Mercurial > dropbear
changeset 290:94ee16f5b8a8
0.48 progress
author | Matt Johnston <matt@ucc.asn.au> |
---|---|
date | Thu, 09 Mar 2006 12:37:38 +0000 |
parents | 497fddd4a94e |
children | 55a99934db87 |
files | CHANGES README TODO libtommath/tommath.tex options.h svr-main.c |
diffstat | 6 files changed, 26 insertions(+), 10824 deletions(-) [+] |
line wrap: on
line diff
--- a/CHANGES Wed Mar 08 15:46:24 2006 +0000 +++ b/CHANGES Thu Mar 09 12:37:38 2006 +0000 @@ -1,3 +1,26 @@ +0.48 - + +- Check that the circular buffer is properly empty before + closing a channel, which could cause truncated transfers + (thanks to Tomas Vanek for helping track it down) + +- Implement per-IP pre-authentication connection limits + (after some poking from Pablo Fernandez) + +- Exit gracefully if trying to connect to as SSH v1 server + (reported by Rushi Lala) + +- Only read /dev/random once at startup when in non-inetd mode + +- Allow ctrl-c to close a dbclient password prompt (may + still have to press enter on some platforms) + +- Merged in uClinux patch for inetd mode + +- Updated to scp from OpenSSH 4.3p2 - fixes a security issue + where use of system() could cause users to execute arbitrary + code through malformed filenames, ref CVE-2006-0225 + 0.47 - Thurs Dec 8 2005 - SECURITY: fix for buffer allocation error in server code, could potentially
--- a/README Wed Mar 08 15:46:24 2006 +0000 +++ b/README Thu Mar 09 12:37:38 2006 +0000 @@ -25,7 +25,7 @@ ssh-rsa AAAAB3NzaC1yc2EAAAABIwAAAIEAwVa6M6cGVmUcLl2cFzkxEoJd06Ub4bVDsYrWvXhvUV+ZAM9uGuewZBDoAqNKJxoIn0Hyd0Nk/yU99UVv6NWV/5YSHtnf35LKds56j7cuzoQpFIdjNwdxAN0PCET/MG8qyskG/2IE2DPNIaJ3Wy+Ws4IZEgdJgPlTYUBWWtCWOGc= [email protected] You must make sure that ~/.ssh, and the key file, are only writable by the -user. +user. Beware of editors that split the key into multiple lines. NOTE: Dropbear ignores authorized_keys options such as those described in the OpenSSH sshd manpage, and will not allow a login for these keys.
--- a/TODO Wed Mar 08 15:46:24 2006 +0000 +++ b/TODO Thu Mar 09 12:37:38 2006 +0000 @@ -6,8 +6,6 @@ - Make options.h generated from configure perhaps? -- Improved queueing of unauthed connections - - handle /etc/environment in AIX - check that there aren't timing issues with valid/invalid user authentication @@ -15,7 +13,6 @@ - Binding to different interfaces -- check PRNG - CTR mode - SSH_MSG_IGNORE sending to improve CBC security - DH Group Exchange possibly, or just add group14 (whatever it's called today)
--- a/libtommath/tommath.tex Wed Mar 08 15:46:24 2006 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,10814 +0,0 @@ -\documentclass[b5paper]{book} -\usepackage{hyperref} -\usepackage{makeidx} -\usepackage{amssymb} -\usepackage{color} -\usepackage{alltt} -\usepackage{graphicx} -\usepackage{layout} -\def\union{\cup} -\def\intersect{\cap} -\def\getsrandom{\stackrel{\rm R}{\gets}} -\def\cross{\times} -\def\cat{\hspace{0.5em} \| \hspace{0.5em}} -\def\catn{$\|$} -\def\divides{\hspace{0.3em} | \hspace{0.3em}} -\def\nequiv{\not\equiv} -\def\approx{\raisebox{0.2ex}{\mbox{\small $\sim$}}} -\def\lcm{{\rm lcm}} -\def\gcd{{\rm gcd}} -\def\log{{\rm log}} -\def\ord{{\rm ord}} -\def\abs{{\mathit abs}} -\def\rep{{\mathit rep}} -\def\mod{{\mathit\ mod\ }} -\renewcommand{\pmod}[1]{\ ({\rm mod\ }{#1})} -\newcommand{\floor}[1]{\left\lfloor{#1}\right\rfloor} -\newcommand{\ceil}[1]{\left\lceil{#1}\right\rceil} -\def\Or{{\rm\ or\ }} -\def\And{{\rm\ and\ }} -\def\iff{\hspace{1em}\Longleftrightarrow\hspace{1em}} -\def\implies{\Rightarrow} -\def\undefined{{\rm ``undefined"}} -\def\Proof{\vspace{1ex}\noindent {\bf Proof:}\hspace{1em}} -\let\oldphi\phi -\def\phi{\varphi} -\def\Pr{{\rm Pr}} -\newcommand{\str}[1]{{\mathbf{#1}}} -\def\F{{\mathbb F}} -\def\N{{\mathbb N}} -\def\Z{{\mathbb Z}} -\def\R{{\mathbb R}} -\def\C{{\mathbb C}} -\def\Q{{\mathbb Q}} -\definecolor{DGray}{gray}{0.5} -\newcommand{\emailaddr}[1]{\mbox{$<${#1}$>$}} -\def\twiddle{\raisebox{0.3ex}{\mbox{\tiny $\sim$}}} -\def\gap{\vspace{0.5ex}} -\makeindex -\begin{document} -\frontmatter -\pagestyle{empty} -\title{Multi--Precision Math} -\author{\mbox{ -%\begin{small} -\begin{tabular}{c} -Tom St Denis \\ -Algonquin College \\ -\\ -Mads Rasmussen \\ -Open Communications Security \\ -\\ -Greg Rose \\ -QUALCOMM Australia \\ -\end{tabular} -%\end{small} -} -} -\maketitle -This text has been placed in the public domain. This text corresponds to the v0.35 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} -When I tell people about my LibTom projects and that I release them as public domain they are often puzzled. -They ask why I did it and especially why I continue to work on them for free. The best I can explain it is ``Because I can.'' -Which seems odd and perhaps too terse for adult conversation. I often qualify it with ``I am able, I am willing.'' which -perhaps explains it better. I am the first to admit there is not anything that special with what I have done. Perhaps -others can see that too and then we would have a society to be proud of. My LibTom projects are what I am doing to give -back to society in the form of tools and knowledge that can help others in their endeavours. - -I started writing this book because it was the most logical task to further my goal of open academia. The LibTomMath source -code itself was written to be easy to follow and learn from. There are times, however, where pure C source code does not -explain the algorithms properly. Hence this book. The book literally starts with the foundation of the library and works -itself outwards to the more complicated algorithms. The use of both pseudo--code and verbatim source code provides a duality -of ``theory'' and ``practice'' that the computer science students of the world shall appreciate. I never deviate too far -from relatively straightforward algebra and I hope that this book can be a valuable learning asset. - -This book and indeed much of the LibTom projects would not exist in their current form if it was not for a plethora -of kind people donating their time, resources and kind words to help support my work. Writing a text of significant -length (along with the source code) is a tiresome and lengthy process. Currently the LibTom project is four years old, -comprises of literally thousands of users and over 100,000 lines of source code, TeX and other material. People like Mads and Greg -were there at the beginning to encourage me to work well. It is amazing how timely validation from others can boost morale to -continue the project. Definitely my parents were there for me by providing room and board during the many months of work in 2003. - -To my many friends whom I have met through the years I thank you for the good times and the words of encouragement. I hope I -honour your kind gestures with this project. - -Open Source. Open Academia. Open Minds. - -\begin{flushright} Tom St Denis \end{flushright} - -\newpage -I found the opportunity to work with Tom appealing for several reasons, not only could I broaden my own horizons, but also -contribute to educate others facing the problem of having to handle big number mathematical calculations. - -This book is Tom's child and he has been caring and fostering the project ever since the beginning with a clear mind of -how he wanted the project to turn out. I have helped by proofreading the text and we have had several discussions about -the layout and language used. - -I hold a masters degree in cryptography from the University of Southern Denmark and have always been interested in the -practical aspects of cryptography. - -Having worked in the security consultancy business for several years in S\~{a}o Paulo, Brazil, I have been in touch with a -great deal of work in which multiple precision mathematics was needed. Understanding the possibilities for speeding up -multiple precision calculations is often very important since we deal with outdated machine architecture where modular -reductions, for example, become painfully slow. - -This text is for people who stop and wonder when first examining algorithms such as RSA for the first time and asks -themselves, ``You tell me this is only secure for large numbers, fine; but how do you implement these numbers?'' - -\begin{flushright} -Mads Rasmussen - -S\~{a}o Paulo - SP - -Brazil -\end{flushright} - -\newpage -It's all because I broke my leg. That just happened to be at about the same time that Tom asked for someone to review the section of the book about -Karatsuba multiplication. I was laid up, alone and immobile, and thought ``Why not?'' I vaguely knew what Karatsuba multiplication was, but not -really, so I thought I could help, learn, and stop myself from watching daytime cable TV, all at once. - -At the time of writing this, I've still not met Tom or Mads in meatspace. I've been following Tom's progress since his first splash on the -sci.crypt Usenet news group. I watched him go from a clueless newbie, to the cryptographic equivalent of a reformed smoker, to a real -contributor to the field, over a period of about two years. I've been impressed with his obvious intelligence, and astounded by his productivity. -Of course, he's young enough to be my own child, so he doesn't have my problems with staying awake. - -When I reviewed that single section of the book, in its very earliest form, I was very pleasantly surprised. So I decided to collaborate more fully, -and at least review all of it, and perhaps write some bits too. There's still a long way to go with it, and I have watched a number of close -friends go through the mill of publication, so I think that the way to go is longer than Tom thinks it is. Nevertheless, it's a good effort, -and I'm pleased to be involved with it. - -\begin{flushright} -Greg Rose, Sydney, Australia, June 2003. -\end{flushright} - -\mainmatter -\pagestyle{headings} -\chapter{Introduction} -\section{Multiple Precision Arithmetic} - -\subsection{What is Multiple Precision Arithmetic?} -When we think of long-hand arithmetic such as addition or multiplication we rarely consider the fact that we instinctively -raise or lower the precision of the numbers we are dealing with. For example, in decimal we almost immediate can -reason that $7$ times $6$ is $42$. However, $42$ has two digits of precision as opposed to one digit we started with. -Further multiplications of say $3$ result in a larger precision result $126$. In these few examples we have multiple -precisions for the numbers we are working with. Despite the various levels of precision a single subset\footnote{With the occasional optimization.} - of algorithms can be designed to accomodate them. - -By way of comparison a fixed or single precision operation would lose precision on various operations. For example, in -the decimal system with fixed precision $6 \cdot 7 = 2$. - -Essentially at the heart of computer based multiple precision arithmetic are the same long-hand algorithms taught in -schools to manually add, subtract, multiply and divide. - -\subsection{The Need for Multiple Precision Arithmetic} -The most prevalent need for multiple precision arithmetic, often referred to as ``bignum'' math, is within the implementation -of public-key cryptography algorithms. Algorithms such as RSA \cite{RSAREF} and Diffie-Hellman \cite{DHREF} require -integers of significant magnitude to resist known cryptanalytic attacks. For example, at the time of this writing a -typical RSA modulus would be at least greater than $10^{309}$. However, modern programming languages such as ISO C \cite{ISOC} and -Java \cite{JAVA} only provide instrinsic support for integers which are relatively small and single precision. - -\begin{figure}[!here] -\begin{center} -\begin{tabular}{|r|c|} -\hline \textbf{Data Type} & \textbf{Range} \\ -\hline char & $-128 \ldots 127$ \\ -\hline short & $-32768 \ldots 32767$ \\ -\hline long & $-2147483648 \ldots 2147483647$ \\ -\hline long long & $-9223372036854775808 \ldots 9223372036854775807$ \\ -\hline -\end{tabular} -\end{center} -\caption{Typical Data Types for the C Programming Language} -\label{fig:ISOC} -\end{figure} - -The largest data type guaranteed to be provided by the ISO C programming -language\footnote{As per the ISO C standard. However, each compiler vendor is allowed to augment the precision as they -see fit.} can only represent values up to $10^{19}$ as shown in figure \ref{fig:ISOC}. On its own the C language is -insufficient to accomodate the magnitude required for the problem at hand. An RSA modulus of magnitude $10^{19}$ could be -trivially factored\footnote{A Pollard-Rho factoring would take only $2^{16}$ time.} on the average desktop computer, -rendering any protocol based on the algorithm insecure. Multiple precision algorithms solve this very problem by -extending the range of representable integers while using single precision data types. - -Most advancements in fast multiple precision arithmetic stem from the need for faster and more efficient cryptographic -primitives. Faster modular reduction and exponentiation algorithms such as Barrett's algorithm, which have appeared in -various cryptographic journals, can render algorithms such as RSA and Diffie-Hellman more efficient. In fact, several -major companies such as RSA Security, Certicom and Entrust have built entire product lines on the implementation and -deployment of efficient algorithms. - -However, cryptography is not the only field of study that can benefit from fast multiple precision integer routines. -Another auxiliary use of multiple precision integers is high precision floating point data types. -The basic IEEE \cite{IEEE} standard floating point type is made up of an integer mantissa $q$, an exponent $e$ and a sign bit $s$. -Numbers are given in the form $n = q \cdot b^e \cdot -1^s$ where $b = 2$ is the most common base for IEEE. Since IEEE -floating point is meant to be implemented in hardware the precision of the mantissa is often fairly small -(\textit{23, 48 and 64 bits}). The mantissa is merely an integer and a multiple precision integer could be used to create -a mantissa of much larger precision than hardware alone can efficiently support. This approach could be useful where -scientific applications must minimize the total output error over long calculations. - -Yet another use for large integers is within arithmetic on polynomials of large characteristic (i.e. $GF(p)[x]$ for large $p$). -In fact the library discussed within this text has already been used to form a polynomial basis library\footnote{See \url{http://poly.libtomcrypt.org} for more details.}. - -\subsection{Benefits of Multiple Precision Arithmetic} -\index{precision} -The benefit of multiple precision representations over single or fixed precision representations is that -no precision is lost while representing the result of an operation which requires excess precision. For example, -the product of two $n$-bit integers requires at least $2n$ bits of precision to be represented faithfully. A multiple -precision algorithm would augment the precision of the destination to accomodate the result while a single precision system -would truncate excess bits to maintain a fixed level of precision. - -It is possible to implement algorithms which require large integers with fixed precision algorithms. For example, elliptic -curve cryptography (\textit{ECC}) is often implemented on smartcards by fixing the precision of the integers to the maximum -size the system will ever need. Such an approach can lead to vastly simpler algorithms which can accomodate the -integers required even if the host platform cannot natively accomodate them\footnote{For example, the average smartcard -processor has an 8 bit accumulator.}. However, as efficient as such an approach may be, the resulting source code is not -normally very flexible. It cannot, at runtime, accomodate inputs of higher magnitude than the designer anticipated. - -Multiple precision algorithms have the most overhead of any style of arithmetic. For the the most part the -overhead can be kept to a minimum with careful planning, but overall, it is not well suited for most memory starved -platforms. However, multiple precision algorithms do offer the most flexibility in terms of the magnitude of the -inputs. That is, the same algorithms based on multiple precision integers can accomodate any reasonable size input -without the designer's explicit forethought. This leads to lower cost of ownership for the code as it only has to -be written and tested once. - -\section{Purpose of This Text} -The purpose of this text is to instruct the reader regarding how to implement efficient multiple precision algorithms. -That is to not only explain a limited subset of the core theory behind the algorithms but also the various ``house keeping'' -elements that are neglected by authors of other texts on the subject. Several well reknowned texts \cite{TAOCPV2,HAC} -give considerably detailed explanations of the theoretical aspects of algorithms and often very little information -regarding the practical implementation aspects. - -In most cases how an algorithm is explained and how it is actually implemented are two very different concepts. For -example, the Handbook of Applied Cryptography (\textit{HAC}), algorithm 14.7 on page 594, gives a relatively simple -algorithm for performing multiple precision integer addition. However, the description lacks any discussion concerning -the fact that the two integer inputs may be of differing magnitudes. As a result the implementation is not as simple -as the text would lead people to believe. Similarly the division routine (\textit{algorithm 14.20, pp. 598}) does not -discuss how to handle sign or handle the dividend's decreasing magnitude in the main loop (\textit{step \#3}). - -Both texts also do not discuss several key optimal algorithms required such as ``Comba'' and Karatsuba multipliers -and fast modular inversion, which we consider practical oversights. These optimal algorithms are vital to achieve -any form of useful performance in non-trivial applications. - -To solve this problem the focus of this text is on the practical aspects of implementing a multiple precision integer -package. As a case study the ``LibTomMath''\footnote{Available at \url{http://math.libtomcrypt.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 $x = (x_{n-1}, \ldots, x_1, x_0)_{ \beta }$ and represent -the integer $x \equiv \sum_{i=0}^{n-1} x_i\beta^i$. The elements of the array $x$ are said to be the radix $\beta$ digits -of the integer. For example, $x = (1,2,3)_{10}$ would represent the integer -$1\cdot 10^2 + 2\cdot10^1 + 3\cdot10^0 = 123$. - -\index{mp\_int} -The term ``mp\_int'' shall refer to a composite structure which contains the digits of the integer it represents, as well -as auxilary data required to manipulate the data. These additional members are discussed further in section -\ref{sec:MPINT}. For the purposes of this text a ``multiple precision integer'' and an ``mp\_int'' are assumed to be -synonymous. When an algorithm is specified to accept an mp\_int variable it is assumed the various auxliary data members -are present as well. An expression of the type \textit{variablename.item} implies that it should evaluate to the -member named ``item'' of the variable. For example, a string of characters may have a member ``length'' which would -evaluate to the number of characters in the string. If the string $a$ equals ``hello'' then it follows that -$a.length = 5$. - -For certain discussions more generic algorithms are presented to help the reader understand the final algorithm used -to solve a given problem. When an algorithm is described as accepting an integer input it is assumed the input is -a plain integer with no additional multiple-precision members. That is, algorithms that use integers as opposed to -mp\_ints as inputs do not concern themselves with the housekeeping operations required such as memory management. These -algorithms will be used to establish the relevant theory which will subsequently be used to describe a multiple -precision algorithm to solve the same problem. - -\subsection{Precision Notation} -The variable $\beta$ represents the radix of a single digit of a multiple precision integer and -must be of the form $q^p$ for $q, p \in \Z^+$. A single precision variable must be able to represent integers in -the range $0 \le x < q \beta$ while a double precision variable must be able to represent integers in the range -$0 \le x < q \beta^2$. The extra radix-$q$ factor allows additions and subtractions to proceed without truncation of the -carry. Since all modern computers are binary, it is assumed that $q$ is two. - -\index{mp\_digit} \index{mp\_word} -Within the source code that will be presented for each algorithm, the data type \textbf{mp\_digit} will represent -a single precision integer type, while, the data type \textbf{mp\_word} will represent a double precision integer type. In -several algorithms (notably the Comba routines) temporary results will be stored in arrays of double precision mp\_words. -For the purposes of this text $x_j$ will refer to the $j$'th digit of a single precision array and $\hat x_j$ will refer to -the $j$'th digit of a double precision array. Whenever an expression is to be assigned to a double precision -variable it is assumed that all single precision variables are promoted to double precision during the evaluation. -Expressions that are assigned to a single precision variable are truncated to fit within the precision of a single -precision data type. - -For example, if $\beta = 10^2$ a single precision data type may represent a value in the -range $0 \le x < 10^3$, while a double precision data type may represent a value in the range $0 \le x < 10^5$. Let -$a = 23$ and $b = 49$ represent two single precision variables. The single precision product shall be written -as $c \leftarrow a \cdot b$ while the double precision product shall be written as $\hat c \leftarrow a \cdot b$. -In this particular case, $\hat c = 1127$ and $c = 127$. The most significant digit of the product would not fit -in a single precision data type and as a result $c \ne \hat c$. - -\subsection{Algorithm Inputs and Outputs} -Within the algorithm descriptions all variables are assumed to be scalars of either single or double precision -as indicated. The only exception to this rule is when variables have been indicated to be of type mp\_int. This -distinction is important as scalars are often used as array indicies and various other counters. - -\subsection{Mathematical Expressions} -The $\lfloor \mbox{ } \rfloor$ brackets imply an expression truncated to an integer not greater than the expression -itself. For example, $\lfloor 5.7 \rfloor = 5$. Similarly the $\lceil \mbox{ } \rceil$ brackets imply an expression -rounded to an integer not less than the expression itself. For example, $\lceil 5.1 \rceil = 6$. Typically when -the $/$ division symbol is used the intention is to perform an integer division with truncation. For example, -$5/2 = 2$ which will often be written as $\lfloor 5/2 \rfloor = 2$ for clarity. When an expression is written as a -fraction a real value division is implied, for example ${5 \over 2} = 2.5$. - -The norm of a multiple precision integer, for example $\vert \vert x \vert \vert$, will be used to represent the number of digits in the representation -of the integer. For example, $\vert \vert 123 \vert \vert = 3$ and $\vert \vert 79452 \vert \vert = 5$. - -\subsection{Work Effort} -\index{big-Oh} -To measure the efficiency of the specified algorithms, a modified big-Oh notation is used. In this system all -single precision operations are considered to have the same cost\footnote{Except where explicitly noted.}. -That is a single precision addition, multiplication and division are assumed to take the same time to -complete. While this is generally not true in practice, it will simplify the discussions considerably. - -Some algorithms have slight advantages over others which is why some constants will not be removed in -the notation. For example, a normal baseline multiplication (section \ref{sec:basemult}) requires $O(n^2)$ work while a -baseline squaring (section \ref{sec:basesquare}) requires $O({{n^2 + n}\over 2})$ work. In standard big-Oh notation these -would both be said to be equivalent to $O(n^2)$. However, -in the context of the this text this is not the case as the magnitude of the inputs will typically be rather small. As a -result small constant factors in the work effort will make an observable difference in algorithm efficiency. - -All of the algorithms presented in this text have a polynomial time work level. That is, of the form -$O(n^k)$ for $n, k \in \Z^{+}$. This will help make useful comparisons in terms of the speed of the algorithms and how -various optimizations will help pay off in the long run. - -\section{Exercises} -Within the more advanced chapters a section will be set aside to give the reader some challenging exercises related to -the discussion at hand. These exercises are not designed to be prize winning problems, but instead to be thought -provoking. Wherever possible the problems are forward minded, stating problems that will be answered in subsequent -chapters. The reader is encouraged to finish the exercises as they appear to get a better understanding of the -subject material. - -That being said, the problems are designed to affirm knowledge of a particular subject matter. Students in particular -are encouraged to verify they can answer the problems correctly before moving on. - -Similar to the exercises of \cite[pp. ix]{TAOCPV2} these exercises are given a scoring system based on the difficulty of -the problem. However, unlike \cite{TAOCPV2} the problems do not get nearly as hard. The scoring of these -exercises ranges from one (the easiest) to five (the hardest). The following table sumarizes the -scoring system used. - -\begin{figure}[here] -\begin{center} -\begin{small} -\begin{tabular}{|c|l|} -\hline $\left [ 1 \right ]$ & An easy problem that should only take the reader a manner of \\ - & minutes to solve. Usually does not involve much computer time \\ - & to solve. \\ -\hline $\left [ 2 \right ]$ & An easy problem that involves a marginal amount of computer \\ - & time usage. Usually requires a program to be written to \\ - & solve the problem. \\ -\hline $\left [ 3 \right ]$ & A moderately hard problem that requires a non-trivial amount \\ - & of work. Usually involves trivial research and development of \\ - & new theory from the perspective of a student. \\ -\hline $\left [ 4 \right ]$ & A moderately hard problem that involves a non-trivial amount \\ - & of work and research, the solution to which will demonstrate \\ - & a higher mastery of the subject matter. \\ -\hline $\left [ 5 \right ]$ & A hard problem that involves concepts that are difficult for a \\ - & novice to solve. Solutions to these problems will demonstrate a \\ - & complete mastery of the given subject. \\ -\hline -\end{tabular} -\end{small} -\end{center} -\caption{Exercise Scoring System} -\end{figure} - -Problems at the first level are meant to be simple questions that the reader can answer quickly without programming a solution or -devising new theory. These problems are quick tests to see if the material is understood. Problems at the second level -are also designed to be easy but will require a program or algorithm to be implemented to arrive at the answer. These -two levels are essentially entry level questions. - -Problems at the third level are meant to be a bit more difficult than the first two levels. The answer is often -fairly obvious but arriving at an exacting solution requires some thought and skill. These problems will almost always -involve devising a new algorithm or implementing a variation of another algorithm previously presented. Readers who can -answer these questions will feel comfortable with the concepts behind the topic at hand. - -Problems at the fourth level are meant to be similar to those of the level three questions except they will require -additional research to be completed. The reader will most likely not know the answer right away, nor will the text provide -the exact details of the answer until a subsequent chapter. - -Problems at the fifth level are meant to be the hardest -problems relative to all the other problems in the chapter. People who can correctly answer fifth level problems have a -mastery of the subject matter at hand. - -Often problems will be tied together. The purpose of this is to start a chain of thought that will be discussed in future chapters. The reader -is encouraged to answer the follow-up problems and try to draw the relevance of problems. - -\section{Introduction to LibTomMath} - -\subsection{What is LibTomMath?} -LibTomMath is a free and open source multiple precision integer library written entirely in portable ISO C. By portable it -is meant that the library does not contain any code that is computer platform dependent or otherwise problematic to use on -any given platform. - -The library has been successfully tested under numerous operating systems including Unix\footnote{All of these -trademarks belong to their respective rightful owners.}, MacOS, Windows, Linux, PalmOS and on standalone hardware such -as the Gameboy Advance. The library is designed to contain enough functionality to be able to develop applications such -as public key cryptosystems and still maintain a relatively small footprint. - -\subsection{Goals of LibTomMath} - -Libraries which obtain the most efficiency are rarely written in a high level programming language such as C. However, -even though this library is written entirely in ISO C, considerable care has been taken to optimize the algorithm implementations within the -library. Specifically the code has been written to work well with the GNU C Compiler (\textit{GCC}) on both x86 and ARM -processors. Wherever possible, highly efficient algorithms, such as Karatsuba multiplication, sliding window -exponentiation and Montgomery reduction have been provided to make the library more efficient. - -Even with the nearly optimal and specialized algorithms that have been included the Application Programing Interface -(\textit{API}) has been kept as simple as possible. Often generic place holder routines will make use of specialized -algorithms automatically without the developer's specific attention. One such example is the generic multiplication -algorithm \textbf{mp\_mul()} which will automatically use Toom--Cook, Karatsuba, Comba or baseline multiplication -based on the magnitude of the inputs and the configuration of the library. - -Making LibTomMath as efficient as possible is not the only goal of the LibTomMath project. Ideally the library should -be source compatible with another popular library which makes it more attractive for developers to use. In this case the -MPI library was used as a API template for all the basic functions. MPI was chosen because it is another library that fits -in the same niche as LibTomMath. Even though LibTomMath uses MPI as the template for the function names and argument -passing conventions, it has been written from scratch by Tom St Denis. - -The project is also meant to act as a learning tool for students, the logic being that no easy-to-follow ``bignum'' -library exists which can be used to teach computer science students how to perform fast and reliable multiple precision -integer arithmetic. To this end the source code has been given quite a few comments and algorithm discussion points. - -\section{Choice of LibTomMath} -LibTomMath was chosen as the case study of this text not only because the author of both projects is one and the same but -for more worthy reasons. Other libraries such as GMP \cite{GMP}, MPI \cite{MPI}, LIP \cite{LIP} and OpenSSL -\cite{OPENSSL} have multiple precision integer arithmetic routines but would not be ideal for this text for -reasons that will be explained in the following sub-sections. - -\subsection{Code Base} -The LibTomMath code base is all portable ISO C source code. This means that there are no platform dependent conditional -segments of code littered throughout the source. This clean and uncluttered approach to the library means that a -developer can more readily discern the true intent of a given section of source code without trying to keep track of -what conditional code will be used. - -The code base of LibTomMath is well organized. Each function is in its own separate source code file -which allows the reader to find a given function very quickly. On average there are $76$ lines of code per source -file which makes the source very easily to follow. By comparison MPI and LIP are single file projects making code tracing -very hard. GMP has many conditional code segments which also hinder tracing. - -When compiled with GCC for the x86 processor and optimized for speed the entire library is approximately $100$KiB\footnote{The notation ``KiB'' means $2^{10}$ octets, similarly ``MiB'' means $2^{20}$ octets.} - which is fairly small compared to GMP (over $250$KiB). LibTomMath is slightly larger than MPI (which compiles to about -$50$KiB) but LibTomMath is also much faster and more complete than MPI. - -\subsection{API Simplicity} -LibTomMath is designed after the MPI library and shares the API design. Quite often programs that use MPI will build -with LibTomMath without change. The function names correlate directly to the action they perform. Almost all of the -functions share the same parameter passing convention. The learning curve is fairly shallow with the API provided -which is an extremely valuable benefit for the student and developer alike. - -The LIP library is an example of a library with an API that is awkward to work with. LIP uses function names that are often ``compressed'' to -illegible short hand. LibTomMath does not share this characteristic. - -The GMP library also does not return error codes. Instead it uses a POSIX.1 \cite{POSIX1} signal system where errors -are signaled to the host application. This happens to be the fastest approach but definitely not the most versatile. In -effect a math error (i.e. invalid input, heap error, etc) can cause a program to stop functioning which is definitely -undersireable in many situations. - -\subsection{Optimizations} -While LibTomMath is certainly not the fastest library (GMP often beats LibTomMath by a factor of two) it does -feature a set of optimal algorithms for tasks such as modular reduction, exponentiation, multiplication and squaring. GMP -and LIP also feature such optimizations while MPI only uses baseline algorithms with no optimizations. GMP lacks a few -of the additional modular reduction optimizations that LibTomMath features\footnote{At the time of this writing GMP -only had Barrett and Montgomery modular reduction algorithms.}. - -LibTomMath is almost always an order of magnitude faster than the MPI library at computationally expensive tasks such as modular -exponentiation. In the grand scheme of ``bignum'' libraries LibTomMath is faster than the average library and usually -slower than the best libraries such as GMP and OpenSSL by only a small factor. - -\subsection{Portability and Stability} -LibTomMath will build ``out of the box'' on any platform equipped with a modern version of the GNU C Compiler -(\textit{GCC}). This means that without changes the library will build without configuration or setting up any -variables. LIP and MPI will build ``out of the box'' as well but have numerous known bugs. Most notably the author of -MPI has recently stopped working on his library and LIP has long since been discontinued. - -GMP requires a configuration script to run and will not build out of the box. GMP and LibTomMath are still in active -development and are very stable across a variety of platforms. - -\subsection{Choice} -LibTomMath is a relatively compact, well documented, highly optimized and portable library which seems only natural for -the case study of this text. Various source files from the LibTomMath project will be included within the text. However, -the reader is encouraged to download their own copy of the library to actually be able to work with the library. - -\chapter{Getting Started} -\section{Library Basics} -The trick to writing any useful library of source code is to build a solid foundation and work outwards from it. First, -a problem along with allowable solution parameters should be identified and analyzed. In this particular case the -inability to accomodate multiple precision integers is the problem. Futhermore, the solution must be written -as portable source code that is reasonably efficient across several different computer platforms. - -After a foundation is formed the remainder of the library can be designed and implemented in a hierarchical fashion. -That is, to implement the lowest level dependencies first and work towards the most abstract functions last. For example, -before implementing a modular exponentiation algorithm one would implement a modular reduction algorithm. -By building outwards from a base foundation instead of using a parallel design methodology the resulting project is -highly modular. Being highly modular is a desirable property of any project as it often means the resulting product -has a small footprint and updates are easy to perform. - -Usually when I start a project I will begin with the header files. I define the data types I think I will need and -prototype the initial functions that are not dependent on other functions (within the library). After I -implement these base functions I prototype more dependent functions and implement them. The process repeats until -I implement all of the functions I require. For example, in the case of LibTomMath I implemented functions such as -mp\_init() well before I implemented mp\_mul() and even further before I implemented mp\_exptmod(). As an example as to -why this design works note that the Karatsuba and Toom-Cook multipliers were written \textit{after} the -dependent function mp\_exptmod() was written. Adding the new multiplication algorithms did not require changes to the -mp\_exptmod() function itself and lowered the total cost of ownership (\textit{so to speak}) and of development -for new algorithms. This methodology allows new algorithms to be tested in a complete framework with relative ease. - -\begin{center} -\begin{figure}[here] -\includegraphics{pics/design_process.ps} -\caption{Design Flow of the First Few Original LibTomMath Functions.} -\label{pic:design_process} -\end{figure} -\end{center} - -Only after the majority of the functions were in place did I pursue a less hierarchical approach to auditing and optimizing -the source code. For example, one day I may audit the multipliers and the next day the polynomial basis functions. - -It only makes sense to begin the text with the preliminary data types and support algorithms required as well. -This chapter discusses the core algorithms of the library which are the dependents for every other algorithm. - -\section{What is a Multiple Precision Integer?} -Recall that most programming languages, in particular ISO C \cite{ISOC}, only have fixed precision data types that on their own cannot -be used to represent values larger than their precision will allow. The purpose of multiple precision algorithms is -to use fixed precision data types to create and manipulate multiple precision integers which may represent values -that are very large. - -As a well known analogy, school children are taught how to form numbers larger than nine by prepending more radix ten digits. In the decimal system -the largest single digit value is $9$. However, by concatenating digits together larger numbers may be represented. Newly prepended digits -(\textit{to the left}) are said to be in a different power of ten column. That is, the number $123$ can be described as having a $1$ in the hundreds -column, $2$ in the tens column and $3$ in the ones column. Or more formally $123 = 1 \cdot 10^2 + 2 \cdot 10^1 + 3 \cdot 10^0$. Computer based -multiple precision arithmetic is essentially the same concept. Larger integers are represented by adjoining fixed -precision computer words with the exception that a different radix is used. - -What most people probably do not think about explicitly are the various other attributes that describe a multiple precision -integer. For example, the integer $154_{10}$ has two immediately obvious properties. First, the integer is positive, -that is the sign of this particular integer is positive as opposed to negative. Second, the integer has three digits in -its representation. There is an additional property that the integer posesses that does not concern pencil-and-paper -arithmetic. The third property is how many digits placeholders are available to hold the integer. - -The human analogy of this third property is ensuring there is enough space on the paper to write the integer. For example, -if one starts writing a large number too far to the right on a piece of paper they will have to erase it and move left. -Similarly, computer algorithms must maintain strict control over memory usage to ensure that the digits of an integer -will not exceed the allowed boundaries. These three properties make up what is known as a multiple precision -integer or mp\_int for short. - -\subsection{The mp\_int Structure} -\label{sec:MPINT} -The mp\_int structure is the ISO C based manifestation of what represents a multiple precision integer. The ISO C standard does not provide for -any such data type but it does provide for making composite data types known as structures. The following is the structure definition -used within LibTomMath. - -\index{mp\_int} -\begin{figure}[here] -\begin{center} -\begin{small} -%\begin{verbatim} -\begin{tabular}{|l|} -\hline -typedef struct \{ \\ -\hspace{3mm}int used, alloc, sign;\\ -\hspace{3mm}mp\_digit *dp;\\ -\} \textbf{mp\_int}; \\ -\hline -\end{tabular} -%\end{verbatim} -\end{small} -\caption{The mp\_int Structure} -\label{fig:mpint} -\end{center} -\end{figure} - -The mp\_int structure (fig. \ref{fig:mpint}) can be broken down as follows. - -\begin{enumerate} -\item The \textbf{used} parameter denotes how many digits of the array \textbf{dp} contain the digits used to represent -a given integer. The \textbf{used} count must be positive (or zero) and may not exceed the \textbf{alloc} count. - -\item The \textbf{alloc} parameter denotes how -many digits are available in the array to use by functions before it has to increase in size. When the \textbf{used} count -of a result would exceed the \textbf{alloc} count all of the algorithms will automatically increase the size of the -array to accommodate the precision of the result. - -\item The pointer \textbf{dp} points to a dynamically allocated array of digits that represent the given multiple -precision integer. It is padded with $(\textbf{alloc} - \textbf{used})$ zero digits. The array is maintained in a least -significant digit order. As a pencil and paper analogy the array is organized such that the right most digits are stored -first starting at the location indexed by zero\footnote{In C all arrays begin at zero.} in the array. For example, -if \textbf{dp} contains $\lbrace a, b, c, \ldots \rbrace$ where \textbf{dp}$_0 = a$, \textbf{dp}$_1 = b$, \textbf{dp}$_2 = c$, $\ldots$ then -it would represent the integer $a + b\beta + c\beta^2 + \ldots$ - -\index{MP\_ZPOS} \index{MP\_NEG} -\item The \textbf{sign} parameter denotes the sign as either zero/positive (\textbf{MP\_ZPOS}) or negative (\textbf{MP\_NEG}). -\end{enumerate} - -\subsubsection{Valid mp\_int Structures} -Several rules are placed on the state of an mp\_int structure and are assumed to be followed for reasons of efficiency. -The only exceptions are when the structure is passed to initialization functions such as mp\_init() and mp\_init\_copy(). - -\begin{enumerate} -\item The value of \textbf{alloc} may not be less than one. That is \textbf{dp} always points to a previously allocated -array of digits. -\item The value of \textbf{used} may not exceed \textbf{alloc} and must be greater than or equal to zero. -\item The value of \textbf{used} implies the digit at index $(used - 1)$ of the \textbf{dp} array is non-zero. That is, -leading zero digits in the most significant positions must be trimmed. - \begin{enumerate} - \item Digits in the \textbf{dp} array at and above the \textbf{used} location must be zero. - \end{enumerate} -\item The value of \textbf{sign} must be \textbf{MP\_ZPOS} if \textbf{used} is zero; -this represents the mp\_int value of zero. -\end{enumerate} - -\section{Argument Passing} -A convention of argument passing must be adopted early on in the development of any library. Making the function -prototypes consistent will help eliminate many headaches in the future as the library grows to significant complexity. -In LibTomMath the multiple precision integer functions accept parameters from left to right as pointers to mp\_int -structures. That means that the source (input) operands are placed on the left and the destination (output) on the right. -Consider the following examples. - -\begin{verbatim} - mp_mul(&a, &b, &c); /* c = a * b */ - mp_add(&a, &b, &a); /* a = a + b */ - mp_sqr(&a, &b); /* b = a * a */ -\end{verbatim} - -The left to right order is a fairly natural way to implement the functions since it lets the developer read aloud the -functions and make sense of them. For example, the first function would read ``multiply a and b and store in c''. - -Certain libraries (\textit{LIP by Lenstra for instance}) accept parameters the other way around, to mimic the order -of assignment expressions. That is, the destination (output) is on the left and arguments (inputs) are on the right. In -truth, it is entirely a matter of preference. In the case of LibTomMath the convention from the MPI library has been -adopted. - -Another very useful design consideration, provided for in LibTomMath, is whether to allow argument sources to also be a -destination. For example, the second example (\textit{mp\_add}) adds $a$ to $b$ and stores in $a$. This is an important -feature to implement since it allows the calling functions to cut down on the number of variables it must maintain. -However, to implement this feature specific care has to be given to ensure the destination is not modified before the -source is fully read. - -\section{Return Values} -A well implemented application, no matter what its purpose, should trap as many runtime errors as possible and return them -to the caller. By catching runtime errors a library can be guaranteed to prevent undefined behaviour. However, the end -developer can still manage to cause a library to crash. For example, by passing an invalid pointer an application may -fault by dereferencing memory not owned by the application. - -In the case of LibTomMath the only errors that are checked for are related to inappropriate inputs (division by zero for -instance) and memory allocation errors. It will not check that the mp\_int passed to any function is valid nor -will it check pointers for validity. Any function that can cause a runtime error will return an error code as an -\textbf{int} data type with one of the following values (fig \ref{fig:errcodes}). - -\index{MP\_OKAY} \index{MP\_VAL} \index{MP\_MEM} -\begin{figure}[here] -\begin{center} -\begin{tabular}{|l|l|} -\hline \textbf{Value} & \textbf{Meaning} \\ -\hline \textbf{MP\_OKAY} & The function was successful \\ -\hline \textbf{MP\_VAL} & One of the input value(s) was invalid \\ -\hline \textbf{MP\_MEM} & The function ran out of heap memory \\ -\hline -\end{tabular} -\end{center} -\caption{LibTomMath Error Codes} -\label{fig:errcodes} -\end{figure} - -When an error is detected within a function it should free any memory it allocated, often during the initialization of -temporary mp\_ints, and return as soon as possible. The goal is to leave the system in the same state it was when the -function was called. Error checking with this style of API is fairly simple. - -\begin{verbatim} - int err; - if ((err = mp_add(&a, &b, &c)) != MP_OKAY) { - printf("Error: %s\n", mp_error_to_string(err)); - exit(EXIT_FAILURE); - } -\end{verbatim} - -The GMP \cite{GMP} library uses C style \textit{signals} to flag errors which is of questionable use. Not all errors are fatal -and it was not deemed ideal by the author of LibTomMath to force developers to have signal handlers for such cases. - -\section{Initialization and Clearing} -The logical starting point when actually writing multiple precision integer functions is the initialization and -clearing of the mp\_int structures. These two algorithms will be used by the majority of the higher level algorithms. - -Given the basic mp\_int structure an initialization routine must first allocate memory to hold the digits of -the integer. Often it is optimal to allocate a sufficiently large pre-set number of digits even though -the initial integer will represent zero. If only a single digit were allocated quite a few subsequent re-allocations -would occur when operations are performed on the integers. There is a tradeoff between how many default digits to allocate -and how many re-allocations are tolerable. Obviously allocating an excessive amount of digits initially will waste -memory and become unmanageable. - -If the memory for the digits has been successfully allocated then the rest of the members of the structure must -be initialized. Since the initial state of an mp\_int is to represent the zero integer, the allocated digits must be set -to zero. The \textbf{used} count set to zero and \textbf{sign} set to \textbf{MP\_ZPOS}. - -\subsection{Initializing an mp\_int} -An mp\_int is said to be initialized if it is set to a valid, preferably default, state such that all of the members of the -structure are set to valid values. The mp\_init algorithm will perform such an action. - -\index{mp\_init} -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_init}. \\ -\textbf{Input}. An mp\_int $a$ \\ -\textbf{Output}. Allocate memory and initialize $a$ to a known valid mp\_int state. \\ -\hline \\ -1. Allocate memory for \textbf{MP\_PREC} digits. \\ -2. If the allocation failed return(\textit{MP\_MEM}) \\ -3. for $n$ from $0$ to $MP\_PREC - 1$ do \\ -\hspace{3mm}3.1 $a_n \leftarrow 0$\\ -4. $a.sign \leftarrow MP\_ZPOS$\\ -5. $a.used \leftarrow 0$\\ -6. $a.alloc \leftarrow MP\_PREC$\\ -7. Return(\textit{MP\_OKAY})\\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_init} -\end{figure} - -\textbf{Algorithm mp\_init.} -The purpose of this function is to initialize an mp\_int structure so that the rest of the library can properly -manipulte it. It is assumed that the input may not have had any of its members previously initialized which is certainly -a valid assumption if the input resides on the stack. - -Before any of the members such as \textbf{sign}, \textbf{used} or \textbf{alloc} are initialized the memory for -the digits is allocated. If this fails the function returns before setting any of the other members. The \textbf{MP\_PREC} -name represents a constant\footnote{Defined in the ``tommath.h'' header file within LibTomMath.} -used to dictate the minimum precision of newly initialized mp\_int integers. Ideally, it is at least equal to the smallest -precision number you'll be working with. - -Allocating a block of digits at first instead of a single digit has the benefit of lowering the number of usually slow -heap operations later functions will have to perform in the future. If \textbf{MP\_PREC} is set correctly the slack -memory and the number of heap operations will be trivial. - -Once the allocation has been made the digits have to be set to zero as well as the \textbf{used}, \textbf{sign} and -\textbf{alloc} members initialized. This ensures that the mp\_int will always represent the default state of zero regardless -of the original condition of the input. - -\textbf{Remark.} -This function introduces the idiosyncrasy that all iterative loops, commonly initiated with the ``for'' keyword, iterate incrementally -when the ``to'' keyword is placed between two expressions. For example, ``for $a$ from $b$ to $c$ do'' means that -a subsequent expression (or body of expressions) are to be evaluated upto $c - b$ times so long as $b \le c$. In each -iteration the variable $a$ is substituted for a new integer that lies inclusively between $b$ and $c$. If $b > c$ occured -the loop would not iterate. By contrast if the ``downto'' keyword were used in place of ``to'' the loop would iterate -decrementally. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_init.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* init a new mp_int */ -018 int mp_init (mp_int * a) -019 \{ -020 int i; -021 -022 /* allocate memory required and clear it */ -023 a->dp = OPT_CAST(mp_digit) XMALLOC (sizeof (mp_digit) * MP_PREC); -024 if (a->dp == NULL) \{ -025 return MP_MEM; -026 \} -027 -028 /* set the digits to zero */ -029 for (i = 0; i < MP_PREC; i++) \{ -030 a->dp[i] = 0; -031 \} -032 -033 /* set the used to zero, allocated digits to the default precision -034 * and sign to positive */ -035 a->used = 0; -036 a->alloc = MP_PREC; -037 a->sign = MP_ZPOS; -038 -039 return MP_OKAY; -040 \} -041 #endif -\end{alltt} -\end{small} - -One immediate observation of this initializtion function is that it does not return a pointer to a mp\_int structure. It -is assumed that the caller has already allocated memory for the mp\_int structure, typically on the application stack. The -call to mp\_init() is used only to initialize the members of the structure to a known default state. - -Here we see (line 23) the memory allocation is performed first. This allows us to exit cleanly and quickly -if there is an error. If the allocation fails the routine will return \textbf{MP\_MEM} to the caller to indicate there -was a memory error. The function XMALLOC is what actually allocates the memory. Technically XMALLOC is not a function -but a macro defined in ``tommath.h``. By default, XMALLOC will evaluate to malloc() which is the C library's built--in -memory allocation routine. - -In order to assure the mp\_int is in a known state the digits must be set to zero. On most platforms this could have been -accomplished by using calloc() instead of malloc(). However, to correctly initialize a integer type to a given value in a -portable fashion you have to actually assign the value. The for loop (line 29) performs this required -operation. - -After the memory has been successfully initialized the remainder of the members are initialized -(lines 33 through 34) to their respective default states. At this point the algorithm has succeeded and -a success code is returned to the calling function. If this function returns \textbf{MP\_OKAY} it is safe to assume the -mp\_int structure has been properly initialized and is safe to use with other functions within the library. - -\subsection{Clearing an mp\_int} -When an mp\_int is no longer required by the application, the memory that has been allocated for its digits must be -returned to the application's memory pool with the mp\_clear algorithm. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_clear}. \\ -\textbf{Input}. An mp\_int $a$ \\ -\textbf{Output}. The memory for $a$ shall be deallocated. \\ -\hline \\ -1. If $a$ has been previously freed then return(\textit{MP\_OKAY}). \\ -2. for $n$ from 0 to $a.used - 1$ do \\ -\hspace{3mm}2.1 $a_n \leftarrow 0$ \\ -3. Free the memory allocated for the digits of $a$. \\ -4. $a.used \leftarrow 0$ \\ -5. $a.alloc \leftarrow 0$ \\ -6. $a.sign \leftarrow MP\_ZPOS$ \\ -7. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_clear} -\end{figure} - -\textbf{Algorithm mp\_clear.} -This algorithm accomplishes two goals. First, it clears the digits and the other mp\_int members. This ensures that -if a developer accidentally re-uses a cleared structure it is less likely to cause problems. The second goal -is to free the allocated memory. - -The logic behind the algorithm is extended by marking cleared mp\_int structures so that subsequent calls to this -algorithm will not try to free the memory multiple times. Cleared mp\_ints are detectable by having a pre-defined invalid -digit pointer \textbf{dp} setting. - -Once an mp\_int has been cleared the mp\_int structure is no longer in a valid state for any other algorithm -with the exception of algorithms mp\_init, mp\_init\_copy, mp\_init\_size and mp\_clear. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_clear.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* clear one (frees) */ -018 void -019 mp_clear (mp_int * a) -020 \{ -021 int i; -022 -023 /* only do anything if a hasn't been freed previously */ -024 if (a->dp != NULL) \{ -025 /* first zero the digits */ -026 for (i = 0; i < a->used; i++) \{ -027 a->dp[i] = 0; -028 \} -029 -030 /* free ram */ -031 XFREE(a->dp); -032 -033 /* reset members to make debugging easier */ -034 a->dp = NULL; -035 a->alloc = a->used = 0; -036 a->sign = MP_ZPOS; -037 \} -038 \} -039 #endif -\end{alltt} -\end{small} - -The algorithm only operates on the mp\_int if it hasn't been previously cleared. The if statement (line 24) -checks to see if the \textbf{dp} member is not \textbf{NULL}. If the mp\_int is a valid mp\_int then \textbf{dp} cannot be -\textbf{NULL} in which case the if statement will evaluate to true. - -The digits of the mp\_int are cleared by the for loop (line 26) which assigns a zero to every digit. Similar to mp\_init() -the digits are assigned zero instead of using block memory operations (such as memset()) since this is more portable. - -The digits are deallocated off the heap via the XFREE macro. Similar to XMALLOC the XFREE macro actually evaluates to -a standard C library function. In this case the free() function. Since free() only deallocates the memory the pointer -still has to be reset to \textbf{NULL} manually (line 34). - -Now that the digits have been cleared and deallocated the other members are set to their final values (lines 35 and 36). - -\section{Maintenance Algorithms} - -The previous sections describes how to initialize and clear an mp\_int structure. To further support operations -that are to be performed on mp\_int structures (such as addition and multiplication) the dependent algorithms must be -able to augment the precision of an mp\_int and -initialize mp\_ints with differing initial conditions. - -These algorithms complete the set of low level algorithms required to work with mp\_int structures in the higher level -algorithms such as addition, multiplication and modular exponentiation. - -\subsection{Augmenting an mp\_int's Precision} -When storing a value in an mp\_int structure, a sufficient number of digits must be available to accomodate the entire -result of an operation without loss of precision. Quite often the size of the array given by the \textbf{alloc} member -is large enough to simply increase the \textbf{used} digit count. However, when the size of the array is too small it -must be re-sized appropriately to accomodate the result. The mp\_grow algorithm will provide this functionality. - -\newpage\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_grow}. \\ -\textbf{Input}. An mp\_int $a$ and an integer $b$. \\ -\textbf{Output}. $a$ is expanded to accomodate $b$ digits. \\ -\hline \\ -1. if $a.alloc \ge b$ then return(\textit{MP\_OKAY}) \\ -2. $u \leftarrow b\mbox{ (mod }MP\_PREC\mbox{)}$ \\ -3. $v \leftarrow b + 2 \cdot MP\_PREC - u$ \\ -4. Re-allocate the array of digits $a$ to size $v$ \\ -5. If the allocation failed then return(\textit{MP\_MEM}). \\ -6. for n from a.alloc to $v - 1$ do \\ -\hspace{+3mm}6.1 $a_n \leftarrow 0$ \\ -7. $a.alloc \leftarrow v$ \\ -8. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_grow} -\end{figure} - -\textbf{Algorithm mp\_grow.} -It is ideal to prevent re-allocations from being performed if they are not required (step one). This is useful to -prevent mp\_ints from growing excessively in code that erroneously calls mp\_grow. - -The requested digit count is padded up to next multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} (steps two and three). -This helps prevent many trivial reallocations that would grow an mp\_int by trivially small values. - -It is assumed that the reallocation (step four) leaves the lower $a.alloc$ digits of the mp\_int intact. This is much -akin to how the \textit{realloc} function from the standard C library works. Since the newly allocated digits are -assumed to contain undefined values they are initially set to zero. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_grow.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* grow as required */ -018 int mp_grow (mp_int * a, int size) -019 \{ -020 int i; -021 mp_digit *tmp; -022 -023 /* if the alloc size is smaller alloc more ram */ -024 if (a->alloc < size) \{ -025 /* ensure there are always at least MP_PREC digits extra on top */ -026 size += (MP_PREC * 2) - (size % MP_PREC); -027 -028 /* reallocate the array a->dp -029 * -030 * We store the return in a temporary variable -031 * in case the operation failed we don't want -032 * to overwrite the dp member of a. -033 */ -034 tmp = OPT_CAST(mp_digit) XREALLOC (a->dp, sizeof (mp_digit) * size); -035 if (tmp == NULL) \{ -036 /* reallocation failed but "a" is still valid [can be freed] */ -037 return MP_MEM; -038 \} -039 -040 /* reallocation succeeded so set a->dp */ -041 a->dp = tmp; -042 -043 /* zero excess digits */ -044 i = a->alloc; -045 a->alloc = size; -046 for (; i < a->alloc; i++) \{ -047 a->dp[i] = 0; -048 \} -049 \} -050 return MP_OKAY; -051 \} -052 #endif -\end{alltt} -\end{small} - -A quick optimization is to first determine if a memory re-allocation is required at all. The if statement (line 24) checks -if the \textbf{alloc} member of the mp\_int is smaller than the requested digit count. If the count is not larger than \textbf{alloc} -the function skips the re-allocation part thus saving time. - -When a re-allocation is performed it is turned into an optimal request to save time in the future. The requested digit count is -padded upwards to 2nd multiple of \textbf{MP\_PREC} larger than \textbf{alloc} (line 26). The XREALLOC function is used -to re-allocate the memory. As per the other functions XREALLOC is actually a macro which evaluates to realloc by default. The realloc -function leaves the base of the allocation intact which means the first \textbf{alloc} digits of the mp\_int are the same as before -the re-allocation. All that is left is to clear the newly allocated digits and return. - -Note that the re-allocation result is actually stored in a temporary pointer $tmp$. This is to allow this function to return -an error with a valid pointer. Earlier releases of the library stored the result of XREALLOC into the mp\_int $a$. That would -result in a memory leak if XREALLOC ever failed. - -\subsection{Initializing Variable Precision mp\_ints} -Occasionally the number of digits required will be known in advance of an initialization, based on, for example, the size -of input mp\_ints to a given algorithm. The purpose of algorithm mp\_init\_size is similar to mp\_init except that it -will allocate \textit{at least} a specified number of digits. - -\begin{figure}[here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_init\_size}. \\ -\textbf{Input}. An mp\_int $a$ and the requested number of digits $b$. \\ -\textbf{Output}. $a$ is initialized to hold at least $b$ digits. \\ -\hline \\ -1. $u \leftarrow b \mbox{ (mod }MP\_PREC\mbox{)}$ \\ -2. $v \leftarrow b + 2 \cdot MP\_PREC - u$ \\ -3. Allocate $v$ digits. \\ -4. for $n$ from $0$ to $v - 1$ do \\ -\hspace{3mm}4.1 $a_n \leftarrow 0$ \\ -5. $a.sign \leftarrow MP\_ZPOS$\\ -6. $a.used \leftarrow 0$\\ -7. $a.alloc \leftarrow v$\\ -8. Return(\textit{MP\_OKAY})\\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_init\_size} -\end{figure} - -\textbf{Algorithm mp\_init\_size.} -This algorithm will initialize an mp\_int structure $a$ like algorithm mp\_init with the exception that the number of -digits allocated can be controlled by the second input argument $b$. The input size is padded upwards so it is a -multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} digits. This padding is used to prevent trivial -allocations from becoming a bottleneck in the rest of the algorithms. - -Like algorithm mp\_init, the mp\_int structure is initialized to a default state representing the integer zero. This -particular algorithm is useful if it is known ahead of time the approximate size of the input. If the approximation is -correct no further memory re-allocations are required to work with the mp\_int. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_init\_size.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* init an mp_init for a given size */ -018 int mp_init_size (mp_int * a, int size) -019 \{ -020 int x; -021 -022 /* pad size so there are always extra digits */ -023 size += (MP_PREC * 2) - (size % MP_PREC); -024 -025 /* alloc mem */ -026 a->dp = OPT_CAST(mp_digit) XMALLOC (sizeof (mp_digit) * size); -027 if (a->dp == NULL) \{ -028 return MP_MEM; -029 \} -030 -031 /* set the members */ -032 a->used = 0; -033 a->alloc = size; -034 a->sign = MP_ZPOS; -035 -036 /* zero the digits */ -037 for (x = 0; x < size; x++) \{ -038 a->dp[x] = 0; -039 \} -040 -041 return MP_OKAY; -042 \} -043 #endif -\end{alltt} -\end{small} - -The number of digits $b$ requested is padded (line 23) 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 28). - -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 32, 33 and 34). If the function -returns succesfully then it is correct to assume that the mp\_int structure is in a valid state for the remainder of the -functions to work with. - -\subsection{Multiple Integer Initializations and Clearings} -Occasionally a function will require a series of mp\_int data types to be made available simultaneously. -The purpose of algorithm mp\_init\_multi is to initialize a variable length array of mp\_int structures in a single -statement. It is essentially a shortcut to multiple initializations. - -\newpage\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_init\_multi}. \\ -\textbf{Input}. Variable length array $V_k$ of mp\_int variables of length $k$. \\ -\textbf{Output}. The array is initialized such that each mp\_int of $V_k$ is ready to use. \\ -\hline \\ -1. for $n$ from 0 to $k - 1$ do \\ -\hspace{+3mm}1.1. Initialize the mp\_int $V_n$ (\textit{mp\_init}) \\ -\hspace{+3mm}1.2. If initialization failed then do \\ -\hspace{+6mm}1.2.1. for $j$ from $0$ to $n$ do \\ -\hspace{+9mm}1.2.1.1. Free the mp\_int $V_j$ (\textit{mp\_clear}) \\ -\hspace{+6mm}1.2.2. Return(\textit{MP\_MEM}) \\ -2. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_init\_multi} -\end{figure} - -\textbf{Algorithm mp\_init\_multi.} -The algorithm will initialize the array of mp\_int variables one at a time. If a runtime error has been detected -(\textit{step 1.2}) all of the previously initialized variables are cleared. The goal is an ``all or nothing'' -initialization which allows for quick recovery from runtime errors. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_init\_multi.c -\vspace{-3mm} -\begin{alltt} -016 #include <stdarg.h> -017 -018 int mp_init_multi(mp_int *mp, ...) -019 \{ -020 mp_err res = MP_OKAY; /* Assume ok until proven otherwise */ -021 int n = 0; /* Number of ok inits */ -022 mp_int* cur_arg = mp; -023 va_list args; -024 -025 va_start(args, mp); /* init args to next argument from caller */ -026 while (cur_arg != NULL) \{ -027 if (mp_init(cur_arg) != MP_OKAY) \{ -028 /* Oops - error! Back-track and mp_clear what we already -029 succeeded in init-ing, then return error. -030 */ -031 va_list clean_args; -032 -033 /* end the current list */ -034 va_end(args); -035 -036 /* now start cleaning up */ -037 cur_arg = mp; -038 va_start(clean_args, mp); -039 while (n--) \{ -040 mp_clear(cur_arg); -041 cur_arg = va_arg(clean_args, mp_int*); -042 \} -043 va_end(clean_args); -044 res = MP_MEM; -045 break; -046 \} -047 n++; -048 cur_arg = va_arg(args, mp_int*); -049 \} -050 va_end(args); -051 return res; /* Assumed ok, if error flagged above. */ -052 \} -053 -054 #endif -\end{alltt} -\end{small} - -This function intializes a variable length list of mp\_int structure pointers. However, instead of having the mp\_int -structures in an actual C array they are simply passed as arguments to the function. This function makes use of the -``...'' argument syntax of the C programming language. The list is terminated with a final \textbf{NULL} argument -appended on the right. - -The function uses the ``stdarg.h'' \textit{va} functions to step portably through the arguments to the function. A count -$n$ of succesfully initialized mp\_int structures is maintained (line 47) such that if a failure does occur, -the algorithm can backtrack and free the previously initialized structures (lines 27 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. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_clamp.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* trim unused digits -018 * -019 * This is used to ensure that leading zero digits are -020 * trimed and the leading "used" digit will be non-zero -021 * Typically very fast. Also fixes the sign if there -022 * are no more leading digits -023 */ -024 void -025 mp_clamp (mp_int * a) -026 \{ -027 /* decrease used while the most significant digit is -028 * zero. -029 */ -030 while (a->used > 0 && a->dp[a->used - 1] == 0) \{ -031 --(a->used); -032 \} -033 -034 /* reset the sign flag if used == 0 */ -035 if (a->used == 0) \{ -036 a->sign = MP_ZPOS; -037 \} -038 \} -039 #endif -\end{alltt} -\end{small} - -Note on line 27 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 30 is used to make sure the \textbf{used} count is decremented and not -the pointer ``a''. - -\section*{Exercises} -\begin{tabular}{cl} -$\left [ 1 \right ]$ & Discuss the relevance of the \textbf{used} member of the mp\_int structure. \\ - & \\ -$\left [ 1 \right ]$ & Discuss the consequences of not using padding when performing allocations. \\ - & \\ -$\left [ 2 \right ]$ & Estimate an ideal value for \textbf{MP\_PREC} when performing 1024-bit RSA \\ - & encryption when $\beta = 2^{28}$. \\ - & \\ -$\left [ 1 \right ]$ & Discuss the relevance of the algorithm mp\_clamp. What does it prevent? \\ - & \\ -$\left [ 1 \right ]$ & Give an example of when the algorithm mp\_init\_copy might be useful. \\ - & \\ -\end{tabular} - - -%%% -% CHAPTER FOUR -%%% - -\chapter{Basic Operations} - -\section{Introduction} -In the previous chapter a series of low level algorithms were established that dealt with initializing and maintaining -mp\_int structures. This chapter will discuss another set of seemingly non-algebraic algorithms which will form the low -level basis of the entire library. While these algorithm are relatively trivial it is important to understand how they -work before proceeding since these algorithms will be used almost intrinsically in the following chapters. - -The algorithms in this chapter deal primarily with more ``programmer'' related tasks such as creating copies of -mp\_int structures, assigning small values to mp\_int structures and comparisons of the values mp\_int structures -represent. - -\section{Assigning Values to mp\_int Structures} -\subsection{Copying an mp\_int} -Assigning the value that a given mp\_int structure represents to another mp\_int structure shall be known as making -a copy for the purposes of this text. The copy of the mp\_int will be a separate entity that represents the same -value as the mp\_int it was copied from. The mp\_copy algorithm provides this functionality. - -\newpage\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_copy}. \\ -\textbf{Input}. An mp\_int $a$ and $b$. \\ -\textbf{Output}. Store a copy of $a$ in $b$. \\ -\hline \\ -1. If $b.alloc < a.used$ then grow $b$ to $a.used$ digits. (\textit{mp\_grow}) \\ -2. for $n$ from 0 to $a.used - 1$ do \\ -\hspace{3mm}2.1 $b_{n} \leftarrow a_{n}$ \\ -3. for $n$ from $a.used$ to $b.used - 1$ do \\ -\hspace{3mm}3.1 $b_{n} \leftarrow 0$ \\ -4. $b.used \leftarrow a.used$ \\ -5. $b.sign \leftarrow a.sign$ \\ -6. return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_copy} -\end{figure} - -\textbf{Algorithm mp\_copy.} -This algorithm copies the mp\_int $a$ such that upon succesful termination of the algorithm the mp\_int $b$ will -represent the same integer as the mp\_int $a$. The mp\_int $b$ shall be a complete and distinct copy of the -mp\_int $a$ meaing that the mp\_int $a$ can be modified and it shall not affect the value of the mp\_int $b$. - -If $b$ does not have enough room for the digits of $a$ it must first have its precision augmented via the mp\_grow -algorithm. The digits of $a$ are copied over the digits of $b$ and any excess digits of $b$ are set to zero (step two -and three). The \textbf{used} and \textbf{sign} members of $a$ are finally copied over the respective members of -$b$. - -\textbf{Remark.} This algorithm also introduces a new idiosyncrasy that will be used throughout the rest of the -text. The error return codes of other algorithms are not explicitly checked in the pseudo-code presented. For example, in -step one of the mp\_copy algorithm the return of mp\_grow is not explicitly checked to ensure it succeeded. Text space is -limited so it is assumed that if a algorithm fails it will clear all temporarily allocated mp\_ints and return -the error code itself. However, the C code presented will demonstrate all of the error handling logic required to -implement the pseudo-code. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_copy.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* copy, b = a */ -018 int -019 mp_copy (mp_int * a, mp_int * b) -020 \{ -021 int res, n; -022 -023 /* if dst == src do nothing */ -024 if (a == b) \{ -025 return MP_OKAY; -026 \} -027 -028 /* grow dest */ -029 if (b->alloc < a->used) \{ -030 if ((res = mp_grow (b, a->used)) != MP_OKAY) \{ -031 return res; -032 \} -033 \} -034 -035 /* zero b and copy the parameters over */ -036 \{ -037 register mp_digit *tmpa, *tmpb; -038 -039 /* pointer aliases */ -040 -041 /* source */ -042 tmpa = a->dp; -043 -044 /* destination */ -045 tmpb = b->dp; -046 -047 /* copy all the digits */ -048 for (n = 0; n < a->used; n++) \{ -049 *tmpb++ = *tmpa++; -050 \} -051 -052 /* clear high digits */ -053 for (; n < b->used; n++) \{ -054 *tmpb++ = 0; -055 \} -056 \} -057 -058 /* copy used count and sign */ -059 b->used = a->used; -060 b->sign = a->sign; -061 return MP_OKAY; -062 \} -063 #endif -\end{alltt} -\end{small} - -Occasionally a dependent algorithm may copy an mp\_int effectively into itself such as when the input and output -mp\_int structures passed to a function are one and the same. For this case it is optimal to return immediately without -copying digits (line 24). - -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 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 and 45) 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 to 50) and then the excess -digits of $b$ are set to zero (lines 53 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. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_init\_copy.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* creates "a" then copies b into it */ -018 int mp_init_copy (mp_int * a, mp_int * b) -019 \{ -020 int res; -021 -022 if ((res = mp_init (a)) != MP_OKAY) \{ -023 return res; -024 \} -025 return mp_copy (b, a); -026 \} -027 #endif -\end{alltt} -\end{small} - -This will initialize \textbf{a} and make it a verbatim copy of the contents of \textbf{b}. Note that -\textbf{a} will have its own memory allocated which means that \textbf{b} may be cleared after the call -and \textbf{a} will be left intact. - -\section{Zeroing an Integer} -Reseting an mp\_int to the default state is a common step in many algorithms. The mp\_zero algorithm will be the algorithm used to -perform this task. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_zero}. \\ -\textbf{Input}. An mp\_int $a$ \\ -\textbf{Output}. Zero the contents of $a$ \\ -\hline \\ -1. $a.used \leftarrow 0$ \\ -2. $a.sign \leftarrow$ MP\_ZPOS \\ -3. for $n$ from 0 to $a.alloc - 1$ do \\ -\hspace{3mm}3.1 $a_n \leftarrow 0$ \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_zero} -\end{figure} - -\textbf{Algorithm mp\_zero.} -This algorithm simply resets a mp\_int to the default state. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_zero.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* set to zero */ -018 void mp_zero (mp_int * a) -019 \{ -020 int n; -021 mp_digit *tmp; -022 -023 a->sign = MP_ZPOS; -024 a->used = 0; -025 -026 tmp = a->dp; -027 for (n = 0; n < a->alloc; n++) \{ -028 *tmp++ = 0; -029 \} -030 \} -031 #endif -\end{alltt} -\end{small} - -After the function is completed, all of the digits are zeroed, the \textbf{used} count is zeroed and the -\textbf{sign} variable is set to \textbf{MP\_ZPOS}. - -\section{Sign Manipulation} -\subsection{Absolute Value} -With the mp\_int representation of an integer, calculating the absolute value is trivial. The mp\_abs algorithm will compute -the absolute value of an mp\_int. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_abs}. \\ -\textbf{Input}. An mp\_int $a$ \\ -\textbf{Output}. Computes $b = \vert a \vert$ \\ -\hline \\ -1. Copy $a$ to $b$. (\textit{mp\_copy}) \\ -2. If the copy failed return(\textit{MP\_MEM}). \\ -3. $b.sign \leftarrow MP\_ZPOS$ \\ -4. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_abs} -\end{figure} - -\textbf{Algorithm mp\_abs.} -This algorithm computes the absolute of an mp\_int input. First it copies $a$ over $b$. This is an example of an -algorithm where the check in mp\_copy that determines if the source and destination are equal proves useful. This allows, -for instance, the developer to pass the same mp\_int as the source and destination to this function without addition -logic to handle it. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_abs.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* b = |a| -018 * -019 * Simple function copies the input and fixes the sign to positive -020 */ -021 int -022 mp_abs (mp_int * a, mp_int * b) -023 \{ -024 int res; -025 -026 /* copy a to b */ -027 if (a != b) \{ -028 if ((res = mp_copy (a, b)) != MP_OKAY) \{ -029 return res; -030 \} -031 \} -032 -033 /* force the sign of b to positive */ -034 b->sign = MP_ZPOS; -035 -036 return MP_OKAY; -037 \} -038 #endif -\end{alltt} -\end{small} - -This fairly trivial algorithm first eliminates non--required duplications (line 27) and then sets the -\textbf{sign} flag to \textbf{MP\_ZPOS}. - -\subsection{Integer Negation} -With the mp\_int representation of an integer, calculating the negation is also trivial. The mp\_neg algorithm will compute -the negative of an mp\_int input. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_neg}. \\ -\textbf{Input}. An mp\_int $a$ \\ -\textbf{Output}. Computes $b = -a$ \\ -\hline \\ -1. Copy $a$ to $b$. (\textit{mp\_copy}) \\ -2. If the copy failed return(\textit{MP\_MEM}). \\ -3. If $a.used = 0$ then return(\textit{MP\_OKAY}). \\ -4. If $a.sign = MP\_ZPOS$ then do \\ -\hspace{3mm}4.1 $b.sign = MP\_NEG$. \\ -5. else do \\ -\hspace{3mm}5.1 $b.sign = MP\_ZPOS$. \\ -6. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_neg} -\end{figure} - -\textbf{Algorithm mp\_neg.} -This algorithm computes the negation of an input. First it copies $a$ over $b$. If $a$ has no used digits then -the algorithm returns immediately. Otherwise it flips the sign flag and stores the result in $b$. Note that if -$a$ had no digits then it must be positive by definition. Had step three been omitted then the algorithm would return -zero as negative. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_neg.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* b = -a */ -018 int mp_neg (mp_int * a, mp_int * b) -019 \{ -020 int res; -021 if (a != b) \{ -022 if ((res = mp_copy (a, b)) != MP_OKAY) \{ -023 return res; -024 \} -025 \} -026 -027 if (mp_iszero(b) != MP_YES) \{ -028 b->sign = (a->sign == MP_ZPOS) ? MP_NEG : MP_ZPOS; -029 \} else \{ -030 b->sign = MP_ZPOS; -031 \} -032 -033 return MP_OKAY; -034 \} -035 #endif -\end{alltt} -\end{small} - -Like mp\_abs() this function avoids non--required duplications (line 21) and then sets the sign. We -have to make sure that only non--zero values get a \textbf{sign} of \textbf{MP\_NEG}. If the mp\_int is zero -than the \textbf{sign} is hard--coded to \textbf{MP\_ZPOS}. - -\section{Small Constants} -\subsection{Setting Small Constants} -Often a mp\_int must be set to a relatively small value such as $1$ or $2$. For these cases the mp\_set algorithm is useful. - -\newpage\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_set}. \\ -\textbf{Input}. An mp\_int $a$ and a digit $b$ \\ -\textbf{Output}. Make $a$ equivalent to $b$ \\ -\hline \\ -1. Zero $a$ (\textit{mp\_zero}). \\ -2. $a_0 \leftarrow b \mbox{ (mod }\beta\mbox{)}$ \\ -3. $a.used \leftarrow \left \lbrace \begin{array}{ll} - 1 & \mbox{if }a_0 > 0 \\ - 0 & \mbox{if }a_0 = 0 - \end{array} \right .$ \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_set} -\end{figure} - -\textbf{Algorithm mp\_set.} -This algorithm sets a mp\_int to a small single digit value. Step number 1 ensures that the integer is reset to the default state. The -single digit is set (\textit{modulo $\beta$}) and the \textbf{used} count is adjusted accordingly. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_set.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* set to a digit */ -018 void mp_set (mp_int * a, mp_digit b) -019 \{ -020 mp_zero (a); -021 a->dp[0] = b & MP_MASK; -022 a->used = (a->dp[0] != 0) ? 1 : 0; -023 \} -024 #endif -\end{alltt} -\end{small} - -First we zero (line 20) the mp\_int to make sure that the other members are initialized for a -small positive constant. mp\_zero() ensures that the \textbf{sign} is positive and the \textbf{used} count -is zero. Next we set the digit and reduce it modulo $\beta$ (line 21). After this step we have to -check if the resulting digit is zero or not. If it is not then we set the \textbf{used} count to one, otherwise -to zero. - -We can quickly reduce modulo $\beta$ since it is of the form $2^k$ and a quick binary AND operation with -$2^k - 1$ will perform the same operation. - -One important limitation of this function is that it will only set one digit. The size of a digit is not fixed, meaning source that uses -this function should take that into account. Only trivially small constants can be set using this function. - -\subsection{Setting Large Constants} -To overcome the limitations of the mp\_set algorithm the mp\_set\_int algorithm is ideal. It accepts a ``long'' -data type as input and will always treat it as a 32-bit integer. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_set\_int}. \\ -\textbf{Input}. An mp\_int $a$ and a ``long'' integer $b$ \\ -\textbf{Output}. Make $a$ equivalent to $b$ \\ -\hline \\ -1. Zero $a$ (\textit{mp\_zero}) \\ -2. for $n$ from 0 to 7 do \\ -\hspace{3mm}2.1 $a \leftarrow a \cdot 16$ (\textit{mp\_mul2d}) \\ -\hspace{3mm}2.2 $u \leftarrow \lfloor b / 2^{4(7 - n)} \rfloor \mbox{ (mod }16\mbox{)}$\\ -\hspace{3mm}2.3 $a_0 \leftarrow a_0 + u$ \\ -\hspace{3mm}2.4 $a.used \leftarrow a.used + 1$ \\ -3. Clamp excess used digits (\textit{mp\_clamp}) \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_set\_int} -\end{figure} - -\textbf{Algorithm mp\_set\_int.} -The algorithm performs eight iterations of a simple loop where in each iteration four bits from the source are added to the -mp\_int. Step 2.1 will multiply the current result by sixteen making room for four more bits in the less significant positions. In step 2.2 the -next four bits from the source are extracted and are added to the mp\_int. The \textbf{used} digit count is -incremented to reflect the addition. The \textbf{used} digit counter is incremented since if any of the leading digits were zero the mp\_int would have -zero digits used and the newly added four bits would be ignored. - -Excess zero digits are trimmed in steps 2.1 and 3 by using higher level algorithms mp\_mul2d and mp\_clamp. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_set\_int.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* set a 32-bit const */ -018 int mp_set_int (mp_int * a, unsigned long b) -019 \{ -020 int x, res; -021 -022 mp_zero (a); -023 -024 /* set four bits at a time */ -025 for (x = 0; x < 8; x++) \{ -026 /* shift the number up four bits */ -027 if ((res = mp_mul_2d (a, 4, a)) != MP_OKAY) \{ -028 return res; -029 \} -030 -031 /* OR in the top four bits of the source */ -032 a->dp[0] |= (b >> 28) & 15; -033 -034 /* shift the source up to the next four bits */ -035 b <<= 4; -036 -037 /* ensure that digits are not clamped off */ -038 a->used += 1; -039 \} -040 mp_clamp (a); -041 return MP_OKAY; -042 \} -043 #endif -\end{alltt} -\end{small} - -This function sets four bits of the number at a time to handle all practical \textbf{DIGIT\_BIT} sizes. The weird -addition on line 38 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 -as well as the call to mp\_clamp() on line 40. Both functions will clamp excess leading digits which keeps -the number of used digits low. - -\section{Comparisons} -\subsection{Unsigned Comparisions} -Comparing a multiple precision integer is performed with the exact same algorithm used to compare two decimal numbers. For example, -to compare $1,234$ to $1,264$ the digits are extracted by their positions. That is we compare $1 \cdot 10^3 + 2 \cdot 10^2 + 3 \cdot 10^1 + 4 \cdot 10^0$ -to $1 \cdot 10^3 + 2 \cdot 10^2 + 6 \cdot 10^1 + 4 \cdot 10^0$ by comparing single digits at a time starting with the highest magnitude -positions. If any leading digit of one integer is greater than a digit in the same position of another integer then obviously it must be greater. - -The first comparision routine that will be developed is the unsigned magnitude compare which will perform a comparison based on the digits of two -mp\_int variables alone. It will ignore the sign of the two inputs. Such a function is useful when an absolute comparison is required or if the -signs are known to agree in advance. - -To facilitate working with the results of the comparison functions three constants are required. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{|r|l|} -\hline \textbf{Constant} & \textbf{Meaning} \\ -\hline \textbf{MP\_GT} & Greater Than \\ -\hline \textbf{MP\_EQ} & Equal To \\ -\hline \textbf{MP\_LT} & Less Than \\ -\hline -\end{tabular} -\end{center} -\caption{Comparison Return Codes} -\end{figure} - -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_cmp\_mag}. \\ -\textbf{Input}. Two mp\_ints $a$ and $b$. \\ -\textbf{Output}. Unsigned comparison results ($a$ to the left of $b$). \\ -\hline \\ -1. If $a.used > b.used$ then return(\textit{MP\_GT}) \\ -2. If $a.used < b.used$ then return(\textit{MP\_LT}) \\ -3. for n from $a.used - 1$ to 0 do \\ -\hspace{+3mm}3.1 if $a_n > b_n$ then return(\textit{MP\_GT}) \\ -\hspace{+3mm}3.2 if $a_n < b_n$ then return(\textit{MP\_LT}) \\ -4. Return(\textit{MP\_EQ}) \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_cmp\_mag} -\end{figure} - -\textbf{Algorithm mp\_cmp\_mag.} -By saying ``$a$ to the left of $b$'' it is meant that the comparison is with respect to $a$, that is if $a$ is greater than $b$ it will return -\textbf{MP\_GT} and similar with respect to when $a = b$ and $a < b$. The first two steps compare the number of digits used in both $a$ and $b$. -Obviously if the digit counts differ there would be an imaginary zero digit in the smaller number where the leading digit of the larger number is. -If both have the same number of digits than the actual digits themselves must be compared starting at the leading digit. - -By step three both inputs must have the same number of digits so its safe to start from either $a.used - 1$ or $b.used - 1$ and count down to -the zero'th digit. If after all of the digits have been compared, no difference is found, the algorithm returns \textbf{MP\_EQ}. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_cmp\_mag.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* compare maginitude of two ints (unsigned) */ -018 int mp_cmp_mag (mp_int * a, mp_int * b) -019 \{ -020 int n; -021 mp_digit *tmpa, *tmpb; -022 -023 /* compare based on # of non-zero digits */ -024 if (a->used > b->used) \{ -025 return MP_GT; -026 \} -027 -028 if (a->used < b->used) \{ -029 return MP_LT; -030 \} -031 -032 /* alias for a */ -033 tmpa = a->dp + (a->used - 1); -034 -035 /* alias for b */ -036 tmpb = b->dp + (a->used - 1); -037 -038 /* compare based on digits */ -039 for (n = 0; n < a->used; ++n, --tmpa, --tmpb) \{ -040 if (*tmpa > *tmpb) \{ -041 return MP_GT; -042 \} -043 -044 if (*tmpa < *tmpb) \{ -045 return MP_LT; -046 \} -047 \} -048 return MP_EQ; -049 \} -050 #endif -\end{alltt} -\end{small} - -The two if statements (lines 24 and 28) compare the number of digits in the two inputs. These two are -performed before all of the digits are compared since it is a very cheap test to perform and can potentially save -considerable time. The implementation given is also not valid without those two statements. $b.alloc$ may be -smaller than $a.used$, meaning that undefined values will be read from $b$ past the end of the array of digits. - - - -\subsection{Signed Comparisons} -Comparing with sign considerations is also fairly critical in several routines (\textit{division for example}). Based on an unsigned magnitude -comparison a trivial signed comparison algorithm can be written. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_cmp}. \\ -\textbf{Input}. Two mp\_ints $a$ and $b$ \\ -\textbf{Output}. Signed Comparison Results ($a$ to the left of $b$) \\ -\hline \\ -1. if $a.sign = MP\_NEG$ and $b.sign = MP\_ZPOS$ then return(\textit{MP\_LT}) \\ -2. if $a.sign = MP\_ZPOS$ and $b.sign = MP\_NEG$ then return(\textit{MP\_GT}) \\ -3. if $a.sign = MP\_NEG$ then \\ -\hspace{+3mm}3.1 Return the unsigned comparison of $b$ and $a$ (\textit{mp\_cmp\_mag}) \\ -4 Otherwise \\ -\hspace{+3mm}4.1 Return the unsigned comparison of $a$ and $b$ \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_cmp} -\end{figure} - -\textbf{Algorithm mp\_cmp.} -The first two steps compare the signs of the two inputs. If the signs do not agree then it can return right away with the appropriate -comparison code. When the signs are equal the digits of the inputs must be compared to determine the correct result. In step -three the unsigned comparision flips the order of the arguments since they are both negative. For instance, if $-a > -b$ then -$\vert a \vert < \vert b \vert$. Step number four will compare the two when they are both positive. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_cmp.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* compare two ints (signed)*/ -018 int -019 mp_cmp (mp_int * a, mp_int * b) -020 \{ -021 /* compare based on sign */ -022 if (a->sign != b->sign) \{ -023 if (a->sign == MP_NEG) \{ -024 return MP_LT; -025 \} else \{ -026 return MP_GT; -027 \} -028 \} -029 -030 /* compare digits */ -031 if (a->sign == MP_NEG) \{ -032 /* if negative compare opposite direction */ -033 return mp_cmp_mag(b, a); -034 \} else \{ -035 return mp_cmp_mag(a, b); -036 \} -037 \} -038 #endif -\end{alltt} -\end{small} - -The two if statements (lines 22 and 23) perform the initial sign comparison. If the signs are not the equal then which ever -has the positive sign is larger. The inputs are compared (line 31) based on magnitudes. If the signs were both -negative then the unsigned comparison is performed in the opposite direction (line 33). Otherwise, the signs are assumed to -be both positive and a forward direction unsigned comparison is performed. - -\section*{Exercises} -\begin{tabular}{cl} -$\left [ 2 \right ]$ & Modify algorithm mp\_set\_int to accept as input a variable length array of bits. \\ - & \\ -$\left [ 3 \right ]$ & Give the probability that algorithm mp\_cmp\_mag will have to compare $k$ digits \\ - & of two random digits (of equal magnitude) before a difference is found. \\ - & \\ -$\left [ 1 \right ]$ & Suggest a simple method to speed up the implementation of mp\_cmp\_mag based \\ - & on the observations made in the previous problem. \\ - & -\end{tabular} - -\chapter{Basic Arithmetic} -\section{Introduction} -At this point algorithms for initialization, clearing, zeroing, copying, comparing and setting small constants have been -established. The next logical set of algorithms to develop are addition, subtraction and digit shifting algorithms. These -algorithms make use of the lower level algorithms and are the cruicial building block for the multiplication algorithms. It is very important -that these algorithms are highly optimized. On their own they are simple $O(n)$ algorithms but they can be called from higher level algorithms -which easily places them at $O(n^2)$ or even $O(n^3)$ work levels. - -All of the algorithms within this chapter make use of the logical bit shift operations denoted by $<<$ and $>>$ for left and right -logical shifts respectively. A logical shift is analogous to sliding the decimal point of radix-10 representations. For example, the real -number $0.9345$ is equivalent to $93.45\%$ which is found by sliding the the decimal two places to the right (\textit{multiplying by $\beta^2 = 10^2$}). -Algebraically a binary logical shift is equivalent to a division or multiplication by a power of two. -For example, $a << k = a \cdot 2^k$ while $a >> k = \lfloor a/2^k \rfloor$. - -One significant difference between a logical shift and the way decimals are shifted is that digits below the zero'th position are removed -from the number. For example, consider $1101_2 >> 1$ using decimal notation this would produce $110.1_2$. However, with a logical shift the -result is $110_2$. - -\section{Addition and Subtraction} -In common twos complement fixed precision arithmetic negative numbers are easily represented by subtraction from the modulus. For example, with 32-bit integers -$a - b\mbox{ (mod }2^{32}\mbox{)}$ is the same as $a + (2^{32} - b) \mbox{ (mod }2^{32}\mbox{)}$ since $2^{32} \equiv 0 \mbox{ (mod }2^{32}\mbox{)}$. -As a result subtraction can be performed with a trivial series of logical operations and an addition. - -However, in multiple precision arithmetic negative numbers are not represented in the same way. Instead a sign flag is used to keep track of the -sign of the integer. As a result signed addition and subtraction are actually implemented as conditional usage of lower level addition or -subtraction algorithms with the sign fixed up appropriately. - -The lower level algorithms will add or subtract integers without regard to the sign flag. That is they will add or subtract the magnitude of -the integers respectively. - -\subsection{Low Level Addition} -An unsigned addition of multiple precision integers is performed with the same long-hand algorithm used to add decimal numbers. That is to add the -trailing digits first and propagate the resulting carry upwards. Since this is a lower level algorithm the name will have a ``s\_'' prefix. -Historically that convention stems from the MPI library where ``s\_'' stood for static functions that were hidden from the developer entirely. - -\newpage -\begin{figure}[!here] -\begin{center} -\begin{small} -\begin{tabular}{l} -\hline Algorithm \textbf{s\_mp\_add}. \\ -\textbf{Input}. Two mp\_ints $a$ and $b$ \\ -\textbf{Output}. The unsigned addition $c = \vert a \vert + \vert b \vert$. \\ -\hline \\ -1. if $a.used > b.used$ then \\ -\hspace{+3mm}1.1 $min \leftarrow b.used$ \\ -\hspace{+3mm}1.2 $max \leftarrow a.used$ \\ -\hspace{+3mm}1.3 $x \leftarrow a$ \\ -2. else \\ -\hspace{+3mm}2.1 $min \leftarrow a.used$ \\ -\hspace{+3mm}2.2 $max \leftarrow b.used$ \\ -\hspace{+3mm}2.3 $x \leftarrow b$ \\ -3. If $c.alloc < max + 1$ then grow $c$ to hold at least $max + 1$ digits (\textit{mp\_grow}) \\ -4. $oldused \leftarrow c.used$ \\ -5. $c.used \leftarrow max + 1$ \\ -6. $u \leftarrow 0$ \\ -7. for $n$ from $0$ to $min - 1$ do \\ -\hspace{+3mm}7.1 $c_n \leftarrow a_n + b_n + u$ \\ -\hspace{+3mm}7.2 $u \leftarrow c_n >> lg(\beta)$ \\ -\hspace{+3mm}7.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ -8. if $min \ne max$ then do \\ -\hspace{+3mm}8.1 for $n$ from $min$ to $max - 1$ do \\ -\hspace{+6mm}8.1.1 $c_n \leftarrow x_n + u$ \\ -\hspace{+6mm}8.1.2 $u \leftarrow c_n >> lg(\beta)$ \\ -\hspace{+6mm}8.1.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ -9. $c_{max} \leftarrow u$ \\ -10. if $olduse > max$ then \\ -\hspace{+3mm}10.1 for $n$ from $max + 1$ to $oldused - 1$ do \\ -\hspace{+6mm}10.1.1 $c_n \leftarrow 0$ \\ -11. Clamp excess digits in $c$. (\textit{mp\_clamp}) \\ -12. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{small} -\end{center} -\caption{Algorithm s\_mp\_add} -\end{figure} - -\textbf{Algorithm s\_mp\_add.} -This algorithm is loosely based on algorithm 14.7 of HAC \cite[pp. 594]{HAC} but has been extended to allow the inputs to have different magnitudes. -Coincidentally the description of algorithm A in Knuth \cite[pp. 266]{TAOCPV2} shares the same deficiency as the algorithm from \cite{HAC}. Even the -MIX pseudo machine code presented by Knuth \cite[pp. 266-267]{TAOCPV2} is incapable of handling inputs which are of different magnitudes. - -The first thing that has to be accomplished is to sort out which of the two inputs is the largest. The addition logic -will simply add all of the smallest input to the largest input and store that first part of the result in the -destination. Then it will apply a simpler addition loop to excess digits of the larger input. - -The first two steps will handle sorting the inputs such that $min$ and $max$ hold the digit counts of the two -inputs. The variable $x$ will be an mp\_int alias for the largest input or the second input $b$ if they have the -same number of digits. After the inputs are sorted the destination $c$ is grown as required to accomodate the sum -of the two inputs. The original \textbf{used} count of $c$ is copied and set to the new used count. - -At this point the first addition loop will go through as many digit positions that both inputs have. The carry -variable $\mu$ is set to zero outside the loop. Inside the loop an ``addition'' step requires three statements to produce -one digit of the summand. First -two digits from $a$ and $b$ are added together along with the carry $\mu$. The carry of this step is extracted and stored -in $\mu$ and finally the digit of the result $c_n$ is truncated within the range $0 \le c_n < \beta$. - -Now all of the digit positions that both inputs have in common have been exhausted. If $min \ne max$ then $x$ is an alias -for one of the inputs that has more digits. A simplified addition loop is then used to essentially copy the remaining digits -and the carry to the destination. - -The final carry is stored in $c_{max}$ and digits above $max$ upto $oldused$ are zeroed which completes the addition. - - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_add.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* low level addition, based on HAC pp.594, Algorithm 14.7 */ -018 int -019 s_mp_add (mp_int * a, mp_int * b, mp_int * c) -020 \{ -021 mp_int *x; -022 int olduse, res, min, max; -023 -024 /* find sizes, we let |a| <= |b| which means we have to sort -025 * them. "x" will point to the input with the most digits -026 */ -027 if (a->used > b->used) \{ -028 min = b->used; -029 max = a->used; -030 x = a; -031 \} else \{ -032 min = a->used; -033 max = b->used; -034 x = b; -035 \} -036 -037 /* init result */ -038 if (c->alloc < max + 1) \{ -039 if ((res = mp_grow (c, max + 1)) != MP_OKAY) \{ -040 return res; -041 \} -042 \} -043 -044 /* get old used digit count and set new one */ -045 olduse = c->used; -046 c->used = max + 1; -047 -048 \{ -049 register mp_digit u, *tmpa, *tmpb, *tmpc; -050 register int i; -051 -052 /* alias for digit pointers */ -053 -054 /* first input */ -055 tmpa = a->dp; -056 -057 /* second input */ -058 tmpb = b->dp; -059 -060 /* destination */ -061 tmpc = c->dp; -062 -063 /* zero the carry */ -064 u = 0; -065 for (i = 0; i < min; i++) \{ -066 /* Compute the sum at one digit, T[i] = A[i] + B[i] + U */ -067 *tmpc = *tmpa++ + *tmpb++ + u; -068 -069 /* U = carry bit of T[i] */ -070 u = *tmpc >> ((mp_digit)DIGIT_BIT); -071 -072 /* take away carry bit from T[i] */ -073 *tmpc++ &= MP_MASK; -074 \} -075 -076 /* now copy higher words if any, that is in A+B -077 * if A or B has more digits add those in -078 */ -079 if (min != max) \{ -080 for (; i < max; i++) \{ -081 /* T[i] = X[i] + U */ -082 *tmpc = x->dp[i] + u; -083 -084 /* U = carry bit of T[i] */ -085 u = *tmpc >> ((mp_digit)DIGIT_BIT); -086 -087 /* take away carry bit from T[i] */ -088 *tmpc++ &= MP_MASK; -089 \} -090 \} -091 -092 /* add carry */ -093 *tmpc++ = u; -094 -095 /* clear digits above oldused */ -096 for (i = c->used; i < olduse; i++) \{ -097 *tmpc++ = 0; -098 \} -099 \} -100 -101 mp_clamp (c); -102 return MP_OKAY; -103 \} -104 #endif -\end{alltt} -\end{small} - -We first sort (lines 27 to 35) the inputs based on magnitude and determine the $min$ and $max$ variables. -Note that $x$ is a pointer to an mp\_int assigned to the largest input, in effect it is a local alias. Next we -grow the destination (37 to 42) ensure that it can accomodate the result of the addition. - -Similar to the implementation of mp\_copy this function uses the braced code and local aliases coding style. The three aliases that are on -lines 55, 58 and 61 represent the two inputs and destination variables respectively. These aliases are used to ensure the -compiler does not have to dereference $a$, $b$ or $c$ (respectively) to access the digits of the respective mp\_int. - -The initial carry $u$ will be cleared (line 64), note that $u$ is of type mp\_digit which ensures type -compatibility within the implementation. The initial addition (line 65 to 74) adds digits from -both inputs until the smallest input runs out of digits. Similarly the conditional addition loop -(line 80 to 90) adds the remaining digits from the larger of the two inputs. The addition is finished -with the final carry being stored in $tmpc$ (line 93). Note the ``++'' operator within the same expression. -After line 93, $tmpc$ will point to the $c.used$'th digit of the mp\_int $c$. This is useful -for the next loop (line 96 to 99) which set any old upper digits to zero. - -\subsection{Low Level Subtraction} -The low level unsigned subtraction algorithm is very similar to the low level unsigned addition algorithm. The principle difference is that the -unsigned subtraction algorithm requires the result to be positive. That is when computing $a - b$ the condition $\vert a \vert \ge \vert b\vert$ must -be met for this algorithm to function properly. Keep in mind this low level algorithm is not meant to be used in higher level algorithms directly. -This algorithm as will be shown can be used to create functional signed addition and subtraction algorithms. - - -For this algorithm a new variable is required to make the description simpler. Recall from section 1.3.1 that a mp\_digit must be able to represent -the range $0 \le x < 2\beta$ for the algorithms to work correctly. However, it is allowable that a mp\_digit represent a larger range of values. For -this algorithm we will assume that the variable $\gamma$ represents the number of bits available in a -mp\_digit (\textit{this implies $2^{\gamma} > \beta$}). - -For example, the default for LibTomMath is to use a ``unsigned long'' for the mp\_digit ``type'' while $\beta = 2^{28}$. In ISO C an ``unsigned long'' -data type must be able to represent $0 \le x < 2^{32}$ meaning that in this case $\gamma \ge 32$. - -\newpage\begin{figure}[!here] -\begin{center} -\begin{small} -\begin{tabular}{l} -\hline Algorithm \textbf{s\_mp\_sub}. \\ -\textbf{Input}. Two mp\_ints $a$ and $b$ ($\vert a \vert \ge \vert b \vert$) \\ -\textbf{Output}. The unsigned subtraction $c = \vert a \vert - \vert b \vert$. \\ -\hline \\ -1. $min \leftarrow b.used$ \\ -2. $max \leftarrow a.used$ \\ -3. If $c.alloc < max$ then grow $c$ to hold at least $max$ digits. (\textit{mp\_grow}) \\ -4. $oldused \leftarrow c.used$ \\ -5. $c.used \leftarrow max$ \\ -6. $u \leftarrow 0$ \\ -7. for $n$ from $0$ to $min - 1$ do \\ -\hspace{3mm}7.1 $c_n \leftarrow a_n - b_n - u$ \\ -\hspace{3mm}7.2 $u \leftarrow c_n >> (\gamma - 1)$ \\ -\hspace{3mm}7.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ -8. if $min < max$ then do \\ -\hspace{3mm}8.1 for $n$ from $min$ to $max - 1$ do \\ -\hspace{6mm}8.1.1 $c_n \leftarrow a_n - u$ \\ -\hspace{6mm}8.1.2 $u \leftarrow c_n >> (\gamma - 1)$ \\ -\hspace{6mm}8.1.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ -9. if $oldused > max$ then do \\ -\hspace{3mm}9.1 for $n$ from $max$ to $oldused - 1$ do \\ -\hspace{6mm}9.1.1 $c_n \leftarrow 0$ \\ -10. Clamp excess digits of $c$. (\textit{mp\_clamp}). \\ -11. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{small} -\end{center} -\caption{Algorithm s\_mp\_sub} -\end{figure} - -\textbf{Algorithm s\_mp\_sub.} -This algorithm performs the unsigned subtraction of two mp\_int variables under the restriction that the result must be positive. That is when -passing variables $a$ and $b$ the condition that $\vert a \vert \ge \vert b \vert$ must be met for the algorithm to function correctly. This -algorithm is loosely based on algorithm 14.9 \cite[pp. 595]{HAC} and is similar to algorithm S in \cite[pp. 267]{TAOCPV2} as well. As was the case -of the algorithm s\_mp\_add both other references lack discussion concerning various practical details such as when the inputs differ in magnitude. - -The initial sorting of the inputs is trivial in this algorithm since $a$ is guaranteed to have at least the same magnitude of $b$. Steps 1 and 2 -set the $min$ and $max$ variables. Unlike the addition routine there is guaranteed to be no carry which means that the final result can be at -most $max$ digits in length as opposed to $max + 1$. Similar to the addition algorithm the \textbf{used} count of $c$ is copied locally and -set to the maximal count for the operation. - -The subtraction loop that begins on step seven is essentially the same as the addition loop of algorithm s\_mp\_add except single precision -subtraction is used instead. Note the use of the $\gamma$ variable to extract the carry (\textit{also known as the borrow}) within the subtraction -loops. Under the assumption that two's complement single precision arithmetic is used this will successfully extract the desired carry. - -For example, consider subtracting $0101_2$ from $0100_2$ where $\gamma = 4$ and $\beta = 2$. The least significant bit will force a carry upwards to -the third bit which will be set to zero after the borrow. After the very first bit has been subtracted $4 - 1 \equiv 0011_2$ will remain, When the -third bit of $0101_2$ is subtracted from the result it will cause another carry. In this case though the carry will be forced to propagate all the -way to the most significant bit. - -Recall that $\beta < 2^{\gamma}$. This means that if a carry does occur just before the $lg(\beta)$'th bit it will propagate all the way to the most -significant bit. Thus, the high order bits of the mp\_digit that are not part of the actual digit will either be all zero, or all one. All that -is needed is a single zero or one bit for the carry. Therefore a single logical shift right by $\gamma - 1$ positions is sufficient to extract the -carry. This method of carry extraction may seem awkward but the reason for it becomes apparent when the implementation is discussed. - -If $b$ has a smaller magnitude than $a$ then step 9 will force the carry and copy operation to propagate through the larger input $a$ into $c$. Step -10 will ensure that any leading digits of $c$ above the $max$'th position are zeroed. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_sub.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* low level subtraction (assumes |a| > |b|), HAC pp.595 Algorithm 14.9 */ -018 int -019 s_mp_sub (mp_int * a, mp_int * b, mp_int * c) -020 \{ -021 int olduse, res, min, max; -022 -023 /* find sizes */ -024 min = b->used; -025 max = a->used; -026 -027 /* init result */ -028 if (c->alloc < max) \{ -029 if ((res = mp_grow (c, max)) != MP_OKAY) \{ -030 return res; -031 \} -032 \} -033 olduse = c->used; -034 c->used = max; -035 -036 \{ -037 register mp_digit u, *tmpa, *tmpb, *tmpc; -038 register int i; -039 -040 /* alias for digit pointers */ -041 tmpa = a->dp; -042 tmpb = b->dp; -043 tmpc = c->dp; -044 -045 /* set carry to zero */ -046 u = 0; -047 for (i = 0; i < min; i++) \{ -048 /* T[i] = A[i] - B[i] - U */ -049 *tmpc = *tmpa++ - *tmpb++ - u; -050 -051 /* U = carry bit of T[i] -052 * Note this saves performing an AND operation since -053 * if a carry does occur it will propagate all the way to the -054 * MSB. As a result a single shift is enough to get the carry -055 */ -056 u = *tmpc >> ((mp_digit)(CHAR_BIT * sizeof (mp_digit) - 1)); -057 -058 /* Clear carry from T[i] */ -059 *tmpc++ &= MP_MASK; -060 \} -061 -062 /* now copy higher words if any, e.g. if A has more digits than B */ -063 for (; i < max; i++) \{ -064 /* T[i] = A[i] - U */ -065 *tmpc = *tmpa++ - u; -066 -067 /* U = carry bit of T[i] */ -068 u = *tmpc >> ((mp_digit)(CHAR_BIT * sizeof (mp_digit) - 1)); -069 -070 /* Clear carry from T[i] */ -071 *tmpc++ &= MP_MASK; -072 \} -073 -074 /* clear digits above used (since we may not have grown result above) */ - -075 for (i = c->used; i < olduse; i++) \{ -076 *tmpc++ = 0; -077 \} -078 \} -079 -080 mp_clamp (c); -081 return MP_OKAY; -082 \} -083 -084 #endif -\end{alltt} -\end{small} - -Like low level addition we ``sort'' the inputs. Except in this case the sorting is hardcoded -(lines 24 and 25). In reality the $min$ and $max$ variables are only aliases and are only -used to make the source code easier to read. Again the pointer alias optimization is used -within this algorithm. The aliases $tmpa$, $tmpb$ and $tmpc$ are initialized -(lines 41, 42 and 43) for $a$, $b$ and $c$ respectively. - -The first subtraction loop (lines 46 through 60) subtract digits from both inputs until the smaller of -the two inputs has been exhausted. As remarked earlier there is an implementation reason for using the ``awkward'' -method of extracting the carry (line 56). The traditional method for extracting the carry would be to shift -by $lg(\beta)$ positions and logically AND the least significant bit. The AND operation is required because all of -the bits above the $\lg(\beta)$'th bit will be set to one after a carry occurs from subtraction. This carry -extraction requires two relatively cheap operations to extract the carry. The other method is to simply shift the -most significant bit to the least significant bit thus extracting the carry with a single cheap operation. This -optimization only works on twos compliment machines which is a safe assumption to make. - -If $a$ has a larger magnitude than $b$ an additional loop (lines 63 through 72) is required to propagate -the carry through $a$ and copy the result to $c$. - -\subsection{High Level Addition} -Now that both lower level addition and subtraction algorithms have been established an effective high level signed addition algorithm can be -established. This high level addition algorithm will be what other algorithms and developers will use to perform addition of mp\_int data -types. - -Recall from section 5.2 that an mp\_int represents an integer with an unsigned mantissa (\textit{the array of digits}) and a \textbf{sign} -flag. A high level addition is actually performed as a series of eight separate cases which can be optimized down to three unique cases. - -\begin{figure}[!here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_add}. \\ -\textbf{Input}. Two mp\_ints $a$ and $b$ \\ -\textbf{Output}. The signed addition $c = a + b$. \\ -\hline \\ -1. if $a.sign = b.sign$ then do \\ -\hspace{3mm}1.1 $c.sign \leftarrow a.sign$ \\ -\hspace{3mm}1.2 $c \leftarrow \vert a \vert + \vert b \vert$ (\textit{s\_mp\_add})\\ -2. else do \\ -\hspace{3mm}2.1 if $\vert a \vert < \vert b \vert$ then do (\textit{mp\_cmp\_mag}) \\ -\hspace{6mm}2.1.1 $c.sign \leftarrow b.sign$ \\ -\hspace{6mm}2.1.2 $c \leftarrow \vert b \vert - \vert a \vert$ (\textit{s\_mp\_sub}) \\ -\hspace{3mm}2.2 else do \\ -\hspace{6mm}2.2.1 $c.sign \leftarrow a.sign$ \\ -\hspace{6mm}2.2.2 $c \leftarrow \vert a \vert - \vert b \vert$ \\ -3. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_add} -\end{figure} - -\textbf{Algorithm mp\_add.} -This algorithm performs the signed addition of two mp\_int variables. There is no reference algorithm to draw upon from -either \cite{TAOCPV2} or \cite{HAC} since they both only provide unsigned operations. The algorithm is fairly -straightforward but restricted since subtraction can only produce positive results. - -\begin{figure}[here] -\begin{small} -\begin{center} -\begin{tabular}{|c|c|c|c|c|} -\hline \textbf{Sign of $a$} & \textbf{Sign of $b$} & \textbf{$\vert a \vert > \vert b \vert $} & \textbf{Unsigned Operation} & \textbf{Result Sign Flag} \\ -\hline $+$ & $+$ & Yes & $c = a + b$ & $a.sign$ \\ -\hline $+$ & $+$ & No & $c = a + b$ & $a.sign$ \\ -\hline $-$ & $-$ & Yes & $c = a + b$ & $a.sign$ \\ -\hline $-$ & $-$ & No & $c = a + b$ & $a.sign$ \\ -\hline &&&&\\ - -\hline $+$ & $-$ & No & $c = b - a$ & $b.sign$ \\ -\hline $-$ & $+$ & No & $c = b - a$ & $b.sign$ \\ - -\hline &&&&\\ - -\hline $+$ & $-$ & Yes & $c = a - b$ & $a.sign$ \\ -\hline $-$ & $+$ & Yes & $c = a - b$ & $a.sign$ \\ - -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Addition Guide Chart} -\label{fig:AddChart} -\end{figure} - -Figure~\ref{fig:AddChart} lists all of the eight possible input combinations and is sorted to show that only three -specific cases need to be handled. The return code of the unsigned operations at step 1.2, 2.1.2 and 2.2.2 are -forwarded to step three to check for errors. This simplifies the description of the algorithm considerably and best -follows how the implementation actually was achieved. - -Also note how the \textbf{sign} is set before the unsigned addition or subtraction is performed. Recall from the descriptions of algorithms -s\_mp\_add and s\_mp\_sub that the mp\_clamp function is used at the end to trim excess digits. The mp\_clamp algorithm will set the \textbf{sign} -to \textbf{MP\_ZPOS} when the \textbf{used} digit count reaches zero. - -For example, consider performing $-a + a$ with algorithm mp\_add. By the description of the algorithm the sign is set to \textbf{MP\_NEG} which would -produce a result of $-0$. However, since the sign is set first then the unsigned addition is performed the subsequent usage of algorithm mp\_clamp -within algorithm s\_mp\_add will force $-0$ to become $0$. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_add.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* high level addition (handles signs) */ -018 int mp_add (mp_int * a, mp_int * b, mp_int * c) -019 \{ -020 int sa, sb, res; -021 -022 /* get sign of both inputs */ -023 sa = a->sign; -024 sb = b->sign; -025 -026 /* handle two cases, not four */ -027 if (sa == sb) \{ -028 /* both positive or both negative */ -029 /* add their magnitudes, copy the sign */ -030 c->sign = sa; -031 res = s_mp_add (a, b, c); -032 \} else \{ -033 /* one positive, the other negative */ -034 /* subtract the one with the greater magnitude from */ -035 /* the one of the lesser magnitude. The result gets */ -036 /* the sign of the one with the greater magnitude. */ -037 if (mp_cmp_mag (a, b) == MP_LT) \{ -038 c->sign = sb; -039 res = s_mp_sub (b, a, c); -040 \} else \{ -041 c->sign = sa; -042 res = s_mp_sub (a, b, c); -043 \} -044 \} -045 return res; -046 \} -047 -048 #endif -\end{alltt} -\end{small} - -The source code follows the algorithm fairly closely. The most notable new source code addition is the usage of the $res$ integer variable which -is used to pass result of the unsigned operations forward. Unlike in the algorithm, the variable $res$ is merely returned as is without -explicitly checking it and returning the constant \textbf{MP\_OKAY}. The observation is this algorithm will succeed or fail only if the lower -level functions do so. Returning their return code is sufficient. - -\subsection{High Level Subtraction} -The high level signed subtraction algorithm is essentially the same as the high level signed addition algorithm. - -\newpage\begin{figure}[!here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_sub}. \\ -\textbf{Input}. Two mp\_ints $a$ and $b$ \\ -\textbf{Output}. The signed subtraction $c = a - b$. \\ -\hline \\ -1. if $a.sign \ne b.sign$ then do \\ -\hspace{3mm}1.1 $c.sign \leftarrow a.sign$ \\ -\hspace{3mm}1.2 $c \leftarrow \vert a \vert + \vert b \vert$ (\textit{s\_mp\_add}) \\ -2. else do \\ -\hspace{3mm}2.1 if $\vert a \vert \ge \vert b \vert$ then do (\textit{mp\_cmp\_mag}) \\ -\hspace{6mm}2.1.1 $c.sign \leftarrow a.sign$ \\ -\hspace{6mm}2.1.2 $c \leftarrow \vert a \vert - \vert b \vert$ (\textit{s\_mp\_sub}) \\ -\hspace{3mm}2.2 else do \\ -\hspace{6mm}2.2.1 $c.sign \leftarrow \left \lbrace \begin{array}{ll} - MP\_ZPOS & \mbox{if }a.sign = MP\_NEG \\ - MP\_NEG & \mbox{otherwise} \\ - \end{array} \right .$ \\ -\hspace{6mm}2.2.2 $c \leftarrow \vert b \vert - \vert a \vert$ \\ -3. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_sub} -\end{figure} - -\textbf{Algorithm mp\_sub.} -This algorithm performs the signed subtraction of two inputs. Similar to algorithm mp\_add there is no reference in either \cite{TAOCPV2} or -\cite{HAC}. Also this algorithm is restricted by algorithm s\_mp\_sub. Chart \ref{fig:SubChart} lists the eight possible inputs and -the operations required. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{|c|c|c|c|c|} -\hline \textbf{Sign of $a$} & \textbf{Sign of $b$} & \textbf{$\vert a \vert \ge \vert b \vert $} & \textbf{Unsigned Operation} & \textbf{Result Sign Flag} \\ -\hline $+$ & $-$ & Yes & $c = a + b$ & $a.sign$ \\ -\hline $+$ & $-$ & No & $c = a + b$ & $a.sign$ \\ -\hline $-$ & $+$ & Yes & $c = a + b$ & $a.sign$ \\ -\hline $-$ & $+$ & No & $c = a + b$ & $a.sign$ \\ -\hline &&&& \\ -\hline $+$ & $+$ & Yes & $c = a - b$ & $a.sign$ \\ -\hline $-$ & $-$ & Yes & $c = a - b$ & $a.sign$ \\ -\hline &&&& \\ -\hline $+$ & $+$ & No & $c = b - a$ & $\mbox{opposite of }a.sign$ \\ -\hline $-$ & $-$ & No & $c = b - a$ & $\mbox{opposite of }a.sign$ \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Subtraction Guide Chart} -\label{fig:SubChart} -\end{figure} - -Similar to the case of algorithm mp\_add the \textbf{sign} is set first before the unsigned addition or subtraction. That is to prevent the -algorithm from producing $-a - -a = -0$ as a result. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_sub.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* high level subtraction (handles signs) */ -018 int -019 mp_sub (mp_int * a, mp_int * b, mp_int * c) -020 \{ -021 int sa, sb, res; -022 -023 sa = a->sign; -024 sb = b->sign; -025 -026 if (sa != sb) \{ -027 /* subtract a negative from a positive, OR */ -028 /* subtract a positive from a negative. */ -029 /* In either case, ADD their magnitudes, */ -030 /* and use the sign of the first number. */ -031 c->sign = sa; -032 res = s_mp_add (a, b, c); -033 \} else \{ -034 /* subtract a positive from a positive, OR */ -035 /* subtract a negative from a negative. */ -036 /* First, take the difference between their */ -037 /* magnitudes, then... */ -038 if (mp_cmp_mag (a, b) != MP_LT) \{ -039 /* Copy the sign from the first */ -040 c->sign = sa; -041 /* The first has a larger or equal magnitude */ -042 res = s_mp_sub (a, b, c); -043 \} else \{ -044 /* The result has the *opposite* sign from */ -045 /* the first number. */ -046 c->sign = (sa == MP_ZPOS) ? MP_NEG : MP_ZPOS; -047 /* The second has a larger magnitude */ -048 res = s_mp_sub (b, a, c); -049 \} -050 \} -051 return res; -052 \} -053 -054 #endif -\end{alltt} -\end{small} - -Much like the implementation of algorithm mp\_add the variable $res$ is used to catch the return code of the unsigned addition or subtraction operations -and forward it to the end of the function. On line 38 the ``not equal to'' \textbf{MP\_LT} expression is used to emulate a -``greater than or equal to'' comparison. - -\section{Bit and Digit Shifting} -It is quite common to think of a multiple precision integer as a polynomial in $x$, that is $y = f(\beta)$ where $f(x) = \sum_{i=0}^{n-1} a_i x^i$. -This notation arises within discussion of Montgomery and Diminished Radix Reduction as well as Karatsuba multiplication and squaring. - -In order to facilitate operations on polynomials in $x$ as above a series of simple ``digit'' algorithms have to be established. That is to shift -the digits left or right as well to shift individual bits of the digits left and right. It is important to note that not all ``shift'' operations -are on radix-$\beta$ digits. - -\subsection{Multiplication by Two} - -In a binary system where the radix is a power of two multiplication by two not only arises often in other algorithms it is a fairly efficient -operation to perform. A single precision logical shift left is sufficient to multiply a single digit by two. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_mul\_2}. \\ -\textbf{Input}. One mp\_int $a$ \\ -\textbf{Output}. $b = 2a$. \\ -\hline \\ -1. If $b.alloc < a.used + 1$ then grow $b$ to hold $a.used + 1$ digits. (\textit{mp\_grow}) \\ -2. $oldused \leftarrow b.used$ \\ -3. $b.used \leftarrow a.used$ \\ -4. $r \leftarrow 0$ \\ -5. for $n$ from 0 to $a.used - 1$ do \\ -\hspace{3mm}5.1 $rr \leftarrow a_n >> (lg(\beta) - 1)$ \\ -\hspace{3mm}5.2 $b_n \leftarrow (a_n << 1) + r \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{3mm}5.3 $r \leftarrow rr$ \\ -6. If $r \ne 0$ then do \\ -\hspace{3mm}6.1 $b_{n + 1} \leftarrow r$ \\ -\hspace{3mm}6.2 $b.used \leftarrow b.used + 1$ \\ -7. If $b.used < oldused - 1$ then do \\ -\hspace{3mm}7.1 for $n$ from $b.used$ to $oldused - 1$ do \\ -\hspace{6mm}7.1.1 $b_n \leftarrow 0$ \\ -8. $b.sign \leftarrow a.sign$ \\ -9. Return(\textit{MP\_OKAY}).\\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_mul\_2} -\end{figure} - -\textbf{Algorithm mp\_mul\_2.} -This algorithm will quickly multiply a mp\_int by two provided $\beta$ is a power of two. Neither \cite{TAOCPV2} nor \cite{HAC} describe such -an algorithm despite the fact it arises often in other algorithms. The algorithm is setup much like the lower level algorithm s\_mp\_add since -it is for all intents and purposes equivalent to the operation $b = \vert a \vert + \vert a \vert$. - -Step 1 and 2 grow the input as required to accomodate the maximum number of \textbf{used} digits in the result. The initial \textbf{used} count -is set to $a.used$ at step 4. Only if there is a final carry will the \textbf{used} count require adjustment. - -Step 6 is an optimization implementation of the addition loop for this specific case. That is since the two values being added together -are the same there is no need to perform two reads from the digits of $a$. Step 6.1 performs a single precision shift on the current digit $a_n$ to -obtain what will be the carry for the next iteration. Step 6.2 calculates the $n$'th digit of the result as single precision shift of $a_n$ plus -the previous carry. Recall from section 4.1 that $a_n << 1$ is equivalent to $a_n \cdot 2$. An iteration of the addition loop is finished with -forwarding the carry to the next iteration. - -Step 7 takes care of any final carry by setting the $a.used$'th digit of the result to the carry and augmenting the \textbf{used} count of $b$. -Step 8 clears any leading digits of $b$ in case it originally had a larger magnitude than $a$. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_mul\_2.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* b = a*2 */ -018 int mp_mul_2(mp_int * a, mp_int * b) -019 \{ -020 int x, res, oldused; -021 -022 /* grow to accomodate result */ -023 if (b->alloc < a->used + 1) \{ -024 if ((res = mp_grow (b, a->used + 1)) != MP_OKAY) \{ -025 return res; -026 \} -027 \} -028 -029 oldused = b->used; -030 b->used = a->used; -031 -032 \{ -033 register mp_digit r, rr, *tmpa, *tmpb; -034 -035 /* alias for source */ -036 tmpa = a->dp; -037 -038 /* alias for dest */ -039 tmpb = b->dp; -040 -041 /* carry */ -042 r = 0; -043 for (x = 0; x < a->used; x++) \{ -044 -045 /* get what will be the *next* carry bit from the -046 * MSB of the current digit -047 */ -048 rr = *tmpa >> ((mp_digit)(DIGIT_BIT - 1)); -049 -050 /* now shift up this digit, add in the carry [from the previous] */ -051 *tmpb++ = ((*tmpa++ << ((mp_digit)1)) | r) & MP_MASK; -052 -053 /* copy the carry that would be from the source -054 * digit into the next iteration -055 */ -056 r = rr; -057 \} -058 -059 /* new leading digit? */ -060 if (r != 0) \{ -061 /* add a MSB which is always 1 at this point */ -062 *tmpb = 1; -063 ++(b->used); -064 \} -065 -066 /* now zero any excess digits on the destination -067 * that we didn't write to -068 */ -069 tmpb = b->dp + b->used; -070 for (x = b->used; x < oldused; x++) \{ -071 *tmpb++ = 0; -072 \} -073 \} -074 b->sign = a->sign; -075 return MP_OKAY; -076 \} -077 #endif -\end{alltt} -\end{small} - -This implementation is essentially an optimized implementation of s\_mp\_add for the case of doubling an input. The only noteworthy difference -is the use of the logical shift operator on line 51 to perform a single precision doubling. - -\subsection{Division by Two} -A division by two can just as easily be accomplished with a logical shift right as multiplication by two can be with a logical shift left. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_div\_2}. \\ -\textbf{Input}. One mp\_int $a$ \\ -\textbf{Output}. $b = a/2$. \\ -\hline \\ -1. If $b.alloc < a.used$ then grow $b$ to hold $a.used$ digits. (\textit{mp\_grow}) \\ -2. If the reallocation failed return(\textit{MP\_MEM}). \\ -3. $oldused \leftarrow b.used$ \\ -4. $b.used \leftarrow a.used$ \\ -5. $r \leftarrow 0$ \\ -6. for $n$ from $b.used - 1$ to $0$ do \\ -\hspace{3mm}6.1 $rr \leftarrow a_n \mbox{ (mod }2\mbox{)}$\\ -\hspace{3mm}6.2 $b_n \leftarrow (a_n >> 1) + (r << (lg(\beta) - 1)) \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{3mm}6.3 $r \leftarrow rr$ \\ -7. If $b.used < oldused - 1$ then do \\ -\hspace{3mm}7.1 for $n$ from $b.used$ to $oldused - 1$ do \\ -\hspace{6mm}7.1.1 $b_n \leftarrow 0$ \\ -8. $b.sign \leftarrow a.sign$ \\ -9. Clamp excess digits of $b$. (\textit{mp\_clamp}) \\ -10. Return(\textit{MP\_OKAY}).\\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_div\_2} -\end{figure} - -\textbf{Algorithm mp\_div\_2.} -This algorithm will divide an mp\_int by two using logical shifts to the right. Like mp\_mul\_2 it uses a modified low level addition -core as the basis of the algorithm. Unlike mp\_mul\_2 the shift operations work from the leading digit to the trailing digit. The algorithm -could be written to work from the trailing digit to the leading digit however, it would have to stop one short of $a.used - 1$ digits to prevent -reading past the end of the array of digits. - -Essentially the loop at step 6 is similar to that of mp\_mul\_2 except the logical shifts go in the opposite direction and the carry is at the -least significant bit not the most significant bit. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_div\_2.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* b = a/2 */ -018 int mp_div_2(mp_int * a, mp_int * b) -019 \{ -020 int x, res, oldused; -021 -022 /* copy */ -023 if (b->alloc < a->used) \{ -024 if ((res = mp_grow (b, a->used)) != MP_OKAY) \{ -025 return res; -026 \} -027 \} -028 -029 oldused = b->used; -030 b->used = a->used; -031 \{ -032 register mp_digit r, rr, *tmpa, *tmpb; -033 -034 /* source alias */ -035 tmpa = a->dp + b->used - 1; -036 -037 /* dest alias */ -038 tmpb = b->dp + b->used - 1; -039 -040 /* carry */ -041 r = 0; -042 for (x = b->used - 1; x >= 0; x--) \{ -043 /* get the carry for the next iteration */ -044 rr = *tmpa & 1; -045 -046 /* shift the current digit, add in carry and store */ -047 *tmpb-- = (*tmpa-- >> 1) | (r << (DIGIT_BIT - 1)); -048 -049 /* forward carry to next iteration */ -050 r = rr; -051 \} -052 -053 /* zero excess digits */ -054 tmpb = b->dp + b->used; -055 for (x = b->used; x < oldused; x++) \{ -056 *tmpb++ = 0; -057 \} -058 \} -059 b->sign = a->sign; -060 mp_clamp (b); -061 return MP_OKAY; -062 \} -063 #endif -\end{alltt} -\end{small} - -\section{Polynomial Basis Operations} -Recall from section 4.3 that any integer can be represented as a polynomial in $x$ as $y = f(\beta)$. Such a representation is also known as -the polynomial basis \cite[pp. 48]{ROSE}. Given such a notation a multiplication or division by $x$ amounts to shifting whole digits a single -place. The need for such operations arises in several other higher level algorithms such as Barrett and Montgomery reduction, integer -division and Karatsuba multiplication. - -Converting from an array of digits to polynomial basis is very simple. Consider the integer $y \equiv (a_2, a_1, a_0)_{\beta}$ and recall that -$y = \sum_{i=0}^{2} a_i \beta^i$. Simply replace $\beta$ with $x$ and the expression is in polynomial basis. For example, $f(x) = 8x + 9$ is the -polynomial basis representation for $89$ using radix ten. That is, $f(10) = 8(10) + 9 = 89$. - -\subsection{Multiplication by $x$} - -Given a polynomial in $x$ such as $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0$ multiplying by $x$ amounts to shifting the coefficients up one -degree. In this case $f(x) \cdot x = a_n x^{n+1} + a_{n-1} x^n + ... + a_0 x$. From a scalar basis point of view multiplying by $x$ is equivalent to -multiplying by the integer $\beta$. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_lshd}. \\ -\textbf{Input}. One mp\_int $a$ and an integer $b$ \\ -\textbf{Output}. $a \leftarrow a \cdot \beta^b$ (equivalent to multiplication by $x^b$). \\ -\hline \\ -1. If $b \le 0$ then return(\textit{MP\_OKAY}). \\ -2. If $a.alloc < a.used + b$ then grow $a$ to at least $a.used + b$ digits. (\textit{mp\_grow}). \\ -3. If the reallocation failed return(\textit{MP\_MEM}). \\ -4. $a.used \leftarrow a.used + b$ \\ -5. $i \leftarrow a.used - 1$ \\ -6. $j \leftarrow a.used - 1 - b$ \\ -7. for $n$ from $a.used - 1$ to $b$ do \\ -\hspace{3mm}7.1 $a_{i} \leftarrow a_{j}$ \\ -\hspace{3mm}7.2 $i \leftarrow i - 1$ \\ -\hspace{3mm}7.3 $j \leftarrow j - 1$ \\ -8. for $n$ from 0 to $b - 1$ do \\ -\hspace{3mm}8.1 $a_n \leftarrow 0$ \\ -9. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_lshd} -\end{figure} - -\textbf{Algorithm mp\_lshd.} -This algorithm multiplies an mp\_int by the $b$'th power of $x$. This is equivalent to multiplying by $\beta^b$. The algorithm differs -from the other algorithms presented so far as it performs the operation in place instead storing the result in a separate location. The -motivation behind this change is due to the way this function is typically used. Algorithms such as mp\_add store the result in an optionally -different third mp\_int because the original inputs are often still required. Algorithm mp\_lshd (\textit{and similarly algorithm mp\_rshd}) is -typically used on values where the original value is no longer required. The algorithm will return success immediately if -$b \le 0$ since the rest of algorithm is only valid when $b > 0$. - -First the destination $a$ is grown as required to accomodate the result. The counters $i$ and $j$ are used to form a \textit{sliding window} over -the digits of $a$ of length $b$. The head of the sliding window is at $i$ (\textit{the leading digit}) and the tail at $j$ (\textit{the trailing digit}). -The loop on step 7 copies the digit from the tail to the head. In each iteration the window is moved down one digit. The last loop on -step 8 sets the lower $b$ digits to zero. - -\newpage -\begin{center} -\begin{figure}[here] -\includegraphics{pics/sliding_window.ps} -\caption{Sliding Window Movement} -\label{pic:sliding_window} -\end{figure} -\end{center} - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_lshd.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* shift left a certain amount of digits */ -018 int mp_lshd (mp_int * a, int b) -019 \{ -020 int x, res; -021 -022 /* if its less than zero return */ -023 if (b <= 0) \{ -024 return MP_OKAY; -025 \} -026 -027 /* grow to fit the new digits */ -028 if (a->alloc < a->used + b) \{ -029 if ((res = mp_grow (a, a->used + b)) != MP_OKAY) \{ -030 return res; -031 \} -032 \} -033 -034 \{ -035 register mp_digit *top, *bottom; -036 -037 /* increment the used by the shift amount then copy upwards */ -038 a->used += b; -039 -040 /* top */ -041 top = a->dp + a->used - 1; -042 -043 /* base */ -044 bottom = a->dp + a->used - 1 - b; -045 -046 /* much like mp_rshd this is implemented using a sliding window -047 * except the window goes the otherway around. Copying from -048 * the bottom to the top. see bn_mp_rshd.c for more info. -049 */ -050 for (x = a->used - 1; x >= b; x--) \{ -051 *top-- = *bottom--; -052 \} -053 -054 /* zero the lower digits */ -055 top = a->dp; -056 for (x = 0; x < b; x++) \{ -057 *top++ = 0; -058 \} -059 \} -060 return MP_OKAY; -061 \} -062 #endif -\end{alltt} -\end{small} - -The if statement (line 23) ensures that the $b$ variable is greater than zero since we do not interpret negative -shift counts properly. The \textbf{used} count is incremented by $b$ before the copy loop begins. This elminates -the need for an additional variable in the for loop. The variable $top$ (line 41) is an alias -for the leading digit while $bottom$ (line 44) is an alias for the trailing edge. The aliases form a -window of exactly $b$ digits over the input. - -\subsection{Division by $x$} - -Division by powers of $x$ is easily achieved by shifting the digits right and removing any that will end up to the right of the zero'th digit. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_rshd}. \\ -\textbf{Input}. One mp\_int $a$ and an integer $b$ \\ -\textbf{Output}. $a \leftarrow a / \beta^b$ (Divide by $x^b$). \\ -\hline \\ -1. If $b \le 0$ then return. \\ -2. If $a.used \le b$ then do \\ -\hspace{3mm}2.1 Zero $a$. (\textit{mp\_zero}). \\ -\hspace{3mm}2.2 Return. \\ -3. $i \leftarrow 0$ \\ -4. $j \leftarrow b$ \\ -5. for $n$ from 0 to $a.used - b - 1$ do \\ -\hspace{3mm}5.1 $a_i \leftarrow a_j$ \\ -\hspace{3mm}5.2 $i \leftarrow i + 1$ \\ -\hspace{3mm}5.3 $j \leftarrow j + 1$ \\ -6. for $n$ from $a.used - b$ to $a.used - 1$ do \\ -\hspace{3mm}6.1 $a_n \leftarrow 0$ \\ -7. $a.used \leftarrow a.used - b$ \\ -8. Return. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_rshd} -\end{figure} - -\textbf{Algorithm mp\_rshd.} -This algorithm divides the input in place by the $b$'th power of $x$. It is analogous to dividing by a $\beta^b$ but much quicker since -it does not require single precision division. This algorithm does not actually return an error code as it cannot fail. - -If the input $b$ is less than one the algorithm quickly returns without performing any work. If the \textbf{used} count is less than or equal -to the shift count $b$ then it will simply zero the input and return. - -After the trivial cases of inputs have been handled the sliding window is setup. Much like the case of algorithm mp\_lshd a sliding window that -is $b$ digits wide is used to copy the digits. Unlike mp\_lshd the window slides in the opposite direction from the trailing to the leading digit. -Also the digits are copied from the leading to the trailing edge. - -Once the window copy is complete the upper digits must be zeroed and the \textbf{used} count decremented. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_rshd.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* shift right a certain amount of digits */ -018 void mp_rshd (mp_int * a, int b) -019 \{ -020 int x; -021 -022 /* if b <= 0 then ignore it */ -023 if (b <= 0) \{ -024 return; -025 \} -026 -027 /* if b > used then simply zero it and return */ -028 if (a->used <= b) \{ -029 mp_zero (a); -030 return; -031 \} -032 -033 \{ -034 register mp_digit *bottom, *top; -035 -036 /* shift the digits down */ -037 -038 /* bottom */ -039 bottom = a->dp; -040 -041 /* top [offset into digits] */ -042 top = a->dp + b; -043 -044 /* this is implemented as a sliding window where -045 * the window is b-digits long and digits from -046 * the top of the window are copied to the bottom -047 * -048 * e.g. -049 -050 b-2 | b-1 | b0 | b1 | b2 | ... | bb | ----> -051 /\symbol{92} | ----> -052 \symbol{92}-------------------/ ----> -053 */ -054 for (x = 0; x < (a->used - b); x++) \{ -055 *bottom++ = *top++; -056 \} -057 -058 /* zero the top digits */ -059 for (; x < a->used; x++) \{ -060 *bottom++ = 0; -061 \} -062 \} -063 -064 /* remove excess digits */ -065 a->used -= b; -066 \} -067 #endif -\end{alltt} -\end{small} - -The only noteworthy element of this routine is the lack of a return type since it cannot fail. Like mp\_lshd() we -form a sliding window except we copy in the other direction. After the window (line 59) we then zero -the upper digits of the input to make sure the result is correct. - -\section{Powers of Two} - -Now that algorithms for moving single bits as well as whole digits exist algorithms for moving the ``in between'' distances are required. For -example, to quickly multiply by $2^k$ for any $k$ without using a full multiplier algorithm would prove useful. Instead of performing single -shifts $k$ times to achieve a multiplication by $2^{\pm k}$ a mixture of whole digit shifting and partial digit shifting is employed. - -\subsection{Multiplication by Power of Two} - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_mul\_2d}. \\ -\textbf{Input}. One mp\_int $a$ and an integer $b$ \\ -\textbf{Output}. $c \leftarrow a \cdot 2^b$. \\ -\hline \\ -1. $c \leftarrow a$. (\textit{mp\_copy}) \\ -2. If $c.alloc < c.used + \lfloor b / lg(\beta) \rfloor + 2$ then grow $c$ accordingly. \\ -3. If the reallocation failed return(\textit{MP\_MEM}). \\ -4. If $b \ge lg(\beta)$ then \\ -\hspace{3mm}4.1 $c \leftarrow c \cdot \beta^{\lfloor b / lg(\beta) \rfloor}$ (\textit{mp\_lshd}). \\ -\hspace{3mm}4.2 If step 4.1 failed return(\textit{MP\_MEM}). \\ -5. $d \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\ -6. If $d \ne 0$ then do \\ -\hspace{3mm}6.1 $mask \leftarrow 2^d$ \\ -\hspace{3mm}6.2 $r \leftarrow 0$ \\ -\hspace{3mm}6.3 for $n$ from $0$ to $c.used - 1$ do \\ -\hspace{6mm}6.3.1 $rr \leftarrow c_n >> (lg(\beta) - d) \mbox{ (mod }mask\mbox{)}$ \\ -\hspace{6mm}6.3.2 $c_n \leftarrow (c_n << d) + r \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{6mm}6.3.3 $r \leftarrow rr$ \\ -\hspace{3mm}6.4 If $r > 0$ then do \\ -\hspace{6mm}6.4.1 $c_{c.used} \leftarrow r$ \\ -\hspace{6mm}6.4.2 $c.used \leftarrow c.used + 1$ \\ -7. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_mul\_2d} -\end{figure} - -\textbf{Algorithm mp\_mul\_2d.} -This algorithm multiplies $a$ by $2^b$ and stores the result in $c$. The algorithm uses algorithm mp\_lshd and a derivative of algorithm mp\_mul\_2 to -quickly compute the product. - -First the algorithm will multiply $a$ by $x^{\lfloor b / lg(\beta) \rfloor}$ which will ensure that the remainder multiplicand is less than -$\beta$. For example, if $b = 37$ and $\beta = 2^{28}$ then this step will multiply by $x$ leaving a multiplication by $2^{37 - 28} = 2^{9}$ -left. - -After the digits have been shifted appropriately at most $lg(\beta) - 1$ shifts are left to perform. Step 5 calculates the number of remaining shifts -required. If it is non-zero a modified shift loop is used to calculate the remaining product. -Essentially the loop is a generic version of 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. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_mul\_2d.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* shift left by a certain bit count */ -018 int mp_mul_2d (mp_int * a, int b, mp_int * c) -019 \{ -020 mp_digit d; -021 int res; -022 -023 /* copy */ -024 if (a != c) \{ -025 if ((res = mp_copy (a, c)) != MP_OKAY) \{ -026 return res; -027 \} -028 \} -029 -030 if (c->alloc < (int)(c->used + b/DIGIT_BIT + 1)) \{ -031 if ((res = mp_grow (c, c->used + b / DIGIT_BIT + 1)) != MP_OKAY) \{ -032 return res; -033 \} -034 \} -035 -036 /* shift by as many digits in the bit count */ -037 if (b >= (int)DIGIT_BIT) \{ -038 if ((res = mp_lshd (c, b / DIGIT_BIT)) != MP_OKAY) \{ -039 return res; -040 \} -041 \} -042 -043 /* shift any bit count < DIGIT_BIT */ -044 d = (mp_digit) (b % DIGIT_BIT); -045 if (d != 0) \{ -046 register mp_digit *tmpc, shift, mask, r, rr; -047 register int x; -048 -049 /* bitmask for carries */ -050 mask = (((mp_digit)1) << d) - 1; -051 -052 /* shift for msbs */ -053 shift = DIGIT_BIT - d; -054 -055 /* alias */ -056 tmpc = c->dp; -057 -058 /* carry */ -059 r = 0; -060 for (x = 0; x < c->used; x++) \{ -061 /* get the higher bits of the current word */ -062 rr = (*tmpc >> shift) & mask; -063 -064 /* shift the current word and OR in the carry */ -065 *tmpc = ((*tmpc << d) | r) & MP_MASK; -066 ++tmpc; -067 -068 /* set the carry to the carry bits of the current word */ -069 r = rr; -070 \} -071 -072 /* set final carry */ -073 if (r != 0) \{ -074 c->dp[(c->used)++] = r; -075 \} -076 \} -077 mp_clamp (c); -078 return MP_OKAY; -079 \} -080 #endif -\end{alltt} -\end{small} - -The shifting is performed in--place which means the first step (line 24) is to copy the input to the -destination. We avoid calling mp\_copy() by making sure the mp\_ints are different. The destination then -has to be grown (line 31) to accomodate the result. - -If the shift count $b$ is larger than $lg(\beta)$ then a call to mp\_lshd() is used to handle all of the multiples -of $lg(\beta)$. Leaving only a remaining shift of $lg(\beta) - 1$ or fewer bits left. Inside the actual shift -loop (lines 45 to 76) we make use of pre--computed values $shift$ and $mask$. These are used to -extract the carry bit(s) to pass into the next iteration of the loop. The $r$ and $rr$ variables form a -chain between consecutive iterations to propagate the carry. - -\subsection{Division by Power of Two} - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_div\_2d}. \\ -\textbf{Input}. One mp\_int $a$ and an integer $b$ \\ -\textbf{Output}. $c \leftarrow \lfloor a / 2^b \rfloor, d \leftarrow a \mbox{ (mod }2^b\mbox{)}$. \\ -\hline \\ -1. If $b \le 0$ then do \\ -\hspace{3mm}1.1 $c \leftarrow a$ (\textit{mp\_copy}) \\ -\hspace{3mm}1.2 $d \leftarrow 0$ (\textit{mp\_zero}) \\ -\hspace{3mm}1.3 Return(\textit{MP\_OKAY}). \\ -2. $c \leftarrow a$ \\ -3. $d \leftarrow a \mbox{ (mod }2^b\mbox{)}$ (\textit{mp\_mod\_2d}) \\ -4. If $b \ge lg(\beta)$ then do \\ -\hspace{3mm}4.1 $c \leftarrow \lfloor c/\beta^{\lfloor b/lg(\beta) \rfloor} \rfloor$ (\textit{mp\_rshd}). \\ -5. $k \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\ -6. If $k \ne 0$ then do \\ -\hspace{3mm}6.1 $mask \leftarrow 2^k$ \\ -\hspace{3mm}6.2 $r \leftarrow 0$ \\ -\hspace{3mm}6.3 for $n$ from $c.used - 1$ to $0$ do \\ -\hspace{6mm}6.3.1 $rr \leftarrow c_n \mbox{ (mod }mask\mbox{)}$ \\ -\hspace{6mm}6.3.2 $c_n \leftarrow (c_n >> k) + (r << (lg(\beta) - k))$ \\ -\hspace{6mm}6.3.3 $r \leftarrow rr$ \\ -7. Clamp excess digits of $c$. (\textit{mp\_clamp}) \\ -8. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_div\_2d} -\end{figure} - -\textbf{Algorithm mp\_div\_2d.} -This algorithm will divide an input $a$ by $2^b$ and produce the quotient and remainder. The algorithm is designed much like algorithm -mp\_mul\_2d by first using whole digit shifts then single precision shifts. This algorithm will also produce the remainder of the division -by using algorithm mp\_mod\_2d. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_div\_2d.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* shift right by a certain bit count (store quotient in c, optional remaind - er in d) */ -018 int mp_div_2d (mp_int * a, int b, mp_int * c, mp_int * d) -019 \{ -020 mp_digit D, r, rr; -021 int x, res; -022 mp_int t; -023 -024 -025 /* if the shift count is <= 0 then we do no work */ -026 if (b <= 0) \{ -027 res = mp_copy (a, c); -028 if (d != NULL) \{ -029 mp_zero (d); -030 \} -031 return res; -032 \} -033 -034 if ((res = mp_init (&t)) != MP_OKAY) \{ -035 return res; -036 \} -037 -038 /* get the remainder */ -039 if (d != NULL) \{ -040 if ((res = mp_mod_2d (a, b, &t)) != MP_OKAY) \{ -041 mp_clear (&t); -042 return res; -043 \} -044 \} -045 -046 /* copy */ -047 if ((res = mp_copy (a, c)) != MP_OKAY) \{ -048 mp_clear (&t); -049 return res; -050 \} -051 -052 /* shift by as many digits in the bit count */ -053 if (b >= (int)DIGIT_BIT) \{ -054 mp_rshd (c, b / DIGIT_BIT); -055 \} -056 -057 /* shift any bit count < DIGIT_BIT */ -058 D = (mp_digit) (b % DIGIT_BIT); -059 if (D != 0) \{ -060 register mp_digit *tmpc, mask, shift; -061 -062 /* mask */ -063 mask = (((mp_digit)1) << D) - 1; -064 -065 /* shift for lsb */ -066 shift = DIGIT_BIT - D; -067 -068 /* alias */ -069 tmpc = c->dp + (c->used - 1); -070 -071 /* carry */ -072 r = 0; -073 for (x = c->used - 1; x >= 0; x--) \{ -074 /* get the lower bits of this word in a temp */ -075 rr = *tmpc & mask; -076 -077 /* shift the current word and mix in the carry bits from the previous - word */ -078 *tmpc = (*tmpc >> D) | (r << shift); -079 --tmpc; -080 -081 /* set the carry to the carry bits of the current word found above */ -082 r = rr; -083 \} -084 \} -085 mp_clamp (c); -086 if (d != NULL) \{ -087 mp_exch (&t, d); -088 \} -089 mp_clear (&t); -090 return MP_OKAY; -091 \} -092 #endif -\end{alltt} -\end{small} - -The implementation of algorithm mp\_div\_2d is slightly different than the algorithm specifies. The remainder $d$ may be optionally -ignored by passing \textbf{NULL} as the pointer to the mp\_int variable. The temporary mp\_int variable $t$ is used to hold the -result of the remainder operation until the end. This allows $d$ and $a$ to represent the same mp\_int without modifying $a$ before -the quotient is obtained. - -The remainder of the source code is essentially the same as the source code for mp\_mul\_2d. The only significant difference is -the direction of the shifts. - -\subsection{Remainder of Division by Power of Two} - -The last algorithm in the series of polynomial basis power of two algorithms is calculating the remainder of division by $2^b$. This -algorithm benefits from the fact that in twos complement arithmetic $a \mbox{ (mod }2^b\mbox{)}$ is the same as $a$ AND $2^b - 1$. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_mod\_2d}. \\ -\textbf{Input}. One mp\_int $a$ and an integer $b$ \\ -\textbf{Output}. $c \leftarrow a \mbox{ (mod }2^b\mbox{)}$. \\ -\hline \\ -1. If $b \le 0$ then do \\ -\hspace{3mm}1.1 $c \leftarrow 0$ (\textit{mp\_zero}) \\ -\hspace{3mm}1.2 Return(\textit{MP\_OKAY}). \\ -2. If $b > a.used \cdot lg(\beta)$ then do \\ -\hspace{3mm}2.1 $c \leftarrow a$ (\textit{mp\_copy}) \\ -\hspace{3mm}2.2 Return the result of step 2.1. \\ -3. $c \leftarrow a$ \\ -4. If step 3 failed return(\textit{MP\_MEM}). \\ -5. for $n$ from $\lceil b / lg(\beta) \rceil$ to $c.used$ do \\ -\hspace{3mm}5.1 $c_n \leftarrow 0$ \\ -6. $k \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\ -7. $c_{\lfloor b / lg(\beta) \rfloor} \leftarrow c_{\lfloor b / lg(\beta) \rfloor} \mbox{ (mod }2^{k}\mbox{)}$. \\ -8. Clamp excess digits of $c$. (\textit{mp\_clamp}) \\ -9. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_mod\_2d} -\end{figure} - -\textbf{Algorithm mp\_mod\_2d.} -This algorithm will quickly calculate the value of $a \mbox{ (mod }2^b\mbox{)}$. First if $b$ is less than or equal to zero the -result is set to zero. If $b$ is greater than the number of bits in $a$ then it simply copies $a$ to $c$ and returns. Otherwise, $a$ -is copied to $b$, leading digits are removed and the remaining leading digit is trimed to the exact bit count. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_mod\_2d.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* calc a value mod 2**b */ -018 int -019 mp_mod_2d (mp_int * a, int b, mp_int * c) -020 \{ -021 int x, res; -022 -023 /* if b is <= 0 then zero the int */ -024 if (b <= 0) \{ -025 mp_zero (c); -026 return MP_OKAY; -027 \} -028 -029 /* if the modulus is larger than the value than return */ -030 if (b >= (int) (a->used * DIGIT_BIT)) \{ -031 res = mp_copy (a, c); -032 return res; -033 \} -034 -035 /* copy */ -036 if ((res = mp_copy (a, c)) != MP_OKAY) \{ -037 return res; -038 \} -039 -040 /* zero digits above the last digit of the modulus */ -041 for (x = (b / DIGIT_BIT) + ((b % DIGIT_BIT) == 0 ? 0 : 1); x < c->used; x+ - +) \{ -042 c->dp[x] = 0; -043 \} -044 /* clear the digit that is not completely outside/inside the modulus */ -045 c->dp[b / DIGIT_BIT] &= -046 (mp_digit) ((((mp_digit) 1) << (((mp_digit) b) % DIGIT_BIT)) - ((mp_digi - t) 1)); -047 mp_clamp (c); -048 return MP_OKAY; -049 \} -050 #endif -\end{alltt} -\end{small} - -We first avoid cases of $b \le 0$ by simply mp\_zero()'ing the destination in such cases. Next if $2^b$ is larger -than the input we just mp\_copy() the input and return right away. After this point we know we must actually -perform some work to produce the remainder. - -Recalling that reducing modulo $2^k$ and a binary ``and'' with $2^k - 1$ are numerically equivalent we can quickly reduce -the number. First we zero any digits above the last digit in $2^b$ (line 41). Next we reduce the -leading digit of both (line 45) and then mp\_clamp(). - -\section*{Exercises} -\begin{tabular}{cl} -$\left [ 3 \right ] $ & Devise an algorithm that performs $a \cdot 2^b$ for generic values of $b$ \\ - & in $O(n)$ time. \\ - &\\ -$\left [ 3 \right ] $ & Devise an efficient algorithm to multiply by small low hamming \\ - & weight values such as $3$, $5$ and $9$. Extend it to handle all values \\ - & upto $64$ with a hamming weight less than three. \\ - &\\ -$\left [ 2 \right ] $ & Modify the preceding algorithm to handle values of the form \\ - & $2^k - 1$ as well. \\ - &\\ -$\left [ 3 \right ] $ & Using only algorithms mp\_mul\_2, mp\_div\_2 and mp\_add create an \\ - & algorithm to multiply two integers in roughly $O(2n^2)$ time for \\ - & any $n$-bit input. Note that the time of addition is ignored in the \\ - & calculation. \\ - & \\ -$\left [ 5 \right ] $ & Improve the previous algorithm to have a working time of at most \\ - & $O \left (2^{(k-1)}n + \left ({2n^2 \over k} \right ) \right )$ for an appropriate choice of $k$. Again ignore \\ - & the cost of addition. \\ - & \\ -$\left [ 2 \right ] $ & Devise a chart to find optimal values of $k$ for the previous problem \\ - & for $n = 64 \ldots 1024$ in steps of $64$. \\ - & \\ -$\left [ 2 \right ] $ & Using only algorithms mp\_abs and mp\_sub devise another method for \\ - & calculating the result of a signed comparison. \\ - & -\end{tabular} - -\chapter{Multiplication and Squaring} -\section{The Multipliers} -For most number theoretic problems including certain public key cryptographic algorithms, the ``multipliers'' form the most important subset of -algorithms of any multiple precision integer package. The set of multiplier algorithms include integer multiplication, squaring and modular reduction -where in each of the algorithms single precision multiplication is the dominant operation performed. This chapter will discuss integer multiplication -and squaring, leaving modular reductions for the subsequent chapter. - -The importance of the multiplier algorithms is for the most part driven by the fact that certain popular public key algorithms are based on modular -exponentiation, that is computing $d \equiv a^b \mbox{ (mod }c\mbox{)}$ for some arbitrary choice of $a$, $b$, $c$ and $d$. During a modular -exponentiation the majority\footnote{Roughly speaking a modular exponentiation will spend about 40\% of the time performing modular reductions, -35\% of the time performing squaring and 25\% of the time performing multiplications.} of the processor time is spent performing single precision -multiplications. - -For centuries general purpose multiplication has required a lengthly $O(n^2)$ process, whereby each digit of one multiplicand has to be multiplied -against every digit of the other multiplicand. Traditional long-hand multiplication is based on this process; while the techniques can differ the -overall algorithm used is essentially the same. Only ``recently'' have faster algorithms been studied. First Karatsuba multiplication was discovered in -1962. This algorithm can multiply two numbers with considerably fewer single precision multiplications when compared to the long-hand approach. -This technique led to the discovery of polynomial basis algorithms (\textit{good reference?}) and subquently Fourier Transform based solutions. - -\section{Multiplication} -\subsection{The Baseline Multiplication} -\label{sec:basemult} -\index{baseline multiplication} -Computing the product of two integers in software can be achieved using a trivial adaptation of the standard $O(n^2)$ long-hand multiplication -algorithm that school children are taught. The algorithm is considered an $O(n^2)$ algorithm since for two $n$-digit inputs $n^2$ single precision -multiplications are required. More specifically for a $m$ and $n$ digit input $m \cdot n$ single precision multiplications are required. To -simplify most discussions, it will be assumed that the inputs have comparable number of digits. - -The ``baseline multiplication'' algorithm is designed to act as the ``catch-all'' algorithm, only to be used when the faster algorithms cannot be -used. This algorithm does not use any particularly interesting optimizations and should ideally be avoided if possible. One important -facet of this algorithm, is that it has been modified to only produce a certain amount of output digits as resolution. The importance of this -modification will become evident during the discussion of Barrett modular reduction. Recall that for a $n$ and $m$ digit input the product -will be at most $n + m$ digits. Therefore, this algorithm can be reduced to a full multiplier by having it produce $n + m$ digits of the product. - -Recall from sub-section 4.2.2 the definition of $\gamma$ as the number of bits in the type \textbf{mp\_digit}. We shall now extend the variable set to -include $\alpha$ which shall represent the number of bits in the type \textbf{mp\_word}. This implies that $2^{\alpha} > 2 \cdot \beta^2$. The -constant $\delta = 2^{\alpha - 2lg(\beta)}$ will represent the maximal weight of any column in a product (\textit{see sub-section 5.2.2 for more information}). - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{s\_mp\_mul\_digs}. \\ -\textbf{Input}. mp\_int $a$, mp\_int $b$ and an integer $digs$ \\ -\textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert \mbox{ (mod }\beta^{digs}\mbox{)}$. \\ -\hline \\ -1. If min$(a.used, b.used) < \delta$ then do \\ -\hspace{3mm}1.1 Calculate $c = \vert a \vert \cdot \vert b \vert$ by the Comba method (\textit{see algorithm~\ref{fig:COMBAMULT}}). \\ -\hspace{3mm}1.2 Return the result of step 1.1 \\ -\\ -Allocate and initialize a temporary mp\_int. \\ -2. Init $t$ to be of size $digs$ \\ -3. If step 2 failed return(\textit{MP\_MEM}). \\ -4. $t.used \leftarrow digs$ \\ -\\ -Compute the product. \\ -5. for $ix$ from $0$ to $a.used - 1$ do \\ -\hspace{3mm}5.1 $u \leftarrow 0$ \\ -\hspace{3mm}5.2 $pb \leftarrow \mbox{min}(b.used, digs - ix)$ \\ -\hspace{3mm}5.3 If $pb < 1$ then goto step 6. \\ -\hspace{3mm}5.4 for $iy$ from $0$ to $pb - 1$ do \\ -\hspace{6mm}5.4.1 $\hat r \leftarrow t_{iy + ix} + a_{ix} \cdot b_{iy} + u$ \\ -\hspace{6mm}5.4.2 $t_{iy + ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{6mm}5.4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ -\hspace{3mm}5.5 if $ix + pb < digs$ then do \\ -\hspace{6mm}5.5.1 $t_{ix + pb} \leftarrow u$ \\ -6. Clamp excess digits of $t$. \\ -7. Swap $c$ with $t$ \\ -8. Clear $t$ \\ -9. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm s\_mp\_mul\_digs} -\end{figure} - -\textbf{Algorithm s\_mp\_mul\_digs.} -This algorithm computes the unsigned product of two inputs $a$ and $b$, limited to an output precision of $digs$ digits. While it may seem -a bit awkward to modify the function from its simple $O(n^2)$ description, the usefulness of partial multipliers will arise in a subsequent -algorithm. The algorithm is loosely based on algorithm 14.12 from \cite[pp. 595]{HAC} and is similar to Algorithm M of Knuth \cite[pp. 268]{TAOCPV2}. -Algorithm s\_mp\_mul\_digs differs from these cited references since it can produce a variable output precision regardless of the precision of the -inputs. - -The first thing this algorithm checks for is whether a Comba multiplier can be used instead. If the minimum digit count of either -input is less than $\delta$, then the Comba method may be used instead. After the Comba method is ruled out, the baseline algorithm begins. A -temporary mp\_int variable $t$ is used to hold the intermediate result of the product. This allows the algorithm to be used to -compute products when either $a = c$ or $b = c$ without overwriting the inputs. - -All of step 5 is the infamous $O(n^2)$ multiplication loop slightly modified to only produce upto $digs$ digits of output. The $pb$ variable -is given the count of digits to read from $b$ inside the nested loop. If $pb \le 1$ then no more output digits can be produced and the algorithm -will exit the loop. The best way to think of the loops are as a series of $pb \times 1$ multiplications. That is, in each pass of the -innermost loop $a_{ix}$ is multiplied against $b$ and the result is added (\textit{with an appropriate shift}) to $t$. - -For example, consider multiplying $576$ by $241$. That is equivalent to computing $10^0(1)(576) + 10^1(4)(576) + 10^2(2)(576)$ which is best -visualized in the following table. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{|c|c|c|c|c|c|l|} -\hline && & 5 & 7 & 6 & \\ -\hline $\times$&& & 2 & 4 & 1 & \\ -\hline &&&&&&\\ - && & 5 & 7 & 6 & $10^0(1)(576)$ \\ - &2 & 3 & 6 & 1 & 6 & $10^1(4)(576) + 10^0(1)(576)$ \\ - 1 & 3 & 8 & 8 & 1 & 6 & $10^2(2)(576) + 10^1(4)(576) + 10^0(1)(576)$ \\ -\hline -\end{tabular} -\end{center} -\caption{Long-Hand Multiplication Diagram} -\end{figure} - -Each row of the product is added to the result after being shifted to the left (\textit{multiplied by a power of the radix}) by the appropriate -count. That is in pass $ix$ of the inner loop the product is added starting at the $ix$'th digit of the reult. - -Step 5.4.1 introduces the hat symbol (\textit{e.g. $\hat r$}) which represents a double precision variable. The multiplication on that step -is assumed to be a double wide output single precision multiplication. That is, two single precision variables are multiplied to produce a -double precision result. The step is somewhat optimized from a long-hand multiplication algorithm because the carry from the addition in step -5.4.1 is propagated through the nested loop. If the carry was not propagated immediately it would overflow the single precision digit -$t_{ix+iy}$ and the result would be lost. - -At step 5.5 the nested loop is finished and any carry that was left over should be forwarded. The carry does not have to be added to the $ix+pb$'th -digit since that digit is assumed to be zero at this point. However, if $ix + pb \ge digs$ the carry is not set as it would make the result -exceed the precision requested. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_mul\_digs.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* multiplies |a| * |b| and only computes upto digs digits of result -018 * HAC pp. 595, Algorithm 14.12 Modified so you can control how -019 * many digits of output are created. -020 */ -021 int s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs) -022 \{ -023 mp_int t; -024 int res, pa, pb, ix, iy; -025 mp_digit u; -026 mp_word r; -027 mp_digit tmpx, *tmpt, *tmpy; -028 -029 /* can we use the fast multiplier? */ -030 if (((digs) < MP_WARRAY) && -031 MIN (a->used, b->used) < -032 (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) \{ -033 return fast_s_mp_mul_digs (a, b, c, digs); -034 \} -035 -036 if ((res = mp_init_size (&t, digs)) != MP_OKAY) \{ -037 return res; -038 \} -039 t.used = digs; -040 -041 /* compute the digits of the product directly */ -042 pa = a->used; -043 for (ix = 0; ix < pa; ix++) \{ -044 /* set the carry to zero */ -045 u = 0; -046 -047 /* limit ourselves to making digs digits of output */ -048 pb = MIN (b->used, digs - ix); -049 -050 /* setup some aliases */ -051 /* copy of the digit from a used within the nested loop */ -052 tmpx = a->dp[ix]; -053 -054 /* an alias for the destination shifted ix places */ -055 tmpt = t.dp + ix; -056 -057 /* an alias for the digits of b */ -058 tmpy = b->dp; -059 -060 /* compute the columns of the output and propagate the carry */ -061 for (iy = 0; iy < pb; iy++) \{ -062 /* compute the column as a mp_word */ -063 r = ((mp_word)*tmpt) + -064 ((mp_word)tmpx) * ((mp_word)*tmpy++) + -065 ((mp_word) u); -066 -067 /* the new column is the lower part of the result */ -068 *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK)); -069 -070 /* get the carry word from the result */ -071 u = (mp_digit) (r >> ((mp_word) DIGIT_BIT)); -072 \} -073 /* set carry if it is placed below digs */ -074 if (ix + iy < digs) \{ -075 *tmpt = u; -076 \} -077 \} -078 -079 mp_clamp (&t); -080 mp_exch (&t, c); -081 -082 mp_clear (&t); -083 return MP_OKAY; -084 \} -085 #endif -\end{alltt} -\end{small} - -First we determine (line 30) if the Comba method can be used first since it's faster. The conditions for -sing the Comba routine are that min$(a.used, b.used) < \delta$ and the number of digits of output is less than -\textbf{MP\_WARRAY}. This new constant is used to control the stack usage in the Comba routines. By default it is -set to $\delta$ but can be reduced when memory is at a premium. - -If we cannot use the Comba method we proceed to setup the baseline routine. We allocate the the destination mp\_int -$t$ (line 36) to the exact size of the output to avoid further re--allocations. At this point we now -begin the $O(n^2)$ loop. - -This implementation of multiplication has the caveat that it can be trimmed to only produce a variable number of -digits as output. In each iteration of the outer loop the $pb$ variable is set (line 48) to the maximum -number of inner loop iterations. - -Inside the inner loop we calculate $\hat r$ as the mp\_word product of the two mp\_digits and the addition of the -carry from the previous iteration. A particularly important observation is that most modern optimizing -C compilers (GCC for instance) can recognize that a $N \times N \rightarrow 2N$ multiplication is all that -is required for the product. In x86 terms for example, this means using the MUL instruction. - -Each digit of the product is stored in turn (line 68) and the carry propagated (line 71) to the -next iteration. - -\subsection{Faster Multiplication by the ``Comba'' Method} - -One of the huge drawbacks of the ``baseline'' algorithms is that at the $O(n^2)$ level the carry must be -computed and propagated upwards. This makes the nested loop very sequential and hard to unroll and implement -in parallel. The ``Comba'' \cite{COMBA} method is named after little known (\textit{in cryptographic venues}) Paul G. -Comba who described a method of implementing fast multipliers that do not require nested carry fixup operations. As an -interesting aside it seems that Paul Barrett describes a similar technique in his 1986 paper \cite{BARRETT} written -five years before. - -At the heart of the Comba technique is once again the long-hand algorithm. Except in this case a slight -twist is placed on how the columns of the result are produced. In the standard long-hand algorithm rows of products -are produced then added together to form the final result. In the baseline algorithm the columns are added together -after each iteration to get the result instantaneously. - -In the Comba algorithm the columns of the result are produced entirely independently of each other. That is at -the $O(n^2)$ level a simple multiplication and addition step is performed. The carries of the columns are propagated -after the nested loop to reduce the amount of work requiored. Succintly the first step of the algorithm is to compute -the product vector $\vec x$ as follows. - -\begin{equation} -\vec x_n = \sum_{i+j = n} a_ib_j, \forall n \in \lbrace 0, 1, 2, \ldots, i + j \rbrace -\end{equation} - -Where $\vec x_n$ is the $n'th$ column of the output vector. Consider the following example which computes the vector $\vec x$ for the multiplication -of $576$ and $241$. - -\newpage\begin{figure}[here] -\begin{small} -\begin{center} -\begin{tabular}{|c|c|c|c|c|c|} - \hline & & 5 & 7 & 6 & First Input\\ - \hline $\times$ & & 2 & 4 & 1 & Second Input\\ -\hline & & $1 \cdot 5 = 5$ & $1 \cdot 7 = 7$ & $1 \cdot 6 = 6$ & First pass \\ - & $4 \cdot 5 = 20$ & $4 \cdot 7+5=33$ & $4 \cdot 6+7=31$ & 6 & Second pass \\ - $2 \cdot 5 = 10$ & $2 \cdot 7 + 20 = 34$ & $2 \cdot 6+33=45$ & 31 & 6 & Third pass \\ -\hline 10 & 34 & 45 & 31 & 6 & Final Result \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Comba Multiplication Diagram} -\end{figure} - -At this point the vector $x = \left < 10, 34, 45, 31, 6 \right >$ is the result of the first step of the Comba multipler. -Now the columns must be fixed by propagating the carry upwards. The resultant vector will have one extra dimension over the input vector which is -congruent to adding a leading zero digit. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Comba Fixup}. \\ -\textbf{Input}. Vector $\vec x$ of dimension $k$ \\ -\textbf{Output}. Vector $\vec x$ such that the carries have been propagated. \\ -\hline \\ -1. for $n$ from $0$ to $k - 1$ do \\ -\hspace{3mm}1.1 $\vec x_{n+1} \leftarrow \vec x_{n+1} + \lfloor \vec x_{n}/\beta \rfloor$ \\ -\hspace{3mm}1.2 $\vec x_{n} \leftarrow \vec x_{n} \mbox{ (mod }\beta\mbox{)}$ \\ -2. Return($\vec x$). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm Comba Fixup} -\end{figure} - -With that algorithm and $k = 5$ and $\beta = 10$ the following vector is produced $\vec x= \left < 1, 3, 8, 8, 1, 6 \right >$. In this case -$241 \cdot 576$ is in fact $138816$ and the procedure succeeded. If the algorithm is correct and as will be demonstrated shortly more -efficient than the baseline algorithm why not simply always use this algorithm? - -\subsubsection{Column Weight.} -At the nested $O(n^2)$ level the Comba method adds the product of two single precision variables to each column of the output -independently. A serious obstacle is if the carry is lost, due to lack of precision before the algorithm has a chance to fix -the carries. For example, in the multiplication of two three-digit numbers the third column of output will be the sum of -three single precision multiplications. If the precision of the accumulator for the output digits is less then $3 \cdot (\beta - 1)^2$ then -an overflow can occur and the carry information will be lost. For any $m$ and $n$ digit inputs the maximum weight of any column is -min$(m, n)$ which is fairly obvious. - -The maximum number of terms in any column of a product is known as the ``column weight'' and strictly governs when the algorithm can be used. Recall -from earlier that a double precision type has $\alpha$ bits of resolution and a single precision digit has $lg(\beta)$ bits of precision. Given these -two quantities we must not violate the following - -\begin{equation} -k \cdot \left (\beta - 1 \right )^2 < 2^{\alpha} -\end{equation} - -Which reduces to - -\begin{equation} -k \cdot \left ( \beta^2 - 2\beta + 1 \right ) < 2^{\alpha} -\end{equation} - -Let $\rho = lg(\beta)$ represent the number of bits in a single precision digit. By further re-arrangement of the equation the final solution is -found. - -\begin{equation} -k < {{2^{\alpha}} \over {\left (2^{2\rho} - 2^{\rho + 1} + 1 \right )}} -\end{equation} - -The defaults for LibTomMath are $\beta = 2^{28}$ and $\alpha = 2^{64}$ which means that $k$ is bounded by $k < 257$. In this configuration -the smaller input may not have more than $256$ digits if the Comba method is to be used. This is quite satisfactory for most applications since -$256$ digits would allow for numbers in the range of $0 \le x < 2^{7168}$ which, is much larger than most public key cryptographic algorithms require. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{fast\_s\_mp\_mul\_digs}. \\ -\textbf{Input}. mp\_int $a$, mp\_int $b$ and an integer $digs$ \\ -\textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert \mbox{ (mod }\beta^{digs}\mbox{)}$. \\ -\hline \\ -Place an array of \textbf{MP\_WARRAY} single precision digits named $W$ on the stack. \\ -1. If $c.alloc < digs$ then grow $c$ to $digs$ digits. (\textit{mp\_grow}) \\ -2. If step 1 failed return(\textit{MP\_MEM}).\\ -\\ -3. $pa \leftarrow \mbox{MIN}(digs, a.used + b.used)$ \\ -\\ -4. $\_ \hat W \leftarrow 0$ \\ -5. for $ix$ from 0 to $pa - 1$ do \\ -\hspace{3mm}5.1 $ty \leftarrow \mbox{MIN}(b.used - 1, ix)$ \\ -\hspace{3mm}5.2 $tx \leftarrow ix - ty$ \\ -\hspace{3mm}5.3 $iy \leftarrow \mbox{MIN}(a.used - tx, ty + 1)$ \\ -\hspace{3mm}5.4 for $iz$ from 0 to $iy - 1$ do \\ -\hspace{6mm}5.4.1 $\_ \hat W \leftarrow \_ \hat W + a_{tx+iy}b_{ty-iy}$ \\ -\hspace{3mm}5.5 $W_{ix} \leftarrow \_ \hat W (\mbox{mod }\beta)$\\ -\hspace{3mm}5.6 $\_ \hat W \leftarrow \lfloor \_ \hat W / \beta \rfloor$ \\ -6. $W_{pa} \leftarrow \_ \hat W (\mbox{mod }\beta)$ \\ -\\ -7. $oldused \leftarrow c.used$ \\ -8. $c.used \leftarrow digs$ \\ -9. for $ix$ from $0$ to $pa$ do \\ -\hspace{3mm}9.1 $c_{ix} \leftarrow W_{ix}$ \\ -10. for $ix$ from $pa + 1$ to $oldused - 1$ do \\ -\hspace{3mm}10.1 $c_{ix} \leftarrow 0$ \\ -\\ -11. Clamp $c$. \\ -12. Return MP\_OKAY. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm fast\_s\_mp\_mul\_digs} -\label{fig:COMBAMULT} -\end{figure} - -\textbf{Algorithm fast\_s\_mp\_mul\_digs.} -This algorithm performs the unsigned multiplication of $a$ and $b$ using the Comba method limited to $digs$ digits of precision. - -The outer loop of this algorithm is more complicated than that of the baseline multiplier. This is because on the inside of the -loop we want to produce one column per pass. This allows the accumulator $\_ \hat W$ to be placed in CPU registers and -reduce the memory bandwidth to two \textbf{mp\_digit} reads per iteration. - -The $ty$ variable is set to the minimum count of $ix$ or the number of digits in $b$. That way if $a$ has more digits than -$b$ this will be limited to $b.used - 1$. The $tx$ variable is set to the to the distance past $b.used$ the variable -$ix$ is. This is used for the immediately subsequent statement where we find $iy$. - -The variable $iy$ is the minimum digits we can read from either $a$ or $b$ before running out. Computing one column at a time -means we have to scan one integer upwards and the other downwards. $a$ starts at $tx$ and $b$ starts at $ty$. In each -pass we are producing the $ix$'th output column and we note that $tx + ty = ix$. As we move $tx$ upwards we have to -move $ty$ downards so the equality remains valid. The $iy$ variable is the number of iterations until -$tx \ge a.used$ or $ty < 0$ occurs. - -After every inner pass we store the lower half of the accumulator into $W_{ix}$ and then propagate the carry of the accumulator -into the next round by dividing $\_ \hat W$ by $\beta$. - -To measure the benefits of the Comba method over the baseline method consider the number of operations that are required. If the -cost in terms of time of a multiply and addition is $p$ and the cost of a carry propagation is $q$ then a baseline multiplication would require -$O \left ((p + q)n^2 \right )$ time to multiply two $n$-digit numbers. The Comba method requires only $O(pn^2 + qn)$ time, however in practice, -the speed increase is actually much more. With $O(n)$ space the algorithm can be reduced to $O(pn + qn)$ time by implementing the $n$ multiply -and addition operations in the nested loop in parallel. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_fast\_s\_mp\_mul\_digs.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* Fast (comba) multiplier -018 * -019 * This is the fast column-array [comba] multiplier. It is -020 * designed to compute the columns of the product first -021 * then handle the carries afterwards. This has the effect -022 * of making the nested loops that compute the columns very -023 * simple and schedulable on super-scalar processors. -024 * -025 * This has been modified to produce a variable number of -026 * digits of output so if say only a half-product is required -027 * you don't have to compute the upper half (a feature -028 * required for fast Barrett reduction). -029 * -030 * Based on Algorithm 14.12 on pp.595 of HAC. -031 * -032 */ -033 int fast_s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs) -034 \{ -035 int olduse, res, pa, ix, iz; -036 mp_digit W[MP_WARRAY]; -037 register mp_word _W; -038 -039 /* grow the destination as required */ -040 if (c->alloc < digs) \{ -041 if ((res = mp_grow (c, digs)) != MP_OKAY) \{ -042 return res; -043 \} -044 \} -045 -046 /* number of output digits to produce */ -047 pa = MIN(digs, a->used + b->used); -048 -049 /* clear the carry */ -050 _W = 0; -051 for (ix = 0; ix < pa; ix++) \{ -052 int tx, ty; -053 int iy; -054 mp_digit *tmpx, *tmpy; -055 -056 /* get offsets into the two bignums */ -057 ty = MIN(b->used-1, ix); -058 tx = ix - ty; -059 -060 /* setup temp aliases */ -061 tmpx = a->dp + tx; -062 tmpy = b->dp + ty; -063 -064 /* this is the number of times the loop will iterrate, essentially -065 while (tx++ < a->used && ty-- >= 0) \{ ... \} -066 */ -067 iy = MIN(a->used-tx, ty+1); -068 -069 /* execute loop */ -070 for (iz = 0; iz < iy; ++iz) \{ -071 _W += ((mp_word)*tmpx++)*((mp_word)*tmpy--); -072 \} -073 -074 /* store term */ -075 W[ix] = ((mp_digit)_W) & MP_MASK; -076 -077 /* make next carry */ -078 _W = _W >> ((mp_word)DIGIT_BIT); -079 \} -080 -081 /* store final carry */ -082 W[ix] = (mp_digit)(_W & MP_MASK); -083 -084 /* setup dest */ -085 olduse = c->used; -086 c->used = pa; -087 -088 \{ -089 register mp_digit *tmpc; -090 tmpc = c->dp; -091 for (ix = 0; ix < pa+1; ix++) \{ -092 /* now extract the previous digit [below the carry] */ -093 *tmpc++ = W[ix]; -094 \} -095 -096 /* clear unused digits [that existed in the old copy of c] */ -097 for (; ix < olduse; ix++) \{ -098 *tmpc++ = 0; -099 \} -100 \} -101 mp_clamp (c); -102 return MP_OKAY; -103 \} -104 #endif -\end{alltt} -\end{small} - -As per the pseudo--code we first calculate $pa$ (line 47) as the number of digits to output. Next we begin the outer loop -to produce the individual columns of the product. We use the two aliases $tmpx$ and $tmpy$ (lines 61, 62) to point -inside the two multiplicands quickly. - -The inner loop (lines 70 to 72) of this implementation is where the tradeoff come into play. Originally this comba -implementation was ``row--major'' which means it adds to each of the columns in each pass. After the outer loop it would then fix -the carries. This was very fast except it had an annoying drawback. You had to read a mp\_word and two mp\_digits and write -one mp\_word per iteration. On processors such as the Athlon XP and P4 this did not matter much since the cache bandwidth -is very high and it can keep the ALU fed with data. It did, however, matter on older and embedded cpus where cache is often -slower and also often doesn't exist. This new algorithm only performs two reads per iteration under the assumption that the -compiler has aliased $\_ \hat W$ to a CPU register. - -After the inner loop we store the current accumulator in $W$ and shift $\_ \hat W$ (lines 75, 78) to forward it as -a carry for the next pass. After the outer loop we use the final carry (line 82) as the last digit of the product. - -\subsection{Polynomial Basis Multiplication} -To break the $O(n^2)$ barrier in multiplication requires a completely different look at integer multiplication. In the following algorithms -the use of polynomial basis representation for two integers $a$ and $b$ as $f(x) = \sum_{i=0}^{n} a_i x^i$ and -$g(x) = \sum_{i=0}^{n} b_i x^i$ respectively, is required. In this system both $f(x)$ and $g(x)$ have $n + 1$ terms and are of the $n$'th degree. - -The product $a \cdot b \equiv f(x)g(x)$ is the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$. The coefficients $w_i$ will -directly yield the desired product when $\beta$ is substituted for $x$. The direct solution to solve for the $2n + 1$ coefficients -requires $O(n^2)$ time and would in practice be slower than the Comba technique. - -However, numerical analysis theory indicates that only $2n + 1$ distinct points in $W(x)$ are required to determine the values of the $2n + 1$ unknown -coefficients. This means by finding $\zeta_y = W(y)$ for $2n + 1$ small values of $y$ the coefficients of $W(x)$ can be found with -Gaussian elimination. This technique is also occasionally refered to as the \textit{interpolation technique} (\textit{references please...}) since in -effect an interpolation based on $2n + 1$ points will yield a polynomial equivalent to $W(x)$. - -The coefficients of the polynomial $W(x)$ are unknown which makes finding $W(y)$ for any value of $y$ impossible. However, since -$W(x) = f(x)g(x)$ the equivalent $\zeta_y = f(y) g(y)$ can be used in its place. The benefit of this technique stems from the -fact that $f(y)$ and $g(y)$ are much smaller than either $a$ or $b$ respectively. As a result finding the $2n + 1$ relations required -by multiplying $f(y)g(y)$ involves multiplying integers that are much smaller than either of the inputs. - -When picking points to gather relations there are always three obvious points to choose, $y = 0, 1$ and $ \infty$. The $\zeta_0$ term -is simply the product $W(0) = w_0 = a_0 \cdot b_0$. The $\zeta_1$ term is the product -$W(1) = \left (\sum_{i = 0}^{n} a_i \right ) \left (\sum_{i = 0}^{n} b_i \right )$. The third point $\zeta_{\infty}$ is less obvious but rather -simple to explain. The $2n + 1$'th coefficient of $W(x)$ is numerically equivalent to the most significant column in an integer multiplication. -The point at $\infty$ is used symbolically to represent the most significant column, that is $W(\infty) = w_{2n} = a_nb_n$. Note that the -points at $y = 0$ and $\infty$ yield the coefficients $w_0$ and $w_{2n}$ directly. - -If more points are required they should be of small values and powers of two such as $2^q$ and the related \textit{mirror points} -$\left (2^q \right )^{2n} \cdot \zeta_{2^{-q}}$ for small values of $q$. The term ``mirror point'' stems from the fact that -$\left (2^q \right )^{2n} \cdot \zeta_{2^{-q}}$ can be calculated in the exact opposite fashion as $\zeta_{2^q}$. For -example, when $n = 2$ and $q = 1$ then following two equations are equivalent to the point $\zeta_{2}$ and its mirror. - -\begin{eqnarray} -\zeta_{2} = f(2)g(2) = (4a_2 + 2a_1 + a_0)(4b_2 + 2b_1 + b_0) \nonumber \\ -16 \cdot \zeta_{1 \over 2} = 4f({1\over 2}) \cdot 4g({1 \over 2}) = (a_2 + 2a_1 + 4a_0)(b_2 + 2b_1 + 4b_0) -\end{eqnarray} - -Using such points will allow the values of $f(y)$ and $g(y)$ to be independently calculated using only left shifts. For example, when $n = 2$ the -polynomial $f(2^q)$ is equal to $2^q((2^qa_2) + a_1) + a_0$. This technique of polynomial representation is known as Horner's method. - -As a general rule of the algorithm when the inputs are split into $n$ parts each there are $2n - 1$ multiplications. Each multiplication is of -multiplicands that have $n$ times fewer digits than the inputs. The asymptotic running time of this algorithm is -$O \left ( k^{lg_n(2n - 1)} \right )$ for $k$ digit inputs (\textit{assuming they have the same number of digits}). Figure~\ref{fig:exponent} -summarizes the exponents for various values of $n$. - -\begin{figure} -\begin{center} -\begin{tabular}{|c|c|c|} -\hline \textbf{Split into $n$ Parts} & \textbf{Exponent} & \textbf{Notes}\\ -\hline $2$ & $1.584962501$ & This is Karatsuba Multiplication. \\ -\hline $3$ & $1.464973520$ & This is Toom-Cook Multiplication. \\ -\hline $4$ & $1.403677461$ &\\ -\hline $5$ & $1.365212389$ &\\ -\hline $10$ & $1.278753601$ &\\ -\hline $100$ & $1.149426538$ &\\ -\hline $1000$ & $1.100270931$ &\\ -\hline $10000$ & $1.075252070$ &\\ -\hline -\end{tabular} -\end{center} -\caption{Asymptotic Running Time of Polynomial Basis Multiplication} -\label{fig:exponent} -\end{figure} - -At first it may seem like a good idea to choose $n = 1000$ since the exponent is approximately $1.1$. However, the overhead -of solving for the 2001 terms of $W(x)$ will certainly consume any savings the algorithm could offer for all but exceedingly large -numbers. - -\subsubsection{Cutoff Point} -The polynomial basis multiplication algorithms all require fewer single precision multiplications than a straight Comba approach. However, -the algorithms incur an overhead (\textit{at the $O(n)$ work level}) since they require a system of equations to be solved. This makes the -polynomial basis approach more costly to use with small inputs. - -Let $m$ represent the number of digits in the multiplicands (\textit{assume both multiplicands have the same number of digits}). There exists a -point $y$ such that when $m < y$ the polynomial basis algorithms are more costly than Comba, when $m = y$ they are roughly the same cost and -when $m > y$ the Comba methods are slower than the polynomial basis algorithms. - -The exact location of $y$ depends on several key architectural elements of the computer platform in question. - -\begin{enumerate} -\item The ratio of clock cycles for single precision multiplication versus other simpler operations such as addition, shifting, etc. For example -on the AMD Athlon the ratio is roughly $17 : 1$ while on the Intel P4 it is $29 : 1$. The higher the ratio in favour of multiplication the lower -the cutoff point $y$ will be. - -\item The complexity of the linear system of equations (\textit{for the coefficients of $W(x)$}) is. Generally speaking as the number of splits -grows the complexity grows substantially. Ideally solving the system will only involve addition, subtraction and shifting of integers. This -directly reflects on the ratio previous mentioned. - -\item To a lesser extent memory bandwidth and function call overheads. Provided the values are in the processor cache this is less of an -influence over the cutoff point. - -\end{enumerate} - -A clean cutoff point separation occurs when a point $y$ is found such that all of the cutoff point conditions are met. For example, if the point -is too low then there will be values of $m$ such that $m > y$ and the Comba method is still faster. Finding the cutoff points is fairly simple when -a high resolution timer is available. - -\subsection{Karatsuba Multiplication} -Karatsuba \cite{KARA} multiplication when originally proposed in 1962 was among the first set of algorithms to break the $O(n^2)$ barrier for -general purpose multiplication. Given two polynomial basis representations $f(x) = ax + b$ and $g(x) = cx + d$, Karatsuba proved with -light algebra \cite{KARAP} that the following polynomial is equivalent to multiplication of the two integers the polynomials represent. - -\begin{equation} -f(x) \cdot g(x) = acx^2 + ((a - b)(c - d) - (ac + bd))x + bd -\end{equation} - -Using the observation that $ac$ and $bd$ could be re-used only three half sized multiplications would be required to produce the product. Applying -this algorithm recursively, the work factor becomes $O(n^{lg(3)})$ which is substantially better than the work factor $O(n^2)$ of the Comba technique. It turns -out what Karatsuba did not know or at least did not publish was that this is simply polynomial basis multiplication with the points -$\zeta_0$, $\zeta_{\infty}$ and $-\zeta_{-1}$. Consider the resultant system of equations. - -\begin{center} -\begin{tabular}{rcrcrcrc} -$\zeta_{0}$ & $=$ & & & & & $w_0$ \\ -$-\zeta_{-1}$ & $=$ & $-w_2$ & $+$ & $w_1$ & $-$ & $w_0$ \\ -$\zeta_{\infty}$ & $=$ & $w_2$ & & & & \\ -\end{tabular} -\end{center} - -By adding the first and last equation to the equation in the middle the term $w_1$ can be isolated and all three coefficients solved for. The simplicity -of this system of equations has made Karatsuba fairly popular. In fact the cutoff point is often fairly low\footnote{With LibTomMath 0.18 it is 70 and 109 digits for the Intel P4 and AMD Athlon respectively.} -making it an ideal algorithm to speed up certain public key cryptosystems such as RSA and Diffie-Hellman. 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. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_karatsuba\_mul.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* c = |a| * |b| using Karatsuba Multiplication using -018 * three half size multiplications -019 * -020 * Let B represent the radix [e.g. 2**DIGIT_BIT] and -021 * let n represent half of the number of digits in -022 * the min(a,b) -023 * -024 * a = a1 * B**n + a0 -025 * b = b1 * B**n + b0 -026 * -027 * Then, a * b => -028 a1b1 * B**2n + ((a1 - a0)(b1 - b0) + a0b0 + a1b1) * B + a0b0 -029 * -030 * Note that a1b1 and a0b0 are used twice and only need to be -031 * computed once. So in total three half size (half # of -032 * digit) multiplications are performed, a0b0, a1b1 and -033 * (a1-b1)(a0-b0) -034 * -035 * Note that a multiplication of half the digits requires -036 * 1/4th the number of single precision multiplications so in -037 * total after one call 25% of the single precision multiplications -038 * are saved. Note also that the call to mp_mul can end up back -039 * in this function if the a0, a1, b0, or b1 are above the threshold. -040 * This is known as divide-and-conquer and leads to the famous -041 * O(N**lg(3)) or O(N**1.584) work which is asymptopically lower than -042 * the standard O(N**2) that the baseline/comba methods use. -043 * Generally though the overhead of this method doesn't pay off -044 * until a certain size (N ~ 80) is reached. -045 */ -046 int mp_karatsuba_mul (mp_int * a, mp_int * b, mp_int * c) -047 \{ -048 mp_int x0, x1, y0, y1, t1, x0y0, x1y1; -049 int B, err; -050 -051 /* default the return code to an error */ -052 err = MP_MEM; -053 -054 /* min # of digits */ -055 B = MIN (a->used, b->used); -056 -057 /* now divide in two */ -058 B = B >> 1; -059 -060 /* init copy all the temps */ -061 if (mp_init_size (&x0, B) != MP_OKAY) -062 goto ERR; -063 if (mp_init_size (&x1, a->used - B) != MP_OKAY) -064 goto X0; -065 if (mp_init_size (&y0, B) != MP_OKAY) -066 goto X1; -067 if (mp_init_size (&y1, b->used - B) != MP_OKAY) -068 goto Y0; -069 -070 /* init temps */ -071 if (mp_init_size (&t1, B * 2) != MP_OKAY) -072 goto Y1; -073 if (mp_init_size (&x0y0, B * 2) != MP_OKAY) -074 goto T1; -075 if (mp_init_size (&x1y1, B * 2) != MP_OKAY) -076 goto X0Y0; -077 -078 /* now shift the digits */ -079 x0.used = y0.used = B; -080 x1.used = a->used - B; -081 y1.used = b->used - B; -082 -083 \{ -084 register int x; -085 register mp_digit *tmpa, *tmpb, *tmpx, *tmpy; -086 -087 /* we copy the digits directly instead of using higher level functions -088 * since we also need to shift the digits -089 */ -090 tmpa = a->dp; -091 tmpb = b->dp; -092 -093 tmpx = x0.dp; -094 tmpy = y0.dp; -095 for (x = 0; x < B; x++) \{ -096 *tmpx++ = *tmpa++; -097 *tmpy++ = *tmpb++; -098 \} -099 -100 tmpx = x1.dp; -101 for (x = B; x < a->used; x++) \{ -102 *tmpx++ = *tmpa++; -103 \} -104 -105 tmpy = y1.dp; -106 for (x = B; x < b->used; x++) \{ -107 *tmpy++ = *tmpb++; -108 \} -109 \} -110 -111 /* only need to clamp the lower words since by definition the -112 * upper words x1/y1 must have a known number of digits -113 */ -114 mp_clamp (&x0); -115 mp_clamp (&y0); -116 -117 /* now calc the products x0y0 and x1y1 */ -118 /* after this x0 is no longer required, free temp [x0==t2]! */ -119 if (mp_mul (&x0, &y0, &x0y0) != MP_OKAY) -120 goto X1Y1; /* x0y0 = x0*y0 */ -121 if (mp_mul (&x1, &y1, &x1y1) != MP_OKAY) -122 goto X1Y1; /* x1y1 = x1*y1 */ -123 -124 /* now calc x1-x0 and y1-y0 */ -125 if (mp_sub (&x1, &x0, &t1) != MP_OKAY) -126 goto X1Y1; /* t1 = x1 - x0 */ -127 if (mp_sub (&y1, &y0, &x0) != MP_OKAY) -128 goto X1Y1; /* t2 = y1 - y0 */ -129 if (mp_mul (&t1, &x0, &t1) != MP_OKAY) -130 goto X1Y1; /* t1 = (x1 - x0) * (y1 - y0) */ -131 -132 /* add x0y0 */ -133 if (mp_add (&x0y0, &x1y1, &x0) != MP_OKAY) -134 goto X1Y1; /* t2 = x0y0 + x1y1 */ -135 if (mp_sub (&x0, &t1, &t1) != MP_OKAY) -136 goto X1Y1; /* t1 = x0y0 + x1y1 - (x1-x0)*(y1-y0) */ -137 -138 /* shift by B */ -139 if (mp_lshd (&t1, B) != MP_OKAY) -140 goto X1Y1; /* t1 = (x0y0 + x1y1 - (x1-x0)*(y1-y0))<<B */ -141 if (mp_lshd (&x1y1, B * 2) != MP_OKAY) -142 goto X1Y1; /* x1y1 = x1y1 << 2*B */ -143 -144 if (mp_add (&x0y0, &t1, &t1) != MP_OKAY) -145 goto X1Y1; /* t1 = x0y0 + t1 */ -146 if (mp_add (&t1, &x1y1, c) != MP_OKAY) -147 goto X1Y1; /* t1 = x0y0 + t1 + x1y1 */ -148 -149 /* Algorithm succeeded set the return code to MP_OKAY */ -150 err = MP_OKAY; -151 -152 X1Y1:mp_clear (&x1y1); -153 X0Y0:mp_clear (&x0y0); -154 T1:mp_clear (&t1); -155 Y1:mp_clear (&y1); -156 Y0:mp_clear (&y0); -157 X1:mp_clear (&x1); -158 X0:mp_clear (&x0); -159 ERR: -160 return err; -161 \} -162 #endif -\end{alltt} -\end{small} - -The new coding element in this routine, not seen in previous routines, is the usage of goto statements. The conventional -wisdom is that goto statements should be avoided. This is generally true, however when every single function call can fail, it makes sense -to handle error recovery with a single piece of code. Lines 61 to 75 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 101 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 106 and 106 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 150 is reached, the algorithm has completed succesfully. The ``error status'' variable $err$ is set to \textbf{MP\_OKAY} so that -the same code that handles errors can be used to clear the temporary variables and return. - -\subsection{Toom-Cook $3$-Way Multiplication} -Toom-Cook $3$-Way \cite{TOOM} multiplication is essentially the polynomial basis algorithm for $n = 2$ except that the points are -chosen such that $\zeta$ is easy to compute and the resulting system of equations easy to reduce. Here, the points $\zeta_{0}$, -$16 \cdot \zeta_{1 \over 2}$, $\zeta_1$, $\zeta_2$ and $\zeta_{\infty}$ make up the five required points to solve for the coefficients -of the $W(x)$. - -With the five relations that Toom-Cook specifies, the following system of equations is formed. - -\begin{center} -\begin{tabular}{rcrcrcrcrcr} -$\zeta_0$ & $=$ & $0w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $1w_0$ \\ -$16 \cdot \zeta_{1 \over 2}$ & $=$ & $1w_4$ & $+$ & $2w_3$ & $+$ & $4w_2$ & $+$ & $8w_1$ & $+$ & $16w_0$ \\ -$\zeta_1$ & $=$ & $1w_4$ & $+$ & $1w_3$ & $+$ & $1w_2$ & $+$ & $1w_1$ & $+$ & $1w_0$ \\ -$\zeta_2$ & $=$ & $16w_4$ & $+$ & $8w_3$ & $+$ & $4w_2$ & $+$ & $2w_1$ & $+$ & $1w_0$ \\ -$\zeta_{\infty}$ & $=$ & $1w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $0w_0$ \\ -\end{tabular} -\end{center} - -A trivial solution to this matrix requires $12$ subtractions, two multiplications by a small power of two, two divisions by a small power -of two, two divisions by three and one multiplication by three. All of these $19$ sub-operations require less than quadratic time, meaning that -the algorithm can be faster than a baseline multiplication. However, the greater complexity of this algorithm places the cutoff point -(\textbf{TOOM\_MUL\_CUTOFF}) where Toom-Cook becomes more efficient much higher than the Karatsuba cutoff point. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_toom\_mul}. \\ -\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ -\textbf{Output}. $c \leftarrow a \cdot b $ \\ -\hline \\ -Split $a$ and $b$ into three pieces. E.g. $a = a_2 \beta^{2k} + a_1 \beta^{k} + a_0$ \\ -1. $k \leftarrow \lfloor \mbox{min}(a.used, b.used) / 3 \rfloor$ \\ -2. $a_0 \leftarrow a \mbox{ (mod }\beta^{k}\mbox{)}$ \\ -3. $a_1 \leftarrow \lfloor a / \beta^k \rfloor$, $a_1 \leftarrow a_1 \mbox{ (mod }\beta^{k}\mbox{)}$ \\ -4. $a_2 \leftarrow \lfloor a / \beta^{2k} \rfloor$, $a_2 \leftarrow a_2 \mbox{ (mod }\beta^{k}\mbox{)}$ \\ -5. $b_0 \leftarrow a \mbox{ (mod }\beta^{k}\mbox{)}$ \\ -6. $b_1 \leftarrow \lfloor a / \beta^k \rfloor$, $b_1 \leftarrow b_1 \mbox{ (mod }\beta^{k}\mbox{)}$ \\ -7. $b_2 \leftarrow \lfloor a / \beta^{2k} \rfloor$, $b_2 \leftarrow b_2 \mbox{ (mod }\beta^{k}\mbox{)}$ \\ -\\ -Find the five equations for $w_0, w_1, ..., w_4$. \\ -8. $w_0 \leftarrow a_0 \cdot b_0$ \\ -9. $w_4 \leftarrow a_2 \cdot b_2$ \\ -10. $tmp_1 \leftarrow 2 \cdot a_0$, $tmp_1 \leftarrow a_1 + tmp_1$, $tmp_1 \leftarrow 2 \cdot tmp_1$, $tmp_1 \leftarrow tmp_1 + a_2$ \\ -11. $tmp_2 \leftarrow 2 \cdot b_0$, $tmp_2 \leftarrow b_1 + tmp_2$, $tmp_2 \leftarrow 2 \cdot tmp_2$, $tmp_2 \leftarrow tmp_2 + b_2$ \\ -12. $w_1 \leftarrow tmp_1 \cdot tmp_2$ \\ -13. $tmp_1 \leftarrow 2 \cdot a_2$, $tmp_1 \leftarrow a_1 + tmp_1$, $tmp_1 \leftarrow 2 \cdot tmp_1$, $tmp_1 \leftarrow tmp_1 + a_0$ \\ -14. $tmp_2 \leftarrow 2 \cdot b_2$, $tmp_2 \leftarrow b_1 + tmp_2$, $tmp_2 \leftarrow 2 \cdot tmp_2$, $tmp_2 \leftarrow tmp_2 + b_0$ \\ -15. $w_3 \leftarrow tmp_1 \cdot tmp_2$ \\ -16. $tmp_1 \leftarrow a_0 + a_1$, $tmp_1 \leftarrow tmp_1 + a_2$, $tmp_2 \leftarrow b_0 + b_1$, $tmp_2 \leftarrow tmp_2 + b_2$ \\ -17. $w_2 \leftarrow tmp_1 \cdot tmp_2$ \\ -\\ -Continued on the next page.\\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_toom\_mul} -\end{figure} - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_toom\_mul} (continued). \\ -\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ -\textbf{Output}. $c \leftarrow a \cdot b $ \\ -\hline \\ -Now solve the system of equations. \\ -18. $w_1 \leftarrow w_4 - w_1$, $w_3 \leftarrow w_3 - w_0$ \\ -19. $w_1 \leftarrow \lfloor w_1 / 2 \rfloor$, $w_3 \leftarrow \lfloor w_3 / 2 \rfloor$ \\ -20. $w_2 \leftarrow w_2 - w_0$, $w_2 \leftarrow w_2 - w_4$ \\ -21. $w_1 \leftarrow w_1 - w_2$, $w_3 \leftarrow w_3 - w_2$ \\ -22. $tmp_1 \leftarrow 8 \cdot w_0$, $w_1 \leftarrow w_1 - tmp_1$, $tmp_1 \leftarrow 8 \cdot w_4$, $w_3 \leftarrow w_3 - tmp_1$ \\ -23. $w_2 \leftarrow 3 \cdot w_2$, $w_2 \leftarrow w_2 - w_1$, $w_2 \leftarrow w_2 - w_3$ \\ -24. $w_1 \leftarrow w_1 - w_2$, $w_3 \leftarrow w_3 - w_2$ \\ -25. $w_1 \leftarrow \lfloor w_1 / 3 \rfloor, w_3 \leftarrow \lfloor w_3 / 3 \rfloor$ \\ -\\ -Now substitute $\beta^k$ for $x$ by shifting $w_0, w_1, ..., w_4$. \\ -26. for $n$ from $1$ to $4$ do \\ -\hspace{3mm}26.1 $w_n \leftarrow w_n \cdot \beta^{nk}$ \\ -27. $c \leftarrow w_0 + w_1$, $c \leftarrow c + w_2$, $c \leftarrow c + w_3$, $c \leftarrow c + w_4$ \\ -28. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_toom\_mul (continued)} -\end{figure} - -\textbf{Algorithm mp\_toom\_mul.} -This algorithm computes the product of two mp\_int variables $a$ and $b$ using the Toom-Cook approach. Compared to the Karatsuba multiplication, this -algorithm has a lower asymptotic running time of approximately $O(n^{1.464})$ but at an obvious cost in overhead. In this -description, several statements have been compounded to save space. The intention is that the statements are executed from left to right across -any given step. - -The two inputs $a$ and $b$ are first split into three $k$-digit integers $a_0, a_1, a_2$ and $b_0, b_1, b_2$ respectively. From these smaller -integers the coefficients of the polynomial basis representations $f(x)$ and $g(x)$ are known and can be used to find the relations required. - -The first two relations $w_0$ and $w_4$ are the points $\zeta_{0}$ and $\zeta_{\infty}$ respectively. The relation $w_1, w_2$ and $w_3$ correspond -to the points $16 \cdot \zeta_{1 \over 2}, \zeta_{2}$ and $\zeta_{1}$ respectively. These are found using logical shifts to independently find -$f(y)$ and $g(y)$ which significantly speeds up the algorithm. - -After the five relations $w_0, w_1, \ldots, w_4$ have been computed, the system they represent must be solved in order for the unknown coefficients -$w_1, w_2$ and $w_3$ to be isolated. The steps 18 through 25 perform the system reduction required as previously described. Each step of -the reduction represents the comparable matrix operation that would be performed had this been performed by pencil. For example, step 18 indicates -that row $1$ must be subtracted from row $4$ and simultaneously row $0$ subtracted from row $3$. - -Once the coeffients have been isolated, the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$ is known. By substituting $\beta^{k}$ for $x$, the integer -result $a \cdot b$ is produced. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_toom\_mul.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* multiplication using the Toom-Cook 3-way algorithm -018 * -019 * Much more complicated than Karatsuba but has a lower -020 * asymptotic running time of O(N**1.464). This algorithm is -021 * only particularly useful on VERY large inputs -022 * (we're talking 1000s of digits here...). -023 */ -024 int mp_toom_mul(mp_int *a, mp_int *b, mp_int *c) -025 \{ -026 mp_int w0, w1, w2, w3, w4, tmp1, tmp2, a0, a1, a2, b0, b1, b2; -027 int res, B; -028 -029 /* init temps */ -030 if ((res = mp_init_multi(&w0, &w1, &w2, &w3, &w4, -031 &a0, &a1, &a2, &b0, &b1, -032 &b2, &tmp1, &tmp2, NULL)) != MP_OKAY) \{ -033 return res; -034 \} -035 -036 /* B */ -037 B = MIN(a->used, b->used) / 3; -038 -039 /* a = a2 * B**2 + a1 * B + a0 */ -040 if ((res = mp_mod_2d(a, DIGIT_BIT * B, &a0)) != MP_OKAY) \{ -041 goto ERR; -042 \} -043 -044 if ((res = mp_copy(a, &a1)) != MP_OKAY) \{ -045 goto ERR; -046 \} -047 mp_rshd(&a1, B); -048 mp_mod_2d(&a1, DIGIT_BIT * B, &a1); -049 -050 if ((res = mp_copy(a, &a2)) != MP_OKAY) \{ -051 goto ERR; -052 \} -053 mp_rshd(&a2, B*2); -054 -055 /* b = b2 * B**2 + b1 * B + b0 */ -056 if ((res = mp_mod_2d(b, DIGIT_BIT * B, &b0)) != MP_OKAY) \{ -057 goto ERR; -058 \} -059 -060 if ((res = mp_copy(b, &b1)) != MP_OKAY) \{ -061 goto ERR; -062 \} -063 mp_rshd(&b1, B); -064 mp_mod_2d(&b1, DIGIT_BIT * B, &b1); -065 -066 if ((res = mp_copy(b, &b2)) != MP_OKAY) \{ -067 goto ERR; -068 \} -069 mp_rshd(&b2, B*2); -070 -071 /* w0 = a0*b0 */ -072 if ((res = mp_mul(&a0, &b0, &w0)) != MP_OKAY) \{ -073 goto ERR; -074 \} -075 -076 /* w4 = a2 * b2 */ -077 if ((res = mp_mul(&a2, &b2, &w4)) != MP_OKAY) \{ -078 goto ERR; -079 \} -080 -081 /* w1 = (a2 + 2(a1 + 2a0))(b2 + 2(b1 + 2b0)) */ -082 if ((res = mp_mul_2(&a0, &tmp1)) != MP_OKAY) \{ -083 goto ERR; -084 \} -085 if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) \{ -086 goto ERR; -087 \} -088 if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) \{ -089 goto ERR; -090 \} -091 if ((res = mp_add(&tmp1, &a2, &tmp1)) != MP_OKAY) \{ -092 goto ERR; -093 \} -094 -095 if ((res = mp_mul_2(&b0, &tmp2)) != MP_OKAY) \{ -096 goto ERR; -097 \} -098 if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) \{ -099 goto ERR; -100 \} -101 if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) \{ -102 goto ERR; -103 \} -104 if ((res = mp_add(&tmp2, &b2, &tmp2)) != MP_OKAY) \{ -105 goto ERR; -106 \} -107 -108 if ((res = mp_mul(&tmp1, &tmp2, &w1)) != MP_OKAY) \{ -109 goto ERR; -110 \} -111 -112 /* w3 = (a0 + 2(a1 + 2a2))(b0 + 2(b1 + 2b2)) */ -113 if ((res = mp_mul_2(&a2, &tmp1)) != MP_OKAY) \{ -114 goto ERR; -115 \} -116 if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) \{ -117 goto ERR; -118 \} -119 if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) \{ -120 goto ERR; -121 \} -122 if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) \{ -123 goto ERR; -124 \} -125 -126 if ((res = mp_mul_2(&b2, &tmp2)) != MP_OKAY) \{ -127 goto ERR; -128 \} -129 if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) \{ -130 goto ERR; -131 \} -132 if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) \{ -133 goto ERR; -134 \} -135 if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) \{ -136 goto ERR; -137 \} -138 -139 if ((res = mp_mul(&tmp1, &tmp2, &w3)) != MP_OKAY) \{ -140 goto ERR; -141 \} -142 -143 -144 /* w2 = (a2 + a1 + a0)(b2 + b1 + b0) */ -145 if ((res = mp_add(&a2, &a1, &tmp1)) != MP_OKAY) \{ -146 goto ERR; -147 \} -148 if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) \{ -149 goto ERR; -150 \} -151 if ((res = mp_add(&b2, &b1, &tmp2)) != MP_OKAY) \{ -152 goto ERR; -153 \} -154 if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) \{ -155 goto ERR; -156 \} -157 if ((res = mp_mul(&tmp1, &tmp2, &w2)) != MP_OKAY) \{ -158 goto ERR; -159 \} -160 -161 /* now solve the matrix -162 -163 0 0 0 0 1 -164 1 2 4 8 16 -165 1 1 1 1 1 -166 16 8 4 2 1 -167 1 0 0 0 0 -168 -169 using 12 subtractions, 4 shifts, -170 2 small divisions and 1 small multiplication -171 */ -172 -173 /* r1 - r4 */ -174 if ((res = mp_sub(&w1, &w4, &w1)) != MP_OKAY) \{ -175 goto ERR; -176 \} -177 /* r3 - r0 */ -178 if ((res = mp_sub(&w3, &w0, &w3)) != MP_OKAY) \{ -179 goto ERR; -180 \} -181 /* r1/2 */ -182 if ((res = mp_div_2(&w1, &w1)) != MP_OKAY) \{ -183 goto ERR; -184 \} -185 /* r3/2 */ -186 if ((res = mp_div_2(&w3, &w3)) != MP_OKAY) \{ -187 goto ERR; -188 \} -189 /* r2 - r0 - r4 */ -190 if ((res = mp_sub(&w2, &w0, &w2)) != MP_OKAY) \{ -191 goto ERR; -192 \} -193 if ((res = mp_sub(&w2, &w4, &w2)) != MP_OKAY) \{ -194 goto ERR; -195 \} -196 /* r1 - r2 */ -197 if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) \{ -198 goto ERR; -199 \} -200 /* r3 - r2 */ -201 if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) \{ -202 goto ERR; -203 \} -204 /* r1 - 8r0 */ -205 if ((res = mp_mul_2d(&w0, 3, &tmp1)) != MP_OKAY) \{ -206 goto ERR; -207 \} -208 if ((res = mp_sub(&w1, &tmp1, &w1)) != MP_OKAY) \{ -209 goto ERR; -210 \} -211 /* r3 - 8r4 */ -212 if ((res = mp_mul_2d(&w4, 3, &tmp1)) != MP_OKAY) \{ -213 goto ERR; -214 \} -215 if ((res = mp_sub(&w3, &tmp1, &w3)) != MP_OKAY) \{ -216 goto ERR; -217 \} -218 /* 3r2 - r1 - r3 */ -219 if ((res = mp_mul_d(&w2, 3, &w2)) != MP_OKAY) \{ -220 goto ERR; -221 \} -222 if ((res = mp_sub(&w2, &w1, &w2)) != MP_OKAY) \{ -223 goto ERR; -224 \} -225 if ((res = mp_sub(&w2, &w3, &w2)) != MP_OKAY) \{ -226 goto ERR; -227 \} -228 /* r1 - r2 */ -229 if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) \{ -230 goto ERR; -231 \} -232 /* r3 - r2 */ -233 if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) \{ -234 goto ERR; -235 \} -236 /* r1/3 */ -237 if ((res = mp_div_3(&w1, &w1, NULL)) != MP_OKAY) \{ -238 goto ERR; -239 \} -240 /* r3/3 */ -241 if ((res = mp_div_3(&w3, &w3, NULL)) != MP_OKAY) \{ -242 goto ERR; -243 \} -244 -245 /* at this point shift W[n] by B*n */ -246 if ((res = mp_lshd(&w1, 1*B)) != MP_OKAY) \{ -247 goto ERR; -248 \} -249 if ((res = mp_lshd(&w2, 2*B)) != MP_OKAY) \{ -250 goto ERR; -251 \} -252 if ((res = mp_lshd(&w3, 3*B)) != MP_OKAY) \{ -253 goto ERR; -254 \} -255 if ((res = mp_lshd(&w4, 4*B)) != MP_OKAY) \{ -256 goto ERR; -257 \} -258 -259 if ((res = mp_add(&w0, &w1, c)) != MP_OKAY) \{ -260 goto ERR; -261 \} -262 if ((res = mp_add(&w2, &w3, &tmp1)) != MP_OKAY) \{ -263 goto ERR; -264 \} -265 if ((res = mp_add(&w4, &tmp1, &tmp1)) != MP_OKAY) \{ -266 goto ERR; -267 \} -268 if ((res = mp_add(&tmp1, c, c)) != MP_OKAY) \{ -269 goto ERR; -270 \} -271 -272 ERR: -273 mp_clear_multi(&w0, &w1, &w2, &w3, &w4, -274 &a0, &a1, &a2, &b0, &b1, -275 &b2, &tmp1, &tmp2, NULL); -276 return res; -277 \} -278 -279 #endif -\end{alltt} -\end{small} - -The first obvious thing to note is that this algorithm is complicated. The complexity is worth it if you are multiplying very -large numbers. For example, a 10,000 digit multiplication takes approximaly 99,282,205 fewer single precision multiplications with -Toom--Cook than a Comba or baseline approach (this is a savings of more than 99$\%$). For most ``crypto'' sized numbers this -algorithm is not practical as Karatsuba has a much lower cutoff point. - -First we split $a$ and $b$ into three roughly equal portions. This has been accomplished (lines 40 to 69) with -combinations of mp\_rshd() and mp\_mod\_2d() function calls. At this point $a = a2 \cdot \beta^2 + a1 \cdot \beta + a0$ and similiarly -for $b$. - -Next we compute the five points $w0, w1, w2, w3$ and $w4$. Recall that $w0$ and $w4$ can be computed directly from the portions so -we get those out of the way first (lines 72 and 77). Next we compute $w1, w2$ and $w3$ using Horners method. - -After this point we solve for the actual values of $w1, w2$ and $w3$ by reducing the $5 \times 5$ system which is relatively -straight forward. - -\subsection{Signed Multiplication} -Now that algorithms to handle multiplications of every useful dimensions have been developed, a rather simple finishing touch is required. So far all -of the multiplication algorithms have been unsigned multiplications which leaves only a signed multiplication algorithm to be established. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_mul}. \\ -\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ -\textbf{Output}. $c \leftarrow a \cdot b$ \\ -\hline \\ -1. If $a.sign = b.sign$ then \\ -\hspace{3mm}1.1 $sign = MP\_ZPOS$ \\ -2. else \\ -\hspace{3mm}2.1 $sign = MP\_ZNEG$ \\ -3. If min$(a.used, b.used) \ge TOOM\_MUL\_CUTOFF$ then \\ -\hspace{3mm}3.1 $c \leftarrow a \cdot b$ using algorithm mp\_toom\_mul \\ -4. else if min$(a.used, b.used) \ge KARATSUBA\_MUL\_CUTOFF$ then \\ -\hspace{3mm}4.1 $c \leftarrow a \cdot b$ using algorithm mp\_karatsuba\_mul \\ -5. else \\ -\hspace{3mm}5.1 $digs \leftarrow a.used + b.used + 1$ \\ -\hspace{3mm}5.2 If $digs < MP\_ARRAY$ and min$(a.used, b.used) \le \delta$ then \\ -\hspace{6mm}5.2.1 $c \leftarrow a \cdot b \mbox{ (mod }\beta^{digs}\mbox{)}$ using algorithm fast\_s\_mp\_mul\_digs. \\ -\hspace{3mm}5.3 else \\ -\hspace{6mm}5.3.1 $c \leftarrow a \cdot b \mbox{ (mod }\beta^{digs}\mbox{)}$ using algorithm s\_mp\_mul\_digs. \\ -6. $c.sign \leftarrow sign$ \\ -7. Return the result of the unsigned multiplication performed. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_mul} -\end{figure} - -\textbf{Algorithm mp\_mul.} -This algorithm performs the signed multiplication of two inputs. It will make use of any of the three unsigned multiplication algorithms -available when the input is of appropriate size. The \textbf{sign} of the result is not set until the end of the algorithm since algorithm -s\_mp\_mul\_digs will clear it. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_mul.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* high level multiplication (handles sign) */ -018 int mp_mul (mp_int * a, mp_int * b, mp_int * c) -019 \{ -020 int res, neg; -021 neg = (a->sign == b->sign) ? MP_ZPOS : MP_NEG; -022 -023 /* use Toom-Cook? */ -024 #ifdef BN_MP_TOOM_MUL_C -025 if (MIN (a->used, b->used) >= TOOM_MUL_CUTOFF) \{ -026 res = mp_toom_mul(a, b, c); -027 \} else -028 #endif -029 #ifdef BN_MP_KARATSUBA_MUL_C -030 /* use Karatsuba? */ -031 if (MIN (a->used, b->used) >= KARATSUBA_MUL_CUTOFF) \{ -032 res = mp_karatsuba_mul (a, b, c); -033 \} else -034 #endif -035 \{ -036 /* can we use the fast multiplier? -037 * -038 * The fast multiplier can be used if the output will -039 * have less than MP_WARRAY digits and the number of -040 * digits won't affect carry propagation -041 */ -042 int digs = a->used + b->used + 1; -043 -044 #ifdef BN_FAST_S_MP_MUL_DIGS_C -045 if ((digs < MP_WARRAY) && -046 MIN(a->used, b->used) <= -047 (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) \{ -048 res = fast_s_mp_mul_digs (a, b, c, digs); -049 \} else -050 #endif -051 #ifdef BN_S_MP_MUL_DIGS_C -052 res = s_mp_mul (a, b, c); /* uses s_mp_mul_digs */ -053 #else -054 res = MP_VAL; -055 #endif -056 -057 \} -058 c->sign = (c->used > 0) ? neg : MP_ZPOS; -059 return res; -060 \} -061 #endif -\end{alltt} -\end{small} - -The implementation is rather simplistic and is not particularly noteworthy. Line 23 computes the sign of the result using the ``?'' -operator from the C programming language. Line 47 computes $\delta$ using the fact that $1 << k$ is equal to $2^k$. - -\section{Squaring} -\label{sec:basesquare} - -Squaring is a special case of multiplication where both multiplicands are equal. At first it may seem like there is no significant optimization -available but in fact there is. Consider the multiplication of $576$ against $241$. In total there will be nine single precision multiplications -performed which are $1\cdot 6$, $1 \cdot 7$, $1 \cdot 5$, $4 \cdot 6$, $4 \cdot 7$, $4 \cdot 5$, $2 \cdot 6$, $2 \cdot 7$ and $2 \cdot 5$. Now consider -the multiplication of $123$ against $123$. The nine products are $3 \cdot 3$, $3 \cdot 2$, $3 \cdot 1$, $2 \cdot 3$, $2 \cdot 2$, $2 \cdot 1$, -$1 \cdot 3$, $1 \cdot 2$ and $1 \cdot 1$. On closer inspection some of the products are equivalent. For example, $3 \cdot 2 = 2 \cdot 3$ -and $3 \cdot 1 = 1 \cdot 3$. - -For any $n$-digit input, there are ${{\left (n^2 + n \right)}\over 2}$ possible unique single precision multiplications required compared to the $n^2$ -required for multiplication. The following diagram gives an example of the operations required. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{ccccc|c} -&&1&2&3&\\ -$\times$ &&1&2&3&\\ -\hline && $3 \cdot 1$ & $3 \cdot 2$ & $3 \cdot 3$ & Row 0\\ - & $2 \cdot 1$ & $2 \cdot 2$ & $2 \cdot 3$ && Row 1 \\ - $1 \cdot 1$ & $1 \cdot 2$ & $1 \cdot 3$ &&& Row 2 \\ -\end{tabular} -\end{center} -\caption{Squaring Optimization Diagram} -\end{figure} - -Starting from zero and numbering the columns from right to left a very simple pattern becomes obvious. For the purposes of this discussion let $x$ -represent the number being squared. The first observation is that in row $k$ the $2k$'th column of the product has a $\left (x_k \right)^2$ term in it. - -The second observation is that every column $j$ in row $k$ where $j \ne 2k$ is part of a double product. Every non-square term of a column will -appear twice hence the name ``double product''. Every odd column is made up entirely of double products. In fact every column is made up of double -products and at most one square (\textit{see the exercise section}). - -The third and final observation is that for row $k$ the first unique non-square term, that is, one that hasn't already appeared in an earlier row, -occurs at column $2k + 1$. For example, on row $1$ of the previous squaring, column one is part of the double product with column one from row zero. -Column two of row one is a square and column three is the first unique column. - -\subsection{The Baseline Squaring Algorithm} -The baseline squaring algorithm is meant to be a catch-all squaring algorithm. It will handle any of the input sizes that the faster routines -will not handle. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{s\_mp\_sqr}. \\ -\textbf{Input}. mp\_int $a$ \\ -\textbf{Output}. $b \leftarrow a^2$ \\ -\hline \\ -1. Init a temporary mp\_int of at least $2 \cdot a.used +1$ digits. (\textit{mp\_init\_size}) \\ -2. If step 1 failed return(\textit{MP\_MEM}) \\ -3. $t.used \leftarrow 2 \cdot a.used + 1$ \\ -4. For $ix$ from 0 to $a.used - 1$ do \\ -\hspace{3mm}Calculate the square. \\ -\hspace{3mm}4.1 $\hat r \leftarrow t_{2ix} + \left (a_{ix} \right )^2$ \\ -\hspace{3mm}4.2 $t_{2ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{3mm}Calculate the double products after the square. \\ -\hspace{3mm}4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ -\hspace{3mm}4.4 For $iy$ from $ix + 1$ to $a.used - 1$ do \\ -\hspace{6mm}4.4.1 $\hat r \leftarrow 2 \cdot a_{ix}a_{iy} + t_{ix + iy} + u$ \\ -\hspace{6mm}4.4.2 $t_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{6mm}4.4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ -\hspace{3mm}Set the last carry. \\ -\hspace{3mm}4.5 While $u > 0$ do \\ -\hspace{6mm}4.5.1 $iy \leftarrow iy + 1$ \\ -\hspace{6mm}4.5.2 $\hat r \leftarrow t_{ix + iy} + u$ \\ -\hspace{6mm}4.5.3 $t_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{6mm}4.5.4 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ -5. Clamp excess digits of $t$. (\textit{mp\_clamp}) \\ -6. Exchange $b$ and $t$. \\ -7. Clear $t$ (\textit{mp\_clear}) \\ -8. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm s\_mp\_sqr} -\end{figure} - -\textbf{Algorithm s\_mp\_sqr.} -This algorithm computes the square of an input using the three observations on squaring. It is based fairly faithfully on algorithm 14.16 of HAC -\cite[pp.596-597]{HAC}. Similar to algorithm s\_mp\_mul\_digs, a temporary mp\_int is allocated to hold the result of the squaring. This allows the -destination mp\_int to be the same as the source mp\_int. - -The outer loop of this algorithm begins on step 4. It is best to think of the outer loop as walking down the rows of the partial results, while -the inner loop computes the columns of the partial result. Step 4.1 and 4.2 compute the square term for each row, and step 4.3 and 4.4 propagate -the carry and compute the double products. - -The requirement that a mp\_word be able to represent the range $0 \le x < 2 \beta^2$ arises from this -very algorithm. The product $a_{ix}a_{iy}$ will lie in the range $0 \le x \le \beta^2 - 2\beta + 1$ which is obviously less than $\beta^2$ meaning that -when it is multiplied by two, it can be properly represented by a mp\_word. - -Similar to algorithm s\_mp\_mul\_digs, after every pass of the inner loop, the destination is correctly set to the sum of all of the partial -results calculated so far. This involves expensive carry propagation which will be eliminated in the next algorithm. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_sqr.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* low level squaring, b = a*a, HAC pp.596-597, Algorithm 14.16 */ -018 int s_mp_sqr (mp_int * a, mp_int * b) -019 \{ -020 mp_int t; -021 int res, ix, iy, pa; -022 mp_word r; -023 mp_digit u, tmpx, *tmpt; -024 -025 pa = a->used; -026 if ((res = mp_init_size (&t, 2*pa + 1)) != MP_OKAY) \{ -027 return res; -028 \} -029 -030 /* default used is maximum possible size */ -031 t.used = 2*pa + 1; -032 -033 for (ix = 0; ix < pa; ix++) \{ -034 /* first calculate the digit at 2*ix */ -035 /* calculate double precision result */ -036 r = ((mp_word) t.dp[2*ix]) + -037 ((mp_word)a->dp[ix])*((mp_word)a->dp[ix]); -038 -039 /* store lower part in result */ -040 t.dp[ix+ix] = (mp_digit) (r & ((mp_word) MP_MASK)); -041 -042 /* get the carry */ -043 u = (mp_digit)(r >> ((mp_word) DIGIT_BIT)); -044 -045 /* left hand side of A[ix] * A[iy] */ -046 tmpx = a->dp[ix]; -047 -048 /* alias for where to store the results */ -049 tmpt = t.dp + (2*ix + 1); -050 -051 for (iy = ix + 1; iy < pa; iy++) \{ -052 /* first calculate the product */ -053 r = ((mp_word)tmpx) * ((mp_word)a->dp[iy]); -054 -055 /* now calculate the double precision result, note we use -056 * addition instead of *2 since it's easier to optimize -057 */ -058 r = ((mp_word) *tmpt) + r + r + ((mp_word) u); -059 -060 /* store lower part */ -061 *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK)); -062 -063 /* get carry */ -064 u = (mp_digit)(r >> ((mp_word) DIGIT_BIT)); -065 \} -066 /* propagate upwards */ -067 while (u != ((mp_digit) 0)) \{ -068 r = ((mp_word) *tmpt) + ((mp_word) u); -069 *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK)); -070 u = (mp_digit)(r >> ((mp_word) DIGIT_BIT)); -071 \} -072 \} -073 -074 mp_clamp (&t); -075 mp_exch (&t, b); -076 mp_clear (&t); -077 return MP_OKAY; -078 \} -079 #endif -\end{alltt} -\end{small} - -Inside the outer loop (line 33) the square term is calculated on line 36. The carry (line 43) has been -extracted from the mp\_word accumulator using a right shift. Aliases for $a_{ix}$ and $t_{ix+iy}$ are initialized -(lines 46 and 49) to simplify the inner loop. The doubling is performed using two -additions (line 58) since it is usually faster than shifting, if not at least as fast. - -The important observation is that the inner loop does not begin at $iy = 0$ like for multiplication. As such the inner loops -get progressively shorter as the algorithm proceeds. This is what leads to the savings compared to using a multiplication to -square a number. - -\subsection{Faster Squaring by the ``Comba'' Method} -A major drawback to the baseline method is the requirement for single precision shifting inside the $O(n^2)$ nested loop. Squaring has an additional -drawback that it must double the product inside the inner loop as well. As for multiplication, the Comba technique can be used to eliminate these -performance hazards. - -The first obvious solution is to make an array of mp\_words which will hold all of the columns. This will indeed eliminate all of the carry -propagation operations from the inner loop. However, the inner product must still be doubled $O(n^2)$ times. The solution stems from the simple fact -that $2a + 2b + 2c = 2(a + b + c)$. That is the sum of all of the double products is equal to double the sum of all the products. For example, -$ab + ba + ac + ca = 2ab + 2ac = 2(ab + ac)$. - -However, we cannot simply double all of the columns, since the squares appear only once per row. The most practical solution is to have two -mp\_word arrays. One array will hold the squares and the other array will hold the double products. With both arrays the doubling and -carry propagation can be moved to a $O(n)$ work level outside the $O(n^2)$ level. In this case, we have an even simpler solution in mind. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{fast\_s\_mp\_sqr}. \\ -\textbf{Input}. mp\_int $a$ \\ -\textbf{Output}. $b \leftarrow a^2$ \\ -\hline \\ -Place an array of \textbf{MP\_WARRAY} mp\_digits named $W$ on the stack. \\ -1. If $b.alloc < 2a.used + 1$ then grow $b$ to $2a.used + 1$ digits. (\textit{mp\_grow}). \\ -2. If step 1 failed return(\textit{MP\_MEM}). \\ -\\ -3. $pa \leftarrow 2 \cdot a.used$ \\ -4. $\hat W1 \leftarrow 0$ \\ -5. for $ix$ from $0$ to $pa - 1$ do \\ -\hspace{3mm}5.1 $\_ \hat W \leftarrow 0$ \\ -\hspace{3mm}5.2 $ty \leftarrow \mbox{MIN}(a.used - 1, ix)$ \\ -\hspace{3mm}5.3 $tx \leftarrow ix - ty$ \\ -\hspace{3mm}5.4 $iy \leftarrow \mbox{MIN}(a.used - tx, ty + 1)$ \\ -\hspace{3mm}5.5 $iy \leftarrow \mbox{MIN}(iy, \lfloor \left (ty - tx + 1 \right )/2 \rfloor)$ \\ -\hspace{3mm}5.6 for $iz$ from $0$ to $iz - 1$ do \\ -\hspace{6mm}5.6.1 $\_ \hat W \leftarrow \_ \hat W + a_{tx + iz}a_{ty - iz}$ \\ -\hspace{3mm}5.7 $\_ \hat W \leftarrow 2 \cdot \_ \hat W + \hat W1$ \\ -\hspace{3mm}5.8 if $ix$ is even then \\ -\hspace{6mm}5.8.1 $\_ \hat W \leftarrow \_ \hat W + \left ( a_{\lfloor ix/2 \rfloor}\right )^2$ \\ -\hspace{3mm}5.9 $W_{ix} \leftarrow \_ \hat W (\mbox{mod }\beta)$ \\ -\hspace{3mm}5.10 $\hat W1 \leftarrow \lfloor \_ \hat W / \beta \rfloor$ \\ -\\ -6. $oldused \leftarrow b.used$ \\ -7. $b.used \leftarrow 2 \cdot a.used$ \\ -8. for $ix$ from $0$ to $pa - 1$ do \\ -\hspace{3mm}8.1 $b_{ix} \leftarrow W_{ix}$ \\ -9. for $ix$ from $pa$ to $oldused - 1$ do \\ -\hspace{3mm}9.1 $b_{ix} \leftarrow 0$ \\ -10. Clamp excess digits from $b$. (\textit{mp\_clamp}) \\ -11. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm fast\_s\_mp\_sqr} -\end{figure} - -\textbf{Algorithm fast\_s\_mp\_sqr.} -This algorithm computes the square of an input using the Comba technique. It is designed to be a replacement for algorithm -s\_mp\_sqr when the number of input digits is less than \textbf{MP\_WARRAY} and less than $\delta \over 2$. -This algorithm is very similar to the Comba multiplier except with a few key differences we shall make note of. - -First, we have an accumulator and carry variables $\_ \hat W$ and $\hat W1$ respectively. This is because the inner loop -products are to be doubled. If we had added the previous carry in we would be doubling too much. Next we perform an -addition MIN condition on $iy$ (step 5.5) to prevent overlapping digits. For example, $a_3 \cdot a_5$ is equal -$a_5 \cdot a_3$. Whereas in the multiplication case we would have $5 < a.used$ and $3 \ge 0$ is maintained since we double the sum -of the products just outside the inner loop we have to avoid doing this. This is also a good thing since we perform -fewer multiplications and the routine ends up being faster. - -Finally the last difference is the addition of the ``square'' term outside the inner loop (step 5.8). We add in the square -only to even outputs and it is the square of the term at the $\lfloor ix / 2 \rfloor$ position. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_fast\_s\_mp\_sqr.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* the jist of squaring... -018 * you do like mult except the offset of the tmpx [one that -019 * starts closer to zero] can't equal the offset of tmpy. -020 * So basically you set up iy like before then you min it with -021 * (ty-tx) so that it never happens. You double all those -022 * you add in the inner loop -023 -024 After that loop you do the squares and add them in. -025 */ -026 -027 int fast_s_mp_sqr (mp_int * a, mp_int * b) -028 \{ -029 int olduse, res, pa, ix, iz; -030 mp_digit W[MP_WARRAY], *tmpx; -031 mp_word W1; -032 -033 /* grow the destination as required */ -034 pa = a->used + a->used; -035 if (b->alloc < pa) \{ -036 if ((res = mp_grow (b, pa)) != MP_OKAY) \{ -037 return res; -038 \} -039 \} -040 -041 /* number of output digits to produce */ -042 W1 = 0; -043 for (ix = 0; ix < pa; ix++) \{ -044 int tx, ty, iy; -045 mp_word _W; -046 mp_digit *tmpy; -047 -048 /* clear counter */ -049 _W = 0; -050 -051 /* get offsets into the two bignums */ -052 ty = MIN(a->used-1, ix); -053 tx = ix - ty; -054 -055 /* setup temp aliases */ -056 tmpx = a->dp + tx; -057 tmpy = a->dp + ty; -058 -059 /* this is the number of times the loop will iterrate, essentially -060 while (tx++ < a->used && ty-- >= 0) \{ ... \} -061 */ -062 iy = MIN(a->used-tx, ty+1); -063 -064 /* now for squaring tx can never equal ty -065 * we halve the distance since they approach at a rate of 2x -066 * and we have to round because odd cases need to be executed -067 */ -068 iy = MIN(iy, (ty-tx+1)>>1); -069 -070 /* execute loop */ -071 for (iz = 0; iz < iy; iz++) \{ -072 _W += ((mp_word)*tmpx++)*((mp_word)*tmpy--); -073 \} -074 -075 /* double the inner product and add carry */ -076 _W = _W + _W + W1; -077 -078 /* even columns have the square term in them */ -079 if ((ix&1) == 0) \{ -080 _W += ((mp_word)a->dp[ix>>1])*((mp_word)a->dp[ix>>1]); -081 \} -082 -083 /* store it */ -084 W[ix] = (mp_digit)(_W & MP_MASK); -085 -086 /* make next carry */ -087 W1 = _W >> ((mp_word)DIGIT_BIT); -088 \} -089 -090 /* setup dest */ -091 olduse = b->used; -092 b->used = a->used+a->used; -093 -094 \{ -095 mp_digit *tmpb; -096 tmpb = b->dp; -097 for (ix = 0; ix < pa; ix++) \{ -098 *tmpb++ = W[ix] & MP_MASK; -099 \} -100 -101 /* clear unused digits [that existed in the old copy of c] */ -102 for (; ix < olduse; ix++) \{ -103 *tmpb++ = 0; -104 \} -105 \} -106 mp_clamp (b); -107 return MP_OKAY; -108 \} -109 #endif -\end{alltt} -\end{small} - -This implementation is essentially a copy of Comba multiplication with the appropriate changes added to make it faster for -the special case of squaring. - -\subsection{Polynomial Basis Squaring} -The same algorithm that performs optimal polynomial basis multiplication can be used to perform polynomial basis squaring. The minor exception -is that $\zeta_y = f(y)g(y)$ is actually equivalent to $\zeta_y = f(y)^2$ since $f(y) = g(y)$. Instead of performing $2n + 1$ -multiplications to find the $\zeta$ relations, squaring operations are performed instead. - -\subsection{Karatsuba Squaring} -Let $f(x) = ax + b$ represent the polynomial basis representation of a number to square. -Let $h(x) = \left ( f(x) \right )^2$ represent the square of the polynomial. The Karatsuba equation can be modified to square a -number with the following equation. - -\begin{equation} -h(x) = a^2x^2 + \left (a^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. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_karatsuba\_sqr.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* Karatsuba squaring, computes b = a*a using three -018 * half size squarings -019 * -020 * See comments of karatsuba_mul for details. It -021 * is essentially the same algorithm but merely -022 * tuned to perform recursive squarings. -023 */ -024 int mp_karatsuba_sqr (mp_int * a, mp_int * b) -025 \{ -026 mp_int x0, x1, t1, t2, x0x0, x1x1; -027 int B, err; -028 -029 err = MP_MEM; -030 -031 /* min # of digits */ -032 B = a->used; -033 -034 /* now divide in two */ -035 B = B >> 1; -036 -037 /* init copy all the temps */ -038 if (mp_init_size (&x0, B) != MP_OKAY) -039 goto ERR; -040 if (mp_init_size (&x1, a->used - B) != MP_OKAY) -041 goto X0; -042 -043 /* init temps */ -044 if (mp_init_size (&t1, a->used * 2) != MP_OKAY) -045 goto X1; -046 if (mp_init_size (&t2, a->used * 2) != MP_OKAY) -047 goto T1; -048 if (mp_init_size (&x0x0, B * 2) != MP_OKAY) -049 goto T2; -050 if (mp_init_size (&x1x1, (a->used - B) * 2) != MP_OKAY) -051 goto X0X0; -052 -053 \{ -054 register int x; -055 register mp_digit *dst, *src; -056 -057 src = a->dp; -058 -059 /* now shift the digits */ -060 dst = x0.dp; -061 for (x = 0; x < B; x++) \{ -062 *dst++ = *src++; -063 \} -064 -065 dst = x1.dp; -066 for (x = B; x < a->used; x++) \{ -067 *dst++ = *src++; -068 \} -069 \} -070 -071 x0.used = B; -072 x1.used = a->used - B; -073 -074 mp_clamp (&x0); -075 -076 /* now calc the products x0*x0 and x1*x1 */ -077 if (mp_sqr (&x0, &x0x0) != MP_OKAY) -078 goto X1X1; /* x0x0 = x0*x0 */ -079 if (mp_sqr (&x1, &x1x1) != MP_OKAY) -080 goto X1X1; /* x1x1 = x1*x1 */ -081 -082 /* now calc (x1-x0)**2 */ -083 if (mp_sub (&x1, &x0, &t1) != MP_OKAY) -084 goto X1X1; /* t1 = x1 - x0 */ -085 if (mp_sqr (&t1, &t1) != MP_OKAY) -086 goto X1X1; /* t1 = (x1 - x0) * (x1 - x0) */ -087 -088 /* add x0y0 */ -089 if (s_mp_add (&x0x0, &x1x1, &t2) != MP_OKAY) -090 goto X1X1; /* t2 = x0x0 + x1x1 */ -091 if (mp_sub (&t2, &t1, &t1) != MP_OKAY) -092 goto X1X1; /* t1 = x0x0 + x1x1 - (x1-x0)*(x1-x0) */ -093 -094 /* shift by B */ -095 if (mp_lshd (&t1, B) != MP_OKAY) -096 goto X1X1; /* t1 = (x0x0 + x1x1 - (x1-x0)*(x1-x0))<<B */ -097 if (mp_lshd (&x1x1, B * 2) != MP_OKAY) -098 goto X1X1; /* x1x1 = x1x1 << 2*B */ -099 -100 if (mp_add (&x0x0, &t1, &t1) != MP_OKAY) -101 goto X1X1; /* t1 = x0x0 + t1 */ -102 if (mp_add (&t1, &x1x1, b) != MP_OKAY) -103 goto X1X1; /* t1 = x0x0 + t1 + x1x1 */ -104 -105 err = MP_OKAY; -106 -107 X1X1:mp_clear (&x1x1); -108 X0X0:mp_clear (&x0x0); -109 T2:mp_clear (&t2); -110 T1:mp_clear (&t1); -111 X1:mp_clear (&x1); -112 X0:mp_clear (&x0); -113 ERR: -114 return err; -115 \} -116 #endif -\end{alltt} -\end{small} - -This implementation is largely based on the implementation of algorithm mp\_karatsuba\_mul. It uses the same inline style to copy and -shift the input into the two halves. The loop from line 53 to line 69 has been modified since only one input exists. The \textbf{used} -count of both $x0$ and $x1$ is fixed up and $x0$ is clamped before the calculations begin. At this point $x1$ and $x0$ are valid equivalents -to the respective halves as if mp\_rshd and mp\_mod\_2d had been used. - -By inlining the copy and shift operations the cutoff point for Karatsuba multiplication can be lowered. On the Athlon the cutoff point -is exactly at the point where Comba squaring can no longer be used (\textit{128 digits}). On slower processors such as the Intel P4 -it is actually below the Comba limit (\textit{at 110 digits}). - -This routine uses the same error trap coding style as mp\_karatsuba\_sqr. As the temporary variables are initialized errors are -redirected to the error trap higher up. If the algorithm completes without error the error code is set to \textbf{MP\_OKAY} and -mp\_clears are executed normally. - -\subsection{Toom-Cook Squaring} -The Toom-Cook squaring algorithm mp\_toom\_sqr is heavily based on the algorithm mp\_toom\_mul with the exception that squarings are used -instead of multiplication to find the five relations. The reader is encouraged to read the description of the latter algorithm and try to -derive their own Toom-Cook squaring algorithm. - -\subsection{High Level Squaring} -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_sqr}. \\ -\textbf{Input}. mp\_int $a$ \\ -\textbf{Output}. $b \leftarrow a^2$ \\ -\hline \\ -1. If $a.used \ge TOOM\_SQR\_CUTOFF$ then \\ -\hspace{3mm}1.1 $b \leftarrow a^2$ using algorithm mp\_toom\_sqr \\ -2. else if $a.used \ge KARATSUBA\_SQR\_CUTOFF$ then \\ -\hspace{3mm}2.1 $b \leftarrow a^2$ using algorithm mp\_karatsuba\_sqr \\ -3. else \\ -\hspace{3mm}3.1 $digs \leftarrow a.used + b.used + 1$ \\ -\hspace{3mm}3.2 If $digs < MP\_ARRAY$ and $a.used \le \delta$ then \\ -\hspace{6mm}3.2.1 $b \leftarrow a^2$ using algorithm fast\_s\_mp\_sqr. \\ -\hspace{3mm}3.3 else \\ -\hspace{6mm}3.3.1 $b \leftarrow a^2$ using algorithm s\_mp\_sqr. \\ -4. $b.sign \leftarrow MP\_ZPOS$ \\ -5. Return the result of the unsigned squaring performed. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_sqr} -\end{figure} - -\textbf{Algorithm mp\_sqr.} -This algorithm computes the square of the input using one of four different algorithms. If the input is very large and has at least -\textbf{TOOM\_SQR\_CUTOFF} or \textbf{KARATSUBA\_SQR\_CUTOFF} digits then either the Toom-Cook or the Karatsuba Squaring algorithm is used. If -neither of the polynomial basis algorithms should be used then either the Comba or baseline algorithm is used. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_sqr.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* computes b = a*a */ -018 int -019 mp_sqr (mp_int * a, mp_int * b) -020 \{ -021 int res; -022 -023 #ifdef BN_MP_TOOM_SQR_C -024 /* use Toom-Cook? */ -025 if (a->used >= TOOM_SQR_CUTOFF) \{ -026 res = mp_toom_sqr(a, b); -027 /* Karatsuba? */ -028 \} else -029 #endif -030 #ifdef BN_MP_KARATSUBA_SQR_C -031 if (a->used >= KARATSUBA_SQR_CUTOFF) \{ -032 res = mp_karatsuba_sqr (a, b); -033 \} else -034 #endif -035 \{ -036 #ifdef BN_FAST_S_MP_SQR_C -037 /* can we use the fast comba multiplier? */ -038 if ((a->used * 2 + 1) < MP_WARRAY && -039 a->used < -040 (1 << (sizeof(mp_word) * CHAR_BIT - 2*DIGIT_BIT - 1))) \{ -041 res = fast_s_mp_sqr (a, b); -042 \} else -043 #endif -044 #ifdef BN_S_MP_SQR_C -045 res = s_mp_sqr (a, b); -046 #else -047 res = MP_VAL; -048 #endif -049 \} -050 b->sign = MP_ZPOS; -051 return res; -052 \} -053 #endif -\end{alltt} -\end{small} - -\section*{Exercises} -\begin{tabular}{cl} -$\left [ 3 \right ] $ & Devise an efficient algorithm for selection of the radix point to handle inputs \\ - & that have different number of digits in Karatsuba multiplication. \\ - & \\ -$\left [ 2 \right ] $ & In section 5.3 the fact that every column of a squaring is made up \\ - & of double products and at most one square is stated. Prove this statement. \\ - & \\ -$\left [ 3 \right ] $ & Prove the equation for Karatsuba squaring. \\ - & \\ -$\left [ 1 \right ] $ & Prove that Karatsuba squaring requires $O \left (n^{lg(3)} \right )$ time. \\ - & \\ -$\left [ 2 \right ] $ & Determine the minimal ratio between addition and multiplication clock cycles \\ - & required for equation $6.7$ to be true. \\ - & \\ -$\left [ 3 \right ] $ & Implement a threaded version of Comba multiplication (and squaring) where you \\ - & compute subsets of the columns in each thread. Determine a cutoff point where \\ - & it is effective and add the logic to mp\_mul() and mp\_sqr(). \\ - &\\ -$\left [ 4 \right ] $ & Same as the previous but also modify the Karatsuba and Toom-Cook. You must \\ - & increase the throughput of mp\_exptmod() for random odd moduli in the range \\ - & $512 \ldots 4096$ bits significantly ($> 2x$) to complete this challenge. \\ - & \\ -\end{tabular} - -\chapter{Modular Reduction} -\section{Basics of Modular Reduction} -\index{modular residue} -Modular reduction is an operation that arises quite often within public key cryptography algorithms and various number theoretic algorithms, -such as factoring. Modular reduction algorithms are the third class of algorithms of the ``multipliers'' set. A number $a$ is said to be \textit{reduced} -modulo another number $b$ by finding the remainder of the division $a/b$. Full integer division with remainder is a topic to be covered -in~\ref{sec:division}. - -Modular reduction is equivalent to solving for $r$ in the following equation. $a = bq + r$ where $q = \lfloor a/b \rfloor$. The result -$r$ is said to be ``congruent to $a$ modulo $b$'' which is also written as $r \equiv a \mbox{ (mod }b\mbox{)}$. In other vernacular $r$ is known as the -``modular residue'' which leads to ``quadratic residue''\footnote{That's fancy talk for $b \equiv a^2 \mbox{ (mod }p\mbox{)}$.} and -other forms of residues. - -Modular reductions are normally used to create either finite groups, rings or fields. The most common usage for performance driven modular reductions -is in modular exponentiation algorithms. That is to compute $d = a^b \mbox{ (mod }c\mbox{)}$ as fast as possible. This operation is used in the -RSA and Diffie-Hellman public key algorithms, for example. Modular multiplication and squaring also appears as a fundamental operation in -elliptic curve cryptographic algorithms. As will be discussed in the subsequent chapter there exist fast algorithms for computing modular -exponentiations without having to perform (\textit{in this example}) $b - 1$ multiplications. These algorithms will produce partial results in the -range $0 \le x < c^2$ which can be taken advantage of to create several efficient algorithms. They have also been used to create redundancy check -algorithms known as CRCs, error correction codes such as Reed-Solomon and solve a variety of number theoeretic problems. - -\section{The Barrett Reduction} -The Barrett reduction algorithm \cite{BARRETT} was inspired by fast division algorithms which multiply by the reciprocal to emulate -division. Barretts observation was that the residue $c$ of $a$ modulo $b$ is equal to - -\begin{equation} -c = a - b \cdot \lfloor a/b \rfloor -\end{equation} - -Since algorithms such as modular exponentiation would be using the same modulus extensively, typical DSP\footnote{It is worth noting that Barrett's paper -targeted the DSP56K processor.} intuition would indicate the next step would be to replace $a/b$ by a multiplication by the reciprocal. However, -DSP intuition on its own will not work as these numbers are considerably larger than the precision of common DSP floating point data types. -It would take another common optimization to optimize the algorithm. - -\subsection{Fixed Point Arithmetic} -The trick used to optimize the above equation is based on a technique of emulating floating point data types with fixed precision integers. Fixed -point arithmetic would become very popular as it greatly optimize the ``3d-shooter'' genre of games in the mid 1990s when floating point units were -fairly slow if not unavailable. The idea behind fixed point arithmetic is to take a normal $k$-bit integer data type and break it into $p$-bit -integer and a $q$-bit fraction part (\textit{where $p+q = k$}). - -In this system a $k$-bit integer $n$ would actually represent $n/2^q$. For example, with $q = 4$ the integer $n = 37$ would actually represent the -value $2.3125$. To multiply two fixed point numbers the integers are multiplied using traditional arithmetic and subsequently normalized by -moving the implied decimal point back to where it should be. For example, with $q = 4$ to multiply the integers $9$ and $5$ they must be converted -to fixed point first by multiplying by $2^q$. Let $a = 9(2^q)$ represent the fixed point representation of $9$ and $b = 5(2^q)$ represent the -fixed point representation of $5$. The product $ab$ is equal to $45(2^{2q})$ which when normalized by dividing by $2^q$ produces $45(2^q)$. - -This technique became popular since a normal integer multiplication and logical shift right are the only required operations to perform a multiplication -of two fixed point numbers. Using fixed point arithmetic, division can be easily approximated by multiplying by the reciprocal. If $2^q$ is -equivalent to one than $2^q/b$ is equivalent to the fixed point approximation of $1/b$ using real arithmetic. Using this fact dividing an integer -$a$ by another integer $b$ can be achieved with the following expression. - -\begin{equation} -\lfloor a / b \rfloor \mbox{ }\approx\mbox{ } \lfloor (a \cdot \lfloor 2^q / b \rfloor)/2^q \rfloor -\end{equation} - -The precision of the division is proportional to the value of $q$. If the divisor $b$ is used frequently as is the case with -modular exponentiation pre-computing $2^q/b$ will allow a division to be performed with a multiplication and a right shift. Both operations -are considerably faster than division on most processors. - -Consider dividing $19$ by $5$. The correct result is $\lfloor 19/5 \rfloor = 3$. With $q = 3$ the reciprocal is $\lfloor 2^q/5 \rfloor = 1$ which -leads to a product of $19$ which when divided by $2^q$ produces $2$. However, with $q = 4$ the reciprocal is $\lfloor 2^q/5 \rfloor = 3$ and -the result of the emulated division is $\lfloor 3 \cdot 19 / 2^q \rfloor = 3$ which is correct. The value of $2^q$ must be close to or ideally -larger than the dividend. In effect if $a$ is the dividend then $q$ should allow $0 \le \lfloor a/2^q \rfloor \le 1$ in order for this approach -to work correctly. Plugging this form of divison into the original equation the following modular residue equation arises. - -\begin{equation} -c = a - b \cdot \lfloor (a \cdot \lfloor 2^q / b \rfloor)/2^q \rfloor -\end{equation} - -Using the notation from \cite{BARRETT} the value of $\lfloor 2^q / b \rfloor$ will be represented by the $\mu$ symbol. Using the $\mu$ -variable also helps re-inforce the idea that it is meant to be computed once and re-used. - -\begin{equation} -c = a - b \cdot \lfloor (a \cdot \mu)/2^q \rfloor -\end{equation} - -Provided that $2^q \ge a$ this algorithm will produce a quotient that is either exactly correct or off by a value of one. In the context of Barrett -reduction the value of $a$ is bound by $0 \le a \le (b - 1)^2$ meaning that $2^q \ge b^2$ is sufficient to ensure the reciprocal will have enough -precision. - -Let $n$ represent the number of digits in $b$. This algorithm requires approximately $2n^2$ single precision multiplications to produce the quotient and -another $n^2$ single precision multiplications to find the residue. In total $3n^2$ single precision multiplications are required to -reduce the number. - -For example, if $b = 1179677$ and $q = 41$ ($2^q > b^2$), then the reciprocal $\mu$ is equal to $\lfloor 2^q / b \rfloor = 1864089$. Consider reducing -$a = 180388626447$ modulo $b$ using the above reduction equation. The quotient using the new formula is $\lfloor (a \cdot \mu) / 2^q \rfloor = 152913$. -By subtracting $152913b$ from $a$ the correct residue $a \equiv 677346 \mbox{ (mod }b\mbox{)}$ is found. - -\subsection{Choosing a Radix Point} -Using the fixed point representation a modular reduction can be performed with $3n^2$ single precision multiplications. If that were the best -that could be achieved a full division\footnote{A division requires approximately $O(2cn^2)$ single precision multiplications for a small value of $c$. -See~\ref{sec:division} for further details.} might as well be used in its place. The key to optimizing the reduction is to reduce the precision of -the initial multiplication that finds the quotient. - -Let $a$ represent the number of which the residue is sought. Let $b$ represent the modulus used to find the residue. Let $m$ represent -the number of digits in $b$. For the purposes of this discussion we will assume that the number of digits in $a$ is $2m$, which is generally true if -two $m$-digit numbers have been multiplied. Dividing $a$ by $b$ is the same as dividing a $2m$ digit integer by a $m$ digit integer. Digits below the -$m - 1$'th digit of $a$ will contribute at most a value of $1$ to the quotient because $\beta^k < b$ for any $0 \le k \le m - 1$. Another way to -express this is by re-writing $a$ as two parts. If $a' \equiv a \mbox{ (mod }b^m\mbox{)}$ and $a'' = a - a'$ then -${a \over b} \equiv {{a' + a''} \over b}$ which is equivalent to ${a' \over b} + {a'' \over b}$. Since $a'$ is bound to be less than $b$ the quotient -is bound by $0 \le {a' \over b} < 1$. - -Since the digits of $a'$ do not contribute much to the quotient the observation is that they might as well be zero. However, if the digits -``might as well be zero'' they might as well not be there in the first place. Let $q_0 = \lfloor a/\beta^{m-1} \rfloor$ represent the input -with the irrelevant digits trimmed. Now the modular reduction is trimmed to the almost equivalent equation - -\begin{equation} -c = a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor -\end{equation} - -Note that the original divisor $2^q$ has been replaced with $\beta^{m+1}$ where in this case $q$ is a multiple of $lg(\beta)$. Also note that the -exponent on the divisor when added to the amount $q_0$ was shifted by equals $2m$. If the optimization had not been performed the divisor -would have the exponent $2m$ so in the end the exponents do ``add up''. Using the above equation the quotient -$\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ can be off from the true quotient by at most two. The original fixed point quotient can be off -by as much as one (\textit{provided the radix point is chosen suitably}) and now that the lower irrelevent digits have been trimmed the quotient -can be off by an additional value of one for a total of at most two. This implies that -$0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$. By first subtracting $b$ times the quotient and then conditionally subtracting -$b$ once or twice the residue is found. - -The quotient is now found using $(m + 1)(m) = m^2 + m$ single precision multiplications and the residue with an additional $m^2$ single -precision multiplications, ignoring the subtractions required. In total $2m^2 + m$ single precision multiplications are required to find the residue. -This is considerably faster than the original attempt. - -For example, let $\beta = 10$ represent the radix of the digits. Let $b = 9999$ represent the modulus which implies $m = 4$. Let $a = 99929878$ -represent the value of which the residue is desired. In this case $q = 8$ since $10^7 < 9999^2$ meaning that $\mu = \lfloor \beta^{q}/b \rfloor = 10001$. -With the new observation the multiplicand for the quotient is equal to $q_0 = \lfloor a / \beta^{m - 1} \rfloor = 99929$. The quotient is then -$\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor = 9993$. Subtracting $9993b$ from $a$ and the correct residue $a \equiv 9871 \mbox{ (mod }b\mbox{)}$ -is found. - -\subsection{Trimming the Quotient} -So far the reduction algorithm has been optimized from $3m^2$ single precision multiplications down to $2m^2 + m$ single precision multiplications. As -it stands now the algorithm is already fairly fast compared to a full integer division algorithm. However, there is still room for -optimization. - -After the first multiplication inside the quotient ($q_0 \cdot \mu$) the value is shifted right by $m + 1$ places effectively nullifying the lower -half of the product. It would be nice to be able to remove those digits from the product to effectively cut down the number of single precision -multiplications. If the number of digits in the modulus $m$ is far less than $\beta$ a full product is not required for the algorithm to work properly. -In fact the lower $m - 2$ digits will not affect the upper half of the product at all and do not need to be computed. - -The value of $\mu$ is a $m$-digit number and $q_0$ is a $m + 1$ digit number. Using a full multiplier $(m + 1)(m) = m^2 + m$ single precision -multiplications would be required. Using a multiplier that will only produce digits at and above the $m - 1$'th digit reduces the number -of single precision multiplications to ${m^2 + m} \over 2$ single precision multiplications. - -\subsection{Trimming the Residue} -After the quotient has been calculated it is used to reduce the input. As previously noted the algorithm is not exact and it can be off by a small -multiple of the modulus, that is $0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$. If $b$ is $m$ digits than the -result of reduction equation is a value of at most $m + 1$ digits (\textit{provided $3 < \beta$}) implying that the upper $m - 1$ digits are -implicitly zero. - -The next optimization arises from this very fact. Instead of computing $b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ using a full -$O(m^2)$ multiplication algorithm only the lower $m+1$ digits of the product have to be computed. Similarly the value of $a$ can -be reduced modulo $\beta^{m+1}$ before the multiple of $b$ is subtracted which simplifes the subtraction as well. A multiplication that produces -only the lower $m+1$ digits requires ${m^2 + 3m - 2} \over 2$ single precision multiplications. - -With both optimizations in place the algorithm is the algorithm Barrett proposed. It requires $m^2 + 2m - 1$ single precision multiplications which -is considerably faster than the straightforward $3m^2$ method. - -\subsection{The Barrett Algorithm} -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_reduce}. \\ -\textbf{Input}. mp\_int $a$, mp\_int $b$ and $\mu = \lfloor \beta^{2m}/b \rfloor, m = \lceil lg_{\beta}(b) \rceil, (0 \le a < b^2, b > 1)$ \\ -\textbf{Output}. $a \mbox{ (mod }b\mbox{)}$ \\ -\hline \\ -Let $m$ represent the number of digits in $b$. \\ -1. Make a copy of $a$ and store it in $q$. (\textit{mp\_init\_copy}) \\ -2. $q \leftarrow \lfloor q / \beta^{m - 1} \rfloor$ (\textit{mp\_rshd}) \\ -\\ -Produce the quotient. \\ -3. $q \leftarrow q \cdot \mu$ (\textit{note: only produce digits at or above $m-1$}) \\ -4. $q \leftarrow \lfloor q / \beta^{m + 1} \rfloor$ \\ -\\ -Subtract the multiple of modulus from the input. \\ -5. $a \leftarrow a \mbox{ (mod }\beta^{m+1}\mbox{)}$ (\textit{mp\_mod\_2d}) \\ -6. $q \leftarrow q \cdot b \mbox{ (mod }\beta^{m+1}\mbox{)}$ (\textit{s\_mp\_mul\_digs}) \\ -7. $a \leftarrow a - q$ (\textit{mp\_sub}) \\ -\\ -Add $\beta^{m+1}$ if a carry occured. \\ -8. If $a < 0$ then (\textit{mp\_cmp\_d}) \\ -\hspace{3mm}8.1 $q \leftarrow 1$ (\textit{mp\_set}) \\ -\hspace{3mm}8.2 $q \leftarrow q \cdot \beta^{m+1}$ (\textit{mp\_lshd}) \\ -\hspace{3mm}8.3 $a \leftarrow a + q$ \\ -\\ -Now subtract the modulus if the residue is too large (e.g. quotient too small). \\ -9. While $a \ge b$ do (\textit{mp\_cmp}) \\ -\hspace{3mm}9.1 $c \leftarrow a - b$ \\ -10. Clear $q$. \\ -11. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_reduce} -\end{figure} - -\textbf{Algorithm mp\_reduce.} -This algorithm will reduce the input $a$ modulo $b$ in place using the Barrett algorithm. It is loosely based on algorithm 14.42 of HAC -\cite[pp. 602]{HAC} which is based on the paper from Paul Barrett \cite{BARRETT}. The algorithm has several restrictions and assumptions which must -be adhered to for the algorithm to work. - -First the modulus $b$ is assumed to be positive and greater than one. If the modulus were less than or equal to one than subtracting -a multiple of it would either accomplish nothing or actually enlarge the input. The input $a$ must be in the range $0 \le a < b^2$ in order -for the quotient to have enough precision. If $a$ is the product of two numbers that were already reduced modulo $b$, this will not be a problem. -Technically the algorithm will still work if $a \ge b^2$ but it will take much longer to finish. The value of $\mu$ is passed as an argument to this -algorithm and is assumed to be calculated and stored before the algorithm is used. - -Recall that the multiplication for the quotient on step 3 must only produce digits at or above the $m-1$'th position. An algorithm called -$s\_mp\_mul\_high\_digs$ which has not been presented is used to accomplish this task. The algorithm is based on $s\_mp\_mul\_digs$ except that -instead of stopping at a given level of precision it starts at a given level of precision. This optimal algorithm can only be used if the number -of digits in $b$ is very much smaller than $\beta$. - -While it is known that -$a \ge b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ only the lower $m+1$ digits are being used to compute the residue, so an implied -``borrow'' from the higher digits might leave a negative result. After the multiple of the modulus has been subtracted from $a$ the residue must be -fixed up in case it is negative. The invariant $\beta^{m+1}$ must be added to the residue to make it positive again. - -The while loop at step 9 will subtract $b$ until the residue is less than $b$. If the algorithm is performed correctly this step is -performed at most twice, and on average once. However, if $a \ge b^2$ than it will iterate substantially more times than it should. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* reduces x mod m, assumes 0 < x < m**2, mu is -018 * precomputed via mp_reduce_setup. -019 * From HAC pp.604 Algorithm 14.42 -020 */ -021 int mp_reduce (mp_int * x, mp_int * m, mp_int * mu) -022 \{ -023 mp_int q; -024 int res, um = m->used; -025 -026 /* q = x */ -027 if ((res = mp_init_copy (&q, x)) != MP_OKAY) \{ -028 return res; -029 \} -030 -031 /* q1 = x / b**(k-1) */ -032 mp_rshd (&q, um - 1); -033 -034 /* according to HAC this optimization is ok */ -035 if (((unsigned long) um) > (((mp_digit)1) << (DIGIT_BIT - 1))) \{ -036 if ((res = mp_mul (&q, mu, &q)) != MP_OKAY) \{ -037 goto CLEANUP; -038 \} -039 \} else \{ -040 #ifdef BN_S_MP_MUL_HIGH_DIGS_C -041 if ((res = s_mp_mul_high_digs (&q, mu, &q, um)) != MP_OKAY) \{ -042 goto CLEANUP; -043 \} -044 #elif defined(BN_FAST_S_MP_MUL_HIGH_DIGS_C) -045 if ((res = fast_s_mp_mul_high_digs (&q, mu, &q, um)) != MP_OKAY) \{ -046 goto CLEANUP; -047 \} -048 #else -049 \{ -050 res = MP_VAL; -051 goto CLEANUP; -052 \} -053 #endif -054 \} -055 -056 /* q3 = q2 / b**(k+1) */ -057 mp_rshd (&q, um + 1); -058 -059 /* x = x mod b**(k+1), quick (no division) */ -060 if ((res = mp_mod_2d (x, DIGIT_BIT * (um + 1), x)) != MP_OKAY) \{ -061 goto CLEANUP; -062 \} -063 -064 /* q = q * m mod b**(k+1), quick (no division) */ -065 if ((res = s_mp_mul_digs (&q, m, &q, um + 1)) != MP_OKAY) \{ -066 goto CLEANUP; -067 \} -068 -069 /* x = x - q */ -070 if ((res = mp_sub (x, &q, x)) != MP_OKAY) \{ -071 goto CLEANUP; -072 \} -073 -074 /* If x < 0, add b**(k+1) to it */ -075 if (mp_cmp_d (x, 0) == MP_LT) \{ -076 mp_set (&q, 1); -077 if ((res = mp_lshd (&q, um + 1)) != MP_OKAY) -078 goto CLEANUP; -079 if ((res = mp_add (x, &q, x)) != MP_OKAY) -080 goto CLEANUP; -081 \} -082 -083 /* Back off if it's too big */ -084 while (mp_cmp (x, m) != MP_LT) \{ -085 if ((res = s_mp_sub (x, m, x)) != MP_OKAY) \{ -086 goto CLEANUP; -087 \} -088 \} -089 -090 CLEANUP: -091 mp_clear (&q); -092 -093 return res; -094 \} -095 #endif -\end{alltt} -\end{small} - -The first multiplication that determines the quotient can be performed by only producing the digits from $m - 1$ and up. This essentially halves -the number of single precision multiplications required. However, the optimization is only safe if $\beta$ is much larger than the number of digits -in the modulus. In the source code this is evaluated on lines 36 to 43 where algorithm s\_mp\_mul\_high\_digs is used when it is -safe to do so. - -\subsection{The Barrett Setup Algorithm} -In order to use algorithm mp\_reduce the value of $\mu$ must be calculated in advance. Ideally this value should be computed once and stored for -future use so that the Barrett algorithm can be used without delay. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_reduce\_setup}. \\ -\textbf{Input}. mp\_int $a$ ($a > 1$) \\ -\textbf{Output}. $\mu \leftarrow \lfloor \beta^{2m}/a \rfloor$ \\ -\hline \\ -1. $\mu \leftarrow 2^{2 \cdot lg(\beta) \cdot m}$ (\textit{mp\_2expt}) \\ -2. $\mu \leftarrow \lfloor \mu / b \rfloor$ (\textit{mp\_div}) \\ -3. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_reduce\_setup} -\end{figure} - -\textbf{Algorithm mp\_reduce\_setup.} -This algorithm computes the reciprocal $\mu$ required for Barrett reduction. First $\beta^{2m}$ is calculated as $2^{2 \cdot lg(\beta) \cdot m}$ which -is equivalent and much faster. The final value is computed by taking the integer quotient of $\lfloor \mu / b \rfloor$. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_setup.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* pre-calculate the value required for Barrett reduction -018 * For a given modulus "b" it calulates the value required in "a" -019 */ -020 int mp_reduce_setup (mp_int * a, mp_int * b) -021 \{ -022 int res; -023 -024 if ((res = mp_2expt (a, b->used * 2 * DIGIT_BIT)) != MP_OKAY) \{ -025 return res; -026 \} -027 return mp_div (a, b, a, NULL); -028 \} -029 #endif -\end{alltt} -\end{small} - -This simple routine calculates the reciprocal $\mu$ required by Barrett reduction. Note the extended usage of algorithm mp\_div where the variable -which would received the remainder is passed as NULL. As will be discussed in~\ref{sec:division} the division routine allows both the quotient and the -remainder to be passed as NULL meaning to ignore the value. - -\section{The Montgomery Reduction} -Montgomery reduction\footnote{Thanks to Niels Ferguson for his insightful explanation of the algorithm.} \cite{MONT} is by far the most interesting -form of reduction in common use. It computes a modular residue which is not actually equal to the residue of the input yet instead equal to a -residue times a constant. However, as perplexing as this may sound the algorithm is relatively simple and very efficient. - -Throughout this entire section the variable $n$ will represent the modulus used to form the residue. As will be discussed shortly the value of -$n$ must be odd. The variable $x$ will represent the quantity of which the residue is sought. Similar to the Barrett algorithm the input -is restricted to $0 \le x < n^2$. To begin the description some simple number theory facts must be established. - -\textbf{Fact 1.} Adding $n$ to $x$ does not change the residue since in effect it adds one to the quotient $\lfloor x / n \rfloor$. Another way -to explain this is that $n$ is (\textit{or multiples of $n$ are}) congruent to zero modulo $n$. Adding zero will not change the value of the residue. - -\textbf{Fact 2.} If $x$ is even then performing a division by two in $\Z$ is congruent to $x \cdot 2^{-1} \mbox{ (mod }n\mbox{)}$. Actually -this is an application of the fact that if $x$ is evenly divisible by any $k \in \Z$ then division in $\Z$ will be congruent to -multiplication by $k^{-1}$ modulo $n$. - -From these two simple facts the following simple algorithm can be derived. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Montgomery Reduction}. \\ -\textbf{Input}. Integer $x$, $n$ and $k$ \\ -\textbf{Output}. $2^{-k}x \mbox{ (mod }n\mbox{)}$ \\ -\hline \\ -1. for $t$ from $1$ to $k$ do \\ -\hspace{3mm}1.1 If $x$ is odd then \\ -\hspace{6mm}1.1.1 $x \leftarrow x + n$ \\ -\hspace{3mm}1.2 $x \leftarrow x/2$ \\ -2. Return $x$. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm Montgomery Reduction} -\end{figure} - -The algorithm reduces the input one bit at a time using the two congruencies stated previously. Inside the loop $n$, which is odd, is -added to $x$ if $x$ is odd. This forces $x$ to be even which allows the division by two in $\Z$ to be congruent to a modular division by two. Since -$x$ is assumed to be initially much larger than $n$ the addition of $n$ will contribute an insignificant magnitude to $x$. Let $r$ represent the -final result of the Montgomery algorithm. If $k > lg(n)$ and $0 \le x < n^2$ then the final result is limited to -$0 \le r < \lfloor x/2^k \rfloor + n$. As a result at most a single subtraction is required to get the residue desired. - -\begin{figure}[here] -\begin{small} -\begin{center} -\begin{tabular}{|c|l|} -\hline \textbf{Step number ($t$)} & \textbf{Result ($x$)} \\ -\hline $1$ & $x + n = 5812$, $x/2 = 2906$ \\ -\hline $2$ & $x/2 = 1453$ \\ -\hline $3$ & $x + n = 1710$, $x/2 = 855$ \\ -\hline $4$ & $x + n = 1112$, $x/2 = 556$ \\ -\hline $5$ & $x/2 = 278$ \\ -\hline $6$ & $x/2 = 139$ \\ -\hline $7$ & $x + n = 396$, $x/2 = 198$ \\ -\hline $8$ & $x/2 = 99$ \\ -\hline -\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 sub-section 6.3.3. - -Step 5 is the main reduction loop of the algorithm. The value of $\mu$ is calculated once per iteration in the outer loop. The inner loop -calculates $x + \mu n \beta^{ix}$ by multiplying $\mu n$ and adding the result to $x$ shifted by $ix$ digits. Both the addition and -multiplication are performed in the same loop to save time and memory. Step 5.4 will handle any additional carries that escape the inner loop. - -Using a quick inspection this algorithm requires $n$ single precision multiplications for the outer loop and $n^2$ single precision multiplications -in the inner loop. In total $n^2 + n$ single precision multiplications which compares favourably to Barrett at $n^2 + 2n - 1$ single precision -multiplications. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_montgomery\_reduce.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* computes xR**-1 == x (mod N) via Montgomery Reduction */ -018 int -019 mp_montgomery_reduce (mp_int * x, mp_int * n, mp_digit rho) -020 \{ -021 int ix, res, digs; -022 mp_digit mu; -023 -024 /* can the fast reduction [comba] method be used? -025 * -026 * Note that unlike in mul you're safely allowed *less* -027 * than the available columns [255 per default] since carries -028 * are fixed up in the inner loop. -029 */ -030 digs = n->used * 2 + 1; -031 if ((digs < MP_WARRAY) && -032 n->used < -033 (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) \{ -034 return fast_mp_montgomery_reduce (x, n, rho); -035 \} -036 -037 /* grow the input as required */ -038 if (x->alloc < digs) \{ -039 if ((res = mp_grow (x, digs)) != MP_OKAY) \{ -040 return res; -041 \} -042 \} -043 x->used = digs; -044 -045 for (ix = 0; ix < n->used; ix++) \{ -046 /* mu = ai * rho mod b -047 * -048 * The value of rho must be precalculated via -049 * montgomery_setup() such that -050 * it equals -1/n0 mod b this allows the -051 * following inner loop to reduce the -052 * input one digit at a time -053 */ -054 mu = (mp_digit) (((mp_word)x->dp[ix]) * ((mp_word)rho) & MP_MASK); -055 -056 /* a = a + mu * m * b**i */ -057 \{ -058 register int iy; -059 register mp_digit *tmpn, *tmpx, u; -060 register mp_word r; -061 -062 /* alias for digits of the modulus */ -063 tmpn = n->dp; -064 -065 /* alias for the digits of x [the input] */ -066 tmpx = x->dp + ix; -067 -068 /* set the carry to zero */ -069 u = 0; -070 -071 /* Multiply and add in place */ -072 for (iy = 0; iy < n->used; iy++) \{ -073 /* compute product and sum */ -074 r = ((mp_word)mu) * ((mp_word)*tmpn++) + -075 ((mp_word) u) + ((mp_word) * tmpx); -076 -077 /* get carry */ -078 u = (mp_digit)(r >> ((mp_word) DIGIT_BIT)); -079 -080 /* fix digit */ -081 *tmpx++ = (mp_digit)(r & ((mp_word) MP_MASK)); -082 \} -083 /* At this point the ix'th digit of x should be zero */ -084 -085 -086 /* propagate carries upwards as required*/ -087 while (u) \{ -088 *tmpx += u; -089 u = *tmpx >> DIGIT_BIT; -090 *tmpx++ &= MP_MASK; -091 \} -092 \} -093 \} -094 -095 /* at this point the n.used'th least -096 * significant digits of x are all zero -097 * which means we can shift x to the -098 * right by n.used digits and the -099 * residue is unchanged. -100 */ -101 -102 /* x = x/b**n.used */ -103 mp_clamp(x); -104 mp_rshd (x, n->used); -105 -106 /* if x >= n then x = x - n */ -107 if (mp_cmp_mag (x, n) != MP_LT) \{ -108 return s_mp_sub (x, n, x); -109 \} -110 -111 return MP_OKAY; -112 \} -113 #endif -\end{alltt} -\end{small} - -This is the baseline implementation of the Montgomery reduction algorithm. Lines 30 to 35 determine if the Comba based -routine can be used instead. Line 48 computes the value of $\mu$ for that particular iteration of the outer loop. - -The multiplication $\mu n \beta^{ix}$ is performed in one step in the inner loop. The alias $tmpx$ refers to the $ix$'th digit of $x$ and -the alias $tmpn$ refers to the modulus $n$. - -\subsection{Faster ``Comba'' Montgomery Reduction} - -The Montgomery reduction requires fewer single precision multiplications than a Barrett reduction, however it is much slower due to the serial -nature of the inner loop. The Barrett reduction algorithm requires two slightly modified multipliers which can be implemented with the Comba -technique. The Montgomery reduction algorithm cannot directly use the Comba technique to any significant advantage since the inner loop calculates -a $k \times 1$ product $k$ times. - -The biggest obstacle is that at the $ix$'th iteration of the outer loop the value of $x_{ix}$ is required to calculate $\mu$. This means the -carries from $0$ to $ix - 1$ must have been propagated upwards to form a valid $ix$'th digit. The solution as it turns out is very simple. -Perform a Comba like multiplier and inside the outer loop just after the inner loop fix up the $ix + 1$'th digit by forwarding the carry. - -With this change in place the Montgomery reduction algorithm can be performed with a Comba style multiplication loop which substantially increases -the speed of the algorithm. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{fast\_mp\_montgomery\_reduce}. \\ -\textbf{Input}. mp\_int $x$, mp\_int $n$ and a digit $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\ -\hspace{11.5mm}($0 \le x < n^2, n > 1, (n, \beta) = 1, \beta^k > n$) \\ -\textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\ -\hline \\ -Place an array of \textbf{MP\_WARRAY} mp\_word variables called $\hat W$ on the stack. \\ -1. if $x.alloc < n.used + 1$ then grow $x$ to $n.used + 1$ digits. \\ -Copy the digits of $x$ into the array $\hat W$ \\ -2. For $ix$ from $0$ to $x.used - 1$ do \\ -\hspace{3mm}2.1 $\hat W_{ix} \leftarrow x_{ix}$ \\ -3. For $ix$ from $x.used$ to $2n.used - 1$ do \\ -\hspace{3mm}3.1 $\hat W_{ix} \leftarrow 0$ \\ -Elimiate the lower $k$ digits. \\ -4. for $ix$ from $0$ to $n.used - 1$ do \\ -\hspace{3mm}4.1 $\mu \leftarrow \hat W_{ix} \cdot \rho \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{3mm}4.2 For $iy$ from $0$ to $n.used - 1$ do \\ -\hspace{6mm}4.2.1 $\hat W_{iy + ix} \leftarrow \hat W_{iy + ix} + \mu \cdot n_{iy}$ \\ -\hspace{3mm}4.3 $\hat W_{ix + 1} \leftarrow \hat W_{ix + 1} + \lfloor \hat W_{ix} / \beta \rfloor$ \\ -Propagate carries upwards. \\ -5. for $ix$ from $n.used$ to $2n.used + 1$ do \\ -\hspace{3mm}5.1 $\hat W_{ix + 1} \leftarrow \hat W_{ix + 1} + \lfloor \hat W_{ix} / \beta \rfloor$ \\ -Shift right and reduce modulo $\beta$ simultaneously. \\ -6. for $ix$ from $0$ to $n.used + 1$ do \\ -\hspace{3mm}6.1 $x_{ix} \leftarrow \hat W_{ix + n.used} \mbox{ (mod }\beta\mbox{)}$ \\ -Zero excess digits and fixup $x$. \\ -7. if $x.used > n.used + 1$ then do \\ -\hspace{3mm}7.1 for $ix$ from $n.used + 1$ to $x.used - 1$ do \\ -\hspace{6mm}7.1.1 $x_{ix} \leftarrow 0$ \\ -8. $x.used \leftarrow n.used + 1$ \\ -9. Clamp excessive digits of $x$. \\ -10. If $x \ge n$ then \\ -\hspace{3mm}10.1 $x \leftarrow x - n$ \\ -11. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm fast\_mp\_montgomery\_reduce} -\end{figure} - -\textbf{Algorithm fast\_mp\_montgomery\_reduce.} -This algorithm will compute the Montgomery reduction of $x$ modulo $n$ using the Comba technique. It is on most computer platforms significantly -faster than algorithm mp\_montgomery\_reduce and algorithm mp\_reduce (\textit{Barrett reduction}). The algorithm has the same restrictions -on the input as the baseline reduction algorithm. An additional two restrictions are imposed on this algorithm. The number of digits $k$ in the -the modulus $n$ must not violate $MP\_WARRAY > 2k +1$ and $n < \delta$. When $\beta = 2^{28}$ this algorithm can be used to reduce modulo -a modulus of at most $3,556$ bits in length. - -As in the other Comba reduction algorithms there is a $\hat W$ array which stores the columns of the product. It is initially filled with the -contents of $x$ with the excess digits zeroed. The reduction loop is very similar the to the baseline loop at heart. The multiplication on step -4.1 can be single precision only since $ab \mbox{ (mod }\beta\mbox{)} \equiv (a \mbox{ mod }\beta)(b \mbox{ mod }\beta)$. Some multipliers such -as those on the ARM processors take a variable length time to complete depending on the number of bytes of result it must produce. By performing -a single precision multiplication instead half the amount of time is spent. - -Also note that digit $\hat W_{ix}$ must have the carry from the $ix - 1$'th digit propagated upwards in order for this to work. That is what step -4.3 will do. In effect over the $n.used$ iterations of the outer loop the $n.used$'th lower columns all have the their carries propagated forwards. Note -how the upper bits of those same words are not reduced modulo $\beta$. This is because those values will be discarded shortly and there is no -point. - -Step 5 will propagate the remainder of the carries upwards. On step 6 the columns are reduced modulo $\beta$ and shifted simultaneously as they are -stored in the destination $x$. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_fast\_mp\_montgomery\_reduce.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* computes xR**-1 == x (mod N) via Montgomery Reduction -018 * -019 * This is an optimized implementation of montgomery_reduce -020 * which uses the comba method to quickly calculate the columns of the -021 * reduction. -022 * -023 * Based on Algorithm 14.32 on pp.601 of HAC. -024 */ -025 int fast_mp_montgomery_reduce (mp_int * x, mp_int * n, mp_digit rho) -026 \{ -027 int ix, res, olduse; -028 mp_word W[MP_WARRAY]; -029 -030 /* get old used count */ -031 olduse = x->used; -032 -033 /* grow a as required */ -034 if (x->alloc < n->used + 1) \{ -035 if ((res = mp_grow (x, n->used + 1)) != MP_OKAY) \{ -036 return res; -037 \} -038 \} -039 -040 /* first we have to get the digits of the input into -041 * an array of double precision words W[...] -042 */ -043 \{ -044 register mp_word *_W; -045 register mp_digit *tmpx; -046 -047 /* alias for the W[] array */ -048 _W = W; -049 -050 /* alias for the digits of x*/ -051 tmpx = x->dp; -052 -053 /* copy the digits of a into W[0..a->used-1] */ -054 for (ix = 0; ix < x->used; ix++) \{ -055 *_W++ = *tmpx++; -056 \} -057 -058 /* zero the high words of W[a->used..m->used*2] */ -059 for (; ix < n->used * 2 + 1; ix++) \{ -060 *_W++ = 0; -061 \} -062 \} -063 -064 /* now we proceed to zero successive digits -065 * from the least significant upwards -066 */ -067 for (ix = 0; ix < n->used; ix++) \{ -068 /* mu = ai * m' mod b -069 * -070 * We avoid a double precision multiplication (which isn't required) -071 * by casting the value down to a mp_digit. Note this requires -072 * that W[ix-1] have the carry cleared (see after the inner loop) -073 */ -074 register mp_digit mu; -075 mu = (mp_digit) (((W[ix] & MP_MASK) * rho) & MP_MASK); -076 -077 /* a = a + mu * m * b**i -078 * -079 * This is computed in place and on the fly. The multiplication -080 * by b**i is handled by offseting which columns the results -081 * are added to. -082 * -083 * Note the comba method normally doesn't handle carries in the -084 * inner loop In this case we fix the carry from the previous -085 * column since the Montgomery reduction requires digits of the -086 * result (so far) [see above] to work. This is -087 * handled by fixing up one carry after the inner loop. The -088 * carry fixups are done in order so after these loops the -089 * first m->used words of W[] have the carries fixed -090 */ -091 \{ -092 register int iy; -093 register mp_digit *tmpn; -094 register mp_word *_W; -095 -096 /* alias for the digits of the modulus */ -097 tmpn = n->dp; -098 -099 /* Alias for the columns set by an offset of ix */ -100 _W = W + ix; -101 -102 /* inner loop */ -103 for (iy = 0; iy < n->used; iy++) \{ -104 *_W++ += ((mp_word)mu) * ((mp_word)*tmpn++); -105 \} -106 \} -107 -108 /* now fix carry for next digit, W[ix+1] */ -109 W[ix + 1] += W[ix] >> ((mp_word) DIGIT_BIT); -110 \} -111 -112 /* now we have to propagate the carries and -113 * shift the words downward [all those least -114 * significant digits we zeroed]. -115 */ -116 \{ -117 register mp_digit *tmpx; -118 register mp_word *_W, *_W1; -119 -120 /* nox fix rest of carries */ -121 -122 /* alias for current word */ -123 _W1 = W + ix; -124 -125 /* alias for next word, where the carry goes */ -126 _W = W + ++ix; -127 -128 for (; ix <= n->used * 2 + 1; ix++) \{ -129 *_W++ += *_W1++ >> ((mp_word) DIGIT_BIT); -130 \} -131 -132 /* copy out, A = A/b**n -133 * -134 * The result is A/b**n but instead of converting from an -135 * array of mp_word to mp_digit than calling mp_rshd -136 * we just copy them in the right order -137 */ -138 -139 /* alias for destination word */ -140 tmpx = x->dp; -141 -142 /* alias for shifted double precision result */ -143 _W = W + n->used; -144 -145 for (ix = 0; ix < n->used + 1; ix++) \{ -146 *tmpx++ = (mp_digit)(*_W++ & ((mp_word) MP_MASK)); -147 \} -148 -149 /* zero oldused digits, if the input a was larger than -150 * m->used+1 we'll have to clear the digits -151 */ -152 for (; ix < olduse; ix++) \{ -153 *tmpx++ = 0; -154 \} -155 \} -156 -157 /* set the max used and clamp */ -158 x->used = n->used + 1; -159 mp_clamp (x); -160 -161 /* if A >= m then A = A - m */ -162 if (mp_cmp_mag (x, n) != MP_LT) \{ -163 return s_mp_sub (x, n, x); -164 \} -165 return MP_OKAY; -166 \} -167 #endif -\end{alltt} -\end{small} - -The $\hat W$ array is first filled with digits of $x$ on line 50 then the rest of the digits are zeroed on line 54. 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 109 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 108 propagates the rest of the carries upwards through the columns. The for loop on line 125 reduces the columns -modulo $\beta$ and shifts them $k$ places at the same time. The alias $\_ \hat W$ actually refers to the array $\hat W$ starting at the $n.used$'th -digit, that is $\_ \hat W_{t} = \hat W_{n.used + t}$. - -\subsection{Montgomery Setup} -To calculate the variable $\rho$ a relatively simple algorithm will be required. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_montgomery\_setup}. \\ -\textbf{Input}. mp\_int $n$ ($n > 1$ and $(n, 2) = 1$) \\ -\textbf{Output}. $\rho \equiv -1/n_0 \mbox{ (mod }\beta\mbox{)}$ \\ -\hline \\ -1. $b \leftarrow n_0$ \\ -2. If $b$ is even return(\textit{MP\_VAL}) \\ -3. $x \leftarrow ((b + 2) \mbox{ AND } 4) << 1) + b$ \\ -4. for $k$ from 0 to $\lceil lg(lg(\beta)) \rceil - 2$ do \\ -\hspace{3mm}4.1 $x \leftarrow x \cdot (2 - bx)$ \\ -5. $\rho \leftarrow \beta - x \mbox{ (mod }\beta\mbox{)}$ \\ -6. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_montgomery\_setup} -\end{figure} - -\textbf{Algorithm mp\_montgomery\_setup.} -This algorithm will calculate the value of $\rho$ required within the Montgomery reduction algorithms. It uses a very interesting trick -to calculate $1/n_0$ when $\beta$ is a power of two. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_montgomery\_setup.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* setups the montgomery reduction stuff */ -018 int -019 mp_montgomery_setup (mp_int * n, mp_digit * rho) -020 \{ -021 mp_digit x, b; -022 -023 /* fast inversion mod 2**k -024 * -025 * Based on the fact that -026 * -027 * XA = 1 (mod 2**n) => (X(2-XA)) A = 1 (mod 2**2n) -028 * => 2*X*A - X*X*A*A = 1 -029 * => 2*(1) - (1) = 1 -030 */ -031 b = n->dp[0]; -032 -033 if ((b & 1) == 0) \{ -034 return MP_VAL; -035 \} -036 -037 x = (((b + 2) & 4) << 1) + b; /* here x*a==1 mod 2**4 */ -038 x *= 2 - b * x; /* here x*a==1 mod 2**8 */ -039 #if !defined(MP_8BIT) -040 x *= 2 - b * x; /* here x*a==1 mod 2**16 */ -041 #endif -042 #if defined(MP_64BIT) || !(defined(MP_8BIT) || defined(MP_16BIT)) -043 x *= 2 - b * x; /* here x*a==1 mod 2**32 */ -044 #endif -045 #ifdef MP_64BIT -046 x *= 2 - b * x; /* here x*a==1 mod 2**64 */ -047 #endif -048 -049 /* rho = -1/m mod b */ -050 *rho = (((mp_word)1 << ((mp_word) DIGIT_BIT)) - x) & MP_MASK; -051 -052 return MP_OKAY; -053 \} -054 #endif -\end{alltt} -\end{small} - -This source code computes the value of $\rho$ required to perform Montgomery reduction. It has been modified to avoid performing excess -multiplications when $\beta$ is not the default 28-bits. - -\section{The Diminished Radix Algorithm} -The Diminished Radix method of modular reduction \cite{DRMET} is a fairly clever technique which can be more efficient than either the Barrett -or Montgomery methods for certain forms of moduli. The technique is based on the following simple congruence. - -\begin{equation} -(x \mbox{ mod } n) + k \lfloor x / n \rfloor \equiv x \mbox{ (mod }(n - k)\mbox{)} -\end{equation} - -This observation was used in the MMB \cite{MMB} block cipher to create a diffusion primitive. It used the fact that if $n = 2^{31}$ and $k=1$ that -then a x86 multiplier could produce the 62-bit product and use the ``shrd'' instruction to perform a double-precision right shift. The proof -of the above equation is very simple. First write $x$ in the product form. - -\begin{equation} -x = qn + r -\end{equation} - -Now reduce both sides modulo $(n - k)$. - -\begin{equation} -x \equiv qk + r \mbox{ (mod }(n-k)\mbox{)} -\end{equation} - -The variable $n$ reduces modulo $n - k$ to $k$. By putting $q = \lfloor x/n \rfloor$ and $r = x \mbox{ mod } n$ -into the equation the original congruence is reproduced, thus concluding the proof. The following algorithm is based on this observation. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Diminished Radix Reduction}. \\ -\textbf{Input}. Integer $x$, $n$, $k$ \\ -\textbf{Output}. $x \mbox{ mod } (n - k)$ \\ -\hline \\ -1. $q \leftarrow \lfloor x / n \rfloor$ \\ -2. $q \leftarrow k \cdot q$ \\ -3. $x \leftarrow x \mbox{ (mod }n\mbox{)}$ \\ -4. $x \leftarrow x + q$ \\ -5. If $x \ge (n - k)$ then \\ -\hspace{3mm}5.1 $x \leftarrow x - (n - k)$ \\ -\hspace{3mm}5.2 Goto step 1. \\ -6. Return $x$ \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm Diminished Radix Reduction} -\label{fig:DR} -\end{figure} - -This algorithm will reduce $x$ modulo $n - k$ and return the residue. If $0 \le x < (n - k)^2$ then the algorithm will loop almost always -once or twice and occasionally three times. For simplicity sake the value of $x$ is bounded by the following simple polynomial. - -\begin{equation} -0 \le x < n^2 + k^2 - 2nk -\end{equation} - -The true bound is $0 \le x < (n - k - 1)^2$ but this has quite a few more terms. The value of $q$ after step 1 is bounded by the following. - -\begin{equation} -q < n - 2k - k^2/n -\end{equation} - -Since $k^2$ is going to be considerably smaller than $n$ that term will always be zero. The value of $x$ after step 3 is bounded trivially as -$0 \le x < n$. By step four the sum $x + q$ is bounded by - -\begin{equation} -0 \le q + x < (k + 1)n - 2k^2 - 1 -\end{equation} - -With a second pass $q$ will be loosely bounded by $0 \le q < k^2$ after step 2 while $x$ will still be loosely bounded by $0 \le x < n$ after step 3. After the second pass it is highly unlike that the -sum in step 4 will exceed $n - k$. In practice fewer than three passes of the algorithm are required to reduce virtually every input in the -range $0 \le x < (n - k - 1)^2$. - -\begin{figure} -\begin{small} -\begin{center} -\begin{tabular}{|l|} -\hline -$x = 123456789, n = 256, k = 3$ \\ -\hline $q \leftarrow \lfloor x/n \rfloor = 482253$ \\ -$q \leftarrow q*k = 1446759$ \\ -$x \leftarrow x \mbox{ mod } n = 21$ \\ -$x \leftarrow x + q = 1446780$ \\ -$x \leftarrow x - (n - k) = 1446527$ \\ -\hline -$q \leftarrow \lfloor x/n \rfloor = 5650$ \\ -$q \leftarrow q*k = 16950$ \\ -$x \leftarrow x \mbox{ mod } n = 127$ \\ -$x \leftarrow x + q = 17077$ \\ -$x \leftarrow x - (n - k) = 16824$ \\ -\hline -$q \leftarrow \lfloor x/n \rfloor = 65$ \\ -$q \leftarrow q*k = 195$ \\ -$x \leftarrow x \mbox{ mod } n = 184$ \\ -$x \leftarrow x + q = 379$ \\ -$x \leftarrow x - (n - k) = 126$ \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Example Diminished Radix Reduction} -\label{fig:EXDR} -\end{figure} - -Figure~\ref{fig:EXDR} demonstrates the reduction of $x = 123456789$ modulo $n - k = 253$ when $n = 256$ and $k = 3$. Note that even while $x$ -is considerably larger than $(n - k - 1)^2 = 63504$ the algorithm still converges on the modular residue exceedingly fast. In this case only -three passes were required to find the residue $x \equiv 126$. - - -\subsection{Choice of Moduli} -On the surface this algorithm looks like a very expensive algorithm. It requires a couple of subtractions followed by multiplication and other -modular reductions. The usefulness of this algorithm becomes exceedingly clear when an appropriate modulus is chosen. - -Division in general is a very expensive operation to perform. The one exception is when the division is by a power of the radix of representation used. -Division by ten for example is simple for pencil and paper mathematics since it amounts to shifting the decimal place to the right. Similarly division -by two (\textit{or powers of two}) is very simple for binary computers to perform. It would therefore seem logical to choose $n$ of the form $2^p$ -which would imply that $\lfloor x / n \rfloor$ is a simple shift of $x$ right $p$ bits. - -However, there is one operation related to division of power of twos that is even faster than this. If $n = \beta^p$ then the division may be -performed by moving whole digits to the right $p$ places. In practice division by $\beta^p$ is much faster than division by $2^p$ for any $p$. -Also with the choice of $n = \beta^p$ reducing $x$ modulo $n$ merely requires zeroing the digits above the $p-1$'th digit of $x$. - -Throughout the next section the term ``restricted modulus'' will refer to a modulus of the form $\beta^p - k$ whereas the term ``unrestricted -modulus'' will refer to a modulus of the form $2^p - k$. The word ``restricted'' in this case refers to the fact that it is based on the -$2^p$ logic except $p$ must be a multiple of $lg(\beta)$. - -\subsection{Choice of $k$} -Now that division and reduction (\textit{step 1 and 3 of figure~\ref{fig:DR}}) have been optimized to simple digit operations the multiplication by $k$ -in step 2 is the most expensive operation. Fortunately the choice of $k$ is not terribly limited. For all intents and purposes it might -as well be a single digit. The smaller the value of $k$ is the faster the algorithm will be. - -\subsection{Restricted Diminished Radix Reduction} -The restricted Diminished Radix algorithm can quickly reduce an input modulo a modulus of the form $n = \beta^p - k$. This algorithm can reduce -an input $x$ within the range $0 \le x < n^2$ using only a couple passes of the algorithm demonstrated in figure~\ref{fig:DR}. The implementation -of this algorithm has been optimized to avoid additional overhead associated with a division by $\beta^p$, the multiplication by $k$ or the addition -of $x$ and $q$. The resulting algorithm is very efficient and can lead to substantial improvements over Barrett and Montgomery reduction when modular -exponentiations are performed. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_dr\_reduce}. \\ -\textbf{Input}. mp\_int $x$, $n$ and a mp\_digit $k = \beta - n_0$ \\ -\hspace{11.5mm}($0 \le x < n^2$, $n > 1$, $0 < k < \beta$) \\ -\textbf{Output}. $x \mbox{ mod } n$ \\ -\hline \\ -1. $m \leftarrow n.used$ \\ -2. If $x.alloc < 2m$ then grow $x$ to $2m$ digits. \\ -3. $\mu \leftarrow 0$ \\ -4. for $i$ from $0$ to $m - 1$ do \\ -\hspace{3mm}4.1 $\hat r \leftarrow k \cdot x_{m+i} + x_{i} + \mu$ \\ -\hspace{3mm}4.2 $x_{i} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{3mm}4.3 $\mu \leftarrow \lfloor \hat r / \beta \rfloor$ \\ -5. $x_{m} \leftarrow \mu$ \\ -6. for $i$ from $m + 1$ to $x.used - 1$ do \\ -\hspace{3mm}6.1 $x_{i} \leftarrow 0$ \\ -7. Clamp excess digits of $x$. \\ -8. If $x \ge n$ then \\ -\hspace{3mm}8.1 $x \leftarrow x - n$ \\ -\hspace{3mm}8.2 Goto step 3. \\ -9. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_dr\_reduce} -\end{figure} - -\textbf{Algorithm mp\_dr\_reduce.} -This algorithm will perform the Dimished Radix reduction of $x$ modulo $n$. It has similar restrictions to that of the Barrett reduction -with the addition that $n$ must be of the form $n = \beta^m - k$ where $0 < k <\beta$. - -This algorithm essentially implements the pseudo-code in figure~\ref{fig:DR} except with a slight optimization. The division by $\beta^m$, multiplication by $k$ -and addition of $x \mbox{ mod }\beta^m$ are all performed simultaneously inside the loop on step 4. The division by $\beta^m$ is emulated by accessing -the term at the $m+i$'th position which is subsequently multiplied by $k$ and added to the term at the $i$'th position. After the loop the $m$'th -digit is set to the carry and the upper digits are zeroed. Steps 5 and 6 emulate the reduction modulo $\beta^m$ that should have happend to -$x$ before the addition of the multiple of the upper half. - -At step 8 if $x$ is still larger than $n$ another pass of the algorithm is required. First $n$ is subtracted from $x$ and then the algorithm resumes -at step 3. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_reduce.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* reduce "x" in place modulo "n" using the Diminished Radix algorithm. -018 * -019 * Based on algorithm from the paper -020 * -021 * "Generating Efficient Primes for Discrete Log Cryptosystems" -022 * Chae Hoon Lim, Pil Joong Lee, -023 * POSTECH Information Research Laboratories -024 * -025 * The modulus must be of a special format [see manual] -026 * -027 * Has been modified to use algorithm 7.10 from the LTM book instead -028 * -029 * Input x must be in the range 0 <= x <= (n-1)**2 -030 */ -031 int -032 mp_dr_reduce (mp_int * x, mp_int * n, mp_digit k) -033 \{ -034 int err, i, m; -035 mp_word r; -036 mp_digit mu, *tmpx1, *tmpx2; -037 -038 /* m = digits in modulus */ -039 m = n->used; -040 -041 /* ensure that "x" has at least 2m digits */ -042 if (x->alloc < m + m) \{ -043 if ((err = mp_grow (x, m + m)) != MP_OKAY) \{ -044 return err; -045 \} -046 \} -047 -048 /* top of loop, this is where the code resumes if -049 * another reduction pass is required. -050 */ -051 top: -052 /* aliases for digits */ -053 /* alias for lower half of x */ -054 tmpx1 = x->dp; -055 -056 /* alias for upper half of x, or x/B**m */ -057 tmpx2 = x->dp + m; -058 -059 /* set carry to zero */ -060 mu = 0; -061 -062 /* compute (x mod B**m) + k * [x/B**m] inline and inplace */ -063 for (i = 0; i < m; i++) \{ -064 r = ((mp_word)*tmpx2++) * ((mp_word)k) + *tmpx1 + mu; -065 *tmpx1++ = (mp_digit)(r & MP_MASK); -066 mu = (mp_digit)(r >> ((mp_word)DIGIT_BIT)); -067 \} -068 -069 /* set final carry */ -070 *tmpx1++ = mu; -071 -072 /* zero words above m */ -073 for (i = m + 1; i < x->used; i++) \{ -074 *tmpx1++ = 0; -075 \} -076 -077 /* clamp, sub and return */ -078 mp_clamp (x); -079 -080 /* if x >= n then subtract and reduce again -081 * Each successive "recursion" makes the input smaller and smaller. -082 */ -083 if (mp_cmp_mag (x, n) != MP_LT) \{ -084 s_mp_sub(x, n, x); -085 goto top; -086 \} -087 return MP_OKAY; -088 \} -089 #endif -\end{alltt} -\end{small} - -The first step is to grow $x$ as required to $2m$ digits since the reduction is performed in place on $x$. The label on line 51 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 63 performs the bulk of the work (\textit{corresponds to step 4 of algorithm 7.11}) -in this algorithm. - -By line 70 the pointer $tmpx1$ points to the $m$'th digit of $x$ which is where the final carry will be placed. Similarly by line 73 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 84 the value of $x$ is known to be greater than or equal to $n$ meaning that an unsigned subtraction can be used -as well. Since the destination of the subtraction is the larger of the inputs the call to algorithm s\_mp\_sub cannot fail and the return code -does not need to be checked. - -\subsubsection{Setup} -To setup the restricted Diminished Radix algorithm the value $k = \beta - n_0$ is required. This algorithm is not really complicated but provided for -completeness. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_dr\_setup}. \\ -\textbf{Input}. mp\_int $n$ \\ -\textbf{Output}. $k = \beta - n_0$ \\ -\hline \\ -1. $k \leftarrow \beta - n_0$ \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_dr\_setup} -\end{figure} - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_setup.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* determines the setup value */ -018 void mp_dr_setup(mp_int *a, mp_digit *d) -019 \{ -020 /* the casts are required if DIGIT_BIT is one less than -021 * the number of bits in a mp_digit [e.g. DIGIT_BIT==31] -022 */ -023 *d = (mp_digit)((((mp_word)1) << ((mp_word)DIGIT_BIT)) - -024 ((mp_word)a->dp[0])); -025 \} -026 -027 #endif -\end{alltt} -\end{small} - -\subsubsection{Modulus Detection} -Another algorithm which will be useful is the ability to detect a restricted Diminished Radix modulus. An integer is said to be -of restricted Diminished Radix form if all of the digits are equal to $\beta - 1$ except the trailing digit which may be any value. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_dr\_is\_modulus}. \\ -\textbf{Input}. mp\_int $n$ \\ -\textbf{Output}. $1$ if $n$ is in D.R form, $0$ otherwise \\ -\hline -1. If $n.used < 2$ then return($0$). \\ -2. for $ix$ from $1$ to $n.used - 1$ do \\ -\hspace{3mm}2.1 If $n_{ix} \ne \beta - 1$ return($0$). \\ -3. Return($1$). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_dr\_is\_modulus} -\end{figure} - -\textbf{Algorithm mp\_dr\_is\_modulus.} -This algorithm determines if a value is in Diminished Radix form. Step 1 rejects obvious cases where fewer than two digits are -in the mp\_int. Step 2 tests all but the first digit to see if they are equal to $\beta - 1$. If the algorithm manages to get to -step 3 then $n$ must be of Diminished Radix form. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_is\_modulus.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* determines if a number is a valid DR modulus */ -018 int mp_dr_is_modulus(mp_int *a) -019 \{ -020 int ix; -021 -022 /* must be at least two digits */ -023 if (a->used < 2) \{ -024 return 0; -025 \} -026 -027 /* must be of the form b**k - a [a <= b] so all -028 * but the first digit must be equal to -1 (mod b). -029 */ -030 for (ix = 1; ix < a->used; ix++) \{ -031 if (a->dp[ix] != MP_MASK) \{ -032 return 0; -033 \} -034 \} -035 return 1; -036 \} -037 -038 #endif -\end{alltt} -\end{small} - -\subsection{Unrestricted Diminished Radix Reduction} -The unrestricted Diminished Radix algorithm allows modular reductions to be performed when the modulus is of the form $2^p - k$. This algorithm -is a straightforward adaptation of algorithm~\ref{fig:DR}. - -In general the restricted Diminished Radix reduction algorithm is much faster since it has considerably lower overhead. However, this new -algorithm is much faster than either Montgomery or Barrett reduction when the moduli are of the appropriate form. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_reduce\_2k}. \\ -\textbf{Input}. mp\_int $a$ and $n$. mp\_digit $k$ \\ -\hspace{11.5mm}($a \ge 0$, $n > 1$, $0 < k < \beta$, $n + k$ is a power of two) \\ -\textbf{Output}. $a \mbox{ (mod }n\mbox{)}$ \\ -\hline -1. $p \leftarrow \lceil lg(n) \rceil$ (\textit{mp\_count\_bits}) \\ -2. While $a \ge n$ do \\ -\hspace{3mm}2.1 $q \leftarrow \lfloor a / 2^p \rfloor$ (\textit{mp\_div\_2d}) \\ -\hspace{3mm}2.2 $a \leftarrow a \mbox{ (mod }2^p\mbox{)}$ (\textit{mp\_mod\_2d}) \\ -\hspace{3mm}2.3 $q \leftarrow q \cdot k$ (\textit{mp\_mul\_d}) \\ -\hspace{3mm}2.4 $a \leftarrow a - q$ (\textit{s\_mp\_sub}) \\ -\hspace{3mm}2.5 If $a \ge n$ then do \\ -\hspace{6mm}2.5.1 $a \leftarrow a - n$ \\ -3. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_reduce\_2k} -\end{figure} - -\textbf{Algorithm mp\_reduce\_2k.} -This algorithm quickly reduces an input $a$ modulo an unrestricted Diminished Radix modulus $n$. Division by $2^p$ is emulated with a right -shift which makes the algorithm fairly inexpensive to use. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_2k.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* reduces a modulo n where n is of the form 2**p - d */ -018 int mp_reduce_2k(mp_int *a, mp_int *n, mp_digit d) -019 \{ -020 mp_int q; -021 int p, res; -022 -023 if ((res = mp_init(&q)) != MP_OKAY) \{ -024 return res; -025 \} -026 -027 p = mp_count_bits(n); -028 top: -029 /* q = a/2**p, a = a mod 2**p */ -030 if ((res = mp_div_2d(a, p, &q, a)) != MP_OKAY) \{ -031 goto ERR; -032 \} -033 -034 if (d != 1) \{ -035 /* q = q * d */ -036 if ((res = mp_mul_d(&q, d, &q)) != MP_OKAY) \{ -037 goto ERR; -038 \} -039 \} -040 -041 /* a = a + q */ -042 if ((res = s_mp_add(a, &q, a)) != MP_OKAY) \{ -043 goto ERR; -044 \} -045 -046 if (mp_cmp_mag(a, n) != MP_LT) \{ -047 s_mp_sub(a, n, a); -048 goto top; -049 \} -050 -051 ERR: -052 mp_clear(&q); -053 return res; -054 \} -055 -056 #endif -\end{alltt} -\end{small} - -The algorithm mp\_count\_bits calculates the number of bits in an mp\_int which is used to find the initial value of $p$. The call to mp\_div\_2d -on line 30 calculates both the quotient $q$ and the remainder $a$ required. By doing both in a single function call the code size -is kept fairly small. The multiplication by $k$ is only performed if $k > 1$. This allows reductions modulo $2^p - 1$ to be performed without -any multiplications. - -The unsigned s\_mp\_add, mp\_cmp\_mag and s\_mp\_sub are used in place of their full sign counterparts since the inputs are only valid if they are -positive. By using the unsigned versions the overhead is kept to a minimum. - -\subsubsection{Unrestricted Setup} -To setup this reduction algorithm the value of $k = 2^p - n$ is required. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_reduce\_2k\_setup}. \\ -\textbf{Input}. mp\_int $n$ \\ -\textbf{Output}. $k = 2^p - n$ \\ -\hline -1. $p \leftarrow \lceil lg(n) \rceil$ (\textit{mp\_count\_bits}) \\ -2. $x \leftarrow 2^p$ (\textit{mp\_2expt}) \\ -3. $x \leftarrow x - n$ (\textit{mp\_sub}) \\ -4. $k \leftarrow x_0$ \\ -5. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_reduce\_2k\_setup} -\end{figure} - -\textbf{Algorithm mp\_reduce\_2k\_setup.} -This algorithm computes the value of $k$ required for the algorithm mp\_reduce\_2k. By making a temporary variable $x$ equal to $2^p$ a subtraction -is sufficient to solve for $k$. Alternatively if $n$ has more than one digit the value of $k$ is simply $\beta - n_0$. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_2k\_setup.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* determines the setup value */ -018 int mp_reduce_2k_setup(mp_int *a, mp_digit *d) -019 \{ -020 int res, p; -021 mp_int tmp; -022 -023 if ((res = mp_init(&tmp)) != MP_OKAY) \{ -024 return res; -025 \} -026 -027 p = mp_count_bits(a); -028 if ((res = mp_2expt(&tmp, p)) != MP_OKAY) \{ -029 mp_clear(&tmp); -030 return res; -031 \} -032 -033 if ((res = s_mp_sub(&tmp, a, &tmp)) != MP_OKAY) \{ -034 mp_clear(&tmp); -035 return res; -036 \} -037 -038 *d = tmp.dp[0]; -039 mp_clear(&tmp); -040 return MP_OKAY; -041 \} -042 #endif -\end{alltt} -\end{small} - -\subsubsection{Unrestricted Detection} -An integer $n$ is a valid unrestricted Diminished Radix modulus if either of the following are true. - -\begin{enumerate} -\item The number has only one digit. -\item The number has more than one digit and every bit from the $\beta$'th to the most significant is one. -\end{enumerate} - -If either condition is true than there is a power of two $2^p$ such that $0 < 2^p - n < \beta$. If the input is only -one digit than it will always be of the correct form. Otherwise all of the bits above the first digit must be one. This arises from the fact -that there will be value of $k$ that when added to the modulus causes a carry in the first digit which propagates all the way to the most -significant bit. The resulting sum will be a power of two. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_reduce\_is\_2k}. \\ -\textbf{Input}. mp\_int $n$ \\ -\textbf{Output}. $1$ if of proper form, $0$ otherwise \\ -\hline -1. If $n.used = 0$ then return($0$). \\ -2. If $n.used = 1$ then return($1$). \\ -3. $p \leftarrow \lceil lg(n) \rceil$ (\textit{mp\_count\_bits}) \\ -4. for $x$ from $lg(\beta)$ to $p$ do \\ -\hspace{3mm}4.1 If the ($x \mbox{ mod }lg(\beta)$)'th bit of the $\lfloor x / lg(\beta) \rfloor$ of $n$ is zero then return($0$). \\ -5. Return($1$). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_reduce\_is\_2k} -\end{figure} - -\textbf{Algorithm mp\_reduce\_is\_2k.} -This algorithm quickly determines if a modulus is of the form required for algorithm mp\_reduce\_2k to function properly. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_is\_2k.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* determines if mp_reduce_2k can be used */ -018 int mp_reduce_is_2k(mp_int *a) -019 \{ -020 int ix, iy, iw; -021 mp_digit iz; -022 -023 if (a->used == 0) \{ -024 return MP_NO; -025 \} else if (a->used == 1) \{ -026 return MP_YES; -027 \} else if (a->used > 1) \{ -028 iy = mp_count_bits(a); -029 iz = 1; -030 iw = 1; -031 -032 /* Test every bit from the second digit up, must be 1 */ -033 for (ix = DIGIT_BIT; ix < iy; ix++) \{ -034 if ((a->dp[iw] & iz) == 0) \{ -035 return MP_NO; -036 \} -037 iz <<= 1; -038 if (iz > (mp_digit)MP_MASK) \{ -039 ++iw; -040 iz = 1; -041 \} -042 \} -043 \} -044 return MP_YES; -045 \} -046 -047 #endif -\end{alltt} -\end{small} - - - -\section{Algorithm Comparison} -So far three very different algorithms for modular reduction have been discussed. Each of the algorithms have their own strengths and weaknesses -that makes having such a selection very useful. The following table sumarizes the three algorithms along with comparisons of work factors. Since -all three algorithms have the restriction that $0 \le x < n^2$ and $n > 1$ those limitations are not included in the table. - -\begin{center} -\begin{small} -\begin{tabular}{|c|c|c|c|c|c|} -\hline \textbf{Method} & \textbf{Work Required} & \textbf{Limitations} & \textbf{$m = 8$} & \textbf{$m = 32$} & \textbf{$m = 64$} \\ -\hline Barrett & $m^2 + 2m - 1$ & None & $79$ & $1087$ & $4223$ \\ -\hline Montgomery & $m^2 + m$ & $n$ must be odd & $72$ & $1056$ & $4160$ \\ -\hline D.R. & $2m$ & $n = \beta^m - k$ & $16$ & $64$ & $128$ \\ -\hline -\end{tabular} -\end{small} -\end{center} - -In theory Montgomery and Barrett reductions would require roughly the same amount of time to complete. However, in practice since Montgomery -reduction can be written as a single function with the Comba technique it is much faster. Barrett reduction suffers from the overhead of -calling the half precision multipliers, addition and division by $\beta$ algorithms. - -For almost every cryptographic algorithm Montgomery reduction is the algorithm of choice. The one set of algorithms where Diminished Radix reduction truly -shines are based on the discrete logarithm problem such as Diffie-Hellman \cite{DH} and ElGamal \cite{ELGAMAL}. In these algorithms -primes of the form $\beta^m - k$ can be found and shared amongst users. These primes will allow the Diminished Radix algorithm to be used in -modular exponentiation to greatly speed up the operation. - - - -\section*{Exercises} -\begin{tabular}{cl} -$\left [ 3 \right ]$ & Prove that the ``trick'' in algorithm mp\_montgomery\_setup actually \\ - & calculates the correct value of $\rho$. \\ - & \\ -$\left [ 2 \right ]$ & Devise an algorithm to reduce modulo $n + k$ for small $k$ quickly. \\ - & \\ -$\left [ 4 \right ]$ & Prove that the pseudo-code algorithm ``Diminished Radix Reduction'' \\ - & (\textit{figure~\ref{fig:DR}}) terminates. Also prove the probability that it will \\ - & terminate within $1 \le k \le 10$ iterations. \\ - & \\ -\end{tabular} - - -\chapter{Exponentiation} -Exponentiation is the operation of raising one variable to the power of another, for example, $a^b$. A variant of exponentiation, computed -in a finite field or ring, is called modular exponentiation. This latter style of operation is typically used in public key -cryptosystems such as RSA and Diffie-Hellman. The ability to quickly compute modular exponentiations is of great benefit to any -such cryptosystem and many methods have been sought to speed it up. - -\section{Exponentiation Basics} -A trivial algorithm would simply multiply $a$ against itself $b - 1$ times to compute the exponentiation desired. However, as $b$ grows in size -the number of multiplications becomes prohibitive. Imagine what would happen if $b$ $\approx$ $2^{1024}$ as is the case when computing an RSA signature -with a $1024$-bit key. Such a calculation could never be completed as it would take simply far too long. - -Fortunately there is a very simple algorithm based on the laws of exponents. Recall that $lg_a(a^b) = b$ and that $lg_a(a^ba^c) = b + c$ which -are two trivial relationships between the base and the exponent. Let $b_i$ represent the $i$'th bit of $b$ starting from the least -significant bit. If $b$ is a $k$-bit integer than the following equation is true. - -\begin{equation} -a^b = \prod_{i=0}^{k-1} a^{2^i \cdot b_i} -\end{equation} - -By taking the base $a$ logarithm of both sides of the equation the following equation is the result. - -\begin{equation} -b = \sum_{i=0}^{k-1}2^i \cdot b_i -\end{equation} - -The term $a^{2^i}$ can be found from the $i - 1$'th term by squaring the term since $\left ( a^{2^i} \right )^2$ is equal to -$a^{2^{i+1}}$. This observation forms the basis of essentially all fast exponentiation algorithms. It requires $k$ squarings and on average -$k \over 2$ multiplications to compute the result. This is indeed quite an improvement over simply multiplying by $a$ a total of $b-1$ times. - -While this current method is a considerable speed up there are further improvements to be made. For example, the $a^{2^i}$ term does not need to -be computed in an auxilary variable. Consider the following equivalent algorithm. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Left to Right Exponentiation}. \\ -\textbf{Input}. Integer $a$, $b$ and $k$ \\ -\textbf{Output}. $c = a^b$ \\ -\hline \\ -1. $c \leftarrow 1$ \\ -2. for $i$ from $k - 1$ to $0$ do \\ -\hspace{3mm}2.1 $c \leftarrow c^2$ \\ -\hspace{3mm}2.2 $c \leftarrow c \cdot a^{b_i}$ \\ -3. Return $c$. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Left to Right Exponentiation} -\label{fig:LTOR} -\end{figure} - -This algorithm starts from the most significant bit and works towards the least significant bit. When the $i$'th bit of $b$ is set $a$ is -multiplied against the current product. In each iteration the product is squared which doubles the exponent of the individual terms of the -product. - -For example, let $b = 101100_2 \equiv 44_{10}$. The following chart demonstrates the actions of the algorithm. - -\newpage\begin{figure} -\begin{center} -\begin{tabular}{|c|c|} -\hline \textbf{Value of $i$} & \textbf{Value of $c$} \\ -\hline - & $1$ \\ -\hline $5$ & $a$ \\ -\hline $4$ & $a^2$ \\ -\hline $3$ & $a^4 \cdot a$ \\ -\hline $2$ & $a^8 \cdot a^2 \cdot a$ \\ -\hline $1$ & $a^{16} \cdot a^4 \cdot a^2$ \\ -\hline $0$ & $a^{32} \cdot a^8 \cdot a^4$ \\ -\hline -\end{tabular} -\end{center} -\caption{Example of Left to Right Exponentiation} -\end{figure} - -When the product $a^{32} \cdot a^8 \cdot a^4$ is simplified it is equal $a^{44}$ which is the desired exponentiation. This particular algorithm is -called ``Left to Right'' because it reads the exponent in that order. All of the exponentiation algorithms that will be presented are of this nature. - -\subsection{Single Digit Exponentiation} -The first algorithm in the series of exponentiation algorithms will be an unbounded algorithm where the exponent is a single digit. It is intended -to be used when a small power of an input is required (\textit{e.g. $a^5$}). It is faster than simply multiplying $b - 1$ times for all values of -$b$ that are greater than three. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_expt\_d}. \\ -\textbf{Input}. mp\_int $a$ and mp\_digit $b$ \\ -\textbf{Output}. $c = a^b$ \\ -\hline \\ -1. $g \leftarrow a$ (\textit{mp\_init\_copy}) \\ -2. $c \leftarrow 1$ (\textit{mp\_set}) \\ -3. for $x$ from 1 to $lg(\beta)$ do \\ -\hspace{3mm}3.1 $c \leftarrow c^2$ (\textit{mp\_sqr}) \\ -\hspace{3mm}3.2 If $b$ AND $2^{lg(\beta) - 1} \ne 0$ then \\ -\hspace{6mm}3.2.1 $c \leftarrow c \cdot g$ (\textit{mp\_mul}) \\ -\hspace{3mm}3.3 $b \leftarrow b << 1$ \\ -4. Clear $g$. \\ -5. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_expt\_d} -\end{figure} - -\textbf{Algorithm mp\_expt\_d.} -This algorithm computes the value of $a$ raised to the power of a single digit $b$. It uses the left to right exponentiation algorithm to -quickly compute the exponentiation. It is loosely based on algorithm 14.79 of HAC \cite[pp. 615]{HAC} with the difference that the -exponent is a fixed width. - -A copy of $a$ is made first to allow destination variable $c$ be the same as the source variable $a$. The result is set to the initial value of -$1$ in the subsequent step. - -Inside the loop the exponent is read from the most significant bit first down to the least significant bit. First $c$ is invariably squared -on step 3.1. In the following step if the most significant bit of $b$ is one the copy of $a$ is multiplied against $c$. The value -of $b$ is shifted left one bit to make the next bit down from the most signficant bit the new most significant bit. In effect each -iteration of the loop moves the bits of the exponent $b$ upwards to the most significant location. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_expt\_d.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* calculate c = a**b using a square-multiply algorithm */ -018 int mp_expt_d (mp_int * a, mp_digit b, mp_int * c) -019 \{ -020 int res, x; -021 mp_int g; -022 -023 if ((res = mp_init_copy (&g, a)) != MP_OKAY) \{ -024 return res; -025 \} -026 -027 /* set initial result */ -028 mp_set (c, 1); -029 -030 for (x = 0; x < (int) DIGIT_BIT; x++) \{ -031 /* square */ -032 if ((res = mp_sqr (c, c)) != MP_OKAY) \{ -033 mp_clear (&g); -034 return res; -035 \} -036 -037 /* if the bit is set multiply */ -038 if ((b & (mp_digit) (((mp_digit)1) << (DIGIT_BIT - 1))) != 0) \{ -039 if ((res = mp_mul (c, &g, c)) != MP_OKAY) \{ -040 mp_clear (&g); -041 return res; -042 \} -043 \} -044 -045 /* shift to next bit */ -046 b <<= 1; -047 \} -048 -049 mp_clear (&g); -050 return MP_OKAY; -051 \} -052 #endif -\end{alltt} -\end{small} - -Line 28 sets the initial value of the result to $1$. Next the loop on line 30 steps through each bit of the exponent starting from -the most significant down towards the least significant. The invariant squaring operation placed on line 32 is performed first. After -the squaring the result $c$ is multiplied by the base $g$ if and only if the most significant bit of the exponent is set. The shift on line -46 moves all of the bits of the exponent upwards towards the most significant location. - -\section{$k$-ary Exponentiation} -When calculating an exponentiation the most time consuming bottleneck is the multiplications which are in general a small factor -slower than squaring. Recall from the previous algorithm that $b_{i}$ refers to the $i$'th bit of the exponent $b$. Suppose instead it referred to -the $i$'th $k$-bit digit of the exponent of $b$. For $k = 1$ the definitions are synonymous and for $k > 1$ algorithm~\ref{fig:KARY} -computes the same exponentiation. A group of $k$ bits from the exponent is called a \textit{window}. That is it is a small window on only a -portion of the entire exponent. Consider the following modification to the basic left to right exponentiation algorithm. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{$k$-ary Exponentiation}. \\ -\textbf{Input}. Integer $a$, $b$, $k$ and $t$ \\ -\textbf{Output}. $c = a^b$ \\ -\hline \\ -1. $c \leftarrow 1$ \\ -2. for $i$ from $t - 1$ to $0$ do \\ -\hspace{3mm}2.1 $c \leftarrow c^{2^k} $ \\ -\hspace{3mm}2.2 Extract the $i$'th $k$-bit word from $b$ and store it in $g$. \\ -\hspace{3mm}2.3 $c \leftarrow c \cdot a^g$ \\ -3. Return $c$. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{$k$-ary Exponentiation} -\label{fig:KARY} -\end{figure} - -The squaring on step 2.1 can be calculated by squaring the value $c$ successively $k$ times. If the values of $a^g$ for $0 < g < 2^k$ have been -precomputed this algorithm requires only $t$ multiplications and $tk$ squarings. The table can be generated with $2^{k - 1} - 1$ squarings and -$2^{k - 1} + 1$ multiplications. This algorithm assumes that the number of bits in the exponent is evenly divisible by $k$. -However, when it is not the remaining $0 < x \le k - 1$ bits can be handled with algorithm~\ref{fig:LTOR}. - -Suppose $k = 4$ and $t = 100$. This modified algorithm will require $109$ multiplications and $408$ squarings to compute the exponentiation. The -original algorithm would on average have required $200$ multiplications and $400$ squrings to compute the same value. The total number of squarings -has increased slightly but the number of multiplications has nearly halved. - -\subsection{Optimal Values of $k$} -An optimal value of $k$ will minimize $2^{k} + \lceil n / k \rceil + n - 1$ for a fixed number of bits in the exponent $n$. The simplest -approach is to brute force search amongst the values $k = 2, 3, \ldots, 8$ for the lowest result. Table~\ref{fig:OPTK} lists optimal values of $k$ -for various exponent sizes and compares the number of multiplication and squarings required against algorithm~\ref{fig:LTOR}. - -\begin{figure}[here] -\begin{center} -\begin{small} -\begin{tabular}{|c|c|c|c|c|c|} -\hline \textbf{Exponent (bits)} & \textbf{Optimal $k$} & \textbf{Work at $k$} & \textbf{Work with ~\ref{fig:LTOR}} \\ -\hline $16$ & $2$ & $27$ & $24$ \\ -\hline $32$ & $3$ & $49$ & $48$ \\ -\hline $64$ & $3$ & $92$ & $96$ \\ -\hline $128$ & $4$ & $175$ & $192$ \\ -\hline $256$ & $4$ & $335$ & $384$ \\ -\hline $512$ & $5$ & $645$ & $768$ \\ -\hline $1024$ & $6$ & $1257$ & $1536$ \\ -\hline $2048$ & $6$ & $2452$ & $3072$ \\ -\hline $4096$ & $7$ & $4808$ & $6144$ \\ -\hline -\end{tabular} -\end{small} -\end{center} -\caption{Optimal Values of $k$ for $k$-ary Exponentiation} -\label{fig:OPTK} -\end{figure} - -\subsection{Sliding-Window Exponentiation} -A simple modification to the previous algorithm is only generate the upper half of the table in the range $2^{k-1} \le g < 2^k$. Essentially -this is a table for all values of $g$ where the most significant bit of $g$ is a one. However, in order for this to be allowed in the -algorithm values of $g$ in the range $0 \le g < 2^{k-1}$ must be avoided. - -Table~\ref{fig:OPTK2} lists optimal values of $k$ for various exponent sizes and compares the work required against algorithm~\ref{fig:KARY}. - -\begin{figure}[here] -\begin{center} -\begin{small} -\begin{tabular}{|c|c|c|c|c|c|} -\hline \textbf{Exponent (bits)} & \textbf{Optimal $k$} & \textbf{Work at $k$} & \textbf{Work with ~\ref{fig:KARY}} \\ -\hline $16$ & $3$ & $24$ & $27$ \\ -\hline $32$ & $3$ & $45$ & $49$ \\ -\hline $64$ & $4$ & $87$ & $92$ \\ -\hline $128$ & $4$ & $167$ & $175$ \\ -\hline $256$ & $5$ & $322$ & $335$ \\ -\hline $512$ & $6$ & $628$ & $645$ \\ -\hline $1024$ & $6$ & $1225$ & $1257$ \\ -\hline $2048$ & $7$ & $2403$ & $2452$ \\ -\hline $4096$ & $8$ & $4735$ & $4808$ \\ -\hline -\end{tabular} -\end{small} -\end{center} -\caption{Optimal Values of $k$ for Sliding Window Exponentiation} -\label{fig:OPTK2} -\end{figure} - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Sliding Window $k$-ary Exponentiation}. \\ -\textbf{Input}. Integer $a$, $b$, $k$ and $t$ \\ -\textbf{Output}. $c = a^b$ \\ -\hline \\ -1. $c \leftarrow 1$ \\ -2. for $i$ from $t - 1$ to $0$ do \\ -\hspace{3mm}2.1 If the $i$'th bit of $b$ is a zero then \\ -\hspace{6mm}2.1.1 $c \leftarrow c^2$ \\ -\hspace{3mm}2.2 else do \\ -\hspace{6mm}2.2.1 $c \leftarrow c^{2^k}$ \\ -\hspace{6mm}2.2.2 Extract the $k$ bits from $(b_{i}b_{i-1}\ldots b_{i-(k-1)})$ and store it in $g$. \\ -\hspace{6mm}2.2.3 $c \leftarrow c \cdot a^g$ \\ -\hspace{6mm}2.2.4 $i \leftarrow i - k$ \\ -3. Return $c$. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Sliding Window $k$-ary Exponentiation} -\end{figure} - -Similar to the previous algorithm this algorithm must have a special handler when fewer than $k$ bits are left in the exponent. While this -algorithm requires the same number of squarings it can potentially have fewer multiplications. The pre-computed table $a^g$ is also half -the size as the previous table. - -Consider the exponent $b = 111101011001000_2 \equiv 31432_{10}$ with $k = 3$ using both algorithms. The first algorithm will divide the exponent up as -the following five $3$-bit words $b \equiv \left ( 111, 101, 011, 001, 000 \right )_{2}$. The second algorithm will break the -exponent as $b \equiv \left ( 111, 101, 0, 110, 0, 100, 0 \right )_{2}$. The single digit $0$ in the second representation are where -a single squaring took place instead of a squaring and multiplication. In total the first method requires $10$ multiplications and $18$ -squarings. The second method requires $8$ multiplications and $18$ squarings. - -In general the sliding window method is never slower than the generic $k$-ary method and often it is slightly faster. - -\section{Modular Exponentiation} - -Modular exponentiation is essentially computing the power of a base within a finite field or ring. For example, computing -$d \equiv a^b \mbox{ (mod }c\mbox{)}$ is a modular exponentiation. Instead of first computing $a^b$ and then reducing it -modulo $c$ the intermediate result is reduced modulo $c$ after every squaring or multiplication operation. - -This guarantees that any intermediate result is bounded by $0 \le d \le c^2 - 2c + 1$ and can be reduced modulo $c$ quickly using -one of the algorithms presented in chapter six. - -Before the actual modular exponentiation algorithm can be written a wrapper algorithm must be written first. This algorithm -will allow the exponent $b$ to be negative which is computed as $c \equiv \left (1 / a \right )^{\vert b \vert} \mbox{(mod }d\mbox{)}$. The -value of $(1/a) \mbox{ mod }c$ is computed using the modular inverse (\textit{see \ref{sec;modinv}}). If no inverse exists the algorithm -terminates with an error. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_exptmod}. \\ -\textbf{Input}. mp\_int $a$, $b$ and $c$ \\ -\textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\ -\hline \\ -1. If $c.sign = MP\_NEG$ return(\textit{MP\_VAL}). \\ -2. If $b.sign = MP\_NEG$ then \\ -\hspace{3mm}2.1 $g' \leftarrow g^{-1} \mbox{ (mod }c\mbox{)}$ \\ -\hspace{3mm}2.2 $x' \leftarrow \vert x \vert$ \\ -\hspace{3mm}2.3 Compute $d \equiv g'^{x'} \mbox{ (mod }c\mbox{)}$ via recursion. \\ -3. if $p$ is odd \textbf{OR} $p$ is a D.R. modulus then \\ -\hspace{3mm}3.1 Compute $y \equiv g^{x} \mbox{ (mod }p\mbox{)}$ via algorithm mp\_exptmod\_fast. \\ -4. else \\ -\hspace{3mm}4.1 Compute $y \equiv g^{x} \mbox{ (mod }p\mbox{)}$ via algorithm s\_mp\_exptmod. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_exptmod} -\end{figure} - -\textbf{Algorithm mp\_exptmod.} -The first algorithm which actually performs modular exponentiation is algorithm s\_mp\_exptmod. It is a sliding window $k$-ary algorithm -which uses Barrett reduction to reduce the product modulo $p$. The second algorithm mp\_exptmod\_fast performs the same operation -except it uses either Montgomery or Diminished Radix reduction. The two latter reduction algorithms are clumped in the same exponentiation -algorithm since their arguments are essentially the same (\textit{two mp\_ints and one mp\_digit}). - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_exptmod.c -\vspace{-3mm} -\begin{alltt} -016 -017 -018 /* this is a shell function that calls either the normal or Montgomery -019 * exptmod functions. Originally the call to the montgomery code was -020 * embedded in the normal function but that wasted alot of stack space -021 * for nothing (since 99% of the time the Montgomery code would be called) -022 */ -023 int mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y) -024 \{ -025 int dr; -026 -027 /* modulus P must be positive */ -028 if (P->sign == MP_NEG) \{ -029 return MP_VAL; -030 \} -031 -032 /* if exponent X is negative we have to recurse */ -033 if (X->sign == MP_NEG) \{ -034 #ifdef BN_MP_INVMOD_C -035 mp_int tmpG, tmpX; -036 int err; -037 -038 /* first compute 1/G mod P */ -039 if ((err = mp_init(&tmpG)) != MP_OKAY) \{ -040 return err; -041 \} -042 if ((err = mp_invmod(G, P, &tmpG)) != MP_OKAY) \{ -043 mp_clear(&tmpG); -044 return err; -045 \} -046 -047 /* now get |X| */ -048 if ((err = mp_init(&tmpX)) != MP_OKAY) \{ -049 mp_clear(&tmpG); -050 return err; -051 \} -052 if ((err = mp_abs(X, &tmpX)) != MP_OKAY) \{ -053 mp_clear_multi(&tmpG, &tmpX, NULL); -054 return err; -055 \} -056 -057 /* and now compute (1/G)**|X| instead of G**X [X < 0] */ -058 err = mp_exptmod(&tmpG, &tmpX, P, Y); -059 mp_clear_multi(&tmpG, &tmpX, NULL); -060 return err; -061 #else -062 /* no invmod */ -063 return MP_VAL; -064 #endif -065 \} -066 -067 /* modified diminished radix reduction */ -068 #if defined(BN_MP_REDUCE_IS_2K_L_C) && defined(BN_MP_REDUCE_2K_L_C) -069 if (mp_reduce_is_2k_l(P) == MP_YES) \{ -070 return s_mp_exptmod(G, X, P, Y, 1); -071 \} -072 #endif -073 -074 #ifdef BN_MP_DR_IS_MODULUS_C -075 /* is it a DR modulus? */ -076 dr = mp_dr_is_modulus(P); -077 #else -078 /* default to no */ -079 dr = 0; -080 #endif -081 -082 #ifdef BN_MP_REDUCE_IS_2K_C -083 /* if not, is it a unrestricted DR modulus? */ -084 if (dr == 0) \{ -085 dr = mp_reduce_is_2k(P) << 1; -086 \} -087 #endif -088 -089 /* if the modulus is odd or dr != 0 use the montgomery method */ -090 #ifdef BN_MP_EXPTMOD_FAST_C -091 if (mp_isodd (P) == 1 || dr != 0) \{ -092 return mp_exptmod_fast (G, X, P, Y, dr); -093 \} else \{ -094 #endif -095 #ifdef BN_S_MP_EXPTMOD_C -096 /* otherwise use the generic Barrett reduction technique */ -097 return s_mp_exptmod (G, X, P, Y, 0); -098 #else -099 /* no exptmod for evens */ -100 return MP_VAL; -101 #endif -102 #ifdef BN_MP_EXPTMOD_FAST_C -103 \} -104 #endif -105 \} -106 -107 #endif -\end{alltt} -\end{small} - -In order to keep the algorithms in a known state the first step on line 28 is to reject any negative modulus as input. If the exponent is -negative the algorithm tries to perform a modular exponentiation with the modular inverse of the base $G$. The temporary variable $tmpG$ is assigned -the modular inverse of $G$ and $tmpX$ is assigned the absolute value of $X$. The algorithm will recuse with these new values with a positive -exponent. - -If the exponent is positive the algorithm resumes the exponentiation. Line 76 determines if the modulus is of the restricted Diminished Radix -form. If it is not line 69 attempts to determine if it is of a unrestricted Diminished Radix form. The integer $dr$ will take on one -of three values. - -\begin{enumerate} -\item $dr = 0$ means that the modulus is not of either restricted or unrestricted Diminished Radix form. -\item $dr = 1$ means that the modulus is of restricted Diminished Radix form. -\item $dr = 2$ means that the modulus is of unrestricted Diminished Radix form. -\end{enumerate} - -Line 69 determines if the fast modular exponentiation algorithm can be used. It is allowed if $dr \ne 0$ or if the modulus is odd. Otherwise, -the slower s\_mp\_exptmod algorithm is used which uses Barrett reduction. - -\subsection{Barrett Modular Exponentiation} - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{s\_mp\_exptmod}. \\ -\textbf{Input}. mp\_int $a$, $b$ and $c$ \\ -\textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\ -\hline \\ -1. $k \leftarrow lg(x)$ \\ -2. $winsize \leftarrow \left \lbrace \begin{array}{ll} - 2 & \mbox{if }k \le 7 \\ - 3 & \mbox{if }7 < k \le 36 \\ - 4 & \mbox{if }36 < k \le 140 \\ - 5 & \mbox{if }140 < k \le 450 \\ - 6 & \mbox{if }450 < k \le 1303 \\ - 7 & \mbox{if }1303 < k \le 3529 \\ - 8 & \mbox{if }3529 < k \\ - \end{array} \right .$ \\ -3. Initialize $2^{winsize}$ mp\_ints in an array named $M$ and one mp\_int named $\mu$ \\ -4. Calculate the $\mu$ required for Barrett Reduction (\textit{mp\_reduce\_setup}). \\ -5. $M_1 \leftarrow g \mbox{ (mod }p\mbox{)}$ \\ -\\ -Setup the table of small powers of $g$. First find $g^{2^{winsize}}$ and then all multiples of it. \\ -6. $k \leftarrow 2^{winsize - 1}$ \\ -7. $M_{k} \leftarrow M_1$ \\ -8. for $ix$ from 0 to $winsize - 2$ do \\ -\hspace{3mm}8.1 $M_k \leftarrow \left ( M_k \right )^2$ (\textit{mp\_sqr}) \\ -\hspace{3mm}8.2 $M_k \leftarrow M_k \mbox{ (mod }p\mbox{)}$ (\textit{mp\_reduce}) \\ -9. for $ix$ from $2^{winsize - 1} + 1$ to $2^{winsize} - 1$ do \\ -\hspace{3mm}9.1 $M_{ix} \leftarrow M_{ix - 1} \cdot M_{1}$ (\textit{mp\_mul}) \\ -\hspace{3mm}9.2 $M_{ix} \leftarrow M_{ix} \mbox{ (mod }p\mbox{)}$ (\textit{mp\_reduce}) \\ -10. $res \leftarrow 1$ \\ -\\ -Start Sliding Window. \\ -11. $mode \leftarrow 0, bitcnt \leftarrow 1, buf \leftarrow 0, digidx \leftarrow x.used - 1, bitcpy \leftarrow 0, bitbuf \leftarrow 0$ \\ -12. Loop \\ -\hspace{3mm}12.1 $bitcnt \leftarrow bitcnt - 1$ \\ -\hspace{3mm}12.2 If $bitcnt = 0$ then do \\ -\hspace{6mm}12.2.1 If $digidx = -1$ goto step 13. \\ -\hspace{6mm}12.2.2 $buf \leftarrow x_{digidx}$ \\ -\hspace{6mm}12.2.3 $digidx \leftarrow digidx - 1$ \\ -\hspace{6mm}12.2.4 $bitcnt \leftarrow lg(\beta)$ \\ -Continued on next page. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm s\_mp\_exptmod} -\end{figure} - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{s\_mp\_exptmod} (\textit{continued}). \\ -\textbf{Input}. mp\_int $a$, $b$ and $c$ \\ -\textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\ -\hline \\ -\hspace{3mm}12.3 $y \leftarrow (buf >> (lg(\beta) - 1))$ AND $1$ \\ -\hspace{3mm}12.4 $buf \leftarrow buf << 1$ \\ -\hspace{3mm}12.5 if $mode = 0$ and $y = 0$ then goto step 12. \\ -\hspace{3mm}12.6 if $mode = 1$ and $y = 0$ then do \\ -\hspace{6mm}12.6.1 $res \leftarrow res^2$ \\ -\hspace{6mm}12.6.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ -\hspace{6mm}12.6.3 Goto step 12. \\ -\hspace{3mm}12.7 $bitcpy \leftarrow bitcpy + 1$ \\ -\hspace{3mm}12.8 $bitbuf \leftarrow bitbuf + (y << (winsize - bitcpy))$ \\ -\hspace{3mm}12.9 $mode \leftarrow 2$ \\ -\hspace{3mm}12.10 If $bitcpy = winsize$ then do \\ -\hspace{6mm}Window is full so perform the squarings and single multiplication. \\ -\hspace{6mm}12.10.1 for $ix$ from $0$ to $winsize -1$ do \\ -\hspace{9mm}12.10.1.1 $res \leftarrow res^2$ \\ -\hspace{9mm}12.10.1.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ -\hspace{6mm}12.10.2 $res \leftarrow res \cdot M_{bitbuf}$ \\ -\hspace{6mm}12.10.3 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ -\hspace{6mm}Reset the window. \\ -\hspace{6mm}12.10.4 $bitcpy \leftarrow 0, bitbuf \leftarrow 0, mode \leftarrow 1$ \\ -\\ -No more windows left. Check for residual bits of exponent. \\ -13. If $mode = 2$ and $bitcpy > 0$ then do \\ -\hspace{3mm}13.1 for $ix$ form $0$ to $bitcpy - 1$ do \\ -\hspace{6mm}13.1.1 $res \leftarrow res^2$ \\ -\hspace{6mm}13.1.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ -\hspace{6mm}13.1.3 $bitbuf \leftarrow bitbuf << 1$ \\ -\hspace{6mm}13.1.4 If $bitbuf$ AND $2^{winsize} \ne 0$ then do \\ -\hspace{9mm}13.1.4.1 $res \leftarrow res \cdot M_{1}$ \\ -\hspace{9mm}13.1.4.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ -14. $y \leftarrow res$ \\ -15. Clear $res$, $mu$ and the $M$ array. \\ -16. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm s\_mp\_exptmod (continued)} -\end{figure} - -\textbf{Algorithm s\_mp\_exptmod.} -This algorithm computes the $x$'th power of $g$ modulo $p$ and stores the result in $y$. It takes advantage of the Barrett reduction -algorithm to keep the product small throughout the algorithm. - -The first two steps determine the optimal window size based on the number of bits in the exponent. The larger the exponent the -larger the window size becomes. After a window size $winsize$ has been chosen an array of $2^{winsize}$ mp\_int variables is allocated. This -table will hold the values of $g^x \mbox{ (mod }p\mbox{)}$ for $2^{winsize - 1} \le x < 2^{winsize}$. - -After the table is allocated the first power of $g$ is found. Since $g \ge p$ is allowed it must be first reduced modulo $p$ to make -the rest of the algorithm more efficient. The first element of the table at $2^{winsize - 1}$ is found by squaring $M_1$ successively $winsize - 2$ -times. The rest of the table elements are found by multiplying the previous element by $M_1$ modulo $p$. - -Now that the table is available the sliding window may begin. The following list describes the functions of all the variables in the window. -\begin{enumerate} -\item The variable $mode$ dictates how the bits of the exponent are interpreted. -\begin{enumerate} - \item When $mode = 0$ the bits are ignored since no non-zero bit of the exponent has been seen yet. For example, if the exponent were simply - $1$ then there would be $lg(\beta) - 1$ zero bits before the first non-zero bit. In this case bits are ignored until a non-zero bit is found. - \item When $mode = 1$ a non-zero bit has been seen before and a new $winsize$-bit window has not been formed yet. In this mode leading $0$ bits - are read and a single squaring is performed. If a non-zero bit is read a new window is created. - \item When $mode = 2$ the algorithm is in the middle of forming a window and new bits are appended to the window from the most significant bit - downwards. -\end{enumerate} -\item The variable $bitcnt$ indicates how many bits are left in the current digit of the exponent left to be read. When it reaches zero a new digit - is fetched from the exponent. -\item The variable $buf$ holds the currently read digit of the exponent. -\item The variable $digidx$ is an index into the exponents digits. It starts at the leading digit $x.used - 1$ and moves towards the trailing digit. -\item The variable $bitcpy$ indicates how many bits are in the currently formed window. When it reaches $winsize$ the window is flushed and - the appropriate operations performed. -\item The variable $bitbuf$ holds the current bits of the window being formed. -\end{enumerate} - -All of step 12 is the window processing loop. It will iterate while there are digits available form the exponent to read. The first step -inside this loop is to extract a new digit if no more bits are available in the current digit. If there are no bits left a new digit is -read and if there are no digits left than the loop terminates. - -After a digit is made available step 12.3 will extract the most significant bit of the current digit and move all other bits in the digit -upwards. In effect the digit is read from most significant bit to least significant bit and since the digits are read from leading to -trailing edges the entire exponent is read from most significant bit to least significant bit. - -At step 12.5 if the $mode$ and currently extracted bit $y$ are both zero the bit is ignored and the next bit is read. This prevents the -algorithm from having to perform trivial squaring and reduction operations before the first non-zero bit is read. Step 12.6 and 12.7-10 handle -the two cases of $mode = 1$ and $mode = 2$ respectively. - -\begin{center} -\begin{figure}[here] -\includegraphics{pics/expt_state.ps} -\caption{Sliding Window State Diagram} -\label{pic:expt_state} -\end{figure} -\end{center} - -By step 13 there are no more digits left in the exponent. However, there may be partial bits in the window left. If $mode = 2$ then -a Left-to-Right algorithm is used to process the remaining few bits. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_exptmod.c -\vspace{-3mm} -\begin{alltt} -016 -017 #ifdef MP_LOW_MEM -018 #define TAB_SIZE 32 -019 #else -020 #define TAB_SIZE 256 -021 #endif -022 -023 int s_mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y, int redmod - e) -024 \{ -025 mp_int M[TAB_SIZE], res, mu; -026 mp_digit buf; -027 int err, bitbuf, bitcpy, bitcnt, mode, digidx, x, y, winsize; -028 int (*redux)(mp_int*,mp_int*,mp_int*); -029 -030 /* find window size */ -031 x = mp_count_bits (X); -032 if (x <= 7) \{ -033 winsize = 2; -034 \} else if (x <= 36) \{ -035 winsize = 3; -036 \} else if (x <= 140) \{ -037 winsize = 4; -038 \} else if (x <= 450) \{ -039 winsize = 5; -040 \} else if (x <= 1303) \{ -041 winsize = 6; -042 \} else if (x <= 3529) \{ -043 winsize = 7; -044 \} else \{ -045 winsize = 8; -046 \} -047 -048 #ifdef MP_LOW_MEM -049 if (winsize > 5) \{ -050 winsize = 5; -051 \} -052 #endif -053 -054 /* init M array */ -055 /* init first cell */ -056 if ((err = mp_init(&M[1])) != MP_OKAY) \{ -057 return err; -058 \} -059 -060 /* now init the second half of the array */ -061 for (x = 1<<(winsize-1); x < (1 << winsize); x++) \{ -062 if ((err = mp_init(&M[x])) != MP_OKAY) \{ -063 for (y = 1<<(winsize-1); y < x; y++) \{ -064 mp_clear (&M[y]); -065 \} -066 mp_clear(&M[1]); -067 return err; -068 \} -069 \} -070 -071 /* create mu, used for Barrett reduction */ -072 if ((err = mp_init (&mu)) != MP_OKAY) \{ -073 goto LBL_M; -074 \} -075 -076 if (redmode == 0) \{ -077 if ((err = mp_reduce_setup (&mu, P)) != MP_OKAY) \{ -078 goto LBL_MU; -079 \} -080 redux = mp_reduce; -081 \} else \{ -082 if ((err = mp_reduce_2k_setup_l (P, &mu)) != MP_OKAY) \{ -083 goto LBL_MU; -084 \} -085 redux = mp_reduce_2k_l; -086 \} -087 -088 /* create M table -089 * -090 * The M table contains powers of the base, -091 * e.g. M[x] = G**x mod P -092 * -093 * The first half of the table is not -094 * computed though accept for M[0] and M[1] -095 */ -096 if ((err = mp_mod (G, P, &M[1])) != MP_OKAY) \{ -097 goto LBL_MU; -098 \} -099 -100 /* compute the value at M[1<<(winsize-1)] by squaring -101 * M[1] (winsize-1) times -102 */ -103 if ((err = mp_copy (&M[1], &M[1 << (winsize - 1)])) != MP_OKAY) \{ -104 goto LBL_MU; -105 \} -106 -107 for (x = 0; x < (winsize - 1); x++) \{ -108 /* square it */ -109 if ((err = mp_sqr (&M[1 << (winsize - 1)], -110 &M[1 << (winsize - 1)])) != MP_OKAY) \{ -111 goto LBL_MU; -112 \} -113 -114 /* reduce modulo P */ -115 if ((err = redux (&M[1 << (winsize - 1)], P, &mu)) != MP_OKAY) \{ -116 goto LBL_MU; -117 \} -118 \} -119 -120 /* create upper table, that is M[x] = M[x-1] * M[1] (mod P) -121 * for x = (2**(winsize - 1) + 1) to (2**winsize - 1) -122 */ -123 for (x = (1 << (winsize - 1)) + 1; x < (1 << winsize); x++) \{ -124 if ((err = mp_mul (&M[x - 1], &M[1], &M[x])) != MP_OKAY) \{ -125 goto LBL_MU; -126 \} -127 if ((err = redux (&M[x], P, &mu)) != MP_OKAY) \{ -128 goto LBL_MU; -129 \} -130 \} -131 -132 /* setup result */ -133 if ((err = mp_init (&res)) != MP_OKAY) \{ -134 goto LBL_MU; -135 \} -136 mp_set (&res, 1); -137 -138 /* set initial mode and bit cnt */ -139 mode = 0; -140 bitcnt = 1; -141 buf = 0; -142 digidx = X->used - 1; -143 bitcpy = 0; -144 bitbuf = 0; -145 -146 for (;;) \{ -147 /* grab next digit as required */ -148 if (--bitcnt == 0) \{ -149 /* if digidx == -1 we are out of digits */ -150 if (digidx == -1) \{ -151 break; -152 \} -153 /* read next digit and reset the bitcnt */ -154 buf = X->dp[digidx--]; -155 bitcnt = (int) DIGIT_BIT; -156 \} -157 -158 /* grab the next msb from the exponent */ -159 y = (buf >> (mp_digit)(DIGIT_BIT - 1)) & 1; -160 buf <<= (mp_digit)1; -161 -162 /* if the bit is zero and mode == 0 then we ignore it -163 * These represent the leading zero bits before the first 1 bit -164 * in the exponent. Technically this opt is not required but it -165 * does lower the # of trivial squaring/reductions used -166 */ -167 if (mode == 0 && y == 0) \{ -168 continue; -169 \} -170 -171 /* if the bit is zero and mode == 1 then we square */ -172 if (mode == 1 && y == 0) \{ -173 if ((err = mp_sqr (&res, &res)) != MP_OKAY) \{ -174 goto LBL_RES; -175 \} -176 if ((err = redux (&res, P, &mu)) != MP_OKAY) \{ -177 goto LBL_RES; -178 \} -179 continue; -180 \} -181 -182 /* else we add it to the window */ -183 bitbuf |= (y << (winsize - ++bitcpy)); -184 mode = 2; -185 -186 if (bitcpy == winsize) \{ -187 /* ok window is filled so square as required and multiply */ -188 /* square first */ -189 for (x = 0; x < winsize; x++) \{ -190 if ((err = mp_sqr (&res, &res)) != MP_OKAY) \{ -191 goto LBL_RES; -192 \} -193 if ((err = redux (&res, P, &mu)) != MP_OKAY) \{ -194 goto LBL_RES; -195 \} -196 \} -197 -198 /* then multiply */ -199 if ((err = mp_mul (&res, &M[bitbuf], &res)) != MP_OKAY) \{ -200 goto LBL_RES; -201 \} -202 if ((err = redux (&res, P, &mu)) != MP_OKAY) \{ -203 goto LBL_RES; -204 \} -205 -206 /* empty window and reset */ -207 bitcpy = 0; -208 bitbuf = 0; -209 mode = 1; -210 \} -211 \} -212 -213 /* if bits remain then square/multiply */ -214 if (mode == 2 && bitcpy > 0) \{ -215 /* square then multiply if the bit is set */ -216 for (x = 0; x < bitcpy; x++) \{ -217 if ((err = mp_sqr (&res, &res)) != MP_OKAY) \{ -218 goto LBL_RES; -219 \} -220 if ((err = redux (&res, P, &mu)) != MP_OKAY) \{ -221 goto LBL_RES; -222 \} -223 -224 bitbuf <<= 1; -225 if ((bitbuf & (1 << winsize)) != 0) \{ -226 /* then multiply */ -227 if ((err = mp_mul (&res, &M[1], &res)) != MP_OKAY) \{ -228 goto LBL_RES; -229 \} -230 if ((err = redux (&res, P, &mu)) != MP_OKAY) \{ -231 goto LBL_RES; -232 \} -233 \} -234 \} -235 \} -236 -237 mp_exch (&res, Y); -238 err = MP_OKAY; -239 LBL_RES:mp_clear (&res); -240 LBL_MU:mp_clear (&mu); -241 LBL_M: -242 mp_clear(&M[1]); -243 for (x = 1<<(winsize-1); x < (1 << winsize); x++) \{ -244 mp_clear (&M[x]); -245 \} -246 return err; -247 \} -248 #endif -\end{alltt} -\end{small} - -Lines 21 through 40 determine the optimal window size based on the length of the exponent in bits. The window divisions are sorted -from smallest to greatest so that in each \textbf{if} statement only one condition must be tested. For example, by the \textbf{if} statement -on line 32 the value of $x$ is already known to be greater than $140$. - -The conditional piece of code beginning on line 48 allows the window size to be restricted to five bits. This logic is used to ensure -the table of precomputed powers of $G$ remains relatively small. - -The for loop on line 61 initializes the $M$ array while lines 62 and 77 compute the value of $\mu$ required for -Barrett reduction. - --- More later. - -\section{Quick Power of Two} -Calculating $b = 2^a$ can be performed much quicker than with any of the previous algorithms. Recall that a logical shift left $m << k$ is -equivalent to $m \cdot 2^k$. By this logic when $m = 1$ a quick power of two can be achieved. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_2expt}. \\ -\textbf{Input}. integer $b$ \\ -\textbf{Output}. $a \leftarrow 2^b$ \\ -\hline \\ -1. $a \leftarrow 0$ \\ -2. If $a.alloc < \lfloor b / lg(\beta) \rfloor + 1$ then grow $a$ appropriately. \\ -3. $a.used \leftarrow \lfloor b / lg(\beta) \rfloor + 1$ \\ -4. $a_{\lfloor b / lg(\beta) \rfloor} \leftarrow 1 << (b \mbox{ mod } lg(\beta))$ \\ -5. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_2expt} -\end{figure} - -\textbf{Algorithm mp\_2expt.} - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_2expt.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* computes a = 2**b -018 * -019 * Simple algorithm which zeroes the int, grows it then just sets one bit -020 * as required. -021 */ -022 int -023 mp_2expt (mp_int * a, int b) -024 \{ -025 int res; -026 -027 /* zero a as per default */ -028 mp_zero (a); -029 -030 /* grow a to accomodate the single bit */ -031 if ((res = mp_grow (a, b / DIGIT_BIT + 1)) != MP_OKAY) \{ -032 return res; -033 \} -034 -035 /* set the used count of where the bit will go */ -036 a->used = b / DIGIT_BIT + 1; -037 -038 /* put the single bit in its place */ -039 a->dp[b / DIGIT_BIT] = ((mp_digit)1) << (b % DIGIT_BIT); -040 -041 return MP_OKAY; -042 \} -043 #endif -\end{alltt} -\end{small} - -\chapter{Higher Level Algorithms} - -This chapter discusses the various higher level algorithms that are required to complete a well rounded multiple precision integer package. These -routines are less performance oriented than the algorithms of chapters five, six and seven but are no less important. - -The first section describes a method of integer division with remainder that is universally well known. It provides the signed division logic -for the package. The subsequent section discusses a set of algorithms which allow a single digit to be the 2nd operand for a variety of operations. -These algorithms serve mostly to simplify other algorithms where small constants are required. The last two sections discuss how to manipulate -various representations of integers. For example, converting from an mp\_int to a string of character. - -\section{Integer Division with Remainder} -\label{sec:division} - -Integer division aside from modular exponentiation is the most intensive algorithm to compute. Like addition, subtraction and multiplication -the basis of this algorithm is the long-hand division algorithm taught to school children. Throughout this discussion several common variables -will be used. Let $x$ represent the divisor and $y$ represent the dividend. Let $q$ represent the integer quotient $\lfloor y / x \rfloor$ and -let $r$ represent the remainder $r = y - x \lfloor y / x \rfloor$. The following simple algorithm will be used to start the discussion. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Radix-$\beta$ Integer Division}. \\ -\textbf{Input}. integer $x$ and $y$ \\ -\textbf{Output}. $q = \lfloor y/x\rfloor, r = y - xq$ \\ -\hline \\ -1. $q \leftarrow 0$ \\ -2. $n \leftarrow \vert \vert y \vert \vert - \vert \vert x \vert \vert$ \\ -3. for $t$ from $n$ down to $0$ do \\ -\hspace{3mm}3.1 Maximize $k$ such that $kx\beta^t$ is less than or equal to $y$ and $(k + 1)x\beta^t$ is greater. \\ -\hspace{3mm}3.2 $q \leftarrow q + k\beta^t$ \\ -\hspace{3mm}3.3 $y \leftarrow y - kx\beta^t$ \\ -4. $r \leftarrow y$ \\ -5. Return($q, r$) \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm Radix-$\beta$ Integer Division} -\label{fig:raddiv} -\end{figure} - -As children we are taught this very simple algorithm for the case of $\beta = 10$. Almost instinctively several optimizations are taught for which -their reason of existing are never explained. For this example let $y = 5471$ represent the dividend and $x = 23$ represent the divisor. - -To find the first digit of the quotient the value of $k$ must be maximized such that $kx\beta^t$ is less than or equal to $y$ and -simultaneously $(k + 1)x\beta^t$ is greater than $y$. Implicitly $k$ is the maximum value the $t$'th digit of the quotient may have. The habitual method -used to find the maximum is to ``eyeball'' the two numbers, typically only the leading digits and quickly estimate a quotient. By only using leading -digits a much simpler division may be used to form an educated guess at what the value must be. In this case $k = \lfloor 54/23\rfloor = 2$ quickly -arises as a possible solution. Indeed $2x\beta^2 = 4600$ is less than $y = 5471$ and simultaneously $(k + 1)x\beta^2 = 6900$ is larger than $y$. -As a result $k\beta^2$ is added to the quotient which now equals $q = 200$ and $4600$ is subtracted from $y$ to give a remainder of $y = 841$. - -Again this process is repeated to produce the quotient digit $k = 3$ which makes the quotient $q = 200 + 3\beta = 230$ and the remainder -$y = 841 - 3x\beta = 181$. Finally the last iteration of the loop produces $k = 7$ which leads to the quotient $q = 230 + 7 = 237$ and the -remainder $y = 181 - 7x = 20$. The final quotient and remainder found are $q = 237$ and $r = y = 20$ which are indeed correct since -$237 \cdot 23 + 20 = 5471$ is true. - -\subsection{Quotient Estimation} -\label{sec:divest} -As alluded to earlier the quotient digit $k$ can be estimated from only the leading digits of both the divisor and dividend. When $p$ leading -digits are used from both the divisor and dividend to form an estimation the accuracy of the estimation rises as $p$ grows. Technically -speaking the estimation is based on assuming the lower $\vert \vert y \vert \vert - p$ and $\vert \vert x \vert \vert - p$ lower digits of the -dividend and divisor are zero. - -The value of the estimation may off by a few values in either direction and in general is fairly correct. A simplification \cite[pp. 271]{TAOCPV2} -of the estimation technique is to use $t + 1$ digits of the dividend and $t$ digits of the divisor, in particularly when $t = 1$. The estimate -using this technique is never too small. For the following proof let $t = \vert \vert y \vert \vert - 1$ and $s = \vert \vert x \vert \vert - 1$ -represent the most significant digits of the dividend and divisor respectively. - -\textbf{Proof.}\textit{ The quotient $\hat k = \lfloor (y_t\beta + y_{t-1}) / x_s \rfloor$ is greater than or equal to -$k = \lfloor y / (x \cdot \beta^{\vert \vert y \vert \vert - \vert \vert x \vert \vert - 1}) \rfloor$. } -The first obvious case is when $\hat k = \beta - 1$ in which case the proof is concluded since the real quotient cannot be larger. For all other -cases $\hat k = \lfloor (y_t\beta + y_{t-1}) / x_s \rfloor$ and $\hat k x_s \ge y_t\beta + y_{t-1} - x_s + 1$. The latter portion of the inequalility -$-x_s + 1$ arises from the fact that a truncated integer division will give the same quotient for at most $x_s - 1$ values. Next a series of -inequalities will prove the hypothesis. - -\begin{equation} -y - \hat k x \le y - \hat k x_s\beta^s -\end{equation} - -This is trivially true since $x \ge x_s\beta^s$. Next we replace $\hat kx_s\beta^s$ by the previous inequality for $\hat kx_s$. - -\begin{equation} -y - \hat k x \le y_t\beta^t + \ldots + y_0 - (y_t\beta^t + y_{t-1}\beta^{t-1} - x_s\beta^t + \beta^s) -\end{equation} - -By simplifying the previous inequality the following inequality is formed. - -\begin{equation} -y - \hat k x \le y_{t-2}\beta^{t-2} + \ldots + y_0 + x_s\beta^s - \beta^s -\end{equation} - -Subsequently, - -\begin{equation} -y_{t-2}\beta^{t-2} + \ldots + y_0 + x_s\beta^s - \beta^s < x_s\beta^s \le x -\end{equation} - -Which proves that $y - \hat kx \le x$ and by consequence $\hat k \ge k$ which concludes the proof. \textbf{QED} - - -\subsection{Normalized Integers} -For the purposes of division a normalized input is when the divisors leading digit $x_n$ is greater than or equal to $\beta / 2$. By multiplying both -$x$ and $y$ by $j = \lfloor (\beta / 2) / x_n \rfloor$ the quotient remains unchanged and the remainder is simply $j$ times the original -remainder. The purpose of normalization is to ensure the leading digit of the divisor is sufficiently large such that the estimated quotient will -lie in the domain of a single digit. Consider the maximum dividend $(\beta - 1) \cdot \beta + (\beta - 1)$ and the minimum divisor $\beta / 2$. - -\begin{equation} -{{\beta^2 - 1} \over { \beta / 2}} \le 2\beta - {2 \over \beta} -\end{equation} - -At most the quotient approaches $2\beta$, however, in practice this will not occur since that would imply the previous quotient digit was too small. - -\subsection{Radix-$\beta$ Division with Remainder} -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_div}. \\ -\textbf{Input}. mp\_int $a, b$ \\ -\textbf{Output}. $c = \lfloor a/b \rfloor$, $d = a - bc$ \\ -\hline \\ -1. If $b = 0$ return(\textit{MP\_VAL}). \\ -2. If $\vert a \vert < \vert b \vert$ then do \\ -\hspace{3mm}2.1 $d \leftarrow a$ \\ -\hspace{3mm}2.2 $c \leftarrow 0$ \\ -\hspace{3mm}2.3 Return(\textit{MP\_OKAY}). \\ -\\ -Setup the quotient to receive the digits. \\ -3. Grow $q$ to $a.used + 2$ digits. \\ -4. $q \leftarrow 0$ \\ -5. $x \leftarrow \vert a \vert , y \leftarrow \vert b \vert$ \\ -6. $sign \leftarrow \left \lbrace \begin{array}{ll} - MP\_ZPOS & \mbox{if }a.sign = b.sign \\ - MP\_NEG & \mbox{otherwise} \\ - \end{array} \right .$ \\ -\\ -Normalize the inputs such that the leading digit of $y$ is greater than or equal to $\beta / 2$. \\ -7. $norm \leftarrow (lg(\beta) - 1) - (\lceil lg(y) \rceil \mbox{ (mod }lg(\beta)\mbox{)})$ \\ -8. $x \leftarrow x \cdot 2^{norm}, y \leftarrow y \cdot 2^{norm}$ \\ -\\ -Find the leading digit of the quotient. \\ -9. $n \leftarrow x.used - 1, t \leftarrow y.used - 1$ \\ -10. $y \leftarrow y \cdot \beta^{n - t}$ \\ -11. While ($x \ge y$) do \\ -\hspace{3mm}11.1 $q_{n - t} \leftarrow q_{n - t} + 1$ \\ -\hspace{3mm}11.2 $x \leftarrow x - y$ \\ -12. $y \leftarrow \lfloor y / \beta^{n-t} \rfloor$ \\ -\\ -Continued on the next page. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_div} -\end{figure} - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_div} (continued). \\ -\textbf{Input}. mp\_int $a, b$ \\ -\textbf{Output}. $c = \lfloor a/b \rfloor$, $d = a - bc$ \\ -\hline \\ -Now find the remainder fo the digits. \\ -13. for $i$ from $n$ down to $(t + 1)$ do \\ -\hspace{3mm}13.1 If $i > x.used$ then jump to the next iteration of this loop. \\ -\hspace{3mm}13.2 If $x_{i} = y_{t}$ then \\ -\hspace{6mm}13.2.1 $q_{i - t - 1} \leftarrow \beta - 1$ \\ -\hspace{3mm}13.3 else \\ -\hspace{6mm}13.3.1 $\hat r \leftarrow x_{i} \cdot \beta + x_{i - 1}$ \\ -\hspace{6mm}13.3.2 $\hat r \leftarrow \lfloor \hat r / y_{t} \rfloor$ \\ -\hspace{6mm}13.3.3 $q_{i - t - 1} \leftarrow \hat r$ \\ -\hspace{3mm}13.4 $q_{i - t - 1} \leftarrow q_{i - t - 1} + 1$ \\ -\\ -Fixup quotient estimation. \\ -\hspace{3mm}13.5 Loop \\ -\hspace{6mm}13.5.1 $q_{i - t - 1} \leftarrow q_{i - t - 1} - 1$ \\ -\hspace{6mm}13.5.2 t$1 \leftarrow 0$ \\ -\hspace{6mm}13.5.3 t$1_0 \leftarrow y_{t - 1}, $ t$1_1 \leftarrow y_t,$ t$1.used \leftarrow 2$ \\ -\hspace{6mm}13.5.4 $t1 \leftarrow t1 \cdot q_{i - t - 1}$ \\ -\hspace{6mm}13.5.5 t$2_0 \leftarrow x_{i - 2}, $ t$2_1 \leftarrow x_{i - 1}, $ t$2_2 \leftarrow x_i, $ t$2.used \leftarrow 3$ \\ -\hspace{6mm}13.5.6 If $\vert t1 \vert > \vert t2 \vert$ then goto step 13.5. \\ -\hspace{3mm}13.6 t$1 \leftarrow y \cdot q_{i - t - 1}$ \\ -\hspace{3mm}13.7 t$1 \leftarrow $ t$1 \cdot \beta^{i - t - 1}$ \\ -\hspace{3mm}13.8 $x \leftarrow x - $ t$1$ \\ -\hspace{3mm}13.9 If $x.sign = MP\_NEG$ then \\ -\hspace{6mm}13.10 t$1 \leftarrow y$ \\ -\hspace{6mm}13.11 t$1 \leftarrow $ t$1 \cdot \beta^{i - t - 1}$ \\ -\hspace{6mm}13.12 $x \leftarrow x + $ t$1$ \\ -\hspace{6mm}13.13 $q_{i - t - 1} \leftarrow q_{i - t - 1} - 1$ \\ -\\ -Finalize the result. \\ -14. Clamp excess digits of $q$ \\ -15. $c \leftarrow q, c.sign \leftarrow sign$ \\ -16. $x.sign \leftarrow a.sign$ \\ -17. $d \leftarrow \lfloor x / 2^{norm} \rfloor$ \\ -18. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_div (continued)} -\end{figure} -\textbf{Algorithm mp\_div.} -This algorithm will calculate quotient and remainder from an integer division given a dividend and divisor. The algorithm is a signed -division and will produce a fully qualified quotient and remainder. - -First the divisor $b$ must be non-zero which is enforced in step one. If the divisor is larger than the dividend than the quotient is implicitly -zero and the remainder is the dividend. - -After the first two trivial cases of inputs are handled the variable $q$ is setup to receive the digits of the quotient. Two unsigned copies of the -divisor $y$ and dividend $x$ are made as well. The core of the division algorithm is an unsigned division and will only work if the values are -positive. Now the two values $x$ and $y$ must be normalized such that the leading digit of $y$ is greater than or equal to $\beta / 2$. -This is performed by shifting both to the left by enough bits to get the desired normalization. - -At this point the division algorithm can begin producing digits of the quotient. Recall that maximum value of the estimation used is -$2\beta - {2 \over \beta}$ which means that a digit of the quotient must be first produced by another means. In this case $y$ is shifted -to the left (\textit{step ten}) so that it has the same number of digits as $x$. The loop on step eleven will subtract multiples of the -shifted copy of $y$ until $x$ is smaller. Since the leading digit of $y$ is greater than or equal to $\beta/2$ this loop will iterate at most two -times to produce the desired leading digit of the quotient. - -Now the remainder of the digits can be produced. The equation $\hat q = \lfloor {{x_i \beta + x_{i-1}}\over y_t} \rfloor$ is used to fairly -accurately approximate the true quotient digit. The estimation can in theory produce an estimation as high as $2\beta - {2 \over \beta}$ but by -induction the upper quotient digit is correct (\textit{as established on step eleven}) and the estimate must be less than $\beta$. - -Recall from section~\ref{sec:divest} that the estimation is never too low but may be too high. The next step of the estimation process is -to refine the estimation. The loop on step 13.5 uses $x_i\beta^2 + x_{i-1}\beta + x_{i-2}$ and $q_{i - t - 1}(y_t\beta + y_{t-1})$ as a higher -order approximation to adjust the quotient digit. - -After both phases of estimation the quotient digit may still be off by a value of one\footnote{This is similar to the error introduced -by optimizing Barrett reduction.}. Steps 13.6 and 13.7 subtract the multiple of the divisor from the dividend (\textit{Similar to step 3.3 of -algorithm~\ref{fig:raddiv}} and then subsequently add a multiple of the divisor if the quotient was too large. - -Now that the quotient has been determine finializing the result is a matter of clamping the quotient, fixing the sizes and de-normalizing the -remainder. An important aspect of this algorithm seemingly overlooked in other descriptions such as that of Algorithm 14.20 HAC \cite[pp. 598]{HAC} -is that when the estimations are being made (\textit{inside the loop on step 13.5}) that the digits $y_{t-1}$, $x_{i-2}$ and $x_{i-1}$ may lie -outside their respective boundaries. For example, if $t = 0$ or $i \le 1$ then the digits would be undefined. In those cases the digits should -respectively be replaced with a zero. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_div.c -\vspace{-3mm} -\begin{alltt} -016 -017 #ifdef BN_MP_DIV_SMALL -018 -019 /* slower bit-bang division... also smaller */ -020 int mp_div(mp_int * a, mp_int * b, mp_int * c, mp_int * d) -021 \{ -022 mp_int ta, tb, tq, q; -023 int res, n, n2; -024 -025 /* is divisor zero ? */ -026 if (mp_iszero (b) == 1) \{ -027 return MP_VAL; -028 \} -029 -030 /* if a < b then q=0, r = a */ -031 if (mp_cmp_mag (a, b) == MP_LT) \{ -032 if (d != NULL) \{ -033 res = mp_copy (a, d); -034 \} else \{ -035 res = MP_OKAY; -036 \} -037 if (c != NULL) \{ -038 mp_zero (c); -039 \} -040 return res; -041 \} -042 -043 /* init our temps */ -044 if ((res = mp_init_multi(&ta, &tb, &tq, &q, NULL) != MP_OKAY)) \{ -045 return res; -046 \} -047 -048 -049 mp_set(&tq, 1); -050 n = mp_count_bits(a) - mp_count_bits(b); -051 if (((res = mp_abs(a, &ta)) != MP_OKAY) || -052 ((res = mp_abs(b, &tb)) != MP_OKAY) || -053 ((res = mp_mul_2d(&tb, n, &tb)) != MP_OKAY) || -054 ((res = mp_mul_2d(&tq, n, &tq)) != MP_OKAY)) \{ -055 goto LBL_ERR; -056 \} -057 -058 while (n-- >= 0) \{ -059 if (mp_cmp(&tb, &ta) != MP_GT) \{ -060 if (((res = mp_sub(&ta, &tb, &ta)) != MP_OKAY) || -061 ((res = mp_add(&q, &tq, &q)) != MP_OKAY)) \{ -062 goto LBL_ERR; -063 \} -064 \} -065 if (((res = mp_div_2d(&tb, 1, &tb, NULL)) != MP_OKAY) || -066 ((res = mp_div_2d(&tq, 1, &tq, NULL)) != MP_OKAY)) \{ -067 goto LBL_ERR; -068 \} -069 \} -070 -071 /* now q == quotient and ta == remainder */ -072 n = a->sign; -073 n2 = (a->sign == b->sign ? MP_ZPOS : MP_NEG); -074 if (c != NULL) \{ -075 mp_exch(c, &q); -076 c->sign = (mp_iszero(c) == MP_YES) ? MP_ZPOS : n2; -077 \} -078 if (d != NULL) \{ -079 mp_exch(d, &ta); -080 d->sign = (mp_iszero(d) == MP_YES) ? MP_ZPOS : n; -081 \} -082 LBL_ERR: -083 mp_clear_multi(&ta, &tb, &tq, &q, NULL); -084 return res; -085 \} -086 -087 #else -088 -089 /* integer signed division. -090 * c*b + d == a [e.g. a/b, c=quotient, d=remainder] -091 * HAC pp.598 Algorithm 14.20 -092 * -093 * Note that the description in HAC is horribly -094 * incomplete. For example, it doesn't consider -095 * the case where digits are removed from 'x' in -096 * the inner loop. It also doesn't consider the -097 * case that y has fewer than three digits, etc.. -098 * -099 * The overall algorithm is as described as -100 * 14.20 from HAC but fixed to treat these cases. -101 */ -102 int mp_div (mp_int * a, mp_int * b, mp_int * c, mp_int * d) -103 \{ -104 mp_int q, x, y, t1, t2; -105 int res, n, t, i, norm, neg; -106 -107 /* is divisor zero ? */ -108 if (mp_iszero (b) == 1) \{ -109 return MP_VAL; -110 \} -111 -112 /* if a < b then q=0, r = a */ -113 if (mp_cmp_mag (a, b) == MP_LT) \{ -114 if (d != NULL) \{ -115 res = mp_copy (a, d); -116 \} else \{ -117 res = MP_OKAY; -118 \} -119 if (c != NULL) \{ -120 mp_zero (c); -121 \} -122 return res; -123 \} -124 -125 if ((res = mp_init_size (&q, a->used + 2)) != MP_OKAY) \{ -126 return res; -127 \} -128 q.used = a->used + 2; -129 -130 if ((res = mp_init (&t1)) != MP_OKAY) \{ -131 goto LBL_Q; -132 \} -133 -134 if ((res = mp_init (&t2)) != MP_OKAY) \{ -135 goto LBL_T1; -136 \} -137 -138 if ((res = mp_init_copy (&x, a)) != MP_OKAY) \{ -139 goto LBL_T2; -140 \} -141 -142 if ((res = mp_init_copy (&y, b)) != MP_OKAY) \{ -143 goto LBL_X; -144 \} -145 -146 /* fix the sign */ -147 neg = (a->sign == b->sign) ? MP_ZPOS : MP_NEG; -148 x.sign = y.sign = MP_ZPOS; -149 -150 /* normalize both x and y, ensure that y >= b/2, [b == 2**DIGIT_BIT] */ -151 norm = mp_count_bits(&y) % DIGIT_BIT; -152 if (norm < (int)(DIGIT_BIT-1)) \{ -153 norm = (DIGIT_BIT-1) - norm; -154 if ((res = mp_mul_2d (&x, norm, &x)) != MP_OKAY) \{ -155 goto LBL_Y; -156 \} -157 if ((res = mp_mul_2d (&y, norm, &y)) != MP_OKAY) \{ -158 goto LBL_Y; -159 \} -160 \} else \{ -161 norm = 0; -162 \} -163 -164 /* note hac does 0 based, so if used==5 then its 0,1,2,3,4, e.g. use 4 */ -165 n = x.used - 1; -166 t = y.used - 1; -167 -168 /* while (x >= y*b**n-t) do \{ q[n-t] += 1; x -= y*b**\{n-t\} \} */ -169 if ((res = mp_lshd (&y, n - t)) != MP_OKAY) \{ /* y = y*b**\{n-t\} */ -170 goto LBL_Y; -171 \} -172 -173 while (mp_cmp (&x, &y) != MP_LT) \{ -174 ++(q.dp[n - t]); -175 if ((res = mp_sub (&x, &y, &x)) != MP_OKAY) \{ -176 goto LBL_Y; -177 \} -178 \} -179 -180 /* reset y by shifting it back down */ -181 mp_rshd (&y, n - t); -182 -183 /* step 3. for i from n down to (t + 1) */ -184 for (i = n; i >= (t + 1); i--) \{ -185 if (i > x.used) \{ -186 continue; -187 \} -188 -189 /* step 3.1 if xi == yt then set q\{i-t-1\} to b-1, -190 * otherwise set q\{i-t-1\} to (xi*b + x\{i-1\})/yt */ -191 if (x.dp[i] == y.dp[t]) \{ -192 q.dp[i - t - 1] = ((((mp_digit)1) << DIGIT_BIT) - 1); -193 \} else \{ -194 mp_word tmp; -195 tmp = ((mp_word) x.dp[i]) << ((mp_word) DIGIT_BIT); -196 tmp |= ((mp_word) x.dp[i - 1]); -197 tmp /= ((mp_word) y.dp[t]); -198 if (tmp > (mp_word) MP_MASK) -199 tmp = MP_MASK; -200 q.dp[i - t - 1] = (mp_digit) (tmp & (mp_word) (MP_MASK)); -201 \} -202 -203 /* while (q\{i-t-1\} * (yt * b + y\{t-1\})) > -204 xi * b**2 + xi-1 * b + xi-2 -205 -206 do q\{i-t-1\} -= 1; -207 */ -208 q.dp[i - t - 1] = (q.dp[i - t - 1] + 1) & MP_MASK; -209 do \{ -210 q.dp[i - t - 1] = (q.dp[i - t - 1] - 1) & MP_MASK; -211 -212 /* find left hand */ -213 mp_zero (&t1); -214 t1.dp[0] = (t - 1 < 0) ? 0 : y.dp[t - 1]; -215 t1.dp[1] = y.dp[t]; -216 t1.used = 2; -217 if ((res = mp_mul_d (&t1, q.dp[i - t - 1], &t1)) != MP_OKAY) \{ -218 goto LBL_Y; -219 \} -220 -221 /* find right hand */ -222 t2.dp[0] = (i - 2 < 0) ? 0 : x.dp[i - 2]; -223 t2.dp[1] = (i - 1 < 0) ? 0 : x.dp[i - 1]; -224 t2.dp[2] = x.dp[i]; -225 t2.used = 3; -226 \} while (mp_cmp_mag(&t1, &t2) == MP_GT); -227 -228 /* step 3.3 x = x - q\{i-t-1\} * y * b**\{i-t-1\} */ -229 if ((res = mp_mul_d (&y, q.dp[i - t - 1], &t1)) != MP_OKAY) \{ -230 goto LBL_Y; -231 \} -232 -233 if ((res = mp_lshd (&t1, i - t - 1)) != MP_OKAY) \{ -234 goto LBL_Y; -235 \} -236 -237 if ((res = mp_sub (&x, &t1, &x)) != MP_OKAY) \{ -238 goto LBL_Y; -239 \} -240 -241 /* if x < 0 then \{ x = x + y*b**\{i-t-1\}; q\{i-t-1\} -= 1; \} */ -242 if (x.sign == MP_NEG) \{ -243 if ((res = mp_copy (&y, &t1)) != MP_OKAY) \{ -244 goto LBL_Y; -245 \} -246 if ((res = mp_lshd (&t1, i - t - 1)) != MP_OKAY) \{ -247 goto LBL_Y; -248 \} -249 if ((res = mp_add (&x, &t1, &x)) != MP_OKAY) \{ -250 goto LBL_Y; -251 \} -252 -253 q.dp[i - t - 1] = (q.dp[i - t - 1] - 1UL) & MP_MASK; -254 \} -255 \} -256 -257 /* now q is the quotient and x is the remainder -258 * [which we have to normalize] -259 */ -260 -261 /* get sign before writing to c */ -262 x.sign = x.used == 0 ? MP_ZPOS : a->sign; -263 -264 if (c != NULL) \{ -265 mp_clamp (&q); -266 mp_exch (&q, c); -267 c->sign = neg; -268 \} -269 -270 if (d != NULL) \{ -271 mp_div_2d (&x, norm, &x, NULL); -272 mp_exch (&x, d); -273 \} -274 -275 res = MP_OKAY; -276 -277 LBL_Y:mp_clear (&y); -278 LBL_X:mp_clear (&x); -279 LBL_T2:mp_clear (&t2); -280 LBL_T1:mp_clear (&t1); -281 LBL_Q:mp_clear (&q); -282 return res; -283 \} -284 -285 #endif -286 -287 #endif -\end{alltt} -\end{small} - -The implementation of this algorithm differs slightly from the pseudo code presented previously. In this algorithm either of the quotient $c$ or -remainder $d$ may be passed as a \textbf{NULL} pointer which indicates their value is not desired. For example, the C code to call the division -algorithm with only the quotient is - -\begin{verbatim} -mp_div(&a, &b, &c, NULL); /* c = [a/b] */ -\end{verbatim} - -Lines 37 and 44 handle the two trivial cases of inputs which are division by zero and dividend smaller than the divisor -respectively. After the two trivial cases all of the temporary variables are initialized. Line 105 determines the sign of -the quotient and line 76 ensures that both $x$ and $y$ are positive. - -The number of bits in the leading digit is calculated on line 105. Implictly an mp\_int with $r$ digits will require $lg(\beta)(r-1) + k$ bits -of precision which when reduced modulo $lg(\beta)$ produces the value of $k$. In this case $k$ is the number of bits in the leading digit which is -exactly what is required. For the algorithm to operate $k$ must equal $lg(\beta) - 1$ and when it does not the inputs must be normalized by shifting -them to the left by $lg(\beta) - 1 - k$ bits. - -Throughout the variables $n$ and $t$ will represent the highest digit of $x$ and $y$ respectively. These are first used to produce the -leading digit of the quotient. The loop beginning on line 183 will produce the remainder of the quotient digits. - -The conditional ``continue'' on line 114 is used to prevent the algorithm from reading past the leading edge of $x$ which can occur when the -algorithm eliminates multiple non-zero digits in a single iteration. This ensures that $x_i$ is always non-zero since by definition the digits -above the $i$'th position $x$ must be zero in order for the quotient to be precise\footnote{Precise as far as integer division is concerned.}. - -Lines 130, 130 and 134 through 134 manually construct the high accuracy estimations by setting the digits of the two mp\_int -variables directly. - -\section{Single Digit Helpers} - -This section briefly describes a series of single digit helper algorithms which come in handy when working with small constants. All of -the helper functions assume the single digit input is positive and will treat them as such. - -\subsection{Single Digit Addition and Subtraction} - -Both addition and subtraction are performed by ``cheating'' and using mp\_set followed by the higher level addition or subtraction -algorithms. As a result these algorithms are subtantially simpler with a slight cost in performance. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_add\_d}. \\ -\textbf{Input}. mp\_int $a$ and a mp\_digit $b$ \\ -\textbf{Output}. $c = a + b$ \\ -\hline \\ -1. $t \leftarrow b$ (\textit{mp\_set}) \\ -2. $c \leftarrow a + t$ \\ -3. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_add\_d} -\end{figure} - -\textbf{Algorithm mp\_add\_d.} -This algorithm initiates a temporary mp\_int with the value of the single digit and uses algorithm mp\_add to add the two values together. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_add\_d.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* single digit addition */ -018 int -019 mp_add_d (mp_int * a, mp_digit b, mp_int * c) -020 \{ -021 int res, ix, oldused; -022 mp_digit *tmpa, *tmpc, mu; -023 -024 /* grow c as required */ -025 if (c->alloc < a->used + 1) \{ -026 if ((res = mp_grow(c, a->used + 1)) != MP_OKAY) \{ -027 return res; -028 \} -029 \} -030 -031 /* if a is negative and |a| >= b, call c = |a| - b */ -032 if (a->sign == MP_NEG && (a->used > 1 || a->dp[0] >= b)) \{ -033 /* temporarily fix sign of a */ -034 a->sign = MP_ZPOS; -035 -036 /* c = |a| - b */ -037 res = mp_sub_d(a, b, c); -038 -039 /* fix sign */ -040 a->sign = c->sign = MP_NEG; -041 -042 return res; -043 \} -044 -045 /* old number of used digits in c */ -046 oldused = c->used; -047 -048 /* sign always positive */ -049 c->sign = MP_ZPOS; -050 -051 /* source alias */ -052 tmpa = a->dp; -053 -054 /* destination alias */ -055 tmpc = c->dp; -056 -057 /* if a is positive */ -058 if (a->sign == MP_ZPOS) \{ -059 /* add digit, after this we're propagating -060 * the carry. -061 */ -062 *tmpc = *tmpa++ + b; -063 mu = *tmpc >> DIGIT_BIT; -064 *tmpc++ &= MP_MASK; -065 -066 /* now handle rest of the digits */ -067 for (ix = 1; ix < a->used; ix++) \{ -068 *tmpc = *tmpa++ + mu; -069 mu = *tmpc >> DIGIT_BIT; -070 *tmpc++ &= MP_MASK; -071 \} -072 /* set final carry */ -073 ix++; -074 *tmpc++ = mu; -075 -076 /* setup size */ -077 c->used = a->used + 1; -078 \} else \{ -079 /* a was negative and |a| < b */ -080 c->used = 1; -081 -082 /* the result is a single digit */ -083 if (a->used == 1) \{ -084 *tmpc++ = b - a->dp[0]; -085 \} else \{ -086 *tmpc++ = b; -087 \} -088 -089 /* setup count so the clearing of oldused -090 * can fall through correctly -091 */ -092 ix = 1; -093 \} -094 -095 /* now zero to oldused */ -096 while (ix++ < oldused) \{ -097 *tmpc++ = 0; -098 \} -099 mp_clamp(c); -100 -101 return MP_OKAY; -102 \} -103 -104 #endif -\end{alltt} -\end{small} - -Clever use of the letter 't'. - -\subsubsection{Subtraction} -The single digit subtraction algorithm mp\_sub\_d is essentially the same except it uses mp\_sub to subtract the digit from the mp\_int. - -\subsection{Single Digit Multiplication} -Single digit multiplication arises enough in division and radix conversion that it ought to be implement as a special case of the baseline -multiplication algorithm. Essentially this algorithm is a modified version of algorithm s\_mp\_mul\_digs where one of the multiplicands -only has one digit. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_mul\_d}. \\ -\textbf{Input}. mp\_int $a$ and a mp\_digit $b$ \\ -\textbf{Output}. $c = ab$ \\ -\hline \\ -1. $pa \leftarrow a.used$ \\ -2. Grow $c$ to at least $pa + 1$ digits. \\ -3. $oldused \leftarrow c.used$ \\ -4. $c.used \leftarrow pa + 1$ \\ -5. $c.sign \leftarrow a.sign$ \\ -6. $\mu \leftarrow 0$ \\ -7. for $ix$ from $0$ to $pa - 1$ do \\ -\hspace{3mm}7.1 $\hat r \leftarrow \mu + a_{ix}b$ \\ -\hspace{3mm}7.2 $c_{ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{3mm}7.3 $\mu \leftarrow \lfloor \hat r / \beta \rfloor$ \\ -8. $c_{pa} \leftarrow \mu$ \\ -9. for $ix$ from $pa + 1$ to $oldused$ do \\ -\hspace{3mm}9.1 $c_{ix} \leftarrow 0$ \\ -10. Clamp excess digits of $c$. \\ -11. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_mul\_d} -\end{figure} -\textbf{Algorithm mp\_mul\_d.} -This algorithm quickly multiplies an mp\_int by a small single digit value. It is specially tailored to the job and has a minimal of overhead. -Unlike the full multiplication algorithms this algorithm does not require any significnat temporary storage or memory allocations. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_mul\_d.c -\vspace{-3mm} -\begin{alltt} -016 -017 /* multiply by a digit */ -018 int -019 mp_mul_d (mp_int * a, mp_digit b, mp_int * c) -020 \{ -021 mp_digit u, *tmpa, *tmpc; -022 mp_word r; -023 int ix, res, olduse; -024 -025 /* make sure c is big enough to hold a*b */ -026 if (c->alloc < a->used + 1) \{ -027 if ((res = mp_grow (c, a->used + 1)) != MP_OKAY) \{ -028 return res; -029 \} -030 \} -031 -032 /* get the original destinations used count */ -033 olduse = c->used; -034 -035 /* set the sign */ -036 c->sign = a->sign; -037 -038 /* alias for a->dp [source] */ -039 tmpa = a->dp; -040 -041 /* alias for c->dp [dest] */ -042 tmpc = c->dp; -043 -044 /* zero carry */ -045 u = 0; -046 -047 /* compute columns */ -048 for (ix = 0; ix < a->used; ix++) \{ -049 /* compute product and carry sum for this term */ -050 r = ((mp_word) u) + ((mp_word)*tmpa++) * ((mp_word)b); -051 -052 /* mask off higher bits to get a single digit */ -053 *tmpc++ = (mp_digit) (r & ((mp_word) MP_MASK)); -054 -055 /* send carry into next iteration */ -056 u = (mp_digit) (r >> ((mp_word) DIGIT_BIT)); -057 \} -058 -059 /* store final carry [if any] and increment ix offset */ -060 *tmpc++ = u; -061 ++ix; -062 -063 /* now zero digits above the top */ -064 while (ix++ < olduse) \{ -065 *tmpc++ = 0; -066 \} -067 -068 /* set used count */ -069 c->used = a->used + 1; -070 mp_clamp(c); -071 -072 return MP_OKAY; -073 \} -074 #endif -\end{alltt} -\end{small} - -In this implementation the destination $c$ may point to the same mp\_int as the source $a$ since the result is written after the digit is -read from the source. This function uses pointer aliases $tmpa$ and $tmpc$ for the digits of $a$ and $c$ respectively. - -\subsection{Single Digit Division} -Like the single digit multiplication algorithm, single digit division is also a fairly common algorithm used in radix conversion. Since the -divisor is only a single digit a specialized variant of the division algorithm can be used to compute the quotient. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_div\_d}. \\ -\textbf{Input}. mp\_int $a$ and a mp\_digit $b$ \\ -\textbf{Output}. $c = \lfloor a / b \rfloor, d = a - cb$ \\ -\hline \\ -1. If $b = 0$ then return(\textit{MP\_VAL}).\\ -2. If $b = 3$ then use algorithm mp\_div\_3 instead. \\ -3. Init $q$ to $a.used$ digits. \\ -4. $q.used \leftarrow a.used$ \\ -5. $q.sign \leftarrow a.sign$ \\ -6. $\hat w \leftarrow 0$ \\ -7. for $ix$ from $a.used - 1$ down to $0$ do \\ -\hspace{3mm}7.1 $\hat w \leftarrow \hat w \beta + a_{ix}$ \\ -\hspace{3mm}7.2 If $\hat w \ge b$ then \\ -\hspace{6mm}7.2.1 $t \leftarrow \lfloor \hat w / b \rfloor$ \\ -\hspace{6mm}7.2.2 $\hat w \leftarrow \hat w \mbox{ (mod }b\mbox{)}$ \\ -\hspace{3mm}7.3 else\\ -\hspace{6mm}7.3.1 $t \leftarrow 0$ \\ -\hspace{3mm}7.4 $q_{ix} \leftarrow t$ \\ -8. $d \leftarrow \hat w$ \\ -9. Clamp excess digits of $q$. \\ -10. $c \leftarrow q$ \\ -11. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_div\_d} -\end{figure} -\textbf{Algorithm mp\_div\_d.} -This algorithm divides the mp\_int $a$ by the single mp\_digit $b$ using an optimized approach. Essentially in every iteration of the -algorithm another digit of the dividend is reduced and another digit of quotient produced. Provided $b < \beta$ the value of $\hat w$ -after step 7.1 will be limited such that $0 \le \lfloor \hat w / b \rfloor < \beta$. - -If the divisor $b$ is equal to three a variant of this algorithm is used which is called mp\_div\_3. It replaces the division by three with -a multiplication by $\lfloor \beta / 3 \rfloor$ and the appropriate shift and residual fixup. In essence it is much like the Barrett reduction -from chapter seven. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_div\_d.c -\vspace{-3mm} -\begin{alltt} -016 -017 static int s_is_power_of_two(mp_digit b, int *p) -018 \{ -019 int x; -020 -021 for (x = 1; x < DIGIT_BIT; x++) \{ -022 if (b == (((mp_digit)1)<<x)) \{ -023 *p = x; -024 return 1; -025 \} -026 \} -027 return 0; -028 \} -029 -030 /* single digit division (based on routine from MPI) */ -031 int mp_div_d (mp_int * a, mp_digit b, mp_int * c, mp_digit * d) -032 \{ -033 mp_int q; -034 mp_word w; -035 mp_digit t; -036 int res, ix; -037 -038 /* cannot divide by zero */ -039 if (b == 0) \{ -040 return MP_VAL; -041 \} -042 -043 /* quick outs */ -044 if (b == 1 || mp_iszero(a) == 1) \{ -045 if (d != NULL) \{ -046 *d = 0; -047 \} -048 if (c != NULL) \{ -049 return mp_copy(a, c); -050 \} -051 return MP_OKAY; -052 \} -053 -054 /* power of two ? */ -055 if (s_is_power_of_two(b, &ix) == 1) \{ -056 if (d != NULL) \{ -057 *d = a->dp[0] & ((((mp_digit)1)<<ix) - 1); -058 \} -059 if (c != NULL) \{ -060 return mp_div_2d(a, ix, c, NULL); -061 \} -062 return MP_OKAY; -063 \} -064 -065 #ifdef BN_MP_DIV_3_C -066 /* three? */ -067 if (b == 3) \{ -068 return mp_div_3(a, c, d); -069 \} -070 #endif -071 -072 /* no easy answer [c'est la vie]. Just division */ -073 if ((res = mp_init_size(&q, a->used)) != MP_OKAY) \{ -074 return res; -075 \} -076 -077 q.used = a->used; -078 q.sign = a->sign; -079 w = 0; -080 for (ix = a->used - 1; ix >= 0; ix--) \{ -081 w = (w << ((mp_word)DIGIT_BIT)) | ((mp_word)a->dp[ix]); -082 -083 if (w >= b) \{ -084 t = (mp_digit)(w / b); -085 w -= ((mp_word)t) * ((mp_word)b); -086 \} else \{ -087 t = 0; -088 \} -089 q.dp[ix] = (mp_digit)t; -090 \} -091 -092 if (d != NULL) \{ -093 *d = (mp_digit)w; -094 \} -095 -096 if (c != NULL) \{ -097 mp_clamp(&q); -098 mp_exch(&q, c); -099 \} -100 mp_clear(&q); -101 -102 return res; -103 \} -104 -105 #endif -\end{alltt} -\end{small} - -Like the implementation of algorithm mp\_div this algorithm allows either of the quotient or remainder to be passed as a \textbf{NULL} pointer to -indicate the respective value is not required. This allows a trivial single digit modular reduction algorithm, mp\_mod\_d to be created. - -The division and remainder on lines 43 and @45,%@ can be replaced often by a single division on most processors. For example, the 32-bit x86 based -processors can divide a 64-bit quantity by a 32-bit quantity and produce the quotient and remainder simultaneously. Unfortunately the GCC -compiler does not recognize that optimization and will actually produce two function calls to find the quotient and remainder respectively. - -\subsection{Single Digit Root Extraction} - -Finding the $n$'th root of an integer is fairly easy as far as numerical analysis is concerned. Algorithms such as the Newton-Raphson approximation -(\ref{eqn:newton}) series will converge very quickly to a root for any continuous function $f(x)$. - -\begin{equation} -x_{i+1} = x_i - {f(x_i) \over f'(x_i)} -\label{eqn:newton} -\end{equation} - -In this case the $n$'th root is desired and $f(x) = x^n - a$ where $a$ is the integer of which the root is desired. The derivative of $f(x)$ is -simply $f'(x) = nx^{n - 1}$. Of particular importance is that this algorithm will be used over the integers not over the a more continuous domain -such as the real numbers. As a result the root found can be abo