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0.30 release of LibTomMath

author | Matt Johnston <matt@ucc.asn.au> |
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date | Tue, 15 Jun 2004 14:42:57 +0000 |

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children | d29b64170cf0 |

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\documentclass[b5paper]{book} \usepackage{hyperref} \usepackage{makeidx} \usepackage{amssymb} \usepackage{color} \usepackage{alltt} \usepackage{graphicx} \usepackage{layout} \def\union{\cup} \def\intersect{\cap} \def\getsrandom{\stackrel{\rm R}{\gets}} \def\cross{\times} \def\cat{\hspace{0.5em} \| \hspace{0.5em}} \def\catn{$\|$} \def\divides{\hspace{0.3em} | \hspace{0.3em}} \def\nequiv{\not\equiv} \def\approx{\raisebox{0.2ex}{\mbox{\small $\sim$}}} \def\lcm{{\rm lcm}} \def\gcd{{\rm gcd}} \def\log{{\rm log}} \def\ord{{\rm ord}} \def\abs{{\mathit abs}} \def\rep{{\mathit rep}} \def\mod{{\mathit\ mod\ }} \renewcommand{\pmod}[1]{\ ({\rm mod\ }{#1})} \newcommand{\floor}[1]{\left\lfloor{#1}\right\rfloor} \newcommand{\ceil}[1]{\left\lceil{#1}\right\rceil} \def\Or{{\rm\ or\ }} \def\And{{\rm\ and\ }} \def\iff{\hspace{1em}\Longleftrightarrow\hspace{1em}} \def\implies{\Rightarrow} \def\undefined{{\rm ``undefined"}} \def\Proof{\vspace{1ex}\noindent {\bf Proof:}\hspace{1em}} \let\oldphi\phi \def\phi{\varphi} \def\Pr{{\rm Pr}} \newcommand{\str}[1]{{\mathbf{#1}}} \def\F{{\mathbb F}} \def\N{{\mathbb N}} \def\Z{{\mathbb Z}} \def\R{{\mathbb R}} \def\C{{\mathbb C}} \def\Q{{\mathbb Q}} \definecolor{DGray}{gray}{0.5} \newcommand{\emailaddr}[1]{\mbox{$<${#1}$>$}} \def\twiddle{\raisebox{0.3ex}{\mbox{\tiny $\sim$}}} \def\gap{\vspace{0.5ex}} \makeindex \begin{document} \frontmatter \pagestyle{empty} \title{Implementing Multiple Precision Arithmetic \\ ~ \\ Draft Edition } \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 v0.30 release of the LibTomMath project. \begin{alltt} Tom St Denis 111 Banning Rd Ottawa, Ontario K2L 1C3 Canada Phone: 1-613-836-3160 Email: [email protected] \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 to the Draft Edition} I started this text in April 2003 to complement my LibTomMath library. That is, explain how to implement the functions contained in LibTomMath. The goal is to have a textbook that any Computer Science student can use when implementing their own multiple precision arithmetic. The plan I wanted to follow was flesh out all the ideas and concepts I had floating around in my head and then work on it afterwards refining a little bit at a time. Chance would have it that I ended up with my summer off from Algonquin College and I was given four months solid to work on the text. Choosing to not waste any time I dove right into the project even before my spring semester was finished. I wrote a bit off and on at first. The moment my exams were finished I jumped into long 12 to 16 hour days. The result after only a couple of months was a ten chapter, three hundred page draft that I quickly had distributed to anyone who wanted to read it. I had Jean-Luc Cooke print copies for me and I brought them to Crypto'03 in Santa Barbara. So far I have managed to grab a certain level of attention having people from around the world ask me for copies of the text was certain rewarding. Now we are past December 2003. By this time I had pictured that I would have at least finished my second draft of the text. Currently I am far off from this goal. I've done partial re-writes of chapters one, two and three but they are not even finished yet. I haven't given up on the project, only had some setbacks. First O'Reilly declined to publish the text then Addison-Wesley and Greg is tried another which I don't know the name of. However, at this point I want to focus my energy onto finishing the book not securing a contract. So why am I writing this text? It seems like a lot of work right? Most certainly it is a lot of work writing a textbook. Even the simplest introductory material has to be lined with references and figures. A lot of the text has to be re-written from point form to prose form to ensure an easier read. Why am I doing all this work for free then? Simple. My philosophy is quite simply ``Open Source. Open Academia. Open Minds'' which means that to achieve a goal of open minds, that is, people willing to accept new ideas and explore the unknown you have to make available material they can access freely without hinderance. I've been writing free software since I was about sixteen but only recently have I hit upon software that people have come to depend upon. I started LibTomCrypt in December 2001 and now several major companies use it as integral portions of their software. Several educational institutions use it as a matter of course and many freelance developers use it as part of their projects. To further my contributions I started the LibTomMath project in December 2002 aimed at providing multiple precision arithmetic routines that students could learn from. That is write routines that are not only easy to understand and follow but provide quite impressive performance considering they are all in standard portable ISO C. The second leg of my philosophy is ``Open Academia'' which is where this textbook comes in. In the end, when all is said and done the text will be useable by educational institutions as a reference on multiple precision arithmetic. At this time I feel I should share a little information about myself. The most common question I was asked at Crypto'03, perhaps just out of professional courtesy, was which school I either taught at or attended. The unfortunate truth is that I neither teach at or attend a school of academic reputation. I'm currently at Algonquin College which is what I'd like to call ``somewhat academic but mostly vocational'' college. In otherwords, job training. I'm a 21 year old computer science student mostly self-taught in the areas I am aware of (which includes a half-dozen computer science fields, a few fields of mathematics and some English). I look forward to teaching someday but I am still far off from that goal. Now it would be improper for me to not introduce the rest of the texts co-authors. While they are only contributing corrections and editorial feedback their support has been tremendously helpful in presenting the concepts laid out in the text so far. Greg has always been there for me. He has tracked my LibTom projects since their inception and even sent cheques to help pay tuition from time to time. His background has provided a wonderful source to bounce ideas off of and improve the quality of my writing. Mads is another fellow who has just ``been there''. I don't even recall what his interest in the LibTom projects is but I'm definitely glad he has been around. His ability to catch logical errors in my written English have saved me on several occasions to say the least. What to expect next? Well this is still a rough draft. I've only had the chance to update a few chapters. However, I've been getting the feeling that people are starting to use my text and I owe them some updated material. My current tenative plan is to edit one chapter every two weeks starting January 4th. It seems insane but my lower course load at college should provide ample time. By Crypto'04 I plan to have a 2nd draft of the text polished and ready to hand out to as many people who will take it. \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. 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.org}} 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 $x = (x_{n-1} ... x_1 x_0)_{ \beta }$ 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} For the purposes of this text 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 variable $\beta$ represents the radix of a single digit of a multiple precision integer and must be of the form $q^p$ for $q, p \in \Z^+$. 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, for all intents and purposes. \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 file. 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. FIGU,design_process,Design Flow of the First Few Original LibTomMath Functions. 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{verbatim} typedef struct { int used, alloc, sign; mp_digit *dp; } mp_int; \end{verbatim} The mp\_int structure can be broken down as follows. \begin{enumerate} \item The \textbf{used} parameter denotes how many digits of the array \textbf{dp} contain the digits used to represent a given integer. The \textbf{used} count must 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. \index{MP\_OKAY} \index{MP\_VAL} \index{MP\_MEM} \begin{center} \begin{tabular}{|l|l|} \hline \textbf{Value} & \textbf{Meaning} \\ \hline \textbf{MP\_OKAY} & The function was successful \\ \hline \textbf{MP\_VAL} & One of the input value(s) was invalid \\ \hline \textbf{MP\_MEM} & The function ran out of heap memory \\ \hline \end{tabular} \end{center} When an error is detected within a function it should free any memory it allocated, 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. \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 \textbf{MP\_PREC} name represents a constant\footnote{Defined in the ``tommath.h'' header file within LibTomMath.} used to dictate the minimum precision of allocated mp\_int integers. Ideally, it is at least equal to $32$ since for most purposes that will be more than enough. Memory for the default number of digits is allocated first. If the allocation fails the algorithm returns immediately with the \textbf{MP\_MEM} error code. If the allocation succeeds the remaining members of the mp\_int structure must be initialized to reflect the default initial state. The allocated digits are all set to zero (step three) to ensure they are in a known state. The \textbf{sign}, \textbf{used} and \textbf{alloc} are subsequently initialized to represent the zero integer. By step seven the algorithm returns a success code and the mp\_int $a$ has been successfully initialized to a valid state representing the integer zero. \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. EXAM,bn_mp_init.c 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. Before any of the other members of the structure are initialized memory from the application heap is allocated with the calloc() function (line @22,[email protected]). The size of the allocated memory is large enough to hold \textbf{MP\_PREC} mp\_digit variables. The calloc() function is used instead\footnote{calloc() will allocate memory in the same manner as malloc() except that it also sets the contents to zero upon successfully allocating the memory.} of malloc() since digits have to be set to zero for the function to finish correctly. The \textbf{OPT\_CAST} token is a macro definition which will turn into a cast from void * to mp\_digit * for C++ compilers. It is not required for C compilers. After the memory has been successfully allocated the remainder of the members are initialized (lines @29,[email protected] through @31,[email protected]) 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 $a$ is freed for reuse. \\ \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 releases the memory allocated for an mp\_int back into the memory pool for reuse. It is designed such that a given mp\_int structure can be cleared multiple times between initializations without attempting to free the memory twice\footnote{In ISO C for example, calling free() twice on the same memory block causes undefinied behaviour.}. The first step determines if the mp\_int structure has been marked as free already. If it has, the algorithm returns success immediately as no further actions are required. Otherwise, the algorithm will proceed to put the structure in a known empty and otherwise invalid state. First the digits of the mp\_int are set to zero. The memory that has been allocated for the digits is then freed. The \textbf{used} and \textbf{alloc} counts are both set to zero and the \textbf{sign} set to \textbf{MP\_ZPOS}. This known fixed state for cleared mp\_int structures will make debuging easier for the end developer. That is, if they spot (via their debugger) an mp\_int they are using that is in this state it will be obvious that they erroneously and prematurely cleared the mp\_int structure. Note that 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. EXAM,bn_mp_clear.c The ``if'' statement (line @21,a->dp != [email protected]) prevents the heap from being corrupted if a user double-frees an mp\_int. This is because once the memory is freed the pointer is set to \textbf{NULL} (line @30,[email protected]). Without the check, code that accidentally calls mp\_clear twice for a given mp\_int structure would try to free the memory allocated for the digits twice. This may cause some C libraries to signal a fault. By setting the pointer to \textbf{NULL} it helps debug code that may inadvertently free the mp\_int before it is truly not needed, because attempts to reference digits should fail immediately. The allocated digits are set to zero before being freed (line @24,[email protected]). This is ideal for cryptographic situations where the integer that the mp\_int represents might need to be kept a secret. \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. EXAM,bn_mp_grow.c The first step is to see if we actually need to perform a re-allocation at all (line @24,a->alloc < [email protected]). If a reallocation must occur the digit count is padded upwards to help prevent many trivial reallocations (line @28,[email protected]). Next the reallocation is performed and the return of realloc() is stored in a temporary pointer named $tmp$ (line @36,[email protected]). The return is stored in a temporary instead of $a.dp$ to prevent the code from losing the original pointer in case the reallocation fails. Had the return been stored in $a.dp$ instead there would be no way to reclaim the heap originally used. If the reallocation fails the function will return \textbf{MP\_MEM} (line @39,[email protected]), otherwise, the value of $tmp$ is assigned to the pointer $a.dp$ and the function continues. A simple for loop from line @48,a->[email protected] to line @50,}@ will zero all digits that were above the old \textbf{alloc} limit to make sure the integer is in a known state. \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. EXAM,bn_mp_init_size.c The number of digits $b$ requested is padded (line @22,MP_[email protected]) 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 @27,[email protected]). The digits are allocated and set to zero at the same time with the calloc() function (line @25,[email protected]). 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 @29,[email protected], @30,[email protected] and @31,[email protected]). 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. EXAM,bn_mp_init_multi.c 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 @47,[email protected]) such that if a failure does occur, the algorithm can backtrack and free the previously initialized structures (lines @27,[email protected] to @46,}@). \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. EXAM,bn_mp_clamp.c Note on line @27,[email protected] how to test for the \textbf{used} count is made on the left of the \&\& operator. In the C programming language the terms to \&\& are evaluated left to right with a boolean short-circuit if any condition fails. This is important since if the \textbf{used} is zero the test on the right would fetch below the array. That is obviously undesirable. The parenthesis on line @28,a->[email protected] 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. EXAM,bn_mp_copy.c 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 @24,a == [email protected]). 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 @29,[email protected] 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 @42,[email protected] and @45,[email protected]) 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 @48,[email protected] to @50,}@) and then the excess digits of $b$ are set to zero (lines @53,[email protected] to @55,}@). 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. EXAM,bn_mp_init_copy.c This will initialize \textbf{a} and make it a verbatim copy of the contents of \textbf{b}. Note that \textbf{a} will have its own memory allocated which means that \textbf{b} may be cleared after the call and \textbf{a} will be left intact. \section{Zeroing an Integer} Reseting an mp\_int to the default state is a common step in many algorithms. The mp\_zero algorithm will be the algorithm used to perform this task. \begin{figure}[here] \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_zero}. \\ \textbf{Input}. An mp\_int $a$ \\ \textbf{Output}. Zero the contents of $a$ \\ \hline \\ 1. $a.used \leftarrow 0$ \\ 2. $a.sign \leftarrow$ MP\_ZPOS \\ 3. for $n$ from 0 to $a.alloc - 1$ do \\ \hspace{3mm}3.1 $a_n \leftarrow 0$ \\ \hline \end{tabular} \end{center} \caption{Algorithm mp\_zero} \end{figure} \textbf{Algorithm mp\_zero.} This algorithm simply resets a mp\_int to the default state. EXAM,bn_mp_zero.c After the function is completed, all of the digits are zeroed, the \textbf{used} count is zeroed and the \textbf{sign} variable is set to \textbf{MP\_ZPOS}. \section{Sign Manipulation} \subsection{Absolute Value} With the mp\_int representation of an integer, calculating the absolute value is trivial. The mp\_abs algorithm will compute the absolute value of an mp\_int. \newpage\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. EXAM,bn_mp_abs.c \subsection{Integer Negation} With the mp\_int representation of an integer, calculating the negation is also trivial. The mp\_neg algorithm will compute the negative of an mp\_int input. \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. EXAM,bn_mp_neg.c \section{Small Constants} \subsection{Setting Small Constants} Often a mp\_int must be set to a relatively small value such as $1$ or $2$. For these cases the mp\_set algorithm is useful. \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. EXAM,bn_mp_set.c Line @21,mp_[email protected] calls mp\_zero() to clear the mp\_int and reset the sign. Line @22,MP_[email protected] copies the digit into the least significant location. Note the usage of a new constant \textbf{MP\_MASK}. This constant is used to quickly reduce an integer modulo $\beta$. Since $\beta$ is of the form $2^k$ for any suitable $k$ it suffices to perform a binary AND with $MP\_MASK = 2^k - 1$ to perform the reduction. Finally line @23,a->[email protected] will set the \textbf{used} member with respect to the digit actually set. This function will always make the integer positive. One important limitation of this function is that it will only set one digit. The size of a digit is not fixed, meaning source that uses this function should take that into account. 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. EXAM,bn_mp_set_int.c This function sets four bits of the number at a time to handle all practical \textbf{DIGIT\_BIT} sizes. The weird addition on line @38,a->[email protected] ensures that the newly added in bits are added to the number of digits. While it may not seem obvious as to why the digit counter does not grow exceedingly large it is because of the shift on line @27,mp_mul_[email protected] as well as the call to mp\_clamp() on line @40,mp_[email protected] Both functions will clamp excess leading digits which keeps the number of used digits low. \section{Comparisons} \subsection{Unsigned Comparisions} Comparing a multiple precision integer is performed with the exact same algorithm used to compare two decimal numbers. For example, to compare $1,234$ to $1,264$ the digits are extracted by their positions. That is we compare $1 \cdot 10^3 + 2 \cdot 10^2 + 3 \cdot 10^1 + 4 \cdot 10^0$ to $1 \cdot 10^3 + 2 \cdot 10^2 + 6 \cdot 10^1 + 4 \cdot 10^0$ by comparing single digits at a time starting with the highest magnitude positions. If any leading digit of one integer is greater than a digit in the same position of another integer then obviously it must be greater. The first comparision routine that will be developed is the unsigned magnitude compare which will perform a comparison based on the digits of two mp\_int variables alone. It will ignore the sign of the two inputs. Such a function is useful when an absolute comparison is required or if the signs are known to agree in advance. To facilitate working with the results of the comparison functions three constants are required. \begin{figure}[here] \begin{center} \begin{tabular}{|r|l|} \hline \textbf{Constant} & \textbf{Meaning} \\ \hline \textbf{MP\_GT} & Greater Than \\ \hline \textbf{MP\_EQ} & Equal To \\ \hline \textbf{MP\_LT} & Less Than \\ \hline \end{tabular} \end{center} \caption{Comparison Return Codes} \end{figure} \begin{figure}[here] \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_cmp\_mag}. \\ \textbf{Input}. Two mp\_ints $a$ and $b$. \\ \textbf{Output}. Unsigned comparison results ($a$ to the left of $b$). \\ \hline \\ 1. If $a.used > b.used$ then return(\textit{MP\_GT}) \\ 2. If $a.used < b.used$ then return(\textit{MP\_LT}) \\ 3. for n from $a.used - 1$ to 0 do \\ \hspace{+3mm}3.1 if $a_n > b_n$ then return(\textit{MP\_GT}) \\ \hspace{+3mm}3.2 if $a_n < b_n$ then return(\textit{MP\_LT}) \\ 4. Return(\textit{MP\_EQ}) \\ \hline \end{tabular} \end{center} \caption{Algorithm mp\_cmp\_mag} \end{figure} \textbf{Algorithm mp\_cmp\_mag.} By saying ``$a$ to the left of $b$'' it is meant that the comparison is with respect to $a$, that is if $a$ is greater than $b$ it will return \textbf{MP\_GT} and similar with respect to when $a = b$ and $a < b$. The first two steps compare the number of digits used in both $a$ and $b$. Obviously if the digit counts differ there would be an imaginary zero digit in the smaller number where the leading digit of the larger number is. If both have the same number of digits than the actual digits themselves must be compared starting at the leading digit. By step three both inputs must have the same number of digits so its safe to start from either $a.used - 1$ or $b.used - 1$ and count down to the zero'th digit. If after all of the digits have been compared, no difference is found, the algorithm returns \textbf{MP\_EQ}. EXAM,bn_mp_cmp_mag.c The two if statements on lines @24,[email protected] and @28,[email protected] compare the number of digits in the two inputs. These two are performed before all of the digits are compared since it is a very cheap test to perform and can potentially save considerable time. The implementation given is also not valid without those two statements. $b.alloc$ may be smaller than $a.used$, meaning that undefined values will be read from $b$ past the end of the array of digits. \subsection{Signed Comparisons} Comparing with sign considerations is also fairly critical in several routines (\textit{division for example}). Based on an unsigned magnitude comparison a trivial signed comparison algorithm can be written. \begin{figure}[here] \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_cmp}. \\ \textbf{Input}. Two mp\_ints $a$ and $b$ \\ \textbf{Output}. Signed Comparison Results ($a$ to the left of $b$) \\ \hline \\ 1. if $a.sign = MP\_NEG$ and $b.sign = MP\_ZPOS$ then return(\textit{MP\_LT}) \\ 2. if $a.sign = MP\_ZPOS$ and $b.sign = MP\_NEG$ then return(\textit{MP\_GT}) \\ 3. if $a.sign = MP\_NEG$ then \\ \hspace{+3mm}3.1 Return the unsigned comparison of $b$ and $a$ (\textit{mp\_cmp\_mag}) \\ 4 Otherwise \\ \hspace{+3mm}4.1 Return the unsigned comparison of $a$ and $b$ \\ \hline \end{tabular} \end{center} \caption{Algorithm mp\_cmp} \end{figure} \textbf{Algorithm mp\_cmp.} The first two steps compare the signs of the two inputs. If the signs do not agree then it can return right away with the appropriate comparison code. When the signs are equal the digits of the inputs must be compared to determine the correct result. In step three the unsigned comparision flips the order of the arguments since they are both negative. For instance, if $-a > -b$ then $\vert a \vert < \vert b \vert$. Step number four will compare the two when they are both positive. EXAM,bn_mp_cmp.c The two if statements on lines @22,[email protected] and @26,[email protected] perform the initial sign comparison. If the signs are not the equal then which ever has the positive sign is larger. At line @30,[email protected], the inputs are compared based on magnitudes. If the signs were both negative then the unsigned comparison is performed in the opposite direction (\textit{line @31,mp_cmp_[email protected]}). 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. MARK,SHIFTS 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. EXAM,bn_s_mp_add.c Lines @27,[email protected] to @35,}@ perform the initial sorting of the inputs and determine the $min$ and $max$ variables. Note that $x$ is a pointer to a mp\_int assigned to the largest input, in effect it is a local alias. Lines @37,[email protected] to @42,}@ ensure that the destination is grown to accomodate the result of the addition. Similar to the implementation of mp\_copy this function uses the braced code and local aliases coding style. The three aliases that are on lines @56,[email protected], @59,[email protected] and @62,[email protected] represent the two inputs and destination variables respectively. These aliases are used to ensure the compiler does not have to dereference $a$, $b$ or $c$ (respectively) to access the digits of the respective mp\_int. The initial carry $u$ is cleared on line @65,u = [email protected], note that $u$ is of type mp\_digit which ensures type compatibility within the implementation. The initial addition loop begins on line @66,[email protected] and ends on line @75,}@. Similarly the conditional addition loop begins on line @81,[email protected] and ends on line @90,}@. The addition is finished with the final carry being stored in $tmpc$ on line @94,[email protected] Note the ``++'' operator on the same line. After line @94,[email protected] $tmpc$ will point to the $c.used$'th digit of the mp\_int $c$. This is useful for the next loop on lines @97,[email protected] to @99,}@ which set any old upper digits to zero. \subsection{Low Level Subtraction} The low level unsigned subtraction algorithm is very similar to the low level unsigned addition algorithm. The principle difference is that the unsigned subtraction algorithm requires the result to be positive. That is when computing $a - b$ the condition $\vert a \vert \ge \vert b\vert$ must be met for this algorithm to function properly. Keep in mind this low level algorithm is not meant to be used in higher level algorithms directly. This algorithm as will be shown can be used to create functional signed addition and subtraction algorithms. MARK,GAMMA For this algorithm a new variable is required to make the description simpler. Recall from section 1.3.1 that a mp\_digit must be able to represent the range $0 \le x < 2\beta$ for the algorithms to work correctly. However, it is allowable that a mp\_digit represent a larger range of values. For this algorithm we will assume that the variable $\gamma$ represents the number of bits available in a mp\_digit (\textit{this implies $2^{\gamma} > \beta$}). For example, the default for LibTomMath is to use a ``unsigned long'' for the mp\_digit ``type'' while $\beta = 2^{28}$. In ISO C an ``unsigned long'' data type must be able to represent $0 \le x < 2^{32}$ meaning that in this case $\gamma = 32$. \newpage\begin{figure}[!here] \begin{center} \begin{small} \begin{tabular}{l} \hline Algorithm \textbf{s\_mp\_sub}. \\ \textbf{Input}. Two mp\_ints $a$ and $b$ ($\vert a \vert \ge \vert b \vert$) \\ \textbf{Output}. The unsigned subtraction $c = \vert a \vert - \vert b \vert$. \\ \hline \\ 1. $min \leftarrow b.used$ \\ 2. $max \leftarrow a.used$ \\ 3. If $c.alloc < max$ then grow $c$ to hold at least $max$ digits. (\textit{mp\_grow}) \\ 4. $oldused \leftarrow c.used$ \\ 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. EXAM,bn_s_mp_sub.c Line @24,[email protected] and @25,[email protected] perform the initial hardcoded sorting of the inputs. In reality the $min$ and $max$ variables are only aliases and are only used to make the source code easier to read. Again the pointer alias optimization is used within this algorithm. Lines @42,[email protected], @43,[email protected] and @44,[email protected] initialize the aliases for $a$, $b$ and $c$ respectively. The first subtraction loop occurs on lines @47,u = [email protected] through @61,}@. The theory behind the subtraction loop is exactly the same as that for the addition loop. As remarked earlier there is an implementation reason for using the ``awkward'' method of extracting the carry (\textit{see line @57, >>@}). The traditional method for extracting the carry would be to shift by $lg(\beta)$ positions and logically AND the least significant bit. The AND operation is required because all of the bits above the $\lg(\beta)$'th bit will be set to one after a carry occurs from subtraction. This carry extraction requires two relatively cheap operations to extract the carry. The other method is to simply shift the most significant bit to the least significant bit thus extracting the carry with a single cheap operation. This optimization only works on twos compliment machines which is a safe assumption to make. If $a$ has a larger magnitude than $b$ an additional loop (\textit{see lines @64,[email protected] 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 $0$. EXAM,bn_mp_add.c The source code follows the algorithm fairly closely. The most notable new source code addition is the usage of the $res$ integer variable which is used to pass result of the unsigned operations forward. Unlike in the algorithm, the variable $res$ is merely returned as is without explicitly checking it and returning the constant \textbf{MP\_OKAY}. The observation is this algorithm will succeed or fail only if the lower level functions do so. Returning their return code is sufficient. \subsection{High Level Subtraction} The high level signed subtraction algorithm is essentially the same as the high level signed addition algorithm. \newpage\begin{figure}[!here] \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_sub}. \\ \textbf{Input}. Two mp\_ints $a$ and $b$ \\ \textbf{Output}. The signed subtraction $c = a - b$. \\ \hline \\ 1. if $a.sign \ne b.sign$ then do \\ \hspace{3mm}1.1 $c.sign \leftarrow a.sign$ \\ \hspace{3mm}1.2 $c \leftarrow \vert a \vert + \vert b \vert$ (\textit{s\_mp\_add}) \\ 2. else do \\ \hspace{3mm}2.1 if $\vert a \vert \ge \vert b \vert$ then do (\textit{mp\_cmp\_mag}) \\ \hspace{6mm}2.1.1 $c.sign \leftarrow a.sign$ \\ \hspace{6mm}2.1.2 $c \leftarrow \vert a \vert - \vert b \vert$ (\textit{s\_mp\_sub}) \\ \hspace{3mm}2.2 else do \\ \hspace{6mm}2.2.1 $c.sign \leftarrow \left \lbrace \begin{array}{ll} MP\_ZPOS & \mbox{if }a.sign = MP\_NEG \\ MP\_NEG & \mbox{otherwise} \\ \end{array} \right .$ \\ \hspace{6mm}2.2.2 $c \leftarrow \vert b \vert - \vert a \vert$ \\ 3. 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. EXAM,bn_mp_sub.c Much like the implementation of algorithm mp\_add the variable $res$ is used to catch the return code of the unsigned addition or subtraction operations and forward it to the end of the function. On line @38, != MP_[email protected] the ``not equal to'' \textbf{MP\_LT} expression is used to emulate a ``greater than or equal to'' comparison. \section{Bit and Digit Shifting} MARK,POLY It is quite common to think of a multiple precision integer as a polynomial in $x$, that is $y = f(\beta)$ where $f(x) = \sum_{i=0}^{n-1} a_i x^i$. This notation arises within discussion of Montgomery and Diminished Radix Reduction as well as Karatsuba multiplication and squaring. In order to facilitate operations on polynomials in $x$ as above a series of simple ``digit'' algorithms have to be established. That is to shift the digits left or right as well to shift individual bits of the digits left and right. It is important to note that not all ``shift'' operations are on radix-$\beta$ digits. \subsection{Multiplication by Two} In a binary system where the radix is a power of two multiplication by two not only arises often in other algorithms it is a fairly efficient operation to perform. A single precision logical shift left is sufficient to multiply a single digit by two. \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_mul\_2}. \\ \textbf{Input}. One mp\_int $a$ \\ \textbf{Output}. $b = 2a$. \\ \hline \\ 1. If $b.alloc < a.used + 1$ then grow $b$ to hold $a.used + 1$ digits. (\textit{mp\_grow}) \\ 2. $oldused \leftarrow b.used$ \\ 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 ~SHIFTS~ that $a_n << 1$ is equivalent to $a_n \cdot 2$. An iteration of the addition loop is finished with forwarding the carry to the next iteration. Step 7 takes care of any final carry by setting the $a.used$'th digit of the result to the carry and augmenting the \textbf{used} count of $b$. Step 8 clears any leading digits of $b$ in case it originally had a larger magnitude than $a$. EXAM,bn_mp_mul_2.c This implementation is essentially an optimized implementation of s\_mp\_add for the case of doubling an input. The only noteworthy difference is the use of the logical shift operator on line @52,<<@ to perform a single precision doubling. \subsection{Division by Two} A division by two can just as easily be accomplished with a logical shift right as multiplication by two can be with a logical shift left. \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_div\_2}. \\ \textbf{Input}. One mp\_int $a$ \\ \textbf{Output}. $b = a/2$. \\ \hline \\ 1. If $b.alloc < a.used$ then grow $b$ to hold $a.used$ digits. (\textit{mp\_grow}) \\ 2. If the reallocation failed return(\textit{MP\_MEM}). \\ 3. $oldused \leftarrow b.used$ \\ 4. $b.used \leftarrow a.used$ \\ 5. $r \leftarrow 0$ \\ 6. for $n$ from $b.used - 1$ to $0$ do \\ \hspace{3mm}6.1 $rr \leftarrow a_n \mbox{ (mod }2\mbox{)}$\\ \hspace{3mm}6.2 $b_n \leftarrow (a_n >> 1) + (r << (lg(\beta) - 1)) \mbox{ (mod }\beta\mbox{)}$ \\ \hspace{3mm}6.3 $r \leftarrow rr$ \\ 7. If $b.used < oldused - 1$ then do \\ \hspace{3mm}7.1 for $n$ from $b.used$ to $oldused - 1$ do \\ \hspace{6mm}7.1.1 $b_n \leftarrow 0$ \\ 8. $b.sign \leftarrow a.sign$ \\ 9. Clamp excess digits of $b$. (\textit{mp\_clamp}) \\ 10. Return(\textit{MP\_OKAY}).\\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_div\_2} \end{figure} \textbf{Algorithm mp\_div\_2.} This algorithm will divide an mp\_int by two using logical shifts to the right. Like mp\_mul\_2 it uses a modified low level addition core as the basis of the algorithm. Unlike mp\_mul\_2 the shift operations work from the leading digit to the trailing digit. The algorithm could be written to work from the trailing digit to the leading digit however, it would have to stop one short of $a.used - 1$ digits to prevent reading past the end of the array of digits. Essentially the loop at step 6 is similar to that of mp\_mul\_2 except the logical shifts go in the opposite direction and the carry is at the least significant bit not the most significant bit. EXAM,bn_mp_div_2.c \section{Polynomial Basis Operations} Recall from ~POLY~ that any integer can be represented as a polynomial in $x$ as $y = f(\beta)$. Such a representation is also known as the polynomial basis \cite[pp. 48]{ROSE}. Given such a notation a multiplication or division by $x$ amounts to shifting whole digits a single place. The need for such operations arises in several other higher level algorithms such as Barrett and Montgomery reduction, integer division and Karatsuba multiplication. Converting from an array of digits to polynomial basis is very simple. Consider the integer $y \equiv (a_2, a_1, a_0)_{\beta}$ and recall that $y = \sum_{i=0}^{2} a_i \beta^i$. Simply replace $\beta$ with $x$ and the expression is in polynomial basis. For example, $f(x) = 8x + 9$ is the polynomial basis representation for $89$ using radix ten. That is, $f(10) = 8(10) + 9 = 89$. \subsection{Multiplication by $x$} Given a polynomial in $x$ such as $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0$ multiplying by $x$ amounts to shifting the coefficients up one degree. In this case $f(x) \cdot x = a_n x^{n+1} + a_{n-1} x^n + ... + a_0 x$. From a scalar basis point of view multiplying by $x$ is equivalent to multiplying by the integer $\beta$. \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_lshd}. \\ \textbf{Input}. One mp\_int $a$ and an integer $b$ \\ \textbf{Output}. $a \leftarrow a \cdot \beta^b$ (equivalent to multiplication by $x^b$). \\ \hline \\ 1. If $b \le 0$ then return(\textit{MP\_OKAY}). \\ 2. If $a.alloc < a.used + b$ then grow $a$ to at least $a.used + b$ digits. (\textit{mp\_grow}). \\ 3. If the reallocation failed return(\textit{MP\_MEM}). \\ 4. $a.used \leftarrow a.used + b$ \\ 5. $i \leftarrow a.used - 1$ \\ 6. $j \leftarrow a.used - 1 - b$ \\ 7. for $n$ from $a.used - 1$ to $b$ do \\ \hspace{3mm}7.1 $a_{i} \leftarrow a_{j}$ \\ \hspace{3mm}7.2 $i \leftarrow i - 1$ \\ \hspace{3mm}7.3 $j \leftarrow j - 1$ \\ 8. for $n$ from 0 to $b - 1$ do \\ \hspace{3mm}8.1 $a_n \leftarrow 0$ \\ 9. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_lshd} \end{figure} \textbf{Algorithm mp\_lshd.} This algorithm multiplies an mp\_int by the $b$'th power of $x$. This is equivalent to multiplying by $\beta^b$. The algorithm differs from the other algorithms presented so far as it performs the operation in place instead storing the result in a separate location. The motivation behind this change is due to the way this function is typically used. Algorithms such as mp\_add store the result in an optionally different third mp\_int because the original inputs are often still required. Algorithm mp\_lshd (\textit{and similarly algorithm mp\_rshd}) is typically used on values where the original value is no longer required. The algorithm will return success immediately if $b \le 0$ since the rest of algorithm is only valid when $b > 0$. First the destination $a$ is grown as required to accomodate the result. The counters $i$ and $j$ are used to form a \textit{sliding window} over the digits of $a$ of length $b$. The head of the sliding window is at $i$ (\textit{the leading digit}) and the tail at $j$ (\textit{the trailing digit}). The loop on step 7 copies the digit from the tail to the head. In each iteration the window is moved down one digit. The last loop on step 8 sets the lower $b$ digits to zero. \newpage FIGU,sliding_window,Sliding Window Movement EXAM,bn_mp_lshd.c The if statement on line @24,[email protected] ensures that the $b$ variable is greater than zero. The \textbf{used} count is incremented by $b$ before the copy loop begins. This elminates the need for an additional variable in the for loop. The variable $top$ on line @42,[email protected] is an alias for the leading digit while $bottom$ on line @45,[email protected] is an alias for the trailing edge. The aliases form a window of exactly $b$ digits over the input. \subsection{Division by $x$} Division by powers of $x$ is easily achieved by shifting the digits right and removing any that will end up to the right of the zero'th digit. \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_rshd}. \\ \textbf{Input}. One mp\_int $a$ and an integer $b$ \\ \textbf{Output}. $a \leftarrow a / \beta^b$ (Divide by $x^b$). \\ \hline \\ 1. If $b \le 0$ then return. \\ 2. If $a.used \le b$ then do \\ \hspace{3mm}2.1 Zero $a$. (\textit{mp\_zero}). \\ \hspace{3mm}2.2 Return. \\ 3. $i \leftarrow 0$ \\ 4. $j \leftarrow b$ \\ 5. for $n$ from 0 to $a.used - b - 1$ do \\ \hspace{3mm}5.1 $a_i \leftarrow a_j$ \\ \hspace{3mm}5.2 $i \leftarrow i + 1$ \\ \hspace{3mm}5.3 $j \leftarrow j + 1$ \\ 6. for $n$ from $a.used - b$ to $a.used - 1$ do \\ \hspace{3mm}6.1 $a_n \leftarrow 0$ \\ 7. $a.used \leftarrow a.used - b$ \\ 8. Return. \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_rshd} \end{figure} \textbf{Algorithm mp\_rshd.} This algorithm divides the input in place by the $b$'th power of $x$. It is analogous to dividing by a $\beta^b$ but much quicker since it does not require single precision division. This algorithm does not actually return an error code as it cannot fail. If the input $b$ is less than one the algorithm quickly returns without performing any work. If the \textbf{used} count is less than or equal to the shift count $b$ then it will simply zero the input and return. After the trivial cases of inputs have been handled the sliding window is setup. Much like the case of algorithm mp\_lshd a sliding window that is $b$ digits wide is used to copy the digits. Unlike mp\_lshd the window slides in the opposite direction from the trailing to the leading digit. Also the digits are copied from the leading to the trailing edge. Once the window copy is complete the upper digits must be zeroed and the \textbf{used} count decremented. EXAM,bn_mp_rshd.c The only noteworthy element of this routine is the lack of a return type. -- Will update later to give it a return type...Tom \section{Powers of Two} Now that algorithms for moving single bits as well as whole digits exist algorithms for moving the ``in between'' distances are required. For example, to quickly multiply by $2^k$ for any $k$ without using a full multiplier algorithm would prove useful. Instead of performing single shifts $k$ times to achieve a multiplication by $2^{\pm k}$ a mixture of whole digit shifting and partial digit shifting is employed. \subsection{Multiplication by Power of Two} \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_mul\_2d}. \\ \textbf{Input}. One mp\_int $a$ and an integer $b$ \\ \textbf{Output}. $c \leftarrow a \cdot 2^b$. \\ \hline \\ 1. $c \leftarrow a$. (\textit{mp\_copy}) \\ 2. If $c.alloc < c.used + \lfloor b / lg(\beta) \rfloor + 2$ then grow $c$ accordingly. \\ 3. If the reallocation failed return(\textit{MP\_MEM}). \\ 4. If $b \ge lg(\beta)$ then \\ \hspace{3mm}4.1 $c \leftarrow c \cdot \beta^{\lfloor b / lg(\beta) \rfloor}$ (\textit{mp\_lshd}). \\ \hspace{3mm}4.2 If step 4.1 failed return(\textit{MP\_MEM}). \\ 5. $d \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\ 6. If $d \ne 0$ then do \\ \hspace{3mm}6.1 $mask \leftarrow 2^d$ \\ \hspace{3mm}6.2 $r \leftarrow 0$ \\ \hspace{3mm}6.3 for $n$ from $0$ to $c.used - 1$ do \\ \hspace{6mm}6.3.1 $rr \leftarrow c_n >> (lg(\beta) - d) \mbox{ (mod }mask\mbox{)}$ \\ \hspace{6mm}6.3.2 $c_n \leftarrow (c_n << d) + r \mbox{ (mod }\beta\mbox{)}$ \\ \hspace{6mm}6.3.3 $r \leftarrow rr$ \\ \hspace{3mm}6.4 If $r > 0$ then do \\ \hspace{6mm}6.4.1 $c_{c.used} \leftarrow r$ \\ \hspace{6mm}6.4.2 $c.used \leftarrow c.used + 1$ \\ 7. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_mul\_2d} \end{figure} \textbf{Algorithm mp\_mul\_2d.} This algorithm multiplies $a$ by $2^b$ and stores the result in $c$. The algorithm uses algorithm mp\_lshd and a derivative of algorithm mp\_mul\_2 to quickly compute the product. First the algorithm will multiply $a$ by $x^{\lfloor b / lg(\beta) \rfloor}$ which will ensure that the remainder multiplicand is less than $\beta$. For example, if $b = 37$ and $\beta = 2^{28}$ then this step will multiply by $x$ leaving a multiplication by $2^{37 - 28} = 2^{9}$ left. After the digits have been shifted appropriately at most $lg(\beta) - 1$ shifts are left to perform. Step 5 calculates the number of remaining shifts required. If it is non-zero a modified shift loop is used to calculate the remaining product. Essentially the loop is a generic version of algorith mp\_mul2 designed to handle any shift count in the range $1 \le x < lg(\beta)$. The $mask$ variable is used to extract the upper $d$ bits to form the carry for the next iteration. This algorithm is loosely measured as a $O(2n)$ algorithm which means that if the input is $n$-digits that it takes $2n$ ``time'' to complete. It is possible to optimize this algorithm down to a $O(n)$ algorithm at a cost of making the algorithm slightly harder to follow. EXAM,bn_mp_mul_2d.c Notes to be revised when code is updated. -- Tom \subsection{Division by Power of Two} \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_div\_2d}. \\ \textbf{Input}. One mp\_int $a$ and an integer $b$ \\ \textbf{Output}. $c \leftarrow \lfloor a / 2^b \rfloor, d \leftarrow a \mbox{ (mod }2^b\mbox{)}$. \\ \hline \\ 1. If $b \le 0$ then do \\ \hspace{3mm}1.1 $c \leftarrow a$ (\textit{mp\_copy}) \\ \hspace{3mm}1.2 $d \leftarrow 0$ (\textit{mp\_zero}) \\ \hspace{3mm}1.3 Return(\textit{MP\_OKAY}). \\ 2. $c \leftarrow a$ \\ 3. $d \leftarrow a \mbox{ (mod }2^b\mbox{)}$ (\textit{mp\_mod\_2d}) \\ 4. If $b \ge lg(\beta)$ then do \\ \hspace{3mm}4.1 $c \leftarrow \lfloor c/\beta^{\lfloor b/lg(\beta) \rfloor} \rfloor$ (\textit{mp\_rshd}). \\ 5. $k \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\ 6. If $k \ne 0$ then do \\ \hspace{3mm}6.1 $mask \leftarrow 2^k$ \\ \hspace{3mm}6.2 $r \leftarrow 0$ \\ \hspace{3mm}6.3 for $n$ from $c.used - 1$ to $0$ do \\ \hspace{6mm}6.3.1 $rr \leftarrow c_n \mbox{ (mod }mask\mbox{)}$ \\ \hspace{6mm}6.3.2 $c_n \leftarrow (c_n >> k) + (r << (lg(\beta) - k))$ \\ \hspace{6mm}6.3.3 $r \leftarrow rr$ \\ 7. Clamp excess digits of $c$. (\textit{mp\_clamp}) \\ 8. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_div\_2d} \end{figure} \textbf{Algorithm mp\_div\_2d.} This algorithm will divide an input $a$ by $2^b$ and produce the quotient and remainder. The algorithm is designed much like algorithm mp\_mul\_2d by first using whole digit shifts then single precision shifts. This algorithm will also produce the remainder of the division by using algorithm mp\_mod\_2d. EXAM,bn_mp_div_2d.c The implementation of algorithm mp\_div\_2d is slightly different than the algorithm specifies. The remainder $d$ may be optionally ignored by passing \textbf{NULL} as the pointer to the mp\_int variable. The temporary mp\_int variable $t$ is used to hold the result of the remainder operation until the end. This allows $d$ and $a$ to represent the same mp\_int without modifying $a$ before the quotient is obtained. The remainder of the source code is essentially the same as the source code for mp\_mul\_2d. (-- Fix this paragraph up later, Tom). \subsection{Remainder of Division by Power of Two} The last algorithm in the series of polynomial basis power of two algorithms is calculating the remainder of division by $2^b$. This algorithm benefits from the fact that in twos complement arithmetic $a \mbox{ (mod }2^b\mbox{)}$ is the same as $a$ AND $2^b - 1$. \begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_mod\_2d}. \\ \textbf{Input}. One mp\_int $a$ and an integer $b$ \\ \textbf{Output}. $c \leftarrow a \mbox{ (mod }2^b\mbox{)}$. \\ \hline \\ 1. If $b \le 0$ then do \\ \hspace{3mm}1.1 $c \leftarrow 0$ (\textit{mp\_zero}) \\ \hspace{3mm}1.2 Return(\textit{MP\_OKAY}). \\ 2. If $b > a.used \cdot lg(\beta)$ then do \\ \hspace{3mm}2.1 $c \leftarrow a$ (\textit{mp\_copy}) \\ \hspace{3mm}2.2 Return the result of step 2.1. \\ 3. $c \leftarrow a$ \\ 4. If step 3 failed return(\textit{MP\_MEM}). \\ 5. for $n$ from $\lceil b / lg(\beta) \rceil$ to $c.used$ do \\ \hspace{3mm}5.1 $c_n \leftarrow 0$ \\ 6. $k \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\ 7. $c_{\lfloor b / lg(\beta) \rfloor} \leftarrow c_{\lfloor b / lg(\beta) \rfloor} \mbox{ (mod }2^{k}\mbox{)}$. \\ 8. Clamp excess digits of $c$. (\textit{mp\_clamp}) \\ 9. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_mod\_2d} \end{figure} \textbf{Algorithm mp\_mod\_2d.} This algorithm will quickly calculate the value of $a \mbox{ (mod }2^b\mbox{)}$. First if $b$ is less than or equal to zero the result is set to zero. If $b$ is greater than the number of bits in $a$ then it simply copies $a$ to $c$ and returns. Otherwise, $a$ is copied to $b$, leading digits are removed and the remaining leading digit is trimed to the exact bit count. EXAM,bn_mp_mod_2d.c -- Add comments later, Tom. \section*{Exercises} \begin{tabular}{cl} $\left [ 3 \right ] $ & Devise an algorithm that performs $a \cdot 2^b$ for generic values of $b$ \\ & in $O(n)$ time. \\ &\\ $\left [ 3 \right ] $ & Devise an efficient algorithm to multiply by small low hamming \\ & weight values such as $3$, $5$ and $9$. Extend it to handle all values \\ & upto $64$ with a hamming weight less than three. \\ &\\ $\left [ 2 \right ] $ & Modify the preceding algorithm to handle values of the form \\ & $2^k - 1$ as well. \\ &\\ $\left [ 3 \right ] $ & Using only algorithms mp\_mul\_2, mp\_div\_2 and mp\_add create an \\ & algorithm to multiply two integers in roughly $O(2n^2)$ time for \\ & any $n$-bit input. Note that the time of addition is ignored in the \\ & calculation. \\ & \\ $\left [ 5 \right ] $ & Improve the previous algorithm to have a working time of at most \\ & $O \left (2^{(k-1)}n + \left ({2n^2 \over k} \right ) \right )$ for an appropriate choice of $k$. Again ignore \\ & the cost of addition. \\ & \\ $\left [ 2 \right ] $ & Devise a chart to find optimal values of $k$ for the previous problem \\ & for $n = 64 \ldots 1024$ in steps of $64$. \\ & \\ $\left [ 2 \right ] $ & Using only algorithms mp\_abs and mp\_sub devise another method for \\ & calculating the result of a signed comparison. \\ & \end{tabular} \chapter{Multiplication and Squaring} \section{The Multipliers} For most number theoretic 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 ~GAMMA~ 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 ~COMBA~ for more information}). \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{s\_mp\_mul\_digs}. \\ \textbf{Input}. mp\_int $a$, mp\_int $b$ and an integer $digs$ \\ \textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert \mbox{ (mod }\beta^{digs}\mbox{)}$. \\ \hline \\ 1. If min$(a.used, b.used) < \delta$ then do \\ \hspace{3mm}1.1 Calculate $c = \vert a \vert \cdot \vert b \vert$ by the Comba method (\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. EXAM,bn_s_mp_mul_digs.c Lines @31,[email protected] to @35,}@ determine if the Comba method can be used first. The conditions for using the Comba routine are that min$(a.used, b.used) < \delta$ and the number of digits of output is less than \textbf{MP\_WARRAY}. This new constant is used to control the stack usage in the Comba routines. By default it is set to $\delta$ but can be reduced when memory is at a premium. Of particular importance is the calculation of the $ix+iy$'th column on lines @64,mp_[email protected], @65,mp_[email protected] and @66,mp_[email protected] Note how all of the variables are cast to the type \textbf{mp\_word}, which is also the type of variable $\hat r$. That is to ensure that double precision operations are used instead of single precision. The multiplication on line @65,) * (@ makes use of a specific GCC optimizer behaviour. On the outset it looks like the compiler will have to use a double precision multiplication to produce the result required. Such an operation would be horribly slow on most processors and drag this to a crawl. However, GCC is smart enough to realize that double wide output single precision multipliers can be used. For example, the instruction ``MUL'' on the x86 processor can multiply two 32-bit values and produce a 64-bit result. \subsection{Faster Multiplication by the ``Comba'' Method} MARK,COMBA One of the huge drawbacks of the ``baseline'' algorithms is that at the $O(n^2)$ level the carry must be computed and propagated upwards. This makes the nested loop very sequential and hard to unroll and implement in parallel. The ``Comba'' \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} double precision digits named $\hat W$ on the stack. \\ 1. If $c.alloc < digs$ then grow $c$ to $digs$ digits. (\textit{mp\_grow}) \\ 2. If step 1 failed return(\textit{MP\_MEM}).\\ \\ Zero the temporary array $\hat W$. \\ 3. for $n$ from $0$ to $digs - 1$ do \\ \hspace{3mm}3.1 $\hat W_n \leftarrow 0$ \\ \\ Compute the columns. \\ 4. for $ix$ from $0$ to $a.used - 1$ do \\ \hspace{3mm}4.1 $pb \leftarrow \mbox{min}(b.used, digs - ix)$ \\ \hspace{3mm}4.2 If $pb < 1$ then goto step 5. \\ \hspace{3mm}4.3 for $iy$ from $0$ to $pb - 1$ do \\ \hspace{6mm}4.3.1 $\hat W_{ix+iy} \leftarrow \hat W_{ix+iy} + a_{ix}b_{iy}$ \\ \\ Propagate the carries upwards. \\ 5. $oldused \leftarrow c.used$ \\ 6. $c.used \leftarrow digs$ \\ 7. If $digs > 1$ then do \\ \hspace{3mm}7.1. for $ix$ from $1$ to $digs - 1$ do \\ \hspace{6mm}7.1.1 $\hat W_{ix} \leftarrow \hat W_{ix} + \lfloor \hat W_{ix-1} / \beta \rfloor$ \\ \hspace{6mm}7.1.2 $c_{ix - 1} \leftarrow \hat W_{ix - 1} \mbox{ (mod }\beta\mbox{)}$ \\ 8. else do \\ \hspace{3mm}8.1 $ix \leftarrow 0$ \\ 9. $c_{ix} \leftarrow \hat W_{ix} \mbox{ (mod }\beta\mbox{)}$ \\ \\ Zero excess digits. \\ 10. If $digs < oldused$ then do \\ \hspace{3mm}10.1 for $n$ from $digs$ to $oldused - 1$ do \\ \hspace{6mm}10.1.1 $c_n \leftarrow 0$ \\ 11. Clamp excessive digits of $c$. (\textit{mp\_clamp}) \\ 12. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm fast\_s\_mp\_mul\_digs} \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 algorithm essentially peforms the same calculation as algorithm s\_mp\_mul\_digs, just much faster. The array $\hat W$ is meant to be on the stack when the algorithm is used. The size of the array does not change which is ideal. Note also that unlike algorithm s\_mp\_mul\_digs no temporary mp\_int is required since the result is calculated directly in $\hat W$. The $O(n^2)$ loop on step four is where the Comba method's advantages begin to show through in comparison to the baseline algorithm. The lack of a carry variable or propagation in this loop allows the loop to be performed with only single precision multiplication and additions. Now that each iteration of the inner loop can be performed independent of the others the inner loop can be performed with a high level of parallelism. 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. EXAM,bn_fast_s_mp_mul_digs.c The memset on line @47,[email protected] clears the initial $\hat W$ array to zero in a single step. Like the slower baseline multiplication implementation a series of aliases (\textit{lines @67, [email protected], @70, [email protected] and @75,_[email protected]}) are used to simplify the inner $O(n^2)$ loop. In this case a new alias $\_\hat W$ has been added which refers to the double precision columns offset by $ix$ in each pass. The inner loop on lines @83,[email protected], @84,mp_[email protected] and @85,}@ is where the algorithm will spend the majority of the time, which is why it has been stripped to the bones of any extra baggage\footnote{Hence the pointer aliases.}. On x86 processors the multiplication and additions amount to at the very least five instructions (\textit{two loads, two additions, one multiply}) while on the ARMv4 processors they amount to only three (\textit{one load, one store, one multiply-add}). For both of the x86 and ARMv4 processors the GCC compiler performs a good job at unrolling the loop and scheduling the instructions so there are very few dependency stalls. In theory the difference between the baseline and comba algorithms is a mere $O(qn)$ time difference. However, in the $O(n^2)$ nested loop of the baseline method there are dependency stalls as the algorithm must wait for the multiplier to finish before propagating the carry to the next digit. As a result fewer of the often multiple execution units\footnote{The AMD Athlon has three execution units and the Intel P4 has four.} can be simultaneously used. \subsection{Polynomial Basis Multiplication} To break the $O(n^2)$ barrier in multiplication requires a completely different look at integer multiplication. In the following algorithms the use of polynomial basis representation for two integers $a$ and $b$ as $f(x) = \sum_{i=0}^{n} a_i x^i$ and $g(x) = \sum_{i=0}^{n} b_i x^i$ respectively, is required. In this system both $f(x)$ and $g(x)$ have $n + 1$ terms and are of the $n$'th degree. The product $a \cdot b \equiv f(x)g(x)$ 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. It is worth noting that the point $\zeta_1$ could be substituted for $-\zeta_{-1}$. In this case the first and third row are subtracted instead of added to the second row. \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_karatsuba\_mul}. \\ \textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ \textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert$ \\ \hline \\ 1. Init the following mp\_int variables: $x0$, $x1$, $y0$, $y1$, $t1$, $x0y0$, $x1y1$.\\ 2. If step 2 failed then return(\textit{MP\_MEM}). \\ \\ Split the input. e.g. $a = x1 \cdot \beta^B + x0$ \\ 3. $B \leftarrow \mbox{min}(a.used, b.used)/2$ \\ 4. $x0 \leftarrow a \mbox{ (mod }\beta^B\mbox{)}$ (\textit{mp\_mod\_2d}) \\ 5. $y0 \leftarrow b \mbox{ (mod }\beta^B\mbox{)}$ \\ 6. $x1 \leftarrow \lfloor a / \beta^B \rfloor$ (\textit{mp\_rshd}) \\ 7. $y1 \leftarrow \lfloor b / \beta^B \rfloor$ \\ \\ Calculate the three products. \\ 8. $x0y0 \leftarrow x0 \cdot y0$ (\textit{mp\_mul}) \\ 9. $x1y1 \leftarrow x1 \cdot y1$ \\ 10. $t1 \leftarrow x1 - x0$ (\textit{mp\_sub}) \\ 11. $x0 \leftarrow y1 - y0$ \\ 12. $t1 \leftarrow t1 \cdot x0$ \\ \\ Calculate the middle term. \\ 13. $x0 \leftarrow x0y0 + x1y1$ \\ 14. $t1 \leftarrow x0 - t1$ \\ \\ Calculate the final product. \\ 15. $t1 \leftarrow t1 \cdot \beta^B$ (\textit{mp\_lshd}) \\ 16. $x1y1 \leftarrow x1y1 \cdot \beta^{2B}$ \\ 17. $t1 \leftarrow x0y0 + t1$ \\ 18. $c \leftarrow t1 + x1y1$ \\ 19. Clear all of the temporary variables. \\ 20. Return(\textit{MP\_OKAY}).\\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_karatsuba\_mul} \end{figure} \textbf{Algorithm mp\_karatsuba\_mul.} This algorithm computes the unsigned product of two inputs using the Karatsuba 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. EXAM,bn_mp_karatsuba_mul.c 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 @61,[email protected] to @75,[email protected] 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 @98,[email protected] copies the lower halves. Since they are both the same magnitude it is simpler to calculate both lower halves in a single loop. The for loop on lines @104,[email protected] and @109,[email protected] 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 @152,[email protected] 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. EXAM,bn_mp_toom_mul.c -- Comments to be added during editing phase. \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. \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_mul}. \\ \textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ \textbf{Output}. $c \leftarrow a \cdot b$ \\ \hline \\ 1. If $a.sign = b.sign$ then \\ \hspace{3mm}1.1 $sign = MP\_ZPOS$ \\ 2. else \\ \hspace{3mm}2.1 $sign = MP\_ZNEG$ \\ 3. If min$(a.used, b.used) \ge TOOM\_MUL\_CUTOFF$ then \\ \hspace{3mm}3.1 $c \leftarrow a \cdot b$ using algorithm mp\_toom\_mul \\ 4. else if min$(a.used, b.used) \ge KARATSUBA\_MUL\_CUTOFF$ then \\ \hspace{3mm}4.1 $c \leftarrow a \cdot b$ using algorithm mp\_karatsuba\_mul \\ 5. else \\ \hspace{3mm}5.1 $digs \leftarrow a.used + b.used + 1$ \\ \hspace{3mm}5.2 If $digs < MP\_ARRAY$ and min$(a.used, b.used) \le \delta$ then \\ \hspace{6mm}5.2.1 $c \leftarrow a \cdot b \mbox{ (mod }\beta^{digs}\mbox{)}$ using algorithm fast\_s\_mp\_mul\_digs. \\ \hspace{3mm}5.3 else \\ \hspace{6mm}5.3.1 $c \leftarrow a \cdot b \mbox{ (mod }\beta^{digs}\mbox{)}$ using algorithm s\_mp\_mul\_digs. \\ 6. $c.sign \leftarrow sign$ \\ 7. Return the result of the unsigned multiplication performed. \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_mul} \end{figure} \textbf{Algorithm mp\_mul.} This algorithm performs the signed multiplication of two inputs. It will make use of any of the three unsigned multiplication algorithms available when the input is of appropriate size. The \textbf{sign} of the result is not set until the end of the algorithm since algorithm s\_mp\_mul\_digs will clear it. EXAM,bn_mp_mul.c The implementation is rather simplistic and is not particularly noteworthy. Line @22,[email protected] computes the sign of the result using the ``?'' operator from the C programming language. Line @37,<<@ computes $\delta$ using the fact that $1 << k$ is equal to $2^k$. \section{Squaring} \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} MARK,SQUARE Starting from zero and numbering the columns from right to left a very simple pattern becomes obvious. For the purposes of this discussion let $x$ represent the number being squared. The first observation is that in row $k$ the $2k$'th column of the product has a $\left (x_k \right)^2$ term in it. The second observation is that every column $j$ in row $k$ where $j \ne 2k$ is part of a double product. Every 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. \newpage\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. EXAM,bn_s_mp_sqr.c Inside the outer loop (\textit{see line @32,[email protected]}) the square term is calculated on line @35,r [email protected] Line @42,>>@ extracts the carry from the square term. Aliases for $a_{ix}$ and $t_{ix+iy}$ are initialized on lines @45,[email protected] and @48,[email protected] respectively. The doubling is performed using two additions (\textit{see line @57,r + [email protected]}) since it is usually faster than shifting,if not at least as fast. \subsection{Faster Squaring by the ``Comba'' Method} A major drawback to the baseline method is the requirement for single precision shifting inside the $O(n^2)$ 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. \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{fast\_s\_mp\_sqr}. \\ \textbf{Input}. mp\_int $a$ \\ \textbf{Output}. $b \leftarrow a^2$ \\ \hline \\ Place two arrays of \textbf{MP\_WARRAY} mp\_words named $\hat W$ and $\hat {X}$ on the stack. \\ 1. If $b.alloc < 2a.used + 1$ then grow $b$ to $2a.used + 1$ digits. (\textit{mp\_grow}). \\ 2. If step 1 failed return(\textit{MP\_MEM}). \\ 3. for $ix$ from $0$ to $2a.used + 1$ do \\ \hspace{3mm}3.1 $\hat W_{ix} \leftarrow 0$ \\ \hspace{3mm}3.2 $\hat {X}_{ix} \leftarrow 0$ \\ 4. for $ix$ from $0$ to $a.used - 1$ do \\ \hspace{3mm}Compute the square.\\ \hspace{3mm}4.1 $\hat {X}_{ix+ix} \leftarrow \left ( a_{ix} \right )^2$ \\ \\ \hspace{3mm}Compute the double products.\\ \hspace{3mm}4.2 for $iy$ from $ix + 1$ to $a.used - 1$ do \\ \hspace{6mm}4.2.1 $\hat W_{ix+iy} \leftarrow \hat W_{ix+iy} + a_{ix}a_{iy}$ \\ 5. $oldused \leftarrow b.used$ \\ 6. $b.used \leftarrow 2a.used + 1$ \\ \\ Double the products and propagate the carries simultaneously. \\ 7. $\hat W_0 \leftarrow 2 \hat W_0 + \hat {X}_0$ \\ 8. for $ix$ from $1$ to $2a.used$ do \\ \hspace{3mm}8.1 $\hat W_{ix} \leftarrow 2 \hat W_{ix} + \hat {X}_{ix}$ \\ \hspace{3mm}8.2 $\hat W_{ix} \leftarrow \hat W_{ix} + \lfloor \hat W_{ix - 1} / \beta \rfloor$ \\ \hspace{3mm}8.3 $b_{ix-1} \leftarrow W_{ix-1} \mbox{ (mod }\beta\mbox{)}$ \\ 9. $b_{2a.used} \leftarrow \hat W_{2a.used} \mbox{ (mod }\beta\mbox{)}$ \\ 10. if $2a.used + 1 < oldused$ then do \\ \hspace{3mm}10.1 for $ix$ from $2a.used + 1$ to $oldused$ do \\ \hspace{6mm}10.1.1 $b_{ix} \leftarrow 0$ \\ 11. Clamp excess digits from $b$. (\textit{mp\_clamp}) \\ 12. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm fast\_s\_mp\_sqr} \end{figure} \textbf{Algorithm fast\_s\_mp\_sqr.} This algorithm computes the square of an input using the Comba technique. It is designed to be a replacement for algorithm s\_mp\_sqr when the number of input digits is less than \textbf{MP\_WARRAY} and less than $\delta \over 2$. This routine requires two arrays of mp\_words to be placed on the stack. The first array $\hat W$ will hold the double products and the second array $\hat X$ will hold the squares. Though only at most $MP\_WARRAY \over 2$ words of $\hat X$ are used, it has proven faster on most processors to simply make it a full size array. The loop on step 3 will zero the two arrays to prepare them for the squaring step. Step 4.1 computes the squares of the product. Note how it simply assigns the value into the $\hat X$ array. The nested loop on step 4.2 computes the doubles of the products. This loop computes the sum of the products for each column. They are not doubled until later. After the squaring loop, the products stored in $\hat W$ musted be doubled and the carries propagated forwards. It makes sense to do both operations at the same time. The expression $\hat W_{ix} \leftarrow 2 \hat W_{ix} + \hat {X}_{ix}$ computes the sum of the double product and the squares in place. EXAM,bn_fast_s_mp_sqr.c -- Write something deep and insightful later, Tom. \subsection{Polynomial Basis Squaring} The same algorithm that performs optimal polynomial basis multiplication can be used to perform polynomial basis squaring. The minor exception is that $\zeta_y = f(y)g(y)$ 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^2 + b^2 - (a - b)^2 \right )x + b^2 \end{equation} Upon closer inspection this equation only requires the calculation of three half-sized squares: $a^2$, $b^2$ and $(a - b)^2$. As in Karatsuba multiplication, this algorithm can be applied recursively on the input and will achieve an asymptotic running time of $O \left ( n^{lg(3)} \right )$. If the asymptotic 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{mp\_sub}) \\ 9. $t1 \leftarrow t1^2$ \\ \\ Compute the middle term. \\ 10. $t2 \leftarrow x0x0 + x1x1$ (\textit{s\_mp\_add}) \\ 11. $t1 \leftarrow t2 - t1$ \\ \\ Compute final product. \\ 12. $t1 \leftarrow t1\beta^B$ (\textit{mp\_lshd}) \\ 13. $x1x1 \leftarrow x1x1\beta^{2B}$ \\ 14. $t1 \leftarrow t1 + x0x0$ \\ 15. $b \leftarrow t1 + x1x1$ \\ 16. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_karatsuba\_sqr} \end{figure} \textbf{Algorithm mp\_karatsuba\_sqr.} This algorithm computes the square of an input $a$ using the Karatsuba technique. This algorithm is very 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 $x1^2 + x0^2 - (x1 - x0)^2 = 2 \cdot x0 \cdot x1$. Now if $5n$ single precision additions and a squaring of $n$-digits is faster than multiplying two $n$-digit numbers and doubling then this method is faster. Assuming no further recursions occur, the difference can be estimated 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. EXAM,bn_mp_karatsuba_sqr.c This implementation is largely based on the implementation of algorithm mp\_karatsuba\_mul. It uses the same inline style to copy and shift the input into the two halves. The loop from line @54,{@ to line @70,}@ has been modified since only one input exists. The \textbf{used} count of both $x0$ and $x1$ is fixed up and $x0$ is clamped before the calculations begin. At this point $x1$ and $x0$ are valid equivalents to the respective halves as if mp\_rshd and mp\_mod\_2d had been used. By inlining the copy and shift operations the cutoff point for Karatsuba multiplication can be lowered. On the Athlon the cutoff point is exactly at the point where Comba squaring can no longer be used (\textit{128 digits}). On slower processors such as the Intel P4 it is actually below the Comba limit (\textit{at 110 digits}). This routine uses the same error trap coding style as mp\_karatsuba\_sqr. As the temporary variables are initialized errors are redirected to the error trap higher up. If the algorithm completes without error the error code is set to \textbf{MP\_OKAY} and mp\_clears are executed normally. \textit{Last paragraph sucks. re-write! -- Tom} \subsection{Toom-Cook Squaring} The Toom-Cook squaring algorithm mp\_toom\_sqr is heavily based on the algorithm mp\_toom\_mul with the 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. EXAM,bn_mp_sqr.c \section*{Exercises} \begin{tabular}{cl} $\left [ 3 \right ] $ & Devise an efficient algorithm for selection of the radix point to handle inputs \\ & that have different number of digits in Karatsuba multiplication. \\ & \\ $\left [ 3 \right ] $ & In ~SQUARE~ the fact that every column of a squaring is made up \\ & of double products and at most one square is stated. Prove this statement. \\ & \\ $\left [ 2 \right ] $ & In the Comba squaring algorithm half of the $\hat X$ variables are not used. \\ & Revise algorithm fast\_s\_mp\_sqr to shrink the $\hat X$ array. \\ & \\ $\left [ 3 \right ] $ & Prove the equation for Karatsuba squaring. \\ & \\ $\left [ 1 \right ] $ & Prove that Karatsuba squaring requires $O \left (n^{lg(3)} \right )$ time. \\ & \\ $\left [ 2 \right ] $ & Determine the minimal ratio between addition and multiplication clock cycles \\ & required for equation $6.7$ to be true. \\ & \\ \end{tabular} \chapter{Modular Reduction} MARK,REDUCTION \section{Basics of Modular Reduction} \index{modular residue} Modular reduction is an operation that arises quite often within public key cryptography algorithms 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. EXAM,bn_mp_reduce.c The first multiplication that determines the quotient can be performed by only producing the digits from $m - 1$ and up. This essentially halves the number of single precision multiplications required. However, the optimization is only safe if $\beta$ is much larger than the number of digits in the modulus. In the source code this is evaluated on lines @36,[email protected] to @44,}@ where algorithm s\_mp\_mul\_high\_digs is used when it is safe to do so. \subsection{The Barrett Setup Algorithm} In order to use algorithm mp\_reduce the value of $\mu$ must be calculated in advance. Ideally this value should be computed once and stored for future use so that the Barrett algorithm can be used without delay. \begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_reduce\_setup}. \\ \textbf{Input}. mp\_int $a$ ($a > 1$) \\ \textbf{Output}. $\mu \leftarrow \lfloor \beta^{2m}/a \rfloor$ \\ \hline \\ 1. $\mu \leftarrow 2^{2 \cdot lg(\beta) \cdot m}$ (\textit{mp\_2expt}) \\ 2. $\mu \leftarrow \lfloor \mu / b \rfloor$ (\textit{mp\_div}) \\ 3. Return(\textit{MP\_OKAY}) \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_reduce\_setup} \end{figure} \textbf{Algorithm mp\_reduce\_setup.} This algorithm computes the reciprocal $\mu$ required for Barrett reduction. First $\beta^{2m}$ is calculated as $2^{2 \cdot lg(\beta) \cdot m}$ which is equivalent and much faster. The final value is computed by taking the integer quotient of $\lfloor \mu / b \rfloor$. EXAM,bn_mp_reduce_setup.c This simple routine calculates the reciprocal $\mu$ required by Barrett reduction. Note the extended usage of algorithm mp\_div where the variable which would received the remainder is passed as NULL. As will be discussed in~\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 \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 $k = 8$. The result of the algorithm $r = 99$ is congruent to the value of $2^{-8} \cdot 5555 \mbox{ (mod }257\mbox{)}$. When $r$ is multiplied by $2^8$ 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 $k$ \\ \textbf{Output}. $2^{-k}x \mbox{ (mod }n\mbox{)}$ \\ \hline \\ 1. for $t$ from $0$ to $k - 1$ do \\ \hspace{3mm}1.1 If the $t$'th bit of $x$ is one then \\ \hspace{6mm}1.1.1 $x \leftarrow x + 2^tn$ \\ 2. Return $x/2^k$. \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm Montgomery Reduction (modified I)} \end{figure} This algorithm is equivalent since $2^tn$ is a multiple of $n$ and the lower $k$ bits of $x$ are zero by step 2. The number of single precision shifts has now been reduced from $2k^2$ to $k^2 + k$ 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 -- & $x/2^k = 99$ & \\ \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 $k = 8$. 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 $k$ \\ \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 ~COMBARED~. Step 5 is the main reduction loop of the algorithm. The value of $\mu$ is calculated once per iteration in the outer loop. The inner loop calculates $x + \mu n \beta^{ix}$ by multiplying $\mu n$ and adding the result to $x$ shifted by $ix$ digits. Both the addition and multiplication are performed in the same loop to save time and memory. Step 5.4 will handle any additional carries that escape the inner loop. Using a quick inspection this algorithm requires $n$ single precision multiplications for the outer loop and $n^2$ single precision multiplications in the inner loop. In total $n^2 + n$ single precision multiplications which compares favourably to Barrett at $n^2 + 2n - 1$ single precision multiplications. EXAM,bn_mp_montgomery_reduce.c This is the baseline implementation of the Montgomery reduction algorithm. Lines @30,[email protected] to @35,}@ determine if the Comba based routine can be used instead. Line @47,[email protected] computes the value of $\mu$ for that particular iteration of the outer loop. The multiplication $\mu n \beta^{ix}$ is performed in one step in the inner loop. The alias $tmpx$ refers to the $ix$'th digit of $x$ and the alias $tmpn$ refers to the modulus $n$. \subsection{Faster ``Comba'' Montgomery Reduction} MARK,COMBARED The Montgomery reduction requires fewer single precision multiplications than a Barrett reduction, however it is much slower due to the serial nature of the inner loop. The Barrett reduction algorithm requires two slightly modified multipliers which can be implemented with the Comba technique. The Montgomery reduction algorithm cannot directly use the Comba technique to any significant advantage since the inner loop calculates a $k \times 1$ product $k$ times. The biggest obstacle is that at the $ix$'th iteration of the outer loop the value of $x_{ix}$ is required to calculate $\mu$. This means the carries from $0$ to $ix - 1$ must have been propagated upwards to form a valid $ix$'th digit. The solution as it turns out is very simple. Perform a Comba like multiplier and inside the outer loop just after the inner loop fix up the $ix + 1$'th digit by forwarding the carry. With this change in place the Montgomery reduction algorithm can be performed with a Comba style multiplication loop which substantially increases the speed of the algorithm. \newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{fast\_mp\_montgomery\_reduce}. \\ \textbf{Input}. mp\_int $x$, mp\_int $n$ and a digit $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\ \hspace{11.5mm}($0 \le x < n^2, n > 1, (n, \beta) = 1, \beta^k > n$) \\ \textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\ \hline \\ Place an array of \textbf{MP\_WARRAY} mp\_word variables called $\hat W$ on the stack. \\ 1. if $x.alloc < n.used + 1$ then grow $x$ to $n.used + 1$ digits. \\ Copy the digits of $x$ into the array $\hat W$ \\ 2. For $ix$ from $0$ to $x.used - 1$ do \\ \hspace{3mm}2.1 $\hat W_{ix} \leftarrow x_{ix}$ \\ 3. For $ix$ from $x.used$ to $2n.used - 1$ do \\ \hspace{3mm}3.1 $\hat W_{ix} \leftarrow 0$ \\ Elimiate the lower $k$ digits. \\ 4. for $ix$ from $0$ to $n.used - 1$ do \\ \hspace{3mm}4.1 $\mu \leftarrow \hat W_{ix} \cdot \rho \mbox{ (mod }\beta\mbox{)}$ \\ \hspace{3mm}4.2 For $iy$ from $0$ to $n.used - 1$ do \\ \hspace{6mm}4.2.1 $\hat W_{iy + ix} \leftarrow \hat W_{iy + ix} + \mu \cdot n_{iy}$ \\ \hspace{3mm}4.3 $\hat W_{ix + 1} \leftarrow \hat W_{ix + 1} + \lfloor \hat W_{ix} / \beta \rfloor$ \\ Propagate carries upwards. \\ 5. for $ix$ from $n.used$ to $2n.used + 1$ do \\ \hspace{3mm}5.1 $\hat W_{ix + 1} \leftarrow \hat W_{ix + 1} + \lfloor \hat W_{ix} / \beta \rfloor$ \\ Shift right and reduce modulo $\beta$ simultaneously. \\ 6. for $ix$ from $0$ to $n.used + 1$ do \\ \hspace{3mm}6.1 $x_{ix} \leftarrow \hat W_{ix + n.used} \mbox{ (mod }\beta\mbox{)}$ \\ Zero excess digits and fixup $x$. \\ 7. if $x.used > n.used + 1$ then do \\ \hspace{3mm}7.1 for $ix$ from $n.used + 1$ to $x.used - 1$ do \\ \hspace{6mm}7.1.1 $x_{ix} \leftarrow 0$ \\ 8. $x.used \leftarrow n.used + 1$ \\ 9. Clamp excessive digits of $x$. \\ 10. If $x \ge n$ then \\ \hspace{3mm}10.1 $x \leftarrow x - n$ \\ 11. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm fast\_mp\_montgomery\_reduce} \end{figure} \textbf{Algorithm fast\_mp\_montgomery\_reduce.} This algorithm will compute the Montgomery reduction of $x$ modulo $n$ using the Comba technique. It is on most computer platforms significantly faster than algorithm mp\_montgomery\_reduce and algorithm mp\_reduce (\textit{Barrett reduction}). The algorithm has the same restrictions on the input as the baseline reduction algorithm. An additional two restrictions are imposed on this algorithm. The number of digits $k$ in the the modulus $n$ must not violate $MP\_WARRAY > 2k +1$ and $n < \delta$. When $\beta = 2^{28}$ this algorithm can be used to reduce modulo a modulus of at most $3,556$ bits in length. As in the other Comba reduction algorithms there is a $\hat W$ array which stores the columns of the product. It is initially filled with the contents of $x$ with the excess digits zeroed. The reduction loop is very similar the to the baseline loop at heart. The multiplication on step 4.1 can be single precision only since $ab \mbox{ (mod }\beta\mbox{)} \equiv (a \mbox{ mod }\beta)(b \mbox{ mod }\beta)$. Some multipliers such as those on the ARM processors take a variable length time to complete depending on the number of bytes of result it must produce. By performing a single precision multiplication instead half the amount of time is spent. Also note that digit $\hat W_{ix}$ must have the carry from the $ix - 1$'th digit propagated upwards in order for this to work. That is what step 4.3 will do. In effect over the $n.used$ iterations of the outer loop the $n.used$'th lower columns all have the their carries propagated forwards. Note how the upper bits of those same words are not reduced modulo $\beta$. This is because those values will be discarded shortly and there is no point. Step 5 will 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 $x$. EXAM,bn_fast_mp_montgomery_reduce.c The $\hat W$ array is first filled with digits of $x$ on line @49,[email protected] then the rest of the digits are zeroed on line @54,[email protected] Both loops share the same alias variables to make the code easier to read. The value of $\mu$ is calculated in an interesting fashion. First the value $\hat W_{ix}$ is reduced modulo $\beta$ and cast to a mp\_digit. This forces the compiler to use a single precision multiplication and prevents any concerns about loss of precision. Line @101,>>@ fixes the carry for the next iteration of the loop by propagating the carry from $\hat W_{ix}$ to $\hat W_{ix+1}$. The for loop on line @113,[email protected] propagates the rest of the carries upwards through the columns. The for loop on line @126,[email protected] 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. EXAM,bn_mp_montgomery_setup.c This source code computes the value of $\rho$ required to perform Montgomery reduction. It has been modified to avoid performing excess multiplications when $\beta$ is not the default 28-bits. \section{The Diminished Radix Algorithm} The Diminished Radix method of modular reduction \cite{DRMET} is a fairly clever technique which 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. EXAM,bn_mp_dr_reduce.c The first step is to grow $x$ as required to $2m$ digits since the reduction is performed in place on $x$. The label on line @49,top:@ is where the algorithm will resume if further reduction passes are required. In theory it could be placed at the top of the function however, the size of the modulus and question of whether $x$ is large enough are invariant after the first pass meaning that it would be a waste of time. The aliases $tmpx1$ and $tmpx2$ refer to the digits of $x$ where the latter is offset by $m$ digits. By reading digits from $x$ offset by $m$ digits a division by $\beta^m$ can be simulated virtually for free. The loop on line @61,[email protected] performs the bulk of the work (\textit{corresponds to step 4 of algorithm 7.11}) in this algorithm. By line @68,[email protected] the pointer $tmpx1$ points to the $m$'th digit of $x$ which is where the final carry will be placed. Similarly by line @71,[email protected] the same pointer will point to the $m+1$'th digit where the zeroes will be placed. Since the algorithm is only valid if both $x$ and $n$ are greater than zero an unsigned comparison suffices to determine if another pass is required. With the same logic at line @82,[email protected] the value of $x$ is known to be greater than or equal to $n$ meaning that an unsigned subtraction can be used as well. Since the destination of the subtraction is the larger of the inputs the call to algorithm s\_mp\_sub cannot fail and the return code does not need to be checked. \subsubsection{Setup} To setup the restricted Diminished Radix algorithm the value $k = \beta - n_0$ is required. This algorithm is not really complicated but provided for completeness. \begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_dr\_setup}. \\ \textbf{Input}. mp\_int $n$ \\ \textbf{Output}. $k = \beta - n_0$ \\ \hline \\ 1. $k \leftarrow \beta - n_0$ \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_dr\_setup} \end{figure} EXAM,bn_mp_dr_setup.c \subsubsection{Modulus Detection} Another algorithm which will be useful is the ability to detect a restricted Diminished Radix modulus. An integer is said to be of restricted Diminished Radix form if all of the digits are equal to $\beta - 1$ except the trailing digit which may be any value. \begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_dr\_is\_modulus}. \\ \textbf{Input}. mp\_int $n$ \\ \textbf{Output}. $1$ if $n$ is in D.R form, $0$ otherwise \\ \hline 1. If $n.used < 2$ then return($0$). \\ 2. for $ix$ from $1$ to $n.used - 1$ do \\ \hspace{3mm}2.1 If $n_{ix} \ne \beta - 1$ return($0$). \\ 3. Return($1$). \\ \hline \end{tabular} \end{center} \end{small} \caption{Algorithm mp\_dr\_is\_modulus} \end{figure} \textbf{Algorithm mp\_dr\_is\_modulus.} This algorithm determines if a value is in Diminished Radix form. Step 1 rejects obvious cases where fewer than two digits are in the mp\_int. Step 2 tests all but the first digit to see if they are equal to $\beta - 1$. If the algorithm manages to get to step 3 then $n$ must be of Diminished Radix form. EXAM,bn_mp_dr_is_modulus.c \subsection{Unrestricted Diminished Radix Reduction} The unrestricted Diminished Radix algorithm allows modular reductions to be performed when the modulus is of the form $2^p - k$. This algorithm is a straightforward adaptation of algorithm~\ref{fig:DR}. In general the restricted Diminished Radix reduction algorithm is much faster since it has considerably lower overhead. However, this new algorithm is much faster than either Montgomery or Barrett reduction when the moduli are of the appropriate form. \begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_reduce\_2k}. \\ \textbf{Input}. mp\_int $a$ and $n$. mp\_digit $k$ \\ \hspace{11.5mm}($a \ge 0$, $n > 1$, $0 < k < \beta$, $n + k$ is a power of two) \\ \textbf{Output}. $a \mbox{ (mod }n\mbox{)}$ \\ \hline 1. $p \leftarrow \lceil lg(n) \rceil$ (\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. EXAM,bn_mp_reduce_2k.c 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,mp_div_[email protected] 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 $\beta - n_0$. EXAM,bn_mp_reduce_2k_setup.c \subsubsection{Unrestricted Detection} An integer $n$ is a valid unrestricted Diminished Radix modulus if either of the following are true. \begin{enumerate} \item The number has only one digit. \item The number has more than one digit and every bit from the $\beta$'th to the most significant is one. \end{enumerate} If either condition is true than there is a power of two $2^p$ 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. EXAM,bn_mp_reduce_is_2k.c \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}. $<