\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 \\ \\ Greg Rose \\ QUALCOMM Australia \\ \end{tabular} %\end{small} } } \maketitle This text in its entirety is copyright \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 redistributable as long as it is packaged along with the LibTomMath library 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 that are considerably larger? Most advancements in fast multiple precision arithmetic stem from the desire for faster cryptographic primitives. However, cryptography is not the only field of study that can benefit from 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 specified. Since IEEE is meant to be implemented in hardware the precision of the mantissa is often fairly small (\textit{23, 48 and 64 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 over the ring of integers denoted by $\Z$ is performed by routines that are collectively and casually referred to 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 of the real value to a given precision is defined loosely as the proximity of the real value to a given representation. 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 learned 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, line up the partial results, 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 \cite[HAC]{HAC} and \cite{TAOCPV2} give considerably detailed explanations of the theoretical aspects of the algorithms and very little regarding the practical aspects. 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 dividend's 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 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 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, $x = (1,2,3)_{10}$ would represent the integer $1\cdot 10^2 + 2\cdot10^1 + 3\cdot10^0 = 123$. 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. The $\lfloor \mbox{ } \rfloor$ brackets represent a value truncated and rounded down to the nearest integer. The $\lceil \mbox{ } \rceil$ brackets represent a value truncated and rounded up to the nearest integer. Typically when the $/$ division symbol is used the intention is to perform an integer division. For example, $5/2 = 2$ which will often be written as $\lfloor 5/2 \rfloor = 2$ for clarity. When a value is presented as a fraction such as $5 \over 2$ a real value division is implied. \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, but to be thought provoking. Wherever possible the problems are forward minded stating problems that will be answered in subsequent chapters. The reader is encouraged to finish the exercises as they appear to get a better understanding of the subject material. 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 LibTomMath?} LibTomMath is a free and open source multiple precision library written in portable ISO C source code. By portable it is meant that the library does not contain any code that is computer platform dependent or otherwise problematic to use on any given platform. The library has been successfully tested under numerous operating systems including 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 applications such as public key cryptosystems. \section{Goals of 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 highly efficient 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 Application Programing Interface (\textit{API}) has been kept as simple as possible. Often generic place holder routines will make use of specialized algorithms automatically without the developer's 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 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 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 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 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 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 an order of magnitude faster than the MPI library at computationally expensive tasks such as modular exponentiation. In the grand scheme of ``bignum'' libraries LibTomMath is faster than the average library and usually slower than the best libraries such as GMP and OpenSSL by 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 begin the design of a multiple precision integer library a primitive data type and a series of primitive algorithms must be established. 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 optimizing multiple precision integers can be developed to form the basis of the entire library of algorithms. \section{What is a Multiple Precision Integer?} Recall that most programming languages (\textit{in particular C}) only have fixed precision data types that on their own cannot be used to represent values larger than their precision alone will allow. The purpose of multiple precision algorithms is to use these fixed precision data types to create multiple precision integers which may represent values that are much larger. As a well known analogy, school children are taught how to form numbers larger than nine by prepending more radix ten digits. In the decimal system the largest value is only $9$ since the digits may only have values from $0$ to $9$. However, by concatenating digits together larger numbers may be represented. Computer based multiple precision arithmetic is essentially the same concept except with a different radix. What most people probably do not think about explicitly are the various other attributes that describe a multiple precision integer. For example, the integer $154_{10}$ has two immediately obvious properties. First, the integer is positive, that is the sign of this particular integer is positive as oppose to negative. Second, the integer has three digits in its representation. There is an additional property that the integer posesses that does not concern pencil-and-paper arithmetic. The third property is how many digits are allowed for the integer. The human analogy of this third property is ensuring there is enough space on the paper to right the integer. Computers must maintain a strict control on memory usage with respect to the digits of a multiple precision integer. These three properties make up what is known as a multiple precision integer or mp\_int for short. \subsection{The mp\_int structure} The mp\_int structure is the ISO C based manifestation of what represents a multiple precision integer. The ISO C standard does not provide for any such data type but it does provide for making composite data types known as structures. The following is the structure definition used within LibTomMath. \index{mp\_int} \begin{verbatim} typedef struct { int used, alloc, sign; mp_digit *dp; } mp_int; \end{verbatim} The mp\_int structure can be broken down as follows. \begin{enumerate} \item The \textbf{used} parameter denotes how many digits of the array \textbf{dp} contain the digits used to represent a given integer. The \textbf{used} count must not exceed the \textbf{alloc} count. \item The array \textbf{dp} holds the digits that represent the given integer. It is padded with $\textbf{alloc} - \textbf{used}$ zero digits. \item The \textbf{alloc} parameter denotes how many digits are available in the array to use by functions before it has to increase in size. When the \textbf{used} count of a result would exceed the \textbf{alloc} count all of the algorithms will automatically increase the size of the array to accommodate the precision of the result. \item The \textbf{sign} parameter denotes the sign as either zero/positive (\textbf{MP\_ZPOS}) or negative (\textbf{MP\_NEG}). \end{enumerate} \section{Argument Passing} A convention of argument passing must be adopted early on in the development of any library. Making the function prototypes consistent will help eliminate many headaches in the future as the library grows to significant complexity. In LibTomMath the multiple precision integer functions accept parameters from left to right as pointers to mp\_int structures. That means that the source 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. In the case of LibTomMath the convention from the MPI library has been adopted. 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 modified 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. In a multiple precision library the only errors that can occur 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 it 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 was not deemed ideal by the author of LibTomMath to force developers to have signal handlers for such cases. \section{Initialization and Clearing} The logical starting point when actually writing multiple precision integer functions is the initialization and clearing of the 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 is 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. Steps 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. Similarly 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 is 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 becoming 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{mp\_init}) \\ 2. If the init of $a$ was unsuccessful return(\textit{MP\_MEM}) \\ 3. Copy $b$ to $a$. (\textit{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 And Clearings} 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{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{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. Similarly to clear a variable length array of mp\_int structures the mp\_clear\_multi algorithm will be used. 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} Note how both lists are terminated with the \textbf{NULL} variable. This indicates to the algorithms to stop fetching parameters off of the stack. If it is not present the functions will most likely cause a segmentation fault. EXAM,bn_mp_multi.c Both routines are implemented in the same source file since they are typically used in conjunction with each other. \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. This is much akin to how the \textit{realloc} function from the standard C library works. Since the newly allocated digits are assumed to contain undefined values they are 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 $i + j$ digits. It is entirely possible that the result is $i + j - 1$ though, with no final carry into the last position. However, suppose the destination had to be first expanded (\textit{via mp\_grow}) to accomodate $i + j - 1$ digits than further expanded to accomodate the final carry. That would be a considerable waste of time since heap operations are relatively slow. The ideal solution is to always assume the result is $i + j$ and fix up the \textbf{used} count after the function terminates. This way a single heap operation (\textit{at most}) is required. However, if the result was not checked there would be an excess high order zero digit. For example, suppose the product of two integers was $x_n = (0x_{n-1}x_{n-2}...x_0)_{\beta}$. The leading zero digit will not contribute to the precision of the result. In fact, through subsequent operations more leading zero digits would accumulate to the point the size of the integer would be prohibitive. As a result even though the precision is very low the representation is excessively large. The mp\_clamp algorithm is designed to solve this very problem. It will trim 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 expected to iterate only once or twice at the most. For example, this will happen in cases where there is not a carry to fill the last position. Step two fixes the sign for when all of the digits are zero to ensure that the mp\_int is valid at all times. EXAM,bn_mp_clamp.c Note on line @27,while@ how to test for the \textbf{used} count is made on the left of the \&\& operator. In the C programming language the terms to \&\& are evaluated left to right with a boolean short-circuit if any condition fails. This is important since if the \textbf{used} is zero the test on the right would fetch below the array. That is obviously undesirable. The parenthesis on line @28,a->used@ is used to make sure the \textbf{used} count is decremented and not the pointer ``a''. \section*{Exercises} \begin{tabular}{cl} $\left [ 1 \right ]$ & Discuss the relevance of the \textbf{used} member of the mp\_int structure. \\ & \\ $\left [ 1 \right ]$ & Discuss the consequences of not using padding when performing allocations. \\ & \\ $\left [ 2 \right ]$ & Estimate an ideal value for \textbf{MP\_PREC} when performing 1024-bit RSA \\ & encryption when $\beta = 2^{28}$. \\ & \\ $\left [ 1 \right ]$ & Discuss the relevance of the algorithm mp\_clamp. What does it prevent? \\ & \\ $\left [ 1 \right ]$ & Give an example of when the algorithm mp\_init\_copy might be useful. \\ & \\ \end{tabular} \chapter{Basic Operations} \section{Copying an Integer} After the various house-keeping routines are in place, simple 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{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 more significant 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 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{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{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{mp\_zero}). \\ 2. $a_0 \leftarrow b \mbox{ (mod }\beta\mbox{)}$ \\ 3. $a.used \leftarrow \left \lbrace \begin{array}{ll} 1 & \mbox{if }a_0 > 0 \\ 0 & \mbox{if }a_0 = 0 \end{array} \right .$ \\ \hline \end{tabular} \end{center} \caption{Algorithm mp\_set} \end{figure} \textbf{Algorithm mp\_set.} This algorithm sets a mp\_int to a small single digit value. Step number 1 ensures that the integer is reset to the default state. The single digit is set (\textit{modulo $\beta$}) and the \textbf{used} count is adjusted accordingly. EXAM,bn_mp_set.c Line @21,mp_zero@ calls mp\_zero() to clear the mp\_int and reset the sign. Line @22,MP_MASK@ copies the digit into the least significant location. Note the usage of a new constant \textbf{MP\_MASK}. This constant is used to quickly reduce an integer modulo $\beta$. Since $\beta$ is of the form $2^k$ for any suitable $k$ it suffices to perform a binary AND with $MP\_MASK = 2^k - 1$ to perform the reduction. Finally line @23,a->used@ will set the \textbf{used} member with respect to the digit actually set. This function will always make the integer positive. One important limitation of this function is that it will only set one digit. The size of a digit is not fixed, meaning source that uses this function should take that into account. Meaning that only trivially small constants can be set using this function. \subsection{Setting Large Constants} To overcome the limitations of the mp\_set algorithm the mp\_set\_int algorithm is 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{mp\_zero}) \\ 2. for $n$ from 0 to 7 do \\ \hspace{3mm}2.1 $a \leftarrow a \cdot 16$ (\textit{mp\_mul2d}) \\ \hspace{3mm}2.2 $u \leftarrow \lfloor b / 2^{4(7 - n)} \rfloor \mbox{ (mod }16\mbox{)}$\\ \hspace{3mm}2.3 $a_0 \leftarrow a_0 + u$ \\ \hspace{3mm}2.4 $a.used \leftarrow a.used + 1$ \\ 3. Clamp excess used digits (\textit{mp\_clamp}) \\ \hline \end{tabular} \end{center} \caption{Algorithm mp\_set\_int} \end{figure} \textbf{Algorithm mp\_set\_int.} The algorithm performs eight iterations of a simple loop where in each iteration four bits from the source are added to the mp\_int. Step 2.1 will multiply the current result by sixteen making room for four more bits in the less significant positions. In step 2.2 the next four bits from the source are extracted and are added to the mp\_int. The \textbf{used} digit count is incremented to reflect the addition. The \textbf{used} digit counter is incremented since if any of the leading digits were zero the mp\_int would have zero digits used and the newly added four bits would be ignored. Excess zero digits are trimmed in steps 2.1 and 3 by using higher level algorithms mp\_mul2d and mp\_clamp. EXAM,bn_mp_set_int.c This function sets four bits of the number at a time to handle all practical \textbf{DIGIT\_BIT} sizes. The weird addition on line @38,a->used@ ensures that the newly added in bits are added to the number of digits. While it may not seem obvious as to why the digit counter does not grow exceedingly large it is because of the shift on line @27,mp_mul_2d@ as well as the call to mp\_clamp() on line @40,mp_clamp@. Both functions will clamp excess leading digits which keeps the number of used digits low. \section{Comparisons} \subsection{Unsigned Comparisions} Comparing a multiple precision integer is performed with the exact same algorithm used to compare two decimal numbers. For example, to compare $1,234$ to $1,264$ the digits are extracted by their positions. That is we compare $1 \cdot 10^3 + 2 \cdot 10^2 + 3 \cdot 10^1 + 4 \cdot 10^0$ to $1 \cdot 10^3 + 2 \cdot 10^2 + 6 \cdot 10^1 + 4 \cdot 10^0$ by comparing single digits at a time starting with the highest magnitude positions. If any leading digit of one integer is greater than a digit in the same position of another integer then obviously it must be greater. The first comparision routine that will be developed is the unsigned magnitude compare which will perform a comparison based on the digits of two mp\_int variables alone. It will ignore the sign of the two inputs. Such a function is useful when an absolute comparison is required or if the signs are known to agree in advance. To facilitate working with the results of the comparison functions three constants are required. \begin{figure}[here] \begin{center} \begin{tabular}{|r|l|} \hline \textbf{Constant} & \textbf{Meaning} \\ \hline \textbf{MP\_GT} & Greater Than \\ \hline \textbf{MP\_EQ} & Equal To \\ \hline \textbf{MP\_LT} & Less Than \\ \hline \end{tabular} \end{center} \caption{Comparison Return Codes} \end{figure} \begin{figure}[here] \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_cmp\_mag}. \\ \textbf{Input}. Two mp\_ints $a$ and $b$. \\ \textbf{Output}. Unsigned comparison results ($a$ to the left of $b$). \\ \hline \\ 1. If $a.used > b.used$ then return(\textit{MP\_GT}) \\ 2. If $a.used < b.used$ then return(\textit{MP\_LT}) \\ 3. for n from $a.used - 1$ to 0 do \\ \hspace{+3mm}3.1 if $a_n > b_n$ then return(\textit{MP\_GT}) \\ \hspace{+3mm}3.2 if $a_n < b_n$ then return(\textit{MP\_LT}) \\ 4. Return(\textit{MP\_EQ}) \\ \hline \end{tabular} \end{center} \caption{Algorithm mp\_cmp\_mag} \end{figure} \textbf{Algorithm mp\_cmp\_mag.} By saying ``$a$ to the left of $b$'' it is meant that the comparison is with respect to $a$, that is if $a$ is greater than $b$ it will return \textbf{MP\_GT} and similar with respect to when $a = b$ and $a < b$. The first two steps compare the number of digits used in both $a$ and $b$. Obviously if the digit counts differ there would be an imaginary zero digit in the smaller number where the leading digit of the larger number is. If both have the same number of digits than the actual digits themselves must be compared starting at the leading digit. By step three both inputs must have the same number of digits so its safe to start from either $a.used - 1$ or $b.used - 1$ and count down to the zero'th digit. If after all of the digits have been compared, no difference is found, the algorithm returns \textbf{MP\_EQ}. EXAM,bn_mp_cmp_mag.c The two if statements on lines @24,if@ and @28,if@ compare the number of digits in the two inputs. These two are performed before all of the digits are compared since it is a very cheap test to perform and can potentially save considerable time. The implementation given is also not valid without those two statements. $b.alloc$ may be smaller than $a.used$, meaning that undefined values will be read from $b$ past the end of the array of digits. \subsection{Signed Comparisons} Comparing with sign considerations is also fairly critical in several routines (\textit{division for example}). Based on an unsigned magnitude comparison a trivial signed comparison algorithm can be written. \begin{figure}[here] \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_cmp}. \\ \textbf{Input}. Two mp\_ints $a$ and $b$ \\ \textbf{Output}. Signed Comparison Results ($a$ to the left of $b$) \\ \hline \\ 1. if $a.sign = MP\_NEG$ and $b.sign = MP\_ZPOS$ then return(\textit{MP\_LT}) \\ 2. if $a.sign = MP\_ZPOS$ and $b.sign = MP\_NEG$ then return(\textit{MP\_GT}) \\ 3. if $a.sign = MP\_NEG$ then \\ \hspace{+3mm}3.1 Return the unsigned comparison of $b$ and $a$ (\textit{mp\_cmp\_mag}) \\ 4 Otherwise \\ \hspace{+3mm}4.1 Return the unsigned comparison of $a$ and $b$ \\ \hline \end{tabular} \end{center} \caption{Algorithm mp\_cmp} \end{figure} \textbf{Algorithm mp\_cmp.} The first two steps compare the signs of the two inputs. If the signs do not agree then it can return right away with the appropriate comparison code. When the signs are equal the digits of the inputs must be compared to determine the correct result. In step three the unsigned comparision flips the order of the arguments since they are both negative. For instance, if $-a > -b$ then $\vert a \vert < \vert b \vert$. Step number four will compare the two when they are both positive. EXAM,bn_mp_cmp.c The two if statements on lines @22,if@ and @26,if@ perform the initial sign comparison. If the signs are not the equal then which ever has the positive sign is larger. At line @30,if@, the inputs are compared based on magnitudes. If the signs were both negative then the unsigned comparison is performed in the opposite direction (\textit{line @31,mp_cmp_mag@}). Otherwise, the signs are assumed to be both positive and a forward direction unsigned comparison is performed. \section*{Exercises} \begin{tabular}{cl} $\left [ 2 \right ]$ & Modify algorithm mp\_set\_int to accept as input a variable length array of bits. \\ & \\ $\left [ 3 \right ]$ & Give the probability that algorithm mp\_cmp\_mag will have to compare $k$ digits \\ & of two random digits (of equal magnitude) before a difference is found. \\ & \\ $\left [ 1 \right ]$ & Suggest a simple method to speed up the implementation of mp\_cmp\_mag based \\ & on the observations made in the previous problem. \\ & \end{tabular} \chapter{Basic Arithmetic} \section{Building Blocks} At this point algorithms for initialization, clearing, zeroing, copying, comparing and setting small constants have been established. The next logical set of algorithms to develop are addition, subtraction and digit shifting algorithms. These algorithms make use of the lower level algorithms and are the cruicial building block for the multiplication algorithms. It is very important that these algorithms are highly optimized. On their own they are simple $O(n)$ algorithms but they can be called from higher level algorithms which easily places them at $O(n^2)$ or even $O(n^3)$ work levels. MARK,SHIFTS All 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{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{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 HAC \cite[pp. 594]{HAC} but has been extended to allow the inputs to have different magnitudes. Coincidentally the description of algorithm A in Knuth \cite[pp. 266]{TAOCPV2} shares the same deficiency as the algorithm from \cite{HAC}. Even the MIX pseudo machine code presented by Knuth \cite[pp. 266-267]{TAOCPV2} is incapable of handling inputs which are of different magnitudes. 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 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 value. 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 $oldused$ so that excess digits can be cleared later, and the new \textbf{used} count is set to $max+1$, so that a carry from the most significant word can be handled. At step 7 the carry variable $u$ is set to zero and the first part 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 by $lg(\beta)$ 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 right or left shift, 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 summand 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 a pointer to a mp\_int assigned to the largest input, in effect it is a local alias. Lines @37,init@ to @42,}@ ensure that the destination is grown to accomodate the result of the addition. Similar to the implementation of mp\_copy this function uses the braced code and local aliases coding style. The three aliases that are on lines @56,tmpa@, @59,tmpb@ and @62,tmpc@ represent the two inputs and destination variables respectively. These aliases are used to ensure the compiler does not have to dereference $a$, $b$ or $c$ (respectively) to access the digits of the respective mp\_int. The initial carry $u$ is cleared on line @65,u = 0@, note that $u$ is of type mp\_digit which ensures type compatibility within the implementation. The initial addition loop begins on line @66,for@ and ends on line @75,}@. Similarly the conditional addition loop begins on line @81,for@ and ends on line @90,}@. The addition is finished with the final carry being stored in $tmpc$ on line @94,tmpc++@. Note the ``++'' operator on the same line. After line @94,tmpc++@ $tmpc$ will point to the $c.used$'th digit of the mp\_int $c$. This is useful for the next loop on lines @97,for@ to @99,}@ which set any old upper digits to zero. \subsection{Low Level Subtraction} The low level unsigned subtraction algorithm is very similar to the low level unsigned addition algorithm. The principle difference is that the unsigned subtraction algorithm requires the result to be positive. That is when computing $a - b$ the condition $\vert a \vert \ge \vert b\vert$ must be met for this algorithm to function properly. Keep in mind this low level algorithm is not meant to be used in higher level algorithms directly. This algorithm as will be shown can be used to create functional signed addition and subtraction algorithms. MARK,GAMMA For this algorithm a new variable is required to make the description simpler. Recall from section 1.3.1 that a mp\_digit must be able to represent the range $0 \le x < 2\beta$ for the algorithms to work correctly. However, it is allowable that a mp\_digit represent a larger range of values. For this algorithm we will assume that the variable $\gamma$ represents the number of bits available in a mp\_digit (\textit{this implies $2^{\gamma} > \beta$}). For example, the default for LibTomMath is to use a ``unsigned long'' for the mp\_digit ``type'' while $\beta = 2^{28}$. In ISO C an ``unsigned long'' data type must be able to represent $0 \le x < 2^{32}$ meaning that in this case $\gamma = 32$. \newpage\begin{figure}[!here] \begin{center} \begin{small} \begin{tabular}{l} \hline Algorithm \textbf{s\_mp\_sub}. \\ \textbf{Input}. Two mp\_ints $a$ and $b$ ($\vert a \vert \ge \vert b \vert$) \\ \textbf{Output}. The unsigned subtraction $c = \vert a \vert - \vert b \vert$. \\ \hline \\ 1. $min \leftarrow b.used$ \\ 2. $max \leftarrow a.used$ \\ 3. If $c.alloc < max$ then grow $c$ to hold at least $max$ digits. (\textit{mp\_grow}) \\ 4. 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{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 opposed to $max + 1$. Similar to the addition algorithm the \textbf{used} count of $c$ is copied locally and set to the maximal count for the operation. The subtraction loop that begins on step 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 (\textit{also known as the borrow}) within the subtraction loops. Under the assumption that two's complement single precision arithmetic is used this will successfully extract the desired carry. For example, consider subtracting $0101_2$ from $0100_2$ where $\gamma = 4$ and $\beta = 2$. The least significant bit will force a carry upwards to the third bit which will be set to zero after the borrow. After the very first bit has been subtracted $4 - 1 \equiv 0011_2$ will remain, When the third bit of $0101_2$ is subtracted from the result it will cause another carry. In this case though the carry will be forced to propagate all the way to the most significant bit. Recall that $\beta < 2^{\gamma}$. This means that if a carry does occur just before the $lg(\beta)$'th bit it will propagate all the way to the most significant bit. Thus, the high order bits of the mp\_digit that are not part of the actual digit will either be all zero, or all one. All that is needed is a single zero or one bit for the carry. Therefore a single logical shift right by $\gamma - 1$ positions is sufficient to extract the carry. This method of carry extraction may seem awkward but the reason for it becomes apparent when the implementation is discussed. If $b$ has a smaller magnitude than $a$ then step 9 will force the carry and copy operation to propagate through the larger input $a$ into $c$. Step 10 will ensure that any leading digits of $c$ above the $max$'th position are zeroed. EXAM,bn_s_mp_sub.c Line @24,min@ and @25,max@ perform the initial hardcoded sorting of the inputs. In reality the $min$ and $max$ variables are only aliases and are only used to make the source code easier to read. Again the pointer alias optimization is used within this algorithm. Lines @42,tmpa@, @43,tmpb@ and @44,tmpc@ initialize the aliases for $a$, $b$ and $c$ respectively. The first subtraction loop occurs on lines @47,u = 0@ through @61,}@. The theory behind the subtraction loop is exactly the same as that for the addition loop. As remarked earlier there is an implementation reason for using the ``awkward'' method of extracting the carry (\textit{see line @57, >>@}). The traditional method for extracting the carry would be to shift by $lg(\beta)$ positions and logically AND the least significant bit. The AND operation is required because all of the bits above the $\lg(\beta)$'th bit will be set to one after a carry occurs from subtraction. This carry extraction requires two relatively cheap operations to extract the carry. The other method is to simply shift the most significant bit to the least significant bit thus extracting the carry with a single cheap operation. This optimization only works on twos compliment machines which is a safe assumption to make. If $a$ has a larger magnitude than $b$ an additional loop (\textit{see lines @64,for@ through @73,}@}) is required to propagate the carry through $a$ and copy the result to $c$. \subsection{High Level Addition} Now that both lower level addition and subtraction algorithms have been established an effective high level signed addition algorithm can be established. This high level addition algorithm will be what other algorithms and developers will use to perform addition of mp\_int data types. Recall from section 5.2 that an mp\_int represents an integer with an unsigned mantissa (\textit{the array of digits}) and a \textbf{sign} flag. A high level addition is actually performed as a series of eight separate cases which can be optimized down to three unique cases. \begin{figure}[!here] \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_add}. \\ \textbf{Input}. Two mp\_ints $a$ and $b$ \\ \textbf{Output}. The signed addition $c = a + b$. \\ \hline \\ 1. if $a.sign = b.sign$ then do \\ \hspace{3mm}1.1 $c.sign \leftarrow a.sign$ \\ \hspace{3mm}1.2 $c \leftarrow \vert a \vert + \vert b \vert$ (\textit{s\_mp\_add})\\ 2. else do \\ \hspace{3mm}2.1 if $\vert a \vert < \vert b \vert$ then do (\textit{mp\_cmp\_mag}) \\ \hspace{6mm}2.1.1 $c.sign \leftarrow b.sign$ \\ \hspace{6mm}2.1.2 $c \leftarrow \vert b \vert - \vert a \vert$ (\textit{s\_mp\_sub}) \\ \hspace{3mm}2.2 else do \\ \hspace{6mm}2.2.1 $c.sign \leftarrow a.sign$ \\ \hspace{6mm}2.2.2 $c \leftarrow \vert a \vert - \vert b \vert$ \\ 3. 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. \begin{figure}[here] \begin{small} \begin{center} \begin{tabular}{|c|c|c|c|c|} \hline \textbf{Sign of $a$} & \textbf{Sign of $b$} & \textbf{$\vert a \vert > \vert b \vert $} & \textbf{Unsigned Operation} & \textbf{Result Sign Flag} \\ \hline $+$ & $+$ & Yes & $c = a + b$ & $a.sign$ \\ \hline $+$ & $+$ & No & $c = a + b$ & $a.sign$ \\ \hline $-$ & $-$ & Yes & $c = a + b$ & $a.sign$ \\ \hline $-$ & $-$ & No & $c = a + b$ & $a.sign$ \\ \hline &&&&\\ \hline $+$ & $-$ & No & $c = b - a$ & $b.sign$ \\ \hline $-$ & $+$ & No & $c = b - a$ & $b.sign$ \\ \hline &&&&\\ \hline $+$ & $-$ & Yes & $c = a - b$ & $a.sign$ \\ \hline $-$ & $+$ & Yes & $c = a - b$ & $a.sign$ \\ \hline \end{tabular} \end{center} \end{small} \caption{Addition Guide Chart} \label{fig:AddChart} \end{figure} Figure~\ref{fig:AddChart} lists all of the eight possible input combinations and is sorted to show that only three specific cases need to be handled. The return code of the unsigned operations at step 1.2, 2.1.2 and 2.2.2 are forwarded to step 3 to check for errors. This simplifies the description of the algorithm considerably and best follows how the implementation actually was achieved. Also note how the \textbf{sign} is set before the unsigned addition or subtraction is performed. Recall from the descriptions of algorithms s\_mp\_add and s\_mp\_sub that the mp\_clamp function is used at the end to trim excess digits. The mp\_clamp algorithm will set the \textbf{sign} to \textbf{MP\_ZPOS} when the \textbf{used} digit count reaches zero. For example, consider performing $-a + a$ with algorithm mp\_add. By the description of the algorithm the sign is set to \textbf{MP\_NEG} which would produce a result of $-0$. However, since the sign is set first then the unsigned addition is performed the subsequent usage of algorithm mp\_clamp within algorithm s\_mp\_add will force $-0$ to become $0$. EXAM,bn_mp_add.c The source code follows the algorithm fairly closely. The most notable new source code addition is the usage of the $res$ integer variable which is used to pass result of the unsigned operations forward. Unlike in the algorithm, the variable $res$ is merely returned as is without explicitly checking it and returning the constant \textbf{MP\_OKAY}. The observation is this algorithm will succeed or fail only if the lower level functions do so. Returning their return code is sufficient. \subsection{High Level Subtraction} The high level signed subtraction algorithm is essentially the same as the high level signed addition algorithm. \newpage\begin{figure}[!here] \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_sub}. \\ \textbf{Input}. Two mp\_ints $a$ and $b$ \\ \textbf{Output}. The signed subtraction $c = a - b$. \\ \hline \\ 1. if $a.sign \ne b.sign$ then do \\ \hspace{3mm}1.1 $c.sign \leftarrow a.sign$ \\ \hspace{3mm}1.2 $c \leftarrow \vert a \vert + \vert b \vert$ (\textit{s\_mp\_add}) \\ 2. else do \\ \hspace{3mm}2.1 if $\vert a \vert \ge \vert b \vert$ then do (\textit{mp\_cmp\_mag}) \\ \hspace{6mm}2.1.1 $c.sign \leftarrow a.sign$ \\ \hspace{6mm}2.1.2 $c \leftarrow \vert a \vert - \vert b \vert$ (\textit{s\_mp\_sub}) \\ \hspace{3mm}2.2 else do \\ \hspace{6mm}2.2.1 $c.sign \leftarrow \left \lbrace \begin{array}{ll} MP\_ZPOS & \mbox{if }a.sign = MP\_NEG \\ MP\_NEG & \mbox{otherwise} \\ \end{array} \right .$ \\ \hspace{6mm}2.2.2 $c \leftarrow \vert b \vert - \vert a \vert$ \\ 3. 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. \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{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 $b_{n + 1} \leftarrow r$ \\ \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 the carry and augmenting the \textbf{used} count of $b$. Step 8 clears any leading digits of $b$ in case it originally had a larger magnitude than $a$. EXAM,bn_mp_mul_2.c This implementation is essentially an optimized implementation of s\_mp\_add for the case of doubling an input. The only noteworthy difference is the use of the logical shift operator on line @52,<<@ to perform a single precision doubling. \subsection{Division by Two} A division by two can just as easily be accomplished with a logical shift right as multiplication by two can be with a logical shift left. \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_div\_2}. \\ \textbf{Input}. One mp\_int $a$ \\ \textbf{Output}. $b = a/2$. \\ \hline \\ 1. If $b.alloc < a.used$ then grow $b$ to hold $a.used$ digits. (\textit{mp\_grow}) \\ 2. If the reallocation failed return(\textit{MP\_MEM}). \\ 3. $oldused \leftarrow b.used$ \\ 4. $b.used \leftarrow a.used$ \\ 5. $r \leftarrow 0$ \\ 6. for $n$ from $b.used - 1$ to $0$ do \\ \hspace{3mm}6.1 $rr \leftarrow a_n \mbox{ (mod }2\mbox{)}$\\ \hspace{3mm}6.2 $b_n \leftarrow (a_n >> 1) + (r << (lg(\beta) - 1)) \mbox{ (mod }\beta\mbox{)}$ \\ \hspace{3mm}6.3 $r \leftarrow rr$ \\ 7. If $b.used < oldused - 1$ then do \\ \hspace{3mm}7.1 for $n$ from $b.used$ to $oldused - 1$ do \\ \hspace{6mm}7.1.1 $b_n \leftarrow 0$ \\ 8. $b.sign \leftarrow a.sign$ \\ 9. Clamp excess digits of $b$. (\textit{mp\_clamp}) \\ 10. Return(\textit{MP\_OKAY}).\\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_div\_2} \end{figure} \textbf{Algorithm mp\_div\_2.} This algorithm will divide an mp\_int by two using logical shifts to the right. Like mp\_mul\_2 it uses a modified low level addition core as the basis of the algorithm. Unlike mp\_mul\_2 the shift operations work from the leading digit to the trailing digit. The algorithm could be written to work from the trailing digit to the leading digit however, it would have to stop one short of $a.used - 1$ digits to prevent reading past the end of the array of digits. Essentially the loop at step 6 is similar to that of mp\_mul\_2 except the logical shifts go in the opposite direction and the carry is at the least significant bit not the most significant bit. EXAM,bn_mp_div_2.c \section{Polynomial Basis Operations} Recall from ~POLY~ that any integer can be represented as a polynomial in $x$ as $y = f(\beta)$. Such a representation is also known as the polynomial basis \cite[pp. 48]{ROSE}. Given such a notation a multiplication or division by $x$ amounts to shifting whole digits a single place. The need for such operations arises in several other higher level algorithms such as Barrett and Montgomery reduction, integer division and Karatsuba multiplication. Converting from an array of digits to polynomial basis is very simple. Consider the integer $y \equiv (a_2, a_1, a_0)_{\beta}$ and recall that $y = \sum_{i=0}^{2} a_i \beta^i$. Simply replace $\beta$ with $x$ and the expression is in polynomial basis. For example, $f(x) = 8x + 9$ is the polynomial basis representation for $89$ using radix ten. That is, $f(10) = 8(10) + 9 = 89$. \subsection{Multiplication by $x$} Given a polynomial in $x$ such as $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0$ multiplying by $x$ amounts to shifting the coefficients up one degree. In this case $f(x) \cdot x = a_n x^{n+1} + a_{n-1} x^n + ... + a_0 x$. From a scalar basis point of view multiplying by $x$ is equivalent to multiplying by the integer $\beta$. \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_lshd}. \\ \textbf{Input}. One mp\_int $a$ and an integer $b$ \\ \textbf{Output}. $a \leftarrow a \cdot \beta^b$ (equivalent to multiplication by $x^b$). \\ \hline \\ 1. If $b \le 0$ then return(\textit{MP\_OKAY}). \\ 2. If $a.alloc < a.used + b$ then grow $a$ to at least $a.used + b$ digits. (\textit{mp\_grow}). \\ 3. If the reallocation failed return(\textit{MP\_MEM}). \\ 4. $a.used \leftarrow a.used + b$ \\ 5. $i \leftarrow a.used - 1$ \\ 6. $j \leftarrow a.used - 1 - b$ \\ 7. for $n$ from $a.used - 1$ to $b$ do \\ \hspace{3mm}7.1 $a_{i} \leftarrow a_{j}$ \\ \hspace{3mm}7.2 $i \leftarrow i - 1$ \\ \hspace{3mm}7.3 $j \leftarrow j - 1$ \\ 8. for $n$ from 0 to $b - 1$ do \\ \hspace{3mm}8.1 $a_n \leftarrow 0$ \\ 9. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_lshd} \end{figure} \textbf{Algorithm mp\_lshd.} This algorithm multiplies an mp\_int by the $b$'th power of $x$. This is equivalent to multiplying by $\beta^b$. The algorithm differs from the other algorithms presented so far as it performs the operation in place instead storing the result in a separate location. The motivation behind this change is due to the way this function is typically used. Algorithms such as mp\_add store the result in an optionally different third mp\_int because the original inputs are often still required. Algorithm mp\_lshd (\textit{and similarly algorithm mp\_rshd}) is typically used on values where the original value is no longer required. The algorithm will return success immediately if $b \le 0$ since the rest of algorithm is only valid when $b > 0$. First the destination $a$ is grown as required to accomodate the result. The counters $i$ and $j$ are used to form a \textit{sliding window} over the digits of $a$ of length $b$. The head of the sliding window is at $i$ (\textit{the leading digit}) and the tail at $j$ (\textit{the trailing digit}). The loop on step 7 copies the digit from the tail to the head. In each iteration the window is moved down one digit. The last loop on step 8 sets the lower $b$ digits to zero. \newpage FIGU,sliding_window,Sliding Window Movement EXAM,bn_mp_lshd.c The if statement on line @24,if@ ensures that the $b$ variable is greater than zero. The \textbf{used} count is incremented by $b$ before the copy loop begins. This elminates the need for an additional variable in the for loop. The variable $top$ on line @42,top@ is an alias for the leading digit while $bottom$ on line @45,bottom@ is an alias for the trailing edge. The aliases form a window of exactly $b$ digits over the input. \subsection{Division by $x$} Division by powers of $x$ is easily achieved by shifting the digits right and removing any that will end up to the right of the zero'th digit. \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_rshd}. \\ \textbf{Input}. One mp\_int $a$ and an integer $b$ \\ \textbf{Output}. $a \leftarrow a / \beta^b$ (Divide by $x^b$). \\ \hline \\ 1. If $b \le 0$ then return. \\ 2. If $a.used \le b$ then do \\ \hspace{3mm}2.1 Zero $a$. (\textit{mp\_zero}). \\ \hspace{3mm}2.2 Return. \\ 3. $i \leftarrow 0$ \\ 4. $j \leftarrow b$ \\ 5. for $n$ from 0 to $a.used - b - 1$ do \\ \hspace{3mm}5.1 $a_i \leftarrow a_j$ \\ \hspace{3mm}5.2 $i \leftarrow i + 1$ \\ \hspace{3mm}5.3 $j \leftarrow j + 1$ \\ 6. for $n$ from $a.used - b$ to $a.used - 1$ do \\ \hspace{3mm}6.1 $a_n \leftarrow 0$ \\ 7. $a.used \leftarrow a.used - b$ \\ 8. Return. \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_rshd} \end{figure} \textbf{Algorithm mp\_rshd.} This algorithm divides the input in place by the $b$'th power of $x$. It is analogous to dividing by a $\beta^b$ but much quicker since it does not require single precision division. This algorithm does not actually return an error code as it cannot fail. If the input $b$ is less than one the algorithm quickly returns without performing any work. If the \textbf{used} count is less than or equal to the shift count $b$ then it will simply zero the input and return. After the trivial cases of inputs have been handled the sliding window is setup. Much like the case of algorithm mp\_lshd a sliding window that is $b$ digits wide is used to copy the digits. Unlike mp\_lshd the window slides in the opposite direction from the trailing to the leading digit. Also the digits are copied from the leading to the trailing edge. Once the window copy is complete the upper digits must be zeroed and the \textbf{used} count decremented. EXAM,bn_mp_rshd.c The only noteworthy element of this routine is the lack of a return type. -- Will update later to give it a return type...Tom \section{Powers of Two} Now that algorithms for moving single bits as well as whole digits exist algorithms for moving the ``in between'' distances are required. For example, to quickly multiply by $2^k$ for any $k$ without using a full multiplier algorithm would prove useful. Instead of performing single shifts $k$ times to achieve a multiplication by $2^{\pm k}$ a mixture of whole digit shifting and partial digit shifting is employed. \subsection{Multiplication by Power of Two} \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_mul\_2d}. \\ \textbf{Input}. One mp\_int $a$ and an integer $b$ \\ \textbf{Output}. $c \leftarrow a \cdot 2^b$. \\ \hline \\ 1. $c \leftarrow a$. (\textit{mp\_copy}) \\ 2. If $c.alloc < c.used + \lfloor b / lg(\beta) \rfloor + 2$ then grow $c$ accordingly. \\ 3. If the reallocation failed return(\textit{MP\_MEM}). \\ 4. If $b \ge lg(\beta)$ then \\ \hspace{3mm}4.1 $c \leftarrow c \cdot \beta^{\lfloor b / lg(\beta) \rfloor}$ (\textit{mp\_lshd}). \\ \hspace{3mm}4.2 If step 4.1 failed return(\textit{MP\_MEM}). \\ 5. $d \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\ 6. If $d \ne 0$ then do \\ \hspace{3mm}6.1 $mask \leftarrow 2^d$ \\ \hspace{3mm}6.2 $r \leftarrow 0$ \\ \hspace{3mm}6.3 for $n$ from $0$ to $c.used - 1$ do \\ \hspace{6mm}6.3.1 $rr \leftarrow c_n >> (lg(\beta) - d) \mbox{ (mod }mask\mbox{)}$ \\ \hspace{6mm}6.3.2 $c_n \leftarrow (c_n << d) + r \mbox{ (mod }\beta\mbox{)}$ \\ \hspace{6mm}6.3.3 $r \leftarrow rr$ \\ \hspace{3mm}6.4 If $r > 0$ then do \\ \hspace{6mm}6.4.1 $c_{c.used} \leftarrow r$ \\ \hspace{6mm}6.4.2 $c.used \leftarrow c.used + 1$ \\ 7. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_mul\_2d} \end{figure} \textbf{Algorithm mp\_mul\_2d.} This algorithm multiplies $a$ by $2^b$ and stores the result in $c$. The algorithm uses algorithm mp\_lshd and a derivative of algorithm mp\_mul\_2 to quickly compute the product. First the algorithm will multiply $a$ by $x^{\lfloor b / lg(\beta) \rfloor}$ which will ensure that the remainder multiplicand is less than $\beta$. For example, if $b = 37$ and $\beta = 2^{28}$ then this step will multiply by $x$ leaving a multiplication by $2^{37 - 28} = 2^{9}$ left. After the digits have been shifted appropriately at most $lg(\beta) - 1$ shifts are left to perform. Step 5 calculates the number of remaining shifts required. If it is non-zero a modified shift loop is used to calculate the remaining product. Essentially the loop is a generic version of algorith mp\_mul2 designed to handle any shift count in the range $1 \le x < lg(\beta)$. The $mask$ variable is used to extract the upper $d$ bits to form the carry for the next iteration. This algorithm is loosely measured as a $O(2n)$ algorithm which means that if the input is $n$-digits that it takes $2n$ ``time'' to complete. It is possible to optimize this algorithm down to a $O(n)$ algorithm at a cost of making the algorithm slightly harder to follow. EXAM,bn_mp_mul_2d.c Notes to be revised when code is updated. -- Tom \subsection{Division by Power of Two} \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_div\_2d}. \\ \textbf{Input}. One mp\_int $a$ and an integer $b$ \\ \textbf{Output}. $c \leftarrow \lfloor a / 2^b \rfloor, d \leftarrow a \mbox{ (mod }2^b\mbox{)}$. \\ \hline \\ 1. If $b \le 0$ then do \\ \hspace{3mm}1.1 $c \leftarrow a$ (\textit{mp\_copy}) \\ \hspace{3mm}1.2 $d \leftarrow 0$ (\textit{mp\_zero}) \\ \hspace{3mm}1.3 Return(\textit{MP\_OKAY}). \\ 2. $c \leftarrow a$ \\ 3. $d \leftarrow a \mbox{ (mod }2^b\mbox{)}$ (\textit{mp\_mod\_2d}) \\ 4. If $b \ge lg(\beta)$ then do \\ \hspace{3mm}4.1 $c \leftarrow \lfloor c/\beta^{\lfloor b/lg(\beta) \rfloor} \rfloor$ (\textit{mp\_rshd}). \\ 5. $k \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\ 6. If $k \ne 0$ then do \\ \hspace{3mm}6.1 $mask \leftarrow 2^k$ \\ \hspace{3mm}6.2 $r \leftarrow 0$ \\ \hspace{3mm}6.3 for $n$ from $c.used - 1$ to $0$ do \\ \hspace{6mm}6.3.1 $rr \leftarrow c_n \mbox{ (mod }mask\mbox{)}$ \\ \hspace{6mm}6.3.2 $c_n \leftarrow (c_n >> k) + (r << (lg(\beta) - k))$ \\ \hspace{6mm}6.3.3 $r \leftarrow rr$ \\ 7. Clamp excess digits of $c$. (\textit{mp\_clamp}) \\ 8. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_div\_2d} \end{figure} \textbf{Algorithm mp\_div\_2d.} This algorithm will divide an input $a$ by $2^b$ and produce the quotient and remainder. The algorithm is designed much like algorithm mp\_mul\_2d by first using whole digit shifts then single precision shifts. This algorithm will also produce the remainder of the division by using algorithm mp\_mod\_2d. EXAM,bn_mp_div_2d.c The implementation of algorithm mp\_div\_2d is slightly different than the algorithm specifies. The remainder $d$ may be optionally ignored by passing \textbf{NULL} as the pointer to the mp\_int variable. The temporary mp\_int variable $t$ is used to hold the result of the remainder operation until the end. This allows $d$ and $a$ to represent the same mp\_int without modifying $a$ before the quotient is obtained. The remainder of the source code is essentially the same as the source code for mp\_mul\_2d. (-- Fix this paragraph up later, Tom). \subsection{Remainder of Division by Power of Two} The last algorithm in the series of polynomial basis power of two algorithms is calculating the remainder of division by $2^b$. This algorithm benefits from the fact that in twos complement arithmetic $a \mbox{ (mod }2^b\mbox{)}$ is the same as $a$ AND $2^b - 1$. \begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_mod\_2d}. \\ \textbf{Input}. One mp\_int $a$ and an integer $b$ \\ \textbf{Output}. $c \leftarrow a \mbox{ (mod }2^b\mbox{)}$. \\ \hline \\ 1. If $b \le 0$ then do \\ \hspace{3mm}1.1 $c \leftarrow 0$ (\textit{mp\_zero}) \\ \hspace{3mm}1.2 Return(\textit{MP\_OKAY}). \\ 2. If $b > a.used \cdot lg(\beta)$ then do \\ \hspace{3mm}2.1 $c \leftarrow a$ (\textit{mp\_copy}) \\ \hspace{3mm}2.2 Return the result of step 2.1. \\ 3. $c \leftarrow a$ \\ 4. If step 3 failed return(\textit{MP\_MEM}). \\ 5. for $n$ from $\lceil b / lg(\beta) \rceil$ to $c.used$ do \\ \hspace{3mm}5.1 $c_n \leftarrow 0$ \\ 6. $k \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\ 7. $c_{\lfloor b / lg(\beta) \rfloor} \leftarrow c_{\lfloor b / lg(\beta) \rfloor} \mbox{ (mod }2^{k}\mbox{)}$. \\ 8. Clamp excess digits of $c$. (\textit{mp\_clamp}) \\ 9. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_mod\_2d} \end{figure} \textbf{Algorithm mp\_mod\_2d.} This algorithm will quickly calculate the value of $a \mbox{ (mod }2^b\mbox{)}$. First if $b$ is less than or equal to zero the result is set to zero. If $b$ is greater than the number of bits in $a$ then it simply copies $a$ to $c$ and returns. Otherwise, $a$ is copied to $b$, leading digits are removed and the remaining leading digit is trimed to the exact bit count. EXAM,bn_mp_mod_2d.c -- Add comments later, Tom. \section*{Exercises} \begin{tabular}{cl} $\left [ 3 \right ] $ & Devise an algorithm that performs $a \cdot 2^b$ for generic values of $b$ \\ & in $O(n)$ time. \\ &\\ $\left [ 3 \right ] $ & Devise an efficient algorithm to multiply by small low hamming \\ & weight values such as $3$, $5$ and $9$. Extend it to handle all values \\ & upto $64$ with a hamming weight less than three. \\ &\\ $\left [ 2 \right ] $ & Modify the preceding algorithm to handle values of the form \\ & $2^k - 1$ as well. \\ &\\ $\left [ 3 \right ] $ & Using only algorithms mp\_mul\_2, mp\_div\_2 and mp\_add create an \\ & algorithm to multiply two integers in roughly $O(2n^2)$ time for \\ & any $n$-bit input. Note that the time of addition is ignored in the \\ & calculation. \\ & \\ $\left [ 5 \right ] $ & Improve the previous algorithm to have a working time of at most \\ & $O \left (2^{(k-1)}n + \left ({2n^2 \over k} \right ) \right )$ for an appropriate choice of $k$. Again ignore \\ & the cost of addition. \\ & \\ $\left [ 2 \right ] $ & Devise a chart to find optimal values of $k$ for the previous problem \\ & for $n = 64 \ldots 1024$ in steps of $64$. \\ & \\ $\left [ 2 \right ] $ & Using only algorithms mp\_abs and mp\_sub devise another method for \\ & calculating the result of a signed comparison. \\ & \end{tabular} \chapter{Multiplication and Squaring} \section{The Multipliers} For most number theoretic 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 $b$ limited to an output precision of $digs$ digits. While it may seem a bit awkward to modify the function from its simple $O(n^2)$ description the usefulness of partial multipliers will arise in a 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. As an interesting aside it seems that Paul Barrett describes a similar technique in his 1986 paper \cite{BARRETT} which was written five years before \cite{COMBA}. 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{mp\_grow}) \\ 2. If step 1 failed return(\textit{MP\_MEM}).\\ \\ Zero the temporary array $\hat W$. \\ 3. for $n$ from $0$ to $digs - 1$ do \\ \hspace{3mm}3.1 $\hat W_n \leftarrow 0$ \\ \\ Compute the columns. \\ 4. for $ix$ from $0$ to $a.used - 1$ do \\ \hspace{3mm}4.1 $pb \leftarrow \mbox{min}(b.used, digs - ix)$ \\ \hspace{3mm}4.2 If $pb < 1$ then goto step 5. \\ \hspace{3mm}4.3 for $iy$ from $0$ to $pb - 1$ do \\ \hspace{6mm}4.3.1 $\hat W_{ix+iy} \leftarrow \hat W_{ix+iy} + a_{ix}b_{iy}$ \\ \\ Propagate the carries upwards. \\ 5. $oldused \leftarrow c.used$ \\ 6. $c.used \leftarrow digs$ \\ 7. If $digs > 1$ then do \\ \hspace{3mm}7.1. for $ix$ from $1$ to $digs - 1$ do \\ \hspace{6mm}7.1.1 $\hat W_{ix} \leftarrow \hat W_{ix} + \lfloor \hat W_{ix-1} / \beta \rfloor$ \\ \hspace{6mm}7.1.2 $c_{ix - 1} \leftarrow \hat W_{ix - 1} \mbox{ (mod }\beta\mbox{)}$ \\ 8. else do \\ \hspace{3mm}8.1 $ix \leftarrow 0$ \\ 9. $c_{ix} \leftarrow \hat W_{ix} \mbox{ (mod }\beta\mbox{)}$ \\ \\ Zero excess digits. \\ 10. If $digs < oldused$ then do \\ \hspace{3mm}10.1 for $n$ from $digs$ to $oldused - 1$ do \\ \hspace{6mm}10.1.1 $c_n \leftarrow 0$ \\ 11. Clamp excessive digits of $c$. (\textit{mp\_clamp}) \\ 12. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm fast\_s\_mp\_mul\_digs} \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{Polynomial Basis Multiplication} To break the $O(n^2)$ barrier in multiplication requires a completely different look at integer multiplication. In the following algorithms the use of polynomial basis representation for two integers $a$ and $b$ as $f(x) = \sum_{i=0}^{n} a_i x^i$ and $g(x) = \sum_{i=0}^{n} b_i x^i$. respectively, is required. In this system both $f(x)$ and $g(x)$ have $n + 1$ terms and are of the $n$'th degree. The product $a \cdot b \equiv f(x) \cdot g(x)$ is the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$. The coefficients $w_i$ will directly yield the desired product when $\beta$ is substituted for $x$. The direct solution to solve for the $2n + 1$ coefficients requires $O(n^2)$ time and is would be in practice slower than the Comba technique. However, numerical analysis theory will indicate that only $2n + 1$ points in $W(x)$ are required to provide $2n + 1$ knowns for the $2n + 1$ unknowns. This means by finding $\zeta_y = W(y)$ for $2n + 1$ small values of $y$ the coefficients of $W(x)$ can be found with trivial Gaussian elimination. Since the polynomial $W(x)$ is unknown the equivalent $\zeta_y = f(y) \cdot g(y)$ is used in its place. The benefit of this technique stems from the fact that $f(y)$ and $g(y)$ are much smaller than either $a$ or $b$ respectively. In fact if both polynomials have $n + 1$ terms then the multiplicands will be $n$ times smaller than the inputs. Even if $2n + 1$ multiplications are required since they are of smaller values the algorithm is still faster. When picking points to gather relations there are always three obvious points to choose, $y = 0, 1$ and $ \infty$. The $\zeta_0$ term is simply the product $W(0) = w_0 = a_0 \cdot b_0$. The $\zeta_1$ term is the product $W(1) = \left (\sum_{i = 0}^{n} a_i \right ) \left (\sum_{i = 0}^{n} b_i \right )$. The third point $\zeta_{\infty}$ is less obvious but rather simple to explain. The $2n + 1$'th coefficient of $W(x)$ is numerically equivalent to the most significant column in an integer multiplication. The point at $\infty$ is used symbolically to represent the most significant column, that is $W(\infty) = w_{2n + 1} = a_nb_n$. Note that the points at $y = 0$ and $\infty$ yield the coefficients $w_0$ and $w_{2n + 1}$ directly. If more points are required they should be of small input values which are powers of two such as $2^q$ and the related \textit{mirror points} $\left (2^q \right )^{2n} \cdot \zeta_{2^{-q}}$ for small values of $q$. Using such points will allow the values of $f(y)$ and $g(y)$ to be independently calculated using only left shifts. As a general rule of the algorithm when the inputs are split into $n$ parts each there are $2n - 1$ multiplications. Each multiplication is of multiplicands that have $n$ times fewer digits than the inputs. The asymptotic running time of this algorithm is $O \left ( k^{lg_n(2n - 1)} \right )$ for $k$ digit inputs (\textit{assuming they have the same number of digits}). The following table summarizes the exponents for various values of $n$. \newpage\begin{figure} \begin{center} \begin{tabular}{|c|c|c|} \hline \textbf{Split into $n$ Parts} & \textbf{Exponent} & \textbf{Notes}\\ \hline $2$ & $1.584962501$ & This is Karatsuba Multiplication. \\ \hline $3$ & $1.464973520$ & This is Toom-Cook Multiplication. \\ \hline $4$ & $1.403677461$ &\\ \hline $5$ & $1.365212389$ &\\ \hline $10$ & $1.278753601$ &\\ \hline $100$ & $1.149426538$ &\\ \hline $1000$ & $1.100270931$ &\\ \hline $10000$ & $1.075252070$ &\\ \hline \end{tabular} \end{center} \caption{Asymptotic Running Time of Polynomial Basis Multiplication} \end{figure} At first it may seem like a good idea to choose $n = 1000$ since afterall the exponent is approximately $1.1$. However, the overhead of solving for the 2001 terms of $W(x)$ will certainly consume any savings the algorithm could offer for all but exceedingly large numbers. \subsubsection{Cutoff Point} The polynomial basis multiplication algorithms all require fewer single precision multiplications than a straight Comba approach. However, the algorithms incur an overhead (\textit{at the $O(n)$ work level}) since they require a system of equations to be solved. This makes them costly to use with small inputs. Let $m$ represent the number of digits in the multiplicands (\textit{assume both multiplicands have the same number of digits}). There exists a point $y$ such that when $m < y$ the polynomial basis algorithms are more costly than Comba, when $m = y$ they are roughly the same cost and when $m > y$ the Comba methods are slower than the polynomial basis algorithms. The exact location of $y$ depends on several key architectural elements of the computer platform in question. \begin{enumerate} \item The ratio of clock cycles for single precision multiplication versus other simpler operations such as addition, shifting, etc. For example on the AMD Athlon the ratio is roughly $17 : 1$ while on the Intel P4 it is $29 : 1$. The higher the ratio in favour of multiplication the lower the cutoff point $y$ will be. \item The complexity of the linear system of equations (\textit{for the coefficients of $W(x)$}) is. Generally speaking as the number of splits grows the complexity grows substantially. Ideally solving the system will only involve addition, subtraction and shifting of integers. This directly reflects on the ratio previous mentioned. \item To a lesser extent memory bandwidth and function call overheads. Provided the values are in the processor cache this is less of an influence over the cutoff point. \end{enumerate} A clean cutoff point separation occurs when a point $y$ is found such that all of the cutoff point conditions are met. For example, if the point is too low then there will be values of $m$ such that $m > y$ and the Comba method is still faster. Finding the cutoff points is fairly simple when a high resolution timer is available. \subsection{Karatsuba Multiplication} Karatsuba multiplication \cite{KARA} when originally proposed in 1962 was among the first set of algorithms to break the $O(n^2)$ barrier for general purpose multiplication. Given two polynomial basis representations $f(x) = ax + b$ and $g(x) = cx + d$ Karatsuba proved with light number theory \cite{KARAP} that the following polynomial is equivalent to multiplication of the two integers the polynomials represent. \begin{equation} f(x) \cdot g(x) = acx^2 + ((a - b)(c - d) + ac + bd)x + bd \end{equation} Using the observation that $ac$ and $bd$ could be re-used only three half sized multiplications would be required to produce the product. Applying this recursively the work factor becomes $O(n^{lg(3)})$ which is substantially better than the work factor $O(n^2)$ of the Comba technique. It turns out what Karatsuba did not know or at least did not publish was that this is simply polynomial basis multiplication with the points $\zeta_0$, $\zeta_{\infty}$ and $-\zeta_{-1}$. Consider the resultant system of equations. \begin{center} \begin{tabular}{rcrcrcrc} $\zeta_{0}$ & $=$ & & & & & $w_0$ \\ $-\zeta_{-1}$ & $=$ & $-w_2$ & $+$ & $w_1$ & $-$ & $w_0$ \\ $\zeta_{\infty}$ & $=$ & $w_2$ & & & & \\ \end{tabular} \end{center} By adding the first and last equation to the equation in the middle the term $w_1$ can be isolated and all three coefficients solved for. The simplicity of this system of equations has made Karatsuba fairly popular. In fact the cutoff point is often fairly low\footnote{With LibTomMath 0.18 it is 70 and 109 for the Intel P4 and AMD Athlon respectively.} making it an ideal algorithm to speed up certain public key cryptosystems such as RSA and Diffie-Hellman. It is worth noting that the point $\zeta_1$ could be substituted for $-\zeta_{-1}$. In this case the first and third row are subtracted instead of added to the second row. \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_karatsuba\_mul}. \\ \textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ \textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert$ \\ \hline \\ 1. Init the following mp\_int variables: $x0$, $x1$, $y0$, $y1$, $t1$, $x0y0$, $x1y1$.\\ 2. If step 2 failed then return(\textit{MP\_MEM}). \\ \\ Split the input. e.g. $a = x1 \cdot \beta^B + x0$ \\ 3. $B \leftarrow \mbox{min}(a.used, b.used)/2$ \\ 4. $x0 \leftarrow a \mbox{ (mod }\beta^B\mbox{)}$ (\textit{mp\_mod\_2d}) \\ 5. $y0 \leftarrow b \mbox{ (mod }\beta^B\mbox{)}$ \\ 6. $x1 \leftarrow \lfloor a / \beta^B \rfloor$ (\textit{mp\_rshd}) \\ 7. $y1 \leftarrow \lfloor b / \beta^B \rfloor$ \\ \\ Calculate the three products. \\ 8. $x0y0 \leftarrow x0 \cdot y0$ (\textit{mp\_mul}) \\ 9. $x1y1 \leftarrow x1 \cdot y1$ \\ 10. $t1 \leftarrow x1 - x0$ (\textit{mp\_sub}) \\ 11. $x0 \leftarrow y1 - y0$ \\ 12. $t1 \leftarrow t1 \cdot x0$ \\ \\ Calculate the middle term. \\ 13. $x0 \leftarrow x0y0 + x1y1$ \\ 14. $t1 \leftarrow x0 - t1$ \\ \\ Calculate the final product. \\ 15. $t1 \leftarrow t1 \cdot \beta^B$ (\textit{mp\_lshd}) \\ 16. $x1y1 \leftarrow x1y1 \cdot \beta^{2B}$ \\ 17. $t1 \leftarrow x0y0 + t1$ \\ 18. $c \leftarrow t1 + x1y1$ \\ 19. Clear all of the temporary variables. \\ 20. Return(\textit{MP\_OKAY}).\\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_karatsuba\_mul} \end{figure} \textbf{Algorithm mp\_karatsuba\_mul.} This algorithm computes the unsigned product of two inputs using the Karatsuba method. It is loosely based on the description from \cite[pp. 294-295]{TAOCPV2}. \index{radix point} In order to split the two inputs into their respective halves a suitable \textit{radix point} must be chosen. The radix point chosen must be used for both of the inputs meaning that it must smaller than the smallest input. Step 3 chooses the radix point $B$ as half of the smallest input \textbf{used} count. After the radix point is chosen the inputs are split into lower and upper halves. Step 4 and 5 compute the lower halves. Step 6 and 7 computer the upper halves. After the halves have been computed the three intermediate half-size products must be computed. Step 8 and 9 compute the trivial products $x0 \cdot y0$ and $x1 \cdot y1$. The mp\_int $x0$ is used as a temporary variable after $x1 - x0$ has been computed. By using $x0$ instead of an additional temporary variable the algorithm can avoid an addition memory allocation operation. The remaining steps 13 through 18 compute the Karatsuba polynomial through a variety of digit shifting and addition operations. EXAM,bn_mp_karatsuba_mul.c The new coding element in this routine that has not been seen in the previous routines yet is the usage of the goto statements. The normal wisdom is that goto statements should be avoided. This is generally true however, when every single function call can fail it makes sense to handle error recovery with a single piece of code. Lines @61,if@ to @75,if@ handle initializing all of the temporary variables required. Note how each of the if statements goes to a different label in case of failure. This allows the routine to correctly free only the temporaries that have been successfully allocated so far. The temporary variables are all initialized using the mp\_init\_size routine since they are expected to be large. This saves the additional reallocation that would have been necessary. Also $x0$, $x1$, $y0$ and $y1$ have to be able to hold at least their respective number of digits for the next section of code. The first algebraic portion of the algorithm is to split the two inputs into their halves. However, instead of using mp\_mod\_2d and mp\_rshd to extract the halves the code has been inlined. To initialize the halves the \textbf{used} and \textbf{sign} members are copied first. The first for loop on line @98,for@ copies the lower halves. Since they are both the same magnitude it is simpler to calculate both lower halves in a single loop. The for loop on lines @104,for@ and @109,for@ calculate the upper halves $x1$ and $y1$ respectively. By inlining the calculation of the halves the Karatsuba multiplier has a slightly lower overhead. As a result it can be used for smaller inputs. When line @152,err@ is reached the algorithm has completed succesfully. The ``error status'' variable $err$ is set to \textbf{MP\_OKAY} so that the same code that handles errors can be used to clear the temporary variables and return. \subsection{Toom-Cook $3$-Way Multiplication} Toom-Cook $3$-Way multiplication \cite{TOOM} is essentially the polynomial basis algorithm for $n = 3$ except that the points are chosen such that $\zeta$ is easy to compute and the resulting system of equations easy to reduce. In this algorithm the points $\zeta_{0}$, $16 \cdot \zeta_{1 \over 2}$, $\zeta_1$, $\zeta_2$ and $\zeta_{\infty}$ make up the five requires points to solve for the coefficients of the product. At first glance the five coefficents are relatively efficient to compute with the exception of $16 \cdot \zeta{1 \over 2}$. This coefficient is related to $\zeta_2 = (4a_2 + 2a_1 + a_0)(4b_2 + 2b_1 + b_0)$ in that the coefficients of two terms are reversed (\textit{or mirrored}). Simply put $16 \cdot \zeta{1 \over 2} = (a_2 + 2a_1 + 4a_0)(b_2 + 2b_1 + 4b_0)$. With the five relations that Toom has chosen the following system of equations is formed. \begin{center} \begin{tabular}{rcrcrcrcrcr} $\zeta_0$ & $=$ & $0w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $1w_0$ \\ $16 \cdot \zeta_{1 \over 2}$ & $=$ & $1w_4$ & $+$ & $2w_3$ & $+$ & $4w_2$ & $+$ & $8w_1$ & $+$ & $16w_0$ \\ $\zeta_1$ & $=$ & $1w_4$ & $+$ & $1w_3$ & $+$ & $1w_2$ & $+$ & $1w_1$ & $+$ & $1w_0$ \\ $\zeta_2$ & $=$ & $16w_4$ & $+$ & $8w_3$ & $+$ & $4w_2$ & $+$ & $2w_1$ & $+$ & $1w_0$ \\ $\zeta_{\infty}$ & $=$ & $1w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $0w_0$ \\ \end{tabular} \end{center} A trivial solution to this matrix requires $12$ subtractions, two multiplications by a small power of two, two divisions by a small power of two, two divisions by three and one multiplication by three. All of these $19$ sub-operations require less than quadratic time meaning that the algorithm overall can be faster than a baseline multiplication. However, the greater complexity of this algorithm places the cutoff point (\textbf{TOOM\_MUL\_CUTOFF}) where Toom-Cook becomes the most efficient algorithm very much higher above the Karatsuba cutoff point. \subsection{Signed Multiplication} Now that algorithms to handle multiplications of every useful dimensions has been developed a rather simple finishing touch is required. So far all of the multiplication algorithms have been unsigned which leaves only a signed multiplication algorithm to be established. \begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_mul}. \\ \textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ \textbf{Output}. $c \leftarrow a \cdot b$ \\ \hline \\ 1. If $a.sign = b.sign$ then \\ \hspace{3mm}1.1 $sign = MP\_ZPOS$ \\ 2. else \\ \hspace{3mm}2.1 $sign = MP\_ZNEG$ \\ 3. If min$(a.used, b.used) \ge TOOM\_MUL\_CUTOFF$ then \\ \hspace{3mm}3.1 $c \leftarrow a \cdot b$ using algorithm mp\_toom\_mul \\ 4. else if min$(a.used, b.used) \ge KARATSUBA\_MUL\_CUTOFF$ then \\ \hspace{3mm}4.1 $c \leftarrow a \cdot b$ using algorithm mp\_karatsuba\_mul \\ 5. else \\ \hspace{3mm}5.1 $digs \leftarrow a.used + b.used + 1$ \\ \hspace{3mm}5.2 If $digs < MP\_ARRAY$ and min$(a.used, b.used) \le \delta$ then \\ \hspace{6mm}5.2.1 $c \leftarrow a \cdot b \mbox{ (mod }\beta^{digs}\mbox{)}$ using algorithm fast\_s\_mp\_mul\_digs. \\ \hspace{3mm}5.3 else \\ \hspace{6mm}5.3.1 $c \leftarrow a \cdot b \mbox{ (mod }\beta^{digs}\mbox{)}$ using algorithm s\_mp\_mul\_digs. \\ 6. $c.sign \leftarrow sign$ \\ 7. Return the result of the unsigned multiplication performed. \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_mul} \end{figure} \textbf{Algorithm mp\_mul.} This algorithm performs the signed multiplication of two inputs. It will make use of any of the three unsigned multiplication algorithms available when the input is of appropriate size. The \textbf{sign} of the result is not set until the end of the algorithm since algorithm s\_mp\_mul\_digs will clear it. EXAM,bn_mp_mul.c The implementation is rather simplistic and is not particularly noteworthy. Line @22,?@ computes the sign of the result using the ``?'' operator from the C programming language. Line @37,<<@ computes $\delta$ using the fact that $1 << k$ is equal to $2^k$. \section{Squaring} Squaring is a special case of multiplication where both multiplicands are equal. At first it may seem like there is no significant optimization available but in fact there is. Consider the multiplication of $576$ against $241$. In total there will be nine single precision multiplications performed which are $1\cdot 6$, $1 \cdot 7$, $1 \cdot 5$, $4 \cdot 6$, $4 \cdot 7$, $4 \cdot 5$, $2 \cdot 6$, $2 \cdot 7$ and $2 \cdot 5$. Now consider the multiplication of $123$ against $123$. The nine products are $3 \cdot 3$, $3 \cdot 2$, $3 \cdot 1$, $2 \cdot 3$, $2 \cdot 2$, $2 \cdot 1$, $1 \cdot 3$, $1 \cdot 2$ and $1 \cdot 1$. On closer inspection some of the products are equivalent. For example, $3 \cdot 2 = 2 \cdot 3$ and $3 \cdot 1 = 1 \cdot 3$. For any $n$-digit input there are ${{\left (n^2 + n \right)}\over 2}$ possible unique single precision multiplications required. The following diagram demonstrates the operations required. \begin{figure}[here] \begin{center} \begin{tabular}{ccccc|c} &&1&2&3&\\ $\times$ &&1&2&3&\\ \hline && $3 \cdot 1$ & $3 \cdot 2$ & $3 \cdot 3$ & Row 0\\ & $2 \cdot 1$ & $2 \cdot 2$ & $2 \cdot 3$ && Row 1 \\ $1 \cdot 1$ & $1 \cdot 2$ & $1 \cdot 3$ &&& Row 2 \\ \end{tabular} \end{center} \caption{Squaring Optimization Diagram} \end{figure} MARK,SQUARE Starting from zero and numbering the columns from right to left a very simple pattern becomes obvious. For the purposes of this discussion let $x$ represent the number being squared. The first observation is that in row $k$ the $2k$'th column of the product has a $\left (x_k \right)^2$ term in it. The second observation is that every column $j$ in row $k$ where $j \ne 2k$ is part of a double product. Every odd column is made up entirely of double products. In fact every column is made up of double products and at most one square (\textit{see the exercise section}). The third and final observation is that for row $k$ the first unique non-square term occurs at column $2k + 1$. For example, on row $1$ of the previous squaring, column one is part of the double product with column one from row zero. Column two of row one is a square and column three is the first unique column. \subsection{The Baseline Squaring Algorithm} The baseline squaring algorithm is meant to be a catch-all squaring algorithm. It will handle any of the input sizes that the faster routines will not handle. \begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{s\_mp\_sqr}. \\ \textbf{Input}. mp\_int $a$ \\ \textbf{Output}. $b \leftarrow a^2$ \\ \hline \\ 1. Init a temporary mp\_int of at least $2 \cdot a.used +1$ digits. (\textit{mp\_init\_size}) \\ 2. If step 1 failed return(\textit{MP\_MEM}) \\ 3. $t.used \leftarrow 2 \cdot a.used + 1$ \\ 4. For $ix$ from 0 to $a.used - 1$ do \\ \hspace{3mm}Calculate the square. \\ \hspace{3mm}4.1 $\hat r \leftarrow t_{2ix} + \left (a_{ix} \right )^2$ \\ \hspace{3mm}4.2 $t_{2ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ \hspace{3mm}Calculate the double products after the square. \\ \hspace{3mm}4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ \hspace{3mm}4.4 For $iy$ from $ix + 1$ to $a.used - 1$ do \\ \hspace{6mm}4.4.1 $\hat r \leftarrow 2 \cdot a_{ix}a_{iy} + t_{ix + iy} + u$ \\ \hspace{6mm}4.4.2 $t_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ \hspace{6mm}4.4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ \hspace{3mm}Set the last carry. \\ \hspace{3mm}4.5 While $u > 0$ do \\ \hspace{6mm}4.5.1 $iy \leftarrow iy + 1$ \\ \hspace{6mm}4.5.2 $\hat r \leftarrow t_{ix + iy} + u$ \\ \hspace{6mm}4.5.3 $t_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ \hspace{6mm}4.5.4 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ 5. Clamp excess digits of $t$. (\textit{mp\_clamp}) \\ 6. Exchange $b$ and $t$. \\ 7. Clear $t$ (\textit{mp\_clear}) \\ 8. Return(\textit{MP\_OKAY}) \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm s\_mp\_sqr} \end{figure} \textbf{Algorithm s\_mp\_sqr.} This algorithm computes the square of an input using the three observations on squaring. It is based fairly faithfully on algorithm 14.16 of \cite[pp.596-597]{HAC}. Similar to algorithm s\_mp\_mul\_digs a temporary mp\_int is allocated to hold the result of the squaring. This allows the destination mp\_int to be the same as the source mp\_int without losing information part way through the squaring. The outer loop of this algorithm begins on step 4. It is best to think of the outer loop as walking down the rows of the partial results while the inner loop computes the columns of the partial result. Step 4.1 and 4.2 compute the square term for each row while step 4.3 and 4.4 propagate the carry and compute the double products. The requirement that a mp\_word be able to represent the range $0 \le x < 2 \beta^2$ arises from this very algorithm. The product $a_{ix}a_{iy}$ will lie in the range $0 \le x \le \beta^2 - 2\beta + 1$ which is obviously less than $\beta^2$ meaning that when it is multiply by two it can be represented by a mp\_word properly. Similar to algorithm s\_mp\_mul\_digs after every pass of the inner loop the destination is correctly set to the sum of all of the partial results calculated so far. This involves expensive carry propagation which will be eliminated shortly. EXAM,bn_s_mp_sqr.c Inside the outer loop (\textit{see line @32,for@}) the square term is calculated on line @35,r =@. Line @42,>>@ extracts the carry from the square term. Aliases for $a_{ix}$ and $t_{ix+iy}$ are initialized on lines @45,tmpx@ and @48,tmpt@ respectively. The doubling is performed using two additions (\textit{see line @57,r + r@}) since it is usually faster than shifting if not at least as fast. \subsection{Faster Squaring by the ``Comba'' Method} A major drawback to the baseline method is the requirement for single precision shifting inside the $O(n^2)$ work level. Squaring has an additional drawback that it must double the product inside the inner loop as well. As for multiplication the Comba technique can be used to eliminate these performance hazards. The first obvious solution is to make an array of mp\_words which will hold all of the columns. This will indeed eliminate all of the carry propagation operations from the inner loop. However, the inner product must still be doubled $O(n^2)$ times. The solution stems from the simple fact that $2a + 2b + 2c = 2(a + b + c)$. That is the sum of all of the double products is equal to double the sum of all the products. For example, $ab + ba + ac + ca = 2ab + 2ac = 2(ab + ac)$. However, we cannot simply double all of the columns since the squares appear only once per row. The most practical solution is to have two mp\_word arrays. One array will hold the squares and the other array will hold the double products. With both arrays the doubling and carry propagation can be moved to a $O(n)$ work level outside the $O(n^2)$ level. \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{fast\_s\_mp\_sqr}. \\ \textbf{Input}. mp\_int $a$ \\ \textbf{Output}. $b \leftarrow a^2$ \\ \hline \\ Place two arrays of \textbf{MP\_WARRAY} mp\_words named $\hat W$ and $\hat {X}$ on the stack. \\ 1. If $b.alloc < 2a.used + 1$ then grow $b$ to $2a.used + 1$ digits. (\textit{mp\_grow}). \\ 2. If step 1 failed return(\textit{MP\_MEM}). \\ 3. for $ix$ from $0$ to $2a.used + 1$ do \\ \hspace{3mm}3.1 $\hat W_{ix} \leftarrow 0$ \\ \hspace{3mm}3.2 $\hat {X}_{ix} \leftarrow 0$ \\ 4. for $ix$ from $0$ to $a.used - 1$ do \\ \hspace{3mm}Compute the square.\\ \hspace{3mm}4.1 $\hat {X}_{ix+ix} \leftarrow \left ( a_ix \right )^2$ \\ \hspace{3mm}Compute the double products.\\ \hspace{3mm}4.2 for $iy$ from $ix + 1$ to $a.used - 1$ do \\ \hspace{6mm}4.2.1 $\hat W_{ix+iy} \leftarrow \hat W_{ix+iy} + a_{ix}a_{iy}$ \\ 5. $oldused \leftarrow b.used$ \\ 6. $b.used \leftarrow 2a.used + 1$ \\ Double the products and propagate the carries simultaneously. \\ 7. $\hat W_0 \leftarrow 2 \hat W_0 + \hat {X}_0$ \\ 8. for $ix$ from $1$ to $2a.used$ do \\ \hspace{3mm}8.1 $\hat W_{ix} \leftarrow 2 \hat W_{ix} + \hat {X}_{ix}$ \\ \hspace{3mm}8.2 $\hat W_{ix} \leftarrow \hat W_{ix} + \lfloor \hat W_{ix - 1} / \beta \rfloor$ \\ \hspace{3mm}8.3 $b_{ix-1} \leftarrow W_{ix-1} \mbox{ (mod }\beta\mbox{)}$ \\ 9. $b_{2a.used} \leftarrow \hat W_{2a.used} \mbox{ (mod }\beta\mbox{)}$ \\ 10. if $2a.used + 1 < oldused$ then do \\ \hspace{3mm}10.1 for $ix$ from $2a.used + 1$ to $oldused$ do \\ \hspace{6mm}10.1.1 $b_{ix} \leftarrow 0$ \\ 11. Clamp excess digits from $b$. (\textit{mp\_clamp}) \\ 12. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm fast\_s\_mp\_sqr} \end{figure} \textbf{Algorithm fast\_s\_mp\_sqr.} This algorithm computes the square of an input using the Comba technique. It is designed to be a replacement for algorithm s\_mp\_sqr when the amount of input digits is less than \textbf{MP\_WARRAY} and less than $\delta \over 2$. This routine requires two arrays of mp\_words to be placed on the stack. The first array $\hat W$ will hold the double products and the second array $\hat X$ will hold the squares. Though only at most $MP\_WARRAY \over 2$ words of $\hat X$ are used it has proven faster on most processors to simply make it a full size array. The loop on step 3 will zero the two arrays to prepare them for the squaring step. Step 4.1 computes the squares of the product. Note how it simply assigns the value into the $\hat X$ array. The nested loop on step 4.2 computes the doubles of the products. In actuality that loop computes the sum of the products for each column. They are not doubled until later. After the squaring loop the products stored in $\hat W$ musted be doubled and the carries propagated forwards. It makes sense to do both operations at the same time. The expression $\hat W_{ix} \leftarrow 2 \hat W_{ix} + \hat {X}_{ix}$ computes the sum of the double product and the squares in place. EXAM,bn_fast_s_mp_sqr.c -- Write something deep and insightful later, Tom. \subsection{Polynomial Basis Squaring} The same algorithm that performs optimal polynomial basis multiplication can be used to perform polynomial basis squaring. The minor exception is that $\zeta_y = f(y) \cdot g(y)$ is actually equivalent to $\zeta_y = f(y)^2$ since $f(y) = g(y)$. That is instead of performing $2n + 1$ multiplications to find the $\zeta$ relations squaring operations are performed instead. \subsection{Karatsuba Squaring} Let $f(x) = ax + b$ represent the polynomial basis representation of a number to square. Let $h(x) = \left ( f(x) \right )^2$ represent the square of the polynomial. The Karatsuba equation can be modified to square a number with the following equation. \begin{equation} h(x) = a^2x^2 + \left (a^2 + b^2 - (a - b)^2 \right )x + b^2 \end{equation} Upon closer inspection this equation only requires the calculation of three half-sized squares: $a^2$, $b^2$ and $(a - b)^2$. As in Karatsuba multiplication this algorithm can be applied recursively on the input and will achieve an asymptotic running time of $O \left ( n^{lg(3)} \right )$. If the asymptotic time of Karatsuba squaring and multiplication is the same why not simply use the multiplication algorithm instead? The answer to this question arises from the cutoff point for squaring. As in multiplication there exists a cutoff point at which the time required for a Comba based squaring and a Karatsuba based squaring meet. Due to the overhead inherent in the Karatsuba method the cutoff point is fairly high. For example, on an Athlon processor with $\beta = 2^{28}$ the cutoff point is around 127 digits. Consider squaring a 200 digit number with this technique. It will be split into two 100 digit halves which are subsequently squared. The 100 digit numbers will not be squared using Karatsuba but instead the faster Comba based squaring algorithm. If Karatsuba multiplication were used instead the 100 digit numbers would be squared with a slower Comba based multiplication. \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_karatsuba\_sqr}. \\ \textbf{Input}. mp\_int $a$ \\ \textbf{Output}. $b \leftarrow a^2$ \\ \hline \\ 1. Initialize the following temporary mp\_ints: $x0$, $x1$, $t1$, $t2$, $x0x0$ and $x1x1$. \\ 2. If any of the initializations on step 1 failed return(\textit{MP\_MEM}). \\ \\ Split the input. e.g. $a = x1\beta^B + x0$ \\ 3. $B \leftarrow a.used / 2$ \\ 4. $x0 \leftarrow a \mbox{ (mod }\beta^B\mbox{)}$ (\textit{mp\_mod\_2d}) \\ 5. $x1 \leftarrow \lfloor a / \beta^B \rfloor$ (\textit{mp\_lshd}) \\ \\ Calculate the three squares. \\ 6. $x0x0 \leftarrow x0^2$ (\textit{mp\_sqr}) \\ 7. $x1x1 \leftarrow x1^2$ \\ 8. $t1 \leftarrow x1 - x0$ (\textit{mp\_sub}) \\ 9. $t1 \leftarrow t1^2$ \\ \\ Compute the middle term. \\ 10. $t2 \leftarrow x0x0 + x1x1$ (\textit{s\_mp\_add}) \\ 11. $t1 \leftarrow t2 - t1$ \\ \\ Compute final product. \\ 12. $t1 \leftarrow t1\beta^B$ (\textit{mp\_lshd}) \\ 13. $x1x1 \leftarrow x1x1\beta^{2B}$ \\ 14. $t1 \leftarrow t1 + x0x0$ \\ 15. $b \leftarrow t1 + x1x1$ \\ 16. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_karatsuba\_sqr} \end{figure} \textbf{Algorithm mp\_karatsuba\_sqr.} This algorithm computes the square of an input $a$ using the Karatsuba technique. This algorithm is very much similar to the Karatsuba based multiplication algorithm. The radix point for squaring is simply the placed above the median of the digits. Step 3, 4 and 5 compute the two halves required using $B$ as the radix point. The first two squares in steps 6 and 7 are rather straightforward while the last square is in a more compact form. By expanding $\left (x1 - x0 \right )^2$ the $x1^2$ and $x0^2$ terms in the middle disappear, that is $x1^2 + x0^2 - (x1 - x0)^2 = 2 \cdot x0 \cdot x1$. Now if $5n$ single precision additions and a squaring of $n$-digits is faster than multiplying two $n$-digit numbers and doubling then this method is faster. Assuming no further recursions occur the difference can be estimated. Let $p$ represent the cost of a single precision addition and $q$ the cost of a single precision multiplication both in terms of time\footnote{Or machine clock cycles.}. The question reduces to whether the following equation is true or not. \begin{equation} 5np +{{q(n^2 + n)} \over 2} \le pn + qn^2 \end{equation} For example, on an AMD Athlon processor $p = {1 \over 3}$ and $q = 6$. This implies that the following inequality should hold. \begin{center} \begin{tabular}{rcl} $5n + 3n^2 + 3n$ & $<$ & ${n \over 3} + 6n^2$ \\ ${25 \over 3} + 3n$ & $<$ & ${1 \over 3} + 6n$ \\ ${25 \over 3}$ & $<$ & $3n$ \\ ${25 \over 9}$ & $<$ & $n$ \\ \end{tabular} \end{center} This results in a cutoff point around $n = 3$. As a consequence it is actually faster to compute the middle term the ``long way'' on processors where multiplication is substantially slower\footnote{On the Athlon there is a 1:17 ratio between clock cycles for addition and multiplication. On the Intel P4 processor this ratio is 1:29 making this method even more beneficial. The only common exception is the ARMv4 processor which has a ratio of 1:7. } than simpler operations such as addition. EXAM,bn_mp_karatsuba_sqr.c This implementation is largely based on the implementation of algorithm mp\_karatsuba\_mul. It uses the same inline style to copy and shift the input into the two halves. The loop from line @54,{@ to line @70,}@ has been modified since only one input exists. The \textbf{used} count of both $x0$ and $x1$ is fixed up and $x0$ is clamped before the calculations begin. At this point $x1$ and $x0$ are valid equivalents to the respective halves as if mp\_rshd and mp\_mod\_2d had been used. By inlining the copy and shift operations the cutoff point for Karatsuba multiplication can be lowered. On the Athlon the cutoff point is exactly at the point where Comba squaring can no longer be used (\textit{128 digits}). On slower processors such as the Intel P4 it is actually below the Comba limit (\textit{at 110 digits}). This routine uses the same error trap coding style as mp\_karatsuba\_sqr. As the temporary variables are initialized errors are redirected to the error trap higher up. If the algorithm completes without error the error code is set to \textbf{MP\_OKAY} and the error traps are executed. \textit{Last paragraph sucks. re-write! -- Tom} \subsection{Toom-Cook Squaring} The Toom-Cook squaring algorithm mp\_toom\_sqr is heavily based on the algorithm mp\_toom\_mul with the minor exception noted. The reader is encouraged to read the description of the latter algorithm and try to derive their own Toom-Cook squaring algorithm. \subsection{Generic Squaring} \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_sqr}. \\ \textbf{Input}. mp\_int $a$ \\ \textbf{Output}. $b \leftarrow a^2$ \\ \hline \\ 1. If $a.used \ge TOOM\_SQR\_CUTOFF$ then \\ \hspace{3mm}1.1 $b \leftarrow a^2$ using algorithm mp\_toom\_sqr \\ 2. else if $a.used \ge KARATSUBA\_SQR\_CUTOFF$ then \\ \hspace{3mm}2.1 $b \leftarrow a^2$ using algorithm mp\_karatsuba\_sqr \\ 3. else \\ \hspace{3mm}3.1 $digs \leftarrow a.used + b.used + 1$ \\ \hspace{3mm}3.2 If $digs < MP\_ARRAY$ and $a.used \le \delta$ then \\ \hspace{6mm}3.2.1 $b \leftarrow a^2$ using algorithm fast\_s\_mp\_sqr. \\ \hspace{3mm}3.3 else \\ \hspace{6mm}3.3.1 $b \leftarrow a^2$ using algorithm s\_mp\_sqr. \\ 4. $b.sign \leftarrow MP\_ZPOS$ \\ 5. Return the result of the unsigned squaring performed. \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_sqr} \end{figure} \textbf{Algorithm mp\_sqr.} This algorithm computes the square of the input using one of four different algorithms. If the input is very large and has at least \textbf{TOOM\_SQR\_CUTOFF} or \textbf{KARATSUBA\_SQR\_CUTOFF} digits then either the Toom-Cook or the Karatsuba Squaring algorithm is used. If neither of the polynomial basis algorithms should be used then either the Comba or baseline algorithm is used. EXAM,bn_mp_sqr.c \section*{Exercises} \begin{tabular}{cl} $\left [ 3 \right ] $ & Devise an efficient algorithm for selection of the radix point to handle inputs \\ & that have different number of digits in Karatsuba multiplication. \\ & \\ $\left [ 3 \right ] $ & In ~SQUARE~ the fact that every column of a squaring is made up \\ & of double products and at most one square is stated. Prove this statement. \\ & \\ $\left [ 2 \right ] $ & In the Comba squaring algorithm half of the $\hat X$ variables are not used. \\ & Revise algorithm fast\_s\_mp\_sqr to shrink the $\hat X$ array. \\ & \\ $\left [ 3 \right ] $ & Prove the equation for Karatsuba squaring. \\ & \\ $\left [ 1 \right ] $ & Prove that Karatsuba squaring requires $O \left (n^{lg(3)} \right )$ time. \\ & \\ $\left [ 2 \right ] $ & Determine the minimal ratio between addition and multiplication clock cycles \\ & required for equation $6.7$ to be true. \\ & \\ \end{tabular} \chapter{Modular Reduction} MARK,REDUCTION \section{Basics of Modular Reduction} \index{modular residue} Modular reduction is an operation that arises quite often within public key cryptography algorithms. A number is said to be reduced modulo another number by finding the remainder of division. If an integer $a$ is reduced modulo $b$ that is to solve the equation $a = bq + p$ then $p$ is the result. To phrase that another way ``$p$ is congruent to $a$ modulo $b$'' which is also written as $p \equiv a \mbox{ (mod }b\mbox{)}$. In other vernacular $p$ is known as the ``modular residue'' which leads to ``quadratic residue''\footnote{That's fancy talk for $b \equiv a^2 \mbox{ (mod }p\mbox{)}$.} and other forms of residues. \index{modulus} Modular reductions are normally used to form finite groups such as fields and rings. For example, in the RSA public key algorithm \cite{RSAPAPER} two private primes $p$ and $q$ are chosen which when multiplied $n = pq$ forms a composite modulus. When operations such as multiplication and squaring are performed on units of the ring $\Z_n$ a finite multiplicative sub-group is formed. This sub-group is the group used to perform RSA operations. Do not worry to much about how RSA works as it is not important for this discussion. The most common usage for performance driven modular reductions is in modular exponentiation algorithms. That is to compute $d = a^b \mbox{ (mod }c\mbox{)}$ as fast as possible. As will be discussed in the subsequent chapter there exists fast algorithms for computing modular exponentiations without having to perform (\textit{in this example}) $b$ multiplications. These algorithms will produce partial results in the range $0 \le x < c^2$ which can be taken advantage of. The obvious line of thinking is to use an integer division routine and just extract the remainder. While this is equivalent to finding the modular residue it turns out that the limited range of the input can be exploited to create several efficient algorithms. \section{The Barrett Reduction} The Barrett reduction algorithm \cite{BARRETT} was inspired by fast division algorithms which multiply by the reciprocal to emulate division. Barretts observation was that the residue $c$ of $a$ modulo $b$ is equal to \begin{equation} c = a - b \cdot \lfloor a/b \rfloor \end{equation} Since algorithms such as modular reduction would be using the same modulus extensively, using typical DSP intuition the next step would be to replace $a/b$ with a multiplication by the reciprocal. However, DSP intuition on its own will not work as these numbers are considerably larger than the precision of common DSP floating point data types. It would take another common optimization to optimize the algorithm. \subsection{Fixed Point Arithmetic} The trick used to optimize the above equation is based on a technique of emulating floating point data types with fixed precision integers. Fixed point arithmetic would be vastly popularlized in the mid 1990s for bringing 3d-games to the mass market. The idea is to take a normal $k$-bit integer data type and break it into $p$-bit integer and a $q$-bit fraction part (\textit{where $p+q = k$}). In this system a $k$-bit integer $n$ would actually represent $n/2^q$. For example, with $q = 4$ the integer $n = 37$ would actually represent the value $2.3125$. To multiply two fixed point numbers the integers are multiplied using traditional arithmetic and subsequently normalized. For example, with $q = 4$ to multiply the integers $9$ and $5$ they must be converted to fixed point first by multiplying by $2^q$. Let $a = 9(2^q)$ represent the fixed point representation of $9$ and $b = 5(2^q)$ represent the fixed point representation of $5$. The product $ab$ is equal to $45(2^{2q})$ which when normalized produces $45(2^q)$. Using fixed point arithmetic division can be easily achieved by multiplying by the reciprocal. If $2^q$ is equivalent to one than $2^q/b$ is equivalent to $1/b$ using real arithmetic. Using this fact dividing an integer $a$ by another integer $b$ can be achieved with the following expression. \begin{equation} \lfloor (a \cdot (\lfloor 2^q / b \rfloor))/2^q \rfloor \end{equation} The precision of the division is proportional to the value of $q$. If the divisor $b$ is used frequently as is the case with modular exponentiation pre-computing $2^q/b$ will allow a division to be performed with a multiplication and a right shift. Both operations are considerably faster than division on most processors. Consider dividing $19$ by $5$. The correct result is $\lfloor 19/5 \rfloor = 3$. With $q = 3$ the reciprocal is $\lfloor 2^q/5 \rfloor = 1$ which leads to a product of $19$ which when divided by $2^q$ produces $2$. However, with $q = 4$ the reciprocal is $\lfloor 2^q/5 \rfloor = 3$ and the result of the emulated division is $\lfloor 3 \cdot 19 / 2^q \rfloor = 3$ which is correct. Plugging this form of divison into the original equation the following modular residue equation arises. \begin{equation} c = a - b \cdot \lfloor (a \cdot (\lfloor 2^q / b \rfloor))/2^q \rfloor \end{equation} Using the notation from \cite{BARRETT} the value of $\lfloor 2^q / b \rfloor$ will be represented by the $\mu$ symbol. Using the $\mu$ variable also helps re-inforce the idea that it is meant to be computed once and re-used. \begin{equation} c = a - b \cdot \lfloor (a \cdot \mu)/2^q \rfloor \end{equation} Provided that $2^q > b^2$ this algorithm will produce a quotient that is either exactly correct or off by a value of one. Let $n$ represent the number of digits in $b$. This algorithm requires approximately $2n^2$ single precision multiplications to produce the quotient and another $n^2$ single precision multiplications to find the residue. In total $3n^2$ single precision multiplications are required to reduce the number. For example, if $b = 1179677$ and $q = 41$ ($2^q > b^2$), then the reciprocal $\mu$ is equal to $\lfloor 2^q / b \rfloor = 1864089$. Consider reducing $a = 180388626447$ modulo $b$ using the above reduction equation. Using long division the quotient $\lfloor a/b \rfloor$ is equal to the quotient found using the fixed point method. In this case the quotient is $\lfloor (a \cdot \mu)/2^q \rfloor = 152913$ and can produce the modular residue $a - 152913b = 677346$. \subsection{Choosing a Radix Point} Using the fixed point representation a modular reduction can be performed with $3n^2$ single precision multiplications. If that were the best that could be achieved a full division might as well be used in its place. The key to optimizing the reduction is to reduce the precision of the initial multiplication that finds the quotient. Let $a$ represent the number of which the residue is sought. Let $b$ represent the modulus used to find the residue. Let $m$ represent the number of digits in $b$. For the purposes of this discussion we will assume that the number of digits in $a$ is $2m$. Dividing $a$ by $b$ is the same as dividing a $2m$ digit integer by a $m$ digit integer. Digits below the $m - 1$'th digit of $a$ will contribute at most a value of $1$ to the quotient because $\beta^k < b$ for any $0 \le k \le m - 1$. Since those digits do not contribute much to the quotient the observation is that they might as well be zero. However, if the digits ``might as well be zero'' they might as well not be there in the first place. Let $q_0 = \lfloor a/\beta^{m-1} \rfloor$ represent the input with the zeroes trimmed. Now the modular reduction is trimmed to the almost equivalent equation \begin{equation} c = a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor \end{equation} Notice how the original divisor $2^q$ has been replaced with $\beta^{m+1}$. Also note how the exponent on the divisor $m+1$ when added to the amount $q_0$ was shifted by ($m-1$) equals $2m$. If the optimization had not been performed the divisor would have the exponent $2m$ so in the end the exponents do ``add up''. By using whole digits the algorithm is much faster since shifting digits is typically slower than simply copying them. Using the above equation the quotient $\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ can be off from the true quotient by at most two implying that $0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$. By first subtracting $b$ times the quotient and then conditionally subtracting $b$ once or twice the residue is found. The quotient is now found using $(m + 1)(m) = m^2 + m$ single precision multiplications and the residue with an additional $m^2$ single precision multiplications. In total $2m^2 + m$ single precision multiplications are required which is considerably faster than the original attempt. For example, let $\beta = 10$ represent the radix of the digits. Let $b = 9999$ represent the modulus which implies $m = 4$. Let $a = 99929878$ represent the value of which the residue is desired. In this case $q = 10$ which means that $\mu = \lfloor \beta^{2m}/b \rfloor = 10001$. With this optimization the multiplicand for the quotient is $q_0 = \lfloor a / \beta^{m - 1} \rfloor = 99929$. The quotient is then $\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor = 9993$. Subtracting $9993b$ from $a$ and the correct residue $9871 \equiv a \mbox{ (mod }b\mbox{)}$ is found. \subsection{Trimming the Quotient} So far the reduction algorithm has been optimized from $3m^2$ single precision multiplications down to $2m^2 + m$ single precision multiplications. As it stands now the algorithm is already fairly fast compared to a full integer division algorithm. However, there is still room for optimization. After the first multiplication inside the quotient ($q_0 \cdot \mu$) the value is shifted right by $m + 1$ places effectively nullifying the lower half of the product. It would be nice to be able to remove those digits from the product to effectively cut down the number of multiplications. If the number of digits in the modulus $m$ is far less than $\beta$ a full product is not required. In fact the lower $m - 2$ digits will not affect the upper half of the product at all and do not need to be computed. The value of $\mu$ is a $m$-digit number and $q_0$ is a $m + 1$ digit number. Using a full multiplier $(m + 1)(m) = m^2 + m$ single precision multiplications would be required. Using a multiplier that will only produce digits at and above the $m - 1$'th digit reduces the number of single precision multiplications to ${m^2 + m} \over 2$ single precision multiplications. \subsection{Trimming the Residue} After the quotient has been calculated it is used to reduce the input. As previously noted the algorithm is not exact and it can be off by a small multiple of the modulus, that is $0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$. If $b$ is $m$ digits than the result of reduction equation is a value of at most $m + 1$ digits (\textit{provided $3 < \beta$}) implying that the upper $m - 1$ digits are implicitly zero. The next optimization arises from this very fact. Instead of computing $b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ using a full $O(m^2)$ multiplication algorithm only the lower $m+1$ digits of the product have to be computed. Similarly the value of $a$ can be reduced modulo $\beta^{m+1}$ before the multiple of $b$ is subtracted which simplifes the subtraction as well. A multiplication that produces only the lower $m+1$ digits requires ${m^2 + 3m - 2} \over 2$ single precision multiplications. With both optimizations in place the algorithm is the algorithm Barrett proposed. It requires $m^2 + 2m - 1$ single precision multiplications which is considerably faster than the straightforward $3m^2$ method. \subsection{The Barrett Algorithm} \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_reduce}. \\ \textbf{Input}. mp\_int $a$, mp\_int $b$ and $\mu = \lfloor \beta^{2m}/b \rfloor$ $(0 \le a < b^2, b > 1)$ \\ \textbf{Output}. $c \leftarrow a \mbox{ (mod }b\mbox{)}$ \\ \hline \\ Let $m$ represent the number of digits in $b$. \\ 1. Make a copy of $a$ and store it in $q$. (\textit{mp\_init\_copy}) \\ 2. $q \leftarrow \lfloor q / \beta^{m - 1} \rfloor$ (\textit{mp\_rshd}) \\ \\ Produce the quotient. \\ 3. $q \leftarrow q \cdot \mu$ (\textit{note: only produce digits at or above $m-1$}) \\ 4. $q \leftarrow \lfloor q / \beta^{m + 1} \rfloor$ \\ \\ Subtract the multiple of modulus from the input. \\ 5. $c \leftarrow a \mbox{ (mod }\beta^{m+1}\mbox{)}$ (\textit{mp\_mod\_2d}) \\ 6. $q \leftarrow q \cdot b \mbox{ (mod }\beta^{m+1}\mbox{)}$ (\textit{s\_mp\_mul\_digs}) \\ 7. $c \leftarrow c - q$ (\textit{mp\_sub}) \\ \\ Add $\beta^{m+1}$ if a carry occured. \\ 8. If $c < 0$ then (\textit{mp\_cmp\_d}) \\ \hspace{3mm}8.1 $q \leftarrow 1$ (\textit{mp\_set}) \\ \hspace{3mm}8.2 $q \leftarrow q \cdot \beta^{m+1}$ (\textit{mp\_lshd}) \\ \hspace{3mm}8.3 $c \leftarrow c + q$ \\ \\ Now subtract the modulus if the residue is too large (e.g. quotient too small). \\ 9. While $c \ge b$ do (\textit{mp\_cmp}) \\ \hspace{3mm}9.1 $c \leftarrow c - b$ \\ 10. Clear $q$. \\ 11. Return(\textit{MP\_OKAY}) \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_reduce} \end{figure} \textbf{Algorithm mp\_reduce.} This algorithm will reduce the input $a$ modulo $b$ in place using the Barrett algorithm. It is loosely based on algorithm 14.42 of \cite[pp. 602]{HAC} which is based on \cite{BARRETT}. The algorithm has several restrictions and assumptions which must be adhered to for the algorithm to work. First the modulus $b$ is assumed to be positive and greater than one. If the modulus were less than or equal to one than subtracting a multiple of it would either accomplish nothing or actually enlarge the input. The input $a$ must be in the range $0 \le a < b^2$ in order for the quotient to have enough precision. Technically the algorithm will still work if $a \ge b^2$ but it will take much longer to finish. The value of $\mu$ is passed as an argument to this algorithm and is assumed to be setup before the algorithm is used. Recall that the multiplication for the quotient on step 3 must only produce digits at or above the $m-1$'th position. An algorithm called $s\_mp\_mul\_high\_digs$ which has not been presented is used to accomplish this task. This optimal algorithm can only be used if the number of digits in $b$ is very much smaller than $\beta$. After the multiple of the modulus has been subtracted from $a$ the residue must be fixed up in case its negative. While it is known that $a \ge b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ only the lower $m+1$ digits are being used to compute the residue. In this case the invariant $\beta^{m+1}$ must be added to the residue to make it positive again. The while loop at step 9 will subtract $b$ until the residue is less than $b$. If the algorithm is performed correctly this step is only performed upto two times. However, if $a \ge b^2$ than it will iterate substantially more times than it should. EXAM,bn_mp_reduce.c The first multiplication that determines the quotient can be performed by only producing the digits from $m - 1$ and up. This essentially halves the number of single precision multiplications required. However, the optimization is only safe if $\beta$ is much larger than the number of digits in the modulus. In the source code this is evaluated on lines @36,if@ to @44,}@ where algorithm s\_mp\_mul\_high\_digs is used when it is safe to do so. \subsection{The Barrett Setup Algorithm} In order to use algorithm mp\_reduce the value of $\mu$ must be calculated in advance. Ideally this value should be computed once and stored for future use so that the Barrett algorithm can be used without delay. \begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_reduce\_setup}. \\ \textbf{Input}. mp\_int $a$ ($a > 1$) \\ \textbf{Output}. $\mu \leftarrow \lfloor \beta^{2m}/a \rfloor$ \\ \hline \\ 1. $\mu \leftarrow 2^{2 \cdot lg(\beta) \cdot m}$ (\textit{mp\_2expt}) \\ 2. $\mu \leftarrow \lfloor \mu / b \rfloor$ (\textit{mp\_div}) \\ 3. Return(\textit{MP\_OKAY}) \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_reduce\_setup} \end{figure} \textbf{Algorithm mp\_reduce\_setup.} This algorithm computes the reciprocal $\mu$ required for Barrett reduction. First $\beta^{2m}$ is calculated as $2^{2 \cdot lg(\beta) \cdot m}$ which is equivalent and much faster. The final value is computed by taking the integer quotient of $\lfloor \mu / b \rfloor$. EXAM,bn_mp_reduce_setup.c This simple routine calculates the reciprocal $\mu$ required by Barrett reduction. Note the extended usage of algorithm mp\_div where the variable which would received the remainder is passed as NULL. As will be discussed in ~DIVISION~ the division routine allows both the quotient and the remainder to be passed as NULL meaning to ignore the value. \section{The Montgomery Reduction} Montgomery reduction\footnote{Thanks to Niels Ferguson for his insightful explanation of the algorithm.} \cite{MONT} is by far the most interesting form of reduction in common use. It computes a modular residue which is not actually equal to the residue of the input yet instead equal to a residue times a constant. However, as perplexing as this may sound the algorithm is relatively simple and very efficient. Throughout this entire section the variable $n$ will represent the modulus used to form the residue. As will be discussed shortly the value of $n$ must be odd. The variable $x$ will represent the quantity of which the residue is sought. Similar to the Barrett algorithm the input is restricted to $0 \le x < n^2$. To begin the description some simple number theory facts must be established. \textbf{Fact 1.} Adding $n$ to $x$ does not change the residue since in effect it adds one to the quotient $\lfloor x / n \rfloor$. \textbf{Fact 2.} If $x$ is even then performing a division by two in $\Z$ is congruent to $x \cdot 2^{-1} \mbox{ (mod }n\mbox{)}$. For example, if $n = 7$ and $x = 6$ then $x/2 = 3$. Using the modular inverse of two the same result is found. That is, $2^{-1} \equiv (n+1)/2 \equiv 4$ and $4 \cdot 6 \equiv 3 \mbox{ (mod }n\mbox{)}$. From these two simple facts the following simple algorithm can be derived. \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{Montgomery Reduction}. \\ \textbf{Input}. Integer $x$, $n$ and $k$ \\ \textbf{Output}. $2^{-k}x \mbox{ (mod }n\mbox{)}$ \\ \hline \\ 1. for $t$ from $1$ to $k$ do \\ \hspace{3mm}1.1 If $x$ is odd then \\ \hspace{6mm}1.1.1 $x \leftarrow x + n$ \\ \hspace{3mm}1.2 $x \leftarrow x/2$ \\ 2. Return $x$. \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm Montgomery Reduction} \end{figure} The algorithm reduces the input one bit at a time using the two congruencies stated previously. Inside the loop $n$, which is odd, is added to $x$ if $x$ is odd. This forces $x$ to be even which allows the division by two in $\Z$ to be congruent to a modular division by two. Let $r$ represent the final result of the Montgomery algorithm. If $k > lg(n)$ and $0 \le x < n^2$ then the final result is limited to $0 \le r < \lfloor x/2^k \rfloor + n$. As a result at most a single subtraction is required to get the residue desired. Let $k = \lfloor lg(n) \rfloor + 1$ represent the number of bits in $n$. The current algorithm requires $2k^2$ single precision shifts and $k^2$ single precision additions. At this rate the algorithm is most certainly slower than Barrett reduction and not terribly useful. Fortunately there exists an alternative representation of the algorithm. \begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{Montgomery Reduction} (modified I). \\ \textbf{Input}. Integer $x$, $n$ and $k$ \\ \textbf{Output}. $2^{-k}x \mbox{ (mod }n\mbox{)}$ \\ \hline \\ 1. for $t$ from $0$ to $k - 1$ do \\ \hspace{3mm}1.1 If the $t$'th bit of $x$ is one then \\ \hspace{6mm}1.1.1 $x \leftarrow x + 2^tn$ \\ 2. Return $x/2^k$. \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm Montgomery Reduction (modified I)} \end{figure} This algorithm is equivalent since $2^tn$ is a multiple of $n$ and the lower $k$ bits of $x$ are zero by step 2. The number of single precision shifts has now been reduced from $2k^2$ to $k^2 + 1$ which is only a small improvement. \subsection{Digit Based Montgomery Reduction} Instead of computing the reduction on a bit-by-bit basis it is actually much faster to compute it on digit-by-digit basis. Consider the previous algorithm re-written to compute the Montgomery reduction in this new fashion. \begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{Montgomery Reduction} (modified II). \\ \textbf{Input}. Integer $x$, $n$ and $k$ \\ \textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\ \hline \\ 1. for $t$ from $0$ to $k - 1$ do \\ \hspace{3mm}1.1 $x \leftarrow x + \mu n \beta^t$ \\ 2. Return $x/\beta^k$. \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm Montgomery Reduction (modified II)} \end{figure} The value $\mu n \beta^t$ is a multiple of the modulus $n$ meaning that it will not change the residue. If the first digit of the value $\mu n \beta^t$ equals the negative (modulo $\beta$) of the $t$'th digit of $x$ then the addition will result in a zero digit. This problem breaks down to solving the following congruency. \begin{center} \begin{tabular}{rcl} $x_t + \mu n_0$ & $\equiv$ & $0 \mbox{ (mod }\beta\mbox{)}$ \\ $\mu n_0$ & $\equiv$ & $-x_t \mbox{ (mod }\beta\mbox{)}$ \\ $\mu$ & $\equiv$ & $-x_t/n_0 \mbox{ (mod }\beta\mbox{)}$ \\ \end{tabular} \end{center} In each iteration of the loop on step 1 a new value of $\mu$ must be calculated. The value of $-1/n_0 \mbox{ (mod }\beta\mbox{)}$ is used extensively in this algorithm and should be precomputed. Let $\rho$ represent the negative of the modular inverse of $n_0$ modulo $\beta$. For example, let $\beta = 10$ represent the radix. Let $n = 17$ represent the modulus which implies $k = 2$ and $\rho \equiv 7$. Let $x = 33$ represent the value to reduce. \newpage\begin{figure} \begin{center} \begin{tabular}{|c|c|c|} \hline \textbf{Step ($t$)} & \textbf{Value of $x$} & \textbf{Value of $\mu$} \\ \hline -- & $33$ & --\\ \hline $0$ & $33 + \mu n = 50$ & $1$ \\ \hline $1$ & $50 + \mu n \beta = 900$ & $5$ \\ \hline \end{tabular} \end{center} \caption{Example of Montgomery Reduction} \end{figure} The final result $900$ is then divided by $\beta^k$ to produce the final result $9$. The first observation is that $9 \nequiv x \mbox{ (mod }n\mbox{)}$ which implies the result is not the modular residue of $x$ modulo $n$. However, recall that the residue is actually multiplied by $\beta^{-k}$ in the algorithm. To get the true residue the value must be multiplied by $\beta^k$. In this case $\beta^k \equiv 15 \mbox{ (mod }n\mbox{)}$ and the correct residue is $9 \cdot 15 \equiv 16 \mbox{ (mod }n\mbox{)}$. \subsection{Baseline Montgomery Reduction} The baseline Montgomery reduction algorithm will produce the residue for any size input. It is designed to be a catch-all algororithm for Montgomery reductions. \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_montgomery\_reduce}. \\ \textbf{Input}. mp\_int $x$, mp\_int $n$ and a digit $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\ \hspace{11.5mm}($0 \le x < n^2, n > 1, (n, \beta) = 1, \beta^k > n$) \\ \textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\ \hline \\ 1. $digs \leftarrow 2n.used + 1$ \\ 2. If $digs < MP\_ARRAY$ and $m.used < \delta$ then \\ \hspace{3mm}2.1 Use algorithm fast\_mp\_montgomery\_reduce instead. \\ \\ Setup $x$ for the reduction. \\ 3. If $x.alloc < digs$ then grow $x$ to $digs$ digits. \\ 4. $x.used \leftarrow digs$ \\ \\ Eliminate the lower $k$ digits. \\ 5. For $ix$ from $0$ to $k - 1$ do \\ \hspace{3mm}5.1 $\mu \leftarrow x_{ix} \cdot \rho \mbox{ (mod }\beta\mbox{)}$ \\ \hspace{3mm}5.2 $u \leftarrow 0$ \\ \hspace{3mm}5.3 For $iy$ from $0$ to $k - 1$ do \\ \hspace{6mm}5.3.1 $\hat r \leftarrow \mu n_{iy} + x_{ix + iy} + u$ \\ \hspace{6mm}5.3.2 $x_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ \hspace{6mm}5.3.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ \hspace{3mm}5.4 While $u > 0$ do \\ \hspace{6mm}5.4.1 $iy \leftarrow iy + 1$ \\ \hspace{6mm}5.4.2 $x_{ix + iy} \leftarrow x_{ix + iy} + u$ \\ \hspace{6mm}5.4.3 $u \leftarrow \lfloor x_{ix+iy} / \beta \rfloor$ \\ \hspace{6mm}5.4.4 $x_{ix + iy} \leftarrow x_{ix+iy} \mbox{ (mod }\beta\mbox{)}$ \\ \\ Divide by $\beta^k$ and fix up as required. \\ 6. $x \leftarrow \lfloor x / \beta^k \rfloor$ \\ 7. If $x \ge n$ then \\ \hspace{3mm}7.1 $x \leftarrow x - n$ \\ 8. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_montgomery\_reduce} \end{figure} \textbf{Algorithm mp\_montgomery\_reduce.} This algorithm reduces the input $x$ modulo $n$ in place using the Montgomery reduction algorithm. The algorithm is loosely based on algorithm 14.32 of \cite[pp.601]{HAC} except it merges the multiplication of $\mu n \beta^t$ with the addition in the inner loop. The restrictions on this algorithm are fairly easy to adapt to. First $0 \le x < n^2$ bounds the input to numbers in the same range as for the Barrett algorithm. Additionally $n > 1$ will ensure a modular inverse $\rho$ exists. $\rho$ must be calculated in advance of this algorithm. Finally the variable $k$ is fixed and a pseudonym for $n.used$. Step 2 decides whether a faster Montgomery algorithm can be used. It is based on the Comba technique meaning that there are limits on the size of the input. This algorithm is discussed in ~COMBARED~. Step 5 is the main reduction loop of the algorithm. The value of $\mu$ is calculated once per iteration in the outer loop. The inner loop calculates $x + \mu n \beta^{ix}$ by multiplying $\mu n$ and adding the result to $x$ shifted by $ix$ digits. Both the addition and multiplication are performed in the same loop to save time and memory. Step 5.4 will handle any additional carries that escape the inner loop. Using a quick inspection this algorithm requires $n$ single precision multiplications for the outer loop and $n^2$ single precision multiplications in the inner loop. In total $n^2 + n$ single precision multiplications which compares favourably to Barrett at $n^2 + 2n - 1$ single precision multiplications. EXAM,bn_mp_montgomery_reduce.c This is the baseline implementation of the Montgomery reduction algorithm. Lines @30,digs@ to @35,}@ determine if the Comba based routine can be used instead. Line @47,mu@ computes the value of $\mu$ for that particular iteration of the outer loop. The multiplication $\mu n \beta^{ix}$ is performed in one step in the inner loop. The alias $tmpx$ refers to the $ix$'th digit of $x$ and the alias $tmpn$ refers to the modulus $n$. \subsection{Faster ``Comba'' Montgomery Reduction} MARK,COMBARED The Montgomery reduction requires fewer single precision multiplications than a Barrett reduction, however it is much slower due to the serial nature of the inner loop. The Barrett reduction algorithm requires two slightly modified multipliers which can be implemented with the Comba technique. The Montgomery reduction algorithm cannot directly use the Comba technique to any significant advantage since the inner loop calculates a $k \times 1$ product $k$ times. The biggest obstacle is that at the $ix$'th iteration of the outer loop the value of $x_{ix}$ is required to calculate $\mu$. This means the carries from $0$ to $ix - 1$ must have been propagated upwards to form a valid $ix$'th digit. The solution as it turns out is very simple. Perform a Comba like multiplier and inside the outer loop just after the inner loop fix up the $ix + 1$'th digit by forwarding the carry. With this change in place the Montgomery reduction algorithm can be performed with a Comba style multiplication loop which substantially increases the speed of the algorithm. \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{fast\_mp\_montgomery\_reduce}. \\ \textbf{Input}. mp\_int $x$, mp\_int $n$ and a digit $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\ \hspace{11.5mm}($0 \le x < n^2, n > 1, (n, \beta) = 1, \beta^k > n$) \\ \textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\ \hline \\ Place an array of \textbf{MP\_WARRAY} mp\_word variables called $\hat W$ on the stack. \\ 1. if $x.alloc < n.used + 1$ then grow $x$ to $n.used + 1$ digits. \\ Copy the digits of $x$ into the array $\hat W$ \\ 2. For $ix$ from $0$ to $x.used - 1$ do \\ \hspace{3mm}2.1 $\hat W_{ix} \leftarrow x_{ix}$ \\ 3. For $ix$ from $x.used$ to $2n.used - 1$ do \\ \hspace{3mm}3.1 $\hat W_{ix} \leftarrow 0$ \\ Elimiate the lower $k$ digits. \\ 4. for $ix$ from $0$ to $n.used - 1$ do \\ \hspace{3mm}4.1 $\mu \leftarrow \hat W_{ix} \cdot \rho \mbox{ (mod }\beta\mbox{)}$ \\ \hspace{3mm}4.2 For $iy$ from $0$ to $n.used - 1$ do \\ \hspace{6mm}4.2.1 $\hat W_{iy + ix} \leftarrow \hat W_{iy + ix} + \mu \cdot n_{iy}$ \\ \hspace{3mm}4.3 $\hat W_{ix + 1} \leftarrow \hat W_{ix + 1} + \lfloor \hat W_{ix} / \beta \rfloor$ \\ Propagate carries upwards. \\ 5. for $ix$ from $n.used$ to $2n.used + 1$ do \\ \hspace{3mm}5.1 $\hat W_{ix + 1} \leftarrow \hat W_{ix + 1} + \lfloor \hat W_{ix} / \beta \rfloor$ \\ Shift right and reduce modulo $\beta$ simultaneously. \\ 6. for $ix$ from $0$ to $n.used + 1$ do \\ \hspace{3mm}6.1 $x_{ix} \leftarrow \hat W_{ix + n.used} \mbox{ (mod }\beta\mbox{)}$ \\ Zero excess digits and fixup $x$. \\ 7. if $x.used > n.used + 1$ then do \\ \hspace{3mm}7.1 for $ix$ from $n.used + 1$ to $x.used - 1$ do \\ \hspace{6mm}7.1.1 $x_{ix} \leftarrow 0$ \\ 8. $x.used \leftarrow n.used + 1$ \\ 9. Clamp excessive digits of $x$. \\ 10. If $x \ge n$ then \\ \hspace{3mm}10.1 $x \leftarrow x - n$ \\ 11. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm fast\_mp\_montgomery\_reduce} \end{figure} \textbf{Algorithm fast\_mp\_montgomery\_reduce.} This algorithm will compute the Montgomery reduction of $x$ modulo $n$ using the Comba technique. It is on most computer platforms significantly faster than algorithm mp\_montgomery\_reduce and algorithm mp\_reduce (\textit{Barrett reduction}). The algorithm has the same restrictions on the input as the baseline reduction algorithm. An additional two restrictions are imposed on this algorithm. The number of digits $k$ in the the modulus $n$ must not violate $MP\_WARRAY > 2k +1$ and $n < \delta$. When $\beta = 2^{28}$ this algorithm can be used to reduce modulo a modulus of at most $3,556$ bits in length. As in the other Comba reduction algorithms there is a $\hat W$ array which stores the columns of the product. It is initially filled with the contents of $x$ with the excess digits zeroed. The reduction loop is very similar the to the baseline loop at heart. The multiplication on step 4.1 can be single precision only since $ab \mbox{ (mod }\beta\mbox{)} \equiv (a \mbox{ mod }\beta)(b \mbox{ mod }\beta)$. Some multipliers such as those on the ARM processors take a variable length time to complete depending on the number of bytes of result it must produce. By performing a single precision multiplication instead half the amount of time is spent. Also note that digit $\hat W_{ix}$ must have the carry from the $ix - 1$'th digit propagated upwards in order for this to work. That is what step 4.3 will do. In effect over the $n.used$ iterations of the outer loop the $n.used$'th lower columns all have the their carries propagated forwards. Note how the upper bits of those same words are not reduced modulo $\beta$. This is because those values will be discarded shortly and there is no point. Step 5 will propgate the remainder of the carries upwards. On step 6 the columns are reduced modulo $\beta$ and shifted simultaneously as they are stored in the destination $x$. EXAM,bn_fast_mp_montgomery_reduce.c The $\hat W$ array is first filled with digits of $x$ on line @49,for@ then the rest of the digits are zeroed on line @54,for@. Both loops share the same alias variables to make the code easier to read. The value of $\mu$ is calculated in an interesting fashion. First the value $\hat W_{ix}$ is reduced modulo $\beta$ and cast to a mp\_digit. This forces the compiler to use a single precision multiplication and prevents any concerns about loss of precision. Line @101,>>@ fixes the carry for the next iteration of the loop by propagating the carry from $\hat W_{ix}$ to $\hat W_{ix+1}$. The for loop on line @113,for@ propagates the rest of the carries upwards through the columns. The for loop on line @126,for@ reduces the columns modulo $\beta$ and shifts them $k$ places at the same time. The alias $\_ \hat W$ actually refers to the array $\hat W$ starting at the $n.used$'th digit, that is $\_ \hat W_{t} = \hat W_{n.used + t}$. \subsection{Montgomery Setup} To calculate the variable $\rho$ a relatively simple algorithm will be required. \begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_montgomery\_setup}. \\ \textbf{Input}. mp\_int $n$ ($n > 1$ and $(n, 2) = 1$) \\ \textbf{Output}. $\rho \equiv -1/n_0 \mbox{ (mod }\beta\mbox{)}$ \\ \hline \\ 1. $b \leftarrow n_0$ \\ 2. If $b$ is even return(\textit{MP\_VAL}) \\ 3. $x \leftarrow ((b + 2) \mbox{ AND } 4) << 1) + b$ \\ 4. for $k$ from 0 to $3$ do \\ \hspace{3mm}4.1 $x \leftarrow x \cdot (2 - bx)$ \\ 5. $\rho \leftarrow \beta - x \mbox{ (mod }\beta\mbox{)}$ \\ 6. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_montgomery\_setup} \end{figure} \textbf{Algorithm mp\_montgomery\_setup.} This algorithm will calculate the value of $\rho$ required within the Montgomery reduction algorithms. It uses a very interesting trick to calculate $1/n_0$ when $\beta$ is a power of two. EXAM,bn_mp_montgomery_setup.c This source code computes the value of $\rho$ required to perform Montgomery reduction. It has been modified to avoid performing excess multiplications when $\beta$ is not the default 28-bits. \section{The Diminished Radix Algorithm} The diminished radix method of modular reduction \cite{DRMET} is a fairly clever technique which is more efficient than either the Barrett or Montgomery methods. The technique is based on a simple congruence. \begin{equation} (x \mbox{ mod } n) + k \lfloor x / n \rfloor \equiv x \mbox{ (mod }(n - k)\mbox{)} \end{equation} This observation was used in the MMB \cite{MMB} block cipher to create a diffusion primitive. It used the fact that if $n = 2^{31}$ and $k=1$ that then a x86 multiplier could produce the 62-bit product and use the ``shrd'' instruction to perform a double-precision right shift. The proof of the above equation is very simple. First write $x$ in the product form. \begin{equation} x = qn + r \end{equation} Now reduce both sides modulo $(n - k)$. \begin{equation} x \equiv qk + r \mbox{ (mod }(n-k)\mbox{)} \end{equation} The variable $n$ reduces as $n \mbox{ mod } (n - k)$ to $k$. By putting $q = \lfloor x/n \rfloor$ and $r = x \mbox{ mod } n$ into the equation the original congruence is reproduced. The following algorithm is based on these observations. \begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{Diminished Radix Reduction}. \\ \textbf{Input}. Integer $x$, $n$, $k$ \\ \textbf{Output}. $x \mbox{ mod } (n - k)$ \\ \hline \\ 1. $q \leftarrow \lfloor x / n \rfloor$ \\ 2. $q \leftarrow k \cdot q$ \\ 3. $x \leftarrow x \mbox{ (mod }n\mbox{)}$ \\ 4. $x \leftarrow x + q$ \\ 5. If $x \ge (n - k)$ then \\ \hspace{3mm}5.1 $x \leftarrow x - (n - k)$ \\ \hspace{3mm}5.2 Goto step 1. \\ 6. Return $x$ \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm Diminished Radix Reduction} \label{fig:DR} \end{figure} This algorithm will reduce $x$ modulo $n - k$ and return the residue. If $0 \le x < (n - k)^2$ then the algorithm will loop almost always once or twice and occasionally three times. For simplicity sake the value of $x$ is bounded by the following simple polynomial. \begin{equation} 0 \le x < n^2 + k^2 - 2nk \end{equation} The true bound is $0 \le x < (n - k - 1)^2$ but this has quite a few more terms. The value of $q$ after step 1 is bounded by the following. \begin{equation} q < n - 2k - k^2/n \end{equation} Since $k^2$ is going to be considerably smaller than $n$ that term will always be zero. The value of $x$ after step 3 is bounded trivially as $0 \le x < n$. By step four the sum $x + q$ is bounded by \begin{equation} 0 \le q + x < (k + 1)n - 2k^2 - 1 \end{equation} As a result at most $k$ subtractions of $n$ are required to produce the residue. With a second pass $q$ will be loosely bounded by $0 \le q < k^2$ after step 2 while $x$ will still be loosely bounded by $0 \le x < n$ after step 3. After the second pass it is highly unlike that the sum in step 4 will exceed $n - k$. In practice fewer than three passes of the algorithm are required to reduce virtually every input in the range $0 \le x < (n - k - 1)^2$. \subsection{Choice of Moduli} On the surface this algorithm looks like a very expensive algorithm. It requires a couple of subtractions followed by multiplication and other modular reductions. The usefulness of this algorithm becomes exceedingly clear when an appropriate moduli is chosen. Division in general is a very expensive operation to perform. The one exception is when the division is by a power of the radix of representation used. Division by ten for example is simple for humans since it amounts to shifting the decimal place. Similarly division by two (\textit{or powers of two}) is very simple for computers to perform. It would therefore seem logical to choose $n$ of the form $2^p$ which would imply that $\lfloor x / n \rfloor$ is a simple shift of $x$ right $p$ bits. However, there is one operation related to division of power of twos that is even faster than this. If $n = \beta^p$ then the division may be performed by moving whole digits to the right $p$ places. In practice division by $\beta^p$ is much faster than division by $2^p$ for any $p$. Also with the choice of $n = \beta^p$ reducing $x$ modulo $n$ requires zeroing the digits above the $p-1$'th digit of $x$. Throughout the next section the term ``restricted modulus'' will refer to a modulus of the form $\beta^p - k$ where as the term ``unrestricted modulus'' will refer to a modulus of the form $2^p - k$. The word ``restricted'' in this case refers to the fact that it is based on the $2^p$ logic except $p$ must be a multiple of $lg(\beta)$. \subsection{Choice of $k$} Now that division and reduction (\textit{step 1 and 3 of figure~\ref{fig:DR}}) have been optimized to simple digit operations the multiplication by $k$ in step 2 is the most expensive operation. Fortunately the choice of $k$ is not terribly limited. For all intents and purposes it might as well be a single digit. \subsection{Restricted Diminished Radix Reduction} The restricted Diminished Radix algorithm can quickly reduce numbers modulo numbers of the form $n = \beta^p - k$. This algorithm can reduce an input $x$ within the range $0 \le x < n^2$ using a couple passes of the algorithm demonstrated in figure~\ref{fig:DR}. The implementation of this algorithm has been optimized to avoid additional overhead associated with a division by $\beta^p$, the multiplication by $k$ or the addition of $x$ and $q$. The resulting algorithm is very efficient and can lead to substantial improvements when modular exponentiations are performed compared to Montgomery based reduction algorithms. \begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_dr\_reduce}. \\ \textbf{Input}. mp\_int $x$, $n$ and a mp\_digit $k = \beta - n_0$ \\ \hspace{11.5mm}($0 \le x < n^2$, $n > 1$, $0 < k \le \beta$) \\ \textbf{Output}. $x \mbox{ mod } n$ \\ \hline \\ 1. $m \leftarrow n.used$ \\ 2. If $x.alloc < 2m$ then grow $x$ to $2m$ digits. \\ 3. $\mu \leftarrow 0$ \\ 4. for $i$ from $0$ to $m - 1$ do \\ \hspace{3mm}4.1 $\hat r \leftarrow k \cdot x_{m+i} + x_{i} + \mu$ \\ \hspace{3mm}4.2 $x_{i} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ \hspace{3mm}4.3 $\mu \leftarrow \lfloor \hat r / \beta \rfloor$ \\ 5. $x_{m} \leftarrow \mu$ \\ 6. for $i$ from $m + 1$ to $x.used - 1$ do \\ \hspace{3mm}6.1 $x_{i} \leftarrow 0$ \\ 7. Clamp excess digits of $x$. \\ 8. If $x \ge n$ then \\ \hspace{3mm}8.1 $x \leftarrow x - n$ \\ \hspace{3mm}8.2 Goto step 3. \\ 9. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_dr\_reduce} \end{figure} \textbf{Algorithm mp\_dr\_reduce.} This algorithm will perform the dimished radix reduction of $x$ modulo $n$. It has similar restrictions to that of the Barrett reduction with the addition that $n$ must be of the form $n = \beta^m - k$ where $0 < k \le \beta$. This algorithm essentially implements the pseudo-code in figure 7.10 except with a slight optimization. The division by $\beta^m$, multiplication by $k$ and addition of $x \mbox{ mod }\beta^m$ are all performed as one step inside the loop on step 4. The division by $\beta^m$ is emulated by accessing the term at the $m+i$'th position which is subsequently multiplied by $k$ and added to the term at the $i$'th position. After the loop the $m$'th digit is set to the carry and the upper digits are zeroed. Step 5 and 6 emulate the reduction modulo $\beta^m$ that should have happend to $x$ before the addition of the multiple of the upper half. At step 8 if $x$ is still larger than $n$ another pass of the algorithm is required. First $n$ is subtracted from $x$ and then the algorithm resumes at step 3. EXAM,bn_mp_dr_reduce.c The first step is to grow $x$ as required to $2m$ digits since the reduction is performed in place on $x$. The label on line @49,top:@ is where the algorithm will resume if further reduction passes are required. In theory it could be placed at the top of the function however, the size of the modulus and question of whether $x$ is large enough are invariant after the first pass meaning that it would be a waste of time. The aliases $tmpx1$ and $tmpx2$ refer to the digits of $x$ where the latter is offset by $m$ digits. By reading digits from $x$ offset by $m$ digits a division by $\beta^m$ can be simulated virtually for free. The loop on line @61,for@ performs the bulk of the work (\textit{corresponds to step 4 of algorithm 7.11}) in this algorithm. By line @68,mu@ the pointer $tmpx1$ points to the $m$'th digit of $x$ which is where the final carry will be placed. Similarly by line @71,for@ the same pointer will point to the $m+1$'th digit where the zeroes will be placed. Since the algorithm is only valid if both $x$ and $n$ are greater than zero an unsigned comparison suffices to determine if another pass is required. With the same logic at line @82,sub@ the value of $x$ is known to be greater than or equal to $n$ meaning that an unsigned subtraction can be used as well. Since the destination of the subtraction is the larger of the inputs the call to algorithm s\_mp\_sub cannot fail and the return code does not need to be checked. \subsubsection{Setup} To setup the restricted Diminished Radix algorithm the value $k = \beta - n_0$ is required. This algorithm is not really complicated but provided for completeness. \begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_dr\_setup}. \\ \textbf{Input}. mp\_int $n$ \\ \textbf{Output}. $k = \beta - n_0$ \\ \hline \\ 1. $k \leftarrow \beta - n_0$ \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_dr\_setup} \end{figure} EXAM,bn_mp_dr_setup.c \subsubsection{Modulus Detection} Another algorithm which will be useful is the ability to detect a restricted Diminished Radix modulus. An integer is said to be of restricted Diminished Radix form if all of the digits are equal to $\beta - 1$ except the trailing digit which may be any value. \begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_dr\_is\_modulus}. \\ \textbf{Input}. mp\_int $n$ \\ \textbf{Output}. $1$ if $n$ is in D.R form, $0$ otherwise \\ \hline 1. If $n.used < 2$ then return($0$). \\ 2. for $ix$ from $1$ to $n.used - 1$ do \\ \hspace{3mm}2.1 If $n_{ix} \ne \beta - 1$ return($0$). \\ 3. Return($1$). \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_dr\_is\_modulus} \end{figure} \textbf{Algorithm mp\_dr\_is\_modulus.} This algorithm determines if a value is in Diminished Radix form. Step 1 rejects obvious cases where fewer than two digits are in the mp\_int. Step 2 tests all but the first digit to see if they are equal to $\beta - 1$. If the algorithm manages to get to step 3 then $n$ must of Diminished Radix form. EXAM,bn_mp_dr_is_modulus.c \subsection{Unrestricted Diminished Radix Reduction} The unrestricted Diminished Radix algorithm allows modular reductions to be performed when the modulus is of the form $2^p - k$. This algorithm is a straightforward adaptation of algorithm~\ref{fig:DR}. In general the restricted Diminished Radix reduction algorithm is much faster since it has considerably lower overhead. However, this new algorithm is much faster than either Montgomery or Barrett reduction when the moduli are of the appropriate form. \begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_reduce\_2k}. \\ \textbf{Input}. mp\_int $a$ and $n$. mp\_digit $k$ \\ \hspace{11.5mm}($a \ge 0$, $n > 1$, $0 < k < \beta$, $n + k$ is a power of two) \\ \textbf{Output}. $a \mbox{ (mod }n\mbox{)}$ \\ \hline 1. $p \leftarrow \lfloor lg(n) \rfloor + 1$ (\textit{mp\_count\_bits}) \\ 2. While $a \ge n$ do \\ \hspace{3mm}2.1 $q \leftarrow \lfloor a / 2^p \rfloor$ (\textit{mp\_div\_2d}) \\ \hspace{3mm}2.2 $a \leftarrow a \mbox{ (mod }2^p\mbox{)}$ (\textit{mp\_mod\_2d}) \\ \hspace{3mm}2.3 $q \leftarrow q \cdot k$ (\textit{mp\_mul\_d}) \\ \hspace{3mm}2.4 $a \leftarrow a - q$ (\textit{s\_mp\_sub}) \\ \hspace{3mm}2.5 If $a \ge n$ then do \\ \hspace{6mm}2.5.1 $a \leftarrow a - n$ \\ 3. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_reduce\_2k} \end{figure} \textbf{Algorithm mp\_reduce\_2k.} This algorithm quickly reduces an input $a$ modulo an unrestricted Diminished Radix modulus $n$. EXAM,bn_mp_reduce_2k.c \subsubsection{Unrestricted Setup} To setup this reduction algorithm the value of $k = 2^p - n$ is required. \begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_reduce\_2k\_setup}. \\ \textbf{Input}. mp\_int $n$ \\ \textbf{Output}. $k = 2^p - n$ \\ \hline 1. $p \leftarrow \lfloor lg(n) \rfloor + 1$ (\textit{mp\_count\_bits}) \\ 2. $x \leftarrow 2^p$ (\textit{mp\_2expt}) \\ 3. $x \leftarrow x - n$ (\textit{mp\_sub}) \\ 4. $k \leftarrow x_0$ \\ 5. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_reduce\_2k\_setup} \end{figure} \textbf{Algorithm mp\_reduce\_2k\_setup.} EXAM,bn_mp_reduce_2k_setup.c \subsubsection{Unrestricted Detection} An integer $n$ is a valid unrestricted Diminished Radix modulus if either of the following are true. \begin{enumerate} \item The number has only one digit. \item The number has more than one digit and every bit from the $\beta$'th to the most significant is one. \end{enumerate} If either condition is true than there is a power of two namely $2^p$ such that $0 < 2^p - n < \beta$. -- Finish this section later, Tom. \section{Algorithm Comparison} So far three very different algorithms for modular reduction have been discussed. Each of the algorithms have their own strengths and weaknesses that makes having such a selection very useful. The following table sumarizes the three algorithms along with comparisons of work factors. Since all three algorithms have the restriction that $0 \le x < n^2$ and $n > 1$ those limitations are not included in the table. \begin{center} \begin{small} \begin{tabular}{|c|c|c|c|c|c|} \hline \textbf{Method} & \textbf{Work Required} & \textbf{Limitations} & \textbf{$m = 8$} & \textbf{$m = 32$} & \textbf{$m = 64$} \\ \hline Barrett & $m^2 + 2m - 1$ & None & $79$ & $1087$ & $4223$ \\ \hline Montgomery & $m^2 + m$ & $n$ must be odd & $72$ & $1056$ & $4160$ \\ \hline D.R. & $2m$ & $n = \beta^m - k$ & $16$ & $64$ & $128$ \\ \hline \end{tabular} \end{small} \end{center} In theory Montgomery and Barrett reductions would require roughly the same amount of time to complete. However, in practice since Montgomery reduction can be written as a single function with the Comba technique it is much faster. Barrett reduction suffers from the overhead of calling the half precision multipliers, addition and division by $\beta$ algorithms. For almost every cryptographic algorithm Montgomery reduction is the algorithm of choice. The one set of algorithms where Diminished Radix reduction truly shines are based on the discrete logarithm problem such as Diffie-Hellman \cite{DH} and ElGamal \cite{ELGAMAL}. In these algorithms primes of the form $\beta^m - k$ can be found and shared amongst users. These primes will allow the Diminished Radix algorithm to be used in modular exponentiation to greatly speed up the operation. \section*{Exercises} \begin{tabular}{cl} $\left [ 3 \right ]$ & Prove that the ``trick'' in algorithm mp\_montgomery\_setup actually \\ & calculates the correct value of $\rho$. \\ & \\ $\left [ 2 \right ]$ & Devise an algorithm to reduce modulo $n + k$ for small $k$ quickly. \\ & \\ $\left [ 4 \right ]$ & Prove that the pseudo-code algorithm ``Diminished Radix Reduction'' \\ & (\textit{figure 7.10}) terminates. Also prove the probability that it will \\ & terminate within $1 \le k \le 10$ iterations. \\ & \\ \end{tabular} \chapter{Exponentiation} Exponentiation is the operation of raising one variable to the power of another, for example, $a^b$. A variant of exponentiation, computed in a finite field or ring, is called modular exponentiation. This latter style of operation is typically used in public key cryptosystems such as RSA and Diffie-Hellman. The ability to quickly compute modular exponentiations is of great benefit to any such cryptosystem and many methods have been sought to speed it up. \section{Exponentiation Basics} A trivial algorithm would simply multiply $a$ against itself $b - 1$ times to compute the exponentiation desired. However, as $b$ grows in size the number of multiplications becomes prohibitive. Imagine what would happen if $b$ $\approx$ $2^{1024}$ as is the case when computing an RSA signature with a $1024$-bit key. Such a calculation could never be completed as it would take simply far too long. Fortunately there is a very simple algorithm based on the laws of exponents. Recall that $lg_a(a^b) = b$ and that $lg_a(a^ba^c) = b + c$ which are two trivial relationships between the base and the exponent. Let $b_i$ represent the $i$'th bit of $b$ starting from the least significant bit. If $b$ is a $k$-bit integer than the following equation is true. \begin{equation} a^b = \prod_{i=0}^{k-1} a^{2^i \cdot b_i} \end{equation} By taking the base $a$ logarithm of both sides of the equation the following equation is the result. \begin{equation} b = \sum_{i=0}^{k-1}2^i \cdot b_i \end{equation} This is indeed true. The term $a^{2^i}$ can be found from the $i - 1$'th term by squaring the term since $\left ( a^{2^i} \right )^2$ is equal to $a^{2^{i+1}}$. This trivial algorithm forms the basis of essentially all fast exponentiation algorithms. It requires $k$ squarings and on average $k \over 2$ multiplications to compute the result. This is indeed quite an improvement over simply multiplying by $a$ a total of $b-1$ times. While this current method is a considerable speed up there are further improvements to be made. For example, the $a^{2^i}$ term does not need to be an auxilary variable. Consider the following algorithm. \begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{Left to Right Exponentiation}. \\ \textbf{Input}. Integer $a$, $b$ and $k$ \\ \textbf{Output}. $c = a^b$ \\ \hline \\ 1. $c \leftarrow 1$ \\ 2. for $i$ from $k - 1$ to $0$ do \\ \hspace{3mm}2.1 $c \leftarrow c^2$ \\ \hspace{3mm}2.2 $c \leftarrow c \cdot a^{b_i}$ \\ 3. Return $c$. \\ \hline \end{tabular} \end{center} \end{small} \caption{Left to Right Exponentiation} \end{figure} This algorithm starts from the most significant bit and works towards the least significant bit. When the $i$'th bit of $b$ is set $a$ is multiplied against the current product. In each iteration the product is squared which doubles the exponent of the individual terms of the product. For example, let $b = 101100_2 \equiv 44_{10}$. The following chart demonstrates the actions of the algorithm. \newpage\begin{figure} \begin{center} \begin{tabular}{|c|c|} \hline \textbf{Value of $i$} & \textbf{Value of $c$} \\ \hline - & $1$ \\ \hline $5$ & $a$ \\ \hline $4$ & $a^2$ \\ \hline $3$ & $a^4 \cdot a$ \\ \hline $2$ & $a^8 \cdot a^2 \cdot a$ \\ \hline $1$ & $a^{16} \cdot a^4 \cdot a^2$ \\ \hline $0$ & $a^{32} \cdot a^8 \cdot a^4$ \\ \hline \end{tabular} \end{center} \caption{Example of Left to Right Exponentiation} \end{figure} When the product $a^{32} \cdot a^8 \cdot a^4$ is simplified it is equal $a^{44}$ which is the desired exponentiation. This particular algorithm is called ``Left to Right'' because it reads the exponent in that order. All of the exponentiation algorithms that will be presented are of this nature. \subsection{Single Digit Exponentiation} The first algorithm in the series of exponentiation algorithms will be an unbounded algorithm where the exponent is a single digit. It is intended to be used when a small power of an input is required (\textit{e.g. $a^5$}). It is faster than simply multiplying $b - 1$ times for all values of $b$ that are greater than three. \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_expt\_d}. \\ \textbf{Input}. mp\_int $a$ and mp\_digit $b$ \\ \textbf{Output}. $c = a^b$ \\ \hline \\ 1. $g \leftarrow a$ (\textit{mp\_init\_copy}) \\ 2. $c \leftarrow 1$ (\textit{mp\_set}) \\ 3. for $x$ from 0 to $lg(\beta) - 1$ do \\ \hspace{3mm}3.1 $c \leftarrow c^2$ (\textit{mp\_sqr}) \\ \hspace{3mm}3.2 If $b$ AND $2^{lg(\beta) - 1} \ne 0$ then \\ \hspace{6mm}3.2.1 $c \leftarrow c \cdot g$ (\textit{mp\_mul}) \\ \hspace{3mm}3.3 $b \leftarrow b << 1$ \\ 4. Clear $g$. \\ 5. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_expt\_d} \end{figure} \textbf{Algorithm mp\_expt\_d.} This algorithm computes the value of $a$ raised to the power of a single digit $b$. It uses the left to right exponentiation algorithm to quickly compute the exponentiation. It is loosely based on algorithm 14.79 of HAC \cite[pp. 615]{HAC} with the difference that the exponent is a fixed width. A copy of $a$ is made on the first step to allow destination variable $c$ be the same as the source variable $a$. The result is set to the initial value of $1$ in the subsequent step. Inside the loop the exponent is read from the most significant bit first downto the least significant bit. First $c$ is invariably squared on step 3.1. In the following step if the most significant bit of $b$ is one the copy of $a$ is multiplied against the result. The value of $b$ is shifted left one bit to make the next bit down from the most signficant bit become the new most significant bit. In effect each iteration of the loop moves the bits of the exponent $b$ upwards to the most significant location. EXAM,bn_mp_expt_d.c -- Some note later. \subsection{$k$-ary Exponentiation} When calculating an exponentiation the most time consuming bottleneck is the multiplications which are in general a small factor slower than squaring. Recall from the previous algorithm that $b_{i}$ refers to the $i$'th bit of the exponent $b$. Suppose it referred to the $i$'th $k$-bit digit of the exponent of $b$. For $k = 1$ the definitions are synonymous and for $k > 1$ the resulting algorithm computes the same exponentiation. A group of $k$ bits from the exponent is called a \textit{window}. That is it is a window on a small portion of the exponent. Consider the following modification to the basic left to right exponentiation algorithm. \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{$k$-ary Exponentiation}. \\ \textbf{Input}. Integer $a$, $b$, $k$ and $t$ \\ \textbf{Output}. $c = a^b$ \\ \hline \\ 1. $c \leftarrow 1$ \\ 2. for $i$ from $t - 1$ to $0$ do \\ \hspace{3mm}2.1 $c \leftarrow c^{2^k} $ \\ \hspace{3mm}2.2 Extract the $i$'th $k$-bit word from $b$ and store it in $g$. \\ \hspace{3mm}2.3 $c \leftarrow c \cdot a^g$ \\ 3. Return $c$. \\ \hline \end{tabular} \end{center} \end{small} \caption{$k$-ary Exponentiation} \end{figure} The squaring on step 2.1 can be calculated by squaring the value $c$ successively $k$ times. If the values of $a^g$ for $0 < g < 2^k$ have been precomputed this algorithm requires only $t$ multiplications and $tk$ squarings. The table can be generated with $2^{k - 1} - 1$ squarings and $2^{k - 1} + 1$ multiplications. This algorithm assumes that the number of bits in the exponent is evenly divisible by $k$. However, when it is not the remaining $0 < x \le k - 1$ bits can be handled with the original left to right style algorithm. Suppose $k = 4$ and $t = 100$. This modified algorithm will require $109$ multiplications and $408$ squarings to compute the exponentiation. The original algorithm would on average have required $200$ multiplications and $400$ squrings to compute the same value. The total number of squarings has increased slightly but the number of multiplications has nearly halved. \subsection{Sliding-Window Exponentiation} A simple modification to the previous algorithm is only generate the upper half of the table in the range $2^{k-1} \le g < 2^k$. Essentially this is a table for all values of $g$ where the most significant bit of $g$ is a one. However, in order for this to be allowed in the algorithm values of $g$ in the range $0 \le g < 2^{k-1}$ must be avoided. \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{Sliding Window $k$-ary Exponentiation}. \\ \textbf{Input}. Integer $a$, $b$, $k$ and $t$ \\ \textbf{Output}. $c = a^b$ \\ \hline \\ 1. $c \leftarrow 1$ \\ 2. for $i$ from $t - 1$ to $0$ do \\ \hspace{3mm}2.1 If the $i$'th bit of $b$ is a zero then \\ \hspace{6mm}2.1.1 $c \leftarrow c^2$ \\ \hspace{3mm}2.2 else do \\ \hspace{6mm}2.2.1 $c \leftarrow c^{2^k}$ \\ \hspace{6mm}2.2.2 Extract the $k$ bits from $(b_{i}b_{i-1}\ldots b_{i-(k-1)})$ and store it in $g$. \\ \hspace{6mm}2.2.3 $c \leftarrow c \cdot a^g$ \\ \hspace{6mm}2.2.4 $i \leftarrow i - k$ \\ 3. Return $c$. \\ \hline \end{tabular} \end{center} \end{small} \caption{Sliding Window $k$-ary Exponentiation} \end{figure} Similar to the previous algorithm this algorithm must have a special handler when fewer than $k$ bits are left in the exponent. While this algorithm requires the same number of squarings it can potentially have fewer multiplications. The pre-computed table $a^g$ is also half the size as the previous table. Consider the exponent $b = 111101011001000_2 \equiv 31432_{10}$ with $k = 3$ using both algorithms. The first algorithm will divide the exponent up as the following five $3$-bit words $b \equiv \left ( 111, 101, 011, 001, 000 \right )_{2}$. The second algorithm will break the exponent as $b \equiv \left ( 111, 101, 0, 110, 0, 100, 0 \right )_{2}$. The single digit $0$ in the second representation are where a single squaring took place instead of a squaring and multiplication. In total the first method requires $10$ multiplications and $18$ squarings. The second method requires $8$ multiplications and $18$ squarings. In general the sliding window method is never slower than the generic $k$-ary method and often it is slightly faster. \section{Modular Exponentiation} Modular exponentiation is essentially computing the power of a base within a finite field or ring. For example, computing $d \equiv a^b \mbox{ (mod }c\mbox{)}$ is a modular exponentiation. Instead of first computing $a^b$ and then reducing it modulo $c$ the intermediate result is reduced modulo $c$ after every squaring or multiplication operation. This guarantees that any intermediate result is bounded by $0 \le d \le c^2 - 2c + 1$ and can be reduced modulo $c$ quickly using any of the three algorithms presented in ~REDUCTION~. Before the actual modular exponentiation algorithm can be written a wrapper algorithm must be written first. This wrapper algorithm will allow the exponent $b$ to be negative which is computed as $c \equiv \left (1 / a \right )^{\vert b \vert} \mbox{(mod }d\mbox{)}$. The value of $(1/a) \mbox{ mod }c$ is computed using the modular inverse (\textit{see ~MODINV~}). If no inverse exists the algorithm terminates with an error. \begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_exptmod}. \\ \textbf{Input}. mp\_int $a$, $b$ and $c$ \\ \textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\ \hline \\ 1. If $c.sign = MP\_NEG$ return(\textit{MP\_VAL}). \\ 2. If $b.sign = MP\_NEG$ then \\ \hspace{3mm}2.1 $g' \leftarrow g^{-1} \mbox{ (mod }c\mbox{)}$ \\ \hspace{3mm}2.2 $x' \leftarrow \vert x \vert$ \\ \hspace{3mm}2.3 Compute $d \equiv g'^{x'} \mbox{ (mod }c\mbox{)}$ via recursion. \\ 3. if ($p$ is odd \textbf{OR} $p$ is a D.R. modulus) \textbf{AND} $p.used > 4$ then \\ \hspace{3mm}3.1 Compute $y \equiv g^{x} \mbox{ (mod }p\mbox{)}$ via algorithm mp\_exptmod\_fast. \\ 4. else \\ \hspace{3mm}4.1 Compute $y \equiv g^{x} \mbox{ (mod }p\mbox{)}$ via algorithm s\_mp\_exptmod. \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_exptmod} \end{figure} \textbf{Algorithm mp\_exptmod.} The first algorithm which actually performs modular exponentiation is algorithm s\_mp\_exptmod. It is a sliding window $k$-ary algorithm which uses Barrett reduction to reduce the product modulo $p$. The second algorithm mp\_exptmod\_fast performs the same operation except it uses either Montgomery or Diminished Radix reduction. The two latter reduction algorithms are clumped in the same exponentiation algorithm since their arguments are essentially the same (\textit{two mp\_ints and one mp\_digit}). EXAM,bn_mp_exptmod.c \subsection{Barrett Modular Exponentiation} \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{s\_mp\_exptmod}. \\ \textbf{Input}. mp\_int $a$, $b$ and $c$ \\ \textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\ \hline \\ 1. $k \leftarrow lg(x)$ \\ 2. $winsize \leftarrow \left \lbrace \begin{array}{ll} 2 & \mbox{if }k \le 7 \\ 3 & \mbox{if }7 < k \le 36 \\ 4 & \mbox{if }36 < k \le 140 \\ 5 & \mbox{if }140 < k \le 450 \\ 6 & \mbox{if }450 < k \le 1303 \\ 7 & \mbox{if }1303 < k \le 3529 \\ 8 & \mbox{if }3529 < k \\ \end{array} \right .$ \\ 3. Initialize $2^{winsize}$ mp\_ints in an array named $M$ and one mp\_int named $\mu$ \\ 4. Calculate the $\mu$ required for Barrett Reduction (\textit{mp\_reduce\_setup}). \\ 5. $M_1 \leftarrow g \mbox{ (mod }p\mbox{)}$ \\ \\ Setup the table of small powers of $g$. First find $g^{2^{winsize}}$ and then all multiples of it. \\ 6. $k \leftarrow 2^{winsize - 1}$ \\ 7. $M_{k} \leftarrow M_1$ \\ 8. for $ix$ from 0 to $winsize - 2$ do \\ \hspace{3mm}8.1 $M_k \leftarrow \left ( M_k \right )^2$ \\ \hspace{3mm}8.2 $M_k \leftarrow M_k \mbox{ (mod }p\mbox{)}$ (\textit{mp\_reduce}) \\ 9. for $ix$ from $2^{winsize - 1} + 1$ to $2^{winsize} - 1$ do \\ \hspace{3mm}9.1 $M_{ix} \leftarrow M_{ix - 1} \cdot M_{1}$ \\ \hspace{3mm}9.2 $M_{ix} \leftarrow M_{ix} \mbox{ (mod }p\mbox{)}$ (\textit{mp\_reduce}) \\ 10. $res \leftarrow 1$ \\ \\ Start Sliding Window. \\ 11. $mode \leftarrow 0, bitcnt \leftarrow 1, buf \leftarrow 0, digidx \leftarrow x.used - 1, bitcpy \leftarrow 0, bitbuf \leftarrow 0$ \\ 12. Loop \\ \hspace{3mm}12.1 $bitcnt \leftarrow bitcnt - 1$ \\ \hspace{3mm}12.2 If $bitcnt = 0$ then do \\ \hspace{6mm}12.2.1 If $digidx = -1$ goto step 13. \\ \hspace{6mm}12.2.2 $buf \leftarrow x_{digidx}$ \\ \hspace{6mm}12.2.3 $digidx \leftarrow digidx - 1$ \\ \hspace{6mm}12.2.4 $bitcnt \leftarrow lg(\beta)$ \\ Continued on next page. \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm s\_mp\_exptmod} \end{figure} \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{s\_mp\_exptmod} (\textit{continued}). \\ \textbf{Input}. mp\_int $a$, $b$ and $c$ \\ \textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\ \hline \\ \hspace{3mm}12.3 $y \leftarrow (buf >> (lg(\beta) - 1))$ AND $1$ \\ \hspace{3mm}12.4 $buf \leftarrow buf << 1$ \\ \hspace{3mm}12.5 if $mode = 0$ and $y = 0$ then goto step 12. \\ \hspace{3mm}12.6 if $mode = 1$ and $y = 0$ then do \\ \hspace{6mm}12.6.1 $res \leftarrow res^2$ \\ \hspace{6mm}12.6.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ \hspace{6mm}12.6.3 Goto step 12. \\ \hspace{3mm}12.7 $bitcpy \leftarrow bitcpy + 1$ \\ \hspace{3mm}12.8 $bitbuf \leftarrow bitbuf + (y << (winsize - bitcpy))$ \\ \hspace{3mm}12.9 $mode \leftarrow 2$ \\ \hspace{3mm}12.10 If $bitcpy = winsize$ then do \\ \hspace{6mm}Window is full so perform the squarings and single multiplication. \\ \hspace{6mm}12.10.1 for $ix$ from $0$ to $winsize -1$ do \\ \hspace{9mm}12.10.1.1 $res \leftarrow res^2$ \\ \hspace{9mm}12.10.1.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ \hspace{6mm}12.10.2 $res \leftarrow res \cdot M_{bitbuf}$ \\ \hspace{6mm}12.10.3 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ \hspace{6mm}Reset the window. \\ \hspace{6mm}12.10.4 $bitcpy \leftarrow 0, bitbuf \leftarrow 0, mode \leftarrow 1$ \\ \\ No more windows left. Check for residual bits of exponent. \\ 13. If $mode = 2$ and $bitcpy > 0$ then do \\ \hspace{3mm}13.1 for $ix$ form $0$ to $bitcpy - 1$ do \\ \hspace{6mm}13.1.1 $res \leftarrow res^2$ \\ \hspace{6mm}13.1.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ \hspace{6mm}13.1.3 $bitbuf \leftarrow bitbuf << 1$ \\ \hspace{6mm}13.1.4 If $bitbuf$ AND $2^{winsize} \ne 0$ then do \\ \hspace{9mm}13.1.4.1 $res \leftarrow res \cdot M_{1}$ \\ \hspace{9mm}13.1.4.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ 14. $y \leftarrow res$ \\ 15. Clear $res$, $mu$ and the $M$ array. \\ 16. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm s\_mp\_exptmod (continued)} \end{figure} \textbf{Algorithm s\_mp\_exptmod.} This algorithm computes the $x$'th power of $g$ modulo $p$ and stores the result in $y$. It takes advantage of the Barrett reduction algorithm to keep the product small throughout the algorithm. The first two steps determine the optimal window size based on the number of bits in the exponent. The larger the exponent the larger the window size becomes. After a window size $winsize$ has been chosen an array of $2^{winsize}$ mp\_int variables is allocated. This table will hold the values of $g^x \mbox{ (mod }p\mbox{)}$ for $2^{winsize - 1} \le x < 2^{winsize}$. After the table is allocated the first power of $g$ is found. Since $g \ge p$ is allowed it must be first reduced modulo $p$ to make the rest of the algorithm more efficient. The first element of the table at $2^{winsize - 1}$ is found by squaring $M_1$ successively $winsize - 2$ times. The rest of the table elements are found by multiplying the previous element by $M_1$ modulo $p$. Now that the table is available the sliding window may begin. The following list describes the functions of all the variables in the window. \begin{enumerate} \item The variable $mode$ dictates how the bits of the exponent are interpreted. \begin{enumerate} \item When $mode = 0$ the bits are ignored since no non-zero bit of the exponent has been seen yet. For example, if the exponent were simply $1$ then there would be $lg(\beta) - 1$ zero bits before the first non-zero bit. In this case bits are ignored until a non-zero bit is found. \item When $mode = 1$ a non-zero bit has been seen before and a new $winsize$-bit window has not been formed yet. In this mode leading $0$ bits are read and a single squaring is performed. If a non-zero bit is read a new window is created. \item When $mode = 2$ the algorithm is in the middle of forming a window and new bits are appended to the window from the most significant bit downards. \end{enumerate} \item The variable $bitcnt$ indicates how many bits are left in the current digit of the exponent left to be read. When it reaches zero a new digit is fetched from the exponent. \item The variable $buf$ holds the currently read digit of the exponent. \item The variable $digidx$ is an index into the exponents digits. It starts at the leading digit $x.used - 1$ and moves towards the trailing digit. \item The variable $bitcpy$ indicates how many bits are in the currently formed window. When it reaches $winsize$ the window is flushed and the appropriate operations performed. \item The variable $bitbuf$ holds the current bits of the window being formed. \end{enumerate} All of step 12 is the window processing loop. It will iterate while there are digits available form the exponent to read. The first step inside this loop is to extract a new digit if no more bits are available in the current digit. If there are no bits left a new digit is read and if there are no digits left than the loop terminates. After a digit is made available step 12.3 will extract the most significant bit of the current digit and move all other bits in the digit upwards. In effect the digit is read from most significant bit to least significant bit and since the digits are read from leading to trailing edges the entire exponent is read from most significant bit to least significant bit. At step 12.5 if the $mode$ and currently extracted bit $y$ are both zero the bit is ignored and the next bit is read. This prevents the algorithm from having todo trivial squaring and reduction operations before the first non-zero bit is read. Step 12.6 and 12.7-10 handle the two cases of $mode = 1$ and $mode = 2$ respectively. FIGU,expt_state,Sliding Window State Diagram By step 13 there are no more digits left in the exponent. However, there may be partial bits in the window left. If $mode = 2$ then a Left-to-Right algorithm is used to process the remaining few bits. EXAM,bn_s_mp_exptmod.c \section{Quick Power of Two} Calculating $b = 2^a$ can be performed much quicker than with any of the previous algorithms. Recall that a logical shift left $m << k$ is equivalent to $m \cdot 2^k$. By this logic when $m = 1$ a quick power of two can be achieved. \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_2expt}. \\ \textbf{Input}. integer $b$ \\ \textbf{Output}. $a \leftarrow 2^b$ \\ \hline \\ 1. $a \leftarrow 0$ \\ 2. If $a.alloc < \lfloor b / lg(\beta) \rfloor + 1$ then grow $a$ appropriately. \\ 3. $a.used \leftarrow \lfloor b / lg(\beta) \rfloor + 1$ \\ 4. $a_{\lfloor b / lg(\beta) \rfloor} \leftarrow 1 << (b \mbox{ mod } lg(\beta))$ \\ 5. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_2expt} \end{figure} \textbf{Algorithm mp\_2expt.} EXAM,bn_mp_2expt.c \chapter{Higher Level Algorithms} \section{Integer Division with Remainder} MARK,DIVISION \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} MARK,MODINV \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 \bibitem[7]{BARRETT} Paul Barrett, \textit{Implementing the Rivest Shamir and Adleman Public Key Encryption Algorithm on a Standard Digital Signal Processor}, Advances in Cryptology, Crypto '86, Springer-Verlag. \bibitem[8]{MONT} P.L.Montgomery. \textit{Modular multiplication without trial division}. Mathematics of Computation, 44(170):519-521, April 1985. \bibitem[9]{DRMET} Chae Hoon Lim and Pil Joong Lee, \textit{Generating Efficient Primes for Discrete Log Cryptosystems}, POSTECH Information Research Laboratories \bibitem[10]{MMB} J. Daemen and R. Govaerts and J. Vandewalle, \textit{Block ciphers based on Modular Arithmetic}, State and {P}rogress in the {R}esearch of {C}ryptography, 1993, pp. 80-89 \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}