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-rw-r--r--.fossil-settings/ignore-glob16
-rw-r--r--generic/tclExecute.c2
-rw-r--r--generic/tclTomMath.decls3
-rw-r--r--generic/tclTomMath.h20
-rw-r--r--libtommath/bn.tex1913
-rw-r--r--libtommath/bn_mp_div_d.c7
-rw-r--r--libtommath/bn_mp_radix_size.c13
-rw-r--r--libtommath/bn_mp_read_radix.c9
-rw-r--r--libtommath/booker.pl267
-rw-r--r--libtommath/changes.txt4
-rw-r--r--libtommath/dep.pl123
-rw-r--r--libtommath/gen.pl19
-rw-r--r--libtommath/makefile.cygwin_dll4
-rw-r--r--libtommath/mess.sh4
-rw-r--r--libtommath/poster.tex35
-rw-r--r--libtommath/pretty.build66
-rw-r--r--libtommath/tommath.h18
-rw-r--r--libtommath/tommath.src6339
-rw-r--r--macosx/Tcl.xcode/project.pbxproj3
-rw-r--r--macosx/Tcl.xcodeproj/project.pbxproj6
-rw-r--r--unix/Makefile.in12
-rwxr-xr-xunix/configure3
-rw-r--r--unix/configure.ac1
-rw-r--r--unix/tclConfig.h.in3
-rw-r--r--win/Makefile.in5
-rw-r--r--win/makefile.vc3
26 files changed, 91 insertions, 8807 deletions
diff --git a/.fossil-settings/ignore-glob b/.fossil-settings/ignore-glob
index 51ca1e2..2f93505 100644
--- a/.fossil-settings/ignore-glob
+++ b/.fossil-settings/ignore-glob
@@ -17,6 +17,22 @@
*/tclsh*
*/tcltest*
*/versions.vc
+libtommath/bn.ilg
+libtommath/bn.ind
+libtommath/pretty.build
+libtommath/tommath.src
+libtommath/*.pdf
+libtommath/*.pl
+libtommath/*.sh
+libtommath/tombc/*
+libtommath/pre_gen/*
+libtommath/pics/*
+libtommath/mtest/*
+libtommath/logs/*
+libtommath/etc/*
+libtommath/demo/*
+libtommath/*.out
+libtommath/*.tex
unix/autoMkindex.tcl
unix/dltest.marker
unix/tcl.pc
diff --git a/generic/tclExecute.c b/generic/tclExecute.c
index 1cfc030..1f78ead 100644
--- a/generic/tclExecute.c
+++ b/generic/tclExecute.c
@@ -8935,7 +8935,7 @@ ExecuteExtendedBinaryMathOp(
}
Tcl_TakeBignumFromObj(NULL, valuePtr, &big1);
mp_init(&bigResult);
- mp_expt_d(&big1, big2.dp[0], &bigResult);
+ mp_expt_d_ex(&big1, big2.dp[0], &bigResult, 1);
mp_clear(&big1);
mp_clear(&big2);
BIG_RESULT(&bigResult);
diff --git a/generic/tclTomMath.decls b/generic/tclTomMath.decls
index 335ca6a..7d095ef 100644
--- a/generic/tclTomMath.decls
+++ b/generic/tclTomMath.decls
@@ -233,8 +233,9 @@ declare 66 {
void TclBNInitBignumFromWideUInt(mp_int *bignum, Tcl_WideUInt initVal)
}
+# Added in libtommath 1.0
declare 67 {
- int TclBN_mp_expt_d_ex(mp_int * a, mp_digit b, mp_int * c, int fast)
+ int TclBN_mp_expt_d_ex(mp_int *a, mp_digit b, mp_int *c, int fast)
}
# Local Variables:
diff --git a/generic/tclTomMath.h b/generic/tclTomMath.h
index 771025e..6b96d2c 100644
--- a/generic/tclTomMath.h
+++ b/generic/tclTomMath.h
@@ -64,8 +64,13 @@ extern "C" {
#endif
#elif defined(MP_64BIT)
/* for GCC only on supported platforms */
+#ifndef CRYPT
+ typedef unsigned long long ulong64;
+ typedef signed long long long64;
+#endif
+
#ifndef MP_DIGIT_DECLARED
- typedef uint64_t mp_digit;
+ typedef ulong64 mp_digit;
#define MP_DIGIT_DECLARED
#endif
#if defined(_WIN32)
@@ -83,11 +88,16 @@ extern "C" {
/* this is the default case, 28-bit digits */
/* this is to make porting into LibTomCrypt easier :-) */
+#ifndef CRYPT
+ typedef unsigned long long ulong64;
+ typedef signed long long long64;
+#endif
+
#ifndef MP_DIGIT_DECLARED
typedef uint32_t mp_digit;
#define MP_DIGIT_DECLARED
#endif
- typedef uint64_t mp_word;
+ typedef ulong64 mp_word;
#ifdef MP_31BIT
/* this is an extension that uses 31-bit digits */
@@ -233,7 +243,7 @@ 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] & 1u) == 0u)) ? MP_YES : MP_NO)
+#define mp_iseven(a) ((((a)->used > 0) && (((a)->dp[0] & 1u) == 0u)) ? MP_YES : MP_NO)
#define mp_isodd(a) ((((a)->used > 0) && (((a)->dp[0] & 1u) == 1u)) ? MP_YES : MP_NO)
#define mp_isneg(a) (((a)->sign != MP_ZPOS) ? MP_YES : MP_NO)
@@ -631,9 +641,9 @@ int mp_exptmod(mp_int *a, mp_int *b, mp_int *c, mp_int *d);
/* number of primes */
#ifdef MP_8BIT
- #define PRIME_SIZE 31
+# define PRIME_SIZE 31
#else
- #define PRIME_SIZE 256
+# define PRIME_SIZE 256
#endif
/* table of first PRIME_SIZE primes */
diff --git a/libtommath/bn.tex b/libtommath/bn.tex
deleted file mode 100644
index 5804318..0000000
--- a/libtommath/bn.tex
+++ /dev/null
@@ -1,1913 +0,0 @@
-\documentclass[synpaper]{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 \\ v1.0}
-\author{Tom St Denis \\ tstdenis82@gmail.com}
-\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 $ = 71.97$ \\
-\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 Five modular reduction algorithms & X & & Faster modular exponentiation for a variety of moduli. \\
-\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. It depends largely on the processor, compiler and the moduli being used.
-
-Essentially the only time you wouldn't use LibTomMath is when blazing speed is the primary concern. However,
-on the other side of the coin LibTomMath offers you a totally free (public domain) well structured math library
-that is very flexible, complete and performs well in resource contrained environments. Fast RSA for example can
-be performed with as little as 8KB of ram for data (again depending on build options).
-
-\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{Long Constants - platform dependant}
-
-\index{mp\_set\_long}
-\begin{alltt}
-int mp_set_long (mp_int * a, unsigned long b);
-\end{alltt}
-
-This will assign the value of the platform-dependant sized variable $b$ to the mp\_int $a$.
-
-To get the ``unsigned long'' copy of an mp\_int the following function can be used.
-
-\index{mp\_get\_long}
-\begin{alltt}
-unsigned long mp_get_long (mp_int * a);
-\end{alltt}
-
-This will return the least significant bits of the mp\_int $a$ that fit into an ``unsigned long''.
-
-\subsection{Long Long Constants}
-
-\index{mp\_set\_long\_long}
-\begin{alltt}
-int mp_set_long_long (mp_int * a, unsigned long long b);
-\end{alltt}
-
-This will assign the value of the 64-bit variable $b$ to the mp\_int $a$.
-
-To get the ``unsigned long long'' copy of an mp\_int the following function can be used.
-
-\index{mp\_get\_long\_long}
-\begin{alltt}
-unsigned long long mp_get_long_long (mp_int * a);
-\end{alltt}
-
-This will return the 64 least significant bits of the mp\_int $a$.
-
-\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_mag(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. The multiplication itself
-is implemented as a right-shift operation of $a$ by $b$ bits.
-
-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. The division itself is implemented as a left-shift
-operation of $a$ by $b$ bits.
-
-\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 because
-of the speed difference.
-
-\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 respectively 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\_ex}
-\begin{alltt}
-int mp_expt_d_ex (mp_int * a, mp_digit b, mp_int * c, int fast)
-\end{alltt}
-This function computes $c = a^b$.
-
-With parameter \textit{fast} set to $0$ the old version of the algorithm is used,
-when \textit{fast} is $1$, a faster but not statically timed version of the algorithm is used.
-
-The old version uses a simple binary left-to-right algorithm.
-It is faster than repeated multiplications by $a$ for all values of $b$ greater than three.
-
-The new version uses a binary right-to-left algorithm.
-
-The difference between the old and the new version is that the old version always
-executes $DIGIT\_BIT$ iterations. The new algorithm executes only $n$ iterations
-where $n$ is equal to the position of the highest bit that is set in $b$.
-
-\index{mp\_expt\_d}
-\begin{alltt}
-int mp_expt_d (mp_int * a, mp_digit b, mp_int * c)
-\end{alltt}
-mp\_expt\_d(a, b, c) is a wrapper function to mp\_expt\_d\_ex(a, b, c, 0).
-
-\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 $\left (a^{1/4} \right)^{1/4}$ or simply
-$\left ( \left ( \left ( a^{1/2} \right )^{1/2} \right )^{1/2} \right )^{1/2}$
-
-\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 square root}
-\index{mp\_sqrtmod\_prime}
-\begin{alltt}
-int mp_sqrtmod_prime(mp_int *n, mp_int *p, mp_int *r)
-\end{alltt}
-
-This will solve the modular equatioon $r^2 = n \mod p$ where $p$ is a prime number greater than 2 (odd prime).
-The result is returned in the third argument $r$, the function returns \textbf{MP\_OKAY} on success,
-other return values indicate failure.
-
-The implementation is split for two different cases:
-
-1. if $p \mod 4 == 3$ we apply \href{http://cacr.uwaterloo.ca/hac/}{Handbook of Applied Cryptography algorithm 3.36} and compute $r$ directly as
-$r = n^{(p+1)/4} \mod p$
-
-2. otherwise we use \href{https://en.wikipedia.org/wiki/Tonelli-Shanks_algorithm}{Tonelli-Shanks algorithm}
-
-The function does not check the primality of parameter $p$ thus it is up to the caller to assure that this parameter
-is a prime number. When $p$ is a composite the function behaviour is undefined, it may even return a false-positive
-\textbf{MP\_OKAY}.
-
-\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_mp_div_d.c b/libtommath/bn_mp_div_d.c
index 4df1d36..7dc0904 100644
--- a/libtommath/bn_mp_div_d.c
+++ b/libtommath/bn_mp_div_d.c
@@ -19,12 +19,11 @@ static int s_is_power_of_two(mp_digit b, int *p)
{
int x;
- /* fast return if no power of two */
+ /* quick out - if (b & (b-1)) isn't zero, b isn't a power of two */
if ((b == 0) || ((b & (b-1)) != 0)) {
- return 0;
+ return 0;
}
-
- for (x = 0; x < DIGIT_BIT; x++) {
+ for (x = 1; x < DIGIT_BIT; x++) {
if (b == (((mp_digit)1)<<x)) {
*p = x;
return 1;
diff --git a/libtommath/bn_mp_radix_size.c b/libtommath/bn_mp_radix_size.c
index e5d7772..9c60401 100644
--- a/libtommath/bn_mp_radix_size.c
+++ b/libtommath/bn_mp_radix_size.c
@@ -66,8 +66,17 @@ int mp_radix_size (mp_int * a, int radix, int *size)
}
mp_clear (&t);
- /* return digs + 1, the 1 is for the NULL byte that would be required. */
- *size = digs + 1;
+ /*
+ * return digs + 1, the 1 is for the NULL byte that would be required.
+ * mp_toradix_n requires a minimum of 3 bytes, so never report less than
+ * that.
+ */
+
+ if ( digs >= 2 ) {
+ *size = digs + 1;
+ } else {
+ *size = 3;
+ }
return MP_OKAY;
}
diff --git a/libtommath/bn_mp_read_radix.c b/libtommath/bn_mp_read_radix.c
index 5c9eb5e..93ccd3b 100644
--- a/libtommath/bn_mp_read_radix.c
+++ b/libtommath/bn_mp_read_radix.c
@@ -48,7 +48,7 @@ int mp_read_radix (mp_int * a, const char *str, int radix)
* this allows numbers like 1AB and 1ab to represent the same value
* [e.g. in hex]
*/
- ch = (radix <= 36) ? (char)toupper((int)*str) : *str;
+ ch = (radix <= 36) ? (char)toupper((unsigned char)*str) : *str;
for (y = 0; y < 64; y++) {
if (ch == mp_s_rmap[y]) {
break;
@@ -72,6 +72,13 @@ int mp_read_radix (mp_int * a, const char *str, int radix)
++str;
}
+ /* if an illegal character was found, fail. */
+
+ if ( *str != '\0' ) {
+ mp_zero( a );
+ return MP_VAL;
+ }
+
/* set the sign only if a != 0 */
if (mp_iszero(a) != MP_YES) {
a->sign = neg;
diff --git a/libtommath/booker.pl b/libtommath/booker.pl
deleted file mode 100644
index c2abae6..0000000
--- a/libtommath/booker.pl
+++ /dev/null
@@ -1,267 +0,0 @@
-#!/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++] = $_;
- last if ($_ =~ /libtom\.org/);
- }
- <SRC>;
- }
-
- $inline = 0;
- while (<SRC>) {
- next if ($_ =~ /\$Source/);
- next if ($_ =~ /\$Revision/);
- next if ($_ =~ /\$Date/);
- $text[$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 == 10) {
- $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 $wroteline lines\n";
-
-close (OUT);
-close (IN);
-
-system('perl -pli -e "s/\s*$//" tommath.tex');
diff --git a/libtommath/changes.txt b/libtommath/changes.txt
index 640b497..d70d589 100644
--- a/libtommath/changes.txt
+++ b/libtommath/changes.txt
@@ -1,6 +1,6 @@
Feb 5th, 2016
-v1.0
- -- Bump to 1.0
+v1.0.0
+ -- Bump to 1.0.0
-- Dirkjan Bussink provided a faster version of mp_expt_d()
-- Moritz Lenz contributed a fix to mp_mod()
and provided mp_get_long() and mp_set_long()
diff --git a/libtommath/dep.pl b/libtommath/dep.pl
deleted file mode 100644
index 0a5d19a..0000000
--- a/libtommath/dep.pl
+++ /dev/null
@@ -1,123 +0,0 @@
-#!/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;
-
-print "Processing $filename\n";
-
- # 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") && !($& eq "mp_min_u32")) {
- 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/gen.pl b/libtommath/gen.pl
deleted file mode 100644
index 57f65ac..0000000
--- a/libtommath/gen.pl
+++ /dev/null
@@ -1,19 +0,0 @@
-#!/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: $!";
-
-system('perl -pli -e "s/\s*$//" mpi.c');
diff --git a/libtommath/makefile.cygwin_dll b/libtommath/makefile.cygwin_dll
index 6feb5b4..59acad3 100644
--- a/libtommath/makefile.cygwin_dll
+++ b/libtommath/makefile.cygwin_dll
@@ -51,7 +51,3 @@ windll: $(OBJECTS)
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
-
-/* $Source: /cvs/libtom/libtommath/makefile.cygwin_dll,v $ */
-/* $Revision: 1.2 $ */
-/* $Date: 2005/05/05 14:38:45 $ */
diff --git a/libtommath/mess.sh b/libtommath/mess.sh
deleted file mode 100644
index bf639ce..0000000
--- a/libtommath/mess.sh
+++ /dev/null
@@ -1,4 +0,0 @@
-#!/bin/bash
-if cvs log $1 >/dev/null 2>/dev/null; then exit 0; else echo "$1 shouldn't be here" ; exit 1; fi
-
-
diff --git a/libtommath/poster.tex b/libtommath/poster.tex
deleted file mode 100644
index e7388f4..0000000
--- a/libtommath/poster.tex
+++ /dev/null
@@ -1,35 +0,0 @@
-\documentclass[landscape,11pt]{article}
-\usepackage{amsmath, amssymb}
-\usepackage{hyperref}
-\begin{document}
-\hspace*{-3in}
-\begin{tabular}{llllll}
-$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/pretty.build b/libtommath/pretty.build
deleted file mode 100644
index a708b8a..0000000
--- a/libtommath/pretty.build
+++ /dev/null
@@ -1,66 +0,0 @@
-#!/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 $lines lines of code.\n";
-print "Doing final make (building archive...)\n";
-!system("make > /dev/null 2>/dev/null") or die "\nERROR: Failed to perform last make command!!!\n";
-print "done.\n"; \ No newline at end of file
diff --git a/libtommath/tommath.h b/libtommath/tommath.h
index 4e6b1a5..8293e01 100644
--- a/libtommath/tommath.h
+++ b/libtommath/tommath.h
@@ -57,7 +57,12 @@ extern "C" {
#endif
#elif defined(MP_64BIT)
/* for GCC only on supported platforms */
- typedef uint64_t mp_digit;
+#ifndef CRYPT
+ typedef unsigned long long ulong64;
+ typedef signed long long long64;
+#endif
+
+ typedef ulong64 mp_digit;
#if defined(_WIN32)
typedef unsigned __int128 mp_word;
#elif defined(__GNUC__)
@@ -73,8 +78,13 @@ extern "C" {
/* this is the default case, 28-bit digits */
/* this is to make porting into LibTomCrypt easier :-) */
+#ifndef CRYPT
+ typedef unsigned long long ulong64;
+ typedef signed long long long64;
+#endif
+
typedef uint32_t mp_digit;
- typedef uint64_t mp_word;
+ typedef ulong64 mp_word;
#ifdef MP_31BIT
/* this is an extension that uses 31-bit digits */
@@ -450,9 +460,9 @@ int mp_exptmod(mp_int *a, mp_int *b, mp_int *c, mp_int *d);
/* number of primes */
#ifdef MP_8BIT
- #define PRIME_SIZE 31
+# define PRIME_SIZE 31
#else
- #define PRIME_SIZE 256
+# define PRIME_SIZE 256
#endif
/* table of first PRIME_SIZE primes */
diff --git a/libtommath/tommath.src b/libtommath/tommath.src
deleted file mode 100644
index 768ed10..0000000
--- a/libtommath/tommath.src
+++ /dev/null
@@ -1,6339 +0,0 @@
-\documentclass[b5paper]{book}
-\usepackage{hyperref}
-\usepackage{makeidx}
-\usepackage{amssymb}
-\usepackage{color}
-\usepackage{alltt}
-\usepackage{graphicx}
-\usepackage{layout}
-\def\union{\cup}
-\def\intersect{\cap}
-\def\getsrandom{\stackrel{\rm R}{\gets}}
-\def\cross{\times}
-\def\cat{\hspace{0.5em} \| \hspace{0.5em}}
-\def\catn{$\|$}
-\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{Multi--Precision Math}
-\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 v0.39 release of the
-LibTomMath project.
-
-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}
-When I tell people about my LibTom projects and that I release them as public domain they are often puzzled.
-They ask why I did it and especially why I continue to work on them for free. The best I can explain it is ``Because I can.''
-Which seems odd and perhaps too terse for adult conversation. I often qualify it with ``I am able, I am willing.'' which
-perhaps explains it better. I am the first to admit there is not anything that special with what I have done. Perhaps
-others can see that too and then we would have a society to be proud of. My LibTom projects are what I am doing to give
-back to society in the form of tools and knowledge that can help others in their endeavours.
-
-I started writing this book because it was the most logical task to further my goal of open academia. The LibTomMath source
-code itself was written to be easy to follow and learn from. There are times, however, where pure C source code does not
-explain the algorithms properly. Hence this book. The book literally starts with the foundation of the library and works
-itself outwards to the more complicated algorithms. The use of both pseudo--code and verbatim source code provides a duality
-of ``theory'' and ``practice'' that the computer science students of the world shall appreciate. I never deviate too far
-from relatively straightforward algebra and I hope that this book can be a valuable learning asset.
-
-This book and indeed much of the LibTom projects would not exist in their current form if it was not for a plethora
-of kind people donating their time, resources and kind words to help support my work. Writing a text of significant
-length (along with the source code) is a tiresome and lengthy process. Currently the LibTom project is four years old,
-comprises of literally thousands of users and over 100,000 lines of source code, TeX and other material. People like Mads and Greg
-were there at the beginning to encourage me to work well. It is amazing how timely validation from others can boost morale to
-continue the project. Definitely my parents were there for me by providing room and board during the many months of work in 2003.
-
-To my many friends whom I have met through the years I thank you for the good times and the words of encouragement. I hope I
-honour your kind gestures with this project.
-
-Open Source. Open Academia. Open Minds.
-
-\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.com}} 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 @24,alloc@) 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 with the malloc() function (line @27,XMALLOC@) and set to zero afterwards (line @38,for@). 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.
-
-\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
-
-This fairly trivial algorithm first eliminates non--required duplications (line @27,a != b@) and then sets the
-\textbf{sign} flag to \textbf{MP\_ZPOS}.
-
-\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
-
-Like mp\_abs() this function avoids non--required duplications (line @21,a != b@) and then sets the sign. We
-have to make sure that only non--zero values get a \textbf{sign} of \textbf{MP\_NEG}. If the mp\_int is zero
-than the \textbf{sign} is hard--coded to \textbf{MP\_ZPOS}.
-
-\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.
-
-\newpage\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
-
-First we zero (line @21,mp_zero@) the mp\_int to make sure that the other members are initialized for a
-small positive constant. mp\_zero() ensures that the \textbf{sign} is positive and the \textbf{used} count
-is zero. Next we set the digit and reduce it modulo $\beta$ (line @22,MP_MASK@). After this step we have to
-check if the resulting digit is zero or not. If it is not then we set the \textbf{used} count to one, otherwise
-to zero.
-
-We can quickly reduce modulo $\beta$ since it is of the form $2^k$ and a quick binary AND operation with
-$2^k - 1$ will perform the same operation.
-
-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 (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 (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. The inputs are compared (line @30,if@) based on magnitudes. If the signs were both
-negative then the unsigned comparison is performed in the opposite direction (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
-
-We first sort (lines @27,if@ to @35,}@) the inputs based on magnitude and determine the $min$ and $max$ variables.
-Note that $x$ is a pointer to an mp\_int assigned to the largest input, in effect it is a local alias. Next we
-grow the destination (@37,init@ to @42,}@) ensure that it can 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 $u$ will be cleared (line @65,u = 0@), note that $u$ is of type mp\_digit which ensures type
-compatibility within the implementation. The initial addition (line @66,for@ to @75,}@) adds digits from
-both inputs until the smallest input runs out of digits. Similarly the conditional addition loop
-(line @81,for@ to @90,}@) adds the remaining digits from the larger of the two inputs. The addition is finished
-with the final carry being stored in $tmpc$ (line @94,tmpc++@). Note the ``++'' operator within the same expression.
-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 (line @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 $\gamma \ge 32$.
-
-\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
-
-Like low level addition we ``sort'' the inputs. Except in this case the sorting is hardcoded
-(lines @24,min@ and @25,max@). 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. The aliases $tmpa$, $tmpb$ and $tmpc$ are initialized
-(lines @42,tmpa@, @43,tmpb@ and @44,tmpc@) for $a$, $b$ and $c$ respectively.
-
-The first subtraction loop (lines @47,u = 0@ through @61,}@) subtract digits from both inputs until the smaller of
-the two inputs has been exhausted. As remarked earlier there is an implementation reason for using the ``awkward''
-method of extracting the carry (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 (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 (line @24,if@) ensures that the $b$ variable is greater than zero since we do not interpret negative
-shift counts properly. 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 $top$ (line @42,top@) is an alias
-for the leading digit while $bottom$ (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 since it cannot fail. Like mp\_lshd() we
-form a sliding window except we copy in the other direction. After the window (line @59,for (;@) we then zero
-the upper digits of the input to make sure the result is correct.
-
-\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 algorithm mp\_mul\_2 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
-
-The shifting is performed in--place which means the first step (line @24,a != c@) is to copy the input to the
-destination. We avoid calling mp\_copy() by making sure the mp\_ints are different. The destination then
-has to be grown (line @31,grow@) to accomodate the result.
-
-If the shift count $b$ is larger than $lg(\beta)$ then a call to mp\_lshd() is used to handle all of the multiples
-of $lg(\beta)$. Leaving only a remaining shift of $lg(\beta) - 1$ or fewer bits left. Inside the actual shift
-loop (lines @45,if@ to @76,}@) we make use of pre--computed values $shift$ and $mask$. These are used to
-extract the carry bit(s) to pass into the next iteration of the loop. The $r$ and $rr$ variables form a
-chain between consecutive iterations to propagate the carry.
-
-\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. The only significant difference is
-the direction of the shifts.
-
-\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
-
-We first avoid cases of $b \le 0$ by simply mp\_zero()'ing the destination in such cases. Next if $2^b$ is larger
-than the input we just mp\_copy() the input and return right away. After this point we know we must actually
-perform some work to produce the remainder.
-
-Recalling that reducing modulo $2^k$ and a binary ``and'' with $2^k - 1$ are numerically equivalent we can quickly reduce
-the number. First we zero any digits above the last digit in $2^b$ (line @41,for@). Next we reduce the
-leading digit of both (line @45,&=@) and then mp\_clamp().
-
-\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
-
-First we determine (line @30,if@) if the Comba method can be used first since it's faster. The conditions for
-sing 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.
-
-If we cannot use the Comba method we proceed to setup the baseline routine. We allocate the the destination mp\_int
-$t$ (line @36,init@) to the exact size of the output to avoid further re--allocations. At this point we now
-begin the $O(n^2)$ loop.
-
-This implementation of multiplication has the caveat that it can be trimmed to only produce a variable number of
-digits as output. In each iteration of the outer loop the $pb$ variable is set (line @48,MIN@) to the maximum
-number of inner loop iterations.
-
-Inside the inner loop we calculate $\hat r$ as the mp\_word product of the two mp\_digits and the addition of the
-carry from the previous iteration. A particularly important observation is that most modern optimizing
-C compilers (GCC for instance) can recognize that a $N \times N \rightarrow 2N$ multiplication is all that
-is required for the product. In x86 terms for example, this means using the MUL instruction.
-
-Each digit of the product is stored in turn (line @68,tmpt@) and the carry propagated (line @71,>>@) to the
-next iteration.
-
-\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} single precision digits named $W$ 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}).\\
-\\
-3. $pa \leftarrow \mbox{MIN}(digs, a.used + b.used)$ \\
-\\
-4. $\_ \hat W \leftarrow 0$ \\
-5. for $ix$ from 0 to $pa - 1$ do \\
-\hspace{3mm}5.1 $ty \leftarrow \mbox{MIN}(b.used - 1, ix)$ \\
-\hspace{3mm}5.2 $tx \leftarrow ix - ty$ \\
-\hspace{3mm}5.3 $iy \leftarrow \mbox{MIN}(a.used - tx, ty + 1)$ \\
-\hspace{3mm}5.4 for $iz$ from 0 to $iy - 1$ do \\
-\hspace{6mm}5.4.1 $\_ \hat W \leftarrow \_ \hat W + a_{tx+iy}b_{ty-iy}$ \\
-\hspace{3mm}5.5 $W_{ix} \leftarrow \_ \hat W (\mbox{mod }\beta)$\\
-\hspace{3mm}5.6 $\_ \hat W \leftarrow \lfloor \_ \hat W / \beta \rfloor$ \\
-\\
-6. $oldused \leftarrow c.used$ \\
-7. $c.used \leftarrow digs$ \\
-8. for $ix$ from $0$ to $pa$ do \\
-\hspace{3mm}8.1 $c_{ix} \leftarrow W_{ix}$ \\
-9. for $ix$ from $pa + 1$ to $oldused - 1$ do \\
-\hspace{3mm}9.1 $c_{ix} \leftarrow 0$ \\
-\\
-10. Clamp $c$. \\
-11. Return 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 outer loop of this algorithm is more complicated than that of the baseline multiplier. This is because on the inside of the
-loop we want to produce one column per pass. This allows the accumulator $\_ \hat W$ to be placed in CPU registers and
-reduce the memory bandwidth to two \textbf{mp\_digit} reads per iteration.
-
-The $ty$ variable is set to the minimum count of $ix$ or the number of digits in $b$. That way if $a$ has more digits than
-$b$ this will be limited to $b.used - 1$. The $tx$ variable is set to the to the distance past $b.used$ the variable
-$ix$ is. This is used for the immediately subsequent statement where we find $iy$.
-
-The variable $iy$ is the minimum digits we can read from either $a$ or $b$ before running out. Computing one column at a time
-means we have to scan one integer upwards and the other downwards. $a$ starts at $tx$ and $b$ starts at $ty$. In each
-pass we are producing the $ix$'th output column and we note that $tx + ty = ix$. As we move $tx$ upwards we have to
-move $ty$ downards so the equality remains valid. The $iy$ variable is the number of iterations until
-$tx \ge a.used$ or $ty < 0$ occurs.
-
-After every inner pass we store the lower half of the accumulator into $W_{ix}$ and then propagate the carry of the accumulator
-into the next round by dividing $\_ \hat W$ by $\beta$.
-
-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
-
-As per the pseudo--code we first calculate $pa$ (line @47,MIN@) as the number of digits to output. Next we begin the outer loop
-to produce the individual columns of the product. We use the two aliases $tmpx$ and $tmpy$ (lines @61,tmpx@, @62,tmpy@) to point
-inside the two multiplicands quickly.
-
-The inner loop (lines @70,for@ to @72,}@) of this implementation is where the tradeoff come into play. Originally this comba
-implementation was ``row--major'' which means it adds to each of the columns in each pass. After the outer loop it would then fix
-the carries. This was very fast except it had an annoying drawback. You had to read a mp\_word and two mp\_digits and write
-one mp\_word per iteration. On processors such as the Athlon XP and P4 this did not matter much since the cache bandwidth
-is very high and it can keep the ALU fed with data. It did, however, matter on older and embedded cpus where cache is often
-slower and also often doesn't exist. This new algorithm only performs two reads per iteration under the assumption that the
-compiler has aliased $\_ \hat W$ to a CPU register.
-
-After the inner loop we store the current accumulator in $W$ and shift $\_ \hat W$ (lines @75,W[ix]@, @78,>>@) to forward it as
-a carry for the next pass. After the outer loop we use the final carry (line @82,W[ix]@) as the last digit of the product.
-
-\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.
-
-\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\_add}) \\
-11. $x0 \leftarrow y1 + y0$ \\
-12. $t1 \leftarrow t1 \cdot x0$ \\
-\\
-Calculate the middle term. \\
-13. $x0 \leftarrow x0y0 + x1y1$ \\
-14. $t1 \leftarrow t1 - x0$ (\textit{s\_mp\_sub}) \\
-\\
-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
-
-The first obvious thing to note is that this algorithm is complicated. The complexity is worth it if you are multiplying very
-large numbers. For example, a 10,000 digit multiplication takes approximaly 99,282,205 fewer single precision multiplications with
-Toom--Cook than a Comba or baseline approach (this is a savings of more than 99$\%$). For most ``crypto'' sized numbers this
-algorithm is not practical as Karatsuba has a much lower cutoff point.
-
-First we split $a$ and $b$ into three roughly equal portions. This has been accomplished (lines @40,mod@ to @69,rshd@) with
-combinations of mp\_rshd() and mp\_mod\_2d() function calls. At this point $a = a2 \cdot \beta^2 + a1 \cdot \beta + a0$ and similiarly
-for $b$.
-
-Next we compute the five points $w0, w1, w2, w3$ and $w4$. Recall that $w0$ and $w4$ can be computed directly from the portions so
-we get those out of the way first (lines @72,mul@ and @77,mul@). Next we compute $w1, w2$ and $w3$ using Horners method.
-
-After this point we solve for the actual values of $w1, w2$ and $w3$ by reducing the $5 \times 5$ system which is relatively
-straight forward.
-
-\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.
-
-\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.
-
-\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 (line @32,for@) the square term is calculated on line @35,r =@. The carry (line @42,>>@) has been
-extracted from the mp\_word accumulator using a right shift. Aliases for $a_{ix}$ and $t_{ix+iy}$ are initialized
-(lines @45,tmpx@ and @48,tmpt@) to simplify the inner loop. The doubling is performed using two
-additions (line @57,r + r@) since it is usually faster than shifting, if not at least as fast.
-
-The important observation is that the inner loop does not begin at $iy = 0$ like for multiplication. As such the inner loops
-get progressively shorter as the algorithm proceeds. This is what leads to the savings compared to using a multiplication to
-square a number.
-
-\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. In this case, we have an even simpler solution in mind.
-
-\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 an array of \textbf{MP\_WARRAY} mp\_digits named $W$ 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. $pa \leftarrow 2 \cdot a.used$ \\
-4. $\hat W1 \leftarrow 0$ \\
-5. for $ix$ from $0$ to $pa - 1$ do \\
-\hspace{3mm}5.1 $\_ \hat W \leftarrow 0$ \\
-\hspace{3mm}5.2 $ty \leftarrow \mbox{MIN}(a.used - 1, ix)$ \\
-\hspace{3mm}5.3 $tx \leftarrow ix - ty$ \\
-\hspace{3mm}5.4 $iy \leftarrow \mbox{MIN}(a.used - tx, ty + 1)$ \\
-\hspace{3mm}5.5 $iy \leftarrow \mbox{MIN}(iy, \lfloor \left (ty - tx + 1 \right )/2 \rfloor)$ \\
-\hspace{3mm}5.6 for $iz$ from $0$ to $iz - 1$ do \\
-\hspace{6mm}5.6.1 $\_ \hat W \leftarrow \_ \hat W + a_{tx + iz}a_{ty - iz}$ \\
-\hspace{3mm}5.7 $\_ \hat W \leftarrow 2 \cdot \_ \hat W + \hat W1$ \\
-\hspace{3mm}5.8 if $ix$ is even then \\
-\hspace{6mm}5.8.1 $\_ \hat W \leftarrow \_ \hat W + \left ( a_{\lfloor ix/2 \rfloor}\right )^2$ \\
-\hspace{3mm}5.9 $W_{ix} \leftarrow \_ \hat W (\mbox{mod }\beta)$ \\
-\hspace{3mm}5.10 $\hat W1 \leftarrow \lfloor \_ \hat W / \beta \rfloor$ \\
-\\
-6. $oldused \leftarrow b.used$ \\
-7. $b.used \leftarrow 2 \cdot a.used$ \\
-8. for $ix$ from $0$ to $pa - 1$ do \\
-\hspace{3mm}8.1 $b_{ix} \leftarrow W_{ix}$ \\
-9. for $ix$ from $pa$ to $oldused - 1$ do \\
-\hspace{3mm}9.1 $b_{ix} \leftarrow 0$ \\
-10. Clamp excess digits from $b$. (\textit{mp\_clamp}) \\
-11. 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 algorithm is very similar to the Comba multiplier except with a few key differences we shall make note of.
-
-First, we have an accumulator and carry variables $\_ \hat W$ and $\hat W1$ respectively. This is because the inner loop
-products are to be doubled. If we had added the previous carry in we would be doubling too much. Next we perform an
-addition MIN condition on $iy$ (step 5.5) to prevent overlapping digits. For example, $a_3 \cdot a_5$ is equal
-$a_5 \cdot a_3$. Whereas in the multiplication case we would have $5 < a.used$ and $3 \ge 0$ is maintained since we double the sum
-of the products just outside the inner loop we have to avoid doing this. This is also a good thing since we perform
-fewer multiplications and the routine ends up being faster.
-
-Finally the last difference is the addition of the ``square'' term outside the inner loop (step 5.8). We add in the square
-only to even outputs and it is the square of the term at the $\lfloor ix / 2 \rfloor$ position.
-
-EXAM,bn_fast_s_mp_sqr.c
-
-This implementation is essentially a copy of Comba multiplication with the appropriate changes added to make it faster for
-the special case of squaring.
-
-\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 + b)^2 - (a^2 + 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{s\_mp\_add}) \\
-9. $t1 \leftarrow t1^2$ \\
-\\
-Compute the middle term. \\
-10. $t2 \leftarrow x0x0 + x1x1$ (\textit{s\_mp\_add}) \\
-11. $t1 \leftarrow t1 - t2$ \\
-\\
-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 $(x0 - x1)^2 - (x1^2 + x0^2) = 2 \cdot x0 \cdot x1$.
-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.
-
-\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 [ 2 \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 [ 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. \\
- & \\
-$\left [ 3 \right ] $ & Implement a threaded version of Comba multiplication (and squaring) where you \\
- & compute subsets of the columns in each thread. Determine a cutoff point where \\
- & it is effective and add the logic to mp\_mul() and mp\_sqr(). \\
- &\\
-$\left [ 4 \right ] $ & Same as the previous but also modify the Karatsuba and Toom-Cook. You must \\
- & increase the throughput of mp\_exptmod() for random odd moduli in the range \\
- & $512 \ldots 4096$ bits significantly ($> 2x$) to complete this challenge. \\
- & \\
-\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.
-
-\newpage\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 $9$ & $x + n = 356$, $x/2 = 178$ \\
-\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 $k = 9$ (note $\beta^k = 512$ which is larger than $n$). The result of
-the algorithm $r = 178$ is congruent to the value of $2^{-9} \cdot 5555 \mbox{ (mod }257\mbox{)}$. When $r$ is multiplied by $2^9$ 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 $k$ ($2^k > n$) \\
-\textbf{Output}. $2^{-k}x \mbox{ (mod }n\mbox{)}$ \\
-\hline \\
-1. for $t$ from $1$ to $k$ 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 $9$ & $x + 2^{7}n = 91136$ & $10110010000000000$ \\
-\hline -- & $x/2^k = 178$ & \\
-\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 $k = 9$.
-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 $k$ ($\beta^k > n$) \\
-\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_ex.c
-
-This describes only the algorithm that is used when the parameter $fast$ is $0$. Line @31,mp_set@ sets the initial value of the result to $1$. Next the loop on line @54,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
-@69,<<@ 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 @31,if@ through @45,}@ 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 @37,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 @60,for@ initializes the $M$ array while lines @71,mp_init@ and @75,mp_reduce@ through @85,}@ initialize the reduction
-function that will be used for this modulus.
-
--- 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 @108,if@ and @113,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 @147,neg@ determines the sign of
-the quotient and line @148,sign@ ensures that both $x$ and $y$ are positive.
-
-The number of bits in the leading digit is calculated on line @151,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 @184,for@ will produce the remainder of the quotient digits.
-
-The conditional ``continue'' on line @186,continue@ 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 @214,t1@, @216,t1@ and @222,t2@ through @225,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 @90,/@ and @91,-@ 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 $a = 0$ then \\
-\hspace{3mm}1.1 $c \leftarrow \vert b \vert $ \\
-\hspace{3mm}1.2 Return(\textit{MP\_OKAY}). \\
-2. If $b = 0$ then \\
-\hspace{3mm}2.1 $c \leftarrow \vert a \vert $ \\
-\hspace{3mm}2.2 Return(\textit{MP\_OKAY}). \\
-3. $u \leftarrow \vert a \vert, v \leftarrow \vert b \vert$ \\
-4. $k \leftarrow 0$ \\
-5. While $u.used > 0$ and $v.used > 0$ and $u_0 \equiv v_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\
-\hspace{3mm}5.1 $k \leftarrow k + 1$ \\
-\hspace{3mm}5.2 $u \leftarrow \lfloor u / 2 \rfloor$ \\
-\hspace{3mm}5.3 $v \leftarrow \lfloor v / 2 \rfloor$ \\
-6. While $u.used > 0$ and $u_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\
-\hspace{3mm}6.1 $u \leftarrow \lfloor u / 2 \rfloor$ \\
-7. While $v.used > 0$ and $v_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\
-\hspace{3mm}7.1 $v \leftarrow \lfloor v / 2 \rfloor$ \\
-8. While $v.used > 0$ \\
-\hspace{3mm}8.1 If $\vert u \vert > \vert v \vert$ then \\
-\hspace{6mm}8.1.1 Swap $u$ and $v$. \\
-\hspace{3mm}8.2 $v \leftarrow \vert v \vert - \vert u \vert$ \\
-\hspace{3mm}8.3 While $v.used > 0$ and $v_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\
-\hspace{6mm}8.3.1 $v \leftarrow \lfloor v / 2 \rfloor$ \\
-9. $c \leftarrow u \cdot 2^k$ \\
-10. 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 two 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 five 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
-six and seven 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 eight 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 8.2 will always produce a positive and even result. Step 8.3 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 @23,zero@ through @29,}@. After those lines the inputs are assumed to be non-zero.
-
-Lines @32,if@ and @36,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 block starting at line @43,common@ removes common factors of two by first counting the number of trailing
-zero bits in both. 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 divisions by a power of two on lines @60,div_2d@ and @67,div_2d@ 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 @72, 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}.
-
-\textit{-- Tom, don't be an ass, cite your source here...!}
-
-\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/macosx/Tcl.xcode/project.pbxproj b/macosx/Tcl.xcode/project.pbxproj
index a99afc5..117cd61 100644
--- a/macosx/Tcl.xcode/project.pbxproj
+++ b/macosx/Tcl.xcode/project.pbxproj
@@ -166,6 +166,7 @@
F9E61D2A090A4891002B3151 /* bn_mp_sqrt.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D42C008F272B3004A47F5 /* bn_mp_sqrt.c */; };
F9E61D2B090A48A4002B3151 /* bn_mp_and.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D426C08F272B3004A47F5 /* bn_mp_and.c */; };
F9E61D2C090A48AC002B3151 /* bn_mp_expt_d.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D427F08F272B3004A47F5 /* bn_mp_expt_d.c */; };
+ F9E61D2C090A48AC002B3151 /* bn_mp_expt_d_ex.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D427F08F272B3004A47F5 /* bn_mp_expt_d_ex.c */; };
F9E61D2D090A48BB002B3151 /* bn_mp_xor.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D42CD08F272B3004A47F5 /* bn_mp_xor.c */; };
F9E61D2E090A48BF002B3151 /* bn_mp_or.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D42A308F272B3004A47F5 /* bn_mp_or.c */; };
F9E61D2F090A48C7002B3151 /* bn_mp_shrink.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D42BC08F272B3004A47F5 /* bn_mp_shrink.c */; };
@@ -1442,6 +1443,7 @@
F96D427A08F272B3004A47F5 /* bn_mp_div_d.c */,
F96D427E08F272B3004A47F5 /* bn_mp_exch.c */,
F96D427F08F272B3004A47F5 /* bn_mp_expt_d.c */,
+ F96D427F08F272B3004A47F5 /* bn_mp_expt_d_ex.c */,
F96D428708F272B3004A47F5 /* bn_mp_grow.c */,
F96D428808F272B3004A47F5 /* bn_mp_init.c */,
F96D428908F272B3004A47F5 /* bn_mp_init_copy.c */,
@@ -2075,6 +2077,7 @@
F96D48F808F272C3004A47F5 /* bn_mp_div_d.c in Sources */,
F96D48FC08F272C3004A47F5 /* bn_mp_exch.c in Sources */,
F9E61D2C090A48AC002B3151 /* bn_mp_expt_d.c in Sources */,
+ F9E61D2C090A48AC002B3151 /* bn_mp_expt_d_ex.c in Sources */,
F96D490508F272C3004A47F5 /* bn_mp_grow.c in Sources */,
F96D490608F272C3004A47F5 /* bn_mp_init.c in Sources */,
F96D490708F272C3004A47F5 /* bn_mp_init_copy.c in Sources */,
diff --git a/macosx/Tcl.xcodeproj/project.pbxproj b/macosx/Tcl.xcodeproj/project.pbxproj
index 393fd31..3be274f 100644
--- a/macosx/Tcl.xcodeproj/project.pbxproj
+++ b/macosx/Tcl.xcodeproj/project.pbxproj
@@ -165,7 +165,8 @@
F9E61D29090A486C002B3151 /* bn_mp_neg.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D42A208F272B3004A47F5 /* bn_mp_neg.c */; };
F9E61D2A090A4891002B3151 /* bn_mp_sqrt.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D42C008F272B3004A47F5 /* bn_mp_sqrt.c */; };
F9E61D2B090A48A4002B3151 /* bn_mp_and.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D426C08F272B3004A47F5 /* bn_mp_and.c */; };
- F9E61D2C090A48AC002B3151 /* bn_mp_expt_d.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D427F08F272B3004A47F5 /* bn_mp_expt_d.c */; };
+ F9E61D2C090A48AC002B3151 /* bn_mp_expt_d.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D427F08F272B3004A47F5 /* bn_mp_expt_d_ex.c */; };
+ F9E61D2C090A48AC002B3151 /* bn_mp_expt_d_ex.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D427F08F272B3004A47F5 /* bn_mp_expt_d.c */; };
F9E61D2D090A48BB002B3151 /* bn_mp_xor.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D42CD08F272B3004A47F5 /* bn_mp_xor.c */; };
F9E61D2E090A48BF002B3151 /* bn_mp_or.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D42A308F272B3004A47F5 /* bn_mp_or.c */; };
F9E61D2F090A48C7002B3151 /* bn_mp_shrink.c in Sources */ = {isa = PBXBuildFile; fileRef = F96D42BC08F272B3004A47F5 /* bn_mp_shrink.c */; };
@@ -579,6 +580,7 @@
F96D427A08F272B3004A47F5 /* bn_mp_div_d.c */ = {isa = PBXFileReference; fileEncoding = 4; lastKnownFileType = sourcecode.c.c; path = bn_mp_div_d.c; sourceTree = "<group>"; };
F96D427E08F272B3004A47F5 /* bn_mp_exch.c */ = {isa = PBXFileReference; fileEncoding = 4; lastKnownFileType = sourcecode.c.c; path = bn_mp_exch.c; sourceTree = "<group>"; };
F96D427F08F272B3004A47F5 /* bn_mp_expt_d.c */ = {isa = PBXFileReference; fileEncoding = 4; lastKnownFileType = sourcecode.c.c; path = bn_mp_expt_d.c; sourceTree = "<group>"; };
+ F96D427F08F272B3004A47F5 /* bn_mp_expt_d_ex.c */ = {isa = PBXFileReference; fileEncoding = 4; lastKnownFileType = sourcecode.c.c; path = bn_mp_expt_d_ex.c; sourceTree = "<group>"; };
F96D428708F272B3004A47F5 /* bn_mp_grow.c */ = {isa = PBXFileReference; fileEncoding = 4; lastKnownFileType = sourcecode.c.c; path = bn_mp_grow.c; sourceTree = "<group>"; };
F96D428808F272B3004A47F5 /* bn_mp_init.c */ = {isa = PBXFileReference; fileEncoding = 4; lastKnownFileType = sourcecode.c.c; path = bn_mp_init.c; sourceTree = "<group>"; };
F96D428908F272B3004A47F5 /* bn_mp_init_copy.c */ = {isa = PBXFileReference; fileEncoding = 4; lastKnownFileType = sourcecode.c.c; path = bn_mp_init_copy.c; sourceTree = "<group>"; };
@@ -1442,6 +1444,7 @@
F96D427A08F272B3004A47F5 /* bn_mp_div_d.c */,
F96D427E08F272B3004A47F5 /* bn_mp_exch.c */,
F96D427F08F272B3004A47F5 /* bn_mp_expt_d.c */,
+ F96D427F08F272B3004A47F5 /* bn_mp_expt_d_ex.c */,
F96D428708F272B3004A47F5 /* bn_mp_grow.c */,
F96D428808F272B3004A47F5 /* bn_mp_init.c */,
F96D428908F272B3004A47F5 /* bn_mp_init_copy.c */,
@@ -2075,6 +2078,7 @@
F96D48F808F272C3004A47F5 /* bn_mp_div_d.c in Sources */,
F96D48FC08F272C3004A47F5 /* bn_mp_exch.c in Sources */,
F9E61D2C090A48AC002B3151 /* bn_mp_expt_d.c in Sources */,
+ F9E61D2C090A48AC002B3151 /* bn_mp_expt_d_ex.c in Sources */,
F96D490508F272C3004A47F5 /* bn_mp_grow.c in Sources */,
F96D490608F272C3004A47F5 /* bn_mp_init.c in Sources */,
F96D490708F272C3004A47F5 /* bn_mp_init_copy.c in Sources */,
diff --git a/unix/Makefile.in b/unix/Makefile.in
index cfc7d78..c4f6136 100644
--- a/unix/Makefile.in
+++ b/unix/Makefile.in
@@ -320,8 +320,8 @@ TOMMATH_OBJS = bncore.o bn_reverse.o bn_fast_s_mp_mul_digs.o \
bn_mp_cnt_lsb.o bn_mp_copy.o \
bn_mp_count_bits.o bn_mp_div.o bn_mp_div_d.o bn_mp_div_2.o \
bn_mp_div_2d.o bn_mp_div_3.o \
- bn_mp_exch.o bn_mp_expt_d_ex.o bn_mp_expt_d.o bn_mp_grow.o \
- bn_mp_init.o bn_mp_init_copy.o bn_mp_init_multi.o bn_mp_init_set.o \
+ bn_mp_exch.o bn_mp_expt_d.o bn_mp_expt_d_ex.o bn_mp_grow.o bn_mp_init.o \
+ bn_mp_init_copy.o bn_mp_init_multi.o bn_mp_init_set.o \
bn_mp_init_set_int.o bn_mp_init_size.o bn_mp_karatsuba_mul.o \
bn_mp_karatsuba_sqr.o \
bn_mp_lshd.o bn_mp_mod.o bn_mp_mod_2d.o bn_mp_mul.o bn_mp_mul_2.o \
@@ -502,8 +502,8 @@ TOMMATH_SRCS = \
$(TOMMATH_DIR)/bn_mp_div_2d.c \
$(TOMMATH_DIR)/bn_mp_div_3.c \
$(TOMMATH_DIR)/bn_mp_exch.c \
- $(TOMMATH_DIR)/bn_mp_expt_d_ex.c \
$(TOMMATH_DIR)/bn_mp_expt_d.c \
+ $(TOMMATH_DIR)/bn_mp_expt_d_ex.c \
$(TOMMATH_DIR)/bn_mp_grow.c \
$(TOMMATH_DIR)/bn_mp_init.c \
$(TOMMATH_DIR)/bn_mp_init_copy.c \
@@ -1418,12 +1418,12 @@ bn_mp_div_3.o: $(TOMMATH_DIR)/bn_mp_div_3.c $(MATHHDRS)
bn_mp_exch.o: $(TOMMATH_DIR)/bn_mp_exch.c $(MATHHDRS)
$(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_exch.c
-bn_mp_expt_d_ex.o: $(TOMMATH_DIR)/bn_mp_expt_d_ex.c $(MATHHDRS)
- $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_expt_d_ex.c
-
bn_mp_expt_d.o: $(TOMMATH_DIR)/bn_mp_expt_d.c $(MATHHDRS)
$(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_expt_d.c
+bn_mp_expt_d_ex.o: $(TOMMATH_DIR)/bn_mp_expt_d_ex.c $(MATHHDRS)
+ $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_expt_d_ex.c
+
bn_mp_grow.o: $(TOMMATH_DIR)/bn_mp_grow.c $(MATHHDRS)
$(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_grow.c
diff --git a/unix/configure b/unix/configure
index e999455..baea1f1 100755
--- a/unix/configure
+++ b/unix/configure
@@ -6857,9 +6857,6 @@ $as_echo "enabled $tcl_ok debugging" >&6; }
-$as_echo "#define TCL_TOMMATH 1" >>confdefs.h
-
-
$as_echo "#define MP_PREC 4" >>confdefs.h
diff --git a/unix/configure.ac b/unix/configure.ac
index 41a1f62..bafb970 100644
--- a/unix/configure.ac
+++ b/unix/configure.ac
@@ -182,7 +182,6 @@ SC_CONFIG_CFLAGS
SC_ENABLE_SYMBOLS(bccdebug)
-AC_DEFINE(TCL_TOMMATH, 1, [Build libtommath?])
AC_DEFINE(MP_PREC, 4, [Default libtommath precision.])
#--------------------------------------------------------------------
diff --git a/unix/tclConfig.h.in b/unix/tclConfig.h.in
index a4ab3e5..adbc80d 100644
--- a/unix/tclConfig.h.in
+++ b/unix/tclConfig.h.in
@@ -394,9 +394,6 @@
/* Are we building with threads enabled? */
#undef TCL_THREADS
-/* Build libtommath? */
-#undef TCL_TOMMATH
-
/* Do we allow unloading of shared libraries? */
#undef TCL_UNLOAD_DLLS
diff --git a/win/Makefile.in b/win/Makefile.in
index 0ab4204..e967ef3 100644
--- a/win/Makefile.in
+++ b/win/Makefile.in
@@ -197,7 +197,7 @@ RM = rm -f
COPY = cp
CC_SWITCHES = ${CFLAGS} ${CFLAGS_WARNING} ${TCL_SHLIB_CFLAGS} \
--I"${ZLIB_DIR_NATIVE}" -I"${GENERIC_DIR_NATIVE}" -DTCL_TOMMATH \
+-I"${ZLIB_DIR_NATIVE}" -I"${GENERIC_DIR_NATIVE}" \
-DMP_PREC=4 -I"${TOMMATH_DIR_NATIVE}" -I"${WIN_DIR_NATIVE}" \
${AC_FLAGS} ${COMPILE_DEBUG_FLAGS} ${NO_DEPRECATED_FLAGS}
@@ -205,7 +205,7 @@ CC_OBJNAME = @CC_OBJNAME@
CC_EXENAME = @CC_EXENAME@
STUB_CC_SWITCHES = ${CFLAGS} ${CFLAGS_WARNING} ${SHLIB_CFLAGS} \
--I"${GENERIC_DIR_NATIVE}" -DTCL_TOMMATH -DMP_PREC=4 -I"${TOMMATH_DIR_NATIVE}" \
+-I"${GENERIC_DIR_NATIVE}" -DMP_PREC=4 -I"${TOMMATH_DIR_NATIVE}" \
-I"${WIN_DIR_NATIVE}" ${AC_FLAGS} \
${COMPILE_DEBUG_FLAGS}
@@ -328,6 +328,7 @@ TOMMATH_OBJS = \
bn_mp_div_3.${OBJEXT} \
bn_mp_exch.${OBJEXT} \
bn_mp_expt_d.${OBJEXT} \
+ bn_mp_expt_d_ex.${OBJEXT} \
bn_mp_grow.${OBJEXT} \
bn_mp_init.${OBJEXT} \
bn_mp_init_copy.${OBJEXT} \
diff --git a/win/makefile.vc b/win/makefile.vc
index f401541..c10b196 100644
--- a/win/makefile.vc
+++ b/win/makefile.vc
@@ -383,6 +383,7 @@ TOMMATHOBJS = \
$(TMP_DIR)\bn_mp_div_3.obj \
$(TMP_DIR)\bn_mp_exch.obj \
$(TMP_DIR)\bn_mp_expt_d.obj \
+ $(TMP_DIR)\bn_mp_expt_d_ex.obj \
$(TMP_DIR)\bn_mp_grow.obj \
$(TMP_DIR)\bn_mp_init.obj \
$(TMP_DIR)\bn_mp_init_copy.obj \
@@ -503,7 +504,7 @@ crt = -MT
!endif
TCL_INCLUDES = -I"$(WINDIR)" -I"$(GENERICDIR)" -I"$(TOMMATHDIR)"
-TCL_DEFINES = -DTCL_TOMMATH -DMP_PREC=4 -Dinline=__inline -DHAVE_ZLIB=1
+TCL_DEFINES = -DMP_PREC=4 -Dinline=__inline -DHAVE_ZLIB=1
BASE_CFLAGS = $(cflags) $(cdebug) $(crt) $(TCL_INCLUDES) $(TCL_DEFINES)
CON_CFLAGS = $(cflags) $(cdebug) $(crt) -DCONSOLE
TCL_CFLAGS = $(BASE_CFLAGS) $(OPTDEFINES)
generated by cgit v0.12 at 2024-11-22 05:04:39 (GMT)