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diff --git a/libtommath/tommath.tex b/libtommath/tommath.tex new file mode 100644 index 0000000..c79a537 --- /dev/null +++ b/libtommath/tommath.tex @@ -0,0 +1,6691 @@ +\documentclass[b5paper]{book} +\usepackage{hyperref} +\usepackage{makeidx} +\usepackage{amssymb} +\usepackage{color} +\usepackage{alltt} +\usepackage{graphicx} +\usepackage{layout} +\def\union{\cup} +\def\intersect{\cap} +\def\getsrandom{\stackrel{\rm R}{\gets}} +\def\cross{\times} +\def\cat{\hspace{0.5em} \| \hspace{0.5em}} +\def\catn{$\|$} +\def\divides{\hspace{0.3em} | \hspace{0.3em}} +\def\nequiv{\not\equiv} +\def\approx{\raisebox{0.2ex}{\mbox{\small $\sim$}}} +\def\lcm{{\rm lcm}} +\def\gcd{{\rm gcd}} +\def\log{{\rm log}} +\def\ord{{\rm ord}} +\def\abs{{\mathit abs}} +\def\rep{{\mathit rep}} +\def\mod{{\mathit\ mod\ }} +\renewcommand{\pmod}[1]{\ ({\rm mod\ }{#1})} +\newcommand{\floor}[1]{\left\lfloor{#1}\right\rfloor} +\newcommand{\ceil}[1]{\left\lceil{#1}\right\rceil} +\def\Or{{\rm\ or\ }} +\def\And{{\rm\ and\ }} +\def\iff{\hspace{1em}\Longleftrightarrow\hspace{1em}} +\def\implies{\Rightarrow} +\def\undefined{{\rm ``undefined"}} +\def\Proof{\vspace{1ex}\noindent {\bf Proof:}\hspace{1em}} +\let\oldphi\phi +\def\phi{\varphi} +\def\Pr{{\rm Pr}} +\newcommand{\str}[1]{{\mathbf{#1}}} +\def\F{{\mathbb F}} +\def\N{{\mathbb N}} +\def\Z{{\mathbb Z}} +\def\R{{\mathbb R}} +\def\C{{\mathbb C}} +\def\Q{{\mathbb Q}} +\definecolor{DGray}{gray}{0.5} +\newcommand{\emailaddr}[1]{\mbox{$<${#1}$>$}} +\def\twiddle{\raisebox{0.3ex}{\mbox{\tiny $\sim$}}} +\def\gap{\vspace{0.5ex}} +\makeindex +\begin{document} +\frontmatter +\pagestyle{empty} +\title{Multi--Precision Math} +\author{\mbox{ +%\begin{small} +\begin{tabular}{c} +Tom St Denis \\ +Algonquin College \\ +\\ +Mads Rasmussen \\ +Open Communications Security \\ +\\ +Greg Rose \\ +QUALCOMM Australia \\ +\end{tabular} +%\end{small} +} +} +\maketitle +This text has been placed in the public domain. This text corresponds to the v0.39 release of the +LibTomMath project. + +\begin{alltt} +Tom St Denis +111 Banning Rd +Ottawa, Ontario +K2L 1C3 +Canada + +Phone: 1-613-836-3160 +Email: tomstdenis@gmail.com +\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*{Prefaces} +When I tell people about my LibTom projects and that I release them as public domain they are often puzzled. +They ask why I did it and especially why I continue to work on them for free. The best I can explain it is ``Because I can.'' +Which seems odd and perhaps too terse for adult conversation. I often qualify it with ``I am able, I am willing.'' which +perhaps explains it better. I am the first to admit there is not anything that special with what I have done. Perhaps +others can see that too and then we would have a society to be proud of. My LibTom projects are what I am doing to give +back to society in the form of tools and knowledge that can help others in their endeavours. + +I started writing this book because it was the most logical task to further my goal of open academia. The LibTomMath source +code itself was written to be easy to follow and learn from. There are times, however, where pure C source code does not +explain the algorithms properly. Hence this book. The book literally starts with the foundation of the library and works +itself outwards to the more complicated algorithms. The use of both pseudo--code and verbatim source code provides a duality +of ``theory'' and ``practice'' that the computer science students of the world shall appreciate. I never deviate too far +from relatively straightforward algebra and I hope that this book can be a valuable learning asset. + +This book and indeed much of the LibTom projects would not exist in their current form if it was not for a plethora +of kind people donating their time, resources and kind words to help support my work. Writing a text of significant +length (along with the source code) is a tiresome and lengthy process. Currently the LibTom project is four years old, +comprises of literally thousands of users and over 100,000 lines of source code, TeX and other material. People like Mads and Greg +were there at the beginning to encourage me to work well. It is amazing how timely validation from others can boost morale to +continue the project. Definitely my parents were there for me by providing room and board during the many months of work in 2003. + +To my many friends whom I have met through the years I thank you for the good times and the words of encouragement. I hope I +honour your kind gestures with this project. + +Open Source. Open Academia. Open Minds. + +\begin{flushright} Tom St Denis \end{flushright} + +\newpage +I found the opportunity to work with Tom appealing for several reasons, not only could I broaden my own horizons, but also +contribute to educate others facing the problem of having to handle big number mathematical calculations. + +This book is Tom's child and he has been caring and fostering the project ever since the beginning with a clear mind of +how he wanted the project to turn out. I have helped by proofreading the text and we have had several discussions about +the layout and language used. + +I hold a masters degree in cryptography from the University of Southern Denmark and have always been interested in the +practical aspects of cryptography. + +Having worked in the security consultancy business for several years in S\~{a}o Paulo, Brazil, I have been in touch with a +great deal of work in which multiple precision mathematics was needed. Understanding the possibilities for speeding up +multiple precision calculations is often very important since we deal with outdated machine architecture where modular +reductions, for example, become painfully slow. + +This text is for people who stop and wonder when first examining algorithms such as RSA for the first time and asks +themselves, ``You tell me this is only secure for large numbers, fine; but how do you implement these numbers?'' + +\begin{flushright} +Mads Rasmussen + +S\~{a}o Paulo - SP + +Brazil +\end{flushright} + +\newpage +It's all because I broke my leg. That just happened to be at about the same time that Tom asked for someone to review the section of the book about +Karatsuba multiplication. I was laid up, alone and immobile, and thought ``Why not?'' I vaguely knew what Karatsuba multiplication was, but not +really, so I thought I could help, learn, and stop myself from watching daytime cable TV, all at once. + +At the time of writing this, I've still not met Tom or Mads in meatspace. I've been following Tom's progress since his first splash on the +sci.crypt Usenet news group. I watched him go from a clueless newbie, to the cryptographic equivalent of a reformed smoker, to a real +contributor to the field, over a period of about two years. I've been impressed with his obvious intelligence, and astounded by his productivity. +Of course, he's young enough to be my own child, so he doesn't have my problems with staying awake. + +When I reviewed that single section of the book, in its very earliest form, I was very pleasantly surprised. So I decided to collaborate more fully, +and at least review all of it, and perhaps write some bits too. There's still a long way to go with it, and I have watched a number of close +friends go through the mill of publication, so I think that the way to go is longer than Tom thinks it is. Nevertheless, it's a good effort, +and I'm pleased to be involved with it. + +\begin{flushright} +Greg Rose, Sydney, Australia, June 2003. +\end{flushright} + +\mainmatter +\pagestyle{headings} +\chapter{Introduction} +\section{Multiple Precision Arithmetic} + +\subsection{What is Multiple Precision Arithmetic?} +When we think of long-hand arithmetic such as addition or multiplication we rarely consider the fact that we instinctively +raise or lower the precision of the numbers we are dealing with. For example, in decimal we almost immediate can +reason that $7$ times $6$ is $42$. However, $42$ has two digits of precision as opposed to one digit we started with. +Further multiplications of say $3$ result in a larger precision result $126$. In these few examples we have multiple +precisions for the numbers we are working with. Despite the various levels of precision a single subset\footnote{With the occasional optimization.} + of algorithms can be designed to accomodate them. + +By way of comparison a fixed or single precision operation would lose precision on various operations. For example, in +the decimal system with fixed precision $6 \cdot 7 = 2$. + +Essentially at the heart of computer based multiple precision arithmetic are the same long-hand algorithms taught in +schools to manually add, subtract, multiply and divide. + +\subsection{The Need for Multiple Precision Arithmetic} +The most prevalent need for multiple precision arithmetic, often referred to as ``bignum'' math, is within the implementation +of public-key cryptography algorithms. Algorithms such as RSA \cite{RSAREF} and Diffie-Hellman \cite{DHREF} require +integers of significant magnitude to resist known cryptanalytic attacks. For example, at the time of this writing a +typical RSA modulus would be at least greater than $10^{309}$. However, modern programming languages such as ISO C \cite{ISOC} and +Java \cite{JAVA} only provide instrinsic support for integers which are relatively small and single precision. + +\begin{figure}[!here] +\begin{center} +\begin{tabular}{|r|c|} +\hline \textbf{Data Type} & \textbf{Range} \\ +\hline char & $-128 \ldots 127$ \\ +\hline short & $-32768 \ldots 32767$ \\ +\hline long & $-2147483648 \ldots 2147483647$ \\ +\hline long long & $-9223372036854775808 \ldots 9223372036854775807$ \\ +\hline +\end{tabular} +\end{center} +\caption{Typical Data Types for the C Programming Language} +\label{fig:ISOC} +\end{figure} + +The largest data type guaranteed to be provided by the ISO C programming +language\footnote{As per the ISO C standard. However, each compiler vendor is allowed to augment the precision as they +see fit.} can only represent values up to $10^{19}$ as shown in figure \ref{fig:ISOC}. On its own the C language is +insufficient to accomodate the magnitude required for the problem at hand. An RSA modulus of magnitude $10^{19}$ could be +trivially factored\footnote{A Pollard-Rho factoring would take only $2^{16}$ time.} on the average desktop computer, +rendering any protocol based on the algorithm insecure. Multiple precision algorithms solve this very problem by +extending the range of representable integers while using single precision data types. + +Most advancements in fast multiple precision arithmetic stem from the need for faster and more efficient cryptographic +primitives. Faster modular reduction and exponentiation algorithms such as Barrett's algorithm, which have appeared in +various cryptographic journals, can render algorithms such as RSA and Diffie-Hellman more efficient. In fact, several +major companies such as RSA Security, Certicom and Entrust have built entire product lines on the implementation and +deployment of efficient algorithms. + +However, cryptography is not the only field of study that can benefit from fast multiple precision integer routines. +Another auxiliary use of multiple precision integers is high precision floating point data types. +The basic IEEE \cite{IEEE} standard floating point type is made up of an integer mantissa $q$, an exponent $e$ and a sign bit $s$. +Numbers are given in the form $n = q \cdot b^e \cdot -1^s$ where $b = 2$ is the most common base for IEEE. Since IEEE +floating point is meant to be implemented in hardware the precision of the mantissa is often fairly small +(\textit{23, 48 and 64 bits}). The mantissa is merely an integer and a multiple precision integer could be used to create +a mantissa of much larger precision than hardware alone can efficiently support. This approach could be useful where +scientific applications must minimize the total output error over long calculations. + +Yet another use for large integers is within arithmetic on polynomials of large characteristic (i.e. $GF(p)[x]$ for large $p$). +In fact the library discussed within this text has already been used to form a polynomial basis library\footnote{See \url{http://poly.libtomcrypt.org} for more details.}. + +\subsection{Benefits of Multiple Precision Arithmetic} +\index{precision} +The benefit of multiple precision representations over single or fixed precision representations is that +no precision is lost while representing the result of an operation which requires excess precision. For example, +the product of two $n$-bit integers requires at least $2n$ bits of precision to be represented faithfully. A multiple +precision algorithm would augment the precision of the destination to accomodate the result while a single precision system +would truncate excess bits to maintain a fixed level of precision. + +It is possible to implement algorithms which require large integers with fixed precision algorithms. For example, elliptic +curve cryptography (\textit{ECC}) is often implemented on smartcards by fixing the precision of the integers to the maximum +size the system will ever need. Such an approach can lead to vastly simpler algorithms which can accomodate the +integers required even if the host platform cannot natively accomodate them\footnote{For example, the average smartcard +processor has an 8 bit accumulator.}. However, as efficient as such an approach may be, the resulting source code is not +normally very flexible. It cannot, at runtime, accomodate inputs of higher magnitude than the designer anticipated. + +Multiple precision algorithms have the most overhead of any style of arithmetic. For the the most part the +overhead can be kept to a minimum with careful planning, but overall, it is not well suited for most memory starved +platforms. However, multiple precision algorithms do offer the most flexibility in terms of the magnitude of the +inputs. That is, the same algorithms based on multiple precision integers can accomodate any reasonable size input +without the designer's explicit forethought. This leads to lower cost of ownership for the code as it only has to +be written and tested once. + +\section{Purpose of This Text} +The purpose of this text is to instruct the reader regarding how to implement efficient multiple precision algorithms. +That is to not only explain a limited subset of the core theory behind the algorithms but also the various ``house keeping'' +elements that are neglected by authors of other texts on the subject. Several well reknowned texts \cite{TAOCPV2,HAC} +give considerably detailed explanations of the theoretical aspects of algorithms and often very little information +regarding the practical implementation aspects. + +In most cases how an algorithm is explained and how it is actually implemented are two very different concepts. For +example, the Handbook of Applied Cryptography (\textit{HAC}), algorithm 14.7 on page 594, gives a relatively simple +algorithm for performing multiple precision integer addition. However, the description lacks any discussion concerning +the fact that the two integer inputs may be of differing magnitudes. As a result the implementation is not as simple +as the text would lead people to believe. Similarly the division routine (\textit{algorithm 14.20, pp. 598}) does not +discuss how to handle sign or handle the dividend's decreasing magnitude in the main loop (\textit{step \#3}). + +Both texts also do not discuss several key optimal algorithms required such as ``Comba'' and Karatsuba multipliers +and fast modular inversion, which we consider practical oversights. These optimal algorithms are vital to achieve +any form of useful performance in non-trivial applications. + +To solve this problem the focus of this text is on the practical aspects of implementing a multiple precision integer +package. As a case study the ``LibTomMath''\footnote{Available at \url{http://math.libtomcrypt.com}} package is used +to demonstrate algorithms with real implementations\footnote{In the ISO C programming language.} that have been field +tested and work very well. The LibTomMath library is freely available on the Internet for all uses and this text +discusses a very large portion of the inner workings of the library. + +The algorithms that are presented will always include at least one ``pseudo-code'' description followed +by the actual C source code that implements the algorithm. The pseudo-code can be used to implement the same +algorithm in other programming languages as the reader sees fit. + +This text shall also serve as a walkthrough of the creation of multiple precision algorithms from scratch. Showing +the reader how the algorithms fit together as well as where to start on various taskings. + +\section{Discussion and Notation} +\subsection{Notation} +A multiple precision integer of $n$-digits shall be denoted as $x = (x_{n-1}, \ldots, x_1, x_0)_{ \beta }$ and represent +the integer $x \equiv \sum_{i=0}^{n-1} x_i\beta^i$. The elements of the array $x$ are said to be the radix $\beta$ digits +of the integer. For example, $x = (1,2,3)_{10}$ would represent the integer +$1\cdot 10^2 + 2\cdot10^1 + 3\cdot10^0 = 123$. + +\index{mp\_int} +The term ``mp\_int'' shall refer to a composite structure which contains the digits of the integer it represents, as well +as auxilary data required to manipulate the data. These additional members are discussed further in section +\ref{sec:MPINT}. For the purposes of this text a ``multiple precision integer'' and an ``mp\_int'' are assumed to be +synonymous. When an algorithm is specified to accept an mp\_int variable it is assumed the various auxliary data members +are present as well. An expression of the type \textit{variablename.item} implies that it should evaluate to the +member named ``item'' of the variable. For example, a string of characters may have a member ``length'' which would +evaluate to the number of characters in the string. If the string $a$ equals ``hello'' then it follows that +$a.length = 5$. + +For certain discussions more generic algorithms are presented to help the reader understand the final algorithm used +to solve a given problem. When an algorithm is described as accepting an integer input it is assumed the input is +a plain integer with no additional multiple-precision members. That is, algorithms that use integers as opposed to +mp\_ints as inputs do not concern themselves with the housekeeping operations required such as memory management. These +algorithms will be used to establish the relevant theory which will subsequently be used to describe a multiple +precision algorithm to solve the same problem. + +\subsection{Precision Notation} +The variable $\beta$ represents the radix of a single digit of a multiple precision integer and +must be of the form $q^p$ for $q, p \in \Z^+$. A single precision variable must be able to represent integers in +the range $0 \le x < q \beta$ while a double precision variable must be able to represent integers in the range +$0 \le x < q \beta^2$. The extra radix-$q$ factor allows additions and subtractions to proceed without truncation of the +carry. Since all modern computers are binary, it is assumed that $q$ is two. + +\index{mp\_digit} \index{mp\_word} +Within the source code that will be presented for each algorithm, the data type \textbf{mp\_digit} will represent +a single precision integer type, while, the data type \textbf{mp\_word} will represent a double precision integer type. In +several algorithms (notably the Comba routines) temporary results will be stored in arrays of double precision mp\_words. +For the purposes of this text $x_j$ will refer to the $j$'th digit of a single precision array and $\hat x_j$ will refer to +the $j$'th digit of a double precision array. Whenever an expression is to be assigned to a double precision +variable it is assumed that all single precision variables are promoted to double precision during the evaluation. +Expressions that are assigned to a single precision variable are truncated to fit within the precision of a single +precision data type. + +For example, if $\beta = 10^2$ a single precision data type may represent a value in the +range $0 \le x < 10^3$, while a double precision data type may represent a value in the range $0 \le x < 10^5$. Let +$a = 23$ and $b = 49$ represent two single precision variables. The single precision product shall be written +as $c \leftarrow a \cdot b$ while the double precision product shall be written as $\hat c \leftarrow a \cdot b$. +In this particular case, $\hat c = 1127$ and $c = 127$. The most significant digit of the product would not fit +in a single precision data type and as a result $c \ne \hat c$. + +\subsection{Algorithm Inputs and Outputs} +Within the algorithm descriptions all variables are assumed to be scalars of either single or double precision +as indicated. The only exception to this rule is when variables have been indicated to be of type mp\_int. This +distinction is important as scalars are often used as array indicies and various other counters. + +\subsection{Mathematical Expressions} +The $\lfloor \mbox{ } \rfloor$ brackets imply an expression truncated to an integer not greater than the expression +itself. For example, $\lfloor 5.7 \rfloor = 5$. Similarly the $\lceil \mbox{ } \rceil$ brackets imply an expression +rounded to an integer not less than the expression itself. For example, $\lceil 5.1 \rceil = 6$. Typically when +the $/$ division symbol is used the intention is to perform an integer division with truncation. For example, +$5/2 = 2$ which will often be written as $\lfloor 5/2 \rfloor = 2$ for clarity. When an expression is written as a +fraction a real value division is implied, for example ${5 \over 2} = 2.5$. + +The norm of a multiple precision integer, for example $\vert \vert x \vert \vert$, will be used to represent the number of digits in the representation +of the integer. For example, $\vert \vert 123 \vert \vert = 3$ and $\vert \vert 79452 \vert \vert = 5$. + +\subsection{Work Effort} +\index{big-Oh} +To measure the efficiency of the specified algorithms, a modified big-Oh notation is used. In this system all +single precision operations are considered to have the same cost\footnote{Except where explicitly noted.}. +That is a single precision addition, multiplication and division are assumed to take the same time to +complete. While this is generally not true in practice, it will simplify the discussions considerably. + +Some algorithms have slight advantages over others which is why some constants will not be removed in +the notation. For example, a normal baseline multiplication (section \ref{sec:basemult}) requires $O(n^2)$ work while a +baseline squaring (section \ref{sec:basesquare}) requires $O({{n^2 + n}\over 2})$ work. In standard big-Oh notation these +would both be said to be equivalent to $O(n^2)$. However, +in the context of the this text this is not the case as the magnitude of the inputs will typically be rather small. As a +result small constant factors in the work effort will make an observable difference in algorithm efficiency. + +All of the algorithms presented in this text have a polynomial time work level. That is, of the form +$O(n^k)$ for $n, k \in \Z^{+}$. This will help make useful comparisons in terms of the speed of the algorithms and how +various optimizations will help pay off in the long run. + +\section{Exercises} +Within the more advanced chapters a section will be set aside to give the reader some challenging exercises related to +the discussion at hand. These exercises are not designed to be prize winning problems, but instead to be thought +provoking. Wherever possible the problems are forward minded, stating problems that will be answered in subsequent +chapters. The reader is encouraged to finish the exercises as they appear to get a better understanding of the +subject material. + +That being said, the problems are designed to affirm knowledge of a particular subject matter. Students in particular +are encouraged to verify they can answer the problems correctly before moving on. + +Similar to the exercises of \cite[pp. ix]{TAOCPV2} these exercises are given a scoring system based on the difficulty of +the problem. However, unlike \cite{TAOCPV2} the problems do not get nearly as hard. The scoring of these +exercises ranges from one (the easiest) to five (the hardest). The following table sumarizes the +scoring system used. + +\begin{figure}[here] +\begin{center} +\begin{small} +\begin{tabular}{|c|l|} +\hline $\left [ 1 \right ]$ & An easy problem that should only take the reader a manner of \\ + & minutes to solve. Usually does not involve much computer time \\ + & to solve. \\ +\hline $\left [ 2 \right ]$ & An easy problem that involves a marginal amount of computer \\ + & time usage. Usually requires a program to be written to \\ + & solve the problem. \\ +\hline $\left [ 3 \right ]$ & A moderately hard problem that requires a non-trivial amount \\ + & of work. Usually involves trivial research and development of \\ + & new theory from the perspective of a student. \\ +\hline $\left [ 4 \right ]$ & A moderately hard problem that involves a non-trivial amount \\ + & of work and research, the solution to which will demonstrate \\ + & a higher mastery of the subject matter. \\ +\hline $\left [ 5 \right ]$ & A hard problem that involves concepts that are difficult for a \\ + & novice to solve. Solutions to these problems will demonstrate a \\ + & complete mastery of the given subject. \\ +\hline +\end{tabular} +\end{small} +\end{center} +\caption{Exercise Scoring System} +\end{figure} + +Problems at the first level are meant to be simple questions that the reader can answer quickly without programming a solution or +devising new theory. These problems are quick tests to see if the material is understood. Problems at the second level +are also designed to be easy but will require a program or algorithm to be implemented to arrive at the answer. These +two levels are essentially entry level questions. + +Problems at the third level are meant to be a bit more difficult than the first two levels. The answer is often +fairly obvious but arriving at an exacting solution requires some thought and skill. These problems will almost always +involve devising a new algorithm or implementing a variation of another algorithm previously presented. Readers who can +answer these questions will feel comfortable with the concepts behind the topic at hand. + +Problems at the fourth level are meant to be similar to those of the level three questions except they will require +additional research to be completed. The reader will most likely not know the answer right away, nor will the text provide +the exact details of the answer until a subsequent chapter. + +Problems at the fifth level are meant to be the hardest +problems relative to all the other problems in the chapter. People who can correctly answer fifth level problems have a +mastery of the subject matter at hand. + +Often problems will be tied together. The purpose of this is to start a chain of thought that will be discussed in future chapters. The reader +is encouraged to answer the follow-up problems and try to draw the relevance of problems. + +\section{Introduction to LibTomMath} + +\subsection{What is LibTomMath?} +LibTomMath is a free and open source multiple precision integer library written entirely in portable ISO C. By portable it +is meant that the library does not contain any code that is computer platform dependent or otherwise problematic to use on +any given platform. + +The library has been successfully tested under numerous operating systems including Unix\footnote{All of these +trademarks belong to their respective rightful owners.}, MacOS, Windows, Linux, PalmOS and on standalone hardware such +as the Gameboy Advance. The library is designed to contain enough functionality to be able to develop applications such +as public key cryptosystems and still maintain a relatively small footprint. + +\subsection{Goals of LibTomMath} + +Libraries which obtain the most efficiency are rarely written in a high level programming language such as C. However, +even though this library is written entirely in ISO C, considerable care has been taken to optimize the algorithm implementations within the +library. Specifically the code has been written to work well with the GNU C Compiler (\textit{GCC}) on both x86 and ARM +processors. Wherever possible, highly efficient algorithms, such as Karatsuba multiplication, sliding window +exponentiation and Montgomery reduction have been provided to make the library more efficient. + +Even with the nearly optimal and specialized algorithms that have been included the Application Programing Interface +(\textit{API}) has been kept as simple as possible. Often generic place holder routines will make use of specialized +algorithms automatically without the developer's specific attention. One such example is the generic multiplication +algorithm \textbf{mp\_mul()} which will automatically use Toom--Cook, Karatsuba, Comba or baseline multiplication +based on the magnitude of the inputs and the configuration of the library. + +Making LibTomMath as efficient as possible is not the only goal of the LibTomMath project. Ideally the library should +be source compatible with another popular library which makes it more attractive for developers to use. In this case the +MPI library was used as a API template for all the basic functions. MPI was chosen because it is another library that fits +in the same niche as LibTomMath. Even though LibTomMath uses MPI as the template for the function names and argument +passing conventions, it has been written from scratch by Tom St Denis. + +The project is also meant to act as a learning tool for students, the logic being that no easy-to-follow ``bignum'' +library exists which can be used to teach computer science students how to perform fast and reliable multiple precision +integer arithmetic. To this end the source code has been given quite a few comments and algorithm discussion points. + +\section{Choice of LibTomMath} +LibTomMath was chosen as the case study of this text not only because the author of both projects is one and the same but +for more worthy reasons. Other libraries such as GMP \cite{GMP}, MPI \cite{MPI}, LIP \cite{LIP} and OpenSSL +\cite{OPENSSL} have multiple precision integer arithmetic routines but would not be ideal for this text for +reasons that will be explained in the following sub-sections. + +\subsection{Code Base} +The LibTomMath code base is all portable ISO C source code. This means that there are no platform dependent conditional +segments of code littered throughout the source. This clean and uncluttered approach to the library means that a +developer can more readily discern the true intent of a given section of source code without trying to keep track of +what conditional code will be used. + +The code base of LibTomMath is well organized. Each function is in its own separate source code file +which allows the reader to find a given function very quickly. On average there are $76$ lines of code per source +file which makes the source very easily to follow. By comparison MPI and LIP are single file projects making code tracing +very hard. GMP has many conditional code segments which also hinder tracing. + +When compiled with GCC for the x86 processor and optimized for speed the entire library is approximately $100$KiB\footnote{The notation ``KiB'' means $2^{10}$ octets, similarly ``MiB'' means $2^{20}$ octets.} + which is fairly small compared to GMP (over $250$KiB). LibTomMath is slightly larger than MPI (which compiles to about +$50$KiB) but LibTomMath is also much faster and more complete than MPI. + +\subsection{API Simplicity} +LibTomMath is designed after the MPI library and shares the API design. Quite often programs that use MPI will build +with LibTomMath without change. The function names correlate directly to the action they perform. Almost all of the +functions share the same parameter passing convention. The learning curve is fairly shallow with the API provided +which is an extremely valuable benefit for the student and developer alike. + +The LIP library is an example of a library with an API that is awkward to work with. LIP uses function names that are often ``compressed'' to +illegible short hand. LibTomMath does not share this characteristic. + +The GMP library also does not return error codes. Instead it uses a POSIX.1 \cite{POSIX1} signal system where errors +are signaled to the host application. This happens to be the fastest approach but definitely not the most versatile. In +effect a math error (i.e. invalid input, heap error, etc) can cause a program to stop functioning which is definitely +undersireable in many situations. + +\subsection{Optimizations} +While LibTomMath is certainly not the fastest library (GMP often beats LibTomMath by a factor of two) it does +feature a set of optimal algorithms for tasks such as modular reduction, exponentiation, multiplication and squaring. GMP +and LIP also feature such optimizations while MPI only uses baseline algorithms with no optimizations. GMP lacks a few +of the additional modular reduction optimizations that LibTomMath features\footnote{At the time of this writing GMP +only had Barrett and Montgomery modular reduction algorithms.}. + +LibTomMath is almost always an order of magnitude faster than the MPI library at computationally expensive tasks such as modular +exponentiation. In the grand scheme of ``bignum'' libraries LibTomMath is faster than the average library and usually +slower than the best libraries such as GMP and OpenSSL by only a small factor. + +\subsection{Portability and Stability} +LibTomMath will build ``out of the box'' on any platform equipped with a modern version of the GNU C Compiler +(\textit{GCC}). This means that without changes the library will build without configuration or setting up any +variables. LIP and MPI will build ``out of the box'' as well but have numerous known bugs. Most notably the author of +MPI has recently stopped working on his library and LIP has long since been discontinued. + +GMP requires a configuration script to run and will not build out of the box. GMP and LibTomMath are still in active +development and are very stable across a variety of platforms. + +\subsection{Choice} +LibTomMath is a relatively compact, well documented, highly optimized and portable library which seems only natural for +the case study of this text. Various source files from the LibTomMath project will be included within the text. However, +the reader is encouraged to download their own copy of the library to actually be able to work with the library. + +\chapter{Getting Started} +\section{Library Basics} +The trick to writing any useful library of source code is to build a solid foundation and work outwards from it. First, +a problem along with allowable solution parameters should be identified and analyzed. In this particular case the +inability to accomodate multiple precision integers is the problem. Futhermore, the solution must be written +as portable source code that is reasonably efficient across several different computer platforms. + +After a foundation is formed the remainder of the library can be designed and implemented in a hierarchical fashion. +That is, to implement the lowest level dependencies first and work towards the most abstract functions last. For example, +before implementing a modular exponentiation algorithm one would implement a modular reduction algorithm. +By building outwards from a base foundation instead of using a parallel design methodology the resulting project is +highly modular. Being highly modular is a desirable property of any project as it often means the resulting product +has a small footprint and updates are easy to perform. + +Usually when I start a project I will begin with the header files. I define the data types I think I will need and +prototype the initial functions that are not dependent on other functions (within the library). After I +implement these base functions I prototype more dependent functions and implement them. The process repeats until +I implement all of the functions I require. For example, in the case of LibTomMath I implemented functions such as +mp\_init() well before I implemented mp\_mul() and even further before I implemented mp\_exptmod(). As an example as to +why this design works note that the Karatsuba and Toom-Cook multipliers were written \textit{after} the +dependent function mp\_exptmod() was written. Adding the new multiplication algorithms did not require changes to the +mp\_exptmod() function itself and lowered the total cost of ownership (\textit{so to speak}) and of development +for new algorithms. This methodology allows new algorithms to be tested in a complete framework with relative ease. + +\begin{center} +\begin{figure}[here] +\includegraphics{pics/design_process.ps} +\caption{Design Flow of the First Few Original LibTomMath Functions.} +\label{pic:design_process} +\end{figure} +\end{center} + +Only after the majority of the functions were in place did I pursue a less hierarchical approach to auditing and optimizing +the source code. For example, one day I may audit the multipliers and the next day the polynomial basis functions. + +It only makes sense to begin the text with the preliminary data types and support algorithms required as well. +This chapter discusses the core algorithms of the library which are the dependents for every other algorithm. + +\section{What is a Multiple Precision Integer?} +Recall that most programming languages, in particular ISO C \cite{ISOC}, only have fixed precision data types that on their own cannot +be used to represent values larger than their precision will allow. The purpose of multiple precision algorithms is +to use fixed precision data types to create and manipulate multiple precision integers which may represent values +that are very large. + +As a well known analogy, school children are taught how to form numbers larger than nine by prepending more radix ten digits. In the decimal system +the largest single digit value is $9$. However, by concatenating digits together larger numbers may be represented. Newly prepended digits +(\textit{to the left}) are said to be in a different power of ten column. That is, the number $123$ can be described as having a $1$ in the hundreds +column, $2$ in the tens column and $3$ in the ones column. Or more formally $123 = 1 \cdot 10^2 + 2 \cdot 10^1 + 3 \cdot 10^0$. Computer based +multiple precision arithmetic is essentially the same concept. Larger integers are represented by adjoining fixed +precision computer words with the exception that a different radix is used. + +What most people probably do not think about explicitly are the various other attributes that describe a multiple precision +integer. For example, the integer $154_{10}$ has two immediately obvious properties. First, the integer is positive, +that is the sign of this particular integer is positive as opposed to negative. Second, the integer has three digits in +its representation. There is an additional property that the integer posesses that does not concern pencil-and-paper +arithmetic. The third property is how many digits placeholders are available to hold the integer. + +The human analogy of this third property is ensuring there is enough space on the paper to write the integer. For example, +if one starts writing a large number too far to the right on a piece of paper they will have to erase it and move left. +Similarly, computer algorithms must maintain strict control over memory usage to ensure that the digits of an integer +will not exceed the allowed boundaries. These three properties make up what is known as a multiple precision +integer or mp\_int for short. + +\subsection{The mp\_int Structure} +\label{sec:MPINT} +The mp\_int structure is the ISO C based manifestation of what represents a multiple precision integer. The ISO C standard does not provide for +any such data type but it does provide for making composite data types known as structures. The following is the structure definition +used within LibTomMath. + +\index{mp\_int} +\begin{figure}[here] +\begin{center} +\begin{small} +%\begin{verbatim} +\begin{tabular}{|l|} +\hline +typedef struct \{ \\ +\hspace{3mm}int used, alloc, sign;\\ +\hspace{3mm}mp\_digit *dp;\\ +\} \textbf{mp\_int}; \\ +\hline +\end{tabular} +%\end{verbatim} +\end{small} +\caption{The mp\_int Structure} +\label{fig:mpint} +\end{center} +\end{figure} + +The mp\_int structure (fig. \ref{fig:mpint}) can be broken down as follows. + +\begin{enumerate} +\item The \textbf{used} parameter denotes how many digits of the array \textbf{dp} contain the digits used to represent +a given integer. The \textbf{used} count must be positive (or zero) and may not exceed the \textbf{alloc} count. + +\item The \textbf{alloc} parameter denotes how +many digits are available in the array to use by functions before it has to increase in size. When the \textbf{used} count +of a result would exceed the \textbf{alloc} count all of the algorithms will automatically increase the size of the +array to accommodate the precision of the result. + +\item The pointer \textbf{dp} points to a dynamically allocated array of digits that represent the given multiple +precision integer. It is padded with $(\textbf{alloc} - \textbf{used})$ zero digits. The array is maintained in a least +significant digit order. As a pencil and paper analogy the array is organized such that the right most digits are stored +first starting at the location indexed by zero\footnote{In C all arrays begin at zero.} in the array. For example, +if \textbf{dp} contains $\lbrace a, b, c, \ldots \rbrace$ where \textbf{dp}$_0 = a$, \textbf{dp}$_1 = b$, \textbf{dp}$_2 = c$, $\ldots$ then +it would represent the integer $a + b\beta + c\beta^2 + \ldots$ + +\index{MP\_ZPOS} \index{MP\_NEG} +\item The \textbf{sign} parameter denotes the sign as either zero/positive (\textbf{MP\_ZPOS}) or negative (\textbf{MP\_NEG}). +\end{enumerate} + +\subsubsection{Valid mp\_int Structures} +Several rules are placed on the state of an mp\_int structure and are assumed to be followed for reasons of efficiency. +The only exceptions are when the structure is passed to initialization functions such as mp\_init() and mp\_init\_copy(). + +\begin{enumerate} +\item The value of \textbf{alloc} may not be less than one. That is \textbf{dp} always points to a previously allocated +array of digits. +\item The value of \textbf{used} may not exceed \textbf{alloc} and must be greater than or equal to zero. +\item The value of \textbf{used} implies the digit at index $(used - 1)$ of the \textbf{dp} array is non-zero. That is, +leading zero digits in the most significant positions must be trimmed. + \begin{enumerate} + \item Digits in the \textbf{dp} array at and above the \textbf{used} location must be zero. + \end{enumerate} +\item The value of \textbf{sign} must be \textbf{MP\_ZPOS} if \textbf{used} is zero; +this represents the mp\_int value of zero. +\end{enumerate} + +\section{Argument Passing} +A convention of argument passing must be adopted early on in the development of any library. Making the function +prototypes consistent will help eliminate many headaches in the future as the library grows to significant complexity. +In LibTomMath the multiple precision integer functions accept parameters from left to right as pointers to mp\_int +structures. That means that the source (input) operands are placed on the left and the destination (output) on the right. +Consider the following examples. + +\begin{verbatim} + mp_mul(&a, &b, &c); /* c = a * b */ + mp_add(&a, &b, &a); /* a = a + b */ + mp_sqr(&a, &b); /* b = a * a */ +\end{verbatim} + +The left to right order is a fairly natural way to implement the functions since it lets the developer read aloud the +functions and make sense of them. For example, the first function would read ``multiply a and b and store in c''. + +Certain libraries (\textit{LIP by Lenstra for instance}) accept parameters the other way around, to mimic the order +of assignment expressions. That is, the destination (output) is on the left and arguments (inputs) are on the right. In +truth, it is entirely a matter of preference. In the case of LibTomMath the convention from the MPI library has been +adopted. + +Another very useful design consideration, provided for in LibTomMath, is whether to allow argument sources to also be a +destination. For example, the second example (\textit{mp\_add}) adds $a$ to $b$ and stores in $a$. This is an important +feature to implement since it allows the calling functions to cut down on the number of variables it must maintain. +However, to implement this feature specific care has to be given to ensure the destination is not modified before the +source is fully read. + +\section{Return Values} +A well implemented application, no matter what its purpose, should trap as many runtime errors as possible and return them +to the caller. By catching runtime errors a library can be guaranteed to prevent undefined behaviour. However, the end +developer can still manage to cause a library to crash. For example, by passing an invalid pointer an application may +fault by dereferencing memory not owned by the application. + +In the case of LibTomMath the only errors that are checked for are related to inappropriate inputs (division by zero for +instance) and memory allocation errors. It will not check that the mp\_int passed to any function is valid nor +will it check pointers for validity. Any function that can cause a runtime error will return an error code as an +\textbf{int} data type with one of the following values (fig \ref{fig:errcodes}). + +\index{MP\_OKAY} \index{MP\_VAL} \index{MP\_MEM} +\begin{figure}[here] +\begin{center} +\begin{tabular}{|l|l|} +\hline \textbf{Value} & \textbf{Meaning} \\ +\hline \textbf{MP\_OKAY} & The function was successful \\ +\hline \textbf{MP\_VAL} & One of the input value(s) was invalid \\ +\hline \textbf{MP\_MEM} & The function ran out of heap memory \\ +\hline +\end{tabular} +\end{center} +\caption{LibTomMath Error Codes} +\label{fig:errcodes} +\end{figure} + +When an error is detected within a function it should free any memory it allocated, often during the initialization of +temporary mp\_ints, and return as soon as possible. The goal is to leave the system in the same state it was when the +function was called. Error checking with this style of API is fairly simple. + +\begin{verbatim} + int err; + if ((err = mp_add(&a, &b, &c)) != MP_OKAY) { + printf("Error: %s\n", mp_error_to_string(err)); + exit(EXIT_FAILURE); + } +\end{verbatim} + +The GMP \cite{GMP} library uses C style \textit{signals} to flag errors which is of questionable use. Not all errors are fatal +and it was not deemed ideal by the author of LibTomMath to force developers to have signal handlers for such cases. + +\section{Initialization and Clearing} +The logical starting point when actually writing multiple precision integer functions is the initialization and +clearing of the mp\_int structures. These two algorithms will be used by the majority of the higher level algorithms. + +Given the basic mp\_int structure an initialization routine must first allocate memory to hold the digits of +the integer. Often it is optimal to allocate a sufficiently large pre-set number of digits even though +the initial integer will represent zero. If only a single digit were allocated quite a few subsequent re-allocations +would occur when operations are performed on the integers. There is a tradeoff between how many default digits to allocate +and how many re-allocations are tolerable. Obviously allocating an excessive amount of digits initially will waste +memory and become unmanageable. + +If the memory for the digits has been successfully allocated then the rest of the members of the structure must +be initialized. Since the initial state of an mp\_int is to represent the zero integer, the allocated digits must be set +to zero. The \textbf{used} count set to zero and \textbf{sign} set to \textbf{MP\_ZPOS}. + +\subsection{Initializing an mp\_int} +An mp\_int is said to be initialized if it is set to a valid, preferably default, state such that all of the members of the +structure are set to valid values. The mp\_init algorithm will perform such an action. + +\index{mp\_init} +\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_init}. \\ +\textbf{Input}. An mp\_int $a$ \\ +\textbf{Output}. Allocate memory and initialize $a$ to a known valid mp\_int state. \\ +\hline \\ +1. Allocate memory for \textbf{MP\_PREC} digits. \\ +2. If the allocation failed return(\textit{MP\_MEM}) \\ +3. for $n$ from $0$ to $MP\_PREC - 1$ do \\ +\hspace{3mm}3.1 $a_n \leftarrow 0$\\ +4. $a.sign \leftarrow MP\_ZPOS$\\ +5. $a.used \leftarrow 0$\\ +6. $a.alloc \leftarrow MP\_PREC$\\ +7. Return(\textit{MP\_OKAY})\\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_init} +\end{figure} + +\textbf{Algorithm mp\_init.} +The purpose of this function is to initialize an mp\_int structure so that the rest of the library can properly +manipulte it. It is assumed that the input may not have had any of its members previously initialized which is certainly +a valid assumption if the input resides on the stack. + +Before any of the members such as \textbf{sign}, \textbf{used} or \textbf{alloc} are initialized the memory for +the digits is allocated. If this fails the function returns before setting any of the other members. The \textbf{MP\_PREC} +name represents a constant\footnote{Defined in the ``tommath.h'' header file within LibTomMath.} +used to dictate the minimum precision of newly initialized mp\_int integers. Ideally, it is at least equal to the smallest +precision number you'll be working with. + +Allocating a block of digits at first instead of a single digit has the benefit of lowering the number of usually slow +heap operations later functions will have to perform in the future. If \textbf{MP\_PREC} is set correctly the slack +memory and the number of heap operations will be trivial. + +Once the allocation has been made the digits have to be set to zero as well as the \textbf{used}, \textbf{sign} and +\textbf{alloc} members initialized. This ensures that the mp\_int will always represent the default state of zero regardless +of the original condition of the input. + +\textbf{Remark.} +This function introduces the idiosyncrasy that all iterative loops, commonly initiated with the ``for'' keyword, iterate incrementally +when the ``to'' keyword is placed between two expressions. For example, ``for $a$ from $b$ to $c$ do'' means that +a subsequent expression (or body of expressions) are to be evaluated upto $c - b$ times so long as $b \le c$. In each +iteration the variable $a$ is substituted for a new integer that lies inclusively between $b$ and $c$. If $b > c$ occured +the loop would not iterate. By contrast if the ``downto'' keyword were used in place of ``to'' the loop would iterate +decrementally. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_init.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +One immediate observation of this initializtion function is that it does not return a pointer to a mp\_int structure. It +is assumed that the caller has already allocated memory for the mp\_int structure, typically on the application stack. The +call to mp\_init() is used only to initialize the members of the structure to a known default state. + +Here we see (line 24) the memory allocation is performed first. This allows us to exit cleanly and quickly +if there is an error. If the allocation fails the routine will return \textbf{MP\_MEM} to the caller to indicate there +was a memory error. The function XMALLOC is what actually allocates the memory. Technically XMALLOC is not a function +but a macro defined in ``tommath.h``. By default, XMALLOC will evaluate to malloc() which is the C library's built--in +memory allocation routine. + +In order to assure the mp\_int is in a known state the digits must be set to zero. On most platforms this could have been +accomplished by using calloc() instead of malloc(). However, to correctly initialize a integer type to a given value in a +portable fashion you have to actually assign the value. The for loop (line 30) performs this required +operation. + +After the memory has been successfully initialized the remainder of the members are initialized +(lines 34 through 35) to their respective default states. At this point the algorithm has succeeded and +a success code is returned to the calling function. If this function returns \textbf{MP\_OKAY} it is safe to assume the +mp\_int structure has been properly initialized and is safe to use with other functions within the library. + +\subsection{Clearing an mp\_int} +When an mp\_int is no longer required by the application, the memory that has been allocated for its digits must be +returned to the application's memory pool with the mp\_clear algorithm. + +\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_clear}. \\ +\textbf{Input}. An mp\_int $a$ \\ +\textbf{Output}. The memory for $a$ shall be deallocated. \\ +\hline \\ +1. If $a$ has been previously freed then return(\textit{MP\_OKAY}). \\ +2. for $n$ from 0 to $a.used - 1$ do \\ +\hspace{3mm}2.1 $a_n \leftarrow 0$ \\ +3. Free the memory allocated for the digits of $a$. \\ +4. $a.used \leftarrow 0$ \\ +5. $a.alloc \leftarrow 0$ \\ +6. $a.sign \leftarrow MP\_ZPOS$ \\ +7. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_clear} +\end{figure} + +\textbf{Algorithm mp\_clear.} +This algorithm accomplishes two goals. First, it clears the digits and the other mp\_int members. This ensures that +if a developer accidentally re-uses a cleared structure it is less likely to cause problems. The second goal +is to free the allocated memory. + +The logic behind the algorithm is extended by marking cleared mp\_int structures so that subsequent calls to this +algorithm will not try to free the memory multiple times. Cleared mp\_ints are detectable by having a pre-defined invalid +digit pointer \textbf{dp} setting. + +Once an mp\_int has been cleared the mp\_int structure is no longer in a valid state for any other algorithm +with the exception of algorithms mp\_init, mp\_init\_copy, mp\_init\_size and mp\_clear. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_clear.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +The algorithm only operates on the mp\_int if it hasn't been previously cleared. The if statement (line 25) +checks to see if the \textbf{dp} member is not \textbf{NULL}. If the mp\_int is a valid mp\_int then \textbf{dp} cannot be +\textbf{NULL} in which case the if statement will evaluate to true. + +The digits of the mp\_int are cleared by the for loop (line 27) which assigns a zero to every digit. Similar to mp\_init() +the digits are assigned zero instead of using block memory operations (such as memset()) since this is more portable. + +The digits are deallocated off the heap via the XFREE macro. Similar to XMALLOC the XFREE macro actually evaluates to +a standard C library function. In this case the free() function. Since free() only deallocates the memory the pointer +still has to be reset to \textbf{NULL} manually (line 35). + +Now that the digits have been cleared and deallocated the other members are set to their final values (lines 36 and 37). + +\section{Maintenance Algorithms} + +The previous sections describes how to initialize and clear an mp\_int structure. To further support operations +that are to be performed on mp\_int structures (such as addition and multiplication) the dependent algorithms must be +able to augment the precision of an mp\_int and +initialize mp\_ints with differing initial conditions. + +These algorithms complete the set of low level algorithms required to work with mp\_int structures in the higher level +algorithms such as addition, multiplication and modular exponentiation. + +\subsection{Augmenting an mp\_int's Precision} +When storing a value in an mp\_int structure, a sufficient number of digits must be available to accomodate the entire +result of an operation without loss of precision. Quite often the size of the array given by the \textbf{alloc} member +is large enough to simply increase the \textbf{used} digit count. However, when the size of the array is too small it +must be re-sized appropriately to accomodate the result. The mp\_grow algorithm will provide this functionality. + +\newpage\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_grow}. \\ +\textbf{Input}. An mp\_int $a$ and an integer $b$. \\ +\textbf{Output}. $a$ is expanded to accomodate $b$ digits. \\ +\hline \\ +1. if $a.alloc \ge b$ then return(\textit{MP\_OKAY}) \\ +2. $u \leftarrow b\mbox{ (mod }MP\_PREC\mbox{)}$ \\ +3. $v \leftarrow b + 2 \cdot MP\_PREC - u$ \\ +4. Re-allocate the array of digits $a$ to size $v$ \\ +5. If the allocation failed then return(\textit{MP\_MEM}). \\ +6. for n from a.alloc to $v - 1$ do \\ +\hspace{+3mm}6.1 $a_n \leftarrow 0$ \\ +7. $a.alloc \leftarrow v$ \\ +8. Return(\textit{MP\_OKAY}) \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_grow} +\end{figure} + +\textbf{Algorithm mp\_grow.} +It is ideal to prevent re-allocations from being performed if they are not required (step one). This is useful to +prevent mp\_ints from growing excessively in code that erroneously calls mp\_grow. + +The requested digit count is padded up to next multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} (steps two and three). +This helps prevent many trivial reallocations that would grow an mp\_int by trivially small values. + +It is assumed that the reallocation (step four) leaves the lower $a.alloc$ digits of the mp\_int intact. This is much +akin to how the \textit{realloc} function from the standard C library works. Since the newly allocated digits are +assumed to contain undefined values they are initially set to zero. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_grow.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +A quick optimization is to first determine if a memory re-allocation is required at all. The if statement (line 24) checks +if the \textbf{alloc} member of the mp\_int is smaller than the requested digit count. If the count is not larger than \textbf{alloc} +the function skips the re-allocation part thus saving time. + +When a re-allocation is performed it is turned into an optimal request to save time in the future. The requested digit count is +padded upwards to 2nd multiple of \textbf{MP\_PREC} larger than \textbf{alloc} (line 25). The XREALLOC function is used +to re-allocate the memory. As per the other functions XREALLOC is actually a macro which evaluates to realloc by default. The realloc +function leaves the base of the allocation intact which means the first \textbf{alloc} digits of the mp\_int are the same as before +the re-allocation. All that is left is to clear the newly allocated digits and return. + +Note that the re-allocation result is actually stored in a temporary pointer $tmp$. This is to allow this function to return +an error with a valid pointer. Earlier releases of the library stored the result of XREALLOC into the mp\_int $a$. That would +result in a memory leak if XREALLOC ever failed. + +\subsection{Initializing Variable Precision mp\_ints} +Occasionally the number of digits required will be known in advance of an initialization, based on, for example, the size +of input mp\_ints to a given algorithm. The purpose of algorithm mp\_init\_size is similar to mp\_init except that it +will allocate \textit{at least} a specified number of digits. + +\begin{figure}[here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_init\_size}. \\ +\textbf{Input}. An mp\_int $a$ and the requested number of digits $b$. \\ +\textbf{Output}. $a$ is initialized to hold at least $b$ digits. \\ +\hline \\ +1. $u \leftarrow b \mbox{ (mod }MP\_PREC\mbox{)}$ \\ +2. $v \leftarrow b + 2 \cdot MP\_PREC - u$ \\ +3. Allocate $v$ digits. \\ +4. for $n$ from $0$ to $v - 1$ do \\ +\hspace{3mm}4.1 $a_n \leftarrow 0$ \\ +5. $a.sign \leftarrow MP\_ZPOS$\\ +6. $a.used \leftarrow 0$\\ +7. $a.alloc \leftarrow v$\\ +8. Return(\textit{MP\_OKAY})\\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_init\_size} +\end{figure} + +\textbf{Algorithm mp\_init\_size.} +This algorithm will initialize an mp\_int structure $a$ like algorithm mp\_init with the exception that the number of +digits allocated can be controlled by the second input argument $b$. The input size is padded upwards so it is a +multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} digits. This padding is used to prevent trivial +allocations from becoming a bottleneck in the rest of the algorithms. + +Like algorithm mp\_init, the mp\_int structure is initialized to a default state representing the integer zero. This +particular algorithm is useful if it is known ahead of time the approximate size of the input. If the approximation is +correct no further memory re-allocations are required to work with the mp\_int. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_init\_size.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +The number of digits $b$ requested is padded (line 24) by first augmenting it to the next multiple of +\textbf{MP\_PREC} and then adding \textbf{MP\_PREC} to the result. If the memory can be successfully allocated the +mp\_int is placed in a default state representing the integer zero. Otherwise, the error code \textbf{MP\_MEM} will be +returned (line 29). + +The digits are allocated and set to zero at the same time with the calloc() function (line @25,XCALLOC@). The +\textbf{used} count is set to zero, the \textbf{alloc} count set to the padded digit count and the \textbf{sign} flag set +to \textbf{MP\_ZPOS} to achieve a default valid mp\_int state (lines 33, 34 and 35). If the function +returns succesfully then it is correct to assume that the mp\_int structure is in a valid state for the remainder of the +functions to work with. + +\subsection{Multiple Integer Initializations and Clearings} +Occasionally a function will require a series of mp\_int data types to be made available simultaneously. +The purpose of algorithm mp\_init\_multi is to initialize a variable length array of mp\_int structures in a single +statement. It is essentially a shortcut to multiple initializations. + +\newpage\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_init\_multi}. \\ +\textbf{Input}. Variable length array $V_k$ of mp\_int variables of length $k$. \\ +\textbf{Output}. The array is initialized such that each mp\_int of $V_k$ is ready to use. \\ +\hline \\ +1. for $n$ from 0 to $k - 1$ do \\ +\hspace{+3mm}1.1. Initialize the mp\_int $V_n$ (\textit{mp\_init}) \\ +\hspace{+3mm}1.2. If initialization failed then do \\ +\hspace{+6mm}1.2.1. for $j$ from $0$ to $n$ do \\ +\hspace{+9mm}1.2.1.1. Free the mp\_int $V_j$ (\textit{mp\_clear}) \\ +\hspace{+6mm}1.2.2. Return(\textit{MP\_MEM}) \\ +2. Return(\textit{MP\_OKAY}) \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_init\_multi} +\end{figure} + +\textbf{Algorithm mp\_init\_multi.} +The algorithm will initialize the array of mp\_int variables one at a time. If a runtime error has been detected +(\textit{step 1.2}) all of the previously initialized variables are cleared. The goal is an ``all or nothing'' +initialization which allows for quick recovery from runtime errors. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_init\_multi.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +This function intializes a variable length list of mp\_int structure pointers. However, instead of having the mp\_int +structures in an actual C array they are simply passed as arguments to the function. This function makes use of the +``...'' argument syntax of the C programming language. The list is terminated with a final \textbf{NULL} argument +appended on the right. + +The function uses the ``stdarg.h'' \textit{va} functions to step portably through the arguments to the function. A count +$n$ of succesfully initialized mp\_int structures is maintained (line 48) such that if a failure does occur, +the algorithm can backtrack and free the previously initialized structures (lines 28 to 47). + + +\subsection{Clamping Excess Digits} +When a function anticipates a result will be $n$ digits it is simpler to assume this is true within the body of +the function instead of checking during the computation. For example, a multiplication of a $i$ digit number by a +$j$ digit produces a result of at most $i + j$ digits. It is entirely possible that the result is $i + j - 1$ +though, with no final carry into the last position. However, suppose the destination had to be first expanded +(\textit{via mp\_grow}) to accomodate $i + j - 1$ digits than further expanded to accomodate the final carry. +That would be a considerable waste of time since heap operations are relatively slow. + +The ideal solution is to always assume the result is $i + j$ and fix up the \textbf{used} count after the function +terminates. This way a single heap operation (\textit{at most}) is required. However, if the result was not checked +there would be an excess high order zero digit. + +For example, suppose the product of two integers was $x_n = (0x_{n-1}x_{n-2}...x_0)_{\beta}$. The leading zero digit +will not contribute to the precision of the result. In fact, through subsequent operations more leading zero digits would +accumulate to the point the size of the integer would be prohibitive. As a result even though the precision is very +low the representation is excessively large. + +The mp\_clamp algorithm is designed to solve this very problem. It will trim high-order zeros by decrementing the +\textbf{used} count until a non-zero most significant digit is found. Also in this system, zero is considered to be a +positive number which means that if the \textbf{used} count is decremented to zero, the sign must be set to +\textbf{MP\_ZPOS}. + +\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_clamp}. \\ +\textbf{Input}. An mp\_int $a$ \\ +\textbf{Output}. Any excess leading zero digits of $a$ are removed \\ +\hline \\ +1. while $a.used > 0$ and $a_{a.used - 1} = 0$ do \\ +\hspace{+3mm}1.1 $a.used \leftarrow a.used - 1$ \\ +2. if $a.used = 0$ then do \\ +\hspace{+3mm}2.1 $a.sign \leftarrow MP\_ZPOS$ \\ +\hline \\ +\end{tabular} +\end{center} +\caption{Algorithm mp\_clamp} +\end{figure} + +\textbf{Algorithm mp\_clamp.} +As can be expected this algorithm is very simple. The loop on step one is expected to iterate only once or twice at +the most. For example, this will happen in cases where there is not a carry to fill the last position. Step two fixes the sign for +when all of the digits are zero to ensure that the mp\_int is valid at all times. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_clamp.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +Note on line 28 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 31 is used to make sure the \textbf{used} count is decremented and not +the pointer ``a''. + +\section*{Exercises} +\begin{tabular}{cl} +$\left [ 1 \right ]$ & Discuss the relevance of the \textbf{used} member of the mp\_int structure. \\ + & \\ +$\left [ 1 \right ]$ & Discuss the consequences of not using padding when performing allocations. \\ + & \\ +$\left [ 2 \right ]$ & Estimate an ideal value for \textbf{MP\_PREC} when performing 1024-bit RSA \\ + & encryption when $\beta = 2^{28}$. \\ + & \\ +$\left [ 1 \right ]$ & Discuss the relevance of the algorithm mp\_clamp. What does it prevent? \\ + & \\ +$\left [ 1 \right ]$ & Give an example of when the algorithm mp\_init\_copy might be useful. \\ + & \\ +\end{tabular} + + +%%% +% CHAPTER FOUR +%%% + +\chapter{Basic Operations} + +\section{Introduction} +In the previous chapter a series of low level algorithms were established that dealt with initializing and maintaining +mp\_int structures. This chapter will discuss another set of seemingly non-algebraic algorithms which will form the low +level basis of the entire library. While these algorithm are relatively trivial it is important to understand how they +work before proceeding since these algorithms will be used almost intrinsically in the following chapters. + +The algorithms in this chapter deal primarily with more ``programmer'' related tasks such as creating copies of +mp\_int structures, assigning small values to mp\_int structures and comparisons of the values mp\_int structures +represent. + +\section{Assigning Values to mp\_int Structures} +\subsection{Copying an mp\_int} +Assigning the value that a given mp\_int structure represents to another mp\_int structure shall be known as making +a copy for the purposes of this text. The copy of the mp\_int will be a separate entity that represents the same +value as the mp\_int it was copied from. The mp\_copy algorithm provides this functionality. + +\newpage\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_copy}. \\ +\textbf{Input}. An mp\_int $a$ and $b$. \\ +\textbf{Output}. Store a copy of $a$ in $b$. \\ +\hline \\ +1. If $b.alloc < a.used$ then grow $b$ to $a.used$ digits. (\textit{mp\_grow}) \\ +2. for $n$ from 0 to $a.used - 1$ do \\ +\hspace{3mm}2.1 $b_{n} \leftarrow a_{n}$ \\ +3. for $n$ from $a.used$ to $b.used - 1$ do \\ +\hspace{3mm}3.1 $b_{n} \leftarrow 0$ \\ +4. $b.used \leftarrow a.used$ \\ +5. $b.sign \leftarrow a.sign$ \\ +6. return(\textit{MP\_OKAY}) \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_copy} +\end{figure} + +\textbf{Algorithm mp\_copy.} +This algorithm copies the mp\_int $a$ such that upon succesful termination of the algorithm the mp\_int $b$ will +represent the same integer as the mp\_int $a$. The mp\_int $b$ shall be a complete and distinct copy of the +mp\_int $a$ meaing that the mp\_int $a$ can be modified and it shall not affect the value of the mp\_int $b$. + +If $b$ does not have enough room for the digits of $a$ it must first have its precision augmented via the mp\_grow +algorithm. The digits of $a$ are copied over the digits of $b$ and any excess digits of $b$ are set to zero (step two +and three). The \textbf{used} and \textbf{sign} members of $a$ are finally copied over the respective members of +$b$. + +\textbf{Remark.} This algorithm also introduces a new idiosyncrasy that will be used throughout the rest of the +text. The error return codes of other algorithms are not explicitly checked in the pseudo-code presented. For example, in +step one of the mp\_copy algorithm the return of mp\_grow is not explicitly checked to ensure it succeeded. Text space is +limited so it is assumed that if a algorithm fails it will clear all temporarily allocated mp\_ints and return +the error code itself. However, the C code presented will demonstrate all of the error handling logic required to +implement the pseudo-code. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_copy.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +Occasionally a dependent algorithm may copy an mp\_int effectively into itself such as when the input and output +mp\_int structures passed to a function are one and the same. For this case it is optimal to return immediately without +copying digits (line 25). + +The mp\_int $b$ must have enough digits to accomodate the used digits of the mp\_int $a$. If $b.alloc$ is less than +$a.used$ the algorithm mp\_grow is used to augment the precision of $b$ (lines 30 to 33). In order to +simplify the inner loop that copies the digits from $a$ to $b$, two aliases $tmpa$ and $tmpb$ point directly at the digits +of the mp\_ints $a$ and $b$ respectively. These aliases (lines 43 and 46) allow the compiler to access the digits without first dereferencing the +mp\_int pointers and then subsequently the pointer to the digits. + +After the aliases are established the digits from $a$ are copied into $b$ (lines 49 to 51) and then the excess +digits of $b$ are set to zero (lines 54 to 56). Both ``for'' loops make use of the pointer aliases and in +fact the alias for $b$ is carried through into the second ``for'' loop to clear the excess digits. This optimization +allows the alias to stay in a machine register fairly easy between the two loops. + +\textbf{Remarks.} The use of pointer aliases is an implementation methodology first introduced in this function that will +be used considerably in other functions. Technically, a pointer alias is simply a short hand alias used to lower the +number of pointer dereferencing operations required to access data. For example, a for loop may resemble + +\begin{alltt} +for (x = 0; x < 100; x++) \{ + a->num[4]->dp[x] = 0; +\} +\end{alltt} + +This could be re-written using aliases as + +\begin{alltt} +mp_digit *tmpa; +a = a->num[4]->dp; +for (x = 0; x < 100; x++) \{ + *a++ = 0; +\} +\end{alltt} + +In this case an alias is used to access the +array of digits within an mp\_int structure directly. It may seem that a pointer alias is strictly not required +as a compiler may optimize out the redundant pointer operations. However, there are two dominant reasons to use aliases. + +The first reason is that most compilers will not effectively optimize pointer arithmetic. For example, some optimizations +may work for the Microsoft Visual C++ compiler (MSVC) and not for the GNU C Compiler (GCC). Also some optimizations may +work for GCC and not MSVC. As such it is ideal to find a common ground for as many compilers as possible. Pointer +aliases optimize the code considerably before the compiler even reads the source code which means the end compiled code +stands a better chance of being faster. + +The second reason is that pointer aliases often can make an algorithm simpler to read. Consider the first ``for'' +loop of the function mp\_copy() re-written to not use pointer aliases. + +\begin{alltt} + /* copy all the digits */ + for (n = 0; n < a->used; n++) \{ + b->dp[n] = a->dp[n]; + \} +\end{alltt} + +Whether this code is harder to read depends strongly on the individual. However, it is quantifiably slightly more +complicated as there are four variables within the statement instead of just two. + +\subsubsection{Nested Statements} +Another commonly used technique in the source routines is that certain sections of code are nested. This is used in +particular with the pointer aliases to highlight code phases. For example, a Comba multiplier (discussed in chapter six) +will typically have three different phases. First the temporaries are initialized, then the columns calculated and +finally the carries are propagated. In this example the middle column production phase will typically be nested as it +uses temporary variables and aliases the most. + +The nesting also simplies the source code as variables that are nested are only valid for their scope. As a result +the various temporary variables required do not propagate into other sections of code. + + +\subsection{Creating a Clone} +Another common operation is to make a local temporary copy of an mp\_int argument. To initialize an mp\_int +and then copy another existing mp\_int into the newly intialized mp\_int will be known as creating a clone. This is +useful within functions that need to modify an argument but do not wish to actually modify the original copy. The +mp\_init\_copy algorithm has been designed to help perform this task. + +\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_init\_copy}. \\ +\textbf{Input}. An mp\_int $a$ and $b$\\ +\textbf{Output}. $a$ is initialized to be a copy of $b$. \\ +\hline \\ +1. Init $a$. (\textit{mp\_init}) \\ +2. Copy $b$ to $a$. (\textit{mp\_copy}) \\ +3. Return the status of the copy operation. \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_init\_copy} +\end{figure} + +\textbf{Algorithm mp\_init\_copy.} +This algorithm will initialize an mp\_int variable and copy another previously initialized mp\_int variable into it. As +such this algorithm will perform two operations in one step. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_init\_copy.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +This will initialize \textbf{a} and make it a verbatim copy of the contents of \textbf{b}. Note that +\textbf{a} will have its own memory allocated which means that \textbf{b} may be cleared after the call +and \textbf{a} will be left intact. + +\section{Zeroing an Integer} +Reseting an mp\_int to the default state is a common step in many algorithms. The mp\_zero algorithm will be the algorithm used to +perform this task. + +\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_zero}. \\ +\textbf{Input}. An mp\_int $a$ \\ +\textbf{Output}. Zero the contents of $a$ \\ +\hline \\ +1. $a.used \leftarrow 0$ \\ +2. $a.sign \leftarrow$ MP\_ZPOS \\ +3. for $n$ from 0 to $a.alloc - 1$ do \\ +\hspace{3mm}3.1 $a_n \leftarrow 0$ \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_zero} +\end{figure} + +\textbf{Algorithm mp\_zero.} +This algorithm simply resets a mp\_int to the default state. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_zero.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +After the function is completed, all of the digits are zeroed, the \textbf{used} count is zeroed and the +\textbf{sign} variable is set to \textbf{MP\_ZPOS}. + +\section{Sign Manipulation} +\subsection{Absolute Value} +With the mp\_int representation of an integer, calculating the absolute value is trivial. The mp\_abs algorithm will compute +the absolute value of an mp\_int. + +\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_abs}. \\ +\textbf{Input}. An mp\_int $a$ \\ +\textbf{Output}. Computes $b = \vert a \vert$ \\ +\hline \\ +1. Copy $a$ to $b$. (\textit{mp\_copy}) \\ +2. If the copy failed return(\textit{MP\_MEM}). \\ +3. $b.sign \leftarrow MP\_ZPOS$ \\ +4. Return(\textit{MP\_OKAY}) \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_abs} +\end{figure} + +\textbf{Algorithm mp\_abs.} +This algorithm computes the absolute of an mp\_int input. First it copies $a$ over $b$. This is an example of an +algorithm where the check in mp\_copy that determines if the source and destination are equal proves useful. This allows, +for instance, the developer to pass the same mp\_int as the source and destination to this function without addition +logic to handle it. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_abs.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +This fairly trivial algorithm first eliminates non--required duplications (line 28) and then sets the +\textbf{sign} flag to \textbf{MP\_ZPOS}. + +\subsection{Integer Negation} +With the mp\_int representation of an integer, calculating the negation is also trivial. The mp\_neg algorithm will compute +the negative of an mp\_int input. + +\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_neg}. \\ +\textbf{Input}. An mp\_int $a$ \\ +\textbf{Output}. Computes $b = -a$ \\ +\hline \\ +1. Copy $a$ to $b$. (\textit{mp\_copy}) \\ +2. If the copy failed return(\textit{MP\_MEM}). \\ +3. If $a.used = 0$ then return(\textit{MP\_OKAY}). \\ +4. If $a.sign = MP\_ZPOS$ then do \\ +\hspace{3mm}4.1 $b.sign = MP\_NEG$. \\ +5. else do \\ +\hspace{3mm}5.1 $b.sign = MP\_ZPOS$. \\ +6. Return(\textit{MP\_OKAY}) \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_neg} +\end{figure} + +\textbf{Algorithm mp\_neg.} +This algorithm computes the negation of an input. First it copies $a$ over $b$. If $a$ has no used digits then +the algorithm returns immediately. Otherwise it flips the sign flag and stores the result in $b$. Note that if +$a$ had no digits then it must be positive by definition. Had step three been omitted then the algorithm would return +zero as negative. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_neg.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +Like mp\_abs() this function avoids non--required duplications (line 22) and then sets the sign. We +have to make sure that only non--zero values get a \textbf{sign} of \textbf{MP\_NEG}. If the mp\_int is zero +than the \textbf{sign} is hard--coded to \textbf{MP\_ZPOS}. + +\section{Small Constants} +\subsection{Setting Small Constants} +Often a mp\_int must be set to a relatively small value such as $1$ or $2$. For these cases the mp\_set algorithm is useful. + +\newpage\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_set}. \\ +\textbf{Input}. An mp\_int $a$ and a digit $b$ \\ +\textbf{Output}. Make $a$ equivalent to $b$ \\ +\hline \\ +1. Zero $a$ (\textit{mp\_zero}). \\ +2. $a_0 \leftarrow b \mbox{ (mod }\beta\mbox{)}$ \\ +3. $a.used \leftarrow \left \lbrace \begin{array}{ll} + 1 & \mbox{if }a_0 > 0 \\ + 0 & \mbox{if }a_0 = 0 + \end{array} \right .$ \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_set} +\end{figure} + +\textbf{Algorithm mp\_set.} +This algorithm sets a mp\_int to a small single digit value. Step number 1 ensures that the integer is reset to the default state. The +single digit is set (\textit{modulo $\beta$}) and the \textbf{used} count is adjusted accordingly. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_set.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +First we zero (line 21) the mp\_int to make sure that the other members are initialized for a +small positive constant. mp\_zero() ensures that the \textbf{sign} is positive and the \textbf{used} count +is zero. Next we set the digit and reduce it modulo $\beta$ (line 22). After this step we have to +check if the resulting digit is zero or not. If it is not then we set the \textbf{used} count to one, otherwise +to zero. + +We can quickly reduce modulo $\beta$ since it is of the form $2^k$ and a quick binary AND operation with +$2^k - 1$ will perform the same operation. + +One important limitation of this function is that it will only set one digit. The size of a digit is not fixed, meaning source that uses +this function should take that into account. Only trivially small constants can be set using this function. + +\subsection{Setting Large Constants} +To overcome the limitations of the mp\_set algorithm the mp\_set\_int algorithm is ideal. It accepts a ``long'' +data type as input and will always treat it as a 32-bit integer. + +\begin{figure}[here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_set\_int}. \\ +\textbf{Input}. An mp\_int $a$ and a ``long'' integer $b$ \\ +\textbf{Output}. Make $a$ equivalent to $b$ \\ +\hline \\ +1. Zero $a$ (\textit{mp\_zero}) \\ +2. for $n$ from 0 to 7 do \\ +\hspace{3mm}2.1 $a \leftarrow a \cdot 16$ (\textit{mp\_mul2d}) \\ +\hspace{3mm}2.2 $u \leftarrow \lfloor b / 2^{4(7 - n)} \rfloor \mbox{ (mod }16\mbox{)}$\\ +\hspace{3mm}2.3 $a_0 \leftarrow a_0 + u$ \\ +\hspace{3mm}2.4 $a.used \leftarrow a.used + 1$ \\ +3. Clamp excess used digits (\textit{mp\_clamp}) \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_set\_int} +\end{figure} + +\textbf{Algorithm mp\_set\_int.} +The algorithm performs eight iterations of a simple loop where in each iteration four bits from the source are added to the +mp\_int. Step 2.1 will multiply the current result by sixteen making room for four more bits in the less significant positions. In step 2.2 the +next four bits from the source are extracted and are added to the mp\_int. The \textbf{used} digit count is +incremented to reflect the addition. The \textbf{used} digit counter is incremented since if any of the leading digits were zero the mp\_int would have +zero digits used and the newly added four bits would be ignored. + +Excess zero digits are trimmed in steps 2.1 and 3 by using higher level algorithms mp\_mul2d and mp\_clamp. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_set\_int.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +This function sets four bits of the number at a time to handle all practical \textbf{DIGIT\_BIT} sizes. The weird +addition on line 39 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 28 +as well as the call to mp\_clamp() on line 41. 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}. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_cmp\_mag.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +The two if statements (lines 25 and 29) 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. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_cmp.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +The two if statements (lines 23 and 24) perform the initial sign comparison. If the signs are not the equal then which ever +has the positive sign is larger. The inputs are compared (line 32) based on magnitudes. If the signs were both +negative then the unsigned comparison is performed in the opposite direction (line 34). Otherwise, the signs are assumed to +be both positive and a forward direction unsigned comparison is performed. + +\section*{Exercises} +\begin{tabular}{cl} +$\left [ 2 \right ]$ & Modify algorithm mp\_set\_int to accept as input a variable length array of bits. \\ + & \\ +$\left [ 3 \right ]$ & Give the probability that algorithm mp\_cmp\_mag will have to compare $k$ digits \\ + & of two random digits (of equal magnitude) before a difference is found. \\ + & \\ +$\left [ 1 \right ]$ & Suggest a simple method to speed up the implementation of mp\_cmp\_mag based \\ + & on the observations made in the previous problem. \\ + & +\end{tabular} + +\chapter{Basic Arithmetic} +\section{Introduction} +At this point algorithms for initialization, clearing, zeroing, copying, comparing and setting small constants have been +established. The next logical set of algorithms to develop are addition, subtraction and digit shifting algorithms. These +algorithms make use of the lower level algorithms and are the cruicial building block for the multiplication algorithms. It is very important +that these algorithms are highly optimized. On their own they are simple $O(n)$ algorithms but they can be called from higher level algorithms +which easily places them at $O(n^2)$ or even $O(n^3)$ work levels. + +All of the algorithms within this chapter make use of the logical bit shift operations denoted by $<<$ and $>>$ for left and right +logical shifts respectively. A logical shift is analogous to sliding the decimal point of radix-10 representations. For example, the real +number $0.9345$ is equivalent to $93.45\%$ which is found by sliding the the decimal two places to the right (\textit{multiplying by $\beta^2 = 10^2$}). +Algebraically a binary logical shift is equivalent to a division or multiplication by a power of two. +For example, $a << k = a \cdot 2^k$ while $a >> k = \lfloor a/2^k \rfloor$. + +One significant difference between a logical shift and the way decimals are shifted is that digits below the zero'th position are removed +from the number. For example, consider $1101_2 >> 1$ using decimal notation this would produce $110.1_2$. However, with a logical shift the +result is $110_2$. + +\section{Addition and Subtraction} +In common twos complement fixed precision arithmetic negative numbers are easily represented by subtraction from the modulus. For example, with 32-bit integers +$a - b\mbox{ (mod }2^{32}\mbox{)}$ is the same as $a + (2^{32} - b) \mbox{ (mod }2^{32}\mbox{)}$ since $2^{32} \equiv 0 \mbox{ (mod }2^{32}\mbox{)}$. +As a result subtraction can be performed with a trivial series of logical operations and an addition. + +However, in multiple precision arithmetic negative numbers are not represented in the same way. Instead a sign flag is used to keep track of the +sign of the integer. As a result signed addition and subtraction are actually implemented as conditional usage of lower level addition or +subtraction algorithms with the sign fixed up appropriately. + +The lower level algorithms will add or subtract integers without regard to the sign flag. That is they will add or subtract the magnitude of +the integers respectively. + +\subsection{Low Level Addition} +An unsigned addition of multiple precision integers is performed with the same long-hand algorithm used to add decimal numbers. That is to add the +trailing digits first and propagate the resulting carry upwards. Since this is a lower level algorithm the name will have a ``s\_'' prefix. +Historically that convention stems from the MPI library where ``s\_'' stood for static functions that were hidden from the developer entirely. + +\newpage +\begin{figure}[!here] +\begin{center} +\begin{small} +\begin{tabular}{l} +\hline Algorithm \textbf{s\_mp\_add}. \\ +\textbf{Input}. Two mp\_ints $a$ and $b$ \\ +\textbf{Output}. The unsigned addition $c = \vert a \vert + \vert b \vert$. \\ +\hline \\ +1. if $a.used > b.used$ then \\ +\hspace{+3mm}1.1 $min \leftarrow b.used$ \\ +\hspace{+3mm}1.2 $max \leftarrow a.used$ \\ +\hspace{+3mm}1.3 $x \leftarrow a$ \\ +2. else \\ +\hspace{+3mm}2.1 $min \leftarrow a.used$ \\ +\hspace{+3mm}2.2 $max \leftarrow b.used$ \\ +\hspace{+3mm}2.3 $x \leftarrow b$ \\ +3. If $c.alloc < max + 1$ then grow $c$ to hold at least $max + 1$ digits (\textit{mp\_grow}) \\ +4. $oldused \leftarrow c.used$ \\ +5. $c.used \leftarrow max + 1$ \\ +6. $u \leftarrow 0$ \\ +7. for $n$ from $0$ to $min - 1$ do \\ +\hspace{+3mm}7.1 $c_n \leftarrow a_n + b_n + u$ \\ +\hspace{+3mm}7.2 $u \leftarrow c_n >> lg(\beta)$ \\ +\hspace{+3mm}7.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ +8. if $min \ne max$ then do \\ +\hspace{+3mm}8.1 for $n$ from $min$ to $max - 1$ do \\ +\hspace{+6mm}8.1.1 $c_n \leftarrow x_n + u$ \\ +\hspace{+6mm}8.1.2 $u \leftarrow c_n >> lg(\beta)$ \\ +\hspace{+6mm}8.1.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ +9. $c_{max} \leftarrow u$ \\ +10. if $olduse > max$ then \\ +\hspace{+3mm}10.1 for $n$ from $max + 1$ to $oldused - 1$ do \\ +\hspace{+6mm}10.1.1 $c_n \leftarrow 0$ \\ +11. Clamp excess digits in $c$. (\textit{mp\_clamp}) \\ +12. Return(\textit{MP\_OKAY}) \\ +\hline +\end{tabular} +\end{small} +\end{center} +\caption{Algorithm s\_mp\_add} +\end{figure} + +\textbf{Algorithm s\_mp\_add.} +This algorithm is loosely based on algorithm 14.7 of HAC \cite[pp. 594]{HAC} but has been extended to allow the inputs to have different magnitudes. +Coincidentally the description of algorithm A in Knuth \cite[pp. 266]{TAOCPV2} shares the same deficiency as the algorithm from \cite{HAC}. Even the +MIX pseudo machine code presented by Knuth \cite[pp. 266-267]{TAOCPV2} is incapable of handling inputs which are of different magnitudes. + +The first thing that has to be accomplished is to sort out which of the two inputs is the largest. The addition logic +will simply add all of the smallest input to the largest input and store that first part of the result in the +destination. Then it will apply a simpler addition loop to excess digits of the larger input. + +The first two steps will handle sorting the inputs such that $min$ and $max$ hold the digit counts of the two +inputs. The variable $x$ will be an mp\_int alias for the largest input or the second input $b$ if they have the +same number of digits. After the inputs are sorted the destination $c$ is grown as required to accomodate the sum +of the two inputs. The original \textbf{used} count of $c$ is copied and set to the new used count. + +At this point the first addition loop will go through as many digit positions that both inputs have. The carry +variable $\mu$ is set to zero outside the loop. Inside the loop an ``addition'' step requires three statements to produce +one digit of the summand. First +two digits from $a$ and $b$ are added together along with the carry $\mu$. The carry of this step is extracted and stored +in $\mu$ and finally the digit of the result $c_n$ is truncated within the range $0 \le c_n < \beta$. + +Now all of the digit positions that both inputs have in common have been exhausted. If $min \ne max$ then $x$ is an alias +for one of the inputs that has more digits. A simplified addition loop is then used to essentially copy the remaining digits +and the carry to the destination. + +The final carry is stored in $c_{max}$ and digits above $max$ upto $oldused$ are zeroed which completes the addition. + + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_add.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +We first sort (lines 28 to 36) the inputs based on magnitude and determine the $min$ and $max$ variables. +Note that $x$ is a pointer to an mp\_int assigned to the largest input, in effect it is a local alias. Next we +grow the destination (38 to 42) ensure that it can accomodate the result of the addition. + +Similar to the implementation of mp\_copy this function uses the braced code and local aliases coding style. The three aliases that are on +lines 56, 59 and 62 represent the two inputs and destination variables respectively. These aliases are used to ensure the +compiler does not have to dereference $a$, $b$ or $c$ (respectively) to access the digits of the respective mp\_int. + +The initial carry $u$ will be cleared (line 65), note that $u$ is of type mp\_digit which ensures type +compatibility within the implementation. The initial addition (line 66 to 75) adds digits from +both inputs until the smallest input runs out of digits. Similarly the conditional addition loop +(line 81 to 90) adds the remaining digits from the larger of the two inputs. The addition is finished +with the final carry being stored in $tmpc$ (line 94). Note the ``++'' operator within the same expression. +After line 94, $tmpc$ will point to the $c.used$'th digit of the mp\_int $c$. This is useful +for the next loop (line 97 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. + + +For this algorithm a new variable is required to make the description simpler. Recall from section 1.3.1 that a mp\_digit must be able to represent +the range $0 \le x < 2\beta$ for the algorithms to work correctly. However, it is allowable that a mp\_digit represent a larger range of values. For +this algorithm we will assume that the variable $\gamma$ represents the number of bits available in a +mp\_digit (\textit{this implies $2^{\gamma} > \beta$}). + +For example, the default for LibTomMath is to use a ``unsigned long'' for the mp\_digit ``type'' while $\beta = 2^{28}$. In ISO C an ``unsigned long'' +data type must be able to represent $0 \le x < 2^{32}$ meaning that in this case $\gamma \ge 32$. + +\newpage\begin{figure}[!here] +\begin{center} +\begin{small} +\begin{tabular}{l} +\hline Algorithm \textbf{s\_mp\_sub}. \\ +\textbf{Input}. Two mp\_ints $a$ and $b$ ($\vert a \vert \ge \vert b \vert$) \\ +\textbf{Output}. The unsigned subtraction $c = \vert a \vert - \vert b \vert$. \\ +\hline \\ +1. $min \leftarrow b.used$ \\ +2. $max \leftarrow a.used$ \\ +3. If $c.alloc < max$ then grow $c$ to hold at least $max$ digits. (\textit{mp\_grow}) \\ +4. $oldused \leftarrow c.used$ \\ +5. $c.used \leftarrow max$ \\ +6. $u \leftarrow 0$ \\ +7. for $n$ from $0$ to $min - 1$ do \\ +\hspace{3mm}7.1 $c_n \leftarrow a_n - b_n - u$ \\ +\hspace{3mm}7.2 $u \leftarrow c_n >> (\gamma - 1)$ \\ +\hspace{3mm}7.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ +8. if $min < max$ then do \\ +\hspace{3mm}8.1 for $n$ from $min$ to $max - 1$ do \\ +\hspace{6mm}8.1.1 $c_n \leftarrow a_n - u$ \\ +\hspace{6mm}8.1.2 $u \leftarrow c_n >> (\gamma - 1)$ \\ +\hspace{6mm}8.1.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ +9. if $oldused > max$ then do \\ +\hspace{3mm}9.1 for $n$ from $max$ to $oldused - 1$ do \\ +\hspace{6mm}9.1.1 $c_n \leftarrow 0$ \\ +10. Clamp excess digits of $c$. (\textit{mp\_clamp}). \\ +11. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{small} +\end{center} +\caption{Algorithm s\_mp\_sub} +\end{figure} + +\textbf{Algorithm s\_mp\_sub.} +This algorithm performs the unsigned subtraction of two mp\_int variables under the restriction that the result must be positive. That is when +passing variables $a$ and $b$ the condition that $\vert a \vert \ge \vert b \vert$ must be met for the algorithm to function correctly. This +algorithm is loosely based on algorithm 14.9 \cite[pp. 595]{HAC} and is similar to algorithm S in \cite[pp. 267]{TAOCPV2} as well. As was the case +of the algorithm s\_mp\_add both other references lack discussion concerning various practical details such as when the inputs differ in magnitude. + +The initial sorting of the inputs is trivial in this algorithm since $a$ is guaranteed to have at least the same magnitude of $b$. Steps 1 and 2 +set the $min$ and $max$ variables. Unlike the addition routine there is guaranteed to be no carry which means that the final result can be at +most $max$ digits in length as opposed to $max + 1$. Similar to the addition algorithm the \textbf{used} count of $c$ is copied locally and +set to the maximal count for the operation. + +The subtraction loop that begins on step seven is essentially the same as the addition loop of algorithm s\_mp\_add except single precision +subtraction is used instead. Note the use of the $\gamma$ variable to extract the carry (\textit{also known as the borrow}) within the subtraction +loops. Under the assumption that two's complement single precision arithmetic is used this will successfully extract the desired carry. + +For example, consider subtracting $0101_2$ from $0100_2$ where $\gamma = 4$ and $\beta = 2$. The least significant bit will force a carry upwards to +the third bit which will be set to zero after the borrow. After the very first bit has been subtracted $4 - 1 \equiv 0011_2$ will remain, When the +third bit of $0101_2$ is subtracted from the result it will cause another carry. In this case though the carry will be forced to propagate all the +way to the most significant bit. + +Recall that $\beta < 2^{\gamma}$. This means that if a carry does occur just before the $lg(\beta)$'th bit it will propagate all the way to the most +significant bit. Thus, the high order bits of the mp\_digit that are not part of the actual digit will either be all zero, or all one. All that +is needed is a single zero or one bit for the carry. Therefore a single logical shift right by $\gamma - 1$ positions is sufficient to extract the +carry. This method of carry extraction may seem awkward but the reason for it becomes apparent when the implementation is discussed. + +If $b$ has a smaller magnitude than $a$ then step 9 will force the carry and copy operation to propagate through the larger input $a$ into $c$. Step +10 will ensure that any leading digits of $c$ above the $max$'th position are zeroed. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_sub.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +Like low level addition we ``sort'' the inputs. Except in this case the sorting is hardcoded +(lines 25 and 26). In reality the $min$ and $max$ variables are only aliases and are only +used to make the source code easier to read. Again the pointer alias optimization is used +within this algorithm. The aliases $tmpa$, $tmpb$ and $tmpc$ are initialized +(lines 42, 43 and 44) for $a$, $b$ and $c$ respectively. + +The first subtraction loop (lines 47 through 61) subtract digits from both inputs until the smaller of +the two inputs has been exhausted. As remarked earlier there is an implementation reason for using the ``awkward'' +method of extracting the carry (line 57). The traditional method for extracting the carry would be to shift +by $lg(\beta)$ positions and logically AND the least significant bit. The AND operation is required because all of +the bits above the $\lg(\beta)$'th bit will be set to one after a carry occurs from subtraction. This carry +extraction requires two relatively cheap operations to extract the carry. The other method is to simply shift the +most significant bit to the least significant bit thus extracting the carry with a single cheap operation. This +optimization only works on twos compliment machines which is a safe assumption to make. + +If $a$ has a larger magnitude than $b$ an additional loop (lines 64 through 73) is required to propagate +the carry through $a$ and copy the result to $c$. + +\subsection{High Level Addition} +Now that both lower level addition and subtraction algorithms have been established an effective high level signed addition algorithm can be +established. This high level addition algorithm will be what other algorithms and developers will use to perform addition of mp\_int data +types. + +Recall from section 5.2 that an mp\_int represents an integer with an unsigned mantissa (\textit{the array of digits}) and a \textbf{sign} +flag. A high level addition is actually performed as a series of eight separate cases which can be optimized down to three unique cases. + +\begin{figure}[!here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_add}. \\ +\textbf{Input}. Two mp\_ints $a$ and $b$ \\ +\textbf{Output}. The signed addition $c = a + b$. \\ +\hline \\ +1. if $a.sign = b.sign$ then do \\ +\hspace{3mm}1.1 $c.sign \leftarrow a.sign$ \\ +\hspace{3mm}1.2 $c \leftarrow \vert a \vert + \vert b \vert$ (\textit{s\_mp\_add})\\ +2. else do \\ +\hspace{3mm}2.1 if $\vert a \vert < \vert b \vert$ then do (\textit{mp\_cmp\_mag}) \\ +\hspace{6mm}2.1.1 $c.sign \leftarrow b.sign$ \\ +\hspace{6mm}2.1.2 $c \leftarrow \vert b \vert - \vert a \vert$ (\textit{s\_mp\_sub}) \\ +\hspace{3mm}2.2 else do \\ +\hspace{6mm}2.2.1 $c.sign \leftarrow a.sign$ \\ +\hspace{6mm}2.2.2 $c \leftarrow \vert a \vert - \vert b \vert$ \\ +3. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_add} +\end{figure} + +\textbf{Algorithm mp\_add.} +This algorithm performs the signed addition of two mp\_int variables. There is no reference algorithm to draw upon from +either \cite{TAOCPV2} or \cite{HAC} since they both only provide unsigned operations. The algorithm is fairly +straightforward but restricted since subtraction can only produce positive results. + +\begin{figure}[here] +\begin{small} +\begin{center} +\begin{tabular}{|c|c|c|c|c|} +\hline \textbf{Sign of $a$} & \textbf{Sign of $b$} & \textbf{$\vert a \vert > \vert b \vert $} & \textbf{Unsigned Operation} & \textbf{Result Sign Flag} \\ +\hline $+$ & $+$ & Yes & $c = a + b$ & $a.sign$ \\ +\hline $+$ & $+$ & No & $c = a + b$ & $a.sign$ \\ +\hline $-$ & $-$ & Yes & $c = a + b$ & $a.sign$ \\ +\hline $-$ & $-$ & No & $c = a + b$ & $a.sign$ \\ +\hline &&&&\\ + +\hline $+$ & $-$ & No & $c = b - a$ & $b.sign$ \\ +\hline $-$ & $+$ & No & $c = b - a$ & $b.sign$ \\ + +\hline &&&&\\ + +\hline $+$ & $-$ & Yes & $c = a - b$ & $a.sign$ \\ +\hline $-$ & $+$ & Yes & $c = a - b$ & $a.sign$ \\ + +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Addition Guide Chart} +\label{fig:AddChart} +\end{figure} + +Figure~\ref{fig:AddChart} lists all of the eight possible input combinations and is sorted to show that only three +specific cases need to be handled. The return code of the unsigned operations at step 1.2, 2.1.2 and 2.2.2 are +forwarded to step three to check for errors. This simplifies the description of the algorithm considerably and best +follows how the implementation actually was achieved. + +Also note how the \textbf{sign} is set before the unsigned addition or subtraction is performed. Recall from the descriptions of algorithms +s\_mp\_add and s\_mp\_sub that the mp\_clamp function is used at the end to trim excess digits. The mp\_clamp algorithm will set the \textbf{sign} +to \textbf{MP\_ZPOS} when the \textbf{used} digit count reaches zero. + +For example, consider performing $-a + a$ with algorithm mp\_add. By the description of the algorithm the sign is set to \textbf{MP\_NEG} which would +produce a result of $-0$. However, since the sign is set first then the unsigned addition is performed the subsequent usage of algorithm mp\_clamp +within algorithm s\_mp\_add will force $-0$ to become $0$. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_add.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +The source code follows the algorithm fairly closely. The most notable new source code addition is the usage of the $res$ integer variable which +is used to pass result of the unsigned operations forward. Unlike in the algorithm, the variable $res$ is merely returned as is without +explicitly checking it and returning the constant \textbf{MP\_OKAY}. The observation is this algorithm will succeed or fail only if the lower +level functions do so. Returning their return code is sufficient. + +\subsection{High Level Subtraction} +The high level signed subtraction algorithm is essentially the same as the high level signed addition algorithm. + +\newpage\begin{figure}[!here] +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_sub}. \\ +\textbf{Input}. Two mp\_ints $a$ and $b$ \\ +\textbf{Output}. The signed subtraction $c = a - b$. \\ +\hline \\ +1. if $a.sign \ne b.sign$ then do \\ +\hspace{3mm}1.1 $c.sign \leftarrow a.sign$ \\ +\hspace{3mm}1.2 $c \leftarrow \vert a \vert + \vert b \vert$ (\textit{s\_mp\_add}) \\ +2. else do \\ +\hspace{3mm}2.1 if $\vert a \vert \ge \vert b \vert$ then do (\textit{mp\_cmp\_mag}) \\ +\hspace{6mm}2.1.1 $c.sign \leftarrow a.sign$ \\ +\hspace{6mm}2.1.2 $c \leftarrow \vert a \vert - \vert b \vert$ (\textit{s\_mp\_sub}) \\ +\hspace{3mm}2.2 else do \\ +\hspace{6mm}2.2.1 $c.sign \leftarrow \left \lbrace \begin{array}{ll} + MP\_ZPOS & \mbox{if }a.sign = MP\_NEG \\ + MP\_NEG & \mbox{otherwise} \\ + \end{array} \right .$ \\ +\hspace{6mm}2.2.2 $c \leftarrow \vert b \vert - \vert a \vert$ \\ +3. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\caption{Algorithm mp\_sub} +\end{figure} + +\textbf{Algorithm mp\_sub.} +This algorithm performs the signed subtraction of two inputs. Similar to algorithm mp\_add there is no reference in either \cite{TAOCPV2} or +\cite{HAC}. Also this algorithm is restricted by algorithm s\_mp\_sub. Chart \ref{fig:SubChart} lists the eight possible inputs and +the operations required. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{|c|c|c|c|c|} +\hline \textbf{Sign of $a$} & \textbf{Sign of $b$} & \textbf{$\vert a \vert \ge \vert b \vert $} & \textbf{Unsigned Operation} & \textbf{Result Sign Flag} \\ +\hline $+$ & $-$ & Yes & $c = a + b$ & $a.sign$ \\ +\hline $+$ & $-$ & No & $c = a + b$ & $a.sign$ \\ +\hline $-$ & $+$ & Yes & $c = a + b$ & $a.sign$ \\ +\hline $-$ & $+$ & No & $c = a + b$ & $a.sign$ \\ +\hline &&&& \\ +\hline $+$ & $+$ & Yes & $c = a - b$ & $a.sign$ \\ +\hline $-$ & $-$ & Yes & $c = a - b$ & $a.sign$ \\ +\hline &&&& \\ +\hline $+$ & $+$ & No & $c = b - a$ & $\mbox{opposite of }a.sign$ \\ +\hline $-$ & $-$ & No & $c = b - a$ & $\mbox{opposite of }a.sign$ \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Subtraction Guide Chart} +\label{fig:SubChart} +\end{figure} + +Similar to the case of algorithm mp\_add the \textbf{sign} is set first before the unsigned addition or subtraction. That is to prevent the +algorithm from producing $-a - -a = -0$ as a result. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_sub.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +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 39 the ``not equal to'' \textbf{MP\_LT} expression is used to emulate a +``greater than or equal to'' comparison. + +\section{Bit and Digit Shifting} +It is quite common to think of a multiple precision integer as a polynomial in $x$, that is $y = f(\beta)$ where $f(x) = \sum_{i=0}^{n-1} a_i x^i$. +This notation arises within discussion of Montgomery and Diminished Radix Reduction as well as Karatsuba multiplication and squaring. + +In order to facilitate operations on polynomials in $x$ as above a series of simple ``digit'' algorithms have to be established. That is to shift +the digits left or right as well to shift individual bits of the digits left and right. It is important to note that not all ``shift'' operations +are on radix-$\beta$ digits. + +\subsection{Multiplication by Two} + +In a binary system where the radix is a power of two multiplication by two not only arises often in other algorithms it is a fairly efficient +operation to perform. A single precision logical shift left is sufficient to multiply a single digit by two. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_mul\_2}. \\ +\textbf{Input}. One mp\_int $a$ \\ +\textbf{Output}. $b = 2a$. \\ +\hline \\ +1. If $b.alloc < a.used + 1$ then grow $b$ to hold $a.used + 1$ digits. (\textit{mp\_grow}) \\ +2. $oldused \leftarrow b.used$ \\ +3. $b.used \leftarrow a.used$ \\ +4. $r \leftarrow 0$ \\ +5. for $n$ from 0 to $a.used - 1$ do \\ +\hspace{3mm}5.1 $rr \leftarrow a_n >> (lg(\beta) - 1)$ \\ +\hspace{3mm}5.2 $b_n \leftarrow (a_n << 1) + r \mbox{ (mod }\beta\mbox{)}$ \\ +\hspace{3mm}5.3 $r \leftarrow rr$ \\ +6. If $r \ne 0$ then do \\ +\hspace{3mm}6.1 $b_{n + 1} \leftarrow r$ \\ +\hspace{3mm}6.2 $b.used \leftarrow b.used + 1$ \\ +7. If $b.used < oldused - 1$ then do \\ +\hspace{3mm}7.1 for $n$ from $b.used$ to $oldused - 1$ do \\ +\hspace{6mm}7.1.1 $b_n \leftarrow 0$ \\ +8. $b.sign \leftarrow a.sign$ \\ +9. Return(\textit{MP\_OKAY}).\\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_mul\_2} +\end{figure} + +\textbf{Algorithm mp\_mul\_2.} +This algorithm will quickly multiply a mp\_int by two provided $\beta$ is a power of two. Neither \cite{TAOCPV2} nor \cite{HAC} describe such +an algorithm despite the fact it arises often in other algorithms. The algorithm is setup much like the lower level algorithm s\_mp\_add since +it is for all intents and purposes equivalent to the operation $b = \vert a \vert + \vert a \vert$. + +Step 1 and 2 grow the input as required to accomodate the maximum number of \textbf{used} digits in the result. The initial \textbf{used} count +is set to $a.used$ at step 4. Only if there is a final carry will the \textbf{used} count require adjustment. + +Step 6 is an optimization implementation of the addition loop for this specific case. That is since the two values being added together +are the same there is no need to perform two reads from the digits of $a$. Step 6.1 performs a single precision shift on the current digit $a_n$ to +obtain what will be the carry for the next iteration. Step 6.2 calculates the $n$'th digit of the result as single precision shift of $a_n$ plus +the previous carry. Recall from section 4.1 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 $a$. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_mul\_2.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +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. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_div\_2.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +\section{Polynomial Basis Operations} +Recall from section 4.3 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 +\begin{center} +\begin{figure}[here] +\includegraphics{pics/sliding_window.ps} +\caption{Sliding Window Movement} +\label{pic:sliding_window} +\end{figure} +\end{center} + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_lshd.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +The if statement (line 24) ensures that the $b$ variable is greater than zero since we do not interpret negative +shift counts properly. The \textbf{used} count is incremented by $b$ before the copy loop begins. This elminates +the need for an additional variable in the for loop. The variable $top$ (line 42) is an alias +for the leading digit while $bottom$ (line 45) 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. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_rshd.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +The only noteworthy element of this routine is the lack of a return type since it cannot fail. Like mp\_lshd() we +form a sliding window except we copy in the other direction. After the window (line 60) we then zero +the upper digits of the input to make sure the result is correct. + +\section{Powers of Two} + +Now that algorithms for moving single bits as well as whole digits exist algorithms for moving the ``in between'' distances are required. For +example, to quickly multiply by $2^k$ for any $k$ without using a full multiplier algorithm would prove useful. Instead of performing single +shifts $k$ times to achieve a multiplication by $2^{\pm k}$ a mixture of whole digit shifting and partial digit shifting is employed. + +\subsection{Multiplication by Power of Two} + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_mul\_2d}. \\ +\textbf{Input}. One mp\_int $a$ and an integer $b$ \\ +\textbf{Output}. $c \leftarrow a \cdot 2^b$. \\ +\hline \\ +1. $c \leftarrow a$. (\textit{mp\_copy}) \\ +2. If $c.alloc < c.used + \lfloor b / lg(\beta) \rfloor + 2$ then grow $c$ accordingly. \\ +3. If the reallocation failed return(\textit{MP\_MEM}). \\ +4. If $b \ge lg(\beta)$ then \\ +\hspace{3mm}4.1 $c \leftarrow c \cdot \beta^{\lfloor b / lg(\beta) \rfloor}$ (\textit{mp\_lshd}). \\ +\hspace{3mm}4.2 If step 4.1 failed return(\textit{MP\_MEM}). \\ +5. $d \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\ +6. If $d \ne 0$ then do \\ +\hspace{3mm}6.1 $mask \leftarrow 2^d$ \\ +\hspace{3mm}6.2 $r \leftarrow 0$ \\ +\hspace{3mm}6.3 for $n$ from $0$ to $c.used - 1$ do \\ +\hspace{6mm}6.3.1 $rr \leftarrow c_n >> (lg(\beta) - d) \mbox{ (mod }mask\mbox{)}$ \\ +\hspace{6mm}6.3.2 $c_n \leftarrow (c_n << d) + r \mbox{ (mod }\beta\mbox{)}$ \\ +\hspace{6mm}6.3.3 $r \leftarrow rr$ \\ +\hspace{3mm}6.4 If $r > 0$ then do \\ +\hspace{6mm}6.4.1 $c_{c.used} \leftarrow r$ \\ +\hspace{6mm}6.4.2 $c.used \leftarrow c.used + 1$ \\ +7. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_mul\_2d} +\end{figure} + +\textbf{Algorithm mp\_mul\_2d.} +This algorithm multiplies $a$ by $2^b$ and stores the result in $c$. The algorithm uses algorithm mp\_lshd and a derivative of algorithm mp\_mul\_2 to +quickly compute the product. + +First the algorithm will multiply $a$ by $x^{\lfloor b / lg(\beta) \rfloor}$ which will ensure that the remainder multiplicand is less than +$\beta$. For example, if $b = 37$ and $\beta = 2^{28}$ then this step will multiply by $x$ leaving a multiplication by $2^{37 - 28} = 2^{9}$ +left. + +After the digits have been shifted appropriately at most $lg(\beta) - 1$ shifts are left to perform. Step 5 calculates the number of remaining shifts +required. If it is non-zero a modified shift loop is used to calculate the remaining product. +Essentially the loop is a generic version of algorithm mp\_mul\_2 designed to handle any shift count in the range $1 \le x < lg(\beta)$. The $mask$ +variable is used to extract the upper $d$ bits to form the carry for the next iteration. + +This algorithm is loosely measured as a $O(2n)$ algorithm which means that if the input is $n$-digits that it takes $2n$ ``time'' to +complete. It is possible to optimize this algorithm down to a $O(n)$ algorithm at a cost of making the algorithm slightly harder to follow. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_mul\_2d.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +The shifting is performed in--place which means the first step (line 25) is to copy the input to the +destination. We avoid calling mp\_copy() by making sure the mp\_ints are different. The destination then +has to be grown (line 32) to accomodate the result. + +If the shift count $b$ is larger than $lg(\beta)$ then a call to mp\_lshd() is used to handle all of the multiples +of $lg(\beta)$. Leaving only a remaining shift of $lg(\beta) - 1$ or fewer bits left. Inside the actual shift +loop (lines 46 to 76) we make use of pre--computed values $shift$ and $mask$. These are used to +extract the carry bit(s) to pass into the next iteration of the loop. The $r$ and $rr$ variables form a +chain between consecutive iterations to propagate the carry. + +\subsection{Division by Power of Two} + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_div\_2d}. \\ +\textbf{Input}. One mp\_int $a$ and an integer $b$ \\ +\textbf{Output}. $c \leftarrow \lfloor a / 2^b \rfloor, d \leftarrow a \mbox{ (mod }2^b\mbox{)}$. \\ +\hline \\ +1. If $b \le 0$ then do \\ +\hspace{3mm}1.1 $c \leftarrow a$ (\textit{mp\_copy}) \\ +\hspace{3mm}1.2 $d \leftarrow 0$ (\textit{mp\_zero}) \\ +\hspace{3mm}1.3 Return(\textit{MP\_OKAY}). \\ +2. $c \leftarrow a$ \\ +3. $d \leftarrow a \mbox{ (mod }2^b\mbox{)}$ (\textit{mp\_mod\_2d}) \\ +4. If $b \ge lg(\beta)$ then do \\ +\hspace{3mm}4.1 $c \leftarrow \lfloor c/\beta^{\lfloor b/lg(\beta) \rfloor} \rfloor$ (\textit{mp\_rshd}). \\ +5. $k \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\ +6. If $k \ne 0$ then do \\ +\hspace{3mm}6.1 $mask \leftarrow 2^k$ \\ +\hspace{3mm}6.2 $r \leftarrow 0$ \\ +\hspace{3mm}6.3 for $n$ from $c.used - 1$ to $0$ do \\ +\hspace{6mm}6.3.1 $rr \leftarrow c_n \mbox{ (mod }mask\mbox{)}$ \\ +\hspace{6mm}6.3.2 $c_n \leftarrow (c_n >> k) + (r << (lg(\beta) - k))$ \\ +\hspace{6mm}6.3.3 $r \leftarrow rr$ \\ +7. Clamp excess digits of $c$. (\textit{mp\_clamp}) \\ +8. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_div\_2d} +\end{figure} + +\textbf{Algorithm mp\_div\_2d.} +This algorithm will divide an input $a$ by $2^b$ and produce the quotient and remainder. The algorithm is designed much like algorithm +mp\_mul\_2d by first using whole digit shifts then single precision shifts. This algorithm will also produce the remainder of the division +by using algorithm mp\_mod\_2d. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_div\_2d.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +The implementation of algorithm mp\_div\_2d is slightly different than the algorithm specifies. The remainder $d$ may be optionally +ignored by passing \textbf{NULL} as the pointer to the mp\_int variable. The temporary mp\_int variable $t$ is used to hold the +result of the remainder operation until the end. This allows $d$ and $a$ to represent the same mp\_int without modifying $a$ before +the quotient is obtained. + +The remainder of the source code is essentially the same as the source code for mp\_mul\_2d. The only significant difference is +the direction of the shifts. + +\subsection{Remainder of Division by Power of Two} + +The last algorithm in the series of polynomial basis power of two algorithms is calculating the remainder of division by $2^b$. This +algorithm benefits from the fact that in twos complement arithmetic $a \mbox{ (mod }2^b\mbox{)}$ is the same as $a$ AND $2^b - 1$. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_mod\_2d}. \\ +\textbf{Input}. One mp\_int $a$ and an integer $b$ \\ +\textbf{Output}. $c \leftarrow a \mbox{ (mod }2^b\mbox{)}$. \\ +\hline \\ +1. If $b \le 0$ then do \\ +\hspace{3mm}1.1 $c \leftarrow 0$ (\textit{mp\_zero}) \\ +\hspace{3mm}1.2 Return(\textit{MP\_OKAY}). \\ +2. If $b > a.used \cdot lg(\beta)$ then do \\ +\hspace{3mm}2.1 $c \leftarrow a$ (\textit{mp\_copy}) \\ +\hspace{3mm}2.2 Return the result of step 2.1. \\ +3. $c \leftarrow a$ \\ +4. If step 3 failed return(\textit{MP\_MEM}). \\ +5. for $n$ from $\lceil b / lg(\beta) \rceil$ to $c.used$ do \\ +\hspace{3mm}5.1 $c_n \leftarrow 0$ \\ +6. $k \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\ +7. $c_{\lfloor b / lg(\beta) \rfloor} \leftarrow c_{\lfloor b / lg(\beta) \rfloor} \mbox{ (mod }2^{k}\mbox{)}$. \\ +8. Clamp excess digits of $c$. (\textit{mp\_clamp}) \\ +9. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_mod\_2d} +\end{figure} + +\textbf{Algorithm mp\_mod\_2d.} +This algorithm will quickly calculate the value of $a \mbox{ (mod }2^b\mbox{)}$. First if $b$ is less than or equal to zero the +result is set to zero. If $b$ is greater than the number of bits in $a$ then it simply copies $a$ to $c$ and returns. Otherwise, $a$ +is copied to $b$, leading digits are removed and the remaining leading digit is trimed to the exact bit count. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_mod\_2d.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +We first avoid cases of $b \le 0$ by simply mp\_zero()'ing the destination in such cases. Next if $2^b$ is larger +than the input we just mp\_copy() the input and return right away. After this point we know we must actually +perform some work to produce the remainder. + +Recalling that reducing modulo $2^k$ and a binary ``and'' with $2^k - 1$ are numerically equivalent we can quickly reduce +the number. First we zero any digits above the last digit in $2^b$ (line 42). Next we reduce the +leading digit of both (line 46) and then mp\_clamp(). + +\section*{Exercises} +\begin{tabular}{cl} +$\left [ 3 \right ] $ & Devise an algorithm that performs $a \cdot 2^b$ for generic values of $b$ \\ + & in $O(n)$ time. \\ + &\\ +$\left [ 3 \right ] $ & Devise an efficient algorithm to multiply by small low hamming \\ + & weight values such as $3$, $5$ and $9$. Extend it to handle all values \\ + & upto $64$ with a hamming weight less than three. \\ + &\\ +$\left [ 2 \right ] $ & Modify the preceding algorithm to handle values of the form \\ + & $2^k - 1$ as well. \\ + &\\ +$\left [ 3 \right ] $ & Using only algorithms mp\_mul\_2, mp\_div\_2 and mp\_add create an \\ + & algorithm to multiply two integers in roughly $O(2n^2)$ time for \\ + & any $n$-bit input. Note that the time of addition is ignored in the \\ + & calculation. \\ + & \\ +$\left [ 5 \right ] $ & Improve the previous algorithm to have a working time of at most \\ + & $O \left (2^{(k-1)}n + \left ({2n^2 \over k} \right ) \right )$ for an appropriate choice of $k$. Again ignore \\ + & the cost of addition. \\ + & \\ +$\left [ 2 \right ] $ & Devise a chart to find optimal values of $k$ for the previous problem \\ + & for $n = 64 \ldots 1024$ in steps of $64$. \\ + & \\ +$\left [ 2 \right ] $ & Using only algorithms mp\_abs and mp\_sub devise another method for \\ + & calculating the result of a signed comparison. \\ + & +\end{tabular} + +\chapter{Multiplication and Squaring} +\section{The Multipliers} +For most number theoretic problems including certain public key cryptographic algorithms, the ``multipliers'' form the most important subset of +algorithms of any multiple precision integer package. The set of multiplier algorithms include integer multiplication, squaring and modular reduction +where in each of the algorithms single precision multiplication is the dominant operation performed. This chapter will discuss integer multiplication +and squaring, leaving modular reductions for the subsequent chapter. + +The importance of the multiplier algorithms is for the most part driven by the fact that certain popular public key algorithms are based on modular +exponentiation, that is computing $d \equiv a^b \mbox{ (mod }c\mbox{)}$ for some arbitrary choice of $a$, $b$, $c$ and $d$. During a modular +exponentiation the majority\footnote{Roughly speaking a modular exponentiation will spend about 40\% of the time performing modular reductions, +35\% of the time performing squaring and 25\% of the time performing multiplications.} of the processor time is spent performing single precision +multiplications. + +For centuries general purpose multiplication has required a lengthly $O(n^2)$ process, whereby each digit of one multiplicand has to be multiplied +against every digit of the other multiplicand. Traditional long-hand multiplication is based on this process; while the techniques can differ the +overall algorithm used is essentially the same. Only ``recently'' have faster algorithms been studied. First Karatsuba multiplication was discovered in +1962. This algorithm can multiply two numbers with considerably fewer single precision multiplications when compared to the long-hand approach. +This technique led to the discovery of polynomial basis algorithms (\textit{good reference?}) and subquently Fourier Transform based solutions. + +\section{Multiplication} +\subsection{The Baseline Multiplication} +\label{sec:basemult} +\index{baseline multiplication} +Computing the product of two integers in software can be achieved using a trivial adaptation of the standard $O(n^2)$ long-hand multiplication +algorithm that school children are taught. The algorithm is considered an $O(n^2)$ algorithm since for two $n$-digit inputs $n^2$ single precision +multiplications are required. More specifically for a $m$ and $n$ digit input $m \cdot n$ single precision multiplications are required. To +simplify most discussions, it will be assumed that the inputs have comparable number of digits. + +The ``baseline multiplication'' algorithm is designed to act as the ``catch-all'' algorithm, only to be used when the faster algorithms cannot be +used. This algorithm does not use any particularly interesting optimizations and should ideally be avoided if possible. One important +facet of this algorithm, is that it has been modified to only produce a certain amount of output digits as resolution. The importance of this +modification will become evident during the discussion of Barrett modular reduction. Recall that for a $n$ and $m$ digit input the product +will be at most $n + m$ digits. Therefore, this algorithm can be reduced to a full multiplier by having it produce $n + m$ digits of the product. + +Recall from sub-section 4.2.2 the definition of $\gamma$ as the number of bits in the type \textbf{mp\_digit}. We shall now extend the variable set to +include $\alpha$ which shall represent the number of bits in the type \textbf{mp\_word}. This implies that $2^{\alpha} > 2 \cdot \beta^2$. The +constant $\delta = 2^{\alpha - 2lg(\beta)}$ will represent the maximal weight of any column in a product (\textit{see sub-section 5.2.2 for more information}). + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{s\_mp\_mul\_digs}. \\ +\textbf{Input}. mp\_int $a$, mp\_int $b$ and an integer $digs$ \\ +\textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert \mbox{ (mod }\beta^{digs}\mbox{)}$. \\ +\hline \\ +1. If min$(a.used, b.used) < \delta$ then do \\ +\hspace{3mm}1.1 Calculate $c = \vert a \vert \cdot \vert b \vert$ by the Comba method (\textit{see algorithm~\ref{fig:COMBAMULT}}). \\ +\hspace{3mm}1.2 Return the result of step 1.1 \\ +\\ +Allocate and initialize a temporary mp\_int. \\ +2. Init $t$ to be of size $digs$ \\ +3. If step 2 failed return(\textit{MP\_MEM}). \\ +4. $t.used \leftarrow digs$ \\ +\\ +Compute the product. \\ +5. for $ix$ from $0$ to $a.used - 1$ do \\ +\hspace{3mm}5.1 $u \leftarrow 0$ \\ +\hspace{3mm}5.2 $pb \leftarrow \mbox{min}(b.used, digs - ix)$ \\ +\hspace{3mm}5.3 If $pb < 1$ then goto step 6. \\ +\hspace{3mm}5.4 for $iy$ from $0$ to $pb - 1$ do \\ +\hspace{6mm}5.4.1 $\hat r \leftarrow t_{iy + ix} + a_{ix} \cdot b_{iy} + u$ \\ +\hspace{6mm}5.4.2 $t_{iy + ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ +\hspace{6mm}5.4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ +\hspace{3mm}5.5 if $ix + pb < digs$ then do \\ +\hspace{6mm}5.5.1 $t_{ix + pb} \leftarrow u$ \\ +6. Clamp excess digits of $t$. \\ +7. Swap $c$ with $t$ \\ +8. Clear $t$ \\ +9. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm s\_mp\_mul\_digs} +\end{figure} + +\textbf{Algorithm s\_mp\_mul\_digs.} +This algorithm computes the unsigned product of two inputs $a$ and $b$, limited to an output precision of $digs$ digits. While it may seem +a bit awkward to modify the function from its simple $O(n^2)$ description, the usefulness of partial multipliers will arise in a subsequent +algorithm. The algorithm is loosely based on algorithm 14.12 from \cite[pp. 595]{HAC} and is similar to Algorithm M of Knuth \cite[pp. 268]{TAOCPV2}. +Algorithm s\_mp\_mul\_digs differs from these cited references since it can produce a variable output precision regardless of the precision of the +inputs. + +The first thing this algorithm checks for is whether a Comba multiplier can be used instead. If the minimum digit count of either +input is less than $\delta$, then the Comba method may be used instead. After the Comba method is ruled out, the baseline algorithm begins. A +temporary mp\_int variable $t$ is used to hold the intermediate result of the product. This allows the algorithm to be used to +compute products when either $a = c$ or $b = c$ without overwriting the inputs. + +All of step 5 is the infamous $O(n^2)$ multiplication loop slightly modified to only produce upto $digs$ digits of output. The $pb$ variable +is given the count of digits to read from $b$ inside the nested loop. If $pb \le 1$ then no more output digits can be produced and the algorithm +will exit the loop. The best way to think of the loops are as a series of $pb \times 1$ multiplications. That is, in each pass of the +innermost loop $a_{ix}$ is multiplied against $b$ and the result is added (\textit{with an appropriate shift}) to $t$. + +For example, consider multiplying $576$ by $241$. That is equivalent to computing $10^0(1)(576) + 10^1(4)(576) + 10^2(2)(576)$ which is best +visualized in the following table. + +\begin{figure}[here] +\begin{center} +\begin{tabular}{|c|c|c|c|c|c|l|} +\hline && & 5 & 7 & 6 & \\ +\hline $\times$&& & 2 & 4 & 1 & \\ +\hline &&&&&&\\ + && & 5 & 7 & 6 & $10^0(1)(576)$ \\ + &2 & 3 & 6 & 1 & 6 & $10^1(4)(576) + 10^0(1)(576)$ \\ + 1 & 3 & 8 & 8 & 1 & 6 & $10^2(2)(576) + 10^1(4)(576) + 10^0(1)(576)$ \\ +\hline +\end{tabular} +\end{center} +\caption{Long-Hand Multiplication Diagram} +\end{figure} + +Each row of the product is added to the result after being shifted to the left (\textit{multiplied by a power of the radix}) by the appropriate +count. That is in pass $ix$ of the inner loop the product is added starting at the $ix$'th digit of the reult. + +Step 5.4.1 introduces the hat symbol (\textit{e.g. $\hat r$}) which represents a double precision variable. The multiplication on that step +is assumed to be a double wide output single precision multiplication. That is, two single precision variables are multiplied to produce a +double precision result. The step is somewhat optimized from a long-hand multiplication algorithm because the carry from the addition in step +5.4.1 is propagated through the nested loop. If the carry was not propagated immediately it would overflow the single precision digit +$t_{ix+iy}$ and the result would be lost. + +At step 5.5 the nested loop is finished and any carry that was left over should be forwarded. The carry does not have to be added to the $ix+pb$'th +digit since that digit is assumed to be zero at this point. However, if $ix + pb \ge digs$ the carry is not set as it would make the result +exceed the precision requested. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_mul\_digs.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +First we determine (line 31) if the Comba method can be used first since it's faster. The conditions for +sing the Comba routine are that min$(a.used, b.used) < \delta$ and the number of digits of output is less than +\textbf{MP\_WARRAY}. This new constant is used to control the stack usage in the Comba routines. By default it is +set to $\delta$ but can be reduced when memory is at a premium. + +If we cannot use the Comba method we proceed to setup the baseline routine. We allocate the the destination mp\_int +$t$ (line 37) to the exact size of the output to avoid further re--allocations. At this point we now +begin the $O(n^2)$ loop. + +This implementation of multiplication has the caveat that it can be trimmed to only produce a variable number of +digits as output. In each iteration of the outer loop the $pb$ variable is set (line 49) to the maximum +number of inner loop iterations. + +Inside the inner loop we calculate $\hat r$ as the mp\_word product of the two mp\_digits and the addition of the +carry from the previous iteration. A particularly important observation is that most modern optimizing +C compilers (GCC for instance) can recognize that a $N \times N \rightarrow 2N$ multiplication is all that +is required for the product. In x86 terms for example, this means using the MUL instruction. + +Each digit of the product is stored in turn (line 69) and the carry propagated (line 72) to the +next iteration. + +\subsection{Faster Multiplication by the ``Comba'' Method} + +One of the huge drawbacks of the ``baseline'' algorithms is that at the $O(n^2)$ level the carry must be +computed and propagated upwards. This makes the nested loop very sequential and hard to unroll and implement +in parallel. The ``Comba'' \cite{COMBA} method is named after little known (\textit{in cryptographic venues}) Paul G. +Comba who described a method of implementing fast multipliers that do not require nested carry fixup operations. As an +interesting aside it seems that Paul Barrett describes a similar technique in his 1986 paper \cite{BARRETT} written +five years before. + +At the heart of the Comba technique is once again the long-hand algorithm. Except in this case a slight +twist is placed on how the columns of the result are produced. In the standard long-hand algorithm rows of products +are produced then added together to form the final result. In the baseline algorithm the columns are added together +after each iteration to get the result instantaneously. + +In the Comba algorithm the columns of the result are produced entirely independently of each other. That is at +the $O(n^2)$ level a simple multiplication and addition step is performed. The carries of the columns are propagated +after the nested loop to reduce the amount of work requiored. Succintly the first step of the algorithm is to compute +the product vector $\vec x$ as follows. + +\begin{equation} +\vec x_n = \sum_{i+j = n} a_ib_j, \forall n \in \lbrace 0, 1, 2, \ldots, i + j \rbrace +\end{equation} + +Where $\vec x_n$ is the $n'th$ column of the output vector. Consider the following example which computes the vector $\vec x$ for the multiplication +of $576$ and $241$. + +\newpage\begin{figure}[here] +\begin{small} +\begin{center} +\begin{tabular}{|c|c|c|c|c|c|} + \hline & & 5 & 7 & 6 & First Input\\ + \hline $\times$ & & 2 & 4 & 1 & Second Input\\ +\hline & & $1 \cdot 5 = 5$ & $1 \cdot 7 = 7$ & $1 \cdot 6 = 6$ & First pass \\ + & $4 \cdot 5 = 20$ & $4 \cdot 7+5=33$ & $4 \cdot 6+7=31$ & 6 & Second pass \\ + $2 \cdot 5 = 10$ & $2 \cdot 7 + 20 = 34$ & $2 \cdot 6+33=45$ & 31 & 6 & Third pass \\ +\hline 10 & 34 & 45 & 31 & 6 & Final Result \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Comba Multiplication Diagram} +\end{figure} + +At this point the vector $x = \left < 10, 34, 45, 31, 6 \right >$ is the result of the first step of the Comba multipler. +Now the columns must be fixed by propagating the carry upwards. The resultant vector will have one extra dimension over the input vector which is +congruent to adding a leading zero digit. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{Comba Fixup}. \\ +\textbf{Input}. Vector $\vec x$ of dimension $k$ \\ +\textbf{Output}. Vector $\vec x$ such that the carries have been propagated. \\ +\hline \\ +1. for $n$ from $0$ to $k - 1$ do \\ +\hspace{3mm}1.1 $\vec x_{n+1} \leftarrow \vec x_{n+1} + \lfloor \vec x_{n}/\beta \rfloor$ \\ +\hspace{3mm}1.2 $\vec x_{n} \leftarrow \vec x_{n} \mbox{ (mod }\beta\mbox{)}$ \\ +2. Return($\vec x$). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm Comba Fixup} +\end{figure} + +With that algorithm and $k = 5$ and $\beta = 10$ the following vector is produced $\vec x= \left < 1, 3, 8, 8, 1, 6 \right >$. In this case +$241 \cdot 576$ is in fact $138816$ and the procedure succeeded. If the algorithm is correct and as will be demonstrated shortly more +efficient than the baseline algorithm why not simply always use this algorithm? + +\subsubsection{Column Weight.} +At the nested $O(n^2)$ level the Comba method adds the product of two single precision variables to each column of the output +independently. A serious obstacle is if the carry is lost, due to lack of precision before the algorithm has a chance to fix +the carries. For example, in the multiplication of two three-digit numbers the third column of output will be the sum of +three single precision multiplications. If the precision of the accumulator for the output digits is less then $3 \cdot (\beta - 1)^2$ then +an overflow can occur and the carry information will be lost. For any $m$ and $n$ digit inputs the maximum weight of any column is +min$(m, n)$ which is fairly obvious. + +The maximum number of terms in any column of a product is known as the ``column weight'' and strictly governs when the algorithm can be used. Recall +from earlier that a double precision type has $\alpha$ bits of resolution and a single precision digit has $lg(\beta)$ bits of precision. Given these +two quantities we must not violate the following + +\begin{equation} +k \cdot \left (\beta - 1 \right )^2 < 2^{\alpha} +\end{equation} + +Which reduces to + +\begin{equation} +k \cdot \left ( \beta^2 - 2\beta + 1 \right ) < 2^{\alpha} +\end{equation} + +Let $\rho = lg(\beta)$ represent the number of bits in a single precision digit. By further re-arrangement of the equation the final solution is +found. + +\begin{equation} +k < {{2^{\alpha}} \over {\left (2^{2\rho} - 2^{\rho + 1} + 1 \right )}} +\end{equation} + +The defaults for LibTomMath are $\beta = 2^{28}$ and $\alpha = 2^{64}$ which means that $k$ is bounded by $k < 257$. In this configuration +the smaller input may not have more than $256$ digits if the Comba method is to be used. This is quite satisfactory for most applications since +$256$ digits would allow for numbers in the range of $0 \le x < 2^{7168}$ which, is much larger than most public key cryptographic algorithms require. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{fast\_s\_mp\_mul\_digs}. \\ +\textbf{Input}. mp\_int $a$, mp\_int $b$ and an integer $digs$ \\ +\textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert \mbox{ (mod }\beta^{digs}\mbox{)}$. \\ +\hline \\ +Place an array of \textbf{MP\_WARRAY} single precision digits named $W$ on the stack. \\ +1. If $c.alloc < digs$ then grow $c$ to $digs$ digits. (\textit{mp\_grow}) \\ +2. If step 1 failed return(\textit{MP\_MEM}).\\ +\\ +3. $pa \leftarrow \mbox{MIN}(digs, a.used + b.used)$ \\ +\\ +4. $\_ \hat W \leftarrow 0$ \\ +5. for $ix$ from 0 to $pa - 1$ do \\ +\hspace{3mm}5.1 $ty \leftarrow \mbox{MIN}(b.used - 1, ix)$ \\ +\hspace{3mm}5.2 $tx \leftarrow ix - ty$ \\ +\hspace{3mm}5.3 $iy \leftarrow \mbox{MIN}(a.used - tx, ty + 1)$ \\ +\hspace{3mm}5.4 for $iz$ from 0 to $iy - 1$ do \\ +\hspace{6mm}5.4.1 $\_ \hat W \leftarrow \_ \hat W + a_{tx+iy}b_{ty-iy}$ \\ +\hspace{3mm}5.5 $W_{ix} \leftarrow \_ \hat W (\mbox{mod }\beta)$\\ +\hspace{3mm}5.6 $\_ \hat W \leftarrow \lfloor \_ \hat W / \beta \rfloor$ \\ +\\ +6. $oldused \leftarrow c.used$ \\ +7. $c.used \leftarrow digs$ \\ +8. for $ix$ from $0$ to $pa$ do \\ +\hspace{3mm}8.1 $c_{ix} \leftarrow W_{ix}$ \\ +9. for $ix$ from $pa + 1$ to $oldused - 1$ do \\ +\hspace{3mm}9.1 $c_{ix} \leftarrow 0$ \\ +\\ +10. Clamp $c$. \\ +11. Return MP\_OKAY. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm fast\_s\_mp\_mul\_digs} +\label{fig:COMBAMULT} +\end{figure} + +\textbf{Algorithm fast\_s\_mp\_mul\_digs.} +This algorithm performs the unsigned multiplication of $a$ and $b$ using the Comba method limited to $digs$ digits of precision. + +The outer loop of this algorithm is more complicated than that of the baseline multiplier. This is because on the inside of the +loop we want to produce one column per pass. This allows the accumulator $\_ \hat W$ to be placed in CPU registers and +reduce the memory bandwidth to two \textbf{mp\_digit} reads per iteration. + +The $ty$ variable is set to the minimum count of $ix$ or the number of digits in $b$. That way if $a$ has more digits than +$b$ this will be limited to $b.used - 1$. The $tx$ variable is set to the to the distance past $b.used$ the variable +$ix$ is. This is used for the immediately subsequent statement where we find $iy$. + +The variable $iy$ is the minimum digits we can read from either $a$ or $b$ before running out. Computing one column at a time +means we have to scan one integer upwards and the other downwards. $a$ starts at $tx$ and $b$ starts at $ty$. In each +pass we are producing the $ix$'th output column and we note that $tx + ty = ix$. As we move $tx$ upwards we have to +move $ty$ downards so the equality remains valid. The $iy$ variable is the number of iterations until +$tx \ge a.used$ or $ty < 0$ occurs. + +After every inner pass we store the lower half of the accumulator into $W_{ix}$ and then propagate the carry of the accumulator +into the next round by dividing $\_ \hat W$ by $\beta$. + +To measure the benefits of the Comba method over the baseline method consider the number of operations that are required. If the +cost in terms of time of a multiply and addition is $p$ and the cost of a carry propagation is $q$ then a baseline multiplication would require +$O \left ((p + q)n^2 \right )$ time to multiply two $n$-digit numbers. The Comba method requires only $O(pn^2 + qn)$ time, however in practice, +the speed increase is actually much more. With $O(n)$ space the algorithm can be reduced to $O(pn + qn)$ time by implementing the $n$ multiply +and addition operations in the nested loop in parallel. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_fast\_s\_mp\_mul\_digs.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +As per the pseudo--code we first calculate $pa$ (line 48) as the number of digits to output. Next we begin the outer loop +to produce the individual columns of the product. We use the two aliases $tmpx$ and $tmpy$ (lines 62, 63) to point +inside the two multiplicands quickly. + +The inner loop (lines 71 to 74) of this implementation is where the tradeoff come into play. Originally this comba +implementation was ``row--major'' which means it adds to each of the columns in each pass. After the outer loop it would then fix +the carries. This was very fast except it had an annoying drawback. You had to read a mp\_word and two mp\_digits and write +one mp\_word per iteration. On processors such as the Athlon XP and P4 this did not matter much since the cache bandwidth +is very high and it can keep the ALU fed with data. It did, however, matter on older and embedded cpus where cache is often +slower and also often doesn't exist. This new algorithm only performs two reads per iteration under the assumption that the +compiler has aliased $\_ \hat W$ to a CPU register. + +After the inner loop we store the current accumulator in $W$ and shift $\_ \hat W$ (lines 77, 80) to forward it as +a carry for the next pass. After the outer loop we use the final carry (line 77) as the last digit of the product. + +\subsection{Polynomial Basis Multiplication} +To break the $O(n^2)$ barrier in multiplication requires a completely different look at integer multiplication. In the following algorithms +the use of polynomial basis representation for two integers $a$ and $b$ as $f(x) = \sum_{i=0}^{n} a_i x^i$ and +$g(x) = \sum_{i=0}^{n} b_i x^i$ respectively, is required. In this system both $f(x)$ and $g(x)$ have $n + 1$ terms and are of the $n$'th degree. + +The product $a \cdot b \equiv f(x)g(x)$ is the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$. The coefficients $w_i$ will +directly yield the desired product when $\beta$ is substituted for $x$. The direct solution to solve for the $2n + 1$ coefficients +requires $O(n^2)$ time and would in practice be slower than the Comba technique. + +However, numerical analysis theory indicates that only $2n + 1$ distinct points in $W(x)$ are required to determine the values of the $2n + 1$ unknown +coefficients. This means by finding $\zeta_y = W(y)$ for $2n + 1$ small values of $y$ the coefficients of $W(x)$ can be found with +Gaussian elimination. This technique is also occasionally refered to as the \textit{interpolation technique} (\textit{references please...}) since in +effect an interpolation based on $2n + 1$ points will yield a polynomial equivalent to $W(x)$. + +The coefficients of the polynomial $W(x)$ are unknown which makes finding $W(y)$ for any value of $y$ impossible. However, since +$W(x) = f(x)g(x)$ the equivalent $\zeta_y = f(y) g(y)$ can be used in its place. The benefit of this technique stems from the +fact that $f(y)$ and $g(y)$ are much smaller than either $a$ or $b$ respectively. As a result finding the $2n + 1$ relations required +by multiplying $f(y)g(y)$ involves multiplying integers that are much smaller than either of the inputs. + +When picking points to gather relations there are always three obvious points to choose, $y = 0, 1$ and $ \infty$. The $\zeta_0$ term +is simply the product $W(0) = w_0 = a_0 \cdot b_0$. The $\zeta_1$ term is the product +$W(1) = \left (\sum_{i = 0}^{n} a_i \right ) \left (\sum_{i = 0}^{n} b_i \right )$. The third point $\zeta_{\infty}$ is less obvious but rather +simple to explain. The $2n + 1$'th coefficient of $W(x)$ is numerically equivalent to the most significant column in an integer multiplication. +The point at $\infty$ is used symbolically to represent the most significant column, that is $W(\infty) = w_{2n} = a_nb_n$. Note that the +points at $y = 0$ and $\infty$ yield the coefficients $w_0$ and $w_{2n}$ directly. + +If more points are required they should be of small values and powers of two such as $2^q$ and the related \textit{mirror points} +$\left (2^q \right )^{2n} \cdot \zeta_{2^{-q}}$ for small values of $q$. The term ``mirror point'' stems from the fact that +$\left (2^q \right )^{2n} \cdot \zeta_{2^{-q}}$ can be calculated in the exact opposite fashion as $\zeta_{2^q}$. For +example, when $n = 2$ and $q = 1$ then following two equations are equivalent to the point $\zeta_{2}$ and its mirror. + +\begin{eqnarray} +\zeta_{2} = f(2)g(2) = (4a_2 + 2a_1 + a_0)(4b_2 + 2b_1 + b_0) \nonumber \\ +16 \cdot \zeta_{1 \over 2} = 4f({1\over 2}) \cdot 4g({1 \over 2}) = (a_2 + 2a_1 + 4a_0)(b_2 + 2b_1 + 4b_0) +\end{eqnarray} + +Using such points will allow the values of $f(y)$ and $g(y)$ to be independently calculated using only left shifts. For example, when $n = 2$ the +polynomial $f(2^q)$ is equal to $2^q((2^qa_2) + a_1) + a_0$. This technique of polynomial representation is known as Horner's method. + +As a general rule of the algorithm when the inputs are split into $n$ parts each there are $2n - 1$ multiplications. Each multiplication is of +multiplicands that have $n$ times fewer digits than the inputs. The asymptotic running time of this algorithm is +$O \left ( k^{lg_n(2n - 1)} \right )$ for $k$ digit inputs (\textit{assuming they have the same number of digits}). Figure~\ref{fig:exponent} +summarizes the exponents for various values of $n$. + +\begin{figure} +\begin{center} +\begin{tabular}{|c|c|c|} +\hline \textbf{Split into $n$ Parts} & \textbf{Exponent} & \textbf{Notes}\\ +\hline $2$ & $1.584962501$ & This is Karatsuba Multiplication. \\ +\hline $3$ & $1.464973520$ & This is Toom-Cook Multiplication. \\ +\hline $4$ & $1.403677461$ &\\ +\hline $5$ & $1.365212389$ &\\ +\hline $10$ & $1.278753601$ &\\ +\hline $100$ & $1.149426538$ &\\ +\hline $1000$ & $1.100270931$ &\\ +\hline $10000$ & $1.075252070$ &\\ +\hline +\end{tabular} +\end{center} +\caption{Asymptotic Running Time of Polynomial Basis Multiplication} +\label{fig:exponent} +\end{figure} + +At first it may seem like a good idea to choose $n = 1000$ since the exponent is approximately $1.1$. However, the overhead +of solving for the 2001 terms of $W(x)$ will certainly consume any savings the algorithm could offer for all but exceedingly large +numbers. + +\subsubsection{Cutoff Point} +The polynomial basis multiplication algorithms all require fewer single precision multiplications than a straight Comba approach. However, +the algorithms incur an overhead (\textit{at the $O(n)$ work level}) since they require a system of equations to be solved. This makes the +polynomial basis approach more costly to use with small inputs. + +Let $m$ represent the number of digits in the multiplicands (\textit{assume both multiplicands have the same number of digits}). There exists a +point $y$ such that when $m < y$ the polynomial basis algorithms are more costly than Comba, when $m = y$ they are roughly the same cost and +when $m > y$ the Comba methods are slower than the polynomial basis algorithms. + +The exact location of $y$ depends on several key architectural elements of the computer platform in question. + +\begin{enumerate} +\item The ratio of clock cycles for single precision multiplication versus other simpler operations such as addition, shifting, etc. For example +on the AMD Athlon the ratio is roughly $17 : 1$ while on the Intel P4 it is $29 : 1$. The higher the ratio in favour of multiplication the lower +the cutoff point $y$ will be. + +\item The complexity of the linear system of equations (\textit{for the coefficients of $W(x)$}) is. Generally speaking as the number of splits +grows the complexity grows substantially. Ideally solving the system will only involve addition, subtraction and shifting of integers. This +directly reflects on the ratio previous mentioned. + +\item To a lesser extent memory bandwidth and function call overheads. Provided the values are in the processor cache this is less of an +influence over the cutoff point. + +\end{enumerate} + +A clean cutoff point separation occurs when a point $y$ is found such that all of the cutoff point conditions are met. For example, if the point +is too low then there will be values of $m$ such that $m > y$ and the Comba method is still faster. Finding the cutoff points is fairly simple when +a high resolution timer is available. + +\subsection{Karatsuba Multiplication} +Karatsuba \cite{KARA} multiplication when originally proposed in 1962 was among the first set of algorithms to break the $O(n^2)$ barrier for +general purpose multiplication. Given two polynomial basis representations $f(x) = ax + b$ and $g(x) = cx + d$, Karatsuba proved with +light algebra \cite{KARAP} that the following polynomial is equivalent to multiplication of the two integers the polynomials represent. + +\begin{equation} +f(x) \cdot g(x) = acx^2 + ((a + b)(c + d) - (ac + bd))x + bd +\end{equation} + +Using the observation that $ac$ and $bd$ could be re-used only three half sized multiplications would be required to produce the product. Applying +this algorithm recursively, the work factor becomes $O(n^{lg(3)})$ which is substantially better than the work factor $O(n^2)$ of the Comba technique. It turns +out what Karatsuba did not know or at least did not publish was that this is simply polynomial basis multiplication with the points +$\zeta_0$, $\zeta_{\infty}$ and $\zeta_{1}$. Consider the resultant system of equations. + +\begin{center} +\begin{tabular}{rcrcrcrc} +$\zeta_{0}$ & $=$ & & & & & $w_0$ \\ +$\zeta_{1}$ & $=$ & $w_2$ & $+$ & $w_1$ & $+$ & $w_0$ \\ +$\zeta_{\infty}$ & $=$ & $w_2$ & & & & \\ +\end{tabular} +\end{center} + +By adding the first and last equation to the equation in the middle the term $w_1$ can be isolated and all three coefficients solved for. The simplicity +of this system of equations has made Karatsuba fairly popular. In fact the cutoff point is often fairly low\footnote{With LibTomMath 0.18 it is 70 and 109 digits for the Intel P4 and AMD Athlon respectively.} +making it an ideal algorithm to speed up certain public key cryptosystems such as RSA and Diffie-Hellman. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_karatsuba\_mul}. \\ +\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ +\textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert$ \\ +\hline \\ +1. Init the following mp\_int variables: $x0$, $x1$, $y0$, $y1$, $t1$, $x0y0$, $x1y1$.\\ +2. If step 2 failed then return(\textit{MP\_MEM}). \\ +\\ +Split the input. e.g. $a = x1 \cdot \beta^B + x0$ \\ +3. $B \leftarrow \mbox{min}(a.used, b.used)/2$ \\ +4. $x0 \leftarrow a \mbox{ (mod }\beta^B\mbox{)}$ (\textit{mp\_mod\_2d}) \\ +5. $y0 \leftarrow b \mbox{ (mod }\beta^B\mbox{)}$ \\ +6. $x1 \leftarrow \lfloor a / \beta^B \rfloor$ (\textit{mp\_rshd}) \\ +7. $y1 \leftarrow \lfloor b / \beta^B \rfloor$ \\ +\\ +Calculate the three products. \\ +8. $x0y0 \leftarrow x0 \cdot y0$ (\textit{mp\_mul}) \\ +9. $x1y1 \leftarrow x1 \cdot y1$ \\ +10. $t1 \leftarrow x1 + x0$ (\textit{mp\_add}) \\ +11. $x0 \leftarrow y1 + y0$ \\ +12. $t1 \leftarrow t1 \cdot x0$ \\ +\\ +Calculate the middle term. \\ +13. $x0 \leftarrow x0y0 + x1y1$ \\ +14. $t1 \leftarrow t1 - x0$ (\textit{s\_mp\_sub}) \\ +\\ +Calculate the final product. \\ +15. $t1 \leftarrow t1 \cdot \beta^B$ (\textit{mp\_lshd}) \\ +16. $x1y1 \leftarrow x1y1 \cdot \beta^{2B}$ \\ +17. $t1 \leftarrow x0y0 + t1$ \\ +18. $c \leftarrow t1 + x1y1$ \\ +19. Clear all of the temporary variables. \\ +20. Return(\textit{MP\_OKAY}).\\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_karatsuba\_mul} +\end{figure} + +\textbf{Algorithm mp\_karatsuba\_mul.} +This algorithm computes the unsigned product of two inputs using the Karatsuba multiplication algorithm. It is loosely based on the description +from Knuth \cite[pp. 294-295]{TAOCPV2}. + +\index{radix point} +In order to split the two inputs into their respective halves, a suitable \textit{radix point} must be chosen. The radix point chosen must +be used for both of the inputs meaning that it must be smaller than the smallest input. Step 3 chooses the radix point $B$ as half of the +smallest input \textbf{used} count. After the radix point is chosen the inputs are split into lower and upper halves. Step 4 and 5 +compute the lower halves. Step 6 and 7 computer the upper halves. + +After the halves have been computed the three intermediate half-size products must be computed. Step 8 and 9 compute the trivial products +$x0 \cdot y0$ and $x1 \cdot y1$. The mp\_int $x0$ is used as a temporary variable after $x1 + x0$ has been computed. By using $x0$ instead +of an additional temporary variable, the algorithm can avoid an addition memory allocation operation. + +The remaining steps 13 through 18 compute the Karatsuba polynomial through a variety of digit shifting and addition operations. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_karatsuba\_mul.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +The new coding element in this routine, not seen in previous routines, is the usage of goto statements. The conventional +wisdom is that goto statements should be avoided. This is generally true, however when every single function call can fail, it makes sense +to handle error recovery with a single piece of code. Lines 62 to 76 handle initializing all of the temporary variables +required. Note how each of the if statements goes to a different label in case of failure. This allows the routine to correctly free only +the temporaries that have been successfully allocated so far. + +The temporary variables are all initialized using the mp\_init\_size routine since they are expected to be large. This saves the +additional reallocation that would have been necessary. Also $x0$, $x1$, $y0$ and $y1$ have to be able to hold at least their respective +number of digits for the next section of code. + +The first algebraic portion of the algorithm is to split the two inputs into their halves. However, instead of using mp\_mod\_2d and mp\_rshd +to extract the halves, the respective code has been placed inline within the body of the function. To initialize the halves, the \textbf{used} and +\textbf{sign} members are copied first. The first for loop on line 96 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 102 and 107 calculate the upper halves $x1$ and +$y1$ respectively. + +By inlining the calculation of the halves, the Karatsuba multiplier has a slightly lower overhead and can be used for smaller magnitude inputs. + +When line 151 is reached, the algorithm has completed succesfully. The ``error status'' variable $err$ is set to \textbf{MP\_OKAY} so that +the same code that handles errors can be used to clear the temporary variables and return. + +\subsection{Toom-Cook $3$-Way Multiplication} +Toom-Cook $3$-Way \cite{TOOM} multiplication is essentially the polynomial basis algorithm for $n = 2$ except that the points are +chosen such that $\zeta$ is easy to compute and the resulting system of equations easy to reduce. Here, the points $\zeta_{0}$, +$16 \cdot \zeta_{1 \over 2}$, $\zeta_1$, $\zeta_2$ and $\zeta_{\infty}$ make up the five required points to solve for the coefficients +of the $W(x)$. + +With the five relations that Toom-Cook specifies, the following system of equations is formed. + +\begin{center} +\begin{tabular}{rcrcrcrcrcr} +$\zeta_0$ & $=$ & $0w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $1w_0$ \\ +$16 \cdot \zeta_{1 \over 2}$ & $=$ & $1w_4$ & $+$ & $2w_3$ & $+$ & $4w_2$ & $+$ & $8w_1$ & $+$ & $16w_0$ \\ +$\zeta_1$ & $=$ & $1w_4$ & $+$ & $1w_3$ & $+$ & $1w_2$ & $+$ & $1w_1$ & $+$ & $1w_0$ \\ +$\zeta_2$ & $=$ & $16w_4$ & $+$ & $8w_3$ & $+$ & $4w_2$ & $+$ & $2w_1$ & $+$ & $1w_0$ \\ +$\zeta_{\infty}$ & $=$ & $1w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $0w_0$ \\ +\end{tabular} +\end{center} + +A trivial solution to this matrix requires $12$ subtractions, two multiplications by a small power of two, two divisions by a small power +of two, two divisions by three and one multiplication by three. All of these $19$ sub-operations require less than quadratic time, meaning that +the algorithm can be faster than a baseline multiplication. However, the greater complexity of this algorithm places the cutoff point +(\textbf{TOOM\_MUL\_CUTOFF}) where Toom-Cook becomes more efficient much higher than the Karatsuba cutoff point. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_toom\_mul}. \\ +\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ +\textbf{Output}. $c \leftarrow a \cdot b $ \\ +\hline \\ +Split $a$ and $b$ into three pieces. E.g. $a = a_2 \beta^{2k} + a_1 \beta^{k} + a_0$ \\ +1. $k \leftarrow \lfloor \mbox{min}(a.used, b.used) / 3 \rfloor$ \\ +2. $a_0 \leftarrow a \mbox{ (mod }\beta^{k}\mbox{)}$ \\ +3. $a_1 \leftarrow \lfloor a / \beta^k \rfloor$, $a_1 \leftarrow a_1 \mbox{ (mod }\beta^{k}\mbox{)}$ \\ +4. $a_2 \leftarrow \lfloor a / \beta^{2k} \rfloor$, $a_2 \leftarrow a_2 \mbox{ (mod }\beta^{k}\mbox{)}$ \\ +5. $b_0 \leftarrow a \mbox{ (mod }\beta^{k}\mbox{)}$ \\ +6. $b_1 \leftarrow \lfloor a / \beta^k \rfloor$, $b_1 \leftarrow b_1 \mbox{ (mod }\beta^{k}\mbox{)}$ \\ +7. $b_2 \leftarrow \lfloor a / \beta^{2k} \rfloor$, $b_2 \leftarrow b_2 \mbox{ (mod }\beta^{k}\mbox{)}$ \\ +\\ +Find the five equations for $w_0, w_1, ..., w_4$. \\ +8. $w_0 \leftarrow a_0 \cdot b_0$ \\ +9. $w_4 \leftarrow a_2 \cdot b_2$ \\ +10. $tmp_1 \leftarrow 2 \cdot a_0$, $tmp_1 \leftarrow a_1 + tmp_1$, $tmp_1 \leftarrow 2 \cdot tmp_1$, $tmp_1 \leftarrow tmp_1 + a_2$ \\ +11. $tmp_2 \leftarrow 2 \cdot b_0$, $tmp_2 \leftarrow b_1 + tmp_2$, $tmp_2 \leftarrow 2 \cdot tmp_2$, $tmp_2 \leftarrow tmp_2 + b_2$ \\ +12. $w_1 \leftarrow tmp_1 \cdot tmp_2$ \\ +13. $tmp_1 \leftarrow 2 \cdot a_2$, $tmp_1 \leftarrow a_1 + tmp_1$, $tmp_1 \leftarrow 2 \cdot tmp_1$, $tmp_1 \leftarrow tmp_1 + a_0$ \\ +14. $tmp_2 \leftarrow 2 \cdot b_2$, $tmp_2 \leftarrow b_1 + tmp_2$, $tmp_2 \leftarrow 2 \cdot tmp_2$, $tmp_2 \leftarrow tmp_2 + b_0$ \\ +15. $w_3 \leftarrow tmp_1 \cdot tmp_2$ \\ +16. $tmp_1 \leftarrow a_0 + a_1$, $tmp_1 \leftarrow tmp_1 + a_2$, $tmp_2 \leftarrow b_0 + b_1$, $tmp_2 \leftarrow tmp_2 + b_2$ \\ +17. $w_2 \leftarrow tmp_1 \cdot tmp_2$ \\ +\\ +Continued on the next page.\\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_toom\_mul} +\end{figure} + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_toom\_mul} (continued). \\ +\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ +\textbf{Output}. $c \leftarrow a \cdot b $ \\ +\hline \\ +Now solve the system of equations. \\ +18. $w_1 \leftarrow w_4 - w_1$, $w_3 \leftarrow w_3 - w_0$ \\ +19. $w_1 \leftarrow \lfloor w_1 / 2 \rfloor$, $w_3 \leftarrow \lfloor w_3 / 2 \rfloor$ \\ +20. $w_2 \leftarrow w_2 - w_0$, $w_2 \leftarrow w_2 - w_4$ \\ +21. $w_1 \leftarrow w_1 - w_2$, $w_3 \leftarrow w_3 - w_2$ \\ +22. $tmp_1 \leftarrow 8 \cdot w_0$, $w_1 \leftarrow w_1 - tmp_1$, $tmp_1 \leftarrow 8 \cdot w_4$, $w_3 \leftarrow w_3 - tmp_1$ \\ +23. $w_2 \leftarrow 3 \cdot w_2$, $w_2 \leftarrow w_2 - w_1$, $w_2 \leftarrow w_2 - w_3$ \\ +24. $w_1 \leftarrow w_1 - w_2$, $w_3 \leftarrow w_3 - w_2$ \\ +25. $w_1 \leftarrow \lfloor w_1 / 3 \rfloor, w_3 \leftarrow \lfloor w_3 / 3 \rfloor$ \\ +\\ +Now substitute $\beta^k$ for $x$ by shifting $w_0, w_1, ..., w_4$. \\ +26. for $n$ from $1$ to $4$ do \\ +\hspace{3mm}26.1 $w_n \leftarrow w_n \cdot \beta^{nk}$ \\ +27. $c \leftarrow w_0 + w_1$, $c \leftarrow c + w_2$, $c \leftarrow c + w_3$, $c \leftarrow c + w_4$ \\ +28. Return(\textit{MP\_OKAY}) \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_toom\_mul (continued)} +\end{figure} + +\textbf{Algorithm mp\_toom\_mul.} +This algorithm computes the product of two mp\_int variables $a$ and $b$ using the Toom-Cook approach. Compared to the Karatsuba multiplication, this +algorithm has a lower asymptotic running time of approximately $O(n^{1.464})$ but at an obvious cost in overhead. In this +description, several statements have been compounded to save space. The intention is that the statements are executed from left to right across +any given step. + +The two inputs $a$ and $b$ are first split into three $k$-digit integers $a_0, a_1, a_2$ and $b_0, b_1, b_2$ respectively. From these smaller +integers the coefficients of the polynomial basis representations $f(x)$ and $g(x)$ are known and can be used to find the relations required. + +The first two relations $w_0$ and $w_4$ are the points $\zeta_{0}$ and $\zeta_{\infty}$ respectively. The relation $w_1, w_2$ and $w_3$ correspond +to the points $16 \cdot \zeta_{1 \over 2}, \zeta_{2}$ and $\zeta_{1}$ respectively. These are found using logical shifts to independently find +$f(y)$ and $g(y)$ which significantly speeds up the algorithm. + +After the five relations $w_0, w_1, \ldots, w_4$ have been computed, the system they represent must be solved in order for the unknown coefficients +$w_1, w_2$ and $w_3$ to be isolated. The steps 18 through 25 perform the system reduction required as previously described. Each step of +the reduction represents the comparable matrix operation that would be performed had this been performed by pencil. For example, step 18 indicates +that row $1$ must be subtracted from row $4$ and simultaneously row $0$ subtracted from row $3$. + +Once the coeffients have been isolated, the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$ is known. By substituting $\beta^{k}$ for $x$, the integer +result $a \cdot b$ is produced. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_toom\_mul.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +The first obvious thing to note is that this algorithm is complicated. The complexity is worth it if you are multiplying very +large numbers. For example, a 10,000 digit multiplication takes approximaly 99,282,205 fewer single precision multiplications with +Toom--Cook than a Comba or baseline approach (this is a savings of more than 99$\%$). For most ``crypto'' sized numbers this +algorithm is not practical as Karatsuba has a much lower cutoff point. + +First we split $a$ and $b$ into three roughly equal portions. This has been accomplished (lines 41 to 70) with +combinations of mp\_rshd() and mp\_mod\_2d() function calls. At this point $a = a2 \cdot \beta^2 + a1 \cdot \beta + a0$ and similiarly +for $b$. + +Next we compute the five points $w0, w1, w2, w3$ and $w4$. Recall that $w0$ and $w4$ can be computed directly from the portions so +we get those out of the way first (lines 73 and 78). Next we compute $w1, w2$ and $w3$ using Horners method. + +After this point we solve for the actual values of $w1, w2$ and $w3$ by reducing the $5 \times 5$ system which is relatively +straight forward. + +\subsection{Signed Multiplication} +Now that algorithms to handle multiplications of every useful dimensions have been developed, a rather simple finishing touch is required. So far all +of the multiplication algorithms have been unsigned multiplications which leaves only a signed multiplication algorithm to be established. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_mul}. \\ +\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ +\textbf{Output}. $c \leftarrow a \cdot b$ \\ +\hline \\ +1. If $a.sign = b.sign$ then \\ +\hspace{3mm}1.1 $sign = MP\_ZPOS$ \\ +2. else \\ +\hspace{3mm}2.1 $sign = MP\_ZNEG$ \\ +3. If min$(a.used, b.used) \ge TOOM\_MUL\_CUTOFF$ then \\ +\hspace{3mm}3.1 $c \leftarrow a \cdot b$ using algorithm mp\_toom\_mul \\ +4. else if min$(a.used, b.used) \ge KARATSUBA\_MUL\_CUTOFF$ then \\ +\hspace{3mm}4.1 $c \leftarrow a \cdot b$ using algorithm mp\_karatsuba\_mul \\ +5. else \\ +\hspace{3mm}5.1 $digs \leftarrow a.used + b.used + 1$ \\ +\hspace{3mm}5.2 If $digs < MP\_ARRAY$ and min$(a.used, b.used) \le \delta$ then \\ +\hspace{6mm}5.2.1 $c \leftarrow a \cdot b \mbox{ (mod }\beta^{digs}\mbox{)}$ using algorithm fast\_s\_mp\_mul\_digs. \\ +\hspace{3mm}5.3 else \\ +\hspace{6mm}5.3.1 $c \leftarrow a \cdot b \mbox{ (mod }\beta^{digs}\mbox{)}$ using algorithm s\_mp\_mul\_digs. \\ +6. $c.sign \leftarrow sign$ \\ +7. Return the result of the unsigned multiplication performed. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_mul} +\end{figure} + +\textbf{Algorithm mp\_mul.} +This algorithm performs the signed multiplication of two inputs. It will make use of any of the three unsigned multiplication algorithms +available when the input is of appropriate size. The \textbf{sign} of the result is not set until the end of the algorithm since algorithm +s\_mp\_mul\_digs will clear it. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_mul.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +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 48 computes $\delta$ using the fact that $1 << k$ is equal to $2^k$. + +\section{Squaring} +\label{sec:basesquare} + +Squaring is a special case of multiplication where both multiplicands are equal. At first it may seem like there is no significant optimization +available but in fact there is. Consider the multiplication of $576$ against $241$. In total there will be nine single precision multiplications +performed which are $1\cdot 6$, $1 \cdot 7$, $1 \cdot 5$, $4 \cdot 6$, $4 \cdot 7$, $4 \cdot 5$, $2 \cdot 6$, $2 \cdot 7$ and $2 \cdot 5$. Now consider +the multiplication of $123$ against $123$. The nine products are $3 \cdot 3$, $3 \cdot 2$, $3 \cdot 1$, $2 \cdot 3$, $2 \cdot 2$, $2 \cdot 1$, +$1 \cdot 3$, $1 \cdot 2$ and $1 \cdot 1$. On closer inspection some of the products are equivalent. For example, $3 \cdot 2 = 2 \cdot 3$ +and $3 \cdot 1 = 1 \cdot 3$. + +For any $n$-digit input, there are ${{\left (n^2 + n \right)}\over 2}$ possible unique single precision multiplications required compared to the $n^2$ +required for multiplication. The following diagram gives an example of the operations required. + +\begin{figure}[here] +\begin{center} +\begin{tabular}{ccccc|c} +&&1&2&3&\\ +$\times$ &&1&2&3&\\ +\hline && $3 \cdot 1$ & $3 \cdot 2$ & $3 \cdot 3$ & Row 0\\ + & $2 \cdot 1$ & $2 \cdot 2$ & $2 \cdot 3$ && Row 1 \\ + $1 \cdot 1$ & $1 \cdot 2$ & $1 \cdot 3$ &&& Row 2 \\ +\end{tabular} +\end{center} +\caption{Squaring Optimization Diagram} +\end{figure} + +Starting from zero and numbering the columns from right to left a very simple pattern becomes obvious. For the purposes of this discussion let $x$ +represent the number being squared. The first observation is that in row $k$ the $2k$'th column of the product has a $\left (x_k \right)^2$ term in it. + +The second observation is that every column $j$ in row $k$ where $j \ne 2k$ is part of a double product. Every non-square term of a column will +appear twice hence the name ``double product''. Every odd column is made up entirely of double products. In fact every column is made up of double +products and at most one square (\textit{see the exercise section}). + +The third and final observation is that for row $k$ the first unique non-square term, that is, one that hasn't already appeared in an earlier row, +occurs at column $2k + 1$. For example, on row $1$ of the previous squaring, column one is part of the double product with column one from row zero. +Column two of row one is a square and column three is the first unique column. + +\subsection{The Baseline Squaring Algorithm} +The baseline squaring algorithm is meant to be a catch-all squaring algorithm. It will handle any of the input sizes that the faster routines +will not handle. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{s\_mp\_sqr}. \\ +\textbf{Input}. mp\_int $a$ \\ +\textbf{Output}. $b \leftarrow a^2$ \\ +\hline \\ +1. Init a temporary mp\_int of at least $2 \cdot a.used +1$ digits. (\textit{mp\_init\_size}) \\ +2. If step 1 failed return(\textit{MP\_MEM}) \\ +3. $t.used \leftarrow 2 \cdot a.used + 1$ \\ +4. For $ix$ from 0 to $a.used - 1$ do \\ +\hspace{3mm}Calculate the square. \\ +\hspace{3mm}4.1 $\hat r \leftarrow t_{2ix} + \left (a_{ix} \right )^2$ \\ +\hspace{3mm}4.2 $t_{2ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ +\hspace{3mm}Calculate the double products after the square. \\ +\hspace{3mm}4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ +\hspace{3mm}4.4 For $iy$ from $ix + 1$ to $a.used - 1$ do \\ +\hspace{6mm}4.4.1 $\hat r \leftarrow 2 \cdot a_{ix}a_{iy} + t_{ix + iy} + u$ \\ +\hspace{6mm}4.4.2 $t_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ +\hspace{6mm}4.4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ +\hspace{3mm}Set the last carry. \\ +\hspace{3mm}4.5 While $u > 0$ do \\ +\hspace{6mm}4.5.1 $iy \leftarrow iy + 1$ \\ +\hspace{6mm}4.5.2 $\hat r \leftarrow t_{ix + iy} + u$ \\ +\hspace{6mm}4.5.3 $t_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ +\hspace{6mm}4.5.4 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ +5. Clamp excess digits of $t$. (\textit{mp\_clamp}) \\ +6. Exchange $b$ and $t$. \\ +7. Clear $t$ (\textit{mp\_clear}) \\ +8. Return(\textit{MP\_OKAY}) \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm s\_mp\_sqr} +\end{figure} + +\textbf{Algorithm s\_mp\_sqr.} +This algorithm computes the square of an input using the three observations on squaring. It is based fairly faithfully on algorithm 14.16 of HAC +\cite[pp.596-597]{HAC}. Similar to algorithm s\_mp\_mul\_digs, a temporary mp\_int is allocated to hold the result of the squaring. This allows the +destination mp\_int to be the same as the source mp\_int. + +The outer loop of this algorithm begins on step 4. It is best to think of the outer loop as walking down the rows of the partial results, while +the inner loop computes the columns of the partial result. Step 4.1 and 4.2 compute the square term for each row, and step 4.3 and 4.4 propagate +the carry and compute the double products. + +The requirement that a mp\_word be able to represent the range $0 \le x < 2 \beta^2$ arises from this +very algorithm. The product $a_{ix}a_{iy}$ will lie in the range $0 \le x \le \beta^2 - 2\beta + 1$ which is obviously less than $\beta^2$ meaning that +when it is multiplied by two, it can be properly represented by a mp\_word. + +Similar to algorithm s\_mp\_mul\_digs, after every pass of the inner loop, the destination is correctly set to the sum of all of the partial +results calculated so far. This involves expensive carry propagation which will be eliminated in the next algorithm. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_sqr.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +Inside the outer loop (line 34) the square term is calculated on line 37. The carry (line 44) has been +extracted from the mp\_word accumulator using a right shift. Aliases for $a_{ix}$ and $t_{ix+iy}$ are initialized +(lines 47 and 50) to simplify the inner loop. The doubling is performed using two +additions (line 59) since it is usually faster than shifting, if not at least as fast. + +The important observation is that the inner loop does not begin at $iy = 0$ like for multiplication. As such the inner loops +get progressively shorter as the algorithm proceeds. This is what leads to the savings compared to using a multiplication to +square a number. + +\subsection{Faster Squaring by the ``Comba'' Method} +A major drawback to the baseline method is the requirement for single precision shifting inside the $O(n^2)$ nested loop. Squaring has an additional +drawback that it must double the product inside the inner loop as well. As for multiplication, the Comba technique can be used to eliminate these +performance hazards. + +The first obvious solution is to make an array of mp\_words which will hold all of the columns. This will indeed eliminate all of the carry +propagation operations from the inner loop. However, the inner product must still be doubled $O(n^2)$ times. The solution stems from the simple fact +that $2a + 2b + 2c = 2(a + b + c)$. That is the sum of all of the double products is equal to double the sum of all the products. For example, +$ab + ba + ac + ca = 2ab + 2ac = 2(ab + ac)$. + +However, we cannot simply double all of the columns, since the squares appear only once per row. The most practical solution is to have two +mp\_word arrays. One array will hold the squares and the other array will hold the double products. With both arrays the doubling and +carry propagation can be moved to a $O(n)$ work level outside the $O(n^2)$ level. In this case, we have an even simpler solution in mind. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{fast\_s\_mp\_sqr}. \\ +\textbf{Input}. mp\_int $a$ \\ +\textbf{Output}. $b \leftarrow a^2$ \\ +\hline \\ +Place an array of \textbf{MP\_WARRAY} mp\_digits named $W$ on the stack. \\ +1. If $b.alloc < 2a.used + 1$ then grow $b$ to $2a.used + 1$ digits. (\textit{mp\_grow}). \\ +2. If step 1 failed return(\textit{MP\_MEM}). \\ +\\ +3. $pa \leftarrow 2 \cdot a.used$ \\ +4. $\hat W1 \leftarrow 0$ \\ +5. for $ix$ from $0$ to $pa - 1$ do \\ +\hspace{3mm}5.1 $\_ \hat W \leftarrow 0$ \\ +\hspace{3mm}5.2 $ty \leftarrow \mbox{MIN}(a.used - 1, ix)$ \\ +\hspace{3mm}5.3 $tx \leftarrow ix - ty$ \\ +\hspace{3mm}5.4 $iy \leftarrow \mbox{MIN}(a.used - tx, ty + 1)$ \\ +\hspace{3mm}5.5 $iy \leftarrow \mbox{MIN}(iy, \lfloor \left (ty - tx + 1 \right )/2 \rfloor)$ \\ +\hspace{3mm}5.6 for $iz$ from $0$ to $iz - 1$ do \\ +\hspace{6mm}5.6.1 $\_ \hat W \leftarrow \_ \hat W + a_{tx + iz}a_{ty - iz}$ \\ +\hspace{3mm}5.7 $\_ \hat W \leftarrow 2 \cdot \_ \hat W + \hat W1$ \\ +\hspace{3mm}5.8 if $ix$ is even then \\ +\hspace{6mm}5.8.1 $\_ \hat W \leftarrow \_ \hat W + \left ( a_{\lfloor ix/2 \rfloor}\right )^2$ \\ +\hspace{3mm}5.9 $W_{ix} \leftarrow \_ \hat W (\mbox{mod }\beta)$ \\ +\hspace{3mm}5.10 $\hat W1 \leftarrow \lfloor \_ \hat W / \beta \rfloor$ \\ +\\ +6. $oldused \leftarrow b.used$ \\ +7. $b.used \leftarrow 2 \cdot a.used$ \\ +8. for $ix$ from $0$ to $pa - 1$ do \\ +\hspace{3mm}8.1 $b_{ix} \leftarrow W_{ix}$ \\ +9. for $ix$ from $pa$ to $oldused - 1$ do \\ +\hspace{3mm}9.1 $b_{ix} \leftarrow 0$ \\ +10. Clamp excess digits from $b$. (\textit{mp\_clamp}) \\ +11. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm fast\_s\_mp\_sqr} +\end{figure} + +\textbf{Algorithm fast\_s\_mp\_sqr.} +This algorithm computes the square of an input using the Comba technique. It is designed to be a replacement for algorithm +s\_mp\_sqr when the number of input digits is less than \textbf{MP\_WARRAY} and less than $\delta \over 2$. +This algorithm is very similar to the Comba multiplier except with a few key differences we shall make note of. + +First, we have an accumulator and carry variables $\_ \hat W$ and $\hat W1$ respectively. This is because the inner loop +products are to be doubled. If we had added the previous carry in we would be doubling too much. Next we perform an +addition MIN condition on $iy$ (step 5.5) to prevent overlapping digits. For example, $a_3 \cdot a_5$ is equal +$a_5 \cdot a_3$. Whereas in the multiplication case we would have $5 < a.used$ and $3 \ge 0$ is maintained since we double the sum +of the products just outside the inner loop we have to avoid doing this. This is also a good thing since we perform +fewer multiplications and the routine ends up being faster. + +Finally the last difference is the addition of the ``square'' term outside the inner loop (step 5.8). We add in the square +only to even outputs and it is the square of the term at the $\lfloor ix / 2 \rfloor$ position. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_fast\_s\_mp\_sqr.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +This implementation is essentially a copy of Comba multiplication with the appropriate changes added to make it faster for +the special case of squaring. + +\subsection{Polynomial Basis Squaring} +The same algorithm that performs optimal polynomial basis multiplication can be used to perform polynomial basis squaring. The minor exception +is that $\zeta_y = f(y)g(y)$ is actually equivalent to $\zeta_y = f(y)^2$ since $f(y) = g(y)$. Instead of performing $2n + 1$ +multiplications to find the $\zeta$ relations, squaring operations are performed instead. + +\subsection{Karatsuba Squaring} +Let $f(x) = ax + b$ represent the polynomial basis representation of a number to square. +Let $h(x) = \left ( f(x) \right )^2$ represent the square of the polynomial. The Karatsuba equation can be modified to square a +number with the following equation. + +\begin{equation} +h(x) = a^2x^2 + \left ((a + b)^2 - (a^2 + b^2) \right )x + b^2 +\end{equation} + +Upon closer inspection this equation only requires the calculation of three half-sized squares: $a^2$, $b^2$ and $(a + b)^2$. As in +Karatsuba multiplication, this algorithm can be applied recursively on the input and will achieve an asymptotic running time of +$O \left ( n^{lg(3)} \right )$. + +If the asymptotic times of Karatsuba squaring and multiplication are the same, why not simply use the multiplication algorithm +instead? The answer to this arises from the cutoff point for squaring. As in multiplication there exists a cutoff point, at which the +time required for a Comba based squaring and a Karatsuba based squaring meet. Due to the overhead inherent in the Karatsuba method, the cutoff +point is fairly high. For example, on an AMD Athlon XP processor with $\beta = 2^{28}$, the cutoff point is around 127 digits. + +Consider squaring a 200 digit number with this technique. It will be split into two 100 digit halves which are subsequently squared. +The 100 digit halves will not be squared using Karatsuba, but instead using the faster Comba based squaring algorithm. If Karatsuba multiplication +were used instead, the 100 digit numbers would be squared with a slower Comba based multiplication. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_karatsuba\_sqr}. \\ +\textbf{Input}. mp\_int $a$ \\ +\textbf{Output}. $b \leftarrow a^2$ \\ +\hline \\ +1. Initialize the following temporary mp\_ints: $x0$, $x1$, $t1$, $t2$, $x0x0$ and $x1x1$. \\ +2. If any of the initializations on step 1 failed return(\textit{MP\_MEM}). \\ +\\ +Split the input. e.g. $a = x1\beta^B + x0$ \\ +3. $B \leftarrow \lfloor a.used / 2 \rfloor$ \\ +4. $x0 \leftarrow a \mbox{ (mod }\beta^B\mbox{)}$ (\textit{mp\_mod\_2d}) \\ +5. $x1 \leftarrow \lfloor a / \beta^B \rfloor$ (\textit{mp\_lshd}) \\ +\\ +Calculate the three squares. \\ +6. $x0x0 \leftarrow x0^2$ (\textit{mp\_sqr}) \\ +7. $x1x1 \leftarrow x1^2$ \\ +8. $t1 \leftarrow x1 + x0$ (\textit{s\_mp\_add}) \\ +9. $t1 \leftarrow t1^2$ \\ +\\ +Compute the middle term. \\ +10. $t2 \leftarrow x0x0 + x1x1$ (\textit{s\_mp\_add}) \\ +11. $t1 \leftarrow t1 - t2$ \\ +\\ +Compute final product. \\ +12. $t1 \leftarrow t1\beta^B$ (\textit{mp\_lshd}) \\ +13. $x1x1 \leftarrow x1x1\beta^{2B}$ \\ +14. $t1 \leftarrow t1 + x0x0$ \\ +15. $b \leftarrow t1 + x1x1$ \\ +16. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_karatsuba\_sqr} +\end{figure} + +\textbf{Algorithm mp\_karatsuba\_sqr.} +This algorithm computes the square of an input $a$ using the Karatsuba technique. This algorithm is very similar to the Karatsuba based +multiplication algorithm with the exception that the three half-size multiplications have been replaced with three half-size squarings. + +The radix point for squaring is simply placed exactly in the middle of the digits when the input has an odd number of digits, otherwise it is +placed just below the middle. Step 3, 4 and 5 compute the two halves required using $B$ +as the radix point. The first two squares in steps 6 and 7 are rather straightforward while the last square is of a more compact form. + +By expanding $\left (x1 + x0 \right )^2$, the $x1^2$ and $x0^2$ terms in the middle disappear, that is $(x0 - x1)^2 - (x1^2 + x0^2) = 2 \cdot x0 \cdot x1$. +Now if $5n$ single precision additions and a squaring of $n$-digits is faster than multiplying two $n$-digit numbers and doubling then +this method is faster. Assuming no further recursions occur, the difference can be estimated with the following inequality. + +Let $p$ represent the cost of a single precision addition and $q$ the cost of a single precision multiplication both in terms of time\footnote{Or +machine clock cycles.}. + +\begin{equation} +5pn +{{q(n^2 + n)} \over 2} \le pn + qn^2 +\end{equation} + +For example, on an AMD Athlon XP processor $p = {1 \over 3}$ and $q = 6$. This implies that the following inequality should hold. +\begin{center} +\begin{tabular}{rcl} +${5n \over 3} + 3n^2 + 3n$ & $<$ & ${n \over 3} + 6n^2$ \\ +${5 \over 3} + 3n + 3$ & $<$ & ${1 \over 3} + 6n$ \\ +${13 \over 9}$ & $<$ & $n$ \\ +\end{tabular} +\end{center} + +This results in a cutoff point around $n = 2$. As a consequence it is actually faster to compute the middle term the ``long way'' on processors +where multiplication is substantially slower\footnote{On the Athlon there is a 1:17 ratio between clock cycles for addition and multiplication. On +the Intel P4 processor this ratio is 1:29 making this method even more beneficial. The only common exception is the ARMv4 processor which has a +ratio of 1:7. } than simpler operations such as addition. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_karatsuba\_sqr.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +This implementation is largely based on the implementation of algorithm mp\_karatsuba\_mul. It uses the same inline style to copy and +shift the input into the two halves. The loop from line 54 to line 70 has been modified since only one input exists. The \textbf{used} +count of both $x0$ and $x1$ is fixed up and $x0$ is clamped before the calculations begin. At this point $x1$ and $x0$ are valid equivalents +to the respective halves as if mp\_rshd and mp\_mod\_2d had been used. + +By inlining the copy and shift operations the cutoff point for Karatsuba multiplication can be lowered. On the Athlon the cutoff point +is exactly at the point where Comba squaring can no longer be used (\textit{128 digits}). On slower processors such as the Intel P4 +it is actually below the Comba limit (\textit{at 110 digits}). + +This routine uses the same error trap coding style as mp\_karatsuba\_sqr. As the temporary variables are initialized errors are +redirected to the error trap higher up. If the algorithm completes without error the error code is set to \textbf{MP\_OKAY} and +mp\_clears are executed normally. + +\subsection{Toom-Cook Squaring} +The Toom-Cook squaring algorithm mp\_toom\_sqr is heavily based on the algorithm mp\_toom\_mul with the exception that squarings are used +instead of multiplication to find the five relations. The reader is encouraged to read the description of the latter algorithm and try to +derive their own Toom-Cook squaring algorithm. + +\subsection{High Level Squaring} +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_sqr}. \\ +\textbf{Input}. mp\_int $a$ \\ +\textbf{Output}. $b \leftarrow a^2$ \\ +\hline \\ +1. If $a.used \ge TOOM\_SQR\_CUTOFF$ then \\ +\hspace{3mm}1.1 $b \leftarrow a^2$ using algorithm mp\_toom\_sqr \\ +2. else if $a.used \ge KARATSUBA\_SQR\_CUTOFF$ then \\ +\hspace{3mm}2.1 $b \leftarrow a^2$ using algorithm mp\_karatsuba\_sqr \\ +3. else \\ +\hspace{3mm}3.1 $digs \leftarrow a.used + b.used + 1$ \\ +\hspace{3mm}3.2 If $digs < MP\_ARRAY$ and $a.used \le \delta$ then \\ +\hspace{6mm}3.2.1 $b \leftarrow a^2$ using algorithm fast\_s\_mp\_sqr. \\ +\hspace{3mm}3.3 else \\ +\hspace{6mm}3.3.1 $b \leftarrow a^2$ using algorithm s\_mp\_sqr. \\ +4. $b.sign \leftarrow MP\_ZPOS$ \\ +5. Return the result of the unsigned squaring performed. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_sqr} +\end{figure} + +\textbf{Algorithm mp\_sqr.} +This algorithm computes the square of the input using one of four different algorithms. If the input is very large and has at least +\textbf{TOOM\_SQR\_CUTOFF} or \textbf{KARATSUBA\_SQR\_CUTOFF} digits then either the Toom-Cook or the Karatsuba Squaring algorithm is used. If +neither of the polynomial basis algorithms should be used then either the Comba or baseline algorithm is used. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_sqr.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +\section*{Exercises} +\begin{tabular}{cl} +$\left [ 3 \right ] $ & Devise an efficient algorithm for selection of the radix point to handle inputs \\ + & that have different number of digits in Karatsuba multiplication. \\ + & \\ +$\left [ 2 \right ] $ & In section 5.3 the fact that every column of a squaring is made up \\ + & of double products and at most one square is stated. Prove this statement. \\ + & \\ +$\left [ 3 \right ] $ & Prove the equation for Karatsuba squaring. \\ + & \\ +$\left [ 1 \right ] $ & Prove that Karatsuba squaring requires $O \left (n^{lg(3)} \right )$ time. \\ + & \\ +$\left [ 2 \right ] $ & Determine the minimal ratio between addition and multiplication clock cycles \\ + & required for equation $6.7$ to be true. \\ + & \\ +$\left [ 3 \right ] $ & Implement a threaded version of Comba multiplication (and squaring) where you \\ + & compute subsets of the columns in each thread. Determine a cutoff point where \\ + & it is effective and add the logic to mp\_mul() and mp\_sqr(). \\ + &\\ +$\left [ 4 \right ] $ & Same as the previous but also modify the Karatsuba and Toom-Cook. You must \\ + & increase the throughput of mp\_exptmod() for random odd moduli in the range \\ + & $512 \ldots 4096$ bits significantly ($> 2x$) to complete this challenge. \\ + & \\ +\end{tabular} + +\chapter{Modular Reduction} +\section{Basics of Modular Reduction} +\index{modular residue} +Modular reduction is an operation that arises quite often within public key cryptography algorithms and various number theoretic algorithms, +such as factoring. Modular reduction algorithms are the third class of algorithms of the ``multipliers'' set. A number $a$ is said to be \textit{reduced} +modulo another number $b$ by finding the remainder of the division $a/b$. Full integer division with remainder is a topic to be covered +in~\ref{sec:division}. + +Modular reduction is equivalent to solving for $r$ in the following equation. $a = bq + r$ where $q = \lfloor a/b \rfloor$. The result +$r$ is said to be ``congruent to $a$ modulo $b$'' which is also written as $r \equiv a \mbox{ (mod }b\mbox{)}$. In other vernacular $r$ is known as the +``modular residue'' which leads to ``quadratic residue''\footnote{That's fancy talk for $b \equiv a^2 \mbox{ (mod }p\mbox{)}$.} and +other forms of residues. + +Modular reductions are normally used to create either finite groups, rings or fields. The most common usage for performance driven modular reductions +is in modular exponentiation algorithms. That is to compute $d = a^b \mbox{ (mod }c\mbox{)}$ as fast as possible. This operation is used in the +RSA and Diffie-Hellman public key algorithms, for example. Modular multiplication and squaring also appears as a fundamental operation in +elliptic curve cryptographic algorithms. As will be discussed in the subsequent chapter there exist fast algorithms for computing modular +exponentiations without having to perform (\textit{in this example}) $b - 1$ multiplications. These algorithms will produce partial results in the +range $0 \le x < c^2$ which can be taken advantage of to create several efficient algorithms. They have also been used to create redundancy check +algorithms known as CRCs, error correction codes such as Reed-Solomon and solve a variety of number theoeretic problems. + +\section{The Barrett Reduction} +The Barrett reduction algorithm \cite{BARRETT} was inspired by fast division algorithms which multiply by the reciprocal to emulate +division. Barretts observation was that the residue $c$ of $a$ modulo $b$ is equal to + +\begin{equation} +c = a - b \cdot \lfloor a/b \rfloor +\end{equation} + +Since algorithms such as modular exponentiation would be using the same modulus extensively, typical DSP\footnote{It is worth noting that Barrett's paper +targeted the DSP56K processor.} intuition would indicate the next step would be to replace $a/b$ by a multiplication by the reciprocal. However, +DSP intuition on its own will not work as these numbers are considerably larger than the precision of common DSP floating point data types. +It would take another common optimization to optimize the algorithm. + +\subsection{Fixed Point Arithmetic} +The trick used to optimize the above equation is based on a technique of emulating floating point data types with fixed precision integers. Fixed +point arithmetic would become very popular as it greatly optimize the ``3d-shooter'' genre of games in the mid 1990s when floating point units were +fairly slow if not unavailable. The idea behind fixed point arithmetic is to take a normal $k$-bit integer data type and break it into $p$-bit +integer and a $q$-bit fraction part (\textit{where $p+q = k$}). + +In this system a $k$-bit integer $n$ would actually represent $n/2^q$. For example, with $q = 4$ the integer $n = 37$ would actually represent the +value $2.3125$. To multiply two fixed point numbers the integers are multiplied using traditional arithmetic and subsequently normalized by +moving the implied decimal point back to where it should be. For example, with $q = 4$ to multiply the integers $9$ and $5$ they must be converted +to fixed point first by multiplying by $2^q$. Let $a = 9(2^q)$ represent the fixed point representation of $9$ and $b = 5(2^q)$ represent the +fixed point representation of $5$. The product $ab$ is equal to $45(2^{2q})$ which when normalized by dividing by $2^q$ produces $45(2^q)$. + +This technique became popular since a normal integer multiplication and logical shift right are the only required operations to perform a multiplication +of two fixed point numbers. Using fixed point arithmetic, division can be easily approximated by multiplying by the reciprocal. If $2^q$ is +equivalent to one than $2^q/b$ is equivalent to the fixed point approximation of $1/b$ using real arithmetic. Using this fact dividing an integer +$a$ by another integer $b$ can be achieved with the following expression. + +\begin{equation} +\lfloor a / b \rfloor \mbox{ }\approx\mbox{ } \lfloor (a \cdot \lfloor 2^q / b \rfloor)/2^q \rfloor +\end{equation} + +The precision of the division is proportional to the value of $q$. If the divisor $b$ is used frequently as is the case with +modular exponentiation pre-computing $2^q/b$ will allow a division to be performed with a multiplication and a right shift. Both operations +are considerably faster than division on most processors. + +Consider dividing $19$ by $5$. The correct result is $\lfloor 19/5 \rfloor = 3$. With $q = 3$ the reciprocal is $\lfloor 2^q/5 \rfloor = 1$ which +leads to a product of $19$ which when divided by $2^q$ produces $2$. However, with $q = 4$ the reciprocal is $\lfloor 2^q/5 \rfloor = 3$ and +the result of the emulated division is $\lfloor 3 \cdot 19 / 2^q \rfloor = 3$ which is correct. The value of $2^q$ must be close to or ideally +larger than the dividend. In effect if $a$ is the dividend then $q$ should allow $0 \le \lfloor a/2^q \rfloor \le 1$ in order for this approach +to work correctly. Plugging this form of divison into the original equation the following modular residue equation arises. + +\begin{equation} +c = a - b \cdot \lfloor (a \cdot \lfloor 2^q / b \rfloor)/2^q \rfloor +\end{equation} + +Using the notation from \cite{BARRETT} the value of $\lfloor 2^q / b \rfloor$ will be represented by the $\mu$ symbol. Using the $\mu$ +variable also helps re-inforce the idea that it is meant to be computed once and re-used. + +\begin{equation} +c = a - b \cdot \lfloor (a \cdot \mu)/2^q \rfloor +\end{equation} + +Provided that $2^q \ge a$ this algorithm will produce a quotient that is either exactly correct or off by a value of one. In the context of Barrett +reduction the value of $a$ is bound by $0 \le a \le (b - 1)^2$ meaning that $2^q \ge b^2$ is sufficient to ensure the reciprocal will have enough +precision. + +Let $n$ represent the number of digits in $b$. This algorithm requires approximately $2n^2$ single precision multiplications to produce the quotient and +another $n^2$ single precision multiplications to find the residue. In total $3n^2$ single precision multiplications are required to +reduce the number. + +For example, if $b = 1179677$ and $q = 41$ ($2^q > b^2$), then the reciprocal $\mu$ is equal to $\lfloor 2^q / b \rfloor = 1864089$. Consider reducing +$a = 180388626447$ modulo $b$ using the above reduction equation. The quotient using the new formula is $\lfloor (a \cdot \mu) / 2^q \rfloor = 152913$. +By subtracting $152913b$ from $a$ the correct residue $a \equiv 677346 \mbox{ (mod }b\mbox{)}$ is found. + +\subsection{Choosing a Radix Point} +Using the fixed point representation a modular reduction can be performed with $3n^2$ single precision multiplications. If that were the best +that could be achieved a full division\footnote{A division requires approximately $O(2cn^2)$ single precision multiplications for a small value of $c$. +See~\ref{sec:division} for further details.} might as well be used in its place. The key to optimizing the reduction is to reduce the precision of +the initial multiplication that finds the quotient. + +Let $a$ represent the number of which the residue is sought. Let $b$ represent the modulus used to find the residue. Let $m$ represent +the number of digits in $b$. For the purposes of this discussion we will assume that the number of digits in $a$ is $2m$, which is generally true if +two $m$-digit numbers have been multiplied. Dividing $a$ by $b$ is the same as dividing a $2m$ digit integer by a $m$ digit integer. Digits below the +$m - 1$'th digit of $a$ will contribute at most a value of $1$ to the quotient because $\beta^k < b$ for any $0 \le k \le m - 1$. Another way to +express this is by re-writing $a$ as two parts. If $a' \equiv a \mbox{ (mod }b^m\mbox{)}$ and $a'' = a - a'$ then +${a \over b} \equiv {{a' + a''} \over b}$ which is equivalent to ${a' \over b} + {a'' \over b}$. Since $a'$ is bound to be less than $b$ the quotient +is bound by $0 \le {a' \over b} < 1$. + +Since the digits of $a'$ do not contribute much to the quotient the observation is that they might as well be zero. However, if the digits +``might as well be zero'' they might as well not be there in the first place. Let $q_0 = \lfloor a/\beta^{m-1} \rfloor$ represent the input +with the irrelevant digits trimmed. Now the modular reduction is trimmed to the almost equivalent equation + +\begin{equation} +c = a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor +\end{equation} + +Note that the original divisor $2^q$ has been replaced with $\beta^{m+1}$ where in this case $q$ is a multiple of $lg(\beta)$. Also note that the +exponent on the divisor when added to the amount $q_0$ was shifted by equals $2m$. If the optimization had not been performed the divisor +would have the exponent $2m$ so in the end the exponents do ``add up''. Using the above equation the quotient +$\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ can be off from the true quotient by at most two. The original fixed point quotient can be off +by as much as one (\textit{provided the radix point is chosen suitably}) and now that the lower irrelevent digits have been trimmed the quotient +can be off by an additional value of one for a total of at most two. This implies that +$0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$. By first subtracting $b$ times the quotient and then conditionally subtracting +$b$ once or twice the residue is found. + +The quotient is now found using $(m + 1)(m) = m^2 + m$ single precision multiplications and the residue with an additional $m^2$ single +precision multiplications, ignoring the subtractions required. In total $2m^2 + m$ single precision multiplications are required to find the residue. +This is considerably faster than the original attempt. + +For example, let $\beta = 10$ represent the radix of the digits. Let $b = 9999$ represent the modulus which implies $m = 4$. Let $a = 99929878$ +represent the value of which the residue is desired. In this case $q = 8$ since $10^7 < 9999^2$ meaning that $\mu = \lfloor \beta^{q}/b \rfloor = 10001$. +With the new observation the multiplicand for the quotient is equal to $q_0 = \lfloor a / \beta^{m - 1} \rfloor = 99929$. The quotient is then +$\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor = 9993$. Subtracting $9993b$ from $a$ and the correct residue $a \equiv 9871 \mbox{ (mod }b\mbox{)}$ +is found. + +\subsection{Trimming the Quotient} +So far the reduction algorithm has been optimized from $3m^2$ single precision multiplications down to $2m^2 + m$ single precision multiplications. As +it stands now the algorithm is already fairly fast compared to a full integer division algorithm. However, there is still room for +optimization. + +After the first multiplication inside the quotient ($q_0 \cdot \mu$) the value is shifted right by $m + 1$ places effectively nullifying the lower +half of the product. It would be nice to be able to remove those digits from the product to effectively cut down the number of single precision +multiplications. If the number of digits in the modulus $m$ is far less than $\beta$ a full product is not required for the algorithm to work properly. +In fact the lower $m - 2$ digits will not affect the upper half of the product at all and do not need to be computed. + +The value of $\mu$ is a $m$-digit number and $q_0$ is a $m + 1$ digit number. Using a full multiplier $(m + 1)(m) = m^2 + m$ single precision +multiplications would be required. Using a multiplier that will only produce digits at and above the $m - 1$'th digit reduces the number +of single precision multiplications to ${m^2 + m} \over 2$ single precision multiplications. + +\subsection{Trimming the Residue} +After the quotient has been calculated it is used to reduce the input. As previously noted the algorithm is not exact and it can be off by a small +multiple of the modulus, that is $0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$. If $b$ is $m$ digits than the +result of reduction equation is a value of at most $m + 1$ digits (\textit{provided $3 < \beta$}) implying that the upper $m - 1$ digits are +implicitly zero. + +The next optimization arises from this very fact. Instead of computing $b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ using a full +$O(m^2)$ multiplication algorithm only the lower $m+1$ digits of the product have to be computed. Similarly the value of $a$ can +be reduced modulo $\beta^{m+1}$ before the multiple of $b$ is subtracted which simplifes the subtraction as well. A multiplication that produces +only the lower $m+1$ digits requires ${m^2 + 3m - 2} \over 2$ single precision multiplications. + +With both optimizations in place the algorithm is the algorithm Barrett proposed. It requires $m^2 + 2m - 1$ single precision multiplications which +is considerably faster than the straightforward $3m^2$ method. + +\subsection{The Barrett Algorithm} +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_reduce}. \\ +\textbf{Input}. mp\_int $a$, mp\_int $b$ and $\mu = \lfloor \beta^{2m}/b \rfloor, m = \lceil lg_{\beta}(b) \rceil, (0 \le a < b^2, b > 1)$ \\ +\textbf{Output}. $a \mbox{ (mod }b\mbox{)}$ \\ +\hline \\ +Let $m$ represent the number of digits in $b$. \\ +1. Make a copy of $a$ and store it in $q$. (\textit{mp\_init\_copy}) \\ +2. $q \leftarrow \lfloor q / \beta^{m - 1} \rfloor$ (\textit{mp\_rshd}) \\ +\\ +Produce the quotient. \\ +3. $q \leftarrow q \cdot \mu$ (\textit{note: only produce digits at or above $m-1$}) \\ +4. $q \leftarrow \lfloor q / \beta^{m + 1} \rfloor$ \\ +\\ +Subtract the multiple of modulus from the input. \\ +5. $a \leftarrow a \mbox{ (mod }\beta^{m+1}\mbox{)}$ (\textit{mp\_mod\_2d}) \\ +6. $q \leftarrow q \cdot b \mbox{ (mod }\beta^{m+1}\mbox{)}$ (\textit{s\_mp\_mul\_digs}) \\ +7. $a \leftarrow a - q$ (\textit{mp\_sub}) \\ +\\ +Add $\beta^{m+1}$ if a carry occured. \\ +8. If $a < 0$ then (\textit{mp\_cmp\_d}) \\ +\hspace{3mm}8.1 $q \leftarrow 1$ (\textit{mp\_set}) \\ +\hspace{3mm}8.2 $q \leftarrow q \cdot \beta^{m+1}$ (\textit{mp\_lshd}) \\ +\hspace{3mm}8.3 $a \leftarrow a + q$ \\ +\\ +Now subtract the modulus if the residue is too large (e.g. quotient too small). \\ +9. While $a \ge b$ do (\textit{mp\_cmp}) \\ +\hspace{3mm}9.1 $c \leftarrow a - b$ \\ +10. Clear $q$. \\ +11. Return(\textit{MP\_OKAY}) \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_reduce} +\end{figure} + +\textbf{Algorithm mp\_reduce.} +This algorithm will reduce the input $a$ modulo $b$ in place using the Barrett algorithm. It is loosely based on algorithm 14.42 of HAC +\cite[pp. 602]{HAC} which is based on the paper from Paul Barrett \cite{BARRETT}. The algorithm has several restrictions and assumptions which must +be adhered to for the algorithm to work. + +First the modulus $b$ is assumed to be positive and greater than one. If the modulus were less than or equal to one than subtracting +a multiple of it would either accomplish nothing or actually enlarge the input. The input $a$ must be in the range $0 \le a < b^2$ in order +for the quotient to have enough precision. If $a$ is the product of two numbers that were already reduced modulo $b$, this will not be a problem. +Technically the algorithm will still work if $a \ge b^2$ but it will take much longer to finish. The value of $\mu$ is passed as an argument to this +algorithm and is assumed to be calculated and stored before the algorithm is used. + +Recall that the multiplication for the quotient on step 3 must only produce digits at or above the $m-1$'th position. An algorithm called +$s\_mp\_mul\_high\_digs$ which has not been presented is used to accomplish this task. The algorithm is based on $s\_mp\_mul\_digs$ except that +instead of stopping at a given level of precision it starts at a given level of precision. This optimal algorithm can only be used if the number +of digits in $b$ is very much smaller than $\beta$. + +While it is known that +$a \ge b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ only the lower $m+1$ digits are being used to compute the residue, so an implied +``borrow'' from the higher digits might leave a negative result. After the multiple of the modulus has been subtracted from $a$ the residue must be +fixed up in case it is negative. The invariant $\beta^{m+1}$ must be added to the residue to make it positive again. + +The while loop at step 9 will subtract $b$ until the residue is less than $b$. If the algorithm is performed correctly this step is +performed at most twice, and on average once. However, if $a \ge b^2$ than it will iterate substantially more times than it should. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +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 to 44 where algorithm s\_mp\_mul\_high\_digs is used when it is +safe to do so. + +\subsection{The Barrett Setup Algorithm} +In order to use algorithm mp\_reduce the value of $\mu$ must be calculated in advance. Ideally this value should be computed once and stored for +future use so that the Barrett algorithm can be used without delay. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_reduce\_setup}. \\ +\textbf{Input}. mp\_int $a$ ($a > 1$) \\ +\textbf{Output}. $\mu \leftarrow \lfloor \beta^{2m}/a \rfloor$ \\ +\hline \\ +1. $\mu \leftarrow 2^{2 \cdot lg(\beta) \cdot m}$ (\textit{mp\_2expt}) \\ +2. $\mu \leftarrow \lfloor \mu / b \rfloor$ (\textit{mp\_div}) \\ +3. Return(\textit{MP\_OKAY}) \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_reduce\_setup} +\end{figure} + +\textbf{Algorithm mp\_reduce\_setup.} +This algorithm computes the reciprocal $\mu$ required for Barrett reduction. First $\beta^{2m}$ is calculated as $2^{2 \cdot lg(\beta) \cdot m}$ which +is equivalent and much faster. The final value is computed by taking the integer quotient of $\lfloor \mu / b \rfloor$. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_setup.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +This simple routine calculates the reciprocal $\mu$ required by Barrett reduction. Note the extended usage of algorithm mp\_div where the variable +which would received the remainder is passed as NULL. As will be discussed in~\ref{sec:division} the division routine allows both the quotient and the +remainder to be passed as NULL meaning to ignore the value. + +\section{The Montgomery Reduction} +Montgomery reduction\footnote{Thanks to Niels Ferguson for his insightful explanation of the algorithm.} \cite{MONT} is by far the most interesting +form of reduction in common use. It computes a modular residue which is not actually equal to the residue of the input yet instead equal to a +residue times a constant. However, as perplexing as this may sound the algorithm is relatively simple and very efficient. + +Throughout this entire section the variable $n$ will represent the modulus used to form the residue. As will be discussed shortly the value of +$n$ must be odd. The variable $x$ will represent the quantity of which the residue is sought. Similar to the Barrett algorithm the input +is restricted to $0 \le x < n^2$. To begin the description some simple number theory facts must be established. + +\textbf{Fact 1.} Adding $n$ to $x$ does not change the residue since in effect it adds one to the quotient $\lfloor x / n \rfloor$. Another way +to explain this is that $n$ is (\textit{or multiples of $n$ are}) congruent to zero modulo $n$. Adding zero will not change the value of the residue. + +\textbf{Fact 2.} If $x$ is even then performing a division by two in $\Z$ is congruent to $x \cdot 2^{-1} \mbox{ (mod }n\mbox{)}$. Actually +this is an application of the fact that if $x$ is evenly divisible by any $k \in \Z$ then division in $\Z$ will be congruent to +multiplication by $k^{-1}$ modulo $n$. + +From these two simple facts the following simple algorithm can be derived. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{Montgomery Reduction}. \\ +\textbf{Input}. Integer $x$, $n$ and $k$ \\ +\textbf{Output}. $2^{-k}x \mbox{ (mod }n\mbox{)}$ \\ +\hline \\ +1. for $t$ from $1$ to $k$ do \\ +\hspace{3mm}1.1 If $x$ is odd then \\ +\hspace{6mm}1.1.1 $x \leftarrow x + n$ \\ +\hspace{3mm}1.2 $x \leftarrow x/2$ \\ +2. Return $x$. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm Montgomery Reduction} +\end{figure} + +The algorithm reduces the input one bit at a time using the two congruencies stated previously. Inside the loop $n$, which is odd, is +added to $x$ if $x$ is odd. This forces $x$ to be even which allows the division by two in $\Z$ to be congruent to a modular division by two. Since +$x$ is assumed to be initially much larger than $n$ the addition of $n$ will contribute an insignificant magnitude to $x$. Let $r$ represent the +final result of the Montgomery algorithm. If $k > lg(n)$ and $0 \le x < n^2$ then the final result is limited to +$0 \le r < \lfloor x/2^k \rfloor + n$. As a result at most a single subtraction is required to get the residue desired. + +\begin{figure}[here] +\begin{small} +\begin{center} +\begin{tabular}{|c|l|} +\hline \textbf{Step number ($t$)} & \textbf{Result ($x$)} \\ +\hline $1$ & $x + n = 5812$, $x/2 = 2906$ \\ +\hline $2$ & $x/2 = 1453$ \\ +\hline $3$ & $x + n = 1710$, $x/2 = 855$ \\ +\hline $4$ & $x + n = 1112$, $x/2 = 556$ \\ +\hline $5$ & $x/2 = 278$ \\ +\hline $6$ & $x/2 = 139$ \\ +\hline $7$ & $x + n = 396$, $x/2 = 198$ \\ +\hline $8$ & $x/2 = 99$ \\ +\hline $9$ & $x + n = 356$, $x/2 = 178$ \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Example of Montgomery Reduction (I)} +\label{fig:MONT1} +\end{figure} + +Consider the example in figure~\ref{fig:MONT1} which reduces $x = 5555$ modulo $n = 257$ when $k = 9$ (note $\beta^k = 512$ which is larger than $n$). The result of +the algorithm $r = 178$ is congruent to the value of $2^{-9} \cdot 5555 \mbox{ (mod }257\mbox{)}$. When $r$ is multiplied by $2^9$ modulo $257$ the correct residue +$r \equiv 158$ is produced. + +Let $k = \lfloor lg(n) \rfloor + 1$ represent the number of bits in $n$. The current algorithm requires $2k^2$ single precision shifts +and $k^2$ single precision additions. At this rate the algorithm is most certainly slower than Barrett reduction and not terribly useful. +Fortunately there exists an alternative representation of the algorithm. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{Montgomery Reduction} (modified I). \\ +\textbf{Input}. Integer $x$, $n$ and $k$ ($2^k > n$) \\ +\textbf{Output}. $2^{-k}x \mbox{ (mod }n\mbox{)}$ \\ +\hline \\ +1. for $t$ from $1$ to $k$ do \\ +\hspace{3mm}1.1 If the $t$'th bit of $x$ is one then \\ +\hspace{6mm}1.1.1 $x \leftarrow x + 2^tn$ \\ +2. Return $x/2^k$. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm Montgomery Reduction (modified I)} +\end{figure} + +This algorithm is equivalent since $2^tn$ is a multiple of $n$ and the lower $k$ bits of $x$ are zero by step 2. The number of single +precision shifts has now been reduced from $2k^2$ to $k^2 + k$ which is only a small improvement. + +\begin{figure}[here] +\begin{small} +\begin{center} +\begin{tabular}{|c|l|r|} +\hline \textbf{Step number ($t$)} & \textbf{Result ($x$)} & \textbf{Result ($x$) in Binary} \\ +\hline -- & $5555$ & $1010110110011$ \\ +\hline $1$ & $x + 2^{0}n = 5812$ & $1011010110100$ \\ +\hline $2$ & $5812$ & $1011010110100$ \\ +\hline $3$ & $x + 2^{2}n = 6840$ & $1101010111000$ \\ +\hline $4$ & $x + 2^{3}n = 8896$ & $10001011000000$ \\ +\hline $5$ & $8896$ & $10001011000000$ \\ +\hline $6$ & $8896$ & $10001011000000$ \\ +\hline $7$ & $x + 2^{6}n = 25344$ & $110001100000000$ \\ +\hline $8$ & $25344$ & $110001100000000$ \\ +\hline $9$ & $x + 2^{7}n = 91136$ & $10110010000000000$ \\ +\hline -- & $x/2^k = 178$ & \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Example of Montgomery Reduction (II)} +\label{fig:MONT2} +\end{figure} + +Figure~\ref{fig:MONT2} demonstrates the modified algorithm reducing $x = 5555$ modulo $n = 257$ with $k = 9$. +With this algorithm a single shift right at the end is the only right shift required to reduce the input instead of $k$ right shifts inside the +loop. Note that for the iterations $t = 2, 5, 6$ and $8$ where the result $x$ is not changed. In those iterations the $t$'th bit of $x$ is +zero and the appropriate multiple of $n$ does not need to be added to force the $t$'th bit of the result to zero. + +\subsection{Digit Based Montgomery Reduction} +Instead of computing the reduction on a bit-by-bit basis it is actually much faster to compute it on digit-by-digit basis. Consider the +previous algorithm re-written to compute the Montgomery reduction in this new fashion. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{Montgomery Reduction} (modified II). \\ +\textbf{Input}. Integer $x$, $n$ and $k$ ($\beta^k > n$) \\ +\textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\ +\hline \\ +1. for $t$ from $0$ to $k - 1$ do \\ +\hspace{3mm}1.1 $x \leftarrow x + \mu n \beta^t$ \\ +2. Return $x/\beta^k$. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm Montgomery Reduction (modified II)} +\end{figure} + +The value $\mu n \beta^t$ is a multiple of the modulus $n$ meaning that it will not change the residue. If the first digit of +the value $\mu n \beta^t$ equals the negative (modulo $\beta$) of the $t$'th digit of $x$ then the addition will result in a zero digit. This +problem breaks down to solving the following congruency. + +\begin{center} +\begin{tabular}{rcl} +$x_t + \mu n_0$ & $\equiv$ & $0 \mbox{ (mod }\beta\mbox{)}$ \\ +$\mu n_0$ & $\equiv$ & $-x_t \mbox{ (mod }\beta\mbox{)}$ \\ +$\mu$ & $\equiv$ & $-x_t/n_0 \mbox{ (mod }\beta\mbox{)}$ \\ +\end{tabular} +\end{center} + +In each iteration of the loop on step 1 a new value of $\mu$ must be calculated. The value of $-1/n_0 \mbox{ (mod }\beta\mbox{)}$ is used +extensively in this algorithm and should be precomputed. Let $\rho$ represent the negative of the modular inverse of $n_0$ modulo $\beta$. + +For example, let $\beta = 10$ represent the radix. Let $n = 17$ represent the modulus which implies $k = 2$ and $\rho \equiv 7$. Let $x = 33$ +represent the value to reduce. + +\newpage\begin{figure} +\begin{center} +\begin{tabular}{|c|c|c|} +\hline \textbf{Step ($t$)} & \textbf{Value of $x$} & \textbf{Value of $\mu$} \\ +\hline -- & $33$ & --\\ +\hline $0$ & $33 + \mu n = 50$ & $1$ \\ +\hline $1$ & $50 + \mu n \beta = 900$ & $5$ \\ +\hline +\end{tabular} +\end{center} +\caption{Example of Montgomery Reduction} +\end{figure} + +The final result $900$ is then divided by $\beta^k$ to produce the final result $9$. The first observation is that $9 \nequiv x \mbox{ (mod }n\mbox{)}$ +which implies the result is not the modular residue of $x$ modulo $n$. However, recall that the residue is actually multiplied by $\beta^{-k}$ in +the algorithm. To get the true residue the value must be multiplied by $\beta^k$. In this case $\beta^k \equiv 15 \mbox{ (mod }n\mbox{)}$ and +the correct residue is $9 \cdot 15 \equiv 16 \mbox{ (mod }n\mbox{)}$. + +\subsection{Baseline Montgomery Reduction} +The baseline Montgomery reduction algorithm will produce the residue for any size input. It is designed to be a catch-all algororithm for +Montgomery reductions. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_montgomery\_reduce}. \\ +\textbf{Input}. mp\_int $x$, mp\_int $n$ and a digit $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\ +\hspace{11.5mm}($0 \le x < n^2, n > 1, (n, \beta) = 1, \beta^k > n$) \\ +\textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\ +\hline \\ +1. $digs \leftarrow 2n.used + 1$ \\ +2. If $digs < MP\_ARRAY$ and $m.used < \delta$ then \\ +\hspace{3mm}2.1 Use algorithm fast\_mp\_montgomery\_reduce instead. \\ +\\ +Setup $x$ for the reduction. \\ +3. If $x.alloc < digs$ then grow $x$ to $digs$ digits. \\ +4. $x.used \leftarrow digs$ \\ +\\ +Eliminate the lower $k$ digits. \\ +5. For $ix$ from $0$ to $k - 1$ do \\ +\hspace{3mm}5.1 $\mu \leftarrow x_{ix} \cdot \rho \mbox{ (mod }\beta\mbox{)}$ \\ +\hspace{3mm}5.2 $u \leftarrow 0$ \\ +\hspace{3mm}5.3 For $iy$ from $0$ to $k - 1$ do \\ +\hspace{6mm}5.3.1 $\hat r \leftarrow \mu n_{iy} + x_{ix + iy} + u$ \\ +\hspace{6mm}5.3.2 $x_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ +\hspace{6mm}5.3.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ +\hspace{3mm}5.4 While $u > 0$ do \\ +\hspace{6mm}5.4.1 $iy \leftarrow iy + 1$ \\ +\hspace{6mm}5.4.2 $x_{ix + iy} \leftarrow x_{ix + iy} + u$ \\ +\hspace{6mm}5.4.3 $u \leftarrow \lfloor x_{ix+iy} / \beta \rfloor$ \\ +\hspace{6mm}5.4.4 $x_{ix + iy} \leftarrow x_{ix+iy} \mbox{ (mod }\beta\mbox{)}$ \\ +\\ +Divide by $\beta^k$ and fix up as required. \\ +6. $x \leftarrow \lfloor x / \beta^k \rfloor$ \\ +7. If $x \ge n$ then \\ +\hspace{3mm}7.1 $x \leftarrow x - n$ \\ +8. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_montgomery\_reduce} +\end{figure} + +\textbf{Algorithm mp\_montgomery\_reduce.} +This algorithm reduces the input $x$ modulo $n$ in place using the Montgomery reduction algorithm. The algorithm is loosely based +on algorithm 14.32 of \cite[pp.601]{HAC} except it merges the multiplication of $\mu n \beta^t$ with the addition in the inner loop. The +restrictions on this algorithm are fairly easy to adapt to. First $0 \le x < n^2$ bounds the input to numbers in the same range as +for the Barrett algorithm. Additionally if $n > 1$ and $n$ is odd there will exist a modular inverse $\rho$. $\rho$ must be calculated in +advance of this algorithm. Finally the variable $k$ is fixed and a pseudonym for $n.used$. + +Step 2 decides whether a faster Montgomery algorithm can be used. It is based on the Comba technique meaning that there are limits on +the size of the input. This algorithm is discussed in sub-section 6.3.3. + +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. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_montgomery\_reduce.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +This is the baseline implementation of the Montgomery reduction algorithm. Lines 31 to 36 determine if the Comba based +routine can be used instead. Line 47 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} + +The Montgomery reduction requires fewer single precision multiplications than a Barrett reduction, however it is much slower due to the serial +nature of the inner loop. The Barrett reduction algorithm requires two slightly modified multipliers which can be implemented with the Comba +technique. The Montgomery reduction algorithm cannot directly use the Comba technique to any significant advantage since the inner loop calculates +a $k \times 1$ product $k$ times. + +The biggest obstacle is that at the $ix$'th iteration of the outer loop the value of $x_{ix}$ is required to calculate $\mu$. This means the +carries from $0$ to $ix - 1$ must have been propagated upwards to form a valid $ix$'th digit. The solution as it turns out is very simple. +Perform a Comba like multiplier and inside the outer loop just after the inner loop fix up the $ix + 1$'th digit by forwarding the carry. + +With this change in place the Montgomery reduction algorithm can be performed with a Comba style multiplication loop which substantially increases +the speed of the algorithm. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{fast\_mp\_montgomery\_reduce}. \\ +\textbf{Input}. mp\_int $x$, mp\_int $n$ and a digit $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\ +\hspace{11.5mm}($0 \le x < n^2, n > 1, (n, \beta) = 1, \beta^k > n$) \\ +\textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\ +\hline \\ +Place an array of \textbf{MP\_WARRAY} mp\_word variables called $\hat W$ on the stack. \\ +1. if $x.alloc < n.used + 1$ then grow $x$ to $n.used + 1$ digits. \\ +Copy the digits of $x$ into the array $\hat W$ \\ +2. For $ix$ from $0$ to $x.used - 1$ do \\ +\hspace{3mm}2.1 $\hat W_{ix} \leftarrow x_{ix}$ \\ +3. For $ix$ from $x.used$ to $2n.used - 1$ do \\ +\hspace{3mm}3.1 $\hat W_{ix} \leftarrow 0$ \\ +Elimiate the lower $k$ digits. \\ +4. for $ix$ from $0$ to $n.used - 1$ do \\ +\hspace{3mm}4.1 $\mu \leftarrow \hat W_{ix} \cdot \rho \mbox{ (mod }\beta\mbox{)}$ \\ +\hspace{3mm}4.2 For $iy$ from $0$ to $n.used - 1$ do \\ +\hspace{6mm}4.2.1 $\hat W_{iy + ix} \leftarrow \hat W_{iy + ix} + \mu \cdot n_{iy}$ \\ +\hspace{3mm}4.3 $\hat W_{ix + 1} \leftarrow \hat W_{ix + 1} + \lfloor \hat W_{ix} / \beta \rfloor$ \\ +Propagate carries upwards. \\ +5. for $ix$ from $n.used$ to $2n.used + 1$ do \\ +\hspace{3mm}5.1 $\hat W_{ix + 1} \leftarrow \hat W_{ix + 1} + \lfloor \hat W_{ix} / \beta \rfloor$ \\ +Shift right and reduce modulo $\beta$ simultaneously. \\ +6. for $ix$ from $0$ to $n.used + 1$ do \\ +\hspace{3mm}6.1 $x_{ix} \leftarrow \hat W_{ix + n.used} \mbox{ (mod }\beta\mbox{)}$ \\ +Zero excess digits and fixup $x$. \\ +7. if $x.used > n.used + 1$ then do \\ +\hspace{3mm}7.1 for $ix$ from $n.used + 1$ to $x.used - 1$ do \\ +\hspace{6mm}7.1.1 $x_{ix} \leftarrow 0$ \\ +8. $x.used \leftarrow n.used + 1$ \\ +9. Clamp excessive digits of $x$. \\ +10. If $x \ge n$ then \\ +\hspace{3mm}10.1 $x \leftarrow x - n$ \\ +11. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm fast\_mp\_montgomery\_reduce} +\end{figure} + +\textbf{Algorithm fast\_mp\_montgomery\_reduce.} +This algorithm will compute the Montgomery reduction of $x$ modulo $n$ using the Comba technique. It is on most computer platforms significantly +faster than algorithm mp\_montgomery\_reduce and algorithm mp\_reduce (\textit{Barrett reduction}). The algorithm has the same restrictions +on the input as the baseline reduction algorithm. An additional two restrictions are imposed on this algorithm. The number of digits $k$ in the +the modulus $n$ must not violate $MP\_WARRAY > 2k +1$ and $n < \delta$. When $\beta = 2^{28}$ this algorithm can be used to reduce modulo +a modulus of at most $3,556$ bits in length. + +As in the other Comba reduction algorithms there is a $\hat W$ array which stores the columns of the product. It is initially filled with the +contents of $x$ with the excess digits zeroed. The reduction loop is very similar the to the baseline loop at heart. The multiplication on step +4.1 can be single precision only since $ab \mbox{ (mod }\beta\mbox{)} \equiv (a \mbox{ mod }\beta)(b \mbox{ mod }\beta)$. Some multipliers such +as those on the ARM processors take a variable length time to complete depending on the number of bytes of result it must produce. By performing +a single precision multiplication instead half the amount of time is spent. + +Also note that digit $\hat W_{ix}$ must have the carry from the $ix - 1$'th digit propagated upwards in order for this to work. That is what step +4.3 will do. In effect over the $n.used$ iterations of the outer loop the $n.used$'th lower columns all have the their carries propagated forwards. Note +how the upper bits of those same words are not reduced modulo $\beta$. This is because those values will be discarded shortly and there is no +point. + +Step 5 will propagate the remainder of the carries upwards. On step 6 the columns are reduced modulo $\beta$ and shifted simultaneously as they are +stored in the destination $x$. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_fast\_mp\_montgomery\_reduce.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +The $\hat W$ array is first filled with digits of $x$ on line 48 then the rest of the digits are zeroed on line 55. 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 110 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 109 propagates the rest of the carries upwards through the columns. The for loop on line 126 reduces the columns +modulo $\beta$ and shifts them $k$ places at the same time. The alias $\_ \hat W$ actually refers to the array $\hat W$ starting at the $n.used$'th +digit, that is $\_ \hat W_{t} = \hat W_{n.used + t}$. + +\subsection{Montgomery Setup} +To calculate the variable $\rho$ a relatively simple algorithm will be required. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_montgomery\_setup}. \\ +\textbf{Input}. mp\_int $n$ ($n > 1$ and $(n, 2) = 1$) \\ +\textbf{Output}. $\rho \equiv -1/n_0 \mbox{ (mod }\beta\mbox{)}$ \\ +\hline \\ +1. $b \leftarrow n_0$ \\ +2. If $b$ is even return(\textit{MP\_VAL}) \\ +3. $x \leftarrow (((b + 2) \mbox{ AND } 4) << 1) + b$ \\ +4. for $k$ from 0 to $\lceil lg(lg(\beta)) \rceil - 2$ do \\ +\hspace{3mm}4.1 $x \leftarrow x \cdot (2 - bx)$ \\ +5. $\rho \leftarrow \beta - x \mbox{ (mod }\beta\mbox{)}$ \\ +6. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_montgomery\_setup} +\end{figure} + +\textbf{Algorithm mp\_montgomery\_setup.} +This algorithm will calculate the value of $\rho$ required within the Montgomery reduction algorithms. It uses a very interesting trick +to calculate $1/n_0$ when $\beta$ is a power of two. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_montgomery\_setup.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +This source code computes the value of $\rho$ required to perform Montgomery reduction. It has been modified to avoid performing excess +multiplications when $\beta$ is not the default 28-bits. + +\section{The Diminished Radix Algorithm} +The Diminished Radix method of modular reduction \cite{DRMET} is a fairly clever technique which can be more efficient than either the Barrett +or Montgomery methods for certain forms of moduli. The technique is based on the following simple congruence. + +\begin{equation} +(x \mbox{ mod } n) + k \lfloor x / n \rfloor \equiv x \mbox{ (mod }(n - k)\mbox{)} +\end{equation} + +This observation was used in the MMB \cite{MMB} block cipher to create a diffusion primitive. It used the fact that if $n = 2^{31}$ and $k=1$ that +then a x86 multiplier could produce the 62-bit product and use the ``shrd'' instruction to perform a double-precision right shift. The proof +of the above equation is very simple. First write $x$ in the product form. + +\begin{equation} +x = qn + r +\end{equation} + +Now reduce both sides modulo $(n - k)$. + +\begin{equation} +x \equiv qk + r \mbox{ (mod }(n-k)\mbox{)} +\end{equation} + +The variable $n$ reduces modulo $n - k$ to $k$. By putting $q = \lfloor x/n \rfloor$ and $r = x \mbox{ mod } n$ +into the equation the original congruence is reproduced, thus concluding the proof. The following algorithm is based on this observation. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{Diminished Radix Reduction}. \\ +\textbf{Input}. Integer $x$, $n$, $k$ \\ +\textbf{Output}. $x \mbox{ mod } (n - k)$ \\ +\hline \\ +1. $q \leftarrow \lfloor x / n \rfloor$ \\ +2. $q \leftarrow k \cdot q$ \\ +3. $x \leftarrow x \mbox{ (mod }n\mbox{)}$ \\ +4. $x \leftarrow x + q$ \\ +5. If $x \ge (n - k)$ then \\ +\hspace{3mm}5.1 $x \leftarrow x - (n - k)$ \\ +\hspace{3mm}5.2 Goto step 1. \\ +6. Return $x$ \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm Diminished Radix Reduction} +\label{fig:DR} +\end{figure} + +This algorithm will reduce $x$ modulo $n - k$ and return the residue. If $0 \le x < (n - k)^2$ then the algorithm will loop almost always +once or twice and occasionally three times. For simplicity sake the value of $x$ is bounded by the following simple polynomial. + +\begin{equation} +0 \le x < n^2 + k^2 - 2nk +\end{equation} + +The true bound is $0 \le x < (n - k - 1)^2$ but this has quite a few more terms. The value of $q$ after step 1 is bounded by the following. + +\begin{equation} +q < n - 2k - k^2/n +\end{equation} + +Since $k^2$ is going to be considerably smaller than $n$ that term will always be zero. The value of $x$ after step 3 is bounded trivially as +$0 \le x < n$. By step four the sum $x + q$ is bounded by + +\begin{equation} +0 \le q + x < (k + 1)n - 2k^2 - 1 +\end{equation} + +With a second pass $q$ will be loosely bounded by $0 \le q < k^2$ after step 2 while $x$ will still be loosely bounded by $0 \le x < n$ after step 3. After the second pass it is highly unlike that the +sum in step 4 will exceed $n - k$. In practice fewer than three passes of the algorithm are required to reduce virtually every input in the +range $0 \le x < (n - k - 1)^2$. + +\begin{figure} +\begin{small} +\begin{center} +\begin{tabular}{|l|} +\hline +$x = 123456789, n = 256, k = 3$ \\ +\hline $q \leftarrow \lfloor x/n \rfloor = 482253$ \\ +$q \leftarrow q*k = 1446759$ \\ +$x \leftarrow x \mbox{ mod } n = 21$ \\ +$x \leftarrow x + q = 1446780$ \\ +$x \leftarrow x - (n - k) = 1446527$ \\ +\hline +$q \leftarrow \lfloor x/n \rfloor = 5650$ \\ +$q \leftarrow q*k = 16950$ \\ +$x \leftarrow x \mbox{ mod } n = 127$ \\ +$x \leftarrow x + q = 17077$ \\ +$x \leftarrow x - (n - k) = 16824$ \\ +\hline +$q \leftarrow \lfloor x/n \rfloor = 65$ \\ +$q \leftarrow q*k = 195$ \\ +$x \leftarrow x \mbox{ mod } n = 184$ \\ +$x \leftarrow x + q = 379$ \\ +$x \leftarrow x - (n - k) = 126$ \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Example Diminished Radix Reduction} +\label{fig:EXDR} +\end{figure} + +Figure~\ref{fig:EXDR} demonstrates the reduction of $x = 123456789$ modulo $n - k = 253$ when $n = 256$ and $k = 3$. Note that even while $x$ +is considerably larger than $(n - k - 1)^2 = 63504$ the algorithm still converges on the modular residue exceedingly fast. In this case only +three passes were required to find the residue $x \equiv 126$. + + +\subsection{Choice of Moduli} +On the surface this algorithm looks like a very expensive algorithm. It requires a couple of subtractions followed by multiplication and other +modular reductions. The usefulness of this algorithm becomes exceedingly clear when an appropriate modulus is chosen. + +Division in general is a very expensive operation to perform. The one exception is when the division is by a power of the radix of representation used. +Division by ten for example is simple for pencil and paper mathematics since it amounts to shifting the decimal place to the right. Similarly division +by two (\textit{or powers of two}) is very simple for binary computers to perform. It would therefore seem logical to choose $n$ of the form $2^p$ +which would imply that $\lfloor x / n \rfloor$ is a simple shift of $x$ right $p$ bits. + +However, there is one operation related to division of power of twos that is even faster than this. If $n = \beta^p$ then the division may be +performed by moving whole digits to the right $p$ places. In practice division by $\beta^p$ is much faster than division by $2^p$ for any $p$. +Also with the choice of $n = \beta^p$ reducing $x$ modulo $n$ merely requires zeroing the digits above the $p-1$'th digit of $x$. + +Throughout the next section the term ``restricted modulus'' will refer to a modulus of the form $\beta^p - k$ whereas the term ``unrestricted +modulus'' will refer to a modulus of the form $2^p - k$. The word ``restricted'' in this case refers to the fact that it is based on the +$2^p$ logic except $p$ must be a multiple of $lg(\beta)$. + +\subsection{Choice of $k$} +Now that division and reduction (\textit{step 1 and 3 of figure~\ref{fig:DR}}) have been optimized to simple digit operations the multiplication by $k$ +in step 2 is the most expensive operation. Fortunately the choice of $k$ is not terribly limited. For all intents and purposes it might +as well be a single digit. The smaller the value of $k$ is the faster the algorithm will be. + +\subsection{Restricted Diminished Radix Reduction} +The restricted Diminished Radix algorithm can quickly reduce an input modulo a modulus of the form $n = \beta^p - k$. This algorithm can reduce +an input $x$ within the range $0 \le x < n^2$ using only a couple passes of the algorithm demonstrated in figure~\ref{fig:DR}. The implementation +of this algorithm has been optimized to avoid additional overhead associated with a division by $\beta^p$, the multiplication by $k$ or the addition +of $x$ and $q$. The resulting algorithm is very efficient and can lead to substantial improvements over Barrett and Montgomery reduction when modular +exponentiations are performed. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_dr\_reduce}. \\ +\textbf{Input}. mp\_int $x$, $n$ and a mp\_digit $k = \beta - n_0$ \\ +\hspace{11.5mm}($0 \le x < n^2$, $n > 1$, $0 < k < \beta$) \\ +\textbf{Output}. $x \mbox{ mod } n$ \\ +\hline \\ +1. $m \leftarrow n.used$ \\ +2. If $x.alloc < 2m$ then grow $x$ to $2m$ digits. \\ +3. $\mu \leftarrow 0$ \\ +4. for $i$ from $0$ to $m - 1$ do \\ +\hspace{3mm}4.1 $\hat r \leftarrow k \cdot x_{m+i} + x_{i} + \mu$ \\ +\hspace{3mm}4.2 $x_{i} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ +\hspace{3mm}4.3 $\mu \leftarrow \lfloor \hat r / \beta \rfloor$ \\ +5. $x_{m} \leftarrow \mu$ \\ +6. for $i$ from $m + 1$ to $x.used - 1$ do \\ +\hspace{3mm}6.1 $x_{i} \leftarrow 0$ \\ +7. Clamp excess digits of $x$. \\ +8. If $x \ge n$ then \\ +\hspace{3mm}8.1 $x \leftarrow x - n$ \\ +\hspace{3mm}8.2 Goto step 3. \\ +9. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_dr\_reduce} +\end{figure} + +\textbf{Algorithm mp\_dr\_reduce.} +This algorithm will perform the Dimished Radix reduction of $x$ modulo $n$. It has similar restrictions to that of the Barrett reduction +with the addition that $n$ must be of the form $n = \beta^m - k$ where $0 < k <\beta$. + +This algorithm essentially implements the pseudo-code in figure~\ref{fig:DR} except with a slight optimization. The division by $\beta^m$, multiplication by $k$ +and addition of $x \mbox{ mod }\beta^m$ are all performed simultaneously inside the loop on step 4. The division by $\beta^m$ is emulated by accessing +the term at the $m+i$'th position which is subsequently multiplied by $k$ and added to the term at the $i$'th position. After the loop the $m$'th +digit is set to the carry and the upper digits are zeroed. Steps 5 and 6 emulate the reduction modulo $\beta^m$ that should have happend to +$x$ before the addition of the multiple of the upper half. + +At step 8 if $x$ is still larger than $n$ another pass of the algorithm is required. First $n$ is subtracted from $x$ and then the algorithm resumes +at step 3. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_reduce.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +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 52 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 64 performs the bulk of the work (\textit{corresponds to step 4 of algorithm 7.11}) +in this algorithm. + +By line 67 the pointer $tmpx1$ points to the $m$'th digit of $x$ which is where the final carry will be placed. Similarly by line 74 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 81 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} + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_setup.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +\subsubsection{Modulus Detection} +Another algorithm which will be useful is the ability to detect a restricted Diminished Radix modulus. An integer is said to be +of restricted Diminished Radix form if all of the digits are equal to $\beta - 1$ except the trailing digit which may be any value. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_dr\_is\_modulus}. \\ +\textbf{Input}. mp\_int $n$ \\ +\textbf{Output}. $1$ if $n$ is in D.R form, $0$ otherwise \\ +\hline +1. If $n.used < 2$ then return($0$). \\ +2. for $ix$ from $1$ to $n.used - 1$ do \\ +\hspace{3mm}2.1 If $n_{ix} \ne \beta - 1$ return($0$). \\ +3. Return($1$). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_dr\_is\_modulus} +\end{figure} + +\textbf{Algorithm mp\_dr\_is\_modulus.} +This algorithm determines if a value is in Diminished Radix form. Step 1 rejects obvious cases where fewer than two digits are +in the mp\_int. Step 2 tests all but the first digit to see if they are equal to $\beta - 1$. If the algorithm manages to get to +step 3 then $n$ must be of Diminished Radix form. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_is\_modulus.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +\subsection{Unrestricted Diminished Radix Reduction} +The unrestricted Diminished Radix algorithm allows modular reductions to be performed when the modulus is of the form $2^p - k$. This algorithm +is a straightforward adaptation of algorithm~\ref{fig:DR}. + +In general the restricted Diminished Radix reduction algorithm is much faster since it has considerably lower overhead. However, this new +algorithm is much faster than either Montgomery or Barrett reduction when the moduli are of the appropriate form. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_reduce\_2k}. \\ +\textbf{Input}. mp\_int $a$ and $n$. mp\_digit $k$ \\ +\hspace{11.5mm}($a \ge 0$, $n > 1$, $0 < k < \beta$, $n + k$ is a power of two) \\ +\textbf{Output}. $a \mbox{ (mod }n\mbox{)}$ \\ +\hline +1. $p \leftarrow \lceil lg(n) \rceil$ (\textit{mp\_count\_bits}) \\ +2. While $a \ge n$ do \\ +\hspace{3mm}2.1 $q \leftarrow \lfloor a / 2^p \rfloor$ (\textit{mp\_div\_2d}) \\ +\hspace{3mm}2.2 $a \leftarrow a \mbox{ (mod }2^p\mbox{)}$ (\textit{mp\_mod\_2d}) \\ +\hspace{3mm}2.3 $q \leftarrow q \cdot k$ (\textit{mp\_mul\_d}) \\ +\hspace{3mm}2.4 $a \leftarrow a - q$ (\textit{s\_mp\_sub}) \\ +\hspace{3mm}2.5 If $a \ge n$ then do \\ +\hspace{6mm}2.5.1 $a \leftarrow a - n$ \\ +3. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_reduce\_2k} +\end{figure} + +\textbf{Algorithm mp\_reduce\_2k.} +This algorithm quickly reduces an input $a$ modulo an unrestricted Diminished Radix modulus $n$. Division by $2^p$ is emulated with a right +shift which makes the algorithm fairly inexpensive to use. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_2k.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +The algorithm mp\_count\_bits calculates the number of bits in an mp\_int which is used to find the initial value of $p$. The call to mp\_div\_2d +on line 31 calculates both the quotient $q$ and the remainder $a$ required. By doing both in a single function call the code size +is kept fairly small. The multiplication by $k$ is only performed if $k > 1$. This allows reductions modulo $2^p - 1$ to be performed without +any multiplications. + +The unsigned s\_mp\_add, mp\_cmp\_mag and s\_mp\_sub are used in place of their full sign counterparts since the inputs are only valid if they are +positive. By using the unsigned versions the overhead is kept to a minimum. + +\subsubsection{Unrestricted Setup} +To setup this reduction algorithm the value of $k = 2^p - n$ is required. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_reduce\_2k\_setup}. \\ +\textbf{Input}. mp\_int $n$ \\ +\textbf{Output}. $k = 2^p - n$ \\ +\hline +1. $p \leftarrow \lceil lg(n) \rceil$ (\textit{mp\_count\_bits}) \\ +2. $x \leftarrow 2^p$ (\textit{mp\_2expt}) \\ +3. $x \leftarrow x - n$ (\textit{mp\_sub}) \\ +4. $k \leftarrow x_0$ \\ +5. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_reduce\_2k\_setup} +\end{figure} + +\textbf{Algorithm mp\_reduce\_2k\_setup.} +This algorithm computes the value of $k$ required for the algorithm mp\_reduce\_2k. By making a temporary variable $x$ equal to $2^p$ a subtraction +is sufficient to solve for $k$. Alternatively if $n$ has more than one digit the value of $k$ is simply $\beta - n_0$. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_2k\_setup.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +\subsubsection{Unrestricted Detection} +An integer $n$ is a valid unrestricted Diminished Radix modulus if either of the following are true. + +\begin{enumerate} +\item The number has only one digit. +\item The number has more than one digit and every bit from the $\beta$'th to the most significant is one. +\end{enumerate} + +If either condition is true than there is a power of two $2^p$ such that $0 < 2^p - n < \beta$. If the input is only +one digit than it will always be of the correct form. Otherwise all of the bits above the first digit must be one. This arises from the fact +that there will be value of $k$ that when added to the modulus causes a carry in the first digit which propagates all the way to the most +significant bit. The resulting sum will be a power of two. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_reduce\_is\_2k}. \\ +\textbf{Input}. mp\_int $n$ \\ +\textbf{Output}. $1$ if of proper form, $0$ otherwise \\ +\hline +1. If $n.used = 0$ then return($0$). \\ +2. If $n.used = 1$ then return($1$). \\ +3. $p \leftarrow \lceil lg(n) \rceil$ (\textit{mp\_count\_bits}) \\ +4. for $x$ from $lg(\beta)$ to $p$ do \\ +\hspace{3mm}4.1 If the ($x \mbox{ mod }lg(\beta)$)'th bit of the $\lfloor x / lg(\beta) \rfloor$ of $n$ is zero then return($0$). \\ +5. Return($1$). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_reduce\_is\_2k} +\end{figure} + +\textbf{Algorithm mp\_reduce\_is\_2k.} +This algorithm quickly determines if a modulus is of the form required for algorithm mp\_reduce\_2k to function properly. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_is\_2k.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + + + +\section{Algorithm Comparison} +So far three very different algorithms for modular reduction have been discussed. Each of the algorithms have their own strengths and weaknesses +that makes having such a selection very useful. The following table sumarizes the three algorithms along with comparisons of work factors. Since +all three algorithms have the restriction that $0 \le x < n^2$ and $n > 1$ those limitations are not included in the table. + +\begin{center} +\begin{small} +\begin{tabular}{|c|c|c|c|c|c|} +\hline \textbf{Method} & \textbf{Work Required} & \textbf{Limitations} & \textbf{$m = 8$} & \textbf{$m = 32$} & \textbf{$m = 64$} \\ +\hline Barrett & $m^2 + 2m - 1$ & None & $79$ & $1087$ & $4223$ \\ +\hline Montgomery & $m^2 + m$ & $n$ must be odd & $72$ & $1056$ & $4160$ \\ +\hline D.R. & $2m$ & $n = \beta^m - k$ & $16$ & $64$ & $128$ \\ +\hline +\end{tabular} +\end{small} +\end{center} + +In theory Montgomery and Barrett reductions would require roughly the same amount of time to complete. However, in practice since Montgomery +reduction can be written as a single function with the Comba technique it is much faster. Barrett reduction suffers from the overhead of +calling the half precision multipliers, addition and division by $\beta$ algorithms. + +For almost every cryptographic algorithm Montgomery reduction is the algorithm of choice. The one set of algorithms where Diminished Radix reduction truly +shines are based on the discrete logarithm problem such as Diffie-Hellman \cite{DH} and ElGamal \cite{ELGAMAL}. In these algorithms +primes of the form $\beta^m - k$ can be found and shared amongst users. These primes will allow the Diminished Radix algorithm to be used in +modular exponentiation to greatly speed up the operation. + + + +\section*{Exercises} +\begin{tabular}{cl} +$\left [ 3 \right ]$ & Prove that the ``trick'' in algorithm mp\_montgomery\_setup actually \\ + & calculates the correct value of $\rho$. \\ + & \\ +$\left [ 2 \right ]$ & Devise an algorithm to reduce modulo $n + k$ for small $k$ quickly. \\ + & \\ +$\left [ 4 \right ]$ & Prove that the pseudo-code algorithm ``Diminished Radix Reduction'' \\ + & (\textit{figure~\ref{fig:DR}}) terminates. Also prove the probability that it will \\ + & terminate within $1 \le k \le 10$ iterations. \\ + & \\ +\end{tabular} + + +\chapter{Exponentiation} +Exponentiation is the operation of raising one variable to the power of another, for example, $a^b$. A variant of exponentiation, computed +in a finite field or ring, is called modular exponentiation. This latter style of operation is typically used in public key +cryptosystems such as RSA and Diffie-Hellman. The ability to quickly compute modular exponentiations is of great benefit to any +such cryptosystem and many methods have been sought to speed it up. + +\section{Exponentiation Basics} +A trivial algorithm would simply multiply $a$ against itself $b - 1$ times to compute the exponentiation desired. However, as $b$ grows in size +the number of multiplications becomes prohibitive. Imagine what would happen if $b$ $\approx$ $2^{1024}$ as is the case when computing an RSA signature +with a $1024$-bit key. Such a calculation could never be completed as it would take simply far too long. + +Fortunately there is a very simple algorithm based on the laws of exponents. Recall that $lg_a(a^b) = b$ and that $lg_a(a^ba^c) = b + c$ which +are two trivial relationships between the base and the exponent. Let $b_i$ represent the $i$'th bit of $b$ starting from the least +significant bit. If $b$ is a $k$-bit integer than the following equation is true. + +\begin{equation} +a^b = \prod_{i=0}^{k-1} a^{2^i \cdot b_i} +\end{equation} + +By taking the base $a$ logarithm of both sides of the equation the following equation is the result. + +\begin{equation} +b = \sum_{i=0}^{k-1}2^i \cdot b_i +\end{equation} + +The term $a^{2^i}$ can be found from the $i - 1$'th term by squaring the term since $\left ( a^{2^i} \right )^2$ is equal to +$a^{2^{i+1}}$. This observation forms the basis of essentially all fast exponentiation algorithms. It requires $k$ squarings and on average +$k \over 2$ multiplications to compute the result. This is indeed quite an improvement over simply multiplying by $a$ a total of $b-1$ times. + +While this current method is a considerable speed up there are further improvements to be made. For example, the $a^{2^i}$ term does not need to +be computed in an auxilary variable. Consider the following equivalent algorithm. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{Left to Right Exponentiation}. \\ +\textbf{Input}. Integer $a$, $b$ and $k$ \\ +\textbf{Output}. $c = a^b$ \\ +\hline \\ +1. $c \leftarrow 1$ \\ +2. for $i$ from $k - 1$ to $0$ do \\ +\hspace{3mm}2.1 $c \leftarrow c^2$ \\ +\hspace{3mm}2.2 $c \leftarrow c \cdot a^{b_i}$ \\ +3. Return $c$. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Left to Right Exponentiation} +\label{fig:LTOR} +\end{figure} + +This algorithm starts from the most significant bit and works towards the least significant bit. When the $i$'th bit of $b$ is set $a$ is +multiplied against the current product. In each iteration the product is squared which doubles the exponent of the individual terms of the +product. + +For example, let $b = 101100_2 \equiv 44_{10}$. The following chart demonstrates the actions of the algorithm. + +\newpage\begin{figure} +\begin{center} +\begin{tabular}{|c|c|} +\hline \textbf{Value of $i$} & \textbf{Value of $c$} \\ +\hline - & $1$ \\ +\hline $5$ & $a$ \\ +\hline $4$ & $a^2$ \\ +\hline $3$ & $a^4 \cdot a$ \\ +\hline $2$ & $a^8 \cdot a^2 \cdot a$ \\ +\hline $1$ & $a^{16} \cdot a^4 \cdot a^2$ \\ +\hline $0$ & $a^{32} \cdot a^8 \cdot a^4$ \\ +\hline +\end{tabular} +\end{center} +\caption{Example of Left to Right Exponentiation} +\end{figure} + +When the product $a^{32} \cdot a^8 \cdot a^4$ is simplified it is equal $a^{44}$ which is the desired exponentiation. This particular algorithm is +called ``Left to Right'' because it reads the exponent in that order. All of the exponentiation algorithms that will be presented are of this nature. + +\subsection{Single Digit Exponentiation} +The first algorithm in the series of exponentiation algorithms will be an unbounded algorithm where the exponent is a single digit. It is intended +to be used when a small power of an input is required (\textit{e.g. $a^5$}). It is faster than simply multiplying $b - 1$ times for all values of +$b$ that are greater than three. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_expt\_d}. \\ +\textbf{Input}. mp\_int $a$ and mp\_digit $b$ \\ +\textbf{Output}. $c = a^b$ \\ +\hline \\ +1. $g \leftarrow a$ (\textit{mp\_init\_copy}) \\ +2. $c \leftarrow 1$ (\textit{mp\_set}) \\ +3. for $x$ from 1 to $lg(\beta)$ do \\ +\hspace{3mm}3.1 $c \leftarrow c^2$ (\textit{mp\_sqr}) \\ +\hspace{3mm}3.2 If $b$ AND $2^{lg(\beta) - 1} \ne 0$ then \\ +\hspace{6mm}3.2.1 $c \leftarrow c \cdot g$ (\textit{mp\_mul}) \\ +\hspace{3mm}3.3 $b \leftarrow b << 1$ \\ +4. Clear $g$. \\ +5. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_expt\_d} +\end{figure} + +\textbf{Algorithm mp\_expt\_d.} +This algorithm computes the value of $a$ raised to the power of a single digit $b$. It uses the left to right exponentiation algorithm to +quickly compute the exponentiation. It is loosely based on algorithm 14.79 of HAC \cite[pp. 615]{HAC} with the difference that the +exponent is a fixed width. + +A copy of $a$ is made first to allow destination variable $c$ be the same as the source variable $a$. The result is set to the initial value of +$1$ in the subsequent step. + +Inside the loop the exponent is read from the most significant bit first down to the least significant bit. First $c$ is invariably squared +on step 3.1. In the following step if the most significant bit of $b$ is one the copy of $a$ is multiplied against $c$. The value +of $b$ is shifted left one bit to make the next bit down from the most signficant bit the new most significant bit. In effect each +iteration of the loop moves the bits of the exponent $b$ upwards to the most significant location. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_expt\_d.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +Line 29 sets the initial value of the result to $1$. Next the loop on line 31 steps through each bit of the exponent starting from +the most significant down towards the least significant. The invariant squaring operation placed on line 33 is performed first. After +the squaring the result $c$ is multiplied by the base $g$ if and only if the most significant bit of the exponent is set. The shift on line +47 moves all of the bits of the exponent upwards towards the most significant location. + +\section{$k$-ary Exponentiation} +When calculating an exponentiation the most time consuming bottleneck is the multiplications which are in general a small factor +slower than squaring. Recall from the previous algorithm that $b_{i}$ refers to the $i$'th bit of the exponent $b$. Suppose instead it referred to +the $i$'th $k$-bit digit of the exponent of $b$. For $k = 1$ the definitions are synonymous and for $k > 1$ algorithm~\ref{fig:KARY} +computes the same exponentiation. A group of $k$ bits from the exponent is called a \textit{window}. That is it is a small window on only a +portion of the entire exponent. Consider the following modification to the basic left to right exponentiation algorithm. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{$k$-ary Exponentiation}. \\ +\textbf{Input}. Integer $a$, $b$, $k$ and $t$ \\ +\textbf{Output}. $c = a^b$ \\ +\hline \\ +1. $c \leftarrow 1$ \\ +2. for $i$ from $t - 1$ to $0$ do \\ +\hspace{3mm}2.1 $c \leftarrow c^{2^k} $ \\ +\hspace{3mm}2.2 Extract the $i$'th $k$-bit word from $b$ and store it in $g$. \\ +\hspace{3mm}2.3 $c \leftarrow c \cdot a^g$ \\ +3. Return $c$. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{$k$-ary Exponentiation} +\label{fig:KARY} +\end{figure} + +The squaring on step 2.1 can be calculated by squaring the value $c$ successively $k$ times. If the values of $a^g$ for $0 < g < 2^k$ have been +precomputed this algorithm requires only $t$ multiplications and $tk$ squarings. The table can be generated with $2^{k - 1} - 1$ squarings and +$2^{k - 1} + 1$ multiplications. This algorithm assumes that the number of bits in the exponent is evenly divisible by $k$. +However, when it is not the remaining $0 < x \le k - 1$ bits can be handled with algorithm~\ref{fig:LTOR}. + +Suppose $k = 4$ and $t = 100$. This modified algorithm will require $109$ multiplications and $408$ squarings to compute the exponentiation. The +original algorithm would on average have required $200$ multiplications and $400$ squrings to compute the same value. The total number of squarings +has increased slightly but the number of multiplications has nearly halved. + +\subsection{Optimal Values of $k$} +An optimal value of $k$ will minimize $2^{k} + \lceil n / k \rceil + n - 1$ for a fixed number of bits in the exponent $n$. The simplest +approach is to brute force search amongst the values $k = 2, 3, \ldots, 8$ for the lowest result. Table~\ref{fig:OPTK} lists optimal values of $k$ +for various exponent sizes and compares the number of multiplication and squarings required against algorithm~\ref{fig:LTOR}. + +\begin{figure}[here] +\begin{center} +\begin{small} +\begin{tabular}{|c|c|c|c|c|c|} +\hline \textbf{Exponent (bits)} & \textbf{Optimal $k$} & \textbf{Work at $k$} & \textbf{Work with ~\ref{fig:LTOR}} \\ +\hline $16$ & $2$ & $27$ & $24$ \\ +\hline $32$ & $3$ & $49$ & $48$ \\ +\hline $64$ & $3$ & $92$ & $96$ \\ +\hline $128$ & $4$ & $175$ & $192$ \\ +\hline $256$ & $4$ & $335$ & $384$ \\ +\hline $512$ & $5$ & $645$ & $768$ \\ +\hline $1024$ & $6$ & $1257$ & $1536$ \\ +\hline $2048$ & $6$ & $2452$ & $3072$ \\ +\hline $4096$ & $7$ & $4808$ & $6144$ \\ +\hline +\end{tabular} +\end{small} +\end{center} +\caption{Optimal Values of $k$ for $k$-ary Exponentiation} +\label{fig:OPTK} +\end{figure} + +\subsection{Sliding-Window Exponentiation} +A simple modification to the previous algorithm is only generate the upper half of the table in the range $2^{k-1} \le g < 2^k$. Essentially +this is a table for all values of $g$ where the most significant bit of $g$ is a one. However, in order for this to be allowed in the +algorithm values of $g$ in the range $0 \le g < 2^{k-1}$ must be avoided. + +Table~\ref{fig:OPTK2} lists optimal values of $k$ for various exponent sizes and compares the work required against algorithm~\ref{fig:KARY}. + +\begin{figure}[here] +\begin{center} +\begin{small} +\begin{tabular}{|c|c|c|c|c|c|} +\hline \textbf{Exponent (bits)} & \textbf{Optimal $k$} & \textbf{Work at $k$} & \textbf{Work with ~\ref{fig:KARY}} \\ +\hline $16$ & $3$ & $24$ & $27$ \\ +\hline $32$ & $3$ & $45$ & $49$ \\ +\hline $64$ & $4$ & $87$ & $92$ \\ +\hline $128$ & $4$ & $167$ & $175$ \\ +\hline $256$ & $5$ & $322$ & $335$ \\ +\hline $512$ & $6$ & $628$ & $645$ \\ +\hline $1024$ & $6$ & $1225$ & $1257$ \\ +\hline $2048$ & $7$ & $2403$ & $2452$ \\ +\hline $4096$ & $8$ & $4735$ & $4808$ \\ +\hline +\end{tabular} +\end{small} +\end{center} +\caption{Optimal Values of $k$ for Sliding Window Exponentiation} +\label{fig:OPTK2} +\end{figure} + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{Sliding Window $k$-ary Exponentiation}. \\ +\textbf{Input}. Integer $a$, $b$, $k$ and $t$ \\ +\textbf{Output}. $c = a^b$ \\ +\hline \\ +1. $c \leftarrow 1$ \\ +2. for $i$ from $t - 1$ to $0$ do \\ +\hspace{3mm}2.1 If the $i$'th bit of $b$ is a zero then \\ +\hspace{6mm}2.1.1 $c \leftarrow c^2$ \\ +\hspace{3mm}2.2 else do \\ +\hspace{6mm}2.2.1 $c \leftarrow c^{2^k}$ \\ +\hspace{6mm}2.2.2 Extract the $k$ bits from $(b_{i}b_{i-1}\ldots b_{i-(k-1)})$ and store it in $g$. \\ +\hspace{6mm}2.2.3 $c \leftarrow c \cdot a^g$ \\ +\hspace{6mm}2.2.4 $i \leftarrow i - k$ \\ +3. Return $c$. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Sliding Window $k$-ary Exponentiation} +\end{figure} + +Similar to the previous algorithm this algorithm must have a special handler when fewer than $k$ bits are left in the exponent. While this +algorithm requires the same number of squarings it can potentially have fewer multiplications. The pre-computed table $a^g$ is also half +the size as the previous table. + +Consider the exponent $b = 111101011001000_2 \equiv 31432_{10}$ with $k = 3$ using both algorithms. The first algorithm will divide the exponent up as +the following five $3$-bit words $b \equiv \left ( 111, 101, 011, 001, 000 \right )_{2}$. The second algorithm will break the +exponent as $b \equiv \left ( 111, 101, 0, 110, 0, 100, 0 \right )_{2}$. The single digit $0$ in the second representation are where +a single squaring took place instead of a squaring and multiplication. In total the first method requires $10$ multiplications and $18$ +squarings. The second method requires $8$ multiplications and $18$ squarings. + +In general the sliding window method is never slower than the generic $k$-ary method and often it is slightly faster. + +\section{Modular Exponentiation} + +Modular exponentiation is essentially computing the power of a base within a finite field or ring. For example, computing +$d \equiv a^b \mbox{ (mod }c\mbox{)}$ is a modular exponentiation. Instead of first computing $a^b$ and then reducing it +modulo $c$ the intermediate result is reduced modulo $c$ after every squaring or multiplication operation. + +This guarantees that any intermediate result is bounded by $0 \le d \le c^2 - 2c + 1$ and can be reduced modulo $c$ quickly using +one of the algorithms presented in chapter six. + +Before the actual modular exponentiation algorithm can be written a wrapper algorithm must be written first. This algorithm +will allow the exponent $b$ to be negative which is computed as $c \equiv \left (1 / a \right )^{\vert b \vert} \mbox{(mod }d\mbox{)}$. The +value of $(1/a) \mbox{ mod }c$ is computed using the modular inverse (\textit{see \ref{sec;modinv}}). If no inverse exists the algorithm +terminates with an error. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_exptmod}. \\ +\textbf{Input}. mp\_int $a$, $b$ and $c$ \\ +\textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\ +\hline \\ +1. If $c.sign = MP\_NEG$ return(\textit{MP\_VAL}). \\ +2. If $b.sign = MP\_NEG$ then \\ +\hspace{3mm}2.1 $g' \leftarrow g^{-1} \mbox{ (mod }c\mbox{)}$ \\ +\hspace{3mm}2.2 $x' \leftarrow \vert x \vert$ \\ +\hspace{3mm}2.3 Compute $d \equiv g'^{x'} \mbox{ (mod }c\mbox{)}$ via recursion. \\ +3. if $p$ is odd \textbf{OR} $p$ is a D.R. modulus then \\ +\hspace{3mm}3.1 Compute $y \equiv g^{x} \mbox{ (mod }p\mbox{)}$ via algorithm mp\_exptmod\_fast. \\ +4. else \\ +\hspace{3mm}4.1 Compute $y \equiv g^{x} \mbox{ (mod }p\mbox{)}$ via algorithm s\_mp\_exptmod. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_exptmod} +\end{figure} + +\textbf{Algorithm mp\_exptmod.} +The first algorithm which actually performs modular exponentiation is algorithm s\_mp\_exptmod. It is a sliding window $k$-ary algorithm +which uses Barrett reduction to reduce the product modulo $p$. The second algorithm mp\_exptmod\_fast performs the same operation +except it uses either Montgomery or Diminished Radix reduction. The two latter reduction algorithms are clumped in the same exponentiation +algorithm since their arguments are essentially the same (\textit{two mp\_ints and one mp\_digit}). + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_exptmod.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +In order to keep the algorithms in a known state the first step on line 29 is to reject any negative modulus as input. If the exponent is +negative the algorithm tries to perform a modular exponentiation with the modular inverse of the base $G$. The temporary variable $tmpG$ is assigned +the modular inverse of $G$ and $tmpX$ is assigned the absolute value of $X$. The algorithm will recuse with these new values with a positive +exponent. + +If the exponent is positive the algorithm resumes the exponentiation. Line 77 determines if the modulus is of the restricted Diminished Radix +form. If it is not line 70 attempts to determine if it is of a unrestricted Diminished Radix form. The integer $dr$ will take on one +of three values. + +\begin{enumerate} +\item $dr = 0$ means that the modulus is not of either restricted or unrestricted Diminished Radix form. +\item $dr = 1$ means that the modulus is of restricted Diminished Radix form. +\item $dr = 2$ means that the modulus is of unrestricted Diminished Radix form. +\end{enumerate} + +Line 69 determines if the fast modular exponentiation algorithm can be used. It is allowed if $dr \ne 0$ or if the modulus is odd. Otherwise, +the slower s\_mp\_exptmod algorithm is used which uses Barrett reduction. + +\subsection{Barrett Modular Exponentiation} + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{s\_mp\_exptmod}. \\ +\textbf{Input}. mp\_int $a$, $b$ and $c$ \\ +\textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\ +\hline \\ +1. $k \leftarrow lg(x)$ \\ +2. $winsize \leftarrow \left \lbrace \begin{array}{ll} + 2 & \mbox{if }k \le 7 \\ + 3 & \mbox{if }7 < k \le 36 \\ + 4 & \mbox{if }36 < k \le 140 \\ + 5 & \mbox{if }140 < k \le 450 \\ + 6 & \mbox{if }450 < k \le 1303 \\ + 7 & \mbox{if }1303 < k \le 3529 \\ + 8 & \mbox{if }3529 < k \\ + \end{array} \right .$ \\ +3. Initialize $2^{winsize}$ mp\_ints in an array named $M$ and one mp\_int named $\mu$ \\ +4. Calculate the $\mu$ required for Barrett Reduction (\textit{mp\_reduce\_setup}). \\ +5. $M_1 \leftarrow g \mbox{ (mod }p\mbox{)}$ \\ +\\ +Setup the table of small powers of $g$. First find $g^{2^{winsize}}$ and then all multiples of it. \\ +6. $k \leftarrow 2^{winsize - 1}$ \\ +7. $M_{k} \leftarrow M_1$ \\ +8. for $ix$ from 0 to $winsize - 2$ do \\ +\hspace{3mm}8.1 $M_k \leftarrow \left ( M_k \right )^2$ (\textit{mp\_sqr}) \\ +\hspace{3mm}8.2 $M_k \leftarrow M_k \mbox{ (mod }p\mbox{)}$ (\textit{mp\_reduce}) \\ +9. for $ix$ from $2^{winsize - 1} + 1$ to $2^{winsize} - 1$ do \\ +\hspace{3mm}9.1 $M_{ix} \leftarrow M_{ix - 1} \cdot M_{1}$ (\textit{mp\_mul}) \\ +\hspace{3mm}9.2 $M_{ix} \leftarrow M_{ix} \mbox{ (mod }p\mbox{)}$ (\textit{mp\_reduce}) \\ +10. $res \leftarrow 1$ \\ +\\ +Start Sliding Window. \\ +11. $mode \leftarrow 0, bitcnt \leftarrow 1, buf \leftarrow 0, digidx \leftarrow x.used - 1, bitcpy \leftarrow 0, bitbuf \leftarrow 0$ \\ +12. Loop \\ +\hspace{3mm}12.1 $bitcnt \leftarrow bitcnt - 1$ \\ +\hspace{3mm}12.2 If $bitcnt = 0$ then do \\ +\hspace{6mm}12.2.1 If $digidx = -1$ goto step 13. \\ +\hspace{6mm}12.2.2 $buf \leftarrow x_{digidx}$ \\ +\hspace{6mm}12.2.3 $digidx \leftarrow digidx - 1$ \\ +\hspace{6mm}12.2.4 $bitcnt \leftarrow lg(\beta)$ \\ +Continued on next page. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm s\_mp\_exptmod} +\end{figure} + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{s\_mp\_exptmod} (\textit{continued}). \\ +\textbf{Input}. mp\_int $a$, $b$ and $c$ \\ +\textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\ +\hline \\ +\hspace{3mm}12.3 $y \leftarrow (buf >> (lg(\beta) - 1))$ AND $1$ \\ +\hspace{3mm}12.4 $buf \leftarrow buf << 1$ \\ +\hspace{3mm}12.5 if $mode = 0$ and $y = 0$ then goto step 12. \\ +\hspace{3mm}12.6 if $mode = 1$ and $y = 0$ then do \\ +\hspace{6mm}12.6.1 $res \leftarrow res^2$ \\ +\hspace{6mm}12.6.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ +\hspace{6mm}12.6.3 Goto step 12. \\ +\hspace{3mm}12.7 $bitcpy \leftarrow bitcpy + 1$ \\ +\hspace{3mm}12.8 $bitbuf \leftarrow bitbuf + (y << (winsize - bitcpy))$ \\ +\hspace{3mm}12.9 $mode \leftarrow 2$ \\ +\hspace{3mm}12.10 If $bitcpy = winsize$ then do \\ +\hspace{6mm}Window is full so perform the squarings and single multiplication. \\ +\hspace{6mm}12.10.1 for $ix$ from $0$ to $winsize -1$ do \\ +\hspace{9mm}12.10.1.1 $res \leftarrow res^2$ \\ +\hspace{9mm}12.10.1.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ +\hspace{6mm}12.10.2 $res \leftarrow res \cdot M_{bitbuf}$ \\ +\hspace{6mm}12.10.3 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ +\hspace{6mm}Reset the window. \\ +\hspace{6mm}12.10.4 $bitcpy \leftarrow 0, bitbuf \leftarrow 0, mode \leftarrow 1$ \\ +\\ +No more windows left. Check for residual bits of exponent. \\ +13. If $mode = 2$ and $bitcpy > 0$ then do \\ +\hspace{3mm}13.1 for $ix$ form $0$ to $bitcpy - 1$ do \\ +\hspace{6mm}13.1.1 $res \leftarrow res^2$ \\ +\hspace{6mm}13.1.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ +\hspace{6mm}13.1.3 $bitbuf \leftarrow bitbuf << 1$ \\ +\hspace{6mm}13.1.4 If $bitbuf$ AND $2^{winsize} \ne 0$ then do \\ +\hspace{9mm}13.1.4.1 $res \leftarrow res \cdot M_{1}$ \\ +\hspace{9mm}13.1.4.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ +14. $y \leftarrow res$ \\ +15. Clear $res$, $mu$ and the $M$ array. \\ +16. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm s\_mp\_exptmod (continued)} +\end{figure} + +\textbf{Algorithm s\_mp\_exptmod.} +This algorithm computes the $x$'th power of $g$ modulo $p$ and stores the result in $y$. It takes advantage of the Barrett reduction +algorithm to keep the product small throughout the algorithm. + +The first two steps determine the optimal window size based on the number of bits in the exponent. The larger the exponent the +larger the window size becomes. After a window size $winsize$ has been chosen an array of $2^{winsize}$ mp\_int variables is allocated. This +table will hold the values of $g^x \mbox{ (mod }p\mbox{)}$ for $2^{winsize - 1} \le x < 2^{winsize}$. + +After the table is allocated the first power of $g$ is found. Since $g \ge p$ is allowed it must be first reduced modulo $p$ to make +the rest of the algorithm more efficient. The first element of the table at $2^{winsize - 1}$ is found by squaring $M_1$ successively $winsize - 2$ +times. The rest of the table elements are found by multiplying the previous element by $M_1$ modulo $p$. + +Now that the table is available the sliding window may begin. The following list describes the functions of all the variables in the window. +\begin{enumerate} +\item The variable $mode$ dictates how the bits of the exponent are interpreted. +\begin{enumerate} + \item When $mode = 0$ the bits are ignored since no non-zero bit of the exponent has been seen yet. For example, if the exponent were simply + $1$ then there would be $lg(\beta) - 1$ zero bits before the first non-zero bit. In this case bits are ignored until a non-zero bit is found. + \item When $mode = 1$ a non-zero bit has been seen before and a new $winsize$-bit window has not been formed yet. In this mode leading $0$ bits + are read and a single squaring is performed. If a non-zero bit is read a new window is created. + \item When $mode = 2$ the algorithm is in the middle of forming a window and new bits are appended to the window from the most significant bit + downwards. +\end{enumerate} +\item The variable $bitcnt$ indicates how many bits are left in the current digit of the exponent left to be read. When it reaches zero a new digit + is fetched from the exponent. +\item The variable $buf$ holds the currently read digit of the exponent. +\item The variable $digidx$ is an index into the exponents digits. It starts at the leading digit $x.used - 1$ and moves towards the trailing digit. +\item The variable $bitcpy$ indicates how many bits are in the currently formed window. When it reaches $winsize$ the window is flushed and + the appropriate operations performed. +\item The variable $bitbuf$ holds the current bits of the window being formed. +\end{enumerate} + +All of step 12 is the window processing loop. It will iterate while there are digits available form the exponent to read. The first step +inside this loop is to extract a new digit if no more bits are available in the current digit. If there are no bits left a new digit is +read and if there are no digits left than the loop terminates. + +After a digit is made available step 12.3 will extract the most significant bit of the current digit and move all other bits in the digit +upwards. In effect the digit is read from most significant bit to least significant bit and since the digits are read from leading to +trailing edges the entire exponent is read from most significant bit to least significant bit. + +At step 12.5 if the $mode$ and currently extracted bit $y$ are both zero the bit is ignored and the next bit is read. This prevents the +algorithm from having to perform trivial squaring and reduction operations before the first non-zero bit is read. Step 12.6 and 12.7-10 handle +the two cases of $mode = 1$ and $mode = 2$ respectively. + +\begin{center} +\begin{figure}[here] +\includegraphics{pics/expt_state.ps} +\caption{Sliding Window State Diagram} +\label{pic:expt_state} +\end{figure} +\end{center} + +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. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_exptmod.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +Lines 32 through 46 determine the optimal window size based on the length of the exponent in bits. The window divisions are sorted +from smallest to greatest so that in each \textbf{if} statement only one condition must be tested. For example, by the \textbf{if} statement +on line 38 the value of $x$ is already known to be greater than $140$. + +The conditional piece of code beginning on line 48 allows the window size to be restricted to five bits. This logic is used to ensure +the table of precomputed powers of $G$ remains relatively small. + +The for loop on line 61 initializes the $M$ array while lines 72 and 77 through 86 initialize the reduction +function that will be used for this modulus. + +-- More later. + +\section{Quick Power of Two} +Calculating $b = 2^a$ can be performed much quicker than with any of the previous algorithms. Recall that a logical shift left $m << k$ is +equivalent to $m \cdot 2^k$. By this logic when $m = 1$ a quick power of two can be achieved. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_2expt}. \\ +\textbf{Input}. integer $b$ \\ +\textbf{Output}. $a \leftarrow 2^b$ \\ +\hline \\ +1. $a \leftarrow 0$ \\ +2. If $a.alloc < \lfloor b / lg(\beta) \rfloor + 1$ then grow $a$ appropriately. \\ +3. $a.used \leftarrow \lfloor b / lg(\beta) \rfloor + 1$ \\ +4. $a_{\lfloor b / lg(\beta) \rfloor} \leftarrow 1 << (b \mbox{ mod } lg(\beta))$ \\ +5. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_2expt} +\end{figure} + +\textbf{Algorithm mp\_2expt.} + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_2expt.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +\chapter{Higher Level Algorithms} + +This chapter discusses the various higher level algorithms that are required to complete a well rounded multiple precision integer package. These +routines are less performance oriented than the algorithms of chapters five, six and seven but are no less important. + +The first section describes a method of integer division with remainder that is universally well known. It provides the signed division logic +for the package. The subsequent section discusses a set of algorithms which allow a single digit to be the 2nd operand for a variety of operations. +These algorithms serve mostly to simplify other algorithms where small constants are required. The last two sections discuss how to manipulate +various representations of integers. For example, converting from an mp\_int to a string of character. + +\section{Integer Division with Remainder} +\label{sec:division} + +Integer division aside from modular exponentiation is the most intensive algorithm to compute. Like addition, subtraction and multiplication +the basis of this algorithm is the long-hand division algorithm taught to school children. Throughout this discussion several common variables +will be used. Let $x$ represent the divisor and $y$ represent the dividend. Let $q$ represent the integer quotient $\lfloor y / x \rfloor$ and +let $r$ represent the remainder $r = y - x \lfloor y / x \rfloor$. The following simple algorithm will be used to start the discussion. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{Radix-$\beta$ Integer Division}. \\ +\textbf{Input}. integer $x$ and $y$ \\ +\textbf{Output}. $q = \lfloor y/x\rfloor, r = y - xq$ \\ +\hline \\ +1. $q \leftarrow 0$ \\ +2. $n \leftarrow \vert \vert y \vert \vert - \vert \vert x \vert \vert$ \\ +3. for $t$ from $n$ down to $0$ do \\ +\hspace{3mm}3.1 Maximize $k$ such that $kx\beta^t$ is less than or equal to $y$ and $(k + 1)x\beta^t$ is greater. \\ +\hspace{3mm}3.2 $q \leftarrow q + k\beta^t$ \\ +\hspace{3mm}3.3 $y \leftarrow y - kx\beta^t$ \\ +4. $r \leftarrow y$ \\ +5. Return($q, r$) \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm Radix-$\beta$ Integer Division} +\label{fig:raddiv} +\end{figure} + +As children we are taught this very simple algorithm for the case of $\beta = 10$. Almost instinctively several optimizations are taught for which +their reason of existing are never explained. For this example let $y = 5471$ represent the dividend and $x = 23$ represent the divisor. + +To find the first digit of the quotient the value of $k$ must be maximized such that $kx\beta^t$ is less than or equal to $y$ and +simultaneously $(k + 1)x\beta^t$ is greater than $y$. Implicitly $k$ is the maximum value the $t$'th digit of the quotient may have. The habitual method +used to find the maximum is to ``eyeball'' the two numbers, typically only the leading digits and quickly estimate a quotient. By only using leading +digits a much simpler division may be used to form an educated guess at what the value must be. In this case $k = \lfloor 54/23\rfloor = 2$ quickly +arises as a possible solution. Indeed $2x\beta^2 = 4600$ is less than $y = 5471$ and simultaneously $(k + 1)x\beta^2 = 6900$ is larger than $y$. +As a result $k\beta^2$ is added to the quotient which now equals $q = 200$ and $4600$ is subtracted from $y$ to give a remainder of $y = 841$. + +Again this process is repeated to produce the quotient digit $k = 3$ which makes the quotient $q = 200 + 3\beta = 230$ and the remainder +$y = 841 - 3x\beta = 181$. Finally the last iteration of the loop produces $k = 7$ which leads to the quotient $q = 230 + 7 = 237$ and the +remainder $y = 181 - 7x = 20$. The final quotient and remainder found are $q = 237$ and $r = y = 20$ which are indeed correct since +$237 \cdot 23 + 20 = 5471$ is true. + +\subsection{Quotient Estimation} +\label{sec:divest} +As alluded to earlier the quotient digit $k$ can be estimated from only the leading digits of both the divisor and dividend. When $p$ leading +digits are used from both the divisor and dividend to form an estimation the accuracy of the estimation rises as $p$ grows. Technically +speaking the estimation is based on assuming the lower $\vert \vert y \vert \vert - p$ and $\vert \vert x \vert \vert - p$ lower digits of the +dividend and divisor are zero. + +The value of the estimation may off by a few values in either direction and in general is fairly correct. A simplification \cite[pp. 271]{TAOCPV2} +of the estimation technique is to use $t + 1$ digits of the dividend and $t$ digits of the divisor, in particularly when $t = 1$. The estimate +using this technique is never too small. For the following proof let $t = \vert \vert y \vert \vert - 1$ and $s = \vert \vert x \vert \vert - 1$ +represent the most significant digits of the dividend and divisor respectively. + +\textbf{Proof.}\textit{ The quotient $\hat k = \lfloor (y_t\beta + y_{t-1}) / x_s \rfloor$ is greater than or equal to +$k = \lfloor y / (x \cdot \beta^{\vert \vert y \vert \vert - \vert \vert x \vert \vert - 1}) \rfloor$. } +The first obvious case is when $\hat k = \beta - 1$ in which case the proof is concluded since the real quotient cannot be larger. For all other +cases $\hat k = \lfloor (y_t\beta + y_{t-1}) / x_s \rfloor$ and $\hat k x_s \ge y_t\beta + y_{t-1} - x_s + 1$. The latter portion of the inequalility +$-x_s + 1$ arises from the fact that a truncated integer division will give the same quotient for at most $x_s - 1$ values. Next a series of +inequalities will prove the hypothesis. + +\begin{equation} +y - \hat k x \le y - \hat k x_s\beta^s +\end{equation} + +This is trivially true since $x \ge x_s\beta^s$. Next we replace $\hat kx_s\beta^s$ by the previous inequality for $\hat kx_s$. + +\begin{equation} +y - \hat k x \le y_t\beta^t + \ldots + y_0 - (y_t\beta^t + y_{t-1}\beta^{t-1} - x_s\beta^t + \beta^s) +\end{equation} + +By simplifying the previous inequality the following inequality is formed. + +\begin{equation} +y - \hat k x \le y_{t-2}\beta^{t-2} + \ldots + y_0 + x_s\beta^s - \beta^s +\end{equation} + +Subsequently, + +\begin{equation} +y_{t-2}\beta^{t-2} + \ldots + y_0 + x_s\beta^s - \beta^s < x_s\beta^s \le x +\end{equation} + +Which proves that $y - \hat kx \le x$ and by consequence $\hat k \ge k$ which concludes the proof. \textbf{QED} + + +\subsection{Normalized Integers} +For the purposes of division a normalized input is when the divisors leading digit $x_n$ is greater than or equal to $\beta / 2$. By multiplying both +$x$ and $y$ by $j = \lfloor (\beta / 2) / x_n \rfloor$ the quotient remains unchanged and the remainder is simply $j$ times the original +remainder. The purpose of normalization is to ensure the leading digit of the divisor is sufficiently large such that the estimated quotient will +lie in the domain of a single digit. Consider the maximum dividend $(\beta - 1) \cdot \beta + (\beta - 1)$ and the minimum divisor $\beta / 2$. + +\begin{equation} +{{\beta^2 - 1} \over { \beta / 2}} \le 2\beta - {2 \over \beta} +\end{equation} + +At most the quotient approaches $2\beta$, however, in practice this will not occur since that would imply the previous quotient digit was too small. + +\subsection{Radix-$\beta$ Division with Remainder} +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_div}. \\ +\textbf{Input}. mp\_int $a, b$ \\ +\textbf{Output}. $c = \lfloor a/b \rfloor$, $d = a - bc$ \\ +\hline \\ +1. If $b = 0$ return(\textit{MP\_VAL}). \\ +2. If $\vert a \vert < \vert b \vert$ then do \\ +\hspace{3mm}2.1 $d \leftarrow a$ \\ +\hspace{3mm}2.2 $c \leftarrow 0$ \\ +\hspace{3mm}2.3 Return(\textit{MP\_OKAY}). \\ +\\ +Setup the quotient to receive the digits. \\ +3. Grow $q$ to $a.used + 2$ digits. \\ +4. $q \leftarrow 0$ \\ +5. $x \leftarrow \vert a \vert , y \leftarrow \vert b \vert$ \\ +6. $sign \leftarrow \left \lbrace \begin{array}{ll} + MP\_ZPOS & \mbox{if }a.sign = b.sign \\ + MP\_NEG & \mbox{otherwise} \\ + \end{array} \right .$ \\ +\\ +Normalize the inputs such that the leading digit of $y$ is greater than or equal to $\beta / 2$. \\ +7. $norm \leftarrow (lg(\beta) - 1) - (\lceil lg(y) \rceil \mbox{ (mod }lg(\beta)\mbox{)})$ \\ +8. $x \leftarrow x \cdot 2^{norm}, y \leftarrow y \cdot 2^{norm}$ \\ +\\ +Find the leading digit of the quotient. \\ +9. $n \leftarrow x.used - 1, t \leftarrow y.used - 1$ \\ +10. $y \leftarrow y \cdot \beta^{n - t}$ \\ +11. While ($x \ge y$) do \\ +\hspace{3mm}11.1 $q_{n - t} \leftarrow q_{n - t} + 1$ \\ +\hspace{3mm}11.2 $x \leftarrow x - y$ \\ +12. $y \leftarrow \lfloor y / \beta^{n-t} \rfloor$ \\ +\\ +Continued on the next page. \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_div} +\end{figure} + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_div} (continued). \\ +\textbf{Input}. mp\_int $a, b$ \\ +\textbf{Output}. $c = \lfloor a/b \rfloor$, $d = a - bc$ \\ +\hline \\ +Now find the remainder fo the digits. \\ +13. for $i$ from $n$ down to $(t + 1)$ do \\ +\hspace{3mm}13.1 If $i > x.used$ then jump to the next iteration of this loop. \\ +\hspace{3mm}13.2 If $x_{i} = y_{t}$ then \\ +\hspace{6mm}13.2.1 $q_{i - t - 1} \leftarrow \beta - 1$ \\ +\hspace{3mm}13.3 else \\ +\hspace{6mm}13.3.1 $\hat r \leftarrow x_{i} \cdot \beta + x_{i - 1}$ \\ +\hspace{6mm}13.3.2 $\hat r \leftarrow \lfloor \hat r / y_{t} \rfloor$ \\ +\hspace{6mm}13.3.3 $q_{i - t - 1} \leftarrow \hat r$ \\ +\hspace{3mm}13.4 $q_{i - t - 1} \leftarrow q_{i - t - 1} + 1$ \\ +\\ +Fixup quotient estimation. \\ +\hspace{3mm}13.5 Loop \\ +\hspace{6mm}13.5.1 $q_{i - t - 1} \leftarrow q_{i - t - 1} - 1$ \\ +\hspace{6mm}13.5.2 t$1 \leftarrow 0$ \\ +\hspace{6mm}13.5.3 t$1_0 \leftarrow y_{t - 1}, $ t$1_1 \leftarrow y_t,$ t$1.used \leftarrow 2$ \\ +\hspace{6mm}13.5.4 $t1 \leftarrow t1 \cdot q_{i - t - 1}$ \\ +\hspace{6mm}13.5.5 t$2_0 \leftarrow x_{i - 2}, $ t$2_1 \leftarrow x_{i - 1}, $ t$2_2 \leftarrow x_i, $ t$2.used \leftarrow 3$ \\ +\hspace{6mm}13.5.6 If $\vert t1 \vert > \vert t2 \vert$ then goto step 13.5. \\ +\hspace{3mm}13.6 t$1 \leftarrow y \cdot q_{i - t - 1}$ \\ +\hspace{3mm}13.7 t$1 \leftarrow $ t$1 \cdot \beta^{i - t - 1}$ \\ +\hspace{3mm}13.8 $x \leftarrow x - $ t$1$ \\ +\hspace{3mm}13.9 If $x.sign = MP\_NEG$ then \\ +\hspace{6mm}13.10 t$1 \leftarrow y$ \\ +\hspace{6mm}13.11 t$1 \leftarrow $ t$1 \cdot \beta^{i - t - 1}$ \\ +\hspace{6mm}13.12 $x \leftarrow x + $ t$1$ \\ +\hspace{6mm}13.13 $q_{i - t - 1} \leftarrow q_{i - t - 1} - 1$ \\ +\\ +Finalize the result. \\ +14. Clamp excess digits of $q$ \\ +15. $c \leftarrow q, c.sign \leftarrow sign$ \\ +16. $x.sign \leftarrow a.sign$ \\ +17. $d \leftarrow \lfloor x / 2^{norm} \rfloor$ \\ +18. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_div (continued)} +\end{figure} +\textbf{Algorithm mp\_div.} +This algorithm will calculate quotient and remainder from an integer division given a dividend and divisor. The algorithm is a signed +division and will produce a fully qualified quotient and remainder. + +First the divisor $b$ must be non-zero which is enforced in step one. If the divisor is larger than the dividend than the quotient is implicitly +zero and the remainder is the dividend. + +After the first two trivial cases of inputs are handled the variable $q$ is setup to receive the digits of the quotient. Two unsigned copies of the +divisor $y$ and dividend $x$ are made as well. The core of the division algorithm is an unsigned division and will only work if the values are +positive. Now the two values $x$ and $y$ must be normalized such that the leading digit of $y$ is greater than or equal to $\beta / 2$. +This is performed by shifting both to the left by enough bits to get the desired normalization. + +At this point the division algorithm can begin producing digits of the quotient. Recall that maximum value of the estimation used is +$2\beta - {2 \over \beta}$ which means that a digit of the quotient must be first produced by another means. In this case $y$ is shifted +to the left (\textit{step ten}) so that it has the same number of digits as $x$. The loop on step eleven will subtract multiples of the +shifted copy of $y$ until $x$ is smaller. Since the leading digit of $y$ is greater than or equal to $\beta/2$ this loop will iterate at most two +times to produce the desired leading digit of the quotient. + +Now the remainder of the digits can be produced. The equation $\hat q = \lfloor {{x_i \beta + x_{i-1}}\over y_t} \rfloor$ is used to fairly +accurately approximate the true quotient digit. The estimation can in theory produce an estimation as high as $2\beta - {2 \over \beta}$ but by +induction the upper quotient digit is correct (\textit{as established on step eleven}) and the estimate must be less than $\beta$. + +Recall from section~\ref{sec:divest} that the estimation is never too low but may be too high. The next step of the estimation process is +to refine the estimation. The loop on step 13.5 uses $x_i\beta^2 + x_{i-1}\beta + x_{i-2}$ and $q_{i - t - 1}(y_t\beta + y_{t-1})$ as a higher +order approximation to adjust the quotient digit. + +After both phases of estimation the quotient digit may still be off by a value of one\footnote{This is similar to the error introduced +by optimizing Barrett reduction.}. Steps 13.6 and 13.7 subtract the multiple of the divisor from the dividend (\textit{Similar to step 3.3 of +algorithm~\ref{fig:raddiv}} and then subsequently add a multiple of the divisor if the quotient was too large. + +Now that the quotient has been determine finializing the result is a matter of clamping the quotient, fixing the sizes and de-normalizing the +remainder. An important aspect of this algorithm seemingly overlooked in other descriptions such as that of Algorithm 14.20 HAC \cite[pp. 598]{HAC} +is that when the estimations are being made (\textit{inside the loop on step 13.5}) that the digits $y_{t-1}$, $x_{i-2}$ and $x_{i-1}$ may lie +outside their respective boundaries. For example, if $t = 0$ or $i \le 1$ then the digits would be undefined. In those cases the digits should +respectively be replaced with a zero. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_div.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +The implementation of this algorithm differs slightly from the pseudo code presented previously. In this algorithm either of the quotient $c$ or +remainder $d$ may be passed as a \textbf{NULL} pointer which indicates their value is not desired. For example, the C code to call the division +algorithm with only the quotient is + +\begin{verbatim} +mp_div(&a, &b, &c, NULL); /* c = [a/b] */ +\end{verbatim} + +Lines 109 and 113 handle the two trivial cases of inputs which are division by zero and dividend smaller than the divisor +respectively. After the two trivial cases all of the temporary variables are initialized. Line 148 determines the sign of +the quotient and line 148 ensures that both $x$ and $y$ are positive. + +The number of bits in the leading digit is calculated on line 151. Implictly an mp\_int with $r$ digits will require $lg(\beta)(r-1) + k$ bits +of precision which when reduced modulo $lg(\beta)$ produces the value of $k$. In this case $k$ is the number of bits in the leading digit which is +exactly what is required. For the algorithm to operate $k$ must equal $lg(\beta) - 1$ and when it does not the inputs must be normalized by shifting +them to the left by $lg(\beta) - 1 - k$ bits. + +Throughout the variables $n$ and $t$ will represent the highest digit of $x$ and $y$ respectively. These are first used to produce the +leading digit of the quotient. The loop beginning on line 184 will produce the remainder of the quotient digits. + +The conditional ``continue'' on line 187 is used to prevent the algorithm from reading past the leading edge of $x$ which can occur when the +algorithm eliminates multiple non-zero digits in a single iteration. This ensures that $x_i$ is always non-zero since by definition the digits +above the $i$'th position $x$ must be zero in order for the quotient to be precise\footnote{Precise as far as integer division is concerned.}. + +Lines 214, 216 and 223 through 225 manually construct the high accuracy estimations by setting the digits of the two mp\_int +variables directly. + +\section{Single Digit Helpers} + +This section briefly describes a series of single digit helper algorithms which come in handy when working with small constants. All of +the helper functions assume the single digit input is positive and will treat them as such. + +\subsection{Single Digit Addition and Subtraction} + +Both addition and subtraction are performed by ``cheating'' and using mp\_set followed by the higher level addition or subtraction +algorithms. As a result these algorithms are subtantially simpler with a slight cost in performance. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_add\_d}. \\ +\textbf{Input}. mp\_int $a$ and a mp\_digit $b$ \\ +\textbf{Output}. $c = a + b$ \\ +\hline \\ +1. $t \leftarrow b$ (\textit{mp\_set}) \\ +2. $c \leftarrow a + t$ \\ +3. Return(\textit{MP\_OKAY}) \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_add\_d} +\end{figure} + +\textbf{Algorithm mp\_add\_d.} +This algorithm initiates a temporary mp\_int with the value of the single digit and uses algorithm mp\_add to add the two values together. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_add\_d.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +Clever use of the letter 't'. + +\subsubsection{Subtraction} +The single digit subtraction algorithm mp\_sub\_d is essentially the same except it uses mp\_sub to subtract the digit from the mp\_int. + +\subsection{Single Digit Multiplication} +Single digit multiplication arises enough in division and radix conversion that it ought to be implement as a special case of the baseline +multiplication algorithm. Essentially this algorithm is a modified version of algorithm s\_mp\_mul\_digs where one of the multiplicands +only has one digit. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_mul\_d}. \\ +\textbf{Input}. mp\_int $a$ and a mp\_digit $b$ \\ +\textbf{Output}. $c = ab$ \\ +\hline \\ +1. $pa \leftarrow a.used$ \\ +2. Grow $c$ to at least $pa + 1$ digits. \\ +3. $oldused \leftarrow c.used$ \\ +4. $c.used \leftarrow pa + 1$ \\ +5. $c.sign \leftarrow a.sign$ \\ +6. $\mu \leftarrow 0$ \\ +7. for $ix$ from $0$ to $pa - 1$ do \\ +\hspace{3mm}7.1 $\hat r \leftarrow \mu + a_{ix}b$ \\ +\hspace{3mm}7.2 $c_{ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ +\hspace{3mm}7.3 $\mu \leftarrow \lfloor \hat r / \beta \rfloor$ \\ +8. $c_{pa} \leftarrow \mu$ \\ +9. for $ix$ from $pa + 1$ to $oldused$ do \\ +\hspace{3mm}9.1 $c_{ix} \leftarrow 0$ \\ +10. Clamp excess digits of $c$. \\ +11. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_mul\_d} +\end{figure} +\textbf{Algorithm mp\_mul\_d.} +This algorithm quickly multiplies an mp\_int by a small single digit value. It is specially tailored to the job and has a minimal of overhead. +Unlike the full multiplication algorithms this algorithm does not require any significnat temporary storage or memory allocations. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_mul\_d.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +In this implementation the destination $c$ may point to the same mp\_int as the source $a$ since the result is written after the digit is +read from the source. This function uses pointer aliases $tmpa$ and $tmpc$ for the digits of $a$ and $c$ respectively. + +\subsection{Single Digit Division} +Like the single digit multiplication algorithm, single digit division is also a fairly common algorithm used in radix conversion. Since the +divisor is only a single digit a specialized variant of the division algorithm can be used to compute the quotient. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_div\_d}. \\ +\textbf{Input}. mp\_int $a$ and a mp\_digit $b$ \\ +\textbf{Output}. $c = \lfloor a / b \rfloor, d = a - cb$ \\ +\hline \\ +1. If $b = 0$ then return(\textit{MP\_VAL}).\\ +2. If $b = 3$ then use algorithm mp\_div\_3 instead. \\ +3. Init $q$ to $a.used$ digits. \\ +4. $q.used \leftarrow a.used$ \\ +5. $q.sign \leftarrow a.sign$ \\ +6. $\hat w \leftarrow 0$ \\ +7. for $ix$ from $a.used - 1$ down to $0$ do \\ +\hspace{3mm}7.1 $\hat w \leftarrow \hat w \beta + a_{ix}$ \\ +\hspace{3mm}7.2 If $\hat w \ge b$ then \\ +\hspace{6mm}7.2.1 $t \leftarrow \lfloor \hat w / b \rfloor$ \\ +\hspace{6mm}7.2.2 $\hat w \leftarrow \hat w \mbox{ (mod }b\mbox{)}$ \\ +\hspace{3mm}7.3 else\\ +\hspace{6mm}7.3.1 $t \leftarrow 0$ \\ +\hspace{3mm}7.4 $q_{ix} \leftarrow t$ \\ +8. $d \leftarrow \hat w$ \\ +9. Clamp excess digits of $q$. \\ +10. $c \leftarrow q$ \\ +11. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_div\_d} +\end{figure} +\textbf{Algorithm mp\_div\_d.} +This algorithm divides the mp\_int $a$ by the single mp\_digit $b$ using an optimized approach. Essentially in every iteration of the +algorithm another digit of the dividend is reduced and another digit of quotient produced. Provided $b < \beta$ the value of $\hat w$ +after step 7.1 will be limited such that $0 \le \lfloor \hat w / b \rfloor < \beta$. + +If the divisor $b$ is equal to three a variant of this algorithm is used which is called mp\_div\_3. It replaces the division by three with +a multiplication by $\lfloor \beta / 3 \rfloor$ and the appropriate shift and residual fixup. In essence it is much like the Barrett reduction +from chapter seven. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_div\_d.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +Like the implementation of algorithm mp\_div this algorithm allows either of the quotient or remainder to be passed as a \textbf{NULL} pointer to +indicate the respective value is not required. This allows a trivial single digit modular reduction algorithm, mp\_mod\_d to be created. + +The division and remainder on lines 44 and @45,%@ can be replaced often by a single division on most processors. For example, the 32-bit x86 based +processors can divide a 64-bit quantity by a 32-bit quantity and produce the quotient and remainder simultaneously. Unfortunately the GCC +compiler does not recognize that optimization and will actually produce two function calls to find the quotient and remainder respectively. + +\subsection{Single Digit Root Extraction} + +Finding the $n$'th root of an integer is fairly easy as far as numerical analysis is concerned. Algorithms such as the Newton-Raphson approximation +(\ref{eqn:newton}) series will converge very quickly to a root for any continuous function $f(x)$. + +\begin{equation} +x_{i+1} = x_i - {f(x_i) \over f'(x_i)} +\label{eqn:newton} +\end{equation} + +In this case the $n$'th root is desired and $f(x) = x^n - a$ where $a$ is the integer of which the root is desired. The derivative of $f(x)$ is +simply $f'(x) = nx^{n - 1}$. Of particular importance is that this algorithm will be used over the integers not over the a more continuous domain +such as the real numbers. As a result the root found can be above the true root by few and must be manually adjusted. Ideally at the end of the +algorithm the $n$'th root $b$ of an integer $a$ is desired such that $b^n \le a$. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_n\_root}. \\ +\textbf{Input}. mp\_int $a$ and a mp\_digit $b$ \\ +\textbf{Output}. $c^b \le a$ \\ +\hline \\ +1. If $b$ is even and $a.sign = MP\_NEG$ return(\textit{MP\_VAL}). \\ +2. $sign \leftarrow a.sign$ \\ +3. $a.sign \leftarrow MP\_ZPOS$ \\ +4. t$2 \leftarrow 2$ \\ +5. Loop \\ +\hspace{3mm}5.1 t$1 \leftarrow $ t$2$ \\ +\hspace{3mm}5.2 t$3 \leftarrow $ t$1^{b - 1}$ \\ +\hspace{3mm}5.3 t$2 \leftarrow $ t$3 $ $\cdot$ t$1$ \\ +\hspace{3mm}5.4 t$2 \leftarrow $ t$2 - a$ \\ +\hspace{3mm}5.5 t$3 \leftarrow $ t$3 \cdot b$ \\ +\hspace{3mm}5.6 t$3 \leftarrow \lfloor $t$2 / $t$3 \rfloor$ \\ +\hspace{3mm}5.7 t$2 \leftarrow $ t$1 - $ t$3$ \\ +\hspace{3mm}5.8 If t$1 \ne $ t$2$ then goto step 5. \\ +6. Loop \\ +\hspace{3mm}6.1 t$2 \leftarrow $ t$1^b$ \\ +\hspace{3mm}6.2 If t$2 > a$ then \\ +\hspace{6mm}6.2.1 t$1 \leftarrow $ t$1 - 1$ \\ +\hspace{6mm}6.2.2 Goto step 6. \\ +7. $a.sign \leftarrow sign$ \\ +8. $c \leftarrow $ t$1$ \\ +9. $c.sign \leftarrow sign$ \\ +10. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_n\_root} +\end{figure} +\textbf{Algorithm mp\_n\_root.} +This algorithm finds the integer $n$'th root of an input using the Newton-Raphson approach. It is partially optimized based on the observation +that the numerator of ${f(x) \over f'(x)}$ can be derived from a partial denominator. That is at first the denominator is calculated by finding +$x^{b - 1}$. This value can then be multiplied by $x$ and have $a$ subtracted from it to find the numerator. This saves a total of $b - 1$ +multiplications by t$1$ inside the loop. + +The initial value of the approximation is t$2 = 2$ which allows the algorithm to start with very small values and quickly converge on the +root. Ideally this algorithm is meant to find the $n$'th root of an input where $n$ is bounded by $2 \le n \le 5$. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_n\_root.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +\section{Random Number Generation} + +Random numbers come up in a variety of activities from public key cryptography to simple simulations and various randomized algorithms. Pollard-Rho +factoring for example, can make use of random values as starting points to find factors of a composite integer. In this case the algorithm presented +is solely for simulations and not intended for cryptographic use. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_rand}. \\ +\textbf{Input}. An integer $b$ \\ +\textbf{Output}. A pseudo-random number of $b$ digits \\ +\hline \\ +1. $a \leftarrow 0$ \\ +2. If $b \le 0$ return(\textit{MP\_OKAY}) \\ +3. Pick a non-zero random digit $d$. \\ +4. $a \leftarrow a + d$ \\ +5. for $ix$ from 1 to $d - 1$ do \\ +\hspace{3mm}5.1 $a \leftarrow a \cdot \beta$ \\ +\hspace{3mm}5.2 Pick a random digit $d$. \\ +\hspace{3mm}5.3 $a \leftarrow a + d$ \\ +6. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_rand} +\end{figure} +\textbf{Algorithm mp\_rand.} +This algorithm produces a pseudo-random integer of $b$ digits. By ensuring that the first digit is non-zero the algorithm also guarantees that the +final result has at least $b$ digits. It relies heavily on a third-part random number generator which should ideally generate uniformly all of +the integers from $0$ to $\beta - 1$. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_rand.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +\section{Formatted Representations} +The ability to emit a radix-$n$ textual representation of an integer is useful for interacting with human parties. For example, the ability to +be given a string of characters such as ``114585'' and turn it into the radix-$\beta$ equivalent would make it easier to enter numbers +into a program. + +\subsection{Reading Radix-n Input} +For the purposes of this text we will assume that a simple lower ASCII map (\ref{fig:ASC}) is used for the values of from $0$ to $63$ to +printable characters. For example, when the character ``N'' is read it represents the integer $23$. The first $16$ characters of the +map are for the common representations up to hexadecimal. After that they match the ``base64'' encoding scheme which are suitable chosen +such that they are printable. While outputting as base64 may not be too helpful for human operators it does allow communication via non binary +mediums. + +\newpage\begin{figure}[here] +\begin{center} +\begin{tabular}{cc|cc|cc|cc} +\hline \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} \\ +\hline +0 & 0 & 1 & 1 & 2 & 2 & 3 & 3 \\ +4 & 4 & 5 & 5 & 6 & 6 & 7 & 7 \\ +8 & 8 & 9 & 9 & 10 & A & 11 & B \\ +12 & C & 13 & D & 14 & E & 15 & F \\ +16 & G & 17 & H & 18 & I & 19 & J \\ +20 & K & 21 & L & 22 & M & 23 & N \\ +24 & O & 25 & P & 26 & Q & 27 & R \\ +28 & S & 29 & T & 30 & U & 31 & V \\ +32 & W & 33 & X & 34 & Y & 35 & Z \\ +36 & a & 37 & b & 38 & c & 39 & d \\ +40 & e & 41 & f & 42 & g & 43 & h \\ +44 & i & 45 & j & 46 & k & 47 & l \\ +48 & m & 49 & n & 50 & o & 51 & p \\ +52 & q & 53 & r & 54 & s & 55 & t \\ +56 & u & 57 & v & 58 & w & 59 & x \\ +60 & y & 61 & z & 62 & $+$ & 63 & $/$ \\ +\hline +\end{tabular} +\end{center} +\caption{Lower ASCII Map} +\label{fig:ASC} +\end{figure} + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_read\_radix}. \\ +\textbf{Input}. A string $str$ of length $sn$ and radix $r$. \\ +\textbf{Output}. The radix-$\beta$ equivalent mp\_int. \\ +\hline \\ +1. If $r < 2$ or $r > 64$ return(\textit{MP\_VAL}). \\ +2. $ix \leftarrow 0$ \\ +3. If $str_0 =$ ``-'' then do \\ +\hspace{3mm}3.1 $ix \leftarrow ix + 1$ \\ +\hspace{3mm}3.2 $sign \leftarrow MP\_NEG$ \\ +4. else \\ +\hspace{3mm}4.1 $sign \leftarrow MP\_ZPOS$ \\ +5. $a \leftarrow 0$ \\ +6. for $iy$ from $ix$ to $sn - 1$ do \\ +\hspace{3mm}6.1 Let $y$ denote the position in the map of $str_{iy}$. \\ +\hspace{3mm}6.2 If $str_{iy}$ is not in the map or $y \ge r$ then goto step 7. \\ +\hspace{3mm}6.3 $a \leftarrow a \cdot r$ \\ +\hspace{3mm}6.4 $a \leftarrow a + y$ \\ +7. If $a \ne 0$ then $a.sign \leftarrow sign$ \\ +8. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_read\_radix} +\end{figure} +\textbf{Algorithm mp\_read\_radix.} +This algorithm will read an ASCII string and produce the radix-$\beta$ mp\_int representation of the same integer. A minus symbol ``-'' may precede the +string to indicate the value is negative, otherwise it is assumed to be positive. The algorithm will read up to $sn$ characters from the input +and will stop when it reads a character it cannot map the algorithm stops reading characters from the string. This allows numbers to be embedded +as part of larger input without any significant problem. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_read\_radix.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +\subsection{Generating Radix-$n$ Output} +Generating radix-$n$ output is fairly trivial with a division and remainder algorithm. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_toradix}. \\ +\textbf{Input}. A mp\_int $a$ and an integer $r$\\ +\textbf{Output}. The radix-$r$ representation of $a$ \\ +\hline \\ +1. If $r < 2$ or $r > 64$ return(\textit{MP\_VAL}). \\ +2. If $a = 0$ then $str = $ ``$0$'' and return(\textit{MP\_OKAY}). \\ +3. $t \leftarrow a$ \\ +4. $str \leftarrow$ ``'' \\ +5. if $t.sign = MP\_NEG$ then \\ +\hspace{3mm}5.1 $str \leftarrow str + $ ``-'' \\ +\hspace{3mm}5.2 $t.sign = MP\_ZPOS$ \\ +6. While ($t \ne 0$) do \\ +\hspace{3mm}6.1 $d \leftarrow t \mbox{ (mod }r\mbox{)}$ \\ +\hspace{3mm}6.2 $t \leftarrow \lfloor t / r \rfloor$ \\ +\hspace{3mm}6.3 Look up $d$ in the map and store the equivalent character in $y$. \\ +\hspace{3mm}6.4 $str \leftarrow str + y$ \\ +7. If $str_0 = $``$-$'' then \\ +\hspace{3mm}7.1 Reverse the digits $str_1, str_2, \ldots str_n$. \\ +8. Otherwise \\ +\hspace{3mm}8.1 Reverse the digits $str_0, str_1, \ldots str_n$. \\ +9. Return(\textit{MP\_OKAY}).\\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_toradix} +\end{figure} +\textbf{Algorithm mp\_toradix.} +This algorithm computes the radix-$r$ representation of an mp\_int $a$. The ``digits'' of the representation are extracted by reducing +successive powers of $\lfloor a / r^k \rfloor$ the input modulo $r$ until $r^k > a$. Note that instead of actually dividing by $r^k$ in +each iteration the quotient $\lfloor a / r \rfloor$ is saved for the next iteration. As a result a series of trivial $n \times 1$ divisions +are required instead of a series of $n \times k$ divisions. One design flaw of this approach is that the digits are produced in the reverse order +(see~\ref{fig:mpradix}). To remedy this flaw the digits must be swapped or simply ``reversed''. + +\begin{figure} +\begin{center} +\begin{tabular}{|c|c|c|} +\hline \textbf{Value of $a$} & \textbf{Value of $d$} & \textbf{Value of $str$} \\ +\hline $1234$ & -- & -- \\ +\hline $123$ & $4$ & ``4'' \\ +\hline $12$ & $3$ & ``43'' \\ +\hline $1$ & $2$ & ``432'' \\ +\hline $0$ & $1$ & ``4321'' \\ +\hline +\end{tabular} +\end{center} +\caption{Example of Algorithm mp\_toradix.} +\label{fig:mpradix} +\end{figure} + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_toradix.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +\chapter{Number Theoretic Algorithms} +This chapter discusses several fundamental number theoretic algorithms such as the greatest common divisor, least common multiple and Jacobi +symbol computation. These algorithms arise as essential components in several key cryptographic algorithms such as the RSA public key algorithm and +various Sieve based factoring algorithms. + +\section{Greatest Common Divisor} +The greatest common divisor of two integers $a$ and $b$, often denoted as $(a, b)$ is the largest integer $k$ that is a proper divisor of +both $a$ and $b$. That is, $k$ is the largest integer such that $0 \equiv a \mbox{ (mod }k\mbox{)}$ and $0 \equiv b \mbox{ (mod }k\mbox{)}$ occur +simultaneously. + +The most common approach (cite) is to reduce one input modulo another. That is if $a$ and $b$ are divisible by some integer $k$ and if $qa + r = b$ then +$r$ is also divisible by $k$. The reduction pattern follows $\left < a , b \right > \rightarrow \left < b, a \mbox{ mod } b \right >$. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{Greatest Common Divisor (I)}. \\ +\textbf{Input}. Two positive integers $a$ and $b$ greater than zero. \\ +\textbf{Output}. The greatest common divisor $(a, b)$. \\ +\hline \\ +1. While ($b > 0$) do \\ +\hspace{3mm}1.1 $r \leftarrow a \mbox{ (mod }b\mbox{)}$ \\ +\hspace{3mm}1.2 $a \leftarrow b$ \\ +\hspace{3mm}1.3 $b \leftarrow r$ \\ +2. Return($a$). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm Greatest Common Divisor (I)} +\label{fig:gcd1} +\end{figure} + +This algorithm will quickly converge on the greatest common divisor since the residue $r$ tends diminish rapidly. However, divisions are +relatively expensive operations to perform and should ideally be avoided. There is another approach based on a similar relationship of +greatest common divisors. The faster approach is based on the observation that if $k$ divides both $a$ and $b$ it will also divide $a - b$. +In particular, we would like $a - b$ to decrease in magnitude which implies that $b \ge a$. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{Greatest Common Divisor (II)}. \\ +\textbf{Input}. Two positive integers $a$ and $b$ greater than zero. \\ +\textbf{Output}. The greatest common divisor $(a, b)$. \\ +\hline \\ +1. While ($b > 0$) do \\ +\hspace{3mm}1.1 Swap $a$ and $b$ such that $a$ is the smallest of the two. \\ +\hspace{3mm}1.2 $b \leftarrow b - a$ \\ +2. Return($a$). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm Greatest Common Divisor (II)} +\label{fig:gcd2} +\end{figure} + +\textbf{Proof} \textit{Algorithm~\ref{fig:gcd2} will return the greatest common divisor of $a$ and $b$.} +The algorithm in figure~\ref{fig:gcd2} will eventually terminate since $b \ge a$ the subtraction in step 1.2 will be a value less than $b$. In other +words in every iteration that tuple $\left < a, b \right >$ decrease in magnitude until eventually $a = b$. Since both $a$ and $b$ are always +divisible by the greatest common divisor (\textit{until the last iteration}) and in the last iteration of the algorithm $b = 0$, therefore, in the +second to last iteration of the algorithm $b = a$ and clearly $(a, a) = a$ which concludes the proof. \textbf{QED}. + +As a matter of practicality algorithm \ref{fig:gcd1} decreases far too slowly to be useful. Specially if $b$ is much larger than $a$ such that +$b - a$ is still very much larger than $a$. A simple addition to the algorithm is to divide $b - a$ by a power of some integer $p$ which does +not divide the greatest common divisor but will divide $b - a$. In this case ${b - a} \over p$ is also an integer and still divisible by +the greatest common divisor. + +However, instead of factoring $b - a$ to find a suitable value of $p$ the powers of $p$ can be removed from $a$ and $b$ that are in common first. +Then inside the loop whenever $b - a$ is divisible by some power of $p$ it can be safely removed. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{Greatest Common Divisor (III)}. \\ +\textbf{Input}. Two positive integers $a$ and $b$ greater than zero. \\ +\textbf{Output}. The greatest common divisor $(a, b)$. \\ +\hline \\ +1. $k \leftarrow 0$ \\ +2. While $a$ and $b$ are both divisible by $p$ do \\ +\hspace{3mm}2.1 $a \leftarrow \lfloor a / p \rfloor$ \\ +\hspace{3mm}2.2 $b \leftarrow \lfloor b / p \rfloor$ \\ +\hspace{3mm}2.3 $k \leftarrow k + 1$ \\ +3. While $a$ is divisible by $p$ do \\ +\hspace{3mm}3.1 $a \leftarrow \lfloor a / p \rfloor$ \\ +4. While $b$ is divisible by $p$ do \\ +\hspace{3mm}4.1 $b \leftarrow \lfloor b / p \rfloor$ \\ +5. While ($b > 0$) do \\ +\hspace{3mm}5.1 Swap $a$ and $b$ such that $a$ is the smallest of the two. \\ +\hspace{3mm}5.2 $b \leftarrow b - a$ \\ +\hspace{3mm}5.3 While $b$ is divisible by $p$ do \\ +\hspace{6mm}5.3.1 $b \leftarrow \lfloor b / p \rfloor$ \\ +6. Return($a \cdot p^k$). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm Greatest Common Divisor (III)} +\label{fig:gcd3} +\end{figure} + +This algorithm is based on the first except it removes powers of $p$ first and inside the main loop to ensure the tuple $\left < a, b \right >$ +decreases more rapidly. The first loop on step two removes powers of $p$ that are in common. A count, $k$, is kept which will present a common +divisor of $p^k$. After step two the remaining common divisor of $a$ and $b$ cannot be divisible by $p$. This means that $p$ can be safely +divided out of the difference $b - a$ so long as the division leaves no remainder. + +In particular the value of $p$ should be chosen such that the division on step 5.3.1 occur often. It also helps that division by $p$ be easy +to compute. The ideal choice of $p$ is two since division by two amounts to a right logical shift. Another important observation is that by +step five both $a$ and $b$ are odd. Therefore, the diffrence $b - a$ must be even which means that each iteration removes one bit from the +largest of the pair. + +\subsection{Complete Greatest Common Divisor} +The algorithms presented so far cannot handle inputs which are zero or negative. The following algorithm can handle all input cases properly +and will produce the greatest common divisor. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_gcd}. \\ +\textbf{Input}. mp\_int $a$ and $b$ \\ +\textbf{Output}. The greatest common divisor $c = (a, b)$. \\ +\hline \\ +1. If $a = 0$ then \\ +\hspace{3mm}1.1 $c \leftarrow \vert b \vert $ \\ +\hspace{3mm}1.2 Return(\textit{MP\_OKAY}). \\ +2. If $b = 0$ then \\ +\hspace{3mm}2.1 $c \leftarrow \vert a \vert $ \\ +\hspace{3mm}2.2 Return(\textit{MP\_OKAY}). \\ +3. $u \leftarrow \vert a \vert, v \leftarrow \vert b \vert$ \\ +4. $k \leftarrow 0$ \\ +5. While $u.used > 0$ and $v.used > 0$ and $u_0 \equiv v_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ +\hspace{3mm}5.1 $k \leftarrow k + 1$ \\ +\hspace{3mm}5.2 $u \leftarrow \lfloor u / 2 \rfloor$ \\ +\hspace{3mm}5.3 $v \leftarrow \lfloor v / 2 \rfloor$ \\ +6. While $u.used > 0$ and $u_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ +\hspace{3mm}6.1 $u \leftarrow \lfloor u / 2 \rfloor$ \\ +7. While $v.used > 0$ and $v_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ +\hspace{3mm}7.1 $v \leftarrow \lfloor v / 2 \rfloor$ \\ +8. While $v.used > 0$ \\ +\hspace{3mm}8.1 If $\vert u \vert > \vert v \vert$ then \\ +\hspace{6mm}8.1.1 Swap $u$ and $v$. \\ +\hspace{3mm}8.2 $v \leftarrow \vert v \vert - \vert u \vert$ \\ +\hspace{3mm}8.3 While $v.used > 0$ and $v_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ +\hspace{6mm}8.3.1 $v \leftarrow \lfloor v / 2 \rfloor$ \\ +9. $c \leftarrow u \cdot 2^k$ \\ +10. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_gcd} +\end{figure} +\textbf{Algorithm mp\_gcd.} +This algorithm will produce the greatest common divisor of two mp\_ints $a$ and $b$. The algorithm was originally based on Algorithm B of +Knuth \cite[pp. 338]{TAOCPV2} but has been modified to be simpler to explain. In theory it achieves the same asymptotic working time as +Algorithm B and in practice this appears to be true. + +The first two steps handle the cases where either one of or both inputs are zero. If either input is zero the greatest common divisor is the +largest input or zero if they are both zero. If the inputs are not trivial than $u$ and $v$ are assigned the absolute values of +$a$ and $b$ respectively and the algorithm will proceed to reduce the pair. + +Step five will divide out any common factors of two and keep track of the count in the variable $k$. After this step, two is no longer a +factor of the remaining greatest common divisor between $u$ and $v$ and can be safely evenly divided out of either whenever they are even. Step +six and seven ensure that the $u$ and $v$ respectively have no more factors of two. At most only one of the while--loops will iterate since +they cannot both be even. + +By step eight both of $u$ and $v$ are odd which is required for the inner logic. First the pair are swapped such that $v$ is equal to +or greater than $u$. This ensures that the subtraction on step 8.2 will always produce a positive and even result. Step 8.3 removes any +factors of two from the difference $u$ to ensure that in the next iteration of the loop both are once again odd. + +After $v = 0$ occurs the variable $u$ has the greatest common divisor of the pair $\left < u, v \right >$ just after step six. The result +must be adjusted by multiplying by the common factors of two ($2^k$) removed earlier. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_gcd.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +This function makes use of the macros mp\_iszero and mp\_iseven. The former evaluates to $1$ if the input mp\_int is equivalent to the +integer zero otherwise it evaluates to $0$. The latter evaluates to $1$ if the input mp\_int represents a non-zero even integer otherwise +it evaluates to $0$. Note that just because mp\_iseven may evaluate to $0$ does not mean the input is odd, it could also be zero. The three +trivial cases of inputs are handled on lines 24 through 30. After those lines the inputs are assumed to be non-zero. + +Lines 32 and 37 make local copies $u$ and $v$ of the inputs $a$ and $b$ respectively. At this point the common factors of two +must be divided out of the two inputs. The block starting at line 44 removes common factors of two by first counting the number of trailing +zero bits in both. The local integer $k$ is used to keep track of how many factors of $2$ are pulled out of both values. It is assumed that +the number of factors will not exceed the maximum value of a C ``int'' data type\footnote{Strictly speaking no array in C may have more than +entries than are accessible by an ``int'' so this is not a limitation.}. + +At this point there are no more common factors of two in the two values. The divisions by a power of two on lines 62 and 68 remove +any independent factors of two such that both $u$ and $v$ are guaranteed to be an odd integer before hitting the main body of the algorithm. The while loop +on line 73 performs the reduction of the pair until $v$ is equal to zero. The unsigned comparison and subtraction algorithms are used in +place of the full signed routines since both values are guaranteed to be positive and the result of the subtraction is guaranteed to be non-negative. + +\section{Least Common Multiple} +The least common multiple of a pair of integers is their product divided by their greatest common divisor. For two integers $a$ and $b$ the +least common multiple is normally denoted as $[ a, b ]$ and numerically equivalent to ${ab} \over {(a, b)}$. For example, if $a = 2 \cdot 2 \cdot 3 = 12$ +and $b = 2 \cdot 3 \cdot 3 \cdot 7 = 126$ the least common multiple is ${126 \over {(12, 126)}} = {126 \over 6} = 21$. + +The least common multiple arises often in coding theory as well as number theory. If two functions have periods of $a$ and $b$ respectively they will +collide, that is be in synchronous states, after only $[ a, b ]$ iterations. This is why, for example, random number generators based on +Linear Feedback Shift Registers (LFSR) tend to use registers with periods which are co-prime (\textit{e.g. the greatest common divisor is one.}). +Similarly in number theory if a composite $n$ has two prime factors $p$ and $q$ then maximal order of any unit of $\Z/n\Z$ will be $[ p - 1, q - 1] $. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_lcm}. \\ +\textbf{Input}. mp\_int $a$ and $b$ \\ +\textbf{Output}. The least common multiple $c = [a, b]$. \\ +\hline \\ +1. $c \leftarrow (a, b)$ \\ +2. $t \leftarrow a \cdot b$ \\ +3. $c \leftarrow \lfloor t / c \rfloor$ \\ +4. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_lcm} +\end{figure} +\textbf{Algorithm mp\_lcm.} +This algorithm computes the least common multiple of two mp\_int inputs $a$ and $b$. It computes the least common multiple directly by +dividing the product of the two inputs by their greatest common divisor. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_lcm.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +\section{Jacobi Symbol Computation} +To explain the Jacobi Symbol we shall first discuss the Legendre function\footnote{Arrg. What is the name of this?} off which the Jacobi symbol is +defined. The Legendre function computes whether or not an integer $a$ is a quadratic residue modulo an odd prime $p$. Numerically it is +equivalent to equation \ref{eqn:legendre}. + +\textit{-- Tom, don't be an ass, cite your source here...!} + +\begin{equation} +a^{(p-1)/2} \equiv \begin{array}{rl} + -1 & \mbox{if }a\mbox{ is a quadratic non-residue.} \\ + 0 & \mbox{if }a\mbox{ divides }p\mbox{.} \\ + 1 & \mbox{if }a\mbox{ is a quadratic residue}. + \end{array} \mbox{ (mod }p\mbox{)} +\label{eqn:legendre} +\end{equation} + +\textbf{Proof.} \textit{Equation \ref{eqn:legendre} correctly identifies the residue status of an integer $a$ modulo a prime $p$.} +An integer $a$ is a quadratic residue if the following equation has a solution. + +\begin{equation} +x^2 \equiv a \mbox{ (mod }p\mbox{)} +\label{eqn:root} +\end{equation} + +Consider the following equation. + +\begin{equation} +0 \equiv x^{p-1} - 1 \equiv \left \lbrace \left (x^2 \right )^{(p-1)/2} - a^{(p-1)/2} \right \rbrace + \left ( a^{(p-1)/2} - 1 \right ) \mbox{ (mod }p\mbox{)} +\label{eqn:rooti} +\end{equation} + +Whether equation \ref{eqn:root} has a solution or not equation \ref{eqn:rooti} is always true. If $a^{(p-1)/2} - 1 \equiv 0 \mbox{ (mod }p\mbox{)}$ +then the quantity in the braces must be zero. By reduction, + +\begin{eqnarray} +\left (x^2 \right )^{(p-1)/2} - a^{(p-1)/2} \equiv 0 \nonumber \\ +\left (x^2 \right )^{(p-1)/2} \equiv a^{(p-1)/2} \nonumber \\ +x^2 \equiv a \mbox{ (mod }p\mbox{)} +\end{eqnarray} + +As a result there must be a solution to the quadratic equation and in turn $a$ must be a quadratic residue. If $a$ does not divide $p$ and $a$ +is not a quadratic residue then the only other value $a^{(p-1)/2}$ may be congruent to is $-1$ since +\begin{equation} +0 \equiv a^{p - 1} - 1 \equiv (a^{(p-1)/2} + 1)(a^{(p-1)/2} - 1) \mbox{ (mod }p\mbox{)} +\end{equation} +One of the terms on the right hand side must be zero. \textbf{QED} + +\subsection{Jacobi Symbol} +The Jacobi symbol is a generalization of the Legendre function for any odd non prime moduli $p$ greater than 2. If $p = \prod_{i=0}^n p_i$ then +the Jacobi symbol $\left ( { a \over p } \right )$ is equal to the following equation. + +\begin{equation} +\left ( { a \over p } \right ) = \left ( { a \over p_0} \right ) \left ( { a \over p_1} \right ) \ldots \left ( { a \over p_n} \right ) +\end{equation} + +By inspection if $p$ is prime the Jacobi symbol is equivalent to the Legendre function. The following facts\footnote{See HAC \cite[pp. 72-74]{HAC} for +further details.} will be used to derive an efficient Jacobi symbol algorithm. Where $p$ is an odd integer greater than two and $a, b \in \Z$ the +following are true. + +\begin{enumerate} +\item $\left ( { a \over p} \right )$ equals $-1$, $0$ or $1$. +\item $\left ( { ab \over p} \right ) = \left ( { a \over p} \right )\left ( { b \over p} \right )$. +\item If $a \equiv b$ then $\left ( { a \over p} \right ) = \left ( { b \over p} \right )$. +\item $\left ( { 2 \over p} \right )$ equals $1$ if $p \equiv 1$ or $7 \mbox{ (mod }8\mbox{)}$. Otherwise, it equals $-1$. +\item $\left ( { a \over p} \right ) \equiv \left ( { p \over a} \right ) \cdot (-1)^{(p-1)(a-1)/4}$. More specifically +$\left ( { a \over p} \right ) = \left ( { p \over a} \right )$ if $p \equiv a \equiv 1 \mbox{ (mod }4\mbox{)}$. +\end{enumerate} + +Using these facts if $a = 2^k \cdot a'$ then + +\begin{eqnarray} +\left ( { a \over p } \right ) = \left ( {{2^k} \over p } \right ) \left ( {a' \over p} \right ) \nonumber \\ + = \left ( {2 \over p } \right )^k \left ( {a' \over p} \right ) +\label{eqn:jacobi} +\end{eqnarray} + +By fact five, + +\begin{equation} +\left ( { a \over p } \right ) = \left ( { p \over a } \right ) \cdot (-1)^{(p-1)(a-1)/4} +\end{equation} + +Subsequently by fact three since $p \equiv (p \mbox{ mod }a) \mbox{ (mod }a\mbox{)}$ then + +\begin{equation} +\left ( { a \over p } \right ) = \left ( { {p \mbox{ mod } a} \over a } \right ) \cdot (-1)^{(p-1)(a-1)/4} +\end{equation} + +By putting both observations into equation \ref{eqn:jacobi} the following simplified equation is formed. + +\begin{equation} +\left ( { a \over p } \right ) = \left ( {2 \over p } \right )^k \left ( {{p\mbox{ mod }a'} \over a'} \right ) \cdot (-1)^{(p-1)(a'-1)/4} +\end{equation} + +The value of $\left ( {{p \mbox{ mod }a'} \over a'} \right )$ can be found by using the same equation recursively. The value of +$\left ( {2 \over p } \right )^k$ equals $1$ if $k$ is even otherwise it equals $\left ( {2 \over p } \right )$. Using this approach the +factors of $p$ do not have to be known. Furthermore, if $(a, p) = 1$ then the algorithm will terminate when the recursion requests the +Jacobi symbol computation of $\left ( {1 \over a'} \right )$ which is simply $1$. + +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_jacobi}. \\ +\textbf{Input}. mp\_int $a$ and $p$, $a \ge 0$, $p \ge 3$, $p \equiv 1 \mbox{ (mod }2\mbox{)}$ \\ +\textbf{Output}. The Jacobi symbol $c = \left ( {a \over p } \right )$. \\ +\hline \\ +1. If $a = 0$ then \\ +\hspace{3mm}1.1 $c \leftarrow 0$ \\ +\hspace{3mm}1.2 Return(\textit{MP\_OKAY}). \\ +2. If $a = 1$ then \\ +\hspace{3mm}2.1 $c \leftarrow 1$ \\ +\hspace{3mm}2.2 Return(\textit{MP\_OKAY}). \\ +3. $a' \leftarrow a$ \\ +4. $k \leftarrow 0$ \\ +5. While $a'.used > 0$ and $a'_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ +\hspace{3mm}5.1 $k \leftarrow k + 1$ \\ +\hspace{3mm}5.2 $a' \leftarrow \lfloor a' / 2 \rfloor$ \\ +6. If $k \equiv 0 \mbox{ (mod }2\mbox{)}$ then \\ +\hspace{3mm}6.1 $s \leftarrow 1$ \\ +7. else \\ +\hspace{3mm}7.1 $r \leftarrow p_0 \mbox{ (mod }8\mbox{)}$ \\ +\hspace{3mm}7.2 If $r = 1$ or $r = 7$ then \\ +\hspace{6mm}7.2.1 $s \leftarrow 1$ \\ +\hspace{3mm}7.3 else \\ +\hspace{6mm}7.3.1 $s \leftarrow -1$ \\ +8. If $p_0 \equiv a'_0 \equiv 3 \mbox{ (mod }4\mbox{)}$ then \\ +\hspace{3mm}8.1 $s \leftarrow -s$ \\ +9. If $a' \ne 1$ then \\ +\hspace{3mm}9.1 $p' \leftarrow p \mbox{ (mod }a'\mbox{)}$ \\ +\hspace{3mm}9.2 $s \leftarrow s \cdot \mbox{mp\_jacobi}(p', a')$ \\ +10. $c \leftarrow s$ \\ +11. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_jacobi} +\end{figure} +\textbf{Algorithm mp\_jacobi.} +This algorithm computes the Jacobi symbol for an arbitrary positive integer $a$ with respect to an odd integer $p$ greater than three. The algorithm +is based on algorithm 2.149 of HAC \cite[pp. 73]{HAC}. + +Step numbers one and two handle the trivial cases of $a = 0$ and $a = 1$ respectively. Step five determines the number of two factors in the +input $a$. If $k$ is even than the term $\left ( { 2 \over p } \right )^k$ must always evaluate to one. If $k$ is odd than the term evaluates to one +if $p_0$ is congruent to one or seven modulo eight, otherwise it evaluates to $-1$. After the the $\left ( { 2 \over p } \right )^k$ term is handled +the $(-1)^{(p-1)(a'-1)/4}$ is computed and multiplied against the current product $s$. The latter term evaluates to one if both $p$ and $a'$ +are congruent to one modulo four, otherwise it evaluates to negative one. + +By step nine if $a'$ does not equal one a recursion is required. Step 9.1 computes $p' \equiv p \mbox{ (mod }a'\mbox{)}$ and will recurse to compute +$\left ( {p' \over a'} \right )$ which is multiplied against the current Jacobi product. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_jacobi.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +As a matter of practicality the variable $a'$ as per the pseudo-code is reprensented by the variable $a1$ since the $'$ symbol is not valid for a C +variable name character. + +The two simple cases of $a = 0$ and $a = 1$ are handled at the very beginning to simplify the algorithm. If the input is non-trivial the algorithm +has to proceed compute the Jacobi. The variable $s$ is used to hold the current Jacobi product. Note that $s$ is merely a C ``int'' data type since +the values it may obtain are merely $-1$, $0$ and $1$. + +After a local copy of $a$ is made all of the factors of two are divided out and the total stored in $k$. Technically only the least significant +bit of $k$ is required, however, it makes the algorithm simpler to follow to perform an addition. In practice an exclusive-or and addition have the same +processor requirements and neither is faster than the other. + +Line 58 through 71 determines the value of $\left ( { 2 \over p } \right )^k$. If the least significant bit of $k$ is zero than +$k$ is even and the value is one. Otherwise, the value of $s$ depends on which residue class $p$ belongs to modulo eight. The value of +$(-1)^{(p-1)(a'-1)/4}$ is compute and multiplied against $s$ on lines 71 through 74. + +Finally, if $a1$ does not equal one the algorithm must recurse and compute $\left ( {p' \over a'} \right )$. + +\textit{-- Comment about default $s$ and such...} + +\section{Modular Inverse} +\label{sec:modinv} +The modular inverse of a number actually refers to the modular multiplicative inverse. Essentially for any integer $a$ such that $(a, p) = 1$ there +exist another integer $b$ such that $ab \equiv 1 \mbox{ (mod }p\mbox{)}$. The integer $b$ is called the multiplicative inverse of $a$ which is +denoted as $b = a^{-1}$. Technically speaking modular inversion is a well defined operation for any finite ring or field not just for rings and +fields of integers. However, the former will be the matter of discussion. + +The simplest approach is to compute the algebraic inverse of the input. That is to compute $b \equiv a^{\Phi(p) - 1}$. If $\Phi(p)$ is the +order of the multiplicative subgroup modulo $p$ then $b$ must be the multiplicative inverse of $a$. The proof of which is trivial. + +\begin{equation} +ab \equiv a \left (a^{\Phi(p) - 1} \right ) \equiv a^{\Phi(p)} \equiv a^0 \equiv 1 \mbox{ (mod }p\mbox{)} +\end{equation} + +However, as simple as this approach may be it has two serious flaws. It requires that the value of $\Phi(p)$ be known which if $p$ is composite +requires all of the prime factors. This approach also is very slow as the size of $p$ grows. + +A simpler approach is based on the observation that solving for the multiplicative inverse is equivalent to solving the linear +Diophantine\footnote{See LeVeque \cite[pp. 40-43]{LeVeque} for more information.} equation. + +\begin{equation} +ab + pq = 1 +\end{equation} + +Where $a$, $b$, $p$ and $q$ are all integers. If such a pair of integers $ \left < b, q \right >$ exist than $b$ is the multiplicative inverse of +$a$ modulo $p$. The extended Euclidean algorithm (Knuth \cite[pp. 342]{TAOCPV2}) can be used to solve such equations provided $(a, p) = 1$. +However, instead of using that algorithm directly a variant known as the binary Extended Euclidean algorithm will be used in its place. The +binary approach is very similar to the binary greatest common divisor algorithm except it will produce a full solution to the Diophantine +equation. + +\subsection{General Case} +\newpage\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_invmod}. \\ +\textbf{Input}. mp\_int $a$ and $b$, $(a, b) = 1$, $p \ge 2$, $0 < a < p$. \\ +\textbf{Output}. The modular inverse $c \equiv a^{-1} \mbox{ (mod }b\mbox{)}$. \\ +\hline \\ +1. If $b \le 0$ then return(\textit{MP\_VAL}). \\ +2. If $b_0 \equiv 1 \mbox{ (mod }2\mbox{)}$ then use algorithm fast\_mp\_invmod. \\ +3. $x \leftarrow \vert a \vert, y \leftarrow b$ \\ +4. If $x_0 \equiv y_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ then return(\textit{MP\_VAL}). \\ +5. $B \leftarrow 0, C \leftarrow 0, A \leftarrow 1, D \leftarrow 1$ \\ +6. While $u.used > 0$ and $u_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ +\hspace{3mm}6.1 $u \leftarrow \lfloor u / 2 \rfloor$ \\ +\hspace{3mm}6.2 If ($A.used > 0$ and $A_0 \equiv 1 \mbox{ (mod }2\mbox{)}$) or ($B.used > 0$ and $B_0 \equiv 1 \mbox{ (mod }2\mbox{)}$) then \\ +\hspace{6mm}6.2.1 $A \leftarrow A + y$ \\ +\hspace{6mm}6.2.2 $B \leftarrow B - x$ \\ +\hspace{3mm}6.3 $A \leftarrow \lfloor A / 2 \rfloor$ \\ +\hspace{3mm}6.4 $B \leftarrow \lfloor B / 2 \rfloor$ \\ +7. While $v.used > 0$ and $v_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ +\hspace{3mm}7.1 $v \leftarrow \lfloor v / 2 \rfloor$ \\ +\hspace{3mm}7.2 If ($C.used > 0$ and $C_0 \equiv 1 \mbox{ (mod }2\mbox{)}$) or ($D.used > 0$ and $D_0 \equiv 1 \mbox{ (mod }2\mbox{)}$) then \\ +\hspace{6mm}7.2.1 $C \leftarrow C + y$ \\ +\hspace{6mm}7.2.2 $D \leftarrow D - x$ \\ +\hspace{3mm}7.3 $C \leftarrow \lfloor C / 2 \rfloor$ \\ +\hspace{3mm}7.4 $D \leftarrow \lfloor D / 2 \rfloor$ \\ +8. If $u \ge v$ then \\ +\hspace{3mm}8.1 $u \leftarrow u - v$ \\ +\hspace{3mm}8.2 $A \leftarrow A - C$ \\ +\hspace{3mm}8.3 $B \leftarrow B - D$ \\ +9. else \\ +\hspace{3mm}9.1 $v \leftarrow v - u$ \\ +\hspace{3mm}9.2 $C \leftarrow C - A$ \\ +\hspace{3mm}9.3 $D \leftarrow D - B$ \\ +10. If $u \ne 0$ goto step 6. \\ +11. If $v \ne 1$ return(\textit{MP\_VAL}). \\ +12. While $C \le 0$ do \\ +\hspace{3mm}12.1 $C \leftarrow C + b$ \\ +13. While $C \ge b$ do \\ +\hspace{3mm}13.1 $C \leftarrow C - b$ \\ +14. $c \leftarrow C$ \\ +15. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\end{figure} +\textbf{Algorithm mp\_invmod.} +This algorithm computes the modular multiplicative inverse of an integer $a$ modulo an integer $b$. This algorithm is a variation of the +extended binary Euclidean algorithm from HAC \cite[pp. 608]{HAC}. It has been modified to only compute the modular inverse and not a complete +Diophantine solution. + +If $b \le 0$ than the modulus is invalid and MP\_VAL is returned. Similarly if both $a$ and $b$ are even then there cannot be a multiplicative +inverse for $a$ and the error is reported. + +The astute reader will observe that steps seven through nine are very similar to the binary greatest common divisor algorithm mp\_gcd. In this case +the other variables to the Diophantine equation are solved. The algorithm terminates when $u = 0$ in which case the solution is + +\begin{equation} +Ca + Db = v +\end{equation} + +If $v$, the greatest common divisor of $a$ and $b$ is not equal to one then the algorithm will report an error as no inverse exists. Otherwise, $C$ +is the modular inverse of $a$. The actual value of $C$ is congruent to, but not necessarily equal to, the ideal modular inverse which should lie +within $1 \le a^{-1} < b$. Step numbers twelve and thirteen adjust the inverse until it is in range. If the original input $a$ is within $0 < a < p$ +then only a couple of additions or subtractions will be required to adjust the inverse. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_invmod.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +\subsubsection{Odd Moduli} + +When the modulus $b$ is odd the variables $A$ and $C$ are fixed and are not required to compute the inverse. In particular by attempting to solve +the Diophantine $Cb + Da = 1$ only $B$ and $D$ are required to find the inverse of $a$. + +The algorithm fast\_mp\_invmod is a direct adaptation of algorithm mp\_invmod with all all steps involving either $A$ or $C$ removed. This +optimization will halve the time required to compute the modular inverse. + +\section{Primality Tests} + +A non-zero integer $a$ is said to be prime if it is not divisible by any other integer excluding one and itself. For example, $a = 7$ is prime +since the integers $2 \ldots 6$ do not evenly divide $a$. By contrast, $a = 6$ is not prime since $a = 6 = 2 \cdot 3$. + +Prime numbers arise in cryptography considerably as they allow finite fields to be formed. The ability to determine whether an integer is prime or +not quickly has been a viable subject in cryptography and number theory for considerable time. The algorithms that will be presented are all +probablistic algorithms in that when they report an integer is composite it must be composite. However, when the algorithms report an integer is +prime the algorithm may be incorrect. + +As will be discussed it is possible to limit the probability of error so well that for practical purposes the probablity of error might as +well be zero. For the purposes of these discussions let $n$ represent the candidate integer of which the primality is in question. + +\subsection{Trial Division} + +Trial division means to attempt to evenly divide a candidate integer by small prime integers. If the candidate can be evenly divided it obviously +cannot be prime. By dividing by all primes $1 < p \le \sqrt{n}$ this test can actually prove whether an integer is prime. However, such a test +would require a prohibitive amount of time as $n$ grows. + +Instead of dividing by every prime, a smaller, more mangeable set of primes may be used instead. By performing trial division with only a subset +of the primes less than $\sqrt{n} + 1$ the algorithm cannot prove if a candidate is prime. However, often it can prove a candidate is not prime. + +The benefit of this test is that trial division by small values is fairly efficient. Specially compared to the other algorithms that will be +discussed shortly. The probability that this approach correctly identifies a composite candidate when tested with all primes upto $q$ is given by +$1 - {1.12 \over ln(q)}$. The graph (\ref{pic:primality}, will be added later) demonstrates the probability of success for the range +$3 \le q \le 100$. + +At approximately $q = 30$ the gain of performing further tests diminishes fairly quickly. At $q = 90$ further testing is generally not going to +be of any practical use. In the case of LibTomMath the default limit $q = 256$ was chosen since it is not too high and will eliminate +approximately $80\%$ of all candidate integers. The constant \textbf{PRIME\_SIZE} is equal to the number of primes in the test base. The +array \_\_prime\_tab is an array of the first \textbf{PRIME\_SIZE} prime numbers. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_prime\_is\_divisible}. \\ +\textbf{Input}. mp\_int $a$ \\ +\textbf{Output}. $c = 1$ if $n$ is divisible by a small prime, otherwise $c = 0$. \\ +\hline \\ +1. for $ix$ from $0$ to $PRIME\_SIZE$ do \\ +\hspace{3mm}1.1 $d \leftarrow n \mbox{ (mod }\_\_prime\_tab_{ix}\mbox{)}$ \\ +\hspace{3mm}1.2 If $d = 0$ then \\ +\hspace{6mm}1.2.1 $c \leftarrow 1$ \\ +\hspace{6mm}1.2.2 Return(\textit{MP\_OKAY}). \\ +2. $c \leftarrow 0$ \\ +3. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_prime\_is\_divisible} +\end{figure} +\textbf{Algorithm mp\_prime\_is\_divisible.} +This algorithm attempts to determine if a candidate integer $n$ is composite by performing trial divisions. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_prime\_is\_divisible.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +The algorithm defaults to a return of $0$ in case an error occurs. The values in the prime table are all specified to be in the range of a +mp\_digit. The table \_\_prime\_tab is defined in the following file. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_prime\_tab.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +Note that there are two possible tables. When an mp\_digit is 7-bits long only the primes upto $127$ may be included, otherwise the primes +upto $1619$ are used. Note that the value of \textbf{PRIME\_SIZE} is a constant dependent on the size of a mp\_digit. + +\subsection{The Fermat Test} +The Fermat test is probably one the oldest tests to have a non-trivial probability of success. It is based on the fact that if $n$ is in +fact prime then $a^{n} \equiv a \mbox{ (mod }n\mbox{)}$ for all $0 < a < n$. The reason being that if $n$ is prime than the order of +the multiplicative sub group is $n - 1$. Any base $a$ must have an order which divides $n - 1$ and as such $a^n$ is equivalent to +$a^1 = a$. + +If $n$ is composite then any given base $a$ does not have to have a period which divides $n - 1$. In which case +it is possible that $a^n \nequiv a \mbox{ (mod }n\mbox{)}$. However, this test is not absolute as it is possible that the order +of a base will divide $n - 1$ which would then be reported as prime. Such a base yields what is known as a Fermat pseudo-prime. Several +integers known as Carmichael numbers will be a pseudo-prime to all valid bases. Fortunately such numbers are extremely rare as $n$ grows +in size. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_prime\_fermat}. \\ +\textbf{Input}. mp\_int $a$ and $b$, $a \ge 2$, $0 < b < a$. \\ +\textbf{Output}. $c = 1$ if $b^a \equiv b \mbox{ (mod }a\mbox{)}$, otherwise $c = 0$. \\ +\hline \\ +1. $t \leftarrow b^a \mbox{ (mod }a\mbox{)}$ \\ +2. If $t = b$ then \\ +\hspace{3mm}2.1 $c = 1$ \\ +3. else \\ +\hspace{3mm}3.1 $c = 0$ \\ +4. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_prime\_fermat} +\end{figure} +\textbf{Algorithm mp\_prime\_fermat.} +This algorithm determines whether an mp\_int $a$ is a Fermat prime to the base $b$ or not. It uses a single modular exponentiation to +determine the result. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_prime\_fermat.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + +\subsection{The Miller-Rabin Test} +The Miller-Rabin (citation) test is another primality test which has tighter error bounds than the Fermat test specifically with sequentially chosen +candidate integers. The algorithm is based on the observation that if $n - 1 = 2^kr$ and if $b^r \nequiv \pm 1$ then after upto $k - 1$ squarings the +value must be equal to $-1$. The squarings are stopped as soon as $-1$ is observed. If the value of $1$ is observed first it means that +some value not congruent to $\pm 1$ when squared equals one which cannot occur if $n$ is prime. + +\begin{figure}[!here] +\begin{small} +\begin{center} +\begin{tabular}{l} +\hline Algorithm \textbf{mp\_prime\_miller\_rabin}. \\ +\textbf{Input}. mp\_int $a$ and $b$, $a \ge 2$, $0 < b < a$. \\ +\textbf{Output}. $c = 1$ if $a$ is a Miller-Rabin prime to the base $a$, otherwise $c = 0$. \\ +\hline +1. $a' \leftarrow a - 1$ \\ +2. $r \leftarrow n1$ \\ +3. $c \leftarrow 0, s \leftarrow 0$ \\ +4. While $r.used > 0$ and $r_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ +\hspace{3mm}4.1 $s \leftarrow s + 1$ \\ +\hspace{3mm}4.2 $r \leftarrow \lfloor r / 2 \rfloor$ \\ +5. $y \leftarrow b^r \mbox{ (mod }a\mbox{)}$ \\ +6. If $y \nequiv \pm 1$ then \\ +\hspace{3mm}6.1 $j \leftarrow 1$ \\ +\hspace{3mm}6.2 While $j \le (s - 1)$ and $y \nequiv a'$ \\ +\hspace{6mm}6.2.1 $y \leftarrow y^2 \mbox{ (mod }a\mbox{)}$ \\ +\hspace{6mm}6.2.2 If $y = 1$ then goto step 8. \\ +\hspace{6mm}6.2.3 $j \leftarrow j + 1$ \\ +\hspace{3mm}6.3 If $y \nequiv a'$ goto step 8. \\ +7. $c \leftarrow 1$\\ +8. Return(\textit{MP\_OKAY}). \\ +\hline +\end{tabular} +\end{center} +\end{small} +\caption{Algorithm mp\_prime\_miller\_rabin} +\end{figure} +\textbf{Algorithm mp\_prime\_miller\_rabin.} +This algorithm performs one trial round of the Miller-Rabin algorithm to the base $b$. It will set $c = 1$ if the algorithm cannot determine +if $b$ is composite or $c = 0$ if $b$ is provably composite. The values of $s$ and $r$ are computed such that $a' = a - 1 = 2^sr$. + +If the value $y \equiv b^r$ is congruent to $\pm 1$ then the algorithm cannot prove if $a$ is composite or not. Otherwise, the algorithm will +square $y$ upto $s - 1$ times stopping only when $y \equiv -1$. If $y^2 \equiv 1$ and $y \nequiv \pm 1$ then the algorithm can report that $a$ +is provably composite. If the algorithm performs $s - 1$ squarings and $y \nequiv -1$ then $a$ is provably composite. If $a$ is not provably +composite then it is \textit{probably} prime. + +\vspace{+3mm}\begin{small} +\hspace{-5.1mm}{\bf File}: bn\_mp\_prime\_miller\_rabin.c +\vspace{-3mm} +\begin{alltt} +\end{alltt} +\end{small} + + + + +\backmatter +\appendix +\begin{thebibliography}{ABCDEF} +\bibitem[1]{TAOCPV2} +Donald Knuth, \textit{The Art of Computer Programming}, Third Edition, Volume Two, Seminumerical Algorithms, Addison-Wesley, 1998 + +\bibitem[2]{HAC} +A. Menezes, P. van Oorschot, S. Vanstone, \textit{Handbook of Applied Cryptography}, CRC Press, 1996 + +\bibitem[3]{ROSE} +Michael Rosing, \textit{Implementing Elliptic Curve Cryptography}, Manning Publications, 1999 + +\bibitem[4]{COMBA} +Paul G. Comba, \textit{Exponentiation Cryptosystems on the IBM PC}. IBM Systems Journal 29(4): 526-538 (1990) + +\bibitem[5]{KARA} +A. Karatsuba, Doklay Akad. Nauk SSSR 145 (1962), pp.293-294 + +\bibitem[6]{KARAP} +Andre Weimerskirch and Christof Paar, \textit{Generalizations of the Karatsuba Algorithm for Polynomial Multiplication}, Submitted to Design, Codes and Cryptography, March 2002 + +\bibitem[7]{BARRETT} +Paul Barrett, \textit{Implementing the Rivest Shamir and Adleman Public Key Encryption Algorithm on a Standard Digital Signal Processor}, Advances in Cryptology, Crypto '86, Springer-Verlag. + +\bibitem[8]{MONT} +P.L.Montgomery. \textit{Modular multiplication without trial division}. Mathematics of Computation, 44(170):519-521, April 1985. + +\bibitem[9]{DRMET} +Chae Hoon Lim and Pil Joong Lee, \textit{Generating Efficient Primes for Discrete Log Cryptosystems}, POSTECH Information Research Laboratories + +\bibitem[10]{MMB} +J. Daemen and R. Govaerts and J. Vandewalle, \textit{Block ciphers based on Modular Arithmetic}, State and {P}rogress in the {R}esearch of {C}ryptography, 1993, pp. 80-89 + +\bibitem[11]{RSAREF} +R.L. Rivest, A. Shamir, L. Adleman, \textit{A Method for Obtaining Digital Signatures and Public-Key Cryptosystems} + +\bibitem[12]{DHREF} +Whitfield Diffie, Martin E. Hellman, \textit{New Directions in Cryptography}, IEEE Transactions on Information Theory, 1976 + +\bibitem[13]{IEEE} +IEEE Standard for Binary Floating-Point Arithmetic (ANSI/IEEE Std 754-1985) + +\bibitem[14]{GMP} +GNU Multiple Precision (GMP), \url{http://www.swox.com/gmp/} + +\bibitem[15]{MPI} +Multiple Precision Integer Library (MPI), Michael Fromberger, \url{http://thayer.dartmouth.edu/~sting/mpi/} + +\bibitem[16]{OPENSSL} +OpenSSL Cryptographic Toolkit, \url{http://openssl.org} + +\bibitem[17]{LIP} +Large Integer Package, \url{http://home.hetnet.nl/~ecstr/LIP.zip} + +\bibitem[18]{ISOC} +JTC1/SC22/WG14, ISO/IEC 9899:1999, ``A draft rationale for the C99 standard.'' + +\bibitem[19]{JAVA} +The Sun Java Website, \url{http://java.sun.com/} + +\end{thebibliography} + +\input{tommath.ind} + +\end{document} |