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\documentclass[b5paper]{book}
\usepackage{hyperref}
\usepackage{makeidx}
\usepackage{amssymb}
\usepackage{color}
\usepackage{alltt}
\usepackage{graphicx}
\usepackage{layout}
\def\union{\cup}
\def\intersect{\cap}
\def\getsrandom{\stackrel{\rm R}{\gets}}
\def\cross{\times}
\def\cat{\hspace{0.5em} \| \hspace{0.5em}}
\def\catn{$\|$}
\def\divides{\hspace{0.3em} | \hspace{0.3em}}
\def\nequiv{\not\equiv}
\def\approx{\raisebox{0.2ex}{\mbox{\small $\sim$}}}
\def\lcm{{\rm lcm}}
\def\gcd{{\rm gcd}}
\def\log{{\rm log}}
\def\ord{{\rm ord}}
\def\abs{{\mathit abs}}
\def\rep{{\mathit rep}}
\def\mod{{\mathit\ mod\ }}
\renewcommand{\pmod}[1]{\ ({\rm mod\ }{#1})}
\newcommand{\floor}[1]{\left\lfloor{#1}\right\rfloor}
\newcommand{\ceil}[1]{\left\lceil{#1}\right\rceil}
\def\Or{{\rm\ or\ }}
\def\And{{\rm\ and\ }}
\def\iff{\hspace{1em}\Longleftrightarrow\hspace{1em}}
\def\implies{\Rightarrow}
\def\undefined{{\rm ``undefined"}}
\def\Proof{\vspace{1ex}\noindent {\bf Proof:}\hspace{1em}}
\let\oldphi\phi
\def\phi{\varphi}
\def\Pr{{\rm Pr}}
\newcommand{\str}[1]{{\mathbf{#1}}}
\def\F{{\mathbb F}}
\def\N{{\mathbb N}}
\def\Z{{\mathbb Z}}
\def\R{{\mathbb R}}
\def\C{{\mathbb C}}
\def\Q{{\mathbb Q}}
\definecolor{DGray}{gray}{0.5}
\newcommand{\emailaddr}[1]{\mbox{$<${#1}$>$}}
\def\twiddle{\raisebox{0.3ex}{\mbox{\tiny $\sim$}}}
\def\gap{\vspace{0.5ex}}
\makeindex
\begin{document}
\frontmatter
\pagestyle{empty}
\title{Multi--Precision Math}
\author{\mbox{
%\begin{small}
\begin{tabular}{c}
Tom St Denis \\
Algonquin College \\
\\
Mads Rasmussen \\
Open Communications Security \\
\\
Greg Rose \\
QUALCOMM Australia \\
\end{tabular}
%\end{small}
}
}
\maketitle
This text has been placed in the public domain. This text corresponds to the v0.37 release of the
LibTomMath project.
\begin{alltt}
Tom St Denis
111 Banning Rd
Ottawa, Ontario
K2L 1C3
Canada
Phone: 1-613-836-3160
Email: tomstdenis@iahu.ca
\end{alltt}
This text is formatted to the international B5 paper size of 176mm wide by 250mm tall using the \LaTeX{}
{\em book} macro package and the Perl {\em booker} package.
\tableofcontents
\listoffigures
\chapter*{Prefaces}
When I tell people about my LibTom projects and that I release them as public domain they are often puzzled.
They ask why I did it and especially why I continue to work on them for free. The best I can explain it is ``Because I can.''
Which seems odd and perhaps too terse for adult conversation. I often qualify it with ``I am able, I am willing.'' which
perhaps explains it better. I am the first to admit there is not anything that special with what I have done. Perhaps
others can see that too and then we would have a society to be proud of. My LibTom projects are what I am doing to give
back to society in the form of tools and knowledge that can help others in their endeavours.
I started writing this book because it was the most logical task to further my goal of open academia. The LibTomMath source
code itself was written to be easy to follow and learn from. There are times, however, where pure C source code does not
explain the algorithms properly. Hence this book. The book literally starts with the foundation of the library and works
itself outwards to the more complicated algorithms. The use of both pseudo--code and verbatim source code provides a duality
of ``theory'' and ``practice'' that the computer science students of the world shall appreciate. I never deviate too far
from relatively straightforward algebra and I hope that this book can be a valuable learning asset.
This book and indeed much of the LibTom projects would not exist in their current form if it was not for a plethora
of kind people donating their time, resources and kind words to help support my work. Writing a text of significant
length (along with the source code) is a tiresome and lengthy process. Currently the LibTom project is four years old,
comprises of literally thousands of users and over 100,000 lines of source code, TeX and other material. People like Mads and Greg
were there at the beginning to encourage me to work well. It is amazing how timely validation from others can boost morale to
continue the project. Definitely my parents were there for me by providing room and board during the many months of work in 2003.
To my many friends whom I have met through the years I thank you for the good times and the words of encouragement. I hope I
honour your kind gestures with this project.
Open Source. Open Academia. Open Minds.
\begin{flushright} Tom St Denis \end{flushright}
\newpage
I found the opportunity to work with Tom appealing for several reasons, not only could I broaden my own horizons, but also
contribute to educate others facing the problem of having to handle big number mathematical calculations.
This book is Tom's child and he has been caring and fostering the project ever since the beginning with a clear mind of
how he wanted the project to turn out. I have helped by proofreading the text and we have had several discussions about
the layout and language used.
I hold a masters degree in cryptography from the University of Southern Denmark and have always been interested in the
practical aspects of cryptography.
Having worked in the security consultancy business for several years in S\~{a}o Paulo, Brazil, I have been in touch with a
great deal of work in which multiple precision mathematics was needed. Understanding the possibilities for speeding up
multiple precision calculations is often very important since we deal with outdated machine architecture where modular
reductions, for example, become painfully slow.
This text is for people who stop and wonder when first examining algorithms such as RSA for the first time and asks
themselves, ``You tell me this is only secure for large numbers, fine; but how do you implement these numbers?''
\begin{flushright}
Mads Rasmussen
S\~{a}o Paulo - SP
Brazil
\end{flushright}
\newpage
It's all because I broke my leg. That just happened to be at about the same time that Tom asked for someone to review the section of the book about
Karatsuba multiplication. I was laid up, alone and immobile, and thought ``Why not?'' I vaguely knew what Karatsuba multiplication was, but not
really, so I thought I could help, learn, and stop myself from watching daytime cable TV, all at once.
At the time of writing this, I've still not met Tom or Mads in meatspace. I've been following Tom's progress since his first splash on the
sci.crypt Usenet news group. I watched him go from a clueless newbie, to the cryptographic equivalent of a reformed smoker, to a real
contributor to the field, over a period of about two years. I've been impressed with his obvious intelligence, and astounded by his productivity.
Of course, he's young enough to be my own child, so he doesn't have my problems with staying awake.
When I reviewed that single section of the book, in its very earliest form, I was very pleasantly surprised. So I decided to collaborate more fully,
and at least review all of it, and perhaps write some bits too. There's still a long way to go with it, and I have watched a number of close
friends go through the mill of publication, so I think that the way to go is longer than Tom thinks it is. Nevertheless, it's a good effort,
and I'm pleased to be involved with it.
\begin{flushright}
Greg Rose, Sydney, Australia, June 2003.
\end{flushright}
\mainmatter
\pagestyle{headings}
\chapter{Introduction}
\section{Multiple Precision Arithmetic}
\subsection{What is Multiple Precision Arithmetic?}
When we think of long-hand arithmetic such as addition or multiplication we rarely consider the fact that we instinctively
raise or lower the precision of the numbers we are dealing with. For example, in decimal we almost immediate can
reason that $7$ times $6$ is $42$. However, $42$ has two digits of precision as opposed to one digit we started with.
Further multiplications of say $3$ result in a larger precision result $126$. In these few examples we have multiple
precisions for the numbers we are working with. Despite the various levels of precision a single subset\footnote{With the occasional optimization.}
of algorithms can be designed to accomodate them.
By way of comparison a fixed or single precision operation would lose precision on various operations. For example, in
the decimal system with fixed precision $6 \cdot 7 = 2$.
Essentially at the heart of computer based multiple precision arithmetic are the same long-hand algorithms taught in
schools to manually add, subtract, multiply and divide.
\subsection{The Need for Multiple Precision Arithmetic}
The most prevalent need for multiple precision arithmetic, often referred to as ``bignum'' math, is within the implementation
of public-key cryptography algorithms. Algorithms such as RSA \cite{RSAREF} and Diffie-Hellman \cite{DHREF} require
integers of significant magnitude to resist known cryptanalytic attacks. For example, at the time of this writing a
typical RSA modulus would be at least greater than $10^{309}$. However, modern programming languages such as ISO C \cite{ISOC} and
Java \cite{JAVA} only provide instrinsic support for integers which are relatively small and single precision.
\begin{figure}[!here]
\begin{center}
\begin{tabular}{|r|c|}
\hline \textbf{Data Type} & \textbf{Range} \\
\hline char & $-128 \ldots 127$ \\
\hline short & $-32768 \ldots 32767$ \\
\hline long & $-2147483648 \ldots 2147483647$ \\
\hline long long & $-9223372036854775808 \ldots 9223372036854775807$ \\
\hline
\end{tabular}
\end{center}
\caption{Typical Data Types for the C Programming Language}
\label{fig:ISOC}
\end{figure}
The largest data type guaranteed to be provided by the ISO C programming
language\footnote{As per the ISO C standard. However, each compiler vendor is allowed to augment the precision as they
see fit.} can only represent values up to $10^{19}$ as shown in figure \ref{fig:ISOC}. On its own the C language is
insufficient to accomodate the magnitude required for the problem at hand. An RSA modulus of magnitude $10^{19}$ could be
trivially factored\footnote{A Pollard-Rho factoring would take only $2^{16}$ time.} on the average desktop computer,
rendering any protocol based on the algorithm insecure. Multiple precision algorithms solve this very problem by
extending the range of representable integers while using single precision data types.
Most advancements in fast multiple precision arithmetic stem from the need for faster and more efficient cryptographic
primitives. Faster modular reduction and exponentiation algorithms such as Barrett's algorithm, which have appeared in
various cryptographic journals, can render algorithms such as RSA and Diffie-Hellman more efficient. In fact, several
major companies such as RSA Security, Certicom and Entrust have built entire product lines on the implementation and
deployment of efficient algorithms.
However, cryptography is not the only field of study that can benefit from fast multiple precision integer routines.
Another auxiliary use of multiple precision integers is high precision floating point data types.
The basic IEEE \cite{IEEE} standard floating point type is made up of an integer mantissa $q$, an exponent $e$ and a sign bit $s$.
Numbers are given in the form $n = q \cdot b^e \cdot -1^s$ where $b = 2$ is the most common base for IEEE. Since IEEE
floating point is meant to be implemented in hardware the precision of the mantissa is often fairly small
(\textit{23, 48 and 64 bits}). The mantissa is merely an integer and a multiple precision integer could be used to create
a mantissa of much larger precision than hardware alone can efficiently support. This approach could be useful where
scientific applications must minimize the total output error over long calculations.
Yet another use for large integers is within arithmetic on polynomials of large characteristic (i.e. $GF(p)[x]$ for large $p$).
In fact the library discussed within this text has already been used to form a polynomial basis library\footnote{See \url{http://poly.libtomcrypt.org} for more details.}.
\subsection{Benefits of Multiple Precision Arithmetic}
\index{precision}
The benefit of multiple precision representations over single or fixed precision representations is that
no precision is lost while representing the result of an operation which requires excess precision. For example,
the product of two $n$-bit integers requires at least $2n$ bits of precision to be represented faithfully. A multiple
precision algorithm would augment the precision of the destination to accomodate the result while a single precision system
would truncate excess bits to maintain a fixed level of precision.
It is possible to implement algorithms which require large integers with fixed precision algorithms. For example, elliptic
curve cryptography (\textit{ECC}) is often implemented on smartcards by fixing the precision of the integers to the maximum
size the system will ever need. Such an approach can lead to vastly simpler algorithms which can accomodate the
integers required even if the host platform cannot natively accomodate them\footnote{For example, the average smartcard
processor has an 8 bit accumulator.}. However, as efficient as such an approach may be, the resulting source code is not
normally very flexible. It cannot, at runtime, accomodate inputs of higher magnitude than the designer anticipated.
Multiple precision algorithms have the most overhead of any style of arithmetic. For the the most part the
overhead can be kept to a minimum with careful planning, but overall, it is not well suited for most memory starved
platforms. However, multiple precision algorithms do offer the most flexibility in terms of the magnitude of the
inputs. That is, the same algorithms based on multiple precision integers can accomodate any reasonable size input
without the designer's explicit forethought. This leads to lower cost of ownership for the code as it only has to
be written and tested once.
\section{Purpose of This Text}
The purpose of this text is to instruct the reader regarding how to implement efficient multiple precision algorithms.
That is to not only explain a limited subset of the core theory behind the algorithms but also the various ``house keeping''
elements that are neglected by authors of other texts on the subject. Several well reknowned texts \cite{TAOCPV2,HAC}
give considerably detailed explanations of the theoretical aspects of algorithms and often very little information
regarding the practical implementation aspects.
In most cases how an algorithm is explained and how it is actually implemented are two very different concepts. For
example, the Handbook of Applied Cryptography (\textit{HAC}), algorithm 14.7 on page 594, gives a relatively simple
algorithm for performing multiple precision integer addition. However, the description lacks any discussion concerning
the fact that the two integer inputs may be of differing magnitudes. As a result the implementation is not as simple
as the text would lead people to believe. Similarly the division routine (\textit{algorithm 14.20, pp. 598}) does not
discuss how to handle sign or handle the dividend's decreasing magnitude in the main loop (\textit{step \#3}).
Both texts also do not discuss several key optimal algorithms required such as ``Comba'' and Karatsuba multipliers
and fast modular inversion, which we consider practical oversights. These optimal algorithms are vital to achieve
any form of useful performance in non-trivial applications.
To solve this problem the focus of this text is on the practical aspects of implementing a multiple precision integer
package. As a case study the ``LibTomMath''\footnote{Available at \url{http://math.libtomcrypt.org}} package is used
to demonstrate algorithms with real implementations\footnote{In the ISO C programming language.} that have been field
tested and work very well. The LibTomMath library is freely available on the Internet for all uses and this text
discusses a very large portion of the inner workings of the library.
The algorithms that are presented will always include at least one ``pseudo-code'' description followed
by the actual C source code that implements the algorithm. The pseudo-code can be used to implement the same
algorithm in other programming languages as the reader sees fit.
This text shall also serve as a walkthrough of the creation of multiple precision algorithms from scratch. Showing
the reader how the algorithms fit together as well as where to start on various taskings.
\section{Discussion and Notation}
\subsection{Notation}
A multiple precision integer of $n$-digits shall be denoted as $x = (x_{n-1}, \ldots, x_1, x_0)_{ \beta }$ and represent
the integer $x \equiv \sum_{i=0}^{n-1} x_i\beta^i$. The elements of the array $x$ are said to be the radix $\beta$ digits
of the integer. For example, $x = (1,2,3)_{10}$ would represent the integer
$1\cdot 10^2 + 2\cdot10^1 + 3\cdot10^0 = 123$.
\index{mp\_int}
The term ``mp\_int'' shall refer to a composite structure which contains the digits of the integer it represents, as well
as auxilary data required to manipulate the data. These additional members are discussed further in section
\ref{sec:MPINT}. For the purposes of this text a ``multiple precision integer'' and an ``mp\_int'' are assumed to be
synonymous. When an algorithm is specified to accept an mp\_int variable it is assumed the various auxliary data members
are present as well. An expression of the type \textit{variablename.item} implies that it should evaluate to the
member named ``item'' of the variable. For example, a string of characters may have a member ``length'' which would
evaluate to the number of characters in the string. If the string $a$ equals ``hello'' then it follows that
$a.length = 5$.
For certain discussions more generic algorithms are presented to help the reader understand the final algorithm used
to solve a given problem. When an algorithm is described as accepting an integer input it is assumed the input is
a plain integer with no additional multiple-precision members. That is, algorithms that use integers as opposed to
mp\_ints as inputs do not concern themselves with the housekeeping operations required such as memory management. These
algorithms will be used to establish the relevant theory which will subsequently be used to describe a multiple
precision algorithm to solve the same problem.
\subsection{Precision Notation}
The variable $\beta$ represents the radix of a single digit of a multiple precision integer and
must be of the form $q^p$ for $q, p \in \Z^+$. A single precision variable must be able to represent integers in
the range $0 \le x < q \beta$ while a double precision variable must be able to represent integers in the range
$0 \le x < q \beta^2$. The extra radix-$q$ factor allows additions and subtractions to proceed without truncation of the
carry. Since all modern computers are binary, it is assumed that $q$ is two.
\index{mp\_digit} \index{mp\_word}
Within the source code that will be presented for each algorithm, the data type \textbf{mp\_digit} will represent
a single precision integer type, while, the data type \textbf{mp\_word} will represent a double precision integer type. In
several algorithms (notably the Comba routines) temporary results will be stored in arrays of double precision mp\_words.
For the purposes of this text $x_j$ will refer to the $j$'th digit of a single precision array and $\hat x_j$ will refer to
the $j$'th digit of a double precision array. Whenever an expression is to be assigned to a double precision
variable it is assumed that all single precision variables are promoted to double precision during the evaluation.
Expressions that are assigned to a single precision variable are truncated to fit within the precision of a single
precision data type.
For example, if $\beta = 10^2$ a single precision data type may represent a value in the
range $0 \le x < 10^3$, while a double precision data type may represent a value in the range $0 \le x < 10^5$. Let
$a = 23$ and $b = 49$ represent two single precision variables. The single precision product shall be written
as $c \leftarrow a \cdot b$ while the double precision product shall be written as $\hat c \leftarrow a \cdot b$.
In this particular case, $\hat c = 1127$ and $c = 127$. The most significant digit of the product would not fit
in a single precision data type and as a result $c \ne \hat c$.
\subsection{Algorithm Inputs and Outputs}
Within the algorithm descriptions all variables are assumed to be scalars of either single or double precision
as indicated. The only exception to this rule is when variables have been indicated to be of type mp\_int. This
distinction is important as scalars are often used as array indicies and various other counters.
\subsection{Mathematical Expressions}
The $\lfloor \mbox{ } \rfloor$ brackets imply an expression truncated to an integer not greater than the expression
itself. For example, $\lfloor 5.7 \rfloor = 5$. Similarly the $\lceil \mbox{ } \rceil$ brackets imply an expression
rounded to an integer not less than the expression itself. For example, $\lceil 5.1 \rceil = 6$. Typically when
the $/$ division symbol is used the intention is to perform an integer division with truncation. For example,
$5/2 = 2$ which will often be written as $\lfloor 5/2 \rfloor = 2$ for clarity. When an expression is written as a
fraction a real value division is implied, for example ${5 \over 2} = 2.5$.
The norm of a multiple precision integer, for example $\vert \vert x \vert \vert$, will be used to represent the number of digits in the representation
of the integer. For example, $\vert \vert 123 \vert \vert = 3$ and $\vert \vert 79452 \vert \vert = 5$.
\subsection{Work Effort}
\index{big-Oh}
To measure the efficiency of the specified algorithms, a modified big-Oh notation is used. In this system all
single precision operations are considered to have the same cost\footnote{Except where explicitly noted.}.
That is a single precision addition, multiplication and division are assumed to take the same time to
complete. While this is generally not true in practice, it will simplify the discussions considerably.
Some algorithms have slight advantages over others which is why some constants will not be removed in
the notation. For example, a normal baseline multiplication (section \ref{sec:basemult}) requires $O(n^2)$ work while a
baseline squaring (section \ref{sec:basesquare}) requires $O({{n^2 + n}\over 2})$ work. In standard big-Oh notation these
would both be said to be equivalent to $O(n^2)$. However,
in the context of the this text this is not the case as the magnitude of the inputs will typically be rather small. As a
result small constant factors in the work effort will make an observable difference in algorithm efficiency.
All of the algorithms presented in this text have a polynomial time work level. That is, of the form
$O(n^k)$ for $n, k \in \Z^{+}$. This will help make useful comparisons in terms of the speed of the algorithms and how
various optimizations will help pay off in the long run.
\section{Exercises}
Within the more advanced chapters a section will be set aside to give the reader some challenging exercises related to
the discussion at hand. These exercises are not designed to be prize winning problems, but instead to be thought
provoking. Wherever possible the problems are forward minded, stating problems that will be answered in subsequent
chapters. The reader is encouraged to finish the exercises as they appear to get a better understanding of the
subject material.
That being said, the problems are designed to affirm knowledge of a particular subject matter. Students in particular
are encouraged to verify they can answer the problems correctly before moving on.
Similar to the exercises of \cite[pp. ix]{TAOCPV2} these exercises are given a scoring system based on the difficulty of
the problem. However, unlike \cite{TAOCPV2} the problems do not get nearly as hard. The scoring of these
exercises ranges from one (the easiest) to five (the hardest). The following table sumarizes the
scoring system used.
\begin{figure}[here]
\begin{center}
\begin{small}
\begin{tabular}{|c|l|}
\hline $\left [ 1 \right ]$ & An easy problem that should only take the reader a manner of \\
& minutes to solve. Usually does not involve much computer time \\
& to solve. \\
\hline $\left [ 2 \right ]$ & An easy problem that involves a marginal amount of computer \\
& time usage. Usually requires a program to be written to \\
& solve the problem. \\
\hline $\left [ 3 \right ]$ & A moderately hard problem that requires a non-trivial amount \\
& of work. Usually involves trivial research and development of \\
& new theory from the perspective of a student. \\
\hline $\left [ 4 \right ]$ & A moderately hard problem that involves a non-trivial amount \\
& of work and research, the solution to which will demonstrate \\
& a higher mastery of the subject matter. \\
\hline $\left [ 5 \right ]$ & A hard problem that involves concepts that are difficult for a \\
& novice to solve. Solutions to these problems will demonstrate a \\
& complete mastery of the given subject. \\
\hline
\end{tabular}
\end{small}
\end{center}
\caption{Exercise Scoring System}
\end{figure}
Problems at the first level are meant to be simple questions that the reader can answer quickly without programming a solution or
devising new theory. These problems are quick tests to see if the material is understood. Problems at the second level
are also designed to be easy but will require a program or algorithm to be implemented to arrive at the answer. These
two levels are essentially entry level questions.
Problems at the third level are meant to be a bit more difficult than the first two levels. The answer is often
fairly obvious but arriving at an exacting solution requires some thought and skill. These problems will almost always
involve devising a new algorithm or implementing a variation of another algorithm previously presented. Readers who can
answer these questions will feel comfortable with the concepts behind the topic at hand.
Problems at the fourth level are meant to be similar to those of the level three questions except they will require
additional research to be completed. The reader will most likely not know the answer right away, nor will the text provide
the exact details of the answer until a subsequent chapter.
Problems at the fifth level are meant to be the hardest
problems relative to all the other problems in the chapter. People who can correctly answer fifth level problems have a
mastery of the subject matter at hand.
Often problems will be tied together. The purpose of this is to start a chain of thought that will be discussed in future chapters. The reader
is encouraged to answer the follow-up problems and try to draw the relevance of problems.
\section{Introduction to LibTomMath}
\subsection{What is LibTomMath?}
LibTomMath is a free and open source multiple precision integer library written entirely in portable ISO C. By portable it
is meant that the library does not contain any code that is computer platform dependent or otherwise problematic to use on
any given platform.
The library has been successfully tested under numerous operating systems including Unix\footnote{All of these
trademarks belong to their respective rightful owners.}, MacOS, Windows, Linux, PalmOS and on standalone hardware such
as the Gameboy Advance. The library is designed to contain enough functionality to be able to develop applications such
as public key cryptosystems and still maintain a relatively small footprint.
\subsection{Goals of LibTomMath}
Libraries which obtain the most efficiency are rarely written in a high level programming language such as C. However,
even though this library is written entirely in ISO C, considerable care has been taken to optimize the algorithm implementations within the
library. Specifically the code has been written to work well with the GNU C Compiler (\textit{GCC}) on both x86 and ARM
processors. Wherever possible, highly efficient algorithms, such as Karatsuba multiplication, sliding window
exponentiation and Montgomery reduction have been provided to make the library more efficient.
Even with the nearly optimal and specialized algorithms that have been included the Application Programing Interface
(\textit{API}) has been kept as simple as possible. Often generic place holder routines will make use of specialized
algorithms automatically without the developer's specific attention. One such example is the generic multiplication
algorithm \textbf{mp\_mul()} which will automatically use Toom--Cook, Karatsuba, Comba or baseline multiplication
based on the magnitude of the inputs and the configuration of the library.
Making LibTomMath as efficient as possible is not the only goal of the LibTomMath project. Ideally the library should
be source compatible with another popular library which makes it more attractive for developers to use. In this case the
MPI library was used as a API template for all the basic functions. MPI was chosen because it is another library that fits
in the same niche as LibTomMath. Even though LibTomMath uses MPI as the template for the function names and argument
passing conventions, it has been written from scratch by Tom St Denis.
The project is also meant to act as a learning tool for students, the logic being that no easy-to-follow ``bignum''
library exists which can be used to teach computer science students how to perform fast and reliable multiple precision
integer arithmetic. To this end the source code has been given quite a few comments and algorithm discussion points.
\section{Choice of LibTomMath}
LibTomMath was chosen as the case study of this text not only because the author of both projects is one and the same but
for more worthy reasons. Other libraries such as GMP \cite{GMP}, MPI \cite{MPI}, LIP \cite{LIP} and OpenSSL
\cite{OPENSSL} have multiple precision integer arithmetic routines but would not be ideal for this text for
reasons that will be explained in the following sub-sections.
\subsection{Code Base}
The LibTomMath code base is all portable ISO C source code. This means that there are no platform dependent conditional
segments of code littered throughout the source. This clean and uncluttered approach to the library means that a
developer can more readily discern the true intent of a given section of source code without trying to keep track of
what conditional code will be used.
The code base of LibTomMath is well organized. Each function is in its own separate source code file
which allows the reader to find a given function very quickly. On average there are $76$ lines of code per source
file which makes the source very easily to follow. By comparison MPI and LIP are single file projects making code tracing
very hard. GMP has many conditional code segments which also hinder tracing.
When compiled with GCC for the x86 processor and optimized for speed the entire library is approximately $100$KiB\footnote{The notation ``KiB'' means $2^{10}$ octets, similarly ``MiB'' means $2^{20}$ octets.}
which is fairly small compared to GMP (over $250$KiB). LibTomMath is slightly larger than MPI (which compiles to about
$50$KiB) but LibTomMath is also much faster and more complete than MPI.
\subsection{API Simplicity}
LibTomMath is designed after the MPI library and shares the API design. Quite often programs that use MPI will build
with LibTomMath without change. The function names correlate directly to the action they perform. Almost all of the
functions share the same parameter passing convention. The learning curve is fairly shallow with the API provided
which is an extremely valuable benefit for the student and developer alike.
The LIP library is an example of a library with an API that is awkward to work with. LIP uses function names that are often ``compressed'' to
illegible short hand. LibTomMath does not share this characteristic.
The GMP library also does not return error codes. Instead it uses a POSIX.1 \cite{POSIX1} signal system where errors
are signaled to the host application. This happens to be the fastest approach but definitely not the most versatile. In
effect a math error (i.e. invalid input, heap error, etc) can cause a program to stop functioning which is definitely
undersireable in many situations.
\subsection{Optimizations}
While LibTomMath is certainly not the fastest library (GMP often beats LibTomMath by a factor of two) it does
feature a set of optimal algorithms for tasks such as modular reduction, exponentiation, multiplication and squaring. GMP
and LIP also feature such optimizations while MPI only uses baseline algorithms with no optimizations. GMP lacks a few
of the additional modular reduction optimizations that LibTomMath features\footnote{At the time of this writing GMP
only had Barrett and Montgomery modular reduction algorithms.}.
LibTomMath is almost always an order of magnitude faster than the MPI library at computationally expensive tasks such as modular
exponentiation. In the grand scheme of ``bignum'' libraries LibTomMath is faster than the average library and usually
slower than the best libraries such as GMP and OpenSSL by only a small factor.
\subsection{Portability and Stability}
LibTomMath will build ``out of the box'' on any platform equipped with a modern version of the GNU C Compiler
(\textit{GCC}). This means that without changes the library will build without configuration or setting up any
variables. LIP and MPI will build ``out of the box'' as well but have numerous known bugs. Most notably the author of
MPI has recently stopped working on his library and LIP has long since been discontinued.
GMP requires a configuration script to run and will not build out of the box. GMP and LibTomMath are still in active
development and are very stable across a variety of platforms.
\subsection{Choice}
LibTomMath is a relatively compact, well documented, highly optimized and portable library which seems only natural for
the case study of this text. Various source files from the LibTomMath project will be included within the text. However,
the reader is encouraged to download their own copy of the library to actually be able to work with the library.
\chapter{Getting Started}
\section{Library Basics}
The trick to writing any useful library of source code is to build a solid foundation and work outwards from it. First,
a problem along with allowable solution parameters should be identified and analyzed. In this particular case the
inability to accomodate multiple precision integers is the problem. Futhermore, the solution must be written
as portable source code that is reasonably efficient across several different computer platforms.
After a foundation is formed the remainder of the library can be designed and implemented in a hierarchical fashion.
That is, to implement the lowest level dependencies first and work towards the most abstract functions last. For example,
before implementing a modular exponentiation algorithm one would implement a modular reduction algorithm.
By building outwards from a base foundation instead of using a parallel design methodology the resulting project is
highly modular. Being highly modular is a desirable property of any project as it often means the resulting product
has a small footprint and updates are easy to perform.
Usually when I start a project I will begin with the header files. I define the data types I think I will need and
prototype the initial functions that are not dependent on other functions (within the library). After I
implement these base functions I prototype more dependent functions and implement them. The process repeats until
I implement all of the functions I require. For example, in the case of LibTomMath I implemented functions such as
mp\_init() well before I implemented mp\_mul() and even further before I implemented mp\_exptmod(). As an example as to
why this design works note that the Karatsuba and Toom-Cook multipliers were written \textit{after} the
dependent function mp\_exptmod() was written. Adding the new multiplication algorithms did not require changes to the
mp\_exptmod() function itself and lowered the total cost of ownership (\textit{so to speak}) and of development
for new algorithms. This methodology allows new algorithms to be tested in a complete framework with relative ease.
\begin{center}
\begin{figure}[here]
\includegraphics{pics/design_process.ps}
\caption{Design Flow of the First Few Original LibTomMath Functions.}
\label{pic:design_process}
\end{figure}
\end{center}
Only after the majority of the functions were in place did I pursue a less hierarchical approach to auditing and optimizing
the source code. For example, one day I may audit the multipliers and the next day the polynomial basis functions.
It only makes sense to begin the text with the preliminary data types and support algorithms required as well.
This chapter discusses the core algorithms of the library which are the dependents for every other algorithm.
\section{What is a Multiple Precision Integer?}
Recall that most programming languages, in particular ISO C \cite{ISOC}, only have fixed precision data types that on their own cannot
be used to represent values larger than their precision will allow. The purpose of multiple precision algorithms is
to use fixed precision data types to create and manipulate multiple precision integers which may represent values
that are very large.
As a well known analogy, school children are taught how to form numbers larger than nine by prepending more radix ten digits. In the decimal system
the largest single digit value is $9$. However, by concatenating digits together larger numbers may be represented. Newly prepended digits
(\textit{to the left}) are said to be in a different power of ten column. That is, the number $123$ can be described as having a $1$ in the hundreds
column, $2$ in the tens column and $3$ in the ones column. Or more formally $123 = 1 \cdot 10^2 + 2 \cdot 10^1 + 3 \cdot 10^0$. Computer based
multiple precision arithmetic is essentially the same concept. Larger integers are represented by adjoining fixed
precision computer words with the exception that a different radix is used.
What most people probably do not think about explicitly are the various other attributes that describe a multiple precision
integer. For example, the integer $154_{10}$ has two immediately obvious properties. First, the integer is positive,
that is the sign of this particular integer is positive as opposed to negative. Second, the integer has three digits in
its representation. There is an additional property that the integer posesses that does not concern pencil-and-paper
arithmetic. The third property is how many digits placeholders are available to hold the integer.
The human analogy of this third property is ensuring there is enough space on the paper to write the integer. For example,
if one starts writing a large number too far to the right on a piece of paper they will have to erase it and move left.
Similarly, computer algorithms must maintain strict control over memory usage to ensure that the digits of an integer
will not exceed the allowed boundaries. These three properties make up what is known as a multiple precision
integer or mp\_int for short.
\subsection{The mp\_int Structure}
\label{sec:MPINT}
The mp\_int structure is the ISO C based manifestation of what represents a multiple precision integer. The ISO C standard does not provide for
any such data type but it does provide for making composite data types known as structures. The following is the structure definition
used within LibTomMath.
\index{mp\_int}
\begin{figure}[here]
\begin{center}
\begin{small}
%\begin{verbatim}
\begin{tabular}{|l|}
\hline
typedef struct \{ \\
\hspace{3mm}int used, alloc, sign;\\
\hspace{3mm}mp\_digit *dp;\\
\} \textbf{mp\_int}; \\
\hline
\end{tabular}
%\end{verbatim}
\end{small}
\caption{The mp\_int Structure}
\label{fig:mpint}
\end{center}
\end{figure}
The mp\_int structure (fig. \ref{fig:mpint}) can be broken down as follows.
\begin{enumerate}
\item The \textbf{used} parameter denotes how many digits of the array \textbf{dp} contain the digits used to represent
a given integer. The \textbf{used} count must be positive (or zero) and may not exceed the \textbf{alloc} count.
\item The \textbf{alloc} parameter denotes how
many digits are available in the array to use by functions before it has to increase in size. When the \textbf{used} count
of a result would exceed the \textbf{alloc} count all of the algorithms will automatically increase the size of the
array to accommodate the precision of the result.
\item The pointer \textbf{dp} points to a dynamically allocated array of digits that represent the given multiple
precision integer. It is padded with $(\textbf{alloc} - \textbf{used})$ zero digits. The array is maintained in a least
significant digit order. As a pencil and paper analogy the array is organized such that the right most digits are stored
first starting at the location indexed by zero\footnote{In C all arrays begin at zero.} in the array. For example,
if \textbf{dp} contains $\lbrace a, b, c, \ldots \rbrace$ where \textbf{dp}$_0 = a$, \textbf{dp}$_1 = b$, \textbf{dp}$_2 = c$, $\ldots$ then
it would represent the integer $a + b\beta + c\beta^2 + \ldots$
\index{MP\_ZPOS} \index{MP\_NEG}
\item The \textbf{sign} parameter denotes the sign as either zero/positive (\textbf{MP\_ZPOS}) or negative (\textbf{MP\_NEG}).
\end{enumerate}
\subsubsection{Valid mp\_int Structures}
Several rules are placed on the state of an mp\_int structure and are assumed to be followed for reasons of efficiency.
The only exceptions are when the structure is passed to initialization functions such as mp\_init() and mp\_init\_copy().
\begin{enumerate}
\item The value of \textbf{alloc} may not be less than one. That is \textbf{dp} always points to a previously allocated
array of digits.
\item The value of \textbf{used} may not exceed \textbf{alloc} and must be greater than or equal to zero.
\item The value of \textbf{used} implies the digit at index $(used - 1)$ of the \textbf{dp} array is non-zero. That is,
leading zero digits in the most significant positions must be trimmed.
\begin{enumerate}
\item Digits in the \textbf{dp} array at and above the \textbf{used} location must be zero.
\end{enumerate}
\item The value of \textbf{sign} must be \textbf{MP\_ZPOS} if \textbf{used} is zero;
this represents the mp\_int value of zero.
\end{enumerate}
\section{Argument Passing}
A convention of argument passing must be adopted early on in the development of any library. Making the function
prototypes consistent will help eliminate many headaches in the future as the library grows to significant complexity.
In LibTomMath the multiple precision integer functions accept parameters from left to right as pointers to mp\_int
structures. That means that the source (input) operands are placed on the left and the destination (output) on the right.
Consider the following examples.
\begin{verbatim}
mp_mul(&a, &b, &c); /* c = a * b */
mp_add(&a, &b, &a); /* a = a + b */
mp_sqr(&a, &b); /* b = a * a */
\end{verbatim}
The left to right order is a fairly natural way to implement the functions since it lets the developer read aloud the
functions and make sense of them. For example, the first function would read ``multiply a and b and store in c''.
Certain libraries (\textit{LIP by Lenstra for instance}) accept parameters the other way around, to mimic the order
of assignment expressions. That is, the destination (output) is on the left and arguments (inputs) are on the right. In
truth, it is entirely a matter of preference. In the case of LibTomMath the convention from the MPI library has been
adopted.
Another very useful design consideration, provided for in LibTomMath, is whether to allow argument sources to also be a
destination. For example, the second example (\textit{mp\_add}) adds $a$ to $b$ and stores in $a$. This is an important
feature to implement since it allows the calling functions to cut down on the number of variables it must maintain.
However, to implement this feature specific care has to be given to ensure the destination is not modified before the
source is fully read.
\section{Return Values}
A well implemented application, no matter what its purpose, should trap as many runtime errors as possible and return them
to the caller. By catching runtime errors a library can be guaranteed to prevent undefined behaviour. However, the end
developer can still manage to cause a library to crash. For example, by passing an invalid pointer an application may
fault by dereferencing memory not owned by the application.
In the case of LibTomMath the only errors that are checked for are related to inappropriate inputs (division by zero for
instance) and memory allocation errors. It will not check that the mp\_int passed to any function is valid nor
will it check pointers for validity. Any function that can cause a runtime error will return an error code as an
\textbf{int} data type with one of the following values (fig \ref{fig:errcodes}).
\index{MP\_OKAY} \index{MP\_VAL} \index{MP\_MEM}
\begin{figure}[here]
\begin{center}
\begin{tabular}{|l|l|}
\hline \textbf{Value} & \textbf{Meaning} \\
\hline \textbf{MP\_OKAY} & The function was successful \\
\hline \textbf{MP\_VAL} & One of the input value(s) was invalid \\
\hline \textbf{MP\_MEM} & The function ran out of heap memory \\
\hline
\end{tabular}
\end{center}
\caption{LibTomMath Error Codes}
\label{fig:errcodes}
\end{figure}
When an error is detected within a function it should free any memory it allocated, often during the initialization of
temporary mp\_ints, and return as soon as possible. The goal is to leave the system in the same state it was when the
function was called. Error checking with this style of API is fairly simple.
\begin{verbatim}
int err;
if ((err = mp_add(&a, &b, &c)) != MP_OKAY) {
printf("Error: %s\n", mp_error_to_string(err));
exit(EXIT_FAILURE);
}
\end{verbatim}
The GMP \cite{GMP} library uses C style \textit{signals} to flag errors which is of questionable use. Not all errors are fatal
and it was not deemed ideal by the author of LibTomMath to force developers to have signal handlers for such cases.
\section{Initialization and Clearing}
The logical starting point when actually writing multiple precision integer functions is the initialization and
clearing of the mp\_int structures. These two algorithms will be used by the majority of the higher level algorithms.
Given the basic mp\_int structure an initialization routine must first allocate memory to hold the digits of
the integer. Often it is optimal to allocate a sufficiently large pre-set number of digits even though
the initial integer will represent zero. If only a single digit were allocated quite a few subsequent re-allocations
would occur when operations are performed on the integers. There is a tradeoff between how many default digits to allocate
and how many re-allocations are tolerable. Obviously allocating an excessive amount of digits initially will waste
memory and become unmanageable.
If the memory for the digits has been successfully allocated then the rest of the members of the structure must
be initialized. Since the initial state of an mp\_int is to represent the zero integer, the allocated digits must be set
to zero. The \textbf{used} count set to zero and \textbf{sign} set to \textbf{MP\_ZPOS}.
\subsection{Initializing an mp\_int}
An mp\_int is said to be initialized if it is set to a valid, preferably default, state such that all of the members of the
structure are set to valid values. The mp\_init algorithm will perform such an action.
\index{mp\_init}
\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_init}. \\
\textbf{Input}. An mp\_int $a$ \\
\textbf{Output}. Allocate memory and initialize $a$ to a known valid mp\_int state. \\
\hline \\
1. Allocate memory for \textbf{MP\_PREC} digits. \\
2. If the allocation failed return(\textit{MP\_MEM}) \\
3. for $n$ from $0$ to $MP\_PREC - 1$ do \\
\hspace{3mm}3.1 $a_n \leftarrow 0$\\
4. $a.sign \leftarrow MP\_ZPOS$\\
5. $a.used \leftarrow 0$\\
6. $a.alloc \leftarrow MP\_PREC$\\
7. Return(\textit{MP\_OKAY})\\
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_init}
\end{figure}
\textbf{Algorithm mp\_init.}
The purpose of this function is to initialize an mp\_int structure so that the rest of the library can properly
manipulte it. It is assumed that the input may not have had any of its members previously initialized which is certainly
a valid assumption if the input resides on the stack.
Before any of the members such as \textbf{sign}, \textbf{used} or \textbf{alloc} are initialized the memory for
the digits is allocated. If this fails the function returns before setting any of the other members. The \textbf{MP\_PREC}
name represents a constant\footnote{Defined in the ``tommath.h'' header file within LibTomMath.}
used to dictate the minimum precision of newly initialized mp\_int integers. Ideally, it is at least equal to the smallest
precision number you'll be working with.
Allocating a block of digits at first instead of a single digit has the benefit of lowering the number of usually slow
heap operations later functions will have to perform in the future. If \textbf{MP\_PREC} is set correctly the slack
memory and the number of heap operations will be trivial.
Once the allocation has been made the digits have to be set to zero as well as the \textbf{used}, \textbf{sign} and
\textbf{alloc} members initialized. This ensures that the mp\_int will always represent the default state of zero regardless
of the original condition of the input.
\textbf{Remark.}
This function introduces the idiosyncrasy that all iterative loops, commonly initiated with the ``for'' keyword, iterate incrementally
when the ``to'' keyword is placed between two expressions. For example, ``for $a$ from $b$ to $c$ do'' means that
a subsequent expression (or body of expressions) are to be evaluated upto $c - b$ times so long as $b \le c$. In each
iteration the variable $a$ is substituted for a new integer that lies inclusively between $b$ and $c$. If $b > c$ occured
the loop would not iterate. By contrast if the ``downto'' keyword were used in place of ``to'' the loop would iterate
decrementally.
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_init.c
\vspace{-3mm}
\begin{alltt}
016
017 /* init a new mp_int */
018 int mp_init (mp_int * a)
019 \{
020 int i;
021
022 /* allocate memory required and clear it */
023 a->dp = OPT_CAST(mp_digit) XMALLOC (sizeof (mp_digit) * MP_PREC);
024 if (a->dp == NULL) \{
025 return MP_MEM;
026 \}
027
028 /* set the digits to zero */
029 for (i = 0; i < MP_PREC; i++) \{
030 a->dp[i] = 0;
031 \}
032
033 /* set the used to zero, allocated digits to the default precision
034 * and sign to positive */
035 a->used = 0;
036 a->alloc = MP_PREC;
037 a->sign = MP_ZPOS;
038
039 return MP_OKAY;
040 \}
041 #endif
042
\end{alltt}
\end{small}
One immediate observation of this initializtion function is that it does not return a pointer to a mp\_int structure. It
is assumed that the caller has already allocated memory for the mp\_int structure, typically on the application stack. The
call to mp\_init() is used only to initialize the members of the structure to a known default state.
Here we see (line 23) the memory allocation is performed first. This allows us to exit cleanly and quickly
if there is an error. If the allocation fails the routine will return \textbf{MP\_MEM} to the caller to indicate there
was a memory error. The function XMALLOC is what actually allocates the memory. Technically XMALLOC is not a function
but a macro defined in ``tommath.h``. By default, XMALLOC will evaluate to malloc() which is the C library's built--in
memory allocation routine.
In order to assure the mp\_int is in a known state the digits must be set to zero. On most platforms this could have been
accomplished by using calloc() instead of malloc(). However, to correctly initialize a integer type to a given value in a
portable fashion you have to actually assign the value. The for loop (line 29) performs this required
operation.
After the memory has been successfully initialized the remainder of the members are initialized
(lines 33 through 34) to their respective default states. At this point the algorithm has succeeded and
a success code is returned to the calling function. If this function returns \textbf{MP\_OKAY} it is safe to assume the
mp\_int structure has been properly initialized and is safe to use with other functions within the library.
\subsection{Clearing an mp\_int}
When an mp\_int is no longer required by the application, the memory that has been allocated for its digits must be
returned to the application's memory pool with the mp\_clear algorithm.
\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_clear}. \\
\textbf{Input}. An mp\_int $a$ \\
\textbf{Output}. The memory for $a$ shall be deallocated. \\
\hline \\
1. If $a$ has been previously freed then return(\textit{MP\_OKAY}). \\
2. for $n$ from 0 to $a.used - 1$ do \\
\hspace{3mm}2.1 $a_n \leftarrow 0$ \\
3. Free the memory allocated for the digits of $a$. \\
4. $a.used \leftarrow 0$ \\
5. $a.alloc \leftarrow 0$ \\
6. $a.sign \leftarrow MP\_ZPOS$ \\
7. Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_clear}
\end{figure}
\textbf{Algorithm mp\_clear.}
This algorithm accomplishes two goals. First, it clears the digits and the other mp\_int members. This ensures that
if a developer accidentally re-uses a cleared structure it is less likely to cause problems. The second goal
is to free the allocated memory.
The logic behind the algorithm is extended by marking cleared mp\_int structures so that subsequent calls to this
algorithm will not try to free the memory multiple times. Cleared mp\_ints are detectable by having a pre-defined invalid
digit pointer \textbf{dp} setting.
Once an mp\_int has been cleared the mp\_int structure is no longer in a valid state for any other algorithm
with the exception of algorithms mp\_init, mp\_init\_copy, mp\_init\_size and mp\_clear.
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_clear.c
\vspace{-3mm}
\begin{alltt}
016
017 /* clear one (frees) */
018 void
019 mp_clear (mp_int * a)
020 \{
021 int i;
022
023 /* only do anything if a hasn't been freed previously */
024 if (a->dp != NULL) \{
025 /* first zero the digits */
026 for (i = 0; i < a->used; i++) \{
027 a->dp[i] = 0;
028 \}
029
030 /* free ram */
031 XFREE(a->dp);
032
033 /* reset members to make debugging easier */
034 a->dp = NULL;
035 a->alloc = a->used = 0;
036 a->sign = MP_ZPOS;
037 \}
038 \}
039 #endif
040
\end{alltt}
\end{small}
The algorithm only operates on the mp\_int if it hasn't been previously cleared. The if statement (line 24)
checks to see if the \textbf{dp} member is not \textbf{NULL}. If the mp\_int is a valid mp\_int then \textbf{dp} cannot be
\textbf{NULL} in which case the if statement will evaluate to true.
The digits of the mp\_int are cleared by the for loop (line 26) which assigns a zero to every digit. Similar to mp\_init()
the digits are assigned zero instead of using block memory operations (such as memset()) since this is more portable.
The digits are deallocated off the heap via the XFREE macro. Similar to XMALLOC the XFREE macro actually evaluates to
a standard C library function. In this case the free() function. Since free() only deallocates the memory the pointer
still has to be reset to \textbf{NULL} manually (line 34).
Now that the digits have been cleared and deallocated the other members are set to their final values (lines 35 and 36).
\section{Maintenance Algorithms}
The previous sections describes how to initialize and clear an mp\_int structure. To further support operations
that are to be performed on mp\_int structures (such as addition and multiplication) the dependent algorithms must be
able to augment the precision of an mp\_int and
initialize mp\_ints with differing initial conditions.
These algorithms complete the set of low level algorithms required to work with mp\_int structures in the higher level
algorithms such as addition, multiplication and modular exponentiation.
\subsection{Augmenting an mp\_int's Precision}
When storing a value in an mp\_int structure, a sufficient number of digits must be available to accomodate the entire
result of an operation without loss of precision. Quite often the size of the array given by the \textbf{alloc} member
is large enough to simply increase the \textbf{used} digit count. However, when the size of the array is too small it
must be re-sized appropriately to accomodate the result. The mp\_grow algorithm will provide this functionality.
\newpage\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_grow}. \\
\textbf{Input}. An mp\_int $a$ and an integer $b$. \\
\textbf{Output}. $a$ is expanded to accomodate $b$ digits. \\
\hline \\
1. if $a.alloc \ge b$ then return(\textit{MP\_OKAY}) \\
2. $u \leftarrow b\mbox{ (mod }MP\_PREC\mbox{)}$ \\
3. $v \leftarrow b + 2 \cdot MP\_PREC - u$ \\
4. Re-allocate the array of digits $a$ to size $v$ \\
5. If the allocation failed then return(\textit{MP\_MEM}). \\
6. for n from a.alloc to $v - 1$ do \\
\hspace{+3mm}6.1 $a_n \leftarrow 0$ \\
7. $a.alloc \leftarrow v$ \\
8. Return(\textit{MP\_OKAY}) \\
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_grow}
\end{figure}
\textbf{Algorithm mp\_grow.}
It is ideal to prevent re-allocations from being performed if they are not required (step one). This is useful to
prevent mp\_ints from growing excessively in code that erroneously calls mp\_grow.
The requested digit count is padded up to next multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} (steps two and three).
This helps prevent many trivial reallocations that would grow an mp\_int by trivially small values.
It is assumed that the reallocation (step four) leaves the lower $a.alloc$ digits of the mp\_int intact. This is much
akin to how the \textit{realloc} function from the standard C library works. Since the newly allocated digits are
assumed to contain undefined values they are initially set to zero.
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_grow.c
\vspace{-3mm}
\begin{alltt}
016
017 /* grow as required */
018 int mp_grow (mp_int * a, int size)
019 \{
020 int i;
021 mp_digit *tmp;
022
023 /* if the alloc size is smaller alloc more ram */
024 if (a->alloc < size) \{
025 /* ensure there are always at least MP_PREC digits extra on top */
026 size += (MP_PREC * 2) - (size % MP_PREC);
027
028 /* reallocate the array a->dp
029 *
030 * We store the return in a temporary variable
031 * in case the operation failed we don't want
032 * to overwrite the dp member of a.
033 */
034 tmp = OPT_CAST(mp_digit) XREALLOC (a->dp, sizeof (mp_digit) * size);
035 if (tmp == NULL) \{
036 /* reallocation failed but "a" is still valid [can be freed] */
037 return MP_MEM;
038 \}
039
040 /* reallocation succeeded so set a->dp */
041 a->dp = tmp;
042
043 /* zero excess digits */
044 i = a->alloc;
045 a->alloc = size;
046 for (; i < a->alloc; i++) \{
047 a->dp[i] = 0;
048 \}
049 \}
050 return MP_OKAY;
051 \}
052 #endif
053
\end{alltt}
\end{small}
A quick optimization is to first determine if a memory re-allocation is required at all. The if statement (line 24) checks
if the \textbf{alloc} member of the mp\_int is smaller than the requested digit count. If the count is not larger than \textbf{alloc}
the function skips the re-allocation part thus saving time.
When a re-allocation is performed it is turned into an optimal request to save time in the future. The requested digit count is
padded upwards to 2nd multiple of \textbf{MP\_PREC} larger than \textbf{alloc} (line 26). The XREALLOC function is used
to re-allocate the memory. As per the other functions XREALLOC is actually a macro which evaluates to realloc by default. The realloc
function leaves the base of the allocation intact which means the first \textbf{alloc} digits of the mp\_int are the same as before
the re-allocation. All that is left is to clear the newly allocated digits and return.
Note that the re-allocation result is actually stored in a temporary pointer $tmp$. This is to allow this function to return
an error with a valid pointer. Earlier releases of the library stored the result of XREALLOC into the mp\_int $a$. That would
result in a memory leak if XREALLOC ever failed.
\subsection{Initializing Variable Precision mp\_ints}
Occasionally the number of digits required will be known in advance of an initialization, based on, for example, the size
of input mp\_ints to a given algorithm. The purpose of algorithm mp\_init\_size is similar to mp\_init except that it
will allocate \textit{at least} a specified number of digits.
\begin{figure}[here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_init\_size}. \\
\textbf{Input}. An mp\_int $a$ and the requested number of digits $b$. \\
\textbf{Output}. $a$ is initialized to hold at least $b$ digits. \\
\hline \\
1. $u \leftarrow b \mbox{ (mod }MP\_PREC\mbox{)}$ \\
2. $v \leftarrow b + 2 \cdot MP\_PREC - u$ \\
3. Allocate $v$ digits. \\
4. for $n$ from $0$ to $v - 1$ do \\
\hspace{3mm}4.1 $a_n \leftarrow 0$ \\
5. $a.sign \leftarrow MP\_ZPOS$\\
6. $a.used \leftarrow 0$\\
7. $a.alloc \leftarrow v$\\
8. Return(\textit{MP\_OKAY})\\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_init\_size}
\end{figure}
\textbf{Algorithm mp\_init\_size.}
This algorithm will initialize an mp\_int structure $a$ like algorithm mp\_init with the exception that the number of
digits allocated can be controlled by the second input argument $b$. The input size is padded upwards so it is a
multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} digits. This padding is used to prevent trivial
allocations from becoming a bottleneck in the rest of the algorithms.
Like algorithm mp\_init, the mp\_int structure is initialized to a default state representing the integer zero. This
particular algorithm is useful if it is known ahead of time the approximate size of the input. If the approximation is
correct no further memory re-allocations are required to work with the mp\_int.
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_init\_size.c
\vspace{-3mm}
\begin{alltt}
016
017 /* init an mp_init for a given size */
018 int mp_init_size (mp_int * a, int size)
019 \{
020 int x;
021
022 /* pad size so there are always extra digits */
023 size += (MP_PREC * 2) - (size % MP_PREC);
024
025 /* alloc mem */
026 a->dp = OPT_CAST(mp_digit) XMALLOC (sizeof (mp_digit) * size);
027 if (a->dp == NULL) \{
028 return MP_MEM;
029 \}
030
031 /* set the members */
032 a->used = 0;
033 a->alloc = size;
034 a->sign = MP_ZPOS;
035
036 /* zero the digits */
037 for (x = 0; x < size; x++) \{
038 a->dp[x] = 0;
039 \}
040
041 return MP_OKAY;
042 \}
043 #endif
044
\end{alltt}
\end{small}
The number of digits $b$ requested is padded (line 23) by first augmenting it to the next multiple of
\textbf{MP\_PREC} and then adding \textbf{MP\_PREC} to the result. If the memory can be successfully allocated the
mp\_int is placed in a default state representing the integer zero. Otherwise, the error code \textbf{MP\_MEM} will be
returned (line 28).
The digits are allocated and set to zero at the same time with the calloc() function (line @25,XCALLOC@). The
\textbf{used} count is set to zero, the \textbf{alloc} count set to the padded digit count and the \textbf{sign} flag set
to \textbf{MP\_ZPOS} to achieve a default valid mp\_int state (lines 32, 33 and 34). If the function
returns succesfully then it is correct to assume that the mp\_int structure is in a valid state for the remainder of the
functions to work with.
\subsection{Multiple Integer Initializations and Clearings}
Occasionally a function will require a series of mp\_int data types to be made available simultaneously.
The purpose of algorithm mp\_init\_multi is to initialize a variable length array of mp\_int structures in a single
statement. It is essentially a shortcut to multiple initializations.
\newpage\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_init\_multi}. \\
\textbf{Input}. Variable length array $V_k$ of mp\_int variables of length $k$. \\
\textbf{Output}. The array is initialized such that each mp\_int of $V_k$ is ready to use. \\
\hline \\
1. for $n$ from 0 to $k - 1$ do \\
\hspace{+3mm}1.1. Initialize the mp\_int $V_n$ (\textit{mp\_init}) \\
\hspace{+3mm}1.2. If initialization failed then do \\
\hspace{+6mm}1.2.1. for $j$ from $0$ to $n$ do \\
\hspace{+9mm}1.2.1.1. Free the mp\_int $V_j$ (\textit{mp\_clear}) \\
\hspace{+6mm}1.2.2. Return(\textit{MP\_MEM}) \\
2. Return(\textit{MP\_OKAY}) \\
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_init\_multi}
\end{figure}
\textbf{Algorithm mp\_init\_multi.}
The algorithm will initialize the array of mp\_int variables one at a time. If a runtime error has been detected
(\textit{step 1.2}) all of the previously initialized variables are cleared. The goal is an ``all or nothing''
initialization which allows for quick recovery from runtime errors.
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_init\_multi.c
\vspace{-3mm}
\begin{alltt}
016 #include <stdarg.h>
017
018 int mp_init_multi(mp_int *mp, ...)
019 \{
020 mp_err res = MP_OKAY; /* Assume ok until proven otherwise */
021 int n = 0; /* Number of ok inits */
022 mp_int* cur_arg = mp;
023 va_list args;
024
025 va_start(args, mp); /* init args to next argument from caller */
026 while (cur_arg != NULL) \{
027 if (mp_init(cur_arg) != MP_OKAY) \{
028 /* Oops - error! Back-track and mp_clear what we already
029 succeeded in init-ing, then return error.
030 */
031 va_list clean_args;
032
033 /* end the current list */
034 va_end(args);
035
036 /* now start cleaning up */
037 cur_arg = mp;
038 va_start(clean_args, mp);
039 while (n--) \{
040 mp_clear(cur_arg);
041 cur_arg = va_arg(clean_args, mp_int*);
042 \}
043 va_end(clean_args);
044 res = MP_MEM;
045 break;
046 \}
047 n++;
048 cur_arg = va_arg(args, mp_int*);
049 \}
050 va_end(args);
051 return res; /* Assumed ok, if error flagged above. */
052 \}
053
054 #endif
055
\end{alltt}
\end{small}
This function intializes a variable length list of mp\_int structure pointers. However, instead of having the mp\_int
structures in an actual C array they are simply passed as arguments to the function. This function makes use of the
``...'' argument syntax of the C programming language. The list is terminated with a final \textbf{NULL} argument
appended on the right.
The function uses the ``stdarg.h'' \textit{va} functions to step portably through the arguments to the function. A count
$n$ of succesfully initialized mp\_int structures is maintained (line 47) such that if a failure does occur,
the algorithm can backtrack and free the previously initialized structures (lines 27 to 46).
\subsection{Clamping Excess Digits}
When a function anticipates a result will be $n$ digits it is simpler to assume this is true within the body of
the function instead of checking during the computation. For example, a multiplication of a $i$ digit number by a
$j$ digit produces a result of at most $i + j$ digits. It is entirely possible that the result is $i + j - 1$
though, with no final carry into the last position. However, suppose the destination had to be first expanded
(\textit{via mp\_grow}) to accomodate $i + j - 1$ digits than further expanded to accomodate the final carry.
That would be a considerable waste of time since heap operations are relatively slow.
The ideal solution is to always assume the result is $i + j$ and fix up the \textbf{used} count after the function
terminates. This way a single heap operation (\textit{at most}) is required. However, if the result was not checked
there would be an excess high order zero digit.
For example, suppose the product of two integers was $x_n = (0x_{n-1}x_{n-2}...x_0)_{\beta}$. The leading zero digit
will not contribute to the precision of the result. In fact, through subsequent operations more leading zero digits would
accumulate to the point the size of the integer would be prohibitive. As a result even though the precision is very
low the representation is excessively large.
The mp\_clamp algorithm is designed to solve this very problem. It will trim high-order zeros by decrementing the
\textbf{used} count until a non-zero most significant digit is found. Also in this system, zero is considered to be a
positive number which means that if the \textbf{used} count is decremented to zero, the sign must be set to
\textbf{MP\_ZPOS}.
\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_clamp}. \\
\textbf{Input}. An mp\_int $a$ \\
\textbf{Output}. Any excess leading zero digits of $a$ are removed \\
\hline \\
1. while $a.used > 0$ and $a_{a.used - 1} = 0$ do \\
\hspace{+3mm}1.1 $a.used \leftarrow a.used - 1$ \\
2. if $a.used = 0$ then do \\
\hspace{+3mm}2.1 $a.sign \leftarrow MP\_ZPOS$ \\
\hline \\
\end{tabular}
\end{center}
\caption{Algorithm mp\_clamp}
\end{figure}
\textbf{Algorithm mp\_clamp.}
As can be expected this algorithm is very simple. The loop on step one is expected to iterate only once or twice at
the most. For example, this will happen in cases where there is not a carry to fill the last position. Step two fixes the sign for
when all of the digits are zero to ensure that the mp\_int is valid at all times.
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_clamp.c
\vspace{-3mm}
\begin{alltt}
016
017 /* trim unused digits
018 *
019 * This is used to ensure that leading zero digits are
020 * trimed and the leading "used" digit will be non-zero
021 * Typically very fast. Also fixes the sign if there
022 * are no more leading digits
023 */
024 void
025 mp_clamp (mp_int * a)
026 \{
027 /* decrease used while the most significant digit is
028 * zero.
029 */
030 while (a->used > 0 && a->dp[a->used - 1] == 0) \{
031 --(a->used);
032 \}
033
034 /* reset the sign flag if used == 0 */
035 if (a->used == 0) \{
036 a->sign = MP_ZPOS;
037 \}
038 \}
039 #endif
040
\end{alltt}
\end{small}
Note on line 27 how to test for the \textbf{used} count is made on the left of the \&\& operator. In the C programming
language the terms to \&\& are evaluated left to right with a boolean short-circuit if any condition fails. This is
important since if the \textbf{used} is zero the test on the right would fetch below the array. That is obviously
undesirable. The parenthesis on line 30 is used to make sure the \textbf{used} count is decremented and not
the pointer ``a''.
\section*{Exercises}
\begin{tabular}{cl}
$\left [ 1 \right ]$ & Discuss the relevance of the \textbf{used} member of the mp\_int structure. \\
& \\
$\left [ 1 \right ]$ & Discuss the consequences of not using padding when performing allocations. \\
& \\
$\left [ 2 \right ]$ & Estimate an ideal value for \textbf{MP\_PREC} when performing 1024-bit RSA \\
& encryption when $\beta = 2^{28}$. \\
& \\
$\left [ 1 \right ]$ & Discuss the relevance of the algorithm mp\_clamp. What does it prevent? \\
& \\
$\left [ 1 \right ]$ & Give an example of when the algorithm mp\_init\_copy might be useful. \\
& \\
\end{tabular}
%%%
% CHAPTER FOUR
%%%
\chapter{Basic Operations}
\section{Introduction}
In the previous chapter a series of low level algorithms were established that dealt with initializing and maintaining
mp\_int structures. This chapter will discuss another set of seemingly non-algebraic algorithms which will form the low
level basis of the entire library. While these algorithm are relatively trivial it is important to understand how they
work before proceeding since these algorithms will be used almost intrinsically in the following chapters.
The algorithms in this chapter deal primarily with more ``programmer'' related tasks such as creating copies of
mp\_int structures, assigning small values to mp\_int structures and comparisons of the values mp\_int structures
represent.
\section{Assigning Values to mp\_int Structures}
\subsection{Copying an mp\_int}
Assigning the value that a given mp\_int structure represents to another mp\_int structure shall be known as making
a copy for the purposes of this text. The copy of the mp\_int will be a separate entity that represents the same
value as the mp\_int it was copied from. The mp\_copy algorithm provides this functionality.
\newpage\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_copy}. \\
\textbf{Input}. An mp\_int $a$ and $b$. \\
\textbf{Output}. Store a copy of $a$ in $b$. \\
\hline \\
1. If $b.alloc < a.used$ then grow $b$ to $a.used$ digits. (\textit{mp\_grow}) \\
2. for $n$ from 0 to $a.used - 1$ do \\
\hspace{3mm}2.1 $b_{n} \leftarrow a_{n}$ \\
3. for $n$ from $a.used$ to $b.used - 1$ do \\
\hspace{3mm}3.1 $b_{n} \leftarrow 0$ \\
4. $b.used \leftarrow a.used$ \\
5. $b.sign \leftarrow a.sign$ \\
6. return(\textit{MP\_OKAY}) \\
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_copy}
\end{figure}
\textbf{Algorithm mp\_copy.}
This algorithm copies the mp\_int $a$ such that upon succesful termination of the algorithm the mp\_int $b$ will
represent the same integer as the mp\_int $a$. The mp\_int $b$ shall be a complete and distinct copy of the
mp\_int $a$ meaing that the mp\_int $a$ can be modified and it shall not affect the value of the mp\_int $b$.
If $b$ does not have enough room for the digits of $a$ it must first have its precision augmented via the mp\_grow
algorithm. The digits of $a$ are copied over the digits of $b$ and any excess digits of $b$ are set to zero (step two
and three). The \textbf{used} and \textbf{sign} members of $a$ are finally copied over the respective members of
$b$.
\textbf{Remark.} This algorithm also introduces a new idiosyncrasy that will be used throughout the rest of the
text. The error return codes of other algorithms are not explicitly checked in the pseudo-code presented. For example, in
step one of the mp\_copy algorithm the return of mp\_grow is not explicitly checked to ensure it succeeded. Text space is
limited so it is assumed that if a algorithm fails it will clear all temporarily allocated mp\_ints and return
the error code itself. However, the C code presented will demonstrate all of the error handling logic required to
implement the pseudo-code.
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_copy.c
\vspace{-3mm}
\begin{alltt}
016
017 /* copy, b = a */
018 int
019 mp_copy (mp_int * a, mp_int * b)
020 \{
021 int res, n;
022
023 /* if dst == src do nothing */
024 if (a == b) \{
025 return MP_OKAY;
026 \}
027
028 /* grow dest */
029 if (b->alloc < a->used) \{
030 if ((res = mp_grow (b, a->used)) != MP_OKAY) \{
031 return res;
032 \}
033 \}
034
035 /* zero b and copy the parameters over */
036 \{
037 register mp_digit *tmpa, *tmpb;
038
039 /* pointer aliases */
040
041 /* source */
042 tmpa = a->dp;
043
044 /* destination */
045 tmpb = b->dp;
046
047 /* copy all the digits */
048 for (n = 0; n < a->used; n++) \{
049 *tmpb++ = *tmpa++;
050 \}
051
052 /* clear high digits */
053 for (; n < b->used; n++) \{
054 *tmpb++ = 0;
055 \}
056 \}
057
058 /* copy used count and sign */
059 b->used = a->used;
060 b->sign = a->sign;
061 return MP_OKAY;
062 \}
063 #endif
064
\end{alltt}
\end{small}
Occasionally a dependent algorithm may copy an mp\_int effectively into itself such as when the input and output
mp\_int structures passed to a function are one and the same. For this case it is optimal to return immediately without
copying digits (line 24).
The mp\_int $b$ must have enough digits to accomodate the used digits of the mp\_int $a$. If $b.alloc$ is less than
$a.used$ the algorithm mp\_grow is used to augment the precision of $b$ (lines 29 to 33). In order to
simplify the inner loop that copies the digits from $a$ to $b$, two aliases $tmpa$ and $tmpb$ point directly at the digits
of the mp\_ints $a$ and $b$ respectively. These aliases (lines 42 and 45) allow the compiler to access the digits without first dereferencing the
mp\_int pointers and then subsequently the pointer to the digits.
After the aliases are established the digits from $a$ are copied into $b$ (lines 48 to 50) and then the excess
digits of $b$ are set to zero (lines 53 to 55). Both ``for'' loops make use of the pointer aliases and in
fact the alias for $b$ is carried through into the second ``for'' loop to clear the excess digits. This optimization
allows the alias to stay in a machine register fairly easy between the two loops.
\textbf{Remarks.} The use of pointer aliases is an implementation methodology first introduced in this function that will
be used considerably in other functions. Technically, a pointer alias is simply a short hand alias used to lower the
number of pointer dereferencing operations required to access data. For example, a for loop may resemble
\begin{alltt}
for (x = 0; x < 100; x++) \{
a->num[4]->dp[x] = 0;
\}
\end{alltt}
This could be re-written using aliases as
\begin{alltt}
mp_digit *tmpa;
a = a->num[4]->dp;
for (x = 0; x < 100; x++) \{
*a++ = 0;
\}
\end{alltt}
In this case an alias is used to access the
array of digits within an mp\_int structure directly. It may seem that a pointer alias is strictly not required
as a compiler may optimize out the redundant pointer operations. However, there are two dominant reasons to use aliases.
The first reason is that most compilers will not effectively optimize pointer arithmetic. For example, some optimizations
may work for the Microsoft Visual C++ compiler (MSVC) and not for the GNU C Compiler (GCC). Also some optimizations may
work for GCC and not MSVC. As such it is ideal to find a common ground for as many compilers as possible. Pointer
aliases optimize the code considerably before the compiler even reads the source code which means the end compiled code
stands a better chance of being faster.
The second reason is that pointer aliases often can make an algorithm simpler to read. Consider the first ``for''
loop of the function mp\_copy() re-written to not use pointer aliases.
\begin{alltt}
/* copy all the digits */
for (n = 0; n < a->used; n++) \{
b->dp[n] = a->dp[n];
\}
\end{alltt}
Whether this code is harder to read depends strongly on the individual. However, it is quantifiably slightly more
complicated as there are four variables within the statement instead of just two.
\subsubsection{Nested Statements}
Another commonly used technique in the source routines is that certain sections of code are nested. This is used in
particular with the pointer aliases to highlight code phases. For example, a Comba multiplier (discussed in chapter six)
will typically have three different phases. First the temporaries are initialized, then the columns calculated and
finally the carries are propagated. In this example the middle column production phase will typically be nested as it
uses temporary variables and aliases the most.
The nesting also simplies the source code as variables that are nested are only valid for their scope. As a result
the various temporary variables required do not propagate into other sections of code.
\subsection{Creating a Clone}
Another common operation is to make a local temporary copy of an mp\_int argument. To initialize an mp\_int
and then copy another existing mp\_int into the newly intialized mp\_int will be known as creating a clone. This is
useful within functions that need to modify an argument but do not wish to actually modify the original copy. The
mp\_init\_copy algorithm has been designed to help perform this task.
\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_init\_copy}. \\
\textbf{Input}. An mp\_int $a$ and $b$\\
\textbf{Output}. $a$ is initialized to be a copy of $b$. \\
\hline \\
1. Init $a$. (\textit{mp\_init}) \\
2. Copy $b$ to $a$. (\textit{mp\_copy}) \\
3. Return the status of the copy operation. \\
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_init\_copy}
\end{figure}
\textbf{Algorithm mp\_init\_copy.}
This algorithm will initialize an mp\_int variable and copy another previously initialized mp\_int variable into it. As
such this algorithm will perform two operations in one step.
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_init\_copy.c
\vspace{-3mm}
\begin{alltt}
016
017 /* creates "a" then copies b into it */
018 int mp_init_copy (mp_int * a, mp_int * b)
019 \{
020 int res;
021
022 if ((res = mp_init (a)) != MP_OKAY) \{
023 return res;
024 \}
025 return mp_copy (b, a);
026 \}
027 #endif
028
\end{alltt}
\end{small}
This will initialize \textbf{a} and make it a verbatim copy of the contents of \textbf{b}. Note that
\textbf{a} will have its own memory allocated which means that \textbf{b} may be cleared after the call
and \textbf{a} will be left intact.
\section{Zeroing an Integer}
Reseting an mp\_int to the default state is a common step in many algorithms. The mp\_zero algorithm will be the algorithm used to
perform this task.
\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_zero}. \\
\textbf{Input}. An mp\_int $a$ \\
\textbf{Output}. Zero the contents of $a$ \\
\hline \\
1. $a.used \leftarrow 0$ \\
2. $a.sign \leftarrow$ MP\_ZPOS \\
3. for $n$ from 0 to $a.alloc - 1$ do \\
\hspace{3mm}3.1 $a_n \leftarrow 0$ \\
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_zero}
\end{figure}
\textbf{Algorithm mp\_zero.}
This algorithm simply resets a mp\_int to the default state.
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_zero.c
\vspace{-3mm}
\begin{alltt}
016
017 /* set to zero */
018 void mp_zero (mp_int * a)
019 \{
020 int n;
021 mp_digit *tmp;
022
023 a->sign = MP_ZPOS;
024 a->used = 0;
025
026 tmp = a->dp;
027 for (n = 0; n < a->alloc; n++) \{
028 *tmp++ = 0;
029 \}
030 \}
031 #endif
032
\end{alltt}
\end{small}
After the function is completed, all of the digits are zeroed, the \textbf{used} count is zeroed and the
\textbf{sign} variable is set to \textbf{MP\_ZPOS}.
\section{Sign Manipulation}
\subsection{Absolute Value}
With the mp\_int representation of an integer, calculating the absolute value is trivial. The mp\_abs algorithm will compute
the absolute value of an mp\_int.
\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_abs}. \\
\textbf{Input}. An mp\_int $a$ \\
\textbf{Output}. Computes $b = \vert a \vert$ \\
\hline \\
1. Copy $a$ to $b$. (\textit{mp\_copy}) \\
2. If the copy failed return(\textit{MP\_MEM}). \\
3. $b.sign \leftarrow MP\_ZPOS$ \\
4. Return(\textit{MP\_OKAY}) \\
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_abs}
\end{figure}
\textbf{Algorithm mp\_abs.}
This algorithm computes the absolute of an mp\_int input. First it copies $a$ over $b$. This is an example of an
algorithm where the check in mp\_copy that determines if the source and destination are equal proves useful. This allows,
for instance, the developer to pass the same mp\_int as the source and destination to this function without addition
logic to handle it.
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_abs.c
\vspace{-3mm}
\begin{alltt}
016
017 /* b = |a|
018 *
019 * Simple function copies the input and fixes the sign to positive
020 */
021 int
022 mp_abs (mp_int * a, mp_int * b)
023 \{
024 int res;
025
026 /* copy a to b */
027 if (a != b) \{
028 if ((res = mp_copy (a, b)) != MP_OKAY) \{
029 return res;
030 \}
031 \}
032
033 /* force the sign of b to positive */
034 b->sign = MP_ZPOS;
035
036 return MP_OKAY;
037 \}
038 #endif
039
\end{alltt}
\end{small}
This fairly trivial algorithm first eliminates non--required duplications (line 27) and then sets the
\textbf{sign} flag to \textbf{MP\_ZPOS}.
\subsection{Integer Negation}
With the mp\_int representation of an integer, calculating the negation is also trivial. The mp\_neg algorithm will compute
the negative of an mp\_int input.
\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_neg}. \\
\textbf{Input}. An mp\_int $a$ \\
\textbf{Output}. Computes $b = -a$ \\
\hline \\
1. Copy $a$ to $b$. (\textit{mp\_copy}) \\
2. If the copy failed return(\textit{MP\_MEM}). \\
3. If $a.used = 0$ then return(\textit{MP\_OKAY}). \\
4. If $a.sign = MP\_ZPOS$ then do \\
\hspace{3mm}4.1 $b.sign = MP\_NEG$. \\
5. else do \\
\hspace{3mm}5.1 $b.sign = MP\_ZPOS$. \\
6. Return(\textit{MP\_OKAY}) \\
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_neg}
\end{figure}
\textbf{Algorithm mp\_neg.}
This algorithm computes the negation of an input. First it copies $a$ over $b$. If $a$ has no used digits then
the algorithm returns immediately. Otherwise it flips the sign flag and stores the result in $b$. Note that if
$a$ had no digits then it must be positive by definition. Had step three been omitted then the algorithm would return
zero as negative.
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_neg.c
\vspace{-3mm}
\begin{alltt}
016
017 /* b = -a */
018 int mp_neg (mp_int * a, mp_int * b)
019 \{
020 int res;
021 if (a != b) \{
022 if ((res = mp_copy (a, b)) != MP_OKAY) \{
023 return res;
024 \}
025 \}
026
027 if (mp_iszero(b) != MP_YES) \{
028 b->sign = (a->sign == MP_ZPOS) ? MP_NEG : MP_ZPOS;
029 \} else \{
030 b->sign = MP_ZPOS;
031 \}
032
033 return MP_OKAY;
034 \}
035 #endif
036
\end{alltt}
\end{small}
Like mp\_abs() this function avoids non--required duplications (line 21) and then sets the sign. We
have to make sure that only non--zero values get a \textbf{sign} of \textbf{MP\_NEG}. If the mp\_int is zero
than the \textbf{sign} is hard--coded to \textbf{MP\_ZPOS}.
\section{Small Constants}
\subsection{Setting Small Constants}
Often a mp\_int must be set to a relatively small value such as $1$ or $2$. For these cases the mp\_set algorithm is useful.
\newpage\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_set}. \\
\textbf{Input}. An mp\_int $a$ and a digit $b$ \\
\textbf{Output}. Make $a$ equivalent to $b$ \\
\hline \\
1. Zero $a$ (\textit{mp\_zero}). \\
2. $a_0 \leftarrow b \mbox{ (mod }\beta\mbox{)}$ \\
3. $a.used \leftarrow \left \lbrace \begin{array}{ll}
1 & \mbox{if }a_0 > 0 \\
0 & \mbox{if }a_0 = 0
\end{array} \right .$ \\
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_set}
\end{figure}
\textbf{Algorithm mp\_set.}
This algorithm sets a mp\_int to a small single digit value. Step number 1 ensures that the integer is reset to the default state. The
single digit is set (\textit{modulo $\beta$}) and the \textbf{used} count is adjusted accordingly.
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_set.c
\vspace{-3mm}
\begin{alltt}
016
017 /* set to a digit */
018 void mp_set (mp_int * a, mp_digit b)
019 \{
020 mp_zero (a);
021 a->dp[0] = b & MP_MASK;
022 a->used = (a->dp[0] != 0) ? 1 : 0;
023 \}
024 #endif
025
\end{alltt}
\end{small}
First we zero (line 20) the mp\_int to make sure that the other members are initialized for a
small positive constant. mp\_zero() ensures that the \textbf{sign} is positive and the \textbf{used} count
is zero. Next we set the digit and reduce it modulo $\beta$ (line 21). After this step we have to
check if the resulting digit is zero or not. If it is not then we set the \textbf{used} count to one, otherwise
to zero.
We can quickly reduce modulo $\beta$ since it is of the form $2^k$ and a quick binary AND operation with
$2^k - 1$ will perform the same operation.
One important limitation of this function is that it will only set one digit. The size of a digit is not fixed, meaning source that uses
this function should take that into account. Only trivially small constants can be set using this function.
\subsection{Setting Large Constants}
To overcome the limitations of the mp\_set algorithm the mp\_set\_int algorithm is ideal. It accepts a ``long''
data type as input and will always treat it as a 32-bit integer.
\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_set\_int}. \\
\textbf{Input}. An mp\_int $a$ and a ``long'' integer $b$ \\
\textbf{Output}. Make $a$ equivalent to $b$ \\
\hline \\
1. Zero $a$ (\textit{mp\_zero}) \\
2. for $n$ from 0 to 7 do \\
\hspace{3mm}2.1 $a \leftarrow a \cdot 16$ (\textit{mp\_mul2d}) \\
\hspace{3mm}2.2 $u \leftarrow \lfloor b / 2^{4(7 - n)} \rfloor \mbox{ (mod }16\mbox{)}$\\
\hspace{3mm}2.3 $a_0 \leftarrow a_0 + u$ \\
\hspace{3mm}2.4 $a.used \leftarrow a.used + 1$ \\
3. Clamp excess used digits (\textit{mp\_clamp}) \\
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_set\_int}
\end{figure}
\textbf{Algorithm mp\_set\_int.}
The algorithm performs eight iterations of a simple loop where in each iteration four bits from the source are added to the
mp\_int. Step 2.1 will multiply the current result by sixteen making room for four more bits in the less significant positions. In step 2.2 the
next four bits from the source are extracted and are added to the mp\_int. The \textbf{used} digit count is
incremented to reflect the addition. The \textbf{used} digit counter is incremented since if any of the leading digits were zero the mp\_int would have
zero digits used and the newly added four bits would be ignored.
Excess zero digits are trimmed in steps 2.1 and 3 by using higher level algorithms mp\_mul2d and mp\_clamp.
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_set\_int.c
\vspace{-3mm}
\begin{alltt}
016
017 /* set a 32-bit const */
018 int mp_set_int (mp_int * a, unsigned long b)
019 \{
020 int x, res;
021
022 mp_zero (a);
023
024 /* set four bits at a time */
025 for (x = 0; x < 8; x++) \{
026 /* shift the number up four bits */
027 if ((res = mp_mul_2d (a, 4, a)) != MP_OKAY) \{
028 return res;
029 \}
030
031 /* OR in the top four bits of the source */
032 a->dp[0] |= (b >> 28) & 15;
033
034 /* shift the source up to the next four bits */
035 b <<= 4;
036
037 /* ensure that digits are not clamped off */
038 a->used += 1;
039 \}
040 mp_clamp (a);
041 return MP_OKAY;
042 \}
043 #endif
044
\end{alltt}
\end{small}
This function sets four bits of the number at a time to handle all practical \textbf{DIGIT\_BIT} sizes. The weird
addition on line 38 ensures that the newly added in bits are added to the number of digits. While it may not
seem obvious as to why the digit counter does not grow exceedingly large it is because of the shift on line 27
as well as the call to mp\_clamp() on line 40. Both functions will clamp excess leading digits which keeps
the number of used digits low.
\section{Comparisons}
\subsection{Unsigned Comparisions}
Comparing a multiple precision integer is performed with the exact same algorithm used to compare two decimal numbers. For example,
to compare $1,234$ to $1,264$ the digits are extracted by their positions. That is we compare $1 \cdot 10^3 + 2 \cdot 10^2 + 3 \cdot 10^1 + 4 \cdot 10^0$
to $1 \cdot 10^3 + 2 \cdot 10^2 + 6 \cdot 10^1 + 4 \cdot 10^0$ by comparing single digits at a time starting with the highest magnitude
positions. If any leading digit of one integer is greater than a digit in the same position of another integer then obviously it must be greater.
The first comparision routine that will be developed is the unsigned magnitude compare which will perform a comparison based on the digits of two
mp\_int variables alone. It will ignore the sign of the two inputs. Such a function is useful when an absolute comparison is required or if the
signs are known to agree in advance.
To facilitate working with the results of the comparison functions three constants are required.
\begin{figure}[here]
\begin{center}
\begin{tabular}{|r|l|}
\hline \textbf{Constant} & \textbf{Meaning} \\
\hline \textbf{MP\_GT} & Greater Than \\
\hline \textbf{MP\_EQ} & Equal To \\
\hline \textbf{MP\_LT} & Less Than \\
\hline
\end{tabular}
\end{center}
\caption{Comparison Return Codes}
\end{figure}
\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_cmp\_mag}. \\
\textbf{Input}. Two mp\_ints $a$ and $b$. \\
\textbf{Output}. Unsigned comparison results ($a$ to the left of $b$). \\
\hline \\
1. If $a.used > b.used$ then return(\textit{MP\_GT}) \\
2. If $a.used < b.used$ then return(\textit{MP\_LT}) \\
3. for n from $a.used - 1$ to 0 do \\
\hspace{+3mm}3.1 if $a_n > b_n$ then return(\textit{MP\_GT}) \\
\hspace{+3mm}3.2 if $a_n < b_n$ then return(\textit{MP\_LT}) \\
4. Return(\textit{MP\_EQ}) \\
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_cmp\_mag}
\end{figure}
\textbf{Algorithm mp\_cmp\_mag.}
By saying ``$a$ to the left of $b$'' it is meant that the comparison is with respect to $a$, that is if $a$ is greater than $b$ it will return
\textbf{MP\_GT} and similar with respect to when $a = b$ and $a < b$. The first two steps compare the number of digits used in both $a$ and $b$.
Obviously if the digit counts differ there would be an imaginary zero digit in the smaller number where the leading digit of the larger number is.
If both have the same number of digits than the actual digits themselves must be compared starting at the leading digit.
By step three both inputs must have the same number of digits so its safe to start from either $a.used - 1$ or $b.used - 1$ and count down to
the zero'th digit. If after all of the digits have been compared, no difference is found, the algorithm returns \textbf{MP\_EQ}.
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_cmp\_mag.c
\vspace{-3mm}
\begin{alltt}
016
017 /* compare maginitude of two ints (unsigned) */
018 int mp_cmp_mag (mp_int * a, mp_int * b)
019 \{
020 int n;
021 mp_digit *tmpa, *tmpb;
022
023 /* compare based on # of non-zero digits */
024 if (a->used > b->used) \{
025 return MP_GT;
026 \}
027
028 if (a->used < b->used) \{
029 return MP_LT;
030 \}
031
032 /* alias for a */
033 tmpa = a->dp + (a->used - 1);
034
035 /* alias for b */
036 tmpb = b->dp + (a->used - 1);
037
038 /* compare based on digits */
039 for (n = 0; n < a->used; ++n, --tmpa, --tmpb) \{
040 if (*tmpa > *tmpb) \{
041 return MP_GT;
042 \}
043
044 if (*tmpa < *tmpb) \{
045 return MP_LT;
046 \}
047 \}
048 return MP_EQ;
049 \}
050 #endif
051
\end{alltt}
\end{small}
The two if statements (lines 24 and 28) compare the number of digits in the two inputs. These two are
performed before all of the digits are compared since it is a very cheap test to perform and can potentially save
considerable time. The implementation given is also not valid without those two statements. $b.alloc$ may be
smaller than $a.used$, meaning that undefined values will be read from $b$ past the end of the array of digits.
\subsection{Signed Comparisons}
Comparing with sign considerations is also fairly critical in several routines (\textit{division for example}). Based on an unsigned magnitude
comparison a trivial signed comparison algorithm can be written.
\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_cmp}. \\
\textbf{Input}. Two mp\_ints $a$ and $b$ \\
\textbf{Output}. Signed Comparison Results ($a$ to the left of $b$) \\
\hline \\
1. if $a.sign = MP\_NEG$ and $b.sign = MP\_ZPOS$ then return(\textit{MP\_LT}) \\
2. if $a.sign = MP\_ZPOS$ and $b.sign = MP\_NEG$ then return(\textit{MP\_GT}) \\
3. if $a.sign = MP\_NEG$ then \\
\hspace{+3mm}3.1 Return the unsigned comparison of $b$ and $a$ (\textit{mp\_cmp\_mag}) \\
4 Otherwise \\
\hspace{+3mm}4.1 Return the unsigned comparison of $a$ and $b$ \\
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_cmp}
\end{figure}
\textbf{Algorithm mp\_cmp.}
The first two steps compare the signs of the two inputs. If the signs do not agree then it can return right away with the appropriate
comparison code. When the signs are equal the digits of the inputs must be compared to determine the correct result. In step
three the unsigned comparision flips the order of the arguments since they are both negative. For instance, if $-a > -b$ then
$\vert a \vert < \vert b \vert$. Step number four will compare the two when they are both positive.
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_cmp.c
\vspace{-3mm}
\begin{alltt}
016
017 /* compare two ints (signed)*/
018 int
019 mp_cmp (mp_int * a, mp_int * b)
020 \{
021 /* compare based on sign */
022 if (a->sign != b->sign) \{
023 if (a->sign == MP_NEG) \{
024 return MP_LT;
025 \} else \{
026 return MP_GT;
027 \}
028 \}
029
030 /* compare digits */
031 if (a->sign == MP_NEG) \{
032 /* if negative compare opposite direction */
033 return mp_cmp_mag(b, a);
034 \} else \{
035 return mp_cmp_mag(a, b);
036 \}
037 \}
038 #endif
039
\end{alltt}
\end{small}
The two if statements (lines 22 and 23) perform the initial sign comparison. If the signs are not the equal then which ever
has the positive sign is larger. The inputs are compared (line 31) based on magnitudes. If the signs were both
negative then the unsigned comparison is performed in the opposite direction (line 33). Otherwise, the signs are assumed to
be both positive and a forward direction unsigned comparison is performed.
\section*{Exercises}
\begin{tabular}{cl}
$\left [ 2 \right ]$ & Modify algorithm mp\_set\_int to accept as input a variable length array of bits. \\
& \\
$\left [ 3 \right ]$ & Give the probability that algorithm mp\_cmp\_mag will have to compare $k$ digits \\
& of two random digits (of equal magnitude) before a difference is found. \\
& \\
$\left [ 1 \right ]$ & Suggest a simple method to speed up the implementation of mp\_cmp\_mag based \\
& on the observations made in the previous problem. \\
&
\end{tabular}
\chapter{Basic Arithmetic}
\section{Introduction}
At this point algorithms for initialization, clearing, zeroing, copying, comparing and setting small constants have been
established. The next logical set of algorithms to develop are addition, subtraction and digit shifting algorithms. These
algorithms make use of the lower level algorithms and are the cruicial building block for the multiplication algorithms. It is very important
that these algorithms are highly optimized. On their own they are simple $O(n)$ algorithms but they can be called from higher level algorithms
which easily places them at $O(n^2)$ or even $O(n^3)$ work levels.
All of the algorithms within this chapter make use of the logical bit shift operations denoted by $<<$ and $>>$ for left and right
logical shifts respectively. A logical shift is analogous to sliding the decimal point of radix-10 representations. For example, the real
number $0.9345$ is equivalent to $93.45\%$ which is found by sliding the the decimal two places to the right (\textit{multiplying by $\beta^2 = 10^2$}).
Algebraically a binary logical shift is equivalent to a division or multiplication by a power of two.
For example, $a << k = a \cdot 2^k$ while $a >> k = \lfloor a/2^k \rfloor$.
One significant difference between a logical shift and the way decimals are shifted is that digits below the zero'th position are removed
from the number. For example, consider $1101_2 >> 1$ using decimal notation this would produce $110.1_2$. However, with a logical shift the
result is $110_2$.
\section{Addition and Subtraction}
In common twos complement fixed precision arithmetic negative numbers are easily represented by subtraction from the modulus. For example, with 32-bit integers
$a - b\mbox{ (mod }2^{32}\mbox{)}$ is the same as $a + (2^{32} - b) \mbox{ (mod }2^{32}\mbox{)}$ since $2^{32} \equiv 0 \mbox{ (mod }2^{32}\mbox{)}$.
As a result subtraction can be performed with a trivial series of logical operations and an addition.
However, in multiple precision arithmetic negative numbers are not represented in the same way. Instead a sign flag is used to keep track of the
sign of the integer. As a result signed addition and subtraction are actually implemented as conditional usage of lower level addition or
subtraction algorithms with the sign fixed up appropriately.
The lower level algorithms will add or subtract integers without regard to the sign flag. That is they will add or subtract the magnitude of
the integers respectively.
\subsection{Low Level Addition}
An unsigned addition of multiple precision integers is performed with the same long-hand algorithm used to add decimal numbers. That is to add the
trailing digits first and propagate the resulting carry upwards. Since this is a lower level algorithm the name will have a ``s\_'' prefix.
Historically that convention stems from the MPI library where ``s\_'' stood for static functions that were hidden from the developer entirely.
\newpage
\begin{figure}[!here]
\begin{center}
\begin{small}
\begin{tabular}{l}
\hline Algorithm \textbf{s\_mp\_add}. \\
\textbf{Input}. Two mp\_ints $a$ and $b$ \\
\textbf{Output}. The unsigned addition $c = \vert a \vert + \vert b \vert$. \\
\hline \\
1. if $a.used > b.used$ then \\
\hspace{+3mm}1.1 $min \leftarrow b.used$ \\
\hspace{+3mm}1.2 $max \leftarrow a.used$ \\
\hspace{+3mm}1.3 $x \leftarrow a$ \\
2. else \\
\hspace{+3mm}2.1 $min \leftarrow a.used$ \\
\hspace{+3mm}2.2 $max \leftarrow b.used$ \\
\hspace{+3mm}2.3 $x \leftarrow b$ \\
3. If $c.alloc < max + 1$ then grow $c$ to hold at least $max + 1$ digits (\textit{mp\_grow}) \\
4. $oldused \leftarrow c.used$ \\
5. $c.used \leftarrow max + 1$ \\
6. $u \leftarrow 0$ \\
7. for $n$ from $0$ to $min - 1$ do \\
\hspace{+3mm}7.1 $c_n \leftarrow a_n + b_n + u$ \\
\hspace{+3mm}7.2 $u \leftarrow c_n >> lg(\beta)$ \\
\hspace{+3mm}7.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\
8. if $min \ne max$ then do \\
\hspace{+3mm}8.1 for $n$ from $min$ to $max - 1$ do \\
\hspace{+6mm}8.1.1 $c_n \leftarrow x_n + u$ \\
\hspace{+6mm}8.1.2 $u \leftarrow c_n >> lg(\beta)$ \\
\hspace{+6mm}8.1.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\
9. $c_{max} \leftarrow u$ \\
10. if $olduse > max$ then \\
\hspace{+3mm}10.1 for $n$ from $max + 1$ to $oldused - 1$ do \\
\hspace{+6mm}10.1.1 $c_n \leftarrow 0$ \\
11. Clamp excess digits in $c$. (\textit{mp\_clamp}) \\
12. Return(\textit{MP\_OKAY}) \\
\hline
\end{tabular}
\end{small}
\end{center}
\caption{Algorithm s\_mp\_add}
\end{figure}
\textbf{Algorithm s\_mp\_add.}
This algorithm is loosely based on algorithm 14.7 of HAC \cite[pp. 594]{HAC} but has been extended to allow the inputs to have different magnitudes.
Coincidentally the description of algorithm A in Knuth \cite[pp. 266]{TAOCPV2} shares the same deficiency as the algorithm from \cite{HAC}. Even the
MIX pseudo machine code presented by Knuth \cite[pp. 266-267]{TAOCPV2} is incapable of handling inputs which are of different magnitudes.
The first thing that has to be accomplished is to sort out which of the two inputs is the largest. The addition logic
will simply add all of the smallest input to the largest input and store that first part of the result in the
destination. Then it will apply a simpler addition loop to excess digits of the larger input.
The first two steps will handle sorting the inputs such that $min$ and $max$ hold the digit counts of the two
inputs. The variable $x$ will be an mp\_int alias for the largest input or the second input $b$ if they have the
same number of digits. After the inputs are sorted the destination $c$ is grown as required to accomodate the sum
of the two inputs. The original \textbf{used} count of $c$ is copied and set to the new used count.
At this point the first addition loop will go through as many digit positions that both inputs have. The carry
variable $\mu$ is set to zero outside the loop. Inside the loop an ``addition'' step requires three statements to produce
one digit of the summand. First
two digits from $a$ and $b$ are added together along with the carry $\mu$. The carry of this step is extracted and stored
in $\mu$ and finally the digit of the result $c_n$ is truncated within the range $0 \le c_n < \beta$.
Now all of the digit positions that both inputs have in common have been exhausted. If $min \ne max$ then $x$ is an alias
for one of the inputs that has more digits. A simplified addition loop is then used to essentially copy the remaining digits
and the carry to the destination.
The final carry is stored in $c_{max}$ and digits above $max$ upto $oldused$ are zeroed which completes the addition.
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_add.c
\vspace{-3mm}
\begin{alltt}
016
017 /* low level addition, based on HAC pp.594, Algorithm 14.7 */
018 int
019 s_mp_add (mp_int * a, mp_int * b, mp_int * c)
020 \{
021 mp_int *x;
022 int olduse, res, min, max;
023
024 /* find sizes, we let |a| <= |b| which means we have to sort
025 * them. "x" will point to the input with the most digits
026 */
027 if (a->used > b->used) \{
028 min = b->used;
029 max = a->used;
030 x = a;
031 \} else \{
032 min = a->used;
033 max = b->used;
034 x = b;
035 \}
036
037 /* init result */
038 if (c->alloc < max + 1) \{
039 if ((res = mp_grow (c, max + 1)) != MP_OKAY) \{
040 return res;
041 \}
042 \}
043
044 /* get old used digit count and set new one */
045 olduse = c->used;
046 c->used = max + 1;
047
048 \{
049 register mp_digit u, *tmpa, *tmpb, *tmpc;
050 register int i;
051
052 /* alias for digit pointers */
053
054 /* first input */
055 tmpa = a->dp;
056
057 /* second input */
058 tmpb = b->dp;
059
060 /* destination */
061 tmpc = c->dp;
062
063 /* zero the carry */
064 u = 0;
065 for (i = 0; i < min; i++) \{
066 /* Compute the sum at one digit, T[i] = A[i] + B[i] + U */
067 *tmpc = *tmpa++ + *tmpb++ + u;
068
069 /* U = carry bit of T[i] */
070 u = *tmpc >> ((mp_digit)DIGIT_BIT);
071
072 /* take away carry bit from T[i] */
073 *tmpc++ &= MP_MASK;
074 \}
075
076 /* now copy higher words if any, that is in A+B
077 * if A or B has more digits add those in
078 */
079 if (min != max) \{
080 for (; i < max; i++) \{
081 /* T[i] = X[i] + U */
082 *tmpc = x->dp[i] + u;
083
084 /* U = carry bit of T[i] */
085 u = *tmpc >> ((mp_digit)DIGIT_BIT);
086
087 /* take away carry bit from T[i] */
088 *tmpc++ &= MP_MASK;
089 \}
090 \}
091
092 /* add carry */
093 *tmpc++ = u;
094
095 /* clear digits above oldused */
096 for (i = c->used; i < olduse; i++) \{
097 *tmpc++ = 0;
098 \}
099 \}
100
101 mp_clamp (c);
102 return MP_OKAY;
103 \}
104 #endif
105
\end{alltt}
\end{small}
We first sort (lines 27 to 35) the inputs based on magnitude and determine the $min$ and $max$ variables.
Note that $x$ is a pointer to an mp\_int assigned to the largest input, in effect it is a local alias. Next we
grow the destination (37 to 42) ensure that it can accomodate the result of the addition.
Similar to the implementation of mp\_copy this function uses the braced code and local aliases coding style. The three aliases that are on
lines 55, 58 and 61 represent the two inputs and destination variables respectively. These aliases are used to ensure the
compiler does not have to dereference $a$, $b$ or $c$ (respectively) to access the digits of the respective mp\_int.
The initial carry $u$ will be cleared (line 64), note that $u$ is of type mp\_digit which ensures type
compatibility within the implementation. The initial addition (line 65 to 74) adds digits from
both inputs until the smallest input runs out of digits. Similarly the conditional addition loop
(line 80 to 90) adds the remaining digits from the larger of the two inputs. The addition is finished
with the final carry being stored in $tmpc$ (line 93). Note the ``++'' operator within the same expression.
After line 93, $tmpc$ will point to the $c.used$'th digit of the mp\_int $c$. This is useful
for the next loop (line 96 to 99) which set any old upper digits to zero.
\subsection{Low Level Subtraction}
The low level unsigned subtraction algorithm is very similar to the low level unsigned addition algorithm. The principle difference is that the
unsigned subtraction algorithm requires the result to be positive. That is when computing $a - b$ the condition $\vert a \vert \ge \vert b\vert$ must
be met for this algorithm to function properly. Keep in mind this low level algorithm is not meant to be used in higher level algorithms directly.
This algorithm as will be shown can be used to create functional signed addition and subtraction algorithms.
For this algorithm a new variable is required to make the description simpler. Recall from section 1.3.1 that a mp\_digit must be able to represent
the range $0 \le x < 2\beta$ for the algorithms to work correctly. However, it is allowable that a mp\_digit represent a larger range of values. For
this algorithm we will assume that the variable $\gamma$ represents the number of bits available in a
mp\_digit (\textit{this implies $2^{\gamma} > \beta$}).
For example, the default for LibTomMath is to use a ``unsigned long'' for the mp\_digit ``type'' while $\beta = 2^{28}$. In ISO C an ``unsigned long''
data type must be able to represent $0 \le x < 2^{32}$ meaning that in this case $\gamma \ge 32$.
\newpage\begin{figure}[!here]
\begin{center}
\begin{small}
\begin{tabular}{l}
\hline Algorithm \textbf{s\_mp\_sub}. \\
\textbf{Input}. Two mp\_ints $a$ and $b$ ($\vert a \vert \ge \vert b \vert$) \\
\textbf{Output}. The unsigned subtraction $c = \vert a \vert - \vert b \vert$. \\
\hline \\
1. $min \leftarrow b.used$ \\
2. $max \leftarrow a.used$ \\
3. If $c.alloc < max$ then grow $c$ to hold at least $max$ digits. (\textit{mp\_grow}) \\
4. $oldused \leftarrow c.used$ \\
5. $c.used \leftarrow max$ \\
6. $u \leftarrow 0$ \\
7. for $n$ from $0$ to $min - 1$ do \\
\hspace{3mm}7.1 $c_n \leftarrow a_n - b_n - u$ \\
\hspace{3mm}7.2 $u \leftarrow c_n >> (\gamma - 1)$ \\
\hspace{3mm}7.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\
8. if $min < max$ then do \\
\hspace{3mm}8.1 for $n$ from $min$ to $max - 1$ do \\
\hspace{6mm}8.1.1 $c_n \leftarrow a_n - u$ \\
\hspace{6mm}8.1.2 $u \leftarrow c_n >> (\gamma - 1)$ \\
\hspace{6mm}8.1.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\
9. if $oldused > max$ then do \\
\hspace{3mm}9.1 for $n$ from $max$ to $oldused - 1$ do \\
\hspace{6mm}9.1.1 $c_n \leftarrow 0$ \\
10. Clamp excess digits of $c$. (\textit{mp\_clamp}). \\
11. Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{small}
\end{center}
\caption{Algorithm s\_mp\_sub}
\end{figure}
\textbf{Algorithm s\_mp\_sub.}
This algorithm performs the unsigned subtraction of two mp\_int variables under the restriction that the result must be positive. That is when
passing variables $a$ and $b$ the condition that $\vert a \vert \ge \vert b \vert$ must be met for the algorithm to function correctly. This
algorithm is loosely based on algorithm 14.9 \cite[pp. 595]{HAC} and is similar to algorithm S in \cite[pp. 267]{TAOCPV2} as well. As was the case
of the algorithm s\_mp\_add both other references lack discussion concerning various practical details such as when the inputs differ in magnitude.
The initial sorting of the inputs is trivial in this algorithm since $a$ is guaranteed to have at least the same magnitude of $b$. Steps 1 and 2
set the $min$ and $max$ variables. Unlike the addition routine there is guaranteed to be no carry which means that the final result can be at
most $max$ digits in length as opposed to $max + 1$. Similar to the addition algorithm the \textbf{used} count of $c$ is copied locally and
set to the maximal count for the operation.
The subtraction loop that begins on step seven is essentially the same as the addition loop of algorithm s\_mp\_add except single precision
subtraction is used instead. Note the use of the $\gamma$ variable to extract the carry (\textit{also known as the borrow}) within the subtraction
loops. Under the assumption that two's complement single precision arithmetic is used this will successfully extract the desired carry.
For example, consider subtracting $0101_2$ from $0100_2$ where $\gamma = 4$ and $\beta = 2$. The least significant bit will force a carry upwards to
the third bit which will be set to zero after the borrow. After the very first bit has been subtracted $4 - 1 \equiv 0011_2$ will remain, When the
third bit of $0101_2$ is subtracted from the result it will cause another carry. In this case though the carry will be forced to propagate all the
way to the most significant bit.
Recall that $\beta < 2^{\gamma}$. This means that if a carry does occur just before the $lg(\beta)$'th bit it will propagate all the way to the most
significant bit. Thus, the high order bits of the mp\_digit that are not part of the actual digit will either be all zero, or all one. All that
is needed is a single zero or one bit for the carry. Therefore a single logical shift right by $\gamma - 1$ positions is sufficient to extract the
carry. This method of carry extraction may seem awkward but the reason for it becomes apparent when the implementation is discussed.
If $b$ has a smaller magnitude than $a$ then step 9 will force the carry and copy operation to propagate through the larger input $a$ into $c$. Step
10 will ensure that any leading digits of $c$ above the $max$'th position are zeroed.
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_sub.c
\vspace{-3mm}
\begin{alltt}
016
017 /* low level subtraction (assumes |a| > |b|), HAC pp.595 Algorithm 14.9 */
018 int
019 s_mp_sub (mp_int * a, mp_int * b, mp_int * c)
020 \{
021 int olduse, res, min, max;
022
023 /* find sizes */
024 min = b->used;
025 max = a->used;
026
027 /* init result */
028 if (c->alloc < max) \{
029 if ((res = mp_grow (c, max)) != MP_OKAY) \{
030 return res;
031 \}
032 \}
033 olduse = c->used;
034 c->used = max;
035
036 \{
037 register mp_digit u, *tmpa, *tmpb, *tmpc;
038 register int i;
039
040 /* alias for digit pointers */
041 tmpa = a->dp;
042 tmpb = b->dp;
043 tmpc = c->dp;
044
045 /* set carry to zero */
046 u = 0;
047 for (i = 0; i < min; i++) \{
048 /* T[i] = A[i] - B[i] - U */
049 *tmpc = *tmpa++ - *tmpb++ - u;
050
051 /* U = carry bit of T[i]
052 * Note this saves performing an AND operation since
053 * if a carry does occur it will propagate all the way to the
054 * MSB. As a result a single shift is enough to get the carry
055 */
056 u = *tmpc >> ((mp_digit)(CHAR_BIT * sizeof (mp_digit) - 1));
057
058 /* Clear carry from T[i] */
059 *tmpc++ &= MP_MASK;
060 \}
061
062 /* now copy higher words if any, e.g. if A has more digits than B */
063 for (; i < max; i++) \{
064 /* T[i] = A[i] - U */
065 *tmpc = *tmpa++ - u;
066
067 /* U = carry bit of T[i] */
068 u = *tmpc >> ((mp_digit)(CHAR_BIT * sizeof (mp_digit) - 1));
069
070 /* Clear carry from T[i] */
071 *tmpc++ &= MP_MASK;
072 \}
073
074 /* clear digits above used (since we may not have grown result above) */
075 for (i = c->used; i < olduse; i++) \{
076 *tmpc++ = 0;
077 \}
078 \}
079
080 mp_clamp (c);
081 return MP_OKAY;
082 \}
083
084 #endif
085
\end{alltt}
\end{small}
Like low level addition we ``sort'' the inputs. Except in this case the sorting is hardcoded
(lines 24 and 25). In reality the $min$ and $max$ variables are only aliases and are only
used to make the source code easier to read. Again the pointer alias optimization is used
within this algorithm. The aliases $tmpa$, $tmpb$ and $tmpc$ are initialized
(lines 41, 42 and 43) for $a$, $b$ and $c$ respectively.
The first subtraction loop (lines 46 through 60) subtract digits from both inputs until the smaller of
the two inputs has been exhausted. As remarked earlier there is an implementation reason for using the ``awkward''
method of extracting the carry (line 56). The traditional method for extracting the carry would be to shift
by $lg(\beta)$ positions and logically AND the least significant bit. The AND operation is required because all of
the bits above the $\lg(\beta)$'th bit will be set to one after a carry occurs from subtraction. This carry
extraction requires two relatively cheap operations to extract the carry. The other method is to simply shift the
most significant bit to the least significant bit thus extracting the carry with a single cheap operation. This
optimization only works on twos compliment machines which is a safe assumption to make.
If $a$ has a larger magnitude than $b$ an additional loop (lines 63 through 72) is required to propagate
the carry through $a$ and copy the result to $c$.
\subsection{High Level Addition}
Now that both lower level addition and subtraction algorithms have been established an effective high level signed addition algorithm can be
established. This high level addition algorithm will be what other algorithms and developers will use to perform addition of mp\_int data
types.
Recall from section 5.2 that an mp\_int represents an integer with an unsigned mantissa (\textit{the array of digits}) and a \textbf{sign}
flag. A high level addition is actually performed as a series of eight separate cases which can be optimized down to three unique cases.
\begin{figure}[!here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_add}. \\
\textbf{Input}. Two mp\_ints $a$ and $b$ \\
\textbf{Output}. The signed addition $c = a + b$. \\
\hline \\
1. if $a.sign = b.sign$ then do \\
\hspace{3mm}1.1 $c.sign \leftarrow a.sign$ \\
\hspace{3mm}1.2 $c \leftarrow \vert a \vert + \vert b \vert$ (\textit{s\_mp\_add})\\
2. else do \\
\hspace{3mm}2.1 if $\vert a \vert < \vert b \vert$ then do (\textit{mp\_cmp\_mag}) \\
\hspace{6mm}2.1.1 $c.sign \leftarrow b.sign$ \\
\hspace{6mm}2.1.2 $c \leftarrow \vert b \vert - \vert a \vert$ (\textit{s\_mp\_sub}) \\
\hspace{3mm}2.2 else do \\
\hspace{6mm}2.2.1 $c.sign \leftarrow a.sign$ \\
\hspace{6mm}2.2.2 $c \leftarrow \vert a \vert - \vert b \vert$ \\
3. Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_add}
\end{figure}
\textbf{Algorithm mp\_add.}
This algorithm performs the signed addition of two mp\_int variables. There is no reference algorithm to draw upon from
either \cite{TAOCPV2} or \cite{HAC} since they both only provide unsigned operations. The algorithm is fairly
straightforward but restricted since subtraction can only produce positive results.
\begin{figure}[here]
\begin{small}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline \textbf{Sign of $a$} & \textbf{Sign of $b$} & \textbf{$\vert a \vert > \vert b \vert $} & \textbf{Unsigned Operation} & \textbf{Result Sign Flag} \\
\hline $+$ & $+$ & Yes & $c = a + b$ & $a.sign$ \\
\hline $+$ & $+$ & No & $c = a + b$ & $a.sign$ \\
\hline $-$ & $-$ & Yes & $c = a + b$ & $a.sign$ \\
\hline $-$ & $-$ & No & $c = a + b$ & $a.sign$ \\
\hline &&&&\\
\hline $+$ & $-$ & No & $c = b - a$ & $b.sign$ \\
\hline $-$ & $+$ & No & $c = b - a$ & $b.sign$ \\
\hline &&&&\\
\hline $+$ & $-$ & Yes & $c = a - b$ & $a.sign$ \\
\hline $-$ & $+$ & Yes & $c = a - b$ & $a.sign$ \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Addition Guide Chart}
\label{fig:AddChart}
\end{figure}
Figure~\ref{fig:AddChart} lists all of the eight possible input combinations and is sorted to show that only three
specific cases need to be handled. The return code of the unsigned operations at step 1.2, 2.1.2 and 2.2.2 are
forwarded to step three to check for errors. This simplifies the description of the algorithm considerably and best
follows how the implementation actually was achieved.
Also note how the \textbf{sign} is set before the unsigned addition or subtraction is performed. Recall from the descriptions of algorithms
s\_mp\_add and s\_mp\_sub that the mp\_clamp function is used at the end to trim excess digits. The mp\_clamp algorithm will set the \textbf{sign}
to \textbf{MP\_ZPOS} when the \textbf{used} digit count reaches zero.
For example, consider performing $-a + a$ with algorithm mp\_add. By the description of the algorithm the sign is set to \textbf{MP\_NEG} which would
produce a result of $-0$. However, since the sign is set first then the unsigned addition is performed the subsequent usage of algorithm mp\_clamp
within algorithm s\_mp\_add will force $-0$ to become $0$.
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_add.c
\vspace{-3mm}
\begin{alltt}
016
017 /* high level addition (handles signs) */
018 int mp_add (mp_int * a, mp_int * b, mp_int * c)
019 \{
020 int sa, sb, res;
021
022 /* get sign of both inputs */
023 sa = a->sign;
024 sb = b->sign;
025
026 /* handle two cases, not four */
027 if (sa == sb) \{
028 /* both positive or both negative */
029 /* add their magnitudes, copy the sign */
030 c->sign = sa;
031 res = s_mp_add (a, b, c);
032 \} else \{
033 /* one positive, the other negative */
034 /* subtract the one with the greater magnitude from */
035 /* the one of the lesser magnitude. The result gets */
036 /* the sign of the one with the greater magnitude. */
037 if (mp_cmp_mag (a, b) == MP_LT) \{
038 c->sign = sb;
039 res = s_mp_sub (b, a, c);
040 \} else \{
041 c->sign = sa;
042 res = s_mp_sub (a, b, c);
043 \}
044 \}
045 return res;
046 \}
047
048 #endif
049
\end{alltt}
\end{small}
The source code follows the algorithm fairly closely. The most notable new source code addition is the usage of the $res$ integer variable which
is used to pass result of the unsigned operations forward. Unlike in the algorithm, the variable $res$ is merely returned as is without
explicitly checking it and returning the constant \textbf{MP\_OKAY}. The observation is this algorithm will succeed or fail only if the lower
level functions do so. Returning their return code is sufficient.
\subsection{High Level Subtraction}
The high level signed subtraction algorithm is essentially the same as the high level signed addition algorithm.
\newpage\begin{figure}[!here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_sub}. \\
\textbf{Input}. Two mp\_ints $a$ and $b$ \\
\textbf{Output}. The signed subtraction $c = a - b$. \\
\hline \\
1. if $a.sign \ne b.sign$ then do \\
\hspace{3mm}1.1 $c.sign \leftarrow a.sign$ \\
\hspace{3mm}1.2 $c \leftarrow \vert a \vert + \vert b \vert$ (\textit{s\_mp\_add}) \\
2. else do \\
\hspace{3mm}2.1 if $\vert a \vert \ge \vert b \vert$ then do (\textit{mp\_cmp\_mag}) \\
\hspace{6mm}2.1.1 $c.sign \leftarrow a.sign$ \\
\hspace{6mm}2.1.2 $c \leftarrow \vert a \vert - \vert b \vert$ (\textit{s\_mp\_sub}) \\
\hspace{3mm}2.2 else do \\
\hspace{6mm}2.2.1 $c.sign \leftarrow \left \lbrace \begin{array}{ll}
MP\_ZPOS & \mbox{if }a.sign = MP\_NEG \\
MP\_NEG & \mbox{otherwise} \\
\end{array} \right .$ \\
\hspace{6mm}2.2.2 $c \leftarrow \vert b \vert - \vert a \vert$ \\
3. Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_sub}
\end{figure}
\textbf{Algorithm mp\_sub.}
This algorithm performs the signed subtraction of two inputs. Similar to algorithm mp\_add there is no reference in either \cite{TAOCPV2} or
\cite{HAC}. Also this algorithm is restricted by algorithm s\_mp\_sub. Chart \ref{fig:SubChart} lists the eight possible inputs and
the operations required.
\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline \textbf{Sign of $a$} & \textbf{Sign of $b$} & \textbf{$\vert a \vert \ge \vert b \vert $} & \textbf{Unsigned Operation} & \textbf{Result Sign Flag} \\
\hline $+$ & $-$ & Yes & $c = a + b$ & $a.sign$ \\
\hline $+$ & $-$ & No & $c = a + b$ & $a.sign$ \\
\hline $-$ & $+$ & Yes & $c = a + b$ & $a.sign$ \\
\hline $-$ & $+$ & No & $c = a + b$ & $a.sign$ \\
\hline &&&& \\
\hline $+$ & $+$ & Yes & $c = a - b$ & $a.sign$ \\
\hline $-$ & $-$ & Yes & $c = a - b$ & $a.sign$ \\
\hline &&&& \\
\hline $+$ & $+$ & No & $c = b - a$ & $\mbox{opposite of }a.sign$ \\
\hline $-$ & $-$ & No & $c = b - a$ & $\mbox{opposite of }a.sign$ \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Subtraction Guide Chart}
\label{fig:SubChart}
\end{figure}
Similar to the case of algorithm mp\_add the \textbf{sign} is set first before the unsigned addition or subtraction. That is to prevent the
algorithm from producing $-a - -a = -0$ as a result.
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_sub.c
\vspace{-3mm}
\begin{alltt}
016
017 /* high level subtraction (handles signs) */
018 int
019 mp_sub (mp_int * a, mp_int * b, mp_int * c)
020 \{
021 int sa, sb, res;
022
023 sa = a->sign;
024 sb = b->sign;
025
026 if (sa != sb) \{
027 /* subtract a negative from a positive, OR */
028 /* subtract a positive from a negative. */
029 /* In either case, ADD their magnitudes, */
030 /* and use the sign of the first number. */
031 c->sign = sa;
032 res = s_mp_add (a, b, c);
033 \} else \{
034 /* subtract a positive from a positive, OR */
035 /* subtract a negative from a negative. */
036 /* First, take the difference between their */
037 /* magnitudes, then... */
038 if (mp_cmp_mag (a, b) != MP_LT) \{
039 /* Copy the sign from the first */
040 c->sign = sa;
041 /* The first has a larger or equal magnitude */
042 res = s_mp_sub (a, b, c);
043 \} else \{
044 /* The result has the *opposite* sign from */
045 /* the first number. */
046 c->sign = (sa == MP_ZPOS) ? MP_NEG : MP_ZPOS;
047 /* The second has a larger magnitude */
048 res = s_mp_sub (b, a, c);
049 \}
050 \}
051 return res;
052 \}
053
054 #endif
055
\end{alltt}
\end{small}
Much like the implementation of algorithm mp\_add the variable $res$ is used to catch the return code of the unsigned addition or subtraction operations
and forward it to the end of the function. On line 38 the ``not equal to'' \textbf{MP\_LT} expression is used to emulate a
``greater than or equal to'' comparison.
\section{Bit and Digit Shifting}
It is quite common to think of a multiple precision integer as a polynomial in $x$, that is $y = f(\beta)$ where $f(x) = \sum_{i=0}^{n-1} a_i x^i$.
This notation arises within discussion of Montgomery and Diminished Radix Reduction as well as Karatsuba multiplication and squaring.
In order to facilitate operations on polynomials in $x$ as above a series of simple ``digit'' algorithms have to be established. That is to shift
the digits left or right as well to shift individual bits of the digits left and right. It is important to note that not all ``shift'' operations
are on radix-$\beta$ digits.
\subsection{Multiplication by Two}
In a binary system where the radix is a power of two multiplication by two not only arises often in other algorithms it is a fairly efficient
operation to perform. A single precision logical shift left is sufficient to multiply a single digit by two.
\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_mul\_2}. \\
\textbf{Input}. One mp\_int $a$ \\
\textbf{Output}. $b = 2a$. \\
\hline \\
1. If $b.alloc < a.used + 1$ then grow $b$ to hold $a.used + 1$ digits. (\textit{mp\_grow}) \\
2. $oldused \leftarrow b.used$ \\
3. $b.used \leftarrow a.used$ \\
4. $r \leftarrow 0$ \\
5. for $n$ from 0 to $a.used - 1$ do \\
\hspace{3mm}5.1 $rr \leftarrow a_n >> (lg(\beta) - 1)$ \\
\hspace{3mm}5.2 $b_n \leftarrow (a_n << 1) + r \mbox{ (mod }\beta\mbox{)}$ \\
\hspace{3mm}5.3 $r \leftarrow rr$ \\
6. If $r \ne 0$ then do \\
\hspace{3mm}6.1 $b_{n + 1} \leftarrow r$ \\
\hspace{3mm}6.2 $b.used \leftarrow b.used + 1$ \\
7. If $b.used < oldused - 1$ then do \\
\hspace{3mm}7.1 for $n$ from $b.used$ to $oldused - 1$ do \\
\hspace{6mm}7.1.1 $b_n \leftarrow 0$ \\
8. $b.sign \leftarrow a.sign$ \\
9. Return(\textit{MP\_OKAY}).\\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_mul\_2}
\end{figure}
\textbf{Algorithm mp\_mul\_2.}
This algorithm will quickly multiply a mp\_int by two provided $\beta$ is a power of two. Neither \cite{TAOCPV2} nor \cite{HAC} describe such
an algorithm despite the fact it arises often in other algorithms. The algorithm is setup much like the lower level algorithm s\_mp\_add since
it is for all intents and purposes equivalent to the operation $b = \vert a \vert + \vert a \vert$.
Step 1 and 2 grow the input as required to accomodate the maximum number of \textbf{used} digits in the result. The initial \textbf{used} count
is set to $a.used$ at step 4. Only if there is a final carry will the \textbf{used} count require adjustment.
Step 6 is an optimization implementation of the addition loop for this specific case. That is since the two values being added together
are the same there is no need to perform two reads from the digits of $a$. Step 6.1 performs a single precision shift on the current digit $a_n$ to
obtain what will be the carry for the next iteration. Step 6.2 calculates the $n$'th digit of the result as single precision shift of $a_n$ plus
the previous carry. Recall from section 4.1 that $a_n << 1$ is equivalent to $a_n \cdot 2$. An iteration of the addition loop is finished with
forwarding the carry to the next iteration.
Step 7 takes care of any final carry by setting the $a.used$'th digit of the result to the carry and augmenting the \textbf{used} count of $b$.
Step 8 clears any leading digits of $b$ in case it originally had a larger magnitude than $a$.
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_mul\_2.c
\vspace{-3mm}
\begin{alltt}
016
017 /* b = a*2 */
018 int mp_mul_2(mp_int * a, mp_int * b)
019 \{
020 int x, res, oldused;
021
022 /* grow to accomodate result */
023 if (b->alloc < a->used + 1) \{
024 if ((res = mp_grow (b, a->used + 1)) != MP_OKAY) \{
025 return res;
026 \}
027 \}
028
029 oldused = b->used;
030 b->used = a->used;
031
032 \{
033 register mp_digit r, rr, *tmpa, *tmpb;
034
035 /* alias for source */
036 tmpa = a->dp;
037
038 /* alias for dest */
039 tmpb = b->dp;
040
041 /* carry */
042 r = 0;
043 for (x = 0; x < a->used; x++) \{
044
045 /* get what will be the *next* carry bit from the
046 * MSB of the current digit
047 */
048 rr = *tmpa >> ((mp_digit)(DIGIT_BIT - 1));
049
050 /* now shift up this digit, add in the carry [from the previous] */
051 *tmpb++ = ((*tmpa++ << ((mp_digit)1)) | r) & MP_MASK;
052
053 /* copy the carry that would be from the source
054 * digit into the next iteration
055 */
056 r = rr;
057 \}
058
059 /* new leading digit? */
060 if (r != 0) \{
061 /* add a MSB which is always 1 at this point */
062 *tmpb = 1;
063 ++(b->used);
064 \}
065
066 /* now zero any excess digits on the destination
067 * that we didn't write to
068 */
069 tmpb = b->dp + b->used;
070 for (x = b->used; x < oldused; x++) \{
071 *tmpb++ = 0;
072 \}
073 \}
074 b->sign = a->sign;
075 return MP_OKAY;
076 \}
077 #endif
078
\end{alltt}
\end{small}
This implementation is essentially an optimized implementation of s\_mp\_add for the case of doubling an input. The only noteworthy difference
is the use of the logical shift operator on line 51 to perform a single precision doubling.
\subsection{Division by Two}
A division by two can just as easily be accomplished with a logical shift right as multiplication by two can be with a logical shift left.
\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_div\_2}. \\
\textbf{Input}. One mp\_int $a$ \\
\textbf{Output}. $b = a/2$. \\
\hline \\
1. If $b.alloc < a.used$ then grow $b$ to hold $a.used$ digits. (\textit{mp\_grow}) \\
2. If the reallocation failed return(\textit{MP\_MEM}). \\
3. $oldused \leftarrow b.used$ \\
4. $b.used \leftarrow a.used$ \\
5. $r \leftarrow 0$ \\
6. for $n$ from $b.used - 1$ to $0$ do \\
\hspace{3mm}6.1 $rr \leftarrow a_n \mbox{ (mod }2\mbox{)}$\\
\hspace{3mm}6.2 $b_n \leftarrow (a_n >> 1) + (r << (lg(\beta) - 1)) \mbox{ (mod }\beta\mbox{)}$ \\
\hspace{3mm}6.3 $r \leftarrow rr$ \\
7. If $b.used < oldused - 1$ then do \\
\hspace{3mm}7.1 for $n$ from $b.used$ to $oldused - 1$ do \\
\hspace{6mm}7.1.1 $b_n \leftarrow 0$ \\
8. $b.sign \leftarrow a.sign$ \\
9. Clamp excess digits of $b$. (\textit{mp\_clamp}) \\
10. Return(\textit{MP\_OKAY}).\\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_div\_2}
\end{figure}
\textbf{Algorithm mp\_div\_2.}
This algorithm will divide an mp\_int by two using logical shifts to the right. Like mp\_mul\_2 it uses a modified low level addition
core as the basis of the algorithm. Unlike mp\_mul\_2 the shift operations work from the leading digit to the trailing digit. The algorithm
could be written to work from the trailing digit to the leading digit however, it would have to stop one short of $a.used - 1$ digits to prevent
reading past the end of the array of digits.
Essentially the loop at step 6 is similar to that of mp\_mul\_2 except the logical shifts go in the opposite direction and the carry is at the
least significant bit not the most significant bit.
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_div\_2.c
\vspace{-3mm}
\begin{alltt}
016
017 /* b = a/2 */
018 int mp_div_2(mp_int * a, mp_int * b)
019 \{
020 int x, res, oldused;
021
022 /* copy */
023 if (b->alloc < a->used) \{
024 if ((res = mp_grow (b, a->used)) != MP_OKAY) \{
025 return res;
026 \}
027 \}
028
029 oldused = b->used;
030 b->used = a->used;
031 \{
032 register mp_digit r, rr, *tmpa, *tmpb;
033
034 /* source alias */
035 tmpa = a->dp + b->used - 1;
036
037 /* dest alias */
038 tmpb = b->dp + b->used - 1;
039
040 /* carry */
041 r = 0;
042 for (x = b->used - 1; x >= 0; x--) \{
043 /* get the carry for the next iteration */
044 rr = *tmpa & 1;
045
046 /* shift the current digit, add in carry and store */
047 *tmpb-- = (*tmpa-- >> 1) | (r << (DIGIT_BIT - 1));
048
049 /* forward carry to next iteration */
050 r = rr;
051 \}
052
053 /* zero excess digits */
054 tmpb = b->dp + b->used;
055 for (x = b->used; x < oldused; x++) \{
056 *tmpb++ = 0;
057 \}
058 \}
059 b->sign = a->sign;
060 mp_clamp (b);
061 return MP_OKAY;
062 \}
063 #endif
064
\end{alltt}
\end{small}
\section{Polynomial Basis Operations}
Recall from section 4.3 that any integer can be represented as a polynomial in $x$ as $y = f(\beta)$. Such a representation is also known as
the polynomial basis \cite[pp. 48]{ROSE}. Given such a notation a multiplication or division by $x$ amounts to shifting whole digits a single
place. The need for such operations arises in several other higher level algorithms such as Barrett and Montgomery reduction, integer
division and Karatsuba multiplication.
Converting from an array of digits to polynomial basis is very simple. Consider the integer $y \equiv (a_2, a_1, a_0)_{\beta}$ and recall that
$y = \sum_{i=0}^{2} a_i \beta^i$. Simply replace $\beta$ with $x$ and the expression is in polynomial basis. For example, $f(x) = 8x + 9$ is the
polynomial basis representation for $89$ using radix ten. That is, $f(10) = 8(10) + 9 = 89$.
\subsection{Multiplication by $x$}
Given a polynomial in $x$ such as $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0$ multiplying by $x$ amounts to shifting the coefficients up one
degree. In this case $f(x) \cdot x = a_n x^{n+1} + a_{n-1} x^n + ... + a_0 x$. From a scalar basis point of view multiplying by $x$ is equivalent to
multiplying by the integer $\beta$.
\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_lshd}. \\
\textbf{Input}. One mp\_int $a$ and an integer $b$ \\
\textbf{Output}. $a \leftarrow a \cdot \beta^b$ (equivalent to multiplication by $x^b$). \\
\hline \\
1. If $b \le 0$ then return(\textit{MP\_OKAY}). \\
2. If $a.alloc < a.used + b$ then grow $a$ to at least $a.used + b$ digits. (\textit{mp\_grow}). \\
3. If the reallocation failed return(\textit{MP\_MEM}). \\
4. $a.used \leftarrow a.used + b$ \\
5. $i \leftarrow a.used - 1$ \\
6. $j \leftarrow a.used - 1 - b$ \\
7. for $n$ from $a.used - 1$ to $b$ do \\
\hspace{3mm}7.1 $a_{i} \leftarrow a_{j}$ \\
\hspace{3mm}7.2 $i \leftarrow i - 1$ \\
\hspace{3mm}7.3 $j \leftarrow j - 1$ \\
8. for $n$ from 0 to $b - 1$ do \\
\hspace{3mm}8.1 $a_n \leftarrow 0$ \\
9. Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_lshd}
\end{figure}
\textbf{Algorithm mp\_lshd.}
This algorithm multiplies an mp\_int by the $b$'th power of $x$. This is equivalent to multiplying by $\beta^b$. The algorithm differs
from the other algorithms presented so far as it performs the operation in place instead storing the result in a separate location. The
motivation behind this change is due to the way this function is typically used. Algorithms such as mp\_add store the result in an optionally
different third mp\_int because the original inputs are often still required. Algorithm mp\_lshd (\textit{and similarly algorithm mp\_rshd}) is
typically used on values where the original value is no longer required. The algorithm will return success immediately if
$b \le 0$ since the rest of algorithm is only valid when $b > 0$.
First the destination $a$ is grown as required to accomodate the result. The counters $i$ and $j$ are used to form a \textit{sliding window} over
the digits of $a$ of length $b$. The head of the sliding window is at $i$ (\textit{the leading digit}) and the tail at $j$ (\textit{the trailing digit}).
The loop on step 7 copies the digit from the tail to the head. In each iteration the window is moved down one digit. The last loop on
step 8 sets the lower $b$ digits to zero.
\newpage
\begin{center}
\begin{figure}[here]
\includegraphics{pics/sliding_window.ps}
\caption{Sliding Window Movement}
\label{pic:sliding_window}
\end{figure}
\end{center}
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_lshd.c
\vspace{-3mm}
\begin{alltt}
016
017 /* shift left a certain amount of digits */
018 int mp_lshd (mp_int * a, int b)
019 \{
020 int x, res;
021
022 /* if its less than zero return */
023 if (b <= 0) \{
024 return MP_OKAY;
025 \}
026
027 /* grow to fit the new digits */
028 if (a->alloc < a->used + b) \{
029 if ((res = mp_grow (a, a->used + b)) != MP_OKAY) \{
030 return res;
031 \}
032 \}
033
034 \{
035 register mp_digit *top, *bottom;
036
037 /* increment the used by the shift amount then copy upwards */
038 a->used += b;
039
040 /* top */
041 top = a->dp + a->used - 1;
042
043 /* base */
044 bottom = a->dp + a->used - 1 - b;
045
046 /* much like mp_rshd this is implemented using a sliding window
047 * except the window goes the otherway around. Copying from
048 * the bottom to the top. see bn_mp_rshd.c for more info.
049 */
050 for (x = a->used - 1; x >= b; x--) \{
051 *top-- = *bottom--;
052 \}
053
054 /* zero the lower digits */
055 top = a->dp;
056 for (x = 0; x < b; x++) \{
057 *top++ = 0;
058 \}
059 \}
060 return MP_OKAY;
061 \}
062 #endif
063
\end{alltt}
\end{small}
The if statement (line 23) ensures that the $b$ variable is greater than zero since we do not interpret negative
shift counts properly. The \textbf{used} count is incremented by $b$ before the copy loop begins. This elminates
the need for an additional variable in the for loop. The variable $top$ (line 41) is an alias
for the leading digit while $bottom$ (line 44) is an alias for the trailing edge. The aliases form a
window of exactly $b$ digits over the input.
\subsection{Division by $x$}
Division by powers of $x$ is easily achieved by shifting the digits right and removing any that will end up to the right of the zero'th digit.
\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_rshd}. \\
\textbf{Input}. One mp\_int $a$ and an integer $b$ \\
\textbf{Output}. $a \leftarrow a / \beta^b$ (Divide by $x^b$). \\
\hline \\
1. If $b \le 0$ then return. \\
2. If $a.used \le b$ then do \\
\hspace{3mm}2.1 Zero $a$. (\textit{mp\_zero}). \\
\hspace{3mm}2.2 Return. \\
3. $i \leftarrow 0$ \\
4. $j \leftarrow b$ \\
5. for $n$ from 0 to $a.used - b - 1$ do \\
\hspace{3mm}5.1 $a_i \leftarrow a_j$ \\
\hspace{3mm}5.2 $i \leftarrow i + 1$ \\
\hspace{3mm}5.3 $j \leftarrow j + 1$ \\
6. for $n$ from $a.used - b$ to $a.used - 1$ do \\
\hspace{3mm}6.1 $a_n \leftarrow 0$ \\
7. $a.used \leftarrow a.used - b$ \\
8. Return. \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_rshd}
\end{figure}
\textbf{Algorithm mp\_rshd.}
This algorithm divides the input in place by the $b$'th power of $x$. It is analogous to dividing by a $\beta^b$ but much quicker since
it does not require single precision division. This algorithm does not actually return an error code as it cannot fail.
If the input $b$ is less than one the algorithm quickly returns without performing any work. If the \textbf{used} count is less than or equal
to the shift count $b$ then it will simply zero the input and return.
After the trivial cases of inputs have been handled the sliding window is setup. Much like the case of algorithm mp\_lshd a sliding window that
is $b$ digits wide is used to copy the digits. Unlike mp\_lshd the window slides in the opposite direction from the trailing to the leading digit.
Also the digits are copied from the leading to the trailing edge.
Once the window copy is complete the upper digits must be zeroed and the \textbf{used} count decremented.
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_rshd.c
\vspace{-3mm}
\begin{alltt}
016
017 /* shift right a certain amount of digits */
018 void mp_rshd (mp_int * a, int b)
019 \{
020 int x;
021
022 /* if b <= 0 then ignore it */
023 if (b <= 0) \{
024 return;
025 \}
026
027 /* if b > used then simply zero it and return */
028 if (a->used <= b) \{
029 mp_zero (a);
030 return;
031 \}
032
033 \{
034 register mp_digit *bottom, *top;
035
036 /* shift the digits down */
037
038 /* bottom */
039 bottom = a->dp;
040
041 /* top [offset into digits] */
042 top = a->dp + b;
043
044 /* this is implemented as a sliding window where
045 * the window is b-digits long and digits from
046 * the top of the window are copied to the bottom
047 *
048 * e.g.
049
050 b-2 | b-1 | b0 | b1 | b2 | ... | bb | ---->
051 /\symbol{92} | ---->
052 \symbol{92}-------------------/ ---->
053 */
054 for (x = 0; x < (a->used - b); x++) \{
055 *bottom++ = *top++;
056 \}
057
058 /* zero the top digits */
059 for (; x < a->used; x++) \{
060 *bottom++ = 0;
061 \}
062 \}
063
064 /* remove excess digits */
065 a->used -= b;
066 \}
067 #endif
068
\end{alltt}
\end{small}
The only noteworthy element of this routine is the lack of a return type since it cannot fail. Like mp\_lshd() we
form a sliding window except we copy in the other direction. After the window (line 59) we then zero
the upper digits of the input to make sure the result is correct.
\section{Powers of Two}
Now that algorithms for moving single bits as well as whole digits exist algorithms for moving the ``in between'' distances are required. For
example, to quickly multiply by $2^k$ for any $k$ without using a full multiplier algorithm would prove useful. Instead of performing single
shifts $k$ times to achieve a multiplication by $2^{\pm k}$ a mixture of whole digit shifting and partial digit shifting is employed.
\subsection{Multiplication by Power of Two}
\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_mul\_2d}. \\
\textbf{Input}. One mp\_int $a$ and an integer $b$ \\
\textbf{Output}. $c \leftarrow a \cdot 2^b$. \\
\hline \\
1. $c \leftarrow a$. (\textit{mp\_copy}) \\
2. If $c.alloc < c.used + \lfloor b / lg(\beta) \rfloor + 2$ then grow $c$ accordingly. \\
3. If the reallocation failed return(\textit{MP\_MEM}). \\
4. If $b \ge lg(\beta)$ then \\
\hspace{3mm}4.1 $c \leftarrow c \cdot \beta^{\lfloor b / lg(\beta) \rfloor}$ (\textit{mp\_lshd}). \\
\hspace{3mm}4.2 If step 4.1 failed return(\textit{MP\_MEM}). \\
5. $d \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\
6. If $d \ne 0$ then do \\
\hspace{3mm}6.1 $mask \leftarrow 2^d$ \\
\hspace{3mm}6.2 $r \leftarrow 0$ \\
\hspace{3mm}6.3 for $n$ from $0$ to $c.used - 1$ do \\
\hspace{6mm}6.3.1 $rr \leftarrow c_n >> (lg(\beta) - d) \mbox{ (mod }mask\mbox{)}$ \\
\hspace{6mm}6.3.2 $c_n \leftarrow (c_n << d) + r \mbox{ (mod }\beta\mbox{)}$ \\
\hspace{6mm}6.3.3 $r \leftarrow rr$ \\
\hspace{3mm}6.4 If $r > 0$ then do \\
\hspace{6mm}6.4.1 $c_{c.used} \leftarrow r$ \\
\hspace{6mm}6.4.2 $c.used \leftarrow c.used + 1$ \\
7. Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_mul\_2d}
\end{figure}
\textbf{Algorithm mp\_mul\_2d.}
This algorithm multiplies $a$ by $2^b$ and stores the result in $c$. The algorithm uses algorithm mp\_lshd and a derivative of algorithm mp\_mul\_2 to
quickly compute the product.
First the algorithm will multiply $a$ by $x^{\lfloor b / lg(\beta) \rfloor}$ which will ensure that the remainder multiplicand is less than
$\beta$. For example, if $b = 37$ and $\beta = 2^{28}$ then this step will multiply by $x$ leaving a multiplication by $2^{37 - 28} = 2^{9}$
left.
After the digits have been shifted appropriately at most $lg(\beta) - 1$ shifts are left to perform. Step 5 calculates the number of remaining shifts
required. If it is non-zero a modified shift loop is used to calculate the remaining product.
Essentially the loop is a generic version of algorith mp\_mul2 designed to handle any shift count in the range $1 \le x < lg(\beta)$. The $mask$
variable is used to extract the upper $d$ bits to form the carry for the next iteration.
This algorithm is loosely measured as a $O(2n)$ algorithm which means that if the input is $n$-digits that it takes $2n$ ``time'' to
complete. It is possible to optimize this algorithm down to a $O(n)$ algorithm at a cost of making the algorithm slightly harder to follow.
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_mul\_2d.c
\vspace{-3mm}
\begin{alltt}
016
017 /* shift left by a certain bit count */
018 int mp_mul_2d (mp_int * a, int b, mp_int * c)
019 \{
020 mp_digit d;
021 int res;
022
023 /* copy */
024 if (a != c) \{
025 if ((res = mp_copy (a, c)) != MP_OKAY) \{
026 return res;
027 \}
028 \}
029
030 if (c->alloc < (int)(c->used + b/DIGIT_BIT + 1)) \{
031 if ((res = mp_grow (c, c->used + b / DIGIT_BIT + 1)) != MP_OKAY) \{
032 return res;
033 \}
034 \}
035
036 /* shift by as many digits in the bit count */
037 if (b >= (int)DIGIT_BIT) \{
038 if ((res = mp_lshd (c, b / DIGIT_BIT)) != MP_OKAY) \{
039 return res;
040 \}
041 \}
042
043 /* shift any bit count < DIGIT_BIT */
044 d = (mp_digit) (b % DIGIT_BIT);
045 if (d != 0) \{
046 register mp_digit *tmpc, shift, mask, r, rr;
047 register int x;
048
049 /* bitmask for carries */
050 mask = (((mp_digit)1) << d) - 1;
051
052 /* shift for msbs */
053 shift = DIGIT_BIT - d;
054
055 /* alias */
056 tmpc = c->dp;
057
058 /* carry */
059 r = 0;
060 for (x = 0; x < c->used; x++) \{
061 /* get the higher bits of the current word */
062 rr = (*tmpc >> shift) & mask;
063
064 /* shift the current word and OR in the carry */
065 *tmpc = ((*tmpc << d) | r) & MP_MASK;
066 ++tmpc;
067
068 /* set the carry to the carry bits of the current word */
069 r = rr;
070 \}
071
072 /* set final carry */
073 if (r != 0) \{
074 c->dp[(c->used)++] = r;
075 \}
076 \}
077 mp_clamp (c);
078 return MP_OKAY;
079 \}
080 #endif
081
\end{alltt}
\end{small}
The shifting is performed in--place which means the first step (line 24) is to copy the input to the
destination. We avoid calling mp\_copy() by making sure the mp\_ints are different. The destination then
has to be grown (line 31) to accomodate the result.
If the shift count $b$ is larger than $lg(\beta)$ then a call to mp\_lshd() is used to handle all of the multiples
of $lg(\beta)$. Leaving only a remaining shift of $lg(\beta) - 1$ or fewer bits left. Inside the actual shift
loop (lines 45 to 76) we make use of pre--computed values $shift$ and $mask$. These are used to
extract the carry bit(s) to pass into the next iteration of the loop. The $r$ and $rr$ variables form a
chain between consecutive iterations to propagate the carry.
\subsection{Division by Power of Two}
\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_div\_2d}. \\
\textbf{Input}. One mp\_int $a$ and an integer $b$ \\
\textbf{Output}. $c \leftarrow \lfloor a / 2^b \rfloor, d \leftarrow a \mbox{ (mod }2^b\mbox{)}$. \\
\hline \\
1. If $b \le 0$ then do \\
\hspace{3mm}1.1 $c \leftarrow a$ (\textit{mp\_copy}) \\
\hspace{3mm}1.2 $d \leftarrow 0$ (\textit{mp\_zero}) \\
\hspace{3mm}1.3 Return(\textit{MP\_OKAY}). \\
2. $c \leftarrow a$ \\
3. $d \leftarrow a \mbox{ (mod }2^b\mbox{)}$ (\textit{mp\_mod\_2d}) \\
4. If $b \ge lg(\beta)$ then do \\
\hspace{3mm}4.1 $c \leftarrow \lfloor c/\beta^{\lfloor b/lg(\beta) \rfloor} \rfloor$ (\textit{mp\_rshd}). \\
5. $k \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\
6. If $k \ne 0$ then do \\
\hspace{3mm}6.1 $mask \leftarrow 2^k$ \\
\hspace{3mm}6.2 $r \leftarrow 0$ \\
\hspace{3mm}6.3 for $n$ from $c.used - 1$ to $0$ do \\
\hspace{6mm}6.3.1 $rr \leftarrow c_n \mbox{ (mod }mask\mbox{)}$ \\
\hspace{6mm}6.3.2 $c_n \leftarrow (c_n >> k) + (r << (lg(\beta) - k))$ \\
\hspace{6mm}6.3.3 $r \leftarrow rr$ \\
7. Clamp excess digits of $c$. (\textit{mp\_clamp}) \\
8. Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_div\_2d}
\end{figure}
\textbf{Algorithm mp\_div\_2d.}
This algorithm will divide an input $a$ by $2^b$ and produce the quotient and remainder. The algorithm is designed much like algorithm
mp\_mul\_2d by first using whole digit shifts then single precision shifts. This algorithm will also produce the remainder of the division
by using algorithm mp\_mod\_2d.
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_div\_2d.c
\vspace{-3mm}
\begin{alltt}
016
017 /* shift right by a certain bit count (store quotient in c, optional remaind
er in d) */
018 int mp_div_2d (mp_int * a, int b, mp_int * c, mp_int * d)
019 \{
020 mp_digit D, r, rr;
021 int x, res;
022 mp_int t;
023
024
025 /* if the shift count is <= 0 then we do no work */
026 if (b <= 0) \{
027 res = mp_copy (a, c);
028 if (d != NULL) \{
029 mp_zero (d);
030 \}
031 return res;
032 \}
033
034 if ((res = mp_init (&t)) != MP_OKAY) \{
035 return res;
036 \}
037
038 /* get the remainder */
039 if (d != NULL) \{
040 if ((res = mp_mod_2d (a, b, &t)) != MP_OKAY) \{
041 mp_clear (&t);
042 return res;
043 \}
044 \}
045
046 /* copy */
047 if ((res = mp_copy (a, c)) != MP_OKAY) \{
048 mp_clear (&t);
049 return res;
050 \}
051
052 /* shift by as many digits in the bit count */
053 if (b >= (int)DIGIT_BIT) \{
054 mp_rshd (c, b / DIGIT_BIT);
055 \}
056
057 /* shift any bit count < DIGIT_BIT */
058 D = (mp_digit) (b % DIGIT_BIT);
059 if (D != 0) \{
060 register mp_digit *tmpc, mask, shift;
061
062 /* mask */
063 mask = (((mp_digit)1) << D) - 1;
064
065 /* shift for lsb */
066 shift = DIGIT_BIT - D;
067
068 /* alias */
069 tmpc = c->dp + (c->used - 1);
070
071 /* carry */
072 r = 0;
073 for (x = c->used - 1; x >= 0; x--) \{
074 /* get the lower bits of this word in a temp */
075 rr = *tmpc & mask;
076
077 /* shift the current word and mix in the carry bits from the previous
word */
078 *tmpc = (*tmpc >> D) | (r << shift);
079 --tmpc;
080
081 /* set the carry to the carry bits of the current word found above */
082 r = rr;
083 \}
084 \}
085 mp_clamp (c);
086 if (d != NULL) \{
087 mp_exch (&t, d);
088 \}
089 mp_clear (&t);
090 return MP_OKAY;
091 \}
092 #endif
093
\end{alltt}
\end{small}
The implementation of algorithm mp\_div\_2d is slightly different than the algorithm specifies. The remainder $d$ may be optionally
ignored by passing \textbf{NULL} as the pointer to the mp\_int variable. The temporary mp\_int variable $t$ is used to hold the
result of the remainder operation until the end. This allows $d$ and $a$ to represent the same mp\_int without modifying $a$ before
the quotient is obtained.
The remainder of the source code is essentially the same as the source code for mp\_mul\_2d. The only significant difference is
the direction of the shifts.
\subsection{Remainder of Division by Power of Two}
The last algorithm in the series of polynomial basis power of two algorithms is calculating the remainder of division by $2^b$. This
algorithm benefits from the fact that in twos complement arithmetic $a \mbox{ (mod }2^b\mbox{)}$ is the same as $a$ AND $2^b - 1$.
\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_mod\_2d}. \\
\textbf{Input}. One mp\_int $a$ and an integer $b$ \\
\textbf{Output}. $c \leftarrow a \mbox{ (mod }2^b\mbox{)}$. \\
\hline \\
1. If $b \le 0$ then do \\
\hspace{3mm}1.1 $c \leftarrow 0$ (\textit{mp\_zero}) \\
\hspace{3mm}1.2 Return(\textit{MP\_OKAY}). \\
2. If $b > a.used \cdot lg(\beta)$ then do \\
\hspace{3mm}2.1 $c \leftarrow a$ (\textit{mp\_copy}) \\
\hspace{3mm}2.2 Return the result of step 2.1. \\
3. $c \leftarrow a$ \\
4. If step 3 failed return(\textit{MP\_MEM}). \\
5. for $n$ from $\lceil b / lg(\beta) \rceil$ to $c.used$ do \\
\hspace{3mm}5.1 $c_n \leftarrow 0$ \\
6. $k \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\
7. $c_{\lfloor b / lg(\beta) \rfloor} \leftarrow c_{\lfloor b / lg(\beta) \rfloor} \mbox{ (mod }2^{k}\mbox{)}$. \\
8. Clamp excess digits of $c$. (\textit{mp\_clamp}) \\
9. Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_mod\_2d}
\end{figure}
\textbf{Algorithm mp\_mod\_2d.}
This algorithm will quickly calculate the value of $a \mbox{ (mod }2^b\mbox{)}$. First if $b$ is less than or equal to zero the
result is set to zero. If $b$ is greater than the number of bits in $a$ then it simply copies $a$ to $c$ and returns. Otherwise, $a$
is copied to $b$, leading digits are removed and the remaining leading digit is trimed to the exact bit count.
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_mod\_2d.c
\vspace{-3mm}
\begin{alltt}
016
017 /* calc a value mod 2**b */
018 int
019 mp_mod_2d (mp_int * a, int b, mp_int * c)
020 \{
021 int x, res;
022
023 /* if b is <= 0 then zero the int */
024 if (b <= 0) \{
025 mp_zero (c);
026 return MP_OKAY;
027 \}
028
029 /* if the modulus is larger than the value than return */
030 if (b >= (int) (a->used * DIGIT_BIT)) \{
031 res = mp_copy (a, c);
032 return res;
033 \}
034
035 /* copy */
036 if ((res = mp_copy (a, c)) != MP_OKAY) \{
037 return res;
038 \}
039
040 /* zero digits above the last digit of the modulus */
041 for (x = (b / DIGIT_BIT) + ((b % DIGIT_BIT) == 0 ? 0 : 1); x < c->used; x+
+) \{
042 c->dp[x] = 0;
043 \}
044 /* clear the digit that is not completely outside/inside the modulus */
045 c->dp[b / DIGIT_BIT] &=
046 (mp_digit) ((((mp_digit) 1) << (((mp_digit) b) % DIGIT_BIT)) - ((mp_digi
t) 1));
047 mp_clamp (c);
048 return MP_OKAY;
049 \}
050 #endif
051
\end{alltt}
\end{small}
We first avoid cases of $b \le 0$ by simply mp\_zero()'ing the destination in such cases. Next if $2^b$ is larger
than the input we just mp\_copy() the input and return right away. After this point we know we must actually
perform some work to produce the remainder.
Recalling that reducing modulo $2^k$ and a binary ``and'' with $2^k - 1$ are numerically equivalent we can quickly reduce
the number. First we zero any digits above the last digit in $2^b$ (line 41). Next we reduce the
leading digit of both (line 45) and then mp\_clamp().
\section*{Exercises}
\begin{tabular}{cl}
$\left [ 3 \right ] $ & Devise an algorithm that performs $a \cdot 2^b$ for generic values of $b$ \\
& in $O(n)$ time. \\
&\\
$\left [ 3 \right ] $ & Devise an efficient algorithm to multiply by small low hamming \\
& weight values such as $3$, $5$ and $9$. Extend it to handle all values \\
& upto $64$ with a hamming weight less than three. \\
&\\
$\left [ 2 \right ] $ & Modify the preceding algorithm to handle values of the form \\
& $2^k - 1$ as well. \\
&\\
$\left [ 3 \right ] $ & Using only algorithms mp\_mul\_2, mp\_div\_2 and mp\_add create an \\
& algorithm to multiply two integers in roughly $O(2n^2)$ time for \\
& any $n$-bit input. Note that the time of addition is ignored in the \\
& calculation. \\
& \\
$\left [ 5 \right ] $ & Improve the previous algorithm to have a working time of at most \\
& $O \left (2^{(k-1)}n + \left ({2n^2 \over k} \right ) \right )$ for an appropriate choice of $k$. Again ignore \\
& the cost of addition. \\
& \\
$\left [ 2 \right ] $ & Devise a chart to find optimal values of $k$ for the previous problem \\
& for $n = 64 \ldots 1024$ in steps of $64$. \\
& \\
$\left [ 2 \right ] $ & Using only algorithms mp\_abs and mp\_sub devise another method for \\
& calculating the result of a signed comparison. \\
&
\end{tabular}
\chapter{Multiplication and Squaring}
\section{The Multipliers}
For most number theoretic problems including certain public key cryptographic algorithms, the ``multipliers'' form the most important subset of
algorithms of any multiple precision integer package. The set of multiplier algorithms include integer multiplication, squaring and modular reduction
where in each of the algorithms single precision multiplication is the dominant operation performed. This chapter will discuss integer multiplication
and squaring, leaving modular reductions for the subsequent chapter.
The importance of the multiplier algorithms is for the most part driven by the fact that certain popular public key algorithms are based on modular
exponentiation, that is computing $d \equiv a^b \mbox{ (mod }c\mbox{)}$ for some arbitrary choice of $a$, $b$, $c$ and $d$. During a modular
exponentiation the majority\footnote{Roughly speaking a modular exponentiation will spend about 40\% of the time performing modular reductions,
35\% of the time performing squaring and 25\% of the time performing multiplications.} of the processor time is spent performing single precision
multiplications.
For centuries general purpose multiplication has required a lengthly $O(n^2)$ process, whereby each digit of one multiplicand has to be multiplied
against every digit of the other multiplicand. Traditional long-hand multiplication is based on this process; while the techniques can differ the
overall algorithm used is essentially the same. Only ``recently'' have faster algorithms been studied. First Karatsuba multiplication was discovered in
1962. This algorithm can multiply two numbers with considerably fewer single precision multiplications when compared to the long-hand approach.
This technique led to the discovery of polynomial basis algorithms (\textit{good reference?}) and subquently Fourier Transform based solutions.
\section{Multiplication}
\subsection{The Baseline Multiplication}
\label{sec:basemult}
\index{baseline multiplication}
Computing the product of two integers in software can be achieved using a trivial adaptation of the standard $O(n^2)$ long-hand multiplication
algorithm that school children are taught. The algorithm is considered an $O(n^2)$ algorithm since for two $n$-digit inputs $n^2$ single precision
multiplications are required. More specifically for a $m$ and $n$ digit input $m \cdot n$ single precision multiplications are required. To
simplify most discussions, it will be assumed that the inputs have comparable number of digits.
The ``baseline multiplication'' algorithm is designed to act as the ``catch-all'' algorithm, only to be used when the faster algorithms cannot be
used. This algorithm does not use any particularly interesting optimizations and should ideally be avoided if possible. One important
facet of this algorithm, is that it has been modified to only produce a certain amount of output digits as resolution. The importance of this
modification will become evident during the discussion of Barrett modular reduction. Recall that for a $n$ and $m$ digit input the product
will be at most $n + m$ digits. Therefore, this algorithm can be reduced to a full multiplier by having it produce $n + m$ digits of the product.
Recall from sub-section 4.2.2 the definition of $\gamma$ as the number of bits in the type \textbf{mp\_digit}. We shall now extend the variable set to
include $\alpha$ which shall represent the number of bits in the type \textbf{mp\_word}. This implies that $2^{\alpha} > 2 \cdot \beta^2$. The
constant $\delta = 2^{\alpha - 2lg(\beta)}$ will represent the maximal weight of any column in a product (\textit{see sub-section 5.2.2 for more information}).
\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{s\_mp\_mul\_digs}. \\
\textbf{Input}. mp\_int $a$, mp\_int $b$ and an integer $digs$ \\
\textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert \mbox{ (mod }\beta^{digs}\mbox{)}$. \\
\hline \\
1. If min$(a.used, b.used) < \delta$ then do \\
\hspace{3mm}1.1 Calculate $c = \vert a \vert \cdot \vert b \vert$ by the Comba method (\textit{see algorithm~\ref{fig:COMBAMULT}}). \\
\hspace{3mm}1.2 Return the result of step 1.1 \\
\\
Allocate and initialize a temporary mp\_int. \\
2. Init $t$ to be of size $digs$ \\