\documentclass[b5paper]{book} \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{\url}[1]{\mbox{$<${#1}$>$}} \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{Multiple-Precision Integer Arithmetic, \\ A Case Study Involving the LibTomMath Project \\ - DRAFT - } \author{\mbox{ %\begin{small} \begin{tabular}{c} Tom St Denis \\ Algonquin College \\ \\ Mads Rasmussen \\ Open Communications Security \\ \\ Gregory Rose \\ Qualcomm \\ \end{tabular} %\end{small} } } \maketitle This text in its entirety is copyrighted \copyright{}2003 by Tom St Denis. It may not be redistributed electronically or otherwise without the sole permission of the author. The text is freely re distributable as long as it is packaged along with the LibTomMath project in a non-commercial project. Contact the author for other redistribution rights. This text corresponds to the v0.17 release of the LibTomMath project. \begin{alltt} Tom St Denis 111 Banning Rd Ottawa, Ontario K2L 1C3 Canada Phone: 1-613-836-3160 Email: tomstdenis@iahu.ca \end{alltt} This text is formatted to the international B5 paper size of 176mm wide by 250mm tall using the \LaTeX{} {\em book} macro package and the Perl {\em booker} package. \tableofcontents \listoffigures \chapter*{Preface} Blah. \mainmatter \pagestyle{headings} \chapter{Introduction} \section{Multiple Precision Arithmetic} \subsection{The Need for Multiple Precision Arithmetic} The most prevalent use for multiple precision arithmetic (\textit{often referred to as bignum math}) is within public key cryptography. Algorithms such as RSA, Diffie-Hellman and Elliptic Curve Cryptography require large integers in order to resist known cryptanalytic attacks. Typical modern programming languages such as C and Java only provide small single-precision data types which are incapable of precisely representing integers which are often hundreds of bits long. For example, consider multiplying $1,234,567$ by $9,876,543$ in C with an ``unsigned long'' data type. With an x86 machine the result is $4,136,875,833$ while the true result is $12,193,254,061,881$. The original inputs were approximately $21$ and $24$ bits respectively. If the C language cannot multiply two relatively small values together precisely how does anyone expect it to multiply two values which are considerably larger? Most advancements in fast multiple precision arithmetic stems from the desire for faster cryptographic primitives. However, cryptography is not the only field of study that can benefit fast large integer routines. Another auxiliary use for multiple precision integers is high precision floating point data types. The basic IEEE standard floating point type is made up of an integer mantissa $q$ and an exponent $e$. Numbers are given in the form $n = q \cdot b^e$ where $b = 2$ is convention. Since IEEE is meant to be implemented in hardware the precision of the mantissa is often fairly small (\textit{roughly 23 bits}). Since the mantissa is merely an integer a large multiple precision integer could be used. In effect very high precision floating point arithmetic could be performed. This would be useful where scientific applications must minimize the total output error over long simulations. \subsection{Multiple Precision Arithmetic} \index{multiple precision} Multiple precision arithmetic attempts to the solve the shortcomings of single precision data types such as those from the C and Java programming languages. In essence multiple precision arithmetic is a set of operations that can be performed on members of an algebraic group whose precision is not fixed. The algorithms when implemented to be multiple precision can allow a developer to work with any practical precision required. Typically the arithmetic is performed over the ring of integers denoted by a $\Z$ and referred to casually as ``bignum'' routines. However, it is possible to have rings of polynomials as well typically denoted by $\Z/p\Z \left [ X \right ]$ which could have variable precision (\textit{or degree}). This text will discuss implementation of the former, however, implementing polynomial basis routines should be relatively easy after reading this text. \subsection{Benefits of Multiple Precision Arithmetic} \index{precision} \index{accuracy} Precision is defined loosely as the proximity to the real value a given representation is. Accuracy is defined as the reproducibility of the result. For example, the calculation $1/3 = 0.25$ is imprecise but can be accurate provided it is reproducible. The benefit of multiple precision representations over single precision representations is that often no precision is lost while representing the result of an operation which requires excess precision. For example, the multiplication of two $n$-bit integers requires at least $2n$ bits to represent the result. A multiple precision system would augment the precision of the destination to accomodate the result while a single precision system would truncate excess bits to maintain a fixed level of precision. Multiple precision representations allow for the precision to be very high (\textit{if not exacting}) but at a cost of modest computer resources. The only reasonable case where a multiple precision system will lose precision is when emulating a floating point data type. However, with multiple precision integer arithmetic no precision is lost. \subsection{Basis of Operations} At the heart of all multiple precision integer operations are the ``long-hand'' algorithms we all learnt as children in grade school. For example, to multiply $1,234$ by $981$ the student is not taught to memorize the times table for $1,234$ instead they are taught how to long-multiply. That is to multiply each column using simple single digit multiplications and add the resulting products by column. The representation that most are familiar with is known as decimal or formally as radix-10. A radix-$n$ representation simply means there are $n$ possible values per digit. For example, binary would be a radix-2 representation. In essence computer based multiple precision arithmetic is very much the same. The most notable difference is the usage of a binary friendly radix. That is to use a radix of the form $2^k$ where $k$ is typically the size of a machine register. Also occasionally more optimal algorithms are used to perform certain operations such as multiplication and squaring instead of traditional long-hand algorithms. \section{Purpose of This Text} The purpose of this text is to instruct the reader regarding how to implement multiple precision algorithms. That is to not only explain the core theoretical algorithms but also the various ``house keeping'' tasks that are neglected by authors of other texts on the subject. Texts such as Knuths' ``The Art of Computer Programming, vol 2.'' and the Handbook of Applied Cryptography (\textit{HAC}) give considerably detailed explanations of the theoretical aspects of the algorithms and very little regarding the practical aspects. That is how an algorithm is explained and how it is actually implemented are two very different realities. For example, algorithm 14.7 on page 594 of HAC lists a relatively simple algorithm for performing multiple precision integer addition. However, what the description lacks is any discussion concerning the fact that the two integer inputs may be of differing magnitudes. Similarly the division routine (\textit{Algorithm 14.20, pp. 598}) does not discuss how to handle sign or handle the dividends decreasing magnitude in the main loop (\textit{Step \#3}). As well as the numerous practical oversights both of the texts do not discuss several key optimal algorithms required such as ``Comba'' and Karatsuba multipliers and fast modular inversion. These optimal algorithms are considerably vital to achieve any form of useful performance in non-trivial applications. To solve this problem the focus of this text is on the practical aspects of implementing the algorithms that constitute a multiple precision integer package with light cursory discussions on the theoretical aspects. As a case study the ``LibTomMath''\footnote{Available freely at http://math.libtomcrypt.org} package is used to demonstrate algorithms with implementations that have been field tested and work very well. \section{Discussion and Notation} \subsection{Notation} A multiple precision integer of $n$-digits shall be denoted as $x = (x_n ... x_1 x_0)_{ \beta }$ to be the multiple precision notation for the integer $x \equiv \sum_{i=0}^{n} x_i\beta^i$. The elements of the array $x$ are said to be the radix $\beta$ digits of the integer. For example, $x = (15,0,7)_{\beta}$ would represent the integer $15\cdot\beta^2 + 0\cdot\beta^1 + 7\cdot\beta^0$. A ``mp\_int'' shall refer to a composite structure which contains the digits of the integer as well as auxilary data required to manipulate the data. These additional members are discussed in ~BASICOP~. For the purposes of this text a ``multiple precision integer'' and a ``mp\_int'' are synonymous. \index{single-precision} \index{double-precision} \index{mp\_digit} \index{mp\_word} For the purposes of this text a single-precision variable must be able to represent integers in the range $0 \le x < 2 \beta$ while a double-precision variable must be able to represent integers in the range $0 \le x < 2 \beta^2$. Within the source code that will be presented the data type \textbf{mp\_digit} will represent a single-precision type while \textbf{mp\_word} will represent a double-precision type. In several algorithms (\textit{notably the Comba routines}) temporary results will be stored in a double-precision arrays. For the purposes of this text $x_j$ will refer to the $j$'th digit of a single-precision array and $\hat x_j$ will refer to the $j$'th digit of a double-precision array. \subsection{Work Effort} \index{big-O} To measure the efficiency of various algorithms a modified big-O notation is used. In this system all single precision operations are considered to have the same cost\footnote{Except where explicitly noted.}. That is a single precision addition, multiplication and division are assumed to take the same time to complete. While this is generally not true in practice it will simplify the discussions considerably. Some algorithms have slight advantages over others which is why some constants will not be removed in the notation. For example, a normal multiplication requires $O(n^2)$ work while a squaring requires $O({{n^2 + n}\over 2})$ work. In standard big-O notation these would be said to be equivalent. However, in the context of the this text the magnitude of the inputs will not approach an infinite size. This means the conventional limit notation wisdom does not apply to the cancellation of constants. Throughout the discussions various ``work levels'' will be discussed. These levels are the $O(1)$, $O(n)$, $O(n^2)$, ..., $O(n^k)$ work efforts. For example, operations at the $O(n^k)$ ``level'' are said to be executed more frequently than operations at the $O(n^m)$ ``level'' when $k > m$. Obviously most optimizations will pay off the most at the higher levels since they represent the bulk of the effort required. \section{Exercises} Within the more advanced chapters a section will be set aside to give the reader some challenging exercises. These exercises are not designed to be prize winning problems yet instead to be thought provoking. Wherever possible the problems are foreward minded stating problems that will be answered in subsequent chapters. The reader is encouraged to finish the exercises as they appear to get a better understanding of the subject material. Similar to the exercises of \cite{TAOCPV2} as explained on pp.\textit{ix} these exercises are given a scoring system. However, unlike \cite{TAOCPV2} the problems do not get nearly as hard as often. The scoring of these exercises ranges from one (\textit{the easiest}) to five (\textit{the hardest}). The following table sumarizes the scoring. \vspace{5mm} \begin{tabular}{cl} $\left [ 1 \right ]$ & An easy problem that should only take the reader a manner of \\ & minutes to solve. Usually does not involve much computer time. \\ & \\ $\left [ 2 \right ]$ & An easy problem that involves a marginal amount of computer \\ & time usage. Usually requires a program to be written to \\ & solve the problem. \\ & \\ $\left [ 3 \right ]$ & A moderately hard problem that requires a non-trivial amount \\ & of work. Usually involves trivial research and development of \\ & new theory from the perspective of a student. \\ & \\ $\left [ 4 \right ]$ & A moderately hard problem that involves a non-trivial amount \\ & of work and research. The solution to which will demonstrate \\ & a higher mastery of the subject matter. \\ & \\ $\left [ 5 \right ]$ & A hard problem that involves concepts that are non-trivial. \\ & Solutions to these problems will demonstrate a complete mastery \\ & of the given subject. \\ & \\ \end{tabular} Essentially problems at the first level are meant to be simple questions that the reader can answer quickly without programming a solution or devising new theory. These problems are quick tests to see if the material is understood. Problems at the second level are also designed to be easy but will require a program or algorithm to be implemented to arrive at the answer. Problems at the third level are meant to be a bit more difficult. Often the answer is fairly obvious but arriving at an exacting solution requires some thought and skill. These problems will almost always involve devising a new algorithm or implementing a variation of another algorithm. Problems at the fourth level are meant to be even more difficult as well as involve some research. The reader will most likely not know the answer right away nor will this text provide the exact details of the answer (\textit{or at least not until a subsequent chapter}). Problems at the fifth level are meant to be the hardest problems relative to all the other problems in the chapter. People who can correctly answer fifth level problems have a mastery of the subject matter at hand. Often problems will be tied together. The purpose of this is to start a chain of thought that will be discussed in future chapters. The reader is encouraged to answer the follow-up problems and try to draw the relevence of problems. \chapter{Introduction to LibTomMath} \section{What is the LibTomMath?} LibTomMath is a free and open source multiple precision number theoretic library written in portable ISO C source code. By portable it is meant that the library does not contain any code that is platform dependent or otherwise problematic to use on any given platform. The library has been successfully tested under numerous operating systems including Solaris, MacOS, Windows, Linux, PalmOS and on standalone hardware such as the Gameboy Advance. The library is designed to contain enough functionality to be able to develop number theoretic applications such as public key cryptosystems. \section{Goals of the LibTomMath} Even though the library is written entirely in portable ISO C considerable care has been taken to optimize the algorithm implementations within the library. Specifically the code has been written to work well with the GNU C Compiler (\textit{GCC}) on both x86 and ARMv4 processors. Wherever possible optimal algorithms (\textit{such as Karatsuba multiplication, sliding window exponentiation and Montgomery reduction.}) have been provided to make the library as efficient as possible. Even with the optimal and sometimes specialized algorithms that have been included the API has been kept as simple as possible. Often generic place holder routines will make use of specialized algorithms automatically without the developers attention. One such example is the generic multiplication algorithm \textbf{mp\_mul()} which will automatically use Karatsuba multiplication if the inputs are of a specific size. Making LibTomMath as efficient as possible is not the only goal of the LibTomMath project. Ideally the library should be source compatible with another popular library which makes it more attractive for developers to use. In this case the MPI library was used as a API template for all the basic functions. The project is also meant to act as a learning tool for students. The logic being that no easy to follow ``bignum'' library exists which can be used to teach computer science students how to perform fast and reliable multiple precision arithmetic. To this end the source code has been given quite a few comments and algorithm discussion points. Often where applicable routines have more comments than lines of code. \section{Choice of LibTomMath} LibTomMath was chosen as the case study of this text not only because the author of both projects is one and the same but for more worthy reasons. Other libraries such as GMP, MPI, LIP and OpenSSL have multiple precision integer arithmetic routines but would not be ideal for this text for numerous reasons as will be explained in the following sub-sections. \subsection{Code Base} The LibTomMath code base is all portable ISO C source code. This means that there are no platform dependent conditional segments of code littered throughout the source. This clean and uncluttered approach to the library means that a developer can more readily ascertain the true intent of a given section of source code without trying to keep track of what conditional code will be used. The code base of LibTomMath is also exceptionally well organized. Each function is in its own separate source code file which allows the reader to find a given function very fast. When compiled with GCC for the x86 processor the entire library is a mere 87,760 bytes (\textit{$116,182$ bytes for ARMv4 processors}). This includes every single function LibTomMath provides from basic arithmetic to various number theoretic functions such as modular exponentiation, various reduction algorithms and Jacobi symbol computation. By comparison MPI which has fewer number theoretic functions than LibTomMath compiled with the same conditions is 45,429 bytes (\textit{$54,536$ for ARMv4}). GMP which has rather large collection of functions with the default configuration on an x86 Athlon is 2,950,688 bytes. Note that while LibTomMath has fewer functions than GMP it has been been used as the sole basis for several public key cryptosystems without having to seek additional outside functions to supplement the library. \subsection{API Simplicity} LibTomMath is designed after the MPI library and shares the API design. Quite often programs that use MPI will build with LibTomMath without change. The function names are relatively straight forward as to what they perform. Almost all of the functions except for a few minor exceptions which as will be discussed are for good reasons share the same parameter passing convention. The learning curve is fairly shallow with the API provided which is an extremely valuable benefit for the student and developer alike. The LIP library is an example of a library with an API that is awkward to work with. LIP uses function names that are often ``compressed'' to illegible short hand. LibTomMath does not share this fault. \subsection{Optimizations} While LibTomMath is certainly not the fastest library (\textit{GMP often beats LibTomMath by a factor of two}) it does feature a set of optimal algorithms for tasks ranging from modular reduction to squaring. GMP and LIP also feature such optimizations while MPI only uses baseline algorithms with no optimizations. LibTomMath is almost always a magnitude faster than the MPI library at computationally expensive tasks such as modular exponentiation. In the grand scheme of ``bignum'' libraries LibTomMath is faster than the average library and usually slower than the best libraries such as GMP and OpenSSL by a small factor. \subsection{Portability and Stability} LibTomMath will build ``out of the box'' on any platform equipped with a modern version of the GNU C Compiler (\textit{GCC}). This means that without changes the library will build without configuration or setting up any variables. LIP and MPI will build ``out of the box'' as well but have numerous known bugs. Most notably the author of MPI is not working on his library anymore. GMP requires a configuration script to run and will not build out of the box. GMP and LibTomMath are still in active development and are very stable across a variety of platforms. \subsection{Choice} LibTomMath is a relatively compact, well documented, highly optimized and portable library which seems only natural for the case study of this text. Various source files from the LibTomMath project will be included within the text. However, the reader is encouraged to download their own copy of the library to actually be able to work with the library. \chapter{Getting Started} MARK,BASICOP \section{Library Basics} To get the ``ball rolling'' so to speak a primitive data type and a series of primitive algorithms must be established. First a data type that will hold the information required to maintain a multiple precision integer must be designed. With this basic data type of a series of low level algorithms for initializing, clearing, growing and clamping integers can be developed to form the basis of the entire package of algorithms. \section{The mp\_int structure} First the data type for storing multiple precision integers must be designed. This data type must be able to hold information to maintain an array of digits, how many are actually used in the representation and the sign. The ISO C standard does not provide for any such data type but it does provide for making composite data types known as structures. The following is the structure definition used within LibTomMath. \index{mp\_int} \begin{verbatim} typedef struct { int used, alloc, sign; mp_digit *dp; } mp_int; \end{verbatim} The \textbf{used} parameter denotes how many digits of the array \textbf{dp} are actually being used. The array \textbf{dp} holds the digits that represent the integer desired. The \textbf{alloc} parameter denotes how many digits are available in the array to use by functions before it has to increase in size. When the \textbf{used} count of a result would exceed the \textbf{alloc} count all LibTomMath routines will automatically increase the size of the array to accommodate the precision of the result. The \textbf{sign} parameter denotes the sign as either zero/positive (\textbf{MP\_ZPOS}) or negative (\textbf{MP\_NEG}). \section{Argument Passing} A convention of arugment passing must be adopted early on in the development of any library. Making the function prototypes consistent will help eliminate many headaches in the future as the library grows to significant complexity. In LibTomMath the multiple precision integer functions accept parameters from left to right as pointers to mp\_int structures. That means that the source operands are placed on the left and the destination on the right. Consider the following examples. \begin{verbatim} mp_mul(&a, &b, &c); /* c = a * b */ mp_add(&a, &b, &a); /* a = a + b */ mp_sqr(&a, &b); /* b = a * a */ \end{verbatim} The left to right order is a fairly natural way to implement the functions since it lets the developer read aloud the functions and make sense of them. For example, the first function would read ``multiply a and b and store in c''. Certain libraries (\textit{LIP by Lenstra for instance}) accept parameters the other way around. That is the destination on the left and arguments on the right. In truth it is entirely a matter of preference. Another very useful design consideration is whether to allow argument sources to also be a destination. For example, the second example (\textit{mp\_add}) adds $a$ to $b$ and stores in $a$. This is an important feature to implement since it allows the higher up functions to cut down on the number of variables. However, to implement this feature specific care has to be given to ensure the destination is not written before the source is fully read. \section{Return Values} A well implemented library, no matter what its purpose, should trap as many runtime errors as possible and return them to the caller. By catching runtime errors a library can be guaranteed to prevent undefined behaviour within reason. In a multiple precision library the only errors that are bound to occur are related to inappropriate inputs (\textit{division by zero for instance}) or memory allocation errors. In LibTomMath any function that can cause a runtime error will return an error as an \textbf{int} data type with one of the following values. \index{MP\_OKAY} \index{MP\_VAL} \index{MP\_MEM} \begin{center} \begin{tabular}{|l|l|} \hline \textbf{Value} & \textbf{Meaning} \\ \hline \textbf{MP\_OKAY} & The function was successful \\ \hline \textbf{MP\_VAL} & One of the input value(s) was invalid \\ \hline \textbf{MP\_MEM} & The function ran out of heap memory \\ \hline \end{tabular} \end{center} When an error is detected within a function it should free any memory they allocated and return as soon as possible. The goal is to leave the system in the same state the system was when the function was called. Error checking with this style of API is fairly simple. \begin{verbatim} int err; if ((err = mp_add(&a, &b, &c)) != MP_OKAY) { printf("Error: %d\n", err); exit(EXIT_FAILURE); } \end{verbatim} The GMP library uses C style \textit{signals} to flag errors which is of questionable use. Not all errors are fatal and it is not ideal to force developers to have signal handlers for such cases. \section{Initialization and Clearing} The logical starting point when actually writing multiple precision integer functions is the initialization and clearing of the integers. These two functions will be used by far the most throughout the algorithms whenever temporary integers are required. Given the basic mp\_int structure an initialization routine must first allocate memory to hold the digits of the integer. Often it is optimal to allocate a sufficiently large pre-set number of digits even considering the initial integer will represent zero. If only a single digit were allocated quite a few re-allocations would occur for the majority of inputs. There exists a tradeoff between how many default digits to allocate and how many re-allocations are tolerable. If the memory for the digits has been successfully allocated then the rest of the members of the structure must be initialized. Since the initial state is to represent a zero integer the digits allocated must all be zeroed. The \textbf{used} count set to zero and \textbf{sign} set to \textbf{MP\_ZPOS}. \subsection{Initializing an mp\_int} To initialize an mp\_int the mp\_init algorithm shall be used. The purpose of this algorithm is to allocate the memory required and initialize the integer to a default representation of zero. \begin{figure}[here] \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_init}. \\ \textbf{Input}. An mp\_int $a$ \\ \textbf{Output}. Allocate memory for the digits and set to a zero state. \\ \hline \\ 1. Allocate memory for \textbf{MP\_PREC} digits. \\ 2. If the allocation failed then return(\textit{MP\_MEM}) \\ 3. for $n$ from $0$ to $MP\_PREC - 1$ do \\ \hspace{3mm}3.1 $a_n \leftarrow 0$\\ 4. $a.sign \leftarrow MP\_ZPOS$\\ 5. $a.used \leftarrow 0$\\ 6. $a.alloc \leftarrow MP\_PREC$\\ 7. Return(\textit{MP\_OKAY})\\ \hline \end{tabular} \end{center} \caption{Algorithm mp\_init} \end{figure} \textbf{Algorithm mp\_init.} The \textbf{MP\_PREC} variable is a simple constant used to dictate minimal precision of allocated integers. It is ideally at least equal to $32$ but can be any reasonable power of two. Step one and two allocate the memory and account for it. If the allocation fails the algorithm returns immediately to signal the failure. Step three will ensure that all the digits are in the default state of zero. Finally steps four through six set the default settings of the \textbf{sign}, \textbf{used} and \textbf{alloc} members of the mp\_int structure. EXAM,bn_mp_init.c The \textbf{OPT\_CAST} type cast on line @22,OPT_CAST@ is designed to allow C++ compilers to build the code out of the box. Microsoft C V5.00 is known to cause problems without the cast. Also note that if the memory allocation fails the other members of the mp\_int will be in an undefined state. The code from line @29,a->used@ to line @31,a->sign@ sets the default state for a mp\_int which is zero, positive and no used digits. \subsection{Clearing an mp\_int} When an mp\_int is no longer required the memory allocated for it can be cleared from the heap with the mp\_clear algorithm. \begin{figure}[here] \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_clear}. \\ \textbf{Input}. An mp\_int $a$ \\ \textbf{Output}. The memory for $a$ is cleared. \\ \hline \\ 1. If $a$ has been previously freed then return(\textit{MP\_OKAY}). \\ 2. Free the digits of $a$ and mark $a$ as freed. \\ 3. $a.used \leftarrow 0$ \\ 4. $a.alloc \leftarrow 0$ \\ 5. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} \caption{Algorithm mp\_clear} \end{figure} \textbf{Algorithm mp\_clear.} In steps one and two the memory for the digits are only free'd if they had not been previously released before. This is more of concern for the implementation since it is used to prevent ``double-free'' errors. It also helps catch code errors where mp\_ints are used after being cleared. Simiarly steps three and four set the \textbf{used} and \textbf{alloc} to known values which would be easy to spot during debugging. For example, if an mp\_int is expected to be non-zero and its \textbf{used} member observed to be zero (\textit{due to being cleared}) then an obvious bug in the code has been spotted. EXAM,bn_mp_clear.c The \textbf{if} statement on line @21,a->dp != NULL@ prevents the heap from being corrupted if a user double-frees an mp\_int. For example, a trivial case of this bug would be as follows. \begin{verbatim} mp_int a; mp_init(&a); mp_clear(&a); mp_clear(&a); \end{verbatim} Without that check the code would try to free the memory allocated for the digits twice which will cause most standard C libraries to cause a fault. Also by setting the pointer to \textbf{NULL} it helps debug code that may inadvertently free the mp\_int before it is truly not needed. The allocated digits are set to zero before being freed on line @24,memset@. This is ideal for cryptographic situations where the mp\_int is a secret parameter. The following snippet is an example of using both the init and clear functions. \begin{small} \begin{verbatim} #include #include #include int main(void) { mp_int num; int err; /* init the bignum */ if ((err = mp_init(&num)) != MP_OKAY) { printf("Error: %d\n", err); return EXIT_FAILURE; } /* do work with it ... */ /* clear up */ mp_clear(&num); return EXIT_SUCCESS; } \end{verbatim} \end{small} \section{Other Initialization Routines} It is often helpful to have specialized initialization algorithms to simplify the design of other algorithms. For example, an initialization followed by a copy is a common operation when temporary copies of integers are required. It is quite beneficial to have a series of simple helper functions available. \subsection{Initializing Variable Sized mp\_int Structures} Occasionally the number of digits required will be known in advance of an initialization. In these cases the mp\_init\_size algorithm can be of use. The purpose of this algorithm is similar to mp\_init except that it will allocate \textit{at least} a specified number of digits. This is ideal to prevent re-allocations when the input size is known. \newpage\begin{figure}[here] \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_init\_size}. \\ \textbf{Input}. An mp\_int $a$ and the requested number of digits $b$\\ \textbf{Output}. $a$ is initialized to hold at least $b$ digits. \\ \hline \\ 1. $u \leftarrow b\mbox{ (mod }MP\_PREC\mbox{)}$ \\ 2. $v \leftarrow b + 2 \cdot MP\_PREC - u$ \\ 3. Allocate $v$ digits. \\ 4. If the allocation failed then return(\textit{MP\_MEM}). \\ 5. for $n$ from $0$ to $v - 1$ do \\ \hspace{3mm}5.1 $a_n \leftarrow 0$ \\ 6. $a.sign \leftarrow MP\_ZPOS$\\ 7. $a.used \leftarrow 0$\\ 8. $a.alloc \leftarrow v$\\ 9. Return(\textit{MP\_OKAY})\\ \hline \end{tabular} \end{center} \caption{Algorithm mp\_init\_size} \end{figure} \textbf{Algorithm mp\_init\_size.} The value of $v$ is calculated to be at least the requested amount of digits $b$ plus additional padding. The padding is calculated to be at least \textbf{MP\_PREC} digits plus enough digits to make the digit count a multiple of \textbf{MP\_PREC}. This padding is used to prevent trivial allocations from becomming a bottleneck in the rest of the algorithms that depend on this. EXAM,bn_mp_init_size.c Line @23,MP_PREC@ will ensure that the number of digits actually allocated is padded up to the next multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC}. This ensures that the number of allocated digit is always greater than the amount requested. As a result it prevents many trivial memory allocations. The value of \textbf{MP\_PREC} is defined in ``tommath.h'' and must be a power of two. \subsection{Creating a Clone} Another common sequence of operations is to make a local temporary copy of an argument. To initialize then copy a mp\_int will be known as creating a clone. This is useful within functions that need to modify an integer argument but do not wish to actually modify the original copy. The mp\_init\_copy algorithm will perform this very task. \begin{figure}[here] \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_init\_copy}. \\ \textbf{Input}. An mp\_int $a$ and $b$\\ \textbf{Output}. $a$ is initialized to be a copy of $b$. \\ \hline \\ 1. Init $a$. (\textit{hint: use mp\_init}) \\ 2. If the init of $a$ was unsuccessful return(\textit{MP\_MEM}) \\ 3. Copy $b$ to $a$. (\textit{hint: use mp\_copy}) \\ 4. Return the status of the copy operation. \\ \hline \end{tabular} \end{center} \caption{Algorithm mp\_init\_copy} \end{figure} \textbf{Algorithm mp\_init\_copy.} This algorithm will initialize a mp\_int variable and copy another previously initialized mp\_int variable into it. The algorithm will detect when the initialization fails and returns the error to the calling algorithm. As such this algorithm will perform two operations in one step. EXAM,bn_mp_init_copy.c This will initialize \textbf{a} and make it a verbatim copy of the contents of \textbf{b}. Note that \textbf{a} will have its own memory allocated which means that \textbf{b} may be cleared after the call and \textbf{a} will be left intact. \subsection{Multiple Integer Initializations} Occasionally a function will require a series of mp\_int data types to be made available. The mp\_init\_multi algorithm is provided to simplify such cases. The purpose of this algorithm is to initialize a variable length array of mp\_int structures at once. As a result algorithms that require multiple integers only has to use one algorithm to initialize all the mp\_int variables. \begin{figure}[here] \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_init\_multi}. \\ \textbf{Input}. Variable length array of mp\_int variables of length $k$. \\ \textbf{Output}. The array is initialized such that each each mp\_int is ready to use. \\ \hline \\ 1. for $n$ from 0 to $k - 1$ do \\ \hspace{+3mm}1.1. Initialize the $n$'th mp\_int (\textit{hint: use mp\_init}) \\ \hspace{+3mm}1.2. If initialization failed then do \\ \hspace{+6mm}1.2.1. for $j$ from $0$ to $n$ do \\ \hspace{+9mm}1.2.1.1. Free the $j$'th mp\_int (\textit{hint: use mp\_clear}) \\ \hspace{+6mm}1.2.2. Return(\textit{MP\_MEM}) \\ 2. Return(\textit{MP\_OKAY}) \\ \hline \end{tabular} \end{center} \caption{Algorithm mp\_init\_multi} \end{figure} \textbf{Algorithm mp\_init\_multi.} The algorithm will initialize the array of mp\_int variables one at a time. As soon as an runtime error is detected (\textit{step 1.2}) all of the previously initialized variables are cleared. The goal is an ``all or nothing'' initialization which allows for quick recovery from runtime errors. \subsection{Multiple Integer Clearing} Similarly to clear a variable length list of mp\_int structures the mp\_clear\_multi algorithm will be used. EXAM,bn_mp_multi.c Consider the following snippet which demonstrates how to use both routines. \begin{small} \begin{verbatim} #include #include #include int main(void) { mp_int num1, num2, num3; int err; if ((err = mp_init_multi(&num1, &num2, &num3, NULL)) !- MP_OKAY) { printf("Error: %d\n", err); return EXIT_FAILURE; } /* at this point num1/num2/num3 are ready */ /* free them */ mp_clear_multi(&num1, &num2, &num3, NULL); return EXIT_SUCCESS; } \end{verbatim} \end{small} \section{Maintenance} A small useful collection of mp\_int maintenance functions will also prove useful. \subsection{Augmenting Integer Precision} When storing a value in an mp\_int sufficient digits must be available to accomodate the entire value without loss of precision. Quite often the size of the array given by the \textbf{alloc} member is large enough to simply increase the \textbf{used} digit count. However, when the size of the array is too small it must be re-sized appropriately to accomodate the result. The mp\_grow algorithm will provide this functionality. \begin{figure}[here] \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_grow}. \\ \textbf{Input}. An mp\_int $a$ and an integer $b$. \\ \textbf{Output}. $a$ is expanded to accomodate $b$ digits. \\ \hline \\ 1. if $a.alloc \ge b$ then return(\textit{MP\_OKAY}) \\ 2. $u \leftarrow b\mbox{ (mod }MP\_PREC\mbox{)}$ \\ 3. $v \leftarrow b + 2 \cdot MP\_PREC - u$ \\ 4. Re-Allocate the array of digits $a$ to size $v$ \\ 5. If the allocation failed then return(\textit{MP\_MEM}). \\ 6. for n from a.alloc to $v - 1$ do \\ \hspace{+3mm}6.1 $a_n \leftarrow 0$ \\ 7. $a.alloc \leftarrow v$ \\ 8. Return(\textit{MP\_OKAY}) \\ \hline \end{tabular} \end{center} \caption{Algorithm mp\_grow} \end{figure} \textbf{Algorithm mp\_grow.} Step one will prevent a re-allocation from being performed if it was not required. This is useful to prevent mp\_ints from growing excessively in code that erroneously calls mp\_grow. Similar to mp\_init\_size the requested digit count is padded to provide more digits than requested. In step four it is assumed that the reallocation leaves the lower $a.alloc$ digits intact. Much akin to how the \textit{realloc} function from the standard C library works. Since the newly allocated digits are assumed to contain undefined values they are also initially zeroed. EXAM,bn_mp_grow.c The first step is to see if we actually need to perform a re-allocation at all. This is tested for on line @24,a->alloc < size@. Similar to mp\_init\_size the same code on line @26,MP_PREC - 1@ was used to resize the digits requested. A simple for loop from line @34,a->alloc@ to line @38,}@ will zero all digits that were above the old \textbf{alloc} limit to make sure the integer is in a known state. \subsection{Clamping Excess Digits} When a function anticipates a result will be $n$ digits it is simpler to assume this is true within the body of the function. For example, a multiplication of a $i$ digit number by a $j$ digit produces a result of at most $i + j + 1$ digits. It is entirely possible that the result is $i + j$ though, with no final carry into the last position. However, suppose the destination had to be first expanded (\textit{via mp\_grow}) to accomodate $i + j$ digits than further expanded to accomodate the final carry. That would be a considerable waste of time since heap operations are relatively slow. The ideal solution is to always assume the result is $i + j + 1$ and fix up the \textbf{used} count after the function terminates. This way a single heap operation (\textit{at most}) is required. However, if the result was not checked there would be an excess high order zero digit. For example, suppose the product of two integers was $x_n = (0x_{n-1}x_{n-2}...x_0)_{\beta}$. The leading zero digit will not contribute to the precision of the result. In fact, through subsequent operations more leading zero digits would accumulate to the point the size of the integer would be prohibitive. As a result even though the precision is very low the representation is excessively large. The mp\_clamp algorithm is designed to solve this very problem. It will trim leading zeros by decrementing the \textbf{used} count until a non-zero leading digit is found. Also in this system, zero is considered to be a positive number which means that if the \textbf{used} count is decremented to zero the sign must be set to \textbf{MP\_ZPOS}. \begin{figure}[here] \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_clamp}. \\ \textbf{Input}. An mp\_int $a$ \\ \textbf{Output}. Any excess leading zero digits of $a$ are removed \\ \hline \\ 1. while $a.used > 0$ and $a_{a.used - 1} = 0$ do \\ \hspace{+3mm}1.1 $a.used \leftarrow a.used - 1$ \\ 2. if $a.used = 0$ then do \\ \hspace{+3mm}2.1 $a.sign \leftarrow MP\_ZPOS$ \\ \hline \\ \end{tabular} \end{center} \caption{Algorithm mp\_clamp} \end{figure} \textbf{Algorithm mp\_clamp.} As can be expected this algorithm is very simple. The loop on step one is indended to be iterate only once or twice at the most. For example, for cases where there is not a carry to fill the last position. Step two fixes the sign for when all of the digits are zero to ensure that the mp\_int is valid at all times. EXAM,bn_mp_clamp.c Note on line @27,while@ how to test for the \textbf{used} count is made on the left of the \&\& operator. In the C programming language the terms to \&\& are evaluated left to right with a boolean short-circuit if any condition fails. This is important since if the \textbf{used} is zero the test on the right would fetch below the array. That is obviously undesirable. The parenthesis on line @28,a->used@ is used to make sure the \textbf{used} count is decremented and not the pointer ``a''. \section*{Exercises} \begin{tabular}{cl} $\left [ 1 \right ]$ & Discuss the relevance of the \textbf{used} member of the mp\_int structure. \\ & \\ $\left [ 1 \right ]$ & Discuss the consequences of not using padding when performing allocations. \\ & \\ $\left [ 2 \right ]$ & Estimate an ideal value for \textbf{MP\_PREC} when performing 1024-bit RSA \\ & encryption when $\beta = 2^{28}$. \\ & \\ $\left [ 1 \right ]$ & Discuss the relevance of the algorithm mp\_clamp. What does it prevent? \\ & \\ $\left [ 1 \right ]$ & Give an example of when the algorithm mp\_init\_copy might be useful. \\ & \\ \end{tabular} \chapter{Basic Operations} \section{Copying an Integer} After the various house-keeping routines are in place, simpl algorithms can be designed to take advantage of them. Being able to make a verbatim copy of an integer is a very useful function to have. To copy an integer the mp\_copy algorithm will be used. \newpage\begin{figure}[here] \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_copy}. \\ \textbf{Input}. An mp\_int $a$ and $b$. \\ \textbf{Output}. Store a copy of $a$ in $b$. \\ \hline \\ 1. Check if $a$ and $b$ point to the same location in memory. \\ 2. If true then return(\textit{MP\_OKAY}). \\ 3. If $b.alloc < a.used$ then grow $b$ to $a.used$ digits. (\textit{hint: use mp\_grow}) \\ 4. If failed to grow then return(\textit{MP\_MEM}). \\ 5. for $n$ from 0 to $a.used - 1$ do \\ \hspace{3mm}5.1 $b_{n} \leftarrow a_{n}$ \\ 6. if $a.used < b.used - 1$ then \\ \hspace{3mm}6.1. for $n$ from $a.used$ to $b.used - 1$ do \\ \hspace{6mm}6.1.1 $b_{n} \leftarrow 0$ \\ 7. $b.used \leftarrow a.used$ \\ 8. $b.sign \leftarrow a.sign$ \\ 9. return(\textit{MP\_OKAY}) \\ \hline \end{tabular} \end{center} \caption{Algorithm mp\_copy} \end{figure} \textbf{Algorithm mp\_copy.} Step 1 and 2 make sure that the two mp\_ints are unique. This allows the user to call the copy function with potentially the same input and not waste time. Step 3 and 4 ensure that the destination is large enough to hold a copy of the input $a$. Note that the \textbf{used} member of $b$ may be smaller than the \textbf{used} member of $a$ but a memory re-allocation is only required if the \textbf{alloc} member of $b$ is smaller. This prevents trivial memory reallocations. Step 5 copies the digits from $a$ to $b$ while step 6 ensures that if initially $\vert b \vert > \vert a \vert$, the leading digits of $b$ will be zeroed. Finally steps 7 and 8 copies the \textbf{used} and \textbf{sign} members over which completes the copy operation. EXAM,bn_mp_copy.c Source lines @23,if dst ==@-@31,}@ do the initial house keeping. That is to see if the input is unique and if so to make sure there is enough room. If not enough space is available it returns the error and leaves the destination variable intact. The inner loop of the copy operation is contained between lines @34,{@ and @50,}@. Many LibTomMath routines are designed with this source code style in mind, making aliases to shorten lengthy pointers (\textit{see line @38,->@ and @39,->@}) for rapid to use. Also the use of nested braces creates a simple way to denote various portions of code that reside on various work levels. Here, the copy loop is at the $O(n)$ level. \section{Zeroing an Integer} Reseting an mp\_int to the default state is a common step in many algorithms. The mp\_zero algorithm will be the algorithm used to perform this task. \begin{figure}[here] \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_zero}. \\ \textbf{Input}. An mp\_int $a$ \\ \textbf{Output}. Zero the contents of $a$ \\ \hline \\ 1. $a.used \leftarrow 0$ \\ 2. $a.sign \leftarrow$ MP\_ZPOS \\ 3. for $n$ from 0 to $a.alloc - 1$ do \\ \hspace{3mm}3.1 $a_n \leftarrow 0$ \\ \hline \end{tabular} \end{center} \caption{Algorithm mp\_zero} \end{figure} \textbf{Algorithm mp\_zero.} This algorithm simply resets a mp\_int to the default state. EXAM,bn_mp_zero.c After the function is completed, all of the digits are zeroed, the \textbf{used} count is zeroed and the \textbf{sign} variable is set to \textbf{MP\_ZPOS}. \section{Sign Manipulation} \subsection{Absolute Value} With the mp\_int representation of an integer, calculating the absolute value is trivial. The mp\_abs algorithm will compute the absolute value of an mp\_int. \begin{figure}[here] \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_abs}. \\ \textbf{Input}. An mp\_int $a$ \\ \textbf{Output}. Computes $b = \vert a \vert$ \\ \hline \\ 1. Copy $a$ to $b$. (\textit{hint: use mp\_copy}) \\ 2. If the copy failed return(\textit{MP\_MEM}). \\ 3. $b.sign \leftarrow MP\_ZPOS$ \\ 4. Return(\textit{MP\_OKAY}) \\ \hline \end{tabular} \end{center} \caption{Algorithm mp\_abs} \end{figure} \textbf{Algorithm mp\_abs.} This algorithm computes the absolute of an mp\_int input. As can be expected the algorithm is very trivial. EXAM,bn_mp_abs.c \subsection{Integer Negation} With the mp\_int representation of an integer, calculating the negation is also trivial. The mp\_neg algorithm will compute the negative of an mp\_int input. \newpage\begin{figure}[here] \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_neg}. \\ \textbf{Input}. An mp\_int $a$ \\ \textbf{Output}. Computes $b = -a$ \\ \hline \\ 1. Copy $a$ to $b$. (\textit{hint: use mp\_copy}) \\ 2. If the copy failed return(\textit{MP\_MEM}). \\ 3. If $a.sign = MP\_ZPOS$ then do \\ \hspace{3mm}3.1 $b.sign = MP\_NEG$. \\ 4. else do \\ \hspace{3mm}4.1 $b.sign = MP\_ZPOS$. \\ 5. Return(\textit{MP\_OKAY}) \\ \hline \end{tabular} \end{center} \caption{Algorithm mp\_neg} \end{figure} \textbf{Algorithm mp\_neg.} This algorithm computes the negation of an input. EXAM,bn_mp_neg.c \section{Small Constants} \subsection{Setting Small Constants} Often a mp\_int must be set to a relatively small value such as $1$ or $2$. For these cases the mp\_set algorithm is useful. \newpage\begin{figure} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_set}. \\ \textbf{Input}. An mp\_int $a$ and a digit $b$ \\ \textbf{Output}. Make $a$ equivalent to $b$ \\ \hline \\ 1. Zero $a$ (\textit{hint: use mp\_zero}). \\ 2. $a_0 \leftarrow b \mbox{ (mod }\beta\mbox{)}$ \\ 3. $a.used \leftarrow \left \lbrace \begin{array}{ll} 1 & \mbox{if }a_0 > 0 \\ 0 & \mbox{if }a_0 = 0 \end{array} \right .$ \\ \hline \end{tabular} \end{center} \caption{Algorithm mp\_set} \end{figure} \textbf{Algorithm mp\_set.} This algorithm sets a mp\_int to a small single digit value. Step number 1 ensures that the integer is reset to the default state. The single digit is set (\textit{modulo $\beta$}) and the \textbf{used} count is adjusted accordingly. EXAM,bn_mp_set.c Line @21,mp_zero@ calls mp\_zero() to clear the mp\_int and reset the sign. Line @22,MP_MASK@ actually copies digit into the least significant location. Note the usage of a new constant \textbf{MP\_MASK}. This constant is used to quickly reduce an integer modulo $\beta$. Since $\beta = 2^k$ it suffices to perform a binary AND with $MP\_MASK = 2^k - 1$ to perform the reduction. Finally line @23,a->used@ will set the \textbf{used} member with respect to the digit actually set. This function will always make the integer positive. One important limitation of this function is that it will only set one digit. The size of a digit is not fixed, meaning source that uses this function should take that into account. The define \textbf{DIGIT\_BIT} in ``tommath.h'' defines how many bits per digit are available. Generally at least seven bits are guaranteed to be available per digit. This means that trivially small constants can be set using this function. \subsection{Setting Large Constants} To overcome the limitations of the mp\_set algorithm the mp\_set\_int algorithm is provided. It accepts a ``long'' data type as input and will always treat it as a 32-bit integer. \begin{figure}[here] \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_set\_int}. \\ \textbf{Input}. An mp\_int $a$ and a ``long'' integer $b$ \\ \textbf{Output}. Make $a$ equivalent to $b$ \\ \hline \\ 1. Zero $a$ (\textit{hint: use mp\_zero}) \\ 2. for $n$ from 0 to 7 do \\ \hspace{3mm}2.1 $a \leftarrow a \cdot 16$ (\textit{hint: use mp\_mul2d}) \\ \hspace{3mm}2.2 $u \leftarrow \lfloor b / 2^{4(7 - n)} \rfloor \mbox{ (mod }16\mbox{)}$\\ \hspace{3mm}2.3 $a_0 \leftarrow a_0 + u$ \\ \hspace{3mm}2.4 $a.used \leftarrow a.used + \lfloor 32 / lg(\beta) \rfloor + 1$ \\ 3. Clamp excess used digits (\textit{hint: use mp\_clamp}) \\ \hline \end{tabular} \end{center} \caption{Algorithm mp\_set\_int} \end{figure} \textbf{Algorithm mp\_set\_int.} The algorithm performs eight iterations of a simple loop where in each iteration four bits from the source are added to the mp\_int. Step 2.1 will multiply the current result by sixteen making room for four more bits. In step 2.2 the next four bits from the source are extracted. The four bits are added to the mp\_int and the \textbf{used} digit count is incremented. The \textbf{used} digit counter is incremented since if any of the leading digits were zero the mp\_int would have zero digits used and the newly added four bits would be ignored. Excess zero digits are trimmed in steps 2.1 and 3 by using higher level algorithms mp\_mul2d and mp\_clamp. EXAM,bn_mp_set_int.c This function sets four bits of the number at a time to handle all practical \textbf{DIGIT\_BIT} sizes. The weird addition on line @38,a->used@ ensures that the newly added in bits are added to the number of digits. While it may not seem obvious as to why the digit counter does not grow exceedingly large it is because of the shift on line @27,mp_mul_2d@ as well as the call to mp\_clamp() on line @40,mp_clamp@. Both functions will clamp excess leading digits which keeps the number of used digits low. \section{Comparisons} \subsection{Unsigned Comparisions} Comparing a multiple precision integer is performed with the exact same algorithm used to compare two decimal numbers. For example, to compare $1,234$ to $1,264$ the digits are extracted by their positions. That is we compare $1 \cdot 10^3 + 2 \cdot 10^2 + 3 \cdot 10^1 + 4 \cdot 10^0$ to $1 \cdot 10^3 + 2 \cdot 10^2 + 6 \cdot 10^1 + 4 \cdot 10^0$ by comparing single digits at a time starting with the highest magnitude positions. If any leading digit of one integer is greater than a digit in the same position of another integer then obviously it must be greater. The first comparision routine that will be developed is the unsigned magnitude compare which will perform a comparison based on the digits of two mp\_int variables alone. It will ignore the sign of the two inputs. Such a function is useful when an absolute comparison is required or if the signs are known to agree in advance. To facilitate working with the results of the comparison functions three constants are required. \begin{figure}[here] \begin{center} \begin{tabular}{|r|l|} \hline \textbf{Constant} & \textbf{Meaning} \\ \hline \textbf{MP\_GT} & Greater Than \\ \hline \textbf{MP\_EQ} & Equal To \\ \hline \textbf{MP\_LT} & Less Than \\ \hline \end{tabular} \end{center} \caption{Comparison Return Codes} \end{figure} \begin{figure}[here] \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_cmp\_mag}. \\ \textbf{Input}. Two mp\_ints $a$ and $b$. \\ \textbf{Output}. Unsigned comparison results ($a$ to the left of $b$). \\ \hline \\ 1. If $a.used > b.used$ then return(\textit{MP\_GT}) \\ 2. If $a.used < b.used$ then return(\textit{MP\_LT}) \\ 3. for n from $a.used - 1$ to 0 do \\ \hspace{+3mm}3.1 if $a_n > b_n$ then return(\textit{MP\_GT}) \\ \hspace{+3mm}3.2 if $a_n < b_n$ then return(\textit{MP\_LT}) \\ 4. Return(\textit{MP\_EQ}) \\ \hline \end{tabular} \end{center} \caption{Algorithm mp\_cmp\_mag} \end{figure} \textbf{Algorithm mp\_cmp\_mag.} By saying ``$a$ to the left of $b$'' it is meant that the comparison is with respect to $a$, that is if $a$ is greater than $b$ it will return \textbf{MP\_GT} and similar with respect to when $a = b$ and $a < b$. The first two steps compare the number of digits used in both $a$ and $b$. Obviously if the digit counts differ there would be an imaginary zero digit in the smaller number where the leading digit of the larger number is. If both have the same number of digits than the actual digits themselves must be compared starting at the leading digit. By step three both inputs must have the same number of digits so its safe to start from either $a.used - 1$ or $b.used - 1$ and count down to the zero'th digit. If after all of the digits have been compared and no difference found the algorithm simply returns \textbf{MP\_EQ}. EXAM,bn_mp_cmp_mag.c The two if statements on lines @24,if@ and @28,if@ compare the number of digits in the two inputs. These two are performed before all of the digits are compared since it is a very cheap test to perform and can potentially save considerable time. The implementation given is also not valid without those two statements. $b.alloc$ may be smaller than $a.used$, meaning that undefined values will be read from $b$ passed the end of the array of digits. \subsection{Signed Comparisons} Comparing with sign considerations is also fairly critical in several routines (\textit{division for example}). Based on an unsigned magnitude comparison a trivial signed comparison algorithm can be written. \newpage\begin{figure}[here] \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_cmp}. \\ \textbf{Input}. Two mp\_ints $a$ and $b$ \\ \textbf{Output}. Signed Comparison Results ($a$ to the left of $b$) \\ \hline \\ 1. if $a.sign = MP\_NEG$ and $b.sign = MP\_ZPOS$ then return(\textit{MP\_LT}) \\ 2. if $a.sign = MP\_ZPOS$ and $b.sign = MP\_NEG$ then return(\textit{MP\_GT}) \\ 3. if $a.sign = MP\_NEG$ then \\ \hspace{+3mm}3.1 Return the unsigned comparison of $b$ and $a$ (\textit{hint: use mp\_cmp\_mag}) \\ 4 Otherwise \\ \hspace{+3mm}4.1 Return the unsigned comparison of $a$ and $b$ \\ \hline \end{tabular} \end{center} \caption{Algorithm mp\_cmp} \end{figure} \textbf{Algorithm mp\_cmp.} The first two steps compare the signs of the two inputs. If the signs do not agree then it can return right away with the appropriate comparison code. When the signs are equal the digits of the inputs must be compared to determine the correct result. In step three the unsigned comparision flips the order of the arguments since they are both negative. For instance, if $-a > -b$ then $\vert a \vert < \vert b \vert$. Step number four will compare the two when they are both positive. EXAM,bn_mp_cmp.c The two if statements on lines @22,if@ and @26,if@ perform the initial sign comparison. If the signs are not the equal then which ever has the positive sign is larger. At line @30,if@, the inputs are compared based on magnitudes. If the signs were both negative then the unsigned comparison is performed in the opposite direction (\textit{line @31,mp_cmp_mag@}). Otherwise, the signs are assumed to be both positive and a forward direction unsigned comparison is performed. \section*{Exercises} \begin{tabular}{cl} $\left [ 2 \right ]$ & Modify algorithm mp\_set\_int to accept as input a variable length array of bits. \\ & \\ $\left [ 3 \right ]$ & Give the probability that algorithm mp\_cmp\_mag will have to compare $k$ digits \\ & of two random digits (of equal magnitude) before a difference is found. \\ & \\ $\left [ 1 \right ]$ & Suggest a simple method to speed up the implementation of mp\_cmp\_mag based \\ & on the observations made in the previous problem. \\ & \end{tabular} \chapter{Basic Arithmetic} \section{Building Blocks} At this point algorithms for initialization, de-initialization, zeroing, copying, comparing and setting small constants have been established. The next logical set of algorithms to develop are the addition, subtraction and digit movement algorithms. These algorithms make use of the lower level algorithms and are the cruicial building block for the multipliers. It is very important that these algorithms are highly optimized. On their own they are simple $O(n)$ algorithms but they can be called from higher level algorithms which easily places them at $O(n^2)$ or even $O(n^3)$ work levels. MARK,SHIFTS All nine algorithms within this chapter make use of the logical bit shift operations denoted by $<<$ and $>>$ for left and right logical shifts respectively. A logical shift is analogous to sliding the decimal point of radix-10 representations. For example, the real number $0.9345$ is equivalent to $93.45\%$ which is found by sliding the the decimal two places to the right (\textit{multiplying by $10^2$}). Mathematically a logical shift is equivalent to a division or multiplication by a power of two. For example, $a << k = a \cdot 2^k$ while $a >> k = \lfloor a/2^k \rfloor$. One significant difference between a logical shift and the way decimals are shifted is that digits below the zero'th position are removed from the number. For example, consider $1101_2 >> 1$ using decimal notation this would produce $110.1_2$. However, with a logical shift the result is $110_2$. \section{Addition and Subtraction} In normal fixed precision arithmetic negative numbers are easily represented by subtraction from the modulus. For example, with 32-bit integers $a - b\mbox{ (mod }2^{32}\mbox{)}$ is the same as $a + (2^{32} - b) \mbox{ (mod }2^{32}\mbox{)}$ since $2^{32} \equiv 0 \mbox{ (mod }2^{32}\mbox{)}$. As a result subtraction can be performed with a trivial series of logical operations and an addition. However, in multiple precision arithmetic negative numbers are not represented in the same way. Instead a sign flag is used to keep track of the sign of the integer. As a result signed addition and subtraction are actually implemented as conditional usage of lower level addition or subtraction algorithms with the sign fixed up appropriately. The lower level algorithms will add or subtract integers without regard to the sign flag. That is they will add or subtract the magnitude of the integers respectively. \subsection{Low Level Addition} An unsigned addition of multiple precision integers is performed with the same long-hand algorithm used to add decimal numbers. That is to add the trailing digits first and propagate the resulting carry upwards. Since this is a lower level algorithm the name will have a ``s\_'' prefix. Historically that convention stems from the MPI library where ``s\_'' stood for static functions that were hidden from the developer entirely. \newpage \begin{figure}[!here] \begin{center} \begin{small} \begin{tabular}{l} \hline Algorithm \textbf{s\_mp\_add}. \\ \textbf{Input}. Two mp\_ints $a$ and $b$ \\ \textbf{Output}. The unsigned addition $c = \vert a \vert + \vert b \vert$. \\ \hline \\ 1. if $a.used > b.used$ then \\ \hspace{+3mm}1.1 $min \leftarrow b.used$ \\ \hspace{+3mm}1.2 $max \leftarrow a.used$ \\ \hspace{+3mm}1.3 $x \leftarrow a$ \\ 2. else \\ \hspace{+3mm}2.1 $min \leftarrow a.used$ \\ \hspace{+3mm}2.2 $max \leftarrow b.used$ \\ \hspace{+3mm}2.3 $x \leftarrow b$ \\ 3. If $c.alloc < max + 1$ then grow $c$ to hold at least $max + 1$ digits (\textit{hint: use mp\_grow}) \\ 4. If failed to grow $c$ return(\textit{MP\_MEM}) \\ 5. $oldused \leftarrow c.used$ \\ 6. $c.used \leftarrow max + 1$ \\ 7. $u \leftarrow 0$ \\ 8. for $n$ from $0$ to $min - 1$ do \\ \hspace{+3mm}8.1 $c_n \leftarrow a_n + b_n + u$ \\ \hspace{+3mm}8.2 $u \leftarrow c_n >> lg(\beta)$ \\ \hspace{+3mm}8.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ 9. if $min \ne max$ then do \\ \hspace{+3mm}9.1 for $n$ from $min$ to $max - 1$ do \\ \hspace{+6mm}9.1.1 $c_n \leftarrow x_n + u$ \\ \hspace{+6mm}9.1.2 $u \leftarrow c_n >> lg(\beta)$ \\ \hspace{+6mm}9.1.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ 10. $c_{max} \leftarrow u$ \\ 11. if $olduse > max$ then \\ \hspace{+3mm}11.1 for $n$ from $max + 1$ to $olduse - 1$ do \\ \hspace{+6mm}11.1.1 $c_n \leftarrow 0$ \\ 12. Clamp excess digits in $c$. (\textit{hint: use mp\_clamp}) \\ 13. Return(\textit{MP\_OKAY}) \\ \hline \end{tabular} \end{small} \end{center} \caption{Algorithm s\_mp\_add} \end{figure} \textbf{Algorithm s\_mp\_add.} This algorithm is loosely based on algorithm 14.7 of \cite[pp. 594]{HAC} but has been extended to allow the inputs to have different magnitudes. Coincidentally the description of algorithm A in \cite[pp. 266]{TAOCPV2} shares the same flaw as that from \cite{HAC}. Even the MIX pseudo machine code presented \cite[pp. 266-267]{TAOCPV2} is incapable of handling inputs which are of different magnitudes. Steps 1 and 2 will sort the two inputs based on their \textbf{used} digit count. This allows the inputs to have varying magnitudes which not only makes it more efficient than the trivial algorithm presented in the other references but more flexible. The variable $min$ is given the lowest digit count while $max$ is given the highest digit count. If both inputs have the same \textbf{used} digit count both $min$ and $max$ are set to the same. The variable $x$ is an \textit{alias} for the largest input and not meant to be a copy of it. After the inputs are sorted steps 3 and 4 will ensure that the destination $c$ can accommodate the result. The old \textbf{used} count from $c$ is copied to $oldused$ and the new count is set to $max + 1$. At step 7 the carry variable $u$ is set to zero and the first leg of the addition loop can begin. The first step of the loop (\textit{8.1}) adds digits from the two inputs together along with the carry variable $u$. The following step extracts the carry bit by shifting the result of the preceding step right $lg(\beta)$ positions. The shift to extract the carry is similar to how carry extraction works with decimal addition. Consider adding $77$ to $65$, the first addition of the first column is $7 + 5$ which produces the result $12$. The trailing digit of the result is $2 \equiv 12 \mbox{ (mod }10\mbox{)}$ and the carry is found by dividing (\textit{and ignoring the remainder}) $12$ by the radix or in this case $10$. The division and multiplication of $10$ is simply a logical shift right or left respectively of the digits. In otherwords the carry can be extracted by shifting one digit to the right. Note that $lg()$ is simply the base two logarithm such that $lg(2^k) = k$. This implies that $lg(\beta)$ is the number of bits in a radix-$\beta$ digit. Therefore, a logical shift right of the single digit by $lg(\beta)$ will extract the carry. The final step of the loop reduces the digit modulo the radix $\beta$ to ensure it is in range. After step 8 the smallest input (\textit{or both if they are the same magnitude}) has been exhausted. Step 9 decides whether the inputs were of equal magnitude. If not than another loop similar to that in step 8 must be executed. The loop at step number 9.1 differs from the previous loop since it only adds the mp\_int $x$ along with the carry. Step 10 finishes the addition phase by copying the final carry to the highest location in the result $c_{max}$. Step 11 ensures that leading digits that were originally present in $c$ are cleared. Finally excess leading digits are clamped and the algorithm returns success. EXAM,bn_s_mp_add.c Lines @27,if@ to @35,}@ perform the initial sorting of the inputs and determine the $min$ and $max$ variables. Note that $x$ is pointer to a mp\_int assigned to the largest input, in effect it is a local alias. Lines @37,init@ to @42,}@ ensure that the destination is grown to accomodate the result of the addition. Similar to the implementation of mp\_copy this function uses the braced code and local aliases coding style. The three aliases on lines @56,tmpa@, @59,tmpb@ and @62,tmpc@ are the for the two inputs and destination respectively. These aliases are used to ensure the compiler does not have to dereference $a$, $b$ or $c$ (respectively) to access the digits of the respective mp\_int. The initial carry $u$ is cleared on line @65,u = 0@, note that $u$ is of type mp\_digit which ensures type compatibility within the implementation. The initial addition loop begins on line @66,for@ and ends on line @75,}@. Similarly the conditional addition loop begins on line @81,for@ and ends on line @90,}@. The addition is finished with the final carry being stored in $tmpc$ on line @94,tmpc++@. Note the ``++'' operator on the same line. After line @94,tmpc++@ $tmpc$ will point to the $c.used$'th digit of the mp\_int $c$. This is useful for the next loop on lines @97,for@ to @99,}@ which set any old upper digits to zero. \subsection{Low Level Subtraction} The low level unsigned subtraction algorithm is very similar to the low level unsigned addition algorithm. The principle difference is that the unsigned subtraction algorithm requires the result to be positive. That is when computing $a - b$ the condition $\vert a \vert \ge \vert b\vert$ must be met for this algorithm to function properly. Keep in mind this low level algorithm is not meant to be used in higher level algorithms directly. This algorithm as will be shown can be used to create functional signed addition and subtraction algorithms. MARK,GAMMA For this algorithm a new variable is required to make the description simpler. Recall from section 1.3.1 that a mp\_digit must be able to represent the range $0 \le x < 2\beta$. It is allowable that a mp\_digit represent a larger range of values. For this algorithm we will assume that the variable $\gamma$ represents the number of bits available in a mp\_digit (\textit{this implies $2^{\gamma} > \beta$}). \newpage\begin{figure}[!here] \begin{center} \begin{small} \begin{tabular}{l} \hline Algorithm \textbf{s\_mp\_sub}. \\ \textbf{Input}. Two mp\_ints $a$ and $b$ ($\vert a \vert \ge \vert b \vert$) \\ \textbf{Output}. The unsigned subtraction $c = \vert a \vert - \vert b \vert$. \\ \hline \\ 1. $min \leftarrow b.used$ \\ 2. $max \leftarrow a.used$ \\ 3. If $c.alloc < max$ then grow $c$ to hold at least $max$ digits. (\textit{hint: use mp\_grow}) \\ 4. If the reallocation failed return(\textit{MP\_MEM}). \\ 5. $oldused \leftarrow c.used$ \\ 6. $c.used \leftarrow max$ \\ 7. $u \leftarrow 0$ \\ 8. for $n$ from $0$ to $min - 1$ do \\ \hspace{3mm}8.1 $c_n \leftarrow a_n - b_n - u$ \\ \hspace{3mm}8.2 $u \leftarrow c_n >> (\gamma - 1)$ \\ \hspace{3mm}8.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ 9. if $min < max$ then do \\ \hspace{3mm}9.1 for $n$ from $min$ to $max - 1$ do \\ \hspace{6mm}9.1.1 $c_n \leftarrow a_n - u$ \\ \hspace{6mm}9.1.2 $u \leftarrow c_n >> (\gamma - 1)$ \\ \hspace{6mm}9.1.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ 10. if $oldused > max$ then do \\ \hspace{3mm}10.1 for $n$ from $max$ to $oldused - 1$ do \\ \hspace{6mm}10.1.1 $c_n \leftarrow 0$ \\ 11. Clamp excess digits of $c$. (\textit{hint: use mp\_clamp}). \\ 12. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{small} \end{center} \caption{Algorithm s\_mp\_sub} \end{figure} \textbf{Algorithm s\_mp\_sub.} This algorithm performs the unsigned subtraction of two mp\_int variables under the restriction that the result must be positive. That is when passing variables $a$ and $b$ the condition that $\vert a \vert \ge \vert b \vert$ must be met for the algorithm to function correctly. This algorithm is loosely based on algorithm 14.9 \cite[pp. 595]{HAC} and is similar to algorithm S in \cite[pp. 267]{TAOCPV2} as well. As was the case of the algorithm s\_mp\_add both other references lack discussion concerning various practical details such as when the inputs differ in magnitude. The initial sorting of the inputs is trivial in this algorithm since $a$ is guaranteed to have at least the same magnitude of $b$. Steps 1 and 2 set the $min$ and $max$ variables. Unlike the addition routine there is guaranteed to be no carry which means that the final result can be at most $max$ digits in length as oppose to $max + 1$. Similar to the addition algorithm the \textbf{used} count of $c$ is copied locally and set to the maximal count for the operation. The subtraction loop that begins on step 8 is essentially the same as the addition loop of algorithm s\_mp\_add except single precision subtraction is used instead. Note the use of the $\gamma$ variable to extract the carry within the subtraction loops. Under the assumption that two's complement single precision arithmetic is used this will successfully extract the carry. For example, consider subtracting $0101_2$ from $0100_2$ where $\gamma = 4$. The least significant bit will force a carry upwards to the third bit which will be set to zero after the borrow. After the very first bit has been subtracted $4 - 1 \equiv 0011_2$ will remain, When the third bit of $0101_2$ is subtracted from the result it will cause another carry. In this case though the carry will be forced to propagate all the way to the most significant bit. Recall that $\beta < 2^{\gamma}$. This means that if a carry does occur it will propagate all the way to the most significant bit. Therefore a single logical shift right by $\gamma - 1$ positions is sufficient to extract the carry. This method of carry extraction may seem awkward but the reason for it becomes apparent when the implementation is discussed. If $b$ has a smaller magnitude than $a$ then step 9 will force the carry and copy operation to propagate through the larger input $a$ into $c$. Step 10 will ensure that any leading digits of $c$ above the $max$'th position are zeroed. EXAM,bn_s_mp_sub.c Line @24,min@ and @25,max@ perform the initial hardcoded sorting. In reality they are only aliases and are only used to make the source easier to read. Again the pointer alias optimization is used within this algorithm. Lines @42,tmpa@, @43,tmpb@ and @44,tmpc@ initialize the aliases for $a$, $b$ and $c$ respectively. The first subtraction loop occurs on lines @47,u = 0@ through @61,}@. The theory behind the subtraction loop is exactly the same as that for the addition loop. As remarked earlier there is an implementation reason for using the ``awkward'' method of extracting the carry (\textit{see line @57, >>@}). The traditional method for extracting the carry would be to shift by $lg(\beta)$ positions and logically AND the least significant bit. The AND operation is required because all of the bits above the $\lg(\beta)$'th bit will be set to one after a carry occurs from subtraction. This carry extraction requires two relatively cheap operations to extract the carry. The other method is to simply shift the most significant bit to the least significant bit thus extracting the carry with a single cheap operation. This optimization only works on twos compliment machines which is a safe assumption to make. If $a$ has a higher magnitude than $b$ an additional loop (\textit{see lines @64,for@ through @73,}@}) is required to propagate the carry through $a$ and copy the result to $c$. \subsection{High Level Addition} Now that both lower level addition and subtraction algorithms have been established an effective high level signed addition algorithm can be established. This high level addition algorithm will be what other algorithms and developers will use to perform addition of mp\_int data types. Recall from section 5.2 that an mp\_int represents an integer with an unsigned mantissa (\textit{the array of digits}) and a \textbf{sign} flag. A high level addition is actually performed as a series of eight seperate cases which can be optimized down to three unique cases. \newpage\begin{figure}[!here] \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_add}. \\ \textbf{Input}. Two mp\_ints $a$ and $b$ \\ \textbf{Output}. The signed addition $c = a + b$. \\ \hline \\ 1. if $a.sign = b.sign$ then do \\ \hspace{3mm}1.1 $c.sign \leftarrow a.sign$ \\ \hspace{3mm}1.2 $c \leftarrow \vert a \vert + \vert b \vert$ (\textit{hint: use s\_mp\_add})\\ 2. else do \\ \hspace{3mm}2.1 if $\vert a \vert < \vert b \vert$ then do (\textit{hint: use mp\_cmp\_mag}) \\ \hspace{6mm}2.1.1 $c.sign \leftarrow b.sign$ \\ \hspace{6mm}2.1.2 $c \leftarrow \vert b \vert - \vert a \vert$ (\textit{hint: use s\_mp\_sub}) \\ \hspace{3mm}2.2 else do \\ \hspace{6mm}2.2.1 $c.sign \leftarrow a.sign$ \\ \hspace{6mm}2.2.2 $c \leftarrow \vert a \vert - \vert b \vert$ \\ 3. If any of the lower level operations failed return(\textit{MP\_MEM}) \\ 4. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} \caption{Algorithm mp\_add} \end{figure} \textbf{Algorithm mp\_add.} This algorithm performs the signed addition of two mp\_int variables. There is no reference algorithm to draw upon from either \cite{TAOCPV2} or \cite{HAC} since they both only provide unsigned operations. The algorithm is fairly straightforward but restricted since subtraction can only produce positive results. Consider the following chart of possible inputs. \begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{|c|c|c|c|c|} \hline \textbf{Sign of $a$} & \textbf{Sign of $b$} & \textbf{$\vert a \vert > \vert b \vert $} & \textbf{Unsigned Operation} & \textbf{Result Sign Flag} \\ \hline $+$ & $+$ & Yes & $c = a + b$ & $a.sign$ \\ \hline $+$ & $+$ & No & $c = a + b$ & $a.sign$ \\ \hline $-$ & $-$ & Yes & $c = a + b$ & $a.sign$ \\ \hline $-$ & $-$ & No & $c = a + b$ & $a.sign$ \\ \hline &&&&\\ \hline $+$ & $-$ & No & $c = b - a$ & $b.sign$ \\ \hline $-$ & $+$ & No & $c = b - a$ & $b.sign$ \\ \hline &&&&\\ \hline $+$ & $-$ & Yes & $c = a - b$ & $a.sign$ \\ \hline $-$ & $+$ & Yes & $c = a - b$ & $a.sign$ \\ \hline \end{tabular} \end{center} \end{small} \caption{Addition Guide Chart} \end{figure} The chart lists all of the eight possible input combinations and is sorted to show that only three specific cases need to be handled. The return code of the unsigned operations at step 1.2, 2.1.2 and 2.2.2 are forwarded to step 3 to check for errors. This simpliies the description of the algorithm considerably and best follows how the implementation actually was achieved. Also note how the \textbf{sign} is set before the unsigned addition or subtraction is performed. Recall from the descriptions of algorithms s\_mp\_add and s\_mp\_sub that the mp\_clamp function is used at the end to trim excess digits. The mp\_clamp algorithm will set the \textbf{sign} to \textbf{MP\_ZPOS} when the \textbf{used} digit count reaches zero. For example, consider performing $-a + a$ with algorithm mp\_add. By the description of the algorithm the sign is set to \textbf{MP\_NEG} which would produce a result of $-0$. However, since the sign is set first then the unsigned addition is performed the subsequent usage of algorithm mp\_clamp within algorithm s\_mp\_add will force $-0$ to become $0$. EXAM,bn_mp_add.c The source code follows the algorithm fairly closely. The most notable new source code addition is the usage of the $res$ integer variable which is used to pass result of the unsigned operations forward. Unlike in the algorithm, the variable $res$ is merely returned as is without explicitly checking it and returning the constant \textbf{MP\_OKAY}. The observation is this algorithm will succeed or fail only if the lower level functions do so. Returning their return code is sufficient. \subsection{High Level Subtraction} The high level signed subtraction algorithm is essentially the same as the high level signed addition algorithm. \begin{figure}[!here] \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_sub}. \\ \textbf{Input}. Two mp\_ints $a$ and $b$ \\ \textbf{Output}. The signed subtraction $c = a - b$. \\ \hline \\ 1. if $a.sign \ne b.sign$ then do \\ \hspace{3mm}1.1 $c.sign \leftarrow a.sign$ \\ \hspace{3mm}1.2 $c \leftarrow \vert a \vert + \vert b \vert$ (\textit{hint: use s\_mp\_add}) \\ 2. else do \\ \hspace{3mm}2.1 if $\vert a \vert \ge \vert b \vert$ then do (\textit{hint: use mp\_cmp\_mag}) \\ \hspace{6mm}2.1.1 $c.sign \leftarrow a.sign$ \\ \hspace{6mm}2.1.2 $c \leftarrow \vert a \vert - \vert b \vert$ (\textit{hint: use s\_mp\_sub}) \\ \hspace{3mm}2.2 else do \\ \hspace{6mm}2.2.1 $c.sign \leftarrow \left \lbrace \begin{array}{ll} MP\_ZPOS & \mbox{if }a.sign = MP\_NEG \\ MP\_NEG & \mbox{otherwise} \\ \end{array} \right .$ \\ \hspace{6mm}2.2.2 $c \leftarrow \vert b \vert - \vert a \vert$ \\ 3. If any of the lower level operations failed return(\textit{MP\_MEM}). \\ 4. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} \caption{Algorithm mp\_sub} \end{figure} \textbf{Algorithm mp\_sub.} This algorithm performs the signed subtraction of two inputs. Similar to algorithm mp\_add there is no reference in either \cite{TAOCPV2} or \cite{HAC}. Also this algorithm is restricted by algorithm s\_mp\_sub. The following chart lists the eight possible inputs and the operations required. \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{|c|c|c|c|c|} \hline \textbf{Sign of $a$} & \textbf{Sign of $b$} & \textbf{$\vert a \vert \ge \vert b \vert $} & \textbf{Unsigned Operation} & \textbf{Result Sign Flag} \\ \hline $+$ & $-$ & Yes & $c = a + b$ & $a.sign$ \\ \hline $+$ & $-$ & No & $c = a + b$ & $a.sign$ \\ \hline $-$ & $+$ & Yes & $c = a + b$ & $a.sign$ \\ \hline $-$ & $+$ & No & $c = a + b$ & $a.sign$ \\ \hline &&&& \\ \hline $+$ & $+$ & Yes & $c = a - b$ & $a.sign$ \\ \hline $-$ & $-$ & Yes & $c = a - b$ & $a.sign$ \\ \hline &&&& \\ \hline $+$ & $+$ & No & $c = b - a$ & $\mbox{opposite of }a.sign$ \\ \hline $-$ & $-$ & No & $c = b - a$ & $\mbox{opposite of }a.sign$ \\ \hline \end{tabular} \end{center} \end{small} \caption{Subtraction Guide Chart} \end{figure} Similar to the case of algorithm mp\_add the \textbf{sign} is set first before the unsigned addition or subtraction. That is to prevent the algorithm from producing $-a - -a = -0$ as a result. EXAM,bn_mp_sub.c Much like the implementation of algorithm mp\_add the variable $res$ is used to catch the return code of the unsigned addition or subtraction operations and forward it to the end of the function. On line @38, != MP_LT@ the ``not equal to'' \textbf{MP\_LT} expression is used to emulate a ``greater than or equal to'' comparison. \section{Bit and Digit Shifting} MARK,POLY It is quite common to think of a multiple precision integer as a polynomial in $x$, that is $y = f(\beta)$ where $f(x) = \sum_{i=0}^{n-1} a_i x^i$. This notation arises within discussion of Montgomery and Diminished Radix Reduction as well as Karatsuba multiplication and squaring. In order to facilitate operations on polynomials in $x$ as above a series of simple ``digit'' algorithms have to be established. That is to shift the digits left or right as well to shift individual bits of the digits left and right. It is important to note that not all ``shift'' operations are on radix-$\beta$ digits. \subsection{Multiplication by Two} In a binary system where the radix is a power of two multiplication by two not only arises often in other algorithms it is a fairly efficient operation to perform. A single precision logical shift left is sufficient to multiply a single digit by two. \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_mul\_2}. \\ \textbf{Input}. One mp\_int $a$ \\ \textbf{Output}. $b = 2a$. \\ \hline \\ 1. If $b.alloc < a.used + 1$ then grow $b$ to hold $a.used + 1$ digits. (\textit{hint: use mp\_grow}) \\ 2. If the reallocation failed return(\textit{MP\_MEM}). \\ 3. $oldused \leftarrow b.used$ \\ 4. $b.used \leftarrow a.used$ \\ 5. $r \leftarrow 0$ \\ 6. for $n$ from 0 to $a.used - 1$ do \\ \hspace{3mm}6.1 $rr \leftarrow a_n >> (lg(\beta) - 1)$ \\ \hspace{3mm}6.2 $b_n \leftarrow (a_n << 1) + r \mbox{ (mod }\beta\mbox{)}$ \\ \hspace{3mm}6.3 $r \leftarrow rr$ \\ 7. If $r \ne 0$ then do \\ \hspace{3mm}7.1 $b_{a.used} = 1$ \\ \hspace{3mm}7.2 $b.used \leftarrow b.used + 1$ \\ 8. If $b.used < oldused - 1$ then do \\ \hspace{3mm}8.1 for $n$ from $b.used$ to $oldused - 1$ do \\ \hspace{6mm}8.1.1 $b_n \leftarrow 0$ \\ 9. $b.sign \leftarrow a.sign$ \\ 10. Return(\textit{MP\_OKAY}).\\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_mul\_2} \end{figure} \textbf{Algorithm mp\_mul\_2.} This algorithm will quickly multiply a mp\_int by two provided $\beta$ is a power of two. Neither \cite{TAOCPV2} nor \cite{HAC} describe such an algorithm despite the fact it arises often in other algorithms. The algorithm is setup much like the lower level algorithm s\_mp\_add since it is for all intents and purposes equivalent to the operation $b = \vert a \vert + \vert a \vert$. Step 1 and 2 grow the input as required to accomodate the maximum number of \textbf{used} digits in the result. The initial \textbf{used} count is set to $a.used$ at step 4. Only if there is a final carry will the \textbf{used} count require adjustment. Step 6 is an optimization implementation of the addition loop for this specific case. That is since the two values being added together are the same there is no need to perform two reads from the digits of $a$. Step 6.1 performs a single precision shift on the current digit $a_n$ to obtain what will be the carry for the next iteration. Step 6.2 calculates the $n$'th digit of the result as single precision shift of $a_n$ plus the previous carry. Recall from ~SHIFTS~ that $a_n << 1$ is equivalent to $a_n \cdot 2$. An iteration of the addition loop is finished with forwarding the carry to the next iteration. Step 7 takes care of any final carry by setting the $a.used$'th digit of the result to one and augmenting the \textbf{used} count. Step 8 clears any original leading digits of $b$. EXAM,bn_mp_mul_2.c This implementation is essentially an optimized implementation of s\_mp\_add for the case of doubling an input. The only noteworthy difference is the use of the logical shift operator on line @52,<<@ to perform a single precision doubling. \subsection{Division by Two} A division by two can just as easily be accomplished with a logical shift right as multiplication by two can be with a logical shift left. \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_div\_2}. \\ \textbf{Input}. One mp\_int $a$ \\ \textbf{Output}. $b = a/2$. \\ \hline \\ 1. If $b.alloc < a.used$ then grow $b$ to hold $a.used$ digits. (\textit{hint: use mp\_grow}) \\ 2. If the reallocation failed return(\textit{MP\_MEM}). \\ 3. $oldused \leftarrow b.used$ \\ 4. $b.used \leftarrow a.used$ \\ 5. $r \leftarrow 0$ \\ 6. for $n$ from $b.used - 1$ to $0$ do \\ \hspace{3mm}6.1 $rr \leftarrow a_n \mbox{ (mod }2\mbox{)}$\\ \hspace{3mm}6.2 $b_n \leftarrow (a_n >> 1) + (r << (lg(\beta) - 1)) \mbox{ (mod }\beta\mbox{)}$ \\ \hspace{3mm}6.3 $r \leftarrow rr$ \\ 7. If $b.used < oldused - 1$ then do \\ \hspace{3mm}7.1 for $n$ from $b.used$ to $oldused - 1$ do \\ \hspace{6mm}7.1.1 $b_n \leftarrow 0$ \\ 8. $b.sign \leftarrow a.sign$ \\ 9. Return(\textit{MP\_OKAY}).\\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_div\_2} \end{figure} \textbf{Algorithm mp\_div\_2.} This algorithm will divide an mp\_int by two using logical shifts to the right. Like mp\_mul\_2 it uses a modified low level addition core as the basis of the algorithm. Unlike mp\_mul\_2 the shift operations work from the leading digit to the trailing digit. The algorithm could be written to work from the trailing digit to the leading digit however, it would have to stop one short of $a.used - 1$ digits to prevent reading passed the end of the array of digits. Essentially the loop at step 6 is similar to that of mp\_mul\_2 except the logical shifts go in the opposite direction and the carry is at the least significant bit not the most significant bit. EXAM,bn_mp_div_2.c \section{Polynomial Basis Operations} Recall from ~POLY~ that any integer can be represented as a polynomial in $x$ as $y = f(\beta)$. Such a representation is also known as the polynomial basis \cite[pp. 48]{ROSE}. Given such a notation a multiplication or division by $x$ amounts to shifting whole digits a single place. The need for such operations arises in several other higher level algorithms such as Barrett and Montgomery reduction, integer division and Karatsuba multiplication. Converting from an array of digits to polynomial basis is very simple. Consider the integer $y \equiv (a_2, a_1, a_0)_{\beta}$ and recall that $y = \sum_{i=0}^{2} a_i \beta^i$. Simply replace $\beta$ with $x$ and the expression is in polynomial basis. For example, $f(x) = 8x + 9$ is the polynomial basis representation for $89$ using radix ten. That is, $f(10) = 8(10) + 9 = 89$. \subsection{Multiplication by $x$} Given a polynomial in $x$ such as $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0$ multiplying by $x$ amounts to shifting the coefficients up one degree. In this case $f(x) \cdot x = a_n x^{n+1} + a_{n-1} x^n + ... + a_0 x$. From a scalar basis point of view multiplying by $x$ is equivalent to multiplying by the integer $\beta$. \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_lshd}. \\ \textbf{Input}. One mp\_int $a$ and an integer $b$ \\ \textbf{Output}. $a \leftarrow a \cdot \beta^b$ (Multiply by $x^b$). \\ \hline \\ 1. If $b \le 0$ then return(\textit{MP\_OKAY}). \\ 2. If $a.alloc < a.used + b$ then grow $a$ to at least $a.used + b$ digits. (\textit{hint: use mp\_grow}). \\ 3. If the reallocation failed return(\textit{MP\_MEM}). \\ 4. $a.used \leftarrow a.used + b$ \\ 5. $i \leftarrow a.used - 1$ \\ 6. $j \leftarrow a.used - 1 - b$ \\ 7. for $n$ from $a.used - 1$ to $b$ do \\ \hspace{3mm}7.1 $a_{i} \leftarrow a_{j}$ \\ \hspace{3mm}7.2 $i \leftarrow i - 1$ \\ \hspace{3mm}7.3 $j \leftarrow j - 1$ \\ 8. for $n$ from 0 to $b - 1$ do \\ \hspace{3mm}8.1 $a_n \leftarrow 0$ \\ 9. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_lshd} \end{figure} \textbf{Algorithm mp\_lshd.} This algorithm multiplies an mp\_int by the $b$'th power of $x$. This is equivalent to multiplying by $\beta^b$. The algorithm differs from the other algorithms presented so far as it performs the operation in place instead storing the result in a seperate location. The algorithm will return success immediately if $b \le 0$ since the rest of algorithm is only valid when $b > 0$. First the destination $a$ is grown as required to accomodate the result. The counters $i$ and $j$ are used to form a \textit{sliding window} over the digits of $a$ of length $b$. The head of the sliding window is at $i$ (\textit{the leading digit}) and the tail at $j$ (\textit{the trailing digit}). The loop on step 7 copies the digit from the tail to the head. In each iteration the window is moved down one digit. The last loop on step 8 sets the lower $b$ digits to zero. \newpage FIGU,sliding_window,Sliding Window Movement EXAM,bn_mp_lshd.c The if statement on line @24,if@ ensures that the $b$ variable is greater than zero. The \textbf{used} count is incremented by $b$ before the copy loop begins. This elminates the need for an additional variable in the for loop. The variable $tmpa$ on line @42,tmpa@ is an alias for the leading digit while $tmpaa$ on line @45,tmpaa@ is an alias for the trailing edge. The aliases form a window of exactly $b$ digits over the input. \subsection{Division by $x$} Division by powers of $x$ is easily achieved by shifting the digits right and removing any that will end up to the right of the zero'th digit. \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_rshd}. \\ \textbf{Input}. One mp\_int $a$ and an integer $b$ \\ \textbf{Output}. $a \leftarrow a / \beta^b$ (Divide by $x^b$). \\ \hline \\ 1. If $b \le 0$ then return. \\ 2. If $a.used \le b$ then do \\ \hspace{3mm}2.1 Zero $a$. (\textit{hint: use mp\_zero}). \\ \hspace{3mm}2.2 Return. \\ 3. $i \leftarrow 0$ \\ 4. $j \leftarrow b$ \\ 5. for $n$ from 0 to $a.used - b - 1$ do \\ \hspace{3mm}5.1 $a_i \leftarrow a_j$ \\ \hspace{3mm}5.2 $i \leftarrow i + 1$ \\ \hspace{3mm}5.3 $j \leftarrow j + 1$ \\ 6. for $n$ from $a.used - b$ to $a.used - 1$ do \\ \hspace{3mm}6.1 $a_n \leftarrow 0$ \\ 7. Clamp excess digits. (\textit{hint: use mp\_clamp}). \\ 8. Return. \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_rshd} \end{figure} \textbf{Algorithm mp\_rshd.} This algorithm divides the input in place by the $b$'th power of $x$. It is analogous to dividing by a $\beta^b$ but much quicker since it does not require single precision division. This algorithm does not actually return an error code as it cannot fail. If the input $b$ is less than one the algorithm quickly returns without performing any work. If the \textbf{used} count is less than or equal to the shift count $b$ then it will simply zero the input and return. After the trivial cases of inputs have been handled the sliding window is setup. Much like the case of algorithm mp\_lshd a sliding window that is $b$ digits wide is used to copy the digits. Unlike mp\_lshd the window slides in the opposite direction from the trailing to the leading digit. Also the digits are copied from the leading to the trailing edge. Once the window copy is complete the upper digits must be zeroed. Finally algorithm mp\_clamp is used to trim excess digits. EXAM,bn_mp_rshd.c The only noteworthy element of this routine is the lack of a return type. This function cannot fail and as such it is more optimal to not return anything. \section{Powers of Two} Now that algorithms for moving single bits as well as whole digits exist algorithms for moving the ``in between'' distances are required. For example, to quickly multiply by $2^k$ for any $k$ without using a full multiplier algorithm would prove useful. Instead of performing single shifts $k$ times to achieve a multiplication by $2^{\pm k}$ a mixture of whole digit shifting and partial digit shifting is employed. \subsection{Multiplication by Power of Two} \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_mul\_2d}. \\ \textbf{Input}. One mp\_int $a$ and an integer $b$ \\ \textbf{Output}. $c \leftarrow a \cdot 2^b$. \\ \hline \\ 1. $c \leftarrow a$. (\textit{hint: use mp\_copy}) \\ 2. If $c.alloc < c.used + \lfloor b / lg(\beta) \rfloor + 2$ then grow $c$ accordingly. \\ 3. If the reallocation failed return(\textit{MP\_MEM}). \\ 4. If $b \ge lg(\beta)$ then \\ \hspace{3mm}4.1 $c \leftarrow c \cdot \beta^{\lfloor b / lg(\beta) \rfloor}$ (\textit{hint: use mp\_lshd}). \\ \hspace{3mm}4.2 If step 4.1 failed return(\textit{MP\_MEM}). \\ 5. $d \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\ 6. If $d \ne 0$ then do \\ \hspace{3mm}6.1 $mask \leftarrow 2^d$ \\ \hspace{3mm}6.2 $r \leftarrow 0$ \\ \hspace{3mm}6.3 for $n$ from $0$ to $c.used - 1$ do \\ \hspace{6mm}6.3.1 $rr \leftarrow c_n >> (lg(\beta) - d) \mbox{ (mod }mask\mbox{)}$ \\ \hspace{6mm}6.3.2 $c_n \leftarrow (c_n << d) + r \mbox{ (mod }\beta\mbox{)}$ \\ \hspace{6mm}6.3.3 $r \leftarrow rr$ \\ \hspace{3mm}6.4 If $r > 0$ then do \\ \hspace{6mm}6.4.1 $c_{c.used} \leftarrow r$ \\ \hspace{6mm}6.4.2 $c.used \leftarrow c.used + 1$ \\ 7. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_mul\_2d} \end{figure} \textbf{Algorithm mp\_mul\_2d.} This algorithm multiplies $a$ by $2^b$ and stores the result in $c$. The algorithm uses algorithm mp\_lshd and a derivative of algorithm mp\_mul\_2 to quickly compute the product. First the algorithm will multiply $a$ by $x^{\lfloor b / lg(\beta) \rfloor}$ which will ensure that the remainder multiplicand is less than $\beta$. For example, if $b = 37$ and $\beta = 2^{28}$ then this step will multiply by $x$ leaving a multiplication by $2^{37 - 28} = 2^{9}$ left. The logarithm of the residue is calculated on step 5. If it is non-zero a modified shift loop is used to calculate the remaining product. Essentially the loop is a generic version of algorith mp\_mul2 designed to handle any shift count in the range $1 \le x < lg(\beta)$. The $mask$ variable is used to extract the upper $d$ bits to form the carry for the next iteration. This algorithm is loosely measured as a $O(2n)$ algorithm which means that if the input is $n$-digits that it takes $2n$ ``time'' to complete. It is possible to optimize this algorithm down to a $O(n)$ algorithm at a cost of making the algorithm slightly harder to follow. EXAM,bn_mp_mul_2d.c Notes to be revised when code is updated. -- Tom \subsection{Division by Power of Two} \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_div\_2d}. \\ \textbf{Input}. One mp\_int $a$ and an integer $b$ \\ \textbf{Output}. $c \leftarrow \lfloor a / 2^b \rfloor, d \leftarrow a \mbox{ (mod }2^b\mbox{)}$. \\ \hline \\ 1. If $b \le 0$ then do \\ \hspace{3mm}1.1 $c \leftarrow a$ (\textit{hint: use mp\_copy}) \\ \hspace{3mm}1.2 $d \leftarrow 0$ (\textit{hint: use mp\_zero}) \\ \hspace{3mm}1.3 Return(\textit{MP\_OKAY}). \\ 2. $c \leftarrow a$ \\ 3. $d \leftarrow a \mbox{ (mod }2^b\mbox{)}$ (\textit{hint: use mp\_mod\_2d}) \\ 4. If $b \ge lg(\beta)$ then do \\ \hspace{3mm}4.1 $c \leftarrow \lfloor c/\beta^{\lfloor b/lg(\beta) \rfloor} \rfloor$ (\textit{hint: use mp\_rshd}). \\ 5. $k \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\ 6. If $k \ne 0$ then do \\ \hspace{3mm}6.1 $mask \leftarrow 2^k$ \\ \hspace{3mm}6.2 $r \leftarrow 0$ \\ \hspace{3mm}6.3 for $n$ from $c.used - 1$ to $0$ do \\ \hspace{6mm}6.3.1 $rr \leftarrow c_n \mbox{ (mod }mask\mbox{)}$ \\ \hspace{6mm}6.3.2 $c_n \leftarrow (c_n >> k) + (r << (lg(\beta) - k))$ \\ \hspace{6mm}6.3.3 $r \leftarrow rr$ \\ 7. Clamp excess digits of $c$. (\textit{hint: use mp\_clamp}) \\ 8. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_div\_2d} \end{figure} \textbf{Algorithm mp\_div\_2d.} This algorithm will divide an input $a$ by $2^b$ and produce the quotient and remainder. The algorithm is designed much like algorithm mp\_mul\_2d by first using whole digit shifts then single precision shifts. This algorithm will also produce the remainder of the division by using algorithm mp\_mod\_2d. EXAM,bn_mp_div_2d.c The implementation of algorithm mp\_div\_2d is slightly different than the algorithm specifies. The remainder $d$ may be optionally ignored by passing \textbf{NULL} as the pointer to the mp\_int variable. The temporary mp\_int variable $t$ is used to hold the result of the remainder operation until the end. This allows $d = a$ to be true without overwriting the input before they are no longer required. The remainder of the source code is essentially the same as the source code for mp\_mul\_2d. (-- Fix this paragraph up later, Tom). \subsection{Remainder of Division by Power of Two} The last algorithm in the series of polynomial basis power of two algorithms is calculating the remainder of division by $2^b$. This algorithm benefits from the fact that in twos complement arithmetic $a \mbox{ (mod }2^b\mbox{)}$ is the same as $a$ AND $2^b - 1$. \begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_mod\_2d}. \\ \textbf{Input}. One mp\_int $a$ and an integer $b$ \\ \textbf{Output}. $c \leftarrow a \mbox{ (mod }2^b\mbox{)}$. \\ \hline \\ 1. If $b \le 0$ then do \\ \hspace{3mm}1.1 $c \leftarrow 0$ (\textit{hint: use mp\_zero}) \\ \hspace{3mm}1.2 Return(\textit{MP\_OKAY}). \\ 2. If $b > a.used \cdot lg(\beta)$ then do \\ \hspace{3mm}2.1 $c \leftarrow a$ (\textit{hint: use mp\_copy}) \\ \hspace{3mm}2.2 Return the result of step 2.1. \\ 3. $c \leftarrow a$ \\ 4. If step 3 failed return(\textit{MP\_MEM}). \\ 5. for $n$ from $\lceil b / lg(\beta) \rceil$ to $c.used$ do \\ \hspace{3mm}5.1 $c_n \leftarrow 0$ \\ 6. $k \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\ 7. $c_{\lfloor b / lg(\beta) \rfloor} \leftarrow c_{\lfloor b / lg(\beta) \rfloor} \mbox{ (mod }2^{k}\mbox{)}$. \\ 8. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_mod\_2d} \end{figure} \textbf{Algorithm mp\_mod\_2d.} This algorithm will quickly calculate the value of $a \mbox{ (mod }2^b\mbox{)}$. First if $b$ is less than or equal to zero the result is set to zero. If $b$ is greater than the number of bits in $a$ then it simply copies $a$ to $c$ and returns. Otherwise, $a$ is copied to $b$, leading digits are removed and the remaining leading digit is trimed to the exact bit count. EXAM,bn_mp_mod_2d.c -- Add comments later, Tom. \section*{Exercises} \begin{tabular}{cl} $\left [ 3 \right ] $ & Devise an algorithm that performs $a \cdot 2^b$ for generic values of $b$ \\ & in $O(n)$ time. \\ &\\ $\left [ 3 \right ] $ & Devise an efficient algorithm to multiply by small low hamming \\ & weight values such as $3$, $5$ and $9$. Extend it to handle all values \\ & upto $64$ with a hamming weight less than three. \\ &\\ $\left [ 2 \right ] $ & Modify the preceding algorithm to handle values of the form \\ & $2^k - 1$ as well. \\ &\\ $\left [ 3 \right ] $ & Using only algorithms mp\_mul\_2, mp\_div\_2 and mp\_add create an \\ & algorithm to multiply two integers in roughly $O(2n^2)$ time for \\ & any $n$-bit input. Note that the time of addition is ignored in the \\ & calculation. \\ & \\ $\left [ 5 \right ] $ & Improve the previous algorithm to have a working time of at most \\ & $O \left (2^{(k-1)}n + \left ({2n^2 \over k} \right ) \right )$ for an appropriate choice of $k$. Again ignore \\ & the cost of addition. \\ & \\ $\left [ 1 \right ] $ & There exists an improvement on the previous algorithm to \\ & slightly reduce the number of additions required. Modify the \\ & previous algorithm to include this improvement. \\ & \\ $\left [ 2 \right ] $ & Devise a chart to find optimal values of $k$ for the previous problem \\ & for $n = 64 \ldots 1024$ in steps of $64$. \\ & \\ $\left [ 2 \right ] $ & Using only algorithms mp\_abs and mp\_sub devise another method for \\ & calculating the result of a signed comparison. \\ & \end{tabular} \chapter{Multiplication and Squaring} \section{The Multipliers} For most number theoretic systems including public key cryptographic algorithms the set of algorithms collectively known as the ``multipliers'' form the most important subset of algorithms of any multiple precision integer package. The set of multipliers include multiplication, squaring and modular reduction algorithms. The importance of these algorithms is driven by the fact that most popular public key algorithms are based on modular exponentiation. That is performing $d \equiv a^b \mbox{ (mod }c\mbox{)}$ for some arbitrary choice of $a$, $b$, $c$ and $d$. Roughly speaking the a modular exponentiation will spend about 40\% of the time in modular reductions, 35\% of the time in squaring and 25\% of the time in multiplications. Only a small trivial amount of time is spent on lower level algorithms such as mp\_clamp, mp\_init, etc... This chapter will discuss only two of the multipliers algorithms, multiplication and squaring. As will be discussed shortly very efficient multiplier algorithms are not always straightforward and deserve a lot of attention. \section{Multiplication} \subsection{The Baseline Multiplication} \index{baseline multiplication} Computing the product of two integers in software can be achieved using a trivial adaptation of the standard $O(n^2)$ long-hand multiplication algorithm school children are taught. The ``baseline multiplication'' algorithm is designed to act as the ``catch-all'' algorithm only called when the faster algorithms cannot be used. This algorithm does not use any particularly interesting optimizations. The first algorithm to review is the unsigned multiplication algorithm from which a signed multiplication algorithm can be established. One important facet of this algorithm to note is that it has been modified to only produce a certain amount of output digits as resolution. Recall that for a $n$ and $m$ digit input the product will be at most $n + m + 1$ digits. Therefore, this algorithm can be reduced to a full multiplier by telling it to produce $n + m + 1$ digits. Recall from ~GAMMA~ the definition of $\gamma$ as the number of bits in the type \textbf{mp\_digit}. We shall now extend this variable set to include $\alpha$ which shall represent the number of bits in the type \textbf{mp\_word}. This implies that $2^{\alpha} > 2 \cdot \beta^2$. The constant $\delta = 2^{\alpha - 2lg(\beta)}$ will represent the maximal weight of any column in a product (\textit{see ~COMBA~ for more information}). \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{s\_mp\_mul\_digs}. \\ \textbf{Input}. mp\_int $a$, mp\_int $b$ and an integer $digs$ \\ \textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert \mbox{ (mod }\beta^{digs}\mbox{)}$. \\ \hline \\ 1. If min$(a.used, b.used) < \delta$ then do \\ \hspace{3mm}1.1 Calculate $c = \vert a \vert \cdot \vert b \vert$ by the Comba method. \\ \hspace{3mm}1.2 Return the result of step 1.1 \\ \\ Allocate and initialize a temporary mp\_int. \\ 2. Init $t$ to be of size $digs$ \\ 3. If step 2 failed return(\textit{MP\_MEM}). \\ 4. $t.used \leftarrow digs$ \\ \\ Compute the product. \\ 5. for $ix$ from $0$ to $a.used - 1$ do \\ \hspace{3mm}5.1 $u \leftarrow 0$ \\ \hspace{3mm}5.2 $pb \leftarrow \mbox{min}(b.used, digs - ix)$ \\ \hspace{3mm}5.3 If $pb < 1$ then goto step 6. \\ \hspace{3mm}5.4 for $iy$ from $0$ to $pb - 1$ do \\ \hspace{6mm}5.4.1 $\hat r \leftarrow t_{iy + ix} + a_{ix} \cdot b_{iy} + u$ \\ \hspace{6mm}5.4.2 $t_{iy + ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ \hspace{6mm}5.4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ \hspace{3mm}5.5 if $ix + iy < digs$ then do \\ \hspace{6mm}5.5.1 $t_{ix + pb} \leftarrow u$ \\ 6. Clamp excess digits of $t$. \\ 7. Swap $c$ with $t$ \\ 8. Clear $t$ \\ 9. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm s\_mp\_mul\_digs} \end{figure} \textbf{Algorithm s\_mp\_mul\_digs.} This algorithm computes the unsigned product of two inputs $a$ and $c$ limited to an output precision of $digs$ digits. While it may seem a bit awkward to modify the function from its simple $O(n^2)$ description the usefulness of partial multipliers will arise in a future algorithm. The algorithm is loosely based on algorithm 14.12 from \cite[pp. 595]{HAC} and is similar to Algorithm M \cite[pp. 268]{TAOCPV2}. The algorithm differs from those cited references because it can produce a variable output precision regardless of the precision of the inputs. The first thing this algorithm checks for is whether a Comba multiplier can be used instead. That is if the minimal digit count of either input is less than $\delta$ the Comba method is used. After the Comba method is ruled out the baseline algorithm begins. A temporary mp\_int variable $t$ is used to hold the intermediate result of the product. This allows the algorithm to be used to compute products when either $a = c$ or $b = c$ without overwriting the inputs. All of step 5 is the infamous $O(n^2)$ multiplication loop slightly modified to only produce upto $digs$ digits of output. The $pb$ variable is given the count of digits to read from $b$ inside the nested loop. If $pb < 0$ then no more output digits can be produced and the algorithm will exit the loop. The best way to think of the loops are as a series of $pb \times 1$ multiplication. That is, in each pass of the innermost loop $a_{ix}$ is multiplied against $b$ and the result is added (\textit{with an appropriate shift}) to $t$. For example, consider multiplying $576$ by $241$. That is equivalent to computing $10^0(1)(576) + 10^1(4)(576) + 10^2(2)(576)$ which is best visualized as the following table. \begin{figure}[here] \begin{center} \begin{tabular}{|c|c|c|c|c|c|c|} \hline && & 5 & 7 & 6 & \\ \hline $\times$&& & 2 & 4 & 1 & \\ \hline &&&&&&\\ && & 5 & 7 & 6 & $10^0(1)(576)$ \\ &2 & 3 & 0 & 4 & 0 & $10^1(4)(576)$ \\ 1 & 1 & 5 & 2 & 0 & 0 & $10^2(2)(576)$ \\ \hline \end{tabular} \end{center} \caption{Long-Hand Multiplication Diagram} \end{figure} Each row of the product is added to the result after being shifted to the left (\textit{multiplied by a power of the radix}) by the appropriate count. That is in pass $ix$ of the inner loop the product is added starting at the $ix$'th digit of the reult. Step 5.4.1 introduces the hat symbol (\textit{e.g. $\hat x$}) which represents a double precision variable. The multiplication on that step is assumed to be a double wide output single precision multiplication. That is, two single precision variables are multiplied to produce a double precision result. The step is somewhat optimized from a long-hand multiplication algorithm because the carry from the addition in step 5.4.1 is forwarded through the nested loop. If the carry was ignored it would overflow the single precision digit $t_{ix+iy}$ and the result would be lost. At step 5.5 the nested loop is finished and any carry that was left over should be forwarded. That is provided $ix + iy < digs$ otherwise the carry is ignored since it will not be part of the result anyways. EXAM,bn_s_mp_mul_digs.c Lines @31,if@ to @35,}@ determine if the Comba method can be used first. The conditions for using the Comba routine are that min$(a.used, b.used) < \delta$ and the number of digits of output is less than \textbf{MP\_WARRAY}. This new constant is used to control the stack usage in the Comba routines. By default it is set to $\delta$ but can be reduced when memory is at a premium. Of particular importance is the calculation of the $ix+iy$'th column on lines @64,mp_word@, @65,mp_word@ and @66,mp_word@. Note how all of the variables are cast to the type \textbf{mp\_word}. That is to ensure that double precision operations are used instead of single precision. The multiplication on line @65,) * (@ is a bit of a GCC optimization. On the outset it looks like the compiler will have to use a double precision multiplication to produce the result required. Such an operation would be horribly slow on most processors and drag this to a crawl. However, GCC is smart enough to realize that double wide output single precision multipliers can be used. For example, the instruction ``MUL'' on the x86 processor can multiply two 32-bit values and produce a 64-bit result. \subsection{Faster Multiplication by the ``Comba'' Method} MARK,COMBA One of the huge drawbacks of the ``baseline'' algorithms is that at the $O(n^2)$ level the carry must be computed and propagated upwards. This makes the nested loop very sequential and hard to unroll and implement in parallel. The ``Comba'' method is named after little known (\textit{in cryptographic venues}) Paul G. Comba where in \cite{COMBA} a method of implementing fast multipliers that do not require nested carry fixup operations was presented. At the heart of algorithm is once again the long-hand algorithm for multiplication. Except in this case a slight twist is placed on how the columns of the result are produced. In the standard long-hand algorithm rows of products are produced then added together to form the final result. In the baseline algorithm the columns are added together to get the result instantaneously. In the Comba algorithm however, the columns of the result are produced entirely independently of each other. That is at the $O(n^2)$ level a simple multiplication and addition step is performed. Or more succintly that \begin{equation} x_n = \sum_{i+j = n} a_ib_j \end{equation} Where $x_n$ is the $n'th$ column of the output vector. To see how this works consider once again multiplying $576$ by $241$. \begin{figure}[here] \begin{small} \begin{center} \begin{tabular}{|c|c|c|c|c|c|} \hline & & 5 & 7 & 6 & First Input\\ \hline $\times$ & & 2 & 4 & 1 & Second Input\\ \hline & & $1 \cdot 5 = 5$ & $1 \cdot 7 = 7$ & $1 \cdot 6 = 6$ & First pass \\ & $4 \cdot 5 = 20$ & $4 \cdot 7+5=33$ & $4 \cdot 6+7=31$ & 6 & Second pass \\ $2 \cdot 5 = 10$ & $2 \cdot 7 + 20 = 34$ & $2 \cdot 6+33=45$ & 31 & 6 & Third pass \\ \hline 10 & 34 & 45 & 31 & 6 & Final Result \\ \hline \end{tabular} \end{center} \end{small} \caption{Comba Multiplication Diagram} \end{figure} At this point the vector $x = \left < 10, 34, 45, 31, 6 \right >$ is the result of the first step of the Comba multipler. Now the columns must be fixed by propagating the carry upwards. The following trivial algorithm will accomplish this. \begin{enumerate} \item for $n$ from 0 to $k - 1$ do \item \hspace{3mm} $x_{n+1} \leftarrow x_{n+1} + \lfloor x_{n}/\beta \rfloor$ \item \hspace{3mm} $x_{n} \leftarrow x_{n} \mbox{ (mod }\beta\mbox{)}$ \end{enumerate} With that algorithm and $k = 5$ and $\beta = 10$ the following vector is produced $y = \left < 1, 3, 8, 8, 1, 6 \right >$. In this case $241 \cdot 576$ is in fact $138816$ and the procedure succeeded. If the algorithm is correct and as will be demonstrated shortly more efficient than the baseline algorithm why not simply always use this algorithm? \subsubsection{Column Weight.} At the nested $O(n^2)$ level the Comba method adds the product of two single precision variables to a each column of the output independently. A serious obstacle is if the carry is lost due to lack of precision before the algorithm has a chance to fix the carries. For example, in the multiplication of two three-digit numbers the third column of output will be the sum of three single precision multiplications. If the precision of the accumulator for the output digits is less then $3 \cdot (\beta - 1)^2$ then an overflow can occur and the carry information will be lost. For any $m$ and $n$ digit input the maximal weight of any column is min$(m, n)$ which is fairly obvious. The maximal number of terms in any column of a product is known as the ``column weight'' and strictly governs when the algorithm can be used. Recall from earlier that a double precision type has $\alpha$ bits of resolution and a single precision digit has $lg(\beta)$ bits of precision. Given these two quantities we may not violate the following \begin{equation} k \cdot \left (\beta - 1 \right )^2 < 2^{\alpha} \end{equation} Which reduces to \begin{equation} k \cdot \left ( \beta^2 - 2\beta + 1 \right ) < 2^{\alpha} \end{equation} Let $\rho = lg(\beta)$ represent the number of bits in a single precision digit. By further re-arrangement of the equation the final solution is found. \begin{equation} k \cdot \left (2^{2\rho} - 2^{\rho + 1} + 1 \right ) < 2^{\alpha} \end{equation} The defaults for LibTomMath are $\beta = 2^{28}, \alpha = 2^{64}$ which simplies to $72057593501057025 \cdot k < 2^{64}$ which when divided out result in $k < 257$. This implies that the smallest input may not have more than $256$ digits if the Comba method is to be used in this configuration. This is quite satisfactory for most applications since $256$ digits would be allow for numbers in the range of $2^{7168}$ which is much larger than the typical $2^{100}$ to $2^{4000}$ range most public key cryptographic algorithms use. \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{fast\_s\_mp\_mul\_digs}. \\ \textbf{Input}. mp\_int $a$, mp\_int $b$ and an integer $digs$ \\ \textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert \mbox{ (mod }\beta^{digs}\mbox{)}$. \\ \hline \\ Place an array of \textbf{MP\_WARRAY} double precision digits named $\hat W$ on the stack. \\ 1. If $c.alloc < digs$ then grow $c$ to $digs$ digits. (\textit{hint: use mp\_grow}) \\ 2. If step 1 failed return(\textit{MP\_MEM}).\\ \\ Zero the temporary array $\hat W$. \\ 3. for $n$ from $0$ to $digs - 1$ do \\ \hspace{3mm}3.1 $\hat W_n \leftarrow 0$ \\ \\ Compute the columns. \\ 4. for $ix$ from $0$ to $a.used - 1$ do \\ \hspace{3mm}4.1 $pb \leftarrow \mbox{min}(b.used, digs - ix)$ \\ \hspace{3mm}4.2 If $pb < 1$ then goto step 5. \\ \hspace{3mm}4.3 for $iy$ from $0$ to $pb - 1$ do \\ \hspace{6mm}4.3.1 $\hat W_{ix+iy} \leftarrow \hat W_{ix+iy} + a_{ix}b_{iy}$ \\ \\ Propagate the carries upwards. \\ 5. $oldused \leftarrow c.used$ \\ 6. $c.used \leftarrow digs$ \\ 7. If $digs > 1$ then do \\ \hspace{3mm}7.1. for $ix$ from $1$ to $digs - 1$ do \\ \hspace{6mm}7.1.1 $\hat W_{ix} \leftarrow \hat W_{ix} + \lfloor \hat W_{ix-1} / \beta \rfloor$ \\ \hspace{6mm}7.1.2 $c_{ix - 1} \leftarrow \hat W_{ix - 1} \mbox{ (mod }\beta\mbox{)}$ \\ 8. else do \\ \hspace{3mm}8.1 $ix \leftarrow 0$ \\ 9. $c_{ix} \leftarrow \hat W_{ix} \mbox{ (mod }\beta\mbox{)}$ \\ \\ Zero excess digits. \\ 10. If $digs < oldused$ then do \\ \hspace{3mm}10.1 for $n$ from $digs$ to $oldused - 1$ do \\ \hspace{6mm}10.1.1 $c_n \leftarrow 0$ \\ 11. Clamp excessive digits of $c$. (\textit{hint: use mp\_clamp}) \\ 12. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm fast\_s\_mp\_mul\_digs} \end{figure} \textbf{Algorithm fast\_s\_mp\_mul\_digs.} This algorithm performs the unsigned multiplication of $a$ and $b$ using the Comba method limited to $digs$ digits of precision. The algorithm essentially peforms the same calculation as algorithm s\_mp\_mul\_digs but much faster. The array $\hat W$ is meant to be on the stack when the algorithm is used. The size of the array does not change which is ideal. Note also that unlike algorithm s\_mp\_mul\_digs no temporary mp\_int is required since the result is calculated in place in $\hat W$. The $O(n^2)$ loop on step four is where the Comba method starts to show through. First there is no carry variable in the loop. Second the double precision multiply and add step does not have a carry fixup of any sort. In fact the nested loop is very simple and can be implemented in parallel. What makes the Comba method so attractive is that the carry propagation only takes place outside the $O(n^2)$ nested loop. For example, if the cost in terms of time of a multiply and add is $p$ and the cost of a carry propagation is $q$ then a baseline multiplication would require $O \left ((p + q)n^2 \right )$ time to multiply two $n$-digit numbers. The Comba method only requires $pn^2 + qn$ time, however, in practice the speed increase is actually much more. With $O(n)$ space the algorithm can be reduced to $O(pn + qn)$ time by implementing the $n$ multiply and add operations in the nested loop in parallel. The carry propagation loop on step 7 is fairly straightforward. It could have been written phased the other direction, that is, to assign to $c_{ix}$ instead of $c_{ix-1}$ in each iteration. However, it would still require pre-caution to make sure that $\hat W_{ix+1}$ is not beyond the \textbf{MP\_WARRAY} words set aside. EXAM,bn_fast_s_mp_mul_digs.c The memset on line @47,memset@ clears the initial $\hat W$ array to zero in a single step. Like the slower baseline multiplication implementation a series of aliases (\textit{lines @67, tmpx@, @70, tmpy@ and @75,_W@}) are used to simplify the inner $O(n^2)$ loop. In this case a new alias $\_\hat W$ has been added which refers to the double precision columns offset by $ix$ in each pass. The inner loop on line @84,mp_word@ is where the algorithm will spend the majority of the time. Which is why it has been stripped to the bones of any extra baggage\footnote{Hence the pointer aliases.}. On x86 processors the multiply and add amounts to at the very least five instructions (\textit{two loads, two additions, one multiply}) while on the ARMv4 processors it amounts to only three (\textit{one load, one store, one multiply-add}). On both the x86 and ARMv4 processors GCC v3.2 does a very good job at unrolling the loop and scheduling it so there are very few dependency stalls. In theory the difference between the baseline and comba algorithms is a mere $O(qn)$ time difference. However, in the $O(n^2)$ nested loop of the baseline method there are dependency stalls as the algorithm must wait for the multiplier to finish before propagating the carry to the next digit. As a result fewer of the often multiple execution units\footnote{The AMD Athlon has three execution units and the Intel P4 has four.} can be simultaneously used. \subsection{Multiplication at New Bounds by Karatsuba Method} So far two methods of multiplication have been presented. Both of the algorithms require asymptotically $O(n^2)$ time to multiply two $n$-digit numbers together. While the Comba method is much faster than the baseline algorithm it still requires far too much time to multiply large inputs together. In fact it was not until \cite{KARA} in 1962 that a faster algorithm had been proposed at all. The idea behind Karatsubas method is that an input can be represented in polynomial basis as two halves then multiplied. For example, if $f(x) = ax + b$ and $g(x) = cx + b$ then the product of the two polynomials $h(x) = f(x)g(x)$ will allow $h(\beta) = (f(\beta))(g(\beta))$. So how does this help? First expand the product $h(x)$. \begin{center} \begin{tabular}{rcl} $h(x)$ & $=$ & $f(x)g(x)$ \\ & $=$ & $(ax + b)(cx + d)$ \\ & $=$ & $acx^2 + adx + bcx + bd$ \\ \end{tabular} \end{center} The next equation is a bit of genius on the part of Karatsuba. He proved that the previous equation is equivalent to \begin{equation} h(x) = acx^2 + ((a - c)(b - d) + bd + ac)x + bd \end{equation} Essentially the proof lies in some fairly light algebraic number theory (\textit{see \cite{KARAP} for details}) that is not important for the discussion. At first glance it appears that the Karatsuba method is actually harder than the straight $O(n^2)$ approach. However, further investigation will prove otherwise. The first important observation is that both $f(x)$ and $g(x)$ are the polynomial basis representation of two-digit numbers. This means that $\left < a, b, c, d \right >$ are single digit values. Using either the baseline or straight polynomial multiplication the old method requires $O \left (4(n/2)^2 \right ) = O(n^2)$ single precision multiplications. Looking closer at Karatsubas equation there are only three unique multiplications required which are $ac$, $bd$ and $(a - c)(b - d)$. As a result only $O \left (3 \cdot (n/2)^2 \right ) = O \left ( {3 \over 4}n^2 \right )$ multiplications are required. So far the algorithm has been discussed from the point of view of ``two-digit'' numbers. However, there is no reason why two digits implies a range of $\beta^2$. It could just as easily represent a range of $\left (\beta^k \right)^2$ as well. For example, the polynomial $f(x) = a_3x^3 + a_2x^2 + a_1x + a_0$ could also be written as $f'(x) = a'_1x + a'_0$ where $f(\beta) = f'(\beta^2)$. Fortunately representing an integer which is already in an array of radix-$\beta$ digits in polynomial basis in terms of a power of $\beta$ is very simple. \subsubsection{Recursion} The Karatsuba multiplication algorithm can be applied to practically any size of input. Therefore, it is possible that the Karatsuba method itself be used for the three multiplications required. For example, when multiplying two four-digit numbers there will be three multiplications of two-digit numbers. In this case the smaller multiplication requires $p(n) = {3 \over 4}n^2$ time to complete while the larger multiplication requires $q(n) = 3 \cdot p(n/2)$ multiplications. By expanding $q(n)$ the following equation is achieved. \begin{center} \begin{tabular}{rcl} $q(n)$ & $=$ & $3 \cdot p(n/2)$ \\ & $=$ & $3 \cdot (3 \cdot ((n/2)/2)^2)$ \\ & $=$ & $9 \cdot (n/4)^2$ \\ & $=$ & ${9 \over 16}n^2$ \\ \end{tabular} \end{center} The generic expression for the multiplicand is simply $\left ( {3 \over 4} \right )^k$ for $k \ge 1$ recurisions. The maximal number of recursions is approximately $lg(n)$. Putting this all in terms of a base $n$ logarithm the asymptotic running time can be deduced. \begin{center} \begin{tabular}{rcl} $lg_n \left ( \left ( {3 \over 4} \right )^{lg_2 n} \cdot n^2 \right )$ & $=$ & $lg_2 n \cdot lg_n \left ( { 3 \over 4 } \right ) + 2$ \\ & $=$ & $\left ( {log N \over log 2} \right ) \cdot \left ( {log \left ( {3 \over 4} \right ) \over log N } \right ) + 2$ \\ & $=$ & ${ log 3 - log 2^2 + 2 \cdot log 2} \over log 2$ \\ & $=$ & $log 3 \over log 2$ \\ \end{tabular} \end{center} Which leads to a running time of $O \left ( n^{lg(3)} \right )$ which is approximately $O(n^{1.584})$. This can lead to impressive savings with fairly moderate sized numbers. For example, when multiplying two 128-digit numbers the Karatsuba method saves $14,197$ (\textit{or $86\%$ of the total}) single precision multiplications. The immediate question becomes why not simply use Karatsuba multiplication all the time and forget about the baseline and Comba algorithms? \subsubsection{Overhead} While the Karatsuba method saves on the number of single precision multiplications required this savings is not entirely free. The product of three half size products must be stored somewhere as well as four additions and two subtractions performed. These operations incur sufficient overhead that often for fairly trivial sized inputs the Karatsuba method is slower. \index{cutoff point} The \textit{cutoff point} for Karatsuba multiplication is the point at which the Karatsuba multiplication and baseline (\textit{or Comba}) meet. For the purposes of this discussion call this value $x$. For any input with $n$ digits such that $n < x$ Karatsuba multiplication will be slower and for $n > x$ it will be faster. Often the break between the two algorithms is not so clean cut in reality. The cleaner the cut the more efficient multiplication will be which is why tuning the multiplication is a very important process. For example, a properly tuned Karatsuba multiplication algorithm can multiply two $4,096$ bit numbers up to five times faster on an Athlon processor compared to the standard baseline algorithm. The exact placement of the value of $x$ depends on several key factors. The cost of allocating storage for the temporary variables, the cost of performing the additions and most importantly the cost of performing a single precision multiplication. With a processor where single precision multiplication is fast\footnote{The AMD Athlon for instance has a six cycle multiplier compared to the Intel P4 which has a 15 cycle multiplier.} the cutoff point will move upwards. Similarly with a slower processor the cutoff point will move downwards. \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_karatsuba\_mul}. \\ \textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ \textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert$ \\ \hline \\ 1. $B \leftarrow \mbox{min}(a.used, b.used)/2$ \\ 2. Init the following mp\_int variables: $x0$, $x1$, $y0$, $y1$, $t1$, $x0y0$, $x1y1$.\\ 3. If step 2 failed then return(\textit{MP\_MEM}). \\ \\ Split the input. e.g. $a = x1 \cdot \beta^B + x0$ \\ 4. $x0 \leftarrow a \mbox{ (mod }\beta^B\mbox{)}$ (\textit{hint: use mp\_mod\_2d}) \\ 5. $y0 \leftarrow b \mbox{ (mod }\beta^B\mbox{)}$ \\ 6. $x1 \leftarrow \lfloor a / \beta^B \rfloor$ (\textit{hint: use mp\_rshd}) \\ 7. $y1 \leftarrow \lfloor b / \beta^B \rfloor$ \\ \\ Calculate the three products. \\ 8. $x0y0 \leftarrow x0 \cdot y0$ (\textit{hint: use mp\_mul}) \\ 9. $x1y1 \leftarrow x1 \cdot y1$ \\ 10. $t1 \leftarrow x1 - x0$ (\textit{hint: use mp\_sub}) \\ 11. $x0 \leftarrow y1 - y0$ \\ 12. $t1 \leftarrow t1 \cdot x0$ \\ \\ Calculate the middle term. \\ 13. $x0 \leftarrow x0y0 + x1y1$ \\ 14. $t1 \leftarrow x0 - t1$ \\ \\ Calculate the final product. \\ 15. $t1 \leftarrow t1 \cdot \beta^B$ (\textit{hint: use mp\_lshd}) \\ 16. $x1y1 \leftarrow x1y1 \cdot \beta^{2B}$ \\ 17. $t1 \leftarrow x0y0 + t1$ \\ 18. $c \leftarrow t1 + x1y1$ \\ 19. Clear all of the temporary variables. \\ 20. Return(\textit{MP\_OKAY}).\\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_karatsuba\_mul} \end{figure} \textbf{Algorithm mp\_karatsuba\_mul.} \section{Squaring} \subsection{The Baseline Squaring Algorithm} \subsection{Faster Squaring by the ``Comba'' Method} \subsection{Karatsuba Squaring} \section{Tuning Algorithms} \subsection{How to Tune Karatsuba Algorithms} \chapter{Modular Reductions} \section{Basics of Modular Reduction} \section{The Barrett Reduction} \section{The Montgomery Reduction} \subsection{Faster ``Comba'' Montgomery Reduction} \subsection{Example Montgomery Algorithms} \section{The Diminished Radix Algorithm} \section{Algorithm Comparison} \chapter{Exponentiation} \section{Single Digit Exponentiation} \section{Modular Exponentiation} \subsection{General Case} \subsection{Odd or Diminished Radix Moduli} \section{Quick Power of Two} \chapter{Higher Level Algorithms} \section{Integer Division with Remainder} \section{Single Digit Helpers} \subsection{Single Digit Addition} \subsection{Single Digit Subtraction} \subsection{Single Digit Multiplication} \subsection{Single Digit Division} \subsection{Single Digit Modulo} \subsection{Single Digit Root Extraction} \section{Random Number Generation} \section{Formatted Output} \subsection{Getting The Output Size} \subsection{Generating Radix-n Output} \subsection{Reading Radix-n Input} \section{Unformatted Output} \subsection{Getting The Output Size} \subsection{Generating Output} \subsection{Reading Input} \chapter{Number Theoretic Algorithms} \section{Greatest Common Divisor} \section{Least Common Multiple} \section{Jacobi Symbol Computation} \section{Modular Inverse} \subsection{General Case} \subsection{Odd Moduli} \section{Primality Tests} \subsection{Trial Division} \subsection{The Fermat Test} \subsection{The Miller-Rabin Test} \subsection{Primality Test in a Bottle} \subsection{The Next Prime} \section{Root Extraction} \backmatter \appendix \begin{thebibliography}{ABCDEF} \bibitem[1]{TAOCPV2} Donald Knuth, \textit{The Art of Computer Programming}, Third Edition, Volume Two, Seminumerical Algorithms, Addison-Wesley, 1998 \bibitem[2]{HAC} A. Menezes, P. van Oorschot, S. Vanstone, \textit{Handbook of Applied Cryptography}, CRC Press, 1996 \bibitem[3]{ROSE} Michael Rosing, \textit{Implementing Elliptic Curve Cryptography}, Manning Publications, 1999 \bibitem[4]{COMBA} Paul G. Comba, \textit{Exponentiation Cryptosystems on the IBM PC}. IBM Systems Journal 29(4): 526-538 (1990) \bibitem[5]{KARA} A. Karatsuba, Doklay Akad. Nauk SSSR 145 (1962), pp.293-294 \bibitem[6]{KARAP} Andre Weimerskirch and Christof Paar, \textit{Generalizations of the Karatsuba Algorithm for Polynomial Multiplication}, Submitted to Design, Codes and Cryptography, March 2002 \end{thebibliography} \input{tommath.ind} \chapter{Appendix} \subsection*{Appendix A -- Source Listing of tommath.h} The following is the source listing of the header file ``tommath.h'' for the LibTomMath project. It contains many of the definitions used throughout the code such as \textbf{mp\_int}, \textbf{MP\_PREC} and so on. The header is presented here for completeness. LIST,tommath.h \end{document}