}. $c \leftarrow a \cdot b$ \\ +\hline \\ +1. If $a.sign = b.sign$ then \\ +\hspace{3mm}1.1 $sign = MP\_ZPOS$ \\ +2. else \\ +\hspace{3mm}2.1 $sign = MP\_ZNEG$ \\ +3. If min$(a.used, b.used) \ge TOOM\_MUL\_CUTOFF$ then \\ +\hspace{3mm}3.1 $c \leftarrow a \cdot b$ using algorithm mp\_toom\_mul \\ +4. else if min$(a.used, b.used) \ge KARATSUBA\_MUL\_CUTOFF$ then \\ +\hspace{3mm}4.1 $c \leftarrow a \cdot b$ using algorithm mp\_karatsuba\_mul \\ +5. else \\ +\hspace{3mm}5.1 $digs \leftarrow a.used + b.used + 1$ \\ +\hspace{3mm}5.2 If $digs < MP\_ARRAY$ and min$(a.used, b.used) \le \delta$ then \\ +\hspace{6mm}5.2.1 $c \leftarrow a \cdot b \mbox{ (mod }\beta^{digs}\mbox{)}$ using algorithm fast\_s\_mp\_mul\_digs. \\ +\hspace{3mm}5.3 else \\ +\hspace{6mm}5.3.1 $c \leftarrow a \cdot b \mbox{ (mod }\beta^{digs}\mbox{)}$ using algorithm s\_mp\_mul\_digs. \\ +6. $c.sign \leftarrow sign$ \\ +7. Return the result of the unsigned multiplication performed. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_mul} +\end{figure} + +\textbf{Algorithm mp\_mul.} +This algorithm performs the signed multiplication of two inputs. It will make use of any of the three unsigned multiplication algorithms +available when the input is of appropriate size. The \textbf{sign} of the result is not set until the end of the algorithm since algorithm +s\_mp\_mul\_digs will clear it. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_mul.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* high level multiplication (handles sign) */ +018 int mp_mul (mp_int * a, mp_int * b, mp_int * c) +019 \{ +020 int res, neg; +021 neg = (a->sign == b->sign) ? MP_ZPOS : MP_NEG; +022 +023 /* use Toom-Cook? */ +024 #ifdef BN_MP_TOOM_MUL_C +025 if (MIN (a->used, b->used) >= TOOM_MUL_CUTOFF) \{ +026 res = mp_toom_mul(a, b, c); +027 \} else +028 #endif +029 #ifdef BN_MP_KARATSUBA_MUL_C +030 /* use Karatsuba? */ +031 if (MIN (a->used, b->used) >= KARATSUBA_MUL_CUTOFF) \{ +032 res = mp_karatsuba_mul (a, b, c); +033 \} else +034 #endif +035 \{ +036 /* can we use the fast multiplier? +037 * +038 * The fast multiplier can be used if the output will +039 * have less than MP_WARRAY digits and the number of +040 * digits won't affect carry propagation +041 */ +042 int digs = a->used + b->used + 1; +043 +044 #ifdef BN_FAST_S_MP_MUL_DIGS_C +045 if ((digs < MP_WARRAY) && +046 MIN(a->used, b->used) <= +047 (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) \{ +048 res = fast_s_mp_mul_digs (a, b, c, digs); +049 \} else +050 #endif +051 #ifdef BN_S_MP_MUL_DIGS_C +052 res = s_mp_mul (a, b, c); /* uses s_mp_mul_digs */ +053 #else +054 res = MP_VAL; +055 #endif +056 +057 \} +058 c->sign = (c->used > 0) ? neg : MP_ZPOS; +059 return res; +060 \} +061 #endif +\end{alltt} +\end{small} + +The implementation is rather simplistic and is not particularly noteworthy. Line 23 computes the sign of the result using the ``?'' +operator from the C programming language. Line 47 computes $\delta$ using the fact that $1 << k$ is equal to $2^k$. + +\section{Squaring} +\label{sec:basesquare} + +Squaring is a special case of multiplication where both multiplicands are equal. At first it may seem like there is no significant optimization +available but in fact there is. Consider the multiplication of $576$ against $241$. In total there will be nine single precision multiplications +performed which are $1\cdot 6$, $1 \cdot 7$, $1 \cdot 5$, $4 \cdot 6$, $4 \cdot 7$, $4 \cdot 5$, $2 \cdot 6$, $2 \cdot 7$ and $2 \cdot 5$. Now consider +the multiplication of $123$ against $123$. The nine products are $3 \cdot 3$, $3 \cdot 2$, $3 \cdot 1$, $2 \cdot 3$, $2 \cdot 2$, $2 \cdot 1$, +$1 \cdot 3$, $1 \cdot 2$ and $1 \cdot 1$. On closer inspection some of the products are equivalent. For example, $3 \cdot 2 = 2 \cdot 3$ +and $3 \cdot 1 = 1 \cdot 3$. + +For any $n$-digit input, there are ${{\left (n^2 + n \right)}\over 2}$ possible unique single precision multiplications required compared to the $n^2$ +required for multiplication. The following diagram gives an example of the operations required. + +\begin{figure}[here] +\begin{center} +\begin{tabular}{ccccc|c} +&&1&2&3&\\ +$\times$ &&1&2&3&\\ +\hline && $3 \cdot 1$ & $3 \cdot 2$ & $3 \cdot 3$ & Row 0\\ + & $2 \cdot 1$ & $2 \cdot 2$ & $2 \cdot 3$ && Row 1 \\ + $1 \cdot 1$ & $1 \cdot 2$ & $1 \cdot 3$ &&& Row 2 \\ +\end{tabular} +\end{center} +\caption{Squaring Optimization Diagram} +\end{figure} + +Starting from zero and numbering the columns from right to left a very simple pattern becomes obvious. For the purposes of this discussion let $x$ +represent the number being squared. The first observation is that in row $k$ the $2k$'th column of the product has a $\left (x_k \right)^2$ term in it. + +The second observation is that every column $j$ in row $k$ where $j \ne 2k$ is part of a double product. Every non-square term of a column will +appear twice hence the name ``double product''. Every odd column is made up entirely of double products. In fact every column is made up of double +products and at most one square (\textit{see the exercise section}). + +The third and final observation is that for row $k$ the first unique non-square term, that is, one that hasn't already appeared in an earlier row, +occurs at column $2k + 1$. For example, on row $1$ of the previous squaring, column one is part of the double product with column one from row zero. +Column two of row one is a square and column three is the first unique column. + +\subsection{The Baseline Squaring Algorithm} +The baseline squaring algorithm is meant to be a catch-all squaring algorithm. It will handle any of the input sizes that the faster routines +will not handle. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{s\_mp\_sqr}. \\ +\textbf{Input}. mp\_int $a$ \\ +\textbf{Output}. $b \leftarrow a^2$ \\ +\hline \\ +1. Init a temporary mp\_int of at least $2 \cdot a.used +1$ digits. (\textit{mp\_init\_size}) \\ +2. If step 1 failed return(\textit{MP\_MEM}) \\ +3. $t.used \leftarrow 2 \cdot a.used + 1$ \\ +4. For $ix$ from 0 to $a.used - 1$ do \\ +\hspace{3mm}Calculate the square. \\ +\hspace{3mm}4.1 $\hat r \leftarrow t_{2ix} + \left (a_{ix} \right )^2$ \\ +\hspace{3mm}4.2 $t_{2ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ +\hspace{3mm}Calculate the double products after the square. \\ +\hspace{3mm}4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ +\hspace{3mm}4.4 For $iy$ from $ix + 1$ to $a.used - 1$ do \\ +\hspace{6mm}4.4.1 $\hat r \leftarrow 2 \cdot a_{ix}a_{iy} + t_{ix + iy} + u$ \\ +\hspace{6mm}4.4.2 $t_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ +\hspace{6mm}4.4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ +\hspace{3mm}Set the last carry. \\ +\hspace{3mm}4.5 While $u > 0$ do \\ +\hspace{6mm}4.5.1 $iy \leftarrow iy + 1$ \\ +\hspace{6mm}4.5.2 $\hat r \leftarrow t_{ix + iy} + u$ \\ +\hspace{6mm}4.5.3 $t_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ +\hspace{6mm}4.5.4 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ +5. Clamp excess digits of $t$. (\textit{mp\_clamp}) \\ +6. Exchange $b$ and $t$. \\ +7. Clear $t$ (\textit{mp\_clear}) \\ +8. Return(\textit{MP\_OKAY}) \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm s\_mp\_sqr} +\end{figure} + +\textbf{Algorithm s\_mp\_sqr.} +This algorithm computes the square of an input using the three observations on squaring. It is based fairly faithfully on algorithm 14.16 of HAC +\cite[pp.596-597]{HAC}. Similar to algorithm s\_mp\_mul\_digs, a temporary mp\_int is allocated to hold the result of the squaring. This allows the +destination mp\_int to be the same as the source mp\_int. + +The outer loop of this algorithm begins on step 4. It is best to think of the outer loop as walking down the rows of the partial results, while +the inner loop computes the columns of the partial result. Step 4.1 and 4.2 compute the square term for each row, and step 4.3 and 4.4 propagate +the carry and compute the double products. + +The requirement that a mp\_word be able to represent the range $0 \le x < 2 \beta^2$ arises from this +very algorithm. The product $a_{ix}a_{iy}$ will lie in the range $0 \le x \le \beta^2 - 2\beta + 1$ which is obviously less than $\beta^2$ meaning that +when it is multiplied by two, it can be properly represented by a mp\_word. + +Similar to algorithm s\_mp\_mul\_digs, after every pass of the inner loop, the destination is correctly set to the sum of all of the partial +results calculated so far. This involves expensive carry propagation which will be eliminated in the next algorithm. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_sqr.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* low level squaring, b = a*a, HAC pp.596-597, Algorithm 14.16 */ +018 int +019 s_mp_sqr (mp_int * a, mp_int * b) +020 \{ +021 mp_int t; +022 int res, ix, iy, pa; +023 mp_word r; +024 mp_digit u, tmpx, *tmpt; +025 +026 pa = a->used; +027 if ((res = mp_init_size (&t, 2*pa + 1)) != MP_OKAY) \{ +028 return res; +029 \} +030 +031 /* default used is maximum possible size */ +032 t.used = 2*pa + 1; +033 +034 for (ix = 0; ix < pa; ix++) \{ +035 /* first calculate the digit at 2*ix */ +036 /* calculate double precision result */ +037 r = ((mp_word) t.dp[2*ix]) + +038 ((mp_word)a->dp[ix])*((mp_word)a->dp[ix]); +039 +040 /* store lower part in result */ +041 t.dp[ix+ix] = (mp_digit) (r & ((mp_word) MP_MASK)); +042 +043 /* get the carry */ +044 u = (mp_digit)(r >> ((mp_word) DIGIT_BIT)); +045 +046 /* left hand side of A[ix] * A[iy] */ +047 tmpx = a->dp[ix]; +048 +049 /* alias for where to store the results */ +050 tmpt = t.dp + (2*ix + 1); +051 +052 for (iy = ix + 1; iy < pa; iy++) \{ +053 /* first calculate the product */ +054 r = ((mp_word)tmpx) * ((mp_word)a->dp[iy]); +055 +056 /* now calculate the double precision result, note we use +057 * addition instead of *2 since it's easier to optimize +058 */ +059 r = ((mp_word) *tmpt) + r + r + ((mp_word) u); +060 +061 /* store lower part */ +062 *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK)); +063 +064 /* get carry */ +065 u = (mp_digit)(r >> ((mp_word) DIGIT_BIT)); +066 \} +067 /* propagate upwards */ +068 while (u != ((mp_digit) 0)) \{ +069 r = ((mp_word) *tmpt) + ((mp_word) u); +070 *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK)); +071 u = (mp_digit)(r >> ((mp_word) DIGIT_BIT)); +072 \} +073 \} +074 +075 mp_clamp (&t); +076 mp_exch (&t, b); +077 mp_clear (&t); +078 return MP_OKAY; +079 \} +080 #endif +\end{alltt} +\end{small} + +Inside the outer loop (\textit{see line 34}) the square term is calculated on line 37. Line 44 extracts the carry from the square +term. Aliases for $a_{ix}$ and $t_{ix+iy}$ are initialized on lines 47 and 50 respectively. The doubling is performed using two +additions (\textit{see line 59}) since it is usually faster than shifting,if not at least as fast. + +\subsection{Faster Squaring by the ``Comba'' Method} +A major drawback to the baseline method is the requirement for single precision shifting inside the $O(n^2)$ nested loop. Squaring has an additional +drawback that it must double the product inside the inner loop as well. As for multiplication, the Comba technique can be used to eliminate these +performance hazards. + +The first obvious solution is to make an array of mp\_words which will hold all of the columns. This will indeed eliminate all of the carry +propagation operations from the inner loop. However, the inner product must still be doubled $O(n^2)$ times. The solution stems from the simple fact +that $2a + 2b + 2c = 2(a + b + c)$. That is the sum of all of the double products is equal to double the sum of all the products. For example, +$ab + ba + ac + ca = 2ab + 2ac = 2(ab + ac)$. + +However, we cannot simply double all of the columns, since the squares appear only once per row. The most practical solution is to have two mp\_word +arrays. One array will hold the squares and the other array will hold the double products. With both arrays the doubling and carry propagation can be +moved to a $O(n)$ work level outside the $O(n^2)$ level. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{fast\_s\_mp\_sqr}. \\ +\textbf{Input}. mp\_int $a$ \\ +\textbf{Output}. $b \leftarrow a^2$ \\ +\hline \\ +Place two arrays of \textbf{MP\_WARRAY} mp\_words named $\hat W$ and $\hat {X}$ on the stack. \\ +1. If $b.alloc < 2a.used + 1$ then grow $b$ to $2a.used + 1$ digits. (\textit{mp\_grow}). \\ +2. If step 1 failed return(\textit{MP\_MEM}). \\ +3. for $ix$ from $0$ to $2a.used + 1$ do \\ +\hspace{3mm}3.1 $\hat W_{ix} \leftarrow 0$ \\ +\hspace{3mm}3.2 $\hat {X}_{ix} \leftarrow 0$ \\ +4. for $ix$ from $0$ to $a.used - 1$ do \\ +\hspace{3mm}Compute the square.\\ +\hspace{3mm}4.1 $\hat {X}_{ix+ix} \leftarrow \left ( a_{ix} \right )^2$ \\ +\\ +\hspace{3mm}Compute the double products.\\ +\hspace{3mm}4.2 for $iy$ from $ix + 1$ to $a.used - 1$ do \\ +\hspace{6mm}4.2.1 $\hat W_{ix+iy} \leftarrow \hat W_{ix+iy} + a_{ix}a_{iy}$ \\ +5. $oldused \leftarrow b.used$ \\ +6. $b.used \leftarrow 2a.used + 1$ \\ +\\ +Double the products and propagate the carries simultaneously. \\ +7. $\hat W_0 \leftarrow 2 \hat W_0 + \hat {X}_0$ \\ +8. for $ix$ from $1$ to $2a.used$ do \\ +\hspace{3mm}8.1 $\hat W_{ix} \leftarrow 2 \hat W_{ix} + \hat {X}_{ix}$ \\ +\hspace{3mm}8.2 $\hat W_{ix} \leftarrow \hat W_{ix} + \lfloor \hat W_{ix - 1} / \beta \rfloor$ \\ +\hspace{3mm}8.3 $b_{ix-1} \leftarrow W_{ix-1} \mbox{ (mod }\beta\mbox{)}$ \\ +9. $b_{2a.used} \leftarrow \hat W_{2a.used} \mbox{ (mod }\beta\mbox{)}$ \\ +10. if $2a.used + 1 < oldused$ then do \\ +\hspace{3mm}10.1 for $ix$ from $2a.used + 1$ to $oldused$ do \\ +\hspace{6mm}10.1.1 $b_{ix} \leftarrow 0$ \\ +11. Clamp excess digits from $b$. (\textit{mp\_clamp}) \\ +12. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm fast\_s\_mp\_sqr} +\end{figure} + +\textbf{Algorithm fast\_s\_mp\_sqr.} +This algorithm computes the square of an input using the Comba technique. It is designed to be a replacement for algorithm s\_mp\_sqr when +the number of input digits is less than \textbf{MP\_WARRAY} and less than $\delta \over 2$. + +This routine requires two arrays of mp\_words to be placed on the stack. The first array $\hat W$ will hold the double products and the second +array $\hat X$ will hold the squares. Though only at most $MP\_WARRAY \over 2$ words of $\hat X$ are used, it has proven faster on most +processors to simply make it a full size array. + +The loop on step 3 will zero the two arrays to prepare them for the squaring step. Step 4.1 computes the squares of the product. Note how +it simply assigns the value into the $\hat X$ array. The nested loop on step 4.2 computes the doubles of the products. This loop +computes the sum of the products for each column. They are not doubled until later. + +After the squaring loop, the products stored in $\hat W$ musted be doubled and the carries propagated forwards. It makes sense to do both +operations at the same time. The expression $\hat W_{ix} \leftarrow 2 \hat W_{ix} + \hat {X}_{ix}$ computes the sum of the double product and the +squares in place. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_fast\_s\_mp\_sqr.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* fast squaring +018 * +019 * This is the comba method where the columns of the product +020 * are computed first then the carries are computed. This +021 * has the effect of making a very simple inner loop that +022 * is executed the most +023 * +024 * W2 represents the outer products and W the inner. +025 * +026 * A further optimizations is made because the inner +027 * products are of the form "A * B * 2". The *2 part does +028 * not need to be computed until the end which is good +029 * because 64-bit shifts are slow! +030 * +031 * Based on Algorithm 14.16 on pp.597 of HAC. +032 * +033 */ +034 /* the jist of squaring... +035 +036 you do like mult except the offset of the tmpx [one that starts closer to ze + ro] +037 can't equal the offset of tmpy. So basically you set up iy like before then + you min it with +038 (ty-tx) so that it never happens. You double all those you add in the inner + loop +039 +040 After that loop you do the squares and add them in. +041 +042 Remove W2 and don't memset W +043 +044 */ +045 +046 int fast_s_mp_sqr (mp_int * a, mp_int * b) +047 \{ +048 int olduse, res, pa, ix, iz; +049 mp_digit W[MP_WARRAY], *tmpx; +050 mp_word W1; +051 +052 /* grow the destination as required */ +053 pa = a->used + a->used; +054 if (b->alloc < pa) \{ +055 if ((res = mp_grow (b, pa)) != MP_OKAY) \{ +056 return res; +057 \} +058 \} +059 +060 /* number of output digits to produce */ +061 W1 = 0; +062 for (ix = 0; ix < pa; ix++) \{ +063 int tx, ty, iy; +064 mp_word _W; +065 mp_digit *tmpy; +066 +067 /* clear counter */ +068 _W = 0; +069 +070 /* get offsets into the two bignums */ +071 ty = MIN(a->used-1, ix); +072 tx = ix - ty; +073 +074 /* setup temp aliases */ +075 tmpx = a->dp + tx; +076 tmpy = a->dp + ty; +077 +078 /* this is the number of times the loop will iterrate, essentially its + +079 while (tx++ < a->used && ty-- >= 0) \{ ... \} +080 */ +081 iy = MIN(a->used-tx, ty+1); +082 +083 /* now for squaring tx can never equal ty +084 * we halve the distance since they approach at a rate of 2x +085 * and we have to round because odd cases need to be executed +086 */ +087 iy = MIN(iy, (ty-tx+1)>>1); +088 +089 /* execute loop */ +090 for (iz = 0; iz < iy; iz++) \{ +091 _W += ((mp_word)*tmpx++)*((mp_word)*tmpy--); +092 \} +093 +094 /* double the inner product and add carry */ +095 _W = _W + _W + W1; +096 +097 /* even columns have the square term in them */ +098 if ((ix&1) == 0) \{ +099 _W += ((mp_word)a->dp[ix>>1])*((mp_word)a->dp[ix>>1]); +100 \} +101 +102 /* store it */ +103 W[ix] = _W; +104 +105 /* make next carry */ +106 W1 = _W >> ((mp_word)DIGIT_BIT); +107 \} +108 +109 /* setup dest */ +110 olduse = b->used; +111 b->used = a->used+a->used; +112 +113 \{ +114 mp_digit *tmpb; +115 tmpb = b->dp; +116 for (ix = 0; ix < pa; ix++) \{ +117 *tmpb++ = W[ix] & MP_MASK; +118 \} +119 +120 /* clear unused digits [that existed in the old copy of c] */ +121 for (; ix < olduse; ix++) \{ +122 *tmpb++ = 0; +123 \} +124 \} +125 mp_clamp (b); +126 return MP_OKAY; +127 \} +128 #endif +\end{alltt} +\end{small} + +-- Write something deep and insightful later, Tom. + +\subsection{Polynomial Basis Squaring} +The same algorithm that performs optimal polynomial basis multiplication can be used to perform polynomial basis squaring. The minor exception +is that $\zeta_y = f(y)g(y)$ is actually equivalent to $\zeta_y = f(y)^2$ since $f(y) = g(y)$. Instead of performing $2n + 1$ +multiplications to find the $\zeta$ relations, squaring operations are performed instead. + +\subsection{Karatsuba Squaring} +Let $f(x) = ax + b$ represent the polynomial basis representation of a number to square. +Let $h(x) = \left ( f(x) \right )^2$ represent the square of the polynomial. The Karatsuba equation can be modified to square a +number with the following equation. + +\begin{equation} +h(x) = a^2x^2 + \left (a^2 + b^2 - (a - b)^2 \right )x + b^2 +\end{equation} + +Upon closer inspection this equation only requires the calculation of three half-sized squares: $a^2$, $b^2$ and $(a - b)^2$. As in +Karatsuba multiplication, this algorithm can be applied recursively on the input and will achieve an asymptotic running time of +$O \left ( n^{lg(3)} \right )$. + +If the asymptotic times of Karatsuba squaring and multiplication are the same, why not simply use the multiplication algorithm +instead? The answer to this arises from the cutoff point for squaring. As in multiplication there exists a cutoff point, at which the +time required for a Comba based squaring and a Karatsuba based squaring meet. Due to the overhead inherent in the Karatsuba method, the cutoff +point is fairly high. For example, on an AMD Athlon XP processor with $\beta = 2^{28}$, the cutoff point is around 127 digits. + +Consider squaring a 200 digit number with this technique. It will be split into two 100 digit halves which are subsequently squared. +The 100 digit halves will not be squared using Karatsuba, but instead using the faster Comba based squaring algorithm. If Karatsuba multiplication +were used instead, the 100 digit numbers would be squared with a slower Comba based multiplication. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_karatsuba\_sqr}. \\ +\textbf{Input}. mp\_int $a$ \\ +\textbf{Output}. $b \leftarrow a^2$ \\ +\hline \\ +1. Initialize the following temporary mp\_ints: $x0$, $x1$, $t1$, $t2$, $x0x0$ and $x1x1$. \\ +2. If any of the initializations on step 1 failed return(\textit{MP\_MEM}). \\ +\\ +Split the input. e.g. $a = x1\beta^B + x0$ \\ +3. $B \leftarrow \lfloor a.used / 2 \rfloor$ \\ +4. $x0 \leftarrow a \mbox{ (mod }\beta^B\mbox{)}$ (\textit{mp\_mod\_2d}) \\ +5. $x1 \leftarrow \lfloor a / \beta^B \rfloor$ (\textit{mp\_lshd}) \\ +\\ +Calculate the three squares. \\ +6. $x0x0 \leftarrow x0^2$ (\textit{mp\_sqr}) \\ +7. $x1x1 \leftarrow x1^2$ \\ +8. $t1 \leftarrow x1 - x0$ (\textit{mp\_sub}) \\ +9. $t1 \leftarrow t1^2$ \\ +\\ +Compute the middle term. \\ +10. $t2 \leftarrow x0x0 + x1x1$ (\textit{s\_mp\_add}) \\ +11. $t1 \leftarrow t2 - t1$ \\ +\\ +Compute final product. \\ +12. $t1 \leftarrow t1\beta^B$ (\textit{mp\_lshd}) \\ +13. $x1x1 \leftarrow x1x1\beta^{2B}$ \\ +14. $t1 \leftarrow t1 + x0x0$ \\ +15. $b \leftarrow t1 + x1x1$ \\ +16. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_karatsuba\_sqr} +\end{figure} + +\textbf{Algorithm mp\_karatsuba\_sqr.} +This algorithm computes the square of an input $a$ using the Karatsuba technique. This algorithm is very similar to the Karatsuba based +multiplication algorithm with the exception that the three half-size multiplications have been replaced with three half-size squarings. + +The radix point for squaring is simply placed exactly in the middle of the digits when the input has an odd number of digits, otherwise it is +placed just below the middle. Step 3, 4 and 5 compute the two halves required using $B$ +as the radix point. The first two squares in steps 6 and 7 are rather straightforward while the last square is of a more compact form. + +By expanding $\left (x1 - x0 \right )^2$, the $x1^2$ and $x0^2$ terms in the middle disappear, that is $x1^2 + x0^2 - (x1 - x0)^2 = 2 \cdot x0 \cdot x1$. +Now if $5n$ single precision additions and a squaring of $n$-digits is faster than multiplying two $n$-digit numbers and doubling then +this method is faster. Assuming no further recursions occur, the difference can be estimated with the following inequality. + +Let $p$ represent the cost of a single precision addition and $q$ the cost of a single precision multiplication both in terms of time\footnote{Or +machine clock cycles.}. + +\begin{equation} +5pn +{{q(n^2 + n)} \over 2} \le pn + qn^2 +\end{equation} + +For example, on an AMD Athlon XP processor $p = {1 \over 3}$ and $q = 6$. This implies that the following inequality should hold. +\begin{center} +\begin{tabular}{rcl} +${5n \over 3} + 3n^2 + 3n$ & $<$ & ${n \over 3} + 6n^2$ \\ +${5 \over 3} + 3n + 3$ & $<$ & ${1 \over 3} + 6n$ \\ +${13 \over 9}$ & $<$ & $n$ \\ +\end{tabular} +\end{center} + +This results in a cutoff point around $n = 2$. As a consequence it is actually faster to compute the middle term the ``long way'' on processors +where multiplication is substantially slower\footnote{On the Athlon there is a 1:17 ratio between clock cycles for addition and multiplication. On +the Intel P4 processor this ratio is 1:29 making this method even more beneficial. The only common exception is the ARMv4 processor which has a +ratio of 1:7. } than simpler operations such as addition. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_karatsuba\_sqr.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* Karatsuba squaring, computes b = a*a using three +018 * half size squarings +019 * +020 * See comments of karatsuba_mul for details. It +021 * is essentially the same algorithm but merely +022 * tuned to perform recursive squarings. +023 */ +024 int mp_karatsuba_sqr (mp_int * a, mp_int * b) +025 \{ +026 mp_int x0, x1, t1, t2, x0x0, x1x1; +027 int B, err; +028 +029 err = MP_MEM; +030 +031 /* min # of digits */ +032 B = a->used; +033 +034 /* now divide in two */ +035 B = B >> 1; +036 +037 /* init copy all the temps */ +038 if (mp_init_size (&x0, B) != MP_OKAY) +039 goto ERR; +040 if (mp_init_size (&x1, a->used - B) != MP_OKAY) +041 goto X0; +042 +043 /* init temps */ +044 if (mp_init_size (&t1, a->used * 2) != MP_OKAY) +045 goto X1; +046 if (mp_init_size (&t2, a->used * 2) != MP_OKAY) +047 goto T1; +048 if (mp_init_size (&x0x0, B * 2) != MP_OKAY) +049 goto T2; +050 if (mp_init_size (&x1x1, (a->used - B) * 2) != MP_OKAY) +051 goto X0X0; +052 +053 \{ +054 register int x; +055 register mp_digit *dst, *src; +056 +057 src = a->dp; +058 +059 /* now shift the digits */ +060 dst = x0.dp; +061 for (x = 0; x < B; x++) \{ +062 *dst++ = *src++; +063 \} +064 +065 dst = x1.dp; +066 for (x = B; x < a->used; x++) \{ +067 *dst++ = *src++; +068 \} +069 \} +070 +071 x0.used = B; +072 x1.used = a->used - B; +073 +074 mp_clamp (&x0); +075 +076 /* now calc the products x0*x0 and x1*x1 */ +077 if (mp_sqr (&x0, &x0x0) != MP_OKAY) +078 goto X1X1; /* x0x0 = x0*x0 */ +079 if (mp_sqr (&x1, &x1x1) != MP_OKAY) +080 goto X1X1; /* x1x1 = x1*x1 */ +081 +082 /* now calc (x1-x0)**2 */ +083 if (mp_sub (&x1, &x0, &t1) != MP_OKAY) +084 goto X1X1; /* t1 = x1 - x0 */ +085 if (mp_sqr (&t1, &t1) != MP_OKAY) +086 goto X1X1; /* t1 = (x1 - x0) * (x1 - x0) */ +087 +088 /* add x0y0 */ +089 if (s_mp_add (&x0x0, &x1x1, &t2) != MP_OKAY) +090 goto X1X1; /* t2 = x0x0 + x1x1 */ +091 if (mp_sub (&t2, &t1, &t1) != MP_OKAY) +092 goto X1X1; /* t1 = x0x0 + x1x1 - (x1-x0)*(x1-x0) */ +093 +094 /* shift by B */ +095 if (mp_lshd (&t1, B) != MP_OKAY) +096 goto X1X1; /* t1 = (x0x0 + x1x1 - (x1-x0)*(x1-x0))<<B */ +097 if (mp_lshd (&x1x1, B * 2) != MP_OKAY) +098 goto X1X1; /* x1x1 = x1x1 << 2*B */ +099 +100 if (mp_add (&x0x0, &t1, &t1) != MP_OKAY) +101 goto X1X1; /* t1 = x0x0 + t1 */ +102 if (mp_add (&t1, &x1x1, b) != MP_OKAY) +103 goto X1X1; /* t1 = x0x0 + t1 + x1x1 */ +104 +105 err = MP_OKAY; +106 +107 X1X1:mp_clear (&x1x1); +108 X0X0:mp_clear (&x0x0); +109 T2:mp_clear (&t2); +110 T1:mp_clear (&t1); +111 X1:mp_clear (&x1); +112 X0:mp_clear (&x0); +113 ERR: +114 return err; +115 \} +116 #endif +\end{alltt} +\end{small} + +This implementation is largely based on the implementation of algorithm mp\_karatsuba\_mul. It uses the same inline style to copy and +shift the input into the two halves. The loop from line 53 to line 69 has been modified since only one input exists. The \textbf{used} +count of both $x0$ and $x1$ is fixed up and $x0$ is clamped before the calculations begin. At this point $x1$ and $x0$ are valid equivalents +to the respective halves as if mp\_rshd and mp\_mod\_2d had been used. + +By inlining the copy and shift operations the cutoff point for Karatsuba multiplication can be lowered. On the Athlon the cutoff point +is exactly at the point where Comba squaring can no longer be used (\textit{128 digits}). On slower processors such as the Intel P4 +it is actually below the Comba limit (\textit{at 110 digits}). + +This routine uses the same error trap coding style as mp\_karatsuba\_sqr. As the temporary variables are initialized errors are redirected to +the error trap higher up. If the algorithm completes without error the error code is set to \textbf{MP\_OKAY} and mp\_clears are executed normally. + +\textit{Last paragraph sucks. re-write! -- Tom} + +\subsection{Toom-Cook Squaring} +The Toom-Cook squaring algorithm mp\_toom\_sqr is heavily based on the algorithm mp\_toom\_mul with the exception that squarings are used +instead of multiplication to find the five relations.. The reader is encouraged to read the description of the latter algorithm and try to +derive their own Toom-Cook squaring algorithm. + +\subsection{High Level Squaring} +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_sqr}. \\ +\textbf{Input}. mp\_int $a$ \\ +\textbf{Output}. $b \leftarrow a^2$ \\ +\hline \\ +1. If $a.used \ge TOOM\_SQR\_CUTOFF$ then \\ +\hspace{3mm}1.1 $b \leftarrow a^2$ using algorithm mp\_toom\_sqr \\ +2. else if $a.used \ge KARATSUBA\_SQR\_CUTOFF$ then \\ +\hspace{3mm}2.1 $b \leftarrow a^2$ using algorithm mp\_karatsuba\_sqr \\ +3. else \\ +\hspace{3mm}3.1 $digs \leftarrow a.used + b.used + 1$ \\ +\hspace{3mm}3.2 If $digs < MP\_ARRAY$ and $a.used \le \delta$ then \\ +\hspace{6mm}3.2.1 $b \leftarrow a^2$ using algorithm fast\_s\_mp\_sqr. \\ +\hspace{3mm}3.3 else \\ +\hspace{6mm}3.3.1 $b \leftarrow a^2$ using algorithm s\_mp\_sqr. \\ +4. $b.sign \leftarrow MP\_ZPOS$ \\ +5. Return the result of the unsigned squaring performed. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_sqr} +\end{figure} + +\textbf{Algorithm mp\_sqr.} +This algorithm computes the square of the input using one of four different algorithms. If the input is very large and has at least +\textbf{TOOM\_SQR\_CUTOFF} or \textbf{KARATSUBA\_SQR\_CUTOFF} digits then either the Toom-Cook or the Karatsuba Squaring algorithm is used. If +neither of the polynomial basis algorithms should be used then either the Comba or baseline algorithm is used. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_sqr.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* computes b = a*a */ +018 int +019 mp_sqr (mp_int * a, mp_int * b) +020 \{ +021 int res; +022 +023 #ifdef BN_MP_TOOM_SQR_C +024 /* use Toom-Cook? */ +025 if (a->used >= TOOM_SQR_CUTOFF) \{ +026 res = mp_toom_sqr(a, b); +027 /* Karatsuba? */ +028 \} else +029 #endif +030 #ifdef BN_MP_KARATSUBA_SQR_C +031 if (a->used >= KARATSUBA_SQR_CUTOFF) \{ +032 res = mp_karatsuba_sqr (a, b); +033 \} else +034 #endif +035 \{ +036 #ifdef BN_FAST_S_MP_SQR_C +037 /* can we use the fast comba multiplier? */ +038 if ((a->used * 2 + 1) < MP_WARRAY && +039 a->used < +040 (1 << (sizeof(mp_word) * CHAR_BIT - 2*DIGIT_BIT - 1))) \{ +041 res = fast_s_mp_sqr (a, b); +042 \} else +043 #endif +044 #ifdef BN_S_MP_SQR_C +045 res = s_mp_sqr (a, b); +046 #else +047 res = MP_VAL; +048 #endif +049 \} +050 b->sign = MP_ZPOS; +051 return res; +052 \} +053 #endif +\end{alltt} +\end{small} + +\section*{Exercises} +\begin{tabular}{cl} +$\left [ 3 \right ] $ & Devise an efficient algorithm for selection of the radix point to handle inputs \\ + & that have different number of digits in Karatsuba multiplication. \\ + & \\ +$\left [ 3 \right ] $ & In section 5.3 the fact that every column of a squaring is made up \\ + & of double products and at most one square is stated. Prove this statement. \\ + & \\ +$\left [ 2 \right ] $ & In the Comba squaring algorithm half of the $\hat X$ variables are not used. \\ + & Revise algorithm fast\_s\_mp\_sqr to shrink the $\hat X$ array. \\ + & \\ +$\left [ 3 \right ] $ & Prove the equation for Karatsuba squaring. \\ + & \\ +$\left [ 1 \right ] $ & Prove that Karatsuba squaring requires $O \left (n^{lg(3)} \right )$ time. \\ + & \\ +$\left [ 2 \right ] $ & Determine the minimal ratio between addition and multiplication clock cycles \\ + & required for equation $6.7$ to be true. \\ + & \\ +\end{tabular} + +\chapter{Modular Reduction} +\section{Basics of Modular Reduction} +\index{modular residue} +Modular reduction is an operation that arises quite often within public key cryptography algorithms and various number theoretic algorithms, +such as factoring. Modular reduction algorithms are the third class of algorithms of the ``multipliers'' set. A number $a$ is said to be \textit{reduced} +modulo another number $b$ by finding the remainder of the division $a/b$. Full integer division with remainder is a topic to be covered +in~\ref{sec:division}. + +Modular reduction is equivalent to solving for $r$ in the following equation. $a = bq + r$ where $q = \lfloor a/b \rfloor$. The result +$r$ is said to be ``congruent to $a$ modulo $b$'' which is also written as $r \equiv a \mbox{ (mod }b\mbox{)}$. In other vernacular $r$ is known as the +``modular residue'' which leads to ``quadratic residue''\footnote{That's fancy talk for $b \equiv a^2 \mbox{ (mod }p\mbox{)}$.} and +other forms of residues. + +Modular reductions are normally used to create either finite groups, rings or fields. The most common usage for performance driven modular reductions +is in modular exponentiation algorithms. That is to compute $d = a^b \mbox{ (mod }c\mbox{)}$ as fast as possible. This operation is used in the +RSA and Diffie-Hellman public key algorithms, for example. Modular multiplication and squaring also appears as a fundamental operation in +Elliptic Curve cryptographic algorithms. As will be discussed in the subsequent chapter there exist fast algorithms for computing modular +exponentiations without having to perform (\textit{in this example}) $b - 1$ multiplications. These algorithms will produce partial results in the +range $0 \le x < c^2$ which can be taken advantage of to create several efficient algorithms. They have also been used to create redundancy check +algorithms known as CRCs, error correction codes such as Reed-Solomon and solve a variety of number theoeretic problems. + +\section{The Barrett Reduction} +The Barrett reduction algorithm \cite{BARRETT} was inspired by fast division algorithms which multiply by the reciprocal to emulate +division. Barretts observation was that the residue $c$ of $a$ modulo $b$ is equal to + +\begin{equation} +c = a - b \cdot \lfloor a/b \rfloor +\end{equation} + +Since algorithms such as modular exponentiation would be using the same modulus extensively, typical DSP\footnote{It is worth noting that Barrett's paper +targeted the DSP56K processor.} intuition would indicate the next step would be to replace $a/b$ by a multiplication by the reciprocal. However, +DSP intuition on its own will not work as these numbers are considerably larger than the precision of common DSP floating point data types. +It would take another common optimization to optimize the algorithm. + +\subsection{Fixed Point Arithmetic} +The trick used to optimize the above equation is based on a technique of emulating floating point data types with fixed precision integers. Fixed +point arithmetic would become very popular as it greatly optimize the ``3d-shooter'' genre of games in the mid 1990s when floating point units were +fairly slow if not unavailable. The idea behind fixed point arithmetic is to take a normal $k$-bit integer data type and break it into $p$-bit +integer and a $q$-bit fraction part (\textit{where $p+q = k$}). + +In this system a $k$-bit integer $n$ would actually represent $n/2^q$. For example, with $q = 4$ the integer $n = 37$ would actually represent the +value $2.3125$. To multiply two fixed point numbers the integers are multiplied using traditional arithmetic and subsequently normalized by +moving the implied decimal point back to where it should be. For example, with $q = 4$ to multiply the integers $9$ and $5$ they must be converted +to fixed point first by multiplying by $2^q$. Let $a = 9(2^q)$ represent the fixed point representation of $9$ and $b = 5(2^q)$ represent the +fixed point representation of $5$. The product $ab$ is equal to $45(2^{2q})$ which when normalized by dividing by $2^q$ produces $45(2^q)$. + +This technique became popular since a normal integer multiplication and logical shift right are the only required operations to perform a multiplication +of two fixed point numbers. Using fixed point arithmetic, division can be easily approximated by multiplying by the reciprocal. If $2^q$ is +equivalent to one than $2^q/b$ is equivalent to the fixed point approximation of $1/b$ using real arithmetic. Using this fact dividing an integer +$a$ by another integer $b$ can be achieved with the following expression. + +\begin{equation} +\lfloor a / b \rfloor \mbox{ }\approx\mbox{ } \lfloor (a \cdot \lfloor 2^q / b \rfloor)/2^q \rfloor +\end{equation} + +The precision of the division is proportional to the value of $q$. If the divisor $b$ is used frequently as is the case with +modular exponentiation pre-computing $2^q/b$ will allow a division to be performed with a multiplication and a right shift. Both operations +are considerably faster than division on most processors. + +Consider dividing $19$ by $5$. The correct result is $\lfloor 19/5 \rfloor = 3$. With $q = 3$ the reciprocal is $\lfloor 2^q/5 \rfloor = 1$ which +leads to a product of $19$ which when divided by $2^q$ produces $2$. However, with $q = 4$ the reciprocal is $\lfloor 2^q/5 \rfloor = 3$ and +the result of the emulated division is $\lfloor 3 \cdot 19 / 2^q \rfloor = 3$ which is correct. The value of $2^q$ must be close to or ideally +larger than the dividend. In effect if $a$ is the dividend then $q$ should allow $0 \le \lfloor a/2^q \rfloor \le 1$ in order for this approach +to work correctly. Plugging this form of divison into the original equation the following modular residue equation arises. + +\begin{equation} +c = a - b \cdot \lfloor (a \cdot \lfloor 2^q / b \rfloor)/2^q \rfloor +\end{equation} + +Using the notation from \cite{BARRETT} the value of $\lfloor 2^q / b \rfloor$ will be represented by the $\mu$ symbol. Using the $\mu$ +variable also helps re-inforce the idea that it is meant to be computed once and re-used. + +\begin{equation} +c = a - b \cdot \lfloor (a \cdot \mu)/2^q \rfloor +\end{equation} + +Provided that $2^q \ge a$ this algorithm will produce a quotient that is either exactly correct or off by a value of one. In the context of Barrett +reduction the value of $a$ is bound by $0 \le a \le (b - 1)^2$ meaning that $2^q \ge b^2$ is sufficient to ensure the reciprocal will have enough +precision. + +Let $n$ represent the number of digits in $b$. This algorithm requires approximately $2n^2$ single precision multiplications to produce the quotient and +another $n^2$ single precision multiplications to find the residue. In total $3n^2$ single precision multiplications are required to +reduce the number. + +For example, if $b = 1179677$ and $q = 41$ ($2^q > b^2$), then the reciprocal $\mu$ is equal to $\lfloor 2^q / b \rfloor = 1864089$. Consider reducing +$a = 180388626447$ modulo $b$ using the above reduction equation. The quotient using the new formula is $\lfloor (a \cdot \mu) / 2^q \rfloor = 152913$. +By subtracting $152913b$ from $a$ the correct residue $a \equiv 677346 \mbox{ (mod }b\mbox{)}$ is found. + +\subsection{Choosing a Radix Point} +Using the fixed point representation a modular reduction can be performed with $3n^2$ single precision multiplications. If that were the best +that could be achieved a full division\footnote{A division requires approximately $O(2cn^2)$ single precision multiplications for a small value of $c$. +See~\ref{sec:division} for further details.} might as well be used in its place. The key to optimizing the reduction is to reduce the precision of +the initial multiplication that finds the quotient. + +Let $a$ represent the number of which the residue is sought. Let $b$ represent the modulus used to find the residue. Let $m$ represent +the number of digits in $b$. For the purposes of this discussion we will assume that the number of digits in $a$ is $2m$, which is generally true if +two $m$-digit numbers have been multiplied. Dividing $a$ by $b$ is the same as dividing a $2m$ digit integer by a $m$ digit integer. Digits below the +$m - 1$'th digit of $a$ will contribute at most a value of $1$ to the quotient because $\beta^k < b$ for any $0 \le k \le m - 1$. Another way to +express this is by re-writing $a$ as two parts. If $a' \equiv a \mbox{ (mod }b^m\mbox{)}$ and $a'' = a - a'$ then +${a \over b} \equiv {{a' + a''} \over b}$ which is equivalent to ${a' \over b} + {a'' \over b}$. Since $a'$ is bound to be less than $b$ the quotient +is bound by $0 \le {a' \over b} < 1$. + +Since the digits of $a'$ do not contribute much to the quotient the observation is that they might as well be zero. However, if the digits +``might as well be zero'' they might as well not be there in the first place. Let $q_0 = \lfloor a/\beta^{m-1} \rfloor$ represent the input +with the irrelevant digits trimmed. Now the modular reduction is trimmed to the almost equivalent equation + +\begin{equation} +c = a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor +\end{equation} + +Note that the original divisor $2^q$ has been replaced with $\beta^{m+1}$ where in this case $q$ is a multiple of $lg(\beta)$. Also note that the +exponent on the divisor when added to the amount $q_0$ was shifted by equals $2m$. If the optimization had not been performed the divisor +would have the exponent $2m$ so in the end the exponents do ``add up''. Using the above equation the quotient +$\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ can be off from the true quotient by at most two. The original fixed point quotient can be off +by as much as one (\textit{provided the radix point is chosen suitably}) and now that the lower irrelevent digits have been trimmed the quotient +can be off by an additional value of one for a total of at most two. This implies that +$0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$. By first subtracting $b$ times the quotient and then conditionally subtracting +$b$ once or twice the residue is found. + +The quotient is now found using $(m + 1)(m) = m^2 + m$ single precision multiplications and the residue with an additional $m^2$ single +precision multiplications, ignoring the subtractions required. In total $2m^2 + m$ single precision multiplications are required to find the residue. +This is considerably faster than the original attempt. + +For example, let $\beta = 10$ represent the radix of the digits. Let $b = 9999$ represent the modulus which implies $m = 4$. Let $a = 99929878$ +represent the value of which the residue is desired. In this case $q = 8$ since $10^7 < 9999^2$ meaning that $\mu = \lfloor \beta^{q}/b \rfloor = 10001$. +With the new observation the multiplicand for the quotient is equal to $q_0 = \lfloor a / \beta^{m - 1} \rfloor = 99929$. The quotient is then +$\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor = 9993$. Subtracting $9993b$ from $a$ and the correct residue $a \equiv 9871 \mbox{ (mod }b\mbox{)}$ +is found. + +\subsection{Trimming the Quotient} +So far the reduction algorithm has been optimized from $3m^2$ single precision multiplications down to $2m^2 + m$ single precision multiplications. As +it stands now the algorithm is already fairly fast compared to a full integer division algorithm. However, there is still room for +optimization. + +After the first multiplication inside the quotient ($q_0 \cdot \mu$) the value is shifted right by $m + 1$ places effectively nullifying the lower +half of the product. It would be nice to be able to remove those digits from the product to effectively cut down the number of single precision +multiplications. If the number of digits in the modulus $m$ is far less than $\beta$ a full product is not required for the algorithm to work properly. +In fact the lower $m - 2$ digits will not affect the upper half of the product at all and do not need to be computed. + +The value of $\mu$ is a $m$-digit number and $q_0$ is a $m + 1$ digit number. Using a full multiplier $(m + 1)(m) = m^2 + m$ single precision +multiplications would be required. Using a multiplier that will only produce digits at and above the $m - 1$'th digit reduces the number +of single precision multiplications to ${m^2 + m} \over 2$ single precision multiplications. + +\subsection{Trimming the Residue} +After the quotient has been calculated it is used to reduce the input. As previously noted the algorithm is not exact and it can be off by a small +multiple of the modulus, that is $0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$. If $b$ is $m$ digits than the +result of reduction equation is a value of at most $m + 1$ digits (\textit{provided $3 < \beta$}) implying that the upper $m - 1$ digits are +implicitly zero. + +The next optimization arises from this very fact. Instead of computing $b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ using a full +$O(m^2)$ multiplication algorithm only the lower $m+1$ digits of the product have to be computed. Similarly the value of $a$ can +be reduced modulo $\beta^{m+1}$ before the multiple of $b$ is subtracted which simplifes the subtraction as well. A multiplication that produces +only the lower $m+1$ digits requires ${m^2 + 3m - 2} \over 2$ single precision multiplications. + +With both optimizations in place the algorithm is the algorithm Barrett proposed. It requires $m^2 + 2m - 1$ single precision multiplications which +is considerably faster than the straightforward $3m^2$ method. + +\subsection{The Barrett Algorithm} +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_reduce}. \\ +\textbf{Input}. mp\_int $a$, mp\_int $b$ and $\mu = \lfloor \beta^{2m}/b \rfloor, m = \lceil lg_{\beta}(b) \rceil, (0 \le a < b^2, b > 1)$ \\ +\textbf{Output}. $a \mbox{ (mod }b\mbox{)}$ \\ +\hline \\ +Let $m$ represent the number of digits in $b$. \\ +1. Make a copy of $a$ and store it in $q$. (\textit{mp\_init\_copy}) \\ +2. $q \leftarrow \lfloor q / \beta^{m - 1} \rfloor$ (\textit{mp\_rshd}) \\ +\\ +Produce the quotient. \\ +3. $q \leftarrow q \cdot \mu$ (\textit{note: only produce digits at or above $m-1$}) \\ +4. $q \leftarrow \lfloor q / \beta^{m + 1} \rfloor$ \\ +\\ +Subtract the multiple of modulus from the input. \\ +5. $a \leftarrow a \mbox{ (mod }\beta^{m+1}\mbox{)}$ (\textit{mp\_mod\_2d}) \\ +6. $q \leftarrow q \cdot b \mbox{ (mod }\beta^{m+1}\mbox{)}$ (\textit{s\_mp\_mul\_digs}) \\ +7. $a \leftarrow a - q$ (\textit{mp\_sub}) \\ +\\ +Add $\beta^{m+1}$ if a carry occured. \\ +8. If $a < 0$ then (\textit{mp\_cmp\_d}) \\ +\hspace{3mm}8.1 $q \leftarrow 1$ (\textit{mp\_set}) \\ +\hspace{3mm}8.2 $q \leftarrow q \cdot \beta^{m+1}$ (\textit{mp\_lshd}) \\ +\hspace{3mm}8.3 $a \leftarrow a + q$ \\ +\\ +Now subtract the modulus if the residue is too large (e.g. quotient too small). \\ +9. While $a \ge b$ do (\textit{mp\_cmp}) \\ +\hspace{3mm}9.1 $c \leftarrow a - b$ \\ +10. Clear $q$. \\ +11. Return(\textit{MP\_OKAY}) \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_reduce} +\end{figure} + +\textbf{Algorithm mp\_reduce.} +This algorithm will reduce the input $a$ modulo $b$ in place using the Barrett algorithm. It is loosely based on algorithm 14.42 of HAC +\cite[pp. 602]{HAC} which is based on the paper from Paul Barrett \cite{BARRETT}. The algorithm has several restrictions and assumptions which must +be adhered to for the algorithm to work. + +First the modulus $b$ is assumed to be positive and greater than one. If the modulus were less than or equal to one than subtracting +a multiple of it would either accomplish nothing or actually enlarge the input. The input $a$ must be in the range $0 \le a < b^2$ in order +for the quotient to have enough precision. If $a$ is the product of two numbers that were already reduced modulo $b$, this will not be a problem. +Technically the algorithm will still work if $a \ge b^2$ but it will take much longer to finish. The value of $\mu$ is passed as an argument to this +algorithm and is assumed to be calculated and stored before the algorithm is used. + +Recall that the multiplication for the quotient on step 3 must only produce digits at or above the $m-1$'th position. An algorithm called +$s\_mp\_mul\_high\_digs$ which has not been presented is used to accomplish this task. The algorithm is based on $s\_mp\_mul\_digs$ except that +instead of stopping at a given level of precision it starts at a given level of precision. This optimal algorithm can only be used if the number +of digits in $b$ is very much smaller than $\beta$. + +While it is known that +$a \ge b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ only the lower $m+1$ digits are being used to compute the residue, so an implied +``borrow'' from the higher digits might leave a negative result. After the multiple of the modulus has been subtracted from $a$ the residue must be +fixed up in case it is negative. The invariant $\beta^{m+1}$ must be added to the residue to make it positive again. + +The while loop at step 9 will subtract $b$ until the residue is less than $b$. If the algorithm is performed correctly this step is +performed at most twice, and on average once. However, if $a \ge b^2$ than it will iterate substantially more times than it should. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* reduces x mod m, assumes 0 < x < m**2, mu is +018 * precomputed via mp_reduce_setup. +019 * From HAC pp.604 Algorithm 14.42 +020 */ +021 int +022 mp_reduce (mp_int * x, mp_int * m, mp_int * mu) +023 \{ +024 mp_int q; +025 int res, um = m->used; +026 +027 /* q = x */ +028 if ((res = mp_init_copy (&q, x)) != MP_OKAY) \{ +029 return res; +030 \} +031 +032 /* q1 = x / b**(k-1) */ +033 mp_rshd (&q, um - 1); +034 +035 /* according to HAC this optimization is ok */ +036 if (((unsigned long) um) > (((mp_digit)1) << (DIGIT_BIT - 1))) \{ +037 if ((res = mp_mul (&q, mu, &q)) != MP_OKAY) \{ +038 goto CLEANUP; +039 \} +040 \} else \{ +041 #ifdef BN_S_MP_MUL_HIGH_DIGS_C +042 if ((res = s_mp_mul_high_digs (&q, mu, &q, um - 1)) != MP_OKAY) \{ +043 goto CLEANUP; +044 \} +045 #elif defined(BN_FAST_S_MP_MUL_HIGH_DIGS_C) +046 if ((res = fast_s_mp_mul_high_digs (&q, mu, &q, um - 1)) != MP_OKAY) \{ +047 goto CLEANUP; +048 \} +049 #else +050 \{ +051 res = MP_VAL; +052 goto CLEANUP; +053 \} +054 #endif +055 \} +056 +057 /* q3 = q2 / b**(k+1) */ +058 mp_rshd (&q, um + 1); +059 +060 /* x = x mod b**(k+1), quick (no division) */ +061 if ((res = mp_mod_2d (x, DIGIT_BIT * (um + 1), x)) != MP_OKAY) \{ +062 goto CLEANUP; +063 \} +064 +065 /* q = q * m mod b**(k+1), quick (no division) */ +066 if ((res = s_mp_mul_digs (&q, m, &q, um + 1)) != MP_OKAY) \{ +067 goto CLEANUP; +068 \} +069 +070 /* x = x - q */ +071 if ((res = mp_sub (x, &q, x)) != MP_OKAY) \{ +072 goto CLEANUP; +073 \} +074 +075 /* If x < 0, add b**(k+1) to it */ +076 if (mp_cmp_d (x, 0) == MP_LT) \{ +077 mp_set (&q, 1); +078 if ((res = mp_lshd (&q, um + 1)) != MP_OKAY) +079 goto CLEANUP; +080 if ((res = mp_add (x, &q, x)) != MP_OKAY) +081 goto CLEANUP; +082 \} +083 +084 /* Back off if it's too big */ +085 while (mp_cmp (x, m) != MP_LT) \{ +086 if ((res = s_mp_sub (x, m, x)) != MP_OKAY) \{ +087 goto CLEANUP; +088 \} +089 \} +090 +091 CLEANUP: +092 mp_clear (&q); +093 +094 return res; +095 \} +096 #endif +\end{alltt} +\end{small} + +The first multiplication that determines the quotient can be performed by only producing the digits from $m - 1$ and up. This essentially halves +the number of single precision multiplications required. However, the optimization is only safe if $\beta$ is much larger than the number of digits +in the modulus. In the source code this is evaluated on lines 36 to 44 where algorithm s\_mp\_mul\_high\_digs is used when it is +safe to do so. + +\subsection{The Barrett Setup Algorithm} +In order to use algorithm mp\_reduce the value of $\mu$ must be calculated in advance. Ideally this value should be computed once and stored for +future use so that the Barrett algorithm can be used without delay. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_reduce\_setup}. \\ +\textbf{Input}. mp\_int $a$ ($a > 1$) \\ +\textbf{Output}. $\mu \leftarrow \lfloor \beta^{2m}/a \rfloor$ \\ +\hline \\ +1. $\mu \leftarrow 2^{2 \cdot lg(\beta) \cdot m}$ (\textit{mp\_2expt}) \\ +2. $\mu \leftarrow \lfloor \mu / b \rfloor$ (\textit{mp\_div}) \\ +3. Return(\textit{MP\_OKAY}) \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_reduce\_setup} +\end{figure} + +\textbf{Algorithm mp\_reduce\_setup.} +This algorithm computes the reciprocal $\mu$ required for Barrett reduction. First $\beta^{2m}$ is calculated as $2^{2 \cdot lg(\beta) \cdot m}$ which +is equivalent and much faster. The final value is computed by taking the integer quotient of $\lfloor \mu / b \rfloor$. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_setup.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* pre-calculate the value required for Barrett reduction +018 * For a given modulus "b" it calulates the value required in "a" +019 */ +020 int mp_reduce_setup (mp_int * a, mp_int * b) +021 \{ +022 int res; +023 +024 if ((res = mp_2expt (a, b->used * 2 * DIGIT_BIT)) != MP_OKAY) \{ +025 return res; +026 \} +027 return mp_div (a, b, a, NULL); +028 \} +029 #endif +\end{alltt} +\end{small} + +This simple routine calculates the reciprocal $\mu$ required by Barrett reduction. Note the extended usage of algorithm mp\_div where the variable +which would received the remainder is passed as NULL. As will be discussed in~\ref{sec:division} the division routine allows both the quotient and the +remainder to be passed as NULL meaning to ignore the value. + +\section{The Montgomery Reduction} +Montgomery reduction\footnote{Thanks to Niels Ferguson for his insightful explanation of the algorithm.} \cite{MONT} is by far the most interesting +form of reduction in common use. It computes a modular residue which is not actually equal to the residue of the input yet instead equal to a +residue times a constant. However, as perplexing as this may sound the algorithm is relatively simple and very efficient. + +Throughout this entire section the variable $n$ will represent the modulus used to form the residue. As will be discussed shortly the value of +$n$ must be odd. The variable $x$ will represent the quantity of which the residue is sought. Similar to the Barrett algorithm the input +is restricted to $0 \le x < n^2$. To begin the description some simple number theory facts must be established. + +\textbf{Fact 1.} Adding $n$ to $x$ does not change the residue since in effect it adds one to the quotient $\lfloor x / n \rfloor$. Another way +to explain this is that $n$ is (\textit{or multiples of $n$ are}) congruent to zero modulo $n$. Adding zero will not change the value of the residue. + +\textbf{Fact 2.} If $x$ is even then performing a division by two in $\Z$ is congruent to $x \cdot 2^{-1} \mbox{ (mod }n\mbox{)}$. Actually +this is an application of the fact that if $x$ is evenly divisible by any $k \in \Z$ then division in $\Z$ will be congruent to +multiplication by $k^{-1}$ modulo $n$. + +From these two simple facts the following simple algorithm can be derived. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{Montgomery Reduction}. \\ +\textbf{Input}. Integer $x$, $n$ and $k$ \\ +\textbf{Output}. $2^{-k}x \mbox{ (mod }n\mbox{)}$ \\ +\hline \\ +1. for $t$ from $1$ to $k$ do \\ +\hspace{3mm}1.1 If $x$ is odd then \\ +\hspace{6mm}1.1.1 $x \leftarrow x + n$ \\ +\hspace{3mm}1.2 $x \leftarrow x/2$ \\ +2. Return $x$. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm Montgomery Reduction} +\end{figure} + +The algorithm reduces the input one bit at a time using the two congruencies stated previously. Inside the loop $n$, which is odd, is +added to $x$ if $x$ is odd. This forces $x$ to be even which allows the division by two in $\Z$ to be congruent to a modular division by two. Since +$x$ is assumed to be initially much larger than $n$ the addition of $n$ will contribute an insignificant magnitude to $x$. Let $r$ represent the +final result of the Montgomery algorithm. If $k > lg(n)$ and $0 \le x < n^2$ then the final result is limited to +$0 \le r < \lfloor x/2^k \rfloor + n$. As a result at most a single subtraction is required to get the residue desired. + +\begin{figure}[here] +\begin{small} +\begin{center} +\begin{tabular}{|c|l|} +\hline \textbf{Step number ($t$)} & \textbf{Result ($x$)} \\ +\hline $1$ & $x + n = 5812$, $x/2 = 2906$ \\ +\hline $2$ & $x/2 = 1453$ \\ +\hline $3$ & $x + n = 1710$, $x/2 = 855$ \\ +\hline $4$ & $x + n = 1112$, $x/2 = 556$ \\ +\hline $5$ & $x/2 = 278$ \\ +\hline $6$ & $x/2 = 139$ \\ +\hline $7$ & $x + n = 396$, $x/2 = 198$ \\ +\hline $8$ & $x/2 = 99$ \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Example of Montgomery Reduction (I)} +\label{fig:MONT1} +\end{figure} + +Consider the example in figure~\ref{fig:MONT1} which reduces $x = 5555$ modulo $n = 257$ when $k = 8$. The result of the algorithm $r = 99$ is +congruent to the value of $2^{-8} \cdot 5555 \mbox{ (mod }257\mbox{)}$. When $r$ is multiplied by $2^8$ modulo $257$ the correct residue +$r \equiv 158$ is produced. + +Let $k = \lfloor lg(n) \rfloor + 1$ represent the number of bits in $n$. The current algorithm requires $2k^2$ single precision shifts +and $k^2$ single precision additions. At this rate the algorithm is most certainly slower than Barrett reduction and not terribly useful. +Fortunately there exists an alternative representation of the algorithm. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{Montgomery Reduction} (modified I). \\ +\textbf{Input}. Integer $x$, $n$ and $k$ \\ +\textbf{Output}. $2^{-k}x \mbox{ (mod }n\mbox{)}$ \\ +\hline \\ +1. for $t$ from $0$ to $k - 1$ do \\ +\hspace{3mm}1.1 If the $t$'th bit of $x$ is one then \\ +\hspace{6mm}1.1.1 $x \leftarrow x + 2^tn$ \\ +2. Return $x/2^k$. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm Montgomery Reduction (modified I)} +\end{figure} + +This algorithm is equivalent since $2^tn$ is a multiple of $n$ and the lower $k$ bits of $x$ are zero by step 2. The number of single +precision shifts has now been reduced from $2k^2$ to $k^2 + k$ which is only a small improvement. + +\begin{figure}[here] +\begin{small} +\begin{center} +\begin{tabular}{|c|l|r|} +\hline \textbf{Step number ($t$)} & \textbf{Result ($x$)} & \textbf{Result ($x$) in Binary} \\ +\hline -- & $5555$ & $1010110110011$ \\ +\hline $1$ & $x + 2^{0}n = 5812$ & $1011010110100$ \\ +\hline $2$ & $5812$ & $1011010110100$ \\ +\hline $3$ & $x + 2^{2}n = 6840$ & $1101010111000$ \\ +\hline $4$ & $x + 2^{3}n = 8896$ & $10001011000000$ \\ +\hline $5$ & $8896$ & $10001011000000$ \\ +\hline $6$ & $8896$ & $10001011000000$ \\ +\hline $7$ & $x + 2^{6}n = 25344$ & $110001100000000$ \\ +\hline $8$ & $25344$ & $110001100000000$ \\ +\hline -- & $x/2^k = 99$ & \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Example of Montgomery Reduction (II)} +\label{fig:MONT2} +\end{figure} + +Figure~\ref{fig:MONT2} demonstrates the modified algorithm reducing $x = 5555$ modulo $n = 257$ with $k = 8$. +With this algorithm a single shift right at the end is the only right shift required to reduce the input instead of $k$ right shifts inside the +loop. Note that for the iterations $t = 2, 5, 6$ and $8$ where the result $x$ is not changed. In those iterations the $t$'th bit of $x$ is +zero and the appropriate multiple of $n$ does not need to be added to force the $t$'th bit of the result to zero. + +\subsection{Digit Based Montgomery Reduction} +Instead of computing the reduction on a bit-by-bit basis it is actually much faster to compute it on digit-by-digit basis. Consider the +previous algorithm re-written to compute the Montgomery reduction in this new fashion. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{Montgomery Reduction} (modified II). \\ +\textbf{Input}. Integer $x$, $n$ and $k$ \\ +\textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\ +\hline \\ +1. for $t$ from $0$ to $k - 1$ do \\ +\hspace{3mm}1.1 $x \leftarrow x + \mu n \beta^t$ \\ +2. Return $x/\beta^k$. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm Montgomery Reduction (modified II)} +\end{figure} + +The value $\mu n \beta^t$ is a multiple of the modulus $n$ meaning that it will not change the residue. If the first digit of +the value $\mu n \beta^t$ equals the negative (modulo $\beta$) of the $t$'th digit of $x$ then the addition will result in a zero digit. This +problem breaks down to solving the following congruency. + +\begin{center} +\begin{tabular}{rcl} +$x_t + \mu n_0$ & $\equiv$ & $0 \mbox{ (mod }\beta\mbox{)}$ \\ +$\mu n_0$ & $\equiv$ & $-x_t \mbox{ (mod }\beta\mbox{)}$ \\ +$\mu$ & $\equiv$ & $-x_t/n_0 \mbox{ (mod }\beta\mbox{)}$ \\ +\end{tabular} +\end{center} + +In each iteration of the loop on step 1 a new value of $\mu$ must be calculated. The value of $-1/n_0 \mbox{ (mod }\beta\mbox{)}$ is used +extensively in this algorithm and should be precomputed. Let $\rho$ represent the negative of the modular inverse of $n_0$ modulo $\beta$. + +For example, let $\beta = 10$ represent the radix. Let $n = 17$ represent the modulus which implies $k = 2$ and $\rho \equiv 7$. Let $x = 33$ +represent the value to reduce. + +\newpage\begin{figure} +\begin{center} +\begin{tabular}{|c|c|c|} +\hline \textbf{Step ($t$)} & \textbf{Value of $x$} & \textbf{Value of $\mu$} \\ +\hline -- & $33$ & --\\ +\hline $0$ & $33 + \mu n = 50$ & $1$ \\ +\hline $1$ & $50 + \mu n \beta = 900$ & $5$ \\ +\hline +\end{tabular} +\end{center} +\caption{Example of Montgomery Reduction} +\end{figure} + +The final result $900$ is then divided by $\beta^k$ to produce the final result $9$. The first observation is that $9 \nequiv x \mbox{ (mod }n\mbox{)}$ +which implies the result is not the modular residue of $x$ modulo $n$. However, recall that the residue is actually multiplied by $\beta^{-k}$ in +the algorithm. To get the true residue the value must be multiplied by $\beta^k$. In this case $\beta^k \equiv 15 \mbox{ (mod }n\mbox{)}$ and +the correct residue is $9 \cdot 15 \equiv 16 \mbox{ (mod }n\mbox{)}$. + +\subsection{Baseline Montgomery Reduction} +The baseline Montgomery reduction algorithm will produce the residue for any size input. It is designed to be a catch-all algororithm for +Montgomery reductions. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_montgomery\_reduce}. \\ +\textbf{Input}. mp\_int $x$, mp\_int $n$ and a digit $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\ +\hspace{11.5mm}($0 \le x < n^2, n > 1, (n, \beta) = 1, \beta^k > n$) \\ +\textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\ +\hline \\ +1. $digs \leftarrow 2n.used + 1$ \\ +2. If $digs < MP\_ARRAY$ and $m.used < \delta$ then \\ +\hspace{3mm}2.1 Use algorithm fast\_mp\_montgomery\_reduce instead. \\ +\\ +Setup $x$ for the reduction. \\ +3. If $x.alloc < digs$ then grow $x$ to $digs$ digits. \\ +4. $x.used \leftarrow digs$ \\ +\\ +Eliminate the lower $k$ digits. \\ +5. For $ix$ from $0$ to $k - 1$ do \\ +\hspace{3mm}5.1 $\mu \leftarrow x_{ix} \cdot \rho \mbox{ (mod }\beta\mbox{)}$ \\ +\hspace{3mm}5.2 $u \leftarrow 0$ \\ +\hspace{3mm}5.3 For $iy$ from $0$ to $k - 1$ do \\ +\hspace{6mm}5.3.1 $\hat r \leftarrow \mu n_{iy} + x_{ix + iy} + u$ \\ +\hspace{6mm}5.3.2 $x_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ +\hspace{6mm}5.3.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ +\hspace{3mm}5.4 While $u > 0$ do \\ +\hspace{6mm}5.4.1 $iy \leftarrow iy + 1$ \\ +\hspace{6mm}5.4.2 $x_{ix + iy} \leftarrow x_{ix + iy} + u$ \\ +\hspace{6mm}5.4.3 $u \leftarrow \lfloor x_{ix+iy} / \beta \rfloor$ \\ +\hspace{6mm}5.4.4 $x_{ix + iy} \leftarrow x_{ix+iy} \mbox{ (mod }\beta\mbox{)}$ \\ +\\ +Divide by $\beta^k$ and fix up as required. \\ +6. $x \leftarrow \lfloor x / \beta^k \rfloor$ \\ +7. If $x \ge n$ then \\ +\hspace{3mm}7.1 $x \leftarrow x - n$ \\ +8. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_montgomery\_reduce} +\end{figure} + +\textbf{Algorithm mp\_montgomery\_reduce.} +This algorithm reduces the input $x$ modulo $n$ in place using the Montgomery reduction algorithm. The algorithm is loosely based +on algorithm 14.32 of \cite[pp.601]{HAC} except it merges the multiplication of $\mu n \beta^t$ with the addition in the inner loop. The +restrictions on this algorithm are fairly easy to adapt to. First $0 \le x < n^2$ bounds the input to numbers in the same range as +for the Barrett algorithm. Additionally if $n > 1$ and $n$ is odd there will exist a modular inverse $\rho$. $\rho$ must be calculated in +advance of this algorithm. Finally the variable $k$ is fixed and a pseudonym for $n.used$. + +Step 2 decides whether a faster Montgomery algorithm can be used. It is based on the Comba technique meaning that there are limits on +the size of the input. This algorithm is discussed in sub-section 6.3.3. + +Step 5 is the main reduction loop of the algorithm. The value of $\mu$ is calculated once per iteration in the outer loop. The inner loop +calculates $x + \mu n \beta^{ix}$ by multiplying $\mu n$ and adding the result to $x$ shifted by $ix$ digits. Both the addition and +multiplication are performed in the same loop to save time and memory. Step 5.4 will handle any additional carries that escape the inner loop. + +Using a quick inspection this algorithm requires $n$ single precision multiplications for the outer loop and $n^2$ single precision multiplications +in the inner loop. In total $n^2 + n$ single precision multiplications which compares favourably to Barrett at $n^2 + 2n - 1$ single precision +multiplications. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_montgomery\_reduce.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* computes xR**-1 == x (mod N) via Montgomery Reduction */ +018 int +019 mp_montgomery_reduce (mp_int * x, mp_int * n, mp_digit rho) +020 \{ +021 int ix, res, digs; +022 mp_digit mu; +023 +024 /* can the fast reduction [comba] method be used? +025 * +026 * Note that unlike in mul you're safely allowed *less* +027 * than the available columns [255 per default] since carries +028 * are fixed up in the inner loop. +029 */ +030 digs = n->used * 2 + 1; +031 if ((digs < MP_WARRAY) && +032 n->used < +033 (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) \{ +034 return fast_mp_montgomery_reduce (x, n, rho); +035 \} +036 +037 /* grow the input as required */ +038 if (x->alloc < digs) \{ +039 if ((res = mp_grow (x, digs)) != MP_OKAY) \{ +040 return res; +041 \} +042 \} +043 x->used = digs; +044 +045 for (ix = 0; ix < n->used; ix++) \{ +046 /* mu = ai * rho mod b +047 * +048 * The value of rho must be precalculated via +049 * montgomery_setup() such that +050 * it equals -1/n0 mod b this allows the +051 * following inner loop to reduce the +052 * input one digit at a time +053 */ +054 mu = (mp_digit) (((mp_word)x->dp[ix]) * ((mp_word)rho) & MP_MASK); +055 +056 /* a = a + mu * m * b**i */ +057 \{ +058 register int iy; +059 register mp_digit *tmpn, *tmpx, u; +060 register mp_word r; +061 +062 /* alias for digits of the modulus */ +063 tmpn = n->dp; +064 +065 /* alias for the digits of x [the input] */ +066 tmpx = x->dp + ix; +067 +068 /* set the carry to zero */ +069 u = 0; +070 +071 /* Multiply and add in place */ +072 for (iy = 0; iy < n->used; iy++) \{ +073 /* compute product and sum */ +074 r = ((mp_word)mu) * ((mp_word)*tmpn++) + +075 ((mp_word) u) + ((mp_word) * tmpx); +076 +077 /* get carry */ +078 u = (mp_digit)(r >> ((mp_word) DIGIT_BIT)); +079 +080 /* fix digit */ +081 *tmpx++ = (mp_digit)(r & ((mp_word) MP_MASK)); +082 \} +083 /* At this point the ix'th digit of x should be zero */ +084 +085 +086 /* propagate carries upwards as required*/ +087 while (u) \{ +088 *tmpx += u; +089 u = *tmpx >> DIGIT_BIT; +090 *tmpx++ &= MP_MASK; +091 \} +092 \} +093 \} +094 +095 /* at this point the n.used'th least +096 * significant digits of x are all zero +097 * which means we can shift x to the +098 * right by n.used digits and the +099 * residue is unchanged. +100 */ +101 +102 /* x = x/b**n.used */ +103 mp_clamp(x); +104 mp_rshd (x, n->used); +105 +106 /* if x >= n then x = x - n */ +107 if (mp_cmp_mag (x, n) != MP_LT) \{ +108 return s_mp_sub (x, n, x); +109 \} +110 +111 return MP_OKAY; +112 \} +113 #endif +\end{alltt} +\end{small} + +This is the baseline implementation of the Montgomery reduction algorithm. Lines 30 to 35 determine if the Comba based +routine can be used instead. Line 48 computes the value of $\mu$ for that particular iteration of the outer loop. + +The multiplication $\mu n \beta^{ix}$ is performed in one step in the inner loop. The alias $tmpx$ refers to the $ix$'th digit of $x$ and +the alias $tmpn$ refers to the modulus $n$. + +\subsection{Faster ``Comba'' Montgomery Reduction} + +The Montgomery reduction requires fewer single precision multiplications than a Barrett reduction, however it is much slower due to the serial +nature of the inner loop. The Barrett reduction algorithm requires two slightly modified multipliers which can be implemented with the Comba +technique. The Montgomery reduction algorithm cannot directly use the Comba technique to any significant advantage since the inner loop calculates +a $k \times 1$ product $k$ times. + +The biggest obstacle is that at the $ix$'th iteration of the outer loop the value of $x_{ix}$ is required to calculate $\mu$. This means the +carries from $0$ to $ix - 1$ must have been propagated upwards to form a valid $ix$'th digit. The solution as it turns out is very simple. +Perform a Comba like multiplier and inside the outer loop just after the inner loop fix up the $ix + 1$'th digit by forwarding the carry. + +With this change in place the Montgomery reduction algorithm can be performed with a Comba style multiplication loop which substantially increases +the speed of the algorithm. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{fast\_mp\_montgomery\_reduce}. \\ +\textbf{Input}. mp\_int $x$, mp\_int $n$ and a digit $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\ +\hspace{11.5mm}($0 \le x < n^2, n > 1, (n, \beta) = 1, \beta^k > n$) \\ +\textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\ +\hline \\ +Place an array of \textbf{MP\_WARRAY} mp\_word variables called $\hat W$ on the stack. \\ +1. if $x.alloc < n.used + 1$ then grow $x$ to $n.used + 1$ digits. \\ +Copy the digits of $x$ into the array $\hat W$ \\ +2. For $ix$ from $0$ to $x.used - 1$ do \\ +\hspace{3mm}2.1 $\hat W_{ix} \leftarrow x_{ix}$ \\ +3. For $ix$ from $x.used$ to $2n.used - 1$ do \\ +\hspace{3mm}3.1 $\hat W_{ix} \leftarrow 0$ \\ +Elimiate the lower $k$ digits. \\ +4. for $ix$ from $0$ to $n.used - 1$ do \\ +\hspace{3mm}4.1 $\mu \leftarrow \hat W_{ix} \cdot \rho \mbox{ (mod }\beta\mbox{)}$ \\ +\hspace{3mm}4.2 For $iy$ from $0$ to $n.used - 1$ do \\ +\hspace{6mm}4.2.1 $\hat W_{iy + ix} \leftarrow \hat W_{iy + ix} + \mu \cdot n_{iy}$ \\ +\hspace{3mm}4.3 $\hat W_{ix + 1} \leftarrow \hat W_{ix + 1} + \lfloor \hat W_{ix} / \beta \rfloor$ \\ +Propagate carries upwards. \\ +5. for $ix$ from $n.used$ to $2n.used + 1$ do \\ +\hspace{3mm}5.1 $\hat W_{ix + 1} \leftarrow \hat W_{ix + 1} + \lfloor \hat W_{ix} / \beta \rfloor$ \\ +Shift right and reduce modulo $\beta$ simultaneously. \\ +6. for $ix$ from $0$ to $n.used + 1$ do \\ +\hspace{3mm}6.1 $x_{ix} \leftarrow \hat W_{ix + n.used} \mbox{ (mod }\beta\mbox{)}$ \\ +Zero excess digits and fixup $x$. \\ +7. if $x.used > n.used + 1$ then do \\ +\hspace{3mm}7.1 for $ix$ from $n.used + 1$ to $x.used - 1$ do \\ +\hspace{6mm}7.1.1 $x_{ix} \leftarrow 0$ \\ +8. $x.used \leftarrow n.used + 1$ \\ +9. Clamp excessive digits of $x$. \\ +10. If $x \ge n$ then \\ +\hspace{3mm}10.1 $x \leftarrow x - n$ \\ +11. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm fast\_mp\_montgomery\_reduce} +\end{figure} + +\textbf{Algorithm fast\_mp\_montgomery\_reduce.} +This algorithm will compute the Montgomery reduction of $x$ modulo $n$ using the Comba technique. It is on most computer platforms significantly +faster than algorithm mp\_montgomery\_reduce and algorithm mp\_reduce (\textit{Barrett reduction}). The algorithm has the same restrictions +on the input as the baseline reduction algorithm. An additional two restrictions are imposed on this algorithm. The number of digits $k$ in the +the modulus $n$ must not violate $MP\_WARRAY > 2k +1$ and $n < \delta$. When $\beta = 2^{28}$ this algorithm can be used to reduce modulo +a modulus of at most $3,556$ bits in length. + +As in the other Comba reduction algorithms there is a $\hat W$ array which stores the columns of the product. It is initially filled with the +contents of $x$ with the excess digits zeroed. The reduction loop is very similar the to the baseline loop at heart. The multiplication on step +4.1 can be single precision only since $ab \mbox{ (mod }\beta\mbox{)} \equiv (a \mbox{ mod }\beta)(b \mbox{ mod }\beta)$. Some multipliers such +as those on the ARM processors take a variable length time to complete depending on the number of bytes of result it must produce. By performing +a single precision multiplication instead half the amount of time is spent. + +Also note that digit $\hat W_{ix}$ must have the carry from the $ix - 1$'th digit propagated upwards in order for this to work. That is what step +4.3 will do. In effect over the $n.used$ iterations of the outer loop the $n.used$'th lower columns all have the their carries propagated forwards. Note +how the upper bits of those same words are not reduced modulo $\beta$. This is because those values will be discarded shortly and there is no +point. + +Step 5 will propagate the remainder of the carries upwards. On step 6 the columns are reduced modulo $\beta$ and shifted simultaneously as they are +stored in the destination $x$. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_fast\_mp\_montgomery\_reduce.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* computes xR**-1 == x (mod N) via Montgomery Reduction +018 * +019 * This is an optimized implementation of montgomery_reduce +020 * which uses the comba method to quickly calculate the columns of the +021 * reduction. +022 * +023 * Based on Algorithm 14.32 on pp.601 of HAC. +024 */ +025 int +026 fast_mp_montgomery_reduce (mp_int * x, mp_int * n, mp_digit rho) +027 \{ +028 int ix, res, olduse; +029 mp_word W[MP_WARRAY]; +030 +031 /* get old used count */ +032 olduse = x->used; +033 +034 /* grow a as required */ +035 if (x->alloc < n->used + 1) \{ +036 if ((res = mp_grow (x, n->used + 1)) != MP_OKAY) \{ +037 return res; +038 \} +039 \} +040 +041 /* first we have to get the digits of the input into +042 * an array of double precision words W[...] +043 */ +044 \{ +045 register mp_word *_W; +046 register mp_digit *tmpx; +047 +048 /* alias for the W[] array */ +049 _W = W; +050 +051 /* alias for the digits of x*/ +052 tmpx = x->dp; +053 +054 /* copy the digits of a into W[0..a->used-1] */ +055 for (ix = 0; ix < x->used; ix++) \{ +056 *_W++ = *tmpx++; +057 \} +058 +059 /* zero the high words of W[a->used..m->used*2] */ +060 for (; ix < n->used * 2 + 1; ix++) \{ +061 *_W++ = 0; +062 \} +063 \} +064 +065 /* now we proceed to zero successive digits +066 * from the least significant upwards +067 */ +068 for (ix = 0; ix < n->used; ix++) \{ +069 /* mu = ai * m' mod b +070 * +071 * We avoid a double precision multiplication (which isn't required) +072 * by casting the value down to a mp_digit. Note this requires +073 * that W[ix-1] have the carry cleared (see after the inner loop) +074 */ +075 register mp_digit mu; +076 mu = (mp_digit) (((W[ix] & MP_MASK) * rho) & MP_MASK); +077 +078 /* a = a + mu * m * b**i +079 * +080 * This is computed in place and on the fly. The multiplication +081 * by b**i is handled by offseting which columns the results +082 * are added to. +083 * +084 * Note the comba method normally doesn't handle carries in the +085 * inner loop In this case we fix the carry from the previous +086 * column since the Montgomery reduction requires digits of the +087 * result (so far) [see above] to work. This is +088 * handled by fixing up one carry after the inner loop. The +089 * carry fixups are done in order so after these loops the +090 * first m->used words of W[] have the carries fixed +091 */ +092 \{ +093 register int iy; +094 register mp_digit *tmpn; +095 register mp_word *_W; +096 +097 /* alias for the digits of the modulus */ +098 tmpn = n->dp; +099 +100 /* Alias for the columns set by an offset of ix */ +101 _W = W + ix; +102 +103 /* inner loop */ +104 for (iy = 0; iy < n->used; iy++) \{ +105 *_W++ += ((mp_word)mu) * ((mp_word)*tmpn++); +106 \} +107 \} +108 +109 /* now fix carry for next digit, W[ix+1] */ +110 W[ix + 1] += W[ix] >> ((mp_word) DIGIT_BIT); +111 \} +112 +113 /* now we have to propagate the carries and +114 * shift the words downward [all those least +115 * significant digits we zeroed]. +116 */ +117 \{ +118 register mp_digit *tmpx; +119 register mp_word *_W, *_W1; +120 +121 /* nox fix rest of carries */ +122 +123 /* alias for current word */ +124 _W1 = W + ix; +125 +126 /* alias for next word, where the carry goes */ +127 _W = W + ++ix; +128 +129 for (; ix <= n->used * 2 + 1; ix++) \{ +130 *_W++ += *_W1++ >> ((mp_word) DIGIT_BIT); +131 \} +132 +133 /* copy out, A = A/b**n +134 * +135 * The result is A/b**n but instead of converting from an +136 * array of mp_word to mp_digit than calling mp_rshd +137 * we just copy them in the right order +138 */ +139 +140 /* alias for destination word */ +141 tmpx = x->dp; +142 +143 /* alias for shifted double precision result */ +144 _W = W + n->used; +145 +146 for (ix = 0; ix < n->used + 1; ix++) \{ +147 *tmpx++ = (mp_digit)(*_W++ & ((mp_word) MP_MASK)); +148 \} +149 +150 /* zero oldused digits, if the input a was larger than +151 * m->used+1 we'll have to clear the digits +152 */ +153 for (; ix < olduse; ix++) \{ +154 *tmpx++ = 0; +155 \} +156 \} +157 +158 /* set the max used and clamp */ +159 x->used = n->used + 1; +160 mp_clamp (x); +161 +162 /* if A >= m then A = A - m */ +163 if (mp_cmp_mag (x, n) != MP_LT) \{ +164 return s_mp_sub (x, n, x); +165 \} +166 return MP_OKAY; +167 \} +168 #endif +\end{alltt} +\end{small} + +The $\hat W$ array is first filled with digits of $x$ on line 48 then the rest of the digits are zeroed on line 55. Both loops share +the same alias variables to make the code easier to read. + +The value of $\mu$ is calculated in an interesting fashion. First the value $\hat W_{ix}$ is reduced modulo $\beta$ and cast to a mp\_digit. This +forces the compiler to use a single precision multiplication and prevents any concerns about loss of precision. Line 110 fixes the carry +for the next iteration of the loop by propagating the carry from $\hat W_{ix}$ to $\hat W_{ix+1}$. + +The for loop on line 109 propagates the rest of the carries upwards through the columns. The for loop on line 126 reduces the columns +modulo $\beta$ and shifts them $k$ places at the same time. The alias $\_ \hat W$ actually refers to the array $\hat W$ starting at the $n.used$'th +digit, that is $\_ \hat W_{t} = \hat W_{n.used + t}$. + +\subsection{Montgomery Setup} +To calculate the variable $\rho$ a relatively simple algorithm will be required. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_montgomery\_setup}. \\ +\textbf{Input}. mp\_int $n$ ($n > 1$ and $(n, 2) = 1$) \\ +\textbf{Output}. $\rho \equiv -1/n_0 \mbox{ (mod }\beta\mbox{)}$ \\ +\hline \\ +1. $b \leftarrow n_0$ \\ +2. If $b$ is even return(\textit{MP\_VAL}) \\ +3. $x \leftarrow ((b + 2) \mbox{ AND } 4) << 1) + b$ \\ +4. for $k$ from 0 to $\lceil lg(lg(\beta)) \rceil - 2$ do \\ +\hspace{3mm}4.1 $x \leftarrow x \cdot (2 - bx)$ \\ +5. $\rho \leftarrow \beta - x \mbox{ (mod }\beta\mbox{)}$ \\ +6. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_montgomery\_setup} +\end{figure} + +\textbf{Algorithm mp\_montgomery\_setup.} +This algorithm will calculate the value of $\rho$ required within the Montgomery reduction algorithms. It uses a very interesting trick +to calculate $1/n_0$ when $\beta$ is a power of two. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_montgomery\_setup.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* setups the montgomery reduction stuff */ +018 int +019 mp_montgomery_setup (mp_int * n, mp_digit * rho) +020 \{ +021 mp_digit x, b; +022 +023 /* fast inversion mod 2**k +024 * +025 * Based on the fact that +026 * +027 * XA = 1 (mod 2**n) => (X(2-XA)) A = 1 (mod 2**2n) +028 * => 2*X*A - X*X*A*A = 1 +029 * => 2*(1) - (1) = 1 +030 */ +031 b = n->dp[0]; +032 +033 if ((b & 1) == 0) \{ +034 return MP_VAL; +035 \} +036 +037 x = (((b + 2) & 4) << 1) + b; /* here x*a==1 mod 2**4 */ +038 x *= 2 - b * x; /* here x*a==1 mod 2**8 */ +039 #if !defined(MP_8BIT) +040 x *= 2 - b * x; /* here x*a==1 mod 2**16 */ +041 #endif +042 #if defined(MP_64BIT) || !(defined(MP_8BIT) || defined(MP_16BIT)) +043 x *= 2 - b * x; /* here x*a==1 mod 2**32 */ +044 #endif +045 #ifdef MP_64BIT +046 x *= 2 - b * x; /* here x*a==1 mod 2**64 */ +047 #endif +048 +049 /* rho = -1/m mod b */ +050 *rho = (((mp_word)1 << ((mp_word) DIGIT_BIT)) - x) & MP_MASK; +051 +052 return MP_OKAY; +053 \} +054 #endif +\end{alltt} +\end{small} + +This source code computes the value of $\rho$ required to perform Montgomery reduction. It has been modified to avoid performing excess +multiplications when $\beta$ is not the default 28-bits. + +\section{The Diminished Radix Algorithm} +The Diminished Radix method of modular reduction \cite{DRMET} is a fairly clever technique which can be more efficient than either the Barrett +or Montgomery methods for certain forms of moduli. The technique is based on the following simple congruence. + +\begin{equation} +(x \mbox{ mod } n) + k \lfloor x / n \rfloor \equiv x \mbox{ (mod }(n - k)\mbox{)} +\end{equation} + +This observation was used in the MMB \cite{MMB} block cipher to create a diffusion primitive. It used the fact that if $n = 2^{31}$ and $k=1$ that +then a x86 multiplier could produce the 62-bit product and use the ``shrd'' instruction to perform a double-precision right shift. The proof +of the above equation is very simple. First write $x$ in the product form. + +\begin{equation} +x = qn + r +\end{equation} + +Now reduce both sides modulo $(n - k)$. + +\begin{equation} +x \equiv qk + r \mbox{ (mod }(n-k)\mbox{)} +\end{equation} + +The variable $n$ reduces modulo $n - k$ to $k$. By putting $q = \lfloor x/n \rfloor$ and $r = x \mbox{ mod } n$ +into the equation the original congruence is reproduced, thus concluding the proof. The following algorithm is based on this observation. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{Diminished Radix Reduction}. \\ +\textbf{Input}. Integer $x$, $n$, $k$ \\ +\textbf{Output}. $x \mbox{ mod } (n - k)$ \\ +\hline \\ +1. $q \leftarrow \lfloor x / n \rfloor$ \\ +2. $q \leftarrow k \cdot q$ \\ +3. $x \leftarrow x \mbox{ (mod }n\mbox{)}$ \\ +4. $x \leftarrow x + q$ \\ +5. If $x \ge (n - k)$ then \\ +\hspace{3mm}5.1 $x \leftarrow x - (n - k)$ \\ +\hspace{3mm}5.2 Goto step 1. \\ +6. Return $x$ \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm Diminished Radix Reduction} +\label{fig:DR} +\end{figure} + +This algorithm will reduce $x$ modulo $n - k$ and return the residue. If $0 \le x < (n - k)^2$ then the algorithm will loop almost always +once or twice and occasionally three times. For simplicity sake the value of $x$ is bounded by the following simple polynomial. + +\begin{equation} +0 \le x < n^2 + k^2 - 2nk +\end{equation} + +The true bound is $0 \le x < (n - k - 1)^2$ but this has quite a few more terms. The value of $q$ after step 1 is bounded by the following. + +\begin{equation} +q < n - 2k - k^2/n +\end{equation} + +Since $k^2$ is going to be considerably smaller than $n$ that term will always be zero. The value of $x$ after step 3 is bounded trivially as +$0 \le x < n$. By step four the sum $x + q$ is bounded by + +\begin{equation} +0 \le q + x < (k + 1)n - 2k^2 - 1 +\end{equation} + +With a second pass $q$ will be loosely bounded by $0 \le q < k^2$ after step 2 while $x$ will still be loosely bounded by $0 \le x < n$ after step 3. After the second pass it is highly unlike that the +sum in step 4 will exceed $n - k$. In practice fewer than three passes of the algorithm are required to reduce virtually every input in the +range $0 \le x < (n - k - 1)^2$. + +\begin{figure} +\begin{small} +\begin{center} +\begin{tabular}{|l|} +\hline +$x = 123456789, n = 256, k = 3$ \\ +\hline $q \leftarrow \lfloor x/n \rfloor = 482253$ \\ +$q \leftarrow q*k = 1446759$ \\ +$x \leftarrow x \mbox{ mod } n = 21$ \\ +$x \leftarrow x + q = 1446780$ \\ +$x \leftarrow x - (n - k) = 1446527$ \\ +\hline +$q \leftarrow \lfloor x/n \rfloor = 5650$ \\ +$q \leftarrow q*k = 16950$ \\ +$x \leftarrow x \mbox{ mod } n = 127$ \\ +$x \leftarrow x + q = 17077$ \\ +$x \leftarrow x - (n - k) = 16824$ \\ +\hline +$q \leftarrow \lfloor x/n \rfloor = 65$ \\ +$q \leftarrow q*k = 195$ \\ +$x \leftarrow x \mbox{ mod } n = 184$ \\ +$x \leftarrow x + q = 379$ \\ +$x \leftarrow x - (n - k) = 126$ \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Example Diminished Radix Reduction} +\label{fig:EXDR} +\end{figure} + +Figure~\ref{fig:EXDR} demonstrates the reduction of $x = 123456789$ modulo $n - k = 253$ when $n = 256$ and $k = 3$. Note that even while $x$ +is considerably larger than $(n - k - 1)^2 = 63504$ the algorithm still converges on the modular residue exceedingly fast. In this case only +three passes were required to find the residue $x \equiv 126$. + + +\subsection{Choice of Moduli} +On the surface this algorithm looks like a very expensive algorithm. It requires a couple of subtractions followed by multiplication and other +modular reductions. The usefulness of this algorithm becomes exceedingly clear when an appropriate modulus is chosen. + +Division in general is a very expensive operation to perform. The one exception is when the division is by a power of the radix of representation used. +Division by ten for example is simple for pencil and paper mathematics since it amounts to shifting the decimal place to the right. Similarly division +by two (\textit{or powers of two}) is very simple for binary computers to perform. It would therefore seem logical to choose $n$ of the form $2^p$ +which would imply that $\lfloor x / n \rfloor$ is a simple shift of $x$ right $p$ bits. + +However, there is one operation related to division of power of twos that is even faster than this. If $n = \beta^p$ then the division may be +performed by moving whole digits to the right $p$ places. In practice division by $\beta^p$ is much faster than division by $2^p$ for any $p$. +Also with the choice of $n = \beta^p$ reducing $x$ modulo $n$ merely requires zeroing the digits above the $p-1$'th digit of $x$. + +Throughout the next section the term ``restricted modulus'' will refer to a modulus of the form $\beta^p - k$ whereas the term ``unrestricted +modulus'' will refer to a modulus of the form $2^p - k$. The word ``restricted'' in this case refers to the fact that it is based on the +$2^p$ logic except $p$ must be a multiple of $lg(\beta)$. + +\subsection{Choice of $k$} +Now that division and reduction (\textit{step 1 and 3 of figure~\ref{fig:DR}}) have been optimized to simple digit operations the multiplication by $k$ +in step 2 is the most expensive operation. Fortunately the choice of $k$ is not terribly limited. For all intents and purposes it might +as well be a single digit. The smaller the value of $k$ is the faster the algorithm will be. + +\subsection{Restricted Diminished Radix Reduction} +The restricted Diminished Radix algorithm can quickly reduce an input modulo a modulus of the form $n = \beta^p - k$. This algorithm can reduce +an input $x$ within the range $0 \le x < n^2$ using only a couple passes of the algorithm demonstrated in figure~\ref{fig:DR}. The implementation +of this algorithm has been optimized to avoid additional overhead associated with a division by $\beta^p$, the multiplication by $k$ or the addition +of $x$ and $q$. The resulting algorithm is very efficient and can lead to substantial improvements over Barrett and Montgomery reduction when modular +exponentiations are performed. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_dr\_reduce}. \\ +\textbf{Input}. mp\_int $x$, $n$ and a mp\_digit $k = \beta - n_0$ \\ +\hspace{11.5mm}($0 \le x < n^2$, $n > 1$, $0 < k < \beta$) \\ +\textbf{Output}. $x \mbox{ mod } n$ \\ +\hline \\ +1. $m \leftarrow n.used$ \\ +2. If $x.alloc < 2m$ then grow $x$ to $2m$ digits. \\ +3. $\mu \leftarrow 0$ \\ +4. for $i$ from $0$ to $m - 1$ do \\ +\hspace{3mm}4.1 $\hat r \leftarrow k \cdot x_{m+i} + x_{i} + \mu$ \\ +\hspace{3mm}4.2 $x_{i} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ +\hspace{3mm}4.3 $\mu \leftarrow \lfloor \hat r / \beta \rfloor$ \\ +5. $x_{m} \leftarrow \mu$ \\ +6. for $i$ from $m + 1$ to $x.used - 1$ do \\ +\hspace{3mm}6.1 $x_{i} \leftarrow 0$ \\ +7. Clamp excess digits of $x$. \\ +8. If $x \ge n$ then \\ +\hspace{3mm}8.1 $x \leftarrow x - n$ \\ +\hspace{3mm}8.2 Goto step 3. \\ +9. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_dr\_reduce} +\end{figure} + +\textbf{Algorithm mp\_dr\_reduce.} +This algorithm will perform the Dimished Radix reduction of $x$ modulo $n$. It has similar restrictions to that of the Barrett reduction +with the addition that $n$ must be of the form $n = \beta^m - k$ where $0 < k <\beta$. + +This algorithm essentially implements the pseudo-code in figure~\ref{fig:DR} except with a slight optimization. The division by $\beta^m$, multiplication by $k$ +and addition of $x \mbox{ mod }\beta^m$ are all performed simultaneously inside the loop on step 4. The division by $\beta^m$ is emulated by accessing +the term at the $m+i$'th position which is subsequently multiplied by $k$ and added to the term at the $i$'th position. After the loop the $m$'th +digit is set to the carry and the upper digits are zeroed. Steps 5 and 6 emulate the reduction modulo $\beta^m$ that should have happend to +$x$ before the addition of the multiple of the upper half. + +At step 8 if $x$ is still larger than $n$ another pass of the algorithm is required. First $n$ is subtracted from $x$ and then the algorithm resumes +at step 3. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_reduce.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* reduce "x" in place modulo "n" using the Diminished Radix algorithm. +018 * +019 * Based on algorithm from the paper +020 * +021 * "Generating Efficient Primes for Discrete Log Cryptosystems" +022 * Chae Hoon Lim, Pil Joong Lee, +023 * POSTECH Information Research Laboratories +024 * +025 * The modulus must be of a special format [see manual] +026 * +027 * Has been modified to use algorithm 7.10 from the LTM book instead +028 * +029 * Input x must be in the range 0 <= x <= (n-1)**2 +030 */ +031 int +032 mp_dr_reduce (mp_int * x, mp_int * n, mp_digit k) +033 \{ +034 int err, i, m; +035 mp_word r; +036 mp_digit mu, *tmpx1, *tmpx2; +037 +038 /* m = digits in modulus */ +039 m = n->used; +040 +041 /* ensure that "x" has at least 2m digits */ +042 if (x->alloc < m + m) \{ +043 if ((err = mp_grow (x, m + m)) != MP_OKAY) \{ +044 return err; +045 \} +046 \} +047 +048 /* top of loop, this is where the code resumes if +049 * another reduction pass is required. +050 */ +051 top: +052 /* aliases for digits */ +053 /* alias for lower half of x */ +054 tmpx1 = x->dp; +055 +056 /* alias for upper half of x, or x/B**m */ +057 tmpx2 = x->dp + m; +058 +059 /* set carry to zero */ +060 mu = 0; +061 +062 /* compute (x mod B**m) + k * [x/B**m] inline and inplace */ +063 for (i = 0; i < m; i++) \{ +064 r = ((mp_word)*tmpx2++) * ((mp_word)k) + *tmpx1 + mu; +065 *tmpx1++ = (mp_digit)(r & MP_MASK); +066 mu = (mp_digit)(r >> ((mp_word)DIGIT_BIT)); +067 \} +068 +069 /* set final carry */ +070 *tmpx1++ = mu; +071 +072 /* zero words above m */ +073 for (i = m + 1; i < x->used; i++) \{ +074 *tmpx1++ = 0; +075 \} +076 +077 /* clamp, sub and return */ +078 mp_clamp (x); +079 +080 /* if x >= n then subtract and reduce again +081 * Each successive "recursion" makes the input smaller and smaller. +082 */ +083 if (mp_cmp_mag (x, n) != MP_LT) \{ +084 s_mp_sub(x, n, x); +085 goto top; +086 \} +087 return MP_OKAY; +088 \} +089 #endif +\end{alltt} +\end{small} + +The first step is to grow $x$ as required to $2m$ digits since the reduction is performed in place on $x$. The label on line 51 is where +the algorithm will resume if further reduction passes are required. In theory it could be placed at the top of the function however, the size of +the modulus and question of whether $x$ is large enough are invariant after the first pass meaning that it would be a waste of time. + +The aliases $tmpx1$ and $tmpx2$ refer to the digits of $x$ where the latter is offset by $m$ digits. By reading digits from $x$ offset by $m$ digits +a division by $\beta^m$ can be simulated virtually for free. The loop on line 63 performs the bulk of the work (\textit{corresponds to step 4 of algorithm 7.11}) +in this algorithm. + +By line 70 the pointer $tmpx1$ points to the $m$'th digit of $x$ which is where the final carry will be placed. Similarly by line 73 the +same pointer will point to the $m+1$'th digit where the zeroes will be placed. + +Since the algorithm is only valid if both $x$ and $n$ are greater than zero an unsigned comparison suffices to determine if another pass is required. +With the same logic at line 84 the value of $x$ is known to be greater than or equal to $n$ meaning that an unsigned subtraction can be used +as well. Since the destination of the subtraction is the larger of the inputs the call to algorithm s\_mp\_sub cannot fail and the return code +does not need to be checked. + +\subsubsection{Setup} +To setup the restricted Diminished Radix algorithm the value $k = \beta - n_0$ is required. This algorithm is not really complicated but provided for +completeness. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_dr\_setup}. \\ +\textbf{Input}. mp\_int $n$ \\ +\textbf{Output}. $k = \beta - n_0$ \\ +\hline \\ +1. $k \leftarrow \beta - n_0$ \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_dr\_setup} +\end{figure} + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_setup.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* determines the setup value */ +018 void mp_dr_setup(mp_int *a, mp_digit *d) +019 \{ +020 /* the casts are required if DIGIT_BIT is one less than +021 * the number of bits in a mp_digit [e.g. DIGIT_BIT==31] +022 */ +023 *d = (mp_digit)((((mp_word)1) << ((mp_word)DIGIT_BIT)) - +024 ((mp_word)a->dp[0])); +025 \} +026 +027 #endif +\end{alltt} +\end{small} + +\subsubsection{Modulus Detection} +Another algorithm which will be useful is the ability to detect a restricted Diminished Radix modulus. An integer is said to be +of restricted Diminished Radix form if all of the digits are equal to $\beta - 1$ except the trailing digit which may be any value. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_dr\_is\_modulus}. \\ +\textbf{Input}. mp\_int $n$ \\ +\textbf{Output}. $1$ if $n$ is in D.R form, $0$ otherwise \\ +\hline +1. If $n.used < 2$ then return($0$). \\ +2. for $ix$ from $1$ to $n.used - 1$ do \\ +\hspace{3mm}2.1 If $n_{ix} \ne \beta - 1$ return($0$). \\ +3. Return($1$). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_dr\_is\_modulus} +\end{figure} + +\textbf{Algorithm mp\_dr\_is\_modulus.} +This algorithm determines if a value is in Diminished Radix form. Step 1 rejects obvious cases where fewer than two digits are +in the mp\_int. Step 2 tests all but the first digit to see if they are equal to $\beta - 1$. If the algorithm manages to get to +step 3 then $n$ must be of Diminished Radix form. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_is\_modulus.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* determines if a number is a valid DR modulus */ +018 int mp_dr_is_modulus(mp_int *a) +019 \{ +020 int ix; +021 +022 /* must be at least two digits */ +023 if (a->used < 2) \{ +024 return 0; +025 \} +026 +027 /* must be of the form b**k - a [a <= b] so all +028 * but the first digit must be equal to -1 (mod b). +029 */ +030 for (ix = 1; ix < a->used; ix++) \{ +031 if (a->dp[ix] != MP_MASK) \{ +032 return 0; +033 \} +034 \} +035 return 1; +036 \} +037 +038 #endif +\end{alltt} +\end{small} + +\subsection{Unrestricted Diminished Radix Reduction} +The unrestricted Diminished Radix algorithm allows modular reductions to be performed when the modulus is of the form $2^p - k$. This algorithm +is a straightforward adaptation of algorithm~\ref{fig:DR}. + +In general the restricted Diminished Radix reduction algorithm is much faster since it has considerably lower overhead. However, this new +algorithm is much faster than either Montgomery or Barrett reduction when the moduli are of the appropriate form. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_reduce\_2k}. \\ +\textbf{Input}. mp\_int $a$ and $n$. mp\_digit $k$ \\ +\hspace{11.5mm}($a \ge 0$, $n > 1$, $0 < k < \beta$, $n + k$ is a power of two) \\ +\textbf{Output}. $a \mbox{ (mod }n\mbox{)}$ \\ +\hline +1. $p \leftarrow \lceil lg(n) \rceil$ (\textit{mp\_count\_bits}) \\ +2. While $a \ge n$ do \\ +\hspace{3mm}2.1 $q \leftarrow \lfloor a / 2^p \rfloor$ (\textit{mp\_div\_2d}) \\ +\hspace{3mm}2.2 $a \leftarrow a \mbox{ (mod }2^p\mbox{)}$ (\textit{mp\_mod\_2d}) \\ +\hspace{3mm}2.3 $q \leftarrow q \cdot k$ (\textit{mp\_mul\_d}) \\ +\hspace{3mm}2.4 $a \leftarrow a - q$ (\textit{s\_mp\_sub}) \\ +\hspace{3mm}2.5 If $a \ge n$ then do \\ +\hspace{6mm}2.5.1 $a \leftarrow a - n$ \\ +3. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_reduce\_2k} +\end{figure} + +\textbf{Algorithm mp\_reduce\_2k.} +This algorithm quickly reduces an input $a$ modulo an unrestricted Diminished Radix modulus $n$. Division by $2^p$ is emulated with a right +shift which makes the algorithm fairly inexpensive to use. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_2k.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* reduces a modulo n where n is of the form 2**p - d */ +018 int +019 mp_reduce_2k(mp_int *a, mp_int *n, mp_digit d) +020 \{ +021 mp_int q; +022 int p, res; +023 +024 if ((res = mp_init(&q)) != MP_OKAY) \{ +025 return res; +026 \} +027 +028 p = mp_count_bits(n); +029 top: +030 /* q = a/2**p, a = a mod 2**p */ +031 if ((res = mp_div_2d(a, p, &q, a)) != MP_OKAY) \{ +032 goto ERR; +033 \} +034 +035 if (d != 1) \{ +036 /* q = q * d */ +037 if ((res = mp_mul_d(&q, d, &q)) != MP_OKAY) \{ +038 goto ERR; +039 \} +040 \} +041 +042 /* a = a + q */ +043 if ((res = s_mp_add(a, &q, a)) != MP_OKAY) \{ +044 goto ERR; +045 \} +046 +047 if (mp_cmp_mag(a, n) != MP_LT) \{ +048 s_mp_sub(a, n, a); +049 goto top; +050 \} +051 +052 ERR: +053 mp_clear(&q); +054 return res; +055 \} +056 +057 #endif +\end{alltt} +\end{small} + +The algorithm mp\_count\_bits calculates the number of bits in an mp\_int which is used to find the initial value of $p$. The call to mp\_div\_2d +on line 31 calculates both the quotient $q$ and the remainder $a$ required. By doing both in a single function call the code size +is kept fairly small. The multiplication by $k$ is only performed if $k > 1$. This allows reductions modulo $2^p - 1$ to be performed without +any multiplications. + +The unsigned s\_mp\_add, mp\_cmp\_mag and s\_mp\_sub are used in place of their full sign counterparts since the inputs are only valid if they are +positive. By using the unsigned versions the overhead is kept to a minimum. + +\subsubsection{Unrestricted Setup} +To setup this reduction algorithm the value of $k = 2^p - n$ is required. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_reduce\_2k\_setup}. \\ +\textbf{Input}. mp\_int $n$ \\ +\textbf{Output}. $k = 2^p - n$ \\ +\hline +1. $p \leftarrow \lceil lg(n) \rceil$ (\textit{mp\_count\_bits}) \\ +2. $x \leftarrow 2^p$ (\textit{mp\_2expt}) \\ +3. $x \leftarrow x - n$ (\textit{mp\_sub}) \\ +4. $k \leftarrow x_0$ \\ +5. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_reduce\_2k\_setup} +\end{figure} + +\textbf{Algorithm mp\_reduce\_2k\_setup.} +This algorithm computes the value of $k$ required for the algorithm mp\_reduce\_2k. By making a temporary variable $x$ equal to $2^p$ a subtraction +is sufficient to solve for $k$. Alternatively if $n$ has more than one digit the value of $k$ is simply $\beta - n_0$. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_2k\_setup.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* determines the setup value */ +018 int +019 mp_reduce_2k_setup(mp_int *a, mp_digit *d) +020 \{ +021 int res, p; +022 mp_int tmp; +023 +024 if ((res = mp_init(&tmp)) != MP_OKAY) \{ +025 return res; +026 \} +027 +028 p = mp_count_bits(a); +029 if ((res = mp_2expt(&tmp, p)) != MP_OKAY) \{ +030 mp_clear(&tmp); +031 return res; +032 \} +033 +034 if ((res = s_mp_sub(&tmp, a, &tmp)) != MP_OKAY) \{ +035 mp_clear(&tmp); +036 return res; +037 \} +038 +039 *d = tmp.dp[0]; +040 mp_clear(&tmp); +041 return MP_OKAY; +042 \} +043 #endif +\end{alltt} +\end{small} + +\subsubsection{Unrestricted Detection} +An integer $n$ is a valid unrestricted Diminished Radix modulus if either of the following are true. + +\begin{enumerate} +\item The number has only one digit. +\item The number has more than one digit and every bit from the $\beta$'th to the most significant is one. +\end{enumerate} + +If either condition is true than there is a power of two $2^p$ such that $0 < 2^p - n < \beta$. If the input is only +one digit than it will always be of the correct form. Otherwise all of the bits above the first digit must be one. This arises from the fact +that there will be value of $k$ that when added to the modulus causes a carry in the first digit which propagates all the way to the most +significant bit. The resulting sum will be a power of two. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_reduce\_is\_2k}. \\ +\textbf{Input}. mp\_int $n$ \\ +\textbf{Output}. $1$ if of proper form, $0$ otherwise \\ +\hline +1. If $n.used = 0$ then return($0$). \\ +2. If $n.used = 1$ then return($1$). \\ +3. $p \leftarrow \lceil lg(n) \rceil$ (\textit{mp\_count\_bits}) \\ +4. for $x$ from $lg(\beta)$ to $p$ do \\ +\hspace{3mm}4.1 If the ($x \mbox{ mod }lg(\beta)$)'th bit of the $\lfloor x / lg(\beta) \rfloor$ of $n$ is zero then return($0$). \\ +5. Return($1$). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_reduce\_is\_2k} +\end{figure} + +\textbf{Algorithm mp\_reduce\_is\_2k.} +This algorithm quickly determines if a modulus is of the form required for algorithm mp\_reduce\_2k to function properly. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_is\_2k.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* determines if mp_reduce_2k can be used */ +018 int mp_reduce_is_2k(mp_int *a) +019 \{ +020 int ix, iy, iw; +021 mp_digit iz; +022 +023 if (a->used == 0) \{ +024 return 0; +025 \} else if (a->used == 1) \{ +026 return 1; +027 \} else if (a->used > 1) \{ +028 iy = mp_count_bits(a); +029 iz = 1; +030 iw = 1; +031 +032 /* Test every bit from the second digit up, must be 1 */ +033 for (ix = DIGIT_BIT; ix < iy; ix++) \{ +034 if ((a->dp[iw] & iz) == 0) \{ +035 return 0; +036 \} +037 iz <<= 1; +038 if (iz > (mp_digit)MP_MASK) \{ +039 ++iw; +040 iz = 1; +041 \} +042 \} +043 \} +044 return 1; +045 \} +046 +047 #endif +\end{alltt} +\end{small} + + + +\section{Algorithm Comparison} +So far three very different algorithms for modular reduction have been discussed. Each of the algorithms have their own strengths and weaknesses +that makes having such a selection very useful. The following table sumarizes the three algorithms along with comparisons of work factors. Since +all three algorithms have the restriction that $0 \le x < n^2$ and $n > 1$ those limitations are not included in the table. + +\begin{center} +\begin{small} +\begin{tabular}{|c|c|c|c|c|c|} +\hline \textbf{Method} & \textbf{Work Required} & \textbf{Limitations} & \textbf{$m = 8$} & \textbf{$m = 32$} & \textbf{$m = 64$} \\ +\hline Barrett & $m^2 + 2m - 1$ & None & $79$ & $1087$ & $4223$ \\ +\hline Montgomery & $m^2 + m$ & $n$ must be odd & $72$ & $1056$ & $4160$ \\ +\hline D.R. & $2m$ & $n = \beta^m - k$ & $16$ & $64$ & $128$ \\ +\hline +\end{tabular} +\end{small} +\end{center} + +In theory Montgomery and Barrett reductions would require roughly the same amount of time to complete. However, in practice since Montgomery +reduction can be written as a single function with the Comba technique it is much faster. Barrett reduction suffers from the overhead of +calling the half precision multipliers, addition and division by $\beta$ algorithms. + +For almost every cryptographic algorithm Montgomery reduction is the algorithm of choice. The one set of algorithms where Diminished Radix reduction truly +shines are based on the discrete logarithm problem such as Diffie-Hellman \cite{DH} and ElGamal \cite{ELGAMAL}. In these algorithms +primes of the form $\beta^m - k$ can be found and shared amongst users. These primes will allow the Diminished Radix algorithm to be used in +modular exponentiation to greatly speed up the operation. + + + +\section*{Exercises} +\begin{tabular}{cl} +$\left [ 3 \right ]$ & Prove that the ``trick'' in algorithm mp\_montgomery\_setup actually \\ + & calculates the correct value of $\rho$. \\ + & \\ +$\left [ 2 \right ]$ & Devise an algorithm to reduce modulo $n + k$ for small $k$ quickly. \\ + & \\ +$\left [ 4 \right ]$ & Prove that the pseudo-code algorithm ``Diminished Radix Reduction'' \\ + & (\textit{figure~\ref{fig:DR}}) terminates. Also prove the probability that it will \\ + & terminate within $1 \le k \le 10$ iterations. \\ + & \\ +\end{tabular} + + +\chapter{Exponentiation} +Exponentiation is the operation of raising one variable to the power of another, for example, $a^b$. A variant of exponentiation, computed +in a finite field or ring, is called modular exponentiation. This latter style of operation is typically used in public key +cryptosystems such as RSA and Diffie-Hellman. The ability to quickly compute modular exponentiations is of great benefit to any +such cryptosystem and many methods have been sought to speed it up. + +\section{Exponentiation Basics} +A trivial algorithm would simply multiply $a$ against itself $b - 1$ times to compute the exponentiation desired. However, as $b$ grows in size +the number of multiplications becomes prohibitive. Imagine what would happen if $b$ $\approx$ $2^{1024}$ as is the case when computing an RSA signature +with a $1024$-bit key. Such a calculation could never be completed as it would take simply far too long. + +Fortunately there is a very simple algorithm based on the laws of exponents. Recall that $lg_a(a^b) = b$ and that $lg_a(a^ba^c) = b + c$ which +are two trivial relationships between the base and the exponent. Let $b_i$ represent the $i$'th bit of $b$ starting from the least +significant bit. If $b$ is a $k$-bit integer than the following equation is true. + +\begin{equation} +a^b = \prod_{i=0}^{k-1} a^{2^i \cdot b_i} +\end{equation} + +By taking the base $a$ logarithm of both sides of the equation the following equation is the result. + +\begin{equation} +b = \sum_{i=0}^{k-1}2^i \cdot b_i +\end{equation} + +The term $a^{2^i}$ can be found from the $i - 1$'th term by squaring the term since $\left ( a^{2^i} \right )^2$ is equal to +$a^{2^{i+1}}$. This observation forms the basis of essentially all fast exponentiation algorithms. It requires $k$ squarings and on average +$k \over 2$ multiplications to compute the result. This is indeed quite an improvement over simply multiplying by $a$ a total of $b-1$ times. + +While this current method is a considerable speed up there are further improvements to be made. For example, the $a^{2^i}$ term does not need to +be computed in an auxilary variable. Consider the following equivalent algorithm. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{Left to Right Exponentiation}. \\ +\textbf{Input}. Integer $a$, $b$ and $k$ \\ +\textbf{Output}. $c = a^b$ \\ +\hline \\ +1. $c \leftarrow 1$ \\ +2. for $i$ from $k - 1$ to $0$ do \\ +\hspace{3mm}2.1 $c \leftarrow c^2$ \\ +\hspace{3mm}2.2 $c \leftarrow c \cdot a^{b_i}$ \\ +3. Return $c$. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Left to Right Exponentiation} +\label{fig:LTOR} +\end{figure} + +This algorithm starts from the most significant bit and works towards the least significant bit. When the $i$'th bit of $b$ is set $a$ is +multiplied against the current product. In each iteration the product is squared which doubles the exponent of the individual terms of the +product. + +For example, let $b = 101100_2 \equiv 44_{10}$. The following chart demonstrates the actions of the algorithm. + +\newpage\begin{figure} +\begin{center} +\begin{tabular}{|c|c|} +\hline \textbf{Value of $i$} & \textbf{Value of $c$} \\ +\hline - & $1$ \\ +\hline $5$ & $a$ \\ +\hline $4$ & $a^2$ \\ +\hline $3$ & $a^4 \cdot a$ \\ +\hline $2$ & $a^8 \cdot a^2 \cdot a$ \\ +\hline $1$ & $a^{16} \cdot a^4 \cdot a^2$ \\ +\hline $0$ & $a^{32} \cdot a^8 \cdot a^4$ \\ +\hline +\end{tabular} +\end{center} +\caption{Example of Left to Right Exponentiation} +\end{figure} + +When the product $a^{32} \cdot a^8 \cdot a^4$ is simplified it is equal $a^{44}$ which is the desired exponentiation. This particular algorithm is +called ``Left to Right'' because it reads the exponent in that order. All of the exponentiation algorithms that will be presented are of this nature. + +\subsection{Single Digit Exponentiation} +The first algorithm in the series of exponentiation algorithms will be an unbounded algorithm where the exponent is a single digit. It is intended +to be used when a small power of an input is required (\textit{e.g. $a^5$}). It is faster than simply multiplying $b - 1$ times for all values of +$b$ that are greater than three. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_expt\_d}. \\ +\textbf{Input}. mp\_int $a$ and mp\_digit $b$ \\ +\textbf{Output}. $c = a^b$ \\ +\hline \\ +1. $g \leftarrow a$ (\textit{mp\_init\_copy}) \\ +2. $c \leftarrow 1$ (\textit{mp\_set}) \\ +3. for $x$ from 1 to $lg(\beta)$ do \\ +\hspace{3mm}3.1 $c \leftarrow c^2$ (\textit{mp\_sqr}) \\ +\hspace{3mm}3.2 If $b$ AND $2^{lg(\beta) - 1} \ne 0$ then \\ +\hspace{6mm}3.2.1 $c \leftarrow c \cdot g$ (\textit{mp\_mul}) \\ +\hspace{3mm}3.3 $b \leftarrow b << 1$ \\ +4. Clear $g$. \\ +5. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_expt\_d} +\end{figure} + +\textbf{Algorithm mp\_expt\_d.} +This algorithm computes the value of $a$ raised to the power of a single digit $b$. It uses the left to right exponentiation algorithm to +quickly compute the exponentiation. It is loosely based on algorithm 14.79 of HAC \cite[pp. 615]{HAC} with the difference that the +exponent is a fixed width. + +A copy of $a$ is made first to allow destination variable $c$ be the same as the source variable $a$. The result is set to the initial value of +$1$ in the subsequent step. + +Inside the loop the exponent is read from the most significant bit first down to the least significant bit. First $c$ is invariably squared +on step 3.1. In the following step if the most significant bit of $b$ is one the copy of $a$ is multiplied against $c$. The value +of $b$ is shifted left one bit to make the next bit down from the most signficant bit the new most significant bit. In effect each +iteration of the loop moves the bits of the exponent $b$ upwards to the most significant location. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_expt\_d.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* calculate c = a**b using a square-multiply algorithm */ +018 int mp_expt_d (mp_int * a, mp_digit b, mp_int * c) +019 \{ +020 int res, x; +021 mp_int g; +022 +023 if ((res = mp_init_copy (&g, a)) != MP_OKAY) \{ +024 return res; +025 \} +026 +027 /* set initial result */ +028 mp_set (c, 1); +029 +030 for (x = 0; x < (int) DIGIT_BIT; x++) \{ +031 /* square */ +032 if ((res = mp_sqr (c, c)) != MP_OKAY) \{ +033 mp_clear (&g); +034 return res; +035 \} +036 +037 /* if the bit is set multiply */ +038 if ((b & (mp_digit) (((mp_digit)1) << (DIGIT_BIT - 1))) != 0) \{ +039 if ((res = mp_mul (c, &g, c)) != MP_OKAY) \{ +040 mp_clear (&g); +041 return res; +042 \} +043 \} +044 +045 /* shift to next bit */ +046 b <<= 1; +047 \} +048 +049 mp_clear (&g); +050 return MP_OKAY; +051 \} +052 #endif +\end{alltt} +\end{small} + +Line 28 sets the initial value of the result to $1$. Next the loop on line 30 steps through each bit of the exponent starting from +the most significant down towards the least significant. The invariant squaring operation placed on line 32 is performed first. After +the squaring the result $c$ is multiplied by the base $g$ if and only if the most significant bit of the exponent is set. The shift on line +46 moves all of the bits of the exponent upwards towards the most significant location. + +\section{$k$-ary Exponentiation} +When calculating an exponentiation the most time consuming bottleneck is the multiplications which are in general a small factor +slower than squaring. Recall from the previous algorithm that $b_{i}$ refers to the $i$'th bit of the exponent $b$. Suppose instead it referred to +the $i$'th $k$-bit digit of the exponent of $b$. For $k = 1$ the definitions are synonymous and for $k > 1$ algorithm~\ref{fig:KARY} +computes the same exponentiation. A group of $k$ bits from the exponent is called a \textit{window}. That is it is a small window on only a +portion of the entire exponent. Consider the following modification to the basic left to right exponentiation algorithm. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{$k$-ary Exponentiation}. \\ +\textbf{Input}. Integer $a$, $b$, $k$ and $t$ \\ +\textbf{Output}. $c = a^b$ \\ +\hline \\ +1. $c \leftarrow 1$ \\ +2. for $i$ from $t - 1$ to $0$ do \\ +\hspace{3mm}2.1 $c \leftarrow c^{2^k} $ \\ +\hspace{3mm}2.2 Extract the $i$'th $k$-bit word from $b$ and store it in $g$. \\ +\hspace{3mm}2.3 $c \leftarrow c \cdot a^g$ \\ +3. Return $c$. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{$k$-ary Exponentiation} +\label{fig:KARY} +\end{figure} + +The squaring on step 2.1 can be calculated by squaring the value $c$ successively $k$ times. If the values of $a^g$ for $0 < g < 2^k$ have been +precomputed this algorithm requires only $t$ multiplications and $tk$ squarings. The table can be generated with $2^{k - 1} - 1$ squarings and +$2^{k - 1} + 1$ multiplications. This algorithm assumes that the number of bits in the exponent is evenly divisible by $k$. +However, when it is not the remaining $0 < x \le k - 1$ bits can be handled with algorithm~\ref{fig:LTOR}. + +Suppose $k = 4$ and $t = 100$. This modified algorithm will require $109$ multiplications and $408$ squarings to compute the exponentiation. The +original algorithm would on average have required $200$ multiplications and $400$ squrings to compute the same value. The total number of squarings +has increased slightly but the number of multiplications has nearly halved. + +\subsection{Optimal Values of $k$} +An optimal value of $k$ will minimize $2^{k} + \lceil n / k \rceil + n - 1$ for a fixed number of bits in the exponent $n$. The simplest +approach is to brute force search amongst the values $k = 2, 3, \ldots, 8$ for the lowest result. Table~\ref{fig:OPTK} lists optimal values of $k$ +for various exponent sizes and compares the number of multiplication and squarings required against algorithm~\ref{fig:LTOR}. + +\begin{figure}[here] +\begin{center} +\begin{small} +\begin{tabular}{|c|c|c|c|c|c|} +\hline \textbf{Exponent (bits)} & \textbf{Optimal $k$} & \textbf{Work at $k$} & \textbf{Work with ~\ref{fig:LTOR}} \\ +\hline $16$ & $2$ & $27$ & $24$ \\ +\hline $32$ & $3$ & $49$ & $48$ \\ +\hline $64$ & $3$ & $92$ & $96$ \\ +\hline $128$ & $4$ & $175$ & $192$ \\ +\hline $256$ & $4$ & $335$ & $384$ \\ +\hline $512$ & $5$ & $645$ & $768$ \\ +\hline $1024$ & $6$ & $1257$ & $1536$ \\ +\hline $2048$ & $6$ & $2452$ & $3072$ \\ +\hline $4096$ & $7$ & $4808$ & $6144$ \\ +\hline +\end{tabular} +\end{small} +\end{center} +\caption{Optimal Values of $k$ for $k$-ary Exponentiation} +\label{fig:OPTK} +\end{figure} + +\subsection{Sliding-Window Exponentiation} +A simple modification to the previous algorithm is only generate the upper half of the table in the range $2^{k-1} \le g < 2^k$. Essentially +this is a table for all values of $g$ where the most significant bit of $g$ is a one. However, in order for this to be allowed in the +algorithm values of $g$ in the range $0 \le g < 2^{k-1}$ must be avoided. + +Table~\ref{fig:OPTK2} lists optimal values of $k$ for various exponent sizes and compares the work required against algorithm~\ref{fig:KARY}. + +\begin{figure}[here] +\begin{center} +\begin{small} +\begin{tabular}{|c|c|c|c|c|c|} +\hline \textbf{Exponent (bits)} & \textbf{Optimal $k$} & \textbf{Work at $k$} & \textbf{Work with ~\ref{fig:KARY}} \\ +\hline $16$ & $3$ & $24$ & $27$ \\ +\hline $32$ & $3$ & $45$ & $49$ \\ +\hline $64$ & $4$ & $87$ & $92$ \\ +\hline $128$ & $4$ & $167$ & $175$ \\ +\hline $256$ & $5$ & $322$ & $335$ \\ +\hline $512$ & $6$ & $628$ & $645$ \\ +\hline $1024$ & $6$ & $1225$ & $1257$ \\ +\hline $2048$ & $7$ & $2403$ & $2452$ \\ +\hline $4096$ & $8$ & $4735$ & $4808$ \\ +\hline +\end{tabular} +\end{small} +\end{center} +\caption{Optimal Values of $k$ for Sliding Window Exponentiation} +\label{fig:OPTK2} +\end{figure} + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{Sliding Window $k$-ary Exponentiation}. \\ +\textbf{Input}. Integer $a$, $b$, $k$ and $t$ \\ +\textbf{Output}. $c = a^b$ \\ +\hline \\ +1. $c \leftarrow 1$ \\ +2. for $i$ from $t - 1$ to $0$ do \\ +\hspace{3mm}2.1 If the $i$'th bit of $b$ is a zero then \\ +\hspace{6mm}2.1.1 $c \leftarrow c^2$ \\ +\hspace{3mm}2.2 else do \\ +\hspace{6mm}2.2.1 $c \leftarrow c^{2^k}$ \\ +\hspace{6mm}2.2.2 Extract the $k$ bits from $(b_{i}b_{i-1}\ldots b_{i-(k-1)})$ and store it in $g$. \\ +\hspace{6mm}2.2.3 $c \leftarrow c \cdot a^g$ \\ +\hspace{6mm}2.2.4 $i \leftarrow i - k$ \\ +3. Return $c$. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Sliding Window $k$-ary Exponentiation} +\end{figure} + +Similar to the previous algorithm this algorithm must have a special handler when fewer than $k$ bits are left in the exponent. While this +algorithm requires the same number of squarings it can potentially have fewer multiplications. The pre-computed table $a^g$ is also half +the size as the previous table. + +Consider the exponent $b = 111101011001000_2 \equiv 31432_{10}$ with $k = 3$ using both algorithms. The first algorithm will divide the exponent up as +the following five $3$-bit words $b \equiv \left ( 111, 101, 011, 001, 000 \right )_{2}$. The second algorithm will break the +exponent as $b \equiv \left ( 111, 101, 0, 110, 0, 100, 0 \right )_{2}$. The single digit $0$ in the second representation are where +a single squaring took place instead of a squaring and multiplication. In total the first method requires $10$ multiplications and $18$ +squarings. The second method requires $8$ multiplications and $18$ squarings. + +In general the sliding window method is never slower than the generic $k$-ary method and often it is slightly faster. + +\section{Modular Exponentiation} + +Modular exponentiation is essentially computing the power of a base within a finite field or ring. For example, computing +$d \equiv a^b \mbox{ (mod }c\mbox{)}$ is a modular exponentiation. Instead of first computing $a^b$ and then reducing it +modulo $c$ the intermediate result is reduced modulo $c$ after every squaring or multiplication operation. + +This guarantees that any intermediate result is bounded by $0 \le d \le c^2 - 2c + 1$ and can be reduced modulo $c$ quickly using +one of the algorithms presented in chapter six. + +Before the actual modular exponentiation algorithm can be written a wrapper algorithm must be written first. This algorithm +will allow the exponent $b$ to be negative which is computed as $c \equiv \left (1 / a \right )^{\vert b \vert} \mbox{(mod }d\mbox{)}$. The +value of $(1/a) \mbox{ mod }c$ is computed using the modular inverse (\textit{see \ref{sec;modinv}}). If no inverse exists the algorithm +terminates with an error. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_exptmod}. \\ +\textbf{Input}. mp\_int $a$, $b$ and $c$ \\ +\textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\ +\hline \\ +1. If $c.sign = MP\_NEG$ return(\textit{MP\_VAL}). \\ +2. If $b.sign = MP\_NEG$ then \\ +\hspace{3mm}2.1 $g' \leftarrow g^{-1} \mbox{ (mod }c\mbox{)}$ \\ +\hspace{3mm}2.2 $x' \leftarrow \vert x \vert$ \\ +\hspace{3mm}2.3 Compute $d \equiv g'^{x'} \mbox{ (mod }c\mbox{)}$ via recursion. \\ +3. if $p$ is odd \textbf{OR} $p$ is a D.R. modulus then \\ +\hspace{3mm}3.1 Compute $y \equiv g^{x} \mbox{ (mod }p\mbox{)}$ via algorithm mp\_exptmod\_fast. \\ +4. else \\ +\hspace{3mm}4.1 Compute $y \equiv g^{x} \mbox{ (mod }p\mbox{)}$ via algorithm s\_mp\_exptmod. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_exptmod} +\end{figure} + +\textbf{Algorithm mp\_exptmod.} +The first algorithm which actually performs modular exponentiation is algorithm s\_mp\_exptmod. It is a sliding window $k$-ary algorithm +which uses Barrett reduction to reduce the product modulo $p$. The second algorithm mp\_exptmod\_fast performs the same operation +except it uses either Montgomery or Diminished Radix reduction. The two latter reduction algorithms are clumped in the same exponentiation +algorithm since their arguments are essentially the same (\textit{two mp\_ints and one mp\_digit}). + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_exptmod.c +\vspace{-3mm} +\begin{alltt} +016 +017 +018 /* this is a shell function that calls either the normal or Montgomery +019 * exptmod functions. Originally the call to the montgomery code was +020 * embedded in the normal function but that wasted alot of stack space +021 * for nothing (since 99% of the time the Montgomery code would be called) +022 */ +023 int mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y) +024 \{ +025 int dr; +026 +027 /* modulus P must be positive */ +028 if (P->sign == MP_NEG) \{ +029 return MP_VAL; +030 \} +031 +032 /* if exponent X is negative we have to recurse */ +033 if (X->sign == MP_NEG) \{ +034 #ifdef BN_MP_INVMOD_C +035 mp_int tmpG, tmpX; +036 int err; +037 +038 /* first compute 1/G mod P */ +039 if ((err = mp_init(&tmpG)) != MP_OKAY) \{ +040 return err; +041 \} +042 if ((err = mp_invmod(G, P, &tmpG)) != MP_OKAY) \{ +043 mp_clear(&tmpG); +044 return err; +045 \} +046 +047 /* now get |X| */ +048 if ((err = mp_init(&tmpX)) != MP_OKAY) \{ +049 mp_clear(&tmpG); +050 return err; +051 \} +052 if ((err = mp_abs(X, &tmpX)) != MP_OKAY) \{ +053 mp_clear_multi(&tmpG, &tmpX, NULL); +054 return err; +055 \} +056 +057 /* and now compute (1/G)**|X| instead of G**X [X < 0] */ +058 err = mp_exptmod(&tmpG, &tmpX, P, Y); +059 mp_clear_multi(&tmpG, &tmpX, NULL); +060 return err; +061 #else +062 /* no invmod */ +063 return MP_VAL; +064 #endif +065 \} +066 +067 #ifdef BN_MP_DR_IS_MODULUS_C +068 /* is it a DR modulus? */ +069 dr = mp_dr_is_modulus(P); +070 #else +071 dr = 0; +072 #endif +073 +074 #ifdef BN_MP_REDUCE_IS_2K_C +075 /* if not, is it a uDR modulus? */ +076 if (dr == 0) \{ +077 dr = mp_reduce_is_2k(P) << 1; +078 \} +079 #endif +080 +081 /* if the modulus is odd or dr != 0 use the fast method */ +082 #ifdef BN_MP_EXPTMOD_FAST_C +083 if (mp_isodd (P) == 1 || dr != 0) \{ +084 return mp_exptmod_fast (G, X, P, Y, dr); +085 \} else \{ +086 #endif +087 #ifdef BN_S_MP_EXPTMOD_C +088 /* otherwise use the generic Barrett reduction technique */ +089 return s_mp_exptmod (G, X, P, Y); +090 #else +091 /* no exptmod for evens */ +092 return MP_VAL; +093 #endif +094 #ifdef BN_MP_EXPTMOD_FAST_C +095 \} +096 #endif +097 \} +098 +099 #endif +\end{alltt} +\end{small} + +In order to keep the algorithms in a known state the first step on line 28 is to reject any negative modulus as input. If the exponent is +negative the algorithm tries to perform a modular exponentiation with the modular inverse of the base $G$. The temporary variable $tmpG$ is assigned +the modular inverse of $G$ and $tmpX$ is assigned the absolute value of $X$. The algorithm will recuse with these new values with a positive +exponent. + +If the exponent is positive the algorithm resumes the exponentiation. Line 69 determines if the modulus is of the restricted Diminished Radix +form. If it is not line 77 attempts to determine if it is of a unrestricted Diminished Radix form. The integer $dr$ will take on one +of three values. + +\begin{enumerate} +\item $dr = 0$ means that the modulus is not of either restricted or unrestricted Diminished Radix form. +\item $dr = 1$ means that the modulus is of restricted Diminished Radix form. +\item $dr = 2$ means that the modulus is of unrestricted Diminished Radix form. +\end{enumerate} + +Line 67 determines if the fast modular exponentiation algorithm can be used. It is allowed if $dr \ne 0$ or if the modulus is odd. Otherwise, +the slower s\_mp\_exptmod algorithm is used which uses Barrett reduction. + +\subsection{Barrett Modular Exponentiation} + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{s\_mp\_exptmod}. \\ +\textbf{Input}. mp\_int $a$, $b$ and $c$ \\ +\textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\ +\hline \\ +1. $k \leftarrow lg(x)$ \\ +2. $winsize \leftarrow \left \lbrace \begin{array}{ll} + 2 & \mbox{if }k \le 7 \\ + 3 & \mbox{if }7 < k \le 36 \\ + 4 & \mbox{if }36 < k \le 140 \\ + 5 & \mbox{if }140 < k \le 450 \\ + 6 & \mbox{if }450 < k \le 1303 \\ + 7 & \mbox{if }1303 < k \le 3529 \\ + 8 & \mbox{if }3529 < k \\ + \end{array} \right .$ \\ +3. Initialize $2^{winsize}$ mp\_ints in an array named $M$ and one mp\_int named $\mu$ \\ +4. Calculate the $\mu$ required for Barrett Reduction (\textit{mp\_reduce\_setup}). \\ +5. $M_1 \leftarrow g \mbox{ (mod }p\mbox{)}$ \\ +\\ +Setup the table of small powers of $g$. First find $g^{2^{winsize}}$ and then all multiples of it. \\ +6. $k \leftarrow 2^{winsize - 1}$ \\ +7. $M_{k} \leftarrow M_1$ \\ +8. for $ix$ from 0 to $winsize - 2$ do \\ +\hspace{3mm}8.1 $M_k \leftarrow \left ( M_k \right )^2$ (\textit{mp\_sqr}) \\ +\hspace{3mm}8.2 $M_k \leftarrow M_k \mbox{ (mod }p\mbox{)}$ (\textit{mp\_reduce}) \\ +9. for $ix$ from $2^{winsize - 1} + 1$ to $2^{winsize} - 1$ do \\ +\hspace{3mm}9.1 $M_{ix} \leftarrow M_{ix - 1} \cdot M_{1}$ (\textit{mp\_mul}) \\ +\hspace{3mm}9.2 $M_{ix} \leftarrow M_{ix} \mbox{ (mod }p\mbox{)}$ (\textit{mp\_reduce}) \\ +10. $res \leftarrow 1$ \\ +\\ +Start Sliding Window. \\ +11. $mode \leftarrow 0, bitcnt \leftarrow 1, buf \leftarrow 0, digidx \leftarrow x.used - 1, bitcpy \leftarrow 0, bitbuf \leftarrow 0$ \\ +12. Loop \\ +\hspace{3mm}12.1 $bitcnt \leftarrow bitcnt - 1$ \\ +\hspace{3mm}12.2 If $bitcnt = 0$ then do \\ +\hspace{6mm}12.2.1 If $digidx = -1$ goto step 13. \\ +\hspace{6mm}12.2.2 $buf \leftarrow x_{digidx}$ \\ +\hspace{6mm}12.2.3 $digidx \leftarrow digidx - 1$ \\ +\hspace{6mm}12.2.4 $bitcnt \leftarrow lg(\beta)$ \\ +Continued on next page. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm s\_mp\_exptmod} +\end{figure} + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{s\_mp\_exptmod} (\textit{continued}). \\ +\textbf{Input}. mp\_int $a$, $b$ and $c$ \\ +\textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\ +\hline \\ +\hspace{3mm}12.3 $y \leftarrow (buf >> (lg(\beta) - 1))$ AND $1$ \\ +\hspace{3mm}12.4 $buf \leftarrow buf << 1$ \\ +\hspace{3mm}12.5 if $mode = 0$ and $y = 0$ then goto step 12. \\ +\hspace{3mm}12.6 if $mode = 1$ and $y = 0$ then do \\ +\hspace{6mm}12.6.1 $res \leftarrow res^2$ \\ +\hspace{6mm}12.6.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ +\hspace{6mm}12.6.3 Goto step 12. \\ +\hspace{3mm}12.7 $bitcpy \leftarrow bitcpy + 1$ \\ +\hspace{3mm}12.8 $bitbuf \leftarrow bitbuf + (y << (winsize - bitcpy))$ \\ +\hspace{3mm}12.9 $mode \leftarrow 2$ \\ +\hspace{3mm}12.10 If $bitcpy = winsize$ then do \\ +\hspace{6mm}Window is full so perform the squarings and single multiplication. \\ +\hspace{6mm}12.10.1 for $ix$ from $0$ to $winsize -1$ do \\ +\hspace{9mm}12.10.1.1 $res \leftarrow res^2$ \\ +\hspace{9mm}12.10.1.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ +\hspace{6mm}12.10.2 $res \leftarrow res \cdot M_{bitbuf}$ \\ +\hspace{6mm}12.10.3 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ +\hspace{6mm}Reset the window. \\ +\hspace{6mm}12.10.4 $bitcpy \leftarrow 0, bitbuf \leftarrow 0, mode \leftarrow 1$ \\ +\\ +No more windows left. Check for residual bits of exponent. \\ +13. If $mode = 2$ and $bitcpy > 0$ then do \\ +\hspace{3mm}13.1 for $ix$ form $0$ to $bitcpy - 1$ do \\ +\hspace{6mm}13.1.1 $res \leftarrow res^2$ \\ +\hspace{6mm}13.1.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ +\hspace{6mm}13.1.3 $bitbuf \leftarrow bitbuf << 1$ \\ +\hspace{6mm}13.1.4 If $bitbuf$ AND $2^{winsize} \ne 0$ then do \\ +\hspace{9mm}13.1.4.1 $res \leftarrow res \cdot M_{1}$ \\ +\hspace{9mm}13.1.4.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ +14. $y \leftarrow res$ \\ +15. Clear $res$, $mu$ and the $M$ array. \\ +16. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm s\_mp\_exptmod (continued)} +\end{figure} + +\textbf{Algorithm s\_mp\_exptmod.} +This algorithm computes the $x$'th power of $g$ modulo $p$ and stores the result in $y$. It takes advantage of the Barrett reduction +algorithm to keep the product small throughout the algorithm. + +The first two steps determine the optimal window size based on the number of bits in the exponent. The larger the exponent the +larger the window size becomes. After a window size $winsize$ has been chosen an array of $2^{winsize}$ mp\_int variables is allocated. This +table will hold the values of $g^x \mbox{ (mod }p\mbox{)}$ for $2^{winsize - 1} \le x < 2^{winsize}$. + +After the table is allocated the first power of $g$ is found. Since $g \ge p$ is allowed it must be first reduced modulo $p$ to make +the rest of the algorithm more efficient. The first element of the table at $2^{winsize - 1}$ is found by squaring $M_1$ successively $winsize - 2$ +times. The rest of the table elements are found by multiplying the previous element by $M_1$ modulo $p$. + +Now that the table is available the sliding window may begin. The following list describes the functions of all the variables in the window. +\begin{enumerate} +\item The variable $mode$ dictates how the bits of the exponent are interpreted. +\begin{enumerate} + \item When $mode = 0$ the bits are ignored since no non-zero bit of the exponent has been seen yet. For example, if the exponent were simply + $1$ then there would be $lg(\beta) - 1$ zero bits before the first non-zero bit. In this case bits are ignored until a non-zero bit is found. + \item When $mode = 1$ a non-zero bit has been seen before and a new $winsize$-bit window has not been formed yet. In this mode leading $0$ bits + are read and a single squaring is performed. If a non-zero bit is read a new window is created. + \item When $mode = 2$ the algorithm is in the middle of forming a window and new bits are appended to the window from the most significant bit + downwards. +\end{enumerate} +\item The variable $bitcnt$ indicates how many bits are left in the current digit of the exponent left to be read. When it reaches zero a new digit + is fetched from the exponent. +\item The variable $buf$ holds the currently read digit of the exponent. +\item The variable $digidx$ is an index into the exponents digits. It starts at the leading digit $x.used - 1$ and moves towards the trailing digit. +\item The variable $bitcpy$ indicates how many bits are in the currently formed window. When it reaches $winsize$ the window is flushed and + the appropriate operations performed. +\item The variable $bitbuf$ holds the current bits of the window being formed. +\end{enumerate} + +All of step 12 is the window processing loop. It will iterate while there are digits available form the exponent to read. The first step +inside this loop is to extract a new digit if no more bits are available in the current digit. If there are no bits left a new digit is +read and if there are no digits left than the loop terminates. + +After a digit is made available step 12.3 will extract the most significant bit of the current digit and move all other bits in the digit +upwards. In effect the digit is read from most significant bit to least significant bit and since the digits are read from leading to +trailing edges the entire exponent is read from most significant bit to least significant bit. + +At step 12.5 if the $mode$ and currently extracted bit $y$ are both zero the bit is ignored and the next bit is read. This prevents the +algorithm from having to perform trivial squaring and reduction operations before the first non-zero bit is read. Step 12.6 and 12.7-10 handle +the two cases of $mode = 1$ and $mode = 2$ respectively. + +\begin{center} +\begin{figure}[here] +\includegraphics{pics/expt_state.ps} +\caption{Sliding Window State Diagram} +\label{pic:expt_state} +\end{figure} +\end{center} + +By step 13 there are no more digits left in the exponent. However, there may be partial bits in the window left. If $mode = 2$ then +a Left-to-Right algorithm is used to process the remaining few bits. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_exptmod.c +\vspace{-3mm} +\begin{alltt} +016 +017 #ifdef MP_LOW_MEM +018 #define TAB_SIZE 32 +019 #else +020 #define TAB_SIZE 256 +021 #endif +022 +023 int s_mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y) +024 \{ +025 mp_int M[TAB_SIZE], res, mu; +026 mp_digit buf; +027 int err, bitbuf, bitcpy, bitcnt, mode, digidx, x, y, winsize; +028 +029 /* find window size */ +030 x = mp_count_bits (X); +031 if (x <= 7) \{ +032 winsize = 2; +033 \} else if (x <= 36) \{ +034 winsize = 3; +035 \} else if (x <= 140) \{ +036 winsize = 4; +037 \} else if (x <= 450) \{ +038 winsize = 5; +039 \} else if (x <= 1303) \{ +040 winsize = 6; +041 \} else if (x <= 3529) \{ +042 winsize = 7; +043 \} else \{ +044 winsize = 8; +045 \} +046 +047 #ifdef MP_LOW_MEM +048 if (winsize > 5) \{ +049 winsize = 5; +050 \} +051 #endif +052 +053 /* init M array */ +054 /* init first cell */ +055 if ((err = mp_init(&M[1])) != MP_OKAY) \{ +056 return err; +057 \} +058 +059 /* now init the second half of the array */ +060 for (x = 1<<(winsize-1); x < (1 << winsize); x++) \{ +061 if ((err = mp_init(&M[x])) != MP_OKAY) \{ +062 for (y = 1<<(winsize-1); y < x; y++) \{ +063 mp_clear (&M[y]); +064 \} +065 mp_clear(&M[1]); +066 return err; +067 \} +068 \} +069 +070 /* create mu, used for Barrett reduction */ +071 if ((err = mp_init (&mu)) != MP_OKAY) \{ +072 goto LBL_M; +073 \} +074 if ((err = mp_reduce_setup (&mu, P)) != MP_OKAY) \{ +075 goto LBL_MU; +076 \} +077 +078 /* create M table +079 * +080 * The M table contains powers of the base, +081 * e.g. M[x] = G**x mod P +082 * +083 * The first half of the table is not +084 * computed though accept for M[0] and M[1] +085 */ +086 if ((err = mp_mod (G, P, &M[1])) != MP_OKAY) \{ +087 goto LBL_MU; +088 \} +089 +090 /* compute the value at M[1<<(winsize-1)] by squaring +091 * M[1] (winsize-1) times +092 */ +093 if ((err = mp_copy (&M[1], &M[1 << (winsize - 1)])) != MP_OKAY) \{ +094 goto LBL_MU; +095 \} +096 +097 for (x = 0; x < (winsize - 1); x++) \{ +098 if ((err = mp_sqr (&M[1 << (winsize - 1)], +099 &M[1 << (winsize - 1)])) != MP_OKAY) \{ +100 goto LBL_MU; +101 \} +102 if ((err = mp_reduce (&M[1 << (winsize - 1)], P, &mu)) != MP_OKAY) \{ +103 goto LBL_MU; +104 \} +105 \} +106 +107 /* create upper table, that is M[x] = M[x-1] * M[1] (mod P) +108 * for x = (2**(winsize - 1) + 1) to (2**winsize - 1) +109 */ +110 for (x = (1 << (winsize - 1)) + 1; x < (1 << winsize); x++) \{ +111 if ((err = mp_mul (&M[x - 1], &M[1], &M[x])) != MP_OKAY) \{ +112 goto LBL_MU; +113 \} +114 if ((err = mp_reduce (&M[x], P, &mu)) != MP_OKAY) \{ +115 goto LBL_MU; +116 \} +117 \} +118 +119 /* setup result */ +120 if ((err = mp_init (&res)) != MP_OKAY) \{ +121 goto LBL_MU; +122 \} +123 mp_set (&res, 1); +124 +125 /* set initial mode and bit cnt */ +126 mode = 0; +127 bitcnt = 1; +128 buf = 0; +129 digidx = X->used - 1; +130 bitcpy = 0; +131 bitbuf = 0; +132 +133 for (;;) \{ +134 /* grab next digit as required */ +135 if (--bitcnt == 0) \{ +136 /* if digidx == -1 we are out of digits */ +137 if (digidx == -1) \{ +138 break; +139 \} +140 /* read next digit and reset the bitcnt */ +141 buf = X->dp[digidx--]; +142 bitcnt = (int) DIGIT_BIT; +143 \} +144 +145 /* grab the next msb from the exponent */ +146 y = (buf >> (mp_digit)(DIGIT_BIT - 1)) & 1; +147 buf <<= (mp_digit)1; +148 +149 /* if the bit is zero and mode == 0 then we ignore it +150 * These represent the leading zero bits before the first 1 bit +151 * in the exponent. Technically this opt is not required but it +152 * does lower the # of trivial squaring/reductions used +153 */ +154 if (mode == 0 && y == 0) \{ +155 continue; +156 \} +157 +158 /* if the bit is zero and mode == 1 then we square */ +159 if (mode == 1 && y == 0) \{ +160 if ((err = mp_sqr (&res, &res)) != MP_OKAY) \{ +161 goto LBL_RES; +162 \} +163 if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) \{ +164 goto LBL_RES; +165 \} +166 continue; +167 \} +168 +169 /* else we add it to the window */ +170 bitbuf |= (y << (winsize - ++bitcpy)); +171 mode = 2; +172 +173 if (bitcpy == winsize) \{ +174 /* ok window is filled so square as required and multiply */ +175 /* square first */ +176 for (x = 0; x < winsize; x++) \{ +177 if ((err = mp_sqr (&res, &res)) != MP_OKAY) \{ +178 goto LBL_RES; +179 \} +180 if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) \{ +181 goto LBL_RES; +182 \} +183 \} +184 +185 /* then multiply */ +186 if ((err = mp_mul (&res, &M[bitbuf], &res)) != MP_OKAY) \{ +187 goto LBL_RES; +188 \} +189 if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) \{ +190 goto LBL_RES; +191 \} +192 +193 /* empty window and reset */ +194 bitcpy = 0; +195 bitbuf = 0; +196 mode = 1; +197 \} +198 \} +199 +200 /* if bits remain then square/multiply */ +201 if (mode == 2 && bitcpy > 0) \{ +202 /* square then multiply if the bit is set */ +203 for (x = 0; x < bitcpy; x++) \{ +204 if ((err = mp_sqr (&res, &res)) != MP_OKAY) \{ +205 goto LBL_RES; +206 \} +207 if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) \{ +208 goto LBL_RES; +209 \} +210 +211 bitbuf <<= 1; +212 if ((bitbuf & (1 << winsize)) != 0) \{ +213 /* then multiply */ +214 if ((err = mp_mul (&res, &M[1], &res)) != MP_OKAY) \{ +215 goto LBL_RES; +216 \} +217 if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) \{ +218 goto LBL_RES; +219 \} +220 \} +221 \} +222 \} +223 +224 mp_exch (&res, Y); +225 err = MP_OKAY; +226 LBL_RES:mp_clear (&res); +227 LBL_MU:mp_clear (&mu); +228 LBL_M: +229 mp_clear(&M[1]); +230 for (x = 1<<(winsize-1); x < (1 << winsize); x++) \{ +231 mp_clear (&M[x]); +232 \} +233 return err; +234 \} +235 #endif +\end{alltt} +\end{small} + +Lines 31 through 41 determine the optimal window size based on the length of the exponent in bits. The window divisions are sorted +from smallest to greatest so that in each \textbf{if} statement only one condition must be tested. For example, by the \textbf{if} statement +on line 33 the value of $x$ is already known to be greater than $140$. + +The conditional piece of code beginning on line 47 allows the window size to be restricted to five bits. This logic is used to ensure +the table of precomputed powers of $G$ remains relatively small. + +The for loop on line 60 initializes the $M$ array while lines 61 and 74 compute the value of $\mu$ required for +Barrett reduction. + +-- More later. + +\section{Quick Power of Two} +Calculating $b = 2^a$ can be performed much quicker than with any of the previous algorithms. Recall that a logical shift left $m << k$ is +equivalent to $m \cdot 2^k$. By this logic when $m = 1$ a quick power of two can be achieved. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_2expt}. \\ +\textbf{Input}. integer $b$ \\ +\textbf{Output}. $a \leftarrow 2^b$ \\ +\hline \\ +1. $a \leftarrow 0$ \\ +2. If $a.alloc < \lfloor b / lg(\beta) \rfloor + 1$ then grow $a$ appropriately. \\ +3. $a.used \leftarrow \lfloor b / lg(\beta) \rfloor + 1$ \\ +4. $a_{\lfloor b / lg(\beta) \rfloor} \leftarrow 1 << (b \mbox{ mod } lg(\beta))$ \\ +5. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_2expt} +\end{figure} + +\textbf{Algorithm mp\_2expt.} + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_2expt.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* computes a = 2**b +018 * +019 * Simple algorithm which zeroes the int, grows it then just sets one bit +020 * as required. +021 */ +022 int +023 mp_2expt (mp_int * a, int b) +024 \{ +025 int res; +026 +027 /* zero a as per default */ +028 mp_zero (a); +029 +030 /* grow a to accomodate the single bit */ +031 if ((res = mp_grow (a, b / DIGIT_BIT + 1)) != MP_OKAY) \{ +032 return res; +033 \} +034 +035 /* set the used count of where the bit will go */ +036 a->used = b / DIGIT_BIT + 1; +037 +038 /* put the single bit in its place */ +039 a->dp[b / DIGIT_BIT] = ((mp_digit)1) << (b % DIGIT_BIT); +040 +041 return MP_OKAY; +042 \} +043 #endif +\end{alltt} +\end{small} + +\chapter{Higher Level Algorithms} + +This chapter discusses the various higher level algorithms that are required to complete a well rounded multiple precision integer package. These +routines are less performance oriented than the algorithms of chapters five, six and seven but are no less important. + +The first section describes a method of integer division with remainder that is universally well known. It provides the signed division logic +for the package. The subsequent section discusses a set of algorithms which allow a single digit to be the 2nd operand for a variety of operations. +These algorithms serve mostly to simplify other algorithms where small constants are required. The last two sections discuss how to manipulate +various representations of integers. For example, converting from an mp\_int to a string of character. + +\section{Integer Division with Remainder} +\label{sec:division} + +Integer division aside from modular exponentiation is the most intensive algorithm to compute. Like addition, subtraction and multiplication +the basis of this algorithm is the long-hand division algorithm taught to school children. Throughout this discussion several common variables +will be used. Let $x$ represent the divisor and $y$ represent the dividend. Let $q$ represent the integer quotient $\lfloor y / x \rfloor$ and +let $r$ represent the remainder $r = y - x \lfloor y / x \rfloor$. The following simple algorithm will be used to start the discussion. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{Radix-$\beta$ Integer Division}. \\ +\textbf{Input}. integer $x$ and $y$ \\ +\textbf{Output}. $q = \lfloor y/x\rfloor, r = y - xq$ \\ +\hline \\ +1. $q \leftarrow 0$ \\ +2. $n \leftarrow \vert \vert y \vert \vert - \vert \vert x \vert \vert$ \\ +3. for $t$ from $n$ down to $0$ do \\ +\hspace{3mm}3.1 Maximize $k$ such that $kx\beta^t$ is less than or equal to $y$ and $(k + 1)x\beta^t$ is greater. \\ +\hspace{3mm}3.2 $q \leftarrow q + k\beta^t$ \\ +\hspace{3mm}3.3 $y \leftarrow y - kx\beta^t$ \\ +4. $r \leftarrow y$ \\ +5. Return($q, r$) \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm Radix-$\beta$ Integer Division} +\label{fig:raddiv} +\end{figure} + +As children we are taught this very simple algorithm for the case of $\beta = 10$. Almost instinctively several optimizations are taught for which +their reason of existing are never explained. For this example let $y = 5471$ represent the dividend and $x = 23$ represent the divisor. + +To find the first digit of the quotient the value of $k$ must be maximized such that $kx\beta^t$ is less than or equal to $y$ and +simultaneously $(k + 1)x\beta^t$ is greater than $y$. Implicitly $k$ is the maximum value the $t$'th digit of the quotient may have. The habitual method +used to find the maximum is to ``eyeball'' the two numbers, typically only the leading digits and quickly estimate a quotient. By only using leading +digits a much simpler division may be used to form an educated guess at what the value must be. In this case $k = \lfloor 54/23\rfloor = 2$ quickly +arises as a possible solution. Indeed $2x\beta^2 = 4600$ is less than $y = 5471$ and simultaneously $(k + 1)x\beta^2 = 6900$ is larger than $y$. +As a result $k\beta^2$ is added to the quotient which now equals $q = 200$ and $4600$ is subtracted from $y$ to give a remainder of $y = 841$. + +Again this process is repeated to produce the quotient digit $k = 3$ which makes the quotient $q = 200 + 3\beta = 230$ and the remainder +$y = 841 - 3x\beta = 181$. Finally the last iteration of the loop produces $k = 7$ which leads to the quotient $q = 230 + 7 = 237$ and the +remainder $y = 181 - 7x = 20$. The final quotient and remainder found are $q = 237$ and $r = y = 20$ which are indeed correct since +$237 \cdot 23 + 20 = 5471$ is true. + +\subsection{Quotient Estimation} +\label{sec:divest} +As alluded to earlier the quotient digit $k$ can be estimated from only the leading digits of both the divisor and dividend. When $p$ leading +digits are used from both the divisor and dividend to form an estimation the accuracy of the estimation rises as $p$ grows. Technically +speaking the estimation is based on assuming the lower $\vert \vert y \vert \vert - p$ and $\vert \vert x \vert \vert - p$ lower digits of the +dividend and divisor are zero. + +The value of the estimation may off by a few values in either direction and in general is fairly correct. A simplification \cite[pp. 271]{TAOCPV2} +of the estimation technique is to use $t + 1$ digits of the dividend and $t$ digits of the divisor, in particularly when $t = 1$. The estimate +using this technique is never too small. For the following proof let $t = \vert \vert y \vert \vert - 1$ and $s = \vert \vert x \vert \vert - 1$ +represent the most significant digits of the dividend and divisor respectively. + +\textbf{Proof.}\textit{ The quotient $\hat k = \lfloor (y_t\beta + y_{t-1}) / x_s \rfloor$ is greater than or equal to +$k = \lfloor y / (x \cdot \beta^{\vert \vert y \vert \vert - \vert \vert x \vert \vert - 1}) \rfloor$. } +The first obvious case is when $\hat k = \beta - 1$ in which case the proof is concluded since the real quotient cannot be larger. For all other +cases $\hat k = \lfloor (y_t\beta + y_{t-1}) / x_s \rfloor$ and $\hat k x_s \ge y_t\beta + y_{t-1} - x_s + 1$. The latter portion of the inequalility +$-x_s + 1$ arises from the fact that a truncated integer division will give the same quotient for at most $x_s - 1$ values. Next a series of +inequalities will prove the hypothesis. + +\begin{equation} +y - \hat k x \le y - \hat k x_s\beta^s +\end{equation} + +This is trivially true since $x \ge x_s\beta^s$. Next we replace $\hat kx_s\beta^s$ by the previous inequality for $\hat kx_s$. + +\begin{equation} +y - \hat k x \le y_t\beta^t + \ldots + y_0 - (y_t\beta^t + y_{t-1}\beta^{t-1} - x_s\beta^t + \beta^s) +\end{equation} + +By simplifying the previous inequality the following inequality is formed. + +\begin{equation} +y - \hat k x \le y_{t-2}\beta^{t-2} + \ldots + y_0 + x_s\beta^s - \beta^s +\end{equation} + +Subsequently, + +\begin{equation} +y_{t-2}\beta^{t-2} + \ldots + y_0 + x_s\beta^s - \beta^s < x_s\beta^s \le x +\end{equation} + +Which proves that $y - \hat kx \le x$ and by consequence $\hat k \ge k$ which concludes the proof. \textbf{QED} + + +\subsection{Normalized Integers} +For the purposes of division a normalized input is when the divisors leading digit $x_n$ is greater than or equal to $\beta / 2$. By multiplying both +$x$ and $y$ by $j = \lfloor (\beta / 2) / x_n \rfloor$ the quotient remains unchanged and the remainder is simply $j$ times the original +remainder. The purpose of normalization is to ensure the leading digit of the divisor is sufficiently large such that the estimated quotient will +lie in the domain of a single digit. Consider the maximum dividend $(\beta - 1) \cdot \beta + (\beta - 1)$ and the minimum divisor $\beta / 2$. + +\begin{equation} +{{\beta^2 - 1} \over { \beta / 2}} \le 2\beta - {2 \over \beta} +\end{equation} + +At most the quotient approaches $2\beta$, however, in practice this will not occur since that would imply the previous quotient digit was too small. + +\subsection{Radix-$\beta$ Division with Remainder} +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_div}. \\ +\textbf{Input}. mp\_int $a, b$ \\ +\textbf{Output}. $c = \lfloor a/b \rfloor$, $d = a - bc$ \\ +\hline \\ +1. If $b = 0$ return(\textit{MP\_VAL}). \\ +2. If $\vert a \vert < \vert b \vert$ then do \\ +\hspace{3mm}2.1 $d \leftarrow a$ \\ +\hspace{3mm}2.2 $c \leftarrow 0$ \\ +\hspace{3mm}2.3 Return(\textit{MP\_OKAY}). \\ +\\ +Setup the quotient to receive the digits. \\ +3. Grow $q$ to $a.used + 2$ digits. \\ +4. $q \leftarrow 0$ \\ +5. $x \leftarrow \vert a \vert , y \leftarrow \vert b \vert$ \\ +6. $sign \leftarrow \left \lbrace \begin{array}{ll} + MP\_ZPOS & \mbox{if }a.sign = b.sign \\ + MP\_NEG & \mbox{otherwise} \\ + \end{array} \right .$ \\ +\\ +Normalize the inputs such that the leading digit of $y$ is greater than or equal to $\beta / 2$. \\ +7. $norm \leftarrow (lg(\beta) - 1) - (\lceil lg(y) \rceil \mbox{ (mod }lg(\beta)\mbox{)})$ \\ +8. $x \leftarrow x \cdot 2^{norm}, y \leftarrow y \cdot 2^{norm}$ \\ +\\ +Find the leading digit of the quotient. \\ +9. $n \leftarrow x.used - 1, t \leftarrow y.used - 1$ \\ +10. $y \leftarrow y \cdot \beta^{n - t}$ \\ +11. While ($x \ge y$) do \\ +\hspace{3mm}11.1 $q_{n - t} \leftarrow q_{n - t} + 1$ \\ +\hspace{3mm}11.2 $x \leftarrow x - y$ \\ +12. $y \leftarrow \lfloor y / \beta^{n-t} \rfloor$ \\ +\\ +Continued on the next page. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_div} +\end{figure} + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_div} (continued). \\ +\textbf{Input}. mp\_int $a, b$ \\ +\textbf{Output}. $c = \lfloor a/b \rfloor$, $d = a - bc$ \\ +\hline \\ +Now find the remainder fo the digits. \\ +13. for $i$ from $n$ down to $(t + 1)$ do \\ +\hspace{3mm}13.1 If $i > x.used$ then jump to the next iteration of this loop. \\ +\hspace{3mm}13.2 If $x_{i} = y_{t}$ then \\ +\hspace{6mm}13.2.1 $q_{i - t - 1} \leftarrow \beta - 1$ \\ +\hspace{3mm}13.3 else \\ +\hspace{6mm}13.3.1 $\hat r \leftarrow x_{i} \cdot \beta + x_{i - 1}$ \\ +\hspace{6mm}13.3.2 $\hat r \leftarrow \lfloor \hat r / y_{t} \rfloor$ \\ +\hspace{6mm}13.3.3 $q_{i - t - 1} \leftarrow \hat r$ \\ +\hspace{3mm}13.4 $q_{i - t - 1} \leftarrow q_{i - t - 1} + 1$ \\ +\\ +Fixup quotient estimation. \\ +\hspace{3mm}13.5 Loop \\ +\hspace{6mm}13.5.1 $q_{i - t - 1} \leftarrow q_{i - t - 1} - 1$ \\ +\hspace{6mm}13.5.2 t$1 \leftarrow 0$ \\ +\hspace{6mm}13.5.3 t$1_0 \leftarrow y_{t - 1}, $ t$1_1 \leftarrow y_t,$ t$1.used \leftarrow 2$ \\ +\hspace{6mm}13.5.4 $t1 \leftarrow t1 \cdot q_{i - t - 1}$ \\ +\hspace{6mm}13.5.5 t$2_0 \leftarrow x_{i - 2}, $ t$2_1 \leftarrow x_{i - 1}, $ t$2_2 \leftarrow x_i, $ t$2.used \leftarrow 3$ \\ +\hspace{6mm}13.5.6 If $\vert t1 \vert > \vert t2 \vert$ then goto step 13.5. \\ +\hspace{3mm}13.6 t$1 \leftarrow y \cdot q_{i - t - 1}$ \\ +\hspace{3mm}13.7 t$1 \leftarrow $ t$1 \cdot \beta^{i - t - 1}$ \\ +\hspace{3mm}13.8 $x \leftarrow x - $ t$1$ \\ +\hspace{3mm}13.9 If $x.sign = MP\_NEG$ then \\ +\hspace{6mm}13.10 t$1 \leftarrow y$ \\ +\hspace{6mm}13.11 t$1 \leftarrow $ t$1 \cdot \beta^{i - t - 1}$ \\ +\hspace{6mm}13.12 $x \leftarrow x + $ t$1$ \\ +\hspace{6mm}13.13 $q_{i - t - 1} \leftarrow q_{i - t - 1} - 1$ \\ +\\ +Finalize the result. \\ +14. Clamp excess digits of $q$ \\ +15. $c \leftarrow q, c.sign \leftarrow sign$ \\ +16. $x.sign \leftarrow a.sign$ \\ +17. $d \leftarrow \lfloor x / 2^{norm} \rfloor$ \\ +18. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_div (continued)} +\end{figure} +\textbf{Algorithm mp\_div.} +This algorithm will calculate quotient and remainder from an integer division given a dividend and divisor. The algorithm is a signed +division and will produce a fully qualified quotient and remainder. + +First the divisor $b$ must be non-zero which is enforced in step one. If the divisor is larger than the dividend than the quotient is implicitly +zero and the remainder is the dividend. + +After the first two trivial cases of inputs are handled the variable $q$ is setup to receive the digits of the quotient. Two unsigned copies of the +divisor $y$ and dividend $x$ are made as well. The core of the division algorithm is an unsigned division and will only work if the values are +positive. Now the two values $x$ and $y$ must be normalized such that the leading digit of $y$ is greater than or equal to $\beta / 2$. +This is performed by shifting both to the left by enough bits to get the desired normalization. + +At this point the division algorithm can begin producing digits of the quotient. Recall that maximum value of the estimation used is +$2\beta - {2 \over \beta}$ which means that a digit of the quotient must be first produced by another means. In this case $y$ is shifted +to the left (\textit{step ten}) so that it has the same number of digits as $x$. The loop on step eleven will subtract multiples of the +shifted copy of $y$ until $x$ is smaller. Since the leading digit of $y$ is greater than or equal to $\beta/2$ this loop will iterate at most two +times to produce the desired leading digit of the quotient. + +Now the remainder of the digits can be produced. The equation $\hat q = \lfloor {{x_i \beta + x_{i-1}}\over y_t} \rfloor$ is used to fairly +accurately approximate the true quotient digit. The estimation can in theory produce an estimation as high as $2\beta - {2 \over \beta}$ but by +induction the upper quotient digit is correct (\textit{as established on step eleven}) and the estimate must be less than $\beta$. + +Recall from section~\ref{sec:divest} that the estimation is never too low but may be too high. The next step of the estimation process is +to refine the estimation. The loop on step 13.5 uses $x_i\beta^2 + x_{i-1}\beta + x_{i-2}$ and $q_{i - t - 1}(y_t\beta + y_{t-1})$ as a higher +order approximation to adjust the quotient digit. + +After both phases of estimation the quotient digit may still be off by a value of one\footnote{This is similar to the error introduced +by optimizing Barrett reduction.}. Steps 13.6 and 13.7 subtract the multiple of the divisor from the dividend (\textit{Similar to step 3.3 of +algorithm~\ref{fig:raddiv}} and then subsequently add a multiple of the divisor if the quotient was too large. + +Now that the quotient has been determine finializing the result is a matter of clamping the quotient, fixing the sizes and de-normalizing the +remainder. An important aspect of this algorithm seemingly overlooked in other descriptions such as that of Algorithm 14.20 HAC \cite[pp. 598]{HAC} +is that when the estimations are being made (\textit{inside the loop on step 13.5}) that the digits $y_{t-1}$, $x_{i-2}$ and $x_{i-1}$ may lie +outside their respective boundaries. For example, if $t = 0$ or $i \le 1$ then the digits would be undefined. In those cases the digits should +respectively be replaced with a zero. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_div.c +\vspace{-3mm} +\begin{alltt} +016 +017 #ifdef BN_MP_DIV_SMALL +018 +019 /* slower bit-bang division... also smaller */ +020 int mp_div(mp_int * a, mp_int * b, mp_int * c, mp_int * d) +021 \{ +022 mp_int ta, tb, tq, q; +023 int res, n, n2; +024 +025 /* is divisor zero ? */ +026 if (mp_iszero (b) == 1) \{ +027 return MP_VAL; +028 \} +029 +030 /* if a < b then q=0, r = a */ +031 if (mp_cmp_mag (a, b) == MP_LT) \{ +032 if (d != NULL) \{ +033 res = mp_copy (a, d); +034 \} else \{ +035 res = MP_OKAY; +036 \} +037 if (c != NULL) \{ +038 mp_zero (c); +039 \} +040 return res; +041 \} +042 +043 /* init our temps */ +044 if ((res = mp_init_multi(&ta, &tb, &tq, &q, NULL) != MP_OKAY)) \{ +045 return res; +046 \} +047 +048 +049 mp_set(&tq, 1); +050 n = mp_count_bits(a) - mp_count_bits(b); +051 if (((res = mp_abs(a, &ta)) != MP_OKAY) || +052 ((res = mp_abs(b, &tb)) != MP_OKAY) || +053 ((res = mp_mul_2d(&tb, n, &tb)) != MP_OKAY) || +054 ((res = mp_mul_2d(&tq, n, &tq)) != MP_OKAY)) \{ +055 goto LBL_ERR; +056 \} +057 +058 while (n-- >= 0) \{ +059 if (mp_cmp(&tb, &ta) != MP_GT) \{ +060 if (((res = mp_sub(&ta, &tb, &ta)) != MP_OKAY) || +061 ((res = mp_add(&q, &tq, &q)) != MP_OKAY)) \{ +062 goto LBL_ERR; +063 \} +064 \} +065 if (((res = mp_div_2d(&tb, 1, &tb, NULL)) != MP_OKAY) || +066 ((res = mp_div_2d(&tq, 1, &tq, NULL)) != MP_OKAY)) \{ +067 goto LBL_ERR; +068 \} +069 \} +070 +071 /* now q == quotient and ta == remainder */ +072 n = a->sign; +073 n2 = (a->sign == b->sign ? MP_ZPOS : MP_NEG); +074 if (c != NULL) \{ +075 mp_exch(c, &q); +076 c->sign = (mp_iszero(c) == MP_YES) ? MP_ZPOS : n2; +077 \} +078 if (d != NULL) \{ +079 mp_exch(d, &ta); +080 d->sign = (mp_iszero(d) == MP_YES) ? MP_ZPOS : n; +081 \} +082 LBL_ERR: +083 mp_clear_multi(&ta, &tb, &tq, &q, NULL); +084 return res; +085 \} +086 +087 #else +088 +089 /* integer signed division. +090 * c*b + d == a [e.g. a/b, c=quotient, d=remainder] +091 * HAC pp.598 Algorithm 14.20 +092 * +093 * Note that the description in HAC is horribly +094 * incomplete. For example, it doesn't consider +095 * the case where digits are removed from 'x' in +096 * the inner loop. It also doesn't consider the +097 * case that y has fewer than three digits, etc.. +098 * +099 * The overall algorithm is as described as +100 * 14.20 from HAC but fixed to treat these cases. +101 */ +102 int mp_div (mp_int * a, mp_int * b, mp_int * c, mp_int * d) +103 \{ +104 mp_int q, x, y, t1, t2; +105 int res, n, t, i, norm, neg; +106 +107 /* is divisor zero ? */ +108 if (mp_iszero (b) == 1) \{ +109 return MP_VAL; +110 \} +111 +112 /* if a < b then q=0, r = a */ +113 if (mp_cmp_mag (a, b) == MP_LT) \{ +114 if (d != NULL) \{ +115 res = mp_copy (a, d); +116 \} else \{ +117 res = MP_OKAY; +118 \} +119 if (c != NULL) \{ +120 mp_zero (c); +121 \} +122 return res; +123 \} +124 +125 if ((res = mp_init_size (&q, a->used + 2)) != MP_OKAY) \{ +126 return res; +127 \} +128 q.used = a->used + 2; +129 +130 if ((res = mp_init (&t1)) != MP_OKAY) \{ +131 goto LBL_Q; +132 \} +133 +134 if ((res = mp_init (&t2)) != MP_OKAY) \{ +135 goto LBL_T1; +136 \} +137 +138 if ((res = mp_init_copy (&x, a)) != MP_OKAY) \{ +139 goto LBL_T2; +140 \} +141 +142 if ((res = mp_init_copy (&y, b)) != MP_OKAY) \{ +143 goto LBL_X; +144 \} +145 +146 /* fix the sign */ +147 neg = (a->sign == b->sign) ? MP_ZPOS : MP_NEG; +148 x.sign = y.sign = MP_ZPOS; +149 +150 /* normalize both x and y, ensure that y >= b/2, [b == 2**DIGIT_BIT] */ +151 norm = mp_count_bits(&y) % DIGIT_BIT; +152 if (norm < (int)(DIGIT_BIT-1)) \{ +153 norm = (DIGIT_BIT-1) - norm; +154 if ((res = mp_mul_2d (&x, norm, &x)) != MP_OKAY) \{ +155 goto LBL_Y; +156 \} +157 if ((res = mp_mul_2d (&y, norm, &y)) != MP_OKAY) \{ +158 goto LBL_Y; +159 \} +160 \} else \{ +161 norm = 0; +162 \} +163 +164 /* note hac does 0 based, so if used==5 then its 0,1,2,3,4, e.g. use 4 */ +165 n = x.used - 1; +166 t = y.used - 1; +167 +168 /* while (x >= y*b**n-t) do \{ q[n-t] += 1; x -= y*b**\{n-t\} \} */ +169 if ((res = mp_lshd (&y, n - t)) != MP_OKAY) \{ /* y = y*b**\{n-t\} */ +170 goto LBL_Y; +171 \} +172 +173 while (mp_cmp (&x, &y) != MP_LT) \{ +174 ++(q.dp[n - t]); +175 if ((res = mp_sub (&x, &y, &x)) != MP_OKAY) \{ +176 goto LBL_Y; +177 \} +178 \} +179 +180 /* reset y by shifting it back down */ +181 mp_rshd (&y, n - t); +182 +183 /* step 3. for i from n down to (t + 1) */ +184 for (i = n; i >= (t + 1); i--) \{ +185 if (i > x.used) \{ +186 continue; +187 \} +188 +189 /* step 3.1 if xi == yt then set q\{i-t-1\} to b-1, +190 * otherwise set q\{i-t-1\} to (xi*b + x\{i-1\})/yt */ +191 if (x.dp[i] == y.dp[t]) \{ +192 q.dp[i - t - 1] = ((((mp_digit)1) << DIGIT_BIT) - 1); +193 \} else \{ +194 mp_word tmp; +195 tmp = ((mp_word) x.dp[i]) << ((mp_word) DIGIT_BIT); +196 tmp |= ((mp_word) x.dp[i - 1]); +197 tmp /= ((mp_word) y.dp[t]); +198 if (tmp > (mp_word) MP_MASK) +199 tmp = MP_MASK; +200 q.dp[i - t - 1] = (mp_digit) (tmp & (mp_word) (MP_MASK)); +201 \} +202 +203 /* while (q\{i-t-1\} * (yt * b + y\{t-1\})) > +204 xi * b**2 + xi-1 * b + xi-2 +205 +206 do q\{i-t-1\} -= 1; +207 */ +208 q.dp[i - t - 1] = (q.dp[i - t - 1] + 1) & MP_MASK; +209 do \{ +210 q.dp[i - t - 1] = (q.dp[i - t - 1] - 1) & MP_MASK; +211 +212 /* find left hand */ +213 mp_zero (&t1); +214 t1.dp[0] = (t - 1 < 0) ? 0 : y.dp[t - 1]; +215 t1.dp[1] = y.dp[t]; +216 t1.used = 2; +217 if ((res = mp_mul_d (&t1, q.dp[i - t - 1], &t1)) != MP_OKAY) \{ +218 goto LBL_Y; +219 \} +220 +221 /* find right hand */ +222 t2.dp[0] = (i - 2 < 0) ? 0 : x.dp[i - 2]; +223 t2.dp[1] = (i - 1 < 0) ? 0 : x.dp[i - 1]; +224 t2.dp[2] = x.dp[i]; +225 t2.used = 3; +226 \} while (mp_cmp_mag(&t1, &t2) == MP_GT); +227 +228 /* step 3.3 x = x - q\{i-t-1\} * y * b**\{i-t-1\} */ +229 if ((res = mp_mul_d (&y, q.dp[i - t - 1], &t1)) != MP_OKAY) \{ +230 goto LBL_Y; +231 \} +232 +233 if ((res = mp_lshd (&t1, i - t - 1)) != MP_OKAY) \{ +234 goto LBL_Y; +235 \} +236 +237 if ((res = mp_sub (&x, &t1, &x)) != MP_OKAY) \{ +238 goto LBL_Y; +239 \} +240 +241 /* if x < 0 then \{ x = x + y*b**\{i-t-1\}; q\{i-t-1\} -= 1; \} */ +242 if (x.sign == MP_NEG) \{ +243 if ((res = mp_copy (&y, &t1)) != MP_OKAY) \{ +244 goto LBL_Y; +245 \} +246 if ((res = mp_lshd (&t1, i - t - 1)) != MP_OKAY) \{ +247 goto LBL_Y; +248 \} +249 if ((res = mp_add (&x, &t1, &x)) != MP_OKAY) \{ +250 goto LBL_Y; +251 \} +252 +253 q.dp[i - t - 1] = (q.dp[i - t - 1] - 1UL) & MP_MASK; +254 \} +255 \} +256 +257 /* now q is the quotient and x is the remainder +258 * [which we have to normalize] +259 */ +260 +261 /* get sign before writing to c */ +262 x.sign = x.used == 0 ? MP_ZPOS : a->sign; +263 +264 if (c != NULL) \{ +265 mp_clamp (&q); +266 mp_exch (&q, c); +267 c->sign = neg; +268 \} +269 +270 if (d != NULL) \{ +271 mp_div_2d (&x, norm, &x, NULL); +272 mp_exch (&x, d); +273 \} +274 +275 res = MP_OKAY; +276 +277 LBL_Y:mp_clear (&y); +278 LBL_X:mp_clear (&x); +279 LBL_T2:mp_clear (&t2); +280 LBL_T1:mp_clear (&t1); +281 LBL_Q:mp_clear (&q); +282 return res; +283 \} +284 +285 #endif +286 +287 #endif +\end{alltt} +\end{small} + +The implementation of this algorithm differs slightly from the pseudo code presented previously. In this algorithm either of the quotient $c$ or +remainder $d$ may be passed as a \textbf{NULL} pointer which indicates their value is not desired. For example, the C code to call the division +algorithm with only the quotient is + +\begin{verbatim} +mp_div(&a, &b, &c, NULL); /* c = [a/b] */ +\end{verbatim} + +Lines 37 and 44 handle the two trivial cases of inputs which are division by zero and dividend smaller than the divisor +respectively. After the two trivial cases all of the temporary variables are initialized. Line 105 determines the sign of +the quotient and line 76 ensures that both $x$ and $y$ are positive. + +The number of bits in the leading digit is calculated on line 105. Implictly an mp\_int with $r$ digits will require $lg(\beta)(r-1) + k$ bits +of precision which when reduced modulo $lg(\beta)$ produces the value of $k$. In this case $k$ is the number of bits in the leading digit which is +exactly what is required. For the algorithm to operate $k$ must equal $lg(\beta) - 1$ and when it does not the inputs must be normalized by shifting +them to the left by $lg(\beta) - 1 - k$ bits. + +Throughout the variables $n$ and $t$ will represent the highest digit of $x$ and $y$ respectively. These are first used to produce the +leading digit of the quotient. The loop beginning on line 183 will produce the remainder of the quotient digits. + +The conditional ``continue'' on line 114 is used to prevent the algorithm from reading past the leading edge of $x$ which can occur when the +algorithm eliminates multiple non-zero digits in a single iteration. This ensures that $x_i$ is always non-zero since by definition the digits +above the $i$'th position $x$ must be zero in order for the quotient to be precise\footnote{Precise as far as integer division is concerned.}. + +Lines 130, 130 and 134 through 134 manually construct the high accuracy estimations by setting the digits of the two mp\_int +variables directly. + +\section{Single Digit Helpers} + +This section briefly describes a series of single digit helper algorithms which come in handy when working with small constants. All of +the helper functions assume the single digit input is positive and will treat them as such. + +\subsection{Single Digit Addition and Subtraction} + +Both addition and subtraction are performed by ``cheating'' and using mp\_set followed by the higher level addition or subtraction +algorithms. As a result these algorithms are subtantially simpler with a slight cost in performance. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_add\_d}. \\ +\textbf{Input}. mp\_int $a$ and a mp\_digit $b$ \\ +\textbf{Output}. $c = a + b$ \\ +\hline \\ +1. $t \leftarrow b$ (\textit{mp\_set}) \\ +2. $c \leftarrow a + t$ \\ +3. Return(\textit{MP\_OKAY}) \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_add\_d} +\end{figure} + +\textbf{Algorithm mp\_add\_d.} +This algorithm initiates a temporary mp\_int with the value of the single digit and uses algorithm mp\_add to add the two values together. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_add\_d.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* single digit addition */ +018 int +019 mp_add_d (mp_int * a, mp_digit b, mp_int * c) +020 \{ +021 int res, ix, oldused; +022 mp_digit *tmpa, *tmpc, mu; +023 +024 /* grow c as required */ +025 if (c->alloc < a->used + 1) \{ +026 if ((res = mp_grow(c, a->used + 1)) != MP_OKAY) \{ +027 return res; +028 \} +029 \} +030 +031 /* if a is negative and |a| >= b, call c = |a| - b */ +032 if (a->sign == MP_NEG && (a->used > 1 || a->dp[0] >= b)) \{ +033 /* temporarily fix sign of a */ +034 a->sign = MP_ZPOS; +035 +036 /* c = |a| - b */ +037 res = mp_sub_d(a, b, c); +038 +039 /* fix sign */ +040 a->sign = c->sign = MP_NEG; +041 +042 return res; +043 \} +044 +045 /* old number of used digits in c */ +046 oldused = c->used; +047 +048 /* sign always positive */ +049 c->sign = MP_ZPOS; +050 +051 /* source alias */ +052 tmpa = a->dp; +053 +054 /* destination alias */ +055 tmpc = c->dp; +056 +057 /* if a is positive */ +058 if (a->sign == MP_ZPOS) \{ +059 /* add digit, after this we're propagating +060 * the carry. +061 */ +062 *tmpc = *tmpa++ + b; +063 mu = *tmpc >> DIGIT_BIT; +064 *tmpc++ &= MP_MASK; +065 +066 /* now handle rest of the digits */ +067 for (ix = 1; ix < a->used; ix++) \{ +068 *tmpc = *tmpa++ + mu; +069 mu = *tmpc >> DIGIT_BIT; +070 *tmpc++ &= MP_MASK; +071 \} +072 /* set final carry */ +073 ix++; +074 *tmpc++ = mu; +075 +076 /* setup size */ +077 c->used = a->used + 1; +078 \} else \{ +079 /* a was negative and |a| < b */ +080 c->used = 1; +081 +082 /* the result is a single digit */ +083 if (a->used == 1) \{ +084 *tmpc++ = b - a->dp[0]; +085 \} else \{ +086 *tmpc++ = b; +087 \} +088 +089 /* setup count so the clearing of oldused +090 * can fall through correctly +091 */ +092 ix = 1; +093 \} +094 +095 /* now zero to oldused */ +096 while (ix++ < oldused) \{ +097 *tmpc++ = 0; +098 \} +099 mp_clamp(c); +100 +101 return MP_OKAY; +102 \} +103 +104 #endif +\end{alltt} +\end{small} + +Clever use of the letter 't'. + +\subsubsection{Subtraction} +The single digit subtraction algorithm mp\_sub\_d is essentially the same except it uses mp\_sub to subtract the digit from the mp\_int. + +\subsection{Single Digit Multiplication} +Single digit multiplication arises enough in division and radix conversion that it ought to be implement as a special case of the baseline +multiplication algorithm. Essentially this algorithm is a modified version of algorithm s\_mp\_mul\_digs where one of the multiplicands +only has one digit. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_mul\_d}. \\ +\textbf{Input}. mp\_int $a$ and a mp\_digit $b$ \\ +\textbf{Output}. $c = ab$ \\ +\hline \\ +1. $pa \leftarrow a.used$ \\ +2. Grow $c$ to at least $pa + 1$ digits. \\ +3. $oldused \leftarrow c.used$ \\ +4. $c.used \leftarrow pa + 1$ \\ +5. $c.sign \leftarrow a.sign$ \\ +6. $\mu \leftarrow 0$ \\ +7. for $ix$ from $0$ to $pa - 1$ do \\ +\hspace{3mm}7.1 $\hat r \leftarrow \mu + a_{ix}b$ \\ +\hspace{3mm}7.2 $c_{ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ +\hspace{3mm}7.3 $\mu \leftarrow \lfloor \hat r / \beta \rfloor$ \\ +8. $c_{pa} \leftarrow \mu$ \\ +9. for $ix$ from $pa + 1$ to $oldused$ do \\ +\hspace{3mm}9.1 $c_{ix} \leftarrow 0$ \\ +10. Clamp excess digits of $c$. \\ +11. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_mul\_d} +\end{figure} +\textbf{Algorithm mp\_mul\_d.} +This algorithm quickly multiplies an mp\_int by a small single digit value. It is specially tailored to the job and has a minimal of overhead. +Unlike the full multiplication algorithms this algorithm does not require any significnat temporary storage or memory allocations. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_mul\_d.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* multiply by a digit */ +018 int +019 mp_mul_d (mp_int * a, mp_digit b, mp_int * c) +020 \{ +021 mp_digit u, *tmpa, *tmpc; +022 mp_word r; +023 int ix, res, olduse; +024 +025 /* make sure c is big enough to hold a*b */ +026 if (c->alloc < a->used + 1) \{ +027 if ((res = mp_grow (c, a->used + 1)) != MP_OKAY) \{ +028 return res; +029 \} +030 \} +031 +032 /* get the original destinations used count */ +033 olduse = c->used; +034 +035 /* set the sign */ +036 c->sign = a->sign; +037 +038 /* alias for a->dp [source] */ +039 tmpa = a->dp; +040 +041 /* alias for c->dp [dest] */ +042 tmpc = c->dp; +043 +044 /* zero carry */ +045 u = 0; +046 +047 /* compute columns */ +048 for (ix = 0; ix < a->used; ix++) \{ +049 /* compute product and carry sum for this term */ +050 r = ((mp_word) u) + ((mp_word)*tmpa++) * ((mp_word)b); +051 +052 /* mask off higher bits to get a single digit */ +053 *tmpc++ = (mp_digit) (r & ((mp_word) MP_MASK)); +054 +055 /* send carry into next iteration */ +056 u = (mp_digit) (r >> ((mp_word) DIGIT_BIT)); +057 \} +058 +059 /* store final carry [if any] */ +060 *tmpc++ = u; +061 +062 /* now zero digits above the top */ +063 while (ix++ < olduse) \{ +064 *tmpc++ = 0; +065 \} +066 +067 /* set used count */ +068 c->used = a->used + 1; +069 mp_clamp(c); +070 +071 return MP_OKAY; +072 \} +073 #endif +\end{alltt} +\end{small} + +In this implementation the destination $c$ may point to the same mp\_int as the source $a$ since the result is written after the digit is +read from the source. This function uses pointer aliases $tmpa$ and $tmpc$ for the digits of $a$ and $c$ respectively. + +\subsection{Single Digit Division} +Like the single digit multiplication algorithm, single digit division is also a fairly common algorithm used in radix conversion. Since the +divisor is only a single digit a specialized variant of the division algorithm can be used to compute the quotient. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_div\_d}. \\ +\textbf{Input}. mp\_int $a$ and a mp\_digit $b$ \\ +\textbf{Output}. $c = \lfloor a / b \rfloor, d = a - cb$ \\ +\hline \\ +1. If $b = 0$ then return(\textit{MP\_VAL}).\\ +2. If $b = 3$ then use algorithm mp\_div\_3 instead. \\ +3. Init $q$ to $a.used$ digits. \\ +4. $q.used \leftarrow a.used$ \\ +5. $q.sign \leftarrow a.sign$ \\ +6. $\hat w \leftarrow 0$ \\ +7. for $ix$ from $a.used - 1$ down to $0$ do \\ +\hspace{3mm}7.1 $\hat w \leftarrow \hat w \beta + a_{ix}$ \\ +\hspace{3mm}7.2 If $\hat w \ge b$ then \\ +\hspace{6mm}7.2.1 $t \leftarrow \lfloor \hat w / b \rfloor$ \\ +\hspace{6mm}7.2.2 $\hat w \leftarrow \hat w \mbox{ (mod }b\mbox{)}$ \\ +\hspace{3mm}7.3 else\\ +\hspace{6mm}7.3.1 $t \leftarrow 0$ \\ +\hspace{3mm}7.4 $q_{ix} \leftarrow t$ \\ +8. $d \leftarrow \hat w$ \\ +9. Clamp excess digits of $q$. \\ +10. $c \leftarrow q$ \\ +11. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_div\_d} +\end{figure} +\textbf{Algorithm mp\_div\_d.} +This algorithm divides the mp\_int $a$ by the single mp\_digit $b$ using an optimized approach. Essentially in every iteration of the +algorithm another digit of the dividend is reduced and another digit of quotient produced. Provided $b < \beta$ the value of $\hat w$ +after step 7.1 will be limited such that $0 \le \lfloor \hat w / b \rfloor < \beta$. + +If the divisor $b$ is equal to three a variant of this algorithm is used which is called mp\_div\_3. It replaces the division by three with +a multiplication by $\lfloor \beta / 3 \rfloor$ and the appropriate shift and residual fixup. In essence it is much like the Barrett reduction +from chapter seven. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_div\_d.c +\vspace{-3mm} +\begin{alltt} +016 +017 static int s_is_power_of_two(mp_digit b, int *p) +018 \{ +019 int x; +020 +021 for (x = 1; x < DIGIT_BIT; x++) \{ +022 if (b == (((mp_digit)1)<<x)) \{ +023 *p = x; +024 return 1; +025 \} +026 \} +027 return 0; +028 \} +029 +030 /* single digit division (based on routine from MPI) */ +031 int mp_div_d (mp_int * a, mp_digit b, mp_int * c, mp_digit * d) +032 \{ +033 mp_int q; +034 mp_word w; +035 mp_digit t; +036 int res, ix; +037 +038 /* cannot divide by zero */ +039 if (b == 0) \{ +040 return MP_VAL; +041 \} +042 +043 /* quick outs */ +044 if (b == 1 || mp_iszero(a) == 1) \{ +045 if (d != NULL) \{ +046 *d = 0; +047 \} +048 if (c != NULL) \{ +049 return mp_copy(a, c); +050 \} +051 return MP_OKAY; +052 \} +053 +054 /* power of two ? */ +055 if (s_is_power_of_two(b, &ix) == 1) \{ +056 if (d != NULL) \{ +057 *d = a->dp[0] & ((((mp_digit)1)<<ix) - 1); +058 \} +059 if (c != NULL) \{ +060 return mp_div_2d(a, ix, c, NULL); +061 \} +062 return MP_OKAY; +063 \} +064 +065 #ifdef BN_MP_DIV_3_C +066 /* three? */ +067 if (b == 3) \{ +068 return mp_div_3(a, c, d); +069 \} +070 #endif +071 +072 /* no easy answer [c'est la vie]. Just division */ +073 if ((res = mp_init_size(&q, a->used)) != MP_OKAY) \{ +074 return res; +075 \} +076 +077 q.used = a->used; +078 q.sign = a->sign; +079 w = 0; +080 for (ix = a->used - 1; ix >= 0; ix--) \{ +081 w = (w << ((mp_word)DIGIT_BIT)) | ((mp_word)a->dp[ix]); +082 +083 if (w >= b) \{ +084 t = (mp_digit)(w / b); +085 w -= ((mp_word)t) * ((mp_word)b); +086 \} else \{ +087 t = 0; +088 \} +089 q.dp[ix] = (mp_digit)t; +090 \} +091 +092 if (d != NULL) \{ +093 *d = (mp_digit)w; +094 \} +095 +096 if (c != NULL) \{ +097 mp_clamp(&q); +098 mp_exch(&q, c); +099 \} +100 mp_clear(&q); +101 +102 return res; +103 \} +104 +105 #endif +\end{alltt} +\end{small} + +Like the implementation of algorithm mp\_div this algorithm allows either of the quotient or remainder to be passed as a \textbf{NULL} pointer to +indicate the respective value is not required. This allows a trivial single digit modular reduction algorithm, mp\_mod\_d to be created. + +The division and remainder on lines 43 and @45,%@ can be replaced often by a single division on most processors. For example, the 32-bit x86 based +processors can divide a 64-bit quantity by a 32-bit quantity and produce the quotient and remainder simultaneously. Unfortunately the GCC +compiler does not recognize that optimization and will actually produce two function calls to find the quotient and remainder respectively. + +\subsection{Single Digit Root Extraction} + +Finding the $n$'th root of an integer is fairly easy as far as numerical analysis is concerned. Algorithms such as the Newton-Raphson approximation +(\ref{eqn:newton}) series will converge very quickly to a root for any continuous function $f(x)$. + +\begin{equation} +x_{i+1} = x_i - {f(x_i) \over f'(x_i)} +\label{eqn:newton} +\end{equation} + +In this case the $n$'th root is desired and $f(x) = x^n - a$ where $a$ is the integer of which the root is desired. The derivative of $f(x)$ is +simply $f'(x) = nx^{n - 1}$. Of particular importance is that this algorithm will be used over the integers not over the a more continuous domain +such as the real numbers. As a result the root found can be above the true root by few and must be manually adjusted. Ideally at the end of the +algorithm the $n$'th root $b$ of an integer $a$ is desired such that $b^n \le a$. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_n\_root}. \\ +\textbf{Input}. mp\_int $a$ and a mp\_digit $b$ \\ +\textbf{Output}. $c^b \le a$ \\ +\hline \\ +1. If $b$ is even and $a.sign = MP\_NEG$ return(\textit{MP\_VAL}). \\ +2. $sign \leftarrow a.sign$ \\ +3. $a.sign \leftarrow MP\_ZPOS$ \\ +4. t$2 \leftarrow 2$ \\ +5. Loop \\ +\hspace{3mm}5.1 t$1 \leftarrow $ t$2$ \\ +\hspace{3mm}5.2 t$3 \leftarrow $ t$1^{b - 1}$ \\ +\hspace{3mm}5.3 t$2 \leftarrow $ t$3 $ $\cdot$ t$1$ \\ +\hspace{3mm}5.4 t$2 \leftarrow $ t$2 - a$ \\ +\hspace{3mm}5.5 t$3 \leftarrow $ t$3 \cdot b$ \\ +\hspace{3mm}5.6 t$3 \leftarrow \lfloor $t$2 / $t$3 \rfloor$ \\ +\hspace{3mm}5.7 t$2 \leftarrow $ t$1 - $ t$3$ \\ +\hspace{3mm}5.8 If t$1 \ne $ t$2$ then goto step 5. \\ +6. Loop \\ +\hspace{3mm}6.1 t$2 \leftarrow $ t$1^b$ \\ +\hspace{3mm}6.2 If t$2 > a$ then \\ +\hspace{6mm}6.2.1 t$1 \leftarrow $ t$1 - 1$ \\ +\hspace{6mm}6.2.2 Goto step 6. \\ +7. $a.sign \leftarrow sign$ \\ +8. $c \leftarrow $ t$1$ \\ +9. $c.sign \leftarrow sign$ \\ +10. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_n\_root} +\end{figure} +\textbf{Algorithm mp\_n\_root.} +This algorithm finds the integer $n$'th root of an input using the Newton-Raphson approach. It is partially optimized based on the observation +that the numerator of ${f(x) \over f'(x)}$ can be derived from a partial denominator. That is at first the denominator is calculated by finding +$x^{b - 1}$. This value can then be multiplied by $x$ and have $a$ subtracted from it to find the numerator. This saves a total of $b - 1$ +multiplications by t$1$ inside the loop. + +The initial value of the approximation is t$2 = 2$ which allows the algorithm to start with very small values and quickly converge on the +root. Ideally this algorithm is meant to find the $n$'th root of an input where $n$ is bounded by $2 \le n \le 5$. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_n\_root.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* find the n'th root of an integer +018 * +019 * Result found such that (c)**b <= a and (c+1)**b > a +020 * +021 * This algorithm uses Newton's approximation +022 * x[i+1] = x[i] - f(x[i])/f'(x[i]) +023 * which will find the root in log(N) time where +024 * each step involves a fair bit. This is not meant to +025 * find huge roots [square and cube, etc]. +026 */ +027 int mp_n_root (mp_int * a, mp_digit b, mp_int * c) +028 \{ +029 mp_int t1, t2, t3; +030 int res, neg; +031 +032 /* input must be positive if b is even */ +033 if ((b & 1) == 0 && a->sign == MP_NEG) \{ +034 return MP_VAL; +035 \} +036 +037 if ((res = mp_init (&t1)) != MP_OKAY) \{ +038 return res; +039 \} +040 +041 if ((res = mp_init (&t2)) != MP_OKAY) \{ +042 goto LBL_T1; +043 \} +044 +045 if ((res = mp_init (&t3)) != MP_OKAY) \{ +046 goto LBL_T2; +047 \} +048 +049 /* if a is negative fudge the sign but keep track */ +050 neg = a->sign; +051 a->sign = MP_ZPOS; +052 +053 /* t2 = 2 */ +054 mp_set (&t2, 2); +055 +056 do \{ +057 /* t1 = t2 */ +058 if ((res = mp_copy (&t2, &t1)) != MP_OKAY) \{ +059 goto LBL_T3; +060 \} +061 +062 /* t2 = t1 - ((t1**b - a) / (b * t1**(b-1))) */ +063 +064 /* t3 = t1**(b-1) */ +065 if ((res = mp_expt_d (&t1, b - 1, &t3)) != MP_OKAY) \{ +066 goto LBL_T3; +067 \} +068 +069 /* numerator */ +070 /* t2 = t1**b */ +071 if ((res = mp_mul (&t3, &t1, &t2)) != MP_OKAY) \{ +072 goto LBL_T3; +073 \} +074 +075 /* t2 = t1**b - a */ +076 if ((res = mp_sub (&t2, a, &t2)) != MP_OKAY) \{ +077 goto LBL_T3; +078 \} +079 +080 /* denominator */ +081 /* t3 = t1**(b-1) * b */ +082 if ((res = mp_mul_d (&t3, b, &t3)) != MP_OKAY) \{ +083 goto LBL_T3; +084 \} +085 +086 /* t3 = (t1**b - a)/(b * t1**(b-1)) */ +087 if ((res = mp_div (&t2, &t3, &t3, NULL)) != MP_OKAY) \{ +088 goto LBL_T3; +089 \} +090 +091 if ((res = mp_sub (&t1, &t3, &t2)) != MP_OKAY) \{ +092 goto LBL_T3; +093 \} +094 \} while (mp_cmp (&t1, &t2) != MP_EQ); +095 +096 /* result can be off by a few so check */ +097 for (;;) \{ +098 if ((res = mp_expt_d (&t1, b, &t2)) != MP_OKAY) \{ +099 goto LBL_T3; +100 \} +101 +102 if (mp_cmp (&t2, a) == MP_GT) \{ +103 if ((res = mp_sub_d (&t1, 1, &t1)) != MP_OKAY) \{ +104 goto LBL_T3; +105 \} +106 \} else \{ +107 break; +108 \} +109 \} +110 +111 /* reset the sign of a first */ +112 a->sign = neg; +113 +114 /* set the result */ +115 mp_exch (&t1, c); +116 +117 /* set the sign of the result */ +118 c->sign = neg; +119 +120 res = MP_OKAY; +121 +122 LBL_T3:mp_clear (&t3); +123 LBL_T2:mp_clear (&t2); +124 LBL_T1:mp_clear (&t1); +125 return res; +126 \} +127 #endif +\end{alltt} +\end{small} + +\section{Random Number Generation} + +Random numbers come up in a variety of activities from public key cryptography to simple simulations and various randomized algorithms. Pollard-Rho +factoring for example, can make use of random values as starting points to find factors of a composite integer. In this case the algorithm presented +is solely for simulations and not intended for cryptographic use. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_rand}. \\ +\textbf{Input}. An integer $b$ \\ +\textbf{Output}. A pseudo-random number of $b$ digits \\ +\hline \\ +1. $a \leftarrow 0$ \\ +2. If $b \le 0$ return(\textit{MP\_OKAY}) \\ +3. Pick a non-zero random digit $d$. \\ +4. $a \leftarrow a + d$ \\ +5. for $ix$ from 1 to $d - 1$ do \\ +\hspace{3mm}5.1 $a \leftarrow a \cdot \beta$ \\ +\hspace{3mm}5.2 Pick a random digit $d$. \\ +\hspace{3mm}5.3 $a \leftarrow a + d$ \\ +6. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_rand} +\end{figure} +\textbf{Algorithm mp\_rand.} +This algorithm produces a pseudo-random integer of $b$ digits. By ensuring that the first digit is non-zero the algorithm also guarantees that the +final result has at least $b$ digits. It relies heavily on a third-part random number generator which should ideally generate uniformly all of +the integers from $0$ to $\beta - 1$. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_rand.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* makes a pseudo-random int of a given size */ +018 int +019 mp_rand (mp_int * a, int digits) +020 \{ +021 int res; +022 mp_digit d; +023 +024 mp_zero (a); +025 if (digits <= 0) \{ +026 return MP_OKAY; +027 \} +028 +029 /* first place a random non-zero digit */ +030 do \{ +031 d = ((mp_digit) abs (rand ())); +032 \} while (d == 0); +033 +034 if ((res = mp_add_d (a, d, a)) != MP_OKAY) \{ +035 return res; +036 \} +037 +038 while (digits-- > 0) \{ +039 if ((res = mp_lshd (a, 1)) != MP_OKAY) \{ +040 return res; +041 \} +042 +043 if ((res = mp_add_d (a, ((mp_digit) abs (rand ())), a)) != MP_OKAY) \{ +044 return res; +045 \} +046 \} +047 +048 return MP_OKAY; +049 \} +050 #endif +\end{alltt} +\end{small} + +\section{Formatted Representations} +The ability to emit a radix-$n$ textual representation of an integer is useful for interacting with human parties. For example, the ability to +be given a string of characters such as ``114585'' and turn it into the radix-$\beta$ equivalent would make it easier to enter numbers +into a program. + +\subsection{Reading Radix-n Input} +For the purposes of this text we will assume that a simple lower ASCII map (\ref{fig:ASC}) is used for the values of from $0$ to $63$ to +printable characters. For example, when the character ``N'' is read it represents the integer $23$. The first $16$ characters of the +map are for the common representations up to hexadecimal. After that they match the ``base64'' encoding scheme which are suitable chosen +such that they are printable. While outputting as base64 may not be too helpful for human operators it does allow communication via non binary +mediums. + +\newpage\begin{figure}[here] +\begin{center} +\begin{tabular}{cc|cc|cc|cc} +\hline \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} \\ +\hline +0 & 0 & 1 & 1 & 2 & 2 & 3 & 3 \\ +4 & 4 & 5 & 5 & 6 & 6 & 7 & 7 \\ +8 & 8 & 9 & 9 & 10 & A & 11 & B \\ +12 & C & 13 & D & 14 & E & 15 & F \\ +16 & G & 17 & H & 18 & I & 19 & J \\ +20 & K & 21 & L & 22 & M & 23 & N \\ +24 & O & 25 & P & 26 & Q & 27 & R \\ +28 & S & 29 & T & 30 & U & 31 & V \\ +32 & W & 33 & X & 34 & Y & 35 & Z \\ +36 & a & 37 & b & 38 & c & 39 & d \\ +40 & e & 41 & f & 42 & g & 43 & h \\ +44 & i & 45 & j & 46 & k & 47 & l \\ +48 & m & 49 & n & 50 & o & 51 & p \\ +52 & q & 53 & r & 54 & s & 55 & t \\ +56 & u & 57 & v & 58 & w & 59 & x \\ +60 & y & 61 & z & 62 & $+$ & 63 & $/$ \\ +\hline +\end{tabular} +\end{center} +\caption{Lower ASCII Map} +\label{fig:ASC} +\end{figure} + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_read\_radix}. \\ +\textbf{Input}. A string $str$ of length $sn$ and radix $r$. \\ +\textbf{Output}. The radix-$\beta$ equivalent mp\_int. \\ +\hline \\ +1. If $r < 2$ or $r > 64$ return(\textit{MP\_VAL}). \\ +2. $ix \leftarrow 0$ \\ +3. If $str_0 =$ ``-'' then do \\ +\hspace{3mm}3.1 $ix \leftarrow ix + 1$ \\ +\hspace{3mm}3.2 $sign \leftarrow MP\_NEG$ \\ +4. else \\ +\hspace{3mm}4.1 $sign \leftarrow MP\_ZPOS$ \\ +5. $a \leftarrow 0$ \\ +6. for $iy$ from $ix$ to $sn - 1$ do \\ +\hspace{3mm}6.1 Let $y$ denote the position in the map of $str_{iy}$. \\ +\hspace{3mm}6.2 If $str_{iy}$ is not in the map or $y \ge r$ then goto step 7. \\ +\hspace{3mm}6.3 $a \leftarrow a \cdot r$ \\ +\hspace{3mm}6.4 $a \leftarrow a + y$ \\ +7. If $a \ne 0$ then $a.sign \leftarrow sign$ \\ +8. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_read\_radix} +\end{figure} +\textbf{Algorithm mp\_read\_radix.} +This algorithm will read an ASCII string and produce the radix-$\beta$ mp\_int representation of the same integer. A minus symbol ``-'' may precede the +string to indicate the value is negative, otherwise it is assumed to be positive. The algorithm will read up to $sn$ characters from the input +and will stop when it reads a character it cannot map the algorithm stops reading characters from the string. This allows numbers to be embedded +as part of larger input without any significant problem. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_read\_radix.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* read a string [ASCII] in a given radix */ +018 int mp_read_radix (mp_int * a, char *str, int radix) +019 \{ +020 int y, res, neg; +021 char ch; +022 +023 /* make sure the radix is ok */ +024 if (radix < 2 || radix > 64) \{ +025 return MP_VAL; +026 \} +027 +028 /* if the leading digit is a +029 * minus set the sign to negative. +030 */ +031 if (*str == '-') \{ +032 ++str; +033 neg = MP_NEG; +034 \} else \{ +035 neg = MP_ZPOS; +036 \} +037 +038 /* set the integer to the default of zero */ +039 mp_zero (a); +040 +041 /* process each digit of the string */ +042 while (*str) \{ +043 /* if the radix < 36 the conversion is case insensitive +044 * this allows numbers like 1AB and 1ab to represent the same value +045 * [e.g. in hex] +046 */ +047 ch = (char) ((radix < 36) ? toupper (*str) : *str); +048 for (y = 0; y < 64; y++) \{ +049 if (ch == mp_s_rmap[y]) \{ +050 break; +051 \} +052 \} +053 +054 /* if the char was found in the map +055 * and is less than the given radix add it +056 * to the number, otherwise exit the loop. +057 */ +058 if (y < radix) \{ +059 if ((res = mp_mul_d (a, (mp_digit) radix, a)) != MP_OKAY) \{ +060 return res; +061 \} +062 if ((res = mp_add_d (a, (mp_digit) y, a)) != MP_OKAY) \{ +063 return res; +064 \} +065 \} else \{ +066 break; +067 \} +068 ++str; +069 \} +070 +071 /* set the sign only if a != 0 */ +072 if (mp_iszero(a) != 1) \{ +073 a->sign = neg; +074 \} +075 return MP_OKAY; +076 \} +077 #endif +\end{alltt} +\end{small} + +\subsection{Generating Radix-$n$ Output} +Generating radix-$n$ output is fairly trivial with a division and remainder algorithm. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_toradix}. \\ +\textbf{Input}. A mp\_int $a$ and an integer $r$\\ +\textbf{Output}. The radix-$r$ representation of $a$ \\ +\hline \\ +1. If $r < 2$ or $r > 64$ return(\textit{MP\_VAL}). \\ +2. If $a = 0$ then $str = $ ``$0$'' and return(\textit{MP\_OKAY}). \\ +3. $t \leftarrow a$ \\ +4. $str \leftarrow$ ``'' \\ +5. if $t.sign = MP\_NEG$ then \\ +\hspace{3mm}5.1 $str \leftarrow str + $ ``-'' \\ +\hspace{3mm}5.2 $t.sign = MP\_ZPOS$ \\ +6. While ($t \ne 0$) do \\ +\hspace{3mm}6.1 $d \leftarrow t \mbox{ (mod }r\mbox{)}$ \\ +\hspace{3mm}6.2 $t \leftarrow \lfloor t / r \rfloor$ \\ +\hspace{3mm}6.3 Look up $d$ in the map and store the equivalent character in $y$. \\ +\hspace{3mm}6.4 $str \leftarrow str + y$ \\ +7. If $str_0 = $``$-$'' then \\ +\hspace{3mm}7.1 Reverse the digits $str_1, str_2, \ldots str_n$. \\ +8. Otherwise \\ +\hspace{3mm}8.1 Reverse the digits $str_0, str_1, \ldots str_n$. \\ +9. Return(\textit{MP\_OKAY}).\\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_toradix} +\end{figure} +\textbf{Algorithm mp\_toradix.} +This algorithm computes the radix-$r$ representation of an mp\_int $a$. The ``digits'' of the representation are extracted by reducing +successive powers of $\lfloor a / r^k \rfloor$ the input modulo $r$ until $r^k > a$. Note that instead of actually dividing by $r^k$ in +each iteration the quotient $\lfloor a / r \rfloor$ is saved for the next iteration. As a result a series of trivial $n \times 1$ divisions +are required instead of a series of $n \times k$ divisions. One design flaw of this approach is that the digits are produced in the reverse order +(see~\ref{fig:mpradix}). To remedy this flaw the digits must be swapped or simply ``reversed''. + +\begin{figure} +\begin{center} +\begin{tabular}{|c|c|c|} +\hline \textbf{Value of $a$} & \textbf{Value of $d$} & \textbf{Value of $str$} \\ +\hline $1234$ & -- & -- \\ +\hline $123$ & $4$ & ``4'' \\ +\hline $12$ & $3$ & ``43'' \\ +\hline $1$ & $2$ & ``432'' \\ +\hline $0$ & $1$ & ``4321'' \\ +\hline +\end{tabular} +\end{center} +\caption{Example of Algorithm mp\_toradix.} +\label{fig:mpradix} +\end{figure} + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_toradix.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* stores a bignum as a ASCII string in a given radix (2..64) */ +018 int mp_toradix (mp_int * a, char *str, int radix) +019 \{ +020 int res, digs; +021 mp_int t; +022 mp_digit d; +023 char *_s = str; +024 +025 /* check range of the radix */ +026 if (radix < 2 || radix > 64) \{ +027 return MP_VAL; +028 \} +029 +030 /* quick out if its zero */ +031 if (mp_iszero(a) == 1) \{ +032 *str++ = '0'; +033 *str = '\symbol{92}0'; +034 return MP_OKAY; +035 \} +036 +037 if ((res = mp_init_copy (&t, a)) != MP_OKAY) \{ +038 return res; +039 \} +040 +041 /* if it is negative output a - */ +042 if (t.sign == MP_NEG) \{ +043 ++_s; +044 *str++ = '-'; +045 t.sign = MP_ZPOS; +046 \} +047 +048 digs = 0; +049 while (mp_iszero (&t) == 0) \{ +050 if ((res = mp_div_d (&t, (mp_digit) radix, &t, &d)) != MP_OKAY) \{ +051 mp_clear (&t); +052 return res; +053 \} +054 *str++ = mp_s_rmap[d]; +055 ++digs; +056 \} +057 +058 /* reverse the digits of the string. In this case _s points +059 * to the first digit [exluding the sign] of the number] +060 */ +061 bn_reverse ((unsigned char *)_s, digs); +062 +063 /* append a NULL so the string is properly terminated */ +064 *str = '\symbol{92}0'; +065 +066 mp_clear (&t); +067 return MP_OKAY; +068 \} +069 +070 #endif +\end{alltt} +\end{small} + +\chapter{Number Theoretic Algorithms} +This chapter discusses several fundamental number theoretic algorithms such as the greatest common divisor, least common multiple and Jacobi +symbol computation. These algorithms arise as essential components in several key cryptographic algorithms such as the RSA public key algorithm and +various Sieve based factoring algorithms. + +\section{Greatest Common Divisor} +The greatest common divisor of two integers $a$ and $b$, often denoted as $(a, b)$ is the largest integer $k$ that is a proper divisor of +both $a$ and $b$. That is, $k$ is the largest integer such that $0 \equiv a \mbox{ (mod }k\mbox{)}$ and $0 \equiv b \mbox{ (mod }k\mbox{)}$ occur +simultaneously. + +The most common approach (cite) is to reduce one input modulo another. That is if $a$ and $b$ are divisible by some integer $k$ and if $qa + r = b$ then +$r$ is also divisible by $k$. The reduction pattern follows $\left < a , b \right > \rightarrow \left < b, a \mbox{ mod } b \right >$. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{Greatest Common Divisor (I)}. \\ +\textbf{Input}. Two positive integers $a$ and $b$ greater than zero. \\ +\textbf{Output}. The greatest common divisor $(a, b)$. \\ +\hline \\ +1. While ($b > 0$) do \\ +\hspace{3mm}1.1 $r \leftarrow a \mbox{ (mod }b\mbox{)}$ \\ +\hspace{3mm}1.2 $a \leftarrow b$ \\ +\hspace{3mm}1.3 $b \leftarrow r$ \\ +2. Return($a$). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm Greatest Common Divisor (I)} +\label{fig:gcd1} +\end{figure} + +This algorithm will quickly converge on the greatest common divisor since the residue $r$ tends diminish rapidly. However, divisions are +relatively expensive operations to perform and should ideally be avoided. There is another approach based on a similar relationship of +greatest common divisors. The faster approach is based on the observation that if $k$ divides both $a$ and $b$ it will also divide $a - b$. +In particular, we would like $a - b$ to decrease in magnitude which implies that $b \ge a$. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{Greatest Common Divisor (II)}. \\ +\textbf{Input}. Two positive integers $a$ and $b$ greater than zero. \\ +\textbf{Output}. The greatest common divisor $(a, b)$. \\ +\hline \\ +1. While ($b > 0$) do \\ +\hspace{3mm}1.1 Swap $a$ and $b$ such that $a$ is the smallest of the two. \\ +\hspace{3mm}1.2 $b \leftarrow b - a$ \\ +2. Return($a$). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm Greatest Common Divisor (II)} +\label{fig:gcd2} +\end{figure} + +\textbf{Proof} \textit{Algorithm~\ref{fig:gcd2} will return the greatest common divisor of $a$ and $b$.} +The algorithm in figure~\ref{fig:gcd2} will eventually terminate since $b \ge a$ the subtraction in step 1.2 will be a value less than $b$. In other +words in every iteration that tuple $\left < a, b \right >$ decrease in magnitude until eventually $a = b$. Since both $a$ and $b$ are always +divisible by the greatest common divisor (\textit{until the last iteration}) and in the last iteration of the algorithm $b = 0$, therefore, in the +second to last iteration of the algorithm $b = a$ and clearly $(a, a) = a$ which concludes the proof. \textbf{QED}. + +As a matter of practicality algorithm \ref{fig:gcd1} decreases far too slowly to be useful. Specially if $b$ is much larger than $a$ such that +$b - a$ is still very much larger than $a$. A simple addition to the algorithm is to divide $b - a$ by a power of some integer $p$ which does +not divide the greatest common divisor but will divide $b - a$. In this case ${b - a} \over p$ is also an integer and still divisible by +the greatest common divisor. + +However, instead of factoring $b - a$ to find a suitable value of $p$ the powers of $p$ can be removed from $a$ and $b$ that are in common first. +Then inside the loop whenever $b - a$ is divisible by some power of $p$ it can be safely removed. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{Greatest Common Divisor (III)}. \\ +\textbf{Input}. Two positive integers $a$ and $b$ greater than zero. \\ +\textbf{Output}. The greatest common divisor $(a, b)$. \\ +\hline \\ +1. $k \leftarrow 0$ \\ +2. While $a$ and $b$ are both divisible by $p$ do \\ +\hspace{3mm}2.1 $a \leftarrow \lfloor a / p \rfloor$ \\ +\hspace{3mm}2.2 $b \leftarrow \lfloor b / p \rfloor$ \\ +\hspace{3mm}2.3 $k \leftarrow k + 1$ \\ +3. While $a$ is divisible by $p$ do \\ +\hspace{3mm}3.1 $a \leftarrow \lfloor a / p \rfloor$ \\ +4. While $b$ is divisible by $p$ do \\ +\hspace{3mm}4.1 $b \leftarrow \lfloor b / p \rfloor$ \\ +5. While ($b > 0$) do \\ +\hspace{3mm}5.1 Swap $a$ and $b$ such that $a$ is the smallest of the two. \\ +\hspace{3mm}5.2 $b \leftarrow b - a$ \\ +\hspace{3mm}5.3 While $b$ is divisible by $p$ do \\ +\hspace{6mm}5.3.1 $b \leftarrow \lfloor b / p \rfloor$ \\ +6. Return($a \cdot p^k$). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm Greatest Common Divisor (III)} +\label{fig:gcd3} +\end{figure} + +This algorithm is based on the first except it removes powers of $p$ first and inside the main loop to ensure the tuple $\left < a, b \right >$ +decreases more rapidly. The first loop on step two removes powers of $p$ that are in common. A count, $k$, is kept which will present a common +divisor of $p^k$. After step two the remaining common divisor of $a$ and $b$ cannot be divisible by $p$. This means that $p$ can be safely +divided out of the difference $b - a$ so long as the division leaves no remainder. + +In particular the value of $p$ should be chosen such that the division on step 5.3.1 occur often. It also helps that division by $p$ be easy +to compute. The ideal choice of $p$ is two since division by two amounts to a right logical shift. Another important observation is that by +step five both $a$ and $b$ are odd. Therefore, the diffrence $b - a$ must be even which means that each iteration removes one bit from the +largest of the pair. + +\subsection{Complete Greatest Common Divisor} +The algorithms presented so far cannot handle inputs which are zero or negative. The following algorithm can handle all input cases properly +and will produce the greatest common divisor. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_gcd}. \\ +\textbf{Input}. mp\_int $a$ and $b$ \\ +\textbf{Output}. The greatest common divisor $c = (a, b)$. \\ +\hline \\ +1. If $a = 0$ and $b \ne 0$ then \\ +\hspace{3mm}1.1 $c \leftarrow b$ \\ +\hspace{3mm}1.2 Return(\textit{MP\_OKAY}). \\ +2. If $a \ne 0$ and $b = 0$ then \\ +\hspace{3mm}2.1 $c \leftarrow a$ \\ +\hspace{3mm}2.2 Return(\textit{MP\_OKAY}). \\ +3. If $a = b = 0$ then \\ +\hspace{3mm}3.1 $c \leftarrow 1$ \\ +\hspace{3mm}3.2 Return(\textit{MP\_OKAY}). \\ +4. $u \leftarrow \vert a \vert, v \leftarrow \vert b \vert$ \\ +5. $k \leftarrow 0$ \\ +6. While $u.used > 0$ and $v.used > 0$ and $u_0 \equiv v_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ +\hspace{3mm}6.1 $k \leftarrow k + 1$ \\ +\hspace{3mm}6.2 $u \leftarrow \lfloor u / 2 \rfloor$ \\ +\hspace{3mm}6.3 $v \leftarrow \lfloor v / 2 \rfloor$ \\ +7. While $u.used > 0$ and $u_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ +\hspace{3mm}7.1 $u \leftarrow \lfloor u / 2 \rfloor$ \\ +8. While $v.used > 0$ and $v_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ +\hspace{3mm}8.1 $v \leftarrow \lfloor v / 2 \rfloor$ \\ +9. While $v.used > 0$ \\ +\hspace{3mm}9.1 If $\vert u \vert > \vert v \vert$ then \\ +\hspace{6mm}9.1.1 Swap $u$ and $v$. \\ +\hspace{3mm}9.2 $v \leftarrow \vert v \vert - \vert u \vert$ \\ +\hspace{3mm}9.3 While $v.used > 0$ and $v_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ +\hspace{6mm}9.3.1 $v \leftarrow \lfloor v / 2 \rfloor$ \\ +10. $c \leftarrow u \cdot 2^k$ \\ +11. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_gcd} +\end{figure} +\textbf{Algorithm mp\_gcd.} +This algorithm will produce the greatest common divisor of two mp\_ints $a$ and $b$. The algorithm was originally based on Algorithm B of +Knuth \cite[pp. 338]{TAOCPV2} but has been modified to be simpler to explain. In theory it achieves the same asymptotic working time as +Algorithm B and in practice this appears to be true. + +The first three steps handle the cases where either one of or both inputs are zero. If either input is zero the greatest common divisor is the +largest input or zero if they are both zero. If the inputs are not trivial than $u$ and $v$ are assigned the absolute values of +$a$ and $b$ respectively and the algorithm will proceed to reduce the pair. + +Step six will divide out any common factors of two and keep track of the count in the variable $k$. After this step two is no longer a +factor of the remaining greatest common divisor between $u$ and $v$ and can be safely evenly divided out of either whenever they are even. Step +seven and eight ensure that the $u$ and $v$ respectively have no more factors of two. At most only one of the while loops will iterate since +they cannot both be even. + +By step nine both of $u$ and $v$ are odd which is required for the inner logic. First the pair are swapped such that $v$ is equal to +or greater than $u$. This ensures that the subtraction on step 9.2 will always produce a positive and even result. Step 9.3 removes any +factors of two from the difference $u$ to ensure that in the next iteration of the loop both are once again odd. + +After $v = 0$ occurs the variable $u$ has the greatest common divisor of the pair $\left < u, v \right >$ just after step six. The result +must be adjusted by multiplying by the common factors of two ($2^k$) removed earlier. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_gcd.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* Greatest Common Divisor using the binary method */ +018 int mp_gcd (mp_int * a, mp_int * b, mp_int * c) +019 \{ +020 mp_int u, v; +021 int k, u_lsb, v_lsb, res; +022 +023 /* either zero than gcd is the largest */ +024 if (mp_iszero (a) == 1 && mp_iszero (b) == 0) \{ +025 return mp_abs (b, c); +026 \} +027 if (mp_iszero (a) == 0 && mp_iszero (b) == 1) \{ +028 return mp_abs (a, c); +029 \} +030 +031 /* optimized. At this point if a == 0 then +032 * b must equal zero too +033 */ +034 if (mp_iszero (a) == 1) \{ +035 mp_zero(c); +036 return MP_OKAY; +037 \} +038 +039 /* get copies of a and b we can modify */ +040 if ((res = mp_init_copy (&u, a)) != MP_OKAY) \{ +041 return res; +042 \} +043 +044 if ((res = mp_init_copy (&v, b)) != MP_OKAY) \{ +045 goto LBL_U; +046 \} +047 +048 /* must be positive for the remainder of the algorithm */ +049 u.sign = v.sign = MP_ZPOS; +050 +051 /* B1. Find the common power of two for u and v */ +052 u_lsb = mp_cnt_lsb(&u); +053 v_lsb = mp_cnt_lsb(&v); +054 k = MIN(u_lsb, v_lsb); +055 +056 if (k > 0) \{ +057 /* divide the power of two out */ +058 if ((res = mp_div_2d(&u, k, &u, NULL)) != MP_OKAY) \{ +059 goto LBL_V; +060 \} +061 +062 if ((res = mp_div_2d(&v, k, &v, NULL)) != MP_OKAY) \{ +063 goto LBL_V; +064 \} +065 \} +066 +067 /* divide any remaining factors of two out */ +068 if (u_lsb != k) \{ +069 if ((res = mp_div_2d(&u, u_lsb - k, &u, NULL)) != MP_OKAY) \{ +070 goto LBL_V; +071 \} +072 \} +073 +074 if (v_lsb != k) \{ +075 if ((res = mp_div_2d(&v, v_lsb - k, &v, NULL)) != MP_OKAY) \{ +076 goto LBL_V; +077 \} +078 \} +079 +080 while (mp_iszero(&v) == 0) \{ +081 /* make sure v is the largest */ +082 if (mp_cmp_mag(&u, &v) == MP_GT) \{ +083 /* swap u and v to make sure v is >= u */ +084 mp_exch(&u, &v); +085 \} +086 +087 /* subtract smallest from largest */ +088 if ((res = s_mp_sub(&v, &u, &v)) != MP_OKAY) \{ +089 goto LBL_V; +090 \} +091 +092 /* Divide out all factors of two */ +093 if ((res = mp_div_2d(&v, mp_cnt_lsb(&v), &v, NULL)) != MP_OKAY) \{ +094 goto LBL_V; +095 \} +096 \} +097 +098 /* multiply by 2**k which we divided out at the beginning */ +099 if ((res = mp_mul_2d (&u, k, c)) != MP_OKAY) \{ +100 goto LBL_V; +101 \} +102 c->sign = MP_ZPOS; +103 res = MP_OKAY; +104 LBL_V:mp_clear (&u); +105 LBL_U:mp_clear (&v); +106 return res; +107 \} +108 #endif +\end{alltt} +\end{small} + +This function makes use of the macros mp\_iszero and mp\_iseven. The former evaluates to $1$ if the input mp\_int is equivalent to the +integer zero otherwise it evaluates to $0$. The latter evaluates to $1$ if the input mp\_int represents a non-zero even integer otherwise +it evaluates to $0$. Note that just because mp\_iseven may evaluate to $0$ does not mean the input is odd, it could also be zero. The three +trivial cases of inputs are handled on lines 24 through 37. After those lines the inputs are assumed to be non-zero. + +Lines 34 and 40 make local copies $u$ and $v$ of the inputs $a$ and $b$ respectively. At this point the common factors of two +must be divided out of the two inputs. The while loop on line 80 iterates so long as both are even. The local integer $k$ is used to +keep track of how many factors of $2$ are pulled out of both values. It is assumed that the number of factors will not exceed the maximum +value of a C ``int'' data type\footnote{Strictly speaking no array in C may have more than entries than are accessible by an ``int'' so this is not +a limitation.}. + +At this point there are no more common factors of two in the two values. The while loops on lines 80 and 80 remove any independent +factors of two such that both $u$ and $v$ are guaranteed to be an odd integer before hitting the main body of the algorithm. The while loop +on line 80 performs the reduction of the pair until $v$ is equal to zero. The unsigned comparison and subtraction algorithms are used in +place of the full signed routines since both values are guaranteed to be positive and the result of the subtraction is guaranteed to be non-negative. + +\section{Least Common Multiple} +The least common multiple of a pair of integers is their product divided by their greatest common divisor. For two integers $a$ and $b$ the +least common multiple is normally denoted as $[ a, b ]$ and numerically equivalent to ${ab} \over {(a, b)}$. For example, if $a = 2 \cdot 2 \cdot 3 = 12$ +and $b = 2 \cdot 3 \cdot 3 \cdot 7 = 126$ the least common multiple is ${126 \over {(12, 126)}} = {126 \over 6} = 21$. + +The least common multiple arises often in coding theory as well as number theory. If two functions have periods of $a$ and $b$ respectively they will +collide, that is be in synchronous states, after only $[ a, b ]$ iterations. This is why, for example, random number generators based on +Linear Feedback Shift Registers (LFSR) tend to use registers with periods which are co-prime (\textit{e.g. the greatest common divisor is one.}). +Similarly in number theory if a composite $n$ has two prime factors $p$ and $q$ then maximal order of any unit of $\Z/n\Z$ will be $[ p - 1, q - 1] $. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_lcm}. \\ +\textbf{Input}. mp\_int $a$ and $b$ \\ +\textbf{Output}. The least common multiple $c = [a, b]$. \\ +\hline \\ +1. $c \leftarrow (a, b)$ \\ +2. $t \leftarrow a \cdot b$ \\ +3. $c \leftarrow \lfloor t / c \rfloor$ \\ +4. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_lcm} +\end{figure} +\textbf{Algorithm mp\_lcm.} +This algorithm computes the least common multiple of two mp\_int inputs $a$ and $b$. It computes the least common multiple directly by +dividing the product of the two inputs by their greatest common divisor. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_lcm.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* computes least common multiple as |a*b|/(a, b) */ +018 int mp_lcm (mp_int * a, mp_int * b, mp_int * c) +019 \{ +020 int res; +021 mp_int t1, t2; +022 +023 +024 if ((res = mp_init_multi (&t1, &t2, NULL)) != MP_OKAY) \{ +025 return res; +026 \} +027 +028 /* t1 = get the GCD of the two inputs */ +029 if ((res = mp_gcd (a, b, &t1)) != MP_OKAY) \{ +030 goto LBL_T; +031 \} +032 +033 /* divide the smallest by the GCD */ +034 if (mp_cmp_mag(a, b) == MP_LT) \{ +035 /* store quotient in t2 such that t2 * b is the LCM */ +036 if ((res = mp_div(a, &t1, &t2, NULL)) != MP_OKAY) \{ +037 goto LBL_T; +038 \} +039 res = mp_mul(b, &t2, c); +040 \} else \{ +041 /* store quotient in t2 such that t2 * a is the LCM */ +042 if ((res = mp_div(b, &t1, &t2, NULL)) != MP_OKAY) \{ +043 goto LBL_T; +044 \} +045 res = mp_mul(a, &t2, c); +046 \} +047 +048 /* fix the sign to positive */ +049 c->sign = MP_ZPOS; +050 +051 LBL_T: +052 mp_clear_multi (&t1, &t2, NULL); +053 return res; +054 \} +055 #endif +\end{alltt} +\end{small} + +\section{Jacobi Symbol Computation} +To explain the Jacobi Symbol we shall first discuss the Legendre function\footnote{Arrg. What is the name of this?} off which the Jacobi symbol is +defined. The Legendre function computes whether or not an integer $a$ is a quadratic residue modulo an odd prime $p$. Numerically it is +equivalent to equation \ref{eqn:legendre}. + +\begin{equation} +a^{(p-1)/2} \equiv \begin{array}{rl} + -1 & \mbox{if }a\mbox{ is a quadratic non-residue.} \\ + 0 & \mbox{if }a\mbox{ divides }p\mbox{.} \\ + 1 & \mbox{if }a\mbox{ is a quadratic residue}. + \end{array} \mbox{ (mod }p\mbox{)} +\label{eqn:legendre} +\end{equation} + +\textbf{Proof.} \textit{Equation \ref{eqn:legendre} correctly identifies the residue status of an integer $a$ modulo a prime $p$.} +An integer $a$ is a quadratic residue if the following equation has a solution. + +\begin{equation} +x^2 \equiv a \mbox{ (mod }p\mbox{)} +\label{eqn:root} +\end{equation} + +Consider the following equation. + +\begin{equation} +0 \equiv x^{p-1} - 1 \equiv \left \lbrace \left (x^2 \right )^{(p-1)/2} - a^{(p-1)/2} \right \rbrace + \left ( a^{(p-1)/2} - 1 \right ) \mbox{ (mod }p\mbox{)} +\label{eqn:rooti} +\end{equation} + +Whether equation \ref{eqn:root} has a solution or not equation \ref{eqn:rooti} is always true. If $a^{(p-1)/2} - 1 \equiv 0 \mbox{ (mod }p\mbox{)}$ +then the quantity in the braces must be zero. By reduction, + +\begin{eqnarray} +\left (x^2 \right )^{(p-1)/2} - a^{(p-1)/2} \equiv 0 \nonumber \\ +\left (x^2 \right )^{(p-1)/2} \equiv a^{(p-1)/2} \nonumber \\ +x^2 \equiv a \mbox{ (mod }p\mbox{)} +\end{eqnarray} + +As a result there must be a solution to the quadratic equation and in turn $a$ must be a quadratic residue. If $a$ does not divide $p$ and $a$ +is not a quadratic residue then the only other value $a^{(p-1)/2}$ may be congruent to is $-1$ since +\begin{equation} +0 \equiv a^{p - 1} - 1 \equiv (a^{(p-1)/2} + 1)(a^{(p-1)/2} - 1) \mbox{ (mod }p\mbox{)} +\end{equation} +One of the terms on the right hand side must be zero. \textbf{QED} + +\subsection{Jacobi Symbol} +The Jacobi symbol is a generalization of the Legendre function for any odd non prime moduli $p$ greater than 2. If $p = \prod_{i=0}^n p_i$ then +the Jacobi symbol $\left ( { a \over p } \right )$ is equal to the following equation. + +\begin{equation} +\left ( { a \over p } \right ) = \left ( { a \over p_0} \right ) \left ( { a \over p_1} \right ) \ldots \left ( { a \over p_n} \right ) +\end{equation} + +By inspection if $p$ is prime the Jacobi symbol is equivalent to the Legendre function. The following facts\footnote{See HAC \cite[pp. 72-74]{HAC} for +further details.} will be used to derive an efficient Jacobi symbol algorithm. Where $p$ is an odd integer greater than two and $a, b \in \Z$ the +following are true. + +\begin{enumerate} +\item $\left ( { a \over p} \right )$ equals $-1$, $0$ or $1$. +\item $\left ( { ab \over p} \right ) = \left ( { a \over p} \right )\left ( { b \over p} \right )$. +\item If $a \equiv b$ then $\left ( { a \over p} \right ) = \left ( { b \over p} \right )$. +\item $\left ( { 2 \over p} \right )$ equals $1$ if $p \equiv 1$ or $7 \mbox{ (mod }8\mbox{)}$. Otherwise, it equals $-1$. +\item $\left ( { a \over p} \right ) \equiv \left ( { p \over a} \right ) \cdot (-1)^{(p-1)(a-1)/4}$. More specifically +$\left ( { a \over p} \right ) = \left ( { p \over a} \right )$ if $p \equiv a \equiv 1 \mbox{ (mod }4\mbox{)}$. +\end{enumerate} + +Using these facts if $a = 2^k \cdot a'$ then + +\begin{eqnarray} +\left ( { a \over p } \right ) = \left ( {{2^k} \over p } \right ) \left ( {a' \over p} \right ) \nonumber \\ + = \left ( {2 \over p } \right )^k \left ( {a' \over p} \right ) +\label{eqn:jacobi} +\end{eqnarray} + +By fact five, + +\begin{equation} +\left ( { a \over p } \right ) = \left ( { p \over a } \right ) \cdot (-1)^{(p-1)(a-1)/4} +\end{equation} + +Subsequently by fact three since $p \equiv (p \mbox{ mod }a) \mbox{ (mod }a\mbox{)}$ then + +\begin{equation} +\left ( { a \over p } \right ) = \left ( { {p \mbox{ mod } a} \over a } \right ) \cdot (-1)^{(p-1)(a-1)/4} +\end{equation} + +By putting both observations into equation \ref{eqn:jacobi} the following simplified equation is formed. + +\begin{equation} +\left ( { a \over p } \right ) = \left ( {2 \over p } \right )^k \left ( {{p\mbox{ mod }a'} \over a'} \right ) \cdot (-1)^{(p-1)(a'-1)/4} +\end{equation} + +The value of $\left ( {{p \mbox{ mod }a'} \over a'} \right )$ can be found by using the same equation recursively. The value of +$\left ( {2 \over p } \right )^k$ equals $1$ if $k$ is even otherwise it equals $\left ( {2 \over p } \right )$. Using this approach the +factors of $p$ do not have to be known. Furthermore, if $(a, p) = 1$ then the algorithm will terminate when the recursion requests the +Jacobi symbol computation of $\left ( {1 \over a'} \right )$ which is simply $1$. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_jacobi}. \\ +\textbf{Input}. mp\_int $a$ and $p$, $a \ge 0$, $p \ge 3$, $p \equiv 1 \mbox{ (mod }2\mbox{)}$ \\ +\textbf{Output}. The Jacobi symbol $c = \left ( {a \over p } \right )$. \\ +\hline \\ +1. If $a = 0$ then \\ +\hspace{3mm}1.1 $c \leftarrow 0$ \\ +\hspace{3mm}1.2 Return(\textit{MP\_OKAY}). \\ +2. If $a = 1$ then \\ +\hspace{3mm}2.1 $c \leftarrow 1$ \\ +\hspace{3mm}2.2 Return(\textit{MP\_OKAY}). \\ +3. $a' \leftarrow a$ \\ +4. $k \leftarrow 0$ \\ +5. While $a'.used > 0$ and $a'_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ +\hspace{3mm}5.1 $k \leftarrow k + 1$ \\ +\hspace{3mm}5.2 $a' \leftarrow \lfloor a' / 2 \rfloor$ \\ +6. If $k \equiv 0 \mbox{ (mod }2\mbox{)}$ then \\ +\hspace{3mm}6.1 $s \leftarrow 1$ \\ +7. else \\ +\hspace{3mm}7.1 $r \leftarrow p_0 \mbox{ (mod }8\mbox{)}$ \\ +\hspace{3mm}7.2 If $r = 1$ or $r = 7$ then \\ +\hspace{6mm}7.2.1 $s \leftarrow 1$ \\ +\hspace{3mm}7.3 else \\ +\hspace{6mm}7.3.1 $s \leftarrow -1$ \\ +8. If $p_0 \equiv a'_0 \equiv 3 \mbox{ (mod }4\mbox{)}$ then \\ +\hspace{3mm}8.1 $s \leftarrow -s$ \\ +9. If $a' \ne 1$ then \\ +\hspace{3mm}9.1 $p' \leftarrow p \mbox{ (mod }a'\mbox{)}$ \\ +\hspace{3mm}9.2 $s \leftarrow s \cdot \mbox{mp\_jacobi}(p', a')$ \\ +10. $c \leftarrow s$ \\ +11. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_jacobi} +\end{figure} +\textbf{Algorithm mp\_jacobi.} +This algorithm computes the Jacobi symbol for an arbitrary positive integer $a$ with respect to an odd integer $p$ greater than three. The algorithm +is based on algorithm 2.149 of HAC \cite[pp. 73]{HAC}. + +Step numbers one and two handle the trivial cases of $a = 0$ and $a = 1$ respectively. Step five determines the number of two factors in the +input $a$. If $k$ is even than the term $\left ( { 2 \over p } \right )^k$ must always evaluate to one. If $k$ is odd than the term evaluates to one +if $p_0$ is congruent to one or seven modulo eight, otherwise it evaluates to $-1$. After the the $\left ( { 2 \over p } \right )^k$ term is handled +the $(-1)^{(p-1)(a'-1)/4}$ is computed and multiplied against the current product $s$. The latter term evaluates to one if both $p$ and $a'$ +are congruent to one modulo four, otherwise it evaluates to negative one. + +By step nine if $a'$ does not equal one a recursion is required. Step 9.1 computes $p' \equiv p \mbox{ (mod }a'\mbox{)}$ and will recurse to compute +$\left ( {p' \over a'} \right )$ which is multiplied against the current Jacobi product. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_jacobi.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* computes the jacobi c = (a | n) (or Legendre if n is prime) +018 * HAC pp. 73 Algorithm 2.149 +019 */ +020 int mp_jacobi (mp_int * a, mp_int * p, int *c) +021 \{ +022 mp_int a1, p1; +023 int k, s, r, res; +024 mp_digit residue; +025 +026 /* if p <= 0 return MP_VAL */ +027 if (mp_cmp_d(p, 0) != MP_GT) \{ +028 return MP_VAL; +029 \} +030 +031 /* step 1. if a == 0, return 0 */ +032 if (mp_iszero (a) == 1) \{ +033 *c = 0; +034 return MP_OKAY; +035 \} +036 +037 /* step 2. if a == 1, return 1 */ +038 if (mp_cmp_d (a, 1) == MP_EQ) \{ +039 *c = 1; +040 return MP_OKAY; +041 \} +042 +043 /* default */ +044 s = 0; +045 +046 /* step 3. write a = a1 * 2**k */ +047 if ((res = mp_init_copy (&a1, a)) != MP_OKAY) \{ +048 return res; +049 \} +050 +051 if ((res = mp_init (&p1)) != MP_OKAY) \{ +052 goto LBL_A1; +053 \} +054 +055 /* divide out larger power of two */ +056 k = mp_cnt_lsb(&a1); +057 if ((res = mp_div_2d(&a1, k, &a1, NULL)) != MP_OKAY) \{ +058 goto LBL_P1; +059 \} +060 +061 /* step 4. if e is even set s=1 */ +062 if ((k & 1) == 0) \{ +063 s = 1; +064 \} else \{ +065 /* else set s=1 if p = 1/7 (mod 8) or s=-1 if p = 3/5 (mod 8) */ +066 residue = p->dp[0] & 7; +067 +068 if (residue == 1 || residue == 7) \{ +069 s = 1; +070 \} else if (residue == 3 || residue == 5) \{ +071 s = -1; +072 \} +073 \} +074 +075 /* step 5. if p == 3 (mod 4) *and* a1 == 3 (mod 4) then s = -s */ +076 if ( ((p->dp[0] & 3) == 3) && ((a1.dp[0] & 3) == 3)) \{ +077 s = -s; +078 \} +079 +080 /* if a1 == 1 we're done */ +081 if (mp_cmp_d (&a1, 1) == MP_EQ) \{ +082 *c = s; +083 \} else \{ +084 /* n1 = n mod a1 */ +085 if ((res = mp_mod (p, &a1, &p1)) != MP_OKAY) \{ +086 goto LBL_P1; +087 \} +088 if ((res = mp_jacobi (&p1, &a1, &r)) != MP_OKAY) \{ +089 goto LBL_P1; +090 \} +091 *c = s * r; +092 \} +093 +094 /* done */ +095 res = MP_OKAY; +096 LBL_P1:mp_clear (&p1); +097 LBL_A1:mp_clear (&a1); +098 return res; +099 \} +100 #endif +\end{alltt} +\end{small} + +As a matter of practicality the variable $a'$ as per the pseudo-code is reprensented by the variable $a1$ since the $'$ symbol is not valid for a C +variable name character. + +The two simple cases of $a = 0$ and $a = 1$ are handled at the very beginning to simplify the algorithm. If the input is non-trivial the algorithm +has to proceed compute the Jacobi. The variable $s$ is used to hold the current Jacobi product. Note that $s$ is merely a C ``int'' data type since +the values it may obtain are merely $-1$, $0$ and $1$. + +After a local copy of $a$ is made all of the factors of two are divided out and the total stored in $k$. Technically only the least significant +bit of $k$ is required, however, it makes the algorithm simpler to follow to perform an addition. In practice an exclusive-or and addition have the same +processor requirements and neither is faster than the other. + +Line 61 through 70 determines the value of $\left ( { 2 \over p } \right )^k$. If the least significant bit of $k$ is zero than +$k$ is even and the value is one. Otherwise, the value of $s$ depends on which residue class $p$ belongs to modulo eight. The value of +$(-1)^{(p-1)(a'-1)/4}$ is compute and multiplied against $s$ on lines 75 through 73. + +Finally, if $a1$ does not equal one the algorithm must recurse and compute $\left ( {p' \over a'} \right )$. + +\textit{-- Comment about default $s$ and such...} + +\section{Modular Inverse} +\label{sec:modinv} +The modular inverse of a number actually refers to the modular multiplicative inverse. Essentially for any integer $a$ such that $(a, p) = 1$ there +exist another integer $b$ such that $ab \equiv 1 \mbox{ (mod }p\mbox{)}$. The integer $b$ is called the multiplicative inverse of $a$ which is +denoted as $b = a^{-1}$. Technically speaking modular inversion is a well defined operation for any finite ring or field not just for rings and +fields of integers. However, the former will be the matter of discussion. + +The simplest approach is to compute the algebraic inverse of the input. That is to compute $b \equiv a^{\Phi(p) - 1}$. If $\Phi(p)$ is the +order of the multiplicative subgroup modulo $p$ then $b$ must be the multiplicative inverse of $a$. The proof of which is trivial. + +\begin{equation} +ab \equiv a \left (a^{\Phi(p) - 1} \right ) \equiv a^{\Phi(p)} \equiv a^0 \equiv 1 \mbox{ (mod }p\mbox{)} +\end{equation} + +However, as simple as this approach may be it has two serious flaws. It requires that the value of $\Phi(p)$ be known which if $p$ is composite +requires all of the prime factors. This approach also is very slow as the size of $p$ grows. + +A simpler approach is based on the observation that solving for the multiplicative inverse is equivalent to solving the linear +Diophantine\footnote{See LeVeque \cite[pp. 40-43]{LeVeque} for more information.} equation. + +\begin{equation} +ab + pq = 1 +\end{equation} + +Where $a$, $b$, $p$ and $q$ are all integers. If such a pair of integers $ \left < b, q \right >$ exist than $b$ is the multiplicative inverse of +$a$ modulo $p$. The extended Euclidean algorithm (Knuth \cite[pp. 342]{TAOCPV2}) can be used to solve such equations provided $(a, p) = 1$. +However, instead of using that algorithm directly a variant known as the binary Extended Euclidean algorithm will be used in its place. The +binary approach is very similar to the binary greatest common divisor algorithm except it will produce a full solution to the Diophantine +equation. + +\subsection{General Case} +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_invmod}. \\ +\textbf{Input}. mp\_int $a$ and $b$, $(a, b) = 1$, $p \ge 2$, $0 < a < p$. \\ +\textbf{Output}. The modular inverse $c \equiv a^{-1} \mbox{ (mod }b\mbox{)}$. \\ +\hline \\ +1. If $b \le 0$ then return(\textit{MP\_VAL}). \\ +2. If $b_0 \equiv 1 \mbox{ (mod }2\mbox{)}$ then use algorithm fast\_mp\_invmod. \\ +3. $x \leftarrow \vert a \vert, y \leftarrow b$ \\ +4. If $x_0 \equiv y_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ then return(\textit{MP\_VAL}). \\ +5. $B \leftarrow 0, C \leftarrow 0, A \leftarrow 1, D \leftarrow 1$ \\ +6. While $u.used > 0$ and $u_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ +\hspace{3mm}6.1 $u \leftarrow \lfloor u / 2 \rfloor$ \\ +\hspace{3mm}6.2 If ($A.used > 0$ and $A_0 \equiv 1 \mbox{ (mod }2\mbox{)}$) or ($B.used > 0$ and $B_0 \equiv 1 \mbox{ (mod }2\mbox{)}$) then \\ +\hspace{6mm}6.2.1 $A \leftarrow A + y$ \\ +\hspace{6mm}6.2.2 $B \leftarrow B - x$ \\ +\hspace{3mm}6.3 $A \leftarrow \lfloor A / 2 \rfloor$ \\ +\hspace{3mm}6.4 $B \leftarrow \lfloor B / 2 \rfloor$ \\ +7. While $v.used > 0$ and $v_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ +\hspace{3mm}7.1 $v \leftarrow \lfloor v / 2 \rfloor$ \\ +\hspace{3mm}7.2 If ($C.used > 0$ and $C_0 \equiv 1 \mbox{ (mod }2\mbox{)}$) or ($D.used > 0$ and $D_0 \equiv 1 \mbox{ (mod }2\mbox{)}$) then \\ +\hspace{6mm}7.2.1 $C \leftarrow C + y$ \\ +\hspace{6mm}7.2.2 $D \leftarrow D - x$ \\ +\hspace{3mm}7.3 $C \leftarrow \lfloor C / 2 \rfloor$ \\ +\hspace{3mm}7.4 $D \leftarrow \lfloor D / 2 \rfloor$ \\ +8. If $u \ge v$ then \\ +\hspace{3mm}8.1 $u \leftarrow u - v$ \\ +\hspace{3mm}8.2 $A \leftarrow A - C$ \\ +\hspace{3mm}8.3 $B \leftarrow B - D$ \\ +9. else \\ +\hspace{3mm}9.1 $v \leftarrow v - u$ \\ +\hspace{3mm}9.2 $C \leftarrow C - A$ \\ +\hspace{3mm}9.3 $D \leftarrow D - B$ \\ +10. If $u \ne 0$ goto step 6. \\ +11. If $v \ne 1$ return(\textit{MP\_VAL}). \\ +12. While $C \le 0$ do \\ +\hspace{3mm}12.1 $C \leftarrow C + b$ \\ +13. While $C \ge b$ do \\ +\hspace{3mm}13.1 $C \leftarrow C - b$ \\ +14. $c \leftarrow C$ \\ +15. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\end{figure} +\textbf{Algorithm mp\_invmod.} +This algorithm computes the modular multiplicative inverse of an integer $a$ modulo an integer $b$. This algorithm is a variation of the +extended binary Euclidean algorithm from HAC \cite[pp. 608]{HAC}. It has been modified to only compute the modular inverse and not a complete +Diophantine solution. + +If $b \le 0$ than the modulus is invalid and MP\_VAL is returned. Similarly if both $a$ and $b$ are even then there cannot be a multiplicative +inverse for $a$ and the error is reported. + +The astute reader will observe that steps seven through nine are very similar to the binary greatest common divisor algorithm mp\_gcd. In this case +the other variables to the Diophantine equation are solved. The algorithm terminates when $u = 0$ in which case the solution is + +\begin{equation} +Ca + Db = v +\end{equation} + +If $v$, the greatest common divisor of $a$ and $b$ is not equal to one then the algorithm will report an error as no inverse exists. Otherwise, $C$ +is the modular inverse of $a$. The actual value of $C$ is congruent to, but not necessarily equal to, the ideal modular inverse which should lie +within $1 \le a^{-1} < b$. Step numbers twelve and thirteen adjust the inverse until it is in range. If the original input $a$ is within $0 < a < p$ +then only a couple of additions or subtractions will be required to adjust the inverse. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_invmod.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* hac 14.61, pp608 */ +018 int mp_invmod (mp_int * a, mp_int * b, mp_int * c) +019 \{ +020 /* b cannot be negative */ +021 if (b->sign == MP_NEG || mp_iszero(b) == 1) \{ +022 return MP_VAL; +023 \} +024 +025 #ifdef BN_FAST_MP_INVMOD_C +026 /* if the modulus is odd we can use a faster routine instead */ +027 if (mp_isodd (b) == 1) \{ +028 return fast_mp_invmod (a, b, c); +029 \} +030 #endif +031 +032 #ifdef BN_MP_INVMOD_SLOW_C +033 return mp_invmod_slow(a, b, c); +034 #endif +035 +036 return MP_VAL; +037 \} +038 #endif +\end{alltt} +\end{small} + +\subsubsection{Odd Moduli} + +When the modulus $b$ is odd the variables $A$ and $C$ are fixed and are not required to compute the inverse. In particular by attempting to solve +the Diophantine $Cb + Da = 1$ only $B$ and $D$ are required to find the inverse of $a$. + +The algorithm fast\_mp\_invmod is a direct adaptation of algorithm mp\_invmod with all all steps involving either $A$ or $C$ removed. This +optimization will halve the time required to compute the modular inverse. + +\section{Primality Tests} + +A non-zero integer $a$ is said to be prime if it is not divisible by any other integer excluding one and itself. For example, $a = 7$ is prime +since the integers $2 \ldots 6$ do not evenly divide $a$. By contrast, $a = 6$ is not prime since $a = 6 = 2 \cdot 3$. + +Prime numbers arise in cryptography considerably as they allow finite fields to be formed. The ability to determine whether an integer is prime or +not quickly has been a viable subject in cryptography and number theory for considerable time. The algorithms that will be presented are all +probablistic algorithms in that when they report an integer is composite it must be composite. However, when the algorithms report an integer is +prime the algorithm may be incorrect. + +As will be discussed it is possible to limit the probability of error so well that for practical purposes the probablity of error might as +well be zero. For the purposes of these discussions let $n$ represent the candidate integer of which the primality is in question. + +\subsection{Trial Division} + +Trial division means to attempt to evenly divide a candidate integer by small prime integers. If the candidate can be evenly divided it obviously +cannot be prime. By dividing by all primes $1 < p \le \sqrt{n}$ this test can actually prove whether an integer is prime. However, such a test +would require a prohibitive amount of time as $n$ grows. + +Instead of dividing by every prime, a smaller, more mangeable set of primes may be used instead. By performing trial division with only a subset +of the primes less than $\sqrt{n} + 1$ the algorithm cannot prove if a candidate is prime. However, often it can prove a candidate is not prime. + +The benefit of this test is that trial division by small values is fairly efficient. Specially compared to the other algorithms that will be +discussed shortly. The probability that this approach correctly identifies a composite candidate when tested with all primes upto $q$ is given by +$1 - {1.12 \over ln(q)}$. The graph (\ref{pic:primality}, will be added later) demonstrates the probability of success for the range +$3 \le q \le 100$. + +At approximately $q = 30$ the gain of performing further tests diminishes fairly quickly. At $q = 90$ further testing is generally not going to +be of any practical use. In the case of LibTomMath the default limit $q = 256$ was chosen since it is not too high and will eliminate +approximately $80\%$ of all candidate integers. The constant \textbf{PRIME\_SIZE} is equal to the number of primes in the test base. The +array \_\_prime\_tab is an array of the first \textbf{PRIME\_SIZE} prime numbers. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_prime\_is\_divisible}. \\ +\textbf{Input}. mp\_int $a$ \\ +\textbf{Output}. $c = 1$ if $n$ is divisible by a small prime, otherwise $c = 0$. \\ +\hline \\ +1. for $ix$ from $0$ to $PRIME\_SIZE$ do \\ +\hspace{3mm}1.1 $d \leftarrow n \mbox{ (mod }\_\_prime\_tab_{ix}\mbox{)}$ \\ +\hspace{3mm}1.2 If $d = 0$ then \\ +\hspace{6mm}1.2.1 $c \leftarrow 1$ \\ +\hspace{6mm}1.2.2 Return(\textit{MP\_OKAY}). \\ +2. $c \leftarrow 0$ \\ +3. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_prime\_is\_divisible} +\end{figure} +\textbf{Algorithm mp\_prime\_is\_divisible.} +This algorithm attempts to determine if a candidate integer $n$ is composite by performing trial divisions. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_prime\_is\_divisible.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* determines if an integers is divisible by one +018 * of the first PRIME_SIZE primes or not +019 * +020 * sets result to 0 if not, 1 if yes +021 */ +022 int mp_prime_is_divisible (mp_int * a, int *result) +023 \{ +024 int err, ix; +025 mp_digit res; +026 +027 /* default to not */ +028 *result = MP_NO; +029 +030 for (ix = 0; ix < PRIME_SIZE; ix++) \{ +031 /* what is a mod LBL_prime_tab[ix] */ +032 if ((err = mp_mod_d (a, ltm_prime_tab[ix], &res)) != MP_OKAY) \{ +033 return err; +034 \} +035 +036 /* is the residue zero? */ +037 if (res == 0) \{ +038 *result = MP_YES; +039 return MP_OKAY; +040 \} +041 \} +042 +043 return MP_OKAY; +044 \} +045 #endif +\end{alltt} +\end{small} + +The algorithm defaults to a return of $0$ in case an error occurs. The values in the prime table are all specified to be in the range of a +mp\_digit. The table \_\_prime\_tab is defined in the following file. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_prime\_tab.c +\vspace{-3mm} +\begin{alltt} +016 const mp_digit ltm_prime_tab[] = \{ +017 0x0002, 0x0003, 0x0005, 0x0007, 0x000B, 0x000D, 0x0011, 0x0013, +018 0x0017, 0x001D, 0x001F, 0x0025, 0x0029, 0x002B, 0x002F, 0x0035, +019 0x003B, 0x003D, 0x0043, 0x0047, 0x0049, 0x004F, 0x0053, 0x0059, +020 0x0061, 0x0065, 0x0067, 0x006B, 0x006D, 0x0071, 0x007F, +021 #ifndef MP_8BIT +022 0x0083, +023 0x0089, 0x008B, 0x0095, 0x0097, 0x009D, 0x00A3, 0x00A7, 0x00AD, +024 0x00B3, 0x00B5, 0x00BF, 0x00C1, 0x00C5, 0x00C7, 0x00D3, 0x00DF, +025 0x00E3, 0x00E5, 0x00E9, 0x00EF, 0x00F1, 0x00FB, 0x0101, 0x0107, +026 0x010D, 0x010F, 0x0115, 0x0119, 0x011B, 0x0125, 0x0133, 0x0137, +027 +028 0x0139, 0x013D, 0x014B, 0x0151, 0x015B, 0x015D, 0x0161, 0x0167, +029 0x016F, 0x0175, 0x017B, 0x017F, 0x0185, 0x018D, 0x0191, 0x0199, +030 0x01A3, 0x01A5, 0x01AF, 0x01B1, 0x01B7, 0x01BB, 0x01C1, 0x01C9, +031 0x01CD, 0x01CF, 0x01D3, 0x01DF, 0x01E7, 0x01EB, 0x01F3, 0x01F7, +032 0x01FD, 0x0209, 0x020B, 0x021D, 0x0223, 0x022D, 0x0233, 0x0239, +033 0x023B, 0x0241, 0x024B, 0x0251, 0x0257, 0x0259, 0x025F, 0x0265, +034 0x0269, 0x026B, 0x0277, 0x0281, 0x0283, 0x0287, 0x028D, 0x0293, +035 0x0295, 0x02A1, 0x02A5, 0x02AB, 0x02B3, 0x02BD, 0x02C5, 0x02CF, +036 +037 0x02D7, 0x02DD, 0x02E3, 0x02E7, 0x02EF, 0x02F5, 0x02F9, 0x0301, +038 0x0305, 0x0313, 0x031D, 0x0329, 0x032B, 0x0335, 0x0337, 0x033B, +039 0x033D, 0x0347, 0x0355, 0x0359, 0x035B, 0x035F, 0x036D, 0x0371, +040 0x0373, 0x0377, 0x038B, 0x038F, 0x0397, 0x03A1, 0x03A9, 0x03AD, +041 0x03B3, 0x03B9, 0x03C7, 0x03CB, 0x03D1, 0x03D7, 0x03DF, 0x03E5, +042 0x03F1, 0x03F5, 0x03FB, 0x03FD, 0x0407, 0x0409, 0x040F, 0x0419, +043 0x041B, 0x0425, 0x0427, 0x042D, 0x043F, 0x0443, 0x0445, 0x0449, +044 0x044F, 0x0455, 0x045D, 0x0463, 0x0469, 0x047F, 0x0481, 0x048B, +045 +046 0x0493, 0x049D, 0x04A3, 0x04A9, 0x04B1, 0x04BD, 0x04C1, 0x04C7, +047 0x04CD, 0x04CF, 0x04D5, 0x04E1, 0x04EB, 0x04FD, 0x04FF, 0x0503, +048 0x0509, 0x050B, 0x0511, 0x0515, 0x0517, 0x051B, 0x0527, 0x0529, +049 0x052F, 0x0551, 0x0557, 0x055D, 0x0565, 0x0577, 0x0581, 0x058F, +050 0x0593, 0x0595, 0x0599, 0x059F, 0x05A7, 0x05AB, 0x05AD, 0x05B3, +051 0x05BF, 0x05C9, 0x05CB, 0x05CF, 0x05D1, 0x05D5, 0x05DB, 0x05E7, +052 0x05F3, 0x05FB, 0x0607, 0x060D, 0x0611, 0x0617, 0x061F, 0x0623, +053 0x062B, 0x062F, 0x063D, 0x0641, 0x0647, 0x0649, 0x064D, 0x0653 +054 #endif +055 \}; +056 #endif +\end{alltt} +\end{small} + +Note that there are two possible tables. When an mp\_digit is 7-bits long only the primes upto $127$ may be included, otherwise the primes +upto $1619$ are used. Note that the value of \textbf{PRIME\_SIZE} is a constant dependent on the size of a mp\_digit. + +\subsection{The Fermat Test} +The Fermat test is probably one the oldest tests to have a non-trivial probability of success. It is based on the fact that if $n$ is in +fact prime then $a^{n} \equiv a \mbox{ (mod }n\mbox{)}$ for all $0 < a < n$. The reason being that if $n$ is prime than the order of +the multiplicative sub group is $n - 1$. Any base $a$ must have an order which divides $n - 1$ and as such $a^n$ is equivalent to +$a^1 = a$. + +If $n$ is composite then any given base $a$ does not have to have a period which divides $n - 1$. In which case +it is possible that $a^n \nequiv a \mbox{ (mod }n\mbox{)}$. However, this test is not absolute as it is possible that the order +of a base will divide $n - 1$ which would then be reported as prime. Such a base yields what is known as a Fermat pseudo-prime. Several +integers known as Carmichael numbers will be a pseudo-prime to all valid bases. Fortunately such numbers are extremely rare as $n$ grows +in size. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_prime\_fermat}. \\ +\textbf{Input}. mp\_int $a$ and $b$, $a \ge 2$, $0 < b < a$. \\ +\textbf{Output}. $c = 1$ if $b^a \equiv b \mbox{ (mod }a\mbox{)}$, otherwise $c = 0$. \\ +\hline \\ +1. $t \leftarrow b^a \mbox{ (mod }a\mbox{)}$ \\ +2. If $t = b$ then \\ +\hspace{3mm}2.1 $c = 1$ \\ +3. else \\ +\hspace{3mm}3.1 $c = 0$ \\ +4. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_prime\_fermat} +\end{figure} +\textbf{Algorithm mp\_prime\_fermat.} +This algorithm determines whether an mp\_int $a$ is a Fermat prime to the base $b$ or not. It uses a single modular exponentiation to +determine the result. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_prime\_fermat.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* performs one Fermat test. +018 * +019 * If "a" were prime then b**a == b (mod a) since the order of +020 * the multiplicative sub-group would be phi(a) = a-1. That means +021 * it would be the same as b**(a mod (a-1)) == b**1 == b (mod a). +022 * +023 * Sets result to 1 if the congruence holds, or zero otherwise. +024 */ +025 int mp_prime_fermat (mp_int * a, mp_int * b, int *result) +026 \{ +027 mp_int t; +028 int err; +029 +030 /* default to composite */ +031 *result = MP_NO; +032 +033 /* ensure b > 1 */ +034 if (mp_cmp_d(b, 1) != MP_GT) \{ +035 return MP_VAL; +036 \} +037 +038 /* init t */ +039 if ((err = mp_init (&t)) != MP_OKAY) \{ +040 return err; +041 \} +042 +043 /* compute t = b**a mod a */ +044 if ((err = mp_exptmod (b, a, a, &t)) != MP_OKAY) \{ +045 goto LBL_T; +046 \} +047 +048 /* is it equal to b? */ +049 if (mp_cmp (&t, b) == MP_EQ) \{ +050 *result = MP_YES; +051 \} +052 +053 err = MP_OKAY; +054 LBL_T:mp_clear (&t); +055 return err; +056 \} +057 #endif +\end{alltt} +\end{small} + +\subsection{The Miller-Rabin Test} +The Miller-Rabin (citation) test is another primality test which has tighter error bounds than the Fermat test specifically with sequentially chosen +candidate integers. The algorithm is based on the observation that if $n - 1 = 2^kr$ and if $b^r \nequiv \pm 1$ then after upto $k - 1$ squarings the +value must be equal to $-1$. The squarings are stopped as soon as $-1$ is observed. If the value of $1$ is observed first it means that +some value not congruent to $\pm 1$ when squared equals one which cannot occur if $n$ is prime. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_prime\_miller\_rabin}. \\ +\textbf{Input}. mp\_int $a$ and $b$, $a \ge 2$, $0 < b < a$. \\ +\textbf{Output}. $c = 1$ if $a$ is a Miller-Rabin prime to the base $a$, otherwise $c = 0$. \\ +\hline +1. $a' \leftarrow a - 1$ \\ +2. $r \leftarrow n1$ \\ +3. $c \leftarrow 0, s \leftarrow 0$ \\ +4. While $r.used > 0$ and $r_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ +\hspace{3mm}4.1 $s \leftarrow s + 1$ \\ +\hspace{3mm}4.2 $r \leftarrow \lfloor r / 2 \rfloor$ \\ +5. $y \leftarrow b^r \mbox{ (mod }a\mbox{)}$ \\ +6. If $y \nequiv \pm 1$ then \\ +\hspace{3mm}6.1 $j \leftarrow 1$ \\ +\hspace{3mm}6.2 While $j \le (s - 1)$ and $y \nequiv a'$ \\ +\hspace{6mm}6.2.1 $y \leftarrow y^2 \mbox{ (mod }a\mbox{)}$ \\ +\hspace{6mm}6.2.2 If $y = 1$ then goto step 8. \\ +\hspace{6mm}6.2.3 $j \leftarrow j + 1$ \\ +\hspace{3mm}6.3 If $y \nequiv a'$ goto step 8. \\ +7. $c \leftarrow 1$\\ +8. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_prime\_miller\_rabin} +\end{figure} +\textbf{Algorithm mp\_prime\_miller\_rabin.} +This algorithm performs one trial round of the Miller-Rabin algorithm to the base $b$. It will set $c = 1$ if the algorithm cannot determine +if $b$ is composite or $c = 0$ if $b$ is provably composite. The values of $s$ and $r$ are computed such that $a' = a - 1 = 2^sr$. + +If the value $y \equiv b^r$ is congruent to $\pm 1$ then the algorithm cannot prove if $a$ is composite or not. Otherwise, the algorithm will +square $y$ upto $s - 1$ times stopping only when $y \equiv -1$. If $y^2 \equiv 1$ and $y \nequiv \pm 1$ then the algorithm can report that $a$ +is provably composite. If the algorithm performs $s - 1$ squarings and $y \nequiv -1$ then $a$ is provably composite. If $a$ is not provably +composite then it is \textit{probably} prime. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_prime\_miller\_rabin.c +\vspace{-3mm} +\begin{alltt} +016 +017 /* Miller-Rabin test of "a" to the base of "b" as described in +018 * HAC pp. 139 Algorithm 4.24 +019 * +020 * Sets result to 0 if definitely composite or 1 if probably prime. +021 * Randomly the chance of error is no more than 1/4 and often +022 * very much lower. +023 */ +024 int mp_prime_miller_rabin (mp_int * a, mp_int * b, int *result) +025 \{ +026 mp_int n1, y, r; +027 int s, j, err; +028 +029 /* default */ +030 *result = MP_NO; +031 +032 /* ensure b > 1 */ +033 if (mp_cmp_d(b, 1) != MP_GT) \{ +034 return MP_VAL; +035 \} +036 +037 /* get n1 = a - 1 */ +038 if ((err = mp_init_copy (&n1, a)) != MP_OKAY) \{ +039 return err; +040 \} +041 if ((err = mp_sub_d (&n1, 1, &n1)) != MP_OKAY) \{ +042 goto LBL_N1; +043 \} +044 +045 /* set 2**s * r = n1 */ +046 if ((err = mp_init_copy (&r, &n1)) != MP_OKAY) \{ +047 goto LBL_N1; +048 \} +049 +050 /* count the number of least significant bits +051 * which are zero +052 */ +053 s = mp_cnt_lsb(&r); +054 +055 /* now divide n - 1 by 2**s */ +056 if ((err = mp_div_2d (&r, s, &r, NULL)) != MP_OKAY) \{ +057 goto LBL_R; +058 \} +059 +060 /* compute y = b**r mod a */ +061 if ((err = mp_init (&y)) != MP_OKAY) \{ +062 goto LBL_R; +063 \} +064 if ((err = mp_exptmod (b, &r, a, &y)) != MP_OKAY) \{ +065 goto LBL_Y; +066 \} +067 +068 /* if y != 1 and y != n1 do */ +069 if (mp_cmp_d (&y, 1) != MP_EQ && mp_cmp (&y, &n1) != MP_EQ) \{ +070 j = 1; +071 /* while j <= s-1 and y != n1 */ +072 while ((j <= (s - 1)) && mp_cmp (&y, &n1) != MP_EQ) \{ +073 if ((err = mp_sqrmod (&y, a, &y)) != MP_OKAY) \{ +074 goto LBL_Y; +075 \} +076 +077 /* if y == 1 then composite */ +078 if (mp_cmp_d (&y, 1) == MP_EQ) \{ +079 goto LBL_Y; +080 \} +081 +082 ++j; +083 \} +084 +085 /* if y != n1 then composite */ +086 if (mp_cmp (&y, &n1) != MP_EQ) \{ +087 goto LBL_Y; +088 \} +089 \} +090 +091 /* probably prime now */ +092 *result = MP_YES; +093 LBL_Y:mp_clear (&y); +094 LBL_R:mp_clear (&r); +095 LBL_N1:mp_clear (&n1); +096 return err; +097 \} +098 #endif +\end{alltt} +\end{small} + + + + +\backmatter +\appendix +\begin{thebibliography}{ABCDEF} +\bibitem[1]{TAOCPV2} +Donald Knuth, \textit{The Art of Computer Programming}, Third Edition, Volume Two, Seminumerical Algorithms, Addison-Wesley, 1998 + +\bibitem[2]{HAC} +A. Menezes, P. van Oorschot, S. Vanstone, \textit{Handbook of Applied Cryptography}, CRC Press, 1996 + +\bibitem[3]{ROSE} +Michael Rosing, \textit{Implementing Elliptic Curve Cryptography}, Manning Publications, 1999 + +\bibitem[4]{COMBA} +Paul G. Comba, \textit{Exponentiation Cryptosystems on the IBM PC}. IBM Systems Journal 29(4): 526-538 (1990) + +\bibitem[5]{KARA} +A. Karatsuba, Doklay Akad. Nauk SSSR 145 (1962), pp.293-294 + +\bibitem[6]{KARAP} +Andre Weimerskirch and Christof Paar, \textit{Generalizations of the Karatsuba Algorithm for Polynomial Multiplication}, Submitted to Design, Codes and Cryptography, March 2002 + +\bibitem[7]{BARRETT} +Paul Barrett, \textit{Implementing the Rivest Shamir and Adleman Public Key Encryption Algorithm on a Standard Digital Signal Processor}, Advances in Cryptology, Crypto '86, Springer-Verlag. + +\bibitem[8]{MONT} +P.L.Montgomery. \textit{Modular multiplication without trial division}. Mathematics of Computation, 44(170):519-521, April 1985. + +\bibitem[9]{DRMET} +Chae Hoon Lim and Pil Joong Lee, \textit{Generating Efficient Primes for Discrete Log Cryptosystems}, POSTECH Information Research Laboratories + +\bibitem[10]{MMB} +J. Daemen and R. Govaerts and J. Vandewalle, \textit{Block ciphers based on Modular Arithmetic}, State and {P}rogress in the {R}esearch of {C}ryptography, 1993, pp. 80-89 + +\bibitem[11]{RSAREF} +R.L. Rivest, A. Shamir, L. Adleman, \textit{A Method for Obtaining Digital Signatures and Public-Key Cryptosystems} + +\bibitem[12]{DHREF} +Whitfield Diffie, Martin E. Hellman, \textit{New Directions in Cryptography}, IEEE Transactions on Information Theory, 1976 + +\bibitem[13]{IEEE} +IEEE Standard for Binary Floating-Point Arithmetic (ANSI/IEEE Std 754-1985) + +\bibitem[14]{GMP} +GNU Multiple Precision (GMP), \url{http://www.swox.com/gmp/} + +\bibitem[15]{MPI} +Multiple Precision Integer Library (MPI), Michael Fromberger, \url{http://thayer.dartmouth.edu/~sting/mpi/} + +\bibitem[16]{OPENSSL} +OpenSSL Cryptographic Toolkit, \url{http://openssl.org} + +\bibitem[17]{LIP} +Large Integer Package, \url{http://home.hetnet.nl/~ecstr/LIP.zip} + +\bibitem[18]{ISOC} +JTC1/SC22/WG14, ISO/IEC 9899:1999, ``A draft rationale for the C99 standard.'' + +\bibitem[19]{JAVA} +The Sun Java Website, \url{http://java.sun.com/} + +\end{thebibliography} + +\input{tommath.ind} + +\end{document} diff --git a/libtommath/tommath_class.h b/libtommath/tommath_class.h new file mode 100644 index 0000000..53bfa31 --- /dev/null +++ b/libtommath/tommath_class.h @@ -0,0 +1,952 @@ +#if !(defined(LTM1) && defined(LTM2) && defined(LTM3)) +#if defined(LTM2) +#define LTM3 +#endif +#if defined(LTM1) +#define LTM2 +#endif +#define LTM1 + +#if defined(LTM_ALL) +#define BN_ERROR_C +#define BN_FAST_MP_INVMOD_C +#define BN_FAST_MP_MONTGOMERY_REDUCE_C +#define BN_FAST_S_MP_MUL_DIGS_C +#define BN_FAST_S_MP_MUL_HIGH_DIGS_C +#define BN_FAST_S_MP_SQR_C +#define BN_MP_2EXPT_C +#define BN_MP_ABS_C +#define BN_MP_ADD_C +#define BN_MP_ADD_D_C +#define BN_MP_ADDMOD_C +#define BN_MP_AND_C +#define BN_MP_CLAMP_C +#define BN_MP_CLEAR_C +#define BN_MP_CLEAR_MULTI_C +#define BN_MP_CMP_C +#define BN_MP_CMP_D_C +#define BN_MP_CMP_MAG_C +#define BN_MP_CNT_LSB_C +#define BN_MP_COPY_C +#define BN_MP_COUNT_BITS_C +#define BN_MP_DIV_C +#define BN_MP_DIV_2_C +#define BN_MP_DIV_2D_C +#define BN_MP_DIV_3_C +#define BN_MP_DIV_D_C +#define BN_MP_DR_IS_MODULUS_C +#define BN_MP_DR_REDUCE_C +#define BN_MP_DR_SETUP_C +#define BN_MP_EXCH_C +#define BN_MP_EXPT_D_C +#define BN_MP_EXPTMOD_C +#define BN_MP_EXPTMOD_FAST_C +#define BN_MP_EXTEUCLID_C +#define BN_MP_FREAD_C +#define BN_MP_FWRITE_C +#define BN_MP_GCD_C +#define BN_MP_GET_INT_C +#define BN_MP_GROW_C +#define BN_MP_INIT_C +#define BN_MP_INIT_COPY_C +#define BN_MP_INIT_MULTI_C +#define BN_MP_INIT_SET_C +#define BN_MP_INIT_SET_INT_C +#define BN_MP_INIT_SIZE_C +#define BN_MP_INVMOD_C +#define BN_MP_INVMOD_SLOW_C +#define BN_MP_IS_SQUARE_C +#define BN_MP_JACOBI_C +#define BN_MP_KARATSUBA_MUL_C +#define BN_MP_KARATSUBA_SQR_C +#define BN_MP_LCM_C +#define BN_MP_LSHD_C +#define BN_MP_MOD_C +#define BN_MP_MOD_2D_C +#define BN_MP_MOD_D_C +#define BN_MP_MONTGOMERY_CALC_NORMALIZATION_C +#define BN_MP_MONTGOMERY_REDUCE_C +#define BN_MP_MONTGOMERY_SETUP_C +#define BN_MP_MUL_C +#define BN_MP_MUL_2_C +#define BN_MP_MUL_2D_C +#define BN_MP_MUL_D_C +#define BN_MP_MULMOD_C +#define BN_MP_N_ROOT_C +#define BN_MP_NEG_C +#define BN_MP_OR_C +#define BN_MP_PRIME_FERMAT_C +#define BN_MP_PRIME_IS_DIVISIBLE_C +#define BN_MP_PRIME_IS_PRIME_C +#define BN_MP_PRIME_MILLER_RABIN_C +#define BN_MP_PRIME_NEXT_PRIME_C +#define BN_MP_PRIME_RABIN_MILLER_TRIALS_C +#define BN_MP_PRIME_RANDOM_EX_C +#define BN_MP_RADIX_SIZE_C +#define BN_MP_RADIX_SMAP_C +#define BN_MP_RAND_C +#define BN_MP_READ_RADIX_C +#define BN_MP_READ_SIGNED_BIN_C +#define BN_MP_READ_UNSIGNED_BIN_C +#define BN_MP_REDUCE_C +#define BN_MP_REDUCE_2K_C +#define BN_MP_REDUCE_2K_SETUP_C +#define BN_MP_REDUCE_IS_2K_C +#define BN_MP_REDUCE_SETUP_C +#define BN_MP_RSHD_C +#define BN_MP_SET_C +#define BN_MP_SET_INT_C +#define BN_MP_SHRINK_C +#define BN_MP_SIGNED_BIN_SIZE_C +#define BN_MP_SQR_C +#define BN_MP_SQRMOD_C +#define BN_MP_SQRT_C +#define BN_MP_SUB_C +#define BN_MP_SUB_D_C +#define BN_MP_SUBMOD_C +#define BN_MP_TO_SIGNED_BIN_C +#define BN_MP_TO_UNSIGNED_BIN_C +#define BN_MP_TOOM_MUL_C +#define BN_MP_TOOM_SQR_C +#define BN_MP_TORADIX_C +#define BN_MP_TORADIX_N_C +#define BN_MP_UNSIGNED_BIN_SIZE_C +#define BN_MP_XOR_C +#define BN_MP_ZERO_C +#define BN_PRIME_TAB_C +#define BN_REVERSE_C +#define BN_S_MP_ADD_C +#define BN_S_MP_EXPTMOD_C +#define BN_S_MP_MUL_DIGS_C +#define BN_S_MP_MUL_HIGH_DIGS_C +#define BN_S_MP_SQR_C +#define BN_S_MP_SUB_C +#define BNCORE_C +#endif + +#if defined(BN_ERROR_C) + #define BN_MP_ERROR_TO_STRING_C +#endif + +#if defined(BN_FAST_MP_INVMOD_C) + #define BN_MP_ISEVEN_C + #define BN_MP_INIT_MULTI_C + #define BN_MP_COPY_C + #define BN_MP_ABS_C + #define BN_MP_SET_C + #define BN_MP_DIV_2_C + #define BN_MP_ISODD_C + #define BN_MP_SUB_C + #define BN_MP_CMP_C + #define BN_MP_ISZERO_C + #define BN_MP_CMP_D_C + #define BN_MP_ADD_C + #define BN_MP_EXCH_C + #define BN_MP_CLEAR_MULTI_C +#endif + +#if defined(BN_FAST_MP_MONTGOMERY_REDUCE_C) + #define BN_MP_GROW_C + #define BN_MP_RSHD_C + #define BN_MP_CLAMP_C + #define BN_MP_CMP_MAG_C + #define BN_S_MP_SUB_C +#endif + +#if defined(BN_FAST_S_MP_MUL_DIGS_C) + #define BN_MP_GROW_C + #define BN_MP_CLAMP_C +#endif + +#if defined(BN_FAST_S_MP_MUL_HIGH_DIGS_C) + #define BN_MP_GROW_C + #define BN_MP_CLAMP_C +#endif + +#if defined(BN_FAST_S_MP_SQR_C) + #define BN_MP_GROW_C + #define BN_MP_CLAMP_C +#endif + +#if defined(BN_MP_2EXPT_C) + #define BN_MP_ZERO_C + #define BN_MP_GROW_C +#endif + +#if defined(BN_MP_ABS_C) + #define BN_MP_COPY_C +#endif + +#if defined(BN_MP_ADD_C) + #define BN_S_MP_ADD_C + #define BN_MP_CMP_MAG_C + #define BN_S_MP_SUB_C +#endif + +#if defined(BN_MP_ADD_D_C) + #define BN_MP_GROW_C + #define BN_MP_SUB_D_C + #define BN_MP_CLAMP_C +#endif + +#if defined(BN_MP_ADDMOD_C) + #define BN_MP_INIT_C + #define BN_MP_ADD_C + #define BN_MP_CLEAR_C + #define BN_MP_MOD_C +#endif + +#if defined(BN_MP_AND_C) + #define BN_MP_INIT_COPY_C + #define BN_MP_CLAMP_C + #define BN_MP_EXCH_C + #define BN_MP_CLEAR_C +#endif + +#if defined(BN_MP_CLAMP_C) +#endif + +#if defined(BN_MP_CLEAR_C) +#endif + +#if defined(BN_MP_CLEAR_MULTI_C) + #define BN_MP_CLEAR_C +#endif + +#if defined(BN_MP_CMP_C) + #define BN_MP_CMP_MAG_C +#endif + +#if defined(BN_MP_CMP_D_C) +#endif + +#if defined(BN_MP_CMP_MAG_C) +#endif + +#if defined(BN_MP_CNT_LSB_C) + #define BN_MP_ISZERO_C +#endif + +#if defined(BN_MP_COPY_C) + #define BN_MP_GROW_C +#endif + +#if defined(BN_MP_COUNT_BITS_C) +#endif + +#if defined(BN_MP_DIV_C) + #define BN_MP_ISZERO_C + #define BN_MP_CMP_MAG_C + #define BN_MP_COPY_C + #define BN_MP_ZERO_C + #define BN_MP_INIT_MULTI_C + #define BN_MP_SET_C + #define BN_MP_COUNT_BITS_C + #define BN_MP_ABS_C + #define BN_MP_MUL_2D_C + #define BN_MP_CMP_C + #define BN_MP_SUB_C + #define BN_MP_ADD_C + #define BN_MP_DIV_2D_C + #define BN_MP_EXCH_C + #define BN_MP_CLEAR_MULTI_C + #define BN_MP_INIT_SIZE_C + #define BN_MP_INIT_C + #define BN_MP_INIT_COPY_C + #define BN_MP_LSHD_C + #define BN_MP_RSHD_C + #define BN_MP_MUL_D_C + #define BN_MP_CLAMP_C + #define BN_MP_CLEAR_C +#endif + +#if defined(BN_MP_DIV_2_C) + #define BN_MP_GROW_C + #define BN_MP_CLAMP_C +#endif + +#if defined(BN_MP_DIV_2D_C) + #define BN_MP_COPY_C + #define BN_MP_ZERO_C + #define BN_MP_INIT_C + #define BN_MP_MOD_2D_C + #define BN_MP_CLEAR_C + #define BN_MP_RSHD_C + #define BN_MP_CLAMP_C + #define BN_MP_EXCH_C +#endif + +#if defined(BN_MP_DIV_3_C) + #define BN_MP_INIT_SIZE_C + #define BN_MP_CLAMP_C + #define BN_MP_EXCH_C + #define BN_MP_CLEAR_C +#endif + +#if defined(BN_MP_DIV_D_C) + #define BN_MP_ISZERO_C + #define BN_MP_COPY_C + #define BN_MP_DIV_2D_C + #define BN_MP_DIV_3_C + #define BN_MP_INIT_SIZE_C + #define BN_MP_CLAMP_C + #define BN_MP_EXCH_C + #define BN_MP_CLEAR_C +#endif + +#if defined(BN_MP_DR_IS_MODULUS_C) +#endif + +#if defined(BN_MP_DR_REDUCE_C) + #define BN_MP_GROW_C + #define BN_MP_CLAMP_C + #define BN_MP_CMP_MAG_C + #define BN_S_MP_SUB_C +#endif + +#if defined(BN_MP_DR_SETUP_C) +#endif + +#if defined(BN_MP_EXCH_C) +#endif + +#if defined(BN_MP_EXPT_D_C) + #define BN_MP_INIT_COPY_C + #define BN_MP_SET_C + #define BN_MP_SQR_C + #define BN_MP_CLEAR_C + #define BN_MP_MUL_C +#endif + +#if defined(BN_MP_EXPTMOD_C) + #define BN_MP_INIT_C + #define BN_MP_INVMOD_C + #define BN_MP_CLEAR_C + #define BN_MP_ABS_C + #define BN_MP_CLEAR_MULTI_C + #define BN_MP_DR_IS_MODULUS_C + #define BN_MP_REDUCE_IS_2K_C + #define BN_MP_ISODD_C + #define BN_MP_EXPTMOD_FAST_C + #define BN_S_MP_EXPTMOD_C +#endif + +#if defined(BN_MP_EXPTMOD_FAST_C) + #define BN_MP_COUNT_BITS_C + #define BN_MP_INIT_C + #define BN_MP_CLEAR_C + #define BN_MP_MONTGOMERY_SETUP_C + #define BN_FAST_MP_MONTGOMERY_REDUCE_C + #define BN_MP_MONTGOMERY_REDUCE_C + #define BN_MP_DR_SETUP_C + #define BN_MP_DR_REDUCE_C + #define BN_MP_REDUCE_2K_SETUP_C + #define BN_MP_REDUCE_2K_C + #define BN_MP_MONTGOMERY_CALC_NORMALIZATION_C + #define BN_MP_MULMOD_C + #define BN_MP_SET_C + #define BN_MP_MOD_C + #define BN_MP_COPY_C + #define BN_MP_SQR_C + #define BN_MP_MUL_C + #define BN_MP_EXCH_C +#endif + +#if defined(BN_MP_EXTEUCLID_C) + #define BN_MP_INIT_MULTI_C + #define BN_MP_SET_C + #define BN_MP_COPY_C + #define BN_MP_ISZERO_C + #define BN_MP_DIV_C + #define BN_MP_MUL_C + #define BN_MP_SUB_C + #define BN_MP_EXCH_C + #define BN_MP_CLEAR_MULTI_C +#endif + +#if defined(BN_MP_FREAD_C) + #define BN_MP_ZERO_C + #define BN_MP_S_RMAP_C + #define BN_MP_MUL_D_C + #define BN_MP_ADD_D_C + #define BN_MP_CMP_D_C +#endif + +#if defined(BN_MP_FWRITE_C) + #define BN_MP_RADIX_SIZE_C + #define BN_MP_TORADIX_C +#endif + +#if defined(BN_MP_GCD_C) + #define BN_MP_ISZERO_C + #define BN_MP_ABS_C + #define BN_MP_ZERO_C + #define BN_MP_INIT_COPY_C + #define BN_MP_CNT_LSB_C + #define BN_MP_DIV_2D_C + #define BN_MP_CMP_MAG_C + #define BN_MP_EXCH_C + #define BN_S_MP_SUB_C + #define BN_MP_MUL_2D_C + #define BN_MP_CLEAR_C +#endif + +#if defined(BN_MP_GET_INT_C) +#endif + +#if defined(BN_MP_GROW_C) +#endif + +#if defined(BN_MP_INIT_C) +#endif + +#if defined(BN_MP_INIT_COPY_C) + #define BN_MP_COPY_C +#endif + +#if defined(BN_MP_INIT_MULTI_C) + #define BN_MP_ERR_C + #define BN_MP_INIT_C + #define BN_MP_CLEAR_C +#endif + +#if defined(BN_MP_INIT_SET_C) + #define BN_MP_INIT_C + #define BN_MP_SET_C +#endif + +#if defined(BN_MP_INIT_SET_INT_C) + #define BN_MP_INIT_C + #define BN_MP_SET_INT_C +#endif + +#if defined(BN_MP_INIT_SIZE_C) + #define BN_MP_INIT_C +#endif + +#if defined(BN_MP_INVMOD_C) + #define BN_MP_ISZERO_C + #define BN_MP_ISODD_C + #define BN_FAST_MP_INVMOD_C + #define BN_MP_INVMOD_SLOW_C +#endif + +#if defined(BN_MP_INVMOD_SLOW_C) + #define BN_MP_ISZERO_C + #define BN_MP_INIT_MULTI_C + #define BN_MP_COPY_C + #define BN_MP_ISEVEN_C + #define BN_MP_SET_C + #define BN_MP_DIV_2_C + #define BN_MP_ISODD_C + #define BN_MP_ADD_C + #define BN_MP_SUB_C + #define BN_MP_CMP_C + #define BN_MP_CMP_D_C + #define BN_MP_CMP_MAG_C + #define BN_MP_EXCH_C + #define BN_MP_CLEAR_MULTI_C +#endif + +#if defined(BN_MP_IS_SQUARE_C) + #define BN_MP_MOD_D_C + #define BN_MP_INIT_SET_INT_C + #define BN_MP_MOD_C + #define BN_MP_GET_INT_C + #define BN_MP_SQRT_C + #define BN_MP_SQR_C + #define BN_MP_CMP_MAG_C + #define BN_MP_CLEAR_C +#endif + +#if defined(BN_MP_JACOBI_C) + #define BN_MP_CMP_D_C + #define BN_MP_ISZERO_C + #define BN_MP_INIT_COPY_C + #define BN_MP_CNT_LSB_C + #define BN_MP_DIV_2D_C + #define BN_MP_MOD_C + #define BN_MP_CLEAR_C +#endif + +#if defined(BN_MP_KARATSUBA_MUL_C) + #define BN_MP_MUL_C + #define BN_MP_INIT_SIZE_C + #define BN_MP_CLAMP_C + #define BN_MP_SUB_C + #define BN_MP_ADD_C + #define BN_MP_LSHD_C + #define BN_MP_CLEAR_C +#endif + +#if defined(BN_MP_KARATSUBA_SQR_C) + #define BN_MP_INIT_SIZE_C + #define BN_MP_CLAMP_C + #define BN_MP_SQR_C + #define BN_MP_SUB_C + #define BN_S_MP_ADD_C + #define BN_MP_LSHD_C + #define BN_MP_ADD_C + #define BN_MP_CLEAR_C +#endif + +#if defined(BN_MP_LCM_C) + #define BN_MP_INIT_MULTI_C + #define BN_MP_GCD_C + #define BN_MP_CMP_MAG_C + #define BN_MP_DIV_C + #define BN_MP_MUL_C + #define BN_MP_CLEAR_MULTI_C +#endif + +#if defined(BN_MP_LSHD_C) + #define BN_MP_GROW_C + #define BN_MP_RSHD_C +#endif + +#if defined(BN_MP_MOD_C) + #define BN_MP_INIT_C + #define BN_MP_DIV_C + #define BN_MP_CLEAR_C + #define BN_MP_ADD_C + #define BN_MP_EXCH_C +#endif + +#if defined(BN_MP_MOD_2D_C) + #define BN_MP_ZERO_C + #define BN_MP_COPY_C + #define BN_MP_CLAMP_C +#endif + +#if defined(BN_MP_MOD_D_C) + #define BN_MP_DIV_D_C +#endif + +#if defined(BN_MP_MONTGOMERY_CALC_NORMALIZATION_C) + #define BN_MP_COUNT_BITS_C + #define BN_MP_2EXPT_C + #define BN_MP_SET_C + #define BN_MP_MUL_2_C + #define BN_MP_CMP_MAG_C + #define BN_S_MP_SUB_C +#endif + +#if defined(BN_MP_MONTGOMERY_REDUCE_C) + #define BN_FAST_MP_MONTGOMERY_REDUCE_C + #define BN_MP_GROW_C + #define BN_MP_CLAMP_C + #define BN_MP_RSHD_C + #define BN_MP_CMP_MAG_C + #define BN_S_MP_SUB_C +#endif + +#if defined(BN_MP_MONTGOMERY_SETUP_C) +#endif + +#if defined(BN_MP_MUL_C) + #define BN_MP_TOOM_MUL_C + #define BN_MP_KARATSUBA_MUL_C + #define BN_FAST_S_MP_MUL_DIGS_C + #define BN_S_MP_MUL_C + #define BN_S_MP_MUL_DIGS_C +#endif + +#if defined(BN_MP_MUL_2_C) + #define BN_MP_GROW_C +#endif + +#if defined(BN_MP_MUL_2D_C) + #define BN_MP_COPY_C + #define BN_MP_GROW_C + #define BN_MP_LSHD_C + #define BN_MP_CLAMP_C +#endif + +#if defined(BN_MP_MUL_D_C) + #define BN_MP_GROW_C + #define BN_MP_CLAMP_C +#endif + +#if defined(BN_MP_MULMOD_C) + #define BN_MP_INIT_C + #define BN_MP_MUL_C + #define BN_MP_CLEAR_C + #define BN_MP_MOD_C +#endif + +#if defined(BN_MP_N_ROOT_C) + #define BN_MP_INIT_C + #define BN_MP_SET_C + #define BN_MP_COPY_C + #define BN_MP_EXPT_D_C + #define BN_MP_MUL_C + #define BN_MP_SUB_C + #define BN_MP_MUL_D_C + #define BN_MP_DIV_C + #define BN_MP_CMP_C + #define BN_MP_SUB_D_C + #define BN_MP_EXCH_C + #define BN_MP_CLEAR_C +#endif + +#if defined(BN_MP_NEG_C) + #define BN_MP_COPY_C + #define BN_MP_ISZERO_C +#endif + +#if defined(BN_MP_OR_C) + #define BN_MP_INIT_COPY_C + #define BN_MP_CLAMP_C + #define BN_MP_EXCH_C + #define BN_MP_CLEAR_C +#endif + +#if defined(BN_MP_PRIME_FERMAT_C) + #define BN_MP_CMP_D_C + #define BN_MP_INIT_C + #define BN_MP_EXPTMOD_C + #define BN_MP_CMP_C + #define BN_MP_CLEAR_C +#endif + +#if defined(BN_MP_PRIME_IS_DIVISIBLE_C) + #define BN_MP_MOD_D_C +#endif + +#if defined(BN_MP_PRIME_IS_PRIME_C) + #define BN_MP_CMP_D_C + #define BN_MP_PRIME_IS_DIVISIBLE_C + #define BN_MP_INIT_C + #define BN_MP_SET_C + #define BN_MP_PRIME_MILLER_RABIN_C + #define BN_MP_CLEAR_C +#endif + +#if defined(BN_MP_PRIME_MILLER_RABIN_C) + #define BN_MP_CMP_D_C + #define BN_MP_INIT_COPY_C + #define BN_MP_SUB_D_C + #define BN_MP_CNT_LSB_C + #define BN_MP_DIV_2D_C + #define BN_MP_EXPTMOD_C + #define BN_MP_CMP_C + #define BN_MP_SQRMOD_C + #define BN_MP_CLEAR_C +#endif + +#if defined(BN_MP_PRIME_NEXT_PRIME_C) + #define BN_MP_CMP_D_C + #define BN_MP_SET_C + #define BN_MP_SUB_D_C + #define BN_MP_ISEVEN_C + #define BN_MP_MOD_D_C + #define BN_MP_INIT_C + #define BN_MP_ADD_D_C + #define BN_MP_PRIME_MILLER_RABIN_C + #define BN_MP_CLEAR_C +#endif + +#if defined(BN_MP_PRIME_RABIN_MILLER_TRIALS_C) +#endif + +#if defined(BN_MP_PRIME_RANDOM_EX_C) + #define BN_MP_READ_UNSIGNED_BIN_C + #define BN_MP_PRIME_IS_PRIME_C + #define BN_MP_SUB_D_C + #define BN_MP_DIV_2_C + #define BN_MP_MUL_2_C + #define BN_MP_ADD_D_C +#endif + +#if defined(BN_MP_RADIX_SIZE_C) + #define BN_MP_COUNT_BITS_C + #define BN_MP_INIT_COPY_C + #define BN_MP_ISZERO_C + #define BN_MP_DIV_D_C + #define BN_MP_CLEAR_C +#endif + +#if defined(BN_MP_RADIX_SMAP_C) + #define BN_MP_S_RMAP_C +#endif + +#if defined(BN_MP_RAND_C) + #define BN_MP_ZERO_C + #define BN_MP_ADD_D_C + #define BN_MP_LSHD_C +#endif + +#if defined(BN_MP_READ_RADIX_C) + #define BN_MP_ZERO_C + #define BN_MP_S_RMAP_C + #define BN_MP_MUL_D_C + #define BN_MP_ADD_D_C + #define BN_MP_ISZERO_C +#endif + +#if defined(BN_MP_READ_SIGNED_BIN_C) + #define BN_MP_READ_UNSIGNED_BIN_C +#endif + +#if defined(BN_MP_READ_UNSIGNED_BIN_C) + #define BN_MP_GROW_C + #define BN_MP_ZERO_C + #define BN_MP_MUL_2D_C + #define BN_MP_CLAMP_C +#endif + +#if defined(BN_MP_REDUCE_C) + #define BN_MP_REDUCE_SETUP_C + #define BN_MP_INIT_COPY_C + #define BN_MP_RSHD_C + #define BN_MP_MUL_C + #define BN_S_MP_MUL_HIGH_DIGS_C + #define BN_FAST_S_MP_MUL_HIGH_DIGS_C + #define BN_MP_MOD_2D_C + #define BN_S_MP_MUL_DIGS_C + #define BN_MP_SUB_C + #define BN_MP_CMP_D_C + #define BN_MP_SET_C + #define BN_MP_LSHD_C + #define BN_MP_ADD_C + #define BN_MP_CMP_C + #define BN_S_MP_SUB_C + #define BN_MP_CLEAR_C +#endif + +#if defined(BN_MP_REDUCE_2K_C) + #define BN_MP_INIT_C + #define BN_MP_COUNT_BITS_C + #define BN_MP_DIV_2D_C + #define BN_MP_MUL_D_C + #define BN_S_MP_ADD_C + #define BN_MP_CMP_MAG_C + #define BN_S_MP_SUB_C + #define BN_MP_CLEAR_C +#endif + +#if defined(BN_MP_REDUCE_2K_SETUP_C) + #define BN_MP_INIT_C + #define BN_MP_COUNT_BITS_C + #define BN_MP_2EXPT_C + #define BN_MP_CLEAR_C + #define BN_S_MP_SUB_C +#endif + +#if defined(BN_MP_REDUCE_IS_2K_C) + #define BN_MP_REDUCE_2K_C + #define BN_MP_COUNT_BITS_C +#endif + +#if defined(BN_MP_REDUCE_SETUP_C) + #define BN_MP_2EXPT_C + #define BN_MP_DIV_C +#endif + +#if defined(BN_MP_RSHD_C) + #define BN_MP_ZERO_C +#endif + +#if defined(BN_MP_SET_C) + #define BN_MP_ZERO_C +#endif + +#if defined(BN_MP_SET_INT_C) + #define BN_MP_ZERO_C + #define BN_MP_MUL_2D_C + #define BN_MP_CLAMP_C +#endif + +#if defined(BN_MP_SHRINK_C) +#endif + +#if defined(BN_MP_SIGNED_BIN_SIZE_C) + #define BN_MP_UNSIGNED_BIN_SIZE_C +#endif + +#if defined(BN_MP_SQR_C) + #define BN_MP_TOOM_SQR_C + #define BN_MP_KARATSUBA_SQR_C + #define BN_FAST_S_MP_SQR_C + #define BN_S_MP_SQR_C +#endif + +#if defined(BN_MP_SQRMOD_C) + #define BN_MP_INIT_C + #define BN_MP_SQR_C + #define BN_MP_CLEAR_C + #define BN_MP_MOD_C +#endif + +#if defined(BN_MP_SQRT_C) + #define BN_MP_N_ROOT_C + #define BN_MP_ISZERO_C + #define BN_MP_ZERO_C + #define BN_MP_INIT_COPY_C + #define BN_MP_RSHD_C + #define BN_MP_DIV_C + #define BN_MP_ADD_C + #define BN_MP_DIV_2_C + #define BN_MP_CMP_MAG_C + #define BN_MP_EXCH_C + #define BN_MP_CLEAR_C +#endif + +#if defined(BN_MP_SUB_C) + #define BN_S_MP_ADD_C + #define BN_MP_CMP_MAG_C + #define BN_S_MP_SUB_C +#endif + +#if defined(BN_MP_SUB_D_C) + #define BN_MP_GROW_C + #define BN_MP_ADD_D_C + #define BN_MP_CLAMP_C +#endif + +#if defined(BN_MP_SUBMOD_C) + #define BN_MP_INIT_C + #define BN_MP_SUB_C + #define BN_MP_CLEAR_C + #define BN_MP_MOD_C +#endif + +#if defined(BN_MP_TO_SIGNED_BIN_C) + #define BN_MP_TO_UNSIGNED_BIN_C +#endif + +#if defined(BN_MP_TO_UNSIGNED_BIN_C) + #define BN_MP_INIT_COPY_C + #define BN_MP_ISZERO_C + #define BN_MP_DIV_2D_C + #define BN_MP_CLEAR_C +#endif + +#if defined(BN_MP_TOOM_MUL_C) + #define BN_MP_INIT_MULTI_C + #define BN_MP_MOD_2D_C + #define BN_MP_COPY_C + #define BN_MP_RSHD_C + #define BN_MP_MUL_C + #define BN_MP_MUL_2_C + #define BN_MP_ADD_C + #define BN_MP_SUB_C + #define BN_MP_DIV_2_C + #define BN_MP_MUL_2D_C + #define BN_MP_MUL_D_C + #define BN_MP_DIV_3_C + #define BN_MP_LSHD_C + #define BN_MP_CLEAR_MULTI_C +#endif + +#if defined(BN_MP_TOOM_SQR_C) + #define BN_MP_INIT_MULTI_C + #define BN_MP_MOD_2D_C + #define BN_MP_COPY_C + #define BN_MP_RSHD_C + #define BN_MP_SQR_C + #define BN_MP_MUL_2_C + #define BN_MP_ADD_C + #define BN_MP_SUB_C + #define BN_MP_DIV_2_C + #define BN_MP_MUL_2D_C + #define BN_MP_MUL_D_C + #define BN_MP_DIV_3_C + #define BN_MP_LSHD_C + #define BN_MP_CLEAR_MULTI_C +#endif + +#if defined(BN_MP_TORADIX_C) + #define BN_MP_ISZERO_C + #define BN_MP_INIT_COPY_C + #define BN_MP_DIV_D_C + #define BN_MP_CLEAR_C + #define BN_MP_S_RMAP_C +#endif + +#if defined(BN_MP_TORADIX_N_C) + #define BN_MP_ISZERO_C + #define BN_MP_INIT_COPY_C + #define BN_MP_DIV_D_C + #define BN_MP_CLEAR_C + #define BN_MP_S_RMAP_C +#endif + +#if defined(BN_MP_UNSIGNED_BIN_SIZE_C) + #define BN_MP_COUNT_BITS_C +#endif + +#if defined(BN_MP_XOR_C) + #define BN_MP_INIT_COPY_C + #define BN_MP_CLAMP_C + #define BN_MP_EXCH_C + #define BN_MP_CLEAR_C +#endif + +#if defined(BN_MP_ZERO_C) +#endif + +#if defined(BN_PRIME_TAB_C) +#endif + +#if defined(BN_REVERSE_C) +#endif + +#if defined(BN_S_MP_ADD_C) + #define BN_MP_GROW_C + #define BN_MP_CLAMP_C +#endif + +#if defined(BN_S_MP_EXPTMOD_C) + #define BN_MP_COUNT_BITS_C + #define BN_MP_INIT_C + #define BN_MP_CLEAR_C + #define BN_MP_REDUCE_SETUP_C + #define BN_MP_MOD_C + #define BN_MP_COPY_C + #define BN_MP_SQR_C + #define BN_MP_REDUCE_C + #define BN_MP_MUL_C + #define BN_MP_SET_C + #define BN_MP_EXCH_C +#endif + +#if defined(BN_S_MP_MUL_DIGS_C) + #define BN_FAST_S_MP_MUL_DIGS_C + #define BN_MP_INIT_SIZE_C + #define BN_MP_CLAMP_C + #define BN_MP_EXCH_C + #define BN_MP_CLEAR_C +#endif + +#if defined(BN_S_MP_MUL_HIGH_DIGS_C) + #define BN_FAST_S_MP_MUL_HIGH_DIGS_C + #define BN_MP_INIT_SIZE_C + #define BN_MP_CLAMP_C + #define BN_MP_EXCH_C + #define BN_MP_CLEAR_C +#endif + +#if defined(BN_S_MP_SQR_C) + #define BN_MP_INIT_SIZE_C + #define BN_MP_CLAMP_C + #define BN_MP_EXCH_C + #define BN_MP_CLEAR_C +#endif + +#if defined(BN_S_MP_SUB_C) + #define BN_MP_GROW_C + #define BN_MP_CLAMP_C +#endif + +#if defined(BNCORE_C) +#endif + +#ifdef LTM3 +#define LTM_LAST +#endif +#include <tommath_superclass.h> +#include <tommath_class.h> +#else +#define LTM_LAST +#endif diff --git a/libtommath/tommath_superclass.h b/libtommath/tommath_superclass.h new file mode 100644 index 0000000..b50ecb0 --- /dev/null +++ b/libtommath/tommath_superclass.h @@ -0,0 +1,72 @@ +/* super class file for PK algos */ + +/* default ... include all MPI */ +#define LTM_ALL + +/* RSA only (does not support DH/DSA/ECC) */ +// #define SC_RSA_1 + +/* For reference.... On an Athlon64 optimizing for speed... + + LTM's mpi.o with all functions [striped] is 142KiB in size. + +*/ + +/* Works for RSA only, mpi.o is 68KiB */ +#ifdef SC_RSA_1 + #define BN_MP_SHRINK_C + #define BN_MP_LCM_C + #define BN_MP_PRIME_RANDOM_EX_C + #define BN_MP_INVMOD_C + #define BN_MP_GCD_C + #define BN_MP_MOD_C + #define BN_MP_MULMOD_C + #define BN_MP_ADDMOD_C + #define BN_MP_EXPTMOD_C + #define BN_MP_SET_INT_C + #define BN_MP_INIT_MULTI_C + #define BN_MP_CLEAR_MULTI_C + #define BN_MP_UNSIGNED_BIN_SIZE_C + #define BN_MP_TO_UNSIGNED_BIN_C + #define BN_MP_MOD_D_C + #define BN_MP_PRIME_RABIN_MILLER_TRIALS_C + #define BN_REVERSE_C + #define BN_PRIME_TAB_C + + /* other modifiers */ + #define BN_MP_DIV_SMALL /* Slower division, not critical */ + + /* here we are on the last pass so we turn things off. The functions classes are still there + * but we remove them specifically from the build. This also invokes tweaks in functions + * like removing support for even moduli, etc... + */ +#ifdef LTM_LAST + #undef BN_MP_TOOM_MUL_C + #undef BN_MP_TOOM_SQR_C + #undef BN_MP_KARATSUBA_MUL_C + #undef BN_MP_KARATSUBA_SQR_C + #undef BN_MP_REDUCE_C + #undef BN_MP_REDUCE_SETUP_C + #undef BN_MP_DR_IS_MODULUS_C + #undef BN_MP_DR_SETUP_C + #undef BN_MP_DR_REDUCE_C + #undef BN_MP_REDUCE_IS_2K_C + #undef BN_MP_REDUCE_2K_SETUP_C + #undef BN_MP_REDUCE_2K_C + #undef BN_S_MP_EXPTMOD_C + #undef BN_MP_DIV_3_C + #undef BN_S_MP_MUL_HIGH_DIGS_C + #undef BN_FAST_S_MP_MUL_HIGH_DIGS_C + #undef BN_FAST_MP_INVMOD_C + + /* To safely undefine these you have to make sure your RSA key won't exceed the Comba threshold + * which is roughly 255 digits [7140 bits for 32-bit machines, 15300 bits for 64-bit machines] + * which means roughly speaking you can handle upto 2536-bit RSA keys with these defined without + * trouble. + */ + #undef BN_S_MP_MUL_DIGS_C + #undef BN_S_MP_SQR_C + #undef BN_MP_MONTGOMERY_REDUCE_C +#endif + +#endif |
