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\begin{document}
\frontmatter
\pagestyle{empty}
\title{Multiple-Precision Integer Arithmetic, \\ A Case Study Involving the LibTomMath Project \\ - DRAFT - }
\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 in its entirety is copyright \copyright{}2003 by Tom St Denis.  It may not be redistributed 
electronically or otherwise without the sole permission of the author.  The text is freely redistributable as long as
it is packaged along with the LibTomMath library in a non-commercial project.  Contact the
author for other redistribution rights.

This text corresponds to the v0.17 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*{Preface}
Blah.

\mainmatter
\pagestyle{headings}
\chapter{Introduction}
\section{Multiple Precision Arithmetic}
\subsection{The Need for Multiple Precision Arithmetic}
The most prevalent use for multiple precision arithmetic (\textit{often referred to as bignum math}) is within public
key cryptography.   Algorithms such as RSA, Diffie-Hellman and Elliptic Curve Cryptography require large integers in order to 
resist known cryptanalytic attacks.  Typical modern programming languages such as C and Java only provide small 
single-precision data types which are incapable of precisely representing integers which are often hundreds of bits long.

For example, consider multiplying $1,234,567$ by $9,876,543$ in C with an ``unsigned long'' data type.  With an 
x86 machine the result is $4,136,875,833$ while the true result is $12,193,254,061,881$.  The original inputs 
were approximately $21$ and $24$ bits respectively.  If the C language cannot multiply two relatively small values 
together precisely how does anyone expect it to multiply two values that are considerably larger?

Most advancements in fast multiple precision arithmetic stem from the desire for faster cryptographic primitives.  However, cryptography
is not the only field of study that can benefit from fast large integer routines.  Another auxiliary use for multiple precision integers is 
high precision floating point data types.  The basic IEEE standard floating point type is made up of an integer mantissa $q$ and an exponent $e$.  
Numbers are given in the form $n = q \cdot b^e$ where $b = 2$ is specified.  Since IEEE is meant to be implemented in 
hardware the precision of the mantissa is often fairly small (\textit{23, 48 and 64 bits}).  Since the mantissa is merely an 
integer a large multiple precision integer could be used.  In effect very high precision floating point arithmetic 
could be performed.  This would be useful where scientific applications must minimize the total output error over long simulations.  

\subsection{Multiple Precision Arithmetic}
\index{multiple precision}
Multiple precision arithmetic attempts to the solve the shortcomings of single precision data types such as those from
the C and Java programming languages.  In essence multiple precision arithmetic is a set of operations that can be 
performed on members of an algebraic group whose precision is not fixed.  The algorithms when implemented to be multiple
precision can allow a developer to work with any practical precision required.

Typically the arithmetic over the ring of integers denoted by $\Z$ is performed by routines that are collectively and 
casually referred to as ``bignum'' routines.  However, it is possible to have rings of polynomials as well typically 
denoted by $\Z/p\Z \left [ X \right ]$ which could have variable precision (\textit{or degree}).  This text will 
discuss implementation of the former, however implementing polynomial basis routines should be relatively easy after reading this text.

\subsection{Benefits of Multiple Precision Arithmetic}
\index{precision} \index{accuracy}
Precision of the real value to a given precision is defined loosely as the proximity of the real value to a given representation.  
Accuracy is defined as the reproducibility of the result.  For example, the calculation $1/3 = 0.25$ is imprecise but can be accurate provided 
it is reproducible.

The benefit of multiple precision representations over single precision representations is that 
often no precision is lost while representing the result of an operation which requires excess precision.  For example, 
the multiplication of two $n$-bit integers requires at least $2n$ bits to represent the result.  A multiple precision 
system 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.

Multiple precision representations allow for the precision to be very high (\textit{if not exacting}) but at a cost of
modest computer resources.  The only reasonable case where a multiple precision system will lose precision is when
emulating a floating point data type.  However, with multiple precision integer arithmetic no precision is lost.

\subsection{Basis of Operations}
At the heart of all multiple precision integer operations are the ``long-hand'' algorithms we all learned as children 
in grade school.  For example, to multiply $1,234$ by $981$ the student is not taught to memorize the times table for 
$1,234$, instead they are taught how to long-multiply.  That is to multiply each column using simple single digit 
multiplications, line up the partial results, and add the resulting products by column.  The representation that most 
are familiar with is known as decimal or formally as radix-10. A radix-$n$ representation simply means there are 
$n$ possible values per digit.  For example, binary would be a radix-2 representation.

In essence computer based multiple precision arithmetic is very much the same.  The most notable difference is the usage
of a binary friendly radix.  That is to use a radix of the form $2^k$ where $k$ is typically the size of a machine 
register.  Also occasionally more optimal algorithms are used to perform certain operations such as multiplication and 
squaring instead of traditional long-hand algorithms.

\section{Purpose of This Text}
The purpose of this text is to instruct the reader regarding how to implement multiple precision algorithms.  That is 
to not only explain the core theoretical algorithms but also the various ``house keeping'' tasks that are neglected by
authors of other texts on the subject.  Texts such as \cite[HAC]{HAC} and \cite{TAOCPV2} give considerably detailed 
explanations of the theoretical aspects of the algorithms and very little regarding the practical aspects.  

How an algorithm is explained and how it is actually implemented are two very different 
realities.  For example, algorithm 14.7 on page 594 of HAC lists a relatively simple algorithm for performing multiple 
precision integer addition.  However, what the description lacks is any discussion concerning the fact that the two 
integer inputs may be of differing magnitudes.  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}).

As well as the numerous practical oversights both of the texts do not discuss several key optimal algorithms required 
such as ``Comba'' and Karatsuba multipliers and fast modular inversion.  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 the algorithms that 
constitute a multiple precision integer package with light discussions on the theoretical aspects.  As a case 
study the ``LibTomMath''\footnote{Available freely at http://math.libtomcrypt.org} package is used to demonstrate 
algorithms with implementations that have been field tested and work very well.

\section{Discussion and Notation}
\subsection{Notation}
A multiple precision integer of $n$-digits shall be denoted as $x = (x_n ... x_1 x_0)_{ \beta }$ to be the 
multiple precision notation for the integer $x \equiv \sum_{i=0}^{n} x_i\beta^i$.  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 $1\cdot 10^2 + 2\cdot10^1 + 3\cdot10^0 = 123$.  

A ``mp\_int'' shall refer to a composite structure which contains the digits of the integer as well as auxilary data
required to manipulate the data.  These additional members are discussed in chapter three.  For the purposes of this text
a ``multiple precision integer'' and a ``mp\_int'' are synonymous.

\index{single-precision} \index{double-precision} \index{mp\_digit} \index{mp\_word}
For the purposes of this text a single-precision variable must be able to represent integers in the range $0 \le x < 2 \beta$ while
a double-precision variable must be able to represent integers in the range $0 \le x < 2 \beta^2$.  Within the source code that will be
presented the data type \textbf{mp\_digit} will represent a single-precision type while \textbf{mp\_word} will represent a 
double-precision type.  In several algorithms (\textit{notably the Comba routines}) temporary results 
will be stored in a double-precision arrays.  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.

The $\lfloor \mbox{ } \rfloor$ brackets represent a value truncated and rounded down to the nearest integer.  The $\lceil \mbox{ } \rceil$ brackets 
represent a value truncated and rounded up to the nearest integer.  Typically when the $/$ division symbol is used the intention is to perform an integer
division.  For example, $5/2 = 2$ which will often be written as $\lfloor 5/2 \rfloor = 2$ for clarity.  When a value is presented as a fraction
such as $5 \over 2$ a real value division is implied.

\subsection{Work Effort}
\index{big-O}
To measure the efficiency of various algorithms a modified big-O 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 multiplication requires $O(n^2)$ work while a squaring requires 
$O({{n^2 + n}\over 2})$ work.  In standard big-O notation these would be said to be equivalent.  However, in the 
context of the this text the magnitude of the inputs will not approach an infinite size.  This means the conventional limit 
notation wisdom does not apply to the cancellation of constants.

Throughout the discussions various ``work levels'' will be discussed.  These levels are the $O(1)$,
$O(n)$, $O(n^2)$, ..., $O(n^k)$ work efforts.  For example, operations at the $O(n^k)$ ``level'' are said to be
executed more frequently than operations at the $O(n^m)$ ``level'' when $k > m$.  Obviously most optimizations will pay
off the most at the higher levels since they represent the bulk of the effort required.  

\section{Exercises}
Within the more advanced chapters a section will be set aside to give the reader some challenging exercises.  These exercises are not 
designed to be prize winning problems, but 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.  

Similar to the exercises of \cite{TAOCPV2} as explained on pp.\textit{ix} these exercises are given a scoring system.  However, unlike 
\cite{TAOCPV2} the problems do not get nearly as hard as often.  The scoring of these exercises ranges from one (\textit{the easiest}) to
five (\textit{the hardest}).  The following table sumarizes the scoring.

\vspace{5mm}
\begin{tabular}{cl}
$\left [ 1 \right ]$ & An easy problem that should only take the reader a manner of \\
                     & minutes to solve.  Usually does not involve much computer time. \\
                     & \\
$\left [ 2 \right ]$ & An easy problem that involves a marginal amount of computer \\
                     & time usage.  Usually requires a program to be written to \\
                     & solve the problem. \\
                     & \\
$\left [ 3 \right ]$ & 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. \\
                     & \\
$\left [ 4 \right ]$ & 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. \\
                     & \\
$\left [ 5 \right ]$ & A hard problem that involves concepts that are non-trivial.  \\
                     & Solutions to these problems will demonstrate a complete mastery \\
                     & of the given subject. \\
                     & \\
\end{tabular}

Essentially 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.  

Problems at the third level are meant to be a bit more difficult.  Often the answer is fairly obvious but arriving at an exacting solution
requires some thought and skill.  These problems will almost always involve devising a new algorithm or implementing a variation of
another algorithm.

Problems at the fourth level are meant to be even more difficult as well as involve some research.  The reader will most likely not know
the answer right away nor will this text provide the exact details of the answer (\textit{or at least not until a subsequent chapter}).  Problems
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 relevence of problems.

\chapter{Introduction to LibTomMath}

\section{What is LibTomMath?}
LibTomMath is a free and open source multiple precision library written in portable ISO C source code.  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 Solaris, 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.

\section{Goals of LibTomMath}

Even though the library is written entirely in portable 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 ARMv4 processors.  Wherever possible highly efficient 
algorithms (\textit{such as Karatsuba multiplication, sliding window exponentiation and Montgomery reduction}) have 
been provided to make the library as efficient as possible.  Even with the optimal and sometimes 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
attention.  One such example is the generic multiplication algorithm \textbf{mp\_mul()} which will automatically use 
Karatsuba multiplication if the inputs are of a specific size.

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.

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 
arithmetic.  To this end the source code has been given quite a few comments and algorithm discussion points.  Often routines have 
more comments than lines of code.

\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, MPI, LIP and OpenSSL have multiple precision 
integer arithmetic routines but would not be ideal for this text for reasons as 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 ascertain 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 also well organized.  Each function is in its own separate source code file 
which allows the reader to find a given function very fast.  When compiled with GCC for the x86 processor the entire 
library is a mere 87,760 bytes (\textit{$116,182$ bytes for ARMv4 processors}).  This includes every single function 
LibTomMath provides from basic arithmetic to various number theoretic functions such as modular exponentiation, various 
reduction algorithms and Jacobi symbol computation.  

By comparison MPI which has fewer functions than LibTomMath compiled with the same conditions is 45,429 bytes 
(\textit{$54,536$ for ARMv4}).  GMP which has rather large collection of functions with the default configuration on an 
x86 Athlon is 2,950,688 bytes.  Note that while LibTomMath has fewer functions than GMP it has been used as the sole basis 
for several public key cryptosystems without having to seek additional outside functions to supplement the library.

\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 are relatively straight forward as to what they perform.  Almost all of the 
functions except for a few minor exceptions which as will be discussed are for good reasons 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 fault.

\subsection{Optimizations}
While LibTomMath is certainly not the fastest library (\textit{GMP often beats LibTomMath by a factor of two}) it does
feature a set of optimal algorithms for tasks ranging from modular reduction to squaring.  GMP and LIP also feature
such optimizations while MPI only uses baseline algorithms with no optimizations.

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 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 is not working on his library anymore.  

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}
To begin the design of a multiple precision integer library a primitive data type and a series of primitive algorithms must be established.  A data
type that will hold the information required to maintain a multiple precision integer must be designed.  With this basic data type of a series
of low level algorithms for initializing, clearing, growing and optimizing multiple precision integers can be developed to form the basis of 
the entire library of algorithms.

\section{What is a Multiple Precision Integer?}
Recall that most programming languages (\textit{in particular C}) only have fixed precision data types that on their own cannot be used
to represent values larger than their precision alone will allow. The purpose of multiple precision algorithms is to use these fixed precision
data types to create multiple precision integers which may represent values that are much larger.  

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 value is only $9$ since the digits may only have values from $0$ to $9$.  However, by concatenating digits together larger numbers 
may be represented.  Computer based multiple precision arithmetic is essentially the same concept except with a different radix.

What most people probably do not think about explicitly are the various other attributes that describe a multiple precision integer.  For example,
the integer $154_{10}$ has two immediately obvious properties.  First, the integer is positive, that is the sign of this particular integer 
is positive as oppose to negative.  Second, the integer has three digits in its representation.  There is an additional property that the integer 
posesses that does not concern pencil-and-paper arithmetic.  The third property is how many digits are allowed for the integer.  

The human analogy of this third property is ensuring there is enough space on the paper to right the integer.  Computers must maintain a
strict control on memory usage with respect to the digits of a multiple precision integer.  These three properties make up what is known
as a multiple precision integer or mp\_int for short.  

\subsection{The mp\_int structure}
The mp\_int structure is the ISO C based manifestation of what represents a multiple precision integer.  The ISO C standard does not provide for 
any such data type but it does provide for making composite data types known as structures.  The following is the structure definition 
used within LibTomMath.

\index{mp\_int}
\begin{verbatim}
typedef struct  {
    int used, alloc, sign;
    mp_digit *dp;
} mp_int;
\end{verbatim}

The mp\_int structure can be broken down as follows.

\begin{enumerate}
\item The \textbf{used} parameter denotes how many digits of the array \textbf{dp} contain the digits used to represent
a given integer.  The \textbf{used} count must not exceed the \textbf{alloc} count.  

\item The array \textbf{dp} holds the digits that represent the given integer.  It is padded with $\textbf{alloc} - \textbf{used}$ zero
digits.

\item The \textbf{alloc} parameter denotes how 
many digits are available in the array to use by functions before it has to increase in size.  When the \textbf{used} count 
of a result would exceed the \textbf{alloc} count all of the algorithms will automatically increase the size of the 
array to accommodate the precision of the result.  

\item The \textbf{sign} parameter denotes the sign as either zero/positive (\textbf{MP\_ZPOS}) or negative (\textbf{MP\_NEG}).  
\end{enumerate}

\section{Argument Passing}
A convention of argument passing must be adopted early on in the development of any library.  Making the function prototypes
consistent will help eliminate many headaches in the future as the library grows to significant complexity.  In LibTomMath the multiple precision 
integer functions accept parameters from left to right as pointers to mp\_int structures.  That means that the source operands are 
placed on the left and the destination on the right.   Consider the following examples.

\begin{verbatim}
   mp_mul(&a, &b, &c);   /* c = a * b */
   mp_add(&a, &b, &a);   /* a = a + b */
   mp_sqr(&a, &b);       /* b = a * a */
\end{verbatim}

The left to right order is a fairly natural way to implement the functions since it lets the developer read aloud the
functions and make sense of them.  For example, the first function would read ``multiply a and b and store in c''.

Certain libraries (\textit{LIP by Lenstra for instance}) accept parameters the other way around.  That is the destination
on the left and arguments on the right.  In truth it is entirely a matter of preference.  In the case of LibTomMath the 
convention from the MPI library has been adopted.  

Another very useful design consideration is whether to allow argument sources to also be a destination.  For example, the
second example (\textit{mp\_add}) adds $a$ to $b$ and stores in $a$.  This is an important feature to implement since it
allows the higher up functions to cut down on the number of variables.  However, to implement this feature specific
care has to be given to ensure the destination is not modified before the source is fully read.

\section{Return Values}
A well implemented library, no matter what its purpose, should trap as many runtime errors as possible and return them to the 
caller.  By catching runtime errors a library can be guaranteed to prevent undefined behaviour.  In a multiple precision 
library the only errors that can occur occur are related to inappropriate inputs (\textit{division by zero for instance}) or 
memory allocation errors.

In LibTomMath any function that can cause a runtime error will return an error as an \textbf{int} data type with one of the 
following values.

\index{MP\_OKAY} \index{MP\_VAL} \index{MP\_MEM}
\begin{center}
\begin{tabular}{|l|l|}
\hline \textbf{Value} & \textbf{Meaning} \\
\hline \textbf{MP\_OKAY} & The function was successful \\
\hline \textbf{MP\_VAL}  & One of the input value(s) was invalid \\
\hline \textbf{MP\_MEM}  & The function ran out of heap memory \\
\hline
\end{tabular}
\end{center}

When an error is detected within a function it should free any memory it allocated and return as soon as possible.  The goal
is to leave the system in the same state the system was when the function was called.  Error checking with this style of API is fairly simple.

\begin{verbatim}
   int err;
   if ((err = mp_add(&a, &b, &c)) != MP_OKAY) {
      printf("Error: %d\n", err);
      exit(EXIT_FAILURE);
   }
\end{verbatim}

The GMP library uses C style \textit{signals} to flag errors which is of questionable use.  Not all errors are fatal 
and it was not deemed ideal by the author of LibTomMath to force developers to have signal handlers for such cases.

\section{Initialization and Clearing}
The logical starting point when actually writing multiple precision integer functions is the initialization and 
clearing of the integers.  These two functions will be used by far the most throughout the algorithms whenever 
temporary integers are required.

Given the basic mp\_int structure an initialization routine must first allocate memory to hold the digits of
the integer.  Often it is optimal to allocate a sufficiently large pre-set number of digits even considering
the initial integer will represent zero.  If only a single digit were allocated quite a few re-allocations
would occur for the majority of inputs.  There is a tradeoff between how many default digits to allocate
and how many re-allocations are tolerable.  

If the memory for the digits has been successfully allocated then the rest of the members of the structure must
be initialized.  Since the initial state is to represent a zero integer the digits allocated must all be zeroed.  The
\textbf{used} count set to zero and \textbf{sign} set to \textbf{MP\_ZPOS}.

\subsection{Initializing an mp\_int}
To initialize an mp\_int the mp\_init algorithm shall be used.  The purpose of this algorithm is to allocate 
the memory required and initialize the integer to a default representation of zero.

\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_init}. \\
\textbf{Input}.   An mp\_int $a$ \\
\textbf{Output}.  Allocate memory for the digits and set to a zero state. \\
\hline \\
1.  Allocate memory for \textbf{MP\_PREC} digits. \\
2.  If the allocation failed then return(\textit{MP\_MEM}) \\
3.  for $n$ from $0$ to $MP\_PREC - 1$ do  \\
\hspace{3mm}3.1  $a_n \leftarrow 0$\\
4.  $a.sign \leftarrow MP\_ZPOS$\\
5.  $a.used \leftarrow 0$\\
6.  $a.alloc \leftarrow MP\_PREC$\\
7.  Return(\textit{MP\_OKAY})\\
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_init}
\end{figure}

\textbf{Algorithm mp\_init.}
The \textbf{MP\_PREC} variable is a simple constant used to dictate minimal precision of allocated integers.  It is ideally at least equal to $32$ but 
can be any reasonable power of two.  Steps one and two allocate the memory and account for it.  If the allocation fails the algorithm returns
immediately to signal the failure.  Step three will ensure that all the digits are in the default state of zero.  Finally steps 
four through six set the default settings of the \textbf{sign}, \textbf{used} and \textbf{alloc} members of the mp\_int structure.

\index{bn\_mp\_init.c}
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_init.c
\vspace{-3mm}
\begin{alltt}
016   
017   /* init a new bigint */
018   int
019   mp_init (mp_int * a)
020   \{
021     /* allocate ram required and clear it */
022     a->dp = OPT_CAST calloc (sizeof (mp_digit), MP_PREC);
023     if (a->dp == NULL) \{
024       return MP_MEM;
025     \}
026   
027     /* set the used to zero, allocated digits to the default precision
028      * and sign to positive */
029     a->used  = 0;
030     a->alloc = MP_PREC;
031     a->sign  = MP_ZPOS;
032   
033     return MP_OKAY;
034   \}
\end{alltt}
\end{small}

The \textbf{OPT\_CAST} type cast on line 22 is designed to allow C++ compilers to build the code out of
the box.  Microsoft C V5.00 is known to cause problems without the cast.  Also note that if the memory
allocation fails the other members of the mp\_int will be in an undefined state.  The code from 
line 29 to line 31 sets the default state for a mp\_int which is zero, positive and no used digits.

\subsection{Clearing an mp\_int}
When an mp\_int is no longer required the memory allocated for it can be cleared from the heap with 
the mp\_clear algorithm.

\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_clear}. \\
\textbf{Input}.   An mp\_int $a$ \\
\textbf{Output}.  The memory for $a$ is cleared. \\
\hline \\
1.  If $a$ has been previously freed then return(\textit{MP\_OKAY}). \\
2.  Free the digits of $a$ and mark $a$ as freed. \\
3.  $a.used \leftarrow 0$ \\
4.  $a.alloc \leftarrow 0$ \\
5.  Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_clear}
\end{figure}

\textbf{Algorithm mp\_clear.}
In steps one and two the memory for the digits are only free'd if they had not been previously released before.  
This is more of concern for the implementation since it is used to prevent ``double-free'' errors.  It also helps catch
code errors where mp\_ints are used after being cleared.  Similarly steps three and four set the 
\textbf{used} and \textbf{alloc} to known values which would be easy to spot during debugging.  For example, if an mp\_int is expected
to be non-zero and its \textbf{used} member is observed to be zero (\textit{due to being cleared}) then an obvious bug in the code has been
spotted.

\index{bn\_mp\_clear.c}
\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     if (a->dp != NULL) \{
022   
023       /* first zero the digits */
024       memset (a->dp, 0, sizeof (mp_digit) * a->used);
025   
026       /* free ram */
027       free (a->dp);
028   
029       /* reset members to make debugging easier */
030       a->dp = NULL;
031       a->alloc = a->used = 0;
032     \}
033   \}
\end{alltt}
\end{small}

The \textbf{if} statement on line 21 prevents the heap from being corrupted if a user double-frees an 
mp\_int.  For example, a trivial case of this bug would be as follows.

\begin{verbatim}
mp_int a;
mp_init(&a);
mp_clear(&a);
mp_clear(&a);
\end{verbatim}

Without that check the code would try to free the memory allocated for the digits twice which will cause most standard C
libraries to cause a fault.  Also by setting the pointer to \textbf{NULL} it helps debug code that may inadvertently 
free the mp\_int before it is truly not needed.  The allocated digits are set to zero before being freed on line 24.  
This is ideal for cryptographic situations where the mp\_int is a secret parameter.

The following snippet is an example of using both the init and clear functions.  

\begin{small}
\begin{verbatim}
#include <tommath.h>
#include <stdio.h>
#include <stdlib.h>
int main(void)
{
   mp_int num;
   int err;
   
   /* init the bignum */
   if ((err = mp_init(&num)) != MP_OKAY) {
      printf("Error: %d\n", err);
      return EXIT_FAILURE;
   }
   
   /* do work with it ... */
   
   /* clear up */
   mp_clear(&num);
   
   return EXIT_SUCCESS;
}
\end{verbatim}
\end{small}

\section{Other Initialization Routines}

It is often helpful to have specialized initialization algorithms to simplify the design of other algorithms.  For example, an 
initialization followed by a copy is a common operation when temporary copies of integers are required.  It is quite
beneficial to have a series of simple helper functions available.

\subsection{Initializing Variable Sized mp\_int Structures}
Occasionally the number of digits required will be known in advance of an initialization.  In these
cases the mp\_init\_size algorithm can be of use.  The purpose of this algorithm is similar to mp\_init except that 
it will allocate \textit{at least} a specified number of digits.  This is ideal to prevent re-allocations when the 
input size is known.

\newpage\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_init\_size}. \\
\textbf{Input}.   An mp\_int $a$ and the requested number of digits $b$\\
\textbf{Output}.  $a$ is initialized to hold at least $b$ digits. \\
\hline \\
1.  $u \leftarrow b\mbox{ (mod }MP\_PREC\mbox{)}$ \\
2.  $v \leftarrow b + 2 \cdot MP\_PREC - u$ \\
3.  Allocate $v$ digits. \\
4.  If the allocation failed then return(\textit{MP\_MEM}). \\
5.  for $n$ from $0$ to $v - 1$ do \\
\hspace{3mm}5.1  $a_n \leftarrow 0$ \\
6.  $a.sign \leftarrow MP\_ZPOS$\\
7.  $a.used \leftarrow 0$\\
8.  $a.alloc \leftarrow v$\\
9.  Return(\textit{MP\_OKAY})\\
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_init\_size}
\end{figure}

\textbf{Algorithm mp\_init\_size.}
The value of $v$ is calculated to be at least the requested amount of digits $b$ plus additional padding.  The padding is calculated
to be at least \textbf{MP\_PREC} digits plus enough digits to make the digit count a multiple of \textbf{MP\_PREC}.  This padding is used to 
prevent trivial allocations from becoming a bottleneck in the rest of the algorithms that depend on this.

\index{bn\_mp\_init\_size.c}
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_init\_size.c
\vspace{-3mm}
\begin{alltt}
016   
017   /* init a mp_init and grow it to a given size */
018   int
019   mp_init_size (mp_int * a, int size)
020   \{
021   
022     /* pad size so there are always extra digits */
023     size += (MP_PREC * 2) - (size & (MP_PREC - 1));    
024     
025     /* alloc mem */
026     a->dp = OPT_CAST calloc (sizeof (mp_digit), size);
027     if (a->dp == NULL) \{
028       return MP_MEM;
029     \}
030     a->used = 0;
031     a->alloc = size;
032     a->sign = MP_ZPOS;
033   
034     return MP_OKAY;
035   \}
\end{alltt}
\end{small}

Line 23 will ensure that the number of digits actually allocated is padded up to the next multiple of 
\textbf{MP\_PREC} plus an additional \textbf{MP\_PREC}.  This ensures that the number of allocated digit is 
always greater than the amount requested.  As a result it prevents many trivial memory allocations.  The value of 
\textbf{MP\_PREC} is defined in ``tommath.h'' and must be a power of two.

\subsection{Creating a Clone}
Another common sequence of operations is to make a local temporary copy of an argument.  To initialize then copy a mp\_int will be known as 
creating a clone.  This is useful within functions that need to modify an integer argument but do not wish to actually modify the original copy.  
The mp\_init\_copy algorithm will perform this very task.

\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_init\_copy}. \\
\textbf{Input}.   An mp\_int $a$ and $b$\\
\textbf{Output}.  $a$ is initialized to be a copy of $b$. \\
\hline \\
1.  Init $a$.  (\textit{mp\_init}) \\
2.  If the init of $a$ was unsuccessful return(\textit{MP\_MEM}) \\
3.  Copy $b$ to $a$.  (\textit{mp\_copy}) \\
4.  Return the status of the copy operation. \\
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_init\_copy}
\end{figure}

\textbf{Algorithm mp\_init\_copy.}
This algorithm will initialize a mp\_int variable and copy another previously initialized mp\_int variable into it.  The algorithm will
detect when the initialization fails and returns the error to the calling algorithm.  As such this algorithm will perform two operations
in one step.  

\index{bn\_mp\_init\_copy.c}
\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
019   mp_init_copy (mp_int * a, mp_int * b)
020   \{
021     int     res;
022   
023     if ((res = mp_init (a)) != MP_OKAY) \{
024       return res;
025     \}
026     return mp_copy (b, a);
027   \}
\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.  

\subsection{Multiple Integer Initializations And Clearings}
Occasionally a function will require a series of mp\_int data types to be made available.  The mp\_init\_multi algorithm
is provided to simplify such cases.  The purpose of this algorithm is to initialize a variable length array of mp\_int 
structures at once.  As a result algorithms that require multiple integers only has to use 
one algorithm to initialize all the mp\_int variables.

\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_init\_multi}. \\
\textbf{Input}.   Variable length array of mp\_int variables of length $k$. \\
\textbf{Output}.  The array is initialized such that each each mp\_int is ready to use. \\
\hline \\
1.  for $n$ from 0 to $k - 1$ do \\
\hspace{+3mm}1.1.  Initialize the $n$'th mp\_int (\textit{mp\_init}) \\
\hspace{+3mm}1.2.  If initialization failed then do \\
\hspace{+6mm}1.2.1.  for $j$ from $0$ to $n$ do \\
\hspace{+9mm}1.2.1.1.  Free the $j$'th mp\_int (\textit{mp\_clear}) \\
\hspace{+6mm}1.2.2.   Return(\textit{MP\_MEM}) \\
2.  Return(\textit{MP\_OKAY}) \\
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_init\_multi}
\end{figure}

\textbf{Algorithm mp\_init\_multi.}
The algorithm will initialize the array of mp\_int variables one at a time.  As soon as an runtime error is detected (\textit{step 1.2}) all of
the previously initialized variables are cleared.  The goal is an ``all or nothing'' initialization which allows for quick recovery from runtime 
errors.

Similarly to clear a variable length array of mp\_int structures the mp\_clear\_multi algorithm will be used.

Consider the following snippet which demonstrates how to use both routines.
\begin{small}
\begin{verbatim}
#include <tommath.h>
#include <stdio.h>
#include <stdlib.h>
int main(void)
{
   mp_int num1, num2, num3;
   int err;
   
   if ((err = mp_init_multi(&num1, &num2, &num3, NULL)) !- MP_OKAY) {
      printf("Error: %d\n", err);
      return EXIT_FAILURE;
   }
   
   /* at this point num1/num2/num3 are ready */
   
   /* free them */
   mp_clear_multi(&num1, &num2, &num3, NULL);
   
   return EXIT_SUCCESS;
}
\end{verbatim}
\end{small}

Note how both lists are terminated with the \textbf{NULL} variable.  This indicates to the algorithms to stop fetching parameters off
of the stack.  If it is not present the functions will most likely cause a segmentation fault.  

\index{bn\_mp\_multi.c}
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_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   void mp_clear_multi(mp_int *mp, ...) 
055   \{
056       mp_int* next_mp = mp;
057       va_list args;
058       va_start(args, mp);
059       while (next_mp != NULL) \{
060           mp_clear(next_mp);
061           next_mp = va_arg(args, mp_int*);
062       \}
063       va_end(args);
064   \}
\end{alltt}
\end{small}

Both routines are implemented in the same source file since they are typically used in conjunction with each other.  

\section{Maintenance}
A small useful collection of mp\_int maintenance functions will also prove useful.  

\subsection{Augmenting Integer Precision}
When storing a value in an mp\_int sufficient digits must be available to accomodate the entire value without
loss of precision.  Quite often the size of the array given by the \textbf{alloc} member is large enough to simply
increase the \textbf{used} digit count.  However, when the size of the array is too small it must be re-sized 
appropriately to accomodate the result.  The mp\_grow algorithm will provide this functionality.

\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_grow}. \\
\textbf{Input}.   An mp\_int $a$ and an integer $b$. \\
\textbf{Output}.  $a$ is expanded to accomodate $b$ digits. \\
\hline \\
1.  if $a.alloc \ge b$ then return(\textit{MP\_OKAY}) \\
2.  $u \leftarrow b\mbox{ (mod }MP\_PREC\mbox{)}$ \\
3.  $v \leftarrow b + 2 \cdot MP\_PREC - u$ \\
4.  Re-Allocate the array of digits $a$ to size $v$ \\
5.  If the allocation failed then return(\textit{MP\_MEM}). \\
6.  for n from a.alloc to $v - 1$ do  \\
\hspace{+3mm}6.1  $a_n \leftarrow 0$ \\
7.  $a.alloc \leftarrow v$ \\
8.  Return(\textit{MP\_OKAY}) \\
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_grow}
\end{figure}

\textbf{Algorithm mp\_grow.}
Step one will prevent a re-allocation from being performed if it was not required.  This is useful to prevent mp\_ints
from growing excessively in code that erroneously calls mp\_grow.  Similar to mp\_init\_size the requested digit count
is padded to provide more digits than requested.  

In step four it is assumed that the reallocation leaves the lower $a.alloc$ digits intact.  This is much akin to how the 
\textit{realloc} function from the standard C library works.  Since the newly allocated digits are assumed to contain
undefined values they are also initially zeroed.

\index{bn\_mp\_grow.c}
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_grow.c
\vspace{-3mm}
\begin{alltt}
016   
017   /* grow as required */
018   int
019   mp_grow (mp_int * a, int size)
020   \{
021     int     i;
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 - 1));     
027   
028       a->dp = OPT_CAST realloc (a->dp, sizeof (mp_digit) * size);
029       if (a->dp == NULL) \{
030         return MP_MEM;
031       \}
032   
033       /* zero excess digits */
034       i        = a->alloc;
035       a->alloc = size;
036       for (; i < a->alloc; i++) \{
037         a->dp[i] = 0;
038       \}
039     \}
040     return MP_OKAY;
041   \}
\end{alltt}
\end{small}

The first step is to see if we actually need to perform a re-allocation at all.  This is tested for on line 
24.  Similar to mp\_init\_size the same code on line 26 was used to resize the 
digits requested.  A simple for loop from line 34 to line 38 will zero all digits that were above the 
old \textbf{alloc} limit to make sure the integer is in a known state.

\subsection{Clamping Excess Digits}
When a function anticipates a result will be $n$ digits it is simpler to assume this is true within the body of 
the function.  For example, a multiplication of a $i$ digit number by a $j$ digit produces a result of at most 
$i + j$ digits.  It is entirely possible that the result is $i + j - 1$ though, with no final carry into the last 
position.  However, suppose the destination had to be first expanded (\textit{via mp\_grow}) to accomodate $i + j - 1$
digits than further expanded to accomodate the final carry.  That would be a considerable waste of time since heap
operations are relatively slow.

The ideal solution is to always assume the result is $i + j$ and fix up the \textbf{used} count after the function
terminates.  This way a single heap operation (\textit{at most}) is required.  However, if the result was not checked
there would be an excess high order zero digit.  

For example, suppose the product of two integers was $x_n = (0x_{n-1}x_{n-2}...x_0)_{\beta}$.  The leading zero digit 
will not contribute to the precision of the result.  In fact, through subsequent operations more leading zero digits would
accumulate to the point the size of the integer would be prohibitive.  As a result even though the precision is very 
low the representation is excessively large.  

The mp\_clamp algorithm is designed to solve this very problem.  It will trim leading zeros by decrementing the 
\textbf{used} count until a non-zero leading digit is found.  Also in this system, zero is considered to be a positive 
number which means that if the \textbf{used} count is decremented to zero the sign must be set to \textbf{MP\_ZPOS}.

\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_clamp}. \\
\textbf{Input}.   An mp\_int $a$ \\
\textbf{Output}.  Any excess leading zero digits of $a$ are removed \\
\hline \\
1.  while $a.used > 0$ and $a_{a.used - 1} = 0$ do \\
\hspace{+3mm}1.1  $a.used \leftarrow a.used - 1$ \\
2.  if $a.used = 0$ then do \\
\hspace{+3mm}2.1  $a.sign \leftarrow MP\_ZPOS$ \\
\hline \\
\end{tabular}
\end{center}
\caption{Algorithm mp\_clamp}
\end{figure}

\textbf{Algorithm mp\_clamp.}
As can be expected this algorithm is very simple.  The loop on step one is expected to iterate only once or twice at
the most.  For example, this will happen in cases where there is not a carry to fill the last position.  Step two fixes the sign for 
when all of the digits are zero to ensure that the mp\_int is valid at all times.

\index{bn\_mp\_clamp.c}
\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     while (a->used > 0 && a->dp[a->used - 1] == 0) \{
028       --(a->used);
029     \}
030     if (a->used == 0) \{
031       a->sign = MP_ZPOS;
032     \}
033   \}
\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 28 is used to make sure the \textbf{used} count is decremented and not
the pointer ``a''.  

\section*{Exercises}
\begin{tabular}{cl}
$\left [ 1 \right ]$ & Discuss the relevance of the \textbf{used} member of the mp\_int structure. \\
                     & \\
$\left [ 1 \right ]$ & Discuss the consequences of not using padding when performing allocations.  \\
                     & \\
$\left [ 2 \right ]$ & Estimate an ideal value for \textbf{MP\_PREC} when performing 1024-bit RSA \\
                     & encryption when $\beta = 2^{28}$.  \\
                     & \\
$\left [ 1 \right ]$ & Discuss the relevance of the algorithm mp\_clamp.  What does it prevent? \\
                     & \\
$\left [ 1 \right ]$ & Give an example of when the algorithm  mp\_init\_copy might be useful. \\
                     & \\
\end{tabular}


\chapter{Basic Operations}
\section{Copying an Integer}
After the various house-keeping routines are in place, simple algorithms can be designed to take advantage of them.  Being able
to make a verbatim copy of an integer is a very useful function to have.  To copy an integer the mp\_copy algorithm will be used.

\newpage\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_copy}. \\
\textbf{Input}.  An mp\_int $a$ and $b$. \\
\textbf{Output}.  Store a copy of $a$ in $b$. \\
\hline \\
1.  Check if $a$ and $b$ point to the same location in memory. \\
2.  If true then return(\textit{MP\_OKAY}). \\
3.  If $b.alloc < a.used$ then grow $b$ to $a.used$ digits.  (\textit{mp\_grow}) \\
4.  If failed to grow then return(\textit{MP\_MEM}). \\
5.  for $n$ from 0 to $a.used - 1$ do \\
\hspace{3mm}5.1  $b_{n} \leftarrow a_{n}$ \\
6.  if $a.used < b.used - 1$ then \\ 
\hspace{3mm}6.1.  for $n$ from $a.used$ to $b.used - 1$ do \\
\hspace{6mm}6.1.1  $b_{n} \leftarrow 0$ \\
7.  $b.used \leftarrow a.used$ \\
8.  $b.sign \leftarrow a.sign$ \\
9.  return(\textit{MP\_OKAY}) \\
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_copy}
\end{figure}

\textbf{Algorithm mp\_copy.}
Step 1 and 2 make sure that the two mp\_ints are unique.  This allows the user to call the copy function with
potentially the same input and not waste time.  Step 3 and 4 ensure that the destination is large enough to
hold a copy of the input $a$.  Note that the \textbf{used} member of $b$ may be smaller than the \textbf{used}
member of $a$ but a memory re-allocation is only required if the \textbf{alloc} member of $b$ is smaller.  This
prevents trivial memory reallocations.

Step 5 copies the digits from $a$ to $b$ while step 6 ensures that if initially $\vert b \vert > \vert a \vert$,
the more significant digits of $b$ will be zeroed.  Finally steps 7 and 8 copies the \textbf{used} and \textbf{sign} members over 
which completes the copy operation.

\index{bn\_mp\_copy.c}
\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 ((res = mp_grow (b, a->used)) != MP_OKAY) \{
030       return res;
031     \}
032   
033     /* zero b and copy the parameters over */
034     \{
035       register mp_digit *tmpa, *tmpb;
036   
037       /* pointer aliases */
038       tmpa = a->dp;
039       tmpb = b->dp;
040   
041       /* copy all the digits */
042       for (n = 0; n < a->used; n++) \{
043         *tmpb++ = *tmpa++;
044       \}
045   
046       /* clear high digits */
047       for (; n < b->used; n++) \{
048         *tmpb++ = 0;
049       \}
050     \}
051     b->used = a->used;
052     b->sign = a->sign;
053     return MP_OKAY;
054   \}
\end{alltt}
\end{small}

Source lines 23-31 do the initial house keeping.  That is to see if the input is unique and if so to 
make sure there is enough room.  If not enough space is available it returns the error and leaves the destination variable
intact.

The inner loop of the copy operation is contained between lines 34 and 50.  Many LibTomMath routines are designed with this source code style
in mind, making aliases to shorten lengthy pointers (\textit{see line 38 and 39}) for rapid use.  Also the
use of nested braces creates a simple way to denote various portions of code that reside on various work levels.  Here, the copy loop is at the 
$O(n)$ level.  

\section{Zeroing an Integer}
Reseting an mp\_int to the default state is a common step in many algorithms.  The mp\_zero algorithm will be the algorithm used to
perform this task.

\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_zero}. \\
\textbf{Input}.   An mp\_int $a$ \\
\textbf{Output}.  Zero the contents of $a$ \\
\hline \\
1.  $a.used \leftarrow 0$ \\
2.  $a.sign \leftarrow$ MP\_ZPOS \\
3.  for $n$ from 0 to $a.alloc - 1$ do \\
\hspace{3mm}3.1  $a_n \leftarrow 0$ \\
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_zero}
\end{figure}

\textbf{Algorithm mp\_zero.}
This algorithm simply resets a mp\_int to the default state.  

\index{bn\_mp\_zero.c}
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_zero.c
\vspace{-3mm}
\begin{alltt}
016   
017   /* set to zero */
018   void
019   mp_zero (mp_int * a)
020   \{
021     a->sign = MP_ZPOS;
022     a->used = 0;
023     memset (a->dp, 0, sizeof (mp_digit) * a->alloc);
024   \}
\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.  As can be expected the algorithm is very trivial.

\index{bn\_mp\_abs.c}
\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     if ((res = mp_copy (a, b)) != MP_OKAY) \{
026       return res;
027     \}
028     b->sign = MP_ZPOS;
029     return MP_OKAY;
030   \}
\end{alltt}
\end{small}

\subsection{Integer Negation}
With the mp\_int representation of an integer, calculating the negation is also trivial.  The mp\_neg algorithm will compute
the negative of an mp\_int input.

\newpage\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_neg}. \\
\textbf{Input}.   An mp\_int $a$ \\
\textbf{Output}.  Computes $b = -a$ \\
\hline \\
1.  Copy $a$ to $b$.  (\textit{mp\_copy}) \\
2.  If the copy failed return(\textit{MP\_MEM}). \\
3.  If $a.sign = MP\_ZPOS$ then do \\
\hspace{3mm}3.1  $b.sign = MP\_NEG$. \\
4.  else do \\
\hspace{3mm}4.1  $b.sign = MP\_ZPOS$. \\
5.  Return(\textit{MP\_OKAY}) \\
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_neg}
\end{figure}

\textbf{Algorithm mp\_neg.}
This algorithm computes the negation of an input.  

\index{bn\_mp\_neg.c}
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_neg.c
\vspace{-3mm}
\begin{alltt}
016   
017   /* b = -a */
018   int
019   mp_neg (mp_int * a, mp_int * b)
020   \{
021     int     res;
022     if ((res = mp_copy (a, b)) != MP_OKAY) \{
023       return res;
024     \}
025     b->sign = (a->sign == MP_ZPOS) ? MP_NEG : MP_ZPOS;
026     return MP_OKAY;
027   \}
\end{alltt}
\end{small}

\section{Small Constants}
\subsection{Setting Small Constants}
Often a mp\_int must be set to a relatively small value such as $1$ or $2$.  For these cases the mp\_set algorithm is useful.

\newpage\begin{figure}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_set}. \\
\textbf{Input}.   An mp\_int $a$ and a digit $b$ \\
\textbf{Output}.  Make $a$ equivalent to $b$ \\
\hline \\
1.  Zero $a$ (\textit{mp\_zero}). \\
2.  $a_0 \leftarrow b \mbox{ (mod }\beta\mbox{)}$ \\
3.  $a.used \leftarrow  \left \lbrace \begin{array}{ll}
                              1 &  \mbox{if }a_0 > 0 \\
                              0 &  \mbox{if }a_0 = 0 
                              \end{array} \right .$ \\
\hline                              
\end{tabular}
\end{center}
\caption{Algorithm mp\_set}
\end{figure}

\textbf{Algorithm mp\_set.}
This algorithm sets a mp\_int to a small single digit value.  Step number 1 ensures that the integer is reset to the default state.  The
single digit is set (\textit{modulo $\beta$}) and the \textbf{used} count is adjusted accordingly.

\index{bn\_mp\_set.c}
\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
019   mp_set (mp_int * a, mp_digit b)
020   \{
021     mp_zero (a);
022     a->dp[0] = b & MP_MASK;
023     a->used = (a->dp[0] != 0) ? 1 : 0;
024   \}
\end{alltt}
\end{small}

Line 21 calls mp\_zero() to clear the mp\_int and reset the sign.  Line 22 copies the digit 
into the least significant location.  Note the usage of a new constant \textbf{MP\_MASK}.  This constant is used to quickly
reduce an integer modulo $\beta$.  Since $\beta$ is of the form $2^k$ for any suitable $k$ it suffices to perform a binary AND with 
$MP\_MASK = 2^k - 1$ to perform the reduction.  Finally line 23 will set the \textbf{used} member with respect to the 
digit actually set. This function will always make the integer positive.

One important limitation of this function is that it will only set one digit.  The size of a digit is not fixed, meaning source that uses 
this function should take that into account.  Meaning that only trivially small constants can be set using this function.

\subsection{Setting Large Constants}
To overcome the limitations of the mp\_set algorithm the mp\_set\_int algorithm is provided.  It accepts a ``long''
data type as input and will always treat it as a 32-bit integer.

\begin{figure}[here]
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_set\_int}. \\
\textbf{Input}.   An mp\_int $a$ and a ``long'' integer $b$ \\
\textbf{Output}.  Make $a$ equivalent to $b$ \\
\hline \\
1.  Zero $a$ (\textit{mp\_zero}) \\
2.  for $n$ from 0 to 7 do \\
\hspace{3mm}2.1  $a \leftarrow a \cdot 16$ (\textit{mp\_mul2d}) \\
\hspace{3mm}2.2  $u \leftarrow \lfloor b / 2^{4(7 - n)} \rfloor \mbox{ (mod }16\mbox{)}$\\
\hspace{3mm}2.3  $a_0 \leftarrow a_0 + u$ \\
\hspace{3mm}2.4  $a.used \leftarrow a.used + 1$ \\
3.  Clamp excess used digits (\textit{mp\_clamp}) \\
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_set\_int}
\end{figure}

\textbf{Algorithm mp\_set\_int.}
The algorithm performs eight iterations of a simple loop where in each iteration four bits from the source are added to the 
mp\_int.  Step 2.1 will multiply the current result by sixteen making room for four more bits in the less significant positions.  In step 2.2 the
next four bits from the source are extracted and are added to the mp\_int. The \textbf{used} digit count is 
incremented to reflect the addition.  The \textbf{used} digit counter is incremented since if any of the leading digits were zero the mp\_int would have
zero digits used and the newly added four bits would be ignored.

Excess zero digits are trimmed in steps 2.1 and 3 by using higher level algorithms mp\_mul2d and mp\_clamp.

\index{bn\_mp\_set\_int.c}
\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
019   mp_set_int (mp_int * a, unsigned int b)
020   \{
021     int     x, res;
022   
023     mp_zero (a);
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   \}
\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}.

\index{bn\_mp\_cmp\_mag.c}
\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
019   mp_cmp_mag (mp_int * a, mp_int * b)
020   \{
021     int     n;
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     /* compare based on digits  */
033     for (n = a->used - 1; n >= 0; n--) \{
034       if (a->dp[n] > b->dp[n]) \{
035         return MP_GT;
036       \} 
037       
038       if (a->dp[n] < b->dp[n]) \{
039         return MP_LT;
040       \}
041     \}
042     return MP_EQ;
043   \}
\end{alltt}
\end{small}

The two if statements on 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.

\index{bn\_mp\_cmp.c}
\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 == MP_NEG && b->sign == MP_ZPOS) \{
023       return MP_LT;
024     \} 
025     
026     if (a->sign == MP_ZPOS && b->sign == MP_NEG) \{
027       return MP_GT;
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   \}
\end{alltt}
\end{small}

The two if statements on lines 22 and 26 perform the initial sign comparison.  If the signs are not the equal then which ever
has the positive sign is larger.   At line 31, the inputs are compared based on magnitudes.  If the signs were both negative then 
the unsigned comparison is performed in the opposite direction (\textit{line 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{Building Blocks}
At this point algorithms for initialization, clearing, zeroing, copying, comparing and setting small constants have been 
established.  The next logical set of algorithms to develop are addition, subtraction and digit shifting algorithms.  These 
algorithms make use of the lower level algorithms and are the cruicial building block for the multiplication algorithms.  It is very important 
that these algorithms are highly optimized.  On their own they are simple $O(n)$ algorithms but they can be called from higher level algorithms 
which easily places them at $O(n^2)$ or even $O(n^3)$ work levels.  

All nine algorithms within this chapter make use of the logical bit shift operations denoted by $<<$ and $>>$ for left and right 
logical shifts respectively.  A logical shift is analogous to sliding the decimal point of radix-10 representations.  For example, the real 
number $0.9345$ is equivalent to $93.45\%$ which is found by sliding the the decimal two places to the right (\textit{multiplying by $10^2$}).  
Mathematically a logical shift is equivalent to a division or multiplication by a power of two.  
For example, $a << k = a \cdot 2^k$ while $a >> k = \lfloor a/2^k \rfloor$.

One significant difference between a logical shift and the way decimals are shifted is that digits below the zero'th position are removed
from the number.  For example, consider $1101_2 >> 1$ using decimal notation this would produce $110.1_2$.  However, with a logical shift the 
result is $110_2$.  

\section{Addition and Subtraction}
In normal fixed precision arithmetic negative numbers are easily represented by subtraction from the modulus.  For example, with 32-bit integers
$a - b\mbox{ (mod }2^{32}\mbox{)}$ is the same as $a + (2^{32} - b) \mbox{ (mod }2^{32}\mbox{)}$  since $2^{32} \equiv 0 \mbox{ (mod }2^{32}\mbox{)}$.  
As a result subtraction can be performed with a trivial series of logical operations and an addition.

However, in multiple precision arithmetic negative numbers are not represented in the same way.  Instead a sign flag is used to keep track of the
sign of the integer.  As a result signed addition and subtraction are actually implemented as conditional usage of lower level addition or 
subtraction algorithms with the sign fixed up appropriately.

The lower level algorithms will add or subtract integers without regard to the sign flag.  That is they will add or subtract the magnitude of
the integers respectively.

\subsection{Low Level Addition}
An unsigned addition of multiple precision integers is performed with the same long-hand algorithm used to add decimal numbers.  That is to add the 
trailing digits first and propagate the resulting carry upwards.  Since this is a lower level algorithm the name will have a ``s\_'' prefix.  
Historically that convention stems from the MPI library where ``s\_'' stood for static functions that were hidden from the developer entirely.

\newpage
\begin{figure}[!here]
\begin{center}
\begin{small}
\begin{tabular}{l}
\hline Algorithm \textbf{s\_mp\_add}. \\
\textbf{Input}.   Two mp\_ints $a$ and $b$ \\
\textbf{Output}.  The unsigned addition $c = \vert a \vert + \vert b \vert$. \\
\hline \\
1.  if $a.used > b.used$ then \\
\hspace{+3mm}1.1  $min \leftarrow b.used$ \\
\hspace{+3mm}1.2  $max \leftarrow a.used$ \\
\hspace{+3mm}1.3  $x   \leftarrow a$ \\
2.  else  \\
\hspace{+3mm}2.1  $min \leftarrow a.used$ \\
\hspace{+3mm}2.2  $max \leftarrow b.used$ \\
\hspace{+3mm}2.3  $x   \leftarrow b$ \\
3.  If $c.alloc < max + 1$ then grow $c$ to hold at least $max + 1$ digits (\textit{mp\_grow}) \\
4.  If failed to grow $c$ return(\textit{MP\_MEM}) \\
5.  $oldused \leftarrow c.used$ \\
6.  $c.used \leftarrow max + 1$ \\
7.  $u \leftarrow 0$ \\
8.  for $n$ from $0$ to $min - 1$ do \\
\hspace{+3mm}8.1  $c_n \leftarrow a_n + b_n + u$ \\
\hspace{+3mm}8.2  $u \leftarrow c_n >> lg(\beta)$ \\
\hspace{+3mm}8.3  $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\
9.  if $min \ne max$ then do \\
\hspace{+3mm}9.1  for $n$ from $min$ to $max - 1$ do \\
\hspace{+6mm}9.1.1  $c_n \leftarrow x_n + u$ \\
\hspace{+6mm}9.1.2  $u \leftarrow c_n >> lg(\beta)$ \\
\hspace{+6mm}9.1.3  $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\
10.  $c_{max} \leftarrow u$ \\
11.  if $olduse > max$ then \\
\hspace{+3mm}11.1  for $n$ from $max + 1$ to $olduse - 1$ do \\
\hspace{+6mm}11.1.1  $c_n \leftarrow 0$ \\
12.  Clamp excess digits in $c$.  (\textit{mp\_clamp}) \\
13.  Return(\textit{MP\_OKAY}) \\
\hline
\end{tabular}
\end{small}
\end{center}
\caption{Algorithm s\_mp\_add}
\end{figure}

\textbf{Algorithm s\_mp\_add.}
This algorithm is loosely based on algorithm 14.7 of HAC \cite[pp. 594]{HAC} but has been extended to allow the inputs to have different magnitudes.  
Coincidentally the description of algorithm A in Knuth \cite[pp. 266]{TAOCPV2} shares the same deficiency as the algorithm from \cite{HAC}.  Even the 
MIX pseudo  machine code presented by Knuth \cite[pp. 266-267]{TAOCPV2} is incapable of handling inputs which are of different magnitudes.

Steps 1 and 2 will sort the two inputs based on their \textbf{used} digit count.  This allows the inputs to have varying magnitudes which not 
only makes it more efficient than the trivial algorithm presented in the references but more flexible.  The variable $min$ is given the lowest 
digit count while $max$ is given the highest digit count.  If both inputs have the same \textbf{used} digit count both $min$ and $max$ are 
set to the same value.  The variable $x$ is an \textit{alias} for the largest input and not meant to be a copy of it.  After the inputs are sorted, 
steps 3 and 4 will ensure that the destination $c$ can accommodate the result.  The old \textbf{used} count from $c$ is copied to 
$oldused$ so that excess digits can be cleared later, and the new \textbf{used} count is set to $max+1$, so that a carry from the most significant 
word can be handled.

At step 7 the carry variable $u$ is set to zero and the first part of the addition loop can begin.  The first step of the loop (\textit{8.1}) adds
digits from the two inputs together along with the carry variable $u$.  The following step extracts the carry bit by shifting the result of the
preceding step right by $lg(\beta)$ positions.  The shift to extract the carry is similar to how carry extraction works with decimal addition.

Consider adding $77$ to $65$, the first addition of the first column is $7 + 5$ which produces the result $12$.  The trailing digit of the result
is $2 \equiv 12 \mbox{ (mod }10\mbox{)}$ and the carry is found by dividing (\textit{and ignoring the remainder}) $12$ by the radix or in this case $10$.  The
division and multiplication of $10$ is simply a logical right or left shift, respectively, of the digits.  In otherwords the carry can be extracted
by shifting one digit to the right.

Note that $lg()$ is simply the base two logarithm such that $lg(2^k) = k$.  This implies that $lg(\beta)$ is the number of bits in a radix-$\beta$ 
digit.  Therefore, a logical shift right of the summand by $lg(\beta)$ will extract the carry.  The final step of the loop reduces the digit 
modulo the radix $\beta$ to ensure it is in range.

After step 8 the smallest input (\textit{or both if they are the same magnitude}) has been exhausted.  Step 9 decides whether
the inputs were of equal magnitude.  If not than another loop similar to that in step 8, must be executed.  The loop at step
number 9.1 differs from the previous loop since it only adds the mp\_int $x$ along with the carry.  

Step 10 finishes the addition phase by copying the final carry to the highest location in the result $c_{max}$.  Step 11 ensures that 
leading digits that were originally present in $c$ are cleared.  Finally excess leading digits are clamped and the algorithm returns success.

\index{bn\_s\_mp\_add.c}
\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   \}
\end{alltt}
\end{small}

Lines 27 to 35 perform the initial sorting of the inputs and determine the $min$ and $max$ variables.  Note that $x$ is a pointer to a 
mp\_int assigned to the largest input, in effect it is a local alias.  Lines 37 to 42 ensure that the destination is grown to 
accomodate the result of the addition. 

Similar to the implementation of mp\_copy this function uses the braced code and local aliases coding style.  The three aliases that are on 
lines 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 $u$ is cleared on line 64, note that $u$ is of type mp\_digit which ensures type compatibility within the 
implementation.  The initial addition loop begins on line 65 and ends on line 74.  Similarly the conditional addition loop
begins on line 80 and ends on line 90.  The addition is finished with the final carry being stored in $tmpc$ on line 93.  
Note the ``++'' operator on the same line.  After line 93 $tmpc$ will point to the $c.used$'th digit of the mp\_int $c$.  This is useful
for the next loop on lines 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 $\gamma = 32$.

\newpage\begin{figure}[!here]
\begin{center}
\begin{small}
\begin{tabular}{l}
\hline Algorithm \textbf{s\_mp\_sub}. \\
\textbf{Input}.   Two mp\_ints $a$ and $b$ ($\vert a \vert \ge \vert b \vert$) \\
\textbf{Output}.  The unsigned subtraction $c = \vert a \vert - \vert b \vert$. \\
\hline \\
1.  $min \leftarrow b.used$ \\
2.  $max \leftarrow a.used$ \\
3.  If $c.alloc < max$ then grow $c$ to hold at least $max$ digits.  (\textit{mp\_grow}) \\
4.  If the reallocation failed return(\textit{MP\_MEM}). \\
5.  $oldused \leftarrow c.used$ \\ 
6.  $c.used \leftarrow max$ \\
7.  $u \leftarrow 0$ \\
8.  for $n$ from $0$ to $min - 1$ do \\
\hspace{3mm}8.1  $c_n \leftarrow a_n - b_n - u$ \\
\hspace{3mm}8.2  $u   \leftarrow c_n >> (\gamma - 1)$ \\
\hspace{3mm}8.3  $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\
9.  if $min < max$ then do \\
\hspace{3mm}9.1  for $n$ from $min$ to $max - 1$ do \\
\hspace{6mm}9.1.1  $c_n \leftarrow a_n - u$ \\
\hspace{6mm}9.1.2  $u   \leftarrow c_n >> (\gamma - 1)$ \\
\hspace{6mm}9.1.3  $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\
10. if $oldused > max$ then do \\
\hspace{3mm}10.1  for $n$ from $max$ to $oldused - 1$ do \\
\hspace{6mm}10.1.1  $c_n \leftarrow 0$ \\
11. Clamp excess digits of $c$.  (\textit{mp\_clamp}). \\
12. Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{small}
\end{center}
\caption{Algorithm s\_mp\_sub}
\end{figure}

\textbf{Algorithm s\_mp\_sub.}
This algorithm performs the unsigned subtraction of two mp\_int variables under the restriction that the result must be positive.  That is when
passing variables $a$ and $b$ the condition that $\vert a \vert \ge \vert b \vert$ must be met for the algorithm to function correctly.  This
algorithm is loosely based on algorithm 14.9 \cite[pp. 595]{HAC} and is similar to algorithm S in \cite[pp. 267]{TAOCPV2} as well.  As was the case
of the algorithm s\_mp\_add both other references lack discussion concerning various practical details such as when the inputs differ in magnitude.

The initial sorting of the inputs is trivial in this algorithm since $a$ is guaranteed to have at least the same magnitude of $b$.  Steps 1 and 2 
set the $min$ and $max$ variables.  Unlike the addition routine there is guaranteed to be no carry which means that the final result can be at 
most $max$ digits in length as opposed to $max + 1$.  Similar to the addition algorithm the \textbf{used} count of $c$ is copied locally and 
set to the maximal count for the operation.

The subtraction loop that begins on step 8 is essentially the same as the addition loop of algorithm s\_mp\_add except single precision 
subtraction is used instead.  Note the use of the $\gamma$ variable to extract the carry (\textit{also known as the borrow}) within the subtraction 
loops.  Under the assumption that two's complement single precision arithmetic is used this will successfully extract the desired carry.  

For example, consider subtracting $0101_2$ from $0100_2$ where $\gamma = 4$ and $\beta = 2$.  The least significant bit will force a carry upwards to 
the third bit which will be set to zero after the borrow.  After the very first bit has been subtracted $4 - 1 \equiv 0011_2$ will remain,  When the 
third bit of $0101_2$ is subtracted from the result it will cause another carry.  In this case though the carry will be forced to propagate all the 
way to the most significant bit.  

Recall that $\beta < 2^{\gamma}$.  This means that if a carry does occur just before the $lg(\beta)$'th bit it will propagate all the way to the most 
significant bit.  Thus, the high order bits of the mp\_digit that are not part of the actual digit will either be all zero, or all one. All that
is needed is a single zero or one bit for the carry.  Therefore a single logical shift right by $\gamma - 1$ positions is sufficient to extract the 
carry.  This method of carry extraction may seem awkward but the reason for it becomes apparent when the implementation is discussed.  

If $b$ has a smaller magnitude than $a$ then step 9 will force the carry and copy operation to propagate through the larger input $a$ into $c$.  Step
10 will ensure that any leading digits of $c$ above the $max$'th position are zeroed.

\index{bn\_s\_mp\_sub.c}
\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   
\end{alltt}
\end{small}

Line 24 and 25 perform the initial hardcoded sorting of the inputs.  In reality the $min$ and $max$ variables are only aliases and are only 
used to make the source code easier to read.  Again the pointer alias optimization is used within this algorithm.  Lines 41, 42 and 43 initialize the aliases for 
$a$, $b$ and $c$ respectively.

The first subtraction loop occurs on lines 46 through 60.  The theory behind the subtraction loop is exactly the same as that for
the addition loop.  As remarked earlier there is an implementation reason for using the ``awkward'' method of extracting the carry 
(\textit{see line 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 (\textit{see 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.  If any of the lower level operations failed return(\textit{MP\_MEM}) \\
4.  Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_add}
\end{figure}

\textbf{Algorithm mp\_add.}
This algorithm performs the signed addition of two mp\_int variables.  There is no reference algorithm to draw upon from either \cite{TAOCPV2} or 
\cite{HAC} since they both only provide unsigned operations.  The algorithm is fairly straightforward but restricted since subtraction can only 
produce positive results.

\begin{figure}[here]
\begin{small}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline \textbf{Sign of $a$} & \textbf{Sign of $b$} & \textbf{$\vert a \vert > \vert b \vert $} & \textbf{Unsigned Operation} & \textbf{Result Sign Flag} \\
\hline $+$ & $+$ & Yes & $c = a + b$ & $a.sign$ \\
\hline $+$ & $+$ & No  & $c = a + b$ & $a.sign$ \\
\hline $-$ & $-$ & Yes & $c = a + b$ & $a.sign$ \\
\hline $-$ & $-$ & No  & $c = a + b$ & $a.sign$ \\
\hline &&&&\\

\hline $+$ & $-$ & No  & $c = b - a$ & $b.sign$ \\
\hline $-$ & $+$ & No  & $c = b - a$ & $b.sign$ \\

\hline &&&&\\

\hline $+$ & $-$ & Yes & $c = a - b$ & $a.sign$ \\
\hline $-$ & $+$ & Yes & $c = a - b$ & $a.sign$ \\

\hline
\end{tabular}
\end{center}
\end{small}
\caption{Addition Guide Chart}
\label{fig:AddChart}
\end{figure}

Figure~\ref{fig:AddChart} lists all of the eight possible input combinations and is sorted to show that only three specific cases need to be handled.  The 
return code of the unsigned operations at step 1.2, 2.1.2 and 2.2.2 are forwarded to step 3 to check for errors.  This simplifies the description
of the algorithm considerably and best follows how the implementation actually was achieved.

Also note how the \textbf{sign} is set before the unsigned addition or subtraction is performed.  Recall from the descriptions of algorithms
s\_mp\_add and s\_mp\_sub that the mp\_clamp function is used at the end to trim excess digits.  The mp\_clamp algorithm will set the \textbf{sign}
to \textbf{MP\_ZPOS} when the \textbf{used} digit count reaches zero.  

For example, consider performing $-a + a$ with algorithm mp\_add.  By the description of the algorithm the sign is set to \textbf{MP\_NEG} which would
produce a result of $-0$.  However, since the sign is set first then the unsigned addition is performed the subsequent usage of algorithm mp\_clamp 
within algorithm s\_mp\_add will force $-0$ to become $0$.  

\index{bn\_mp\_add.c}
\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
019   mp_add (mp_int * a, mp_int * b, mp_int * c)
020   \{
021     int     sa, sb, res;
022   
023     /* get sign of both inputs */
024     sa = a->sign;
025     sb = b->sign;
026   
027     /* handle two cases, not four */
028     if (sa == sb) \{
029       /* both positive or both negative */
030       /* add their magnitudes, copy the sign */
031       c->sign = sa;
032       res = s_mp_add (a, b, c);
033     \} else \{
034       /* one positive, the other negative */
035       /* subtract the one with the greater magnitude from */
036       /* the one of the lesser magnitude.  The result gets */
037       /* the sign of the one with the greater magnitude. */
038       if (mp_cmp_mag (a, b) == MP_LT) \{
039         c->sign = sb;
040         res = s_mp_sub (b, a, c);
041       \} else \{
042         c->sign = sa;
043         res = s_mp_sub (a, b, c);
044       \}
045     \}
046     return res;
047   \}
048   
\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.  If any of the lower level operations failed return(\textit{MP\_MEM}). \\
4.  Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{center}
\caption{Algorithm mp\_sub}
\end{figure}

\textbf{Algorithm mp\_sub.}
This algorithm performs the signed subtraction of two inputs.  Similar to algorithm mp\_add there is no reference in either \cite{TAOCPV2} or 
\cite{HAC}.  Also this algorithm is restricted by algorithm s\_mp\_sub.  The following chart lists the eight possible inputs and
the operations required.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline \textbf{Sign of $a$} & \textbf{Sign of $b$} & \textbf{$\vert a \vert \ge \vert b \vert $} & \textbf{Unsigned Operation} & \textbf{Result Sign Flag} \\
\hline $+$ & $-$ & Yes & $c = a + b$ & $a.sign$ \\
\hline $+$ & $-$ & No  & $c = a + b$ & $a.sign$ \\
\hline $-$ & $+$ & Yes & $c = a + b$ & $a.sign$ \\
\hline $-$ & $+$ & No  & $c = a + b$ & $a.sign$ \\
\hline &&&& \\
\hline $+$ & $+$ & Yes & $c = a - b$ & $a.sign$ \\
\hline $-$ & $-$ & Yes & $c = a - b$ & $a.sign$ \\
\hline &&&& \\
\hline $+$ & $+$ & No  & $c = b - a$ & $\mbox{opposite of }a.sign$ \\
\hline $-$ & $-$ & No  & $c = b - a$ & $\mbox{opposite of }a.sign$ \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Subtraction Guide Chart}
\end{figure}

Similar to the case of algorithm mp\_add the \textbf{sign} is set first before the unsigned addition or subtraction.  That is to prevent the 
algorithm from producing $-a - -a = -0$ as a result.  

\index{bn\_mp\_sub.c}
\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   
\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.  If the reallocation failed return(\textit{MP\_MEM}). \\
3.  $oldused \leftarrow b.used$ \\
4.  $b.used \leftarrow a.used$ \\
5.  $r \leftarrow 0$ \\
6.  for $n$ from 0 to $a.used - 1$ do \\
\hspace{3mm}6.1  $rr \leftarrow a_n >> (lg(\beta) - 1)$ \\
\hspace{3mm}6.2  $b_n \leftarrow (a_n << 1) + r \mbox{ (mod }\beta\mbox{)}$ \\
\hspace{3mm}6.3  $r \leftarrow rr$ \\
7.  If $r \ne 0$ then do \\
\hspace{3mm}7.1  $b_{n + 1} \leftarrow r$ \\
\hspace{3mm}7.2  $b.used \leftarrow b.used + 1$ \\
8.  If $b.used < oldused - 1$ then do \\
\hspace{3mm}8.1  for $n$ from $b.used$ to $oldused - 1$ do \\
\hspace{6mm}8.1.1  $b_n \leftarrow 0$ \\
9.  $b.sign \leftarrow a.sign$ \\
10.  Return(\textit{MP\_OKAY}).\\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_mul\_2}
\end{figure}

\textbf{Algorithm mp\_mul\_2.}
This algorithm will quickly multiply a mp\_int by two provided $\beta$ is a power of two.  Neither \cite{TAOCPV2} nor \cite{HAC} describe such 
an algorithm despite the fact it arises often in other algorithms.  The algorithm is setup much like the lower level algorithm s\_mp\_add since 
it is for all intents and purposes equivalent to the operation $b = \vert a \vert + \vert a \vert$.  

Step 1 and 2 grow the input as required to accomodate the maximum number of \textbf{used} digits in the result.  The initial \textbf{used} count
is set to $a.used$ at step 4.  Only if there is a final carry will the \textbf{used} count require adjustment.

Step 6 is an optimization implementation of the addition loop for this specific case.  That is since the two values being added together 
are the same there is no need to perform two reads from the digits of $a$.  Step 6.1 performs a single precision shift on the current digit $a_n$ to
obtain what will be the carry for the next iteration.  Step 6.2 calculates the $n$'th digit of the result as single precision shift of $a_n$ plus
the previous carry.  Recall from section 5.1 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 $a$.

\index{bn\_mp\_mul\_2.c}
\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
019   mp_mul_2 (mp_int * a, mp_int * b)
020   \{
021     int     x, res, oldused;
022   
023     /* grow to accomodate result */
024     if (b->alloc < a->used + 1) \{
025       if ((res = mp_grow (b, a->used + 1)) != MP_OKAY) \{
026         return res;
027       \}
028     \}
029   
030     oldused = b->used;
031     b->used = a->used;
032   
033     \{
034       register mp_digit r, rr, *tmpa, *tmpb;
035   
036       /* alias for source */
037       tmpa = a->dp;
038       
039       /* alias for dest */
040       tmpb = b->dp;
041   
042       /* carry */
043       r = 0;
044       for (x = 0; x < a->used; x++) \{
045       
046         /* get what will be the *next* carry bit from the 
047          * MSB of the current digit 
048          */
049         rr = *tmpa >> ((mp_digit)(DIGIT_BIT - 1));
050         
051         /* now shift up this digit, add in the carry [from the previous] */
052         *tmpb++ = ((*tmpa++ << ((mp_digit)1)) | r) & MP_MASK;
053         
054         /* copy the carry that would be from the source 
055          * digit into the next iteration 
056          */
057         r = rr;
058       \}
059   
060       /* new leading digit? */
061       if (r != 0) \{
062         /* add a MSB which is always 1 at this point */
063         *tmpb = 1;
064         ++b->used;
065       \}
066   
067       /* now zero any excess digits on the destination 
068        * that we didn't write to 
069        */
070       tmpb = b->dp + b->used;
071       for (x = b->used; x < oldused; x++) \{
072         *tmpb++ = 0;
073       \}
074     \}
075     b->sign = a->sign;
076     return MP_OKAY;
077   \}
\end{alltt}
\end{small}

This implementation is essentially an optimized implementation of s\_mp\_add for the case of doubling an input.  The only noteworthy difference
is the use of the logical shift operator on line 52 to perform a single precision doubling.  

\subsection{Division by Two}
A division by two can just as easily be accomplished with a logical shift right as multiplication by two can be with a logical shift left.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_div\_2}. \\
\textbf{Input}.   One mp\_int $a$ \\
\textbf{Output}.  $b = a/2$. \\
\hline \\
1.  If $b.alloc < a.used$ then grow $b$ to hold $a.used$ digits.  (\textit{mp\_grow}) \\
2.  If the reallocation failed return(\textit{MP\_MEM}). \\
3.  $oldused \leftarrow b.used$ \\
4.  $b.used \leftarrow a.used$ \\
5.  $r \leftarrow 0$ \\
6.  for $n$ from $b.used - 1$ to $0$ do \\
\hspace{3mm}6.1  $rr \leftarrow a_n \mbox{ (mod }2\mbox{)}$\\
\hspace{3mm}6.2  $b_n \leftarrow (a_n >> 1) + (r << (lg(\beta) - 1)) \mbox{ (mod }\beta\mbox{)}$ \\
\hspace{3mm}6.3  $r \leftarrow rr$ \\
7.  If $b.used < oldused - 1$ then do \\
\hspace{3mm}7.1  for $n$ from $b.used$ to $oldused - 1$ do \\
\hspace{6mm}7.1.1  $b_n \leftarrow 0$ \\
8.  $b.sign \leftarrow a.sign$ \\
9.  Clamp excess digits of $b$.  (\textit{mp\_clamp}) \\
10.  Return(\textit{MP\_OKAY}).\\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_div\_2}
\end{figure}

\textbf{Algorithm mp\_div\_2.}
This algorithm will divide an mp\_int by two using logical shifts to the right.  Like mp\_mul\_2 it uses a modified low level addition
core as the basis of the algorithm.  Unlike mp\_mul\_2 the shift operations work from the leading digit to the trailing digit.  The algorithm
could be written to work from the trailing digit to the leading digit however, it would have to stop one short of $a.used - 1$ digits to prevent
reading past the end of the array of digits.

Essentially the loop at step 6 is similar to that of mp\_mul\_2 except the logical shifts go in the opposite direction and the carry is at the 
least significant bit not the most significant bit.  

\index{bn\_mp\_div\_2.c}
\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
019   mp_div_2 (mp_int * a, mp_int * b)
020   \{
021     int     x, res, oldused;
022   
023     /* copy */
024     if (b->alloc < a->used) \{
025       if ((res = mp_grow (b, a->used)) != MP_OKAY) \{
026         return res;
027       \}
028     \}
029   
030     oldused = b->used;
031     b->used = a->used;
032     \{
033       register mp_digit r, rr, *tmpa, *tmpb;
034   
035       /* source alias */
036       tmpa = a->dp + b->used - 1;
037   
038       /* dest alias */
039       tmpb = b->dp + b->used - 1;
040   
041       /* carry */
042       r = 0;
043       for (x = b->used - 1; x >= 0; x--) \{
044         /* get the carry for the next iteration */
045         rr = *tmpa & 1;
046   
047         /* shift the current digit, add in carry and store */
048         *tmpb-- = (*tmpa-- >> 1) | (r << (DIGIT_BIT - 1));
049   
050         /* forward carry to next iteration */
051         r = rr;
052       \}
053   
054       /* zero excess digits */
055       tmpb = b->dp + b->used;
056       for (x = b->used; x < oldused; x++) \{
057         *tmpb++ = 0;
058       \}
059     \}
060     b->sign = a->sign;
061     mp_clamp (b);
062     return MP_OKAY;
063   \}
\end{alltt}
\end{small}

\section{Polynomial Basis Operations}
Recall from section 5.3 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}
\end{figure}
\end{center}

\index{bn\_mp\_lshd.c}
\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
019   mp_lshd (mp_int * a, int b)
020   \{
021     int     x, res;
022   
023     /* if its less than zero return */
024     if (b <= 0) \{
025       return MP_OKAY;
026     \}
027   
028     /* grow to fit the new digits */
029     if (a->alloc < a->used + b) \{
030        if ((res = mp_grow (a, a->used + b)) != MP_OKAY) \{
031          return res;
032        \}
033     \}
034   
035     \{
036       register mp_digit *top, *bottom;
037   
038       /* increment the used by the shift amount then copy upwards */
039       a->used += b;
040   
041       /* top */
042       top = a->dp + a->used - 1;
043   
044       /* base */
045       bottom = a->dp + a->used - 1 - b;
046   
047       /* much like mp_rshd this is implemented using a sliding window
048        * except the window goes the otherway around.  Copying from
049        * the bottom to the top.  see bn_mp_rshd.c for more info.
050        */
051       for (x = a->used - 1; x >= b; x--) \{
052         *top-- = *bottom--;
053       \}
054   
055       /* zero the lower digits */
056       top = a->dp;
057       for (x = 0; x < b; x++) \{
058         *top++ = 0;
059       \}
060     \}
061     return MP_OKAY;
062   \}
\end{alltt}
\end{small}

The if statement on line 24 ensures that the $b$ variable is greater than zero.  The \textbf{used} count is incremented by $b$ before
the copy loop begins.  This elminates the need for an additional variable in the for loop.  The variable $top$ on line 42 is an alias
for the leading digit while $bottom$ on line 45 is an alias for the trailing edge.  The aliases form a window of exactly $b$ digits
over the input.  

\subsection{Division by $x$}

Division by powers of $x$ is easily achieved by shifting the digits right and removing any that will end up to the right of the zero'th digit.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_rshd}. \\
\textbf{Input}.   One mp\_int $a$ and an integer $b$ \\
\textbf{Output}.  $a \leftarrow a / \beta^b$ (Divide by $x^b$). \\
\hline \\
1.  If $b \le 0$ then return. \\
2.  If $a.used \le b$ then do \\
\hspace{3mm}2.1  Zero $a$.  (\textit{mp\_zero}). \\
\hspace{3mm}2.2  Return. \\
3.  $i \leftarrow 0$ \\
4.  $j \leftarrow b$ \\
5.  for $n$ from 0 to $a.used - b - 1$ do \\
\hspace{3mm}5.1  $a_i \leftarrow a_j$ \\
\hspace{3mm}5.2  $i \leftarrow i + 1$ \\
\hspace{3mm}5.3  $j \leftarrow j + 1$ \\
6.  for $n$ from $a.used - b$ to $a.used - 1$ do \\
\hspace{3mm}6.1  $a_n \leftarrow 0$ \\
7.  $a.used \leftarrow a.used - b$ \\
8.  Return. \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_rshd}
\end{figure}

\textbf{Algorithm mp\_rshd.}
This algorithm divides the input in place by the $b$'th power of $x$.  It is analogous to dividing by a $\beta^b$ but much quicker since
it does not require single precision division.  This algorithm does not actually return an error code as it cannot fail.  

If the input $b$ is less than one the algorithm quickly returns without performing any work.  If the \textbf{used} count is less than or equal
to the shift count $b$ then it will simply zero the input and return.

After the trivial cases of inputs have been handled the sliding window is setup.  Much like the case of algorithm mp\_lshd a sliding window that
is $b$ digits wide is used to copy the digits.  Unlike mp\_lshd the window slides in the opposite direction from the trailing to the leading digit.  
Also the digits are copied from the leading to the trailing edge.

Once the window copy is complete the upper digits must be zeroed and the \textbf{used} count decremented.

\index{bn\_mp\_rshd.c}
\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
019   mp_rshd (mp_int * a, int b)
020   \{
021     int     x;
022   
023     /* if b <= 0 then ignore it */
024     if (b <= 0) \{
025       return;
026     \}
027   
028     /* if b > used then simply zero it and return */
029     if (a->used <= b) \{
030       mp_zero (a);
031       return;
032     \}
033   
034     \{
035       register mp_digit *bottom, *top;
036   
037       /* shift the digits down */
038   
039       /* bottom */
040       bottom = a->dp;
041   
042       /* top [offset into digits] */
043       top = a->dp + b;
044   
045       /* this is implemented as a sliding window where 
046        * the window is b-digits long and digits from 
047        * the top of the window are copied to the bottom
048        *
049        * e.g.
050   
051        b-2 | b-1 | b0 | b1 | b2 | ... | bb |   ---->
052                    /\symbol{92}                   |      ---->
053                     \symbol{92}-------------------/      ---->
054        */
055       for (x = 0; x < (a->used - b); x++) \{
056         *bottom++ = *top++;
057       \}
058   
059       /* zero the top digits */
060       for (; x < a->used; x++) \{
061         *bottom++ = 0;
062       \}
063     \}
064     
065     /* remove excess digits */
066     a->used -= b;
067   \}
\end{alltt}
\end{small}

The only noteworthy element of this routine is the lack of a return type.  

-- Will update later to give it a return type...Tom

\section{Powers of Two}

Now that algorithms for moving single bits as well as whole digits exist algorithms for moving the ``in between'' distances are required.  For 
example, to quickly multiply by $2^k$ for any $k$ without using a full multiplier algorithm would prove useful.  Instead of performing single
shifts $k$ times to achieve a multiplication by $2^{\pm k}$ a mixture of whole digit shifting and partial digit shifting is employed.  

\subsection{Multiplication by Power of Two}

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_mul\_2d}. \\
\textbf{Input}.   One mp\_int $a$ and an integer $b$ \\
\textbf{Output}.  $c \leftarrow a \cdot 2^b$. \\
\hline \\
1.  $c \leftarrow a$.  (\textit{mp\_copy}) \\
2.  If $c.alloc < c.used + \lfloor b / lg(\beta) \rfloor + 2$ then grow $c$ accordingly. \\
3.  If the reallocation failed return(\textit{MP\_MEM}). \\
4.  If $b \ge lg(\beta)$ then \\
\hspace{3mm}4.1  $c \leftarrow c \cdot \beta^{\lfloor b / lg(\beta) \rfloor}$ (\textit{mp\_lshd}). \\
\hspace{3mm}4.2  If step 4.1 failed return(\textit{MP\_MEM}). \\
5.  $d \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\
6.  If $d \ne 0$ then do \\
\hspace{3mm}6.1  $mask \leftarrow 2^d$ \\
\hspace{3mm}6.2  $r \leftarrow 0$ \\
\hspace{3mm}6.3  for $n$ from $0$ to $c.used - 1$ do \\
\hspace{6mm}6.3.1  $rr \leftarrow c_n >> (lg(\beta) - d) \mbox{ (mod }mask\mbox{)}$ \\
\hspace{6mm}6.3.2  $c_n \leftarrow (c_n << d) + r \mbox{ (mod }\beta\mbox{)}$ \\
\hspace{6mm}6.3.3  $r \leftarrow rr$ \\
\hspace{3mm}6.4  If $r > 0$ then do \\
\hspace{6mm}6.4.1  $c_{c.used} \leftarrow r$ \\
\hspace{6mm}6.4.2  $c.used \leftarrow c.used + 1$ \\
7.  Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_mul\_2d}
\end{figure}

\textbf{Algorithm mp\_mul\_2d.}
This algorithm multiplies $a$ by $2^b$ and stores the result in $c$.  The algorithm uses algorithm mp\_lshd and a derivative of algorithm mp\_mul\_2 to
quickly compute the product.

First the algorithm will multiply $a$ by $x^{\lfloor b / lg(\beta) \rfloor}$ which will ensure that the remainder multiplicand is less than 
$\beta$.  For example, if $b = 37$ and $\beta = 2^{28}$ then this step will multiply by $x$ leaving a multiplication by $2^{37 - 28} = 2^{9}$ 
left.

After the digits have been shifted appropriately at most $lg(\beta) - 1$ shifts are left to perform.  Step 5 calculates the number of remaining shifts 
required.  If it is non-zero a modified shift loop is used to calculate the remaining product.  
Essentially the loop is a generic version of algorith mp\_mul2 designed to handle any shift count in the range $1 \le x < lg(\beta)$.  The $mask$
variable is used to extract the upper $d$ bits to form the carry for the next iteration.  

This algorithm is loosely measured as a $O(2n)$ algorithm which means that if the input is $n$-digits that it takes $2n$ ``time'' to 
complete.  It is possible to optimize this algorithm down to a $O(n)$ algorithm at a cost of making the algorithm slightly harder to follow.

\index{bn\_mp\_mul\_2d.c}
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_mul\_2d.c
\vspace{-3mm}
\begin{alltt}
016   
017   /* NOTE:  This routine requires updating.  For instance the c->used = c->all
      oc bit
018      is wrong.  We should just shift c->used digits then set the carry as c->d
      p[c->used] = carry
019    
020      To be fixed for LTM 0.18
021    */
022   
023   /* shift left by a certain bit count */
024   int
025   mp_mul_2d (mp_int * a, int b, mp_int * c)
026   \{
027     mp_digit d;
028     int      res;
029   
030     /* copy */
031     if (a != c) \{
032        if ((res = mp_copy (a, c)) != MP_OKAY) \{
033          return res;
034        \}
035     \}
036   
037     if (c->alloc < (int)(c->used + b/DIGIT_BIT + 2)) \{
038        if ((res = mp_grow (c, c->used + b / DIGIT_BIT + 2)) != MP_OKAY) \{
039          return res;
040        \}
041     \}
042   
043     /* shift by as many digits in the bit count */
044     if (b >= (int)DIGIT_BIT) \{
045       if ((res = mp_lshd (c, b / DIGIT_BIT)) != MP_OKAY) \{
046         return res;
047       \}
048     \}
049     c->used = c->alloc;
050   
051     /* shift any bit count < DIGIT_BIT */
052     d = (mp_digit) (b % DIGIT_BIT);
053     if (d != 0) \{
054       register mp_digit *tmpc, mask, r, rr;
055       register int x;
056   
057       /* bitmask for carries */
058       mask = (((mp_digit)1) << d) - 1;
059   
060       /* alias */
061       tmpc = c->dp;
062   
063       /* carry */
064       r    = 0;
065       for (x = 0; x < c->used; x++) \{
066         /* get the higher bits of the current word */
067         rr = (*tmpc >> (DIGIT_BIT - d)) & mask;
068   
069         /* shift the current word and OR in the carry */
070         *tmpc = ((*tmpc << d) | r) & MP_MASK;
071         ++tmpc;
072   
073         /* set the carry to the carry bits of the current word */
074         r = rr;
075       \}
076     \}
077     mp_clamp (c);
078     return MP_OKAY;
079   \}
\end{alltt}
\end{small}

Notes to be revised when code is updated. -- Tom

\subsection{Division by Power of Two}

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_div\_2d}. \\
\textbf{Input}.   One mp\_int $a$ and an integer $b$ \\
\textbf{Output}.  $c \leftarrow \lfloor a / 2^b \rfloor, d \leftarrow a \mbox{ (mod }2^b\mbox{)}$. \\
\hline \\
1.  If $b \le 0$ then do \\
\hspace{3mm}1.1  $c \leftarrow a$ (\textit{mp\_copy}) \\
\hspace{3mm}1.2  $d \leftarrow 0$ (\textit{mp\_zero}) \\
\hspace{3mm}1.3  Return(\textit{MP\_OKAY}). \\
2.  $c \leftarrow a$ \\
3.  $d \leftarrow a \mbox{ (mod }2^b\mbox{)}$ (\textit{mp\_mod\_2d}) \\
4.  If $b \ge lg(\beta)$ then do \\
\hspace{3mm}4.1  $c \leftarrow \lfloor c/\beta^{\lfloor b/lg(\beta) \rfloor} \rfloor$ (\textit{mp\_rshd}). \\
5.  $k \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\
6.  If $k \ne 0$ then do \\
\hspace{3mm}6.1  $mask \leftarrow 2^k$ \\
\hspace{3mm}6.2  $r \leftarrow 0$ \\
\hspace{3mm}6.3  for $n$ from $c.used - 1$ to $0$ do \\
\hspace{6mm}6.3.1  $rr \leftarrow c_n \mbox{ (mod }mask\mbox{)}$ \\
\hspace{6mm}6.3.2  $c_n \leftarrow (c_n >> k) + (r << (lg(\beta) - k))$ \\
\hspace{6mm}6.3.3  $r \leftarrow rr$ \\
7.  Clamp excess digits of $c$.  (\textit{mp\_clamp}) \\
8.  Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_div\_2d}
\end{figure}

\textbf{Algorithm mp\_div\_2d.}
This algorithm will divide an input $a$ by $2^b$ and produce the quotient and remainder.  The algorithm is designed much like algorithm 
mp\_mul\_2d by first using whole digit shifts then single precision shifts.  This algorithm will also produce the remainder of the division
by using algorithm mp\_mod\_2d.

\index{bn\_mp\_div\_2d.c}
\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
019   mp_div_2d (mp_int * a, int b, mp_int * c, mp_int * d)
020   \{
021     mp_digit D, r, rr;
022     int     x, res;
023     mp_int  t;
024   
025   
026     /* if the shift count is <= 0 then we do no work */
027     if (b <= 0) \{
028       res = mp_copy (a, c);
029       if (d != NULL) \{
030         mp_zero (d);
031       \}
032       return res;
033     \}
034   
035     if ((res = mp_init (&t)) != MP_OKAY) \{
036       return res;
037     \}
038   
039     /* get the remainder */
040     if (d != NULL) \{
041       if ((res = mp_mod_2d (a, b, &t)) != MP_OKAY) \{
042         mp_clear (&t);
043         return res;
044       \}
045     \}
046   
047     /* copy */
048     if ((res = mp_copy (a, c)) != MP_OKAY) \{
049       mp_clear (&t);
050       return res;
051     \}
052   
053     /* shift by as many digits in the bit count */
054     if (b >= (int)DIGIT_BIT) \{
055       mp_rshd (c, b / DIGIT_BIT);
056     \}
057   
058     /* shift any bit count < DIGIT_BIT */
059     D = (mp_digit) (b % DIGIT_BIT);
060     if (D != 0) \{
061       register mp_digit *tmpc, mask;
062   
063       /* mask */
064       mask = (((mp_digit)1) << D) - 1;
065   
066       /* alias */
067       tmpc = c->dp + (c->used - 1);
068   
069       /* carry */
070       r = 0;
071       for (x = c->used - 1; x >= 0; x--) \{
072         /* get the lower  bits of this word in a temp */
073         rr = *tmpc & mask;
074   
075         /* shift the current word and mix in the carry bits from the previous 
      word */
076         *tmpc = (*tmpc >> D) | (r << (DIGIT_BIT - D));
077         --tmpc;
078   
079         /* set the carry to the carry bits of the current word found above */
080         r = rr;
081       \}
082     \}
083     mp_clamp (c);
084     if (d != NULL) \{
085       mp_exch (&t, d);
086     \}
087     mp_clear (&t);
088     return MP_OKAY;
089   \}
\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.  (-- Fix this paragraph up later, Tom).

\subsection{Remainder of Division by Power of Two}

The last algorithm in the series of polynomial basis power of two algorithms is calculating the remainder of division by $2^b$.  This
algorithm benefits from the fact that in twos complement arithmetic $a \mbox{ (mod }2^b\mbox{)}$ is the same as $a$ AND $2^b - 1$.  

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_mod\_2d}. \\
\textbf{Input}.   One mp\_int $a$ and an integer $b$ \\
\textbf{Output}.  $c \leftarrow a \mbox{ (mod }2^b\mbox{)}$. \\
\hline \\
1.  If $b \le 0$ then do \\
\hspace{3mm}1.1  $c \leftarrow 0$ (\textit{mp\_zero}) \\
\hspace{3mm}1.2  Return(\textit{MP\_OKAY}). \\
2.  If $b > a.used \cdot lg(\beta)$ then do \\
\hspace{3mm}2.1  $c \leftarrow a$ (\textit{mp\_copy}) \\
\hspace{3mm}2.2  Return the result of step 2.1. \\
3.  $c \leftarrow a$ \\
4.  If step 3 failed return(\textit{MP\_MEM}). \\
5.  for $n$ from $\lceil b / lg(\beta) \rceil$ to $c.used$ do \\
\hspace{3mm}5.1  $c_n \leftarrow 0$ \\
6.  $k \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\
7.  $c_{\lfloor b / lg(\beta) \rfloor} \leftarrow c_{\lfloor b / lg(\beta) \rfloor} \mbox{ (mod }2^{k}\mbox{)}$. \\
8.  Clamp excess digits of $c$.  (\textit{mp\_clamp}) \\
9.  Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_mod\_2d}
\end{figure}

\textbf{Algorithm mp\_mod\_2d.}
This algorithm will quickly calculate the value of $a \mbox{ (mod }2^b\mbox{)}$.  First if $b$ is less than or equal to zero the 
result is set to zero.  If $b$ is greater than the number of bits in $a$ then it simply copies $a$ to $c$ and returns.  Otherwise, $a$ 
is copied to $b$, leading digits are removed and the remaining leading digit is trimed to the exact bit count.

\index{bn\_mp\_mod\_2d.c}
\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   
024     /* if b is <= 0 then zero the int */
025     if (b <= 0) \{
026       mp_zero (c);
027       return MP_OKAY;
028     \}
029   
030     /* if the modulus is larger than the value than return */
031     if (b > (int) (a->used * DIGIT_BIT)) \{
032       res = mp_copy (a, c);
033       return res;
034     \}
035   
036     /* copy */
037     if ((res = mp_copy (a, c)) != MP_OKAY) \{
038       return res;
039     \}
040   
041     /* zero digits above the last digit of the modulus */
042     for (x = (b / DIGIT_BIT) + ((b % DIGIT_BIT) == 0 ? 0 : 1); x < c->used; x+
      +) \{
043       c->dp[x] = 0;
044     \}
045     /* clear the digit that is not completely outside/inside the modulus */
046     c->dp[b / DIGIT_BIT] &=
047       (mp_digit) ((((mp_digit) 1) << (((mp_digit) b) % DIGIT_BIT)) - ((mp_digi
      t) 1));
048     mp_clamp (c);
049     return MP_OKAY;
050   \}
\end{alltt}
\end{small}

-- Add comments later, Tom.

\section*{Exercises}
\begin{tabular}{cl}
$\left [ 3 \right ] $ & Devise an algorithm that performs $a \cdot 2^b$ for generic values of $b$ \\
                      & in $O(n)$ time. \\
                      &\\
$\left [ 3 \right ] $ & Devise an efficient algorithm to multiply by small low hamming  \\
                      & weight values such as $3$, $5$ and $9$.  Extend it to handle all values \\
                      & upto $64$ with a hamming weight less than three. \\
                      &\\
$\left [ 2 \right ] $ & Modify the preceding algorithm to handle values of the form \\
                      & $2^k - 1$ as well. \\
                      &\\
$\left [ 3 \right ] $ & Using only algorithms mp\_mul\_2, mp\_div\_2 and mp\_add create an \\
                      & algorithm to multiply two integers in roughly $O(2n^2)$ time for \\
                      & any $n$-bit input.  Note that the time of addition is ignored in the \\
                      & calculation.  \\
                      & \\
$\left [ 5 \right ] $ & Improve the previous algorithm to have a working time of at most \\
                      & $O \left (2^{(k-1)}n + \left ({2n^2 \over k} \right ) \right )$ for an appropriate choice of $k$.  Again ignore \\
                      & the cost of addition. \\
                      & \\
$\left [ 2 \right ] $ & Devise a chart to find optimal values of $k$ for the previous problem \\
                      & for $n = 64 \ldots 1024$ in steps of $64$. \\
                      & \\
$\left [ 2 \right ] $ & Using only algorithms mp\_abs and mp\_sub devise another method for \\
                      & calculating the result of a signed comparison. \\
                      &
\end{tabular}

\chapter{Multiplication and Squaring}
\section{The Multipliers}
For most number theoretic systems including public key cryptographic algorithms the set of algorithms collectively known as the
``multipliers'' form the most important subset of algorithms of any multiple precision integer package.  The set of multipliers 
include multiplication, squaring and modular reduction algorithms.  

The importance of these algorithms is driven by the fact that most popular public key algorithms are based on modular 
exponentiation.  That is performing $d \equiv a^b \mbox{ (mod }c\mbox{)}$ for some arbitrary choice of $a$, $b$, $c$ and $d$.  Roughly
speaking the a modular exponentiation will spend about 40\% of the time in modular reductions, 35\% of the time in squaring and 25\% of
the time in multiplications.  Only a small trivial amount of time is spent on lower level algorithms such as mp\_clamp, mp\_init, etc...

This chapter will discuss only two of the multipliers algorithms, multiplication and squaring.  As will be discussed shortly very efficient
multiplier algorithms are not always straightforward and deserve a lot of attention.

\section{Multiplication}
\subsection{The Baseline Multiplication}
\index{baseline multiplication}
Computing the product of two integers in software can be achieved using a trivial adaptation of the standard $O(n^2)$ long-hand multiplication
algorithm school children are taught.  The ``baseline multiplication'' algorithm is designed to act as the ``catch-all'' algorithm only called
when the faster algorithms cannot be used.  This algorithm does not use any particularly interesting optimizations.

The first algorithm to review is the unsigned multiplication algorithm from which a signed multiplication algorithm can be established.  One important 
facet of this algorithm to note is that it has been modified to only produce a certain amount of output digits as resolution.  Recall that for
a $n$ and $m$ digit input the product will be at most $n + m + 1$ digits.  Therefore, this algorithm can be reduced to a full multiplier by
telling it to produce $n + m + 1$ digits.  

Recall from sub-section 5.2.2 the definition of $\gamma$ as the number of bits in the type \textbf{mp\_digit}.  We shall now extend this 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 6.2.2 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.  \\
\hspace{3mm}1.2  Return the result of step 1.1 \\
\\
Allocate and initialize a temporary mp\_int. \\
2.  Init $t$ to be of size $digs$ \\
3.  If step 2 failed return(\textit{MP\_MEM}). \\
4.  $t.used \leftarrow digs$ \\
\\
Compute the product. \\
5.  for $ix$ from $0$ to $a.used - 1$ do \\
\hspace{3mm}5.1  $u \leftarrow 0$ \\
\hspace{3mm}5.2  $pb \leftarrow \mbox{min}(b.used, digs - ix)$ \\
\hspace{3mm}5.3  If $pb < 1$ then goto step 6. \\
\hspace{3mm}5.4  for $iy$ from $0$ to $pb - 1$ do \\
\hspace{6mm}5.4.1  $\hat r \leftarrow t_{iy + ix} + a_{ix} \cdot b_{iy} + u$ \\
\hspace{6mm}5.4.2  $t_{iy + ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\
\hspace{6mm}5.4.3  $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\
\hspace{3mm}5.5  if $ix + iy < digs$ then do \\
\hspace{6mm}5.5.1  $t_{ix + pb} \leftarrow u$ \\
6.  Clamp excess digits of $t$. \\
7.  Swap $c$ with $t$ \\
8.  Clear $t$ \\
9.  Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm s\_mp\_mul\_digs}
\end{figure}



\textbf{Algorithm s\_mp\_mul\_digs.}
This algorithm computes the unsigned product of two inputs $a$ and $b$ limited to an output precision of $digs$ digits.  While it may seem
a bit awkward to modify the function from its simple $O(n^2)$ description the usefulness of partial multipliers will arise in a future 
algorithm.  The algorithm is loosely based on algorithm 14.12 from \cite[pp. 595]{HAC} and is similar to Algorithm M \cite[pp. 268]{TAOCPV2}.  The
algorithm differs from those cited references because it can produce a variable output precision regardless of the precision of the inputs.

The first thing this algorithm checks for is whether a Comba multiplier can be used instead.   That is if the minimal digit count of either
input is less than $\delta$ the Comba method is used.    After the Comba method is ruled out the baseline algorithm begins.  A 
temporary mp\_int variable $t$ is used to hold the intermediate result of the product.  This allows the algorithm to be used to 
compute products when either $a = c$ or $b = c$ without overwriting the inputs.  

All of step 5 is the infamous $O(n^2)$ multiplication loop slightly modified to only produce upto $digs$ digits of output.  The $pb$ variable
is given the count of digits to read from $b$ inside the nested loop.  If $pb < 0$ then no more output digits can be produced and the algorithm
will exit the loop.  The best way to think of the loops are as a series of $pb \times 1$ multiplication.    That is, in each pass of the 
innermost loop $a_{ix}$ is multiplied against $b$ and the result is added (\textit{with an appropriate shift}) to $t$.  

For example, consider multiplying $576$ by $241$.  That is equivalent to computing $10^0(1)(576) + 10^1(4)(576) + 10^2(2)(576)$ which is best
visualized as the following table.

\begin{figure}[here]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline   &&          & 5 & 7 & 6 & \\
\hline   $\times$&&  & 2 & 4 & 1 & \\
\hline &&&&&&\\
  &&          & 5 & 7 & 6 & $10^0(1)(576)$ \\
  &2 &   3    & 0 & 4 & 0 & $10^1(4)(576)$ \\
  1 & 1 & 5 & 2 & 0 & 0 &  $10^2(2)(576)$ \\
\hline  
\end{tabular}
\end{center}
\caption{Long-Hand Multiplication Diagram}
\end{figure}

Each row of the product is added to the result after being shifted to the left (\textit{multiplied by a power of the radix}) by the appropriate 
count.  That is in pass $ix$ of the inner loop the product is added starting at the $ix$'th digit of the reult.

Step 5.4.1 introduces the hat symbol (\textit{e.g. $\hat x$}) which represents a double precision variable.  The multiplication on that step
is assumed to be a double wide output single precision multiplication.  That is, two single precision variables are multiplied to produce a
double precision result.  The step is somewhat optimized from a long-hand multiplication algorithm because the carry from the addition in step
5.4.1 is forwarded through the nested loop.  If the carry was ignored it would overflow the single precision digit $t_{ix+iy}$ and the result
would be lost.  

At step 5.5 the nested loop is finished and any carry that was left over should be forwarded.  That is provided $ix + iy < digs$ otherwise the
carry is ignored since it will not be part of the result anyways.  

\index{bn\_s\_mp\_mul\_digs.c}
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_mul\_digs.c
\vspace{-3mm}
\begin{alltt}
016   
017   /* multiplies |a| * |b| and only computes upto digs digits of result
018    * HAC pp. 595, Algorithm 14.12  Modified so you can control how 
019    * many digits of output are created.
020    */
021   int
022   s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs)
023   \{
024     mp_int  t;
025     int     res, pa, pb, ix, iy;
026     mp_digit u;
027     mp_word r;
028     mp_digit tmpx, *tmpt, *tmpy;
029   
030     /* can we use the fast multiplier? */
031     if (((digs) < MP_WARRAY) &&
032         MIN (a->used, b->used) < 
033             (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) \{
034       return fast_s_mp_mul_digs (a, b, c, digs);
035     \}
036   
037     if ((res = mp_init_size (&t, digs)) != MP_OKAY) \{
038       return res;
039     \}
040     t.used = digs;
041   
042     /* compute the digits of the product directly */
043     pa = a->used;
044     for (ix = 0; ix < pa; ix++) \{
045       /* set the carry to zero */
046       u = 0;
047   
048       /* limit ourselves to making digs digits of output */
049       pb = MIN (b->used, digs - ix);
050   
051       /* setup some aliases */
052       /* copy of the digit from a used within the nested loop */
053       tmpx = a->dp[ix];
054       
055       /* an alias for the destination shifted ix places */
056       tmpt = t.dp + ix;
057       
058       /* an alias for the digits of b */
059       tmpy = b->dp;
060   
061       /* compute the columns of the output and propagate the carry */
062       for (iy = 0; iy < pb; iy++) \{
063         /* compute the column as a mp_word */
064         r = ((mp_word) *tmpt) + 
065             ((mp_word) tmpx) * ((mp_word) * tmpy++) + 
066             ((mp_word) u);
067   
068         /* the new column is the lower part of the result */
069         *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK));
070   
071         /* get the carry word from the result */
072         u = (mp_digit) (r >> ((mp_word) DIGIT_BIT));
073       \}
074       /* set carry if it is placed below digs */
075       if (ix + iy < digs) \{
076         *tmpt = u;
077       \}
078     \}
079   
080     mp_clamp (&t);
081     mp_exch (&t, c);
082   
083     mp_clear (&t);
084     return MP_OKAY;
085   \}
\end{alltt}
\end{small}

Lines 31 to 35 determine if the Comba method can be used first.  The conditions for using the Comba routine are that min$(a.used, b.used) < \delta$ and
the number of digits of output is less than \textbf{MP\_WARRAY}.  This new constant is used to control the stack usage in the Comba routines.  By
default it is set to $\delta$ but can be reduced when memory is at a premium.

Of particular importance is the calculation of the $ix+iy$'th column on lines 64, 65 and 66.  Note how all of the
variables are cast to the type \textbf{mp\_word}.  That is to ensure that double precision operations are used instead of single precision.  The
multiplication on line 65 is a bit of a GCC optimization.  On the outset it looks like the compiler will have to use a double precision
multiplication to produce the result required.  Such an operation would be horribly slow on most processors and drag this to a crawl.  However,
GCC is smart enough to realize that double wide output single precision multipliers can be used.  For example, the instruction ``MUL'' on the
x86 processor can multiply two 32-bit values and produce a 64-bit result.  

\subsection{Faster Multiplication by the ``Comba'' Method}

One of the huge drawbacks of the ``baseline'' algorithms is that at the $O(n^2)$ level the carry must be computed and propagated upwards.  This
makes the nested loop very sequential and hard to unroll and implement in parallel.  The ``Comba'' method is named after little known 
(\textit{in cryptographic venues}) Paul G. Comba where in \cite{COMBA} a method of implementing fast multipliers that do not require nested 
carry fixup operations was presented.  As an interesting aside it seems that Paul Barrett  describes a similar technique in
his 1986 paper \cite{BARRETT} which was written five years before \cite{COMBA}.  

At the heart of algorithm is once again the long-hand algorithm for multiplication.  Except in this case a slight twist is placed on how
the columns of the result are produced.  In the standard long-hand algorithm rows of products are produced then added together to form the 
final result.  In the baseline algorithm the columns are added together to get the result instantaneously.  

In the Comba algorithm however, the columns of the result are produced entirely independently of each other.  That is at the $O(n^2)$ level a 
simple multiplication and addition step is performed.  Or more succintly that 

\begin{equation}
x_n = \sum_{i+j = n} a_ib_j
\end{equation}

Where $x_n$ is the $n'th$ column of the output vector.  To see how this works consider once again multiplying $576$ by $241$.  

\begin{figure}[here]
\begin{small}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
  \hline &          & 5 & 7 & 6 & First Input\\
  \hline $\times$ & & 2 & 4 & 1 & Second Input\\
\hline            &                        & $1 \cdot 5 = 5$   & $1 \cdot 7 = 7$   & $1 \cdot 6 = 6$ & First pass \\
                  &  $4 \cdot 5 = 20$      & $4 \cdot 7+5=33$  & $4 \cdot 6+7=31$  & 6               & Second pass \\
   $2 \cdot 5 = 10$ &  $2 \cdot 7 + 20 = 34$ & $2 \cdot 6+33=45$ & 31                & 6             & Third pass \\
\hline 10 & 34 & 45 & 31 & 6 & Final Result \\   
\hline   
\end{tabular}
\end{center}
\end{small}
\caption{Comba Multiplication Diagram}
\end{figure}

At this point the vector $x = \left < 10, 34, 45, 31, 6 \right >$ is the result of the first step of the Comba multipler.  
Now the columns must be fixed by propagating the carry upwards.  The following trivial algorithm will accomplish this.

\begin{enumerate}
    \item for $n$ from 0 to $k - 1$ do
    \item \hspace{3mm} $x_{n+1} \leftarrow x_{n+1} + \lfloor x_{n}/\beta \rfloor$ 
    \item \hspace{3mm} $x_{n} \leftarrow x_{n} \mbox{ (mod }\beta\mbox{)}$
\end{enumerate}

With that algorithm and $k = 5$ and $\beta = 10$ the following vector is produced $y = \left < 1, 3, 8, 8, 1, 6 \right >$.  In this case 
$241 \cdot 576$ is in fact $138816$ and the procedure succeeded.  If the algorithm is correct and as will be demonstrated shortly more
efficient than the baseline algorithm why not simply always use this algorithm?

\subsubsection{Column Weight.}
At the nested $O(n^2)$ level the Comba method adds the product of two single precision variables to a each column of the output 
independently.  A serious obstacle is if the carry is lost due to lack of precision before the algorithm has a chance to fix
the carries.  For example, in the multiplication of two three-digit numbers the third column of output will be the sum of
three single precision multiplications.  If the precision of the accumulator for the output digits is less then $3 \cdot (\beta - 1)^2$ then
an overflow can occur and the carry information will be lost.  For any $m$ and $n$ digit input the maximal weight of any column is 
min$(m, n)$ which is fairly obvious.

The maximal number of terms in any column of a product is known as the ``column weight'' and strictly governs when the algorithm can be used.  Recall
from earlier that a double precision type has $\alpha$ bits of resolution and a single precision digit has $lg(\beta)$ bits of precision.  Given these
two quantities we may not violate the following

\begin{equation}
k \cdot \left (\beta - 1 \right )^2 < 2^{\alpha}
\end{equation}

Which reduces to 

\begin{equation}
k \cdot \left ( \beta^2 - 2\beta + 1 \right ) < 2^{\alpha}
\end{equation}

Let $\rho = lg(\beta)$ represent the number of bits in a single precision digit.  By further re-arrangement of the equation the final solution is
found.

\begin{equation}
k \cdot \left (2^{2\rho} - 2^{\rho + 1} + 1 \right ) < 2^{\alpha}
\end{equation}

The defaults for LibTomMath are $\beta = 2^{28}, \alpha = 2^{64}$ which simplies to $72057593501057025 \cdot k < 2^{64}$ which when divided out
result in $k < 257$.  This implies that the smallest input may not have more than $256$ digits if the Comba method is to be used in
this configuration.  This is quite satisfactory for most applications since $256$ digits would be allow for numbers in the range of $2^{7168}$ 
which is much larger than the typical $2^{100}$ to $2^{4000}$ range most public key cryptographic algorithms use.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{fast\_s\_mp\_mul\_digs}. \\
\textbf{Input}.   mp\_int $a$, mp\_int $b$ and an integer $digs$ \\
\textbf{Output}.  $c \leftarrow \vert a \vert \cdot \vert b \vert \mbox{ (mod }\beta^{digs}\mbox{)}$. \\
\hline \\
Place an array of \textbf{MP\_WARRAY} double precision digits named $\hat W$ on the stack. \\
1.  If $c.alloc < digs$ then grow $c$ to $digs$ digits. (\textit{mp\_grow}) \\
2.  If step 1 failed return(\textit{MP\_MEM}).\\
\\
Zero the temporary array $\hat W$. \\
3.  for $n$ from $0$ to $digs - 1$ do \\
\hspace{3mm}3.1  $\hat W_n \leftarrow 0$ \\
\\
Compute the columns. \\
4.  for $ix$ from $0$ to $a.used - 1$ do \\
\hspace{3mm}4.1  $pb \leftarrow \mbox{min}(b.used, digs - ix)$ \\
\hspace{3mm}4.2  If $pb < 1$ then goto step 5. \\
\hspace{3mm}4.3  for $iy$ from $0$ to $pb - 1$ do \\
\hspace{6mm}4.3.1  $\hat W_{ix+iy} \leftarrow \hat W_{ix+iy} + a_{ix}b_{iy}$ \\
\\
Propagate the carries upwards. \\
5.  $oldused \leftarrow c.used$ \\
6.  $c.used \leftarrow digs$ \\
7.  If $digs > 1$ then do \\
\hspace{3mm}7.1.  for $ix$ from $1$ to $digs - 1$ do \\
\hspace{6mm}7.1.1  $\hat W_{ix} \leftarrow \hat W_{ix} + \lfloor \hat W_{ix-1} / \beta \rfloor$ \\
\hspace{6mm}7.1.2  $c_{ix - 1} \leftarrow \hat W_{ix - 1} \mbox{ (mod }\beta\mbox{)}$ \\
8.  else do \\
\hspace{3mm}8.1  $ix \leftarrow 0$ \\
9.  $c_{ix} \leftarrow \hat W_{ix} \mbox{ (mod }\beta\mbox{)}$ \\
\\
Zero excess digits. \\
10.  If $digs < oldused$ then do \\
\hspace{3mm}10.1  for $n$ from $digs$ to $oldused - 1$ do \\
\hspace{6mm}10.1.1  $c_n \leftarrow 0$ \\
11.  Clamp excessive digits of $c$.  (\textit{mp\_clamp}) \\
12.  Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm fast\_s\_mp\_mul\_digs}
\end{figure}

\textbf{Algorithm fast\_s\_mp\_mul\_digs.}
This algorithm performs the unsigned multiplication of $a$ and $b$ using the Comba method limited to $digs$ digits of precision.  The algorithm
essentially peforms the same calculation as algorithm s\_mp\_mul\_digs but much faster.

The array $\hat W$ is meant to be on the stack when the algorithm is used.  The size of the array does not change which is ideal.  Note also that 
unlike algorithm s\_mp\_mul\_digs no temporary mp\_int is required since the result is calculated in place in $\hat W$.  

The $O(n^2)$ loop on step four is where the Comba method starts to show through.  First there is no carry variable in the loop.  Second the
double precision multiply and add step does not have a carry fixup of any sort.  In fact the nested loop is very simple and can be implemented
in parallel.  

What makes the Comba method so attractive is that the carry propagation only takes place outside the $O(n^2)$ nested loop.  For example, if the 
cost in terms of time of a multiply and add is $p$ and the cost of a carry propagation is $q$ then a baseline multiplication would require 
$O \left ((p + q)n^2 \right )$ time to multiply two $n$-digit numbers.  The Comba method only requires $pn^2 + qn$ time, however, in practice 
the speed increase is actually much more.  With $O(n)$ space the algorithm can be reduced to $O(pn + qn)$ time by implementing the $n$ multiply
and add operations in the nested loop in parallel.  

The carry propagation loop on step 7 is fairly straightforward.  It could have been written phased the other direction, that is, to assign
to $c_{ix}$ instead of $c_{ix-1}$ in each iteration.  However, it would still require pre-caution to make sure that $\hat W_{ix+1}$ is not beyond
the \textbf{MP\_WARRAY} words set aside.  

\index{bn\_fast\_s\_mp\_mul\_digs.c}
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_fast\_s\_mp\_mul\_digs.c
\vspace{-3mm}
\begin{alltt}
016   
017   /* Fast (comba) multiplier
018    *
019    * This is the fast column-array [comba] multiplier.  It is 
020    * designed to compute the columns of the product first 
021    * then handle the carries afterwards.  This has the effect 
022    * of making the nested loops that compute the columns very
023    * simple and schedulable on super-scalar processors.
024    *
025    * This has been modified to produce a variable number of 
026    * digits of output so if say only a half-product is required 
027    * you don't have to compute the upper half (a feature 
028    * required for fast Barrett reduction).
029    *
030    * Based on Algorithm 14.12 on pp.595 of HAC.
031    *
032    */
033   int
034   fast_s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs)
035   \{
036     int     olduse, res, pa, ix;
037     mp_word W[MP_WARRAY];
038   
039     /* grow the destination as required */
040     if (c->alloc < digs) \{
041       if ((res = mp_grow (c, digs)) != MP_OKAY) \{
042         return res;
043       \}
044     \}
045   
046     /* clear temp buf (the columns) */
047     memset (W, 0, sizeof (mp_word) * digs);
048   
049     /* calculate the columns */
050     pa = a->used;
051     for (ix = 0; ix < pa; ix++) \{
052       /* this multiplier has been modified to allow you to 
053        * control how many digits of output are produced.  
054        * So at most we want to make upto "digs" digits of output.
055        *
056        * this adds products to distinct columns (at ix+iy) of W
057        * note that each step through the loop is not dependent on
058        * the previous which means the compiler can easily unroll
059        * the loop without scheduling problems
060        */
061       \{
062         register mp_digit tmpx, *tmpy;
063         register mp_word *_W;
064         register int iy, pb;
065   
066         /* alias for the the word on the left e.g. A[ix] * A[iy] */
067         tmpx = a->dp[ix];
068   
069         /* alias for the right side */
070         tmpy = b->dp;
071   
072         /* alias for the columns, each step through the loop adds a new
073            term to each column
074          */
075         _W = W + ix;
076   
077         /* the number of digits is limited by their placement.  E.g.
078            we avoid multiplying digits that will end up above the # of
079            digits of precision requested
080          */
081         pb = MIN (b->used, digs - ix);
082   
083         for (iy = 0; iy < pb; iy++) \{
084           *_W++ += ((mp_word) tmpx) * ((mp_word) * tmpy++);
085         \}
086       \}
087   
088     \}
089   
090     /* setup dest */
091     olduse = c->used;
092     c->used = digs;
093   
094     \{
095       register mp_digit *tmpc;
096   
097       /* At this point W[] contains the sums of each column.  To get the
098        * correct result we must take the extra bits from each column and
099        * carry them down
100        *
101        * Note that while this adds extra code to the multiplier it 
102        * saves time since the carry propagation is removed from the 
103        * above nested loop.This has the effect of reducing the work 
104        * from N*(N+N*c)==N**2 + c*N**2 to N**2 + N*c where c is the 
105        * cost of the shifting.  On very small numbers this is slower 
106        * but on most cryptographic size numbers it is faster.
107        */
108       tmpc = c->dp;
109       for (ix = 1; ix < digs; ix++) \{
110         W[ix] += (W[ix - 1] >> ((mp_word) DIGIT_BIT));
111         *tmpc++ = (mp_digit) (W[ix - 1] & ((mp_word) MP_MASK));
112       \}
113       *tmpc++ = (mp_digit) (W[digs - 1] & ((mp_word) MP_MASK));
114   
115       /* clear unused */
116       for (; ix < olduse; ix++) \{
117         *tmpc++ = 0;
118       \}
119     \}
120   
121     mp_clamp (c);
122     return MP_OKAY;
123   \}
\end{alltt}
\end{small}

The memset on line 47 clears the initial $\hat W$ array to zero in a single step. Like the slower baseline multiplication
implementation a series of aliases (\textit{lines 67, 70 and 75}) are used to simplify the inner $O(n^2)$ loop.  
In this case a new alias $\_\hat W$ has been added which refers to the double precision columns offset by $ix$ in each pass.  

The inner loop on line 84 is where the algorithm will spend the majority of the time.  Which is why it has been stripped to the 
bones of any extra baggage\footnote{Hence the pointer aliases.}.  On x86 processors the multiply and add amounts to at the very least five
instructions (\textit{two loads, two additions, one multiply}) while on the ARMv4 processors it amounts to only three (\textit{one load, one store,
one multiply-add}).   On both the x86 and ARMv4 processors GCC v3.2 does a very good job at unrolling the loop and scheduling it so there 
are very few dependency stalls.

In theory the difference between the baseline and comba algorithms is a mere $O(qn)$ time difference.  However, in the $O(n^2)$ nested loop of the
baseline method there are dependency stalls as the algorithm must wait for the multiplier to finish before propagating the carry to the next 
digit.  As a result fewer of the often multiple execution units\footnote{The AMD Athlon has three execution units and the Intel P4 has four.} can
be simultaneously used.  

\subsection{Polynomial Basis Multiplication}
To break the $O(n^2)$ barrier in multiplication requires a completely different look at integer multiplication.  In the following algorithms
the use of polynomial basis representation for two integers $a$ and $b$ as $f(x) = \sum_{i=0}^{n} a_i x^i$ and  
$g(x) = \sum_{i=0}^{n} b_i x^i$. respectively, is required.  In this system both $f(x)$ and $g(x)$ have $n + 1$ terms and are of the $n$'th degree.
 
The product $a \cdot b \equiv f(x) \cdot g(x)$ is the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$.  The coefficients $w_i$ will
directly yield the desired product when $\beta$ is substituted for $x$.  The direct solution to solve for the $2n + 1$ coefficients
requires $O(n^2)$ time and is would be in practice slower than the Comba technique.

However, numerical analysis theory will indicate that only $2n + 1$ points in $W(x)$ are required to provide $2n + 1$ knowns for the $2n + 1$ unknowns.  
This means by finding $\zeta_y = W(y)$ for $2n + 1$ small values of $y$ the coefficients of $W(x)$ can be found with trivial Gaussian elimination.  
Since the polynomial $W(x)$ is unknown the equivalent $\zeta_y = f(y) \cdot g(y)$ is used in its place.  

The benefit of this technique stems from the fact that $f(y)$ and $g(y)$ are much smaller than either $a$ or $b$ respectively.  In fact if 
both polynomials have $n + 1$ terms then the multiplicands will be $n$  times smaller than the inputs.  Even if $2n + 1$ multiplications are required
since they are of smaller values the algorithm is still faster.

When picking points to gather relations there are always three obvious points to choose, $y = 0, 1$ and $ \infty$.  The $\zeta_0$ term
is simply the product $W(0) = w_0 = a_0 \cdot b_0$.  The $\zeta_1$ term is the product 
$W(1) = \left (\sum_{i = 0}^{n} a_i \right ) \left (\sum_{i = 0}^{n} b_i \right )$.  The third point $\zeta_{\infty}$ is less obvious but rather
simple to explain.  The $2n + 1$'th coefficient of $W(x)$ is numerically equivalent to the most significant column in an integer multiplication.  
The point at $\infty$ is used symbolically to represent the most significant column, that is $W(\infty) = w_{2n + 1} = a_nb_n$.  Note that the 
points at $y = 0$ and $\infty$ yield the coefficients $w_0$ and $w_{2n + 1}$ directly.

If more points are required they should be of small input values which are powers of two such as 
$2^q$ and the related \textit{mirror points} $\left (2^q \right )^{2n}  \cdot \zeta_{2^{-q}}$ for small values of $q$.  Using such 
points will allow the values of $f(y)$ and $g(y)$ to be independently calculated using only left shifts.

As a general rule of the algorithm when the inputs are split into $n$ parts each there are $2n - 1$ multiplications.  Each multiplication is of 
multiplicands that have $n$ times fewer digits than the inputs.  The asymptotic running time of this algorithm is 
$O \left ( k^{lg_n(2n - 1)} \right )$ for $k$ digit inputs (\textit{assuming they have the same number of digits}).  The following table 
summarizes the exponents for various values of $n$.

\newpage\begin{figure}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline \textbf{Split into $n$ Parts} & \textbf{Exponent}  & \textbf{Notes}\\
\hline $2$ & $1.584962501$ & This is Karatsuba Multiplication. \\
\hline $3$ & $1.464973520$ & This is Toom-Cook Multiplication. \\
\hline $4$ & $1.403677461$ &\\
\hline $5$ & $1.365212389$ &\\
\hline $10$ & $1.278753601$ &\\
\hline $100$ & $1.149426538$ &\\
\hline $1000$ & $1.100270931$ &\\
\hline $10000$ & $1.075252070$ &\\
\hline
\end{tabular}
\end{center}
\caption{Asymptotic Running Time of Polynomial Basis Multiplication}
\end{figure}

At first it may seem like a good idea to choose $n = 1000$ since afterall the exponent is approximately $1.1$.  However, the overhead
of solving for the 2001 terms of $W(x)$ will certainly consume any savings the algorithm could offer for all but exceedingly large
numbers.  

\subsubsection{Cutoff Point}
The polynomial basis multiplication algorithms all require fewer single precision multiplications than a straight Comba approach.  However, 
the algorithms incur an overhead (\textit{at the $O(n)$ work level}) since they require a system of equations to be solved.  This makes them costly to 
use with small inputs.  

Let $m$ represent the number of digits in the multiplicands (\textit{assume both multiplicands have the same number of digits}).  There exists a 
point $y$ such that when $m < y$ the polynomial basis algorithms are more costly than Comba, when $m = y$ they are roughly the same cost and 
when $m > y$ the Comba methods are slower than the polynomial basis algorithms.  

The exact location of $y$ depends on several key architectural elements of the computer platform in question.

\begin{enumerate}
\item  The ratio of clock cycles for single precision multiplication versus other simpler operations such as addition, shifting, etc.  For example
on the AMD Athlon the ratio is roughly $17 : 1$ while on the Intel P4 it is $29 : 1$.  The higher the ratio in favour of multiplication the lower
the cutoff point $y$ will be.  

\item  The complexity of the linear system of equations (\textit{for the coefficients of $W(x)$}) is.  Generally speaking as the number of splits
grows the complexity grows substantially.  Ideally solving the system will only involve addition, subtraction and shifting of integers.  This
directly reflects on the ratio previous mentioned.

\item  To a lesser extent memory bandwidth and function call overheads.  Provided the values are in the processor cache this is less of an
influence over the cutoff point.

\end{enumerate}

A clean cutoff point separation occurs when a point $y$ is found such that all of the cutoff point conditions are met.  For example, if the point
is too low then there will be values of $m$ such that $m > y$ and the Comba method is still faster.  Finding the cutoff points is fairly simple when
a high resolution timer is available.  

\subsection{Karatsuba Multiplication}
Karatsuba multiplication \cite{KARA} when originally proposed in 1962 was among the first set of algorithms to break the $O(n^2)$ barrier for
general purpose multiplication.  Given two polynomial basis representations $f(x) = ax + b$ and $g(x) = cx + d$ Karatsuba proved with 
light number theory \cite{KARAP} that the following polynomial is equivalent to multiplication of the two integers the polynomials represent.

\begin{equation}
f(x) \cdot g(x) = acx^2 + ((a - b)(c - d) + ac + bd)x + bd
\end{equation}

Using the observation that $ac$ and $bd$ could be re-used only three half sized multiplications would be required to produce the product.  Applying
this recursively the work factor becomes $O(n^{lg(3)})$ which is substantially better than the work factor $O(n^2)$ of the Comba technique.  It turns 
out what Karatsuba did not know or at least did not publish was that this is simply polynomial basis multiplication with the points 
$\zeta_0$, $\zeta_{\infty}$ and $-\zeta_{-1}$.  Consider the resultant system of equations.

\begin{center}
\begin{tabular}{rcrcrcrc}
$\zeta_{0}$ &      $=$ &  &  &  & & $w_0$ \\
$-\zeta_{-1}$ &    $=$ & $-w_2$ & $+$ & $w_1$ & $-$ & $w_0$ \\
$\zeta_{\infty}$ & $=$ & $w_2$ &  & &  & \\
\end{tabular}
\end{center}

By adding the first and last equation to the equation in the middle the term $w_1$ can be isolated and all three coefficients solved for.  The simplicity
of this system of equations has made Karatsuba fairly popular.  In fact the cutoff point is often fairly low\footnote{With LibTomMath 0.18 it is 70 and 109 for the Intel P4 and AMD Athlon respectively.}
making it an ideal algorithm to speed up certain public key cryptosystems such as RSA and Diffie-Hellman.  It is worth noting that the point 
$\zeta_1$ could be substituted for $-\zeta_{-1}$.  In this case the first and third row are subtracted instead of added to the second row.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_karatsuba\_mul}. \\
\textbf{Input}.   mp\_int $a$ and mp\_int $b$ \\
\textbf{Output}.  $c \leftarrow \vert a \vert \cdot \vert b \vert$ \\
\hline \\
1.  Init the following mp\_int variables: $x0$, $x1$, $y0$, $y1$, $t1$, $x0y0$, $x1y1$.\\
2.  If step 2 failed then return(\textit{MP\_MEM}). \\
\\
Split the input.  e.g. $a = x1 \cdot \beta^B + x0$ \\
3.  $B \leftarrow \mbox{min}(a.used, b.used)/2$ \\
4.  $x0 \leftarrow a \mbox{ (mod }\beta^B\mbox{)}$ (\textit{mp\_mod\_2d}) \\
5.  $y0 \leftarrow b \mbox{ (mod }\beta^B\mbox{)}$ \\
6.  $x1 \leftarrow \lfloor a / \beta^B \rfloor$ (\textit{mp\_rshd}) \\
7.  $y1 \leftarrow \lfloor b / \beta^B \rfloor$ \\
\\
Calculate the three products. \\
8.  $x0y0 \leftarrow x0 \cdot y0$ (\textit{mp\_mul}) \\
9.  $x1y1 \leftarrow x1 \cdot y1$ \\
10.  $t1 \leftarrow x1 - x0$ (\textit{mp\_sub}) \\
11.  $x0 \leftarrow y1 - y0$ \\
12.  $t1 \leftarrow t1 \cdot x0$ \\
\\
Calculate the middle term. \\
13.  $x0 \leftarrow x0y0 + x1y1$ \\
14.  $t1 \leftarrow x0 - t1$ \\
\\
Calculate the final product. \\
15.  $t1 \leftarrow t1 \cdot \beta^B$ (\textit{mp\_lshd}) \\
16.  $x1y1 \leftarrow x1y1 \cdot \beta^{2B}$ \\
17.  $t1 \leftarrow x0y0 + t1$ \\
18.  $c \leftarrow t1 + x1y1$ \\
19.  Clear all of the temporary variables. \\
20.  Return(\textit{MP\_OKAY}).\\
\hline 
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_karatsuba\_mul}
\end{figure}

\textbf{Algorithm mp\_karatsuba\_mul.}
This algorithm computes the unsigned product of two inputs using the Karatsuba method.  It is loosely based on the description
from \cite[pp. 294-295]{TAOCPV2}.  

\index{radix point}
In order to split the two inputs into their respective halves a suitable \textit{radix point} must be chosen.  The radix point chosen must
be used for both of the inputs meaning that it must smaller than the smallest input.  Step 3 chooses the radix point $B$ as half of the 
smallest input \textbf{used} count.  After the radix point is chosen the inputs are split into lower and upper halves.  Step 4 and 5 
compute the lower halves.  Step 6 and 7 computer the upper halves.  

After the halves have been computed the three intermediate half-size products must be computed.  Step 8 and 9 compute the trivial products
$x0 \cdot y0$ and $x1 \cdot y1$.  The mp\_int $x0$ is used as a temporary variable after $x1 - x0$ has been computed.  By using $x0$ instead
of an additional temporary variable the algorithm can avoid an addition memory allocation operation.

The remaining steps 13 through 18 compute the Karatsuba polynomial through a variety of digit shifting and addition operations.

\index{bn\_mp\_karatsuba\_mul.c}
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_karatsuba\_mul.c
\vspace{-3mm}
\begin{alltt}
016   
017   /* c = |a| * |b| using Karatsuba Multiplication using 
018    * three half size multiplications
019    *
020    * Let B represent the radix [e.g. 2**DIGIT_BIT] and 
021    * let n represent half of the number of digits in 
022    * the min(a,b)
023    *
024    * a = a1 * B**n + a0
025    * b = b1 * B**n + b0
026    *
027    * Then, a * b => 
028      a1b1 * B**2n + ((a1 - a0)(b1 - b0) + a0b0 + a1b1) * B + a0b0
029    *
030    * Note that a1b1 and a0b0 are used twice and only need to be 
031    * computed once.  So in total three half size (half # of 
032    * digit) multiplications are performed, a0b0, a1b1 and 
033    * (a1-b1)(a0-b0)
034    *
035    * Note that a multiplication of half the digits requires
036    * 1/4th the number of single precision multiplications so in 
037    * total after one call 25% of the single precision multiplications 
038    * are saved.  Note also that the call to mp_mul can end up back 
039    * in this function if the a0, a1, b0, or b1 are above the threshold.  
040    * This is known as divide-and-conquer and leads to the famous 
041    * O(N**lg(3)) or O(N**1.584) work which is asymptopically lower than 
042    * the standard O(N**2) that the baseline/comba methods use.  
043    * Generally though the overhead of this method doesn't pay off 
044    * until a certain size (N ~ 80) is reached.
045    */
046   int
047   mp_karatsuba_mul (mp_int * a, mp_int * b, mp_int * c)
048   \{
049     mp_int  x0, x1, y0, y1, t1, x0y0, x1y1;
050     int     B, err;
051   
052     err = MP_MEM;
053   
054     /* min # of digits */
055     B = MIN (a->used, b->used);
056   
057     /* now divide in two */
058     B = B / 2;
059   
060     /* init copy all the temps */
061     if (mp_init_size (&x0, B) != MP_OKAY)
062       goto ERR;
063     if (mp_init_size (&x1, a->used - B) != MP_OKAY)
064       goto X0;
065     if (mp_init_size (&y0, B) != MP_OKAY)
066       goto X1;
067     if (mp_init_size (&y1, b->used - B) != MP_OKAY)
068       goto Y0;
069   
070     /* init temps */
071     if (mp_init_size (&t1, B * 2) != MP_OKAY)
072       goto Y1;
073     if (mp_init_size (&x0y0, B * 2) != MP_OKAY)
074       goto T1;
075     if (mp_init_size (&x1y1, B * 2) != MP_OKAY)
076       goto X0Y0;
077   
078     /* now shift the digits */
079     x0.sign = x1.sign = a->sign;
080     y0.sign = y1.sign = b->sign;
081   
082     x0.used = y0.used = B;
083     x1.used = a->used - B;
084     y1.used = b->used - B;
085   
086     \{
087       register int x;
088       register mp_digit *tmpa, *tmpb, *tmpx, *tmpy;
089   
090       /* we copy the digits directly instead of using higher level functions
091        * since we also need to shift the digits
092        */
093       tmpa = a->dp;
094       tmpb = b->dp;
095   
096       tmpx = x0.dp;
097       tmpy = y0.dp;
098       for (x = 0; x < B; x++) \{
099         *tmpx++ = *tmpa++;
100         *tmpy++ = *tmpb++;
101       \}
102   
103       tmpx = x1.dp;
104       for (x = B; x < a->used; x++) \{
105         *tmpx++ = *tmpa++;
106       \}
107   
108       tmpy = y1.dp;
109       for (x = B; x < b->used; x++) \{
110         *tmpy++ = *tmpb++;
111       \}
112     \}
113   
114     /* only need to clamp the lower words since by definition the 
115      * upper words x1/y1 must have a known number of digits
116      */
117     mp_clamp (&x0);
118     mp_clamp (&y0);
119   
120     /* now calc the products x0y0 and x1y1 */
121     /* after this x0 is no longer required, free temp [x0==t2]! */
122     if (mp_mul (&x0, &y0, &x0y0) != MP_OKAY)  
123       goto X1Y1;          /* x0y0 = x0*y0 */
124     if (mp_mul (&x1, &y1, &x1y1) != MP_OKAY)
125       goto X1Y1;          /* x1y1 = x1*y1 */
126   
127     /* now calc x1-x0 and y1-y0 */
128     if (mp_sub (&x1, &x0, &t1) != MP_OKAY)
129       goto X1Y1;          /* t1 = x1 - x0 */
130     if (mp_sub (&y1, &y0, &x0) != MP_OKAY)
131       goto X1Y1;          /* t2 = y1 - y0 */
132     if (mp_mul (&t1, &x0, &t1) != MP_OKAY)
133       goto X1Y1;          /* t1 = (x1 - x0) * (y1 - y0) */
134   
135     /* add x0y0 */
136     if (mp_add (&x0y0, &x1y1, &x0) != MP_OKAY)
137       goto X1Y1;          /* t2 = x0y0 + x1y1 */
138     if (mp_sub (&x0, &t1, &t1) != MP_OKAY)
139       goto X1Y1;          /* t1 = x0y0 + x1y1 - (x1-x0)*(y1-y0) */
140   
141     /* shift by B */
142     if (mp_lshd (&t1, B) != MP_OKAY)
143       goto X1Y1;          /* t1 = (x0y0 + x1y1 - (x1-x0)*(y1-y0))<<B */
144     if (mp_lshd (&x1y1, B * 2) != MP_OKAY)
145       goto X1Y1;          /* x1y1 = x1y1 << 2*B */
146   
147     if (mp_add (&x0y0, &t1, &t1) != MP_OKAY)
148       goto X1Y1;          /* t1 = x0y0 + t1 */
149     if (mp_add (&t1, &x1y1, c) != MP_OKAY)
150       goto X1Y1;          /* t1 = x0y0 + t1 + x1y1 */
151   
152     err = MP_OKAY;
153   
154   X1Y1:mp_clear (&x1y1);
155   X0Y0:mp_clear (&x0y0);
156   T1:mp_clear (&t1);
157   Y1:mp_clear (&y1);
158   Y0:mp_clear (&y0);
159   X1:mp_clear (&x1);
160   X0:mp_clear (&x0);
161   ERR:
162     return err;
163   \}
\end{alltt}
\end{small}

The new coding element in this routine that has not been seen in the previous routines yet is the usage of the goto statements.  The normal
wisdom is that goto statements should be avoided.  This is generally true however, when every single function call can fail it makes sense
to handle error recovery with a single piece of code.  Lines 61 to 75 handle initializing all of the temporary variables 
required.  Note how each of the if statements goes to a different label in case of failure.  This allows the routine to correctly free only
the temporaries that have been successfully allocated so far.

The temporary variables are all initialized using the mp\_init\_size routine since they are expected to be large.  This saves the 
additional reallocation that would have been necessary.  Also $x0$, $x1$, $y0$ and $y1$ have to be able to hold at least their respective
number of digits for the next section of code.

The first algebraic portion of the algorithm is to split the two inputs into their halves.  However, instead of using mp\_mod\_2d and mp\_rshd
to extract the halves the code has been inlined.  To initialize the halves the \textbf{used} and \textbf{sign} members are copied first.  The first
for loop on line 98 copies the lower halves.  Since they are both the same magnitude it is simpler to calculate both lower halves in a single
loop.  The for loop on lines 104 and 109 calculate the upper halves $x1$ and $y1$ respectively.

By inlining the calculation of the halves the Karatsuba multiplier has a slightly lower overhead.  As a result it can be used for smaller 
inputs.

When line 152 is reached the algorithm has completed succesfully.  The ``error status'' variable $err$ is set to \textbf{MP\_OKAY} so that
the same code that handles errors can be used to clear the temporary variables and return.  

\subsection{Toom-Cook $3$-Way Multiplication}
Toom-Cook $3$-Way multiplication \cite{TOOM} is essentially the polynomial basis algorithm for $n = 3$ except that the points  are 
chosen such that $\zeta$ is easy to compute and the resulting system of equations easy to reduce.  In this algorithm the points $\zeta_{0}$, 
$16 \cdot \zeta_{1 \over 2}$, $\zeta_1$, $\zeta_2$ and $\zeta_{\infty}$ make up the five requires points to solve for the coefficients of the
product.  

At first glance the five coefficents are relatively efficient to compute with the exception of $16 \cdot \zeta{1 \over 2}$.  This coefficient
is related to $\zeta_2 = (4a_2 + 2a_1 + a_0)(4b_2 + 2b_1 + b_0)$ in that the coefficients  of two terms are reversed (\textit{or mirrored}).  
Simply put $16 \cdot \zeta{1 \over 2} = (a_2 + 2a_1 + 4a_0)(b_2 + 2b_1 + 4b_0)$.  

With the five relations that Toom has chosen the following system of equations is formed.

\begin{center}
\begin{tabular}{rcrcrcrcrcr}
$\zeta_0$                    & $=$ & $0w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $1w_0$  \\
$16 \cdot \zeta_{1 \over 2}$ & $=$ & $1w_4$ & $+$ & $2w_3$ & $+$ & $4w_2$ & $+$ & $8w_1$ & $+$ & $16w_0$  \\
$\zeta_1$                    & $=$ & $1w_4$ & $+$ & $1w_3$ & $+$ & $1w_2$ & $+$ & $1w_1$ & $+$ & $1w_0$  \\
$\zeta_2$                    & $=$ & $16w_4$ & $+$ & $8w_3$ & $+$ & $4w_2$ & $+$ & $2w_1$ & $+$ & $1w_0$  \\
$\zeta_{\infty}$             & $=$ & $1w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $0w_0$  \\
\end{tabular}
\end{center}

A trivial solution to this matrix requires $12$ subtractions, two multiplications by a small power of two, two divisions by a small power
of two, two divisions by three and one multiplication by three.  All of these $19$ sub-operations require less than quadratic time meaning that
the algorithm overall can be faster than a baseline multiplication.  However, the greater complexity of this algorithm places the cutoff point
(\textbf{TOOM\_MUL\_CUTOFF}) where Toom-Cook becomes the most efficient algorithm very much higher above the Karatsuba cutoff point.  

\subsection{Signed Multiplication}
Now that algorithms to handle multiplications of every useful dimensions has been developed a rather simple finishing touch is required.  So far all
of the multiplication algorithms have been unsigned which leaves only a signed multiplication algorithm to be established.  

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_mul}. \\
\textbf{Input}.   mp\_int $a$ and mp\_int $b$ \\
\textbf{Output}.  $c \leftarrow a \cdot b$ \\
\hline \\
1.  If $a.sign = b.sign$ then \\
\hspace{3mm}1.1  $sign = MP\_ZPOS$ \\
2.  else \\
\hspace{3mm}2.1  $sign = MP\_ZNEG$ \\
3.  If min$(a.used, b.used) \ge TOOM\_MUL\_CUTOFF$ then  \\
\hspace{3mm}3.1  $c \leftarrow a \cdot b$ using algorithm mp\_toom\_mul \\
4.  else if min$(a.used, b.used) \ge KARATSUBA\_MUL\_CUTOFF$ then \\
\hspace{3mm}4.1  $c \leftarrow a \cdot b$ using algorithm mp\_karatsuba\_mul \\
5.  else \\
\hspace{3mm}5.1  $digs \leftarrow a.used + b.used + 1$ \\
\hspace{3mm}5.2  If $digs < MP\_ARRAY$ and min$(a.used, b.used) \le \delta$ then \\
\hspace{6mm}5.2.1  $c \leftarrow a \cdot b \mbox{ (mod }\beta^{digs}\mbox{)}$ using algorithm fast\_s\_mp\_mul\_digs.  \\
\hspace{3mm}5.3  else \\
\hspace{6mm}5.3.1  $c \leftarrow a \cdot b \mbox{ (mod }\beta^{digs}\mbox{)}$ using algorithm s\_mp\_mul\_digs.  \\
6.  $c.sign \leftarrow sign$ \\
7.  Return the result of the unsigned multiplication performed. \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_mul}
\end{figure}

\textbf{Algorithm mp\_mul.}
This algorithm performs the signed multiplication of two inputs.  It will make use of any of the three unsigned multiplication algorithms 
available when the input is of appropriate size.  The \textbf{sign} of the result is not set until the end of the algorithm since algorithm
s\_mp\_mul\_digs will clear it.  

\index{bn\_mp\_mul.c}
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_mul.c
\vspace{-3mm}
\begin{alltt}
016   
017   /* high level multiplication (handles sign) */
018   int
019   mp_mul (mp_int * a, mp_int * b, mp_int * c)
020   \{
021     int     res, neg;
022     neg = (a->sign == b->sign) ? MP_ZPOS : MP_NEG;
023     
024     if (MIN (a->used, b->used) >= TOOM_MUL_CUTOFF) \{
025       res = mp_toom_mul(a, b, c);
026     \} else if (MIN (a->used, b->used) >= KARATSUBA_MUL_CUTOFF) \{
027       res = mp_karatsuba_mul (a, b, c);
028     \} else \{
029   
030       /* can we use the fast multiplier?
031        *
032        * The fast multiplier can be used if the output will 
033        * have less than MP_WARRAY digits and the number of 
034        * digits won't affect carry propagation
035        */
036       int     digs = a->used + b->used + 1;
037   
038       if ((digs < MP_WARRAY) &&
039           MIN(a->used, b->used) <= 
040           (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) \{
041         res = fast_s_mp_mul_digs (a, b, c, digs);
042       \} else \{
043         res = s_mp_mul (a, b, c);
044       \}
045   
046     \}
047     c->sign = neg;
048     return res;
049   \}
\end{alltt}
\end{small}

The implementation is rather simplistic and is not particularly noteworthy.  Line 22 computes the sign of the result using the ``?'' 
operator from the C programming language.  Line 40 computes $\delta$ using the fact that $1 << k$ is equal to $2^k$.  

\section{Squaring}

Squaring is a special case of multiplication where both multiplicands are equal.  At first it may seem like there is no significant optimization
available but in fact there is.  Consider the multiplication of $576$ against $241$.  In total there will be nine single precision multiplications
performed which are $1\cdot 6$, $1 \cdot 7$, $1 \cdot 5$, $4 \cdot 6$, $4 \cdot 7$, $4 \cdot 5$, $2 \cdot  6$, $2 \cdot 7$ and $2 \cdot 5$.  Now consider 
the multiplication of $123$ against $123$.  The nine products are $3 \cdot 3$, $3 \cdot 2$, $3 \cdot 1$, $2 \cdot 3$, $2 \cdot 2$, $2 \cdot 1$, 
$1 \cdot 3$, $1 \cdot 2$ and $1 \cdot 1$.  On closer inspection some of the products are equivalent.  For example, $3 \cdot 2 = 2 \cdot 3$ 
and $3 \cdot 1 = 1 \cdot 3$. 

For any $n$-digit input there are ${{\left (n^2 + n \right)}\over 2}$ possible unique single precision multiplications required.  The following
diagram demonstrates the operations required.

\begin{figure}[here]
\begin{center}
\begin{tabular}{ccccc|c}
&&1&2&3&\\
$\times$ &&1&2&3&\\
\hline && $3 \cdot 1$ & $3 \cdot 2$ & $3 \cdot 3$ & Row 0\\
       & $2 \cdot 1$  & $2 \cdot 2$ & $2 \cdot 3$ && Row 1 \\
         $1 \cdot 1$  & $1 \cdot 2$ & $1 \cdot 3$ &&& Row 2 \\
\end{tabular}
\end{center}
\caption{Squaring Optimization Diagram}
\end{figure}

Starting from zero and numbering the columns from right to left a very simple pattern becomes obvious.  For the purposes of this discussion let $x$
represent the number being squared.  The first observation is that in row $k$ the $2k$'th column of the product has a $\left (x_k \right)^2$ term in it.  

The second observation is that every column $j$ in row $k$ where $j \ne 2k$ is part of a double product.  Every odd column is made up entirely of 
double products.  In fact every column is made up of double products and at most one square (\textit{see the exercise section}).  

The third and final observation is that for row $k$ the first unique non-square term occurs at column $2k + 1$.  For example, on row $1$ of the 
previous squaring, column one is part of the double product with column one from row zero. Column two of row one is a square and column three is
the first unique column.

\subsection{The Baseline Squaring Algorithm}
The baseline squaring algorithm is meant to be a catch-all squaring algorithm.  It will handle any of the input sizes that the faster routines
will not handle.  

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{s\_mp\_sqr}. \\
\textbf{Input}.   mp\_int $a$ \\
\textbf{Output}.  $b \leftarrow a^2$ \\
\hline \\
1.  Init a temporary mp\_int of at least $2 \cdot a.used +1$ digits.  (\textit{mp\_init\_size}) \\
2.  If step 1 failed return(\textit{MP\_MEM}) \\
3.  $t.used \leftarrow 2 \cdot a.used + 1$ \\
4.  For $ix$ from 0 to $a.used - 1$ do \\
\hspace{3mm}Calculate the square. \\
\hspace{3mm}4.1  $\hat r \leftarrow t_{2ix} + \left (a_{ix} \right )^2$ \\
\hspace{3mm}4.2  $t_{2ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\
\hspace{3mm}Calculate the double products after the square. \\
\hspace{3mm}4.3  $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\
\hspace{3mm}4.4  For $iy$ from $ix + 1$ to $a.used - 1$ do \\
\hspace{6mm}4.4.1  $\hat r \leftarrow 2 \cdot a_{ix}a_{iy} + t_{ix + iy} + u$ \\
\hspace{6mm}4.4.2  $t_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\
\hspace{6mm}4.4.3  $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\
\hspace{3mm}Set the last carry. \\
\hspace{3mm}4.5  While $u > 0$ do \\
\hspace{6mm}4.5.1  $iy \leftarrow iy + 1$ \\
\hspace{6mm}4.5.2  $\hat r \leftarrow t_{ix + iy} + u$ \\
\hspace{6mm}4.5.3  $t_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\
\hspace{6mm}4.5.4  $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\
5.  Clamp excess digits of $t$.  (\textit{mp\_clamp}) \\
6.  Exchange $b$ and $t$. \\
7.  Clear $t$ (\textit{mp\_clear}) \\
8.  Return(\textit{MP\_OKAY}) \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm s\_mp\_sqr}
\end{figure}

\textbf{Algorithm s\_mp\_sqr.}
This algorithm computes the square of an input using the three observations on squaring.  It is based fairly faithfully on  algorithm 14.16 of 
\cite[pp.596-597]{HAC}.  Similar to algorithm s\_mp\_mul\_digs a temporary mp\_int is allocated to hold the result of the squaring.  This allows the 
destination mp\_int to be the same as the source mp\_int without losing information part way through the squaring.

The outer loop of this algorithm begins on step 4. It is best to think of the outer loop as walking down the rows of the partial results while
the inner loop computes the columns of the partial result.  Step 4.1 and 4.2 compute the square term for each row while step 4.3 and 4.4 propagate
the carry and compute the double products.  

The requirement that a mp\_word be able to represent the range $0 \le x < 2 \beta^2$ arises from this
very algorithm.  The product $a_{ix}a_{iy}$ will lie in the range $0 \le x \le \beta^2 - 2\beta + 1$ which is obviously less than $\beta^2$ meaning that
when it is multiply by two it can be represented by a mp\_word properly.

Similar to algorithm s\_mp\_mul\_digs after every pass of the inner loop the destination is correctly set to the sum of all of the partial 
results calculated so far.  This involves expensive carry propagation which will be eliminated shortly.  

\index{bn\_s\_mp\_sqr.c}
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_sqr.c
\vspace{-3mm}
\begin{alltt}
016   
017   /* low level squaring, b = a*a, HAC pp.596-597, Algorithm 14.16 */
018   int
019   s_mp_sqr (mp_int * a, mp_int * b)
020   \{
021     mp_int  t;
022     int     res, ix, iy, pa;
023     mp_word r;
024     mp_digit u, tmpx, *tmpt;
025   
026     pa = a->used;
027     if ((res = mp_init_size (&t, pa + pa + 1)) != MP_OKAY) \{
028       return res;
029     \}
030     t.used = pa + pa + 1;
031   
032     for (ix = 0; ix < pa; ix++) \{
033       /* first calculate the digit at 2*ix */
034       /* calculate double precision result */
035       r = ((mp_word) t.dp[ix + ix]) + 
036           ((mp_word) a->dp[ix]) * ((mp_word) a->dp[ix]);
037   
038       /* store lower part in result */
039       t.dp[ix + ix] = (mp_digit) (r & ((mp_word) MP_MASK));
040   
041       /* get the carry */
042       u = (r >> ((mp_word) DIGIT_BIT));
043   
044       /* left hand side of A[ix] * A[iy] */
045       tmpx = a->dp[ix];
046   
047       /* alias for where to store the results */
048       tmpt = t.dp + (ix + ix + 1);
049       
050       for (iy = ix + 1; iy < pa; iy++) \{
051         /* first calculate the product */
052         r = ((mp_word) tmpx) * ((mp_word) a->dp[iy]);
053   
054         /* now calculate the double precision result, note we use
055          * addition instead of *2 since its easier to optimize
056          */
057         r = ((mp_word) * tmpt) + r + r + ((mp_word) u);
058   
059         /* store lower part */
060         *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK));
061   
062         /* get carry */
063         u = (r >> ((mp_word) DIGIT_BIT));
064       \}
065       /* propagate upwards */
066       while (u != ((mp_digit) 0)) \{
067         r = ((mp_word) * tmpt) + ((mp_word) u);
068         *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK));
069         u = (r >> ((mp_word) DIGIT_BIT));
070       \}
071     \}
072   
073     mp_clamp (&t);
074     mp_exch (&t, b);
075     mp_clear (&t);
076     return MP_OKAY;
077   \}
\end{alltt}
\end{small}

Inside the outer loop (\textit{see line 32}) the square term is calculated on line 35.  Line 42 extracts the carry from the square
term.  Aliases for $a_{ix}$ and $t_{ix+iy}$ are initialized on lines 45 and 48 respectively.  The doubling is performed using two
additions (\textit{see line 57}) since it is usually faster than shifting if not at least as fast.  

\subsection{Faster Squaring by the ``Comba'' Method}
A major drawback to the baseline method is the requirement for single precision shifting inside the $O(n^2)$ work level.  Squaring has an additional
drawback that it must double the product inside the inner loop as well.  As for multiplication the Comba technique can be used to eliminate these
performance hazards.

The first obvious solution is to make an array of mp\_words which will hold all of the columns.  This will indeed eliminate all of the carry
propagation operations from the inner loop.  However, the inner product must still be doubled $O(n^2)$ times.  The solution stems from the simple fact
that $2a + 2b + 2c = 2(a + b + c)$.  That is the sum of all of the double products is equal to double the sum of all the products.  For example,
$ab + ba + ac + ca = 2ab + 2ac = 2(ab + ac)$.  

However, we cannot simply double all of the columns since the squares appear only once per row.  The most practical solution is to have two mp\_word
arrays.  One array will hold the squares and the other array will hold the double products.  With both arrays the doubling and carry propagation can be 
moved to a $O(n)$ work level outside the $O(n^2)$ level.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{fast\_s\_mp\_sqr}. \\
\textbf{Input}.   mp\_int $a$ \\
\textbf{Output}.  $b \leftarrow a^2$ \\
\hline \\
Place two arrays of \textbf{MP\_WARRAY} mp\_words named $\hat W$ and $\hat {X}$ on the stack. \\
1.  If $b.alloc < 2a.used + 1$ then grow $b$ to $2a.used + 1$ digits.  (\textit{mp\_grow}). \\
2.  If step 1 failed return(\textit{MP\_MEM}). \\
3.  for $ix$ from $0$ to $2a.used + 1$ do \\
\hspace{3mm}3.1  $\hat W_{ix} \leftarrow 0$ \\
\hspace{3mm}3.2  $\hat {X}_{ix} \leftarrow 0$ \\
4.  for $ix$ from $0$ to $a.used - 1$ do \\
\hspace{3mm}Compute the square.\\
\hspace{3mm}4.1  $\hat {X}_{ix+ix} \leftarrow \left ( a_ix \right )^2$ \\
\hspace{3mm}Compute the double products.\\
\hspace{3mm}4.2  for $iy$ from $ix + 1$ to $a.used - 1$ do \\
\hspace{6mm}4.2.1  $\hat W_{ix+iy} \leftarrow \hat W_{ix+iy} + a_{ix}a_{iy}$ \\
5.  $oldused \leftarrow b.used$ \\
6.  $b.used \leftarrow 2a.used + 1$ \\
Double the products and propagate the carries simultaneously. \\
7.  $\hat W_0 \leftarrow 2 \hat W_0 + \hat {X}_0$ \\
8.  for $ix$ from $1$ to $2a.used$ do \\
\hspace{3mm}8.1 $\hat W_{ix} \leftarrow 2 \hat W_{ix} + \hat {X}_{ix}$ \\
\hspace{3mm}8.2 $\hat W_{ix} \leftarrow \hat W_{ix} + \lfloor \hat W_{ix - 1} / \beta \rfloor$ \\
\hspace{3mm}8.3 $b_{ix-1} \leftarrow W_{ix-1} \mbox{ (mod }\beta\mbox{)}$ \\
9.  $b_{2a.used} \leftarrow \hat W_{2a.used} \mbox{ (mod }\beta\mbox{)}$ \\
10.  if $2a.used + 1 < oldused$ then do \\
\hspace{3mm}10.1  for $ix$ from $2a.used + 1$ to $oldused$ do \\
\hspace{6mm}10.1.1  $b_{ix} \leftarrow 0$ \\
11.  Clamp excess digits from $b$.  (\textit{mp\_clamp}) \\
12.  Return(\textit{MP\_OKAY}). \\ 
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm fast\_s\_mp\_sqr}
\end{figure}

\textbf{Algorithm fast\_s\_mp\_sqr.}
This algorithm computes the square of an input using the Comba technique.  It is designed to be a replacement for algorithm s\_mp\_sqr when
the amount of input digits is less than \textbf{MP\_WARRAY} and less than $\delta \over 2$.  

This routine requires two arrays of mp\_words to be placed on the stack.  The first array $\hat W$ will hold the double products and the second
array $\hat X$ will hold the squares.  Though only at most $MP\_WARRAY \over 2$ words of $\hat X$ are used it has proven faster on most 
processors to simply make it a full size array.

The loop on step 3 will zero the two arrays to prepare them for the squaring step.  Step 4.1 computes the squares of the product.  Note how 
it simply assigns the value into the $\hat X$ array.  The nested loop on step 4.2 computes the doubles of the products.  In actuality that loop
computes the sum of the products for each column.  They are not doubled until later.

After the squaring loop the products stored in $\hat W$ musted be doubled and the carries propagated forwards.  It makes sense to do both
operations at the same time.  The expression $\hat W_{ix} \leftarrow 2 \hat W_{ix} + \hat {X}_{ix}$ computes the sum of the double product and the
squares in place.  

\index{bn\_fast\_s\_mp\_sqr.c}
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_fast\_s\_mp\_sqr.c
\vspace{-3mm}
\begin{alltt}
016   
017   /* fast squaring
018    *
019    * This is the comba method where the columns of the product 
020    * are computed first then the carries are computed.  This 
021    * has the effect of making a very simple inner loop that 
022    * is executed the most
023    *
024    * W2 represents the outer products and W the inner.
025    *
026    * A further optimizations is made because the inner 
027    * products are of the form "A * B * 2".  The *2 part does 
028    * not need to be computed until the end which is good 
029    * because 64-bit shifts are slow!
030    *
031    * Based on Algorithm 14.16 on pp.597 of HAC.
032    *
033    */
034   int
035   fast_s_mp_sqr (mp_int * a, mp_int * b)
036   \{
037     int     olduse, newused, res, ix, pa;
038     mp_word W2[MP_WARRAY], W[MP_WARRAY];
039   
040     /* calculate size of product and allocate as required */
041     pa = a->used;
042     newused = pa + pa + 1;
043     if (b->alloc < newused) \{
044       if ((res = mp_grow (b, newused)) != MP_OKAY) \{
045         return res;
046       \}
047     \}
048   
049     /* zero temp buffer (columns)
050      * Note that there are two buffers.  Since squaring requires
051      * a outter and inner product and the inner product requires
052      * computing a product and doubling it (a relatively expensive
053      * op to perform n**2 times if you don't have to) the inner and
054      * outer products are computed in different buffers.  This way
055      * the inner product can be doubled using n doublings instead of
056      * n**2
057      */
058     memset (W, 0, newused * sizeof (mp_word));
059     memset (W2, 0, newused * sizeof (mp_word));
060   
061     /* This computes the inner product.  To simplify the inner N**2 loop
062      * the multiplication by two is done afterwards in the N loop.
063      */
064     for (ix = 0; ix < pa; ix++) \{
065       /* compute the outer product
066        *
067        * Note that every outer product is computed
068        * for a particular column only once which means that
069        * there is no need todo a double precision addition
070        */
071       W2[ix + ix] = ((mp_word) a->dp[ix]) * ((mp_word) a->dp[ix]);
072   
073       \{
074         register mp_digit tmpx, *tmpy;
075         register mp_word *_W;
076         register int iy;
077   
078         /* copy of left side */
079         tmpx = a->dp[ix];
080   
081         /* alias for right side */
082         tmpy = a->dp + (ix + 1);
083   
084         /* the column to store the result in */
085         _W = W + (ix + ix + 1);
086   
087         /* inner products */
088         for (iy = ix + 1; iy < pa; iy++) \{
089             *_W++ += ((mp_word) tmpx) * ((mp_word) * tmpy++);
090         \}
091       \}
092     \}
093   
094     /* setup dest */
095     olduse  = b->used;
096     b->used = newused;
097   
098     /* now compute digits */
099     \{
100       register mp_digit *tmpb;
101   
102       /* double first value, since the inner products are 
103        * half of what they should be 
104        */
105       W[0] += W[0] + W2[0];
106   
107       tmpb = b->dp;
108       for (ix = 1; ix < newused; ix++) \{
109         /* double/add next digit */
110         W[ix] += W[ix] + W2[ix];
111   
112         W[ix] = W[ix] + (W[ix - 1] >> ((mp_word) DIGIT_BIT));
113         *tmpb++ = (mp_digit) (W[ix - 1] & ((mp_word) MP_MASK));
114       \}
115       /* set the last value.  Note even if the carry is zero 
116        * this is required since the next step will not zero 
117        * it if b originally had a value at b->dp[2*a.used]
118        */
119       *tmpb++ = (mp_digit) (W[(newused) - 1] & ((mp_word) MP_MASK));
120   
121       /* clear high digits */
122       for (; ix < olduse; ix++) \{
123         *tmpb++ = 0;
124       \}
125     \}
126   
127     mp_clamp (b);
128     return MP_OKAY;
129   \}
\end{alltt}
\end{small}

-- Write something deep and insightful later, Tom.

\subsection{Polynomial Basis Squaring}
The same algorithm that performs optimal polynomial basis multiplication can be used to perform polynomial basis squaring.  The minor exception
is that $\zeta_y = f(y) \cdot g(y)$ is actually equivalent to $\zeta_y = f(y)^2$ since $f(y) = g(y)$.  That is instead of performing $2n + 1$
multiplications to find the $\zeta$ relations squaring operations are performed instead.  

\subsection{Karatsuba Squaring}
Let $f(x) = ax + b$ represent the polynomial basis representation of a number to square.  
Let $h(x) = \left ( f(x) \right )^2$ represent the square of the polynomial.  The Karatsuba equation can be modified to square a 
number with the following equation.

\begin{equation}
h(x) = a^2x^2 + \left (a^2 + b^2 - (a - b)^2 \right )x + b^2
\end{equation}

Upon closer inspection this equation only requires the calculation of three half-sized squares: $a^2$, $b^2$ and $(a - b)^2$.  As in 
Karatsuba multiplication this algorithm can be applied recursively on the input and will achieve an asymptotic running time of 
$O \left ( n^{lg(3)} \right )$.

If the asymptotic time of Karatsuba squaring and multiplication is the same why not simply use the multiplication algorithm instead?  The answer
to this question arises from the cutoff point for squaring.  As in multiplication there exists a cutoff point at which the time required for a 
Comba based squaring and a Karatsuba based squaring meet.  Due to the overhead inherent in the Karatsuba method the cutoff point is fairly 
high.  For example, on an Athlon processor with $\beta = 2^{28}$ the cutoff point is around 127 digits.  

Consider squaring a 200 digit number with this technique.  It will be split into two 100 digit halves which are subsequently squared.  
The 100 digit numbers will not be squared using Karatsuba but instead the faster Comba based squaring algorithm.  If Karatsuba multiplication
were used instead the 100 digit numbers would be squared with a slower Comba based multiplication.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_karatsuba\_sqr}. \\
\textbf{Input}.   mp\_int $a$ \\
\textbf{Output}.  $b \leftarrow a^2$ \\
\hline \\
1.  Initialize the following temporary mp\_ints:  $x0$, $x1$, $t1$, $t2$, $x0x0$ and $x1x1$. \\
2.  If any of the initializations on step 1 failed return(\textit{MP\_MEM}). \\
\\
Split the input.  e.g. $a = x1\beta^B + x0$ \\
3.  $B \leftarrow a.used / 2$ \\
4.  $x0 \leftarrow a \mbox{ (mod }\beta^B\mbox{)}$ (\textit{mp\_mod\_2d}) \\
5.  $x1 \leftarrow \lfloor a / \beta^B \rfloor$ (\textit{mp\_lshd}) \\
\\
Calculate the three squares. \\
6.  $x0x0 \leftarrow x0^2$ (\textit{mp\_sqr}) \\
7.  $x1x1 \leftarrow x1^2$ \\
8.  $t1 \leftarrow x1 - x0$ (\textit{mp\_sub}) \\
9.  $t1 \leftarrow t1^2$ \\
\\
Compute the middle term. \\
10.  $t2 \leftarrow x0x0 + x1x1$ (\textit{s\_mp\_add}) \\
11.  $t1 \leftarrow t2 - t1$ \\
\\
Compute final product. \\
12.  $t1 \leftarrow t1\beta^B$ (\textit{mp\_lshd}) \\
13.  $x1x1 \leftarrow x1x1\beta^{2B}$ \\
14.  $t1 \leftarrow t1 + x0x0$ \\
15.  $b \leftarrow t1 + x1x1$ \\
16.  Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_karatsuba\_sqr}
\end{figure}

\textbf{Algorithm mp\_karatsuba\_sqr.}
This algorithm computes the square of an input $a$ using the Karatsuba technique.  This algorithm is very much similar to the Karatsuba based
multiplication algorithm.  

The radix point for squaring is simply the placed above the median of the digits.  Step 3, 4 and 5 compute the two halves required using $B$
as the radix point.  The first two squares in steps 6 and 7 are rather straightforward while the last square is in a more compact form.

By expanding $\left (x1 - x0 \right )^2$ the $x1^2$ and $x0^2$ terms in the middle disappear, that is $x1^2 + x0^2 - (x1 - x0)^2 = 2 \cdot x0 \cdot x1$.
Now if $5n$ single precision additions and a squaring of $n$-digits is faster than multiplying two $n$-digit numbers and doubling then
this method is faster.  Assuming no further recursions occur the difference can be estimated.

Let $p$ represent the cost of a single precision addition and $q$ the cost of a single precision multiplication both in terms of time\footnote{Or
machine clock cycles.}.  The question reduces to whether the following equation is true or not.

\begin{equation}
5np +{{q(n^2 + n)} \over 2} \le pn + qn^2
\end{equation}

For example, on an AMD Athlon processor $p = {1 \over 3}$ and $q = 6$.  This implies that the following inequality should hold.
\begin{center}
\begin{tabular}{rcl}
$5n + 3n^2 + 3n$     & $<$ & ${n \over 3} + 6n^2$ \\
${25 \over 3} + 3n$  & $<$ & ${1 \over 3} + 6n$ \\
${25 \over 3}$       & $<$ & $3n$ \\
${25 \over 9}$       & $<$ & $n$ \\
\end{tabular}
\end{center}

This results in a cutoff point around $n = 3$.  As a consequence it is actually faster to compute the middle term the ``long way'' on processors
where multiplication is substantially slower\footnote{On the Athlon there is a 1:17 ratio between clock cycles for addition and multiplication.  On
the Intel P4 processor this ratio is 1:29 making this method even more beneficial.  The only common exception is the ARMv4 processor which has a
ratio of 1:7.  } than simpler operations such as addition.  

\index{bn\_mp\_karatsuba\_sqr.c}
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_karatsuba\_sqr.c
\vspace{-3mm}
\begin{alltt}
016   
017   /* Karatsuba squaring, computes b = a*a using three 
018    * half size squarings
019    *
020    * See comments of mp_karatsuba_mul for details.  It 
021    * is essentially the same algorithm but merely 
022    * tuned to perform recursive squarings.
023    */
024   int
025   mp_karatsuba_sqr (mp_int * a, mp_int * b)
026   \{
027     mp_int  x0, x1, t1, t2, x0x0, x1x1;
028     int     B, err;
029   
030     err = MP_MEM;
031   
032     /* min # of digits */
033     B = a->used;
034   
035     /* now divide in two */
036     B = B / 2;
037   
038     /* init copy all the temps */
039     if (mp_init_size (&x0, B) != MP_OKAY)
040       goto ERR;
041     if (mp_init_size (&x1, a->used - B) != MP_OKAY)
042       goto X0;
043   
044     /* init temps */
045     if (mp_init_size (&t1, a->used * 2) != MP_OKAY)
046       goto X1;
047     if (mp_init_size (&t2, a->used * 2) != MP_OKAY)
048       goto T1;
049     if (mp_init_size (&x0x0, B * 2) != MP_OKAY)
050       goto T2;
051     if (mp_init_size (&x1x1, (a->used - B) * 2) != MP_OKAY)
052       goto X0X0;
053   
054     \{
055       register int x;
056       register mp_digit *dst, *src;
057   
058       src = a->dp;
059   
060       /* now shift the digits */
061       dst = x0.dp;
062       for (x = 0; x < B; x++) \{
063         *dst++ = *src++;
064       \}
065   
066       dst = x1.dp;
067       for (x = B; x < a->used; x++) \{
068         *dst++ = *src++;
069       \}
070     \}
071   
072     x0.used = B;
073     x1.used = a->used - B;
074   
075     mp_clamp (&x0);
076   
077     /* now calc the products x0*x0 and x1*x1 */
078     if (mp_sqr (&x0, &x0x0) != MP_OKAY)
079       goto X1X1;           /* x0x0 = x0*x0 */
080     if (mp_sqr (&x1, &x1x1) != MP_OKAY)
081       goto X1X1;           /* x1x1 = x1*x1 */
082   
083     /* now calc (x1-x0)**2 */
084     if (mp_sub (&x1, &x0, &t1) != MP_OKAY)
085       goto X1X1;           /* t1 = x1 - x0 */
086     if (mp_sqr (&t1, &t1) != MP_OKAY)
087       goto X1X1;           /* t1 = (x1 - x0) * (x1 - x0) */
088   
089     /* add x0y0 */
090     if (s_mp_add (&x0x0, &x1x1, &t2) != MP_OKAY)
091       goto X1X1;           /* t2 = x0x0 + x1x1 */
092     if (mp_sub (&t2, &t1, &t1) != MP_OKAY)
093       goto X1X1;           /* t1 = x0x0 + x1x1 - (x1-x0)*(x1-x0) */
094   
095     /* shift by B */
096     if (mp_lshd (&t1, B) != MP_OKAY)
097       goto X1X1;           /* t1 = (x0x0 + x1x1 - (x1-x0)*(x1-x0))<<B */
098     if (mp_lshd (&x1x1, B * 2) != MP_OKAY)
099       goto X1X1;           /* x1x1 = x1x1 << 2*B */
100   
101     if (mp_add (&x0x0, &t1, &t1) != MP_OKAY)
102       goto X1X1;           /* t1 = x0x0 + t1 */
103     if (mp_add (&t1, &x1x1, b) != MP_OKAY)
104       goto X1X1;           /* t1 = x0x0 + t1 + x1x1 */
105   
106     err = MP_OKAY;
107   
108   X1X1:mp_clear (&x1x1);
109   X0X0:mp_clear (&x0x0);
110   T2:mp_clear (&t2);
111   T1:mp_clear (&t1);
112   X1:mp_clear (&x1);
113   X0:mp_clear (&x0);
114   ERR:
115     return err;
116   \}
\end{alltt}
\end{small}

This implementation is largely based on the implementation of algorithm mp\_karatsuba\_mul.  It uses the same inline style to copy and 
shift the input into the two halves.  The loop from line 54 to line 70 has been modified since only one input exists.  The \textbf{used}
count of both $x0$ and $x1$ is fixed up and $x0$ is clamped before the calculations begin.  At this point $x1$ and $x0$ are valid equivalents
to the respective halves as if mp\_rshd and mp\_mod\_2d had been used.  

By inlining the copy and shift operations the cutoff point for Karatsuba multiplication can be lowered.  On the Athlon the cutoff point
is exactly at the point where Comba squaring can no longer be used (\textit{128 digits}).  On slower processors such as the Intel P4
it is actually below the Comba limit (\textit{at 110 digits}).

This routine uses the same error trap coding style as mp\_karatsuba\_sqr.  As the temporary variables are initialized errors are redirected to
the error trap higher up.  If the algorithm completes without error the error code is set to \textbf{MP\_OKAY} and the error traps are 
executed.

\textit{Last paragraph sucks.  re-write! -- Tom}

\subsection{Toom-Cook Squaring}
The Toom-Cook squaring algorithm mp\_toom\_sqr is heavily based on the algorithm mp\_toom\_mul with the minor exception noted.  The reader is 
encouraged to read the description of the latter algorithm and try to derive their own Toom-Cook squaring algorithm.  

\subsection{Generic Squaring}
\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_sqr}. \\
\textbf{Input}.   mp\_int $a$ \\
\textbf{Output}.  $b \leftarrow a^2$ \\
\hline \\
1.  If $a.used \ge TOOM\_SQR\_CUTOFF$ then  \\
\hspace{3mm}1.1  $b \leftarrow a^2$ using algorithm mp\_toom\_sqr \\
2.  else if $a.used \ge KARATSUBA\_SQR\_CUTOFF$ then \\
\hspace{3mm}2.1  $b \leftarrow a^2$ using algorithm mp\_karatsuba\_sqr \\
3.  else \\
\hspace{3mm}3.1  $digs \leftarrow a.used + b.used + 1$ \\
\hspace{3mm}3.2  If $digs < MP\_ARRAY$ and $a.used \le \delta$ then \\
\hspace{6mm}3.2.1  $b \leftarrow a^2$ using algorithm fast\_s\_mp\_sqr.  \\
\hspace{3mm}3.3  else \\
\hspace{6mm}3.3.1  $b \leftarrow a^2$ using algorithm s\_mp\_sqr.  \\
4.  $b.sign \leftarrow MP\_ZPOS$ \\
5.  Return the result of the unsigned squaring performed. \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_sqr}
\end{figure}

\textbf{Algorithm mp\_sqr.}
This algorithm computes the square of the input using one of four different algorithms.  If the input is very large and has at least
\textbf{TOOM\_SQR\_CUTOFF} or \textbf{KARATSUBA\_SQR\_CUTOFF} digits then either the Toom-Cook or the Karatsuba Squaring algorithm is used.  If
neither of the polynomial basis algorithms should be used then either the Comba or baseline algorithm is used.  

\index{bn\_mp\_sqr.c}
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_sqr.c
\vspace{-3mm}
\begin{alltt}
016   
017   /* computes b = a*a */
018   int
019   mp_sqr (mp_int * a, mp_int * b)
020   \{
021     int     res;
022     if (a->used >= TOOM_SQR_CUTOFF) \{
023       res = mp_toom_sqr(a, b);
024     \} else if (a->used >= KARATSUBA_SQR_CUTOFF) \{
025       res = mp_karatsuba_sqr (a, b);
026     \} else \{
027   
028       /* can we use the fast multiplier? */
029       if ((a->used * 2 + 1) < MP_WARRAY && 
030            a->used < 
031            (1 << (sizeof(mp_word) * CHAR_BIT - 2*DIGIT_BIT - 1))) \{
032         res = fast_s_mp_sqr (a, b);
033       \} else \{
034         res = s_mp_sqr (a, b);
035       \}
036     \}
037     b->sign = MP_ZPOS;
038     return res;
039   \}
\end{alltt}
\end{small}

\section*{Exercises}
\begin{tabular}{cl}
$\left [ 3 \right ] $ & Devise an efficient algorithm for selection of the radix point to handle inputs \\
                      & that have different number of digits in Karatsuba multiplication. \\
                      & \\
$\left [ 3 \right ] $ & In section 6.3 the fact that every column of a squaring is made up \\
                      & of double products and at most one square is stated.  Prove this statement. \\
                      & \\                      
$\left [ 2 \right ] $ & In the Comba squaring algorithm half of the $\hat X$ variables are not used. \\
                      & Revise algorithm fast\_s\_mp\_sqr to shrink the $\hat X$ array. \\
                      & \\
$\left [ 3 \right ] $ & Prove the equation for Karatsuba squaring. \\
                      & \\
$\left [ 1 \right ] $ & Prove that Karatsuba squaring requires $O \left (n^{lg(3)} \right )$ time. \\
                      & \\ 
$\left [ 2 \right ] $ & Determine the minimal ratio between addition and multiplication clock cycles \\
                      & required for equation $6.7$ to be true.  \\
                      & \\
\end{tabular}

\chapter{Modular Reduction}
\section{Basics of Modular Reduction}
\index{modular residue}
Modular reduction is an operation that arises quite often within public key cryptography algorithms.  A number is said to be reduced modulo another
number by finding the remainder of division.  If an integer $a$ is reduced modulo $b$ that is to solve the equation $a = bq + p$ then $p$ is the 
result.  To phrase that another way ``$p$ is congruent to $a$ modulo $b$'' which is also written as $p \equiv a \mbox{ (mod }b\mbox{)}$.  In
other vernacular $p$ is known as the ``modular residue'' which leads to ``quadratic residue''\footnote{That's fancy talk for $b \equiv a^2 \mbox{ (mod }p\mbox{)}$.} and
other forms of residues.  

\index{modulus}
Modular reductions are normally used to form finite groups such as fields and rings.  For example, in the RSA public key algorithm \cite{RSAPAPER} 
two private primes $p$ and $q$ are chosen which when multiplied $n = pq$ forms a composite modulus.  When operations such as multiplication and
squaring are performed on units of the ring $\Z_n$ a finite multiplicative sub-group is formed.  This sub-group is the group used to perform RSA
operations.  Do not worry to much about how RSA works as it is not important for this discussion.   

The most common usage for performance driven modular reductions is in modular exponentiation algorithms.  That is to compute 
$d = a^b \mbox{ (mod }c\mbox{)}$ as fast as possible.  As will be discussed in the subsequent chapter there exists fast algorithms for computing
modular exponentiations without having to perform (\textit{in this example}) $b$ multiplications.  These algorithms will produce partial
results in the range $0 \le x < c^2$ which can be taken advantage of.  

The obvious line of thinking is to use an integer division routine and just extract the remainder.  While this is equivalent to finding the 
modular residue it turns out that the limited range of the input can be exploited to create several efficient algorithms.  

\section{The Barrett Reduction}
The Barrett reduction algorithm \cite{BARRETT} was inspired by fast division algorithms which multiply by the reciprocal to emulate
division.  Barretts observation was that the residue $c$ of $a$ modulo $b$ is equal to 

\begin{equation}
c = a - b \cdot \lfloor a/b \rfloor
\end{equation}

Since algorithms such as modular reduction would be using the same modulus extensively, using typical DSP intuition the next step would be to 
replace $a/b$ with a multiplication by the reciprocal.  However, DSP intuition on its own will not work as these numbers are considerably
larger than the precision of common DSP floating point data types.  It would take another common optimization to optimize the algorithm.

\subsection{Fixed Point Arithmetic}
The trick used to optimize the above equation is based on a technique of emulating floating point data types with fixed precision integers.  Fixed
point arithmetic would be vastly popularlized in the mid 1990s for bringing 3d-games to the mass market.  The idea is to take a normal $k$-bit 
integer data type and break it into $p$-bit integer and a $q$-bit fraction part (\textit{where $p+q = k$}).  

In this system a $k$-bit integer $n$ would actually represent $n/2^q$.  For example, with $q = 4$ the integer $n = 37$ would actually represent the
value $2.3125$.  To multiply two fixed point numbers the integers are multiplied using traditional arithmetic and subsequently normalized. For example, 
with $q = 4$ to multiply the integers $9$ and $5$ they must be converted to fixed point first by multiplying by $2^q$.  Let $a = 9(2^q)$ 
represent the fixed point representation of $9$ and $b = 5(2^q)$ represent the fixed point representation of $5$.  The product $ab$ is equal to
$45(2^{2q})$ which when normalized produces $45(2^q)$.  

Using fixed point arithmetic division can be easily achieved by multiplying by the reciprocal.  If $2^q$ is equivalent to one than $2^q/b$ is 
equivalent to $1/b$ using real arithmetic.  Using this fact dividing an integer $a$ by another integer $b$ can be achieved with the following
expression.

\begin{equation}
\lfloor (a \cdot (\lfloor 2^q / b \rfloor))/2^q \rfloor
\end{equation}

The precision of the division is proportional to the value of $q$.  If the divisor $b$ is used frequently as is the case with 
modular exponentiation pre-computing $2^q/b$ will allow a division to be performed with a multiplication and a right shift.  Both operations
are considerably faster than division on most processors.  

Consider dividing $19$ by $5$.  The correct result is $\lfloor 19/5 \rfloor = 3$.  With $q = 3$ the reciprocal is $\lfloor 2^q/5 \rfloor = 1$ which
leads to a product of $19$ which when divided by $2^q$ produces $2$.  However, with $q = 4$ the reciprocal is $\lfloor 2^q/5 \rfloor = 3$ and
the result of the emulated division is $\lfloor 3 \cdot 19 / 2^q \rfloor = 3$ which is correct.  

Plugging this form of divison into the original equation the following modular residue equation arises.

\begin{equation}
c = a - b \cdot \lfloor (a \cdot (\lfloor 2^q / b \rfloor))/2^q \rfloor
\end{equation}

Using the notation from \cite{BARRETT} the value of $\lfloor 2^q / b \rfloor$ will be represented by the $\mu$ symbol.  Using the $\mu$
variable also helps re-inforce the idea that it is meant to be computed once and re-used.

\begin{equation}
c = a - b \cdot \lfloor (a \cdot \mu)/2^q \rfloor
\end{equation}

Provided that $2^q > b^2$ this algorithm will produce a quotient that is either exactly correct or off by a value of one.  Let $n$ represent
the number of digits in $b$.  This algorithm requires approximately $2n^2$ single precision multiplications to produce the quotient and 
another $n^2$ single precision multiplications to find the residue.  In total $3n^2$ single precision multiplications are required to 
reduce the number.  

For example, if $b = 1179677$ and $q = 41$ ($2^q > b^2$), then the reciprocal $\mu$ is equal to $\lfloor 2^q / b \rfloor = 1864089$.  Consider reducing
$a = 180388626447$ modulo $b$ using the above reduction equation.  Using long division the quotient $\lfloor a/b \rfloor$ is equal
to the quotient found using the fixed point method.  In this case the quotient is $\lfloor (a \cdot \mu)/2^q \rfloor = 152913$ and can
produce the modular residue $a - 152913b = 677346$.

\subsection{Choosing a Radix Point}
Using the fixed point representation a modular reduction can be performed with $3n^2$ single precision multiplications.  If that were the best
that could be achieved a full division might as well be used in its place.  The key to optimizing the reduction is to reduce the precision of
the initial multiplication that finds the quotient.  

Let $a$ represent the number of which the residue is sought.  Let $b$ represent the modulus used to find the residue.  Let $m$ represent
the number of digits in $b$.  For the purposes of this discussion we will assume that the number of digits in $a$ is $2m$.  Dividing $a$ by 
$b$ is the same as dividing a $2m$ digit integer by a $m$ digit integer.  Digits below the $m - 1$'th digit of $a$ will contribute at most a value
of $1$ to the quotient because $\beta^k < b$ for any $0 \le k \le m - 1$.  

Since those digits do not contribute much to the quotient the observation is that they might as well be zero.  However, if the digits 
``might as well be zero'' they might as well not be there in the first place.  Let $q_0 = \lfloor a/\beta^{m-1} \rfloor$ represent the input
with the zeroes trimmed.  Now the modular reduction is trimmed to the almost equivalent equation

\begin{equation}
c = a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor
\end{equation}

Notice how the original divisor $2^q$ has been replaced with $\beta^{m+1}$. Also note how the exponent on the divisor $m+1$ when added to the amount $q_0$
was shifted by ($m-1$) equals $2m$.  If the optimization had not been performed the divisor would have the exponent $2m$ so in the end the exponents
do ``add up''. By using whole digits the algorithm is much faster since shifting digits is typically slower than simply copying them.  Using the 
above equation the quotient $\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ can be off from the true quotient by at most two implying that 
$0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$.  By first subtracting $b$ times the quotient and then conditionally 
subtracting $b$ once or twice the residue is found.

The quotient is now found using $(m + 1)(m) = m^2 + m$ single precision multiplications and the residue with an additional $m^2$ single
precision multiplications.  In total $2m^2 + m$ single precision multiplications are required which is considerably faster than the original
attempt.

For example, let $\beta = 10$ represent the radix of the digits.  Let $b = 9999$ represent the modulus which implies $m = 4$. Let $a = 99929878$ 
represent the value of which the residue is desired.  In this case $q = 10$ which means that $\mu = \lfloor \beta^{2m}/b \rfloor = 10001$.  
With this optimization the multiplicand for the quotient is $q_0 = \lfloor a / \beta^{m - 1} \rfloor = 99929$.  The quotient is then 
$\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor = 9993$.  Subtracting $9993b$ from $a$ and the correct residue $9871 \equiv a \mbox{ (mod }b\mbox{)}$ 
is found.  

\subsection{Trimming the Quotient}
So far the reduction algorithm has been optimized from $3m^2$ single precision multiplications down to $2m^2 + m$ single precision multiplications.  As 
it stands now the algorithm is already fairly fast compared to a full integer division algorithm.  However, there is still room for
optimization.  

After the first multiplication inside the quotient ($q_0 \cdot \mu$) the value is shifted right by $m + 1$ places effectively nullifying the lower
half of the product.  It would be nice to be able to remove those digits from the product to effectively cut down the number of multiplications.  
If the number of digits in the modulus $m$ is far less than $\beta$ a full product is not required.  In fact the lower $m - 2$ digits will not
affect the upper half of the product at all and do not need to be computed.  

The value of $\mu$ is a $m$-digit number and $q_0$ is a $m + 1$ digit number.  Using a full multiplier $(m + 1)(m) = m^2 + m$ single precision
multiplications would be required.  Using a multiplier that will only produce digits at and above the $m - 1$'th digit reduces the number
of single precision multiplications to ${m^2 + m} \over 2$ single precision multiplications.  

\subsection{Trimming the Residue}
After the quotient has been calculated it is used to reduce the input.  As previously noted the algorithm is not exact and it can be off by a small
multiple of the modulus, that is $0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$.  If $b$ is $m$ digits than the 
result of reduction equation is a value of at most $m + 1$ digits (\textit{provided $3 < \beta$}) implying that the upper $m - 1$ digits are
implicitly zero.  

The next optimization arises from this very fact.  Instead of computing $b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ using a full
$O(m^2)$ multiplication algorithm only the lower $m+1$ digits of the product have to be computed.  Similarly the value of $a$ can
be reduced modulo $\beta^{m+1}$ before the multiple of $b$ is subtracted which simplifes the subtraction as well.  A multiplication that produces 
only the lower $m+1$ digits requires ${m^2 + 3m - 2} \over 2$ single precision multiplications.  

With both optimizations in place the algorithm is the algorithm Barrett proposed.  It requires $m^2 + 2m - 1$ single precision multiplications which
is considerably faster than the straightforward $3m^2$ method.  

\subsection{The Barrett Algorithm}
\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_reduce}. \\
\textbf{Input}.   mp\_int $a$, mp\_int $b$ and $\mu = \lfloor \beta^{2m}/b \rfloor$ $(0 \le a < b^2, b > 1)$ \\
\textbf{Output}.  $c \leftarrow a \mbox{ (mod }b\mbox{)}$ \\
\hline \\
Let $m$ represent the number of digits in $b$.  \\
1.  Make a copy of $a$ and store it in $q$.  (\textit{mp\_init\_copy}) \\
2.  $q \leftarrow \lfloor q / \beta^{m - 1} \rfloor$ (\textit{mp\_rshd}) \\
\\
Produce the quotient. \\
3.  $q \leftarrow q \cdot \mu$  (\textit{note: only produce digits at or above $m-1$}) \\
4.  $q \leftarrow \lfloor q / \beta^{m + 1} \rfloor$ \\
\\
Subtract the multiple of modulus from the input. \\
5.  $c \leftarrow a \mbox{ (mod }\beta^{m+1}\mbox{)}$ (\textit{mp\_mod\_2d}) \\
6.  $q \leftarrow q \cdot b \mbox{ (mod }\beta^{m+1}\mbox{)}$ (\textit{s\_mp\_mul\_digs}) \\
7.  $c \leftarrow c - q$ (\textit{mp\_sub}) \\
\\
Add $\beta^{m+1}$ if a carry occured. \\
8.  If $c < 0$ then (\textit{mp\_cmp\_d}) \\
\hspace{3mm}8.1  $q \leftarrow 1$ (\textit{mp\_set}) \\
\hspace{3mm}8.2  $q \leftarrow q \cdot \beta^{m+1}$ (\textit{mp\_lshd}) \\
\hspace{3mm}8.3  $c \leftarrow c + q$ \\
\\
Now subtract the modulus if the residue is too large (e.g. quotient too small). \\
9.  While $c \ge b$ do (\textit{mp\_cmp}) \\
\hspace{3mm}9.1  $c \leftarrow c - b$ \\
10.  Clear $q$. \\
11.  Return(\textit{MP\_OKAY}) \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_reduce}
\end{figure}

\textbf{Algorithm mp\_reduce.}
This algorithm will reduce the input $a$ modulo $b$ in place using the Barrett algorithm.  It is loosely based on algorithm 14.42 of 
\cite[pp.  602]{HAC} which is based on \cite{BARRETT}.  The algorithm has several restrictions and assumptions which must be adhered to
for the algorithm to work.

First the modulus $b$ is assumed to be positive and greater than one.  If the modulus were less than or equal to one than subtracting
a multiple of it would either accomplish nothing or actually enlarge the input.  The input $a$ must be in the range $0 \le a < b^2$ in order
for the quotient to have enough precision.  Technically the algorithm will still work if $a \ge b^2$ but it will take much longer to finish.  The
value of $\mu$ is passed as an argument to this algorithm and is assumed to be setup before the algorithm is used.  

Recall that the multiplication for the quotient on step 3 must only produce digits at or above the $m-1$'th position.  An algorithm called 
$s\_mp\_mul\_high\_digs$ which has not been presented is used to accomplish this task.  This optimal algorithm can only be used if the number
of digits in $b$ is very much smaller than $\beta$.  

After the multiple of the modulus has been subtracted from $a$ the residue must be fixed up in case its negative.  While it is known that 
$a \ge b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ only the lower $m+1$ digits are being used to compute the residue.  In this case 
the invariant $\beta^{m+1}$ must be added to the residue to make it positive again.  

The while loop at step 9 will subtract $b$ until the residue is less than $b$.  If the algorithm is performed correctly this step is only
performed upto two times.  However, if $a \ge b^2$ than it will iterate substantially more times than it should.

\index{bn\_mp\_reduce.c}
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce.c
\vspace{-3mm}
\begin{alltt}
016   
017   /* reduces x mod m, assumes 0 < x < m**2, mu is 
018    * precomputed via mp_reduce_setup.
019    * From HAC pp.604 Algorithm 14.42
020    */
021   int
022   mp_reduce (mp_int * x, mp_int * m, mp_int * mu)
023   \{
024     mp_int  q;
025     int     res, um = m->used;
026   
027     /* q = x */
028     if ((res = mp_init_copy (&q, x)) != MP_OKAY) \{
029       return res;
030     \}
031   
032     /* q1 = x / b**(k-1)  */
033     mp_rshd (&q, um - 1);         
034   
035     /* according to HAC this is optimization is ok */
036     if (((unsigned long) m->used) > (((mp_digit)1) << (DIGIT_BIT - 1))) \{
037       if ((res = mp_mul (&q, mu, &q)) != MP_OKAY) \{
038         goto CLEANUP;
039       \}
040     \} else \{
041       if ((res = s_mp_mul_high_digs (&q, mu, &q, um - 1)) != MP_OKAY) \{
042         goto CLEANUP;
043       \}
044     \}
045   
046     /* q3 = q2 / b**(k+1) */
047     mp_rshd (&q, um + 1);         
048   
049     /* x = x mod b**(k+1), quick (no division) */
050     if ((res = mp_mod_2d (x, DIGIT_BIT * (um + 1), x)) != MP_OKAY) \{
051       goto CLEANUP;
052     \}
053   
054     /* q = q * m mod b**(k+1), quick (no division) */
055     if ((res = s_mp_mul_digs (&q, m, &q, um + 1)) != MP_OKAY) \{
056       goto CLEANUP;
057     \}
058   
059     /* x = x - q */
060     if ((res = mp_sub (x, &q, x)) != MP_OKAY) \{
061       goto CLEANUP;
062     \}
063   
064     /* If x < 0, add b**(k+1) to it */
065     if (mp_cmp_d (x, 0) == MP_LT) \{
066       mp_set (&q, 1);
067       if ((res = mp_lshd (&q, um + 1)) != MP_OKAY)
068         goto CLEANUP;
069       if ((res = mp_add (x, &q, x)) != MP_OKAY)
070         goto CLEANUP;
071     \}
072   
073     /* Back off if it's too big */
074     while (mp_cmp (x, m) != MP_LT) \{
075       if ((res = s_mp_sub (x, m, x)) != MP_OKAY) \{
076         break;
077       \}
078     \}
079     
080   CLEANUP:
081     mp_clear (&q);
082   
083     return res;
084   \}
\end{alltt}
\end{small}

The first multiplication that determines the quotient can be performed by only producing the digits from $m - 1$ and up.  This essentially halves
the number of single precision multiplications required.  However, the optimization is only safe if $\beta$ is much larger than the number of digits
in the modulus.  In the source code this is evaluated on lines 36 to 44 where algorithm s\_mp\_mul\_high\_digs is used when it is
safe to do so.  

\subsection{The Barrett Setup Algorithm}
In order to use algorithm mp\_reduce the value of $\mu$ must be calculated in advance.  Ideally this value should be computed once and stored for
future use so that the Barrett algorithm can be used without delay.  

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_reduce\_setup}. \\
\textbf{Input}.   mp\_int $a$ ($a > 1$)  \\
\textbf{Output}.  $\mu \leftarrow \lfloor \beta^{2m}/a \rfloor$ \\
\hline \\
1.  $\mu \leftarrow 2^{2 \cdot lg(\beta) \cdot  m}$ (\textit{mp\_2expt}) \\
2.  $\mu \leftarrow \lfloor \mu / b \rfloor$ (\textit{mp\_div}) \\
3.  Return(\textit{MP\_OKAY}) \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_reduce\_setup}
\end{figure}

\textbf{Algorithm mp\_reduce\_setup.}
This algorithm computes the reciprocal $\mu$ required for Barrett reduction.  First $\beta^{2m}$ is calculated as $2^{2 \cdot lg(\beta) \cdot  m}$ which
is equivalent and much faster.  The final value is computed by taking the integer quotient of $\lfloor \mu / b \rfloor$.

\index{bn\_mp\_reduce\_setup.c}
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_setup.c
\vspace{-3mm}
\begin{alltt}
016   
017   /* pre-calculate the value required for Barrett reduction
018    * For a given modulus "b" it calulates the value required in "a"
019    */
020   int
021   mp_reduce_setup (mp_int * a, mp_int * b)
022   \{
023     int     res;
024     
025     if ((res = mp_2expt (a, b->used * 2 * DIGIT_BIT)) != MP_OKAY) \{
026       return res;
027     \}
028     return mp_div (a, b, a, NULL);
029   \}
\end{alltt}
\end{small}

This simple routine calculates the reciprocal $\mu$ required by Barrett reduction.  Note the extended usage of algorithm mp\_div where the variable
which would received the remainder is passed as NULL.  As will be discussed in section 9.1 the division routine allows both the quotient and the 
remainder to be passed as NULL meaning to ignore the value.  

\section{The Montgomery Reduction}
Montgomery reduction\footnote{Thanks to Niels Ferguson for his insightful explanation of the algorithm.} \cite{MONT} is by far the most interesting 
form of reduction in common use.  It computes a modular residue which is not actually equal to the residue of the input yet instead equal to a 
residue times a constant.  However, as perplexing as this may sound the algorithm is relatively simple and very efficient.  

Throughout this entire section the variable $n$ will represent the modulus used to form the residue.  As will be discussed shortly the value of
$n$ must be odd.  The variable $x$ will represent the quantity of which the residue is sought.  Similar to the Barrett algorithm the input
is restricted to $0 \le x < n^2$.  To begin the description some simple number theory facts must be established.

\textbf{Fact 1.}  Adding $n$ to $x$ does not change the residue since in effect it adds one to the quotient $\lfloor x / n \rfloor$.

\textbf{Fact 2.}  If $x$ is even then performing a division by two in $\Z$ is congruent to $x \cdot 2^{-1} \mbox{ (mod }n\mbox{)}$.  For example,
if $n = 7$ and $x = 6$ then $x/2 = 3$.  Using the modular inverse of two the same result is found.  That is, $2^{-1} \equiv (n+1)/2 \equiv 4$ and
$4 \cdot 6 \equiv 3 \mbox{ (mod }n\mbox{)}$.  

From these two simple facts the following simple algorithm can be derived.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{Montgomery Reduction}. \\
\textbf{Input}.   Integer $x$, $n$ and $k$ \\
\textbf{Output}.  $2^{-k}x \mbox{ (mod }n\mbox{)}$ \\
\hline \\
1.  for $t$ from $1$ to $k$ do \\
\hspace{3mm}1.1  If $x$ is odd then \\
\hspace{6mm}1.1.1  $x \leftarrow x + n$ \\
\hspace{3mm}1.2  $x \leftarrow x/2$ \\
2.  Return $x$. \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm Montgomery Reduction}
\end{figure}

The algorithm reduces the input one bit at a time using the two congruencies stated previously.  Inside the loop $n$, which is odd, is
added to $x$ if $x$ is odd.  This forces $x$ to be even which allows the division by two in $\Z$ to be congruent to a modular division by two. 

Let $r$ represent the final result of the Montgomery algorithm.  If $k > lg(n)$ and $0 \le x < n^2$ then the final result is limited to 
$0 \le r < \lfloor x/2^k \rfloor + n$.  As a result at most a single subtraction is required to get the residue desired.

Let $k = \lfloor lg(n) \rfloor + 1$ represent the number of bits in $n$.  The current algorithm requires $2k^2$ single precision shifts
and $k^2$ single precision additions.  At this rate the algorithm is most certainly slower than Barrett reduction and not terribly useful.  
Fortunately there exists an alternative representation of the algorithm.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{Montgomery Reduction} (modified I). \\
\textbf{Input}.   Integer $x$, $n$ and $k$ \\
\textbf{Output}.  $2^{-k}x \mbox{ (mod }n\mbox{)}$ \\
\hline \\
1.  for $t$ from $0$ to $k - 1$ do \\
\hspace{3mm}1.1  If the $t$'th bit of $x$ is one then \\
\hspace{6mm}1.1.1  $x \leftarrow x + 2^tn$ \\
2.  Return $x/2^k$. \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm Montgomery Reduction (modified I)}
\end{figure}

This algorithm is equivalent since $2^tn$ is a multiple of $n$ and the lower $k$ bits of $x$ are zero by step 2.  The number of single
precision shifts has now been reduced from $2k^2$ to $k^2 + 1$ which is only a small improvement.

\subsection{Digit Based Montgomery Reduction}
Instead of computing the reduction on a bit-by-bit basis it is actually much faster to compute it on digit-by-digit basis.  Consider the
previous algorithm re-written to compute the Montgomery reduction in this new fashion.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{Montgomery Reduction} (modified II). \\
\textbf{Input}.   Integer $x$, $n$ and $k$ \\
\textbf{Output}.  $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\
\hline \\
1.  for $t$ from $0$ to $k - 1$ do \\
\hspace{3mm}1.1  $x \leftarrow x + \mu n \beta^t$ \\
2.  Return $x/\beta^k$. \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm Montgomery Reduction (modified II)}
\end{figure}

The value $\mu n \beta^t$ is a multiple of the modulus $n$ meaning that it will not change the residue.  If the first digit of 
the value $\mu n \beta^t$ equals the negative (modulo $\beta$) of the $t$'th digit of $x$ then the addition will result in a zero digit.  This
problem breaks down to solving the following congruency.  

\begin{center}
\begin{tabular}{rcl}
$x_t + \mu n_0$ & $\equiv$ & $0 \mbox{ (mod }\beta\mbox{)}$ \\
$\mu n_0$ & $\equiv$ & $-x_t \mbox{ (mod }\beta\mbox{)}$ \\
$\mu$ & $\equiv$ & $-x_t/n_0 \mbox{ (mod }\beta\mbox{)}$ \\
\end{tabular}
\end{center}

In each iteration of the loop on step 1 a new value of $\mu$ must be calculated.  The value of $-1/n_0 \mbox{ (mod }\beta\mbox{)}$ is used 
extensively in this algorithm and should be precomputed.  Let $\rho$ represent the negative of the modular inverse of $n_0$ modulo $\beta$.  

For example, let $\beta = 10$ represent the radix.  Let $n = 17$ represent the modulus which implies $k = 2$ and $\rho \equiv 7$.  Let $x = 33$ 
represent the value to reduce.

\newpage\begin{figure}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline \textbf{Step ($t$)} & \textbf{Value of $x$} & \textbf{Value of $\mu$} \\
\hline --                 & $33$ & --\\
\hline $0$                 & $33 + \mu n = 50$ & $1$ \\
\hline $1$                 & $50 + \mu n \beta = 900$ & $5$ \\
\hline
\end{tabular}
\end{center}
\caption{Example of Montgomery Reduction}
\end{figure}

The final result $900$ is then divided by $\beta^k$ to produce the final result $9$.  The first observation is that $9 \nequiv x \mbox{ (mod }n\mbox{)}$ 
which implies the result is not the modular residue of $x$ modulo $n$.  However, recall that the residue is actually multiplied by $\beta^{-k}$ in
the algorithm.  To get the true residue the value must be multiplied by $\beta^k$.  In this case $\beta^k \equiv 15 \mbox{ (mod }n\mbox{)}$ and
the correct residue is $9 \cdot 15 \equiv 16 \mbox{ (mod }n\mbox{)}$.  

\subsection{Baseline Montgomery Reduction}
The baseline Montgomery reduction algorithm will produce the residue for any size input.  It is designed to be a catch-all algororithm for 
Montgomery reductions.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_montgomery\_reduce}. \\
\textbf{Input}.   mp\_int $x$, mp\_int $n$ and a digit $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\
\hspace{11.5mm}($0 \le x < n^2, n > 1, (n, \beta) = 1, \beta^k > n$) \\
\textbf{Output}.  $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\
\hline \\
1.  $digs \leftarrow 2n.used + 1$ \\
2.  If $digs < MP\_ARRAY$ and $m.used < \delta$ then \\
\hspace{3mm}2.1  Use algorithm fast\_mp\_montgomery\_reduce instead. \\
\\
Setup $x$ for the reduction. \\
3.  If $x.alloc < digs$ then grow $x$ to $digs$ digits. \\
4.  $x.used \leftarrow digs$ \\
\\
Eliminate the lower $k$ digits. \\
5.  For $ix$ from $0$ to $k - 1$ do \\
\hspace{3mm}5.1  $\mu \leftarrow x_{ix} \cdot \rho \mbox{ (mod }\beta\mbox{)}$ \\
\hspace{3mm}5.2  $u \leftarrow 0$ \\
\hspace{3mm}5.3  For $iy$ from $0$ to $k - 1$ do \\
\hspace{6mm}5.3.1  $\hat r \leftarrow \mu n_{iy} + x_{ix + iy} + u$ \\
\hspace{6mm}5.3.2  $x_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\
\hspace{6mm}5.3.3  $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\
\hspace{3mm}5.4  While $u > 0$ do \\
\hspace{6mm}5.4.1  $iy \leftarrow iy + 1$ \\
\hspace{6mm}5.4.2  $x_{ix + iy} \leftarrow x_{ix + iy} + u$ \\
\hspace{6mm}5.4.3  $u \leftarrow \lfloor x_{ix+iy} / \beta \rfloor$ \\
\hspace{6mm}5.4.4  $x_{ix + iy} \leftarrow x_{ix+iy} \mbox{ (mod }\beta\mbox{)}$ \\
\\
Divide by $\beta^k$ and fix up as required. \\
6.  $x \leftarrow \lfloor x / \beta^k \rfloor$ \\
7.  If $x \ge n$ then \\
\hspace{3mm}7.1  $x \leftarrow x - n$ \\
8.  Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_montgomery\_reduce}
\end{figure}

\textbf{Algorithm mp\_montgomery\_reduce.}
This algorithm reduces the input $x$ modulo $n$ in place using the Montgomery reduction algorithm.  The algorithm is loosely based
on algorithm 14.32 of \cite[pp.601]{HAC} except it merges the multiplication of $\mu n \beta^t$ with the addition in the inner loop.  The
restrictions on this algorithm are fairly easy to adapt to.  First $0 \le x < n^2$ bounds the input to numbers in the same range as 
for the Barrett algorithm.  Additionally $n > 1$ will ensure a modular inverse $\rho$ exists.  $\rho$ must be calculated in
advance of this algorithm.  Finally the variable $k$ is fixed and a pseudonym for $n.used$.  

Step 2 decides whether a faster Montgomery algorithm can be used.  It is based on the Comba technique meaning that there are limits on
the size of the input.  This algorithm is discussed in sub-section 7.3.3.

Step 5 is the main reduction loop of the algorithm.  The value of $\mu$ is calculated once per iteration in the outer loop.  The inner loop
calculates $x + \mu n \beta^{ix}$ by multiplying $\mu n$ and adding the result to $x$ shifted by $ix$ digits.  Both the addition and
multiplication are performed in the same loop to save time and memory.  Step 5.4 will handle any additional carries that escape the inner loop.

Using a quick inspection this algorithm requires $n$ single precision multiplications for the outer loop and $n^2$ single precision multiplications 
in the inner loop.  In total $n^2 + n$ single precision multiplications which compares favourably to Barrett at $n^2 + 2n - 1$ single precision
multiplications.  

\index{bn\_mp\_montgomery\_reduce.c}
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_montgomery\_reduce.c
\vspace{-3mm}
\begin{alltt}
016   
017   /* computes xR**-1 == x (mod N) via Montgomery Reduction */
018   int
019   mp_montgomery_reduce (mp_int * x, mp_int * n, mp_digit rho)
020   \{
021     int     ix, res, digs;
022     mp_digit mu;
023   
024     /* can the fast reduction [comba] method be used?
025      *
026      * Note that unlike in mp_mul you're safely allowed *less*
027      * than the available columns [255 per default] since carries
028      * are fixed up in the inner loop.
029      */
030     digs = n->used * 2 + 1;
031     if ((digs < MP_WARRAY) && 
032         n->used < 
033         (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) \{
034       return fast_mp_montgomery_reduce (x, n, rho);
035     \}
036   
037     /* grow the input as required */
038     if (x->alloc < digs) \{
039       if ((res = mp_grow (x, digs)) != MP_OKAY) \{
040         return res;
041       \}
042     \}
043     x->used = digs;
044   
045     for (ix = 0; ix < n->used; ix++) \{
046       /* mu = ai * m' mod b */
047       mu = (x->dp[ix] * rho) & MP_MASK;
048   
049       /* a = a + mu * m * b**i */
050       \{
051         register int iy;
052         register mp_digit *tmpn, *tmpx, u;
053         register mp_word r;
054   
055         /* aliases */
056         tmpn = n->dp;
057         tmpx = x->dp + ix;
058   
059         /* set the carry to zero */
060         u = 0;
061         
062         /* Multiply and add in place */
063         for (iy = 0; iy < n->used; iy++) \{
064           r = ((mp_word) mu) * ((mp_word) * tmpn++) + 
065               ((mp_word) u) + ((mp_word) * tmpx);
066           u = (r >> ((mp_word) DIGIT_BIT));
067           *tmpx++ = (r & ((mp_word) MP_MASK));
068         \}
069         /* propagate carries */
070         while (u) \{
071           *tmpx += u;
072           u = *tmpx >> DIGIT_BIT;
073           *tmpx++ &= MP_MASK;
074         \}
075       \}
076     \}
077   
078     /* x = x/b**n.used */
079     mp_rshd (x, n->used);
080   
081     /* if A >= m then A = A - m */
082     if (mp_cmp_mag (x, n) != MP_LT) \{
083       return s_mp_sub (x, n, x);
084     \}
085   
086     return MP_OKAY;
087   \}
\end{alltt}
\end{small}

This is the baseline implementation of the Montgomery reduction algorithm.  Lines 30 to 35 determine if the Comba based
routine can be used instead.  Line 47 computes the value of $\mu$ for that particular iteration of the outer loop.  

The multiplication $\mu n \beta^{ix}$ is performed in one step in the inner loop.  The alias $tmpx$ refers to the $ix$'th digit of $x$ and
the alias $tmpn$ refers to the modulus $n$.  

\subsection{Faster ``Comba'' Montgomery Reduction}

The Montgomery reduction requires fewer single precision multiplications than a Barrett reduction, however it is much slower due to the serial
nature of the inner loop.  The Barrett reduction algorithm requires two slightly modified multipliers which can be implemented with the Comba
technique.  The Montgomery reduction algorithm cannot directly use the Comba technique to any significant advantage since the inner loop calculates
a $k \times 1$ product $k$ times. 

The biggest obstacle is that at the $ix$'th iteration of the outer loop the value of $x_{ix}$ is required to calculate $\mu$.  This means the 
carries from $0$ to $ix - 1$ must have been propagated upwards to form a valid $ix$'th digit.  The solution as it turns out is very simple.  
Perform a Comba like multiplier and inside the outer loop just after the inner loop fix up the $ix + 1$'th digit by forwarding the carry.  

With this change in place the Montgomery reduction algorithm can be performed with a Comba style multiplication loop which substantially increases
the speed of the algorithm.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{fast\_mp\_montgomery\_reduce}. \\
\textbf{Input}.   mp\_int $x$, mp\_int $n$ and a digit $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\
\hspace{11.5mm}($0 \le x < n^2, n > 1, (n, \beta) = 1, \beta^k > n$) \\
\textbf{Output}.  $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\
\hline \\
Place an array of \textbf{MP\_WARRAY} mp\_word variables called $\hat W$ on the stack. \\
1.  if $x.alloc < n.used + 1$ then grow $x$ to $n.used + 1$ digits. \\
Copy the digits of $x$ into the array $\hat W$ \\
2.  For $ix$ from $0$ to $x.used - 1$ do \\
\hspace{3mm}2.1  $\hat W_{ix} \leftarrow x_{ix}$ \\
3.  For $ix$ from $x.used$ to $2n.used - 1$ do \\
\hspace{3mm}3.1  $\hat W_{ix} \leftarrow 0$ \\
Elimiate the lower $k$ digits. \\
4.  for $ix$ from $0$ to $n.used - 1$ do \\
\hspace{3mm}4.1  $\mu \leftarrow \hat W_{ix} \cdot \rho \mbox{ (mod }\beta\mbox{)}$ \\
\hspace{3mm}4.2  For $iy$ from $0$ to $n.used - 1$ do \\
\hspace{6mm}4.2.1  $\hat W_{iy + ix} \leftarrow \hat W_{iy + ix} + \mu \cdot n_{iy}$ \\
\hspace{3mm}4.3  $\hat W_{ix + 1} \leftarrow \hat W_{ix + 1} + \lfloor \hat W_{ix} / \beta \rfloor$ \\
Propagate carries upwards. \\
5.  for $ix$ from $n.used$ to $2n.used + 1$ do \\
\hspace{3mm}5.1  $\hat W_{ix + 1} \leftarrow \hat W_{ix + 1} + \lfloor \hat W_{ix} / \beta \rfloor$ \\
Shift right and reduce modulo $\beta$ simultaneously. \\
6.  for $ix$ from $0$ to $n.used + 1$ do \\
\hspace{3mm}6.1  $x_{ix} \leftarrow \hat W_{ix + n.used} \mbox{ (mod }\beta\mbox{)}$ \\
Zero excess digits and fixup $x$. \\
7.  if $x.used > n.used + 1$ then do \\
\hspace{3mm}7.1  for $ix$ from $n.used + 1$ to $x.used - 1$ do \\
\hspace{6mm}7.1.1  $x_{ix} \leftarrow 0$ \\
8.  $x.used \leftarrow n.used + 1$ \\
9.  Clamp excessive digits of $x$. \\
10.  If $x \ge n$ then \\
\hspace{3mm}10.1  $x \leftarrow x - n$ \\
11.  Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm fast\_mp\_montgomery\_reduce}
\end{figure}

\textbf{Algorithm fast\_mp\_montgomery\_reduce.}
This algorithm will compute the Montgomery reduction of $x$ modulo $n$ using the Comba technique.  It is on most computer platforms significantly
faster than algorithm mp\_montgomery\_reduce and algorithm mp\_reduce (\textit{Barrett reduction}).  The algorithm has the same restrictions
on the input as the baseline reduction algorithm.  An additional two restrictions are imposed on this algorithm.  The number of digits $k$ in the 
the modulus $n$ must not violate $MP\_WARRAY > 2k +1$ and $n < \delta$.   When $\beta = 2^{28}$ this algorithm can be used to reduce modulo
a modulus of at most $3,556$ bits in length.  

As in the other Comba reduction algorithms there is a $\hat W$ array which stores the columns of the product.  It is initially filled with the
contents of $x$ with the excess digits zeroed.  The reduction loop is very similar the to the baseline loop at heart.  The multiplication on step
4.1 can be single precision only since $ab \mbox{ (mod }\beta\mbox{)} \equiv (a \mbox{ mod }\beta)(b \mbox{ mod }\beta)$.  Some multipliers such
as those on the ARM processors take a variable length time to complete depending on the number of bytes of result it must produce.  By performing
a single precision multiplication instead half the amount of time is spent.

Also note that digit $\hat W_{ix}$ must have the carry from the $ix - 1$'th digit propagated upwards in order for this to work.  That is what step
4.3 will do.  In effect over the $n.used$ iterations of the outer loop the $n.used$'th lower columns all have the their carries propagated forwards.  Note
how the upper bits of those same words are not reduced modulo $\beta$.  This is because those values will be discarded shortly and there is no
point.

Step 5 will propgate the remainder of the carries upwards.  On step 6 the columns are reduced modulo $\beta$ and shifted simultaneously as they are
stored in the destination $x$.  

\index{bn\_fast\_mp\_montgomery\_reduce.c}
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_fast\_mp\_montgomery\_reduce.c
\vspace{-3mm}
\begin{alltt}
016   
017   /* computes xR**-1 == x (mod N) via Montgomery Reduction 
018    * 
019    * This is an optimized implementation of mp_montgomery_reduce 
020    * which uses the comba method to quickly calculate the columns of the
021    * reduction.  
022    *
023    * Based on Algorithm 14.32 on pp.601 of HAC.
024   */
025   int
026   fast_mp_montgomery_reduce (mp_int * x, mp_int * n, mp_digit rho)
027   \{
028     int     ix, res, olduse;
029     mp_word W[MP_WARRAY];
030   
031     /* get old used count */
032     olduse = x->used;
033   
034     /* grow a as required */
035     if (x->alloc < n->used + 1) \{
036       if ((res = mp_grow (x, n->used + 1)) != MP_OKAY) \{
037         return res;
038       \}
039     \}
040   
041     \{
042       register mp_word *_W;
043       register mp_digit *tmpx;
044   
045       _W = W;
046       tmpx = x->dp;
047   
048       /* copy the digits of a into W[0..a->used-1] */
049       for (ix = 0; ix < x->used; ix++) \{
050         *_W++ = *tmpx++;
051       \}
052   
053       /* zero the high words of W[a->used..m->used*2] */
054       for (; ix < n->used * 2 + 1; ix++) \{
055         *_W++ = 0;
056       \}
057     \}
058   
059     for (ix = 0; ix < n->used; ix++) \{
060       /* mu = ai * m' mod b
061        *
062        * We avoid a double precision multiplication (which isn't required)
063        * by casting the value down to a mp_digit.  Note this requires 
064        * that W[ix-1] have  the carry cleared (see after the inner loop)
065        */
066       register mp_digit mu;
067       mu = (((mp_digit) (W[ix] & MP_MASK)) * rho) & MP_MASK;
068   
069       /* a = a + mu * m * b**i
070        *
071        * This is computed in place and on the fly.  The multiplication
072        * by b**i is handled by offseting which columns the results
073        * are added to.
074        *
075        * Note the comba method normally doesn't handle carries in the 
076        * inner loop In this case we fix the carry from the previous 
077        * column since the Montgomery reduction requires digits of the 
078        * result (so far) [see above] to work.  This is
079        * handled by fixing up one carry after the inner loop.  The 
080        * carry fixups are done in order so after these loops the 
081        * first m->used words of W[] have the carries fixed
082        */
083       \{
084         register int iy;
085         register mp_digit *tmpn;
086         register mp_word *_W;
087   
088         /* alias for the digits of the modulus */
089         tmpn = n->dp;
090   
091         /* Alias for the columns set by an offset of ix */
092         _W = W + ix;
093   
094         /* inner loop */
095         for (iy = 0; iy < n->used; iy++) \{
096             *_W++ += ((mp_word) mu) * ((mp_word) * tmpn++);
097         \}
098       \}
099   
100       /* now fix carry for next digit, W[ix+1] */
101       W[ix + 1] += W[ix] >> ((mp_word) DIGIT_BIT);
102     \}
103   
104   
105     \{
106       register mp_digit *tmpx;
107       register mp_word *_W, *_W1;
108   
109       /* nox fix rest of carries */
110       _W1 = W + ix;
111       _W = W + ++ix;
112   
113       for (; ix <= n->used * 2 + 1; ix++) \{
114         *_W++ += *_W1++ >> ((mp_word) DIGIT_BIT);
115       \}
116   
117       /* copy out, A = A/b**n
118        *
119        * The result is A/b**n but instead of converting from an 
120        * array of mp_word to mp_digit than calling mp_rshd 
121        * we just copy them in the right order
122        */
123       tmpx = x->dp;
124       _W = W + n->used;
125   
126       for (ix = 0; ix < n->used + 1; ix++) \{
127         *tmpx++ = *_W++ & ((mp_word) MP_MASK);
128       \}
129   
130       /* zero oldused digits, if the input a was larger than
131        * m->used+1 we'll have to clear the digits */
132       for (; ix < olduse; ix++) \{
133         *tmpx++ = 0;
134       \}
135     \}
136   
137     /* set the max used and clamp */
138     x->used = n->used + 1;
139     mp_clamp (x);
140   
141     /* if A >= m then A = A - m */
142     if (mp_cmp_mag (x, n) != MP_LT) \{
143       return s_mp_sub (x, n, x);
144     \}
145     return MP_OKAY;
146   \}
\end{alltt}
\end{small}

The $\hat W$ array is first filled with digits of $x$ on line 49 then the rest of the digits are zeroed on line 54.  Both loops share
the same alias variables to make the code easier to read.  

The value of $\mu$ is calculated in an interesting fashion.  First the value $\hat W_{ix}$ is reduced modulo $\beta$ and cast to a mp\_digit.  This
forces the compiler to use a single precision multiplication and prevents any concerns about loss of precision.   Line 101 fixes the carry 
for the next iteration of the loop by propagating the carry from $\hat W_{ix}$ to $\hat W_{ix+1}$.

The for loop on line 113 propagates the rest of the carries upwards through the columns.  The for loop on line 126 reduces the columns
modulo $\beta$ and shifts them $k$ places at the same time.  The alias $\_ \hat W$ actually refers to the array $\hat W$ starting at the $n.used$'th
digit, that is $\_ \hat W_{t} = \hat W_{n.used + t}$.  

\subsection{Montgomery Setup}
To calculate the variable $\rho$ a relatively simple algorithm will be required.  

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_montgomery\_setup}. \\
\textbf{Input}.   mp\_int $n$ ($n > 1$ and $(n, 2) = 1$) \\
\textbf{Output}.  $\rho \equiv -1/n_0 \mbox{ (mod }\beta\mbox{)}$ \\
\hline \\
1.  $b \leftarrow n_0$ \\
2.  If $b$ is even return(\textit{MP\_VAL}) \\
3.  $x \leftarrow ((b + 2) \mbox{ AND } 4) << 1) + b$ \\
4.  for $k$ from 0 to $3$ do \\
\hspace{3mm}4.1  $x \leftarrow x \cdot (2 - bx)$ \\
5.  $\rho \leftarrow \beta - x \mbox{ (mod }\beta\mbox{)}$ \\
6.  Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_montgomery\_setup} 
\end{figure}

\textbf{Algorithm mp\_montgomery\_setup.}
This algorithm will calculate the value of $\rho$ required within the Montgomery reduction algorithms.  It uses a very interesting trick 
to calculate $1/n_0$ when $\beta$ is a power of two.  

\index{bn\_mp\_montgomery\_setup.c}
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_montgomery\_setup.c
\vspace{-3mm}
\begin{alltt}
016   
017   /* setups the montgomery reduction stuff */
018   int
019   mp_montgomery_setup (mp_int * n, mp_digit * rho)
020   \{
021     mp_digit x, b;
022   
023   /* fast inversion mod 2**k
024    *
025    * Based on the fact that
026    *
027    * XA = 1 (mod 2**n)  =>  (X(2-XA)) A = 1 (mod 2**2n)
028    *                    =>  2*X*A - X*X*A*A = 1
029    *                    =>  2*(1) - (1)     = 1
030    */
031     b = n->dp[0];
032   
033     if ((b & 1) == 0) \{
034       return MP_VAL;
035     \}
036   
037     x = (((b + 2) & 4) << 1) + b; /* here x*a==1 mod 2**4 */
038     x *= 2 - b * x;               /* here x*a==1 mod 2**8 */
039   #if !defined(MP_8BIT)
040     x *= 2 - b * x;               /* here x*a==1 mod 2**16 */
041   #endif
042   #if defined(MP_64BIT) || !(defined(MP_8BIT) || defined(MP_16BIT))
043     x *= 2 - b * x;               /* here x*a==1 mod 2**32 */
044   #endif
045   #ifdef MP_64BIT
046     x *= 2 - b * x;               /* here x*a==1 mod 2**64 */
047   #endif
048   
049     /* rho = -1/m mod b */
050     *rho = (((mp_digit) 1 << ((mp_digit) DIGIT_BIT)) - x) & MP_MASK;
051   
052     return MP_OKAY;
053   \}
\end{alltt}
\end{small}

This source code computes the value of $\rho$ required to perform Montgomery reduction.  It has been modified to avoid performing excess
multiplications when $\beta$ is not the default 28-bits.  

\section{The Diminished Radix Algorithm}
The diminished radix method of modular reduction \cite{DRMET} is a fairly clever technique which is more efficient than either the Barrett
or Montgomery methods.  The technique is based on a simple congruence.

\begin{equation}
(x \mbox{ mod } n) + k \lfloor x / n \rfloor \equiv x \mbox{ (mod }(n - k)\mbox{)}
\end{equation}

This observation was used in the MMB \cite{MMB} block cipher to create a diffusion primitive.  It used the fact that if $n = 2^{31}$ and $k=1$ that 
then a x86 multiplier could produce the 62-bit product and use  the ``shrd'' instruction to perform a double-precision right shift.  The proof
of the above equation is very simple.  First write $x$ in the product form.

\begin{equation}
x = qn + r
\end{equation}

Now reduce both sides modulo $(n - k)$.

\begin{equation}
x \equiv qk + r  \mbox{ (mod }(n-k)\mbox{)}
\end{equation}

The variable $n$ reduces as $n \mbox{ mod } (n - k)$ to $k$.  By putting $q = \lfloor x/n \rfloor$ and $r = x \mbox{ mod } n$ 
into the equation the original congruence is reproduced.  The following algorithm is based on these observations.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{Diminished Radix Reduction}. \\
\textbf{Input}.   Integer $x$, $n$, $k$ \\
\textbf{Output}.  $x \mbox{ mod } (n - k)$ \\
\hline \\
1.  $q \leftarrow \lfloor x / n \rfloor$ \\
2.  $q \leftarrow k \cdot q$ \\
3.  $x \leftarrow x \mbox{ (mod }n\mbox{)}$ \\
4.  $x \leftarrow x + q$ \\
5.  If $x \ge (n - k)$ then \\
\hspace{3mm}5.1  $x \leftarrow x - (n - k)$ \\
\hspace{3mm}5.2  Goto step 1. \\
6.  Return $x$ \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm Diminished Radix Reduction}
\label{fig:DR}
\end{figure}

This algorithm will reduce $x$ modulo $n - k$ and return the residue.  If $0 \le x < (n - k)^2$ then the algorithm will loop almost always
once or twice and occasionally three times.  For simplicity sake the value of $x$ is bounded by the following simple polynomial.

\begin{equation} 
0 \le x < n^2 + k^2 - 2nk
\end{equation}

The true bound is  $0 \le x < (n - k - 1)^2$ but this has quite a few more terms.  The value of $q$ after step 1 is bounded by the following.

\begin{equation}
q < n - 2k - k^2/n
\end{equation}

Since $k^2$ is going to be considerably smaller than $n$ that term will always be zero.  The value of $x$ after step 3 is bounded trivially as
$0 \le x < n$.  By step four the sum $x + q$ is bounded by 

\begin{equation}
0 \le q + x < (k + 1)n - 2k^2 - 1
\end{equation}

As a result at most $k$ subtractions of $n$ are required to produce the residue.  With a second pass $q$ will be loosely bounded by $0 \le q < k^2$
after step 2 while $x$ will still be loosely bounded by $0 \le x < n$ after step 3.  After the second pass it is highly unlike that the
sum in step 4 will exceed $n - k$.  In practice fewer than three passes of the algorithm are required to reduce virtually every input in the 
range $0 \le x < (n - k - 1)^2$.  

\subsection{Choice of Moduli}
On the surface this algorithm looks like a very expensive algorithm.  It requires a couple of subtractions followed by multiplication and other
modular reductions.  The usefulness of this algorithm becomes exceedingly clear when an appropriate moduli is chosen.

Division in general is a very expensive operation to perform.  The one exception is when the division is by a power of the radix of representation used.  
Division by ten for example is simple for humans since it amounts to shifting the decimal place.  Similarly division by two 
(\textit{or powers of two}) is very simple for computers to perform.  It would therefore seem logical to choose $n$ of the form $2^p$ 
which would imply that $\lfloor x / n \rfloor$ is a simple shift of $x$ right $p$ bits.  

However, there is one operation related to division of power of twos that is even faster than this.  If $n = \beta^p$ then the division may be 
performed by moving whole digits to the right $p$ places.  In practice division by $\beta^p$ is much faster than division by $2^p$ for any $p$.  
Also with the choice of $n = \beta^p$ reducing $x$ modulo $n$ requires zeroing the digits above the $p-1$'th digit of $x$.  

Throughout the next section the term ``restricted modulus'' will refer to a modulus of the form $\beta^p - k$ where as the term ``unrestricted
modulus'' will refer to a modulus of the form $2^p - k$.  The word ``restricted'' in this case refers to the fact that it is based on the 
$2^p$ logic except $p$ must be a multiple of $lg(\beta)$.  

\subsection{Choice of $k$}
Now that division and reduction (\textit{step 1 and 3 of figure~\ref{fig:DR}}) have been optimized to simple digit operations the multiplication by $k$
in step 2 is the most expensive operation.  Fortunately the choice of $k$ is not terribly limited.  For all intents and purposes it might
as well be a single digit.  

\subsection{Restricted Diminished Radix Reduction}
The restricted Diminished Radix algorithm can quickly reduce numbers modulo numbers of the form $n = \beta^p - k$.  This algorithm can reduce 
an input $x$ within the range $0 \le x < n^2$ using a couple passes of the algorithm demonstrated in figure~\ref{fig:DR}.  The implementation
of this algorithm has been optimized to avoid additional overhead associated with a division by $\beta^p$, the
multiplication by $k$ or the addition of $x$ and $q$.  The resulting algorithm is very efficient and can lead to substantial improvements when
modular exponentiations are performed compared to Montgomery based reduction algorithms.  

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_dr\_reduce}. \\
\textbf{Input}.   mp\_int $x$, $n$ and a mp\_digit $k = \beta - n_0$ \\
\hspace{11.5mm}($0 \le x < n^2$, $n > 1$, $0 < k \le \beta$) \\
\textbf{Output}.  $x \mbox{ mod } n$ \\
\hline \\
1.  $m \leftarrow n.used$ \\
2.  If $x.alloc < 2m$ then grow $x$ to $2m$ digits. \\
3.  $\mu \leftarrow 0$ \\
4.  for $i$ from $0$ to $m - 1$ do \\
\hspace{3mm}4.1  $\hat r \leftarrow k \cdot x_{m+i} + x_{i} + \mu$ \\
\hspace{3mm}4.2  $x_{i} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\
\hspace{3mm}4.3  $\mu \leftarrow \lfloor \hat r / \beta \rfloor$ \\
5.  $x_{m} \leftarrow \mu$ \\
6.  for $i$ from $m + 1$ to $x.used - 1$ do \\
\hspace{3mm}6.1  $x_{i} \leftarrow 0$ \\
7.  Clamp excess digits of $x$. \\
8.  If $x \ge n$ then \\
\hspace{3mm}8.1  $x \leftarrow x - n$ \\
\hspace{3mm}8.2  Goto step 3. \\
9.  Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_dr\_reduce}
\end{figure}

\textbf{Algorithm mp\_dr\_reduce.}
This algorithm will perform the dimished radix reduction of $x$ modulo $n$.  It has similar restrictions to that of the Barrett reduction
with the addition that $n$ must be of the form $n = \beta^m - k$ where $0 < k \le \beta$.  

This algorithm essentially implements the pseudo-code in figure 7.10 except with a slight optimization.  The division by $\beta^m$, multiplication by $k$
and addition of $x \mbox{ mod }\beta^m$ are all performed as one step inside the loop on step 4.  The division by $\beta^m$ is emulated by accessing
the term at the $m+i$'th position which is subsequently multiplied by $k$ and added to the term at the $i$'th position.  After the loop the $m$'th
digit is set to the carry and the upper digits are zeroed.  Step 5 and 6 emulate the reduction modulo $\beta^m$ that should have happend to 
$x$ before the addition of the multiple of the upper half.  

At step 8 if $x$ is still larger than $n$ another pass of the algorithm is required.  First $n$ is subtracted from $x$ and then the algorithm resumes
at step 3.  

\index{bn\_mp\_dr\_reduce.c}
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_reduce.c
\vspace{-3mm}
\begin{alltt}
016   
017   /* reduce "x" in place modulo "n" using the Diminished Radix algorithm.
018    *
019    * Based on algorithm from the paper
020    *
021    * "Generating Efficient Primes for Discrete Log Cryptosystems"
022    *                 Chae Hoon Lim, Pil Loong Lee,
023    *          POSTECH Information Research Laboratories
024    *
025    * The modulus must be of a special format [see manual]
026    *
027    * Has been modified to use algorithm 7.10 from the LTM book instead
028    */
029   int
030   mp_dr_reduce (mp_int * x, mp_int * n, mp_digit k)
031   \{
032     int      err, i, m;
033     mp_word  r;
034     mp_digit mu, *tmpx1, *tmpx2;
035     
036     /* m = digits in modulus */
037     m = n->used;
038     
039     /* ensure that "x" has at least 2m digits */
040     if (x->alloc < m + m) \{
041       if ((err = mp_grow (x, m + m)) != MP_OKAY) \{
042         return err;
043       \}
044     \}
045   
046   /* top of loop, this is where the code resumes if 
047    * another reduction pass is required.
048    */
049   top:
050     /* aliases for digits */
051     /* alias for lower half of x */
052     tmpx1 = x->dp;
053     
054     /* alias for upper half of x, or x/B**m */
055     tmpx2 = x->dp + m;
056     
057     /* set carry to zero */
058     mu = 0;
059     
060     /* compute (x mod B**m) + mp * [x/B**m] inline and inplace */
061     for (i = 0; i < m; i++) \{
062         r         = ((mp_word)*tmpx2++) * ((mp_word)k) + *tmpx1 + mu;
063         *tmpx1++  = r & MP_MASK;
064         mu        = r >> ((mp_word)DIGIT_BIT);
065     \}
066     
067     /* set final carry */
068     *tmpx1++ = mu;
069     
070     /* zero words above m */
071     for (i = m + 1; i < x->used; i++) \{
072         *tmpx1++ = 0;
073     \}
074   
075     /* clamp, sub and return */
076     mp_clamp (x);
077   
078     /* if x >= n then subtract and reduce again 
079      * Each successive "recursion" makes the input smaller and smaller.
080      */
081     if (mp_cmp_mag (x, n) != MP_LT) \{
082       s_mp_sub(x, n, x);
083       goto top;
084     \}
085     return MP_OKAY;
086   \}
\end{alltt}
\end{small}

The first step is to grow $x$ as required to $2m$ digits since the reduction is performed in place on $x$.  The label on line 49 is where
the algorithm will resume if further reduction passes are required.  In theory it could be placed at the top of the function however, the size of
the modulus and question of whether $x$ is large enough are invariant after the first pass meaning that it would be a waste of time.  

The aliases $tmpx1$ and $tmpx2$ refer to the digits of $x$ where the latter is offset by $m$ digits.  By reading digits from $x$ offset by $m$ digits
a division by $\beta^m$ can be simulated virtually for free.  The loop on line 61 performs the bulk of the work (\textit{corresponds to step 4 of algorithm 7.11})
in this algorithm.

By line 68 the pointer $tmpx1$ points to the $m$'th digit of $x$ which is where the final carry will be placed.  Similarly by line 71 the 
same pointer will point to the $m+1$'th digit where the zeroes will be placed.  

Since the algorithm is only valid if both $x$ and $n$ are greater than zero an unsigned comparison suffices to determine if another pass is required.  
With the same logic at line 82 the value of $x$ is known to be greater than or equal to $n$ meaning that an unsigned subtraction can be used
as well.  Since the destination of the subtraction is the larger of the inputs the call to algorithm s\_mp\_sub cannot fail and the return code
does not need to be checked.

\subsubsection{Setup}
To setup the restricted Diminished Radix algorithm the value $k = \beta - n_0$ is required.  This algorithm is not really complicated but provided for
completeness.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_dr\_setup}. \\
\textbf{Input}.   mp\_int $n$ \\
\textbf{Output}.  $k = \beta - n_0$ \\
\hline \\
1.  $k \leftarrow \beta - n_0$ \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_dr\_setup}
\end{figure}

\index{bn\_mp\_dr\_setup.c}
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_setup.c
\vspace{-3mm}
\begin{alltt}
016   
017   /* determines the setup value */
018   void mp_dr_setup(mp_int *a, mp_digit *d)
019   \{
020      /* the casts are required if DIGIT_BIT is one less than
021       * the number of bits in a mp_digit [e.g. DIGIT_BIT==31]
022       */
023      *d = (mp_digit)((((mp_word)1) << ((mp_word)DIGIT_BIT)) - 
024           ((mp_word)a->dp[0]));
025   \}
026   
\end{alltt}
\end{small}

\subsubsection{Modulus Detection}
Another algorithm which will be useful is the ability to detect a restricted Diminished Radix modulus.  An integer is said to be
of restricted Diminished Radix form if all of the digits are equal to $\beta - 1$ except the trailing digit which may be any value.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_dr\_is\_modulus}. \\
\textbf{Input}.   mp\_int $n$ \\
\textbf{Output}.  $1$ if $n$ is in D.R form, $0$ otherwise \\
\hline
1.  If $n.used < 2$ then return($0$). \\
2.  for $ix$ from $1$ to $n.used - 1$ do \\
\hspace{3mm}2.1  If $n_{ix} \ne \beta - 1$ return($0$). \\
3.  Return($1$). \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_dr\_is\_modulus}
\end{figure}

\textbf{Algorithm mp\_dr\_is\_modulus.}
This algorithm determines if a value is in Diminished Radix form.  Step 1 rejects obvious cases where fewer than two digits are
in the mp\_int.  Step 2 tests all but the first digit to see if they are equal to $\beta - 1$.  If the algorithm manages to get to
step 3 then $n$ must of Diminished Radix form.

\index{bn\_mp\_dr\_is\_modulus.c}
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_is\_modulus.c
\vspace{-3mm}
\begin{alltt}
016   
017   /* determines if a number is a valid DR modulus */
018   int mp_dr_is_modulus(mp_int *a)
019   \{
020      int ix;
021   
022      /* must be at least two digits */
023      if (a->used < 2) \{
024         return 0;
025      \}
026   
027      for (ix = 1; ix < a->used; ix++) \{
028          if (a->dp[ix] != MP_MASK) \{
029             return 0;
030          \}
031      \}
032      return 1;
033   \}
034   
\end{alltt}
\end{small}

\subsection{Unrestricted Diminished Radix Reduction}
The unrestricted Diminished Radix algorithm allows modular reductions to be performed when the modulus is of the form $2^p - k$.  This algorithm
is a straightforward adaptation of algorithm~\ref{fig:DR}.

In general the restricted Diminished Radix reduction algorithm is much faster since it has considerably lower overhead.  However, this new
algorithm is much faster than either Montgomery or Barrett reduction when the moduli are of the appropriate form.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_reduce\_2k}. \\
\textbf{Input}.   mp\_int $a$ and $n$.  mp\_digit $k$  \\
\hspace{11.5mm}($a \ge 0$, $n > 1$, $0 < k < \beta$, $n + k$ is a power of two) \\
\textbf{Output}.  $a \mbox{ (mod }n\mbox{)}$ \\
\hline
1.  $p \leftarrow \lfloor lg(n) \rfloor + 1$  (\textit{mp\_count\_bits}) \\
2.  While $a \ge n$ do \\
\hspace{3mm}2.1  $q \leftarrow \lfloor a / 2^p \rfloor$ (\textit{mp\_div\_2d}) \\
\hspace{3mm}2.2  $a \leftarrow a \mbox{ (mod }2^p\mbox{)}$ (\textit{mp\_mod\_2d}) \\
\hspace{3mm}2.3  $q \leftarrow q \cdot k$ (\textit{mp\_mul\_d}) \\
\hspace{3mm}2.4  $a \leftarrow a - q$ (\textit{s\_mp\_sub}) \\
\hspace{3mm}2.5  If $a \ge n$ then do \\
\hspace{6mm}2.5.1  $a \leftarrow a - n$ \\
3.  Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_reduce\_2k}
\end{figure}

\textbf{Algorithm mp\_reduce\_2k.}
This algorithm quickly reduces an input $a$ modulo an unrestricted Diminished Radix modulus $n$.

\index{bn\_mp\_reduce\_2k.c}
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_2k.c
\vspace{-3mm}
\begin{alltt}
016   
017   /* reduces a modulo n where n is of the form 2**p - k */
018   int
019   mp_reduce_2k(mp_int *a, mp_int *n, mp_digit k)
020   \{
021      mp_int q;
022      int    p, res;
023      
024      if ((res = mp_init(&q)) != MP_OKAY) \{
025         return res;
026      \}
027      
028      p = mp_count_bits(n);    
029   top:
030      /* q = a/2**p, a = a mod 2**p */
031      if ((res = mp_div_2d(a, p, &q, a)) != MP_OKAY) \{
032         goto ERR;
033      \}
034      
035      if (k != 1) \{
036         /* q = q * k */
037         if ((res = mp_mul_d(&q, k, &q)) != MP_OKAY) \{ 
038            goto ERR;
039         \}
040      \}
041      
042      /* a = a + q */
043      if ((res = s_mp_add(a, &q, a)) != MP_OKAY) \{
044         goto ERR;
045      \}
046      
047      if (mp_cmp_mag(a, n) != MP_LT) \{
048         s_mp_sub(a, n, a);
049         goto top;
050      \}
051      
052   ERR:
053      mp_clear(&q);
054      return res;
055   \}
056   
\end{alltt}
\end{small}

\subsubsection{Unrestricted Setup}
To setup this reduction algorithm the value of $k = 2^p - n$ is required.  

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_reduce\_2k\_setup}. \\
\textbf{Input}.   mp\_int $n$   \\
\textbf{Output}.  $k = 2^p - n$ \\
\hline
1.  $p \leftarrow \lfloor lg(n) \rfloor + 1$  (\textit{mp\_count\_bits}) \\
2.  $x \leftarrow 2^p$ (\textit{mp\_2expt}) \\
3.  $x \leftarrow x - n$ (\textit{mp\_sub}) \\
4.  $k \leftarrow x_0$ \\
5.  Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_reduce\_2k\_setup}
\end{figure}

\textbf{Algorithm mp\_reduce\_2k\_setup.}

\index{bn\_mp\_reduce\_2k\_setup.c}
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_2k\_setup.c
\vspace{-3mm}
\begin{alltt}
016   
017   /* determines the setup value */
018   int 
019   mp_reduce_2k_setup(mp_int *a, mp_digit *d)
020   \{
021      int res, p;
022      mp_int tmp;
023      
024      if ((res = mp_init(&tmp)) != MP_OKAY) \{
025         return res;
026      \}
027      
028      p = mp_count_bits(a);
029      if ((res = mp_2expt(&tmp, p)) != MP_OKAY) \{
030         mp_clear(&tmp);
031         return res;
032      \}
033      
034      if ((res = s_mp_sub(&tmp, a, &tmp)) != MP_OKAY) \{
035         mp_clear(&tmp);
036         return res;
037      \}
038      
039      *d = tmp.dp[0];
040      mp_clear(&tmp);
041      return MP_OKAY;
042   \}
\end{alltt}
\end{small}

\subsubsection{Unrestricted Detection}
An integer $n$ is a valid unrestricted Diminished Radix modulus if either of the following are true.

\begin{enumerate}
\item  The number has only one digit.
\item  The number has more than one digit and every bit from the $\beta$'th to the most significant is one.
\end{enumerate}

If either condition is true than there is a power of two namely $2^p$ such that $0 < 2^p - n < \beta$.  

-- Finish this section later, Tom.

\section{Algorithm Comparison}
So far three very different algorithms for modular reduction have been discussed.  Each of the algorithms have their own strengths and weaknesses
that makes having such a selection very useful.  The following table sumarizes the three algorithms along with comparisons of work factors.  Since
all three algorithms have the restriction that $0 \le x < n^2$ and $n > 1$ those limitations are not included in the table.  

\begin{center}
\begin{small}
\begin{tabular}{|c|c|c|c|c|c|}
\hline \textbf{Method} & \textbf{Work Required} & \textbf{Limitations} & \textbf{$m = 8$} & \textbf{$m = 32$} & \textbf{$m = 64$} \\
\hline Barrett    & $m^2 + 2m - 1$ & None              & $79$ & $1087$ & $4223$ \\
\hline Montgomery & $m^2 + m$      & $n$ must be odd   & $72$ & $1056$ & $4160$ \\
\hline D.R.       & $2m$           & $n = \beta^m - k$ & $16$ & $64$   & $128$  \\
\hline
\end{tabular}
\end{small}
\end{center}

In theory Montgomery and Barrett reductions would require roughly the same amount of time to complete.  However, in practice since Montgomery
reduction can be written as a single function with the Comba technique it is much faster.  Barrett reduction suffers from the overhead of
calling the half precision multipliers, addition and division by $\beta$ algorithms.

For almost every cryptographic algorithm Montgomery reduction is the algorithm of choice.  The one set of algorithms where Diminished Radix reduction truly
shines are based on the discrete logarithm problem such as Diffie-Hellman \cite{DH} and ElGamal \cite{ELGAMAL}.  In these algorithms
primes of the form $\beta^m - k$ can be found and shared amongst users.  These primes will allow the Diminished Radix algorithm to be used in
modular exponentiation to greatly speed up the operation.



\section*{Exercises}
\begin{tabular}{cl}
$\left [ 3 \right ]$ & Prove that the ``trick'' in algorithm mp\_montgomery\_setup actually \\
                     & calculates the correct value of $\rho$. \\
                     & \\
$\left [ 2 \right ]$ & Devise an algorithm to reduce modulo $n + k$ for small $k$ quickly.  \\
                     & \\
$\left [ 4 \right ]$ & Prove that the pseudo-code algorithm ``Diminished Radix Reduction'' \\
                     & (\textit{figure 7.10}) terminates.  Also prove the probability that it will \\
                     & terminate within $1 \le k \le 10$ iterations. \\
                     & \\
\end{tabular}                     


\chapter{Exponentiation}
Exponentiation is the operation of raising one variable to the power of another, for example, $a^b$.  A variant of exponentiation, computed
in a finite field or ring, is called modular exponentiation.  This latter style of operation is typically used in public key 
cryptosystems such as RSA and Diffie-Hellman.  The ability to quickly compute modular exponentiations is of great benefit to any
such cryptosystem and many methods have been sought to speed it up.

\section{Exponentiation Basics}
A trivial algorithm would simply multiply $a$ against itself $b - 1$ times to compute the exponentiation desired.  However, as $b$ grows in size
the number of multiplications becomes prohibitive.  Imagine what would happen if $b$ $\approx$ $2^{1024}$ as is the case when computing an RSA signature
with a $1024$-bit key.  Such a calculation could never be completed as it would take simply far too long.

Fortunately there is a very simple algorithm based on the laws of exponents.  Recall that $lg_a(a^b) = b$ and that $lg_a(a^ba^c) = b + c$ which
are two trivial relationships between the base and the exponent.  Let $b_i$ represent the $i$'th bit of $b$ starting from the least 
significant bit.  If $b$ is a $k$-bit integer than the following equation is true.

\begin{equation}
a^b = \prod_{i=0}^{k-1} a^{2^i \cdot b_i}
\end{equation}

By taking the base $a$ logarithm of both sides of the equation the following equation is the result.

\begin{equation}
b = \sum_{i=0}^{k-1}2^i \cdot b_i
\end{equation}

This is indeed true.  The term $a^{2^i}$ can be found from the $i - 1$'th term by squaring the term since $\left ( a^{2^i} \right )^2$ is equal to
$a^{2^{i+1}}$.  This trivial algorithm forms the basis of essentially all fast exponentiation algorithms.  It requires $k$ squarings and on average
$k \over 2$ multiplications to compute the result.  This is indeed quite an improvement over simply multiplying by $a$ a total of $b-1$ times.

While this current method is a considerable speed up there are further improvements to be made.  For example, the $a^{2^i}$ term does not need to 
be an auxilary variable.  Consider the following algorithm.

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{Left to Right Exponentiation}. \\
\textbf{Input}.   Integer $a$, $b$ and $k$ \\
\textbf{Output}.  $c = a^b$ \\
\hline \\
1.  $c \leftarrow 1$ \\
2.  for $i$ from $k - 1$ to $0$ do \\
\hspace{3mm}2.1  $c \leftarrow c^2$ \\
\hspace{3mm}2.2  $c \leftarrow c \cdot a^{b_i}$ \\
3.  Return $c$. \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Left to Right Exponentiation}
\end{figure}

This algorithm starts from the most significant bit and works towards the least significant bit.  When the $i$'th bit of $b$ is set $a$ is
multiplied against the current product.  In each iteration the product is squared which doubles the exponent of the individual terms of the
product.  

For example, let $b = 101100_2 \equiv 44_{10}$.  The following chart demonstrates the actions of the algorithm.

\newpage\begin{figure}
\begin{center}
\begin{tabular}{|c|c|}
\hline \textbf{Value of $i$} & \textbf{Value of $c$} \\
\hline - & $1$ \\
\hline $5$ & $a$ \\
\hline $4$ & $a^2$ \\
\hline $3$ & $a^4 \cdot a$ \\
\hline $2$ & $a^8 \cdot a^2 \cdot a$ \\
\hline $1$ & $a^{16} \cdot a^4 \cdot a^2$ \\
\hline $0$ & $a^{32} \cdot a^8 \cdot a^4$ \\
\hline
\end{tabular}
\end{center}
\caption{Example of Left to Right Exponentiation}
\end{figure}

When the product $a^{32} \cdot a^8 \cdot a^4$ is simplified it is equal $a^{44}$ which is the desired exponentiation.  This particular algorithm is 
called ``Left to Right'' because it reads the exponent in that order.  All of the exponentiation algorithms that will be presented are of this nature.  

\subsection{Single Digit Exponentiation}
The first algorithm in the series of exponentiation algorithms will be an unbounded algorithm where the exponent is a single digit.  It is intended 
to be used when a small power of an input is required (\textit{e.g. $a^5$}).  It is faster than simply multiplying $b - 1$ times for all values of 
$b$ that are greater than three.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_expt\_d}. \\
\textbf{Input}.   mp\_int $a$ and mp\_digit $b$ \\
\textbf{Output}.  $c = a^b$ \\
\hline \\
1.  $g \leftarrow a$ (\textit{mp\_init\_copy}) \\
2.  $c \leftarrow 1$ (\textit{mp\_set}) \\
3.  for $x$ from 0 to $lg(\beta) - 1$ do \\
\hspace{3mm}3.1  $c \leftarrow c^2$ (\textit{mp\_sqr}) \\
\hspace{3mm}3.2  If $b$ AND $2^{lg(\beta) - 1} \ne 0$ then \\
\hspace{6mm}3.2.1  $c \leftarrow c \cdot g$ (\textit{mp\_mul}) \\
\hspace{3mm}3.3  $b \leftarrow b << 1$ \\
4.  Clear $g$. \\
5.  Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_expt\_d}
\end{figure}

\textbf{Algorithm mp\_expt\_d.}
This algorithm computes the value of $a$ raised to the power of a single digit $b$.  It uses the left to right exponentiation algorithm to
quickly compute the exponentiation.  It is loosely based on algorithm 14.79 of HAC \cite[pp. 615]{HAC} with the difference that the 
exponent is a fixed width.  

A copy of $a$ is made on the first step to allow destination variable $c$ be the same as the source variable $a$.  The result 
is set to the initial value of $1$ in the subsequent step.

Inside the loop the exponent is read from the most significant bit first downto the least significant bit.  First $c$ is invariably squared
on step 3.1.  In the following step if the most significant bit of $b$ is one the copy of $a$ is multiplied against the result.  The value
of $b$ is shifted left one bit to make the next bit down from the most signficant bit become the new most significant bit.  In effect each
iteration of the loop moves the bits of the exponent $b$ upwards to the most significant location.

\index{bn\_mp\_expt\_d.c}
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_expt\_d.c
\vspace{-3mm}
\begin{alltt}
016   
017   /* calculate c = a**b  using a square-multiply algorithm */
018   int
019   mp_expt_d (mp_int * a, mp_digit b, mp_int * c)
020   \{
021     int     res, x;
022     mp_int  g;
023   
024     if ((res = mp_init_copy (&g, a)) != MP_OKAY) \{
025       return res;
026     \}
027   
028     /* set initial result */
029     mp_set (c, 1);
030   
031     for (x = 0; x < (int) DIGIT_BIT; x++) \{
032       /* square */
033       if ((res = mp_sqr (c, c)) != MP_OKAY) \{
034         mp_clear (&g);
035         return res;
036       \}
037   
038       /* if the bit is set multiply */
039       if ((b & (mp_digit) (((mp_digit)1) << (DIGIT_BIT - 1))) != 0) \{
040         if ((res = mp_mul (c, &g, c)) != MP_OKAY) \{
041            mp_clear (&g);
042            return res;
043         \}
044       \}
045   
046       /* shift to next bit */
047       b <<= 1;
048     \}
049   
050     mp_clear (&g);
051     return MP_OKAY;
052   \}
\end{alltt}
\end{small}

-- Some note later.

\subsection{$k$-ary Exponentiation}
When calculating an exponentiation the most time consuming bottleneck is the multiplications which are in general a small factor
slower than squaring.  Recall from the previous algorithm that $b_{i}$ refers to the $i$'th bit of the exponent $b$.  Suppose it referred to
the $i$'th $k$-bit digit of the exponent of $b$.  For $k = 1$ the definitions are synonymous and for $k > 1$ the resulting algorithm
computes the same exponentiation.  A group of $k$ bits from the exponent is called a \textit{window}.  That is it is a window on a small
portion of the exponent.  Consider the following modification to the basic left to right exponentiation algorithm.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{$k$-ary Exponentiation}. \\
\textbf{Input}.   Integer $a$, $b$, $k$ and $t$ \\
\textbf{Output}.  $c = a^b$ \\
\hline \\
1.  $c \leftarrow 1$ \\
2.  for $i$ from $t - 1$ to $0$ do \\
\hspace{3mm}2.1  $c \leftarrow c^{2^k} $ \\
\hspace{3mm}2.2  Extract the $i$'th $k$-bit word from $b$ and store it in $g$. \\
\hspace{3mm}2.3  $c \leftarrow c \cdot a^g$ \\
3.  Return $c$. \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{$k$-ary Exponentiation}
\end{figure}

The squaring on step 2.1 can be calculated by squaring the value $c$ successively $k$ times.  If the values of $a^g$ for $0 < g < 2^k$ have been
precomputed this algorithm requires only $t$ multiplications and $tk$ squarings.  The table can be generated with $2^{k - 1} - 1$ squarings and
$2^{k - 1} + 1$ multiplications.  This algorithm assumes that the number of bits in the exponent is evenly divisible by $k$.  
However, when it is not the remaining $0 < x \le k - 1$ bits can be handled with the original left to right style algorithm.

Suppose $k = 4$ and $t = 100$.  This modified algorithm will require $109$ multiplications and $408$ squarings to compute the exponentiation.  The
original algorithm would on average have required $200$ multiplications and $400$ squrings to compute the same value.  The total number of squarings
has increased slightly but the number of multiplications has nearly halved.

\subsection{Sliding-Window Exponentiation}
A simple modification to the previous algorithm is only generate the upper half of the table in the range $2^{k-1} \le g < 2^k$.  Essentially
this is a table for all values of $g$ where the most significant bit of $g$ is a one.  However, in order for this to be allowed in the 
algorithm values of $g$ in the range $0 \le g < 2^{k-1}$ must be avoided.  

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{Sliding Window $k$-ary Exponentiation}. \\
\textbf{Input}.   Integer $a$, $b$, $k$ and $t$ \\
\textbf{Output}.  $c = a^b$ \\
\hline \\
1.  $c \leftarrow 1$ \\
2.  for $i$ from $t - 1$ to $0$ do \\
\hspace{3mm}2.1  If the $i$'th bit of $b$ is a zero then \\
\hspace{6mm}2.1.1   $c \leftarrow c^2$ \\
\hspace{3mm}2.2  else do \\
\hspace{6mm}2.2.1  $c \leftarrow c^{2^k}$ \\
\hspace{6mm}2.2.2  Extract the $k$ bits from $(b_{i}b_{i-1}\ldots b_{i-(k-1)})$ and store it in $g$. \\
\hspace{6mm}2.2.3  $c \leftarrow c \cdot a^g$ \\
\hspace{6mm}2.2.4  $i \leftarrow i - k$ \\
3.  Return $c$. \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Sliding Window $k$-ary Exponentiation}
\end{figure}

Similar to the previous algorithm this algorithm must have a special handler when fewer than $k$ bits are left in the exponent.  While this
algorithm requires the same number of squarings it can potentially have fewer multiplications.  The pre-computed table $a^g$ is also half
the size as the previous table.  

Consider the exponent $b = 111101011001000_2 \equiv 31432_{10}$ with $k = 3$ using both algorithms.  The first algorithm will divide the exponent up as 
the following five $3$-bit words $b \equiv \left ( 111, 101, 011, 001, 000 \right )_{2}$.  The second algorithm will break the 
exponent as $b \equiv \left ( 111, 101, 0, 110, 0, 100, 0 \right )_{2}$.  The single digit $0$ in the second representation are where
a single squaring took place instead of a squaring and multiplication.  In total the first method requires $10$ multiplications and $18$ 
squarings.  The second method requires $8$ multiplications and $18$ squarings.  

In general the sliding window method is never slower than the generic $k$-ary method and often it is slightly faster.  

\section{Modular Exponentiation}

Modular exponentiation is essentially computing the power of a base within a finite field or ring.  For example, computing 
$d \equiv a^b \mbox{ (mod }c\mbox{)}$ is a modular exponentiation.  Instead of first computing $a^b$ and then reducing it 
modulo $c$ the intermediate result is reduced modulo $c$ after every squaring or multiplication operation.  

This guarantees that any intermediate result is bounded by $0 \le d \le c^2 - 2c + 1$ and can be reduced modulo $c$ quickly using
any of the three algorithms presented in chapter seven.  

Before the actual modular exponentiation algorithm can be written a wrapper algorithm must be written first.  This wrapper algorithm
will allow the exponent $b$ to be negative which is computed as $c \equiv \left (1 / a \right )^{\vert b \vert} \mbox{(mod }d\mbox{)}$. The
value of $(1/a) \mbox{ mod }c$ is computed using the modular inverse (\textit{see section 10.4}).  If no inverse exists the algorithm
terminates with an error.  

\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_exptmod}. \\
\textbf{Input}.   mp\_int $a$, $b$ and $c$ \\
\textbf{Output}.  $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\
\hline \\
1.  If $c.sign = MP\_NEG$ return(\textit{MP\_VAL}). \\
2.  If $b.sign = MP\_NEG$ then \\
\hspace{3mm}2.1  $g' \leftarrow g^{-1} \mbox{ (mod }c\mbox{)}$ \\
\hspace{3mm}2.2  $x' \leftarrow \vert x \vert$ \\
\hspace{3mm}2.3  Compute $d \equiv g'^{x'} \mbox{ (mod }c\mbox{)}$ via recursion. \\
3.  if ($p$ is odd \textbf{OR} $p$ is a D.R. modulus) \textbf{AND} $p.used  > 4$ then \\
\hspace{3mm}3.1  Compute $y \equiv g^{x} \mbox{ (mod }p\mbox{)}$ via algorithm mp\_exptmod\_fast. \\
4.  else \\
\hspace{3mm}4.1  Compute $y \equiv g^{x} \mbox{ (mod }p\mbox{)}$ via algorithm s\_mp\_exptmod. \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_exptmod}
\end{figure}

\textbf{Algorithm mp\_exptmod.}
The first algorithm which actually performs modular exponentiation is algorithm s\_mp\_exptmod.  It is a sliding window $k$-ary algorithm 
which uses Barrett reduction to reduce the product modulo $p$.  The second algorithm mp\_exptmod\_fast performs the same operation 
except it uses either Montgomery or Diminished Radix reduction.  The two latter reduction algorithms are clumped in the same exponentiation
algorithm since their arguments are essentially the same (\textit{two mp\_ints and one mp\_digit}).  

\index{bn\_mp\_exptmod.c}
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_exptmod.c
\vspace{-3mm}
\begin{alltt}
016   
017   
018   /* this is a shell function that calls either the normal or Montgomery
019    * exptmod functions.  Originally the call to the montgomery code was
020    * embedded in the normal function but that wasted alot of stack space
021    * for nothing (since 99% of the time the Montgomery code would be called)
022    */
023   int
024   mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y)
025   \{
026     int dr;
027   
028     /* modulus P must be positive */
029     if (P->sign == MP_NEG) \{
030        return MP_VAL;
031     \}
032   
033     /* if exponent X is negative we have to recurse */
034     if (X->sign == MP_NEG) \{
035        mp_int tmpG, tmpX;
036        int err;
037   
038        /* first compute 1/G mod P */
039        if ((err = mp_init(&tmpG)) != MP_OKAY) \{
040           return err;
041        \}
042        if ((err = mp_invmod(G, P, &tmpG)) != MP_OKAY) \{
043           mp_clear(&tmpG);
044           return err;
045        \}
046   
047        /* now get |X| */
048        if ((err = mp_init(&tmpX)) != MP_OKAY) \{
049           mp_clear(&tmpG);
050           return err;
051        \}
052        if ((err = mp_abs(X, &tmpX)) != MP_OKAY) \{
053           mp_clear_multi(&tmpG, &tmpX, NULL);
054           return err;
055        \}
056   
057        /* and now compute (1/G)**|X| instead of G**X [X < 0] */
058        err = mp_exptmod(&tmpG, &tmpX, P, Y);
059        mp_clear_multi(&tmpG, &tmpX, NULL);
060        return err;
061     \}
062   
063     dr = mp_dr_is_modulus(P);
064     if (dr == 0) \{
065        dr = mp_reduce_is_2k(P) << 1;
066     \}
067     
068     /* if the modulus is odd use the fast method */
069     if ((mp_isodd (P) == 1 || dr !=  0) && P->used > 4) \{
070       return mp_exptmod_fast (G, X, P, Y, dr);
071     \} else \{
072       return s_mp_exptmod (G, X, P, Y);
073     \}
074   \}
075   
\end{alltt}
\end{small}

\subsection{Barrett Modular Exponentiation}

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{s\_mp\_exptmod}. \\
\textbf{Input}.   mp\_int $a$, $b$ and $c$ \\
\textbf{Output}.  $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\
\hline \\
1.  $k \leftarrow lg(x)$ \\
2.  $winsize \leftarrow  \left \lbrace \begin{array}{ll}
                              2 &  \mbox{if }k \le 7 \\
                              3 &  \mbox{if }7 < k \le 36 \\
                              4 &  \mbox{if }36 < k \le 140 \\
                              5 &  \mbox{if }140 < k \le 450 \\
                              6 &  \mbox{if }450 < k \le 1303 \\
                              7 &  \mbox{if }1303 < k \le 3529 \\
                              8 &  \mbox{if }3529 < k \\
                              \end{array} \right .$ \\
3.  Initialize $2^{winsize}$ mp\_ints in an array named $M$ and one mp\_int named $\mu$ \\
4.  Calculate the $\mu$ required for Barrett Reduction (\textit{mp\_reduce\_setup}). \\
5.  $M_1 \leftarrow g \mbox{ (mod }p\mbox{)}$ \\
\\
Setup the table of small powers of $g$.  First find $g^{2^{winsize}}$ and then all multiples of it. \\
6.  $k \leftarrow 2^{winsize - 1}$ \\
7.  $M_{k} \leftarrow M_1$ \\
8.  for $ix$ from 0 to $winsize - 2$ do \\
\hspace{3mm}8.1  $M_k \leftarrow \left ( M_k \right )^2$ \\
\hspace{3mm}8.2  $M_k \leftarrow M_k \mbox{ (mod }p\mbox{)}$ (\textit{mp\_reduce}) \\
9.  for $ix$ from $2^{winsize - 1} + 1$ to $2^{winsize} - 1$ do \\
\hspace{3mm}9.1  $M_{ix} \leftarrow M_{ix - 1} \cdot M_{1}$ \\
\hspace{3mm}9.2  $M_{ix} \leftarrow M_{ix} \mbox{ (mod }p\mbox{)}$ (\textit{mp\_reduce}) \\
10.  $res \leftarrow 1$ \\
\\
Start Sliding Window. \\
11.  $mode \leftarrow 0, bitcnt \leftarrow 1, buf \leftarrow 0, digidx \leftarrow x.used - 1, bitcpy \leftarrow 0, bitbuf \leftarrow 0$ \\
12.  Loop \\
\hspace{3mm}12.1  $bitcnt \leftarrow bitcnt - 1$ \\
\hspace{3mm}12.2  If $bitcnt = 0$ then do \\
\hspace{6mm}12.2.1  If $digidx = -1$ goto step 13. \\
\hspace{6mm}12.2.2  $buf \leftarrow x_{digidx}$ \\
\hspace{6mm}12.2.3  $digidx \leftarrow digidx - 1$ \\
\hspace{6mm}12.2.4  $bitcnt \leftarrow lg(\beta)$ \\
Continued on next page. \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm s\_mp\_exptmod}
\end{figure}

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{s\_mp\_exptmod} (\textit{continued}). \\
\textbf{Input}.   mp\_int $a$, $b$ and $c$ \\
\textbf{Output}.  $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\
\hline \\
\hspace{3mm}12.3  $y \leftarrow (buf >> (lg(\beta) - 1))$ AND $1$ \\
\hspace{3mm}12.4  $buf \leftarrow buf << 1$ \\
\hspace{3mm}12.5  if $mode = 0$ and $y = 0$ then goto step 12. \\
\hspace{3mm}12.6  if $mode = 1$ and $y = 0$ then do \\
\hspace{6mm}12.6.1  $res \leftarrow res^2$ \\
\hspace{6mm}12.6.2  $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\
\hspace{6mm}12.6.3  Goto step 12. \\
\hspace{3mm}12.7  $bitcpy \leftarrow bitcpy + 1$ \\
\hspace{3mm}12.8  $bitbuf \leftarrow bitbuf + (y << (winsize - bitcpy))$ \\
\hspace{3mm}12.9  $mode \leftarrow 2$ \\
\hspace{3mm}12.10  If $bitcpy = winsize$ then do \\
\hspace{6mm}Window is full so perform the squarings and single multiplication. \\
\hspace{6mm}12.10.1  for $ix$ from $0$ to $winsize -1$ do \\
\hspace{9mm}12.10.1.1  $res \leftarrow res^2$ \\
\hspace{9mm}12.10.1.2  $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\
\hspace{6mm}12.10.2  $res \leftarrow res \cdot M_{bitbuf}$ \\
\hspace{6mm}12.10.3  $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\
\hspace{6mm}Reset the window. \\
\hspace{6mm}12.10.4  $bitcpy \leftarrow 0, bitbuf \leftarrow 0, mode \leftarrow 1$ \\
\\
No more windows left.  Check for residual bits of exponent. \\
13.  If $mode = 2$ and $bitcpy > 0$ then do \\
\hspace{3mm}13.1  for $ix$ form $0$ to $bitcpy - 1$ do \\
\hspace{6mm}13.1.1  $res \leftarrow res^2$ \\
\hspace{6mm}13.1.2  $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\
\hspace{6mm}13.1.3  $bitbuf \leftarrow bitbuf << 1$ \\
\hspace{6mm}13.1.4  If $bitbuf$ AND $2^{winsize} \ne 0$ then do \\
\hspace{9mm}13.1.4.1  $res \leftarrow res \cdot M_{1}$ \\
\hspace{9mm}13.1.4.2  $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\
14.  $y \leftarrow res$ \\
15.  Clear $res$, $mu$ and the $M$ array. \\
16.  Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm s\_mp\_exptmod (continued)}
\end{figure}

\textbf{Algorithm s\_mp\_exptmod.}
This algorithm computes the $x$'th power of $g$ modulo $p$ and stores the result in $y$.  It takes advantage of the Barrett reduction
algorithm to keep the product small throughout the algorithm.

The first two steps determine the optimal window size based on the number of bits in the exponent.  The larger the exponent the 
larger the window size becomes.  After a window size $winsize$ has been chosen an array of $2^{winsize}$ mp\_int variables is allocated.  This
table will hold the values of $g^x \mbox{ (mod }p\mbox{)}$ for $2^{winsize - 1} \le x < 2^{winsize}$.  

After the table is allocated the first power of $g$ is found.  Since $g \ge p$ is allowed it must be first reduced modulo $p$ to make
the rest of the algorithm more efficient.  The first element of the table at $2^{winsize - 1}$ is found by squaring $M_1$ successively $winsize - 2$
times.  The rest of the table elements are found by multiplying the previous element by $M_1$ modulo $p$.

Now that the table is available the sliding window may begin.  The following list describes the functions of all the variables in the window.
\begin{enumerate}
\item The variable $mode$ dictates how the bits of the exponent are interpreted.  
\begin{enumerate}
   \item When $mode = 0$ the bits are ignored since no non-zero bit of the exponent has been seen yet.  For example, if the exponent were simply 
         $1$ then there would be $lg(\beta) - 1$ zero bits before the first non-zero bit.  In this case bits are ignored until a non-zero bit is found.  
   \item When $mode = 1$ a non-zero bit has been seen before and a new $winsize$-bit window has not been formed yet.  In this mode leading $0$ bits 
         are read and a single squaring is performed.  If a non-zero bit is read a new window is created.  
   \item When $mode = 2$ the algorithm is in the middle of forming a window and new bits are appended to the window from the most significant bit
         downards.
\end{enumerate}
\item The variable $bitcnt$ indicates how many bits are left in the current digit of the exponent left to be read.  When it reaches zero a new digit
      is fetched from the exponent.
\item The variable $buf$ holds the currently read digit of the exponent. 
\item The variable $digidx$ is an index into the exponents digits.  It starts at the leading digit $x.used - 1$ and moves towards the trailing digit.
\item The variable $bitcpy$ indicates how many bits are in the currently formed window.  When it reaches $winsize$ the window is flushed and
      the appropriate operations performed.
\item The variable $bitbuf$ holds the current bits of the window being formed.  
\end{enumerate}

All of step 12 is the window processing loop.  It will iterate while there are digits available form the exponent to read.  The first step
inside this loop is to extract a new digit if no more bits are available in the current digit.  If there are no bits left a new digit is
read and if there are no digits left than the loop terminates.  

After a digit is made available step 12.3 will extract the most significant bit of the current digit and move all other bits in the digit
upwards.  In effect the digit is read from most significant bit to least significant bit and since the digits are read from leading to 
trailing edges the entire exponent is read from most significant bit to least significant bit.

At step 12.5 if the $mode$ and currently extracted bit $y$ are both zero the bit is ignored and the next bit is read.  This prevents the 
algorithm from having todo trivial squaring and reduction operations before the first non-zero bit is read.  Step 12.6 and 12.7-10 handle
the two cases of $mode = 1$ and $mode = 2$ respectively.  

\begin{center}
\begin{figure}[here]
\includegraphics{pics/expt_state.ps}
\caption{Sliding Window State Diagram}
\end{figure}
\end{center}

By step 13 there are no more digits left in the exponent.  However, there may be partial bits in the window left.  If $mode = 2$ then 
a Left-to-Right algorithm is used to process the remaining few bits.  

\index{bn\_s\_mp\_exptmod.c}
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_exptmod.c
\vspace{-3mm}
\begin{alltt}
016   
017   int
018   s_mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y)
019   \{
020     mp_int  M[256], res, mu;
021     mp_digit buf;
022     int     err, bitbuf, bitcpy, bitcnt, mode, digidx, x, y, winsize;
023   
024     /* find window size */
025     x = mp_count_bits (X);
026     if (x <= 7) \{
027       winsize = 2;
028     \} else if (x <= 36) \{
029       winsize = 3;
030     \} else if (x <= 140) \{
031       winsize = 4;
032     \} else if (x <= 450) \{
033       winsize = 5;
034     \} else if (x <= 1303) \{
035       winsize = 6;
036     \} else if (x <= 3529) \{
037       winsize = 7;
038     \} else \{
039       winsize = 8;
040     \}
041   
042   #ifdef MP_LOW_MEM
043       if (winsize > 5) \{
044          winsize = 5;
045       \}
046   #endif
047   
048     /* init M array */
049     for (x = 0; x < (1 << winsize); x++) \{
050       if ((err = mp_init_size (&M[x], 1)) != MP_OKAY) \{
051         for (y = 0; y < x; y++) \{
052           mp_clear (&M[y]);
053         \}
054         return err;
055       \}
056     \}
057   
058     /* create mu, used for Barrett reduction */
059     if ((err = mp_init (&mu)) != MP_OKAY) \{
060       goto __M;
061     \}
062     if ((err = mp_reduce_setup (&mu, P)) != MP_OKAY) \{
063       goto __MU;
064     \}
065   
066     /* create M table
067      *
068      * The M table contains powers of the input base, e.g. M[x] = G**x mod P
069      *
070      * The first half of the table is not computed though accept for M[0] and 
      M[1]
071      */
072     if ((err = mp_mod (G, P, &M[1])) != MP_OKAY) \{
073       goto __MU;
074     \}
075   
076     /* compute the value at M[1<<(winsize-1)] by squaring M[1] (winsize-1) tim
      es */
077     if ((err = mp_copy (&M[1], &M[1 << (winsize - 1)])) != MP_OKAY) \{
078       goto __MU;
079     \}
080   
081     for (x = 0; x < (winsize - 1); x++) \{
082       if ((err = mp_sqr (&M[1 << (winsize - 1)], &M[1 << (winsize - 1)])) != M
      P_OKAY) \{
083         goto __MU;
084       \}
085       if ((err = mp_reduce (&M[1 << (winsize - 1)], P, &mu)) != MP_OKAY) \{
086         goto __MU;
087       \}
088     \}
089   
090     /* create upper table */
091     for (x = (1 << (winsize - 1)) + 1; x < (1 << winsize); x++) \{
092       if ((err = mp_mul (&M[x - 1], &M[1], &M[x])) != MP_OKAY) \{
093         goto __MU;
094       \}
095       if ((err = mp_reduce (&M[x], P, &mu)) != MP_OKAY) \{
096         goto __MU;
097       \}
098     \}
099   
100     /* setup result */
101     if ((err = mp_init (&res)) != MP_OKAY) \{
102       goto __MU;
103     \}
104     mp_set (&res, 1);
105   
106     /* set initial mode and bit cnt */
107     mode   = 0;
108     bitcnt = 1;
109     buf    = 0;
110     digidx = X->used - 1;
111     bitcpy = bitbuf = 0;
112   
113     for (;;) \{
114       /* grab next digit as required */
115       if (--bitcnt == 0) \{
116         if (digidx == -1) \{
117           break;
118         \}
119         buf = X->dp[digidx--];
120         bitcnt = (int) DIGIT_BIT;
121       \}
122   
123       /* grab the next msb from the exponent */
124       y = (buf >> (mp_digit)(DIGIT_BIT - 1)) & 1;
125       buf <<= (mp_digit)1;
126   
127       /* if the bit is zero and mode == 0 then we ignore it
128        * These represent the leading zero bits before the first 1 bit
129        * in the exponent.  Technically this opt is not required but it
130        * does lower the # of trivial squaring/reductions used
131        */
132       if (mode == 0 && y == 0)
133         continue;
134   
135       /* if the bit is zero and mode == 1 then we square */
136       if (mode == 1 && y == 0) \{
137         if ((err = mp_sqr (&res, &res)) != MP_OKAY) \{
138           goto __RES;
139         \}
140         if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) \{
141           goto __RES;
142         \}
143         continue;
144       \}
145   
146       /* else we add it to the window */
147       bitbuf |= (y << (winsize - ++bitcpy));
148       mode = 2;
149   
150       if (bitcpy == winsize) \{
151         /* ok window is filled so square as required and multiply  */
152         /* square first */
153         for (x = 0; x < winsize; x++) \{
154           if ((err = mp_sqr (&res, &res)) != MP_OKAY) \{
155             goto __RES;
156           \}
157           if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) \{
158             goto __RES;
159           \}
160         \}
161   
162         /* then multiply */
163         if ((err = mp_mul (&res, &M[bitbuf], &res)) != MP_OKAY) \{
164           goto __MU;
165         \}
166         if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) \{
167           goto __MU;
168         \}
169   
170         /* empty window and reset */
171         bitcpy = bitbuf = 0;
172         mode = 1;
173       \}
174     \}
175   
176     /* if bits remain then square/multiply */
177     if (mode == 2 && bitcpy > 0) \{
178       /* square then multiply if the bit is set */
179       for (x = 0; x < bitcpy; x++) \{
180         if ((err = mp_sqr (&res, &res)) != MP_OKAY) \{
181           goto __RES;
182         \}
183         if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) \{
184           goto __RES;
185         \}
186   
187         bitbuf <<= 1;
188         if ((bitbuf & (1 << winsize)) != 0) \{
189           /* then multiply */
190           if ((err = mp_mul (&res, &M[1], &res)) != MP_OKAY) \{
191             goto __RES;
192           \}
193           if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) \{
194             goto __RES;
195           \}
196         \}
197       \}
198     \}
199   
200     mp_exch (&res, Y);
201     err = MP_OKAY;
202   __RES:mp_clear (&res);
203   __MU:mp_clear (&mu);
204   __M:
205     for (x = 0; x < (1 << winsize); x++) \{
206       mp_clear (&M[x]);
207     \}
208     return err;
209   \}
\end{alltt}
\end{small}

\section{Quick Power of Two}
Calculating $b = 2^a$ can be performed much quicker than with any of the previous algorithms.  Recall that a logical shift left $m << k$ is
equivalent to $m \cdot 2^k$.  By this logic when $m = 1$ a quick power of two can be achieved.

\newpage\begin{figure}[!here]
\begin{small}
\begin{center}
\begin{tabular}{l}
\hline Algorithm \textbf{mp\_2expt}. \\
\textbf{Input}.   integer $b$ \\
\textbf{Output}.  $a \leftarrow 2^b$ \\
\hline \\
1.  $a \leftarrow 0$ \\
2.  If $a.alloc < \lfloor b / lg(\beta) \rfloor + 1$ then grow $a$ appropriately. \\
3.  $a.used \leftarrow \lfloor b / lg(\beta) \rfloor + 1$ \\
4.  $a_{\lfloor b / lg(\beta) \rfloor} \leftarrow 1 << (b \mbox{ mod } lg(\beta))$ \\
5.  Return(\textit{MP\_OKAY}). \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Algorithm mp\_2expt}
\end{figure}

\textbf{Algorithm mp\_2expt.}

\index{bn\_mp\_2expt.c}
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: bn\_mp\_2expt.c
\vspace{-3mm}
\begin{alltt}
016   
017   /* computes a = 2**b 
018    *
019    * Simple algorithm which zeroes the int, grows it then just sets one bit
020    * as required.
021    */
022   int
023   mp_2expt (mp_int * a, int b)
024   \{
025     int     res;
026   
027     mp_zero (a);
028     if ((res = mp_grow (a, b / DIGIT_BIT + 1)) != MP_OKAY) \{
029       return res;
030     \}
031     a->used = b / DIGIT_BIT + 1;
032     a->dp[b / DIGIT_BIT] = 1 << (b % DIGIT_BIT);
033   
034     return MP_OKAY;
035   \}
\end{alltt}
\end{small}


\chapter{Higher Level Algorithms}
\section{Integer Division with Remainder}

\section{Single Digit Helpers}
\subsection{Single Digit Addition}
\subsection{Single Digit Subtraction}
\subsection{Single Digit Multiplication}
\subsection{Single Digit Division}
\subsection{Single Digit Modulo}
\subsection{Single Digit Root Extraction}
\section{Random Number Generation}
\section{Formatted Output}
\subsection{Getting The Output Size}
\subsection{Generating Radix-n Output}
\subsection{Reading Radix-n Input}
\section{Unformatted Output}
\subsection{Getting The Output Size}
\subsection{Generating Output}
\subsection{Reading Input}

\chapter{Number Theoretic Algorithms}
\section{Greatest Common Divisor}
\section{Least Common Multiple}
\section{Jacobi Symbol Computation}
\section{Modular Inverse}
\subsection{General Case}
\subsection{Odd Moduli}
\section{Primality Tests}
\subsection{Trial Division}
\subsection{The Fermat Test}
\subsection{The Miller-Rabin Test}
\subsection{Primality Test in a Bottle}
\subsection{The Next Prime}
\section{Root Extraction}

\backmatter
\appendix
\begin{thebibliography}{ABCDEF}
\bibitem[1]{TAOCPV2}
Donald Knuth, \textit{The Art of Computer Programming}, Third Edition, Volume Two, Seminumerical Algorithms, Addison-Wesley, 1998

\bibitem[2]{HAC}
A. Menezes, P. van Oorschot, S. Vanstone, \textit{Handbook of Applied Cryptography}, CRC Press, 1996

\bibitem[3]{ROSE}
Michael Rosing, \textit{Implementing Elliptic Curve Cryptography}, Manning Publications, 1999

\bibitem[4]{COMBA}
Paul G. Comba, \textit{Exponentiation Cryptosystems on the IBM PC}. IBM Systems Journal 29(4): 526-538 (1990)

\bibitem[5]{KARA}
A. Karatsuba, Doklay Akad. Nauk SSSR 145 (1962), pp.293-294

\bibitem[6]{KARAP}
Andre Weimerskirch and Christof Paar, \textit{Generalizations of the Karatsuba Algorithm for Polynomial Multiplication}, Submitted to Design, Codes and Cryptography, March 2002

\bibitem[7]{BARRETT}
Paul Barrett, \textit{Implementing the Rivest Shamir and Adleman Public Key Encryption Algorithm on a Standard Digital Signal Processor}, Advances in Cryptology, Crypto '86, Springer-Verlag.

\bibitem[8]{MONT}
P.L.Montgomery. \textit{Modular multiplication without trial division}. Mathematics of Computation, 44(170):519-521, April 1985.

\bibitem[9]{DRMET}
Chae Hoon Lim and Pil Joong Lee, \textit{Generating Efficient Primes for Discrete Log Cryptosystems}, POSTECH Information Research Laboratories

\bibitem[10]{MMB}
J. Daemen and R. Govaerts and J. Vandewalle, \textit{Block ciphers based on Modular Arithmetic}, State and {P}rogress in the {R}esearch of {C}ryptography, 1993, pp. 80-89

\end{thebibliography}

\input{tommath.ind}

\chapter{Appendix}
\subsection*{Appendix A -- Source Listing of tommath.h}

The following is the source listing of the header file ``tommath.h'' for the LibTomMath project.  It contains many of 
the definitions used throughout the code such as \textbf{mp\_int}, \textbf{MP\_PREC} and so on.  The header is 
presented here for completeness.

\index{tommath.h}
\vspace{+3mm}\begin{small}
\hspace{-5.1mm}{\bf File}: tommath.h
\vspace{-3mm}
\begin{alltt}
001   /* LibTomMath, multiple-precision integer library -- Tom St Denis
002    *
003    * LibTomMath is library that provides for multiple-precision
004    * integer arithmetic as well as number theoretic functionality.
005    *
006    * The library is designed directly after the MPI library by
007    * Michael Fromberger but has been written from scratch with
008    * additional optimizations in place.
009    *
010    * The library is free for all purposes without any express
011    * guarantee it works.
012    *
013    * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org
014    */
015   #ifndef BN_H_
016   #define BN_H_
017   
018   #include <stdio.h>
019   #include <string.h>
020   #include <stdlib.h>
021   #include <ctype.h>
022   #include <limits.h>
023   
024   #undef MIN
025   #define MIN(x,y) ((x)<(y)?(x):(y))
026   #undef MAX
027   #define MAX(x,y) ((x)>(y)?(x):(y))
028   
029   #ifdef __cplusplus
030   extern "C" \{
031   
032   /* C++ compilers don't like assigning void * to mp_digit * */
033   #define  OPT_CAST  (mp_digit *)
034   
035   #else
036   
037   /* C on the other hand doesn't care */
038   #define  OPT_CAST
039   
040   #endif
041   
042   /* some default configurations.
043    *
044    * A "mp_digit" must be able to hold DIGIT_BIT + 1 bits
045    * A "mp_word" must be able to hold 2*DIGIT_BIT + 1 bits
046    *
047    * At the very least a mp_digit must be able to hold 7 bits
048    * [any size beyond that is ok provided it doesn't overflow the data type]
049    */
050   #ifdef MP_8BIT
051      typedef unsigned char      mp_digit;
052      typedef unsigned short     mp_word;
053   #elif defined(MP_16BIT)
054      typedef unsigned short     mp_digit;
055      typedef unsigned long      mp_word;
056   #elif defined(MP_64BIT)
057      /* for GCC only on supported platforms */
058   #ifndef CRYPT
059      typedef unsigned long long ulong64;
060      typedef signed long long   long64;
061   #endif
062   
063      typedef ulong64            mp_digit;
064      typedef unsigned long      mp_word __attribute__ ((mode(TI)));
065   
066      #define DIGIT_BIT          60
067   #else
068      /* this is the default case, 28-bit digits */
069      
070      /* this is to make porting into LibTomCrypt easier :-) */
071   #ifndef CRYPT
072      #if defined(_MSC_VER) || defined(__BORLANDC__) 
073         typedef unsigned __int64   ulong64;
074         typedef signed __int64     long64;
075      #else
076         typedef unsigned long long ulong64;
077         typedef signed long long   long64;
078      #endif
079   #endif
080   
081      typedef unsigned long      mp_digit;
082      typedef ulong64            mp_word;
083   
084   #ifdef MP_31BIT   
085      #define DIGIT_BIT          31
086   #else
087      #define DIGIT_BIT          28
088   #endif   
089   #endif
090   
091   /* otherwise the bits per digit is calculated automatically from the size of
       a mp_digit */
092   #ifndef DIGIT_BIT
093      #define DIGIT_BIT     ((CHAR_BIT * sizeof(mp_digit) - 1))  /* bits per di
      git */
094   #endif
095   
096   
097   #define MP_DIGIT_BIT     DIGIT_BIT
098   #define MP_MASK          ((((mp_digit)1)<<((mp_digit)DIGIT_BIT))-((mp_digit)
      1))
099   #define MP_DIGIT_MAX     MP_MASK
100   
101   /* equalities */
102   #define MP_LT        -1   /* less than */
103   #define MP_EQ         0   /* equal to */
104   #define MP_GT         1   /* greater than */
105   
106   #define MP_ZPOS       0   /* positive integer */
107   #define MP_NEG        1   /* negative */
108   
109   #define MP_OKAY       0   /* ok result */
110   #define MP_MEM        -2  /* out of mem */
111   #define MP_VAL        -3  /* invalid input */
112   #define MP_RANGE      MP_VAL
113   
114   typedef int           mp_err;
115   
116   /* you'll have to tune these... */
117   extern int KARATSUBA_MUL_CUTOFF,
118              KARATSUBA_SQR_CUTOFF,
119              TOOM_MUL_CUTOFF,
120              TOOM_SQR_CUTOFF;
121   
122   /* various build options */
123   #define MP_PREC                 64      /* default digits of precision (must
       be power of two) */
124   
125   /* define this to use lower memory usage routines (exptmods mostly) */
126   /* #define MP_LOW_MEM */
127   
128   /* size of comba arrays, should be at least 2 * 2**(BITS_PER_WORD - BITS_PER
      _DIGIT*2) */
129   #define MP_WARRAY               (1 << (sizeof(mp_word) * CHAR_BIT - 2 * DIGI
      T_BIT + 1))
130   
131   typedef struct  \{
132       int used, alloc, sign;
133       mp_digit *dp;
134   \} mp_int;
135   
136   #define USED(m)    ((m)->used)
137   #define DIGIT(m,k) ((m)->dp[k])
138   #define SIGN(m)    ((m)->sign)
139   
140   /* ---> init and deinit bignum functions <--- */
141   
142   /* init a bignum */
143   int mp_init(mp_int *a);
144   
145   /* free a bignum */
146   void mp_clear(mp_int *a);
147   
148   /* init a null terminated series of arguments */
149   int mp_init_multi(mp_int *mp, ...);
150   
151   /* clear a null terminated series of arguments */
152   void mp_clear_multi(mp_int *mp, ...);
153   
154   /* exchange two ints */
155   void mp_exch(mp_int *a, mp_int *b);
156   
157   /* shrink ram required for a bignum */
158   int mp_shrink(mp_int *a);
159   
160   /* grow an int to a given size */
161   int mp_grow(mp_int *a, int size);
162   
163   /* init to a given number of digits */
164   int mp_init_size(mp_int *a, int size);
165   
166   /* ---> Basic Manipulations <--- */
167   
168   #define mp_iszero(a) (((a)->used == 0) ? 1 : 0)
169   #define mp_iseven(a) (((a)->used == 0 || (((a)->dp[0] & 1) == 0)) ? 1 : 0)
170   #define mp_isodd(a)  (((a)->used > 0 && (((a)->dp[0] & 1) == 1)) ? 1 : 0)
171   
172   /* set to zero */
173   void mp_zero(mp_int *a);
174   
175   /* set to a digit */
176   void mp_set(mp_int *a, mp_digit b);
177   
178   /* set a 32-bit const */
179   int mp_set_int(mp_int *a, unsigned int b);
180   
181   /* copy, b = a */
182   int mp_copy(mp_int *a, mp_int *b);
183   
184   /* inits and copies, a = b */
185   int mp_init_copy(mp_int *a, mp_int *b);
186   
187   /* trim unused digits */
188   void mp_clamp(mp_int *a);
189   
190   /* ---> digit manipulation <--- */
191   
192   /* right shift by "b" digits */
193   void mp_rshd(mp_int *a, int b);
194   
195   /* left shift by "b" digits */
196   int mp_lshd(mp_int *a, int b);
197   
198   /* c = a / 2**b */
199   int mp_div_2d(mp_int *a, int b, mp_int *c, mp_int *d);
200   
201   /* b = a/2 */
202   int mp_div_2(mp_int *a, mp_int *b);
203   
204   /* c = a * 2**b */
205   int mp_mul_2d(mp_int *a, int b, mp_int *c);
206   
207   /* b = a*2 */
208   int mp_mul_2(mp_int *a, mp_int *b);
209   
210   /* c = a mod 2**d */
211   int mp_mod_2d(mp_int *a, int b, mp_int *c);
212   
213   /* computes a = 2**b */
214   int mp_2expt(mp_int *a, int b);
215   
216   /* makes a pseudo-random int of a given size */
217   int mp_rand(mp_int *a, int digits);
218   
219   /* ---> binary operations <--- */
220   /* c = a XOR b  */
221   int mp_xor(mp_int *a, mp_int *b, mp_int *c);
222   
223   /* c = a OR b */
224   int mp_or(mp_int *a, mp_int *b, mp_int *c);
225   
226   /* c = a AND b */
227   int mp_and(mp_int *a, mp_int *b, mp_int *c);
228   
229   /* ---> Basic arithmetic <--- */
230   
231   /* b = -a */
232   int mp_neg(mp_int *a, mp_int *b);
233   
234   /* b = |a| */
235   int mp_abs(mp_int *a, mp_int *b);
236   
237   /* compare a to b */
238   int mp_cmp(mp_int *a, mp_int *b);
239   
240   /* compare |a| to |b| */
241   int mp_cmp_mag(mp_int *a, mp_int *b);
242   
243   /* c = a + b */
244   int mp_add(mp_int *a, mp_int *b, mp_int *c);
245   
246   /* c = a - b */
247   int mp_sub(mp_int *a, mp_int *b, mp_int *c);
248   
249   /* c = a * b */
250   int mp_mul(mp_int *a, mp_int *b, mp_int *c);
251   
252   /* b = a*a  */
253   int mp_sqr(mp_int *a, mp_int *b);
254   
255   /* a/b => cb + d == a */
256   int mp_div(mp_int *a, mp_int *b, mp_int *c, mp_int *d);
257   
258   /* c = a mod b, 0 <= c < b  */
259   int mp_mod(mp_int *a, mp_int *b, mp_int *c);
260   
261   /* ---> single digit functions <--- */
262   
263   /* compare against a single digit */
264   int mp_cmp_d(mp_int *a, mp_digit b);
265   
266   /* c = a + b */
267   int mp_add_d(mp_int *a, mp_digit b, mp_int *c);
268   
269   /* c = a - b */
270   int mp_sub_d(mp_int *a, mp_digit b, mp_int *c);
271   
272   /* c = a * b */
273   int mp_mul_d(mp_int *a, mp_digit b, mp_int *c);
274   
275   /* a/b => cb + d == a */
276   int mp_div_d(mp_int *a, mp_digit b, mp_int *c, mp_digit *d);
277   
278   /* a/3 => 3c + d == a */
279   int mp_div_3(mp_int *a, mp_int *c, mp_digit *d);
280   
281   /* c = a**b */
282   int mp_expt_d(mp_int *a, mp_digit b, mp_int *c);
283   
284   /* c = a mod b, 0 <= c < b  */
285   int mp_mod_d(mp_int *a, mp_digit b, mp_digit *c);
286   
287   /* ---> number theory <--- */
288   
289   /* d = a + b (mod c) */
290   int mp_addmod(mp_int *a, mp_int *b, mp_int *c, mp_int *d);
291   
292   /* d = a - b (mod c) */
293   int mp_submod(mp_int *a, mp_int *b, mp_int *c, mp_int *d);
294   
295   /* d = a * b (mod c) */
296   int mp_mulmod(mp_int *a, mp_int *b, mp_int *c, mp_int *d);
297   
298   /* c = a * a (mod b) */
299   int mp_sqrmod(mp_int *a, mp_int *b, mp_int *c);
300   
301   /* c = 1/a (mod b) */
302   int mp_invmod(mp_int *a, mp_int *b, mp_int *c);
303   
304   /* c = (a, b) */
305   int mp_gcd(mp_int *a, mp_int *b, mp_int *c);
306   
307   /* c = [a, b] or (a*b)/(a, b) */
308   int mp_lcm(mp_int *a, mp_int *b, mp_int *c);
309   
310   /* finds one of the b'th root of a, such that |c|**b <= |a|
311    *
312    * returns error if a < 0 and b is even
313    */
314   int mp_n_root(mp_int *a, mp_digit b, mp_int *c);
315   
316   /* shortcut for square root */
317   #define mp_sqrt(a, b) mp_n_root(a, 2, b)
318   
319   /* computes the jacobi c = (a | n) (or Legendre if b is prime)  */
320   int mp_jacobi(mp_int *a, mp_int *n, int *c);
321   
322   /* used to setup the Barrett reduction for a given modulus b */
323   int mp_reduce_setup(mp_int *a, mp_int *b);
324   
325   /* Barrett Reduction, computes a (mod b) with a precomputed value c
326    *
327    * Assumes that 0 < a <= b*b, note if 0 > a > -(b*b) then you can merely
328    * compute the reduction as -1 * mp_reduce(mp_abs(a)) [pseudo code].
329    */
330   int mp_reduce(mp_int *a, mp_int *b, mp_int *c);
331   
332   /* setups the montgomery reduction */
333   int mp_montgomery_setup(mp_int *a, mp_digit *mp);
334   
335   /* computes a = B**n mod b without division or multiplication useful for
336    * normalizing numbers in a Montgomery system.
337    */
338   int mp_montgomery_calc_normalization(mp_int *a, mp_int *b);
339   
340   /* computes x/R == x (mod N) via Montgomery Reduction */
341   int mp_montgomery_reduce(mp_int *a, mp_int *m, mp_digit mp);
342   
343   /* returns 1 if a is a valid DR modulus */
344   int mp_dr_is_modulus(mp_int *a);
345   
346   /* sets the value of "d" required for mp_dr_reduce */
347   void mp_dr_setup(mp_int *a, mp_digit *d);
348   
349   /* reduces a modulo b using the Diminished Radix method */
350   int mp_dr_reduce(mp_int *a, mp_int *b, mp_digit mp);
351   
352   /* returns true if a can be reduced with mp_reduce_2k */
353   int mp_reduce_is_2k(mp_int *a);
354   
355   /* determines k value for 2k reduction */
356   int mp_reduce_2k_setup(mp_int *a, mp_digit *d);
357   
358   /* reduces a modulo b where b is of the form 2**p - k [0 <= a] */
359   int mp_reduce_2k(mp_int *a, mp_int *n, mp_digit k);
360   
361   /* d = a**b (mod c) */
362   int mp_exptmod(mp_int *a, mp_int *b, mp_int *c, mp_int *d);
363   
364   /* ---> Primes <--- */
365   
366   /* number of primes */
367   #ifdef MP_8BIT
368      #define PRIME_SIZE      31
369   #else
370      #define PRIME_SIZE      256
371   #endif
372   
373   /* table of first PRIME_SIZE primes */
374   extern const mp_digit __prime_tab[];
375   
376   /* result=1 if a is divisible by one of the first PRIME_SIZE primes */
377   int mp_prime_is_divisible(mp_int *a, int *result);
378   
379   /* performs one Fermat test of "a" using base "b".
380    * Sets result to 0 if composite or 1 if probable prime
381    */
382   int mp_prime_fermat(mp_int *a, mp_int *b, int *result);
383   
384   /* performs one Miller-Rabin test of "a" using base "b".
385    * Sets result to 0 if composite or 1 if probable prime
386    */
387   int mp_prime_miller_rabin(mp_int *a, mp_int *b, int *result);
388   
389   /* performs t rounds of Miller-Rabin on "a" using the first
390    * t prime bases.  Also performs an initial sieve of trial
391    * division.  Determines if "a" is prime with probability
392    * of error no more than (1/4)**t.
393    *
394    * Sets result to 1 if probably prime, 0 otherwise
395    */
396   int mp_prime_is_prime(mp_int *a, int t, int *result);
397   
398   /* finds the next prime after the number "a" using "t" trials
399    * of Miller-Rabin.
400    */
401   int mp_prime_next_prime(mp_int *a, int t);
402   
403   
404   /* ---> radix conversion <--- */
405   int mp_count_bits(mp_int *a);
406   
407   int mp_unsigned_bin_size(mp_int *a);
408   int mp_read_unsigned_bin(mp_int *a, unsigned char *b, int c);
409   int mp_to_unsigned_bin(mp_int *a, unsigned char *b);
410   
411   int mp_signed_bin_size(mp_int *a);
412   int mp_read_signed_bin(mp_int *a, unsigned char *b, int c);
413   int mp_to_signed_bin(mp_int *a, unsigned char *b);
414   
415   int mp_read_radix(mp_int *a, char *str, int radix);
416   int mp_toradix(mp_int *a, char *str, int radix);
417   int mp_radix_size(mp_int *a, int radix);
418   
419   int mp_fread(mp_int *a, int radix, FILE *stream);
420   int mp_fwrite(mp_int *a, int radix, FILE *stream);
421   
422   #define mp_read_raw(mp, str, len) mp_read_signed_bin((mp), (str), (len))
423   #define mp_raw_size(mp)           mp_signed_bin_size(mp)
424   #define mp_toraw(mp, str)         mp_to_signed_bin((mp), (str))
425   #define mp_read_mag(mp, str, len) mp_read_unsigned_bin((mp), (str), (len))
426   #define mp_mag_size(mp)           mp_unsigned_bin_size(mp)
427   #define mp_tomag(mp, str)         mp_to_unsigned_bin((mp), (str))
428   
429   #define mp_tobinary(M, S)  mp_toradix((M), (S), 2)
430   #define mp_tooctal(M, S)   mp_toradix((M), (S), 8)
431   #define mp_todecimal(M, S) mp_toradix((M), (S), 10)
432   #define mp_tohex(M, S)     mp_toradix((M), (S), 16)
433   
434   /* lowlevel functions, do not call! */
435   int s_mp_add(mp_int *a, mp_int *b, mp_int *c);
436   int s_mp_sub(mp_int *a, mp_int *b, mp_int *c);
437   #define s_mp_mul(a, b, c) s_mp_mul_digs(a, b, c, (a)->used + (b)->used + 1)
438   int fast_s_mp_mul_digs(mp_int *a, mp_int *b, mp_int *c, int digs);
439   int s_mp_mul_digs(mp_int *a, mp_int *b, mp_int *c, int digs);
440   int fast_s_mp_mul_high_digs(mp_int *a, mp_int *b, mp_int *c, int digs);
441   int s_mp_mul_high_digs(mp_int *a, mp_int *b, mp_int *c, int digs);
442   int fast_s_mp_sqr(mp_int *a, mp_int *b);
443   int s_mp_sqr(mp_int *a, mp_int *b);
444   int mp_karatsuba_mul(mp_int *a, mp_int *b, mp_int *c);
445   int mp_toom_mul(mp_int *a, mp_int *b, mp_int *c);
446   int mp_karatsuba_sqr(mp_int *a, mp_int *b);
447   int mp_toom_sqr(mp_int *a, mp_int *b);
448   int fast_mp_invmod(mp_int *a, mp_int *b, mp_int *c);
449   int fast_mp_montgomery_reduce(mp_int *a, mp_int *m, mp_digit mp);
450   int mp_exptmod_fast(mp_int *G, mp_int *X, mp_int *P, mp_int *Y, int mode);
451   int s_mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y);
452   void bn_reverse(unsigned char *s, int len);
453   
454   #ifdef __cplusplus
455      \}
456   #endif
457   
458   #endif
459   
\end{alltt}
\end{small}

\end{document}