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diff --git a/libtommath/bn.tex b/libtommath/bn.tex new file mode 100644 index 0000000..38ece04 --- /dev/null +++ b/libtommath/bn.tex @@ -0,0 +1,1835 @@ +\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 \\ v0.40} +\author{Tom St Denis \\ tomstdenis@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{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. + +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 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} +\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 $\left (a^{1/4} \right)^{1/4}$ 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 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} |