4675 lines
253 KiB
Plaintext
4675 lines
253 KiB
Plaintext
\documentclass[b5paper]{book}
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\usepackage{hyperref}
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\usepackage{makeidx}
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\usepackage{amssymb}
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\usepackage{color}
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\usepackage{alltt}
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\usepackage{graphicx}
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\usepackage{layout}
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\def\union{\cup}
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\def\intersect{\cap}
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\def\getsrandom{\stackrel{\rm R}{\gets}}
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\def\cross{\times}
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\def\cat{\hspace{0.5em} \| \hspace{0.5em}}
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\def\catn{$\|$}
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\def\divides{\hspace{0.3em} | \hspace{0.3em}}
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\def\nequiv{\not\equiv}
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\def\approx{\raisebox{0.2ex}{\mbox{\small $\sim$}}}
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\def\lcm{{\rm lcm}}
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\def\gcd{{\rm gcd}}
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\def\log{{\rm log}}
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\def\ord{{\rm ord}}
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\def\abs{{\mathit abs}}
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\def\rep{{\mathit rep}}
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\def\mod{{\mathit\ mod\ }}
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\renewcommand{\pmod}[1]{\ ({\rm mod\ }{#1})}
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\newcommand{\floor}[1]{\left\lfloor{#1}\right\rfloor}
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\newcommand{\ceil}[1]{\left\lceil{#1}\right\rceil}
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\def\Or{{\rm\ or\ }}
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\def\And{{\rm\ and\ }}
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\def\iff{\hspace{1em}\Longleftrightarrow\hspace{1em}}
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\def\implies{\Rightarrow}
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\def\undefined{{\rm ``undefined"}}
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\def\Proof{\vspace{1ex}\noindent {\bf Proof:}\hspace{1em}}
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\let\oldphi\phi
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\def\phi{\varphi}
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\def\Pr{{\rm Pr}}
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\newcommand{\str}[1]{{\mathbf{#1}}}
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\def\F{{\mathbb F}}
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\def\N{{\mathbb N}}
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\def\Z{{\mathbb Z}}
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\def\R{{\mathbb R}}
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\def\C{{\mathbb C}}
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\def\Q{{\mathbb Q}}
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\definecolor{DGray}{gray}{0.5}
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\newcommand{\emailaddr}[1]{\mbox{$<${#1}$>$}}
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\def\twiddle{\raisebox{0.3ex}{\mbox{\tiny $\sim$}}}
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\def\gap{\vspace{0.5ex}}
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\makeindex
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\begin{document}
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\frontmatter
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\pagestyle{empty}
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\title{Multiple-Precision Integer Arithmetic, \\ A Case Study Involving the LibTomMath Project \\ - DRAFT - }
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\author{\mbox{
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%\begin{small}
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\begin{tabular}{c}
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Tom St Denis \\
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Algonquin College \\
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\\
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Mads Rasmussen \\
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Open Communications Security \\
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\\
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Greg Rose \\
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QUALCOMM Australia \\
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\end{tabular}
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%\end{small}
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}
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}
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\maketitle
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This text in its entirety is copyright \copyright{}2003 by Tom St Denis. It may not be redistributed
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electronically or otherwise without the sole permission of the author. The text is freely redistributable as long as
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it is packaged along with the LibTomMath library in a non-commercial project. Contact the
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author for other redistribution rights.
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This text corresponds to the v0.17 release of the LibTomMath project.
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\begin{alltt}
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Tom St Denis
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111 Banning Rd
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Ottawa, Ontario
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K2L 1C3
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Canada
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Phone: 1-613-836-3160
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Email: tomstdenis@iahu.ca
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\end{alltt}
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This text is formatted to the international B5 paper size of 176mm wide by 250mm tall using the \LaTeX{}
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{\em book} macro package and the Perl {\em booker} package.
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\tableofcontents
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\listoffigures
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\chapter*{Preface}
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Blah.
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\mainmatter
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\pagestyle{headings}
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\chapter{Introduction}
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\section{Multiple Precision Arithmetic}
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\subsection{The Need for Multiple Precision Arithmetic}
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The most prevalent use for multiple precision arithmetic (\textit{often referred to as bignum math}) is within public
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key cryptography. Algorithms such as RSA, Diffie-Hellman and Elliptic Curve Cryptography require large integers in order to
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resist known cryptanalytic attacks. Typical modern programming languages such as C and Java only provide small
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single-precision data types which are incapable of precisely representing integers which are often hundreds of bits long.
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For example, consider multiplying $1,234,567$ by $9,876,543$ in C with an ``unsigned long'' data type. With an
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x86 machine the result is $4,136,875,833$ while the true result is $12,193,254,061,881$. The original inputs
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were approximately $21$ and $24$ bits respectively. If the C language cannot multiply two relatively small values
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together precisely how does anyone expect it to multiply two values that are considerably larger?
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Most advancements in fast multiple precision arithmetic stem from the desire for faster cryptographic primitives. However, cryptography
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is not the only field of study that can benefit from fast large integer routines. Another auxiliary use for multiple precision integers is
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high precision floating point data types. The basic IEEE standard floating point type is made up of an integer mantissa $q$ and an exponent $e$.
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Numbers are given in the form $n = q \cdot b^e$ where $b = 2$ is specified. Since IEEE is meant to be implemented in
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hardware the precision of the mantissa is often fairly small (\textit{23, 48 and 64 bits}). Since the mantissa is merely an
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integer a large multiple precision integer could be used. In effect very high precision floating point arithmetic
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could be performed. This would be useful where scientific applications must minimize the total output error over long simulations.
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\subsection{Multiple Precision Arithmetic}
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\index{multiple precision}
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Multiple precision arithmetic attempts to the solve the shortcomings of single precision data types such as those from
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the C and Java programming languages. In essence multiple precision arithmetic is a set of operations that can be
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performed on members of an algebraic group whose precision is not fixed. The algorithms when implemented to be multiple
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precision can allow a developer to work with any practical precision required.
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Typically the arithmetic over the ring of integers denoted by $\Z$ is performed by routines that are collectively and
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casually referred to as ``bignum'' routines. However, it is possible to have rings of polynomials as well typically
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denoted by $\Z/p\Z \left [ X \right ]$ which could have variable precision (\textit{or degree}). This text will
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discuss implementation of the former, however implementing polynomial basis routines should be relatively easy after reading this text.
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\subsection{Benefits of Multiple Precision Arithmetic}
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\index{precision} \index{accuracy}
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Precision of the real value to a given precision is defined loosely as the proximity of the real value to a given representation.
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Accuracy is defined as the reproducibility of the result. For example, the calculation $1/3 = 0.25$ is imprecise but can be accurate provided
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it is reproducible.
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The benefit of multiple precision representations over single precision representations is that
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often no precision is lost while representing the result of an operation which requires excess precision. For example,
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the multiplication of two $n$-bit integers requires at least $2n$ bits to represent the result. A multiple precision
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system would augment the precision of the destination to accomodate the result while a single precision system would
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truncate excess bits to maintain a fixed level of precision.
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Multiple precision representations allow for the precision to be very high (\textit{if not exacting}) but at a cost of
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modest computer resources. The only reasonable case where a multiple precision system will lose precision is when
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emulating a floating point data type. However, with multiple precision integer arithmetic no precision is lost.
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\subsection{Basis of Operations}
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At the heart of all multiple precision integer operations are the ``long-hand'' algorithms we all learned as children
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in grade school. For example, to multiply $1,234$ by $981$ the student is not taught to memorize the times table for
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$1,234$, instead they are taught how to long-multiply. That is to multiply each column using simple single digit
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multiplications, line up the partial results, and add the resulting products by column. The representation that most
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are familiar with is known as decimal or formally as radix-10. A radix-$n$ representation simply means there are
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$n$ possible values per digit. For example, binary would be a radix-2 representation.
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In essence computer based multiple precision arithmetic is very much the same. The most notable difference is the usage
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of a binary friendly radix. That is to use a radix of the form $2^k$ where $k$ is typically the size of a machine
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register. Also occasionally more optimal algorithms are used to perform certain operations such as multiplication and
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squaring instead of traditional long-hand algorithms.
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\section{Purpose of This Text}
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The purpose of this text is to instruct the reader regarding how to implement multiple precision algorithms. That is
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to not only explain the core theoretical algorithms but also the various ``house keeping'' tasks that are neglected by
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authors of other texts on the subject. Texts such as \cite[HAC]{HAC} and \cite{TAOCPV2} give considerably detailed
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explanations of the theoretical aspects of the algorithms and very little regarding the practical aspects.
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How an algorithm is explained and how it is actually implemented are two very different
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realities. For example, algorithm 14.7 on page 594 of HAC lists a relatively simple algorithm for performing multiple
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precision integer addition. However, what the description lacks is any discussion concerning the fact that the two
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integer inputs may be of differing magnitudes. Similarly the division routine (\textit{Algorithm 14.20, pp. 598})
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does not discuss how to handle sign or handle the dividend's decreasing magnitude in the main loop (\textit{Step \#3}).
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As well as the numerous practical oversights both of the texts do not discuss several key optimal algorithms required
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such as ``Comba'' and Karatsuba multipliers and fast modular inversion. These optimal algorithms are vital to achieve
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any form of useful performance in non-trivial applications.
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To solve this problem the focus of this text is on the practical aspects of implementing the algorithms that
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constitute a multiple precision integer package with light discussions on the theoretical aspects. As a case
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study the ``LibTomMath''\footnote{Available freely at http://math.libtomcrypt.org} package is used to demonstrate
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algorithms with implementations that have been field tested and work very well.
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\section{Discussion and Notation}
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\subsection{Notation}
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A multiple precision integer of $n$-digits shall be denoted as $x = (x_n ... x_1 x_0)_{ \beta }$ to be the
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multiple precision notation for the integer $x \equiv \sum_{i=0}^{n} x_i\beta^i$. The elements of the array $x$ are
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said to be the radix $\beta$ digits of the integer. For example, $x = (1,2,3)_{10}$ would represent the
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integer $1\cdot 10^2 + 2\cdot10^1 + 3\cdot10^0 = 123$.
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A ``mp\_int'' shall refer to a composite structure which contains the digits of the integer as well as auxilary data
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required to manipulate the data. These additional members are discussed in ~BASICOP~. For the purposes of this text
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a ``multiple precision integer'' and a ``mp\_int'' are synonymous.
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\index{single-precision} \index{double-precision} \index{mp\_digit} \index{mp\_word}
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For the purposes of this text a single-precision variable must be able to represent integers in the range $0 \le x < 2 \beta$ while
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a double-precision variable must be able to represent integers in the range $0 \le x < 2 \beta^2$. Within the source code that will be
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presented the data type \textbf{mp\_digit} will represent a single-precision type while \textbf{mp\_word} will represent a
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double-precision type. In several algorithms (\textit{notably the Comba routines}) temporary results
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will be stored in a double-precision arrays. For the purposes of this text $x_j$ will refer to the
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$j$'th digit of a single-precision array and $\hat x_j$ will refer to the $j$'th digit of a double-precision
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array.
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The $\lfloor \mbox{ } \rfloor$ brackets represent a value truncated and rounded down to the nearest integer. The $\lceil \mbox{ } \rceil$ brackets
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represent a value truncated and rounded up to the nearest integer. Typically when the $/$ division symbol is used the intention is to perform an integer
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division. For example, $5/2 = 2$ which will often be written as $\lfloor 5/2 \rfloor = 2$ for clarity. When a value is presented as a fraction
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such as $5 \over 2$ a real value division is implied.
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\subsection{Work Effort}
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\index{big-O}
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To measure the efficiency of various algorithms a modified big-O notation is used. In this system all
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single precision operations are considered to have the same cost\footnote{Except where explicitly noted.}.
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That is a single precision addition, multiplication and division are assumed to take the same time to
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complete. While this is generally not true in practice it will simplify the discussions considerably.
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Some algorithms have slight advantages over others which is why some constants will not be removed in
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the notation. For example, a normal multiplication requires $O(n^2)$ work while a squaring requires
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$O({{n^2 + n}\over 2})$ work. In standard big-O notation these would be said to be equivalent. However, in the
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context of the this text the magnitude of the inputs will not approach an infinite size. This means the conventional limit
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notation wisdom does not apply to the cancellation of constants.
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Throughout the discussions various ``work levels'' will be discussed. These levels are the $O(1)$,
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$O(n)$, $O(n^2)$, ..., $O(n^k)$ work efforts. For example, operations at the $O(n^k)$ ``level'' are said to be
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executed more frequently than operations at the $O(n^m)$ ``level'' when $k > m$. Obviously most optimizations will pay
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off the most at the higher levels since they represent the bulk of the effort required.
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\section{Exercises}
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Within the more advanced chapters a section will be set aside to give the reader some challenging exercises. These exercises are not
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designed to be prize winning problems, but to be thought provoking. Wherever possible the problems are forward minded stating
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problems that will be answered in subsequent chapters. The reader is encouraged to finish the exercises as they appear to get a
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better understanding of the subject material.
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Similar to the exercises of \cite{TAOCPV2} as explained on pp.\textit{ix} these exercises are given a scoring system. However, unlike
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\cite{TAOCPV2} the problems do not get nearly as hard as often. The scoring of these exercises ranges from one (\textit{the easiest}) to
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five (\textit{the hardest}). The following table sumarizes the scoring.
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\vspace{5mm}
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\begin{tabular}{cl}
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$\left [ 1 \right ]$ & An easy problem that should only take the reader a manner of \\
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& minutes to solve. Usually does not involve much computer time. \\
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& \\
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$\left [ 2 \right ]$ & An easy problem that involves a marginal amount of computer \\
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& time usage. Usually requires a program to be written to \\
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& solve the problem. \\
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& \\
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$\left [ 3 \right ]$ & A moderately hard problem that requires a non-trivial amount \\
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& of work. Usually involves trivial research and development of \\
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& new theory from the perspective of a student. \\
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& \\
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$\left [ 4 \right ]$ & A moderately hard problem that involves a non-trivial amount \\
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& of work and research. The solution to which will demonstrate \\
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& a higher mastery of the subject matter. \\
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& \\
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$\left [ 5 \right ]$ & A hard problem that involves concepts that are non-trivial. \\
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& Solutions to these problems will demonstrate a complete mastery \\
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& of the given subject. \\
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& \\
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\end{tabular}
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Essentially problems at the first level are meant to be simple questions that the reader can answer quickly without programming a solution or
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devising new theory. These problems are quick tests to see if the material is understood. Problems at the second level are also
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designed to be easy but will require a program or algorithm to be implemented to arrive at the answer.
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Problems at the third level are meant to be a bit more difficult. Often the answer is fairly obvious but arriving at an exacting solution
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requires some thought and skill. These problems will almost always involve devising a new algorithm or implementing a variation of
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another algorithm.
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Problems at the fourth level are meant to be even more difficult as well as involve some research. The reader will most likely not know
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the answer right away nor will this text provide the exact details of the answer (\textit{or at least not until a subsequent chapter}). Problems
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at the fifth level are meant to be the hardest problems relative to all the other problems in the chapter. People who can correctly
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answer fifth level problems have a mastery of the subject matter at hand.
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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
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is encouraged to answer the follow-up problems and try to draw the relevence of problems.
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\chapter{Introduction to LibTomMath}
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\section{What is LibTomMath?}
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LibTomMath is a free and open source multiple precision library written in portable ISO C source code. By portable it is
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meant that the library does not contain any code that is computer platform dependent or otherwise problematic to use on any
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given platform. The library has been successfully tested under numerous operating systems including Solaris, MacOS, Windows,
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Linux, PalmOS and on standalone hardware such as the Gameboy Advance. The library is designed to contain enough
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functionality to be able to develop applications such as public key cryptosystems.
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\section{Goals of LibTomMath}
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Even though the library is written entirely in portable ISO C considerable care has been taken to
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optimize the algorithm implementations within the library. Specifically the code has been written to work well with
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the GNU C Compiler (\textit{GCC}) on both x86 and ARMv4 processors. Wherever possible highly efficient
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algorithms (\textit{such as Karatsuba multiplication, sliding window exponentiation and Montgomery reduction}) have
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been provided to make the library as efficient as possible. Even with the optimal and sometimes specialized
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algorithms that have been included the Application Programing Interface (\textit{API}) has been kept as simple as possible.
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Often generic place holder routines will make use of specialized algorithms automatically without the developer's
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attention. One such example is the generic multiplication algorithm \textbf{mp\_mul()} which will automatically use
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Karatsuba multiplication if the inputs are of a specific size.
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Making LibTomMath as efficient as possible is not the only goal of the LibTomMath project. Ideally the library should
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be source compatible with another popular library which makes it more attractive for developers to use. In this case the
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MPI library was used as a API template for all the basic functions.
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The project is also meant to act as a learning tool for students. The logic being that no easy-to-follow ``bignum''
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library exists which can be used to teach computer science students how to perform fast and reliable multiple precision
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arithmetic. To this end the source code has been given quite a few comments and algorithm discussion points. Often routines have
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more comments than lines of code.
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\section{Choice of LibTomMath}
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LibTomMath was chosen as the case study of this text not only because the author of both projects is one and the same but
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for more worthy reasons. Other libraries such as GMP, MPI, LIP and OpenSSL have multiple precision
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integer arithmetic routines but would not be ideal for this text for reasons as will be explained in the
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following sub-sections.
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\subsection{Code Base}
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The LibTomMath code base is all portable ISO C source code. This means that there are no platform dependent conditional
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segments of code littered throughout the source. This clean and uncluttered approach to the library means that a
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developer can more readily ascertain the true intent of a given section of source code without trying to keep track of
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what conditional code will be used.
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The code base of LibTomMath is also well organized. Each function is in its own separate source code file
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which allows the reader to find a given function very fast. When compiled with GCC for the x86 processor the entire
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library is a mere 87,760 bytes (\textit{$116,182$ bytes for ARMv4 processors}). This includes every single function
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LibTomMath provides from basic arithmetic to various number theoretic functions such as modular exponentiation, various
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reduction algorithms and Jacobi symbol computation.
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By comparison MPI which has fewer functions than LibTomMath compiled with the same conditions is 45,429 bytes
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(\textit{$54,536$ for ARMv4}). GMP which has rather large collection of functions with the default configuration on an
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x86 Athlon is 2,950,688 bytes. Note that while LibTomMath has fewer functions than GMP it has been used as the sole basis
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for several public key cryptosystems without having to seek additional outside functions to supplement the library.
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\subsection{API Simplicity}
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LibTomMath is designed after the MPI library and shares the API design. Quite often programs that use MPI will build
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with LibTomMath without change. The function names are relatively straight forward as to what they perform. Almost all of the
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functions except for a few minor exceptions which as will be discussed are for good reasons share the same parameter passing
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convention. The learning curve is fairly shallow with the API provided which is an extremely valuable benefit for the
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student and developer alike.
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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
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illegible short hand. LibTomMath does not share this fault.
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\subsection{Optimizations}
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While LibTomMath is certainly not the fastest library (\textit{GMP often beats LibTomMath by a factor of two}) it does
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feature a set of optimal algorithms for tasks ranging from modular reduction to squaring. GMP and LIP also feature
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such optimizations while MPI only uses baseline algorithms with no optimizations.
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LibTomMath is almost always an order of magnitude faster than the MPI library at computationally expensive tasks such as modular
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exponentiation. In the grand scheme of ``bignum'' libraries LibTomMath is faster than the average library and usually
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slower than the best libraries such as GMP and OpenSSL by a small factor.
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\subsection{Portability and Stability}
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LibTomMath will build ``out of the box'' on any platform equipped with a modern version of the GNU C Compiler
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(\textit{GCC}). This means that without changes the library will build without configuration or setting up any
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variables. LIP and MPI will build ``out of the box'' as well but have numerous known bugs. Most notably the author of
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MPI is not working on his library anymore.
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GMP requires a configuration script to run and will not build out of the box. GMP and LibTomMath are still in active
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development and are very stable across a variety of platforms.
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\subsection{Choice}
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LibTomMath is a relatively compact, well documented, highly optimized and portable library which seems only natural for
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the case study of this text. Various source files from the LibTomMath project will be included within the text. However, the
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reader is encouraged to download their own copy of the library to actually be able to work with the library.
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\chapter{Getting Started}
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MARK,BASICOP
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\section{Library Basics}
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To begin the design of a multiple precision integer library a primitive data type and a series of primitive algorithms must be established. A data
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type that will hold the information required to maintain a multiple precision integer must be designed. With this basic data type of a series
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of low level algorithms for initializing, clearing, growing and optimizing multiple precision integers can be developed to form the basis of
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the entire library of algorithms.
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\section{What is a Multiple Precision Integer?}
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Recall that most programming languages (\textit{in particular C}) only have fixed precision data types that on their own cannot be used
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to represent values larger than their precision alone will allow. The purpose of multiple precision algorithms is to use these fixed precision
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data types to create multiple precision integers which may represent values that are much larger.
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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
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the largest value is only $9$ since the digits may only have values from $0$ to $9$. However, by concatenating digits together larger numbers
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may be represented. Computer based multiple precision arithmetic is essentially the same concept except with a different radix.
|
|
|
|
What most people probably do not think about explicitly are the various other attributes that describe a multiple precision integer. For example,
|
|
the integer $154_{10}$ has two immediately obvious properties. First, the integer is positive, that is the sign of this particular integer
|
|
is positive as oppose to negative. Second, the integer has three digits in its representation. There is an additional property that the integer
|
|
posesses that does not concern pencil-and-paper arithmetic. The third property is how many digits are allowed for the integer.
|
|
|
|
The human analogy of this third property is ensuring there is enough space on the paper to right the integer. Computers must maintain a
|
|
strict control on memory usage with respect to the digits of a multiple precision integer. These three properties make up what is known
|
|
as a multiple precision integer or mp\_int for short.
|
|
|
|
\subsection{The mp\_int structure}
|
|
The mp\_int structure is the ISO C based manifestation of what represents a multiple precision integer. The ISO C standard does not provide for
|
|
any such data type but it does provide for making composite data types known as structures. The following is the structure definition
|
|
used within LibTomMath.
|
|
|
|
\index{mp\_int}
|
|
\begin{verbatim}
|
|
typedef struct {
|
|
int used, alloc, sign;
|
|
mp_digit *dp;
|
|
} mp_int;
|
|
\end{verbatim}
|
|
|
|
The mp\_int structure can be broken down as follows.
|
|
|
|
\begin{enumerate}
|
|
\item The \textbf{used} parameter denotes how many digits of the array \textbf{dp} contain the digits used to represent
|
|
a given integer. The \textbf{used} count must not exceed the \textbf{alloc} count.
|
|
|
|
\item The array \textbf{dp} holds the digits that represent the given integer. It is padded with $\textbf{alloc} - \textbf{used}$ zero
|
|
digits.
|
|
|
|
\item The \textbf{alloc} parameter denotes how
|
|
many digits are available in the array to use by functions before it has to increase in size. When the \textbf{used} count
|
|
of a result would exceed the \textbf{alloc} count all of the algorithms will automatically increase the size of the
|
|
array to accommodate the precision of the result.
|
|
|
|
\item The \textbf{sign} parameter denotes the sign as either zero/positive (\textbf{MP\_ZPOS}) or negative (\textbf{MP\_NEG}).
|
|
\end{enumerate}
|
|
|
|
\section{Argument Passing}
|
|
A convention of argument passing must be adopted early on in the development of any library. Making the function prototypes
|
|
consistent will help eliminate many headaches in the future as the library grows to significant complexity. In LibTomMath the multiple precision
|
|
integer functions accept parameters from left to right as pointers to mp\_int structures. That means that the source operands are
|
|
placed on the left and the destination on the right. Consider the following examples.
|
|
|
|
\begin{verbatim}
|
|
mp_mul(&a, &b, &c); /* c = a * b */
|
|
mp_add(&a, &b, &a); /* a = a + b */
|
|
mp_sqr(&a, &b); /* b = a * a */
|
|
\end{verbatim}
|
|
|
|
The left to right order is a fairly natural way to implement the functions since it lets the developer read aloud the
|
|
functions and make sense of them. For example, the first function would read ``multiply a and b and store in c''.
|
|
|
|
Certain libraries (\textit{LIP by Lenstra for instance}) accept parameters the other way around. That is the destination
|
|
on the left and arguments on the right. In truth it is entirely a matter of preference. In the case of LibTomMath the
|
|
convention from the MPI library has been adopted.
|
|
|
|
Another very useful design consideration is whether to allow argument sources to also be a destination. For example, the
|
|
second example (\textit{mp\_add}) adds $a$ to $b$ and stores in $a$. This is an important feature to implement since it
|
|
allows the higher up functions to cut down on the number of variables. However, to implement this feature specific
|
|
care has to be given to ensure the destination is not modified before the source is fully read.
|
|
|
|
\section{Return Values}
|
|
A well implemented library, no matter what its purpose, should trap as many runtime errors as possible and return them to the
|
|
caller. By catching runtime errors a library can be guaranteed to prevent undefined behaviour. In a multiple precision
|
|
library the only errors that can occur occur are related to inappropriate inputs (\textit{division by zero for instance}) or
|
|
memory allocation errors.
|
|
|
|
In LibTomMath any function that can cause a runtime error will return an error as an \textbf{int} data type with one of the
|
|
following values.
|
|
|
|
\index{MP\_OKAY} \index{MP\_VAL} \index{MP\_MEM}
|
|
\begin{center}
|
|
\begin{tabular}{|l|l|}
|
|
\hline \textbf{Value} & \textbf{Meaning} \\
|
|
\hline \textbf{MP\_OKAY} & The function was successful \\
|
|
\hline \textbf{MP\_VAL} & One of the input value(s) was invalid \\
|
|
\hline \textbf{MP\_MEM} & The function ran out of heap memory \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
|
|
When an error is detected within a function it should free any memory it allocated and return as soon as possible. The goal
|
|
is to leave the system in the same state the system was when the function was called. Error checking with this style of API is fairly simple.
|
|
|
|
\begin{verbatim}
|
|
int err;
|
|
if ((err = mp_add(&a, &b, &c)) != MP_OKAY) {
|
|
printf("Error: %d\n", err);
|
|
exit(EXIT_FAILURE);
|
|
}
|
|
\end{verbatim}
|
|
|
|
The GMP library uses C style \textit{signals} to flag errors which is of questionable use. Not all errors are fatal
|
|
and it was not deemed ideal by the author of LibTomMath to force developers to have signal handlers for such cases.
|
|
|
|
\section{Initialization and Clearing}
|
|
The logical starting point when actually writing multiple precision integer functions is the initialization and
|
|
clearing of the integers. These two functions will be used by far the most throughout the algorithms whenever
|
|
temporary integers are required.
|
|
|
|
Given the basic mp\_int structure an initialization routine must first allocate memory to hold the digits of
|
|
the integer. Often it is optimal to allocate a sufficiently large pre-set number of digits even considering
|
|
the initial integer will represent zero. If only a single digit were allocated quite a few re-allocations
|
|
would occur for the majority of inputs. There is a tradeoff between how many default digits to allocate
|
|
and how many re-allocations are tolerable.
|
|
|
|
If the memory for the digits has been successfully allocated then the rest of the members of the structure must
|
|
be initialized. Since the initial state is to represent a zero integer the digits allocated must all be zeroed. The
|
|
\textbf{used} count set to zero and \textbf{sign} set to \textbf{MP\_ZPOS}.
|
|
|
|
\subsection{Initializing an mp\_int}
|
|
To initialize an mp\_int the mp\_init algorithm shall be used. The purpose of this algorithm is to allocate
|
|
the memory required and initialize the integer to a default representation of zero.
|
|
|
|
\begin{figure}[here]
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_init}. \\
|
|
\textbf{Input}. An mp\_int $a$ \\
|
|
\textbf{Output}. Allocate memory for the digits and set to a zero state. \\
|
|
\hline \\
|
|
1. Allocate memory for \textbf{MP\_PREC} digits. \\
|
|
2. If the allocation failed then return(\textit{MP\_MEM}) \\
|
|
3. for $n$ from $0$ to $MP\_PREC - 1$ do \\
|
|
\hspace{3mm}3.1 $a_n \leftarrow 0$\\
|
|
4. $a.sign \leftarrow MP\_ZPOS$\\
|
|
5. $a.used \leftarrow 0$\\
|
|
6. $a.alloc \leftarrow MP\_PREC$\\
|
|
7. Return(\textit{MP\_OKAY})\\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\caption{Algorithm mp\_init}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_init.}
|
|
The \textbf{MP\_PREC} variable is a simple constant used to dictate minimal precision of allocated integers. It is ideally at least equal to $32$ but
|
|
can be any reasonable power of two. Steps one and two allocate the memory and account for it. If the allocation fails the algorithm returns
|
|
immediately to signal the failure. Step three will ensure that all the digits are in the default state of zero. Finally steps
|
|
four through six set the default settings of the \textbf{sign}, \textbf{used} and \textbf{alloc} members of the mp\_int structure.
|
|
|
|
EXAM,bn_mp_init.c
|
|
|
|
The \textbf{OPT\_CAST} type cast on line @22,OPT_CAST@ is designed to allow C++ compilers to build the code out of
|
|
the box. Microsoft C V5.00 is known to cause problems without the cast. Also note that if the memory
|
|
allocation fails the other members of the mp\_int will be in an undefined state. The code from
|
|
line @29,a->used@ to line @31,a->sign@ sets the default state for a mp\_int which is zero, positive and no used digits.
|
|
|
|
\subsection{Clearing an mp\_int}
|
|
When an mp\_int is no longer required the memory allocated for it can be cleared from the heap with
|
|
the mp\_clear algorithm.
|
|
|
|
\begin{figure}[here]
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_clear}. \\
|
|
\textbf{Input}. An mp\_int $a$ \\
|
|
\textbf{Output}. The memory for $a$ is cleared. \\
|
|
\hline \\
|
|
1. If $a$ has been previously freed then return(\textit{MP\_OKAY}). \\
|
|
2. Free the digits of $a$ and mark $a$ as freed. \\
|
|
3. $a.used \leftarrow 0$ \\
|
|
4. $a.alloc \leftarrow 0$ \\
|
|
5. Return(\textit{MP\_OKAY}). \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\caption{Algorithm mp\_clear}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_clear.}
|
|
In steps one and two the memory for the digits are only free'd if they had not been previously released before.
|
|
This is more of concern for the implementation since it is used to prevent ``double-free'' errors. It also helps catch
|
|
code errors where mp\_ints are used after being cleared. Similarly steps three and four set the
|
|
\textbf{used} and \textbf{alloc} to known values which would be easy to spot during debugging. For example, if an mp\_int is expected
|
|
to be non-zero and its \textbf{used} member is observed to be zero (\textit{due to being cleared}) then an obvious bug in the code has been
|
|
spotted.
|
|
|
|
EXAM,bn_mp_clear.c
|
|
|
|
The \textbf{if} statement on line @21,a->dp != NULL@ prevents the heap from being corrupted if a user double-frees an
|
|
mp\_int. For example, a trivial case of this bug would be as follows.
|
|
|
|
\begin{verbatim}
|
|
mp_int a;
|
|
mp_init(&a);
|
|
mp_clear(&a);
|
|
mp_clear(&a);
|
|
\end{verbatim}
|
|
|
|
Without that check the code would try to free the memory allocated for the digits twice which will cause most standard C
|
|
libraries to cause a fault. Also by setting the pointer to \textbf{NULL} it helps debug code that may inadvertently
|
|
free the mp\_int before it is truly not needed. The allocated digits are set to zero before being freed on line @24,memset@.
|
|
This is ideal for cryptographic situations where the mp\_int is a secret parameter.
|
|
|
|
The following snippet is an example of using both the init and clear functions.
|
|
|
|
\begin{small}
|
|
\begin{verbatim}
|
|
#include <tommath.h>
|
|
#include <stdio.h>
|
|
#include <stdlib.h>
|
|
int main(void)
|
|
{
|
|
mp_int num;
|
|
int err;
|
|
|
|
/* init the bignum */
|
|
if ((err = mp_init(&num)) != MP_OKAY) {
|
|
printf("Error: %d\n", err);
|
|
return EXIT_FAILURE;
|
|
}
|
|
|
|
/* do work with it ... */
|
|
|
|
/* clear up */
|
|
mp_clear(&num);
|
|
|
|
return EXIT_SUCCESS;
|
|
}
|
|
\end{verbatim}
|
|
\end{small}
|
|
|
|
\section{Other Initialization Routines}
|
|
|
|
It is often helpful to have specialized initialization algorithms to simplify the design of other algorithms. For example, an
|
|
initialization followed by a copy is a common operation when temporary copies of integers are required. It is quite
|
|
beneficial to have a series of simple helper functions available.
|
|
|
|
\subsection{Initializing Variable Sized mp\_int Structures}
|
|
Occasionally the number of digits required will be known in advance of an initialization. In these
|
|
cases the mp\_init\_size algorithm can be of use. The purpose of this algorithm is similar to mp\_init except that
|
|
it will allocate \textit{at least} a specified number of digits. This is ideal to prevent re-allocations when the
|
|
input size is known.
|
|
|
|
\newpage\begin{figure}[here]
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_init\_size}. \\
|
|
\textbf{Input}. An mp\_int $a$ and the requested number of digits $b$\\
|
|
\textbf{Output}. $a$ is initialized to hold at least $b$ digits. \\
|
|
\hline \\
|
|
1. $u \leftarrow b\mbox{ (mod }MP\_PREC\mbox{)}$ \\
|
|
2. $v \leftarrow b + 2 \cdot MP\_PREC - u$ \\
|
|
3. Allocate $v$ digits. \\
|
|
4. If the allocation failed then return(\textit{MP\_MEM}). \\
|
|
5. for $n$ from $0$ to $v - 1$ do \\
|
|
\hspace{3mm}5.1 $a_n \leftarrow 0$ \\
|
|
6. $a.sign \leftarrow MP\_ZPOS$\\
|
|
7. $a.used \leftarrow 0$\\
|
|
8. $a.alloc \leftarrow v$\\
|
|
9. Return(\textit{MP\_OKAY})\\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\caption{Algorithm mp\_init\_size}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_init\_size.}
|
|
The value of $v$ is calculated to be at least the requested amount of digits $b$ plus additional padding. The padding is calculated
|
|
to be at least \textbf{MP\_PREC} digits plus enough digits to make the digit count a multiple of \textbf{MP\_PREC}. This padding is used to
|
|
prevent trivial allocations from becoming a bottleneck in the rest of the algorithms that depend on this.
|
|
|
|
EXAM,bn_mp_init_size.c
|
|
|
|
Line @23,MP_PREC@ will ensure that the number of digits actually allocated is padded up to the next multiple of
|
|
\textbf{MP\_PREC} plus an additional \textbf{MP\_PREC}. This ensures that the number of allocated digit is
|
|
always greater than the amount requested. As a result it prevents many trivial memory allocations. The value of
|
|
\textbf{MP\_PREC} is defined in ``tommath.h'' and must be a power of two.
|
|
|
|
\subsection{Creating a Clone}
|
|
Another common sequence of operations is to make a local temporary copy of an argument. To initialize then copy a mp\_int will be known as
|
|
creating a clone. This is useful within functions that need to modify an integer argument but do not wish to actually modify the original copy.
|
|
The mp\_init\_copy algorithm will perform this very task.
|
|
|
|
\begin{figure}[here]
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_init\_copy}. \\
|
|
\textbf{Input}. An mp\_int $a$ and $b$\\
|
|
\textbf{Output}. $a$ is initialized to be a copy of $b$. \\
|
|
\hline \\
|
|
1. Init $a$. (\textit{mp\_init}) \\
|
|
2. If the init of $a$ was unsuccessful return(\textit{MP\_MEM}) \\
|
|
3. Copy $b$ to $a$. (\textit{mp\_copy}) \\
|
|
4. Return the status of the copy operation. \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\caption{Algorithm mp\_init\_copy}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_init\_copy.}
|
|
This algorithm will initialize a mp\_int variable and copy another previously initialized mp\_int variable into it. The algorithm will
|
|
detect when the initialization fails and returns the error to the calling algorithm. As such this algorithm will perform two operations
|
|
in one step.
|
|
|
|
EXAM,bn_mp_init_copy.c
|
|
|
|
This will initialize \textbf{a} and make it a verbatim copy of the contents of \textbf{b}. Note that
|
|
\textbf{a} will have its own memory allocated which means that \textbf{b} may be cleared after the call
|
|
and \textbf{a} will be left intact.
|
|
|
|
\subsection{Multiple Integer Initializations And Clearings}
|
|
Occasionally a function will require a series of mp\_int data types to be made available. The mp\_init\_multi algorithm
|
|
is provided to simplify such cases. The purpose of this algorithm is to initialize a variable length array of mp\_int
|
|
structures at once. As a result algorithms that require multiple integers only has to use
|
|
one algorithm to initialize all the mp\_int variables.
|
|
|
|
\begin{figure}[here]
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_init\_multi}. \\
|
|
\textbf{Input}. Variable length array of mp\_int variables of length $k$. \\
|
|
\textbf{Output}. The array is initialized such that each each mp\_int is ready to use. \\
|
|
\hline \\
|
|
1. for $n$ from 0 to $k - 1$ do \\
|
|
\hspace{+3mm}1.1. Initialize the $n$'th mp\_int (\textit{mp\_init}) \\
|
|
\hspace{+3mm}1.2. If initialization failed then do \\
|
|
\hspace{+6mm}1.2.1. for $j$ from $0$ to $n$ do \\
|
|
\hspace{+9mm}1.2.1.1. Free the $j$'th mp\_int (\textit{mp\_clear}) \\
|
|
\hspace{+6mm}1.2.2. Return(\textit{MP\_MEM}) \\
|
|
2. Return(\textit{MP\_OKAY}) \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\caption{Algorithm mp\_init\_multi}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_init\_multi.}
|
|
The algorithm will initialize the array of mp\_int variables one at a time. As soon as an runtime error is detected (\textit{step 1.2}) all of
|
|
the previously initialized variables are cleared. The goal is an ``all or nothing'' initialization which allows for quick recovery from runtime
|
|
errors.
|
|
|
|
Similarly to clear a variable length array of mp\_int structures the mp\_clear\_multi algorithm will be used.
|
|
|
|
Consider the following snippet which demonstrates how to use both routines.
|
|
\begin{small}
|
|
\begin{verbatim}
|
|
#include <tommath.h>
|
|
#include <stdio.h>
|
|
#include <stdlib.h>
|
|
int main(void)
|
|
{
|
|
mp_int num1, num2, num3;
|
|
int err;
|
|
|
|
if ((err = mp_init_multi(&num1, &num2, &num3, NULL)) !- MP_OKAY) {
|
|
printf("Error: %d\n", err);
|
|
return EXIT_FAILURE;
|
|
}
|
|
|
|
/* at this point num1/num2/num3 are ready */
|
|
|
|
/* free them */
|
|
mp_clear_multi(&num1, &num2, &num3, NULL);
|
|
|
|
return EXIT_SUCCESS;
|
|
}
|
|
\end{verbatim}
|
|
\end{small}
|
|
|
|
Note how both lists are terminated with the \textbf{NULL} variable. This indicates to the algorithms to stop fetching parameters off
|
|
of the stack. If it is not present the functions will most likely cause a segmentation fault.
|
|
|
|
EXAM,bn_mp_multi.c
|
|
|
|
Both routines are implemented in the same source file since they are typically used in conjunction with each other.
|
|
|
|
\section{Maintenance}
|
|
A small useful collection of mp\_int maintenance functions will also prove useful.
|
|
|
|
\subsection{Augmenting Integer Precision}
|
|
When storing a value in an mp\_int sufficient digits must be available to accomodate the entire value without
|
|
loss of precision. Quite often the size of the array given by the \textbf{alloc} member is large enough to simply
|
|
increase the \textbf{used} digit count. However, when the size of the array is too small it must be re-sized
|
|
appropriately to accomodate the result. The mp\_grow algorithm will provide this functionality.
|
|
|
|
\begin{figure}[here]
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_grow}. \\
|
|
\textbf{Input}. An mp\_int $a$ and an integer $b$. \\
|
|
\textbf{Output}. $a$ is expanded to accomodate $b$ digits. \\
|
|
\hline \\
|
|
1. if $a.alloc \ge b$ then return(\textit{MP\_OKAY}) \\
|
|
2. $u \leftarrow b\mbox{ (mod }MP\_PREC\mbox{)}$ \\
|
|
3. $v \leftarrow b + 2 \cdot MP\_PREC - u$ \\
|
|
4. Re-Allocate the array of digits $a$ to size $v$ \\
|
|
5. If the allocation failed then return(\textit{MP\_MEM}). \\
|
|
6. for n from a.alloc to $v - 1$ do \\
|
|
\hspace{+3mm}6.1 $a_n \leftarrow 0$ \\
|
|
7. $a.alloc \leftarrow v$ \\
|
|
8. Return(\textit{MP\_OKAY}) \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\caption{Algorithm mp\_grow}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_grow.}
|
|
Step one will prevent a re-allocation from being performed if it was not required. This is useful to prevent mp\_ints
|
|
from growing excessively in code that erroneously calls mp\_grow. Similar to mp\_init\_size the requested digit count
|
|
is padded to provide more digits than requested.
|
|
|
|
In step four it is assumed that the reallocation leaves the lower $a.alloc$ digits intact. This is much akin to how the
|
|
\textit{realloc} function from the standard C library works. Since the newly allocated digits are assumed to contain
|
|
undefined values they are also initially zeroed.
|
|
|
|
EXAM,bn_mp_grow.c
|
|
|
|
The first step is to see if we actually need to perform a re-allocation at all. This is tested for on line
|
|
@24,a->alloc < size@. Similar to mp\_init\_size the same code on line @26,MP_PREC - 1@ was used to resize the
|
|
digits requested. A simple for loop from line @34,a->alloc@ to line @38,}@ will zero all digits that were above the
|
|
old \textbf{alloc} limit to make sure the integer is in a known state.
|
|
|
|
\subsection{Clamping Excess Digits}
|
|
When a function anticipates a result will be $n$ digits it is simpler to assume this is true within the body of
|
|
the function. For example, a multiplication of a $i$ digit number by a $j$ digit produces a result of at most
|
|
$i + j$ digits. It is entirely possible that the result is $i + j - 1$ though, with no final carry into the last
|
|
position. However, suppose the destination had to be first expanded (\textit{via mp\_grow}) to accomodate $i + j - 1$
|
|
digits than further expanded to accomodate the final carry. That would be a considerable waste of time since heap
|
|
operations are relatively slow.
|
|
|
|
The ideal solution is to always assume the result is $i + j$ and fix up the \textbf{used} count after the function
|
|
terminates. This way a single heap operation (\textit{at most}) is required. However, if the result was not checked
|
|
there would be an excess high order zero digit.
|
|
|
|
For example, suppose the product of two integers was $x_n = (0x_{n-1}x_{n-2}...x_0)_{\beta}$. The leading zero digit
|
|
will not contribute to the precision of the result. In fact, through subsequent operations more leading zero digits would
|
|
accumulate to the point the size of the integer would be prohibitive. As a result even though the precision is very
|
|
low the representation is excessively large.
|
|
|
|
The mp\_clamp algorithm is designed to solve this very problem. It will trim leading zeros by decrementing the
|
|
\textbf{used} count until a non-zero leading digit is found. Also in this system, zero is considered to be a positive
|
|
number which means that if the \textbf{used} count is decremented to zero the sign must be set to \textbf{MP\_ZPOS}.
|
|
|
|
\begin{figure}[here]
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_clamp}. \\
|
|
\textbf{Input}. An mp\_int $a$ \\
|
|
\textbf{Output}. Any excess leading zero digits of $a$ are removed \\
|
|
\hline \\
|
|
1. while $a.used > 0$ and $a_{a.used - 1} = 0$ do \\
|
|
\hspace{+3mm}1.1 $a.used \leftarrow a.used - 1$ \\
|
|
2. if $a.used = 0$ then do \\
|
|
\hspace{+3mm}2.1 $a.sign \leftarrow MP\_ZPOS$ \\
|
|
\hline \\
|
|
\end{tabular}
|
|
\end{center}
|
|
\caption{Algorithm mp\_clamp}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_clamp.}
|
|
As can be expected this algorithm is very simple. The loop on step one is expected to iterate only once or twice at
|
|
the most. For example, this will happen in cases where there is not a carry to fill the last position. Step two fixes the sign for
|
|
when all of the digits are zero to ensure that the mp\_int is valid at all times.
|
|
|
|
EXAM,bn_mp_clamp.c
|
|
|
|
Note on line @27,while@ how to test for the \textbf{used} count is made on the left of the \&\& operator. In the C programming
|
|
language the terms to \&\& are evaluated left to right with a boolean short-circuit if any condition fails. This is
|
|
important since if the \textbf{used} is zero the test on the right would fetch below the array. That is obviously
|
|
undesirable. The parenthesis on line @28,a->used@ is used to make sure the \textbf{used} count is decremented and not
|
|
the pointer ``a''.
|
|
|
|
\section*{Exercises}
|
|
\begin{tabular}{cl}
|
|
$\left [ 1 \right ]$ & Discuss the relevance of the \textbf{used} member of the mp\_int structure. \\
|
|
& \\
|
|
$\left [ 1 \right ]$ & Discuss the consequences of not using padding when performing allocations. \\
|
|
& \\
|
|
$\left [ 2 \right ]$ & Estimate an ideal value for \textbf{MP\_PREC} when performing 1024-bit RSA \\
|
|
& encryption when $\beta = 2^{28}$. \\
|
|
& \\
|
|
$\left [ 1 \right ]$ & Discuss the relevance of the algorithm mp\_clamp. What does it prevent? \\
|
|
& \\
|
|
$\left [ 1 \right ]$ & Give an example of when the algorithm mp\_init\_copy might be useful. \\
|
|
& \\
|
|
\end{tabular}
|
|
|
|
|
|
\chapter{Basic Operations}
|
|
\section{Copying an Integer}
|
|
After the various house-keeping routines are in place, simple algorithms can be designed to take advantage of them. Being able
|
|
to make a verbatim copy of an integer is a very useful function to have. To copy an integer the mp\_copy algorithm will be used.
|
|
|
|
\newpage\begin{figure}[here]
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_copy}. \\
|
|
\textbf{Input}. An mp\_int $a$ and $b$. \\
|
|
\textbf{Output}. Store a copy of $a$ in $b$. \\
|
|
\hline \\
|
|
1. Check if $a$ and $b$ point to the same location in memory. \\
|
|
2. If true then return(\textit{MP\_OKAY}). \\
|
|
3. If $b.alloc < a.used$ then grow $b$ to $a.used$ digits. (\textit{mp\_grow}) \\
|
|
4. If failed to grow then return(\textit{MP\_MEM}). \\
|
|
5. for $n$ from 0 to $a.used - 1$ do \\
|
|
\hspace{3mm}5.1 $b_{n} \leftarrow a_{n}$ \\
|
|
6. if $a.used < b.used - 1$ then \\
|
|
\hspace{3mm}6.1. for $n$ from $a.used$ to $b.used - 1$ do \\
|
|
\hspace{6mm}6.1.1 $b_{n} \leftarrow 0$ \\
|
|
7. $b.used \leftarrow a.used$ \\
|
|
8. $b.sign \leftarrow a.sign$ \\
|
|
9. return(\textit{MP\_OKAY}) \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\caption{Algorithm mp\_copy}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_copy.}
|
|
Step 1 and 2 make sure that the two mp\_ints are unique. This allows the user to call the copy function with
|
|
potentially the same input and not waste time. Step 3 and 4 ensure that the destination is large enough to
|
|
hold a copy of the input $a$. Note that the \textbf{used} member of $b$ may be smaller than the \textbf{used}
|
|
member of $a$ but a memory re-allocation is only required if the \textbf{alloc} member of $b$ is smaller. This
|
|
prevents trivial memory reallocations.
|
|
|
|
Step 5 copies the digits from $a$ to $b$ while step 6 ensures that if initially $\vert b \vert > \vert a \vert$,
|
|
the more significant digits of $b$ will be zeroed. Finally steps 7 and 8 copies the \textbf{used} and \textbf{sign} members over
|
|
which completes the copy operation.
|
|
|
|
EXAM,bn_mp_copy.c
|
|
|
|
Source lines @23,if dst ==@-@31,}@ do the initial house keeping. That is to see if the input is unique and if so to
|
|
make sure there is enough room. If not enough space is available it returns the error and leaves the destination variable
|
|
intact.
|
|
|
|
The inner loop of the copy operation is contained between lines @34,{@ and @50,}@. Many LibTomMath routines are designed with this source code style
|
|
in mind, making aliases to shorten lengthy pointers (\textit{see line @38,->@ and @39,->@}) for rapid use. Also the
|
|
use of nested braces creates a simple way to denote various portions of code that reside on various work levels. Here, the copy loop is at the
|
|
$O(n)$ level.
|
|
|
|
\section{Zeroing an Integer}
|
|
Reseting an mp\_int to the default state is a common step in many algorithms. The mp\_zero algorithm will be the algorithm used to
|
|
perform this task.
|
|
|
|
\begin{figure}[here]
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_zero}. \\
|
|
\textbf{Input}. An mp\_int $a$ \\
|
|
\textbf{Output}. Zero the contents of $a$ \\
|
|
\hline \\
|
|
1. $a.used \leftarrow 0$ \\
|
|
2. $a.sign \leftarrow$ MP\_ZPOS \\
|
|
3. for $n$ from 0 to $a.alloc - 1$ do \\
|
|
\hspace{3mm}3.1 $a_n \leftarrow 0$ \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\caption{Algorithm mp\_zero}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_zero.}
|
|
This algorithm simply resets a mp\_int to the default state.
|
|
|
|
EXAM,bn_mp_zero.c
|
|
|
|
After the function is completed, all of the digits are zeroed, the \textbf{used} count is zeroed and the
|
|
\textbf{sign} variable is set to \textbf{MP\_ZPOS}.
|
|
|
|
\section{Sign Manipulation}
|
|
\subsection{Absolute Value}
|
|
With the mp\_int representation of an integer, calculating the absolute value is trivial. The mp\_abs algorithm will compute
|
|
the absolute value of an mp\_int.
|
|
|
|
\begin{figure}[here]
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_abs}. \\
|
|
\textbf{Input}. An mp\_int $a$ \\
|
|
\textbf{Output}. Computes $b = \vert a \vert$ \\
|
|
\hline \\
|
|
1. Copy $a$ to $b$. (\textit{mp\_copy}) \\
|
|
2. If the copy failed return(\textit{MP\_MEM}). \\
|
|
3. $b.sign \leftarrow MP\_ZPOS$ \\
|
|
4. Return(\textit{MP\_OKAY}) \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\caption{Algorithm mp\_abs}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_abs.}
|
|
This algorithm computes the absolute of an mp\_int input. As can be expected the algorithm is very trivial.
|
|
|
|
EXAM,bn_mp_abs.c
|
|
|
|
\subsection{Integer Negation}
|
|
With the mp\_int representation of an integer, calculating the negation is also trivial. The mp\_neg algorithm will compute
|
|
the negative of an mp\_int input.
|
|
|
|
\newpage\begin{figure}[here]
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_neg}. \\
|
|
\textbf{Input}. An mp\_int $a$ \\
|
|
\textbf{Output}. Computes $b = -a$ \\
|
|
\hline \\
|
|
1. Copy $a$ to $b$. (\textit{mp\_copy}) \\
|
|
2. If the copy failed return(\textit{MP\_MEM}). \\
|
|
3. If $a.sign = MP\_ZPOS$ then do \\
|
|
\hspace{3mm}3.1 $b.sign = MP\_NEG$. \\
|
|
4. else do \\
|
|
\hspace{3mm}4.1 $b.sign = MP\_ZPOS$. \\
|
|
5. Return(\textit{MP\_OKAY}) \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\caption{Algorithm mp\_neg}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_neg.}
|
|
This algorithm computes the negation of an input.
|
|
|
|
EXAM,bn_mp_neg.c
|
|
|
|
\section{Small Constants}
|
|
\subsection{Setting Small Constants}
|
|
Often a mp\_int must be set to a relatively small value such as $1$ or $2$. For these cases the mp\_set algorithm is useful.
|
|
|
|
\newpage\begin{figure}
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_set}. \\
|
|
\textbf{Input}. An mp\_int $a$ and a digit $b$ \\
|
|
\textbf{Output}. Make $a$ equivalent to $b$ \\
|
|
\hline \\
|
|
1. Zero $a$ (\textit{mp\_zero}). \\
|
|
2. $a_0 \leftarrow b \mbox{ (mod }\beta\mbox{)}$ \\
|
|
3. $a.used \leftarrow \left \lbrace \begin{array}{ll}
|
|
1 & \mbox{if }a_0 > 0 \\
|
|
0 & \mbox{if }a_0 = 0
|
|
\end{array} \right .$ \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\caption{Algorithm mp\_set}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_set.}
|
|
This algorithm sets a mp\_int to a small single digit value. Step number 1 ensures that the integer is reset to the default state. The
|
|
single digit is set (\textit{modulo $\beta$}) and the \textbf{used} count is adjusted accordingly.
|
|
|
|
EXAM,bn_mp_set.c
|
|
|
|
Line @21,mp_zero@ calls mp\_zero() to clear the mp\_int and reset the sign. Line @22,MP_MASK@ copies the digit
|
|
into the least significant location. Note the usage of a new constant \textbf{MP\_MASK}. This constant is used to quickly
|
|
reduce an integer modulo $\beta$. Since $\beta$ is of the form $2^k$ for any suitable $k$ it suffices to perform a binary AND with
|
|
$MP\_MASK = 2^k - 1$ to perform the reduction. Finally line @23,a->used@ will set the \textbf{used} member with respect to the
|
|
digit actually set. This function will always make the integer positive.
|
|
|
|
One important limitation of this function is that it will only set one digit. The size of a digit is not fixed, meaning source that uses
|
|
this function should take that into account. Meaning that only trivially small constants can be set using this function.
|
|
|
|
\subsection{Setting Large Constants}
|
|
To overcome the limitations of the mp\_set algorithm the mp\_set\_int algorithm is provided. It accepts a ``long''
|
|
data type as input and will always treat it as a 32-bit integer.
|
|
|
|
\begin{figure}[here]
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_set\_int}. \\
|
|
\textbf{Input}. An mp\_int $a$ and a ``long'' integer $b$ \\
|
|
\textbf{Output}. Make $a$ equivalent to $b$ \\
|
|
\hline \\
|
|
1. Zero $a$ (\textit{mp\_zero}) \\
|
|
2. for $n$ from 0 to 7 do \\
|
|
\hspace{3mm}2.1 $a \leftarrow a \cdot 16$ (\textit{mp\_mul2d}) \\
|
|
\hspace{3mm}2.2 $u \leftarrow \lfloor b / 2^{4(7 - n)} \rfloor \mbox{ (mod }16\mbox{)}$\\
|
|
\hspace{3mm}2.3 $a_0 \leftarrow a_0 + u$ \\
|
|
\hspace{3mm}2.4 $a.used \leftarrow a.used + 1$ \\
|
|
3. Clamp excess used digits (\textit{mp\_clamp}) \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\caption{Algorithm mp\_set\_int}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_set\_int.}
|
|
The algorithm performs eight iterations of a simple loop where in each iteration four bits from the source are added to the
|
|
mp\_int. Step 2.1 will multiply the current result by sixteen making room for four more bits in the less significant positions. In step 2.2 the
|
|
next four bits from the source are extracted and are added to the mp\_int. The \textbf{used} digit count is
|
|
incremented to reflect the addition. The \textbf{used} digit counter is incremented since if any of the leading digits were zero the mp\_int would have
|
|
zero digits used and the newly added four bits would be ignored.
|
|
|
|
Excess zero digits are trimmed in steps 2.1 and 3 by using higher level algorithms mp\_mul2d and mp\_clamp.
|
|
|
|
EXAM,bn_mp_set_int.c
|
|
|
|
This function sets four bits of the number at a time to handle all practical \textbf{DIGIT\_BIT} sizes. The weird
|
|
addition on line @38,a->used@ ensures that the newly added in bits are added to the number of digits. While it may not
|
|
seem obvious as to why the digit counter does not grow exceedingly large it is because of the shift on line @27,mp_mul_2d@
|
|
as well as the call to mp\_clamp() on line @40,mp_clamp@. Both functions will clamp excess leading digits which keeps
|
|
the number of used digits low.
|
|
|
|
\section{Comparisons}
|
|
\subsection{Unsigned Comparisions}
|
|
Comparing a multiple precision integer is performed with the exact same algorithm used to compare two decimal numbers. For example,
|
|
to compare $1,234$ to $1,264$ the digits are extracted by their positions. That is we compare $1 \cdot 10^3 + 2 \cdot 10^2 + 3 \cdot 10^1 + 4 \cdot 10^0$
|
|
to $1 \cdot 10^3 + 2 \cdot 10^2 + 6 \cdot 10^1 + 4 \cdot 10^0$ by comparing single digits at a time starting with the highest magnitude
|
|
positions. If any leading digit of one integer is greater than a digit in the same position of another integer then obviously it must be greater.
|
|
|
|
The first comparision routine that will be developed is the unsigned magnitude compare which will perform a comparison based on the digits of two
|
|
mp\_int variables alone. It will ignore the sign of the two inputs. Such a function is useful when an absolute comparison is required or if the
|
|
signs are known to agree in advance.
|
|
|
|
To facilitate working with the results of the comparison functions three constants are required.
|
|
|
|
\begin{figure}[here]
|
|
\begin{center}
|
|
\begin{tabular}{|r|l|}
|
|
\hline \textbf{Constant} & \textbf{Meaning} \\
|
|
\hline \textbf{MP\_GT} & Greater Than \\
|
|
\hline \textbf{MP\_EQ} & Equal To \\
|
|
\hline \textbf{MP\_LT} & Less Than \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\caption{Comparison Return Codes}
|
|
\end{figure}
|
|
|
|
\begin{figure}[here]
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_cmp\_mag}. \\
|
|
\textbf{Input}. Two mp\_ints $a$ and $b$. \\
|
|
\textbf{Output}. Unsigned comparison results ($a$ to the left of $b$). \\
|
|
\hline \\
|
|
1. If $a.used > b.used$ then return(\textit{MP\_GT}) \\
|
|
2. If $a.used < b.used$ then return(\textit{MP\_LT}) \\
|
|
3. for n from $a.used - 1$ to 0 do \\
|
|
\hspace{+3mm}3.1 if $a_n > b_n$ then return(\textit{MP\_GT}) \\
|
|
\hspace{+3mm}3.2 if $a_n < b_n$ then return(\textit{MP\_LT}) \\
|
|
4. Return(\textit{MP\_EQ}) \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\caption{Algorithm mp\_cmp\_mag}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_cmp\_mag.}
|
|
By saying ``$a$ to the left of $b$'' it is meant that the comparison is with respect to $a$, that is if $a$ is greater than $b$ it will return
|
|
\textbf{MP\_GT} and similar with respect to when $a = b$ and $a < b$. The first two steps compare the number of digits used in both $a$ and $b$.
|
|
Obviously if the digit counts differ there would be an imaginary zero digit in the smaller number where the leading digit of the larger number is.
|
|
If both have the same number of digits than the actual digits themselves must be compared starting at the leading digit.
|
|
|
|
By step three both inputs must have the same number of digits so its safe to start from either $a.used - 1$ or $b.used - 1$ and count down to
|
|
the zero'th digit. If after all of the digits have been compared, no difference is found, the algorithm returns \textbf{MP\_EQ}.
|
|
|
|
EXAM,bn_mp_cmp_mag.c
|
|
|
|
The two if statements on lines @24,if@ and @28,if@ compare the number of digits in the two inputs. These two are performed before all of the digits
|
|
are compared since it is a very cheap test to perform and can potentially save considerable time. The implementation given is also not valid
|
|
without those two statements. $b.alloc$ may be smaller than $a.used$, meaning that undefined values will be read from $b$ past the end of the
|
|
array of digits.
|
|
|
|
\subsection{Signed Comparisons}
|
|
Comparing with sign considerations is also fairly critical in several routines (\textit{division for example}). Based on an unsigned magnitude
|
|
comparison a trivial signed comparison algorithm can be written.
|
|
|
|
\begin{figure}[here]
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_cmp}. \\
|
|
\textbf{Input}. Two mp\_ints $a$ and $b$ \\
|
|
\textbf{Output}. Signed Comparison Results ($a$ to the left of $b$) \\
|
|
\hline \\
|
|
1. if $a.sign = MP\_NEG$ and $b.sign = MP\_ZPOS$ then return(\textit{MP\_LT}) \\
|
|
2. if $a.sign = MP\_ZPOS$ and $b.sign = MP\_NEG$ then return(\textit{MP\_GT}) \\
|
|
3. if $a.sign = MP\_NEG$ then \\
|
|
\hspace{+3mm}3.1 Return the unsigned comparison of $b$ and $a$ (\textit{mp\_cmp\_mag}) \\
|
|
4 Otherwise \\
|
|
\hspace{+3mm}4.1 Return the unsigned comparison of $a$ and $b$ \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\caption{Algorithm mp\_cmp}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_cmp.}
|
|
The first two steps compare the signs of the two inputs. If the signs do not agree then it can return right away with the appropriate
|
|
comparison code. When the signs are equal the digits of the inputs must be compared to determine the correct result. In step
|
|
three the unsigned comparision flips the order of the arguments since they are both negative. For instance, if $-a > -b$ then
|
|
$\vert a \vert < \vert b \vert$. Step number four will compare the two when they are both positive.
|
|
|
|
EXAM,bn_mp_cmp.c
|
|
|
|
The two if statements on lines @22,if@ and @26,if@ perform the initial sign comparison. If the signs are not the equal then which ever
|
|
has the positive sign is larger. At line @30,if@, the inputs are compared based on magnitudes. If the signs were both negative then
|
|
the unsigned comparison is performed in the opposite direction (\textit{line @31,mp_cmp_mag@}). Otherwise, the signs are assumed to
|
|
be both positive and a forward direction unsigned comparison is performed.
|
|
|
|
\section*{Exercises}
|
|
\begin{tabular}{cl}
|
|
$\left [ 2 \right ]$ & Modify algorithm mp\_set\_int to accept as input a variable length array of bits. \\
|
|
& \\
|
|
$\left [ 3 \right ]$ & Give the probability that algorithm mp\_cmp\_mag will have to compare $k$ digits \\
|
|
& of two random digits (of equal magnitude) before a difference is found. \\
|
|
& \\
|
|
$\left [ 1 \right ]$ & Suggest a simple method to speed up the implementation of mp\_cmp\_mag based \\
|
|
& on the observations made in the previous problem. \\
|
|
&
|
|
\end{tabular}
|
|
|
|
\chapter{Basic Arithmetic}
|
|
\section{Building Blocks}
|
|
At this point algorithms for initialization, clearing, zeroing, copying, comparing and setting small constants have been
|
|
established. The next logical set of algorithms to develop are addition, subtraction and digit shifting algorithms. These
|
|
algorithms make use of the lower level algorithms and are the cruicial building block for the multiplication algorithms. It is very important
|
|
that these algorithms are highly optimized. On their own they are simple $O(n)$ algorithms but they can be called from higher level algorithms
|
|
which easily places them at $O(n^2)$ or even $O(n^3)$ work levels.
|
|
|
|
MARK,SHIFTS
|
|
All nine algorithms within this chapter make use of the logical bit shift operations denoted by $<<$ and $>>$ for left and right
|
|
logical shifts respectively. A logical shift is analogous to sliding the decimal point of radix-10 representations. For example, the real
|
|
number $0.9345$ is equivalent to $93.45\%$ which is found by sliding the the decimal two places to the right (\textit{multiplying by $10^2$}).
|
|
Mathematically a logical shift is equivalent to a division or multiplication by a power of two.
|
|
For example, $a << k = a \cdot 2^k$ while $a >> k = \lfloor a/2^k \rfloor$.
|
|
|
|
One significant difference between a logical shift and the way decimals are shifted is that digits below the zero'th position are removed
|
|
from the number. For example, consider $1101_2 >> 1$ using decimal notation this would produce $110.1_2$. However, with a logical shift the
|
|
result is $110_2$.
|
|
|
|
\section{Addition and Subtraction}
|
|
In normal fixed precision arithmetic negative numbers are easily represented by subtraction from the modulus. For example, with 32-bit integers
|
|
$a - b\mbox{ (mod }2^{32}\mbox{)}$ is the same as $a + (2^{32} - b) \mbox{ (mod }2^{32}\mbox{)}$ since $2^{32} \equiv 0 \mbox{ (mod }2^{32}\mbox{)}$.
|
|
As a result subtraction can be performed with a trivial series of logical operations and an addition.
|
|
|
|
However, in multiple precision arithmetic negative numbers are not represented in the same way. Instead a sign flag is used to keep track of the
|
|
sign of the integer. As a result signed addition and subtraction are actually implemented as conditional usage of lower level addition or
|
|
subtraction algorithms with the sign fixed up appropriately.
|
|
|
|
The lower level algorithms will add or subtract integers without regard to the sign flag. That is they will add or subtract the magnitude of
|
|
the integers respectively.
|
|
|
|
\subsection{Low Level Addition}
|
|
An unsigned addition of multiple precision integers is performed with the same long-hand algorithm used to add decimal numbers. That is to add the
|
|
trailing digits first and propagate the resulting carry upwards. Since this is a lower level algorithm the name will have a ``s\_'' prefix.
|
|
Historically that convention stems from the MPI library where ``s\_'' stood for static functions that were hidden from the developer entirely.
|
|
|
|
\newpage
|
|
\begin{figure}[!here]
|
|
\begin{center}
|
|
\begin{small}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{s\_mp\_add}. \\
|
|
\textbf{Input}. Two mp\_ints $a$ and $b$ \\
|
|
\textbf{Output}. The unsigned addition $c = \vert a \vert + \vert b \vert$. \\
|
|
\hline \\
|
|
1. if $a.used > b.used$ then \\
|
|
\hspace{+3mm}1.1 $min \leftarrow b.used$ \\
|
|
\hspace{+3mm}1.2 $max \leftarrow a.used$ \\
|
|
\hspace{+3mm}1.3 $x \leftarrow a$ \\
|
|
2. else \\
|
|
\hspace{+3mm}2.1 $min \leftarrow a.used$ \\
|
|
\hspace{+3mm}2.2 $max \leftarrow b.used$ \\
|
|
\hspace{+3mm}2.3 $x \leftarrow b$ \\
|
|
3. If $c.alloc < max + 1$ then grow $c$ to hold at least $max + 1$ digits (\textit{mp\_grow}) \\
|
|
4. If failed to grow $c$ return(\textit{MP\_MEM}) \\
|
|
5. $oldused \leftarrow c.used$ \\
|
|
6. $c.used \leftarrow max + 1$ \\
|
|
7. $u \leftarrow 0$ \\
|
|
8. for $n$ from $0$ to $min - 1$ do \\
|
|
\hspace{+3mm}8.1 $c_n \leftarrow a_n + b_n + u$ \\
|
|
\hspace{+3mm}8.2 $u \leftarrow c_n >> lg(\beta)$ \\
|
|
\hspace{+3mm}8.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\
|
|
9. if $min \ne max$ then do \\
|
|
\hspace{+3mm}9.1 for $n$ from $min$ to $max - 1$ do \\
|
|
\hspace{+6mm}9.1.1 $c_n \leftarrow x_n + u$ \\
|
|
\hspace{+6mm}9.1.2 $u \leftarrow c_n >> lg(\beta)$ \\
|
|
\hspace{+6mm}9.1.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\
|
|
10. $c_{max} \leftarrow u$ \\
|
|
11. if $olduse > max$ then \\
|
|
\hspace{+3mm}11.1 for $n$ from $max + 1$ to $olduse - 1$ do \\
|
|
\hspace{+6mm}11.1.1 $c_n \leftarrow 0$ \\
|
|
12. Clamp excess digits in $c$. (\textit{mp\_clamp}) \\
|
|
13. Return(\textit{MP\_OKAY}) \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{small}
|
|
\end{center}
|
|
\caption{Algorithm s\_mp\_add}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm s\_mp\_add.}
|
|
This algorithm is loosely based on algorithm 14.7 of HAC \cite[pp. 594]{HAC} but has been extended to allow the inputs to have different magnitudes.
|
|
Coincidentally the description of algorithm A in Knuth \cite[pp. 266]{TAOCPV2} shares the same deficiency as the algorithm from \cite{HAC}. Even the
|
|
MIX pseudo machine code presented by Knuth \cite[pp. 266-267]{TAOCPV2} is incapable of handling inputs which are of different magnitudes.
|
|
|
|
Steps 1 and 2 will sort the two inputs based on their \textbf{used} digit count. This allows the inputs to have varying magnitudes which not
|
|
only makes it more efficient than the trivial algorithm presented in the references but more flexible. The variable $min$ is given the lowest
|
|
digit count while $max$ is given the highest digit count. If both inputs have the same \textbf{used} digit count both $min$ and $max$ are
|
|
set to the same value. The variable $x$ is an \textit{alias} for the largest input and not meant to be a copy of it. After the inputs are sorted,
|
|
steps 3 and 4 will ensure that the destination $c$ can accommodate the result. The old \textbf{used} count from $c$ is copied to
|
|
$oldused$ so that excess digits can be cleared later, and the new \textbf{used} count is set to $max+1$, so that a carry from the most significant
|
|
word can be handled.
|
|
|
|
At step 7 the carry variable $u$ is set to zero and the first part of the addition loop can begin. The first step of the loop (\textit{8.1}) adds
|
|
digits from the two inputs together along with the carry variable $u$. The following step extracts the carry bit by shifting the result of the
|
|
preceding step right by $lg(\beta)$ positions. The shift to extract the carry is similar to how carry extraction works with decimal addition.
|
|
|
|
Consider adding $77$ to $65$, the first addition of the first column is $7 + 5$ which produces the result $12$. The trailing digit of the result
|
|
is $2 \equiv 12 \mbox{ (mod }10\mbox{)}$ and the carry is found by dividing (\textit{and ignoring the remainder}) $12$ by the radix or in this case $10$. The
|
|
division and multiplication of $10$ is simply a logical right or left shift, respectively, of the digits. In otherwords the carry can be extracted
|
|
by shifting one digit to the right.
|
|
|
|
Note that $lg()$ is simply the base two logarithm such that $lg(2^k) = k$. This implies that $lg(\beta)$ is the number of bits in a radix-$\beta$
|
|
digit. Therefore, a logical shift right of the summand by $lg(\beta)$ will extract the carry. The final step of the loop reduces the digit
|
|
modulo the radix $\beta$ to ensure it is in range.
|
|
|
|
After step 8 the smallest input (\textit{or both if they are the same magnitude}) has been exhausted. Step 9 decides whether
|
|
the inputs were of equal magnitude. If not than another loop similar to that in step 8, must be executed. The loop at step
|
|
number 9.1 differs from the previous loop since it only adds the mp\_int $x$ along with the carry.
|
|
|
|
Step 10 finishes the addition phase by copying the final carry to the highest location in the result $c_{max}$. Step 11 ensures that
|
|
leading digits that were originally present in $c$ are cleared. Finally excess leading digits are clamped and the algorithm returns success.
|
|
|
|
EXAM,bn_s_mp_add.c
|
|
|
|
Lines @27,if@ to @35,}@ perform the initial sorting of the inputs and determine the $min$ and $max$ variables. Note that $x$ is a pointer to a
|
|
mp\_int assigned to the largest input, in effect it is a local alias. Lines @37,init@ to @42,}@ ensure that the destination is grown to
|
|
accomodate the result of the addition.
|
|
|
|
Similar to the implementation of mp\_copy this function uses the braced code and local aliases coding style. The three aliases that are on
|
|
lines @56,tmpa@, @59,tmpb@ and @62,tmpc@ represent the two inputs and destination variables respectively. These aliases are used to ensure the
|
|
compiler does not have to dereference $a$, $b$ or $c$ (respectively) to access the digits of the respective mp\_int.
|
|
|
|
The initial carry $u$ is cleared on line @65,u = 0@, note that $u$ is of type mp\_digit which ensures type compatibility within the
|
|
implementation. The initial addition loop begins on line @66,for@ and ends on line @75,}@. Similarly the conditional addition loop
|
|
begins on line @81,for@ and ends on line @90,}@. The addition is finished with the final carry being stored in $tmpc$ on line @94,tmpc++@.
|
|
Note the ``++'' operator on the same line. After line @94,tmpc++@ $tmpc$ will point to the $c.used$'th digit of the mp\_int $c$. This is useful
|
|
for the next loop on lines @97,for@ to @99,}@ which set any old upper digits to zero.
|
|
|
|
\subsection{Low Level Subtraction}
|
|
The low level unsigned subtraction algorithm is very similar to the low level unsigned addition algorithm. The principle difference is that the
|
|
unsigned subtraction algorithm requires the result to be positive. That is when computing $a - b$ the condition $\vert a \vert \ge \vert b\vert$ must
|
|
be met for this algorithm to function properly. Keep in mind this low level algorithm is not meant to be used in higher level algorithms directly.
|
|
This algorithm as will be shown can be used to create functional signed addition and subtraction algorithms.
|
|
|
|
MARK,GAMMA
|
|
|
|
For this algorithm a new variable is required to make the description simpler. Recall from section 1.3.1 that a mp\_digit must be able to represent
|
|
the range $0 \le x < 2\beta$ for the algorithms to work correctly. However, it is allowable that a mp\_digit represent a larger range of values. For
|
|
this algorithm we will assume that the variable $\gamma$ represents the number of bits available in a
|
|
mp\_digit (\textit{this implies $2^{\gamma} > \beta$}).
|
|
|
|
For example, the default for LibTomMath is to use a ``unsigned long'' for the mp\_digit ``type'' while $\beta = 2^{28}$. In ISO C an ``unsigned long''
|
|
data type must be able to represent $0 \le x < 2^{32}$ meaning that in this case $\gamma = 32$.
|
|
|
|
\newpage\begin{figure}[!here]
|
|
\begin{center}
|
|
\begin{small}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{s\_mp\_sub}. \\
|
|
\textbf{Input}. Two mp\_ints $a$ and $b$ ($\vert a \vert \ge \vert b \vert$) \\
|
|
\textbf{Output}. The unsigned subtraction $c = \vert a \vert - \vert b \vert$. \\
|
|
\hline \\
|
|
1. $min \leftarrow b.used$ \\
|
|
2. $max \leftarrow a.used$ \\
|
|
3. If $c.alloc < max$ then grow $c$ to hold at least $max$ digits. (\textit{mp\_grow}) \\
|
|
4. If the reallocation failed return(\textit{MP\_MEM}). \\
|
|
5. $oldused \leftarrow c.used$ \\
|
|
6. $c.used \leftarrow max$ \\
|
|
7. $u \leftarrow 0$ \\
|
|
8. for $n$ from $0$ to $min - 1$ do \\
|
|
\hspace{3mm}8.1 $c_n \leftarrow a_n - b_n - u$ \\
|
|
\hspace{3mm}8.2 $u \leftarrow c_n >> (\gamma - 1)$ \\
|
|
\hspace{3mm}8.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\
|
|
9. if $min < max$ then do \\
|
|
\hspace{3mm}9.1 for $n$ from $min$ to $max - 1$ do \\
|
|
\hspace{6mm}9.1.1 $c_n \leftarrow a_n - u$ \\
|
|
\hspace{6mm}9.1.2 $u \leftarrow c_n >> (\gamma - 1)$ \\
|
|
\hspace{6mm}9.1.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\
|
|
10. if $oldused > max$ then do \\
|
|
\hspace{3mm}10.1 for $n$ from $max$ to $oldused - 1$ do \\
|
|
\hspace{6mm}10.1.1 $c_n \leftarrow 0$ \\
|
|
11. Clamp excess digits of $c$. (\textit{mp\_clamp}). \\
|
|
12. Return(\textit{MP\_OKAY}). \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{small}
|
|
\end{center}
|
|
\caption{Algorithm s\_mp\_sub}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm s\_mp\_sub.}
|
|
This algorithm performs the unsigned subtraction of two mp\_int variables under the restriction that the result must be positive. That is when
|
|
passing variables $a$ and $b$ the condition that $\vert a \vert \ge \vert b \vert$ must be met for the algorithm to function correctly. This
|
|
algorithm is loosely based on algorithm 14.9 \cite[pp. 595]{HAC} and is similar to algorithm S in \cite[pp. 267]{TAOCPV2} as well. As was the case
|
|
of the algorithm s\_mp\_add both other references lack discussion concerning various practical details such as when the inputs differ in magnitude.
|
|
|
|
The initial sorting of the inputs is trivial in this algorithm since $a$ is guaranteed to have at least the same magnitude of $b$. Steps 1 and 2
|
|
set the $min$ and $max$ variables. Unlike the addition routine there is guaranteed to be no carry which means that the final result can be at
|
|
most $max$ digits in length as opposed to $max + 1$. Similar to the addition algorithm the \textbf{used} count of $c$ is copied locally and
|
|
set to the maximal count for the operation.
|
|
|
|
The subtraction loop that begins on step 8 is essentially the same as the addition loop of algorithm s\_mp\_add except single precision
|
|
subtraction is used instead. Note the use of the $\gamma$ variable to extract the carry (\textit{also known as the borrow}) within the subtraction
|
|
loops. Under the assumption that two's complement single precision arithmetic is used this will successfully extract the desired carry.
|
|
|
|
For example, consider subtracting $0101_2$ from $0100_2$ where $\gamma = 4$ and $\beta = 2$. The least significant bit will force a carry upwards to
|
|
the third bit which will be set to zero after the borrow. After the very first bit has been subtracted $4 - 1 \equiv 0011_2$ will remain, When the
|
|
third bit of $0101_2$ is subtracted from the result it will cause another carry. In this case though the carry will be forced to propagate all the
|
|
way to the most significant bit.
|
|
|
|
Recall that $\beta < 2^{\gamma}$. This means that if a carry does occur just before the $lg(\beta)$'th bit it will propagate all the way to the most
|
|
significant bit. Thus, the high order bits of the mp\_digit that are not part of the actual digit will either be all zero, or all one. All that
|
|
is needed is a single zero or one bit for the carry. Therefore a single logical shift right by $\gamma - 1$ positions is sufficient to extract the
|
|
carry. This method of carry extraction may seem awkward but the reason for it becomes apparent when the implementation is discussed.
|
|
|
|
If $b$ has a smaller magnitude than $a$ then step 9 will force the carry and copy operation to propagate through the larger input $a$ into $c$. Step
|
|
10 will ensure that any leading digits of $c$ above the $max$'th position are zeroed.
|
|
|
|
EXAM,bn_s_mp_sub.c
|
|
|
|
Line @24,min@ and @25,max@ perform the initial hardcoded sorting of the inputs. In reality the $min$ and $max$ variables are only aliases and are only
|
|
used to make the source code easier to read. Again the pointer alias optimization is used within this algorithm. Lines @42,tmpa@, @43,tmpb@ and @44,tmpc@ initialize the aliases for
|
|
$a$, $b$ and $c$ respectively.
|
|
|
|
The first subtraction loop occurs on lines @47,u = 0@ through @61,}@. The theory behind the subtraction loop is exactly the same as that for
|
|
the addition loop. As remarked earlier there is an implementation reason for using the ``awkward'' method of extracting the carry
|
|
(\textit{see line @57, >>@}). The traditional method for extracting the carry would be to shift by $lg(\beta)$ positions and logically AND
|
|
the least significant bit. The AND operation is required because all of the bits above the $\lg(\beta)$'th bit will be set to one after a carry
|
|
occurs from subtraction. This carry extraction requires two relatively cheap operations to extract the carry. The other method is to simply
|
|
shift the most significant bit to the least significant bit thus extracting the carry with a single cheap operation. This optimization only works on
|
|
twos compliment machines which is a safe assumption to make.
|
|
|
|
If $a$ has a larger magnitude than $b$ an additional loop (\textit{see lines @64,for@ through @73,}@}) is required to propagate the carry through
|
|
$a$ and copy the result to $c$.
|
|
|
|
\subsection{High Level Addition}
|
|
Now that both lower level addition and subtraction algorithms have been established an effective high level signed addition algorithm can be
|
|
established. This high level addition algorithm will be what other algorithms and developers will use to perform addition of mp\_int data
|
|
types.
|
|
|
|
Recall from section 5.2 that an mp\_int represents an integer with an unsigned mantissa (\textit{the array of digits}) and a \textbf{sign}
|
|
flag. A high level addition is actually performed as a series of eight separate cases which can be optimized down to three unique cases.
|
|
|
|
\begin{figure}[!here]
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_add}. \\
|
|
\textbf{Input}. Two mp\_ints $a$ and $b$ \\
|
|
\textbf{Output}. The signed addition $c = a + b$. \\
|
|
\hline \\
|
|
1. if $a.sign = b.sign$ then do \\
|
|
\hspace{3mm}1.1 $c.sign \leftarrow a.sign$ \\
|
|
\hspace{3mm}1.2 $c \leftarrow \vert a \vert + \vert b \vert$ (\textit{s\_mp\_add})\\
|
|
2. else do \\
|
|
\hspace{3mm}2.1 if $\vert a \vert < \vert b \vert$ then do (\textit{mp\_cmp\_mag}) \\
|
|
\hspace{6mm}2.1.1 $c.sign \leftarrow b.sign$ \\
|
|
\hspace{6mm}2.1.2 $c \leftarrow \vert b \vert - \vert a \vert$ (\textit{s\_mp\_sub}) \\
|
|
\hspace{3mm}2.2 else do \\
|
|
\hspace{6mm}2.2.1 $c.sign \leftarrow a.sign$ \\
|
|
\hspace{6mm}2.2.2 $c \leftarrow \vert a \vert - \vert b \vert$ \\
|
|
3. If any of the lower level operations failed return(\textit{MP\_MEM}) \\
|
|
4. Return(\textit{MP\_OKAY}). \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\caption{Algorithm mp\_add}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_add.}
|
|
This algorithm performs the signed addition of two mp\_int variables. There is no reference algorithm to draw upon from either \cite{TAOCPV2} or
|
|
\cite{HAC} since they both only provide unsigned operations. The algorithm is fairly straightforward but restricted since subtraction can only
|
|
produce positive results.
|
|
|
|
\begin{figure}[here]
|
|
\begin{small}
|
|
\begin{center}
|
|
\begin{tabular}{|c|c|c|c|c|}
|
|
\hline \textbf{Sign of $a$} & \textbf{Sign of $b$} & \textbf{$\vert a \vert > \vert b \vert $} & \textbf{Unsigned Operation} & \textbf{Result Sign Flag} \\
|
|
\hline $+$ & $+$ & Yes & $c = a + b$ & $a.sign$ \\
|
|
\hline $+$ & $+$ & No & $c = a + b$ & $a.sign$ \\
|
|
\hline $-$ & $-$ & Yes & $c = a + b$ & $a.sign$ \\
|
|
\hline $-$ & $-$ & No & $c = a + b$ & $a.sign$ \\
|
|
\hline &&&&\\
|
|
|
|
\hline $+$ & $-$ & No & $c = b - a$ & $b.sign$ \\
|
|
\hline $-$ & $+$ & No & $c = b - a$ & $b.sign$ \\
|
|
|
|
\hline &&&&\\
|
|
|
|
\hline $+$ & $-$ & Yes & $c = a - b$ & $a.sign$ \\
|
|
\hline $-$ & $+$ & Yes & $c = a - b$ & $a.sign$ \\
|
|
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{small}
|
|
\caption{Addition Guide Chart}
|
|
\label{fig:AddChart}
|
|
\end{figure}
|
|
|
|
Figure~\ref{fig:AddChart} lists all of the eight possible input combinations and is sorted to show that only three specific cases need to be handled. The
|
|
return code of the unsigned operations at step 1.2, 2.1.2 and 2.2.2 are forwarded to step 3 to check for errors. This simplifies the description
|
|
of the algorithm considerably and best follows how the implementation actually was achieved.
|
|
|
|
Also note how the \textbf{sign} is set before the unsigned addition or subtraction is performed. Recall from the descriptions of algorithms
|
|
s\_mp\_add and s\_mp\_sub that the mp\_clamp function is used at the end to trim excess digits. The mp\_clamp algorithm will set the \textbf{sign}
|
|
to \textbf{MP\_ZPOS} when the \textbf{used} digit count reaches zero.
|
|
|
|
For example, consider performing $-a + a$ with algorithm mp\_add. By the description of the algorithm the sign is set to \textbf{MP\_NEG} which would
|
|
produce a result of $-0$. However, since the sign is set first then the unsigned addition is performed the subsequent usage of algorithm mp\_clamp
|
|
within algorithm s\_mp\_add will force $-0$ to become $0$.
|
|
|
|
EXAM,bn_mp_add.c
|
|
|
|
The source code follows the algorithm fairly closely. The most notable new source code addition is the usage of the $res$ integer variable which
|
|
is used to pass result of the unsigned operations forward. Unlike in the algorithm, the variable $res$ is merely returned as is without
|
|
explicitly checking it and returning the constant \textbf{MP\_OKAY}. The observation is this algorithm will succeed or fail only if the lower
|
|
level functions do so. Returning their return code is sufficient.
|
|
|
|
\subsection{High Level Subtraction}
|
|
The high level signed subtraction algorithm is essentially the same as the high level signed addition algorithm.
|
|
|
|
\newpage\begin{figure}[!here]
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_sub}. \\
|
|
\textbf{Input}. Two mp\_ints $a$ and $b$ \\
|
|
\textbf{Output}. The signed subtraction $c = a - b$. \\
|
|
\hline \\
|
|
1. if $a.sign \ne b.sign$ then do \\
|
|
\hspace{3mm}1.1 $c.sign \leftarrow a.sign$ \\
|
|
\hspace{3mm}1.2 $c \leftarrow \vert a \vert + \vert b \vert$ (\textit{s\_mp\_add}) \\
|
|
2. else do \\
|
|
\hspace{3mm}2.1 if $\vert a \vert \ge \vert b \vert$ then do (\textit{mp\_cmp\_mag}) \\
|
|
\hspace{6mm}2.1.1 $c.sign \leftarrow a.sign$ \\
|
|
\hspace{6mm}2.1.2 $c \leftarrow \vert a \vert - \vert b \vert$ (\textit{s\_mp\_sub}) \\
|
|
\hspace{3mm}2.2 else do \\
|
|
\hspace{6mm}2.2.1 $c.sign \leftarrow \left \lbrace \begin{array}{ll}
|
|
MP\_ZPOS & \mbox{if }a.sign = MP\_NEG \\
|
|
MP\_NEG & \mbox{otherwise} \\
|
|
\end{array} \right .$ \\
|
|
\hspace{6mm}2.2.2 $c \leftarrow \vert b \vert - \vert a \vert$ \\
|
|
3. If any of the lower level operations failed return(\textit{MP\_MEM}). \\
|
|
4. Return(\textit{MP\_OKAY}). \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\caption{Algorithm mp\_sub}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_sub.}
|
|
This algorithm performs the signed subtraction of two inputs. Similar to algorithm mp\_add there is no reference in either \cite{TAOCPV2} or
|
|
\cite{HAC}. Also this algorithm is restricted by algorithm s\_mp\_sub. The following chart lists the eight possible inputs and
|
|
the operations required.
|
|
|
|
\begin{figure}[!here]
|
|
\begin{small}
|
|
\begin{center}
|
|
\begin{tabular}{|c|c|c|c|c|}
|
|
\hline \textbf{Sign of $a$} & \textbf{Sign of $b$} & \textbf{$\vert a \vert \ge \vert b \vert $} & \textbf{Unsigned Operation} & \textbf{Result Sign Flag} \\
|
|
\hline $+$ & $-$ & Yes & $c = a + b$ & $a.sign$ \\
|
|
\hline $+$ & $-$ & No & $c = a + b$ & $a.sign$ \\
|
|
\hline $-$ & $+$ & Yes & $c = a + b$ & $a.sign$ \\
|
|
\hline $-$ & $+$ & No & $c = a + b$ & $a.sign$ \\
|
|
\hline &&&& \\
|
|
\hline $+$ & $+$ & Yes & $c = a - b$ & $a.sign$ \\
|
|
\hline $-$ & $-$ & Yes & $c = a - b$ & $a.sign$ \\
|
|
\hline &&&& \\
|
|
\hline $+$ & $+$ & No & $c = b - a$ & $\mbox{opposite of }a.sign$ \\
|
|
\hline $-$ & $-$ & No & $c = b - a$ & $\mbox{opposite of }a.sign$ \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{small}
|
|
\caption{Subtraction Guide Chart}
|
|
\end{figure}
|
|
|
|
Similar to the case of algorithm mp\_add the \textbf{sign} is set first before the unsigned addition or subtraction. That is to prevent the
|
|
algorithm from producing $-a - -a = -0$ as a result.
|
|
|
|
EXAM,bn_mp_sub.c
|
|
|
|
Much like the implementation of algorithm mp\_add the variable $res$ is used to catch the return code of the unsigned addition or subtraction operations
|
|
and forward it to the end of the function. On line @38, != MP_LT@ the ``not equal to'' \textbf{MP\_LT} expression is used to emulate a
|
|
``greater than or equal to'' comparison.
|
|
|
|
\section{Bit and Digit Shifting}
|
|
MARK,POLY
|
|
It is quite common to think of a multiple precision integer as a polynomial in $x$, that is $y = f(\beta)$ where $f(x) = \sum_{i=0}^{n-1} a_i x^i$.
|
|
This notation arises within discussion of Montgomery and Diminished Radix Reduction as well as Karatsuba multiplication and squaring.
|
|
|
|
In order to facilitate operations on polynomials in $x$ as above a series of simple ``digit'' algorithms have to be established. That is to shift
|
|
the digits left or right as well to shift individual bits of the digits left and right. It is important to note that not all ``shift'' operations
|
|
are on radix-$\beta$ digits.
|
|
|
|
\subsection{Multiplication by Two}
|
|
|
|
In a binary system where the radix is a power of two multiplication by two not only arises often in other algorithms it is a fairly efficient
|
|
operation to perform. A single precision logical shift left is sufficient to multiply a single digit by two.
|
|
|
|
\newpage\begin{figure}[!here]
|
|
\begin{small}
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_mul\_2}. \\
|
|
\textbf{Input}. One mp\_int $a$ \\
|
|
\textbf{Output}. $b = 2a$. \\
|
|
\hline \\
|
|
1. If $b.alloc < a.used + 1$ then grow $b$ to hold $a.used + 1$ digits. (\textit{mp\_grow}) \\
|
|
2. If the reallocation failed return(\textit{MP\_MEM}). \\
|
|
3. $oldused \leftarrow b.used$ \\
|
|
4. $b.used \leftarrow a.used$ \\
|
|
5. $r \leftarrow 0$ \\
|
|
6. for $n$ from 0 to $a.used - 1$ do \\
|
|
\hspace{3mm}6.1 $rr \leftarrow a_n >> (lg(\beta) - 1)$ \\
|
|
\hspace{3mm}6.2 $b_n \leftarrow (a_n << 1) + r \mbox{ (mod }\beta\mbox{)}$ \\
|
|
\hspace{3mm}6.3 $r \leftarrow rr$ \\
|
|
7. If $r \ne 0$ then do \\
|
|
\hspace{3mm}7.1 $b_{n + 1} \leftarrow r$ \\
|
|
\hspace{3mm}7.2 $b.used \leftarrow b.used + 1$ \\
|
|
8. If $b.used < oldused - 1$ then do \\
|
|
\hspace{3mm}8.1 for $n$ from $b.used$ to $oldused - 1$ do \\
|
|
\hspace{6mm}8.1.1 $b_n \leftarrow 0$ \\
|
|
9. $b.sign \leftarrow a.sign$ \\
|
|
10. Return(\textit{MP\_OKAY}).\\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{small}
|
|
\caption{Algorithm mp\_mul\_2}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_mul\_2.}
|
|
This algorithm will quickly multiply a mp\_int by two provided $\beta$ is a power of two. Neither \cite{TAOCPV2} nor \cite{HAC} describe such
|
|
an algorithm despite the fact it arises often in other algorithms. The algorithm is setup much like the lower level algorithm s\_mp\_add since
|
|
it is for all intents and purposes equivalent to the operation $b = \vert a \vert + \vert a \vert$.
|
|
|
|
Step 1 and 2 grow the input as required to accomodate the maximum number of \textbf{used} digits in the result. The initial \textbf{used} count
|
|
is set to $a.used$ at step 4. Only if there is a final carry will the \textbf{used} count require adjustment.
|
|
|
|
Step 6 is an optimization implementation of the addition loop for this specific case. That is since the two values being added together
|
|
are the same there is no need to perform two reads from the digits of $a$. Step 6.1 performs a single precision shift on the current digit $a_n$ to
|
|
obtain what will be the carry for the next iteration. Step 6.2 calculates the $n$'th digit of the result as single precision shift of $a_n$ plus
|
|
the previous carry. Recall from ~SHIFTS~ that $a_n << 1$ is equivalent to $a_n \cdot 2$. An iteration of the addition loop is finished with
|
|
forwarding the carry to the next iteration.
|
|
|
|
Step 7 takes care of any final carry by setting the $a.used$'th digit of the result to the carry and augmenting the \textbf{used} count of $b$.
|
|
Step 8 clears any leading digits of $b$ in case it originally had a larger magnitude than $a$.
|
|
|
|
EXAM,bn_mp_mul_2.c
|
|
|
|
This implementation is essentially an optimized implementation of s\_mp\_add for the case of doubling an input. The only noteworthy difference
|
|
is the use of the logical shift operator on line @52,<<@ to perform a single precision doubling.
|
|
|
|
\subsection{Division by Two}
|
|
A division by two can just as easily be accomplished with a logical shift right as multiplication by two can be with a logical shift left.
|
|
|
|
\newpage\begin{figure}[!here]
|
|
\begin{small}
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_div\_2}. \\
|
|
\textbf{Input}. One mp\_int $a$ \\
|
|
\textbf{Output}. $b = a/2$. \\
|
|
\hline \\
|
|
1. If $b.alloc < a.used$ then grow $b$ to hold $a.used$ digits. (\textit{mp\_grow}) \\
|
|
2. If the reallocation failed return(\textit{MP\_MEM}). \\
|
|
3. $oldused \leftarrow b.used$ \\
|
|
4. $b.used \leftarrow a.used$ \\
|
|
5. $r \leftarrow 0$ \\
|
|
6. for $n$ from $b.used - 1$ to $0$ do \\
|
|
\hspace{3mm}6.1 $rr \leftarrow a_n \mbox{ (mod }2\mbox{)}$\\
|
|
\hspace{3mm}6.2 $b_n \leftarrow (a_n >> 1) + (r << (lg(\beta) - 1)) \mbox{ (mod }\beta\mbox{)}$ \\
|
|
\hspace{3mm}6.3 $r \leftarrow rr$ \\
|
|
7. If $b.used < oldused - 1$ then do \\
|
|
\hspace{3mm}7.1 for $n$ from $b.used$ to $oldused - 1$ do \\
|
|
\hspace{6mm}7.1.1 $b_n \leftarrow 0$ \\
|
|
8. $b.sign \leftarrow a.sign$ \\
|
|
9. Clamp excess digits of $b$. (\textit{mp\_clamp}) \\
|
|
10. Return(\textit{MP\_OKAY}).\\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{small}
|
|
\caption{Algorithm mp\_div\_2}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_div\_2.}
|
|
This algorithm will divide an mp\_int by two using logical shifts to the right. Like mp\_mul\_2 it uses a modified low level addition
|
|
core as the basis of the algorithm. Unlike mp\_mul\_2 the shift operations work from the leading digit to the trailing digit. The algorithm
|
|
could be written to work from the trailing digit to the leading digit however, it would have to stop one short of $a.used - 1$ digits to prevent
|
|
reading past the end of the array of digits.
|
|
|
|
Essentially the loop at step 6 is similar to that of mp\_mul\_2 except the logical shifts go in the opposite direction and the carry is at the
|
|
least significant bit not the most significant bit.
|
|
|
|
EXAM,bn_mp_div_2.c
|
|
|
|
\section{Polynomial Basis Operations}
|
|
Recall from ~POLY~ that any integer can be represented as a polynomial in $x$ as $y = f(\beta)$. Such a representation is also known as
|
|
the polynomial basis \cite[pp. 48]{ROSE}. Given such a notation a multiplication or division by $x$ amounts to shifting whole digits a single
|
|
place. The need for such operations arises in several other higher level algorithms such as Barrett and Montgomery reduction, integer
|
|
division and Karatsuba multiplication.
|
|
|
|
Converting from an array of digits to polynomial basis is very simple. Consider the integer $y \equiv (a_2, a_1, a_0)_{\beta}$ and recall that
|
|
$y = \sum_{i=0}^{2} a_i \beta^i$. Simply replace $\beta$ with $x$ and the expression is in polynomial basis. For example, $f(x) = 8x + 9$ is the
|
|
polynomial basis representation for $89$ using radix ten. That is, $f(10) = 8(10) + 9 = 89$.
|
|
|
|
\subsection{Multiplication by $x$}
|
|
|
|
Given a polynomial in $x$ such as $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0$ multiplying by $x$ amounts to shifting the coefficients up one
|
|
degree. In this case $f(x) \cdot x = a_n x^{n+1} + a_{n-1} x^n + ... + a_0 x$. From a scalar basis point of view multiplying by $x$ is equivalent to
|
|
multiplying by the integer $\beta$.
|
|
|
|
\newpage\begin{figure}[!here]
|
|
\begin{small}
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_lshd}. \\
|
|
\textbf{Input}. One mp\_int $a$ and an integer $b$ \\
|
|
\textbf{Output}. $a \leftarrow a \cdot \beta^b$ (equivalent to multiplication by $x^b$). \\
|
|
\hline \\
|
|
1. If $b \le 0$ then return(\textit{MP\_OKAY}). \\
|
|
2. If $a.alloc < a.used + b$ then grow $a$ to at least $a.used + b$ digits. (\textit{mp\_grow}). \\
|
|
3. If the reallocation failed return(\textit{MP\_MEM}). \\
|
|
4. $a.used \leftarrow a.used + b$ \\
|
|
5. $i \leftarrow a.used - 1$ \\
|
|
6. $j \leftarrow a.used - 1 - b$ \\
|
|
7. for $n$ from $a.used - 1$ to $b$ do \\
|
|
\hspace{3mm}7.1 $a_{i} \leftarrow a_{j}$ \\
|
|
\hspace{3mm}7.2 $i \leftarrow i - 1$ \\
|
|
\hspace{3mm}7.3 $j \leftarrow j - 1$ \\
|
|
8. for $n$ from 0 to $b - 1$ do \\
|
|
\hspace{3mm}8.1 $a_n \leftarrow 0$ \\
|
|
9. Return(\textit{MP\_OKAY}). \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{small}
|
|
\caption{Algorithm mp\_lshd}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_lshd.}
|
|
This algorithm multiplies an mp\_int by the $b$'th power of $x$. This is equivalent to multiplying by $\beta^b$. The algorithm differs
|
|
from the other algorithms presented so far as it performs the operation in place instead storing the result in a separate location. The
|
|
motivation behind this change is due to the way this function is typically used. Algorithms such as mp\_add store the result in an optionally
|
|
different third mp\_int because the original inputs are often still required. Algorithm mp\_lshd (\textit{and similarly algorithm mp\_rshd}) is
|
|
typically used on values where the original value is no longer required. The algorithm will return success immediately if
|
|
$b \le 0$ since the rest of algorithm is only valid when $b > 0$.
|
|
|
|
First the destination $a$ is grown as required to accomodate the result. The counters $i$ and $j$ are used to form a \textit{sliding window} over
|
|
the digits of $a$ of length $b$. The head of the sliding window is at $i$ (\textit{the leading digit}) and the tail at $j$ (\textit{the trailing digit}).
|
|
The loop on step 7 copies the digit from the tail to the head. In each iteration the window is moved down one digit. The last loop on
|
|
step 8 sets the lower $b$ digits to zero.
|
|
|
|
\newpage
|
|
FIGU,sliding_window,Sliding Window Movement
|
|
|
|
EXAM,bn_mp_lshd.c
|
|
|
|
The if statement on line @24,if@ ensures that the $b$ variable is greater than zero. The \textbf{used} count is incremented by $b$ before
|
|
the copy loop begins. This elminates the need for an additional variable in the for loop. The variable $top$ on line @42,top@ is an alias
|
|
for the leading digit while $bottom$ on line @45,bottom@ is an alias for the trailing edge. The aliases form a window of exactly $b$ digits
|
|
over the input.
|
|
|
|
\subsection{Division by $x$}
|
|
|
|
Division by powers of $x$ is easily achieved by shifting the digits right and removing any that will end up to the right of the zero'th digit.
|
|
|
|
\newpage\begin{figure}[!here]
|
|
\begin{small}
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_rshd}. \\
|
|
\textbf{Input}. One mp\_int $a$ and an integer $b$ \\
|
|
\textbf{Output}. $a \leftarrow a / \beta^b$ (Divide by $x^b$). \\
|
|
\hline \\
|
|
1. If $b \le 0$ then return. \\
|
|
2. If $a.used \le b$ then do \\
|
|
\hspace{3mm}2.1 Zero $a$. (\textit{mp\_zero}). \\
|
|
\hspace{3mm}2.2 Return. \\
|
|
3. $i \leftarrow 0$ \\
|
|
4. $j \leftarrow b$ \\
|
|
5. for $n$ from 0 to $a.used - b - 1$ do \\
|
|
\hspace{3mm}5.1 $a_i \leftarrow a_j$ \\
|
|
\hspace{3mm}5.2 $i \leftarrow i + 1$ \\
|
|
\hspace{3mm}5.3 $j \leftarrow j + 1$ \\
|
|
6. for $n$ from $a.used - b$ to $a.used - 1$ do \\
|
|
\hspace{3mm}6.1 $a_n \leftarrow 0$ \\
|
|
7. $a.used \leftarrow a.used - b$ \\
|
|
8. Return. \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{small}
|
|
\caption{Algorithm mp\_rshd}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_rshd.}
|
|
This algorithm divides the input in place by the $b$'th power of $x$. It is analogous to dividing by a $\beta^b$ but much quicker since
|
|
it does not require single precision division. This algorithm does not actually return an error code as it cannot fail.
|
|
|
|
If the input $b$ is less than one the algorithm quickly returns without performing any work. If the \textbf{used} count is less than or equal
|
|
to the shift count $b$ then it will simply zero the input and return.
|
|
|
|
After the trivial cases of inputs have been handled the sliding window is setup. Much like the case of algorithm mp\_lshd a sliding window that
|
|
is $b$ digits wide is used to copy the digits. Unlike mp\_lshd the window slides in the opposite direction from the trailing to the leading digit.
|
|
Also the digits are copied from the leading to the trailing edge.
|
|
|
|
Once the window copy is complete the upper digits must be zeroed and the \textbf{used} count decremented.
|
|
|
|
EXAM,bn_mp_rshd.c
|
|
|
|
The only noteworthy element of this routine is the lack of a return type.
|
|
|
|
-- Will update later to give it a return type...Tom
|
|
|
|
\section{Powers of Two}
|
|
|
|
Now that algorithms for moving single bits as well as whole digits exist algorithms for moving the ``in between'' distances are required. For
|
|
example, to quickly multiply by $2^k$ for any $k$ without using a full multiplier algorithm would prove useful. Instead of performing single
|
|
shifts $k$ times to achieve a multiplication by $2^{\pm k}$ a mixture of whole digit shifting and partial digit shifting is employed.
|
|
|
|
\subsection{Multiplication by Power of Two}
|
|
|
|
\newpage\begin{figure}[!here]
|
|
\begin{small}
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_mul\_2d}. \\
|
|
\textbf{Input}. One mp\_int $a$ and an integer $b$ \\
|
|
\textbf{Output}. $c \leftarrow a \cdot 2^b$. \\
|
|
\hline \\
|
|
1. $c \leftarrow a$. (\textit{mp\_copy}) \\
|
|
2. If $c.alloc < c.used + \lfloor b / lg(\beta) \rfloor + 2$ then grow $c$ accordingly. \\
|
|
3. If the reallocation failed return(\textit{MP\_MEM}). \\
|
|
4. If $b \ge lg(\beta)$ then \\
|
|
\hspace{3mm}4.1 $c \leftarrow c \cdot \beta^{\lfloor b / lg(\beta) \rfloor}$ (\textit{mp\_lshd}). \\
|
|
\hspace{3mm}4.2 If step 4.1 failed return(\textit{MP\_MEM}). \\
|
|
5. $d \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\
|
|
6. If $d \ne 0$ then do \\
|
|
\hspace{3mm}6.1 $mask \leftarrow 2^d$ \\
|
|
\hspace{3mm}6.2 $r \leftarrow 0$ \\
|
|
\hspace{3mm}6.3 for $n$ from $0$ to $c.used - 1$ do \\
|
|
\hspace{6mm}6.3.1 $rr \leftarrow c_n >> (lg(\beta) - d) \mbox{ (mod }mask\mbox{)}$ \\
|
|
\hspace{6mm}6.3.2 $c_n \leftarrow (c_n << d) + r \mbox{ (mod }\beta\mbox{)}$ \\
|
|
\hspace{6mm}6.3.3 $r \leftarrow rr$ \\
|
|
\hspace{3mm}6.4 If $r > 0$ then do \\
|
|
\hspace{6mm}6.4.1 $c_{c.used} \leftarrow r$ \\
|
|
\hspace{6mm}6.4.2 $c.used \leftarrow c.used + 1$ \\
|
|
7. Return(\textit{MP\_OKAY}). \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{small}
|
|
\caption{Algorithm mp\_mul\_2d}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_mul\_2d.}
|
|
This algorithm multiplies $a$ by $2^b$ and stores the result in $c$. The algorithm uses algorithm mp\_lshd and a derivative of algorithm mp\_mul\_2 to
|
|
quickly compute the product.
|
|
|
|
First the algorithm will multiply $a$ by $x^{\lfloor b / lg(\beta) \rfloor}$ which will ensure that the remainder multiplicand is less than
|
|
$\beta$. For example, if $b = 37$ and $\beta = 2^{28}$ then this step will multiply by $x$ leaving a multiplication by $2^{37 - 28} = 2^{9}$
|
|
left.
|
|
|
|
After the digits have been shifted appropriately at most $lg(\beta) - 1$ shifts are left to perform. Step 5 calculates the number of remaining shifts
|
|
required. If it is non-zero a modified shift loop is used to calculate the remaining product.
|
|
Essentially the loop is a generic version of algorith mp\_mul2 designed to handle any shift count in the range $1 \le x < lg(\beta)$. The $mask$
|
|
variable is used to extract the upper $d$ bits to form the carry for the next iteration.
|
|
|
|
This algorithm is loosely measured as a $O(2n)$ algorithm which means that if the input is $n$-digits that it takes $2n$ ``time'' to
|
|
complete. It is possible to optimize this algorithm down to a $O(n)$ algorithm at a cost of making the algorithm slightly harder to follow.
|
|
|
|
EXAM,bn_mp_mul_2d.c
|
|
|
|
Notes to be revised when code is updated. -- Tom
|
|
|
|
\subsection{Division by Power of Two}
|
|
|
|
\newpage\begin{figure}[!here]
|
|
\begin{small}
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_div\_2d}. \\
|
|
\textbf{Input}. One mp\_int $a$ and an integer $b$ \\
|
|
\textbf{Output}. $c \leftarrow \lfloor a / 2^b \rfloor, d \leftarrow a \mbox{ (mod }2^b\mbox{)}$. \\
|
|
\hline \\
|
|
1. If $b \le 0$ then do \\
|
|
\hspace{3mm}1.1 $c \leftarrow a$ (\textit{mp\_copy}) \\
|
|
\hspace{3mm}1.2 $d \leftarrow 0$ (\textit{mp\_zero}) \\
|
|
\hspace{3mm}1.3 Return(\textit{MP\_OKAY}). \\
|
|
2. $c \leftarrow a$ \\
|
|
3. $d \leftarrow a \mbox{ (mod }2^b\mbox{)}$ (\textit{mp\_mod\_2d}) \\
|
|
4. If $b \ge lg(\beta)$ then do \\
|
|
\hspace{3mm}4.1 $c \leftarrow \lfloor c/\beta^{\lfloor b/lg(\beta) \rfloor} \rfloor$ (\textit{mp\_rshd}). \\
|
|
5. $k \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\
|
|
6. If $k \ne 0$ then do \\
|
|
\hspace{3mm}6.1 $mask \leftarrow 2^k$ \\
|
|
\hspace{3mm}6.2 $r \leftarrow 0$ \\
|
|
\hspace{3mm}6.3 for $n$ from $c.used - 1$ to $0$ do \\
|
|
\hspace{6mm}6.3.1 $rr \leftarrow c_n \mbox{ (mod }mask\mbox{)}$ \\
|
|
\hspace{6mm}6.3.2 $c_n \leftarrow (c_n >> k) + (r << (lg(\beta) - k))$ \\
|
|
\hspace{6mm}6.3.3 $r \leftarrow rr$ \\
|
|
7. Clamp excess digits of $c$. (\textit{mp\_clamp}) \\
|
|
8. Return(\textit{MP\_OKAY}). \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{small}
|
|
\caption{Algorithm mp\_div\_2d}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_div\_2d.}
|
|
This algorithm will divide an input $a$ by $2^b$ and produce the quotient and remainder. The algorithm is designed much like algorithm
|
|
mp\_mul\_2d by first using whole digit shifts then single precision shifts. This algorithm will also produce the remainder of the division
|
|
by using algorithm mp\_mod\_2d.
|
|
|
|
EXAM,bn_mp_div_2d.c
|
|
|
|
The implementation of algorithm mp\_div\_2d is slightly different than the algorithm specifies. The remainder $d$ may be optionally
|
|
ignored by passing \textbf{NULL} as the pointer to the mp\_int variable. The temporary mp\_int variable $t$ is used to hold the
|
|
result of the remainder operation until the end. This allows $d$ and $a$ to represent the same mp\_int without modifying $a$ before
|
|
the quotient is obtained.
|
|
|
|
The remainder of the source code is essentially the same as the source code for mp\_mul\_2d. (-- Fix this paragraph up later, Tom).
|
|
|
|
\subsection{Remainder of Division by Power of Two}
|
|
|
|
The last algorithm in the series of polynomial basis power of two algorithms is calculating the remainder of division by $2^b$. This
|
|
algorithm benefits from the fact that in twos complement arithmetic $a \mbox{ (mod }2^b\mbox{)}$ is the same as $a$ AND $2^b - 1$.
|
|
|
|
\begin{figure}[!here]
|
|
\begin{small}
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_mod\_2d}. \\
|
|
\textbf{Input}. One mp\_int $a$ and an integer $b$ \\
|
|
\textbf{Output}. $c \leftarrow a \mbox{ (mod }2^b\mbox{)}$. \\
|
|
\hline \\
|
|
1. If $b \le 0$ then do \\
|
|
\hspace{3mm}1.1 $c \leftarrow 0$ (\textit{mp\_zero}) \\
|
|
\hspace{3mm}1.2 Return(\textit{MP\_OKAY}). \\
|
|
2. If $b > a.used \cdot lg(\beta)$ then do \\
|
|
\hspace{3mm}2.1 $c \leftarrow a$ (\textit{mp\_copy}) \\
|
|
\hspace{3mm}2.2 Return the result of step 2.1. \\
|
|
3. $c \leftarrow a$ \\
|
|
4. If step 3 failed return(\textit{MP\_MEM}). \\
|
|
5. for $n$ from $\lceil b / lg(\beta) \rceil$ to $c.used$ do \\
|
|
\hspace{3mm}5.1 $c_n \leftarrow 0$ \\
|
|
6. $k \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\
|
|
7. $c_{\lfloor b / lg(\beta) \rfloor} \leftarrow c_{\lfloor b / lg(\beta) \rfloor} \mbox{ (mod }2^{k}\mbox{)}$. \\
|
|
8. Clamp excess digits of $c$. (\textit{mp\_clamp}) \\
|
|
9. Return(\textit{MP\_OKAY}). \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{small}
|
|
\caption{Algorithm mp\_mod\_2d}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_mod\_2d.}
|
|
This algorithm will quickly calculate the value of $a \mbox{ (mod }2^b\mbox{)}$. First if $b$ is less than or equal to zero the
|
|
result is set to zero. If $b$ is greater than the number of bits in $a$ then it simply copies $a$ to $c$ and returns. Otherwise, $a$
|
|
is copied to $b$, leading digits are removed and the remaining leading digit is trimed to the exact bit count.
|
|
|
|
EXAM,bn_mp_mod_2d.c
|
|
|
|
-- Add comments later, Tom.
|
|
|
|
\section*{Exercises}
|
|
\begin{tabular}{cl}
|
|
$\left [ 3 \right ] $ & Devise an algorithm that performs $a \cdot 2^b$ for generic values of $b$ \\
|
|
& in $O(n)$ time. \\
|
|
&\\
|
|
$\left [ 3 \right ] $ & Devise an efficient algorithm to multiply by small low hamming \\
|
|
& weight values such as $3$, $5$ and $9$. Extend it to handle all values \\
|
|
& upto $64$ with a hamming weight less than three. \\
|
|
&\\
|
|
$\left [ 2 \right ] $ & Modify the preceding algorithm to handle values of the form \\
|
|
& $2^k - 1$ as well. \\
|
|
&\\
|
|
$\left [ 3 \right ] $ & Using only algorithms mp\_mul\_2, mp\_div\_2 and mp\_add create an \\
|
|
& algorithm to multiply two integers in roughly $O(2n^2)$ time for \\
|
|
& any $n$-bit input. Note that the time of addition is ignored in the \\
|
|
& calculation. \\
|
|
& \\
|
|
$\left [ 5 \right ] $ & Improve the previous algorithm to have a working time of at most \\
|
|
& $O \left (2^{(k-1)}n + \left ({2n^2 \over k} \right ) \right )$ for an appropriate choice of $k$. Again ignore \\
|
|
& the cost of addition. \\
|
|
& \\
|
|
$\left [ 2 \right ] $ & Devise a chart to find optimal values of $k$ for the previous problem \\
|
|
& for $n = 64 \ldots 1024$ in steps of $64$. \\
|
|
& \\
|
|
$\left [ 2 \right ] $ & Using only algorithms mp\_abs and mp\_sub devise another method for \\
|
|
& calculating the result of a signed comparison. \\
|
|
&
|
|
\end{tabular}
|
|
|
|
\chapter{Multiplication and Squaring}
|
|
\section{The Multipliers}
|
|
For most number theoretic 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}
|
|
\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 $O(n^2)$ algoritn since for two $n$-digit inputs $n^2$ single precision
|
|
multiplications are required. More specifically for a $m$ and $n$ digit input $m \cdot n$ single precision multiplications are required. To
|
|
simplify most discussions, it will be assumed that the inputs have comparable number of digits.
|
|
|
|
The ``baseline multiplication'' algorithm is designed to act as the ``catch-all'' algorithm, only to be used when the faster algorithms cannot be
|
|
used. This algorithm does not use any particularly interesting optimizations and should ideally be avoided if possible. One important
|
|
facet of this algorithm, is that it has been modified to only produce a certain amount of output digits as resolution. The importance of this
|
|
modification will become evident during the discussion of Barrett modular reduction. Recall that for a $n$ and $m$ digit input the product
|
|
will be at most $n + m$ digits. Therefore, this algorithm can be reduced to a full multiplier by having it produce $n + m$ digits of the product.
|
|
|
|
Recall from ~GAMMA~ the definition of $\gamma$ as the number of bits in the type \textbf{mp\_digit}. We shall now extend the variable set to
|
|
include $\alpha$ which shall represent the number of bits in the type \textbf{mp\_word}. This implies that $2^{\alpha} > 2 \cdot \beta^2$. The
|
|
constant $\delta = 2^{\alpha - 2lg(\beta)}$ will represent the maximal weight of any column in a product (\textit{see ~COMBA~ for more information}).
|
|
|
|
\newpage\begin{figure}[!here]
|
|
\begin{small}
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{s\_mp\_mul\_digs}. \\
|
|
\textbf{Input}. mp\_int $a$, mp\_int $b$ and an integer $digs$ \\
|
|
\textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert \mbox{ (mod }\beta^{digs}\mbox{)}$. \\
|
|
\hline \\
|
|
1. If min$(a.used, b.used) < \delta$ then do \\
|
|
\hspace{3mm}1.1 Calculate $c = \vert a \vert \cdot \vert b \vert$ by the Comba method (\textit{see algorithm~\ref{fig:COMBAMULT}}). \\
|
|
\hspace{3mm}1.2 Return the result of step 1.1 \\
|
|
\\
|
|
Allocate and initialize a temporary mp\_int. \\
|
|
2. Init $t$ to be of size $digs$ \\
|
|
3. If step 2 failed return(\textit{MP\_MEM}). \\
|
|
4. $t.used \leftarrow digs$ \\
|
|
\\
|
|
Compute the product. \\
|
|
5. for $ix$ from $0$ to $a.used - 1$ do \\
|
|
\hspace{3mm}5.1 $u \leftarrow 0$ \\
|
|
\hspace{3mm}5.2 $pb \leftarrow \mbox{min}(b.used, digs - ix)$ \\
|
|
\hspace{3mm}5.3 If $pb < 1$ then goto step 6. \\
|
|
\hspace{3mm}5.4 for $iy$ from $0$ to $pb - 1$ do \\
|
|
\hspace{6mm}5.4.1 $\hat r \leftarrow t_{iy + ix} + a_{ix} \cdot b_{iy} + u$ \\
|
|
\hspace{6mm}5.4.2 $t_{iy + ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\
|
|
\hspace{6mm}5.4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\
|
|
\hspace{3mm}5.5 if $ix + pb < digs$ then do \\
|
|
\hspace{6mm}5.5.1 $t_{ix + pb} \leftarrow u$ \\
|
|
6. Clamp excess digits of $t$. \\
|
|
7. Swap $c$ with $t$ \\
|
|
8. Clear $t$ \\
|
|
9. Return(\textit{MP\_OKAY}). \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{small}
|
|
\caption{Algorithm s\_mp\_mul\_digs}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm s\_mp\_mul\_digs.}
|
|
This algorithm computes the unsigned product of two inputs $a$ and $b$, limited to an output precision of $digs$ digits. While it may seem
|
|
a bit awkward to modify the function from its simple $O(n^2)$ description, the usefulness of partial multipliers will arise in a subsequent
|
|
algorithm. The algorithm is loosely based on algorithm 14.12 from \cite[pp. 595]{HAC} and is similar to Algorithm M of Knuth \cite[pp. 268]{TAOCPV2}.
|
|
Algorithm s\_mp\_mul\_digs differs from these cited references since it can produce a variable output precision regardless of the precision of the
|
|
inputs.
|
|
|
|
The first thing this algorithm checks for is whether a Comba multiplier can be used instead. If the minimum digit count of either
|
|
input is less than $\delta$, then the Comba method may be used instead. After the Comba method is ruled out, the baseline algorithm begins. A
|
|
temporary mp\_int variable $t$ is used to hold the intermediate result of the product. This allows the algorithm to be used to
|
|
compute products when either $a = c$ or $b = c$ without overwriting the inputs.
|
|
|
|
All of step 5 is the infamous $O(n^2)$ multiplication loop slightly modified to only produce upto $digs$ digits of output. The $pb$ variable
|
|
is given the count of digits to read from $b$ inside the nested loop. If $pb \le 1$ then no more output digits can be produced and the algorithm
|
|
will exit the loop. The best way to think of the loops are as a series of $pb \times 1$ multiplications. That is, in each pass of the
|
|
innermost loop $a_{ix}$ is multiplied against $b$ and the result is added (\textit{with an appropriate shift}) to $t$.
|
|
|
|
For example, consider multiplying $576$ by $241$. That is equivalent to computing $10^0(1)(576) + 10^1(4)(576) + 10^2(2)(576)$ which is best
|
|
visualized in the following table.
|
|
|
|
\begin{figure}[here]
|
|
\begin{center}
|
|
\begin{tabular}{|c|c|c|c|c|c|l|}
|
|
\hline && & 5 & 7 & 6 & \\
|
|
\hline $\times$&& & 2 & 4 & 1 & \\
|
|
\hline &&&&&&\\
|
|
&& & 5 & 7 & 6 & $10^0(1)(576)$ \\
|
|
&2 & 3 & 6 & 1 & 6 & $10^1(4)(576) + 10^0(1)(576)$ \\
|
|
1 & 3 & 8 & 8 & 1 & 6 & $10^2(2)(576) + 10^1(4)(576) + 10^0(1)(576)$ \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\caption{Long-Hand Multiplication Diagram}
|
|
\end{figure}
|
|
|
|
Each row of the product is added to the result after being shifted to the left (\textit{multiplied by a power of the radix}) by the appropriate
|
|
count. That is in pass $ix$ of the inner loop the product is added starting at the $ix$'th digit of the reult.
|
|
|
|
Step 5.4.1 introduces the hat symbol (\textit{e.g. $\hat r$}) which represents a double precision variable. The multiplication on that step
|
|
is assumed to be a double wide output single precision multiplication. That is, two single precision variables are multiplied to produce a
|
|
double precision result. The step is somewhat optimized from a long-hand multiplication algorithm because the carry from the addition in step
|
|
5.4.1 is propagated through the nested loop. If the carry was not propagated immediately it would overflow the single precision digit
|
|
$t_{ix+iy}$ and the result would be lost.
|
|
|
|
At step 5.5 the nested loop is finished and any carry that was left over should be forwarded. The carry does not have to be added to the $ix+pb$'th
|
|
digit since that digit is assumed to be zero at this point. However, if $ix + pb \ge digs$ the carry is not set as it would make the result
|
|
exceed the precision requested.
|
|
|
|
EXAM,bn_s_mp_mul_digs.c
|
|
|
|
Lines @31,if@ to @35,}@ determine if the Comba method can be used first. The conditions for using the Comba routine are that min$(a.used, b.used) < \delta$ and
|
|
the number of digits of output is less than \textbf{MP\_WARRAY}. This new constant is used to control
|
|
the stack usage in the Comba routines. By default it is set to $\delta$ but can be reduced when memory is at a premium.
|
|
|
|
Of particular importance is the calculation of the $ix+iy$'th column on lines @64,mp_word@, @65,mp_word@ and @66,mp_word@. Note how all of the
|
|
variables are cast to the type \textbf{mp\_word}, which is also the type of variable $\hat r$. That is to ensure that double precision operations
|
|
are used instead of single precision. The multiplication on line @65,) * (@ makes use of a specific GCC optimizer behaviour. On the outset it looks like
|
|
the compiler will have to use a double precision multiplication to produce the result required. Such an operation would be horribly slow on most
|
|
processors and drag this to a crawl. However, GCC is smart enough to realize that double wide output single precision multipliers can be used. For
|
|
example, the instruction ``MUL'' on the x86 processor can multiply two 32-bit values and produce a 64-bit result.
|
|
|
|
\subsection{Faster Multiplication by the ``Comba'' Method}
|
|
MARK,COMBA
|
|
|
|
One of the huge drawbacks of the ``baseline'' algorithms is that at the $O(n^2)$ level the carry must be computed and propagated upwards. This
|
|
makes the nested loop very sequential and hard to unroll and implement in parallel. The ``Comba'' \cite{COMBA} method is named after little known
|
|
(\textit{in cryptographic venues}) Paul G. Comba who described a method of implementing fast multipliers that do not require nested
|
|
carry fixup operations. As an interesting aside it seems that Paul Barrett describes a similar technique in
|
|
his 1986 paper \cite{BARRETT} written five years before.
|
|
|
|
At the heart of the Comba technique is once again the long-hand algorithm. Except in this case a slight twist is placed on how
|
|
the columns of the result are produced. In the standard long-hand algorithm rows of products are produced then added together to form the
|
|
final result. In the baseline algorithm the columns are added together after each iteration to get the result instantaneously.
|
|
|
|
In the Comba algorithm the columns of the result are produced entirely independently of each other. That is at the $O(n^2)$ level a
|
|
simple multiplication and addition step is performed. The carries of the columns are propagated after the nested loop to reduce the amount
|
|
of work requiored. Succintly the first step of the algorithm is to compute the product vector $\vec x$ as follows.
|
|
|
|
\begin{equation}
|
|
\vec x_n = \sum_{i+j = n} a_ib_j, \forall n \in \lbrace 0, 1, 2, \ldots, i + j \rbrace
|
|
\end{equation}
|
|
|
|
Where $\vec x_n$ is the $n'th$ column of the output vector. Consider the following example which computes the vector $\vec x$ for the multiplication
|
|
of $576$ and $241$.
|
|
|
|
\newpage\begin{figure}[here]
|
|
\begin{small}
|
|
\begin{center}
|
|
\begin{tabular}{|c|c|c|c|c|c|}
|
|
\hline & & 5 & 7 & 6 & First Input\\
|
|
\hline $\times$ & & 2 & 4 & 1 & Second Input\\
|
|
\hline & & $1 \cdot 5 = 5$ & $1 \cdot 7 = 7$ & $1 \cdot 6 = 6$ & First pass \\
|
|
& $4 \cdot 5 = 20$ & $4 \cdot 7+5=33$ & $4 \cdot 6+7=31$ & 6 & Second pass \\
|
|
$2 \cdot 5 = 10$ & $2 \cdot 7 + 20 = 34$ & $2 \cdot 6+33=45$ & 31 & 6 & Third pass \\
|
|
\hline 10 & 34 & 45 & 31 & 6 & Final Result \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{small}
|
|
\caption{Comba Multiplication Diagram}
|
|
\end{figure}
|
|
|
|
At this point the vector $x = \left < 10, 34, 45, 31, 6 \right >$ is the result of the first step of the Comba multipler.
|
|
Now the columns must be fixed by propagating the carry upwards. The resultant vector will have one extra dimension over the input vector which is
|
|
congruent to adding a leading zero digit.
|
|
|
|
\begin{figure}[!here]
|
|
\begin{small}
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{Comba Fixup}. \\
|
|
\textbf{Input}. Vector $\vec x$ of dimension $k$ \\
|
|
\textbf{Output}. Vector $\vec x$ such that the carries have been propagated. \\
|
|
\hline \\
|
|
1. for $n$ from $0$ to $k - 1$ do \\
|
|
\hspace{3mm}1.1 $\vec x_{n+1} \leftarrow \vec x_{n+1} + \lfloor \vec x_{n}/\beta \rfloor$ \\
|
|
\hspace{3mm}1.2 $\vec x_{n} \leftarrow \vec x_{n} \mbox{ (mod }\beta\mbox{)}$ \\
|
|
2. Return($\vec x$). \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{small}
|
|
\caption{Algorithm Comba Fixup}
|
|
\end{figure}
|
|
|
|
With that algorithm and $k = 5$ and $\beta = 10$ the following vector is produced $\vec x= \left < 1, 3, 8, 8, 1, 6 \right >$. In this case
|
|
$241 \cdot 576$ is in fact $138816$ and the procedure succeeded. If the algorithm is correct and as will be demonstrated shortly more
|
|
efficient than the baseline algorithm why not simply always use this algorithm?
|
|
|
|
\subsubsection{Column Weight.}
|
|
At the nested $O(n^2)$ level the Comba method adds the product of two single precision variables to each column of the output
|
|
independently. A serious obstacle is if the carry is lost, due to lack of precision before the algorithm has a chance to fix
|
|
the carries. For example, in the multiplication of two three-digit numbers the third column of output will be the sum of
|
|
three single precision multiplications. If the precision of the accumulator for the output digits is less then $3 \cdot (\beta - 1)^2$ then
|
|
an overflow can occur and the carry information will be lost. For any $m$ and $n$ digit inputs the maximum weight of any column is
|
|
min$(m, n)$ which is fairly obvious.
|
|
|
|
The maximum number of terms in any column of a product is known as the ``column weight'' and strictly governs when the algorithm can be used. Recall
|
|
from earlier that a double precision type has $\alpha$ bits of resolution and a single precision digit has $lg(\beta)$ bits of precision. Given these
|
|
two quantities we must not violate the following
|
|
|
|
\begin{equation}
|
|
k \cdot \left (\beta - 1 \right )^2 < 2^{\alpha}
|
|
\end{equation}
|
|
|
|
Which reduces to
|
|
|
|
\begin{equation}
|
|
k \cdot \left ( \beta^2 - 2\beta + 1 \right ) < 2^{\alpha}
|
|
\end{equation}
|
|
|
|
Let $\rho = lg(\beta)$ represent the number of bits in a single precision digit. By further re-arrangement of the equation the final solution is
|
|
found.
|
|
|
|
\begin{equation}
|
|
k < {{2^{\alpha}} \over {\left (2^{2\rho} - 2^{\rho + 1} + 1 \right )}}
|
|
\end{equation}
|
|
|
|
The defaults for LibTomMath are $\beta = 2^{28}$ and $\alpha = 2^{64}$ which means that $k$ is bounded by $k < 257$. In this configuration
|
|
the smaller input may not have more than $256$ digits if the Comba method is to be used. This is quite satisfactory for most applications since
|
|
$256$ digits would allow for numbers in the range of $0 \le x < 2^{7168}$ which, is much larger than most public key cryptographic algorithms require.
|
|
|
|
\newpage\begin{figure}[!here]
|
|
\begin{small}
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{fast\_s\_mp\_mul\_digs}. \\
|
|
\textbf{Input}. mp\_int $a$, mp\_int $b$ and an integer $digs$ \\
|
|
\textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert \mbox{ (mod }\beta^{digs}\mbox{)}$. \\
|
|
\hline \\
|
|
Place an array of \textbf{MP\_WARRAY} double precision digits named $\hat W$ on the stack. \\
|
|
1. If $c.alloc < digs$ then grow $c$ to $digs$ digits. (\textit{mp\_grow}) \\
|
|
2. If step 1 failed return(\textit{MP\_MEM}).\\
|
|
\\
|
|
Zero the temporary array $\hat W$. \\
|
|
3. for $n$ from $0$ to $digs - 1$ do \\
|
|
\hspace{3mm}3.1 $\hat W_n \leftarrow 0$ \\
|
|
\\
|
|
Compute the columns. \\
|
|
4. for $ix$ from $0$ to $a.used - 1$ do \\
|
|
\hspace{3mm}4.1 $pb \leftarrow \mbox{min}(b.used, digs - ix)$ \\
|
|
\hspace{3mm}4.2 If $pb < 1$ then goto step 5. \\
|
|
\hspace{3mm}4.3 for $iy$ from $0$ to $pb - 1$ do \\
|
|
\hspace{6mm}4.3.1 $\hat W_{ix+iy} \leftarrow \hat W_{ix+iy} + a_{ix}b_{iy}$ \\
|
|
\\
|
|
Propagate the carries upwards. \\
|
|
5. $oldused \leftarrow c.used$ \\
|
|
6. $c.used \leftarrow digs$ \\
|
|
7. If $digs > 1$ then do \\
|
|
\hspace{3mm}7.1. for $ix$ from $1$ to $digs - 1$ do \\
|
|
\hspace{6mm}7.1.1 $\hat W_{ix} \leftarrow \hat W_{ix} + \lfloor \hat W_{ix-1} / \beta \rfloor$ \\
|
|
\hspace{6mm}7.1.2 $c_{ix - 1} \leftarrow \hat W_{ix - 1} \mbox{ (mod }\beta\mbox{)}$ \\
|
|
8. else do \\
|
|
\hspace{3mm}8.1 $ix \leftarrow 0$ \\
|
|
9. $c_{ix} \leftarrow \hat W_{ix} \mbox{ (mod }\beta\mbox{)}$ \\
|
|
\\
|
|
Zero excess digits. \\
|
|
10. If $digs < oldused$ then do \\
|
|
\hspace{3mm}10.1 for $n$ from $digs$ to $oldused - 1$ do \\
|
|
\hspace{6mm}10.1.1 $c_n \leftarrow 0$ \\
|
|
11. Clamp excessive digits of $c$. (\textit{mp\_clamp}) \\
|
|
12. Return(\textit{MP\_OKAY}). \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{small}
|
|
\caption{Algorithm fast\_s\_mp\_mul\_digs}
|
|
\label{fig:COMBAMULT}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm fast\_s\_mp\_mul\_digs.}
|
|
This algorithm performs the unsigned multiplication of $a$ and $b$ using the Comba method limited to $digs$ digits of precision. The algorithm
|
|
essentially peforms the same calculation as algorithm s\_mp\_mul\_digs, just much faster.
|
|
|
|
The array $\hat W$ is meant to be on the stack when the algorithm is used. The size of the array does not change which is ideal. Note also that
|
|
unlike algorithm s\_mp\_mul\_digs no temporary mp\_int is required since the result is calculated directly in $\hat W$.
|
|
|
|
The $O(n^2)$ loop on step four is where the Comba method's advantages begin to show through in comparison to the baseline algorithm. The lack of
|
|
a carry variable or propagation in this loop allows the loop to be performed with only single precision multiplication and additions. Now that each
|
|
iteration of the inner loop can be performed independent of the others the inner loop can be performed with a high level of parallelism.
|
|
|
|
To measure the benefits of the Comba method over the baseline method consider the number of operations that are required. If the
|
|
cost in terms of time of a multiply and addition is $p$ and the cost of a carry propagation is $q$ then a baseline multiplication would require
|
|
$O \left ((p + q)n^2 \right )$ time to multiply two $n$-digit numbers. The Comba method requires only $O(pn^2 + qn)$ time, however in practice,
|
|
the speed increase is actually much more. With $O(n)$ space the algorithm can be reduced to $O(pn + qn)$ time by implementing the $n$ multiply
|
|
and addition operations in the nested loop in parallel.
|
|
|
|
EXAM,bn_fast_s_mp_mul_digs.c
|
|
|
|
The memset on line @47,memset@ clears the initial $\hat W$ array to zero in a single step. Like the slower baseline multiplication
|
|
implementation a series of aliases (\textit{lines @67, tmpx@, @70, tmpy@ and @75,_W@}) are used to simplify the inner $O(n^2)$ loop.
|
|
In this case a new alias $\_\hat W$ has been added which refers to the double precision columns offset by $ix$ in each pass.
|
|
|
|
The inner loop on lines @83,for@, @84,mp_word@ and @85,}@ is where the algorithm will spend the majority of the time, which is why it has been
|
|
stripped to the bones of any extra baggage\footnote{Hence the pointer aliases.}. On x86 processors the multiplication and additions amount to at the
|
|
very least five instructions (\textit{two loads, two additions, one multiply}) while on the ARMv4 processors they amount to only three
|
|
(\textit{one load, one store, one multiply-add}). For both of the x86 and ARMv4 processors the GCC compiler performs a good job at unrolling the loop
|
|
and scheduling the instructions so there are very few dependency stalls.
|
|
|
|
In theory the difference between the baseline and comba algorithms is a mere $O(qn)$ time difference. However, in the $O(n^2)$ nested loop of the
|
|
baseline method there are dependency stalls as the algorithm must wait for the multiplier to finish before propagating the carry to the next
|
|
digit. As a result fewer of the often multiple execution units\footnote{The AMD Athlon has three execution units and the Intel P4 has four.} can
|
|
be simultaneously used.
|
|
|
|
\subsection{Polynomial Basis Multiplication}
|
|
To break the $O(n^2)$ barrier in multiplication requires a completely different look at integer multiplication. In the following algorithms
|
|
the use of polynomial basis representation for two integers $a$ and $b$ as $f(x) = \sum_{i=0}^{n} a_i x^i$ and
|
|
$g(x) = \sum_{i=0}^{n} b_i x^i$ respectively, is required. In this system both $f(x)$ and $g(x)$ have $n + 1$ terms and are of the $n$'th degree.
|
|
|
|
The product $a \cdot b \equiv f(x)g(x)$ is the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$. The coefficients $w_i$ will
|
|
directly yield the desired product when $\beta$ is substituted for $x$. The direct solution to solve for the $2n + 1$ coefficients
|
|
requires $O(n^2)$ time and would in practice be slower than the Comba technique.
|
|
|
|
However, numerical analysis theory indicates that only $2n + 1$ distinct points in $W(x)$ are required to determine the values of the $2n + 1$ unknown
|
|
coefficients. This means by finding $\zeta_y = W(y)$ for $2n + 1$ small values of $y$ the coefficients of $W(x)$ can be found with
|
|
Gaussian elimination. This technique is also occasionally refered to as the \textit{interpolation technique} (\textit{references please...}) since in
|
|
effect an interpolation based on $2n + 1$ points will yield a polynomial equivalent to $W(x)$.
|
|
|
|
The coefficients of the polynomial $W(x)$ are unknown which makes finding $W(y)$ for any value of $y$ impossible. However, since
|
|
$W(x) = f(x)g(x)$ the equivalent $\zeta_y = f(y) g(y)$ can be used in its place. The benefit of this technique stems from the
|
|
fact that $f(y)$ and $g(y)$ are much smaller than either $a$ or $b$ respectively. As a result finding the $2n + 1$ relations required
|
|
by multiplying $f(y)g(y)$ involves multiplying integers that are much smaller than either of the inputs.
|
|
|
|
When picking points to gather relations there are always three obvious points to choose, $y = 0, 1$ and $ \infty$. The $\zeta_0$ term
|
|
is simply the product $W(0) = w_0 = a_0 \cdot b_0$. The $\zeta_1$ term is the product
|
|
$W(1) = \left (\sum_{i = 0}^{n} a_i \right ) \left (\sum_{i = 0}^{n} b_i \right )$. The third point $\zeta_{\infty}$ is less obvious but rather
|
|
simple to explain. The $2n + 1$'th coefficient of $W(x)$ is numerically equivalent to the most significant column in an integer multiplication.
|
|
The point at $\infty$ is used symbolically to represent the most significant column, that is $W(\infty) = w_{2n} = a_nb_n$. Note that the
|
|
points at $y = 0$ and $\infty$ yield the coefficients $w_0$ and $w_{2n}$ directly.
|
|
|
|
If more points are required they should be of small values and powers of two such as $2^q$ and the related \textit{mirror points}
|
|
$\left (2^q \right )^{2n} \cdot \zeta_{2^{-q}}$ for small values of $q$. The term ``mirror point'' stems from the fact that
|
|
$\left (2^q \right )^{2n} \cdot \zeta_{2^{-q}}$ can be calculated in the exact opposite fashion as $\zeta_{2^q}$. For
|
|
example, when $n = 2$ and $q = 1$ then following two equations are equivalent to the point $\zeta_{2}$ and its mirror.
|
|
|
|
\begin{eqnarray}
|
|
\zeta_{2} = f(2)g(2) = (4a_2 + 2a_1 + a_0)(4b_2 + 2b_1 + b_0) \nonumber \\
|
|
16 \cdot \zeta_{1 \over 2} = 4f({1\over 2}) \cdot 4g({1 \over 2}) = (a_2 + 2a_1 + 4a_0)(b_2 + 2b_1 + 4b_0)
|
|
\end{eqnarray}
|
|
|
|
Using such points will allow the values of $f(y)$ and $g(y)$ to be independently calculated using only left shifts. For example, when $n = 2$ the
|
|
polynomial $f(2^q)$ is equal to $2^q((2^qa_2) + a_1) + a_0$. This technique of polynomial representation is known as Horner's method.
|
|
|
|
As a general rule of the algorithm when the inputs are split into $n$ parts each there are $2n - 1$ multiplications. Each multiplication is of
|
|
multiplicands that have $n$ times fewer digits than the inputs. The asymptotic running time of this algorithm is
|
|
$O \left ( k^{lg_n(2n - 1)} \right )$ for $k$ digit inputs (\textit{assuming they have the same number of digits}). Figure~\ref{fig:exponent}
|
|
summarizes the exponents for various values of $n$.
|
|
|
|
\begin{figure}
|
|
\begin{center}
|
|
\begin{tabular}{|c|c|c|}
|
|
\hline \textbf{Split into $n$ Parts} & \textbf{Exponent} & \textbf{Notes}\\
|
|
\hline $2$ & $1.584962501$ & This is Karatsuba Multiplication. \\
|
|
\hline $3$ & $1.464973520$ & This is Toom-Cook Multiplication. \\
|
|
\hline $4$ & $1.403677461$ &\\
|
|
\hline $5$ & $1.365212389$ &\\
|
|
\hline $10$ & $1.278753601$ &\\
|
|
\hline $100$ & $1.149426538$ &\\
|
|
\hline $1000$ & $1.100270931$ &\\
|
|
\hline $10000$ & $1.075252070$ &\\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\caption{Asymptotic Running Time of Polynomial Basis Multiplication}
|
|
\label{fig:exponent}
|
|
\end{figure}
|
|
|
|
At first it may seem like a good idea to choose $n = 1000$ since the exponent is approximately $1.1$. However, the overhead
|
|
of solving for the 2001 terms of $W(x)$ will certainly consume any savings the algorithm could offer for all but exceedingly large
|
|
numbers.
|
|
|
|
\subsubsection{Cutoff Point}
|
|
The polynomial basis multiplication algorithms all require fewer single precision multiplications than a straight Comba approach. However,
|
|
the algorithms incur an overhead (\textit{at the $O(n)$ work level}) since they require a system of equations to be solved. This makes the
|
|
polynomial basis approach more costly to use with small inputs.
|
|
|
|
Let $m$ represent the number of digits in the multiplicands (\textit{assume both multiplicands have the same number of digits}). There exists a
|
|
point $y$ such that when $m < y$ the polynomial basis algorithms are more costly than Comba, when $m = y$ they are roughly the same cost and
|
|
when $m > y$ the Comba methods are slower than the polynomial basis algorithms.
|
|
|
|
The exact location of $y$ depends on several key architectural elements of the computer platform in question.
|
|
|
|
\begin{enumerate}
|
|
\item The ratio of clock cycles for single precision multiplication versus other simpler operations such as addition, shifting, etc. For example
|
|
on the AMD Athlon the ratio is roughly $17 : 1$ while on the Intel P4 it is $29 : 1$. The higher the ratio in favour of multiplication the lower
|
|
the cutoff point $y$ will be.
|
|
|
|
\item The complexity of the linear system of equations (\textit{for the coefficients of $W(x)$}) is. Generally speaking as the number of splits
|
|
grows the complexity grows substantially. Ideally solving the system will only involve addition, subtraction and shifting of integers. This
|
|
directly reflects on the ratio previous mentioned.
|
|
|
|
\item To a lesser extent memory bandwidth and function call overheads. Provided the values are in the processor cache this is less of an
|
|
influence over the cutoff point.
|
|
|
|
\end{enumerate}
|
|
|
|
A clean cutoff point separation occurs when a point $y$ is found such that all of the cutoff point conditions are met. For example, if the point
|
|
is too low then there will be values of $m$ such that $m > y$ and the Comba method is still faster. Finding the cutoff points is fairly simple when
|
|
a high resolution timer is available.
|
|
|
|
\subsection{Karatsuba Multiplication}
|
|
Karatsuba \cite{KARA} multiplication when originally proposed in 1962 was among the first set of algorithms to break the $O(n^2)$ barrier for
|
|
general purpose multiplication. Given two polynomial basis representations $f(x) = ax + b$ and $g(x) = cx + d$, Karatsuba proved with
|
|
light algebra \cite{KARAP} that the following polynomial is equivalent to multiplication of the two integers the polynomials represent.
|
|
|
|
\begin{equation}
|
|
f(x) \cdot g(x) = acx^2 + ((a - b)(c - d) + ac + bd)x + bd
|
|
\end{equation}
|
|
|
|
Using the observation that $ac$ and $bd$ could be re-used only three half sized multiplications would be required to produce the product. Applying
|
|
this algorithm recursively, the work factor becomes $O(n^{lg(3)})$ which is substantially better than the work factor $O(n^2)$ of the Comba technique. It turns
|
|
out what Karatsuba did not know or at least did not publish was that this is simply polynomial basis multiplication with the points
|
|
$\zeta_0$, $\zeta_{\infty}$ and $-\zeta_{-1}$. Consider the resultant system of equations.
|
|
|
|
\begin{center}
|
|
\begin{tabular}{rcrcrcrc}
|
|
$\zeta_{0}$ & $=$ & & & & & $w_0$ \\
|
|
$-\zeta_{-1}$ & $=$ & $-w_2$ & $+$ & $w_1$ & $-$ & $w_0$ \\
|
|
$\zeta_{\infty}$ & $=$ & $w_2$ & & & & \\
|
|
\end{tabular}
|
|
\end{center}
|
|
|
|
By adding the first and last equation to the equation in the middle the term $w_1$ can be isolated and all three coefficients solved for. The simplicity
|
|
of this system of equations has made Karatsuba fairly popular. In fact the cutoff point is often fairly low\footnote{With LibTomMath 0.18 it is 70 and 109 digits for the Intel P4 and AMD Athlon respectively.}
|
|
making it an ideal algorithm to speed up certain public key cryptosystems such as RSA and Diffie-Hellman. It is worth noting that the point
|
|
$\zeta_1$ could be substituted for $-\zeta_{-1}$. In this case the first and third row are subtracted instead of added to the second row.
|
|
|
|
\newpage\begin{figure}[!here]
|
|
\begin{small}
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_karatsuba\_mul}. \\
|
|
\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\
|
|
\textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert$ \\
|
|
\hline \\
|
|
1. Init the following mp\_int variables: $x0$, $x1$, $y0$, $y1$, $t1$, $x0y0$, $x1y1$.\\
|
|
2. If step 2 failed then return(\textit{MP\_MEM}). \\
|
|
\\
|
|
Split the input. e.g. $a = x1 \cdot \beta^B + x0$ \\
|
|
3. $B \leftarrow \mbox{min}(a.used, b.used)/2$ \\
|
|
4. $x0 \leftarrow a \mbox{ (mod }\beta^B\mbox{)}$ (\textit{mp\_mod\_2d}) \\
|
|
5. $y0 \leftarrow b \mbox{ (mod }\beta^B\mbox{)}$ \\
|
|
6. $x1 \leftarrow \lfloor a / \beta^B \rfloor$ (\textit{mp\_rshd}) \\
|
|
7. $y1 \leftarrow \lfloor b / \beta^B \rfloor$ \\
|
|
\\
|
|
Calculate the three products. \\
|
|
8. $x0y0 \leftarrow x0 \cdot y0$ (\textit{mp\_mul}) \\
|
|
9. $x1y1 \leftarrow x1 \cdot y1$ \\
|
|
10. $t1 \leftarrow x1 - x0$ (\textit{mp\_sub}) \\
|
|
11. $x0 \leftarrow y1 - y0$ \\
|
|
12. $t1 \leftarrow t1 \cdot x0$ \\
|
|
\\
|
|
Calculate the middle term. \\
|
|
13. $x0 \leftarrow x0y0 + x1y1$ \\
|
|
14. $t1 \leftarrow x0 - t1$ \\
|
|
\\
|
|
Calculate the final product. \\
|
|
15. $t1 \leftarrow t1 \cdot \beta^B$ (\textit{mp\_lshd}) \\
|
|
16. $x1y1 \leftarrow x1y1 \cdot \beta^{2B}$ \\
|
|
17. $t1 \leftarrow x0y0 + t1$ \\
|
|
18. $c \leftarrow t1 + x1y1$ \\
|
|
19. Clear all of the temporary variables. \\
|
|
20. Return(\textit{MP\_OKAY}).\\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{small}
|
|
\caption{Algorithm mp\_karatsuba\_mul}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_karatsuba\_mul.}
|
|
This algorithm computes the unsigned product of two inputs using the Karatsuba multiplication algorithm. It is loosely based on the description
|
|
from Knuth \cite[pp. 294-295]{TAOCPV2}.
|
|
|
|
\index{radix point}
|
|
In order to split the two inputs into their respective halves, a suitable \textit{radix point} must be chosen. The radix point chosen must
|
|
be used for both of the inputs meaning that it must be smaller than the smallest input. Step 3 chooses the radix point $B$ as half of the
|
|
smallest input \textbf{used} count. After the radix point is chosen the inputs are split into lower and upper halves. Step 4 and 5
|
|
compute the lower halves. Step 6 and 7 computer the upper halves.
|
|
|
|
After the halves have been computed the three intermediate half-size products must be computed. Step 8 and 9 compute the trivial products
|
|
$x0 \cdot y0$ and $x1 \cdot y1$. The mp\_int $x0$ is used as a temporary variable after $x1 - x0$ has been computed. By using $x0$ instead
|
|
of an additional temporary variable, the algorithm can avoid an addition memory allocation operation.
|
|
|
|
The remaining steps 13 through 18 compute the Karatsuba polynomial through a variety of digit shifting and addition operations.
|
|
|
|
EXAM,bn_mp_karatsuba_mul.c
|
|
|
|
The new coding element in this routine, not seen in previous routines, is the usage of goto statements. The conventional
|
|
wisdom is that goto statements should be avoided. This is generally true, however when every single function call can fail, it makes sense
|
|
to handle error recovery with a single piece of code. Lines @61,if@ to @75,if@ handle initializing all of the temporary variables
|
|
required. Note how each of the if statements goes to a different label in case of failure. This allows the routine to correctly free only
|
|
the temporaries that have been successfully allocated so far.
|
|
|
|
The temporary variables are all initialized using the mp\_init\_size routine since they are expected to be large. This saves the
|
|
additional reallocation that would have been necessary. Also $x0$, $x1$, $y0$ and $y1$ have to be able to hold at least their respective
|
|
number of digits for the next section of code.
|
|
|
|
The first algebraic portion of the algorithm is to split the two inputs into their halves. However, instead of using mp\_mod\_2d and mp\_rshd
|
|
to extract the halves, the respective code has been placed inline within the body of the function. To initialize the halves, the \textbf{used} and
|
|
\textbf{sign} members are copied first. The first for loop on line @98,for@ copies the lower halves. Since they are both the same magnitude it
|
|
is simpler to calculate both lower halves in a single loop. The for loop on lines @104,for@ and @109,for@ calculate the upper halves $x1$ and
|
|
$y1$ respectively.
|
|
|
|
By inlining the calculation of the halves, the Karatsuba multiplier has a slightly lower overhead and can be used for smaller magnitude inputs.
|
|
|
|
When line @152,err@ is reached, the algorithm has completed succesfully. The ``error status'' variable $err$ is set to \textbf{MP\_OKAY} so that
|
|
the same code that handles errors can be used to clear the temporary variables and return.
|
|
|
|
\subsection{Toom-Cook $3$-Way Multiplication}
|
|
Toom-Cook $3$-Way \cite{TOOM} multiplication is essentially the polynomial basis algorithm for $n = 3$ except that the points are
|
|
chosen such that $\zeta$ is easy to compute and the resulting system of equations easy to reduce. Here, the points $\zeta_{0}$,
|
|
$16 \cdot \zeta_{1 \over 2}$, $\zeta_1$, $\zeta_2$ and $\zeta_{\infty}$ make up the five required points to solve for the coefficients
|
|
of the $W(x)$.
|
|
|
|
With the five relations that Toom-Cook specifies, the following system of equations is formed.
|
|
|
|
\begin{center}
|
|
\begin{tabular}{rcrcrcrcrcr}
|
|
$\zeta_0$ & $=$ & $0w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $1w_0$ \\
|
|
$16 \cdot \zeta_{1 \over 2}$ & $=$ & $1w_4$ & $+$ & $2w_3$ & $+$ & $4w_2$ & $+$ & $8w_1$ & $+$ & $16w_0$ \\
|
|
$\zeta_1$ & $=$ & $1w_4$ & $+$ & $1w_3$ & $+$ & $1w_2$ & $+$ & $1w_1$ & $+$ & $1w_0$ \\
|
|
$\zeta_2$ & $=$ & $16w_4$ & $+$ & $8w_3$ & $+$ & $4w_2$ & $+$ & $2w_1$ & $+$ & $1w_0$ \\
|
|
$\zeta_{\infty}$ & $=$ & $1w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $0w_0$ \\
|
|
\end{tabular}
|
|
\end{center}
|
|
|
|
A trivial solution to this matrix requires $12$ subtractions, two multiplications by a small power of two, two divisions by a small power
|
|
of two, two divisions by three and one multiplication by three. All of these $19$ sub-operations require less than quadratic time, meaning that
|
|
the algorithm can be faster than a baseline multiplication. However, the greater complexity of this algorithm places the cutoff point
|
|
(\textbf{TOOM\_MUL\_CUTOFF}) where Toom-Cook becomes more efficient much higher than the Karatsuba cutoff point.
|
|
|
|
\begin{figure}[!here]
|
|
\begin{small}
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_toom\_mul}. \\
|
|
\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\
|
|
\textbf{Output}. $c \leftarrow a \cdot b $ \\
|
|
\hline \\
|
|
Split $a$ and $b$ into three pieces. E.g. $a = a_2 \beta^{2k} + a_1 \beta^{k} + a_0$ \\
|
|
1. $k \leftarrow \lfloor \mbox{min}(a.used, b.used) / 3 \rfloor$ \\
|
|
2. $a_0 \leftarrow a \mbox{ (mod }\beta^{k}\mbox{)}$ \\
|
|
3. $a_1 \leftarrow \lfloor a / \beta^k \rfloor$, $a_1 \leftarrow a_1 \mbox{ (mod }\beta^{k}\mbox{)}$ \\
|
|
4. $a_2 \leftarrow \lfloor a / \beta^{2k} \rfloor$, $a_2 \leftarrow a_2 \mbox{ (mod }\beta^{k}\mbox{)}$ \\
|
|
5. $b_0 \leftarrow a \mbox{ (mod }\beta^{k}\mbox{)}$ \\
|
|
6. $b_1 \leftarrow \lfloor a / \beta^k \rfloor$, $b_1 \leftarrow b_1 \mbox{ (mod }\beta^{k}\mbox{)}$ \\
|
|
7. $b_2 \leftarrow \lfloor a / \beta^{2k} \rfloor$, $b_2 \leftarrow b_2 \mbox{ (mod }\beta^{k}\mbox{)}$ \\
|
|
\\
|
|
Find the five equations for $w_0, w_1, ..., w_4$. \\
|
|
8. $w_0 \leftarrow a_0 \cdot b_0$ \\
|
|
9. $w_4 \leftarrow a_2 \cdot b_2$ \\
|
|
10. $tmp_1 \leftarrow 2 \cdot a_0$, $tmp_1 \leftarrow a_1 + tmp_1$, $tmp_1 \leftarrow 2 \cdot tmp_1$, $tmp_1 \leftarrow tmp_1 + a_2$ \\
|
|
11. $tmp_2 \leftarrow 2 \cdot b_0$, $tmp_2 \leftarrow b_1 + tmp_2$, $tmp_2 \leftarrow 2 \cdot tmp_2$, $tmp_2 \leftarrow tmp_2 + b_2$ \\
|
|
12. $w_1 \leftarrow tmp_1 \cdot tmp_2$ \\
|
|
13. $tmp_1 \leftarrow 2 \cdot a_2$, $tmp_1 \leftarrow a_1 + tmp_1$, $tmp_1 \leftarrow 2 \cdot tmp_1$, $tmp_1 \leftarrow tmp_1 + a_0$ \\
|
|
14. $tmp_2 \leftarrow 2 \cdot b_2$, $tmp_2 \leftarrow b_1 + tmp_2$, $tmp_2 \leftarrow 2 \cdot tmp_2$, $tmp_2 \leftarrow tmp_2 + b_0$ \\
|
|
15. $w_3 \leftarrow tmp_1 \cdot tmp_2$ \\
|
|
16. $tmp_1 \leftarrow a_0 + a_1$, $tmp_1 \leftarrow tmp_1 + a_2$, $tmp_2 \leftarrow b_0 + b_1$, $tmp_2 \leftarrow tmp_2 + b_2$ \\
|
|
17. $w_2 \leftarrow tmp_1 \cdot tmp_2$ \\
|
|
\\
|
|
Continued on the next page.\\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{small}
|
|
\caption{Algorithm mp\_toom\_mul}
|
|
\end{figure}
|
|
|
|
\newpage\begin{figure}[!here]
|
|
\begin{small}
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_toom\_mul} (continued). \\
|
|
\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\
|
|
\textbf{Output}. $c \leftarrow a \cdot b $ \\
|
|
\hline \\
|
|
Now solve the system of equations. \\
|
|
18. $w_1 \leftarrow w_4 - w_1$, $w_3 \leftarrow w_3 - w_0$ \\
|
|
19. $w_1 \leftarrow \lfloor w_1 / 2 \rfloor$, $w_3 \leftarrow \lfloor w_3 / 2 \rfloor$ \\
|
|
20. $w_2 \leftarrow w_2 - w_0$, $w_2 \leftarrow w_2 - w_4$ \\
|
|
21. $w_1 \leftarrow w_1 - w_2$, $w_3 \leftarrow w_3 - w_2$ \\
|
|
22. $tmp_1 \leftarrow 8 \cdot w_0$, $w_1 \leftarrow w_1 - tmp_1$, $tmp_1 \leftarrow 8 \cdot w_4$, $w_3 \leftarrow w_3 - tmp_1$ \\
|
|
23. $w_2 \leftarrow 3 \cdot w_2$, $w_2 \leftarrow w_2 - w_1$, $w_2 \leftarrow w_2 - w_3$ \\
|
|
24. $w_1 \leftarrow w_1 - w_2$, $w_3 \leftarrow w_3 - w_2$ \\
|
|
25. $w_1 \leftarrow \lfloor w_1 / 3 \rfloor, w_3 \leftarrow \lfloor w_3 / 3 \rfloor$ \\
|
|
\\
|
|
Now substitute $\beta^k$ for $x$ by shifting $w_0, w_1, ..., w_4$. \\
|
|
26. for $n$ from $1$ to $4$ do \\
|
|
\hspace{3mm}26.1 $w_n \leftarrow w_n \cdot \beta^{nk}$ \\
|
|
27. $c \leftarrow w_0 + w_1$, $c \leftarrow c + w_2$, $c \leftarrow c + w_3$, $c \leftarrow c + w_4$ \\
|
|
28. Return(\textit{MP\_OKAY}) \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{small}
|
|
\caption{Algorithm mp\_toom\_mul (continued)}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_toom\_mul.}
|
|
This algorithm computes the product of two mp\_int variables $a$ and $b$ using the Toom-Cook approach. Compared to the Karatsuba multiplication, this
|
|
algorithm has a lower asymptotic running time of approximately $O(n^{1.464})$ but at an obvious cost in overhead. In this
|
|
description, several statements have been compounded to save space. The intention is that the statements are executed from left to right across
|
|
any given step.
|
|
|
|
The two inputs $a$ and $b$ are first split into three $k$-digit integers $a_0, a_1, a_2$ and $b_0, b_1, b_2$ respectively. From these smaller
|
|
integers the coefficients of the polynomial basis representations $f(x)$ and $g(x)$ are known and can be used to find the relations required.
|
|
|
|
The first two relations $w_0$ and $w_4$ are the points $\zeta_{0}$ and $\zeta_{\infty}$ respectively. The relation $w_1, w_2$ and $w_3$ correspond
|
|
to the points $16 \cdot \zeta_{1 \over 2}, \zeta_{2}$ and $\zeta_{1}$ respectively. These are found using logical shifts to independently find
|
|
$f(y)$ and $g(y)$ which significantly speeds up the algorithm.
|
|
|
|
After the five relations $w_0, w_1, \ldots, w_4$ have been computed, the system they represent must be solved in order for the unknown coefficients
|
|
$w_1, w_2$ and $w_3$ to be isolated. The steps 18 through 25 perform the system reduction required as previously described. Each step of
|
|
the reduction represents the comparable matrix operation that would be performed had this been performed by pencil. For example, step 18 indicates
|
|
that row $1$ must be subtracted from row $4$ and simultaneously row $0$ subtracted from row $3$.
|
|
|
|
Once the coeffients have been isolated, the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$ is known. By substituting $\beta^{k}$ for $x$, the integer
|
|
result $a \cdot b$ is produced.
|
|
|
|
EXAM,bn_mp_toom_mul.c
|
|
|
|
-- Comments to be added during editing phase.
|
|
|
|
\subsection{Signed Multiplication}
|
|
Now that algorithms to handle multiplications of every useful dimensions have been developed, a rather simple finishing touch is required. So far all
|
|
of the multiplication algorithms have been unsigned multiplications which leaves only a signed multiplication algorithm to be established.
|
|
|
|
\newpage\begin{figure}[!here]
|
|
\begin{small}
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_mul}. \\
|
|
\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\
|
|
\textbf{Output}. $c \leftarrow a \cdot b$ \\
|
|
\hline \\
|
|
1. If $a.sign = b.sign$ then \\
|
|
\hspace{3mm}1.1 $sign = MP\_ZPOS$ \\
|
|
2. else \\
|
|
\hspace{3mm}2.1 $sign = MP\_ZNEG$ \\
|
|
3. If min$(a.used, b.used) \ge TOOM\_MUL\_CUTOFF$ then \\
|
|
\hspace{3mm}3.1 $c \leftarrow a \cdot b$ using algorithm mp\_toom\_mul \\
|
|
4. else if min$(a.used, b.used) \ge KARATSUBA\_MUL\_CUTOFF$ then \\
|
|
\hspace{3mm}4.1 $c \leftarrow a \cdot b$ using algorithm mp\_karatsuba\_mul \\
|
|
5. else \\
|
|
\hspace{3mm}5.1 $digs \leftarrow a.used + b.used + 1$ \\
|
|
\hspace{3mm}5.2 If $digs < MP\_ARRAY$ and min$(a.used, b.used) \le \delta$ then \\
|
|
\hspace{6mm}5.2.1 $c \leftarrow a \cdot b \mbox{ (mod }\beta^{digs}\mbox{)}$ using algorithm fast\_s\_mp\_mul\_digs. \\
|
|
\hspace{3mm}5.3 else \\
|
|
\hspace{6mm}5.3.1 $c \leftarrow a \cdot b \mbox{ (mod }\beta^{digs}\mbox{)}$ using algorithm s\_mp\_mul\_digs. \\
|
|
6. $c.sign \leftarrow sign$ \\
|
|
7. Return the result of the unsigned multiplication performed. \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{small}
|
|
\caption{Algorithm mp\_mul}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_mul.}
|
|
This algorithm performs the signed multiplication of two inputs. It will make use of any of the three unsigned multiplication algorithms
|
|
available when the input is of appropriate size. The \textbf{sign} of the result is not set until the end of the algorithm since algorithm
|
|
s\_mp\_mul\_digs will clear it.
|
|
|
|
EXAM,bn_mp_mul.c
|
|
|
|
The implementation is rather simplistic and is not particularly noteworthy. Line @22,?@ computes the sign of the result using the ``?''
|
|
operator from the C programming language. Line @37,<<@ computes $\delta$ using the fact that $1 << k$ is equal to $2^k$.
|
|
|
|
\section{Squaring}
|
|
|
|
Squaring is a special case of multiplication where both multiplicands are equal. At first it may seem like there is no significant optimization
|
|
available but in fact there is. Consider the multiplication of $576$ against $241$. In total there will be nine single precision multiplications
|
|
performed which are $1\cdot 6$, $1 \cdot 7$, $1 \cdot 5$, $4 \cdot 6$, $4 \cdot 7$, $4 \cdot 5$, $2 \cdot 6$, $2 \cdot 7$ and $2 \cdot 5$. Now consider
|
|
the multiplication of $123$ against $123$. The nine products are $3 \cdot 3$, $3 \cdot 2$, $3 \cdot 1$, $2 \cdot 3$, $2 \cdot 2$, $2 \cdot 1$,
|
|
$1 \cdot 3$, $1 \cdot 2$ and $1 \cdot 1$. On closer inspection some of the products are equivalent. For example, $3 \cdot 2 = 2 \cdot 3$
|
|
and $3 \cdot 1 = 1 \cdot 3$.
|
|
|
|
For any $n$-digit input, there are ${{\left (n^2 + n \right)}\over 2}$ possible unique single precision multiplications required compared to the $n^2$
|
|
required for multiplication. The following diagram gives an example of the operations required.
|
|
|
|
\begin{figure}[here]
|
|
\begin{center}
|
|
\begin{tabular}{ccccc|c}
|
|
&&1&2&3&\\
|
|
$\times$ &&1&2&3&\\
|
|
\hline && $3 \cdot 1$ & $3 \cdot 2$ & $3 \cdot 3$ & Row 0\\
|
|
& $2 \cdot 1$ & $2 \cdot 2$ & $2 \cdot 3$ && Row 1 \\
|
|
$1 \cdot 1$ & $1 \cdot 2$ & $1 \cdot 3$ &&& Row 2 \\
|
|
\end{tabular}
|
|
\end{center}
|
|
\caption{Squaring Optimization Diagram}
|
|
\end{figure}
|
|
|
|
MARK,SQUARE
|
|
Starting from zero and numbering the columns from right to left a very simple pattern becomes obvious. For the purposes of this discussion let $x$
|
|
represent the number being squared. The first observation is that in row $k$ the $2k$'th column of the product has a $\left (x_k \right)^2$ term in it.
|
|
|
|
The second observation is that every column $j$ in row $k$ where $j \ne 2k$ is part of a double product. Every non-square term of a column will
|
|
appear twice hence the name ``double product''. Every odd column is made up entirely of double products. In fact every column is made up of double
|
|
products and at most one square (\textit{see the exercise section}).
|
|
|
|
The third and final observation is that for row $k$ the first unique non-square term, that is, one that hasn't already appeared in an earlier row,
|
|
occurs at column $2k + 1$. For example, on row $1$ of the previous squaring, column one is part of the double product with column one from row zero.
|
|
Column two of row one is a square and column three is the first unique column.
|
|
|
|
\subsection{The Baseline Squaring Algorithm}
|
|
The baseline squaring algorithm is meant to be a catch-all squaring algorithm. It will handle any of the input sizes that the faster routines
|
|
will not handle.
|
|
|
|
\newpage\begin{figure}[!here]
|
|
\begin{small}
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{s\_mp\_sqr}. \\
|
|
\textbf{Input}. mp\_int $a$ \\
|
|
\textbf{Output}. $b \leftarrow a^2$ \\
|
|
\hline \\
|
|
1. Init a temporary mp\_int of at least $2 \cdot a.used +1$ digits. (\textit{mp\_init\_size}) \\
|
|
2. If step 1 failed return(\textit{MP\_MEM}) \\
|
|
3. $t.used \leftarrow 2 \cdot a.used + 1$ \\
|
|
4. For $ix$ from 0 to $a.used - 1$ do \\
|
|
\hspace{3mm}Calculate the square. \\
|
|
\hspace{3mm}4.1 $\hat r \leftarrow t_{2ix} + \left (a_{ix} \right )^2$ \\
|
|
\hspace{3mm}4.2 $t_{2ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\
|
|
\hspace{3mm}Calculate the double products after the square. \\
|
|
\hspace{3mm}4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\
|
|
\hspace{3mm}4.4 For $iy$ from $ix + 1$ to $a.used - 1$ do \\
|
|
\hspace{6mm}4.4.1 $\hat r \leftarrow 2 \cdot a_{ix}a_{iy} + t_{ix + iy} + u$ \\
|
|
\hspace{6mm}4.4.2 $t_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\
|
|
\hspace{6mm}4.4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\
|
|
\hspace{3mm}Set the last carry. \\
|
|
\hspace{3mm}4.5 While $u > 0$ do \\
|
|
\hspace{6mm}4.5.1 $iy \leftarrow iy + 1$ \\
|
|
\hspace{6mm}4.5.2 $\hat r \leftarrow t_{ix + iy} + u$ \\
|
|
\hspace{6mm}4.5.3 $t_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\
|
|
\hspace{6mm}4.5.4 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\
|
|
5. Clamp excess digits of $t$. (\textit{mp\_clamp}) \\
|
|
6. Exchange $b$ and $t$. \\
|
|
7. Clear $t$ (\textit{mp\_clear}) \\
|
|
8. Return(\textit{MP\_OKAY}) \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{small}
|
|
\caption{Algorithm s\_mp\_sqr}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm s\_mp\_sqr.}
|
|
This algorithm computes the square of an input using the three observations on squaring. It is based fairly faithfully on algorithm 14.16 of HAC
|
|
\cite[pp.596-597]{HAC}. Similar to algorithm s\_mp\_mul\_digs, a temporary mp\_int is allocated to hold the result of the squaring. This allows the
|
|
destination mp\_int to be the same as the source mp\_int.
|
|
|
|
The outer loop of this algorithm begins on step 4. It is best to think of the outer loop as walking down the rows of the partial results, while
|
|
the inner loop computes the columns of the partial result. Step 4.1 and 4.2 compute the square term for each row, and step 4.3 and 4.4 propagate
|
|
the carry and compute the double products.
|
|
|
|
The requirement that a mp\_word be able to represent the range $0 \le x < 2 \beta^2$ arises from this
|
|
very algorithm. The product $a_{ix}a_{iy}$ will lie in the range $0 \le x \le \beta^2 - 2\beta + 1$ which is obviously less than $\beta^2$ meaning that
|
|
when it is multiplied by two, it can be properly represented by a mp\_word.
|
|
|
|
Similar to algorithm s\_mp\_mul\_digs, after every pass of the inner loop, the destination is correctly set to the sum of all of the partial
|
|
results calculated so far. This involves expensive carry propagation which will be eliminated in the next algorithm.
|
|
|
|
EXAM,bn_s_mp_sqr.c
|
|
|
|
Inside the outer loop (\textit{see line @32,for@}) the square term is calculated on line @35,r =@. Line @42,>>@ extracts the carry from the square
|
|
term. Aliases for $a_{ix}$ and $t_{ix+iy}$ are initialized on lines @45,tmpx@ and @48,tmpt@ respectively. The doubling is performed using two
|
|
additions (\textit{see line @57,r + r@}) since it is usually faster than shifting,if not at least as fast.
|
|
|
|
\subsection{Faster Squaring by the ``Comba'' Method}
|
|
A major drawback to the baseline method is the requirement for single precision shifting inside the $O(n^2)$ nested loop. Squaring has an additional
|
|
drawback that it must double the product inside the inner loop as well. As for multiplication, the Comba technique can be used to eliminate these
|
|
performance hazards.
|
|
|
|
The first obvious solution is to make an array of mp\_words which will hold all of the columns. This will indeed eliminate all of the carry
|
|
propagation operations from the inner loop. However, the inner product must still be doubled $O(n^2)$ times. The solution stems from the simple fact
|
|
that $2a + 2b + 2c = 2(a + b + c)$. That is the sum of all of the double products is equal to double the sum of all the products. For example,
|
|
$ab + ba + ac + ca = 2ab + 2ac = 2(ab + ac)$.
|
|
|
|
However, we cannot simply double all of the columns, since the squares appear only once per row. The most practical solution is to have two mp\_word
|
|
arrays. One array will hold the squares and the other array will hold the double products. With both arrays the doubling and carry propagation can be
|
|
moved to a $O(n)$ work level outside the $O(n^2)$ level.
|
|
|
|
\newpage\begin{figure}[!here]
|
|
\begin{small}
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{fast\_s\_mp\_sqr}. \\
|
|
\textbf{Input}. mp\_int $a$ \\
|
|
\textbf{Output}. $b \leftarrow a^2$ \\
|
|
\hline \\
|
|
Place two arrays of \textbf{MP\_WARRAY} mp\_words named $\hat W$ and $\hat {X}$ on the stack. \\
|
|
1. If $b.alloc < 2a.used + 1$ then grow $b$ to $2a.used + 1$ digits. (\textit{mp\_grow}). \\
|
|
2. If step 1 failed return(\textit{MP\_MEM}). \\
|
|
3. for $ix$ from $0$ to $2a.used + 1$ do \\
|
|
\hspace{3mm}3.1 $\hat W_{ix} \leftarrow 0$ \\
|
|
\hspace{3mm}3.2 $\hat {X}_{ix} \leftarrow 0$ \\
|
|
4. for $ix$ from $0$ to $a.used - 1$ do \\
|
|
\hspace{3mm}Compute the square.\\
|
|
\hspace{3mm}4.1 $\hat {X}_{ix+ix} \leftarrow \left ( a_ix \right )^2$ \\
|
|
\\
|
|
\hspace{3mm}Compute the double products.\\
|
|
\hspace{3mm}4.2 for $iy$ from $ix + 1$ to $a.used - 1$ do \\
|
|
\hspace{6mm}4.2.1 $\hat W_{ix+iy} \leftarrow \hat W_{ix+iy} + a_{ix}a_{iy}$ \\
|
|
5. $oldused \leftarrow b.used$ \\
|
|
6. $b.used \leftarrow 2a.used + 1$ \\
|
|
\\
|
|
Double the products and propagate the carries simultaneously. \\
|
|
7. $\hat W_0 \leftarrow 2 \hat W_0 + \hat {X}_0$ \\
|
|
8. for $ix$ from $1$ to $2a.used$ do \\
|
|
\hspace{3mm}8.1 $\hat W_{ix} \leftarrow 2 \hat W_{ix} + \hat {X}_{ix}$ \\
|
|
\hspace{3mm}8.2 $\hat W_{ix} \leftarrow \hat W_{ix} + \lfloor \hat W_{ix - 1} / \beta \rfloor$ \\
|
|
\hspace{3mm}8.3 $b_{ix-1} \leftarrow W_{ix-1} \mbox{ (mod }\beta\mbox{)}$ \\
|
|
9. $b_{2a.used} \leftarrow \hat W_{2a.used} \mbox{ (mod }\beta\mbox{)}$ \\
|
|
10. if $2a.used + 1 < oldused$ then do \\
|
|
\hspace{3mm}10.1 for $ix$ from $2a.used + 1$ to $oldused$ do \\
|
|
\hspace{6mm}10.1.1 $b_{ix} \leftarrow 0$ \\
|
|
11. Clamp excess digits from $b$. (\textit{mp\_clamp}) \\
|
|
12. Return(\textit{MP\_OKAY}). \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{small}
|
|
\caption{Algorithm fast\_s\_mp\_sqr}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm fast\_s\_mp\_sqr.}
|
|
This algorithm computes the square of an input using the Comba technique. It is designed to be a replacement for algorithm s\_mp\_sqr when
|
|
the number of input digits is less than \textbf{MP\_WARRAY} and less than $\delta \over 2$.
|
|
|
|
This routine requires two arrays of mp\_words to be placed on the stack. The first array $\hat W$ will hold the double products and the second
|
|
array $\hat X$ will hold the squares. Though only at most $MP\_WARRAY \over 2$ words of $\hat X$ are used, it has proven faster on most
|
|
processors to simply make it a full size array.
|
|
|
|
The loop on step 3 will zero the two arrays to prepare them for the squaring step. Step 4.1 computes the squares of the product. Note how
|
|
it simply assigns the value into the $\hat X$ array. The nested loop on step 4.2 computes the doubles of the products. This loop
|
|
computes the sum of the products for each column. They are not doubled until later.
|
|
|
|
After the squaring loop, the products stored in $\hat W$ musted be doubled and the carries propagated forwards. It makes sense to do both
|
|
operations at the same time. The expression $\hat W_{ix} \leftarrow 2 \hat W_{ix} + \hat {X}_{ix}$ computes the sum of the double product and the
|
|
squares in place.
|
|
|
|
EXAM,bn_fast_s_mp_sqr.c
|
|
|
|
-- Write something deep and insightful later, Tom.
|
|
|
|
\subsection{Polynomial Basis Squaring}
|
|
The same algorithm that performs optimal polynomial basis multiplication can be used to perform polynomial basis squaring. The minor exception
|
|
is that $\zeta_y = f(y)g(y)$ is actually equivalent to $\zeta_y = f(y)^2$ since $f(y) = g(y)$. Instead of performing $2n + 1$
|
|
multiplications to find the $\zeta$ relations, squaring operations are performed instead.
|
|
|
|
\subsection{Karatsuba Squaring}
|
|
Let $f(x) = ax + b$ represent the polynomial basis representation of a number to square.
|
|
Let $h(x) = \left ( f(x) \right )^2$ represent the square of the polynomial. The Karatsuba equation can be modified to square a
|
|
number with the following equation.
|
|
|
|
\begin{equation}
|
|
h(x) = a^2x^2 + \left (a^2 + b^2 - (a - b)^2 \right )x + b^2
|
|
\end{equation}
|
|
|
|
Upon closer inspection this equation only requires the calculation of three half-sized squares: $a^2$, $b^2$ and $(a - b)^2$. As in
|
|
Karatsuba multiplication, this algorithm can be applied recursively on the input and will achieve an asymptotic running time of
|
|
$O \left ( n^{lg(3)} \right )$.
|
|
|
|
You might ask yourself, if the asymptotic time of Karatsuba squaring and multiplication is the same, why not simply use the multiplication algorithm
|
|
instead? The answer to this arises from the cutoff point for squaring. As in multiplication there exists a cutoff point, at which the
|
|
time required for a Comba based squaring and a Karatsuba based squaring meet. Due to the overhead inherent in the Karatsuba method, the cutoff
|
|
point is fairly high. For example, on an AMD Athlon XP processor with $\beta = 2^{28}$, the cutoff point is around 127 digits.
|
|
|
|
Consider squaring a 200 digit number with this technique. It will be split into two 100 digit halves which are subsequently squared.
|
|
The 100 digit halves will not be squared using Karatsuba, but instead using the faster Comba based squaring algorithm. If Karatsuba multiplication
|
|
were used instead, the 100 digit numbers would be squared with a slower Comba based multiplication.
|
|
|
|
\newpage\begin{figure}[!here]
|
|
\begin{small}
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_karatsuba\_sqr}. \\
|
|
\textbf{Input}. mp\_int $a$ \\
|
|
\textbf{Output}. $b \leftarrow a^2$ \\
|
|
\hline \\
|
|
1. Initialize the following temporary mp\_ints: $x0$, $x1$, $t1$, $t2$, $x0x0$ and $x1x1$. \\
|
|
2. If any of the initializations on step 1 failed return(\textit{MP\_MEM}). \\
|
|
\\
|
|
Split the input. e.g. $a = x1\beta^B + x0$ \\
|
|
3. $B \leftarrow \lfloor a.used / 2 \rfloor$ \\
|
|
4. $x0 \leftarrow a \mbox{ (mod }\beta^B\mbox{)}$ (\textit{mp\_mod\_2d}) \\
|
|
5. $x1 \leftarrow \lfloor a / \beta^B \rfloor$ (\textit{mp\_lshd}) \\
|
|
\\
|
|
Calculate the three squares. \\
|
|
6. $x0x0 \leftarrow x0^2$ (\textit{mp\_sqr}) \\
|
|
7. $x1x1 \leftarrow x1^2$ \\
|
|
8. $t1 \leftarrow x1 - x0$ (\textit{mp\_sub}) \\
|
|
9. $t1 \leftarrow t1^2$ \\
|
|
\\
|
|
Compute the middle term. \\
|
|
10. $t2 \leftarrow x0x0 + x1x1$ (\textit{s\_mp\_add}) \\
|
|
11. $t1 \leftarrow t2 - t1$ \\
|
|
\\
|
|
Compute final product. \\
|
|
12. $t1 \leftarrow t1\beta^B$ (\textit{mp\_lshd}) \\
|
|
13. $x1x1 \leftarrow x1x1\beta^{2B}$ \\
|
|
14. $t1 \leftarrow t1 + x0x0$ \\
|
|
15. $b \leftarrow t1 + x1x1$ \\
|
|
16. Return(\textit{MP\_OKAY}). \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{small}
|
|
\caption{Algorithm mp\_karatsuba\_sqr}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_karatsuba\_sqr.}
|
|
This algorithm computes the square of an input $a$ using the Karatsuba technique. This algorithm is very similar to the Karatsuba based
|
|
multiplication algorithm with the exception that the three half-size multiplications have been replaced with three half-size squarings.
|
|
|
|
The radix point for squaring is simply placed exactly in the middle of the digits when the input has an odd number of digits, otherwise it is
|
|
placed just below the middle. Step 3, 4 and 5 compute the two halves required using $B$
|
|
as the radix point. The first two squares in steps 6 and 7 are rather straightforward while the last square is of a more compact form.
|
|
|
|
By expanding $\left (x1 - x0 \right )^2$, the $x1^2$ and $x0^2$ terms in the middle disappear, that is $x1^2 + x0^2 - (x1 - x0)^2 = 2 \cdot x0 \cdot x1$.
|
|
Now if $5n$ single precision additions and a squaring of $n$-digits is faster than multiplying two $n$-digit numbers and doubling then
|
|
this method is faster. Assuming no further recursions occur, the difference can be estimated with the following inequality.
|
|
|
|
Let $p$ represent the cost of a single precision addition and $q$ the cost of a single precision multiplication both in terms of time\footnote{Or
|
|
machine clock cycles.}.
|
|
|
|
\begin{equation}
|
|
5pn +{{q(n^2 + n)} \over 2} \le pn + qn^2
|
|
\end{equation}
|
|
|
|
For example, on an AMD Athlon XP processor $p = {1 \over 3}$ and $q = 6$. This implies that the following inequality should hold.
|
|
\begin{center}
|
|
\begin{tabular}{rcl}
|
|
${5n \over 3} + 3n^2 + 3n$ & $<$ & ${n \over 3} + 6n^2$ \\
|
|
${5 \over 3} + 3n + 3$ & $<$ & ${1 \over 3} + 6n$ \\
|
|
${13 \over 9}$ & $<$ & $n$ \\
|
|
\end{tabular}
|
|
\end{center}
|
|
|
|
This results in a cutoff point around $n = 2$. As a consequence it is actually faster to compute the middle term the ``long way'' on processors
|
|
where multiplication is substantially slower\footnote{On the Athlon there is a 1:17 ratio between clock cycles for addition and multiplication. On
|
|
the Intel P4 processor this ratio is 1:29 making this method even more beneficial. The only common exception is the ARMv4 processor which has a
|
|
ratio of 1:7. } than simpler operations such as addition.
|
|
|
|
EXAM,bn_mp_karatsuba_sqr.c
|
|
|
|
This implementation is largely based on the implementation of algorithm mp\_karatsuba\_mul. It uses the same inline style to copy and
|
|
shift the input into the two halves. The loop from line @54,{@ to line @70,}@ has been modified since only one input exists. The \textbf{used}
|
|
count of both $x0$ and $x1$ is fixed up and $x0$ is clamped before the calculations begin. At this point $x1$ and $x0$ are valid equivalents
|
|
to the respective halves as if mp\_rshd and mp\_mod\_2d had been used.
|
|
|
|
By inlining the copy and shift operations the cutoff point for Karatsuba multiplication can be lowered. On the Athlon the cutoff point
|
|
is exactly at the point where Comba squaring can no longer be used (\textit{128 digits}). On slower processors such as the Intel P4
|
|
it is actually below the Comba limit (\textit{at 110 digits}).
|
|
|
|
This routine uses the same error trap coding style as mp\_karatsuba\_sqr. As the temporary variables are initialized errors are redirected to
|
|
the error trap higher up. If the algorithm completes without error the error code is set to \textbf{MP\_OKAY} and mp\_clears are executed normally.
|
|
|
|
\textit{Last paragraph sucks. re-write! -- Tom}
|
|
|
|
\subsection{Toom-Cook Squaring}
|
|
The Toom-Cook squaring algorithm mp\_toom\_sqr is heavily based on the algorithm mp\_toom\_mul with the exception that squarings are used
|
|
instead of multiplication to find the five relations.. The reader is encouraged to read the description of the latter algorithm and try to
|
|
derive their own Toom-Cook squaring algorithm.
|
|
|
|
\subsection{High Level Squaring}
|
|
\newpage\begin{figure}[!here]
|
|
\begin{small}
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_sqr}. \\
|
|
\textbf{Input}. mp\_int $a$ \\
|
|
\textbf{Output}. $b \leftarrow a^2$ \\
|
|
\hline \\
|
|
1. If $a.used \ge TOOM\_SQR\_CUTOFF$ then \\
|
|
\hspace{3mm}1.1 $b \leftarrow a^2$ using algorithm mp\_toom\_sqr \\
|
|
2. else if $a.used \ge KARATSUBA\_SQR\_CUTOFF$ then \\
|
|
\hspace{3mm}2.1 $b \leftarrow a^2$ using algorithm mp\_karatsuba\_sqr \\
|
|
3. else \\
|
|
\hspace{3mm}3.1 $digs \leftarrow a.used + b.used + 1$ \\
|
|
\hspace{3mm}3.2 If $digs < MP\_ARRAY$ and $a.used \le \delta$ then \\
|
|
\hspace{6mm}3.2.1 $b \leftarrow a^2$ using algorithm fast\_s\_mp\_sqr. \\
|
|
\hspace{3mm}3.3 else \\
|
|
\hspace{6mm}3.3.1 $b \leftarrow a^2$ using algorithm s\_mp\_sqr. \\
|
|
4. $b.sign \leftarrow MP\_ZPOS$ \\
|
|
5. Return the result of the unsigned squaring performed. \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{small}
|
|
\caption{Algorithm mp\_sqr}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_sqr.}
|
|
This algorithm computes the square of the input using one of four different algorithms. If the input is very large and has at least
|
|
\textbf{TOOM\_SQR\_CUTOFF} or \textbf{KARATSUBA\_SQR\_CUTOFF} digits then either the Toom-Cook or the Karatsuba Squaring algorithm is used. If
|
|
neither of the polynomial basis algorithms should be used then either the Comba or baseline algorithm is used.
|
|
|
|
EXAM,bn_mp_sqr.c
|
|
|
|
\section*{Exercises}
|
|
\begin{tabular}{cl}
|
|
$\left [ 3 \right ] $ & Devise an efficient algorithm for selection of the radix point to handle inputs \\
|
|
& that have different number of digits in Karatsuba multiplication. \\
|
|
& \\
|
|
$\left [ 3 \right ] $ & In ~SQUARE~ the fact that every column of a squaring is made up \\
|
|
& of double products and at most one square is stated. Prove this statement. \\
|
|
& \\
|
|
$\left [ 2 \right ] $ & In the Comba squaring algorithm half of the $\hat X$ variables are not used. \\
|
|
& Revise algorithm fast\_s\_mp\_sqr to shrink the $\hat X$ array. \\
|
|
& \\
|
|
$\left [ 3 \right ] $ & Prove the equation for Karatsuba squaring. \\
|
|
& \\
|
|
$\left [ 1 \right ] $ & Prove that Karatsuba squaring requires $O \left (n^{lg(3)} \right )$ time. \\
|
|
& \\
|
|
$\left [ 2 \right ] $ & Determine the minimal ratio between addition and multiplication clock cycles \\
|
|
& required for equation $6.7$ to be true. \\
|
|
& \\
|
|
\end{tabular}
|
|
|
|
\chapter{Modular Reduction}
|
|
MARK,REDUCTION
|
|
\section{Basics of Modular Reduction}
|
|
\index{modular residue}
|
|
Modular reduction is an operation that arises quite often within public key cryptography algorithms and various number theoretic algorithms,
|
|
such as factoring. Modular reduction algorithms are the third class of algorithms of the ``multipliers'' set. A number $a$ is said to be reduced
|
|
modulo another number $b$ by finding the remainder of the division $a/b$.
|
|
|
|
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.
|
|
|
|
\index{modulus}
|
|
Modular reductions are normally used to form finite groups such as fields and rings. For example, in the RSA public key algorithm \cite{RSAPAPER}
|
|
two private primes $p$ and $q$ are chosen which when multiplied $n = pq$ forms a composite modulus. When operations such as multiplication and
|
|
squaring are performed on units of the ring $\Z_n$ a finite multiplicative sub-group is formed.
|
|
|
|
Modular reductions have a variety of other useful properties. For example, a number $x$ is a square if and only if it is a quadratic
|
|
residue modulo a prime. With a finite set of primes $B = \left < p_0, p_1, \ldots, p_n \right >$ a quick test for whether $x$ is square or not can
|
|
be performed\footnote{Provided none of the primes from $B$ divide $x$.}. Consider the figure~\ref{fig:QR} with the candiate $x = 955621$ a simple
|
|
set of modular reductions modulo $3, 5, \ldots, 11$ may detect whether $x$ is a square or not. In this case $955621 \equiv 7 \mbox{ (mod }11\mbox{)}$
|
|
and since $7$ is not a quadratic residue modulo $11$ the number $955621$ is not a square.
|
|
|
|
\begin{figure}
|
|
\begin{center}
|
|
\begin{tabular}{|c|l|}
|
|
\hline \textbf{Prime} & \textbf{Quadratic Residues} \\
|
|
\hline $3$ & $1$ \\
|
|
\hline $5$ & $1, 4$ \\
|
|
\hline $7$ & $1, 2, 4$ \\
|
|
\hline $11$ & $1, 3, 4, 5, 9$ \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\caption{Quadratic Residues for primes less than $13$}
|
|
\label{fig:QR}
|
|
\end{figure}
|
|
|
|
The most common usage for performance driven modular reductions is in modular exponentiation algorithms. That is to compute
|
|
$d = a^b \mbox{ (mod }c\mbox{)}$ as fast as possible. As will be discussed in the subsequent chapter there exists fast algorithms for computing
|
|
modular exponentiations without having to perform (\textit{in this example}) $b - 1$ 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.
|
|
|
|
\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 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 vastly popularlize the ``3d-shooter'' genre of games in the mid 1990s when floating point units were fairly slow. 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. For example,
|
|
with $q = 4$ to multiply the integers $9$ and $5$ they must be converted to fixed point first by multiplying by $2^q$. Let $a = 9(2^q)$
|
|
represent the fixed point representation of $9$ and $b = 5(2^q)$ represent the fixed point representation of $5$. The product $ab$ is equal to
|
|
$45(2^{2q})$ which when normalized produces $45(2^q)$.
|
|
|
|
Using fixed point arithmetic division can be easily achieved by multiplying by the reciprocal. If $2^q$ is equivalent to one than $2^q/b$ is
|
|
equivalent to $1/b$ using real arithmetic. Using this fact dividing an integer $a$ by another integer $b$ can be achieved with the following
|
|
expression.
|
|
|
|
\begin{equation}
|
|
\lfloor (a \cdot (\lfloor 2^q / b \rfloor))/2^q \rfloor
|
|
\end{equation}
|
|
|
|
The precision of the division is proportional to the value of $q$. If the divisor $b$ is used frequently as is the case with
|
|
modular exponentiation pre-computing $2^q/b$ will allow a division to be performed with a multiplication and a right shift. Both operations
|
|
are considerably faster than division on most processors.
|
|
|
|
Consider dividing $19$ by $5$. The correct result is $\lfloor 19/5 \rfloor = 3$. With $q = 3$ the reciprocal is $\lfloor 2^q/5 \rfloor = 1$ which
|
|
leads to a product of $19$ which when divided by $2^q$ produces $2$. However, with $q = 4$ the reciprocal is $\lfloor 2^q/5 \rfloor = 3$ and
|
|
the result of the emulated division is $\lfloor 3 \cdot 19 / 2^q \rfloor = 3$ which is correct.
|
|
|
|
Plugging this form of divison into the original equation the following modular residue equation arises.
|
|
|
|
\begin{equation}
|
|
c = a - b \cdot \lfloor (a \cdot (\lfloor 2^q / b \rfloor))/2^q \rfloor
|
|
\end{equation}
|
|
|
|
Using the notation from \cite{BARRETT} the value of $\lfloor 2^q / b \rfloor$ will be represented by the $\mu$ symbol. Using the $\mu$
|
|
variable also helps re-inforce the idea that it is meant to be computed once and re-used.
|
|
|
|
\begin{equation}
|
|
c = a - b \cdot \lfloor (a \cdot \mu)/2^q \rfloor
|
|
\end{equation}
|
|
|
|
Provided that $2^q > b^2$ this algorithm will produce a quotient that is either exactly correct or off by a value of one. Let $n$ represent
|
|
the number of digits in $b$. This algorithm requires approximately $2n^2$ single precision multiplications to produce the quotient and
|
|
another $n^2$ single precision multiplications to find the residue. In total $3n^2$ single precision multiplications are required to
|
|
reduce the number.
|
|
|
|
For example, if $b = 1179677$ and $q = 41$ ($2^q > b^2$), then the reciprocal $\mu$ is equal to $\lfloor 2^q / b \rfloor = 1864089$. Consider reducing
|
|
$a = 180388626447$ modulo $b$ using the above reduction equation. 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 might as well be used in its place. The key to optimizing the reduction is to reduce the precision of
|
|
the initial multiplication that finds the quotient.
|
|
|
|
Let $a$ represent the number of which the residue is sought. Let $b$ represent the modulus used to find the residue. Let $m$ represent
|
|
the number of digits in $b$. For the purposes of this discussion we will assume that the number of digits in $a$ is $2m$. Dividing $a$ by
|
|
$b$ is the same as dividing a $2m$ digit integer by a $m$ digit integer. Digits below the $m - 1$'th digit of $a$ will contribute at most a value
|
|
of $1$ to the quotient because $\beta^k < b$ for any $0 \le k \le m - 1$.
|
|
|
|
Since those digits do not contribute much to the quotient the observation is that they might as well be zero. However, if the digits
|
|
``might as well be zero'' they might as well not be there in the first place. Let $q_0 = \lfloor a/\beta^{m-1} \rfloor$ represent the input
|
|
with the zeroes trimmed. Now the modular reduction is trimmed to the almost equivalent equation
|
|
|
|
\begin{equation}
|
|
c = a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor
|
|
\end{equation}
|
|
|
|
Note that the original divisor $2^q$ has been replaced with $\beta^{m+1}$. 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 implying that $0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$. By first subtracting $b$ times the quotient and then
|
|
conditionally subtracting $b$ once or twice the residue is found.
|
|
|
|
The quotient is now found using $(m + 1)(m) = m^2 + m$ single precision multiplications and the residue with an additional $m^2$ single
|
|
precision multiplications. In total $2m^2 + m$ single precision multiplications are required which is considerably faster than the original
|
|
attempt.
|
|
|
|
For example, let $\beta = 10$ represent the radix of the digits. Let $b = 9999$ represent the modulus which implies $m = 4$. Let $a = 99929878$
|
|
represent the value of which the residue is desired. In this case $q = 8$ 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 $\mu = \lfloor \beta^{2m}/b \rfloor$ $(0 \le a < b^2, b > 1)$ \\
|
|
\textbf{Output}. $c \leftarrow a \mbox{ (mod }b\mbox{)}$ \\
|
|
\hline \\
|
|
Let $m$ represent the number of digits in $b$. \\
|
|
1. Make a copy of $a$ and store it in $q$. (\textit{mp\_init\_copy}) \\
|
|
2. $q \leftarrow \lfloor q / \beta^{m - 1} \rfloor$ (\textit{mp\_rshd}) \\
|
|
\\
|
|
Produce the quotient. \\
|
|
3. $q \leftarrow q \cdot \mu$ (\textit{note: only produce digits at or above $m-1$}) \\
|
|
4. $q \leftarrow \lfloor q / \beta^{m + 1} \rfloor$ \\
|
|
\\
|
|
Subtract the multiple of modulus from the input. \\
|
|
5. $c \leftarrow a \mbox{ (mod }\beta^{m+1}\mbox{)}$ (\textit{mp\_mod\_2d}) \\
|
|
6. $q \leftarrow q \cdot b \mbox{ (mod }\beta^{m+1}\mbox{)}$ (\textit{s\_mp\_mul\_digs}) \\
|
|
7. $c \leftarrow c - q$ (\textit{mp\_sub}) \\
|
|
\\
|
|
Add $\beta^{m+1}$ if a carry occured. \\
|
|
8. If $c < 0$ then (\textit{mp\_cmp\_d}) \\
|
|
\hspace{3mm}8.1 $q \leftarrow 1$ (\textit{mp\_set}) \\
|
|
\hspace{3mm}8.2 $q \leftarrow q \cdot \beta^{m+1}$ (\textit{mp\_lshd}) \\
|
|
\hspace{3mm}8.3 $c \leftarrow c + q$ \\
|
|
\\
|
|
Now subtract the modulus if the residue is too large (e.g. quotient too small). \\
|
|
9. While $c \ge b$ do (\textit{mp\_cmp}) \\
|
|
\hspace{3mm}9.1 $c \leftarrow c - b$ \\
|
|
10. Clear $q$. \\
|
|
11. Return(\textit{MP\_OKAY}) \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{small}
|
|
\caption{Algorithm mp\_reduce}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_reduce.}
|
|
This algorithm will reduce the input $a$ modulo $b$ in place using the Barrett algorithm. It is loosely based on algorithm 14.42 of 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. Technically the algorithm will still work if $a \ge b^2$ but it will take much longer to finish. The
|
|
value of $\mu$ is passed as an argument to this algorithm and is assumed to be setup before the algorithm is used.
|
|
|
|
Recall that the multiplication for the quotient on step 3 must only produce digits at or above the $m-1$'th position. An algorithm called
|
|
$s\_mp\_mul\_high\_digs$ which has not been presented is used to accomplish this task. This optimal algorithm can only be used if the number
|
|
of digits in $b$ is very much smaller than $\beta$.
|
|
|
|
After the multiple of the modulus has been subtracted from $a$ the residue must be fixed up in case its negative. While it is known that
|
|
$a \ge b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ only the lower $m+1$ digits are being used to compute the residue. In this case
|
|
the invariant $\beta^{m+1}$ must be added to the residue to make it positive again.
|
|
|
|
The while loop at step 9 will subtract $b$ until the residue is less than $b$. If the algorithm is performed correctly this step is only
|
|
performed upto two times. However, if $a \ge b^2$ than it will iterate substantially more times than it should.
|
|
|
|
EXAM,bn_mp_reduce.c
|
|
|
|
The first multiplication that determines the quotient can be performed by only producing the digits from $m - 1$ and up. This essentially halves
|
|
the number of single precision multiplications required. However, the optimization is only safe if $\beta$ is much larger than the number of digits
|
|
in the modulus. In the source code this is evaluated on lines @36,if@ to @44,}@ where algorithm s\_mp\_mul\_high\_digs is used when it is
|
|
safe to do so.
|
|
|
|
\subsection{The Barrett Setup Algorithm}
|
|
In order to use algorithm mp\_reduce the value of $\mu$ must be calculated in advance. Ideally this value should be computed once and stored for
|
|
future use so that the Barrett algorithm can be used without delay.
|
|
|
|
\begin{figure}[!here]
|
|
\begin{small}
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_reduce\_setup}. \\
|
|
\textbf{Input}. mp\_int $a$ ($a > 1$) \\
|
|
\textbf{Output}. $\mu \leftarrow \lfloor \beta^{2m}/a \rfloor$ \\
|
|
\hline \\
|
|
1. $\mu \leftarrow 2^{2 \cdot lg(\beta) \cdot m}$ (\textit{mp\_2expt}) \\
|
|
2. $\mu \leftarrow \lfloor \mu / b \rfloor$ (\textit{mp\_div}) \\
|
|
3. Return(\textit{MP\_OKAY}) \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{small}
|
|
\caption{Algorithm mp\_reduce\_setup}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_reduce\_setup.}
|
|
This algorithm computes the reciprocal $\mu$ required for Barrett reduction. First $\beta^{2m}$ is calculated as $2^{2 \cdot lg(\beta) \cdot m}$ which
|
|
is equivalent and much faster. The final value is computed by taking the integer quotient of $\lfloor \mu / b \rfloor$.
|
|
|
|
EXAM,bn_mp_reduce_setup.c
|
|
|
|
This simple routine calculates the reciprocal $\mu$ required by Barrett reduction. Note the extended usage of algorithm mp\_div where the variable
|
|
which would received the remainder is passed as NULL. As will be discussed in ~DIVISION~ the division routine allows both the quotient and the
|
|
remainder to be passed as NULL meaning to ignore the value.
|
|
|
|
\section{The Montgomery Reduction}
|
|
Montgomery reduction\footnote{Thanks to Niels Ferguson for his insightful explanation of the algorithm.} \cite{MONT} is by far the most interesting
|
|
form of reduction in common use. It computes a modular residue which is not actually equal to the residue of the input yet instead equal to a
|
|
residue times a constant. However, as perplexing as this may sound the algorithm is relatively simple and very efficient.
|
|
|
|
Throughout this entire section the variable $n$ will represent the modulus used to form the residue. As will be discussed shortly the value of
|
|
$n$ must be odd. The variable $x$ will represent the quantity of which the residue is sought. Similar to the Barrett algorithm the input
|
|
is restricted to $0 \le x < n^2$. To begin the description some simple number theory facts must be established.
|
|
|
|
\textbf{Fact 1.} Adding $n$ to $x$ does not change the residue since in effect it adds one to the quotient $\lfloor x / n \rfloor$. Another way
|
|
to explain this is that $n$ (\textit{or multiples of $n$}) is 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 final result $r = 99$ which is actually
|
|
$2^{-8} \cdot 5555 \mbox{ (mod }257\mbox{)}$ can reveal the residue $x \equiv 158$ by multiplying by $2^8$ modulo $n$.
|
|
|
|
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|}
|
|
\hline \textbf{Step number ($t$)} & \textbf{Result ($x$)} \\
|
|
\hline $1$ & $x + 2^{0}n = 5812$ \\
|
|
\hline $2$ & $5812$ \\
|
|
\hline $3$ & $x + 2^{2}n = 6840$ \\
|
|
\hline $4$ & $x + 2^{3}n = 8896$ \\
|
|
\hline $5$ & $8896$ \\
|
|
\hline $6$ & $8896$ \\
|
|
\hline $7$ & $x + 2^{6}n = 25344$ \\
|
|
\hline $8$ & $25344$ \\
|
|
\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 $x = 4093$ 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 $n > 1$ will ensure a modular inverse $\rho$ exists. $\rho$ must be calculated in
|
|
advance of this algorithm. Finally the variable $k$ is fixed and a pseudonym for $n.used$.
|
|
|
|
Step 2 decides whether a faster Montgomery algorithm can be used. It is based on the Comba technique meaning that there are limits on
|
|
the size of the input. This algorithm is discussed in ~COMBARED~.
|
|
|
|
Step 5 is the main reduction loop of the algorithm. The value of $\mu$ is calculated once per iteration in the outer loop. The inner loop
|
|
calculates $x + \mu n \beta^{ix}$ by multiplying $\mu n$ and adding the result to $x$ shifted by $ix$ digits. Both the addition and
|
|
multiplication are performed in the same loop to save time and memory. Step 5.4 will handle any additional carries that escape the inner loop.
|
|
|
|
Using a quick inspection this algorithm requires $n$ single precision multiplications for the outer loop and $n^2$ single precision multiplications
|
|
in the inner loop. In total $n^2 + n$ single precision multiplications which compares favourably to Barrett at $n^2 + 2n - 1$ single precision
|
|
multiplications.
|
|
|
|
EXAM,bn_mp_montgomery_reduce.c
|
|
|
|
This is the baseline implementation of the Montgomery reduction algorithm. Lines @30,digs@ to @35,}@ determine if the Comba based
|
|
routine can be used instead. Line @47,mu@ computes the value of $\mu$ for that particular iteration of the outer loop.
|
|
|
|
The multiplication $\mu n \beta^{ix}$ is performed in one step in the inner loop. The alias $tmpx$ refers to the $ix$'th digit of $x$ and
|
|
the alias $tmpn$ refers to the modulus $n$.
|
|
|
|
\subsection{Faster ``Comba'' Montgomery Reduction}
|
|
MARK,COMBARED
|
|
|
|
The Montgomery reduction requires fewer single precision multiplications than a Barrett reduction, however it is much slower due to the serial
|
|
nature of the inner loop. The Barrett reduction algorithm requires two slightly modified multipliers which can be implemented with the Comba
|
|
technique. The Montgomery reduction algorithm cannot directly use the Comba technique to any significant advantage since the inner loop calculates
|
|
a $k \times 1$ product $k$ times.
|
|
|
|
The biggest obstacle is that at the $ix$'th iteration of the outer loop the value of $x_{ix}$ is required to calculate $\mu$. This means the
|
|
carries from $0$ to $ix - 1$ must have been propagated upwards to form a valid $ix$'th digit. The solution as it turns out is very simple.
|
|
Perform a Comba like multiplier and inside the outer loop just after the inner loop fix up the $ix + 1$'th digit by forwarding the carry.
|
|
|
|
With this change in place the Montgomery reduction algorithm can be performed with a Comba style multiplication loop which substantially increases
|
|
the speed of the algorithm.
|
|
|
|
\newpage\begin{figure}[!here]
|
|
\begin{small}
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{fast\_mp\_montgomery\_reduce}. \\
|
|
\textbf{Input}. mp\_int $x$, mp\_int $n$ and a digit $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\
|
|
\hspace{11.5mm}($0 \le x < n^2, n > 1, (n, \beta) = 1, \beta^k > n$) \\
|
|
\textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\
|
|
\hline \\
|
|
Place an array of \textbf{MP\_WARRAY} mp\_word variables called $\hat W$ on the stack. \\
|
|
1. if $x.alloc < n.used + 1$ then grow $x$ to $n.used + 1$ digits. \\
|
|
Copy the digits of $x$ into the array $\hat W$ \\
|
|
2. For $ix$ from $0$ to $x.used - 1$ do \\
|
|
\hspace{3mm}2.1 $\hat W_{ix} \leftarrow x_{ix}$ \\
|
|
3. For $ix$ from $x.used$ to $2n.used - 1$ do \\
|
|
\hspace{3mm}3.1 $\hat W_{ix} \leftarrow 0$ \\
|
|
Elimiate the lower $k$ digits. \\
|
|
4. for $ix$ from $0$ to $n.used - 1$ do \\
|
|
\hspace{3mm}4.1 $\mu \leftarrow \hat W_{ix} \cdot \rho \mbox{ (mod }\beta\mbox{)}$ \\
|
|
\hspace{3mm}4.2 For $iy$ from $0$ to $n.used - 1$ do \\
|
|
\hspace{6mm}4.2.1 $\hat W_{iy + ix} \leftarrow \hat W_{iy + ix} + \mu \cdot n_{iy}$ \\
|
|
\hspace{3mm}4.3 $\hat W_{ix + 1} \leftarrow \hat W_{ix + 1} + \lfloor \hat W_{ix} / \beta \rfloor$ \\
|
|
Propagate carries upwards. \\
|
|
5. for $ix$ from $n.used$ to $2n.used + 1$ do \\
|
|
\hspace{3mm}5.1 $\hat W_{ix + 1} \leftarrow \hat W_{ix + 1} + \lfloor \hat W_{ix} / \beta \rfloor$ \\
|
|
Shift right and reduce modulo $\beta$ simultaneously. \\
|
|
6. for $ix$ from $0$ to $n.used + 1$ do \\
|
|
\hspace{3mm}6.1 $x_{ix} \leftarrow \hat W_{ix + n.used} \mbox{ (mod }\beta\mbox{)}$ \\
|
|
Zero excess digits and fixup $x$. \\
|
|
7. if $x.used > n.used + 1$ then do \\
|
|
\hspace{3mm}7.1 for $ix$ from $n.used + 1$ to $x.used - 1$ do \\
|
|
\hspace{6mm}7.1.1 $x_{ix} \leftarrow 0$ \\
|
|
8. $x.used \leftarrow n.used + 1$ \\
|
|
9. Clamp excessive digits of $x$. \\
|
|
10. If $x \ge n$ then \\
|
|
\hspace{3mm}10.1 $x \leftarrow x - n$ \\
|
|
11. Return(\textit{MP\_OKAY}). \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{small}
|
|
\caption{Algorithm fast\_mp\_montgomery\_reduce}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm fast\_mp\_montgomery\_reduce.}
|
|
This algorithm will compute the Montgomery reduction of $x$ modulo $n$ using the Comba technique. It is on most computer platforms significantly
|
|
faster than algorithm mp\_montgomery\_reduce and algorithm mp\_reduce (\textit{Barrett reduction}). The algorithm has the same restrictions
|
|
on the input as the baseline reduction algorithm. An additional two restrictions are imposed on this algorithm. The number of digits $k$ in the
|
|
the modulus $n$ must not violate $MP\_WARRAY > 2k +1$ and $n < \delta$. When $\beta = 2^{28}$ this algorithm can be used to reduce modulo
|
|
a modulus of at most $3,556$ bits in length.
|
|
|
|
As in the other Comba reduction algorithms there is a $\hat W$ array which stores the columns of the product. It is initially filled with the
|
|
contents of $x$ with the excess digits zeroed. The reduction loop is very similar the to the baseline loop at heart. The multiplication on step
|
|
4.1 can be single precision only since $ab \mbox{ (mod }\beta\mbox{)} \equiv (a \mbox{ mod }\beta)(b \mbox{ mod }\beta)$. Some multipliers such
|
|
as those on the ARM processors take a variable length time to complete depending on the number of bytes of result it must produce. By performing
|
|
a single precision multiplication instead half the amount of time is spent.
|
|
|
|
Also note that digit $\hat W_{ix}$ must have the carry from the $ix - 1$'th digit propagated upwards in order for this to work. That is what step
|
|
4.3 will do. In effect over the $n.used$ iterations of the outer loop the $n.used$'th lower columns all have the their carries propagated forwards. Note
|
|
how the upper bits of those same words are not reduced modulo $\beta$. This is because those values will be discarded shortly and there is no
|
|
point.
|
|
|
|
Step 5 will propgate the remainder of the carries upwards. On step 6 the columns are reduced modulo $\beta$ and shifted simultaneously as they are
|
|
stored in the destination $x$.
|
|
|
|
EXAM,bn_fast_mp_montgomery_reduce.c
|
|
|
|
The $\hat W$ array is first filled with digits of $x$ on line @49,for@ then the rest of the digits are zeroed on line @54,for@. Both loops share
|
|
the same alias variables to make the code easier to read.
|
|
|
|
The value of $\mu$ is calculated in an interesting fashion. First the value $\hat W_{ix}$ is reduced modulo $\beta$ and cast to a mp\_digit. This
|
|
forces the compiler to use a single precision multiplication and prevents any concerns about loss of precision. Line @101,>>@ fixes the carry
|
|
for the next iteration of the loop by propagating the carry from $\hat W_{ix}$ to $\hat W_{ix+1}$.
|
|
|
|
The for loop on line @113,for@ propagates the rest of the carries upwards through the columns. The for loop on line @126,for@ reduces the columns
|
|
modulo $\beta$ and shifts them $k$ places at the same time. The alias $\_ \hat W$ actually refers to the array $\hat W$ starting at the $n.used$'th
|
|
digit, that is $\_ \hat W_{t} = \hat W_{n.used + t}$.
|
|
|
|
\subsection{Montgomery Setup}
|
|
To calculate the variable $\rho$ a relatively simple algorithm will be required.
|
|
|
|
\begin{figure}[!here]
|
|
\begin{small}
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_montgomery\_setup}. \\
|
|
\textbf{Input}. mp\_int $n$ ($n > 1$ and $(n, 2) = 1$) \\
|
|
\textbf{Output}. $\rho \equiv -1/n_0 \mbox{ (mod }\beta\mbox{)}$ \\
|
|
\hline \\
|
|
1. $b \leftarrow n_0$ \\
|
|
2. If $b$ is even return(\textit{MP\_VAL}) \\
|
|
3. $x \leftarrow ((b + 2) \mbox{ AND } 4) << 1) + b$ \\
|
|
4. for $k$ from 0 to $3$ do \\
|
|
\hspace{3mm}4.1 $x \leftarrow x \cdot (2 - bx)$ \\
|
|
5. $\rho \leftarrow \beta - x \mbox{ (mod }\beta\mbox{)}$ \\
|
|
6. Return(\textit{MP\_OKAY}). \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{small}
|
|
\caption{Algorithm mp\_montgomery\_setup}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_montgomery\_setup.}
|
|
This algorithm will calculate the value of $\rho$ required within the Montgomery reduction algorithms. It uses a very interesting trick
|
|
to calculate $1/n_0$ when $\beta$ is a power of two.
|
|
|
|
EXAM,bn_mp_montgomery_setup.c
|
|
|
|
This source code computes the value of $\rho$ required to perform Montgomery reduction. It has been modified to avoid performing excess
|
|
multiplications when $\beta$ is not the default 28-bits.
|
|
|
|
\section{The Diminished Radix Algorithm}
|
|
The Diminished Radix method of modular reduction \cite{DRMET} is a fairly clever technique which 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 moduli is chosen.
|
|
|
|
Division in general is a very expensive operation to perform. The one exception is when the division is by a power of the radix of representation used.
|
|
Division by ten for example is simple for 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 $n$ requires zeroing the digits above the $p-1$'th digit of $x$.
|
|
|
|
Throughout the next section the term ``restricted modulus'' will refer to a modulus of the form $\beta^p - k$ where as the term ``unrestricted
|
|
modulus'' will refer to a modulus of the form $2^p - k$. The word ``restricted'' in this case refers to the fact that it is based on the
|
|
$2^p$ logic except $p$ must be a multiple of $lg(\beta)$.
|
|
|
|
\subsection{Choice of $k$}
|
|
Now that division and reduction (\textit{step 1 and 3 of figure~\ref{fig:DR}}) have been optimized to simple digit operations the multiplication by $k$
|
|
in step 2 is the most expensive operation. Fortunately the choice of $k$ is not terribly limited. For all intents and purposes it might
|
|
as well be a single digit. The smaller the value of $k$ is the faster the algorithm will be.
|
|
|
|
\subsection{Restricted Diminished Radix Reduction}
|
|
The restricted Diminished Radix algorithm can quickly reduce an input modulo a modulus of the form $n = \beta^p - k$. This algorithm can reduce
|
|
an input $x$ within the range $0 \le x < n^2$ using only a couple passes of the algorithm demonstrated in figure~\ref{fig:DR}. The implementation
|
|
of this algorithm has been optimized to avoid additional overhead associated with a division by $\beta^p$, the multiplication by $k$ or the addition
|
|
of $x$ and $q$. The resulting algorithm is very efficient and can lead to substantial improvements over Barrett and Montgomery reduction when modular
|
|
exponentiations are performed.
|
|
|
|
\newpage\begin{figure}[!here]
|
|
\begin{small}
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_dr\_reduce}. \\
|
|
\textbf{Input}. mp\_int $x$, $n$ and a mp\_digit $k = \beta - n_0$ \\
|
|
\hspace{11.5mm}($0 \le x < n^2$, $n > 1$, $0 < k < \beta$) \\
|
|
\textbf{Output}. $x \mbox{ mod } n$ \\
|
|
\hline \\
|
|
1. $m \leftarrow n.used$ \\
|
|
2. If $x.alloc < 2m$ then grow $x$ to $2m$ digits. \\
|
|
3. $\mu \leftarrow 0$ \\
|
|
4. for $i$ from $0$ to $m - 1$ do \\
|
|
\hspace{3mm}4.1 $\hat r \leftarrow k \cdot x_{m+i} + x_{i} + \mu$ \\
|
|
\hspace{3mm}4.2 $x_{i} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\
|
|
\hspace{3mm}4.3 $\mu \leftarrow \lfloor \hat r / \beta \rfloor$ \\
|
|
5. $x_{m} \leftarrow \mu$ \\
|
|
6. for $i$ from $m + 1$ to $x.used - 1$ do \\
|
|
\hspace{3mm}6.1 $x_{i} \leftarrow 0$ \\
|
|
7. Clamp excess digits of $x$. \\
|
|
8. If $x \ge n$ then \\
|
|
\hspace{3mm}8.1 $x \leftarrow x - n$ \\
|
|
\hspace{3mm}8.2 Goto step 3. \\
|
|
9. Return(\textit{MP\_OKAY}). \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{small}
|
|
\caption{Algorithm mp\_dr\_reduce}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_dr\_reduce.}
|
|
This algorithm will perform the Dimished Radix reduction of $x$ modulo $n$. It has similar restrictions to that of the Barrett reduction
|
|
with the addition that $n$ must be of the form $n = \beta^m - k$ where $0 < k <\beta$.
|
|
|
|
This algorithm essentially implements the pseudo-code in figure~\ref{fig:DR} except with a slight optimization. The division by $\beta^m$, multiplication by $k$
|
|
and addition of $x \mbox{ mod }\beta^m$ are all performed simultaneously inside the loop on step 4. The division by $\beta^m$ is emulated by accessing
|
|
the term at the $m+i$'th position which is subsequently multiplied by $k$ and added to the term at the $i$'th position. After the loop the $m$'th
|
|
digit is set to the carry and the upper digits are zeroed. Steps 5 and 6 emulate the reduction modulo $\beta^m$ that should have happend to
|
|
$x$ before the addition of the multiple of the upper half.
|
|
|
|
At step 8 if $x$ is still larger than $n$ another pass of the algorithm is required. First $n$ is subtracted from $x$ and then the algorithm resumes
|
|
at step 3.
|
|
|
|
EXAM,bn_mp_dr_reduce.c
|
|
|
|
The first step is to grow $x$ as required to $2m$ digits since the reduction is performed in place on $x$. The label on line @49,top:@ is where
|
|
the algorithm will resume if further reduction passes are required. In theory it could be placed at the top of the function however, the size of
|
|
the modulus and question of whether $x$ is large enough are invariant after the first pass meaning that it would be a waste of time.
|
|
|
|
The aliases $tmpx1$ and $tmpx2$ refer to the digits of $x$ where the latter is offset by $m$ digits. By reading digits from $x$ offset by $m$ digits
|
|
a division by $\beta^m$ can be simulated virtually for free. The loop on line @61,for@ performs the bulk of the work (\textit{corresponds to step 4 of algorithm 7.11})
|
|
in this algorithm.
|
|
|
|
By line @68,mu@ the pointer $tmpx1$ points to the $m$'th digit of $x$ which is where the final carry will be placed. Similarly by line @71,for@ the
|
|
same pointer will point to the $m+1$'th digit where the zeroes will be placed.
|
|
|
|
Since the algorithm is only valid if both $x$ and $n$ are greater than zero an unsigned comparison suffices to determine if another pass is required.
|
|
With the same logic at line @82,sub@ the value of $x$ is known to be greater than or equal to $n$ meaning that an unsigned subtraction can be used
|
|
as well. Since the destination of the subtraction is the larger of the inputs the call to algorithm s\_mp\_sub cannot fail and the return code
|
|
does not need to be checked.
|
|
|
|
\subsubsection{Setup}
|
|
To setup the restricted Diminished Radix algorithm the value $k = \beta - n_0$ is required. This algorithm is not really complicated but provided for
|
|
completeness.
|
|
|
|
\begin{figure}[!here]
|
|
\begin{small}
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_dr\_setup}. \\
|
|
\textbf{Input}. mp\_int $n$ \\
|
|
\textbf{Output}. $k = \beta - n_0$ \\
|
|
\hline \\
|
|
1. $k \leftarrow \beta - n_0$ \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{small}
|
|
\caption{Algorithm mp\_dr\_setup}
|
|
\end{figure}
|
|
|
|
EXAM,bn_mp_dr_setup.c
|
|
|
|
\subsubsection{Modulus Detection}
|
|
Another algorithm which will be useful is the ability to detect a restricted Diminished Radix modulus. An integer is said to be
|
|
of restricted Diminished Radix form if all of the digits are equal to $\beta - 1$ except the trailing digit which may be any value.
|
|
|
|
\begin{figure}[!here]
|
|
\begin{small}
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_dr\_is\_modulus}. \\
|
|
\textbf{Input}. mp\_int $n$ \\
|
|
\textbf{Output}. $1$ if $n$ is in D.R form, $0$ otherwise \\
|
|
\hline
|
|
1. If $n.used < 2$ then return($0$). \\
|
|
2. for $ix$ from $1$ to $n.used - 1$ do \\
|
|
\hspace{3mm}2.1 If $n_{ix} \ne \beta - 1$ return($0$). \\
|
|
3. Return($1$). \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{small}
|
|
\caption{Algorithm mp\_dr\_is\_modulus}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_dr\_is\_modulus.}
|
|
This algorithm determines if a value is in Diminished Radix form. Step 1 rejects obvious cases where fewer than two digits are
|
|
in the mp\_int. Step 2 tests all but the first digit to see if they are equal to $\beta - 1$. If the algorithm manages to get to
|
|
step 3 then $n$ must of Diminished Radix form.
|
|
|
|
EXAM,bn_mp_dr_is_modulus.c
|
|
|
|
\subsection{Unrestricted Diminished Radix Reduction}
|
|
The unrestricted Diminished Radix algorithm allows modular reductions to be performed when the modulus is of the form $2^p - k$. This algorithm
|
|
is a straightforward adaptation of algorithm~\ref{fig:DR}.
|
|
|
|
In general the restricted Diminished Radix reduction algorithm is much faster since it has considerably lower overhead. However, this new
|
|
algorithm is much faster than either Montgomery or Barrett reduction when the moduli are of the appropriate form.
|
|
|
|
\begin{figure}[!here]
|
|
\begin{small}
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_reduce\_2k}. \\
|
|
\textbf{Input}. mp\_int $a$ and $n$. mp\_digit $k$ \\
|
|
\hspace{11.5mm}($a \ge 0$, $n > 1$, $0 < k < \beta$, $n + k$ is a power of two) \\
|
|
\textbf{Output}. $a \mbox{ (mod }n\mbox{)}$ \\
|
|
\hline
|
|
1. $p \leftarrow \lceil lg(n) \rceil$ (\textit{mp\_count\_bits}) \\
|
|
2. While $a \ge n$ do \\
|
|
\hspace{3mm}2.1 $q \leftarrow \lfloor a / 2^p \rfloor$ (\textit{mp\_div\_2d}) \\
|
|
\hspace{3mm}2.2 $a \leftarrow a \mbox{ (mod }2^p\mbox{)}$ (\textit{mp\_mod\_2d}) \\
|
|
\hspace{3mm}2.3 $q \leftarrow q \cdot k$ (\textit{mp\_mul\_d}) \\
|
|
\hspace{3mm}2.4 $a \leftarrow a - q$ (\textit{s\_mp\_sub}) \\
|
|
\hspace{3mm}2.5 If $a \ge n$ then do \\
|
|
\hspace{6mm}2.5.1 $a \leftarrow a - n$ \\
|
|
3. Return(\textit{MP\_OKAY}). \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{small}
|
|
\caption{Algorithm mp\_reduce\_2k}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_reduce\_2k.}
|
|
This algorithm quickly reduces an input $a$ modulo an unrestricted Diminished Radix modulus $n$. Division by $2^p$ is emulated with a right
|
|
shift which makes the algorithm fairly inexpensive to use.
|
|
|
|
EXAM,bn_mp_reduce_2k.c
|
|
|
|
The algorithm mp\_count\_bits calculates the number of bits in an mp\_int which is used to find the initial value of $p$. The call to mp\_div\_2d
|
|
on line @31,mp_div_2d@ calculates both the quotient $q$ and the remainder $a$ required. By doing both in a single function call the code size
|
|
is kept fairly small. The multiplication by $k$ is only performed if $k > 1$. This allows reductions modulo $2^p - 1$ to be performed without
|
|
any multiplications.
|
|
|
|
The unsigned s\_mp\_add, mp\_cmp\_mag and s\_mp\_sub are used in place of their full sign counterparts since the inputs are only valid if they are
|
|
positive. By using the unsigned versions the overhead is kept to a minimum.
|
|
|
|
\subsubsection{Unrestricted Setup}
|
|
To setup this reduction algorithm the value of $k = 2^p - n$ is required.
|
|
|
|
\begin{figure}[!here]
|
|
\begin{small}
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{mp\_reduce\_2k\_setup}. \\
|
|
\textbf{Input}. mp\_int $n$ \\
|
|
\textbf{Output}. $k = 2^p - n$ \\
|
|
\hline
|
|
1. $p \leftarrow \lceil lg(n) \rceil$ (\textit{mp\_count\_bits}) \\
|
|
2. $x \leftarrow 2^p$ (\textit{mp\_2expt}) \\
|
|
3. $x \leftarrow x - n$ (\textit{mp\_sub}) \\
|
|
4. $k \leftarrow x_0$ \\
|
|
5. Return(\textit{MP\_OKAY}). \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{small}
|
|
\caption{Algorithm mp\_reduce\_2k\_setup}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_reduce\_2k\_setup.}
|
|
This algorithm computes the value of $k$ required for the algorithm mp\_reduce\_2k. By making a temporary variable $x$ equal to $2^p$ a subtraction
|
|
is sufficient to solve for $k$. Alternatively if $n$ has more than one digit the value of $k$ is simply $\beta - n_0$.
|
|
|
|
EXAM,bn_mp_reduce_2k_setup.c
|
|
|
|
\subsubsection{Unrestricted Detection}
|
|
An integer $n$ is a valid unrestricted Diminished Radix modulus if either of the following are true.
|
|
|
|
\begin{enumerate}
|
|
\item The number has only one digit.
|
|
\item The number has more than one digit and every bit from the $\beta$'th to the most significant is one.
|
|
\end{enumerate}
|
|
|
|
If either condition is true than there is a power of two namely $2^p$ 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. $p \leftarrow \rceil lg(n) \lceil$ (\textit{mp\_count\_bits}) \\
|
|
4. for $x$ from $lg(\beta)$ to $p$ do \\
|
|
\hspace{3mm}4.1 If the ($x \mbox{ mod }lg(\beta)$)'th bit of the $\lfloor x / lg(\beta) \rfloor$ of $n$ is zero then return($0$). \\
|
|
5. Return($1$). \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{small}
|
|
\caption{Algorithm mp\_reduce\_is\_2k}
|
|
\end{figure}
|
|
|
|
\textbf{Algorithm mp\_reduce\_is\_2k.}
|
|
This algorithm quickly determines if a modulus is of the form required for algorithm mp\_reduce\_2k to function properly.
|
|
|
|
EXAM,bn_mp_reduce_is_2k.c
|
|
|
|
|
|
|
|
\section{Algorithm Comparison}
|
|
So far three very different algorithms for modular reduction have been discussed. Each of the algorithms have their own strengths and weaknesses
|
|
that makes having such a selection very useful. The following table sumarizes the three algorithms along with comparisons of work factors. Since
|
|
all three algorithms have the restriction that $0 \le x < n^2$ and $n > 1$ those limitations are not included in the table.
|
|
|
|
\begin{center}
|
|
\begin{small}
|
|
\begin{tabular}{|c|c|c|c|c|c|}
|
|
\hline \textbf{Method} & \textbf{Work Required} & \textbf{Limitations} & \textbf{$m = 8$} & \textbf{$m = 32$} & \textbf{$m = 64$} \\
|
|
\hline Barrett & $m^2 + 2m - 1$ & None & $79$ & $1087$ & $4223$ \\
|
|
\hline Montgomery & $m^2 + m$ & $n$ must be odd & $72$ & $1056$ & $4160$ \\
|
|
\hline D.R. & $2m$ & $n = \beta^m - k$ & $16$ & $64$ & $128$ \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{small}
|
|
\end{center}
|
|
|
|
In theory Montgomery and Barrett reductions would require roughly the same amount of time to complete. However, in practice since Montgomery
|
|
reduction can be written as a single function with the Comba technique it is much faster. Barrett reduction suffers from the overhead of
|
|
calling the half precision multipliers, addition and division by $\beta$ algorithms.
|
|
|
|
For almost every cryptographic algorithm Montgomery reduction is the algorithm of choice. The one set of algorithms where Diminished Radix reduction truly
|
|
shines are based on the discrete logarithm problem such as Diffie-Hellman \cite{DH} and ElGamal \cite{ELGAMAL}. In these algorithms
|
|
primes of the form $\beta^m - k$ can be found and shared amongst users. These primes will allow the Diminished Radix algorithm to be used in
|
|
modular exponentiation to greatly speed up the operation.
|
|
|
|
|
|
|
|
\section*{Exercises}
|
|
\begin{tabular}{cl}
|
|
$\left [ 3 \right ]$ & Prove that the ``trick'' in algorithm mp\_montgomery\_setup actually \\
|
|
& calculates the correct value of $\rho$. \\
|
|
& \\
|
|
$\left [ 2 \right ]$ & Devise an algorithm to reduce modulo $n + k$ for small $k$ quickly. \\
|
|
& \\
|
|
$\left [ 4 \right ]$ & Prove that the pseudo-code algorithm ``Diminished Radix Reduction'' \\
|
|
& (\textit{figure~\ref{fig:DR}}) terminates. Also prove the probability that it will \\
|
|
& terminate within $1 \le k \le 10$ iterations. \\
|
|
& \\
|
|
\end{tabular}
|
|
|
|
|
|
\chapter{Exponentiation}
|
|
Exponentiation is the operation of raising one variable to the power of another, for example, $a^b$. A variant of exponentiation, computed
|
|
in a finite field or ring, is called modular exponentiation. This latter style of operation is typically used in public key
|
|
cryptosystems such as RSA and Diffie-Hellman. The ability to quickly compute modular exponentiations is of great benefit to any
|
|
such cryptosystem and many methods have been sought to speed it up.
|
|
|
|
\section{Exponentiation Basics}
|
|
A trivial algorithm would simply multiply $a$ against itself $b - 1$ times to compute the exponentiation desired. However, as $b$ grows in size
|
|
the number of multiplications becomes prohibitive. Imagine what would happen if $b$ $\approx$ $2^{1024}$ as is the case when computing an RSA signature
|
|
with a $1024$-bit key. Such a calculation could never be completed as it would take simply far too long.
|
|
|
|
Fortunately there is a very simple algorithm based on the laws of exponents. Recall that $lg_a(a^b) = b$ and that $lg_a(a^ba^c) = b + c$ which
|
|
are two trivial relationships between the base and the exponent. Let $b_i$ represent the $i$'th bit of $b$ starting from the least
|
|
significant bit. If $b$ is a $k$-bit integer than the following equation is true.
|
|
|
|
\begin{equation}
|
|
a^b = \prod_{i=0}^{k-1} a^{2^i \cdot b_i}
|
|
\end{equation}
|
|
|
|
By taking the base $a$ logarithm of both sides of the equation the following equation is the result.
|
|
|
|
\begin{equation}
|
|
b = \sum_{i=0}^{k-1}2^i \cdot b_i
|
|
\end{equation}
|
|
|
|
The term $a^{2^i}$ can be found from the $i - 1$'th term by squaring the term since $\left ( a^{2^i} \right )^2$ is equal to
|
|
$a^{2^{i+1}}$. This observation forms the basis of essentially all fast exponentiation algorithms. It requires $k$ squarings and on average
|
|
$k \over 2$ multiplications to compute the result. This is indeed quite an improvement over simply multiplying by $a$ a total of $b-1$ times.
|
|
|
|
While this current method is a considerable speed up there are further improvements to be made. For example, the $a^{2^i}$ term does not need to
|
|
be computed in an auxilary variable. Consider the following equivalent algorithm.
|
|
|
|
\begin{figure}[!here]
|
|
\begin{small}
|
|
\begin{center}
|
|
\begin{tabular}{l}
|
|
\hline Algorithm \textbf{Left to Right Exponentiation}. \\
|
|
\textbf{Input}. Integer $a$, $b$ and $k$ \\
|
|
\textbf{Output}. $c = a^b$ \\
|
|
\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.
|
|
|
|
EXAM,bn_mp_expt_d.c
|
|
|
|
-- Some note later.
|
|
|
|
\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 ~REDUCTION~.
|
|
|
|
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 ~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}).
|
|
|
|
EXAM,bn_mp_exptmod.c
|
|
|
|
In order to keep the algorithms in a known state the first step on line @29,if@ 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 @63,dr_@ determines if the modulus is of the restricted Diminished Radix
|
|
form. If it is not line @65,reduce@ 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,if@ 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.
|
|
|
|
FIGU,expt_state,Sliding Window State Diagram
|
|
|
|
By step 13 there are no more digits left in the exponent. However, there may be partial bits in the window left. If $mode = 2$ then
|
|
a Left-to-Right algorithm is used to process the remaining few bits.
|
|
|
|
EXAM,bn_s_mp_exptmod.c
|
|
|
|
Lines @26,if@ 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,if@ the value of $x$ is already known to be greater than $140$.
|
|
|
|
The conditional piece of code beginning on line @42,define@ 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 @49,for@ initializes the $M$ array while lines @59,mp_init@ and @62,mp_reduce@ 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.}
|
|
|
|
EXAM,bn_mp_2expt.c
|
|
|
|
\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}
|
|
MARK,DIVISION
|
|
|
|
Integer division aside from modular exponentiation is most intensive algorithm to compute.
|
|
|
|
|
|
\section{Single Digit Helpers}
|
|
\subsection{Single Digit Addition}
|
|
\subsection{Single Digit Subtraction}
|
|
\subsection{Single Digit Multiplication}
|
|
\subsection{Single Digit Division}
|
|
\subsection{Single Digit Modulo}
|
|
\subsection{Single Digit Root Extraction}
|
|
\section{Random Number Generation}
|
|
\section{Formatted Output}
|
|
\subsection{Getting The Output Size}
|
|
\subsection{Generating Radix-n Output}
|
|
\subsection{Reading Radix-n Input}
|
|
\section{Unformatted Output}
|
|
\subsection{Getting The Output Size}
|
|
\subsection{Generating Output}
|
|
\subsection{Reading Input}
|
|
|
|
\chapter{Number Theoretic Algorithms}
|
|
\section{Greatest Common Divisor}
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\section{Least Common Multiple}
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\section{Jacobi Symbol Computation}
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\section{Modular Inverse}
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MARK,MODINV
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\subsection{General Case}
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\subsection{Odd Moduli}
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\section{Primality Tests}
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\subsection{Trial Division}
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\subsection{The Fermat Test}
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\subsection{The Miller-Rabin Test}
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\subsection{Primality Test in a Bottle}
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\subsection{The Next Prime}
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\section{Root Extraction}
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\backmatter
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\appendix
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\begin{thebibliography}{ABCDEF}
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\bibitem[1]{TAOCPV2}
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Donald Knuth, \textit{The Art of Computer Programming}, Third Edition, Volume Two, Seminumerical Algorithms, Addison-Wesley, 1998
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\bibitem[2]{HAC}
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A. Menezes, P. van Oorschot, S. Vanstone, \textit{Handbook of Applied Cryptography}, CRC Press, 1996
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\bibitem[3]{ROSE}
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Michael Rosing, \textit{Implementing Elliptic Curve Cryptography}, Manning Publications, 1999
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\bibitem[4]{COMBA}
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Paul G. Comba, \textit{Exponentiation Cryptosystems on the IBM PC}. IBM Systems Journal 29(4): 526-538 (1990)
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\bibitem[5]{KARA}
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A. Karatsuba, Doklay Akad. Nauk SSSR 145 (1962), pp.293-294
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\bibitem[6]{KARAP}
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Andre Weimerskirch and Christof Paar, \textit{Generalizations of the Karatsuba Algorithm for Polynomial Multiplication}, Submitted to Design, Codes and Cryptography, March 2002
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|
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\bibitem[7]{BARRETT}
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Paul Barrett, \textit{Implementing the Rivest Shamir and Adleman Public Key Encryption Algorithm on a Standard Digital Signal Processor}, Advances in Cryptology, Crypto '86, Springer-Verlag.
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\bibitem[8]{MONT}
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P.L.Montgomery. \textit{Modular multiplication without trial division}. Mathematics of Computation, 44(170):519-521, April 1985.
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\bibitem[9]{DRMET}
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Chae Hoon Lim and Pil Joong Lee, \textit{Generating Efficient Primes for Discrete Log Cryptosystems}, POSTECH Information Research Laboratories
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\bibitem[10]{MMB}
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J. Daemen and R. Govaerts and J. Vandewalle, \textit{Block ciphers based on Modular Arithmetic}, State and {P}rogress in the {R}esearch of {C}ryptography, 1993, pp. 80-89
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\end{thebibliography}
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\input{tommath.ind}
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\chapter{Appendix}
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\subsection*{Appendix A -- Source Listing of tommath.h}
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The following is the source listing of the header file ``tommath.h'' for the LibTomMath project. It contains many of
|
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the definitions used throughout the code such as \textbf{mp\_int}, \textbf{MP\_PREC} and so on. The header is
|
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presented here for completeness.
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LIST,tommath.h
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\end{document} |