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authorKevin B Kenny <kennykb@acm.org>2005-01-19 22:41:26 (GMT)
committerKevin B Kenny <kennykb@acm.org>2005-01-19 22:41:26 (GMT)
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+\documentclass[b5paper]{book}
+\usepackage{hyperref}
+\usepackage{makeidx}
+\usepackage{amssymb}
+\usepackage{color}
+\usepackage{alltt}
+\usepackage{graphicx}
+\usepackage{layout}
+\def\union{\cup}
+\def\intersect{\cap}
+\def\getsrandom{\stackrel{\rm R}{\gets}}
+\def\cross{\times}
+\def\cat{\hspace{0.5em} \| \hspace{0.5em}}
+\def\catn{$\|$}
+\def\divides{\hspace{0.3em} | \hspace{0.3em}}
+\def\nequiv{\not\equiv}
+\def\approx{\raisebox{0.2ex}{\mbox{\small $\sim$}}}
+\def\lcm{{\rm lcm}}
+\def\gcd{{\rm gcd}}
+\def\log{{\rm log}}
+\def\ord{{\rm ord}}
+\def\abs{{\mathit abs}}
+\def\rep{{\mathit rep}}
+\def\mod{{\mathit\ mod\ }}
+\renewcommand{\pmod}[1]{\ ({\rm mod\ }{#1})}
+\newcommand{\floor}[1]{\left\lfloor{#1}\right\rfloor}
+\newcommand{\ceil}[1]{\left\lceil{#1}\right\rceil}
+\def\Or{{\rm\ or\ }}
+\def\And{{\rm\ and\ }}
+\def\iff{\hspace{1em}\Longleftrightarrow\hspace{1em}}
+\def\implies{\Rightarrow}
+\def\undefined{{\rm ``undefined"}}
+\def\Proof{\vspace{1ex}\noindent {\bf Proof:}\hspace{1em}}
+\let\oldphi\phi
+\def\phi{\varphi}
+\def\Pr{{\rm Pr}}
+\newcommand{\str}[1]{{\mathbf{#1}}}
+\def\F{{\mathbb F}}
+\def\N{{\mathbb N}}
+\def\Z{{\mathbb Z}}
+\def\R{{\mathbb R}}
+\def\C{{\mathbb C}}
+\def\Q{{\mathbb Q}}
+\definecolor{DGray}{gray}{0.5}
+\newcommand{\emailaddr}[1]{\mbox{$<${#1}$>$}}
+\def\twiddle{\raisebox{0.3ex}{\mbox{\tiny $\sim$}}}
+\def\gap{\vspace{0.5ex}}
+\makeindex
+\begin{document}
+\frontmatter
+\pagestyle{empty}
+\title{LibTomMath User Manual \\ v0.33}
+\author{Tom St Denis \\ tomstdenis@iahu.ca}
+\maketitle
+This text, the library and the accompanying textbook are all hereby placed in the public domain. This book has been
+formatted for B5 [176x250] paper using the \LaTeX{} {\em book} macro package.
+
+\vspace{10cm}
+
+\begin{flushright}Open Source. Open Academia. Open Minds.
+
+\mbox{ }
+
+Tom St Denis,
+
+Ontario, Canada
+\end{flushright}
+
+\tableofcontents
+\listoffigures
+\mainmatter
+\pagestyle{headings}
+\chapter{Introduction}
+\section{What is LibTomMath?}
+LibTomMath is a library of source code which provides a series of efficient and carefully written functions for manipulating
+large integer numbers. It was written in portable ISO C source code so that it will build on any platform with a conforming
+C compiler.
+
+In a nutshell the library was written from scratch with verbose comments to help instruct computer science students how
+to implement ``bignum'' math. However, the resulting code has proven to be very useful. It has been used by numerous
+universities, commercial and open source software developers. It has been used on a variety of platforms ranging from
+Linux and Windows based x86 to ARM based Gameboys and PPC based MacOS machines.
+
+\section{License}
+As of the v0.25 the library source code has been placed in the public domain with every new release. As of the v0.28
+release the textbook ``Implementing Multiple Precision Arithmetic'' has been placed in the public domain with every new
+release as well. This textbook is meant to compliment the project by providing a more solid walkthrough of the development
+algorithms used in the library.
+
+Since both\footnote{Note that the MPI files under mtest/ are copyrighted by Michael Fromberger. They are not required to use LibTomMath.} are in the
+public domain everyone is entitled to do with them as they see fit.
+
+\section{Building LibTomMath}
+
+LibTomMath is meant to be very ``GCC friendly'' as it comes with a makefile well suited for GCC. However, the library will
+also build in MSVC, Borland C out of the box. For any other ISO C compiler a makefile will have to be made by the end
+developer.
+
+\subsection{Static Libraries}
+To build as a static library for GCC issue the following
+\begin{alltt}
+make
+\end{alltt}
+
+command. This will build the library and archive the object files in ``libtommath.a''. Now you link against
+that and include ``tommath.h'' within your programs. Alternatively to build with MSVC issue the following
+\begin{alltt}
+nmake -f makefile.msvc
+\end{alltt}
+
+This will build the library and archive the object files in ``tommath.lib''. This has been tested with MSVC
+version 6.00 with service pack 5.
+
+\subsection{Shared Libraries}
+To build as a shared library for GCC issue the following
+\begin{alltt}
+make -f makefile.shared
+\end{alltt}
+This requires the ``libtool'' package (common on most Linux/BSD systems). It will build LibTomMath as both shared
+and static then install (by default) into /usr/lib as well as install the header files in /usr/include. The shared
+library (resource) will be called ``libtommath.la'' while the static library called ``libtommath.a''. Generally
+you use libtool to link your application against the shared object.
+
+There is limited support for making a ``DLL'' in windows via the ``makefile.cygwin\_dll'' makefile. It requires
+Cygwin to work with since it requires the auto-export/import functionality. The resulting DLL and import library
+``libtommath.dll.a'' can be used to link LibTomMath dynamically to any Windows program using Cygwin.
+
+\subsection{Testing}
+To build the library and the test harness type
+
+\begin{alltt}
+make test
+\end{alltt}
+
+This will build the library, ``test'' and ``mtest/mtest''. The ``test'' program will accept test vectors and verify the
+results. ``mtest/mtest'' will generate test vectors using the MPI library by Michael Fromberger\footnote{A copy of MPI
+is included in the package}. Simply pipe mtest into test using
+
+\begin{alltt}
+mtest/mtest | test
+\end{alltt}
+
+If you do not have a ``/dev/urandom'' style RNG source you will have to write your own PRNG and simply pipe that into
+mtest. For example, if your PRNG program is called ``myprng'' simply invoke
+
+\begin{alltt}
+myprng | mtest/mtest | test
+\end{alltt}
+
+This will output a row of numbers that are increasing. Each column is a different test (such as addition, multiplication, etc)
+that is being performed. The numbers represent how many times the test was invoked. If an error is detected the program
+will exit with a dump of the relevent numbers it was working with.
+
+\section{Build Configuration}
+LibTomMath can configured at build time in three phases we shall call ``depends'', ``tweaks'' and ``trims''.
+Each phase changes how the library is built and they are applied one after another respectively.
+
+To make the system more powerful you can tweak the build process. Classes are defined in the file
+``tommath\_superclass.h''. By default, the symbol ``LTM\_ALL'' shall be defined which simply
+instructs the system to build all of the functions. This is how LibTomMath used to be packaged. This will give you
+access to every function LibTomMath offers.
+
+However, there are cases where such a build is not optional. For instance, you want to perform RSA operations. You
+don't need the vast majority of the library to perform these operations. Aside from LTM\_ALL there is
+another pre--defined class ``SC\_RSA\_1'' which works in conjunction with the RSA from LibTomCrypt. Additional
+classes can be defined base on the need of the user.
+
+\subsection{Build Depends}
+In the file tommath\_class.h you will see a large list of C ``defines'' followed by a series of ``ifdefs''
+which further define symbols. All of the symbols (technically they're macros $\ldots$) represent a given C source
+file. For instance, BN\_MP\_ADD\_C represents the file ``bn\_mp\_add.c''. When a define has been enabled the
+function in the respective file will be compiled and linked into the library. Accordingly when the define
+is absent the file will not be compiled and not contribute any size to the library.
+
+You will also note that the header tommath\_class.h is actually recursively included (it includes itself twice).
+This is to help resolve as many dependencies as possible. In the last pass the symbol LTM\_LAST will be defined.
+This is useful for ``trims''.
+
+\subsection{Build Tweaks}
+A tweak is an algorithm ``alternative''. For example, to provide tradeoffs (usually between size and space).
+They can be enabled at any pass of the configuration phase.
+
+\begin{small}
+\begin{center}
+\begin{tabular}{|l|l|}
+\hline \textbf{Define} & \textbf{Purpose} \\
+\hline BN\_MP\_DIV\_SMALL & Enables a slower, smaller and equally \\
+ & functional mp\_div() function \\
+\hline
+\end{tabular}
+\end{center}
+\end{small}
+
+\subsection{Build Trims}
+A trim is a manner of removing functionality from a function that is not required. For instance, to perform
+RSA cryptography you only require exponentiation with odd moduli so even moduli support can be safely removed.
+Build trims are meant to be defined on the last pass of the configuration which means they are to be defined
+only if LTM\_LAST has been defined.
+
+\subsubsection{Moduli Related}
+\begin{small}
+\begin{center}
+\begin{tabular}{|l|l|}
+\hline \textbf{Restriction} & \textbf{Undefine} \\
+\hline Exponentiation with odd moduli only & BN\_S\_MP\_EXPTMOD\_C \\
+ & BN\_MP\_REDUCE\_C \\
+ & BN\_MP\_REDUCE\_SETUP\_C \\
+ & BN\_S\_MP\_MUL\_HIGH\_DIGS\_C \\
+ & BN\_FAST\_S\_MP\_MUL\_HIGH\_DIGS\_C \\
+\hline Exponentiation with random odd moduli & (The above plus the following) \\
+ & BN\_MP\_REDUCE\_2K\_C \\
+ & BN\_MP\_REDUCE\_2K\_SETUP\_C \\
+ & BN\_MP\_REDUCE\_IS\_2K\_C \\
+ & BN\_MP\_DR\_IS\_MODULUS\_C \\
+ & BN\_MP\_DR\_REDUCE\_C \\
+ & BN\_MP\_DR\_SETUP\_C \\
+\hline Modular inverse odd moduli only & BN\_MP\_INVMOD\_SLOW\_C \\
+\hline Modular inverse (both, smaller/slower) & BN\_FAST\_MP\_INVMOD\_C \\
+\hline
+\end{tabular}
+\end{center}
+\end{small}
+
+\subsubsection{Operand Size Related}
+\begin{small}
+\begin{center}
+\begin{tabular}{|l|l|}
+\hline \textbf{Restriction} & \textbf{Undefine} \\
+\hline Moduli $\le 2560$ bits & BN\_MP\_MONTGOMERY\_REDUCE\_C \\
+ & BN\_S\_MP\_MUL\_DIGS\_C \\
+ & BN\_S\_MP\_MUL\_HIGH\_DIGS\_C \\
+ & BN\_S\_MP\_SQR\_C \\
+\hline Polynomial Schmolynomial & BN\_MP\_KARATSUBA\_MUL\_C \\
+ & BN\_MP\_KARATSUBA\_SQR\_C \\
+ & BN\_MP\_TOOM\_MUL\_C \\
+ & BN\_MP\_TOOM\_SQR\_C \\
+
+\hline
+\end{tabular}
+\end{center}
+\end{small}
+
+
+\section{Purpose of LibTomMath}
+Unlike GNU MP (GMP) Library, LIP, OpenSSL or various other commercial kits (Miracl), LibTomMath was not written with
+bleeding edge performance in mind. First and foremost LibTomMath was written to be entirely open. Not only is the
+source code public domain (unlike various other GPL/etc licensed code), not only is the code freely downloadable but the
+source code is also accessible for computer science students attempting to learn ``BigNum'' or multiple precision
+arithmetic techniques.
+
+LibTomMath was written to be an instructive collection of source code. This is why there are many comments, only one
+function per source file and often I use a ``middle-road'' approach where I don't cut corners for an extra 2\% speed
+increase.
+
+Source code alone cannot really teach how the algorithms work which is why I also wrote a textbook that accompanies
+the library (beat that!).
+
+So you may be thinking ``should I use LibTomMath?'' and the answer is a definite maybe. Let me tabulate what I think
+are the pros and cons of LibTomMath by comparing it to the math routines from GnuPG\footnote{GnuPG v1.2.3 versus LibTomMath v0.28}.
+
+\newpage\begin{figure}[here]
+\begin{small}
+\begin{center}
+\begin{tabular}{|l|c|c|l|}
+\hline \textbf{Criteria} & \textbf{Pro} & \textbf{Con} & \textbf{Notes} \\
+\hline Few lines of code per file & X & & GnuPG $ = 300.9$, LibTomMath $ = 76.04$ \\
+\hline Commented function prototypes & X && GnuPG function names are cryptic. \\
+\hline Speed && X & LibTomMath is slower. \\
+\hline Totally free & X & & GPL has unfavourable restrictions.\\
+\hline Large function base & X & & GnuPG is barebones. \\
+\hline Four modular reduction algorithms & X & & Faster modular exponentiation. \\
+\hline Portable & X & & GnuPG requires configuration to build. \\
+\hline
+\end{tabular}
+\end{center}
+\end{small}
+\caption{LibTomMath Valuation}
+\end{figure}
+
+It may seem odd to compare LibTomMath to GnuPG since the math in GnuPG is only a small portion of the entire application.
+However, LibTomMath was written with cryptography in mind. It provides essentially all of the functions a cryptosystem
+would require when working with large integers.
+
+So it may feel tempting to just rip the math code out of GnuPG (or GnuMP where it was taken from originally) in your
+own application but I think there are reasons not to. While LibTomMath is slower than libraries such as GnuMP it is
+not normally significantly slower. On x86 machines the difference is normally a factor of two when performing modular
+exponentiations.
+
+Essentially the only time you wouldn't use LibTomMath is when blazing speed is the primary concern.
+
+\chapter{Getting Started with LibTomMath}
+\section{Building Programs}
+In order to use LibTomMath you must include ``tommath.h'' and link against the appropriate library file (typically
+libtommath.a). There is no library initialization required and the entire library is thread safe.
+
+\section{Return Codes}
+There are three possible return codes a function may return.
+
+\index{MP\_OKAY}\index{MP\_YES}\index{MP\_NO}\index{MP\_VAL}\index{MP\_MEM}
+\begin{figure}[here!]
+\begin{center}
+\begin{small}
+\begin{tabular}{|l|l|}
+\hline \textbf{Code} & \textbf{Meaning} \\
+\hline MP\_OKAY & The function succeeded. \\
+\hline MP\_VAL & The function input was invalid. \\
+\hline MP\_MEM & Heap memory exhausted. \\
+\hline &\\
+\hline MP\_YES & Response is yes. \\
+\hline MP\_NO & Response is no. \\
+\hline
+\end{tabular}
+\end{small}
+\end{center}
+\caption{Return Codes}
+\end{figure}
+
+The last two codes listed are not actually ``return'ed'' by a function. They are placed in an integer (the caller must
+provide the address of an integer it can store to) which the caller can access. To convert one of the three return codes
+to a string use the following function.
+
+\index{mp\_error\_to\_string}
+\begin{alltt}
+char *mp_error_to_string(int code);
+\end{alltt}
+
+This will return a pointer to a string which describes the given error code. It will not work for the return codes
+MP\_YES and MP\_NO.
+
+\section{Data Types}
+The basic ``multiple precision integer'' type is known as the ``mp\_int'' within LibTomMath. This data type is used to
+organize all of the data required to manipulate the integer it represents. Within LibTomMath it has been prototyped
+as the following.
+
+\index{mp\_int}
+\begin{alltt}
+typedef struct \{
+ int used, alloc, sign;
+ mp_digit *dp;
+\} mp_int;
+\end{alltt}
+
+Where ``mp\_digit'' is a data type that represents individual digits of the integer. By default, an mp\_digit is the
+ISO C ``unsigned long'' data type and each digit is $28-$bits long. The mp\_digit type can be configured to suit other
+platforms by defining the appropriate macros.
+
+All LTM functions that use the mp\_int type will expect a pointer to mp\_int structure. You must allocate memory to
+hold the structure itself by yourself (whether off stack or heap it doesn't matter). The very first thing that must be
+done to use an mp\_int is that it must be initialized.
+
+\section{Function Organization}
+
+The arithmetic functions of the library are all organized to have the same style prototype. That is source operands
+are passed on the left and the destination is on the right. For instance,
+
+\begin{alltt}
+mp_add(&a, &b, &c); /* c = a + b */
+mp_mul(&a, &a, &c); /* c = a * a */
+mp_div(&a, &b, &c, &d); /* c = [a/b], d = a mod b */
+\end{alltt}
+
+Another feature of the way the functions have been implemented is that source operands can be destination operands as well.
+For instance,
+
+\begin{alltt}
+mp_add(&a, &b, &b); /* b = a + b */
+mp_div(&a, &b, &a, &c); /* a = [a/b], c = a mod b */
+\end{alltt}
+
+This allows operands to be re-used which can make programming simpler.
+
+\section{Initialization}
+\subsection{Single Initialization}
+A single mp\_int can be initialized with the ``mp\_init'' function.
+
+\index{mp\_init}
+\begin{alltt}
+int mp_init (mp_int * a);
+\end{alltt}
+
+This function expects a pointer to an mp\_int structure and will initialize the members of the structure so the mp\_int
+represents the default integer which is zero. If the functions returns MP\_OKAY then the mp\_int is ready to be used
+by the other LibTomMath functions.
+
+\begin{small} \begin{alltt}
+int main(void)
+\{
+ mp_int number;
+ int result;
+
+ if ((result = mp_init(&number)) != MP_OKAY) \{
+ printf("Error initializing the number. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ /* use the number */
+
+ return EXIT_SUCCESS;
+\}
+\end{alltt} \end{small}
+
+\subsection{Single Free}
+When you are finished with an mp\_int it is ideal to return the heap it used back to the system. The following function
+provides this functionality.
+
+\index{mp\_clear}
+\begin{alltt}
+void mp_clear (mp_int * a);
+\end{alltt}
+
+The function expects a pointer to a previously initialized mp\_int structure and frees the heap it uses. It sets the
+pointer\footnote{The ``dp'' member.} within the mp\_int to \textbf{NULL} which is used to prevent double free situations.
+Is is legal to call mp\_clear() twice on the same mp\_int in a row.
+
+\begin{small} \begin{alltt}
+int main(void)
+\{
+ mp_int number;
+ int result;
+
+ if ((result = mp_init(&number)) != MP_OKAY) \{
+ printf("Error initializing the number. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ /* use the number */
+
+ /* We're done with it. */
+ mp_clear(&number);
+
+ return EXIT_SUCCESS;
+\}
+\end{alltt} \end{small}
+
+\subsection{Multiple Initializations}
+Certain algorithms require more than one large integer. In these instances it is ideal to initialize all of the mp\_int
+variables in an ``all or nothing'' fashion. That is, they are either all initialized successfully or they are all
+not initialized.
+
+The mp\_init\_multi() function provides this functionality.
+
+\index{mp\_init\_multi} \index{mp\_clear\_multi}
+\begin{alltt}
+int mp_init_multi(mp_int *mp, ...);
+\end{alltt}
+
+It accepts a \textbf{NULL} terminated list of pointers to mp\_int structures. It will attempt to initialize them all
+at once. If the function returns MP\_OKAY then all of the mp\_int variables are ready to use, otherwise none of them
+are available for use. A complementary mp\_clear\_multi() function allows multiple mp\_int variables to be free'd
+from the heap at the same time.
+
+\begin{small} \begin{alltt}
+int main(void)
+\{
+ mp_int num1, num2, num3;
+ int result;
+
+ if ((result = mp_init_multi(&num1,
+ &num2,
+ &num3, NULL)) != MP\_OKAY) \{
+ printf("Error initializing the numbers. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ /* use the numbers */
+
+ /* We're done with them. */
+ mp_clear_multi(&num1, &num2, &num3, NULL);
+
+ return EXIT_SUCCESS;
+\}
+\end{alltt} \end{small}
+
+\subsection{Other Initializers}
+To initialized and make a copy of an mp\_int the mp\_init\_copy() function has been provided.
+
+\index{mp\_init\_copy}
+\begin{alltt}
+int mp_init_copy (mp_int * a, mp_int * b);
+\end{alltt}
+
+This function will initialize $a$ and make it a copy of $b$ if all goes well.
+
+\begin{small} \begin{alltt}
+int main(void)
+\{
+ mp_int num1, num2;
+ int result;
+
+ /* initialize and do work on num1 ... */
+
+ /* We want a copy of num1 in num2 now */
+ if ((result = mp_init_copy(&num2, &num1)) != MP_OKAY) \{
+ printf("Error initializing the copy. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ /* now num2 is ready and contains a copy of num1 */
+
+ /* We're done with them. */
+ mp_clear_multi(&num1, &num2, NULL);
+
+ return EXIT_SUCCESS;
+\}
+\end{alltt} \end{small}
+
+Another less common initializer is mp\_init\_size() which allows the user to initialize an mp\_int with a given
+default number of digits. By default, all initializers allocate \textbf{MP\_PREC} digits. This function lets
+you override this behaviour.
+
+\index{mp\_init\_size}
+\begin{alltt}
+int mp_init_size (mp_int * a, int size);
+\end{alltt}
+
+The $size$ parameter must be greater than zero. If the function succeeds the mp\_int $a$ will be initialized
+to have $size$ digits (which are all initially zero).
+
+\begin{small} \begin{alltt}
+int main(void)
+\{
+ mp_int number;
+ int result;
+
+ /* we need a 60-digit number */
+ if ((result = mp_init_size(&number, 60)) != MP_OKAY) \{
+ printf("Error initializing the number. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ /* use the number */
+
+ return EXIT_SUCCESS;
+\}
+\end{alltt} \end{small}
+
+\section{Maintenance Functions}
+
+\subsection{Reducing Memory Usage}
+When an mp\_int is in a state where it won't be changed again\footnote{A Diffie-Hellman modulus for instance.} excess
+digits can be removed to return memory to the heap with the mp\_shrink() function.
+
+\index{mp\_shrink}
+\begin{alltt}
+int mp_shrink (mp_int * a);
+\end{alltt}
+
+This will remove excess digits of the mp\_int $a$. If the operation fails the mp\_int should be intact without the
+excess digits being removed. Note that you can use a shrunk mp\_int in further computations, however, such operations
+will require heap operations which can be slow. It is not ideal to shrink mp\_int variables that you will further
+modify in the system (unless you are seriously low on memory).
+
+\begin{small} \begin{alltt}
+int main(void)
+\{
+ mp_int number;
+ int result;
+
+ if ((result = mp_init(&number)) != MP_OKAY) \{
+ printf("Error initializing the number. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ /* use the number [e.g. pre-computation] */
+
+ /* We're done with it for now. */
+ if ((result = mp_shrink(&number)) != MP_OKAY) \{
+ printf("Error shrinking the number. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ /* use it .... */
+
+
+ /* we're done with it. */
+ mp_clear(&number);
+
+ return EXIT_SUCCESS;
+\}
+\end{alltt} \end{small}
+
+\subsection{Adding additional digits}
+
+Within the mp\_int structure are two parameters which control the limitations of the array of digits that represent
+the integer the mp\_int is meant to equal. The \textit{used} parameter dictates how many digits are significant, that is,
+contribute to the value of the mp\_int. The \textit{alloc} parameter dictates how many digits are currently available in
+the array. If you need to perform an operation that requires more digits you will have to mp\_grow() the mp\_int to
+your desired size.
+
+\index{mp\_grow}
+\begin{alltt}
+int mp_grow (mp_int * a, int size);
+\end{alltt}
+
+This will grow the array of digits of $a$ to $size$. If the \textit{alloc} parameter is already bigger than
+$size$ the function will not do anything.
+
+\begin{small} \begin{alltt}
+int main(void)
+\{
+ mp_int number;
+ int result;
+
+ if ((result = mp_init(&number)) != MP_OKAY) \{
+ printf("Error initializing the number. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ /* use the number */
+
+ /* We need to add 20 digits to the number */
+ if ((result = mp_grow(&number, number.alloc + 20)) != MP_OKAY) \{
+ printf("Error growing the number. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+
+ /* use the number */
+
+ /* we're done with it. */
+ mp_clear(&number);
+
+ return EXIT_SUCCESS;
+\}
+\end{alltt} \end{small}
+
+\chapter{Basic Operations}
+\section{Small Constants}
+Setting mp\_ints to small constants is a relatively common operation. To accomodate these instances there are two
+small constant assignment functions. The first function is used to set a single digit constant while the second sets
+an ISO C style ``unsigned long'' constant. The reason for both functions is efficiency. Setting a single digit is quick but the
+domain of a digit can change (it's always at least $0 \ldots 127$).
+
+\subsection{Single Digit}
+
+Setting a single digit can be accomplished with the following function.
+
+\index{mp\_set}
+\begin{alltt}
+void mp_set (mp_int * a, mp_digit b);
+\end{alltt}
+
+This will zero the contents of $a$ and make it represent an integer equal to the value of $b$. Note that this
+function has a return type of \textbf{void}. It cannot cause an error so it is safe to assume the function
+succeeded.
+
+\begin{small} \begin{alltt}
+int main(void)
+\{
+ mp_int number;
+ int result;
+
+ if ((result = mp_init(&number)) != MP_OKAY) \{
+ printf("Error initializing the number. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ /* set the number to 5 */
+ mp_set(&number, 5);
+
+ /* we're done with it. */
+ mp_clear(&number);
+
+ return EXIT_SUCCESS;
+\}
+\end{alltt} \end{small}
+
+\subsection{Long Constants}
+
+To set a constant that is the size of an ISO C ``unsigned long'' and larger than a single digit the following function
+can be used.
+
+\index{mp\_set\_int}
+\begin{alltt}
+int mp_set_int (mp_int * a, unsigned long b);
+\end{alltt}
+
+This will assign the value of the 32-bit variable $b$ to the mp\_int $a$. Unlike mp\_set() this function will always
+accept a 32-bit input regardless of the size of a single digit. However, since the value may span several digits
+this function can fail if it runs out of heap memory.
+
+To get the ``unsigned long'' copy of an mp\_int the following function can be used.
+
+\index{mp\_get\_int}
+\begin{alltt}
+unsigned long mp_get_int (mp_int * a);
+\end{alltt}
+
+This will return the 32 least significant bits of the mp\_int $a$.
+
+\begin{small} \begin{alltt}
+int main(void)
+\{
+ mp_int number;
+ int result;
+
+ if ((result = mp_init(&number)) != MP_OKAY) \{
+ printf("Error initializing the number. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ /* set the number to 654321 (note this is bigger than 127) */
+ if ((result = mp_set_int(&number, 654321)) != MP_OKAY) \{
+ printf("Error setting the value of the number. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ printf("number == \%lu", mp_get_int(&number));
+
+ /* we're done with it. */
+ mp_clear(&number);
+
+ return EXIT_SUCCESS;
+\}
+\end{alltt} \end{small}
+
+This should output the following if the program succeeds.
+
+\begin{alltt}
+number == 654321
+\end{alltt}
+
+\subsection{Initialize and Setting Constants}
+To both initialize and set small constants the following two functions are available.
+\index{mp\_init\_set} \index{mp\_init\_set\_int}
+\begin{alltt}
+int mp_init_set (mp_int * a, mp_digit b);
+int mp_init_set_int (mp_int * a, unsigned long b);
+\end{alltt}
+
+Both functions work like the previous counterparts except they first mp\_init $a$ before setting the values.
+
+\begin{alltt}
+int main(void)
+\{
+ mp_int number1, number2;
+ int result;
+
+ /* initialize and set a single digit */
+ if ((result = mp_init_set(&number1, 100)) != MP_OKAY) \{
+ printf("Error setting number1: \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ /* initialize and set a long */
+ if ((result = mp_init_set_int(&number2, 1023)) != MP_OKAY) \{
+ printf("Error setting number2: \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ /* display */
+ printf("Number1, Number2 == \%lu, \%lu",
+ mp_get_int(&number1), mp_get_int(&number2));
+
+ /* clear */
+ mp_clear_multi(&number1, &number2, NULL);
+
+ return EXIT_SUCCESS;
+\}
+\end{alltt}
+
+If this program succeeds it shall output.
+\begin{alltt}
+Number1, Number2 == 100, 1023
+\end{alltt}
+
+\section{Comparisons}
+
+Comparisons in LibTomMath are always performed in a ``left to right'' fashion. There are three possible return codes
+for any comparison.
+
+\index{MP\_GT} \index{MP\_EQ} \index{MP\_LT}
+\begin{figure}[here]
+\begin{center}
+\begin{tabular}{|c|c|}
+\hline \textbf{Result Code} & \textbf{Meaning} \\
+\hline MP\_GT & $a > b$ \\
+\hline MP\_EQ & $a = b$ \\
+\hline MP\_LT & $a < b$ \\
+\hline
+\end{tabular}
+\end{center}
+\caption{Comparison Codes for $a, b$}
+\label{fig:CMP}
+\end{figure}
+
+In figure \ref{fig:CMP} two integers $a$ and $b$ are being compared. In this case $a$ is said to be ``to the left'' of
+$b$.
+
+\subsection{Unsigned comparison}
+
+An unsigned comparison considers only the digits themselves and not the associated \textit{sign} flag of the
+mp\_int structures. This is analogous to an absolute comparison. The function mp\_cmp\_mag() will compare two
+mp\_int variables based on their digits only.
+
+\index{mp\_cmp\_mag}
+\begin{alltt}
+int mp_cmp(mp_int * a, mp_int * b);
+\end{alltt}
+This will compare $a$ to $b$ placing $a$ to the left of $b$. This function cannot fail and will return one of the
+three compare codes listed in figure \ref{fig:CMP}.
+
+\begin{small} \begin{alltt}
+int main(void)
+\{
+ mp_int number1, number2;
+ int result;
+
+ if ((result = mp_init_multi(&number1, &number2, NULL)) != MP_OKAY) \{
+ printf("Error initializing the numbers. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ /* set the number1 to 5 */
+ mp_set(&number1, 5);
+
+ /* set the number2 to -6 */
+ mp_set(&number2, 6);
+ if ((result = mp_neg(&number2, &number2)) != MP_OKAY) \{
+ printf("Error negating number2. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ switch(mp_cmp_mag(&number1, &number2)) \{
+ case MP_GT: printf("|number1| > |number2|"); break;
+ case MP_EQ: printf("|number1| = |number2|"); break;
+ case MP_LT: printf("|number1| < |number2|"); break;
+ \}
+
+ /* we're done with it. */
+ mp_clear_multi(&number1, &number2, NULL);
+
+ return EXIT_SUCCESS;
+\}
+\end{alltt} \end{small}
+
+If this program\footnote{This function uses the mp\_neg() function which is discussed in section \ref{sec:NEG}.} completes
+successfully it should print the following.
+
+\begin{alltt}
+|number1| < |number2|
+\end{alltt}
+
+This is because $\vert -6 \vert = 6$ and obviously $5 < 6$.
+
+\subsection{Signed comparison}
+
+To compare two mp\_int variables based on their signed value the mp\_cmp() function is provided.
+
+\index{mp\_cmp}
+\begin{alltt}
+int mp_cmp(mp_int * a, mp_int * b);
+\end{alltt}
+
+This will compare $a$ to the left of $b$. It will first compare the signs of the two mp\_int variables. If they
+differ it will return immediately based on their signs. If the signs are equal then it will compare the digits
+individually. This function will return one of the compare conditions codes listed in figure \ref{fig:CMP}.
+
+\begin{small} \begin{alltt}
+int main(void)
+\{
+ mp_int number1, number2;
+ int result;
+
+ if ((result = mp_init_multi(&number1, &number2, NULL)) != MP_OKAY) \{
+ printf("Error initializing the numbers. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ /* set the number1 to 5 */
+ mp_set(&number1, 5);
+
+ /* set the number2 to -6 */
+ mp_set(&number2, 6);
+ if ((result = mp_neg(&number2, &number2)) != MP_OKAY) \{
+ printf("Error negating number2. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ switch(mp_cmp(&number1, &number2)) \{
+ case MP_GT: printf("number1 > number2"); break;
+ case MP_EQ: printf("number1 = number2"); break;
+ case MP_LT: printf("number1 < number2"); break;
+ \}
+
+ /* we're done with it. */
+ mp_clear_multi(&number1, &number2, NULL);
+
+ return EXIT_SUCCESS;
+\}
+\end{alltt} \end{small}
+
+If this program\footnote{This function uses the mp\_neg() function which is discussed in section \ref{sec:NEG}.} completes
+successfully it should print the following.
+
+\begin{alltt}
+number1 > number2
+\end{alltt}
+
+\subsection{Single Digit}
+
+To compare a single digit against an mp\_int the following function has been provided.
+
+\index{mp\_cmp\_d}
+\begin{alltt}
+int mp_cmp_d(mp_int * a, mp_digit b);
+\end{alltt}
+
+This will compare $a$ to the left of $b$ using a signed comparison. Note that it will always treat $b$ as
+positive. This function is rather handy when you have to compare against small values such as $1$ (which often
+comes up in cryptography). The function cannot fail and will return one of the tree compare condition codes
+listed in figure \ref{fig:CMP}.
+
+
+\begin{small} \begin{alltt}
+int main(void)
+\{
+ mp_int number;
+ int result;
+
+ if ((result = mp_init(&number)) != MP_OKAY) \{
+ printf("Error initializing the number. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ /* set the number to 5 */
+ mp_set(&number, 5);
+
+ switch(mp_cmp_d(&number, 7)) \{
+ case MP_GT: printf("number > 7"); break;
+ case MP_EQ: printf("number = 7"); break;
+ case MP_LT: printf("number < 7"); break;
+ \}
+
+ /* we're done with it. */
+ mp_clear(&number);
+
+ return EXIT_SUCCESS;
+\}
+\end{alltt} \end{small}
+
+If this program functions properly it will print out the following.
+
+\begin{alltt}
+number < 7
+\end{alltt}
+
+\section{Logical Operations}
+
+Logical operations are operations that can be performed either with simple shifts or boolean operators such as
+AND, XOR and OR directly. These operations are very quick.
+
+\subsection{Multiplication by two}
+
+Multiplications and divisions by any power of two can be performed with quick logical shifts either left or
+right depending on the operation.
+
+When multiplying or dividing by two a special case routine can be used which are as follows.
+\index{mp\_mul\_2} \index{mp\_div\_2}
+\begin{alltt}
+int mp_mul_2(mp_int * a, mp_int * b);
+int mp_div_2(mp_int * a, mp_int * b);
+\end{alltt}
+
+The former will assign twice $a$ to $b$ while the latter will assign half $a$ to $b$. These functions are fast
+since the shift counts and maskes are hardcoded into the routines.
+
+\begin{small} \begin{alltt}
+int main(void)
+\{
+ mp_int number;
+ int result;
+
+ if ((result = mp_init(&number)) != MP_OKAY) \{
+ printf("Error initializing the number. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ /* set the number to 5 */
+ mp_set(&number, 5);
+
+ /* multiply by two */
+ if ((result = mp\_mul\_2(&number, &number)) != MP_OKAY) \{
+ printf("Error multiplying the number. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+ switch(mp_cmp_d(&number, 7)) \{
+ case MP_GT: printf("2*number > 7"); break;
+ case MP_EQ: printf("2*number = 7"); break;
+ case MP_LT: printf("2*number < 7"); break;
+ \}
+
+ /* now divide by two */
+ if ((result = mp\_div\_2(&number, &number)) != MP_OKAY) \{
+ printf("Error dividing the number. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+ switch(mp_cmp_d(&number, 7)) \{
+ case MP_GT: printf("2*number/2 > 7"); break;
+ case MP_EQ: printf("2*number/2 = 7"); break;
+ case MP_LT: printf("2*number/2 < 7"); break;
+ \}
+
+ /* we're done with it. */
+ mp_clear(&number);
+
+ return EXIT_SUCCESS;
+\}
+\end{alltt} \end{small}
+
+If this program is successful it will print out the following text.
+
+\begin{alltt}
+2*number > 7
+2*number/2 < 7
+\end{alltt}
+
+Since $10 > 7$ and $5 < 7$. To multiply by a power of two the following function can be used.
+
+\index{mp\_mul\_2d}
+\begin{alltt}
+int mp_mul_2d(mp_int * a, int b, mp_int * c);
+\end{alltt}
+
+This will multiply $a$ by $2^b$ and store the result in ``c''. If the value of $b$ is less than or equal to
+zero the function will copy $a$ to ``c'' without performing any further actions.
+
+To divide by a power of two use the following.
+
+\index{mp\_div\_2d}
+\begin{alltt}
+int mp_div_2d (mp_int * a, int b, mp_int * c, mp_int * d);
+\end{alltt}
+Which will divide $a$ by $2^b$, store the quotient in ``c'' and the remainder in ``d'. If $b \le 0$ then the
+function simply copies $a$ over to ``c'' and zeroes $d$. The variable $d$ may be passed as a \textbf{NULL}
+value to signal that the remainder is not desired.
+
+\subsection{Polynomial Basis Operations}
+
+Strictly speaking the organization of the integers within the mp\_int structures is what is known as a
+``polynomial basis''. This simply means a field element is stored by divisions of a radix. For example, if
+$f(x) = \sum_{i=0}^{k} y_ix^k$ for any vector $\vec y$ then the array of digits in $\vec y$ are said to be
+the polynomial basis representation of $z$ if $f(\beta) = z$ for a given radix $\beta$.
+
+To multiply by the polynomial $g(x) = x$ all you have todo is shift the digits of the basis left one place. The
+following function provides this operation.
+
+\index{mp\_lshd}
+\begin{alltt}
+int mp_lshd (mp_int * a, int b);
+\end{alltt}
+
+This will multiply $a$ in place by $x^b$ which is equivalent to shifting the digits left $b$ places and inserting zeroes
+in the least significant digits. Similarly to divide by a power of $x$ the following function is provided.
+
+\index{mp\_rshd}
+\begin{alltt}
+void mp_rshd (mp_int * a, int b)
+\end{alltt}
+This will divide $a$ in place by $x^b$ and discard the remainder. This function cannot fail as it performs the operations
+in place and no new digits are required to complete it.
+
+\subsection{AND, OR and XOR Operations}
+
+While AND, OR and XOR operations are not typical ``bignum functions'' they can be useful in several instances. The
+three functions are prototyped as follows.
+
+\index{mp\_or} \index{mp\_and} \index{mp\_xor}
+\begin{alltt}
+int mp_or (mp_int * a, mp_int * b, mp_int * c);
+int mp_and (mp_int * a, mp_int * b, mp_int * c);
+int mp_xor (mp_int * a, mp_int * b, mp_int * c);
+\end{alltt}
+
+Which compute $c = a \odot b$ where $\odot$ is one of OR, AND or XOR.
+
+\section{Addition and Subtraction}
+
+To compute an addition or subtraction the following two functions can be used.
+
+\index{mp\_add} \index{mp\_sub}
+\begin{alltt}
+int mp_add (mp_int * a, mp_int * b, mp_int * c);
+int mp_sub (mp_int * a, mp_int * b, mp_int * c)
+\end{alltt}
+
+Which perform $c = a \odot b$ where $\odot$ is one of signed addition or subtraction. The operations are fully sign
+aware.
+
+\section{Sign Manipulation}
+\subsection{Negation}
+\label{sec:NEG}
+Simple integer negation can be performed with the following.
+
+\index{mp\_neg}
+\begin{alltt}
+int mp_neg (mp_int * a, mp_int * b);
+\end{alltt}
+
+Which assigns $-a$ to $b$.
+
+\subsection{Absolute}
+Simple integer absolutes can be performed with the following.
+
+\index{mp\_neg}
+\begin{alltt}
+int mp_abs (mp_int * a, mp_int * b);
+\end{alltt}
+
+Which assigns $\vert a \vert$ to $b$.
+
+\section{Integer Division and Remainder}
+To perform a complete and general integer division with remainder use the following function.
+
+\index{mp\_div}
+\begin{alltt}
+int mp_div (mp_int * a, mp_int * b, mp_int * c, mp_int * d);
+\end{alltt}
+
+This divides $a$ by $b$ and stores the quotient in $c$ and $d$. The signed quotient is computed such that
+$bc + d = a$. Note that either of $c$ or $d$ can be set to \textbf{NULL} if their value is not required. If
+$b$ is zero the function returns \textbf{MP\_VAL}.
+
+
+\chapter{Multiplication and Squaring}
+\section{Multiplication}
+A full signed integer multiplication can be performed with the following.
+\index{mp\_mul}
+\begin{alltt}
+int mp_mul (mp_int * a, mp_int * b, mp_int * c);
+\end{alltt}
+Which assigns the full signed product $ab$ to $c$. This function actually breaks into one of four cases which are
+specific multiplication routines optimized for given parameters. First there are the Toom-Cook multiplications which
+should only be used with very large inputs. This is followed by the Karatsuba multiplications which are for moderate
+sized inputs. Then followed by the Comba and baseline multipliers.
+
+Fortunately for the developer you don't really need to know this unless you really want to fine tune the system. mp\_mul()
+will determine on its own\footnote{Some tweaking may be required.} what routine to use automatically when it is called.
+
+\begin{alltt}
+int main(void)
+\{
+ mp_int number1, number2;
+ int result;
+
+ /* Initialize the numbers */
+ if ((result = mp_init_multi(&number1,
+ &number2, NULL)) != MP_OKAY) \{
+ printf("Error initializing the numbers. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ /* set the terms */
+ if ((result = mp_set_int(&number, 257)) != MP_OKAY) \{
+ printf("Error setting number1. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ if ((result = mp_set_int(&number2, 1023)) != MP_OKAY) \{
+ printf("Error setting number2. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ /* multiply them */
+ if ((result = mp_mul(&number1, &number2,
+ &number1)) != MP_OKAY) \{
+ printf("Error multiplying terms. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ /* display */
+ printf("number1 * number2 == \%lu", mp_get_int(&number1));
+
+ /* free terms and return */
+ mp_clear_multi(&number1, &number2, NULL);
+
+ return EXIT_SUCCESS;
+\}
+\end{alltt}
+
+If this program succeeds it shall output the following.
+
+\begin{alltt}
+number1 * number2 == 262911
+\end{alltt}
+
+\section{Squaring}
+Since squaring can be performed faster than multiplication it is performed it's own function instead of just using
+mp\_mul().
+
+\index{mp\_sqr}
+\begin{alltt}
+int mp_sqr (mp_int * a, mp_int * b);
+\end{alltt}
+
+Will square $a$ and store it in $b$. Like the case of multiplication there are four different squaring
+algorithms all which can be called from mp\_sqr(). It is ideal to use mp\_sqr over mp\_mul when squaring terms.
+
+\section{Tuning Polynomial Basis Routines}
+
+Both of the Toom-Cook and Karatsuba multiplication algorithms are faster than the traditional $O(n^2)$ approach that
+the Comba and baseline algorithms use. At $O(n^{1.464973})$ and $O(n^{1.584962})$ running times respectfully they require
+considerably less work. For example, a 10000-digit multiplication would take roughly 724,000 single precision
+multiplications with Toom-Cook or 100,000,000 single precision multiplications with the standard Comba (a factor
+of 138).
+
+So why not always use Karatsuba or Toom-Cook? The simple answer is that they have so much overhead that they're not
+actually faster than Comba until you hit distinct ``cutoff'' points. For Karatsuba with the default configuration,
+GCC 3.3.1 and an Athlon XP processor the cutoff point is roughly 110 digits (about 70 for the Intel P4). That is, at
+110 digits Karatsuba and Comba multiplications just about break even and for 110+ digits Karatsuba is faster.
+
+Toom-Cook has incredible overhead and is probably only useful for very large inputs. So far no known cutoff points
+exist and for the most part I just set the cutoff points very high to make sure they're not called.
+
+A demo program in the ``etc/'' directory of the project called ``tune.c'' can be used to find the cutoff points. This
+can be built with GCC as follows
+
+\begin{alltt}
+make XXX
+\end{alltt}
+Where ``XXX'' is one of the following entries from the table \ref{fig:tuning}.
+
+\begin{figure}[here]
+\begin{center}
+\begin{small}
+\begin{tabular}{|l|l|}
+\hline \textbf{Value of XXX} & \textbf{Meaning} \\
+\hline tune & Builds portable tuning application \\
+\hline tune86 & Builds x86 (pentium and up) program for COFF \\
+\hline tune86c & Builds x86 program for Cygwin \\
+\hline tune86l & Builds x86 program for Linux (ELF format) \\
+\hline
+\end{tabular}
+\end{small}
+\end{center}
+\caption{Build Names for Tuning Programs}
+\label{fig:tuning}
+\end{figure}
+
+When the program is running it will output a series of measurements for different cutoff points. It will first find
+good Karatsuba squaring and multiplication points. Then it proceeds to find Toom-Cook points. Note that the Toom-Cook
+tuning takes a very long time as the cutoff points are likely to be very high.
+
+\chapter{Modular Reduction}
+
+Modular reduction is process of taking the remainder of one quantity divided by another. Expressed
+as (\ref{eqn:mod}) the modular reduction is equivalent to the remainder of $b$ divided by $c$.
+
+\begin{equation}
+a \equiv b \mbox{ (mod }c\mbox{)}
+\label{eqn:mod}
+\end{equation}
+
+Of particular interest to cryptography are reductions where $b$ is limited to the range $0 \le b < c^2$ since particularly
+fast reduction algorithms can be written for the limited range.
+
+Note that one of the four optimized reduction algorithms are automatically chosen in the modular exponentiation
+algorithm mp\_exptmod when an appropriate modulus is detected.
+
+\section{Straight Division}
+In order to effect an arbitrary modular reduction the following algorithm is provided.
+
+\index{mp\_mod}
+\begin{alltt}
+int mp_mod(mp_int *a, mp_int *b, mp_int *c);
+\end{alltt}
+
+This reduces $a$ modulo $b$ and stores the result in $c$. The sign of $c$ shall agree with the sign
+of $b$. This algorithm accepts an input $a$ of any range and is not limited by $0 \le a < b^2$.
+
+\section{Barrett Reduction}
+
+Barrett reduction is a generic optimized reduction algorithm that requires pre--computation to achieve
+a decent speedup over straight division. First a $mu$ value must be precomputed with the following function.
+
+\index{mp\_reduce\_setup}
+\begin{alltt}
+int mp_reduce_setup(mp_int *a, mp_int *b);
+\end{alltt}
+
+Given a modulus in $b$ this produces the required $mu$ value in $a$. For any given modulus this only has to
+be computed once. Modular reduction can now be performed with the following.
+
+\index{mp\_reduce}
+\begin{alltt}
+int mp_reduce(mp_int *a, mp_int *b, mp_int *c);
+\end{alltt}
+
+This will reduce $a$ in place modulo $b$ with the precomputed $mu$ value in $c$. $a$ must be in the range
+$0 \le a < b^2$.
+
+\begin{alltt}
+int main(void)
+\{
+ mp_int a, b, c, mu;
+ int result;
+
+ /* initialize a,b to desired values, mp_init mu,
+ * c and set c to 1...we want to compute a^3 mod b
+ */
+
+ /* get mu value */
+ if ((result = mp_reduce_setup(&mu, b)) != MP_OKAY) \{
+ printf("Error getting mu. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ /* square a to get c = a^2 */
+ if ((result = mp_sqr(&a, &c)) != MP_OKAY) \{
+ printf("Error squaring. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ /* now reduce `c' modulo b */
+ if ((result = mp_reduce(&c, &b, &mu)) != MP_OKAY) \{
+ printf("Error reducing. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ /* multiply a to get c = a^3 */
+ if ((result = mp_mul(&a, &c, &c)) != MP_OKAY) \{
+ printf("Error reducing. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ /* now reduce `c' modulo b */
+ if ((result = mp_reduce(&c, &b, &mu)) != MP_OKAY) \{
+ printf("Error reducing. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ /* c now equals a^3 mod b */
+
+ return EXIT_SUCCESS;
+\}
+\end{alltt}
+
+This program will calculate $a^3 \mbox{ mod }b$ if all the functions succeed.
+
+\section{Montgomery Reduction}
+
+Montgomery is a specialized reduction algorithm for any odd moduli. Like Barrett reduction a pre--computation
+step is required. This is accomplished with the following.
+
+\index{mp\_montgomery\_setup}
+\begin{alltt}
+int mp_montgomery_setup(mp_int *a, mp_digit *mp);
+\end{alltt}
+
+For the given odd moduli $a$ the precomputation value is placed in $mp$. The reduction is computed with the
+following.
+
+\index{mp\_montgomery\_reduce}
+\begin{alltt}
+int mp_montgomery_reduce(mp_int *a, mp_int *m, mp_digit mp);
+\end{alltt}
+This reduces $a$ in place modulo $m$ with the pre--computed value $mp$. $a$ must be in the range
+$0 \le a < b^2$.
+
+Montgomery reduction is faster than Barrett reduction for moduli smaller than the ``comba'' limit. With the default
+setup for instance, the limit is $127$ digits ($3556$--bits). Note that this function is not limited to
+$127$ digits just that it falls back to a baseline algorithm after that point.
+
+An important observation is that this reduction does not return $a \mbox{ mod }m$ but $aR^{-1} \mbox{ mod }m$
+where $R = \beta^n$, $n$ is the n number of digits in $m$ and $\beta$ is radix used (default is $2^{28}$).
+
+To quickly calculate $R$ the following function was provided.
+
+\index{mp\_montgomery\_calc\_normalization}
+\begin{alltt}
+int mp_montgomery_calc_normalization(mp_int *a, mp_int *b);
+\end{alltt}
+Which calculates $a = R$ for the odd moduli $b$ without using multiplication or division.
+
+The normal modus operandi for Montgomery reductions is to normalize the integers before entering the system. For
+example, to calculate $a^3 \mbox { mod }b$ using Montgomery reduction the value of $a$ can be normalized by
+multiplying it by $R$. Consider the following code snippet.
+
+\begin{alltt}
+int main(void)
+\{
+ mp_int a, b, c, R;
+ mp_digit mp;
+ int result;
+
+ /* initialize a,b to desired values,
+ * mp_init R, c and set c to 1....
+ */
+
+ /* get normalization */
+ if ((result = mp_montgomery_calc_normalization(&R, b)) != MP_OKAY) \{
+ printf("Error getting norm. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ /* get mp value */
+ if ((result = mp_montgomery_setup(&c, &mp)) != MP_OKAY) \{
+ printf("Error setting up montgomery. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ /* normalize `a' so now a is equal to aR */
+ if ((result = mp_mulmod(&a, &R, &b, &a)) != MP_OKAY) \{
+ printf("Error computing aR. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ /* square a to get c = a^2R^2 */
+ if ((result = mp_sqr(&a, &c)) != MP_OKAY) \{
+ printf("Error squaring. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ /* now reduce `c' back down to c = a^2R^2 * R^-1 == a^2R */
+ if ((result = mp_montgomery_reduce(&c, &b, mp)) != MP_OKAY) \{
+ printf("Error reducing. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ /* multiply a to get c = a^3R^2 */
+ if ((result = mp_mul(&a, &c, &c)) != MP_OKAY) \{
+ printf("Error reducing. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ /* now reduce `c' back down to c = a^3R^2 * R^-1 == a^3R */
+ if ((result = mp_montgomery_reduce(&c, &b, mp)) != MP_OKAY) \{
+ printf("Error reducing. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ /* now reduce (again) `c' back down to c = a^3R * R^-1 == a^3 */
+ if ((result = mp_montgomery_reduce(&c, &b, mp)) != MP_OKAY) \{
+ printf("Error reducing. \%s",
+ mp_error_to_string(result));
+ return EXIT_FAILURE;
+ \}
+
+ /* c now equals a^3 mod b */
+
+ return EXIT_SUCCESS;
+\}
+\end{alltt}
+
+This particular example does not look too efficient but it demonstrates the point of the algorithm. By
+normalizing the inputs the reduced results are always of the form $aR$ for some variable $a$. This allows
+a single final reduction to correct for the normalization and the fast reduction used within the algorithm.
+
+For more details consider examining the file \textit{bn\_mp\_exptmod\_fast.c}.
+
+\section{Restricted Dimminished Radix}
+
+``Dimminished Radix'' reduction refers to reduction with respect to moduli that are ameniable to simple
+digit shifting and small multiplications. In this case the ``restricted'' variant refers to moduli of the
+form $\beta^k - p$ for some $k \ge 0$ and $0 < p < \beta$ where $\beta$ is the radix (default to $2^{28}$).
+
+As in the case of Montgomery reduction there is a pre--computation phase required for a given modulus.
+
+\index{mp\_dr\_setup}
+\begin{alltt}
+void mp_dr_setup(mp_int *a, mp_digit *d);
+\end{alltt}
+
+This computes the value required for the modulus $a$ and stores it in $d$. This function cannot fail
+and does not return any error codes. After the pre--computation a reduction can be performed with the
+following.
+
+\index{mp\_dr\_reduce}
+\begin{alltt}
+int mp_dr_reduce(mp_int *a, mp_int *b, mp_digit mp);
+\end{alltt}
+
+This reduces $a$ in place modulo $b$ with the pre--computed value $mp$. $b$ must be of a restricted
+dimminished radix form and $a$ must be in the range $0 \le a < b^2$. Dimminished radix reductions are
+much faster than both Barrett and Montgomery reductions as they have a much lower asymtotic running time.
+
+Since the moduli are restricted this algorithm is not particularly useful for something like Rabin, RSA or
+BBS cryptographic purposes. This reduction algorithm is useful for Diffie-Hellman and ECC where fixed
+primes are acceptable.
+
+Note that unlike Montgomery reduction there is no normalization process. The result of this function is
+equal to the correct residue.
+
+\section{Unrestricted Dimminshed Radix}
+
+Unrestricted reductions work much like the restricted counterparts except in this case the moduli is of the
+form $2^k - p$ for $0 < p < \beta$. In this sense the unrestricted reductions are more flexible as they
+can be applied to a wider range of numbers.
+
+\index{mp\_reduce\_2k\_setup}
+\begin{alltt}
+int mp_reduce_2k_setup(mp_int *a, mp_digit *d);
+\end{alltt}
+
+This will compute the required $d$ value for the given moduli $a$.
+
+\index{mp\_reduce\_2k}
+\begin{alltt}
+int mp_reduce_2k(mp_int *a, mp_int *n, mp_digit d);
+\end{alltt}
+
+This will reduce $a$ in place modulo $n$ with the pre--computed value $d$. From my experience this routine is
+slower than mp\_dr\_reduce but faster for most moduli sizes than the Montgomery reduction.
+
+\chapter{Exponentiation}
+\section{Single Digit Exponentiation}
+\index{mp\_expt\_d}
+\begin{alltt}
+int mp_expt_d (mp_int * a, mp_digit b, mp_int * c)
+\end{alltt}
+This computes $c = a^b$ using a simple binary left-to-right algorithm. It is faster than repeated multiplications by
+$a$ for all values of $b$ greater than three.
+
+\section{Modular Exponentiation}
+\index{mp\_exptmod}
+\begin{alltt}
+int mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y)
+\end{alltt}
+This computes $Y \equiv G^X \mbox{ (mod }P\mbox{)}$ using a variable width sliding window algorithm. This function
+will automatically detect the fastest modular reduction technique to use during the operation. For negative values of
+$X$ the operation is performed as $Y \equiv (G^{-1} \mbox{ mod }P)^{\vert X \vert} \mbox{ (mod }P\mbox{)}$ provided that
+$gcd(G, P) = 1$.
+
+This function is actually a shell around the two internal exponentiation functions. This routine will automatically
+detect when Barrett, Montgomery, Restricted and Unrestricted Dimminished Radix based exponentiation can be used. Generally
+moduli of the a ``restricted dimminished radix'' form lead to the fastest modular exponentiations. Followed by Montgomery
+and the other two algorithms.
+
+\section{Root Finding}
+\index{mp\_n\_root}
+\begin{alltt}
+int mp_n_root (mp_int * a, mp_digit b, mp_int * c)
+\end{alltt}
+This computes $c = a^{1/b}$ such that $c^b \le a$ and $(c+1)^b > a$. The implementation of this function is not
+ideal for values of $b$ greater than three. It will work but become very slow. So unless you are working with very small
+numbers (less than 1000 bits) I'd avoid $b > 3$ situations. Will return a positive root only for even roots and return
+a root with the sign of the input for odd roots. For example, performing $4^{1/2}$ will return $2$ whereas $(-8)^{1/3}$
+will return $-2$.
+
+This algorithm uses the ``Newton Approximation'' method and will converge on the correct root fairly quickly. Since
+the algorithm requires raising $a$ to the power of $b$ it is not ideal to attempt to find roots for large
+values of $b$. If particularly large roots are required then a factor method could be used instead. For example,
+$a^{1/16}$ is equivalent to $\left (a^{1/4} \right)^{1/4}$.
+
+\chapter{Prime Numbers}
+\section{Trial Division}
+\index{mp\_prime\_is\_divisible}
+\begin{alltt}
+int mp_prime_is_divisible (mp_int * a, int *result)
+\end{alltt}
+This will attempt to evenly divide $a$ by a list of primes\footnote{Default is the first 256 primes.} and store the
+outcome in ``result''. That is if $result = 0$ then $a$ is not divisible by the primes, otherwise it is. Note that
+if the function does not return \textbf{MP\_OKAY} the value in ``result'' should be considered undefined\footnote{Currently
+the default is to set it to zero first.}.
+
+\section{Fermat Test}
+\index{mp\_prime\_fermat}
+\begin{alltt}
+int mp_prime_fermat (mp_int * a, mp_int * b, int *result)
+\end{alltt}
+Performs a Fermat primality test to the base $b$. That is it computes $b^a \mbox{ mod }a$ and tests whether the value is
+equal to $b$ or not. If the values are equal then $a$ is probably prime and $result$ is set to one. Otherwise $result$
+is set to zero.
+
+\section{Miller-Rabin Test}
+\index{mp\_prime\_miller\_rabin}
+\begin{alltt}
+int mp_prime_miller_rabin (mp_int * a, mp_int * b, int *result)
+\end{alltt}
+Performs a Miller-Rabin test to the base $b$ of $a$. This test is much stronger than the Fermat test and is very hard to
+fool (besides with Carmichael numbers). If $a$ passes the test (therefore is probably prime) $result$ is set to one.
+Otherwise $result$ is set to zero.
+
+Note that is suggested that you use the Miller-Rabin test instead of the Fermat test since all of the failures of
+Miller-Rabin are a subset of the failures of the Fermat test.
+
+\subsection{Required Number of Tests}
+Generally to ensure a number is very likely to be prime you have to perform the Miller-Rabin with at least a half-dozen
+or so unique bases. However, it has been proven that the probability of failure goes down as the size of the input goes up.
+This is why a simple function has been provided to help out.
+
+\index{mp\_prime\_rabin\_miller\_trials}
+\begin{alltt}
+int mp_prime_rabin_miller_trials(int size)
+\end{alltt}
+This returns the number of trials required for a $2^{-96}$ (or lower) probability of failure for a given ``size'' expressed
+in bits. This comes in handy specially since larger numbers are slower to test. For example, a 512-bit number would
+require ten tests whereas a 1024-bit number would only require four tests.
+
+You should always still perform a trial division before a Miller-Rabin test though.
+
+\section{Primality Testing}
+\index{mp\_prime\_is\_prime}
+\begin{alltt}
+int mp_prime_is_prime (mp_int * a, int t, int *result)
+\end{alltt}
+This will perform a trial division followed by $t$ rounds of Miller-Rabin tests on $a$ and store the result in $result$.
+If $a$ passes all of the tests $result$ is set to one, otherwise it is set to zero. Note that $t$ is bounded by
+$1 \le t < PRIME\_SIZE$ where $PRIME\_SIZE$ is the number of primes in the prime number table (by default this is $256$).
+
+\section{Next Prime}
+\index{mp\_prime\_next\_prime}
+\begin{alltt}
+int mp_prime_next_prime(mp_int *a, int t, int bbs_style)
+\end{alltt}
+This finds the next prime after $a$ that passes mp\_prime\_is\_prime() with $t$ tests. Set $bbs\_style$ to one if you
+want only the next prime congruent to $3 \mbox{ mod } 4$, otherwise set it to zero to find any next prime.
+
+\section{Random Primes}
+\index{mp\_prime\_random}
+\begin{alltt}
+int mp_prime_random(mp_int *a, int t, int size, int bbs,
+ ltm_prime_callback cb, void *dat)
+\end{alltt}
+This will find a prime greater than $256^{size}$ which can be ``bbs\_style'' or not depending on $bbs$ and must pass
+$t$ rounds of tests. The ``ltm\_prime\_callback'' is a typedef for
+
+\begin{alltt}
+typedef int ltm_prime_callback(unsigned char *dst, int len, void *dat);
+\end{alltt}
+
+Which is a function that must read $len$ bytes (and return the amount stored) into $dst$. The $dat$ variable is simply
+copied from the original input. It can be used to pass RNG context data to the callback. The function
+mp\_prime\_random() is more suitable for generating primes which must be secret (as in the case of RSA) since there
+is no skew on the least significant bits.
+
+\textit{Note:} As of v0.30 of the LibTomMath library this function has been deprecated. It is still available
+but users are encouraged to use the new mp\_prime\_random\_ex() function instead.
+
+\subsection{Extended Generation}
+\index{mp\_prime\_random\_ex}
+\begin{alltt}
+int mp_prime_random_ex(mp_int *a, int t,
+ int size, int flags,
+ ltm_prime_callback cb, void *dat);
+\end{alltt}
+This will generate a prime in $a$ using $t$ tests of the primality testing algorithms. The variable $size$
+specifies the bit length of the prime desired. The variable $flags$ specifies one of several options available
+(see fig. \ref{fig:primeopts}) which can be OR'ed together. The callback parameters are used as in
+mp\_prime\_random().
+
+\begin{figure}[here]
+\begin{center}
+\begin{small}
+\begin{tabular}{|r|l|}
+\hline \textbf{Flag} & \textbf{Meaning} \\
+\hline LTM\_PRIME\_BBS & Make the prime congruent to $3$ modulo $4$ \\
+\hline LTM\_PRIME\_SAFE & Make a prime $p$ such that $(p - 1)/2$ is also prime. \\
+ & This option implies LTM\_PRIME\_BBS as well. \\
+\hline LTM\_PRIME\_2MSB\_OFF & Makes sure that the bit adjacent to the most significant bit \\
+ & Is forced to zero. \\
+\hline LTM\_PRIME\_2MSB\_ON & Makes sure that the bit adjacent to the most significant bit \\
+ & Is forced to one. \\
+\hline
+\end{tabular}
+\end{small}
+\end{center}
+\caption{Primality Generation Options}
+\label{fig:primeopts}
+\end{figure}
+
+\chapter{Input and Output}
+\section{ASCII Conversions}
+\subsection{To ASCII}
+\index{mp\_toradix}
+\begin{alltt}
+int mp_toradix (mp_int * a, char *str, int radix);
+\end{alltt}
+This still store $a$ in ``str'' as a base-``radix'' string of ASCII chars. This function appends a NUL character
+to terminate the string. Valid values of ``radix'' line in the range $[2, 64]$. To determine the size (exact) required
+by the conversion before storing any data use the following function.
+
+\index{mp\_radix\_size}
+\begin{alltt}
+int mp_radix_size (mp_int * a, int radix, int *size)
+\end{alltt}
+This stores in ``size'' the number of characters (including space for the NUL terminator) required. Upon error this
+function returns an error code and ``size'' will be zero.
+
+\subsection{From ASCII}
+\index{mp\_read\_radix}
+\begin{alltt}
+int mp_read_radix (mp_int * a, char *str, int radix);
+\end{alltt}
+This will read the base-``radix'' NUL terminated string from ``str'' into $a$. It will stop reading when it reads a
+character it does not recognize (which happens to include th NUL char... imagine that...). A single leading $-$ sign
+can be used to denote a negative number.
+
+\section{Binary Conversions}
+
+Converting an mp\_int to and from binary is another keen idea.
+
+\index{mp\_unsigned\_bin\_size}
+\begin{alltt}
+int mp_unsigned_bin_size(mp_int *a);
+\end{alltt}
+
+This will return the number of bytes (octets) required to store the unsigned copy of the integer $a$.
+
+\index{mp\_to\_unsigned\_bin}
+\begin{alltt}
+int mp_to_unsigned_bin(mp_int *a, unsigned char *b);
+\end{alltt}
+This will store $a$ into the buffer $b$ in big--endian format. Fortunately this is exactly what DER (or is it ASN?)
+requires. It does not store the sign of the integer.
+
+\index{mp\_read\_unsigned\_bin}
+\begin{alltt}
+int mp_read_unsigned_bin(mp_int *a, unsigned char *b, int c);
+\end{alltt}
+This will read in an unsigned big--endian array of bytes (octets) from $b$ of length $c$ into $a$. The resulting
+integer $a$ will always be positive.
+
+For those who acknowledge the existence of negative numbers (heretic!) there are ``signed'' versions of the
+previous functions.
+
+\begin{alltt}
+int mp_signed_bin_size(mp_int *a);
+int mp_read_signed_bin(mp_int *a, unsigned char *b, int c);
+int mp_to_signed_bin(mp_int *a, unsigned char *b);
+\end{alltt}
+They operate essentially the same as the unsigned copies except they prefix the data with zero or non--zero
+byte depending on the sign. If the sign is zpos (e.g. not negative) the prefix is zero, otherwise the prefix
+is non--zero.
+
+\chapter{Algebraic Functions}
+\section{Extended Euclidean Algorithm}
+\index{mp\_exteuclid}
+\begin{alltt}
+int mp_exteuclid(mp_int *a, mp_int *b,
+ mp_int *U1, mp_int *U2, mp_int *U3);
+\end{alltt}
+
+This finds the triple U1/U2/U3 using the Extended Euclidean algorithm such that the following equation holds.
+
+\begin{equation}
+a \cdot U1 + b \cdot U2 = U3
+\end{equation}
+
+Any of the U1/U2/U3 paramters can be set to \textbf{NULL} if they are not desired.
+
+\section{Greatest Common Divisor}
+\index{mp\_gcd}
+\begin{alltt}
+int mp_gcd (mp_int * a, mp_int * b, mp_int * c)
+\end{alltt}
+This will compute the greatest common divisor of $a$ and $b$ and store it in $c$.
+
+\section{Least Common Multiple}
+\index{mp\_lcm}
+\begin{alltt}
+int mp_lcm (mp_int * a, mp_int * b, mp_int * c)
+\end{alltt}
+This will compute the least common multiple of $a$ and $b$ and store it in $c$.
+
+\section{Jacobi Symbol}
+\index{mp\_jacobi}
+\begin{alltt}
+int mp_jacobi (mp_int * a, mp_int * p, int *c)
+\end{alltt}
+This will compute the Jacobi symbol for $a$ with respect to $p$. If $p$ is prime this essentially computes the Legendre
+symbol. The result is stored in $c$ and can take on one of three values $\lbrace -1, 0, 1 \rbrace$. If $p$ is prime
+then the result will be $-1$ when $a$ is not a quadratic residue modulo $p$. The result will be $0$ if $a$ divides $p$
+and the result will be $1$ if $a$ is a quadratic residue modulo $p$.
+
+\section{Modular Inverse}
+\index{mp\_invmod}
+\begin{alltt}
+int mp_invmod (mp_int * a, mp_int * b, mp_int * c)
+\end{alltt}
+Computes the multiplicative inverse of $a$ modulo $b$ and stores the result in $c$ such that $ac \equiv 1 \mbox{ (mod }b\mbox{)}$.
+
+\section{Single Digit Functions}
+
+For those using small numbers (\textit{snicker snicker}) there are several ``helper'' functions
+
+\index{mp\_add\_d} \index{mp\_sub\_d} \index{mp\_mul\_d} \index{mp\_div\_d} \index{mp\_mod\_d}
+\begin{alltt}
+int mp_add_d(mp_int *a, mp_digit b, mp_int *c);
+int mp_sub_d(mp_int *a, mp_digit b, mp_int *c);
+int mp_mul_d(mp_int *a, mp_digit b, mp_int *c);
+int mp_div_d(mp_int *a, mp_digit b, mp_int *c, mp_digit *d);
+int mp_mod_d(mp_int *a, mp_digit b, mp_digit *c);
+\end{alltt}
+
+These work like the full mp\_int capable variants except the second parameter $b$ is a mp\_digit. These
+functions fairly handy if you have to work with relatively small numbers since you will not have to allocate
+an entire mp\_int to store a number like $1$ or $2$.
+
+\input{bn.ind}
+
+\end{document}