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\title{Multi--Precision Math}
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Open Communications Security \\
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This text has been placed in the public domain. This text corresponds to the v0.39 release of the
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Email: tomstdenis@gmail.com
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This text is formatted to the international B5 paper size of 176mm wide by 250mm tall using the \LaTeX{}
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{\em book} macro package and the Perl {\em booker} package.
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When I tell people about my LibTom projects and that I release them as public domain they are often puzzled.
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They ask why I did it and especially why I continue to work on them for free. The best I can explain it is ``Because I can.''
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Which seems odd and perhaps too terse for adult conversation. I often qualify it with ``I am able, I am willing.'' which
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perhaps explains it better. I am the first to admit there is not anything that special with what I have done. Perhaps
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others can see that too and then we would have a society to be proud of. My LibTom projects are what I am doing to give
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back to society in the form of tools and knowledge that can help others in their endeavours.
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I started writing this book because it was the most logical task to further my goal of open academia. The LibTomMath source
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code itself was written to be easy to follow and learn from. There are times, however, where pure C source code does not
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explain the algorithms properly. Hence this book. The book literally starts with the foundation of the library and works
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itself outwards to the more complicated algorithms. The use of both pseudo--code and verbatim source code provides a duality
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of ``theory'' and ``practice'' that the computer science students of the world shall appreciate. I never deviate too far
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from relatively straightforward algebra and I hope that this book can be a valuable learning asset.
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This book and indeed much of the LibTom projects would not exist in their current form if it was not for a plethora
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of kind people donating their time, resources and kind words to help support my work. Writing a text of significant
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length (along with the source code) is a tiresome and lengthy process. Currently the LibTom project is four years old,
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comprises of literally thousands of users and over 100,000 lines of source code, TeX and other material. People like Mads and Greg
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were there at the beginning to encourage me to work well. It is amazing how timely validation from others can boost morale to
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continue the project. Definitely my parents were there for me by providing room and board during the many months of work in 2003.
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To my many friends whom I have met through the years I thank you for the good times and the words of encouragement. I hope I
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honour your kind gestures with this project.
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Open Source. Open Academia. Open Minds.
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\begin{flushright} Tom St Denis \end{flushright}
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I found the opportunity to work with Tom appealing for several reasons, not only could I broaden my own horizons, but also
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contribute to educate others facing the problem of having to handle big number mathematical calculations.
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This book is Tom's child and he has been caring and fostering the project ever since the beginning with a clear mind of
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how he wanted the project to turn out. I have helped by proofreading the text and we have had several discussions about
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the layout and language used.
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I hold a masters degree in cryptography from the University of Southern Denmark and have always been interested in the
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practical aspects of cryptography.
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Having worked in the security consultancy business for several years in S\~{a}o Paulo, Brazil, I have been in touch with a
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great deal of work in which multiple precision mathematics was needed. Understanding the possibilities for speeding up
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multiple precision calculations is often very important since we deal with outdated machine architecture where modular
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reductions, for example, become painfully slow.
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This text is for people who stop and wonder when first examining algorithms such as RSA for the first time and asks
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themselves, ``You tell me this is only secure for large numbers, fine; but how do you implement these numbers?''
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It's all because I broke my leg. That just happened to be at about the same time that Tom asked for someone to review the section of the book about
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Karatsuba multiplication. I was laid up, alone and immobile, and thought ``Why not?'' I vaguely knew what Karatsuba multiplication was, but not
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really, so I thought I could help, learn, and stop myself from watching daytime cable TV, all at once.
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At the time of writing this, I've still not met Tom or Mads in meatspace. I've been following Tom's progress since his first splash on the
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sci.crypt Usenet news group. I watched him go from a clueless newbie, to the cryptographic equivalent of a reformed smoker, to a real
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contributor to the field, over a period of about two years. I've been impressed with his obvious intelligence, and astounded by his productivity.
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Of course, he's young enough to be my own child, so he doesn't have my problems with staying awake.
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When I reviewed that single section of the book, in its very earliest form, I was very pleasantly surprised. So I decided to collaborate more fully,
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and at least review all of it, and perhaps write some bits too. There's still a long way to go with it, and I have watched a number of close
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friends go through the mill of publication, so I think that the way to go is longer than Tom thinks it is. Nevertheless, it's a good effort,
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and I'm pleased to be involved with it.
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Greg Rose, Sydney, Australia, June 2003.
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\chapter{Introduction}
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\section{Multiple Precision Arithmetic}
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\subsection{What is Multiple Precision Arithmetic?}
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When we think of long-hand arithmetic such as addition or multiplication we rarely consider the fact that we instinctively
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raise or lower the precision of the numbers we are dealing with. For example, in decimal we almost immediate can
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reason that $7$ times $6$ is $42$. However, $42$ has two digits of precision as opposed to one digit we started with.
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Further multiplications of say $3$ result in a larger precision result $126$. In these few examples we have multiple
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precisions for the numbers we are working with. Despite the various levels of precision a single subset\footnote{With the occasional optimization.}
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of algorithms can be designed to accomodate them.
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By way of comparison a fixed or single precision operation would lose precision on various operations. For example, in
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the decimal system with fixed precision $6 \cdot 7 = 2$.
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Essentially at the heart of computer based multiple precision arithmetic are the same long-hand algorithms taught in
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schools to manually add, subtract, multiply and divide.
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\subsection{The Need for Multiple Precision Arithmetic}
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The most prevalent need for multiple precision arithmetic, often referred to as ``bignum'' math, is within the implementation
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of public-key cryptography algorithms. Algorithms such as RSA \cite{RSAREF} and Diffie-Hellman \cite{DHREF} require
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integers of significant magnitude to resist known cryptanalytic attacks. For example, at the time of this writing a
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typical RSA modulus would be at least greater than $10^{309}$. However, modern programming languages such as ISO C \cite{ISOC} and
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Java \cite{JAVA} only provide instrinsic support for integers which are relatively small and single precision.
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\begin{figure}[!here]
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\begin{tabular}{|r|c|}
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\hline \textbf{Data Type} & \textbf{Range} \\
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\hline char & $-128 \ldots 127$ \\
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\hline short & $-32768 \ldots 32767$ \\
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\hline long & $-2147483648 \ldots 2147483647$ \\
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\hline long long & $-9223372036854775808 \ldots 9223372036854775807$ \\
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\caption{Typical Data Types for the C Programming Language}
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The largest data type guaranteed to be provided by the ISO C programming
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language\footnote{As per the ISO C standard. However, each compiler vendor is allowed to augment the precision as they
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see fit.} can only represent values up to $10^{19}$ as shown in figure \ref{fig:ISOC}. On its own the C language is
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insufficient to accomodate the magnitude required for the problem at hand. An RSA modulus of magnitude $10^{19}$ could be
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trivially factored\footnote{A Pollard-Rho factoring would take only $2^{16}$ time.} on the average desktop computer,
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rendering any protocol based on the algorithm insecure. Multiple precision algorithms solve this very problem by
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extending the range of representable integers while using single precision data types.
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Most advancements in fast multiple precision arithmetic stem from the need for faster and more efficient cryptographic
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primitives. Faster modular reduction and exponentiation algorithms such as Barrett's algorithm, which have appeared in
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various cryptographic journals, can render algorithms such as RSA and Diffie-Hellman more efficient. In fact, several
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major companies such as RSA Security, Certicom and Entrust have built entire product lines on the implementation and
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deployment of efficient algorithms.
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However, cryptography is not the only field of study that can benefit from fast multiple precision integer routines.
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Another auxiliary use of multiple precision integers is high precision floating point data types.
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The basic IEEE \cite{IEEE} standard floating point type is made up of an integer mantissa $q$, an exponent $e$ and a sign bit $s$.
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Numbers are given in the form $n = q \cdot b^e \cdot -1^s$ where $b = 2$ is the most common base for IEEE. Since IEEE
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floating point is meant to be implemented in hardware the precision of the mantissa is often fairly small
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(\textit{23, 48 and 64 bits}). The mantissa is merely an integer and a multiple precision integer could be used to create
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a mantissa of much larger precision than hardware alone can efficiently support. This approach could be useful where
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scientific applications must minimize the total output error over long calculations.
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Yet another use for large integers is within arithmetic on polynomials of large characteristic (i.e. $GF(p)[x]$ for large $p$).
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In fact the library discussed within this text has already been used to form a polynomial basis library\footnote{See \url{http://poly.libtomcrypt.org} for more details.}.
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\subsection{Benefits of Multiple Precision Arithmetic}
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The benefit of multiple precision representations over single or fixed precision representations is that
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no precision is lost while representing the result of an operation which requires excess precision. For example,
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the product of two $n$-bit integers requires at least $2n$ bits of precision to be represented faithfully. A multiple
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precision algorithm would augment the precision of the destination to accomodate the result while a single precision system
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would truncate excess bits to maintain a fixed level of precision.
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It is possible to implement algorithms which require large integers with fixed precision algorithms. For example, elliptic
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curve cryptography (\textit{ECC}) is often implemented on smartcards by fixing the precision of the integers to the maximum
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size the system will ever need. Such an approach can lead to vastly simpler algorithms which can accomodate the
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integers required even if the host platform cannot natively accomodate them\footnote{For example, the average smartcard
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processor has an 8 bit accumulator.}. However, as efficient as such an approach may be, the resulting source code is not
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normally very flexible. It cannot, at runtime, accomodate inputs of higher magnitude than the designer anticipated.
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Multiple precision algorithms have the most overhead of any style of arithmetic. For the the most part the
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overhead can be kept to a minimum with careful planning, but overall, it is not well suited for most memory starved
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platforms. However, multiple precision algorithms do offer the most flexibility in terms of the magnitude of the
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inputs. That is, the same algorithms based on multiple precision integers can accomodate any reasonable size input
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without the designer's explicit forethought. This leads to lower cost of ownership for the code as it only has to
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be written and tested once.
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\section{Purpose of This Text}
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The purpose of this text is to instruct the reader regarding how to implement efficient multiple precision algorithms.
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That is to not only explain a limited subset of the core theory behind the algorithms but also the various ``house keeping''
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elements that are neglected by authors of other texts on the subject. Several well reknowned texts \cite{TAOCPV2,HAC}
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give considerably detailed explanations of the theoretical aspects of algorithms and often very little information
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regarding the practical implementation aspects.
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In most cases how an algorithm is explained and how it is actually implemented are two very different concepts. For
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example, the Handbook of Applied Cryptography (\textit{HAC}), algorithm 14.7 on page 594, gives a relatively simple
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algorithm for performing multiple precision integer addition. However, the description lacks any discussion concerning
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the fact that the two integer inputs may be of differing magnitudes. As a result the implementation is not as simple
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as the text would lead people to believe. Similarly the division routine (\textit{algorithm 14.20, pp. 598}) does not
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discuss how to handle sign or handle the dividend's decreasing magnitude in the main loop (\textit{step \#3}).
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Both texts also do not discuss several key optimal algorithms required such as ``Comba'' and Karatsuba multipliers
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and fast modular inversion, which we consider practical oversights. These optimal algorithms are vital to achieve
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any form of useful performance in non-trivial applications.
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To solve this problem the focus of this text is on the practical aspects of implementing a multiple precision integer
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package. As a case study the ``LibTomMath''\footnote{Available at \url{http://math.libtomcrypt.com}} package is used
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to demonstrate algorithms with real implementations\footnote{In the ISO C programming language.} that have been field
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tested and work very well. The LibTomMath library is freely available on the Internet for all uses and this text
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discusses a very large portion of the inner workings of the library.
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The algorithms that are presented will always include at least one ``pseudo-code'' description followed
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by the actual C source code that implements the algorithm. The pseudo-code can be used to implement the same
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algorithm in other programming languages as the reader sees fit.
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This text shall also serve as a walkthrough of the creation of multiple precision algorithms from scratch. Showing
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the reader how the algorithms fit together as well as where to start on various taskings.
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\section{Discussion and Notation}
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\subsection{Notation}
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A multiple precision integer of $n$-digits shall be denoted as $x = (x_{n-1}, \ldots, x_1, x_0)_{ \beta }$ and represent
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the integer $x \equiv \sum_{i=0}^{n-1} x_i\beta^i$. The elements of the array $x$ are said to be the radix $\beta$ digits
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of the integer. For example, $x = (1,2,3)_{10}$ would represent the integer
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$1\cdot 10^2 + 2\cdot10^1 + 3\cdot10^0 = 123$.
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The term ``mp\_int'' shall refer to a composite structure which contains the digits of the integer it represents, as well
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as auxilary data required to manipulate the data. These additional members are discussed further in section
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\ref{sec:MPINT}. For the purposes of this text a ``multiple precision integer'' and an ``mp\_int'' are assumed to be
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synonymous. When an algorithm is specified to accept an mp\_int variable it is assumed the various auxliary data members
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are present as well. An expression of the type \textit{variablename.item} implies that it should evaluate to the
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member named ``item'' of the variable. For example, a string of characters may have a member ``length'' which would
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evaluate to the number of characters in the string. If the string $a$ equals ``hello'' then it follows that
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For certain discussions more generic algorithms are presented to help the reader understand the final algorithm used
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to solve a given problem. When an algorithm is described as accepting an integer input it is assumed the input is
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a plain integer with no additional multiple-precision members. That is, algorithms that use integers as opposed to
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mp\_ints as inputs do not concern themselves with the housekeeping operations required such as memory management. These
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algorithms will be used to establish the relevant theory which will subsequently be used to describe a multiple
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precision algorithm to solve the same problem.
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\subsection{Precision Notation}
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The variable $\beta$ represents the radix of a single digit of a multiple precision integer and
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must be of the form $q^p$ for $q, p \in \Z^+$. A single precision variable must be able to represent integers in
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the range $0 \le x < q \beta$ while a double precision variable must be able to represent integers in the range
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$0 \le x < q \beta^2$. The extra radix-$q$ factor allows additions and subtractions to proceed without truncation of the
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carry. Since all modern computers are binary, it is assumed that $q$ is two.
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\index{mp\_digit} \index{mp\_word}
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Within the source code that will be presented for each algorithm, the data type \textbf{mp\_digit} will represent
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a single precision integer type, while, the data type \textbf{mp\_word} will represent a double precision integer type. In
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several algorithms (notably the Comba routines) temporary results will be stored in arrays of double precision mp\_words.
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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
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the $j$'th digit of a double precision array. Whenever an expression is to be assigned to a double precision
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variable it is assumed that all single precision variables are promoted to double precision during the evaluation.
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Expressions that are assigned to a single precision variable are truncated to fit within the precision of a single
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For example, if $\beta = 10^2$ a single precision data type may represent a value in the
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range $0 \le x < 10^3$, while a double precision data type may represent a value in the range $0 \le x < 10^5$. Let
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$a = 23$ and $b = 49$ represent two single precision variables. The single precision product shall be written
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as $c \leftarrow a \cdot b$ while the double precision product shall be written as $\hat c \leftarrow a \cdot b$.
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In this particular case, $\hat c = 1127$ and $c = 127$. The most significant digit of the product would not fit
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in a single precision data type and as a result $c \ne \hat c$.
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\subsection{Algorithm Inputs and Outputs}
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Within the algorithm descriptions all variables are assumed to be scalars of either single or double precision
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as indicated. The only exception to this rule is when variables have been indicated to be of type mp\_int. This
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distinction is important as scalars are often used as array indicies and various other counters.
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\subsection{Mathematical Expressions}
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The $\lfloor \mbox{ } \rfloor$ brackets imply an expression truncated to an integer not greater than the expression
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itself. For example, $\lfloor 5.7 \rfloor = 5$. Similarly the $\lceil \mbox{ } \rceil$ brackets imply an expression
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rounded to an integer not less than the expression itself. For example, $\lceil 5.1 \rceil = 6$. Typically when
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the $/$ division symbol is used the intention is to perform an integer division with truncation. For example,
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$5/2 = 2$ which will often be written as $\lfloor 5/2 \rfloor = 2$ for clarity. When an expression is written as a
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fraction a real value division is implied, for example ${5 \over 2} = 2.5$.
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The norm of a multiple precision integer, for example $\vert \vert x \vert \vert$, will be used to represent the number of digits in the representation
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of the integer. For example, $\vert \vert 123 \vert \vert = 3$ and $\vert \vert 79452 \vert \vert = 5$.
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\subsection{Work Effort}
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To measure the efficiency of the specified algorithms, a modified big-Oh notation is used. In this system all
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single precision operations are considered to have the same cost\footnote{Except where explicitly noted.}.
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That is a single precision addition, multiplication and division are assumed to take the same time to
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complete. While this is generally not true in practice, it will simplify the discussions considerably.
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Some algorithms have slight advantages over others which is why some constants will not be removed in
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the notation. For example, a normal baseline multiplication (section \ref{sec:basemult}) requires $O(n^2)$ work while a
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baseline squaring (section \ref{sec:basesquare}) requires $O({{n^2 + n}\over 2})$ work. In standard big-Oh notation these
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would both be said to be equivalent to $O(n^2)$. However,
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in the context of the this text this is not the case as the magnitude of the inputs will typically be rather small. As a
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result small constant factors in the work effort will make an observable difference in algorithm efficiency.
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All of the algorithms presented in this text have a polynomial time work level. That is, of the form
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$O(n^k)$ for $n, k \in \Z^{+}$. This will help make useful comparisons in terms of the speed of the algorithms and how
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various optimizations will help pay off in the long run.
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Within the more advanced chapters a section will be set aside to give the reader some challenging exercises related to
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the discussion at hand. These exercises are not designed to be prize winning problems, but instead to be thought
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provoking. Wherever possible the problems are forward minded, stating problems that will be answered in subsequent
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chapters. The reader is encouraged to finish the exercises as they appear to get a better understanding of the
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That being said, the problems are designed to affirm knowledge of a particular subject matter. Students in particular
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are encouraged to verify they can answer the problems correctly before moving on.
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Similar to the exercises of \cite[pp. ix]{TAOCPV2} these exercises are given a scoring system based on the difficulty of
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the problem. However, unlike \cite{TAOCPV2} the problems do not get nearly as hard. The scoring of these
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exercises ranges from one (the easiest) to five (the hardest). The following table sumarizes the
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\begin{tabular}{|c|l|}
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\hline $\left [ 1 \right ]$ & An easy problem that should only take the reader a manner of \\
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& minutes to solve. Usually does not involve much computer time \\
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\hline $\left [ 2 \right ]$ & An easy problem that involves a marginal amount of computer \\
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& time usage. Usually requires a program to be written to \\
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& solve the problem. \\
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\hline $\left [ 3 \right ]$ & A moderately hard problem that requires a non-trivial amount \\
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& of work. Usually involves trivial research and development of \\
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& new theory from the perspective of a student. \\
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\hline $\left [ 4 \right ]$ & A moderately hard problem that involves a non-trivial amount \\
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& of work and research, the solution to which will demonstrate \\
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& a higher mastery of the subject matter. \\
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\hline $\left [ 5 \right ]$ & A hard problem that involves concepts that are difficult for a \\
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& novice to solve. Solutions to these problems will demonstrate a \\
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& complete mastery of the given subject. \\
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\caption{Exercise Scoring System}
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Problems at the first level are meant to be simple questions that the reader can answer quickly without programming a solution or
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devising new theory. These problems are quick tests to see if the material is understood. Problems at the second level
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are also designed to be easy but will require a program or algorithm to be implemented to arrive at the answer. These
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two levels are essentially entry level questions.
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Problems at the third level are meant to be a bit more difficult than the first two levels. The answer is often
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fairly obvious but arriving at an exacting solution requires some thought and skill. These problems will almost always
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involve devising a new algorithm or implementing a variation of another algorithm previously presented. Readers who can
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answer these questions will feel comfortable with the concepts behind the topic at hand.
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Problems at the fourth level are meant to be similar to those of the level three questions except they will require
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additional research to be completed. The reader will most likely not know the answer right away, nor will the text provide
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the exact details of the answer until a subsequent chapter.
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Problems at the fifth level are meant to be the hardest
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problems relative to all the other problems in the chapter. People who can correctly answer fifth level problems have a
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mastery of the subject matter at hand.
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Often problems will be tied together. The purpose of this is to start a chain of thought that will be discussed in future chapters. The reader
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is encouraged to answer the follow-up problems and try to draw the relevance of problems.
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\section{Introduction to LibTomMath}
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\subsection{What is LibTomMath?}
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LibTomMath is a free and open source multiple precision integer library written entirely in portable ISO C. By portable it
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is meant that the library does not contain any code that is computer platform dependent or otherwise problematic to use on
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The library has been successfully tested under numerous operating systems including Unix\footnote{All of these
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trademarks belong to their respective rightful owners.}, MacOS, Windows, Linux, PalmOS and on standalone hardware such
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as the Gameboy Advance. The library is designed to contain enough functionality to be able to develop applications such
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as public key cryptosystems and still maintain a relatively small footprint.
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\subsection{Goals of LibTomMath}
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Libraries which obtain the most efficiency are rarely written in a high level programming language such as C. However,
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even though this library is written entirely in ISO C, considerable care has been taken to optimize the algorithm implementations within the
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library. Specifically the code has been written to work well with the GNU C Compiler (\textit{GCC}) on both x86 and ARM
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processors. Wherever possible, highly efficient algorithms, such as Karatsuba multiplication, sliding window
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exponentiation and Montgomery reduction have been provided to make the library more efficient.
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Even with the nearly optimal and specialized algorithms that have been included the Application Programing Interface
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(\textit{API}) has been kept as simple as possible. Often generic place holder routines will make use of specialized
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algorithms automatically without the developer's specific attention. One such example is the generic multiplication
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algorithm \textbf{mp\_mul()} which will automatically use Toom--Cook, Karatsuba, Comba or baseline multiplication
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based on the magnitude of the inputs and the configuration of the library.
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Making LibTomMath as efficient as possible is not the only goal of the LibTomMath project. Ideally the library should
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be source compatible with another popular library which makes it more attractive for developers to use. In this case the
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MPI library was used as a API template for all the basic functions. MPI was chosen because it is another library that fits
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in the same niche as LibTomMath. Even though LibTomMath uses MPI as the template for the function names and argument
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passing conventions, it has been written from scratch by Tom St Denis.
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The project is also meant to act as a learning tool for students, the logic being that no easy-to-follow ``bignum''
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library exists which can be used to teach computer science students how to perform fast and reliable multiple precision
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integer arithmetic. To this end the source code has been given quite a few comments and algorithm discussion points.
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\section{Choice of LibTomMath}
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LibTomMath was chosen as the case study of this text not only because the author of both projects is one and the same but
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for more worthy reasons. Other libraries such as GMP \cite{GMP}, MPI \cite{MPI}, LIP \cite{LIP} and OpenSSL
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\cite{OPENSSL} have multiple precision integer arithmetic routines but would not be ideal for this text for
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reasons that will be explained in the following sub-sections.
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\subsection{Code Base}
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The LibTomMath code base is all portable ISO C source code. This means that there are no platform dependent conditional
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segments of code littered throughout the source. This clean and uncluttered approach to the library means that a
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developer can more readily discern the true intent of a given section of source code without trying to keep track of
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what conditional code will be used.
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The code base of LibTomMath is well organized. Each function is in its own separate source code file
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which allows the reader to find a given function very quickly. On average there are $76$ lines of code per source
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file which makes the source very easily to follow. By comparison MPI and LIP are single file projects making code tracing
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very hard. GMP has many conditional code segments which also hinder tracing.
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When compiled with GCC for the x86 processor and optimized for speed the entire library is approximately $100$KiB\footnote{The notation ``KiB'' means $2^{10}$ octets, similarly ``MiB'' means $2^{20}$ octets.}
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which is fairly small compared to GMP (over $250$KiB). LibTomMath is slightly larger than MPI (which compiles to about
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$50$KiB) but LibTomMath is also much faster and more complete than MPI.
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\subsection{API Simplicity}
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LibTomMath is designed after the MPI library and shares the API design. Quite often programs that use MPI will build
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with LibTomMath without change. The function names correlate directly to the action they perform. Almost all of the
487
functions share the same parameter passing convention. The learning curve is fairly shallow with the API provided
488
which is an extremely valuable benefit for the student and developer alike.
490
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
491
illegible short hand. LibTomMath does not share this characteristic.
493
The GMP library also does not return error codes. Instead it uses a POSIX.1 \cite{POSIX1} signal system where errors
494
are signaled to the host application. This happens to be the fastest approach but definitely not the most versatile. In
495
effect a math error (i.e. invalid input, heap error, etc) can cause a program to stop functioning which is definitely
496
undersireable in many situations.
498
\subsection{Optimizations}
499
While LibTomMath is certainly not the fastest library (GMP often beats LibTomMath by a factor of two) it does
500
feature a set of optimal algorithms for tasks such as modular reduction, exponentiation, multiplication and squaring. GMP
501
and LIP also feature such optimizations while MPI only uses baseline algorithms with no optimizations. GMP lacks a few
502
of the additional modular reduction optimizations that LibTomMath features\footnote{At the time of this writing GMP
503
only had Barrett and Montgomery modular reduction algorithms.}.
505
LibTomMath is almost always an order of magnitude faster than the MPI library at computationally expensive tasks such as modular
506
exponentiation. In the grand scheme of ``bignum'' libraries LibTomMath is faster than the average library and usually
507
slower than the best libraries such as GMP and OpenSSL by only a small factor.
509
\subsection{Portability and Stability}
510
LibTomMath will build ``out of the box'' on any platform equipped with a modern version of the GNU C Compiler
511
(\textit{GCC}). This means that without changes the library will build without configuration or setting up any
512
variables. LIP and MPI will build ``out of the box'' as well but have numerous known bugs. Most notably the author of
513
MPI has recently stopped working on his library and LIP has long since been discontinued.
515
GMP requires a configuration script to run and will not build out of the box. GMP and LibTomMath are still in active
516
development and are very stable across a variety of platforms.
519
LibTomMath is a relatively compact, well documented, highly optimized and portable library which seems only natural for
520
the case study of this text. Various source files from the LibTomMath project will be included within the text. However,
521
the reader is encouraged to download their own copy of the library to actually be able to work with the library.
523
\chapter{Getting Started}
524
\section{Library Basics}
525
The trick to writing any useful library of source code is to build a solid foundation and work outwards from it. First,
526
a problem along with allowable solution parameters should be identified and analyzed. In this particular case the
527
inability to accomodate multiple precision integers is the problem. Futhermore, the solution must be written
528
as portable source code that is reasonably efficient across several different computer platforms.
530
After a foundation is formed the remainder of the library can be designed and implemented in a hierarchical fashion.
531
That is, to implement the lowest level dependencies first and work towards the most abstract functions last. For example,
532
before implementing a modular exponentiation algorithm one would implement a modular reduction algorithm.
533
By building outwards from a base foundation instead of using a parallel design methodology the resulting project is
534
highly modular. Being highly modular is a desirable property of any project as it often means the resulting product
535
has a small footprint and updates are easy to perform.
537
Usually when I start a project I will begin with the header files. I define the data types I think I will need and
538
prototype the initial functions that are not dependent on other functions (within the library). After I
539
implement these base functions I prototype more dependent functions and implement them. The process repeats until
540
I implement all of the functions I require. For example, in the case of LibTomMath I implemented functions such as
541
mp\_init() well before I implemented mp\_mul() and even further before I implemented mp\_exptmod(). As an example as to
542
why this design works note that the Karatsuba and Toom-Cook multipliers were written \textit{after} the
543
dependent function mp\_exptmod() was written. Adding the new multiplication algorithms did not require changes to the
544
mp\_exptmod() function itself and lowered the total cost of ownership (\textit{so to speak}) and of development
545
for new algorithms. This methodology allows new algorithms to be tested in a complete framework with relative ease.
549
\includegraphics{pics/design_process.ps}
550
\caption{Design Flow of the First Few Original LibTomMath Functions.}
551
\label{pic:design_process}
555
Only after the majority of the functions were in place did I pursue a less hierarchical approach to auditing and optimizing
556
the source code. For example, one day I may audit the multipliers and the next day the polynomial basis functions.
558
It only makes sense to begin the text with the preliminary data types and support algorithms required as well.
559
This chapter discusses the core algorithms of the library which are the dependents for every other algorithm.
561
\section{What is a Multiple Precision Integer?}
562
Recall that most programming languages, in particular ISO C \cite{ISOC}, only have fixed precision data types that on their own cannot
563
be used to represent values larger than their precision will allow. The purpose of multiple precision algorithms is
564
to use fixed precision data types to create and manipulate multiple precision integers which may represent values
567
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
568
the largest single digit value is $9$. However, by concatenating digits together larger numbers may be represented. Newly prepended digits
569
(\textit{to the left}) are said to be in a different power of ten column. That is, the number $123$ can be described as having a $1$ in the hundreds
570
column, $2$ in the tens column and $3$ in the ones column. Or more formally $123 = 1 \cdot 10^2 + 2 \cdot 10^1 + 3 \cdot 10^0$. Computer based
571
multiple precision arithmetic is essentially the same concept. Larger integers are represented by adjoining fixed
572
precision computer words with the exception that a different radix is used.
574
What most people probably do not think about explicitly are the various other attributes that describe a multiple precision
575
integer. For example, the integer $154_{10}$ has two immediately obvious properties. First, the integer is positive,
576
that is the sign of this particular integer is positive as opposed to negative. Second, the integer has three digits in
577
its representation. There is an additional property that the integer posesses that does not concern pencil-and-paper
578
arithmetic. The third property is how many digits placeholders are available to hold the integer.
580
The human analogy of this third property is ensuring there is enough space on the paper to write the integer. For example,
581
if one starts writing a large number too far to the right on a piece of paper they will have to erase it and move left.
582
Similarly, computer algorithms must maintain strict control over memory usage to ensure that the digits of an integer
583
will not exceed the allowed boundaries. These three properties make up what is known as a multiple precision
584
integer or mp\_int for short.
586
\subsection{The mp\_int Structure}
588
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
589
any such data type but it does provide for making composite data types known as structures. The following is the structure definition
590
used within LibTomMath.
600
\hspace{3mm}int used, alloc, sign;\\
601
\hspace{3mm}mp\_digit *dp;\\
602
\} \textbf{mp\_int}; \\
607
\caption{The mp\_int Structure}
612
The mp\_int structure (fig. \ref{fig:mpint}) can be broken down as follows.
615
\item The \textbf{used} parameter denotes how many digits of the array \textbf{dp} contain the digits used to represent
616
a given integer. The \textbf{used} count must be positive (or zero) and may not exceed the \textbf{alloc} count.
618
\item The \textbf{alloc} parameter denotes how
619
many digits are available in the array to use by functions before it has to increase in size. When the \textbf{used} count
620
of a result would exceed the \textbf{alloc} count all of the algorithms will automatically increase the size of the
621
array to accommodate the precision of the result.
623
\item The pointer \textbf{dp} points to a dynamically allocated array of digits that represent the given multiple
624
precision integer. It is padded with $(\textbf{alloc} - \textbf{used})$ zero digits. The array is maintained in a least
625
significant digit order. As a pencil and paper analogy the array is organized such that the right most digits are stored
626
first starting at the location indexed by zero\footnote{In C all arrays begin at zero.} in the array. For example,
627
if \textbf{dp} contains $\lbrace a, b, c, \ldots \rbrace$ where \textbf{dp}$_0 = a$, \textbf{dp}$_1 = b$, \textbf{dp}$_2 = c$, $\ldots$ then
628
it would represent the integer $a + b\beta + c\beta^2 + \ldots$
630
\index{MP\_ZPOS} \index{MP\_NEG}
631
\item The \textbf{sign} parameter denotes the sign as either zero/positive (\textbf{MP\_ZPOS}) or negative (\textbf{MP\_NEG}).
634
\subsubsection{Valid mp\_int Structures}
635
Several rules are placed on the state of an mp\_int structure and are assumed to be followed for reasons of efficiency.
636
The only exceptions are when the structure is passed to initialization functions such as mp\_init() and mp\_init\_copy().
639
\item The value of \textbf{alloc} may not be less than one. That is \textbf{dp} always points to a previously allocated
641
\item The value of \textbf{used} may not exceed \textbf{alloc} and must be greater than or equal to zero.
642
\item The value of \textbf{used} implies the digit at index $(used - 1)$ of the \textbf{dp} array is non-zero. That is,
643
leading zero digits in the most significant positions must be trimmed.
645
\item Digits in the \textbf{dp} array at and above the \textbf{used} location must be zero.
647
\item The value of \textbf{sign} must be \textbf{MP\_ZPOS} if \textbf{used} is zero;
648
this represents the mp\_int value of zero.
651
\section{Argument Passing}
652
A convention of argument passing must be adopted early on in the development of any library. Making the function
653
prototypes consistent will help eliminate many headaches in the future as the library grows to significant complexity.
654
In LibTomMath the multiple precision integer functions accept parameters from left to right as pointers to mp\_int
655
structures. That means that the source (input) operands are placed on the left and the destination (output) on the right.
656
Consider the following examples.
659
mp_mul(&a, &b, &c); /* c = a * b */
660
mp_add(&a, &b, &a); /* a = a + b */
661
mp_sqr(&a, &b); /* b = a * a */
664
The left to right order is a fairly natural way to implement the functions since it lets the developer read aloud the
665
functions and make sense of them. For example, the first function would read ``multiply a and b and store in c''.
667
Certain libraries (\textit{LIP by Lenstra for instance}) accept parameters the other way around, to mimic the order
668
of assignment expressions. That is, the destination (output) is on the left and arguments (inputs) are on the right. In
669
truth, it is entirely a matter of preference. In the case of LibTomMath the convention from the MPI library has been
672
Another very useful design consideration, provided for in LibTomMath, is whether to allow argument sources to also be a
673
destination. For example, the second example (\textit{mp\_add}) adds $a$ to $b$ and stores in $a$. This is an important
674
feature to implement since it allows the calling functions to cut down on the number of variables it must maintain.
675
However, to implement this feature specific care has to be given to ensure the destination is not modified before the
676
source is fully read.
678
\section{Return Values}
679
A well implemented application, no matter what its purpose, should trap as many runtime errors as possible and return them
680
to the caller. By catching runtime errors a library can be guaranteed to prevent undefined behaviour. However, the end
681
developer can still manage to cause a library to crash. For example, by passing an invalid pointer an application may
682
fault by dereferencing memory not owned by the application.
684
In the case of LibTomMath the only errors that are checked for are related to inappropriate inputs (division by zero for
685
instance) and memory allocation errors. It will not check that the mp\_int passed to any function is valid nor
686
will it check pointers for validity. Any function that can cause a runtime error will return an error code as an
687
\textbf{int} data type with one of the following values (fig \ref{fig:errcodes}).
689
\index{MP\_OKAY} \index{MP\_VAL} \index{MP\_MEM}
692
\begin{tabular}{|l|l|}
693
\hline \textbf{Value} & \textbf{Meaning} \\
694
\hline \textbf{MP\_OKAY} & The function was successful \\
695
\hline \textbf{MP\_VAL} & One of the input value(s) was invalid \\
696
\hline \textbf{MP\_MEM} & The function ran out of heap memory \\
700
\caption{LibTomMath Error Codes}
704
When an error is detected within a function it should free any memory it allocated, often during the initialization of
705
temporary mp\_ints, and return as soon as possible. The goal is to leave the system in the same state it was when the
706
function was called. Error checking with this style of API is fairly simple.
710
if ((err = mp_add(&a, &b, &c)) != MP_OKAY) {
711
printf("Error: %s\n", mp_error_to_string(err));
716
The GMP \cite{GMP} library uses C style \textit{signals} to flag errors which is of questionable use. Not all errors are fatal
717
and it was not deemed ideal by the author of LibTomMath to force developers to have signal handlers for such cases.
719
\section{Initialization and Clearing}
720
The logical starting point when actually writing multiple precision integer functions is the initialization and
721
clearing of the mp\_int structures. These two algorithms will be used by the majority of the higher level algorithms.
723
Given the basic mp\_int structure an initialization routine must first allocate memory to hold the digits of
724
the integer. Often it is optimal to allocate a sufficiently large pre-set number of digits even though
725
the initial integer will represent zero. If only a single digit were allocated quite a few subsequent re-allocations
726
would occur when operations are performed on the integers. There is a tradeoff between how many default digits to allocate
727
and how many re-allocations are tolerable. Obviously allocating an excessive amount of digits initially will waste
728
memory and become unmanageable.
730
If the memory for the digits has been successfully allocated then the rest of the members of the structure must
731
be initialized. Since the initial state of an mp\_int is to represent the zero integer, the allocated digits must be set
732
to zero. The \textbf{used} count set to zero and \textbf{sign} set to \textbf{MP\_ZPOS}.
734
\subsection{Initializing an mp\_int}
735
An mp\_int is said to be initialized if it is set to a valid, preferably default, state such that all of the members of the
736
structure are set to valid values. The mp\_init algorithm will perform such an action.
742
\hline Algorithm \textbf{mp\_init}. \\
743
\textbf{Input}. An mp\_int $a$ \\
744
\textbf{Output}. Allocate memory and initialize $a$ to a known valid mp\_int state. \\
746
1. Allocate memory for \textbf{MP\_PREC} digits. \\
747
2. If the allocation failed return(\textit{MP\_MEM}) \\
748
3. for $n$ from $0$ to $MP\_PREC - 1$ do \\
749
\hspace{3mm}3.1 $a_n \leftarrow 0$\\
750
4. $a.sign \leftarrow MP\_ZPOS$\\
751
5. $a.used \leftarrow 0$\\
752
6. $a.alloc \leftarrow MP\_PREC$\\
753
7. Return(\textit{MP\_OKAY})\\
757
\caption{Algorithm mp\_init}
760
\textbf{Algorithm mp\_init.}
761
The purpose of this function is to initialize an mp\_int structure so that the rest of the library can properly
762
manipulte it. It is assumed that the input may not have had any of its members previously initialized which is certainly
763
a valid assumption if the input resides on the stack.
765
Before any of the members such as \textbf{sign}, \textbf{used} or \textbf{alloc} are initialized the memory for
766
the digits is allocated. If this fails the function returns before setting any of the other members. The \textbf{MP\_PREC}
767
name represents a constant\footnote{Defined in the ``tommath.h'' header file within LibTomMath.}
768
used to dictate the minimum precision of newly initialized mp\_int integers. Ideally, it is at least equal to the smallest
769
precision number you'll be working with.
771
Allocating a block of digits at first instead of a single digit has the benefit of lowering the number of usually slow
772
heap operations later functions will have to perform in the future. If \textbf{MP\_PREC} is set correctly the slack
773
memory and the number of heap operations will be trivial.
775
Once the allocation has been made the digits have to be set to zero as well as the \textbf{used}, \textbf{sign} and
776
\textbf{alloc} members initialized. This ensures that the mp\_int will always represent the default state of zero regardless
777
of the original condition of the input.
780
This function introduces the idiosyncrasy that all iterative loops, commonly initiated with the ``for'' keyword, iterate incrementally
781
when the ``to'' keyword is placed between two expressions. For example, ``for $a$ from $b$ to $c$ do'' means that
782
a subsequent expression (or body of expressions) are to be evaluated upto $c - b$ times so long as $b \le c$. In each
783
iteration the variable $a$ is substituted for a new integer that lies inclusively between $b$ and $c$. If $b > c$ occured
784
the loop would not iterate. By contrast if the ``downto'' keyword were used in place of ``to'' the loop would iterate
787
\vspace{+3mm}\begin{small}
788
\hspace{-5.1mm}{\bf File}: bn\_mp\_init.c
794
One immediate observation of this initializtion function is that it does not return a pointer to a mp\_int structure. It
795
is assumed that the caller has already allocated memory for the mp\_int structure, typically on the application stack. The
796
call to mp\_init() is used only to initialize the members of the structure to a known default state.
798
Here we see (line 24) the memory allocation is performed first. This allows us to exit cleanly and quickly
799
if there is an error. If the allocation fails the routine will return \textbf{MP\_MEM} to the caller to indicate there
800
was a memory error. The function XMALLOC is what actually allocates the memory. Technically XMALLOC is not a function
801
but a macro defined in ``tommath.h``. By default, XMALLOC will evaluate to malloc() which is the C library's built--in
802
memory allocation routine.
804
In order to assure the mp\_int is in a known state the digits must be set to zero. On most platforms this could have been
805
accomplished by using calloc() instead of malloc(). However, to correctly initialize a integer type to a given value in a
806
portable fashion you have to actually assign the value. The for loop (line 30) performs this required
809
After the memory has been successfully initialized the remainder of the members are initialized
810
(lines 34 through 35) to their respective default states. At this point the algorithm has succeeded and
811
a success code is returned to the calling function. If this function returns \textbf{MP\_OKAY} it is safe to assume the
812
mp\_int structure has been properly initialized and is safe to use with other functions within the library.
814
\subsection{Clearing an mp\_int}
815
When an mp\_int is no longer required by the application, the memory that has been allocated for its digits must be
816
returned to the application's memory pool with the mp\_clear algorithm.
821
\hline Algorithm \textbf{mp\_clear}. \\
822
\textbf{Input}. An mp\_int $a$ \\
823
\textbf{Output}. The memory for $a$ shall be deallocated. \\
825
1. If $a$ has been previously freed then return(\textit{MP\_OKAY}). \\
826
2. for $n$ from 0 to $a.used - 1$ do \\
827
\hspace{3mm}2.1 $a_n \leftarrow 0$ \\
828
3. Free the memory allocated for the digits of $a$. \\
829
4. $a.used \leftarrow 0$ \\
830
5. $a.alloc \leftarrow 0$ \\
831
6. $a.sign \leftarrow MP\_ZPOS$ \\
832
7. Return(\textit{MP\_OKAY}). \\
836
\caption{Algorithm mp\_clear}
839
\textbf{Algorithm mp\_clear.}
840
This algorithm accomplishes two goals. First, it clears the digits and the other mp\_int members. This ensures that
841
if a developer accidentally re-uses a cleared structure it is less likely to cause problems. The second goal
842
is to free the allocated memory.
844
The logic behind the algorithm is extended by marking cleared mp\_int structures so that subsequent calls to this
845
algorithm will not try to free the memory multiple times. Cleared mp\_ints are detectable by having a pre-defined invalid
846
digit pointer \textbf{dp} setting.
848
Once an mp\_int has been cleared the mp\_int structure is no longer in a valid state for any other algorithm
849
with the exception of algorithms mp\_init, mp\_init\_copy, mp\_init\_size and mp\_clear.
851
\vspace{+3mm}\begin{small}
852
\hspace{-5.1mm}{\bf File}: bn\_mp\_clear.c
858
The algorithm only operates on the mp\_int if it hasn't been previously cleared. The if statement (line 25)
859
checks to see if the \textbf{dp} member is not \textbf{NULL}. If the mp\_int is a valid mp\_int then \textbf{dp} cannot be
860
\textbf{NULL} in which case the if statement will evaluate to true.
862
The digits of the mp\_int are cleared by the for loop (line 27) which assigns a zero to every digit. Similar to mp\_init()
863
the digits are assigned zero instead of using block memory operations (such as memset()) since this is more portable.
865
The digits are deallocated off the heap via the XFREE macro. Similar to XMALLOC the XFREE macro actually evaluates to
866
a standard C library function. In this case the free() function. Since free() only deallocates the memory the pointer
867
still has to be reset to \textbf{NULL} manually (line 35).
869
Now that the digits have been cleared and deallocated the other members are set to their final values (lines 36 and 37).
871
\section{Maintenance Algorithms}
873
The previous sections describes how to initialize and clear an mp\_int structure. To further support operations
874
that are to be performed on mp\_int structures (such as addition and multiplication) the dependent algorithms must be
875
able to augment the precision of an mp\_int and
876
initialize mp\_ints with differing initial conditions.
878
These algorithms complete the set of low level algorithms required to work with mp\_int structures in the higher level
879
algorithms such as addition, multiplication and modular exponentiation.
881
\subsection{Augmenting an mp\_int's Precision}
882
When storing a value in an mp\_int structure, a sufficient number of digits must be available to accomodate the entire
883
result of an operation without loss of precision. Quite often the size of the array given by the \textbf{alloc} member
884
is large enough to simply increase the \textbf{used} digit count. However, when the size of the array is too small it
885
must be re-sized appropriately to accomodate the result. The mp\_grow algorithm will provide this functionality.
887
\newpage\begin{figure}[here]
890
\hline Algorithm \textbf{mp\_grow}. \\
891
\textbf{Input}. An mp\_int $a$ and an integer $b$. \\
892
\textbf{Output}. $a$ is expanded to accomodate $b$ digits. \\
894
1. if $a.alloc \ge b$ then return(\textit{MP\_OKAY}) \\
895
2. $u \leftarrow b\mbox{ (mod }MP\_PREC\mbox{)}$ \\
896
3. $v \leftarrow b + 2 \cdot MP\_PREC - u$ \\
897
4. Re-allocate the array of digits $a$ to size $v$ \\
898
5. If the allocation failed then return(\textit{MP\_MEM}). \\
899
6. for n from a.alloc to $v - 1$ do \\
900
\hspace{+3mm}6.1 $a_n \leftarrow 0$ \\
901
7. $a.alloc \leftarrow v$ \\
902
8. Return(\textit{MP\_OKAY}) \\
906
\caption{Algorithm mp\_grow}
909
\textbf{Algorithm mp\_grow.}
910
It is ideal to prevent re-allocations from being performed if they are not required (step one). This is useful to
911
prevent mp\_ints from growing excessively in code that erroneously calls mp\_grow.
913
The requested digit count is padded up to next multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} (steps two and three).
914
This helps prevent many trivial reallocations that would grow an mp\_int by trivially small values.
916
It is assumed that the reallocation (step four) leaves the lower $a.alloc$ digits of the mp\_int intact. This is much
917
akin to how the \textit{realloc} function from the standard C library works. Since the newly allocated digits are
918
assumed to contain undefined values they are initially set to zero.
920
\vspace{+3mm}\begin{small}
921
\hspace{-5.1mm}{\bf File}: bn\_mp\_grow.c
927
A quick optimization is to first determine if a memory re-allocation is required at all. The if statement (line 24) checks
928
if the \textbf{alloc} member of the mp\_int is smaller than the requested digit count. If the count is not larger than \textbf{alloc}
929
the function skips the re-allocation part thus saving time.
931
When a re-allocation is performed it is turned into an optimal request to save time in the future. The requested digit count is
932
padded upwards to 2nd multiple of \textbf{MP\_PREC} larger than \textbf{alloc} (line 25). The XREALLOC function is used
933
to re-allocate the memory. As per the other functions XREALLOC is actually a macro which evaluates to realloc by default. The realloc
934
function leaves the base of the allocation intact which means the first \textbf{alloc} digits of the mp\_int are the same as before
935
the re-allocation. All that is left is to clear the newly allocated digits and return.
937
Note that the re-allocation result is actually stored in a temporary pointer $tmp$. This is to allow this function to return
938
an error with a valid pointer. Earlier releases of the library stored the result of XREALLOC into the mp\_int $a$. That would
939
result in a memory leak if XREALLOC ever failed.
941
\subsection{Initializing Variable Precision mp\_ints}
942
Occasionally the number of digits required will be known in advance of an initialization, based on, for example, the size
943
of input mp\_ints to a given algorithm. The purpose of algorithm mp\_init\_size is similar to mp\_init except that it
944
will allocate \textit{at least} a specified number of digits.
950
\hline Algorithm \textbf{mp\_init\_size}. \\
951
\textbf{Input}. An mp\_int $a$ and the requested number of digits $b$. \\
952
\textbf{Output}. $a$ is initialized to hold at least $b$ digits. \\
954
1. $u \leftarrow b \mbox{ (mod }MP\_PREC\mbox{)}$ \\
955
2. $v \leftarrow b + 2 \cdot MP\_PREC - u$ \\
956
3. Allocate $v$ digits. \\
957
4. for $n$ from $0$ to $v - 1$ do \\
958
\hspace{3mm}4.1 $a_n \leftarrow 0$ \\
959
5. $a.sign \leftarrow MP\_ZPOS$\\
960
6. $a.used \leftarrow 0$\\
961
7. $a.alloc \leftarrow v$\\
962
8. Return(\textit{MP\_OKAY})\\
967
\caption{Algorithm mp\_init\_size}
970
\textbf{Algorithm mp\_init\_size.}
971
This algorithm will initialize an mp\_int structure $a$ like algorithm mp\_init with the exception that the number of
972
digits allocated can be controlled by the second input argument $b$. The input size is padded upwards so it is a
973
multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} digits. This padding is used to prevent trivial
974
allocations from becoming a bottleneck in the rest of the algorithms.
976
Like algorithm mp\_init, the mp\_int structure is initialized to a default state representing the integer zero. This
977
particular algorithm is useful if it is known ahead of time the approximate size of the input. If the approximation is
978
correct no further memory re-allocations are required to work with the mp\_int.
980
\vspace{+3mm}\begin{small}
981
\hspace{-5.1mm}{\bf File}: bn\_mp\_init\_size.c
987
The number of digits $b$ requested is padded (line 24) by first augmenting it to the next multiple of
988
\textbf{MP\_PREC} and then adding \textbf{MP\_PREC} to the result. If the memory can be successfully allocated the
989
mp\_int is placed in a default state representing the integer zero. Otherwise, the error code \textbf{MP\_MEM} will be
992
The digits are allocated and set to zero at the same time with the calloc() function (line @25,XCALLOC@). The
993
\textbf{used} count is set to zero, the \textbf{alloc} count set to the padded digit count and the \textbf{sign} flag set
994
to \textbf{MP\_ZPOS} to achieve a default valid mp\_int state (lines 33, 34 and 35). If the function
995
returns succesfully then it is correct to assume that the mp\_int structure is in a valid state for the remainder of the
996
functions to work with.
998
\subsection{Multiple Integer Initializations and Clearings}
999
Occasionally a function will require a series of mp\_int data types to be made available simultaneously.
1000
The purpose of algorithm mp\_init\_multi is to initialize a variable length array of mp\_int structures in a single
1001
statement. It is essentially a shortcut to multiple initializations.
1003
\newpage\begin{figure}[here]
1006
\hline Algorithm \textbf{mp\_init\_multi}. \\
1007
\textbf{Input}. Variable length array $V_k$ of mp\_int variables of length $k$. \\
1008
\textbf{Output}. The array is initialized such that each mp\_int of $V_k$ is ready to use. \\
1010
1. for $n$ from 0 to $k - 1$ do \\
1011
\hspace{+3mm}1.1. Initialize the mp\_int $V_n$ (\textit{mp\_init}) \\
1012
\hspace{+3mm}1.2. If initialization failed then do \\
1013
\hspace{+6mm}1.2.1. for $j$ from $0$ to $n$ do \\
1014
\hspace{+9mm}1.2.1.1. Free the mp\_int $V_j$ (\textit{mp\_clear}) \\
1015
\hspace{+6mm}1.2.2. Return(\textit{MP\_MEM}) \\
1016
2. Return(\textit{MP\_OKAY}) \\
1020
\caption{Algorithm mp\_init\_multi}
1023
\textbf{Algorithm mp\_init\_multi.}
1024
The algorithm will initialize the array of mp\_int variables one at a time. If a runtime error has been detected
1025
(\textit{step 1.2}) all of the previously initialized variables are cleared. The goal is an ``all or nothing''
1026
initialization which allows for quick recovery from runtime errors.
1028
\vspace{+3mm}\begin{small}
1029
\hspace{-5.1mm}{\bf File}: bn\_mp\_init\_multi.c
1035
This function intializes a variable length list of mp\_int structure pointers. However, instead of having the mp\_int
1036
structures in an actual C array they are simply passed as arguments to the function. This function makes use of the
1037
``...'' argument syntax of the C programming language. The list is terminated with a final \textbf{NULL} argument
1038
appended on the right.
1040
The function uses the ``stdarg.h'' \textit{va} functions to step portably through the arguments to the function. A count
1041
$n$ of succesfully initialized mp\_int structures is maintained (line 48) such that if a failure does occur,
1042
the algorithm can backtrack and free the previously initialized structures (lines 28 to 47).
1045
\subsection{Clamping Excess Digits}
1046
When a function anticipates a result will be $n$ digits it is simpler to assume this is true within the body of
1047
the function instead of checking during the computation. For example, a multiplication of a $i$ digit number by a
1048
$j$ digit produces a result of at most $i + j$ digits. It is entirely possible that the result is $i + j - 1$
1049
though, with no final carry into the last position. However, suppose the destination had to be first expanded
1050
(\textit{via mp\_grow}) to accomodate $i + j - 1$ digits than further expanded to accomodate the final carry.
1051
That would be a considerable waste of time since heap operations are relatively slow.
1053
The ideal solution is to always assume the result is $i + j$ and fix up the \textbf{used} count after the function
1054
terminates. This way a single heap operation (\textit{at most}) is required. However, if the result was not checked
1055
there would be an excess high order zero digit.
1057
For example, suppose the product of two integers was $x_n = (0x_{n-1}x_{n-2}...x_0)_{\beta}$. The leading zero digit
1058
will not contribute to the precision of the result. In fact, through subsequent operations more leading zero digits would
1059
accumulate to the point the size of the integer would be prohibitive. As a result even though the precision is very
1060
low the representation is excessively large.
1062
The mp\_clamp algorithm is designed to solve this very problem. It will trim high-order zeros by decrementing the
1063
\textbf{used} count until a non-zero most significant digit is found. Also in this system, zero is considered to be a
1064
positive number which means that if the \textbf{used} count is decremented to zero, the sign must be set to
1067
\begin{figure}[here]
1070
\hline Algorithm \textbf{mp\_clamp}. \\
1071
\textbf{Input}. An mp\_int $a$ \\
1072
\textbf{Output}. Any excess leading zero digits of $a$ are removed \\
1074
1. while $a.used > 0$ and $a_{a.used - 1} = 0$ do \\
1075
\hspace{+3mm}1.1 $a.used \leftarrow a.used - 1$ \\
1076
2. if $a.used = 0$ then do \\
1077
\hspace{+3mm}2.1 $a.sign \leftarrow MP\_ZPOS$ \\
1081
\caption{Algorithm mp\_clamp}
1084
\textbf{Algorithm mp\_clamp.}
1085
As can be expected this algorithm is very simple. The loop on step one is expected to iterate only once or twice at
1086
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
1087
when all of the digits are zero to ensure that the mp\_int is valid at all times.
1089
\vspace{+3mm}\begin{small}
1090
\hspace{-5.1mm}{\bf File}: bn\_mp\_clamp.c
1096
Note on line 28 how to test for the \textbf{used} count is made on the left of the \&\& operator. In the C programming
1097
language the terms to \&\& are evaluated left to right with a boolean short-circuit if any condition fails. This is
1098
important since if the \textbf{used} is zero the test on the right would fetch below the array. That is obviously
1099
undesirable. The parenthesis on line 31 is used to make sure the \textbf{used} count is decremented and not
1102
\section*{Exercises}
1104
$\left [ 1 \right ]$ & Discuss the relevance of the \textbf{used} member of the mp\_int structure. \\
1106
$\left [ 1 \right ]$ & Discuss the consequences of not using padding when performing allocations. \\
1108
$\left [ 2 \right ]$ & Estimate an ideal value for \textbf{MP\_PREC} when performing 1024-bit RSA \\
1109
& encryption when $\beta = 2^{28}$. \\
1111
$\left [ 1 \right ]$ & Discuss the relevance of the algorithm mp\_clamp. What does it prevent? \\
1113
$\left [ 1 \right ]$ & Give an example of when the algorithm mp\_init\_copy might be useful. \\
1122
\chapter{Basic Operations}
1124
\section{Introduction}
1125
In the previous chapter a series of low level algorithms were established that dealt with initializing and maintaining
1126
mp\_int structures. This chapter will discuss another set of seemingly non-algebraic algorithms which will form the low
1127
level basis of the entire library. While these algorithm are relatively trivial it is important to understand how they
1128
work before proceeding since these algorithms will be used almost intrinsically in the following chapters.
1130
The algorithms in this chapter deal primarily with more ``programmer'' related tasks such as creating copies of
1131
mp\_int structures, assigning small values to mp\_int structures and comparisons of the values mp\_int structures
1134
\section{Assigning Values to mp\_int Structures}
1135
\subsection{Copying an mp\_int}
1136
Assigning the value that a given mp\_int structure represents to another mp\_int structure shall be known as making
1137
a copy for the purposes of this text. The copy of the mp\_int will be a separate entity that represents the same
1138
value as the mp\_int it was copied from. The mp\_copy algorithm provides this functionality.
1140
\newpage\begin{figure}[here]
1143
\hline Algorithm \textbf{mp\_copy}. \\
1144
\textbf{Input}. An mp\_int $a$ and $b$. \\
1145
\textbf{Output}. Store a copy of $a$ in $b$. \\
1147
1. If $b.alloc < a.used$ then grow $b$ to $a.used$ digits. (\textit{mp\_grow}) \\
1148
2. for $n$ from 0 to $a.used - 1$ do \\
1149
\hspace{3mm}2.1 $b_{n} \leftarrow a_{n}$ \\
1150
3. for $n$ from $a.used$ to $b.used - 1$ do \\
1151
\hspace{3mm}3.1 $b_{n} \leftarrow 0$ \\
1152
4. $b.used \leftarrow a.used$ \\
1153
5. $b.sign \leftarrow a.sign$ \\
1154
6. return(\textit{MP\_OKAY}) \\
1158
\caption{Algorithm mp\_copy}
1161
\textbf{Algorithm mp\_copy.}
1162
This algorithm copies the mp\_int $a$ such that upon succesful termination of the algorithm the mp\_int $b$ will
1163
represent the same integer as the mp\_int $a$. The mp\_int $b$ shall be a complete and distinct copy of the
1164
mp\_int $a$ meaing that the mp\_int $a$ can be modified and it shall not affect the value of the mp\_int $b$.
1166
If $b$ does not have enough room for the digits of $a$ it must first have its precision augmented via the mp\_grow
1167
algorithm. The digits of $a$ are copied over the digits of $b$ and any excess digits of $b$ are set to zero (step two
1168
and three). The \textbf{used} and \textbf{sign} members of $a$ are finally copied over the respective members of
1171
\textbf{Remark.} This algorithm also introduces a new idiosyncrasy that will be used throughout the rest of the
1172
text. The error return codes of other algorithms are not explicitly checked in the pseudo-code presented. For example, in
1173
step one of the mp\_copy algorithm the return of mp\_grow is not explicitly checked to ensure it succeeded. Text space is
1174
limited so it is assumed that if a algorithm fails it will clear all temporarily allocated mp\_ints and return
1175
the error code itself. However, the C code presented will demonstrate all of the error handling logic required to
1176
implement the pseudo-code.
1178
\vspace{+3mm}\begin{small}
1179
\hspace{-5.1mm}{\bf File}: bn\_mp\_copy.c
1185
Occasionally a dependent algorithm may copy an mp\_int effectively into itself such as when the input and output
1186
mp\_int structures passed to a function are one and the same. For this case it is optimal to return immediately without
1187
copying digits (line 25).
1189
The mp\_int $b$ must have enough digits to accomodate the used digits of the mp\_int $a$. If $b.alloc$ is less than
1190
$a.used$ the algorithm mp\_grow is used to augment the precision of $b$ (lines 30 to 33). In order to
1191
simplify the inner loop that copies the digits from $a$ to $b$, two aliases $tmpa$ and $tmpb$ point directly at the digits
1192
of the mp\_ints $a$ and $b$ respectively. These aliases (lines 43 and 46) allow the compiler to access the digits without first dereferencing the
1193
mp\_int pointers and then subsequently the pointer to the digits.
1195
After the aliases are established the digits from $a$ are copied into $b$ (lines 49 to 51) and then the excess
1196
digits of $b$ are set to zero (lines 54 to 56). Both ``for'' loops make use of the pointer aliases and in
1197
fact the alias for $b$ is carried through into the second ``for'' loop to clear the excess digits. This optimization
1198
allows the alias to stay in a machine register fairly easy between the two loops.
1200
\textbf{Remarks.} The use of pointer aliases is an implementation methodology first introduced in this function that will
1201
be used considerably in other functions. Technically, a pointer alias is simply a short hand alias used to lower the
1202
number of pointer dereferencing operations required to access data. For example, a for loop may resemble
1205
for (x = 0; x < 100; x++) \{
1206
a->num[4]->dp[x] = 0;
1210
This could be re-written using aliases as
1215
for (x = 0; x < 100; x++) \{
1220
In this case an alias is used to access the
1221
array of digits within an mp\_int structure directly. It may seem that a pointer alias is strictly not required
1222
as a compiler may optimize out the redundant pointer operations. However, there are two dominant reasons to use aliases.
1224
The first reason is that most compilers will not effectively optimize pointer arithmetic. For example, some optimizations
1225
may work for the Microsoft Visual C++ compiler (MSVC) and not for the GNU C Compiler (GCC). Also some optimizations may
1226
work for GCC and not MSVC. As such it is ideal to find a common ground for as many compilers as possible. Pointer
1227
aliases optimize the code considerably before the compiler even reads the source code which means the end compiled code
1228
stands a better chance of being faster.
1230
The second reason is that pointer aliases often can make an algorithm simpler to read. Consider the first ``for''
1231
loop of the function mp\_copy() re-written to not use pointer aliases.
1234
/* copy all the digits */
1235
for (n = 0; n < a->used; n++) \{
1236
b->dp[n] = a->dp[n];
1240
Whether this code is harder to read depends strongly on the individual. However, it is quantifiably slightly more
1241
complicated as there are four variables within the statement instead of just two.
1243
\subsubsection{Nested Statements}
1244
Another commonly used technique in the source routines is that certain sections of code are nested. This is used in
1245
particular with the pointer aliases to highlight code phases. For example, a Comba multiplier (discussed in chapter six)
1246
will typically have three different phases. First the temporaries are initialized, then the columns calculated and
1247
finally the carries are propagated. In this example the middle column production phase will typically be nested as it
1248
uses temporary variables and aliases the most.
1250
The nesting also simplies the source code as variables that are nested are only valid for their scope. As a result
1251
the various temporary variables required do not propagate into other sections of code.
1254
\subsection{Creating a Clone}
1255
Another common operation is to make a local temporary copy of an mp\_int argument. To initialize an mp\_int
1256
and then copy another existing mp\_int into the newly intialized mp\_int will be known as creating a clone. This is
1257
useful within functions that need to modify an argument but do not wish to actually modify the original copy. The
1258
mp\_init\_copy algorithm has been designed to help perform this task.
1260
\begin{figure}[here]
1263
\hline Algorithm \textbf{mp\_init\_copy}. \\
1264
\textbf{Input}. An mp\_int $a$ and $b$\\
1265
\textbf{Output}. $a$ is initialized to be a copy of $b$. \\
1267
1. Init $a$. (\textit{mp\_init}) \\
1268
2. Copy $b$ to $a$. (\textit{mp\_copy}) \\
1269
3. Return the status of the copy operation. \\
1273
\caption{Algorithm mp\_init\_copy}
1276
\textbf{Algorithm mp\_init\_copy.}
1277
This algorithm will initialize an mp\_int variable and copy another previously initialized mp\_int variable into it. As
1278
such this algorithm will perform two operations in one step.
1280
\vspace{+3mm}\begin{small}
1281
\hspace{-5.1mm}{\bf File}: bn\_mp\_init\_copy.c
1287
This will initialize \textbf{a} and make it a verbatim copy of the contents of \textbf{b}. Note that
1288
\textbf{a} will have its own memory allocated which means that \textbf{b} may be cleared after the call
1289
and \textbf{a} will be left intact.
1291
\section{Zeroing an Integer}
1292
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
1295
\begin{figure}[here]
1298
\hline Algorithm \textbf{mp\_zero}. \\
1299
\textbf{Input}. An mp\_int $a$ \\
1300
\textbf{Output}. Zero the contents of $a$ \\
1302
1. $a.used \leftarrow 0$ \\
1303
2. $a.sign \leftarrow$ MP\_ZPOS \\
1304
3. for $n$ from 0 to $a.alloc - 1$ do \\
1305
\hspace{3mm}3.1 $a_n \leftarrow 0$ \\
1309
\caption{Algorithm mp\_zero}
1312
\textbf{Algorithm mp\_zero.}
1313
This algorithm simply resets a mp\_int to the default state.
1315
\vspace{+3mm}\begin{small}
1316
\hspace{-5.1mm}{\bf File}: bn\_mp\_zero.c
1322
After the function is completed, all of the digits are zeroed, the \textbf{used} count is zeroed and the
1323
\textbf{sign} variable is set to \textbf{MP\_ZPOS}.
1325
\section{Sign Manipulation}
1326
\subsection{Absolute Value}
1327
With the mp\_int representation of an integer, calculating the absolute value is trivial. The mp\_abs algorithm will compute
1328
the absolute value of an mp\_int.
1330
\begin{figure}[here]
1333
\hline Algorithm \textbf{mp\_abs}. \\
1334
\textbf{Input}. An mp\_int $a$ \\
1335
\textbf{Output}. Computes $b = \vert a \vert$ \\
1337
1. Copy $a$ to $b$. (\textit{mp\_copy}) \\
1338
2. If the copy failed return(\textit{MP\_MEM}). \\
1339
3. $b.sign \leftarrow MP\_ZPOS$ \\
1340
4. Return(\textit{MP\_OKAY}) \\
1344
\caption{Algorithm mp\_abs}
1347
\textbf{Algorithm mp\_abs.}
1348
This algorithm computes the absolute of an mp\_int input. First it copies $a$ over $b$. This is an example of an
1349
algorithm where the check in mp\_copy that determines if the source and destination are equal proves useful. This allows,
1350
for instance, the developer to pass the same mp\_int as the source and destination to this function without addition
1353
\vspace{+3mm}\begin{small}
1354
\hspace{-5.1mm}{\bf File}: bn\_mp\_abs.c
1360
This fairly trivial algorithm first eliminates non--required duplications (line 28) and then sets the
1361
\textbf{sign} flag to \textbf{MP\_ZPOS}.
1363
\subsection{Integer Negation}
1364
With the mp\_int representation of an integer, calculating the negation is also trivial. The mp\_neg algorithm will compute
1365
the negative of an mp\_int input.
1367
\begin{figure}[here]
1370
\hline Algorithm \textbf{mp\_neg}. \\
1371
\textbf{Input}. An mp\_int $a$ \\
1372
\textbf{Output}. Computes $b = -a$ \\
1374
1. Copy $a$ to $b$. (\textit{mp\_copy}) \\
1375
2. If the copy failed return(\textit{MP\_MEM}). \\
1376
3. If $a.used = 0$ then return(\textit{MP\_OKAY}). \\
1377
4. If $a.sign = MP\_ZPOS$ then do \\
1378
\hspace{3mm}4.1 $b.sign = MP\_NEG$. \\
1380
\hspace{3mm}5.1 $b.sign = MP\_ZPOS$. \\
1381
6. Return(\textit{MP\_OKAY}) \\
1385
\caption{Algorithm mp\_neg}
1388
\textbf{Algorithm mp\_neg.}
1389
This algorithm computes the negation of an input. First it copies $a$ over $b$. If $a$ has no used digits then
1390
the algorithm returns immediately. Otherwise it flips the sign flag and stores the result in $b$. Note that if
1391
$a$ had no digits then it must be positive by definition. Had step three been omitted then the algorithm would return
1394
\vspace{+3mm}\begin{small}
1395
\hspace{-5.1mm}{\bf File}: bn\_mp\_neg.c
1401
Like mp\_abs() this function avoids non--required duplications (line 22) and then sets the sign. We
1402
have to make sure that only non--zero values get a \textbf{sign} of \textbf{MP\_NEG}. If the mp\_int is zero
1403
than the \textbf{sign} is hard--coded to \textbf{MP\_ZPOS}.
1405
\section{Small Constants}
1406
\subsection{Setting Small Constants}
1407
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.
1409
\newpage\begin{figure}[here]
1412
\hline Algorithm \textbf{mp\_set}. \\
1413
\textbf{Input}. An mp\_int $a$ and a digit $b$ \\
1414
\textbf{Output}. Make $a$ equivalent to $b$ \\
1416
1. Zero $a$ (\textit{mp\_zero}). \\
1417
2. $a_0 \leftarrow b \mbox{ (mod }\beta\mbox{)}$ \\
1418
3. $a.used \leftarrow \left \lbrace \begin{array}{ll}
1419
1 & \mbox{if }a_0 > 0 \\
1420
0 & \mbox{if }a_0 = 0
1421
\end{array} \right .$ \\
1425
\caption{Algorithm mp\_set}
1428
\textbf{Algorithm mp\_set.}
1429
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
1430
single digit is set (\textit{modulo $\beta$}) and the \textbf{used} count is adjusted accordingly.
1432
\vspace{+3mm}\begin{small}
1433
\hspace{-5.1mm}{\bf File}: bn\_mp\_set.c
1439
First we zero (line 21) the mp\_int to make sure that the other members are initialized for a
1440
small positive constant. mp\_zero() ensures that the \textbf{sign} is positive and the \textbf{used} count
1441
is zero. Next we set the digit and reduce it modulo $\beta$ (line 22). After this step we have to
1442
check if the resulting digit is zero or not. If it is not then we set the \textbf{used} count to one, otherwise
1445
We can quickly reduce modulo $\beta$ since it is of the form $2^k$ and a quick binary AND operation with
1446
$2^k - 1$ will perform the same operation.
1448
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
1449
this function should take that into account. Only trivially small constants can be set using this function.
1451
\subsection{Setting Large Constants}
1452
To overcome the limitations of the mp\_set algorithm the mp\_set\_int algorithm is ideal. It accepts a ``long''
1453
data type as input and will always treat it as a 32-bit integer.
1455
\begin{figure}[here]
1458
\hline Algorithm \textbf{mp\_set\_int}. \\
1459
\textbf{Input}. An mp\_int $a$ and a ``long'' integer $b$ \\
1460
\textbf{Output}. Make $a$ equivalent to $b$ \\
1462
1. Zero $a$ (\textit{mp\_zero}) \\
1463
2. for $n$ from 0 to 7 do \\
1464
\hspace{3mm}2.1 $a \leftarrow a \cdot 16$ (\textit{mp\_mul2d}) \\
1465
\hspace{3mm}2.2 $u \leftarrow \lfloor b / 2^{4(7 - n)} \rfloor \mbox{ (mod }16\mbox{)}$\\
1466
\hspace{3mm}2.3 $a_0 \leftarrow a_0 + u$ \\
1467
\hspace{3mm}2.4 $a.used \leftarrow a.used + 1$ \\
1468
3. Clamp excess used digits (\textit{mp\_clamp}) \\
1472
\caption{Algorithm mp\_set\_int}
1475
\textbf{Algorithm mp\_set\_int.}
1476
The algorithm performs eight iterations of a simple loop where in each iteration four bits from the source are added to the
1477
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
1478
next four bits from the source are extracted and are added to the mp\_int. The \textbf{used} digit count is
1479
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
1480
zero digits used and the newly added four bits would be ignored.
1482
Excess zero digits are trimmed in steps 2.1 and 3 by using higher level algorithms mp\_mul2d and mp\_clamp.
1484
\vspace{+3mm}\begin{small}
1485
\hspace{-5.1mm}{\bf File}: bn\_mp\_set\_int.c
1491
This function sets four bits of the number at a time to handle all practical \textbf{DIGIT\_BIT} sizes. The weird
1492
addition on line 39 ensures that the newly added in bits are added to the number of digits. While it may not
1493
seem obvious as to why the digit counter does not grow exceedingly large it is because of the shift on line 28
1494
as well as the call to mp\_clamp() on line 41. Both functions will clamp excess leading digits which keeps
1495
the number of used digits low.
1497
\section{Comparisons}
1498
\subsection{Unsigned Comparisions}
1499
Comparing a multiple precision integer is performed with the exact same algorithm used to compare two decimal numbers. For example,
1500
to compare $1,234$ to $1,264$ the digits are extracted by their positions. That is we compare $1 \cdot 10^3 + 2 \cdot 10^2 + 3 \cdot 10^1 + 4 \cdot 10^0$
1501
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
1502
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.
1504
The first comparision routine that will be developed is the unsigned magnitude compare which will perform a comparison based on the digits of two
1505
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
1506
signs are known to agree in advance.
1508
To facilitate working with the results of the comparison functions three constants are required.
1510
\begin{figure}[here]
1512
\begin{tabular}{|r|l|}
1513
\hline \textbf{Constant} & \textbf{Meaning} \\
1514
\hline \textbf{MP\_GT} & Greater Than \\
1515
\hline \textbf{MP\_EQ} & Equal To \\
1516
\hline \textbf{MP\_LT} & Less Than \\
1520
\caption{Comparison Return Codes}
1523
\begin{figure}[here]
1526
\hline Algorithm \textbf{mp\_cmp\_mag}. \\
1527
\textbf{Input}. Two mp\_ints $a$ and $b$. \\
1528
\textbf{Output}. Unsigned comparison results ($a$ to the left of $b$). \\
1530
1. If $a.used > b.used$ then return(\textit{MP\_GT}) \\
1531
2. If $a.used < b.used$ then return(\textit{MP\_LT}) \\
1532
3. for n from $a.used - 1$ to 0 do \\
1533
\hspace{+3mm}3.1 if $a_n > b_n$ then return(\textit{MP\_GT}) \\
1534
\hspace{+3mm}3.2 if $a_n < b_n$ then return(\textit{MP\_LT}) \\
1535
4. Return(\textit{MP\_EQ}) \\
1539
\caption{Algorithm mp\_cmp\_mag}
1542
\textbf{Algorithm mp\_cmp\_mag.}
1543
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
1544
\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 $b$.
1545
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.
1546
If both have the same number of digits than the actual digits themselves must be compared starting at the leading digit.
1548
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
1549
the zero'th digit. If after all of the digits have been compared, no difference is found, the algorithm returns \textbf{MP\_EQ}.
1551
\vspace{+3mm}\begin{small}
1552
\hspace{-5.1mm}{\bf File}: bn\_mp\_cmp\_mag.c
1558
The two if statements (lines 25 and 29) compare the number of digits in the two inputs. These two are
1559
performed before all of the digits are compared since it is a very cheap test to perform and can potentially save
1560
considerable time. The implementation given is also not valid without those two statements. $b.alloc$ may be
1561
smaller than $a.used$, meaning that undefined values will be read from $b$ past the end of the array of digits.
1565
\subsection{Signed Comparisons}
1566
Comparing with sign considerations is also fairly critical in several routines (\textit{division for example}). Based on an unsigned magnitude
1567
comparison a trivial signed comparison algorithm can be written.
1569
\begin{figure}[here]
1572
\hline Algorithm \textbf{mp\_cmp}. \\
1573
\textbf{Input}. Two mp\_ints $a$ and $b$ \\
1574
\textbf{Output}. Signed Comparison Results ($a$ to the left of $b$) \\
1576
1. if $a.sign = MP\_NEG$ and $b.sign = MP\_ZPOS$ then return(\textit{MP\_LT}) \\
1577
2. if $a.sign = MP\_ZPOS$ and $b.sign = MP\_NEG$ then return(\textit{MP\_GT}) \\
1578
3. if $a.sign = MP\_NEG$ then \\
1579
\hspace{+3mm}3.1 Return the unsigned comparison of $b$ and $a$ (\textit{mp\_cmp\_mag}) \\
1581
\hspace{+3mm}4.1 Return the unsigned comparison of $a$ and $b$ \\
1585
\caption{Algorithm mp\_cmp}
1588
\textbf{Algorithm mp\_cmp.}
1589
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
1590
comparison code. When the signs are equal the digits of the inputs must be compared to determine the correct result. In step
1591
three the unsigned comparision flips the order of the arguments since they are both negative. For instance, if $-a > -b$ then
1592
$\vert a \vert < \vert b \vert$. Step number four will compare the two when they are both positive.
1594
\vspace{+3mm}\begin{small}
1595
\hspace{-5.1mm}{\bf File}: bn\_mp\_cmp.c
1601
The two if statements (lines 23 and 24) perform the initial sign comparison. If the signs are not the equal then which ever
1602
has the positive sign is larger. The inputs are compared (line 32) based on magnitudes. If the signs were both
1603
negative then the unsigned comparison is performed in the opposite direction (line 34). Otherwise, the signs are assumed to
1604
be both positive and a forward direction unsigned comparison is performed.
1606
\section*{Exercises}
1608
$\left [ 2 \right ]$ & Modify algorithm mp\_set\_int to accept as input a variable length array of bits. \\
1610
$\left [ 3 \right ]$ & Give the probability that algorithm mp\_cmp\_mag will have to compare $k$ digits \\
1611
& of two random digits (of equal magnitude) before a difference is found. \\
1613
$\left [ 1 \right ]$ & Suggest a simple method to speed up the implementation of mp\_cmp\_mag based \\
1614
& on the observations made in the previous problem. \\
1618
\chapter{Basic Arithmetic}
1619
\section{Introduction}
1620
At this point algorithms for initialization, clearing, zeroing, copying, comparing and setting small constants have been
1621
established. The next logical set of algorithms to develop are addition, subtraction and digit shifting algorithms. These
1622
algorithms make use of the lower level algorithms and are the cruicial building block for the multiplication algorithms. It is very important
1623
that these algorithms are highly optimized. On their own they are simple $O(n)$ algorithms but they can be called from higher level algorithms
1624
which easily places them at $O(n^2)$ or even $O(n^3)$ work levels.
1626
All of the algorithms within this chapter make use of the logical bit shift operations denoted by $<<$ and $>>$ for left and right
1627
logical shifts respectively. A logical shift is analogous to sliding the decimal point of radix-10 representations. For example, the real
1628
number $0.9345$ is equivalent to $93.45\%$ which is found by sliding the the decimal two places to the right (\textit{multiplying by $\beta^2 = 10^2$}).
1629
Algebraically a binary logical shift is equivalent to a division or multiplication by a power of two.
1630
For example, $a << k = a \cdot 2^k$ while $a >> k = \lfloor a/2^k \rfloor$.
1632
One significant difference between a logical shift and the way decimals are shifted is that digits below the zero'th position are removed
1633
from the number. For example, consider $1101_2 >> 1$ using decimal notation this would produce $110.1_2$. However, with a logical shift the
1636
\section{Addition and Subtraction}
1637
In common twos complement fixed precision arithmetic negative numbers are easily represented by subtraction from the modulus. For example, with 32-bit integers
1638
$a - b\mbox{ (mod }2^{32}\mbox{)}$ is the same as $a + (2^{32} - b) \mbox{ (mod }2^{32}\mbox{)}$ since $2^{32} \equiv 0 \mbox{ (mod }2^{32}\mbox{)}$.
1639
As a result subtraction can be performed with a trivial series of logical operations and an addition.
1641
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
1642
sign of the integer. As a result signed addition and subtraction are actually implemented as conditional usage of lower level addition or
1643
subtraction algorithms with the sign fixed up appropriately.
1645
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
1646
the integers respectively.
1648
\subsection{Low Level Addition}
1649
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
1650
trailing digits first and propagate the resulting carry upwards. Since this is a lower level algorithm the name will have a ``s\_'' prefix.
1651
Historically that convention stems from the MPI library where ``s\_'' stood for static functions that were hidden from the developer entirely.
1654
\begin{figure}[!here]
1658
\hline Algorithm \textbf{s\_mp\_add}. \\
1659
\textbf{Input}. Two mp\_ints $a$ and $b$ \\
1660
\textbf{Output}. The unsigned addition $c = \vert a \vert + \vert b \vert$. \\
1662
1. if $a.used > b.used$ then \\
1663
\hspace{+3mm}1.1 $min \leftarrow b.used$ \\
1664
\hspace{+3mm}1.2 $max \leftarrow a.used$ \\
1665
\hspace{+3mm}1.3 $x \leftarrow a$ \\
1667
\hspace{+3mm}2.1 $min \leftarrow a.used$ \\
1668
\hspace{+3mm}2.2 $max \leftarrow b.used$ \\
1669
\hspace{+3mm}2.3 $x \leftarrow b$ \\
1670
3. If $c.alloc < max + 1$ then grow $c$ to hold at least $max + 1$ digits (\textit{mp\_grow}) \\
1671
4. $oldused \leftarrow c.used$ \\
1672
5. $c.used \leftarrow max + 1$ \\
1673
6. $u \leftarrow 0$ \\
1674
7. for $n$ from $0$ to $min - 1$ do \\
1675
\hspace{+3mm}7.1 $c_n \leftarrow a_n + b_n + u$ \\
1676
\hspace{+3mm}7.2 $u \leftarrow c_n >> lg(\beta)$ \\
1677
\hspace{+3mm}7.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\
1678
8. if $min \ne max$ then do \\
1679
\hspace{+3mm}8.1 for $n$ from $min$ to $max - 1$ do \\
1680
\hspace{+6mm}8.1.1 $c_n \leftarrow x_n + u$ \\
1681
\hspace{+6mm}8.1.2 $u \leftarrow c_n >> lg(\beta)$ \\
1682
\hspace{+6mm}8.1.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\
1683
9. $c_{max} \leftarrow u$ \\
1684
10. if $olduse > max$ then \\
1685
\hspace{+3mm}10.1 for $n$ from $max + 1$ to $oldused - 1$ do \\
1686
\hspace{+6mm}10.1.1 $c_n \leftarrow 0$ \\
1687
11. Clamp excess digits in $c$. (\textit{mp\_clamp}) \\
1688
12. Return(\textit{MP\_OKAY}) \\
1693
\caption{Algorithm s\_mp\_add}
1696
\textbf{Algorithm s\_mp\_add.}
1697
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.
1698
Coincidentally the description of algorithm A in Knuth \cite[pp. 266]{TAOCPV2} shares the same deficiency as the algorithm from \cite{HAC}. Even the
1699
MIX pseudo machine code presented by Knuth \cite[pp. 266-267]{TAOCPV2} is incapable of handling inputs which are of different magnitudes.
1701
The first thing that has to be accomplished is to sort out which of the two inputs is the largest. The addition logic
1702
will simply add all of the smallest input to the largest input and store that first part of the result in the
1703
destination. Then it will apply a simpler addition loop to excess digits of the larger input.
1705
The first two steps will handle sorting the inputs such that $min$ and $max$ hold the digit counts of the two
1706
inputs. The variable $x$ will be an mp\_int alias for the largest input or the second input $b$ if they have the
1707
same number of digits. After the inputs are sorted the destination $c$ is grown as required to accomodate the sum
1708
of the two inputs. The original \textbf{used} count of $c$ is copied and set to the new used count.
1710
At this point the first addition loop will go through as many digit positions that both inputs have. The carry
1711
variable $\mu$ is set to zero outside the loop. Inside the loop an ``addition'' step requires three statements to produce
1712
one digit of the summand. First
1713
two digits from $a$ and $b$ are added together along with the carry $\mu$. The carry of this step is extracted and stored
1714
in $\mu$ and finally the digit of the result $c_n$ is truncated within the range $0 \le c_n < \beta$.
1716
Now all of the digit positions that both inputs have in common have been exhausted. If $min \ne max$ then $x$ is an alias
1717
for one of the inputs that has more digits. A simplified addition loop is then used to essentially copy the remaining digits
1718
and the carry to the destination.
1720
The final carry is stored in $c_{max}$ and digits above $max$ upto $oldused$ are zeroed which completes the addition.
1723
\vspace{+3mm}\begin{small}
1724
\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_add.c
1730
We first sort (lines 28 to 36) the inputs based on magnitude and determine the $min$ and $max$ variables.
1731
Note that $x$ is a pointer to an mp\_int assigned to the largest input, in effect it is a local alias. Next we
1732
grow the destination (38 to 42) ensure that it can accomodate the result of the addition.
1734
Similar to the implementation of mp\_copy this function uses the braced code and local aliases coding style. The three aliases that are on
1735
lines 56, 59 and 62 represent the two inputs and destination variables respectively. These aliases are used to ensure the
1736
compiler does not have to dereference $a$, $b$ or $c$ (respectively) to access the digits of the respective mp\_int.
1738
The initial carry $u$ will be cleared (line 65), note that $u$ is of type mp\_digit which ensures type
1739
compatibility within the implementation. The initial addition (line 66 to 75) adds digits from
1740
both inputs until the smallest input runs out of digits. Similarly the conditional addition loop
1741
(line 81 to 90) adds the remaining digits from the larger of the two inputs. The addition is finished
1742
with the final carry being stored in $tmpc$ (line 94). Note the ``++'' operator within the same expression.
1743
After line 94, $tmpc$ will point to the $c.used$'th digit of the mp\_int $c$. This is useful
1744
for the next loop (line 97 to 99) which set any old upper digits to zero.
1746
\subsection{Low Level Subtraction}
1747
The low level unsigned subtraction algorithm is very similar to the low level unsigned addition algorithm. The principle difference is that the
1748
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
1749
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.
1750
This algorithm as will be shown can be used to create functional signed addition and subtraction algorithms.
1753
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
1754
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
1755
this algorithm we will assume that the variable $\gamma$ represents the number of bits available in a
1756
mp\_digit (\textit{this implies $2^{\gamma} > \beta$}).
1758
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''
1759
data type must be able to represent $0 \le x < 2^{32}$ meaning that in this case $\gamma \ge 32$.
1761
\newpage\begin{figure}[!here]
1765
\hline Algorithm \textbf{s\_mp\_sub}. \\
1766
\textbf{Input}. Two mp\_ints $a$ and $b$ ($\vert a \vert \ge \vert b \vert$) \\
1767
\textbf{Output}. The unsigned subtraction $c = \vert a \vert - \vert b \vert$. \\
1769
1. $min \leftarrow b.used$ \\
1770
2. $max \leftarrow a.used$ \\
1771
3. If $c.alloc < max$ then grow $c$ to hold at least $max$ digits. (\textit{mp\_grow}) \\
1772
4. $oldused \leftarrow c.used$ \\
1773
5. $c.used \leftarrow max$ \\
1774
6. $u \leftarrow 0$ \\
1775
7. for $n$ from $0$ to $min - 1$ do \\
1776
\hspace{3mm}7.1 $c_n \leftarrow a_n - b_n - u$ \\
1777
\hspace{3mm}7.2 $u \leftarrow c_n >> (\gamma - 1)$ \\
1778
\hspace{3mm}7.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\
1779
8. if $min < max$ then do \\
1780
\hspace{3mm}8.1 for $n$ from $min$ to $max - 1$ do \\
1781
\hspace{6mm}8.1.1 $c_n \leftarrow a_n - u$ \\
1782
\hspace{6mm}8.1.2 $u \leftarrow c_n >> (\gamma - 1)$ \\
1783
\hspace{6mm}8.1.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\
1784
9. if $oldused > max$ then do \\
1785
\hspace{3mm}9.1 for $n$ from $max$ to $oldused - 1$ do \\
1786
\hspace{6mm}9.1.1 $c_n \leftarrow 0$ \\
1787
10. Clamp excess digits of $c$. (\textit{mp\_clamp}). \\
1788
11. Return(\textit{MP\_OKAY}). \\
1793
\caption{Algorithm s\_mp\_sub}
1796
\textbf{Algorithm s\_mp\_sub.}
1797
This algorithm performs the unsigned subtraction of two mp\_int variables under the restriction that the result must be positive. That is when
1798
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
1799
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
1800
of the algorithm s\_mp\_add both other references lack discussion concerning various practical details such as when the inputs differ in magnitude.
1802
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
1803
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
1804
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
1805
set to the maximal count for the operation.
1807
The subtraction loop that begins on step seven is essentially the same as the addition loop of algorithm s\_mp\_add except single precision
1808
subtraction is used instead. Note the use of the $\gamma$ variable to extract the carry (\textit{also known as the borrow}) within the subtraction
1809
loops. Under the assumption that two's complement single precision arithmetic is used this will successfully extract the desired carry.
1811
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
1812
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
1813
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
1814
way to the most significant bit.
1816
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
1817
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
1818
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
1819
carry. This method of carry extraction may seem awkward but the reason for it becomes apparent when the implementation is discussed.
1821
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
1822
10 will ensure that any leading digits of $c$ above the $max$'th position are zeroed.
1824
\vspace{+3mm}\begin{small}
1825
\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_sub.c
1831
Like low level addition we ``sort'' the inputs. Except in this case the sorting is hardcoded
1832
(lines 25 and 26). In reality the $min$ and $max$ variables are only aliases and are only
1833
used to make the source code easier to read. Again the pointer alias optimization is used
1834
within this algorithm. The aliases $tmpa$, $tmpb$ and $tmpc$ are initialized
1835
(lines 42, 43 and 44) for $a$, $b$ and $c$ respectively.
1837
The first subtraction loop (lines 47 through 61) subtract digits from both inputs until the smaller of
1838
the two inputs has been exhausted. As remarked earlier there is an implementation reason for using the ``awkward''
1839
method of extracting the carry (line 57). The traditional method for extracting the carry would be to shift
1840
by $lg(\beta)$ positions and logically AND the least significant bit. The AND operation is required because all of
1841
the bits above the $\lg(\beta)$'th bit will be set to one after a carry occurs from subtraction. This carry
1842
extraction requires two relatively cheap operations to extract the carry. The other method is to simply shift the
1843
most significant bit to the least significant bit thus extracting the carry with a single cheap operation. This
1844
optimization only works on twos compliment machines which is a safe assumption to make.
1846
If $a$ has a larger magnitude than $b$ an additional loop (lines 64 through 73) is required to propagate
1847
the carry through $a$ and copy the result to $c$.
1849
\subsection{High Level Addition}
1850
Now that both lower level addition and subtraction algorithms have been established an effective high level signed addition algorithm can be
1851
established. This high level addition algorithm will be what other algorithms and developers will use to perform addition of mp\_int data
1854
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}
1855
flag. A high level addition is actually performed as a series of eight separate cases which can be optimized down to three unique cases.
1857
\begin{figure}[!here]
1860
\hline Algorithm \textbf{mp\_add}. \\
1861
\textbf{Input}. Two mp\_ints $a$ and $b$ \\
1862
\textbf{Output}. The signed addition $c = a + b$. \\
1864
1. if $a.sign = b.sign$ then do \\
1865
\hspace{3mm}1.1 $c.sign \leftarrow a.sign$ \\
1866
\hspace{3mm}1.2 $c \leftarrow \vert a \vert + \vert b \vert$ (\textit{s\_mp\_add})\\
1868
\hspace{3mm}2.1 if $\vert a \vert < \vert b \vert$ then do (\textit{mp\_cmp\_mag}) \\
1869
\hspace{6mm}2.1.1 $c.sign \leftarrow b.sign$ \\
1870
\hspace{6mm}2.1.2 $c \leftarrow \vert b \vert - \vert a \vert$ (\textit{s\_mp\_sub}) \\
1871
\hspace{3mm}2.2 else do \\
1872
\hspace{6mm}2.2.1 $c.sign \leftarrow a.sign$ \\
1873
\hspace{6mm}2.2.2 $c \leftarrow \vert a \vert - \vert b \vert$ \\
1874
3. Return(\textit{MP\_OKAY}). \\
1878
\caption{Algorithm mp\_add}
1881
\textbf{Algorithm mp\_add.}
1882
This algorithm performs the signed addition of two mp\_int variables. There is no reference algorithm to draw upon from
1883
either \cite{TAOCPV2} or \cite{HAC} since they both only provide unsigned operations. The algorithm is fairly
1884
straightforward but restricted since subtraction can only produce positive results.
1886
\begin{figure}[here]
1889
\begin{tabular}{|c|c|c|c|c|}
1890
\hline \textbf{Sign of $a$} & \textbf{Sign of $b$} & \textbf{$\vert a \vert > \vert b \vert $} & \textbf{Unsigned Operation} & \textbf{Result Sign Flag} \\
1891
\hline $+$ & $+$ & Yes & $c = a + b$ & $a.sign$ \\
1892
\hline $+$ & $+$ & No & $c = a + b$ & $a.sign$ \\
1893
\hline $-$ & $-$ & Yes & $c = a + b$ & $a.sign$ \\
1894
\hline $-$ & $-$ & No & $c = a + b$ & $a.sign$ \\
1897
\hline $+$ & $-$ & No & $c = b - a$ & $b.sign$ \\
1898
\hline $-$ & $+$ & No & $c = b - a$ & $b.sign$ \\
1902
\hline $+$ & $-$ & Yes & $c = a - b$ & $a.sign$ \\
1903
\hline $-$ & $+$ & Yes & $c = a - b$ & $a.sign$ \\
1909
\caption{Addition Guide Chart}
1910
\label{fig:AddChart}
1913
Figure~\ref{fig:AddChart} lists all of the eight possible input combinations and is sorted to show that only three
1914
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
1915
forwarded to step three to check for errors. This simplifies the description of the algorithm considerably and best
1916
follows how the implementation actually was achieved.
1918
Also note how the \textbf{sign} is set before the unsigned addition or subtraction is performed. Recall from the descriptions of algorithms
1919
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}
1920
to \textbf{MP\_ZPOS} when the \textbf{used} digit count reaches zero.
1922
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
1923
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
1924
within algorithm s\_mp\_add will force $-0$ to become $0$.
1926
\vspace{+3mm}\begin{small}
1927
\hspace{-5.1mm}{\bf File}: bn\_mp\_add.c
1933
The source code follows the algorithm fairly closely. The most notable new source code addition is the usage of the $res$ integer variable which
1934
is used to pass result of the unsigned operations forward. Unlike in the algorithm, the variable $res$ is merely returned as is without
1935
explicitly checking it and returning the constant \textbf{MP\_OKAY}. The observation is this algorithm will succeed or fail only if the lower
1936
level functions do so. Returning their return code is sufficient.
1938
\subsection{High Level Subtraction}
1939
The high level signed subtraction algorithm is essentially the same as the high level signed addition algorithm.
1941
\newpage\begin{figure}[!here]
1944
\hline Algorithm \textbf{mp\_sub}. \\
1945
\textbf{Input}. Two mp\_ints $a$ and $b$ \\
1946
\textbf{Output}. The signed subtraction $c = a - b$. \\
1948
1. if $a.sign \ne b.sign$ then do \\
1949
\hspace{3mm}1.1 $c.sign \leftarrow a.sign$ \\
1950
\hspace{3mm}1.2 $c \leftarrow \vert a \vert + \vert b \vert$ (\textit{s\_mp\_add}) \\
1952
\hspace{3mm}2.1 if $\vert a \vert \ge \vert b \vert$ then do (\textit{mp\_cmp\_mag}) \\
1953
\hspace{6mm}2.1.1 $c.sign \leftarrow a.sign$ \\
1954
\hspace{6mm}2.1.2 $c \leftarrow \vert a \vert - \vert b \vert$ (\textit{s\_mp\_sub}) \\
1955
\hspace{3mm}2.2 else do \\
1956
\hspace{6mm}2.2.1 $c.sign \leftarrow \left \lbrace \begin{array}{ll}
1957
MP\_ZPOS & \mbox{if }a.sign = MP\_NEG \\
1958
MP\_NEG & \mbox{otherwise} \\
1959
\end{array} \right .$ \\
1960
\hspace{6mm}2.2.2 $c \leftarrow \vert b \vert - \vert a \vert$ \\
1961
3. Return(\textit{MP\_OKAY}). \\
1965
\caption{Algorithm mp\_sub}
1968
\textbf{Algorithm mp\_sub.}
1969
This algorithm performs the signed subtraction of two inputs. Similar to algorithm mp\_add there is no reference in either \cite{TAOCPV2} or
1970
\cite{HAC}. Also this algorithm is restricted by algorithm s\_mp\_sub. Chart \ref{fig:SubChart} lists the eight possible inputs and
1971
the operations required.
1973
\begin{figure}[!here]
1976
\begin{tabular}{|c|c|c|c|c|}
1977
\hline \textbf{Sign of $a$} & \textbf{Sign of $b$} & \textbf{$\vert a \vert \ge \vert b \vert $} & \textbf{Unsigned Operation} & \textbf{Result Sign Flag} \\
1978
\hline $+$ & $-$ & Yes & $c = a + b$ & $a.sign$ \\
1979
\hline $+$ & $-$ & No & $c = a + b$ & $a.sign$ \\
1980
\hline $-$ & $+$ & Yes & $c = a + b$ & $a.sign$ \\
1981
\hline $-$ & $+$ & No & $c = a + b$ & $a.sign$ \\
1983
\hline $+$ & $+$ & Yes & $c = a - b$ & $a.sign$ \\
1984
\hline $-$ & $-$ & Yes & $c = a - b$ & $a.sign$ \\
1986
\hline $+$ & $+$ & No & $c = b - a$ & $\mbox{opposite of }a.sign$ \\
1987
\hline $-$ & $-$ & No & $c = b - a$ & $\mbox{opposite of }a.sign$ \\
1992
\caption{Subtraction Guide Chart}
1993
\label{fig:SubChart}
1996
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
1997
algorithm from producing $-a - -a = -0$ as a result.
1999
\vspace{+3mm}\begin{small}
2000
\hspace{-5.1mm}{\bf File}: bn\_mp\_sub.c
2006
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
2007
and forward it to the end of the function. On line 39 the ``not equal to'' \textbf{MP\_LT} expression is used to emulate a
2008
``greater than or equal to'' comparison.
2010
\section{Bit and Digit Shifting}
2011
It is quite common to think of a multiple precision integer as a polynomial in $x$, that is $y = f(\beta)$ where $f(x) = \sum_{i=0}^{n-1} a_i x^i$.
2012
This notation arises within discussion of Montgomery and Diminished Radix Reduction as well as Karatsuba multiplication and squaring.
2014
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
2015
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
2016
are on radix-$\beta$ digits.
2018
\subsection{Multiplication by Two}
2020
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
2021
operation to perform. A single precision logical shift left is sufficient to multiply a single digit by two.
2023
\newpage\begin{figure}[!here]
2027
\hline Algorithm \textbf{mp\_mul\_2}. \\
2028
\textbf{Input}. One mp\_int $a$ \\
2029
\textbf{Output}. $b = 2a$. \\
2031
1. If $b.alloc < a.used + 1$ then grow $b$ to hold $a.used + 1$ digits. (\textit{mp\_grow}) \\
2032
2. $oldused \leftarrow b.used$ \\
2033
3. $b.used \leftarrow a.used$ \\
2034
4. $r \leftarrow 0$ \\
2035
5. for $n$ from 0 to $a.used - 1$ do \\
2036
\hspace{3mm}5.1 $rr \leftarrow a_n >> (lg(\beta) - 1)$ \\
2037
\hspace{3mm}5.2 $b_n \leftarrow (a_n << 1) + r \mbox{ (mod }\beta\mbox{)}$ \\
2038
\hspace{3mm}5.3 $r \leftarrow rr$ \\
2039
6. If $r \ne 0$ then do \\
2040
\hspace{3mm}6.1 $b_{n + 1} \leftarrow r$ \\
2041
\hspace{3mm}6.2 $b.used \leftarrow b.used + 1$ \\
2042
7. If $b.used < oldused - 1$ then do \\
2043
\hspace{3mm}7.1 for $n$ from $b.used$ to $oldused - 1$ do \\
2044
\hspace{6mm}7.1.1 $b_n \leftarrow 0$ \\
2045
8. $b.sign \leftarrow a.sign$ \\
2046
9. Return(\textit{MP\_OKAY}).\\
2051
\caption{Algorithm mp\_mul\_2}
2054
\textbf{Algorithm mp\_mul\_2.}
2055
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
2056
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
2057
it is for all intents and purposes equivalent to the operation $b = \vert a \vert + \vert a \vert$.
2059
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
2060
is set to $a.used$ at step 4. Only if there is a final carry will the \textbf{used} count require adjustment.
2062
Step 6 is an optimization implementation of the addition loop for this specific case. That is since the two values being added together
2063
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
2064
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
2065
the previous carry. Recall from section 4.1 that $a_n << 1$ is equivalent to $a_n \cdot 2$. An iteration of the addition loop is finished with
2066
forwarding the carry to the next iteration.
2068
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 $b$.
2069
Step 8 clears any leading digits of $b$ in case it originally had a larger magnitude than $a$.
2071
\vspace{+3mm}\begin{small}
2072
\hspace{-5.1mm}{\bf File}: bn\_mp\_mul\_2.c
2078
This implementation is essentially an optimized implementation of s\_mp\_add for the case of doubling an input. The only noteworthy difference
2079
is the use of the logical shift operator on line 52 to perform a single precision doubling.
2081
\subsection{Division by Two}
2082
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.
2084
\newpage\begin{figure}[!here]
2088
\hline Algorithm \textbf{mp\_div\_2}. \\
2089
\textbf{Input}. One mp\_int $a$ \\
2090
\textbf{Output}. $b = a/2$. \\
2092
1. If $b.alloc < a.used$ then grow $b$ to hold $a.used$ digits. (\textit{mp\_grow}) \\
2093
2. If the reallocation failed return(\textit{MP\_MEM}). \\
2094
3. $oldused \leftarrow b.used$ \\
2095
4. $b.used \leftarrow a.used$ \\
2096
5. $r \leftarrow 0$ \\
2097
6. for $n$ from $b.used - 1$ to $0$ do \\
2098
\hspace{3mm}6.1 $rr \leftarrow a_n \mbox{ (mod }2\mbox{)}$\\
2099
\hspace{3mm}6.2 $b_n \leftarrow (a_n >> 1) + (r << (lg(\beta) - 1)) \mbox{ (mod }\beta\mbox{)}$ \\
2100
\hspace{3mm}6.3 $r \leftarrow rr$ \\
2101
7. If $b.used < oldused - 1$ then do \\
2102
\hspace{3mm}7.1 for $n$ from $b.used$ to $oldused - 1$ do \\
2103
\hspace{6mm}7.1.1 $b_n \leftarrow 0$ \\
2104
8. $b.sign \leftarrow a.sign$ \\
2105
9. Clamp excess digits of $b$. (\textit{mp\_clamp}) \\
2106
10. Return(\textit{MP\_OKAY}).\\
2111
\caption{Algorithm mp\_div\_2}
2114
\textbf{Algorithm mp\_div\_2.}
2115
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
2116
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
2117
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
2118
reading past the end of the array of digits.
2120
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
2121
least significant bit not the most significant bit.
2123
\vspace{+3mm}\begin{small}
2124
\hspace{-5.1mm}{\bf File}: bn\_mp\_div\_2.c
2130
\section{Polynomial Basis Operations}
2131
Recall from section 4.3 that any integer can be represented as a polynomial in $x$ as $y = f(\beta)$. Such a representation is also known as
2132
the polynomial basis \cite[pp. 48]{ROSE}. Given such a notation a multiplication or division by $x$ amounts to shifting whole digits a single
2133
place. The need for such operations arises in several other higher level algorithms such as Barrett and Montgomery reduction, integer
2134
division and Karatsuba multiplication.
2136
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
2137
$y = \sum_{i=0}^{2} a_i \beta^i$. Simply replace $\beta$ with $x$ and the expression is in polynomial basis. For example, $f(x) = 8x + 9$ is the
2138
polynomial basis representation for $89$ using radix ten. That is, $f(10) = 8(10) + 9 = 89$.
2140
\subsection{Multiplication by $x$}
2142
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
2143
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
2144
multiplying by the integer $\beta$.
2146
\newpage\begin{figure}[!here]
2150
\hline Algorithm \textbf{mp\_lshd}. \\
2151
\textbf{Input}. One mp\_int $a$ and an integer $b$ \\
2152
\textbf{Output}. $a \leftarrow a \cdot \beta^b$ (equivalent to multiplication by $x^b$). \\
2154
1. If $b \le 0$ then return(\textit{MP\_OKAY}). \\
2155
2. If $a.alloc < a.used + b$ then grow $a$ to at least $a.used + b$ digits. (\textit{mp\_grow}). \\
2156
3. If the reallocation failed return(\textit{MP\_MEM}). \\
2157
4. $a.used \leftarrow a.used + b$ \\
2158
5. $i \leftarrow a.used - 1$ \\
2159
6. $j \leftarrow a.used - 1 - b$ \\
2160
7. for $n$ from $a.used - 1$ to $b$ do \\
2161
\hspace{3mm}7.1 $a_{i} \leftarrow a_{j}$ \\
2162
\hspace{3mm}7.2 $i \leftarrow i - 1$ \\
2163
\hspace{3mm}7.3 $j \leftarrow j - 1$ \\
2164
8. for $n$ from 0 to $b - 1$ do \\
2165
\hspace{3mm}8.1 $a_n \leftarrow 0$ \\
2166
9. Return(\textit{MP\_OKAY}). \\
2171
\caption{Algorithm mp\_lshd}
2174
\textbf{Algorithm mp\_lshd.}
2175
This algorithm multiplies an mp\_int by the $b$'th power of $x$. This is equivalent to multiplying by $\beta^b$. The algorithm differs
2176
from the other algorithms presented so far as it performs the operation in place instead storing the result in a separate location. The
2177
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
2178
different third mp\_int because the original inputs are often still required. Algorithm mp\_lshd (\textit{and similarly algorithm mp\_rshd}) is
2179
typically used on values where the original value is no longer required. The algorithm will return success immediately if
2180
$b \le 0$ since the rest of algorithm is only valid when $b > 0$.
2182
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
2183
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}).
2184
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
2185
step 8 sets the lower $b$ digits to zero.
2189
\begin{figure}[here]
2190
\includegraphics{pics/sliding_window.ps}
2191
\caption{Sliding Window Movement}
2192
\label{pic:sliding_window}
2196
\vspace{+3mm}\begin{small}
2197
\hspace{-5.1mm}{\bf File}: bn\_mp\_lshd.c
2203
The if statement (line 24) ensures that the $b$ variable is greater than zero since we do not interpret negative
2204
shift counts properly. The \textbf{used} count is incremented by $b$ before the copy loop begins. This elminates
2205
the need for an additional variable in the for loop. The variable $top$ (line 42) is an alias
2206
for the leading digit while $bottom$ (line 45) is an alias for the trailing edge. The aliases form a
2207
window of exactly $b$ digits over the input.
2209
\subsection{Division by $x$}
2211
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.
2213
\newpage\begin{figure}[!here]
2217
\hline Algorithm \textbf{mp\_rshd}. \\
2218
\textbf{Input}. One mp\_int $a$ and an integer $b$ \\
2219
\textbf{Output}. $a \leftarrow a / \beta^b$ (Divide by $x^b$). \\
2221
1. If $b \le 0$ then return. \\
2222
2. If $a.used \le b$ then do \\
2223
\hspace{3mm}2.1 Zero $a$. (\textit{mp\_zero}). \\
2224
\hspace{3mm}2.2 Return. \\
2225
3. $i \leftarrow 0$ \\
2226
4. $j \leftarrow b$ \\
2227
5. for $n$ from 0 to $a.used - b - 1$ do \\
2228
\hspace{3mm}5.1 $a_i \leftarrow a_j$ \\
2229
\hspace{3mm}5.2 $i \leftarrow i + 1$ \\
2230
\hspace{3mm}5.3 $j \leftarrow j + 1$ \\
2231
6. for $n$ from $a.used - b$ to $a.used - 1$ do \\
2232
\hspace{3mm}6.1 $a_n \leftarrow 0$ \\
2233
7. $a.used \leftarrow a.used - b$ \\
2239
\caption{Algorithm mp\_rshd}
2242
\textbf{Algorithm mp\_rshd.}
2243
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
2244
it does not require single precision division. This algorithm does not actually return an error code as it cannot fail.
2246
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
2247
to the shift count $b$ then it will simply zero the input and return.
2249
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
2250
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.
2251
Also the digits are copied from the leading to the trailing edge.
2253
Once the window copy is complete the upper digits must be zeroed and the \textbf{used} count decremented.
2255
\vspace{+3mm}\begin{small}
2256
\hspace{-5.1mm}{\bf File}: bn\_mp\_rshd.c
2262
The only noteworthy element of this routine is the lack of a return type since it cannot fail. Like mp\_lshd() we
2263
form a sliding window except we copy in the other direction. After the window (line 60) we then zero
2264
the upper digits of the input to make sure the result is correct.
2266
\section{Powers of Two}
2268
Now that algorithms for moving single bits as well as whole digits exist algorithms for moving the ``in between'' distances are required. For
2269
example, to quickly multiply by $2^k$ for any $k$ without using a full multiplier algorithm would prove useful. Instead of performing single
2270
shifts $k$ times to achieve a multiplication by $2^{\pm k}$ a mixture of whole digit shifting and partial digit shifting is employed.
2272
\subsection{Multiplication by Power of Two}
2274
\newpage\begin{figure}[!here]
2278
\hline Algorithm \textbf{mp\_mul\_2d}. \\
2279
\textbf{Input}. One mp\_int $a$ and an integer $b$ \\
2280
\textbf{Output}. $c \leftarrow a \cdot 2^b$. \\
2282
1. $c \leftarrow a$. (\textit{mp\_copy}) \\
2283
2. If $c.alloc < c.used + \lfloor b / lg(\beta) \rfloor + 2$ then grow $c$ accordingly. \\
2284
3. If the reallocation failed return(\textit{MP\_MEM}). \\
2285
4. If $b \ge lg(\beta)$ then \\
2286
\hspace{3mm}4.1 $c \leftarrow c \cdot \beta^{\lfloor b / lg(\beta) \rfloor}$ (\textit{mp\_lshd}). \\
2287
\hspace{3mm}4.2 If step 4.1 failed return(\textit{MP\_MEM}). \\
2288
5. $d \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\
2289
6. If $d \ne 0$ then do \\
2290
\hspace{3mm}6.1 $mask \leftarrow 2^d$ \\
2291
\hspace{3mm}6.2 $r \leftarrow 0$ \\
2292
\hspace{3mm}6.3 for $n$ from $0$ to $c.used - 1$ do \\
2293
\hspace{6mm}6.3.1 $rr \leftarrow c_n >> (lg(\beta) - d) \mbox{ (mod }mask\mbox{)}$ \\
2294
\hspace{6mm}6.3.2 $c_n \leftarrow (c_n << d) + r \mbox{ (mod }\beta\mbox{)}$ \\
2295
\hspace{6mm}6.3.3 $r \leftarrow rr$ \\
2296
\hspace{3mm}6.4 If $r > 0$ then do \\
2297
\hspace{6mm}6.4.1 $c_{c.used} \leftarrow r$ \\
2298
\hspace{6mm}6.4.2 $c.used \leftarrow c.used + 1$ \\
2299
7. Return(\textit{MP\_OKAY}). \\
2304
\caption{Algorithm mp\_mul\_2d}
2307
\textbf{Algorithm mp\_mul\_2d.}
2308
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
2309
quickly compute the product.
2311
First the algorithm will multiply $a$ by $x^{\lfloor b / lg(\beta) \rfloor}$ which will ensure that the remainder multiplicand is less than
2312
$\beta$. For example, if $b = 37$ and $\beta = 2^{28}$ then this step will multiply by $x$ leaving a multiplication by $2^{37 - 28} = 2^{9}$
2315
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
2316
required. If it is non-zero a modified shift loop is used to calculate the remaining product.
2317
Essentially the loop is a generic version of algorithm mp\_mul\_2 designed to handle any shift count in the range $1 \le x < lg(\beta)$. The $mask$
2318
variable is used to extract the upper $d$ bits to form the carry for the next iteration.
2320
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
2321
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.
2323
\vspace{+3mm}\begin{small}
2324
\hspace{-5.1mm}{\bf File}: bn\_mp\_mul\_2d.c
2330
The shifting is performed in--place which means the first step (line 25) is to copy the input to the
2331
destination. We avoid calling mp\_copy() by making sure the mp\_ints are different. The destination then
2332
has to be grown (line 32) to accomodate the result.
2334
If the shift count $b$ is larger than $lg(\beta)$ then a call to mp\_lshd() is used to handle all of the multiples
2335
of $lg(\beta)$. Leaving only a remaining shift of $lg(\beta) - 1$ or fewer bits left. Inside the actual shift
2336
loop (lines 46 to 76) we make use of pre--computed values $shift$ and $mask$. These are used to
2337
extract the carry bit(s) to pass into the next iteration of the loop. The $r$ and $rr$ variables form a
2338
chain between consecutive iterations to propagate the carry.
2340
\subsection{Division by Power of Two}
2342
\newpage\begin{figure}[!here]
2346
\hline Algorithm \textbf{mp\_div\_2d}. \\
2347
\textbf{Input}. One mp\_int $a$ and an integer $b$ \\
2348
\textbf{Output}. $c \leftarrow \lfloor a / 2^b \rfloor, d \leftarrow a \mbox{ (mod }2^b\mbox{)}$. \\
2350
1. If $b \le 0$ then do \\
2351
\hspace{3mm}1.1 $c \leftarrow a$ (\textit{mp\_copy}) \\
2352
\hspace{3mm}1.2 $d \leftarrow 0$ (\textit{mp\_zero}) \\
2353
\hspace{3mm}1.3 Return(\textit{MP\_OKAY}). \\
2354
2. $c \leftarrow a$ \\
2355
3. $d \leftarrow a \mbox{ (mod }2^b\mbox{)}$ (\textit{mp\_mod\_2d}) \\
2356
4. If $b \ge lg(\beta)$ then do \\
2357
\hspace{3mm}4.1 $c \leftarrow \lfloor c/\beta^{\lfloor b/lg(\beta) \rfloor} \rfloor$ (\textit{mp\_rshd}). \\
2358
5. $k \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\
2359
6. If $k \ne 0$ then do \\
2360
\hspace{3mm}6.1 $mask \leftarrow 2^k$ \\
2361
\hspace{3mm}6.2 $r \leftarrow 0$ \\
2362
\hspace{3mm}6.3 for $n$ from $c.used - 1$ to $0$ do \\
2363
\hspace{6mm}6.3.1 $rr \leftarrow c_n \mbox{ (mod }mask\mbox{)}$ \\
2364
\hspace{6mm}6.3.2 $c_n \leftarrow (c_n >> k) + (r << (lg(\beta) - k))$ \\
2365
\hspace{6mm}6.3.3 $r \leftarrow rr$ \\
2366
7. Clamp excess digits of $c$. (\textit{mp\_clamp}) \\
2367
8. Return(\textit{MP\_OKAY}). \\
2372
\caption{Algorithm mp\_div\_2d}
2375
\textbf{Algorithm mp\_div\_2d.}
2376
This algorithm will divide an input $a$ by $2^b$ and produce the quotient and remainder. The algorithm is designed much like algorithm
2377
mp\_mul\_2d by first using whole digit shifts then single precision shifts. This algorithm will also produce the remainder of the division
2378
by using algorithm mp\_mod\_2d.
2380
\vspace{+3mm}\begin{small}
2381
\hspace{-5.1mm}{\bf File}: bn\_mp\_div\_2d.c
2387
The implementation of algorithm mp\_div\_2d is slightly different than the algorithm specifies. The remainder $d$ may be optionally
2388
ignored by passing \textbf{NULL} as the pointer to the mp\_int variable. The temporary mp\_int variable $t$ is used to hold the
2389
result of the remainder operation until the end. This allows $d$ and $a$ to represent the same mp\_int without modifying $a$ before
2390
the quotient is obtained.
2392
The remainder of the source code is essentially the same as the source code for mp\_mul\_2d. The only significant difference is
2393
the direction of the shifts.
2395
\subsection{Remainder of Division by Power of Two}
2397
The last algorithm in the series of polynomial basis power of two algorithms is calculating the remainder of division by $2^b$. This
2398
algorithm benefits from the fact that in twos complement arithmetic $a \mbox{ (mod }2^b\mbox{)}$ is the same as $a$ AND $2^b - 1$.
2400
\begin{figure}[!here]
2404
\hline Algorithm \textbf{mp\_mod\_2d}. \\
2405
\textbf{Input}. One mp\_int $a$ and an integer $b$ \\
2406
\textbf{Output}. $c \leftarrow a \mbox{ (mod }2^b\mbox{)}$. \\
2408
1. If $b \le 0$ then do \\
2409
\hspace{3mm}1.1 $c \leftarrow 0$ (\textit{mp\_zero}) \\
2410
\hspace{3mm}1.2 Return(\textit{MP\_OKAY}). \\
2411
2. If $b > a.used \cdot lg(\beta)$ then do \\
2412
\hspace{3mm}2.1 $c \leftarrow a$ (\textit{mp\_copy}) \\
2413
\hspace{3mm}2.2 Return the result of step 2.1. \\
2414
3. $c \leftarrow a$ \\
2415
4. If step 3 failed return(\textit{MP\_MEM}). \\
2416
5. for $n$ from $\lceil b / lg(\beta) \rceil$ to $c.used$ do \\
2417
\hspace{3mm}5.1 $c_n \leftarrow 0$ \\
2418
6. $k \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\
2419
7. $c_{\lfloor b / lg(\beta) \rfloor} \leftarrow c_{\lfloor b / lg(\beta) \rfloor} \mbox{ (mod }2^{k}\mbox{)}$. \\
2420
8. Clamp excess digits of $c$. (\textit{mp\_clamp}) \\
2421
9. Return(\textit{MP\_OKAY}). \\
2426
\caption{Algorithm mp\_mod\_2d}
2429
\textbf{Algorithm mp\_mod\_2d.}
2430
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
2431
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, $a$
2432
is copied to $b$, leading digits are removed and the remaining leading digit is trimed to the exact bit count.
2434
\vspace{+3mm}\begin{small}
2435
\hspace{-5.1mm}{\bf File}: bn\_mp\_mod\_2d.c
2441
We first avoid cases of $b \le 0$ by simply mp\_zero()'ing the destination in such cases. Next if $2^b$ is larger
2442
than the input we just mp\_copy() the input and return right away. After this point we know we must actually
2443
perform some work to produce the remainder.
2445
Recalling that reducing modulo $2^k$ and a binary ``and'' with $2^k - 1$ are numerically equivalent we can quickly reduce
2446
the number. First we zero any digits above the last digit in $2^b$ (line 42). Next we reduce the
2447
leading digit of both (line 46) and then mp\_clamp().
2449
\section*{Exercises}
2451
$\left [ 3 \right ] $ & Devise an algorithm that performs $a \cdot 2^b$ for generic values of $b$ \\
2452
& in $O(n)$ time. \\
2454
$\left [ 3 \right ] $ & Devise an efficient algorithm to multiply by small low hamming \\
2455
& weight values such as $3$, $5$ and $9$. Extend it to handle all values \\
2456
& upto $64$ with a hamming weight less than three. \\
2458
$\left [ 2 \right ] $ & Modify the preceding algorithm to handle values of the form \\
2459
& $2^k - 1$ as well. \\
2461
$\left [ 3 \right ] $ & Using only algorithms mp\_mul\_2, mp\_div\_2 and mp\_add create an \\
2462
& algorithm to multiply two integers in roughly $O(2n^2)$ time for \\
2463
& any $n$-bit input. Note that the time of addition is ignored in the \\
2466
$\left [ 5 \right ] $ & Improve the previous algorithm to have a working time of at most \\
2467
& $O \left (2^{(k-1)}n + \left ({2n^2 \over k} \right ) \right )$ for an appropriate choice of $k$. Again ignore \\
2468
& the cost of addition. \\
2470
$\left [ 2 \right ] $ & Devise a chart to find optimal values of $k$ for the previous problem \\
2471
& for $n = 64 \ldots 1024$ in steps of $64$. \\
2473
$\left [ 2 \right ] $ & Using only algorithms mp\_abs and mp\_sub devise another method for \\
2474
& calculating the result of a signed comparison. \\
2478
\chapter{Multiplication and Squaring}
2479
\section{The Multipliers}
2480
For most number theoretic problems including certain public key cryptographic algorithms, the ``multipliers'' form the most important subset of
2481
algorithms of any multiple precision integer package. The set of multiplier algorithms include integer multiplication, squaring and modular reduction
2482
where in each of the algorithms single precision multiplication is the dominant operation performed. This chapter will discuss integer multiplication
2483
and squaring, leaving modular reductions for the subsequent chapter.
2485
The importance of the multiplier algorithms is for the most part driven by the fact that certain popular public key algorithms are based on modular
2486
exponentiation, that is computing $d \equiv a^b \mbox{ (mod }c\mbox{)}$ for some arbitrary choice of $a$, $b$, $c$ and $d$. During a modular
2487
exponentiation the majority\footnote{Roughly speaking a modular exponentiation will spend about 40\% of the time performing modular reductions,
2488
35\% of the time performing squaring and 25\% of the time performing multiplications.} of the processor time is spent performing single precision
2491
For centuries general purpose multiplication has required a lengthly $O(n^2)$ process, whereby each digit of one multiplicand has to be multiplied
2492
against every digit of the other multiplicand. Traditional long-hand multiplication is based on this process; while the techniques can differ the
2493
overall algorithm used is essentially the same. Only ``recently'' have faster algorithms been studied. First Karatsuba multiplication was discovered in
2494
1962. This algorithm can multiply two numbers with considerably fewer single precision multiplications when compared to the long-hand approach.
2495
This technique led to the discovery of polynomial basis algorithms (\textit{good reference?}) and subquently Fourier Transform based solutions.
2497
\section{Multiplication}
2498
\subsection{The Baseline Multiplication}
2499
\label{sec:basemult}
2500
\index{baseline multiplication}
2501
Computing the product of two integers in software can be achieved using a trivial adaptation of the standard $O(n^2)$ long-hand multiplication
2502
algorithm that school children are taught. The algorithm is considered an $O(n^2)$ algorithm since for two $n$-digit inputs $n^2$ single precision
2503
multiplications are required. More specifically for a $m$ and $n$ digit input $m \cdot n$ single precision multiplications are required. To
2504
simplify most discussions, it will be assumed that the inputs have comparable number of digits.
2506
The ``baseline multiplication'' algorithm is designed to act as the ``catch-all'' algorithm, only to be used when the faster algorithms cannot be
2507
used. This algorithm does not use any particularly interesting optimizations and should ideally be avoided if possible. One important
2508
facet of this algorithm, is that it has been modified to only produce a certain amount of output digits as resolution. The importance of this
2509
modification will become evident during the discussion of Barrett modular reduction. Recall that for a $n$ and $m$ digit input the product
2510
will be at most $n + m$ digits. Therefore, this algorithm can be reduced to a full multiplier by having it produce $n + m$ digits of the product.
2512
Recall from sub-section 4.2.2 the definition of $\gamma$ as the number of bits in the type \textbf{mp\_digit}. We shall now extend the variable set to
2513
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
2514
constant $\delta = 2^{\alpha - 2lg(\beta)}$ will represent the maximal weight of any column in a product (\textit{see sub-section 5.2.2 for more information}).
2516
\newpage\begin{figure}[!here]
2520
\hline Algorithm \textbf{s\_mp\_mul\_digs}. \\
2521
\textbf{Input}. mp\_int $a$, mp\_int $b$ and an integer $digs$ \\
2522
\textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert \mbox{ (mod }\beta^{digs}\mbox{)}$. \\
2524
1. If min$(a.used, b.used) < \delta$ then do \\
2525
\hspace{3mm}1.1 Calculate $c = \vert a \vert \cdot \vert b \vert$ by the Comba method (\textit{see algorithm~\ref{fig:COMBAMULT}}). \\
2526
\hspace{3mm}1.2 Return the result of step 1.1 \\
2528
Allocate and initialize a temporary mp\_int. \\
2529
2. Init $t$ to be of size $digs$ \\
2530
3. If step 2 failed return(\textit{MP\_MEM}). \\
2531
4. $t.used \leftarrow digs$ \\
2533
Compute the product. \\
2534
5. for $ix$ from $0$ to $a.used - 1$ do \\
2535
\hspace{3mm}5.1 $u \leftarrow 0$ \\
2536
\hspace{3mm}5.2 $pb \leftarrow \mbox{min}(b.used, digs - ix)$ \\
2537
\hspace{3mm}5.3 If $pb < 1$ then goto step 6. \\
2538
\hspace{3mm}5.4 for $iy$ from $0$ to $pb - 1$ do \\
2539
\hspace{6mm}5.4.1 $\hat r \leftarrow t_{iy + ix} + a_{ix} \cdot b_{iy} + u$ \\
2540
\hspace{6mm}5.4.2 $t_{iy + ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\
2541
\hspace{6mm}5.4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\
2542
\hspace{3mm}5.5 if $ix + pb < digs$ then do \\
2543
\hspace{6mm}5.5.1 $t_{ix + pb} \leftarrow u$ \\
2544
6. Clamp excess digits of $t$. \\
2545
7. Swap $c$ with $t$ \\
2547
9. Return(\textit{MP\_OKAY}). \\
2552
\caption{Algorithm s\_mp\_mul\_digs}
2555
\textbf{Algorithm s\_mp\_mul\_digs.}
2556
This algorithm computes the unsigned product of two inputs $a$ and $b$, limited to an output precision of $digs$ digits. While it may seem
2557
a bit awkward to modify the function from its simple $O(n^2)$ description, the usefulness of partial multipliers will arise in a subsequent
2558
algorithm. The algorithm is loosely based on algorithm 14.12 from \cite[pp. 595]{HAC} and is similar to Algorithm M of Knuth \cite[pp. 268]{TAOCPV2}.
2559
Algorithm s\_mp\_mul\_digs differs from these cited references since it can produce a variable output precision regardless of the precision of the
2562
The first thing this algorithm checks for is whether a Comba multiplier can be used instead. If the minimum digit count of either
2563
input is less than $\delta$, then the Comba method may be used instead. After the Comba method is ruled out, the baseline algorithm begins. A
2564
temporary mp\_int variable $t$ is used to hold the intermediate result of the product. This allows the algorithm to be used to
2565
compute products when either $a = c$ or $b = c$ without overwriting the inputs.
2567
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
2568
is given the count of digits to read from $b$ inside the nested loop. If $pb \le 1$ then no more output digits can be produced and the algorithm
2569
will exit the loop. The best way to think of the loops are as a series of $pb \times 1$ multiplications. That is, in each pass of the
2570
innermost loop $a_{ix}$ is multiplied against $b$ and the result is added (\textit{with an appropriate shift}) to $t$.
2572
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
2573
visualized in the following table.
2575
\begin{figure}[here]
2577
\begin{tabular}{|c|c|c|c|c|c|l|}
2578
\hline && & 5 & 7 & 6 & \\
2579
\hline $\times$&& & 2 & 4 & 1 & \\
2581
&& & 5 & 7 & 6 & $10^0(1)(576)$ \\
2582
&2 & 3 & 6 & 1 & 6 & $10^1(4)(576) + 10^0(1)(576)$ \\
2583
1 & 3 & 8 & 8 & 1 & 6 & $10^2(2)(576) + 10^1(4)(576) + 10^0(1)(576)$ \\
2587
\caption{Long-Hand Multiplication Diagram}
2590
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
2591
count. That is in pass $ix$ of the inner loop the product is added starting at the $ix$'th digit of the reult.
2593
Step 5.4.1 introduces the hat symbol (\textit{e.g. $\hat r$}) which represents a double precision variable. The multiplication on that step
2594
is assumed to be a double wide output single precision multiplication. That is, two single precision variables are multiplied to produce a
2595
double precision result. The step is somewhat optimized from a long-hand multiplication algorithm because the carry from the addition in step
2596
5.4.1 is propagated through the nested loop. If the carry was not propagated immediately it would overflow the single precision digit
2597
$t_{ix+iy}$ and the result would be lost.
2599
At step 5.5 the nested loop is finished and any carry that was left over should be forwarded. The carry does not have to be added to the $ix+pb$'th
2600
digit since that digit is assumed to be zero at this point. However, if $ix + pb \ge digs$ the carry is not set as it would make the result
2601
exceed the precision requested.
2603
\vspace{+3mm}\begin{small}
2604
\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_mul\_digs.c
2610
First we determine (line 31) if the Comba method can be used first since it's faster. The conditions for
2611
sing the Comba routine are that min$(a.used, b.used) < \delta$ and the number of digits of output is less than
2612
\textbf{MP\_WARRAY}. This new constant is used to control the stack usage in the Comba routines. By default it is
2613
set to $\delta$ but can be reduced when memory is at a premium.
2615
If we cannot use the Comba method we proceed to setup the baseline routine. We allocate the the destination mp\_int
2616
$t$ (line 37) to the exact size of the output to avoid further re--allocations. At this point we now
2617
begin the $O(n^2)$ loop.
2619
This implementation of multiplication has the caveat that it can be trimmed to only produce a variable number of
2620
digits as output. In each iteration of the outer loop the $pb$ variable is set (line 49) to the maximum
2621
number of inner loop iterations.
2623
Inside the inner loop we calculate $\hat r$ as the mp\_word product of the two mp\_digits and the addition of the
2624
carry from the previous iteration. A particularly important observation is that most modern optimizing
2625
C compilers (GCC for instance) can recognize that a $N \times N \rightarrow 2N$ multiplication is all that
2626
is required for the product. In x86 terms for example, this means using the MUL instruction.
2628
Each digit of the product is stored in turn (line 69) and the carry propagated (line 72) to the
2631
\subsection{Faster Multiplication by the ``Comba'' Method}
2633
One of the huge drawbacks of the ``baseline'' algorithms is that at the $O(n^2)$ level the carry must be
2634
computed and propagated upwards. This makes the nested loop very sequential and hard to unroll and implement
2635
in parallel. The ``Comba'' \cite{COMBA} method is named after little known (\textit{in cryptographic venues}) Paul G.
2636
Comba who described a method of implementing fast multipliers that do not require nested carry fixup operations. As an
2637
interesting aside it seems that Paul Barrett describes a similar technique in his 1986 paper \cite{BARRETT} written
2640
At the heart of the Comba technique is once again the long-hand algorithm. Except in this case a slight
2641
twist is placed on how the columns of the result are produced. In the standard long-hand algorithm rows of products
2642
are produced then added together to form the final result. In the baseline algorithm the columns are added together
2643
after each iteration to get the result instantaneously.
2645
In the Comba algorithm the columns of the result are produced entirely independently of each other. That is at
2646
the $O(n^2)$ level a simple multiplication and addition step is performed. The carries of the columns are propagated
2647
after the nested loop to reduce the amount of work requiored. Succintly the first step of the algorithm is to compute
2648
the product vector $\vec x$ as follows.
2651
\vec x_n = \sum_{i+j = n} a_ib_j, \forall n \in \lbrace 0, 1, 2, \ldots, i + j \rbrace
2654
Where $\vec x_n$ is the $n'th$ column of the output vector. Consider the following example which computes the vector $\vec x$ for the multiplication
2657
\newpage\begin{figure}[here]
2660
\begin{tabular}{|c|c|c|c|c|c|}
2661
\hline & & 5 & 7 & 6 & First Input\\
2662
\hline $\times$ & & 2 & 4 & 1 & Second Input\\
2663
\hline & & $1 \cdot 5 = 5$ & $1 \cdot 7 = 7$ & $1 \cdot 6 = 6$ & First pass \\
2664
& $4 \cdot 5 = 20$ & $4 \cdot 7+5=33$ & $4 \cdot 6+7=31$ & 6 & Second pass \\
2665
$2 \cdot 5 = 10$ & $2 \cdot 7 + 20 = 34$ & $2 \cdot 6+33=45$ & 31 & 6 & Third pass \\
2666
\hline 10 & 34 & 45 & 31 & 6 & Final Result \\
2671
\caption{Comba Multiplication Diagram}
2674
At this point the vector $x = \left < 10, 34, 45, 31, 6 \right >$ is the result of the first step of the Comba multipler.
2675
Now the columns must be fixed by propagating the carry upwards. The resultant vector will have one extra dimension over the input vector which is
2676
congruent to adding a leading zero digit.
2678
\begin{figure}[!here]
2682
\hline Algorithm \textbf{Comba Fixup}. \\
2683
\textbf{Input}. Vector $\vec x$ of dimension $k$ \\
2684
\textbf{Output}. Vector $\vec x$ such that the carries have been propagated. \\
2686
1. for $n$ from $0$ to $k - 1$ do \\
2687
\hspace{3mm}1.1 $\vec x_{n+1} \leftarrow \vec x_{n+1} + \lfloor \vec x_{n}/\beta \rfloor$ \\
2688
\hspace{3mm}1.2 $\vec x_{n} \leftarrow \vec x_{n} \mbox{ (mod }\beta\mbox{)}$ \\
2689
2. Return($\vec x$). \\
2694
\caption{Algorithm Comba Fixup}
2697
With that algorithm and $k = 5$ and $\beta = 10$ the following vector is produced $\vec x= \left < 1, 3, 8, 8, 1, 6 \right >$. In this case
2698
$241 \cdot 576$ is in fact $138816$ and the procedure succeeded. If the algorithm is correct and as will be demonstrated shortly more
2699
efficient than the baseline algorithm why not simply always use this algorithm?
2701
\subsubsection{Column Weight.}
2702
At the nested $O(n^2)$ level the Comba method adds the product of two single precision variables to each column of the output
2703
independently. A serious obstacle is if the carry is lost, due to lack of precision before the algorithm has a chance to fix
2704
the carries. For example, in the multiplication of two three-digit numbers the third column of output will be the sum of
2705
three single precision multiplications. If the precision of the accumulator for the output digits is less then $3 \cdot (\beta - 1)^2$ then
2706
an overflow can occur and the carry information will be lost. For any $m$ and $n$ digit inputs the maximum weight of any column is
2707
min$(m, n)$ which is fairly obvious.
2709
The maximum number of terms in any column of a product is known as the ``column weight'' and strictly governs when the algorithm can be used. Recall
2710
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
2711
two quantities we must not violate the following
2714
k \cdot \left (\beta - 1 \right )^2 < 2^{\alpha}
2720
k \cdot \left ( \beta^2 - 2\beta + 1 \right ) < 2^{\alpha}
2723
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
2727
k < {{2^{\alpha}} \over {\left (2^{2\rho} - 2^{\rho + 1} + 1 \right )}}
2730
The defaults for LibTomMath are $\beta = 2^{28}$ and $\alpha = 2^{64}$ which means that $k$ is bounded by $k < 257$. In this configuration
2731
the smaller input may not have more than $256$ digits if the Comba method is to be used. This is quite satisfactory for most applications since
2732
$256$ digits would allow for numbers in the range of $0 \le x < 2^{7168}$ which, is much larger than most public key cryptographic algorithms require.
2734
\newpage\begin{figure}[!here]
2738
\hline Algorithm \textbf{fast\_s\_mp\_mul\_digs}. \\
2739
\textbf{Input}. mp\_int $a$, mp\_int $b$ and an integer $digs$ \\
2740
\textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert \mbox{ (mod }\beta^{digs}\mbox{)}$. \\
2742
Place an array of \textbf{MP\_WARRAY} single precision digits named $W$ on the stack. \\
2743
1. If $c.alloc < digs$ then grow $c$ to $digs$ digits. (\textit{mp\_grow}) \\
2744
2. If step 1 failed return(\textit{MP\_MEM}).\\
2746
3. $pa \leftarrow \mbox{MIN}(digs, a.used + b.used)$ \\
2748
4. $\_ \hat W \leftarrow 0$ \\
2749
5. for $ix$ from 0 to $pa - 1$ do \\
2750
\hspace{3mm}5.1 $ty \leftarrow \mbox{MIN}(b.used - 1, ix)$ \\
2751
\hspace{3mm}5.2 $tx \leftarrow ix - ty$ \\
2752
\hspace{3mm}5.3 $iy \leftarrow \mbox{MIN}(a.used - tx, ty + 1)$ \\
2753
\hspace{3mm}5.4 for $iz$ from 0 to $iy - 1$ do \\
2754
\hspace{6mm}5.4.1 $\_ \hat W \leftarrow \_ \hat W + a_{tx+iy}b_{ty-iy}$ \\
2755
\hspace{3mm}5.5 $W_{ix} \leftarrow \_ \hat W (\mbox{mod }\beta)$\\
2756
\hspace{3mm}5.6 $\_ \hat W \leftarrow \lfloor \_ \hat W / \beta \rfloor$ \\
2758
6. $oldused \leftarrow c.used$ \\
2759
7. $c.used \leftarrow digs$ \\
2760
8. for $ix$ from $0$ to $pa$ do \\
2761
\hspace{3mm}8.1 $c_{ix} \leftarrow W_{ix}$ \\
2762
9. for $ix$ from $pa + 1$ to $oldused - 1$ do \\
2763
\hspace{3mm}9.1 $c_{ix} \leftarrow 0$ \\
2766
11. Return MP\_OKAY. \\
2771
\caption{Algorithm fast\_s\_mp\_mul\_digs}
2772
\label{fig:COMBAMULT}
2775
\textbf{Algorithm fast\_s\_mp\_mul\_digs.}
2776
This algorithm performs the unsigned multiplication of $a$ and $b$ using the Comba method limited to $digs$ digits of precision.
2778
The outer loop of this algorithm is more complicated than that of the baseline multiplier. This is because on the inside of the
2779
loop we want to produce one column per pass. This allows the accumulator $\_ \hat W$ to be placed in CPU registers and
2780
reduce the memory bandwidth to two \textbf{mp\_digit} reads per iteration.
2782
The $ty$ variable is set to the minimum count of $ix$ or the number of digits in $b$. That way if $a$ has more digits than
2783
$b$ this will be limited to $b.used - 1$. The $tx$ variable is set to the to the distance past $b.used$ the variable
2784
$ix$ is. This is used for the immediately subsequent statement where we find $iy$.
2786
The variable $iy$ is the minimum digits we can read from either $a$ or $b$ before running out. Computing one column at a time
2787
means we have to scan one integer upwards and the other downwards. $a$ starts at $tx$ and $b$ starts at $ty$. In each
2788
pass we are producing the $ix$'th output column and we note that $tx + ty = ix$. As we move $tx$ upwards we have to
2789
move $ty$ downards so the equality remains valid. The $iy$ variable is the number of iterations until
2790
$tx \ge a.used$ or $ty < 0$ occurs.
2792
After every inner pass we store the lower half of the accumulator into $W_{ix}$ and then propagate the carry of the accumulator
2793
into the next round by dividing $\_ \hat W$ by $\beta$.
2795
To measure the benefits of the Comba method over the baseline method consider the number of operations that are required. If the
2796
cost in terms of time of a multiply and addition is $p$ and the cost of a carry propagation is $q$ then a baseline multiplication would require
2797
$O \left ((p + q)n^2 \right )$ time to multiply two $n$-digit numbers. The Comba method requires only $O(pn^2 + qn)$ time, however in practice,
2798
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
2799
and addition operations in the nested loop in parallel.
2801
\vspace{+3mm}\begin{small}
2802
\hspace{-5.1mm}{\bf File}: bn\_fast\_s\_mp\_mul\_digs.c
2808
As per the pseudo--code we first calculate $pa$ (line 48) as the number of digits to output. Next we begin the outer loop
2809
to produce the individual columns of the product. We use the two aliases $tmpx$ and $tmpy$ (lines 62, 63) to point
2810
inside the two multiplicands quickly.
2812
The inner loop (lines 71 to 74) of this implementation is where the tradeoff come into play. Originally this comba
2813
implementation was ``row--major'' which means it adds to each of the columns in each pass. After the outer loop it would then fix
2814
the carries. This was very fast except it had an annoying drawback. You had to read a mp\_word and two mp\_digits and write
2815
one mp\_word per iteration. On processors such as the Athlon XP and P4 this did not matter much since the cache bandwidth
2816
is very high and it can keep the ALU fed with data. It did, however, matter on older and embedded cpus where cache is often
2817
slower and also often doesn't exist. This new algorithm only performs two reads per iteration under the assumption that the
2818
compiler has aliased $\_ \hat W$ to a CPU register.
2820
After the inner loop we store the current accumulator in $W$ and shift $\_ \hat W$ (lines 77, 80) to forward it as
2821
a carry for the next pass. After the outer loop we use the final carry (line 77) as the last digit of the product.
2823
\subsection{Polynomial Basis Multiplication}
2824
To break the $O(n^2)$ barrier in multiplication requires a completely different look at integer multiplication. In the following algorithms
2825
the use of polynomial basis representation for two integers $a$ and $b$ as $f(x) = \sum_{i=0}^{n} a_i x^i$ and
2826
$g(x) = \sum_{i=0}^{n} b_i x^i$ respectively, is required. In this system both $f(x)$ and $g(x)$ have $n + 1$ terms and are of the $n$'th degree.
2828
The product $a \cdot b \equiv f(x)g(x)$ is the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$. The coefficients $w_i$ will
2829
directly yield the desired product when $\beta$ is substituted for $x$. The direct solution to solve for the $2n + 1$ coefficients
2830
requires $O(n^2)$ time and would in practice be slower than the Comba technique.
2832
However, numerical analysis theory indicates that only $2n + 1$ distinct points in $W(x)$ are required to determine the values of the $2n + 1$ unknown
2833
coefficients. This means by finding $\zeta_y = W(y)$ for $2n + 1$ small values of $y$ the coefficients of $W(x)$ can be found with
2834
Gaussian elimination. This technique is also occasionally refered to as the \textit{interpolation technique} (\textit{references please...}) since in
2835
effect an interpolation based on $2n + 1$ points will yield a polynomial equivalent to $W(x)$.
2837
The coefficients of the polynomial $W(x)$ are unknown which makes finding $W(y)$ for any value of $y$ impossible. However, since
2838
$W(x) = f(x)g(x)$ the equivalent $\zeta_y = f(y) g(y)$ can be used in its place. The benefit of this technique stems from the
2839
fact that $f(y)$ and $g(y)$ are much smaller than either $a$ or $b$ respectively. As a result finding the $2n + 1$ relations required
2840
by multiplying $f(y)g(y)$ involves multiplying integers that are much smaller than either of the inputs.
2842
When picking points to gather relations there are always three obvious points to choose, $y = 0, 1$ and $ \infty$. The $\zeta_0$ term
2843
is simply the product $W(0) = w_0 = a_0 \cdot b_0$. The $\zeta_1$ term is the product
2844
$W(1) = \left (\sum_{i = 0}^{n} a_i \right ) \left (\sum_{i = 0}^{n} b_i \right )$. The third point $\zeta_{\infty}$ is less obvious but rather
2845
simple to explain. The $2n + 1$'th coefficient of $W(x)$ is numerically equivalent to the most significant column in an integer multiplication.
2846
The point at $\infty$ is used symbolically to represent the most significant column, that is $W(\infty) = w_{2n} = a_nb_n$. Note that the
2847
points at $y = 0$ and $\infty$ yield the coefficients $w_0$ and $w_{2n}$ directly.
2849
If more points are required they should be of small values and powers of two such as $2^q$ and the related \textit{mirror points}
2850
$\left (2^q \right )^{2n} \cdot \zeta_{2^{-q}}$ for small values of $q$. The term ``mirror point'' stems from the fact that
2851
$\left (2^q \right )^{2n} \cdot \zeta_{2^{-q}}$ can be calculated in the exact opposite fashion as $\zeta_{2^q}$. For
2852
example, when $n = 2$ and $q = 1$ then following two equations are equivalent to the point $\zeta_{2}$ and its mirror.
2855
\zeta_{2} = f(2)g(2) = (4a_2 + 2a_1 + a_0)(4b_2 + 2b_1 + b_0) \nonumber \\
2856
16 \cdot \zeta_{1 \over 2} = 4f({1\over 2}) \cdot 4g({1 \over 2}) = (a_2 + 2a_1 + 4a_0)(b_2 + 2b_1 + 4b_0)
2859
Using such points will allow the values of $f(y)$ and $g(y)$ to be independently calculated using only left shifts. For example, when $n = 2$ the
2860
polynomial $f(2^q)$ is equal to $2^q((2^qa_2) + a_1) + a_0$. This technique of polynomial representation is known as Horner's method.
2862
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
2863
multiplicands that have $n$ times fewer digits than the inputs. The asymptotic running time of this algorithm is
2864
$O \left ( k^{lg_n(2n - 1)} \right )$ for $k$ digit inputs (\textit{assuming they have the same number of digits}). Figure~\ref{fig:exponent}
2865
summarizes the exponents for various values of $n$.
2869
\begin{tabular}{|c|c|c|}
2870
\hline \textbf{Split into $n$ Parts} & \textbf{Exponent} & \textbf{Notes}\\
2871
\hline $2$ & $1.584962501$ & This is Karatsuba Multiplication. \\
2872
\hline $3$ & $1.464973520$ & This is Toom-Cook Multiplication. \\
2873
\hline $4$ & $1.403677461$ &\\
2874
\hline $5$ & $1.365212389$ &\\
2875
\hline $10$ & $1.278753601$ &\\
2876
\hline $100$ & $1.149426538$ &\\
2877
\hline $1000$ & $1.100270931$ &\\
2878
\hline $10000$ & $1.075252070$ &\\
2882
\caption{Asymptotic Running Time of Polynomial Basis Multiplication}
2883
\label{fig:exponent}
2886
At first it may seem like a good idea to choose $n = 1000$ since the exponent is approximately $1.1$. However, the overhead
2887
of solving for the 2001 terms of $W(x)$ will certainly consume any savings the algorithm could offer for all but exceedingly large
2890
\subsubsection{Cutoff Point}
2891
The polynomial basis multiplication algorithms all require fewer single precision multiplications than a straight Comba approach. However,
2892
the algorithms incur an overhead (\textit{at the $O(n)$ work level}) since they require a system of equations to be solved. This makes the
2893
polynomial basis approach more costly to use with small inputs.
2895
Let $m$ represent the number of digits in the multiplicands (\textit{assume both multiplicands have the same number of digits}). There exists a
2896
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
2897
when $m > y$ the Comba methods are slower than the polynomial basis algorithms.
2899
The exact location of $y$ depends on several key architectural elements of the computer platform in question.
2902
\item The ratio of clock cycles for single precision multiplication versus other simpler operations such as addition, shifting, etc. For example
2903
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
2904
the cutoff point $y$ will be.
2906
\item The complexity of the linear system of equations (\textit{for the coefficients of $W(x)$}) is. Generally speaking as the number of splits
2907
grows the complexity grows substantially. Ideally solving the system will only involve addition, subtraction and shifting of integers. This
2908
directly reflects on the ratio previous mentioned.
2910
\item To a lesser extent memory bandwidth and function call overheads. Provided the values are in the processor cache this is less of an
2911
influence over the cutoff point.
2915
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
2916
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
2917
a high resolution timer is available.
2919
\subsection{Karatsuba Multiplication}
2920
Karatsuba \cite{KARA} multiplication when originally proposed in 1962 was among the first set of algorithms to break the $O(n^2)$ barrier for
2921
general purpose multiplication. Given two polynomial basis representations $f(x) = ax + b$ and $g(x) = cx + d$, Karatsuba proved with
2922
light algebra \cite{KARAP} that the following polynomial is equivalent to multiplication of the two integers the polynomials represent.
2925
f(x) \cdot g(x) = acx^2 + ((a + b)(c + d) - (ac + bd))x + bd
2928
Using the observation that $ac$ and $bd$ could be re-used only three half sized multiplications would be required to produce the product. Applying
2929
this algorithm recursively, the work factor becomes $O(n^{lg(3)})$ which is substantially better than the work factor $O(n^2)$ of the Comba technique. It turns
2930
out what Karatsuba did not know or at least did not publish was that this is simply polynomial basis multiplication with the points
2931
$\zeta_0$, $\zeta_{\infty}$ and $\zeta_{1}$. Consider the resultant system of equations.
2934
\begin{tabular}{rcrcrcrc}
2935
$\zeta_{0}$ & $=$ & & & & & $w_0$ \\
2936
$\zeta_{1}$ & $=$ & $w_2$ & $+$ & $w_1$ & $+$ & $w_0$ \\
2937
$\zeta_{\infty}$ & $=$ & $w_2$ & & & & \\
2941
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
2942
of this system of equations has made Karatsuba fairly popular. In fact the cutoff point is often fairly low\footnote{With LibTomMath 0.18 it is 70 and 109 digits for the Intel P4 and AMD Athlon respectively.}
2943
making it an ideal algorithm to speed up certain public key cryptosystems such as RSA and Diffie-Hellman.
2945
\newpage\begin{figure}[!here]
2949
\hline Algorithm \textbf{mp\_karatsuba\_mul}. \\
2950
\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\
2951
\textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert$ \\
2953
1. Init the following mp\_int variables: $x0$, $x1$, $y0$, $y1$, $t1$, $x0y0$, $x1y1$.\\
2954
2. If step 2 failed then return(\textit{MP\_MEM}). \\
2956
Split the input. e.g. $a = x1 \cdot \beta^B + x0$ \\
2957
3. $B \leftarrow \mbox{min}(a.used, b.used)/2$ \\
2958
4. $x0 \leftarrow a \mbox{ (mod }\beta^B\mbox{)}$ (\textit{mp\_mod\_2d}) \\
2959
5. $y0 \leftarrow b \mbox{ (mod }\beta^B\mbox{)}$ \\
2960
6. $x1 \leftarrow \lfloor a / \beta^B \rfloor$ (\textit{mp\_rshd}) \\
2961
7. $y1 \leftarrow \lfloor b / \beta^B \rfloor$ \\
2963
Calculate the three products. \\
2964
8. $x0y0 \leftarrow x0 \cdot y0$ (\textit{mp\_mul}) \\
2965
9. $x1y1 \leftarrow x1 \cdot y1$ \\
2966
10. $t1 \leftarrow x1 + x0$ (\textit{mp\_add}) \\
2967
11. $x0 \leftarrow y1 + y0$ \\
2968
12. $t1 \leftarrow t1 \cdot x0$ \\
2970
Calculate the middle term. \\
2971
13. $x0 \leftarrow x0y0 + x1y1$ \\
2972
14. $t1 \leftarrow t1 - x0$ (\textit{s\_mp\_sub}) \\
2974
Calculate the final product. \\
2975
15. $t1 \leftarrow t1 \cdot \beta^B$ (\textit{mp\_lshd}) \\
2976
16. $x1y1 \leftarrow x1y1 \cdot \beta^{2B}$ \\
2977
17. $t1 \leftarrow x0y0 + t1$ \\
2978
18. $c \leftarrow t1 + x1y1$ \\
2979
19. Clear all of the temporary variables. \\
2980
20. Return(\textit{MP\_OKAY}).\\
2985
\caption{Algorithm mp\_karatsuba\_mul}
2988
\textbf{Algorithm mp\_karatsuba\_mul.}
2989
This algorithm computes the unsigned product of two inputs using the Karatsuba multiplication algorithm. It is loosely based on the description
2990
from Knuth \cite[pp. 294-295]{TAOCPV2}.
2993
In order to split the two inputs into their respective halves, a suitable \textit{radix point} must be chosen. The radix point chosen must
2994
be used for both of the inputs meaning that it must be smaller than the smallest input. Step 3 chooses the radix point $B$ as half of the
2995
smallest input \textbf{used} count. After the radix point is chosen the inputs are split into lower and upper halves. Step 4 and 5
2996
compute the lower halves. Step 6 and 7 computer the upper halves.
2998
After the halves have been computed the three intermediate half-size products must be computed. Step 8 and 9 compute the trivial products
2999
$x0 \cdot y0$ and $x1 \cdot y1$. The mp\_int $x0$ is used as a temporary variable after $x1 + x0$ has been computed. By using $x0$ instead
3000
of an additional temporary variable, the algorithm can avoid an addition memory allocation operation.
3002
The remaining steps 13 through 18 compute the Karatsuba polynomial through a variety of digit shifting and addition operations.
3004
\vspace{+3mm}\begin{small}
3005
\hspace{-5.1mm}{\bf File}: bn\_mp\_karatsuba\_mul.c
3011
The new coding element in this routine, not seen in previous routines, is the usage of goto statements. The conventional
3012
wisdom is that goto statements should be avoided. This is generally true, however when every single function call can fail, it makes sense
3013
to handle error recovery with a single piece of code. Lines 62 to 76 handle initializing all of the temporary variables
3014
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
3015
the temporaries that have been successfully allocated so far.
3017
The temporary variables are all initialized using the mp\_init\_size routine since they are expected to be large. This saves the
3018
additional reallocation that would have been necessary. Also $x0$, $x1$, $y0$ and $y1$ have to be able to hold at least their respective
3019
number of digits for the next section of code.
3021
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
3022
to extract the halves, the respective code has been placed inline within the body of the function. To initialize the halves, the \textbf{used} and
3023
\textbf{sign} members are copied first. The first for loop on line 96 copies the lower halves. Since they are both the same magnitude it
3024
is simpler to calculate both lower halves in a single loop. The for loop on lines 102 and 107 calculate the upper halves $x1$ and
3027
By inlining the calculation of the halves, the Karatsuba multiplier has a slightly lower overhead and can be used for smaller magnitude inputs.
3029
When line 151 is reached, the algorithm has completed succesfully. The ``error status'' variable $err$ is set to \textbf{MP\_OKAY} so that
3030
the same code that handles errors can be used to clear the temporary variables and return.
3032
\subsection{Toom-Cook $3$-Way Multiplication}
3033
Toom-Cook $3$-Way \cite{TOOM} multiplication is essentially the polynomial basis algorithm for $n = 2$ except that the points are
3034
chosen such that $\zeta$ is easy to compute and the resulting system of equations easy to reduce. Here, the points $\zeta_{0}$,
3035
$16 \cdot \zeta_{1 \over 2}$, $\zeta_1$, $\zeta_2$ and $\zeta_{\infty}$ make up the five required points to solve for the coefficients
3038
With the five relations that Toom-Cook specifies, the following system of equations is formed.
3041
\begin{tabular}{rcrcrcrcrcr}
3042
$\zeta_0$ & $=$ & $0w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $1w_0$ \\
3043
$16 \cdot \zeta_{1 \over 2}$ & $=$ & $1w_4$ & $+$ & $2w_3$ & $+$ & $4w_2$ & $+$ & $8w_1$ & $+$ & $16w_0$ \\
3044
$\zeta_1$ & $=$ & $1w_4$ & $+$ & $1w_3$ & $+$ & $1w_2$ & $+$ & $1w_1$ & $+$ & $1w_0$ \\
3045
$\zeta_2$ & $=$ & $16w_4$ & $+$ & $8w_3$ & $+$ & $4w_2$ & $+$ & $2w_1$ & $+$ & $1w_0$ \\
3046
$\zeta_{\infty}$ & $=$ & $1w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $0w_0$ \\
3050
A trivial solution to this matrix requires $12$ subtractions, two multiplications by a small power of two, two divisions by a small power
3051
of two, two divisions by three and one multiplication by three. All of these $19$ sub-operations require less than quadratic time, meaning that
3052
the algorithm can be faster than a baseline multiplication. However, the greater complexity of this algorithm places the cutoff point
3053
(\textbf{TOOM\_MUL\_CUTOFF}) where Toom-Cook becomes more efficient much higher than the Karatsuba cutoff point.
3055
\begin{figure}[!here]
3059
\hline Algorithm \textbf{mp\_toom\_mul}. \\
3060
\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\
3061
\textbf{Output}. $c \leftarrow a \cdot b $ \\
3063
Split $a$ and $b$ into three pieces. E.g. $a = a_2 \beta^{2k} + a_1 \beta^{k} + a_0$ \\
3064
1. $k \leftarrow \lfloor \mbox{min}(a.used, b.used) / 3 \rfloor$ \\
3065
2. $a_0 \leftarrow a \mbox{ (mod }\beta^{k}\mbox{)}$ \\
3066
3. $a_1 \leftarrow \lfloor a / \beta^k \rfloor$, $a_1 \leftarrow a_1 \mbox{ (mod }\beta^{k}\mbox{)}$ \\
3067
4. $a_2 \leftarrow \lfloor a / \beta^{2k} \rfloor$, $a_2 \leftarrow a_2 \mbox{ (mod }\beta^{k}\mbox{)}$ \\
3068
5. $b_0 \leftarrow a \mbox{ (mod }\beta^{k}\mbox{)}$ \\
3069
6. $b_1 \leftarrow \lfloor a / \beta^k \rfloor$, $b_1 \leftarrow b_1 \mbox{ (mod }\beta^{k}\mbox{)}$ \\
3070
7. $b_2 \leftarrow \lfloor a / \beta^{2k} \rfloor$, $b_2 \leftarrow b_2 \mbox{ (mod }\beta^{k}\mbox{)}$ \\
3072
Find the five equations for $w_0, w_1, ..., w_4$. \\
3073
8. $w_0 \leftarrow a_0 \cdot b_0$ \\
3074
9. $w_4 \leftarrow a_2 \cdot b_2$ \\
3075
10. $tmp_1 \leftarrow 2 \cdot a_0$, $tmp_1 \leftarrow a_1 + tmp_1$, $tmp_1 \leftarrow 2 \cdot tmp_1$, $tmp_1 \leftarrow tmp_1 + a_2$ \\
3076
11. $tmp_2 \leftarrow 2 \cdot b_0$, $tmp_2 \leftarrow b_1 + tmp_2$, $tmp_2 \leftarrow 2 \cdot tmp_2$, $tmp_2 \leftarrow tmp_2 + b_2$ \\
3077
12. $w_1 \leftarrow tmp_1 \cdot tmp_2$ \\
3078
13. $tmp_1 \leftarrow 2 \cdot a_2$, $tmp_1 \leftarrow a_1 + tmp_1$, $tmp_1 \leftarrow 2 \cdot tmp_1$, $tmp_1 \leftarrow tmp_1 + a_0$ \\
3079
14. $tmp_2 \leftarrow 2 \cdot b_2$, $tmp_2 \leftarrow b_1 + tmp_2$, $tmp_2 \leftarrow 2 \cdot tmp_2$, $tmp_2 \leftarrow tmp_2 + b_0$ \\
3080
15. $w_3 \leftarrow tmp_1 \cdot tmp_2$ \\
3081
16. $tmp_1 \leftarrow a_0 + a_1$, $tmp_1 \leftarrow tmp_1 + a_2$, $tmp_2 \leftarrow b_0 + b_1$, $tmp_2 \leftarrow tmp_2 + b_2$ \\
3082
17. $w_2 \leftarrow tmp_1 \cdot tmp_2$ \\
3084
Continued on the next page.\\
3089
\caption{Algorithm mp\_toom\_mul}
3092
\newpage\begin{figure}[!here]
3096
\hline Algorithm \textbf{mp\_toom\_mul} (continued). \\
3097
\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\
3098
\textbf{Output}. $c \leftarrow a \cdot b $ \\
3100
Now solve the system of equations. \\
3101
18. $w_1 \leftarrow w_4 - w_1$, $w_3 \leftarrow w_3 - w_0$ \\
3102
19. $w_1 \leftarrow \lfloor w_1 / 2 \rfloor$, $w_3 \leftarrow \lfloor w_3 / 2 \rfloor$ \\
3103
20. $w_2 \leftarrow w_2 - w_0$, $w_2 \leftarrow w_2 - w_4$ \\
3104
21. $w_1 \leftarrow w_1 - w_2$, $w_3 \leftarrow w_3 - w_2$ \\
3105
22. $tmp_1 \leftarrow 8 \cdot w_0$, $w_1 \leftarrow w_1 - tmp_1$, $tmp_1 \leftarrow 8 \cdot w_4$, $w_3 \leftarrow w_3 - tmp_1$ \\
3106
23. $w_2 \leftarrow 3 \cdot w_2$, $w_2 \leftarrow w_2 - w_1$, $w_2 \leftarrow w_2 - w_3$ \\
3107
24. $w_1 \leftarrow w_1 - w_2$, $w_3 \leftarrow w_3 - w_2$ \\
3108
25. $w_1 \leftarrow \lfloor w_1 / 3 \rfloor, w_3 \leftarrow \lfloor w_3 / 3 \rfloor$ \\
3110
Now substitute $\beta^k$ for $x$ by shifting $w_0, w_1, ..., w_4$. \\
3111
26. for $n$ from $1$ to $4$ do \\
3112
\hspace{3mm}26.1 $w_n \leftarrow w_n \cdot \beta^{nk}$ \\
3113
27. $c \leftarrow w_0 + w_1$, $c \leftarrow c + w_2$, $c \leftarrow c + w_3$, $c \leftarrow c + w_4$ \\
3114
28. Return(\textit{MP\_OKAY}) \\
3119
\caption{Algorithm mp\_toom\_mul (continued)}
3122
\textbf{Algorithm mp\_toom\_mul.}
3123
This algorithm computes the product of two mp\_int variables $a$ and $b$ using the Toom-Cook approach. Compared to the Karatsuba multiplication, this
3124
algorithm has a lower asymptotic running time of approximately $O(n^{1.464})$ but at an obvious cost in overhead. In this
3125
description, several statements have been compounded to save space. The intention is that the statements are executed from left to right across
3128
The two inputs $a$ and $b$ are first split into three $k$-digit integers $a_0, a_1, a_2$ and $b_0, b_1, b_2$ respectively. From these smaller
3129
integers the coefficients of the polynomial basis representations $f(x)$ and $g(x)$ are known and can be used to find the relations required.
3131
The first two relations $w_0$ and $w_4$ are the points $\zeta_{0}$ and $\zeta_{\infty}$ respectively. The relation $w_1, w_2$ and $w_3$ correspond
3132
to the points $16 \cdot \zeta_{1 \over 2}, \zeta_{2}$ and $\zeta_{1}$ respectively. These are found using logical shifts to independently find
3133
$f(y)$ and $g(y)$ which significantly speeds up the algorithm.
3135
After the five relations $w_0, w_1, \ldots, w_4$ have been computed, the system they represent must be solved in order for the unknown coefficients
3136
$w_1, w_2$ and $w_3$ to be isolated. The steps 18 through 25 perform the system reduction required as previously described. Each step of
3137
the reduction represents the comparable matrix operation that would be performed had this been performed by pencil. For example, step 18 indicates
3138
that row $1$ must be subtracted from row $4$ and simultaneously row $0$ subtracted from row $3$.
3140
Once the coeffients have been isolated, the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$ is known. By substituting $\beta^{k}$ for $x$, the integer
3141
result $a \cdot b$ is produced.
3143
\vspace{+3mm}\begin{small}
3144
\hspace{-5.1mm}{\bf File}: bn\_mp\_toom\_mul.c
3150
The first obvious thing to note is that this algorithm is complicated. The complexity is worth it if you are multiplying very
3151
large numbers. For example, a 10,000 digit multiplication takes approximaly 99,282,205 fewer single precision multiplications with
3152
Toom--Cook than a Comba or baseline approach (this is a savings of more than 99$\%$). For most ``crypto'' sized numbers this
3153
algorithm is not practical as Karatsuba has a much lower cutoff point.
3155
First we split $a$ and $b$ into three roughly equal portions. This has been accomplished (lines 41 to 70) with
3156
combinations of mp\_rshd() and mp\_mod\_2d() function calls. At this point $a = a2 \cdot \beta^2 + a1 \cdot \beta + a0$ and similiarly
3159
Next we compute the five points $w0, w1, w2, w3$ and $w4$. Recall that $w0$ and $w4$ can be computed directly from the portions so
3160
we get those out of the way first (lines 73 and 78). Next we compute $w1, w2$ and $w3$ using Horners method.
3162
After this point we solve for the actual values of $w1, w2$ and $w3$ by reducing the $5 \times 5$ system which is relatively
3165
\subsection{Signed Multiplication}
3166
Now that algorithms to handle multiplications of every useful dimensions have been developed, a rather simple finishing touch is required. So far all
3167
of the multiplication algorithms have been unsigned multiplications which leaves only a signed multiplication algorithm to be established.
3169
\begin{figure}[!here]
3173
\hline Algorithm \textbf{mp\_mul}. \\
3174
\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\
3175
\textbf{Output}. $c \leftarrow a \cdot b$ \\
3177
1. If $a.sign = b.sign$ then \\
3178
\hspace{3mm}1.1 $sign = MP\_ZPOS$ \\
3180
\hspace{3mm}2.1 $sign = MP\_ZNEG$ \\
3181
3. If min$(a.used, b.used) \ge TOOM\_MUL\_CUTOFF$ then \\
3182
\hspace{3mm}3.1 $c \leftarrow a \cdot b$ using algorithm mp\_toom\_mul \\
3183
4. else if min$(a.used, b.used) \ge KARATSUBA\_MUL\_CUTOFF$ then \\
3184
\hspace{3mm}4.1 $c \leftarrow a \cdot b$ using algorithm mp\_karatsuba\_mul \\
3186
\hspace{3mm}5.1 $digs \leftarrow a.used + b.used + 1$ \\
3187
\hspace{3mm}5.2 If $digs < MP\_ARRAY$ and min$(a.used, b.used) \le \delta$ then \\
3188
\hspace{6mm}5.2.1 $c \leftarrow a \cdot b \mbox{ (mod }\beta^{digs}\mbox{)}$ using algorithm fast\_s\_mp\_mul\_digs. \\
3189
\hspace{3mm}5.3 else \\
3190
\hspace{6mm}5.3.1 $c \leftarrow a \cdot b \mbox{ (mod }\beta^{digs}\mbox{)}$ using algorithm s\_mp\_mul\_digs. \\
3191
6. $c.sign \leftarrow sign$ \\
3192
7. Return the result of the unsigned multiplication performed. \\
3197
\caption{Algorithm mp\_mul}
3200
\textbf{Algorithm mp\_mul.}
3201
This algorithm performs the signed multiplication of two inputs. It will make use of any of the three unsigned multiplication algorithms
3202
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
3203
s\_mp\_mul\_digs will clear it.
3205
\vspace{+3mm}\begin{small}
3206
\hspace{-5.1mm}{\bf File}: bn\_mp\_mul.c
3212
The implementation is rather simplistic and is not particularly noteworthy. Line 22 computes the sign of the result using the ``?''
3213
operator from the C programming language. Line 48 computes $\delta$ using the fact that $1 << k$ is equal to $2^k$.
3216
\label{sec:basesquare}
3218
Squaring is a special case of multiplication where both multiplicands are equal. At first it may seem like there is no significant optimization
3219
available but in fact there is. Consider the multiplication of $576$ against $241$. In total there will be nine single precision multiplications
3220
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
3221
the multiplication of $123$ against $123$. The nine products are $3 \cdot 3$, $3 \cdot 2$, $3 \cdot 1$, $2 \cdot 3$, $2 \cdot 2$, $2 \cdot 1$,
3222
$1 \cdot 3$, $1 \cdot 2$ and $1 \cdot 1$. On closer inspection some of the products are equivalent. For example, $3 \cdot 2 = 2 \cdot 3$
3223
and $3 \cdot 1 = 1 \cdot 3$.
3225
For any $n$-digit input, there are ${{\left (n^2 + n \right)}\over 2}$ possible unique single precision multiplications required compared to the $n^2$
3226
required for multiplication. The following diagram gives an example of the operations required.
3228
\begin{figure}[here]
3230
\begin{tabular}{ccccc|c}
3233
\hline && $3 \cdot 1$ & $3 \cdot 2$ & $3 \cdot 3$ & Row 0\\
3234
& $2 \cdot 1$ & $2 \cdot 2$ & $2 \cdot 3$ && Row 1 \\
3235
$1 \cdot 1$ & $1 \cdot 2$ & $1 \cdot 3$ &&& Row 2 \\
3238
\caption{Squaring Optimization Diagram}
3241
Starting from zero and numbering the columns from right to left a very simple pattern becomes obvious. For the purposes of this discussion let $x$
3242
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.
3244
The second observation is that every column $j$ in row $k$ where $j \ne 2k$ is part of a double product. Every non-square term of a column will
3245
appear twice hence the name ``double product''. Every odd column is made up entirely of double products. In fact every column is made up of double
3246
products and at most one square (\textit{see the exercise section}).
3248
The third and final observation is that for row $k$ the first unique non-square term, that is, one that hasn't already appeared in an earlier row,
3249
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.
3250
Column two of row one is a square and column three is the first unique column.
3252
\subsection{The Baseline Squaring Algorithm}
3253
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
3256
\begin{figure}[!here]
3260
\hline Algorithm \textbf{s\_mp\_sqr}. \\
3261
\textbf{Input}. mp\_int $a$ \\
3262
\textbf{Output}. $b \leftarrow a^2$ \\
3264
1. Init a temporary mp\_int of at least $2 \cdot a.used +1$ digits. (\textit{mp\_init\_size}) \\
3265
2. If step 1 failed return(\textit{MP\_MEM}) \\
3266
3. $t.used \leftarrow 2 \cdot a.used + 1$ \\
3267
4. For $ix$ from 0 to $a.used - 1$ do \\
3268
\hspace{3mm}Calculate the square. \\
3269
\hspace{3mm}4.1 $\hat r \leftarrow t_{2ix} + \left (a_{ix} \right )^2$ \\
3270
\hspace{3mm}4.2 $t_{2ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\
3271
\hspace{3mm}Calculate the double products after the square. \\
3272
\hspace{3mm}4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\
3273
\hspace{3mm}4.4 For $iy$ from $ix + 1$ to $a.used - 1$ do \\
3274
\hspace{6mm}4.4.1 $\hat r \leftarrow 2 \cdot a_{ix}a_{iy} + t_{ix + iy} + u$ \\
3275
\hspace{6mm}4.4.2 $t_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\
3276
\hspace{6mm}4.4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\
3277
\hspace{3mm}Set the last carry. \\
3278
\hspace{3mm}4.5 While $u > 0$ do \\
3279
\hspace{6mm}4.5.1 $iy \leftarrow iy + 1$ \\
3280
\hspace{6mm}4.5.2 $\hat r \leftarrow t_{ix + iy} + u$ \\
3281
\hspace{6mm}4.5.3 $t_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\
3282
\hspace{6mm}4.5.4 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\
3283
5. Clamp excess digits of $t$. (\textit{mp\_clamp}) \\
3284
6. Exchange $b$ and $t$. \\
3285
7. Clear $t$ (\textit{mp\_clear}) \\
3286
8. Return(\textit{MP\_OKAY}) \\
3291
\caption{Algorithm s\_mp\_sqr}
3294
\textbf{Algorithm s\_mp\_sqr.}
3295
This algorithm computes the square of an input using the three observations on squaring. It is based fairly faithfully on algorithm 14.16 of HAC
3296
\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
3297
destination mp\_int to be the same as the source mp\_int.
3299
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
3300
the inner loop computes the columns of the partial result. Step 4.1 and 4.2 compute the square term for each row, and step 4.3 and 4.4 propagate
3301
the carry and compute the double products.
3303
The requirement that a mp\_word be able to represent the range $0 \le x < 2 \beta^2$ arises from this
3304
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
3305
when it is multiplied by two, it can be properly represented by a mp\_word.
3307
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
3308
results calculated so far. This involves expensive carry propagation which will be eliminated in the next algorithm.
3310
\vspace{+3mm}\begin{small}
3311
\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_sqr.c
3317
Inside the outer loop (line 34) the square term is calculated on line 37. The carry (line 44) has been
3318
extracted from the mp\_word accumulator using a right shift. Aliases for $a_{ix}$ and $t_{ix+iy}$ are initialized
3319
(lines 47 and 50) to simplify the inner loop. The doubling is performed using two
3320
additions (line 59) since it is usually faster than shifting, if not at least as fast.
3322
The important observation is that the inner loop does not begin at $iy = 0$ like for multiplication. As such the inner loops
3323
get progressively shorter as the algorithm proceeds. This is what leads to the savings compared to using a multiplication to
3326
\subsection{Faster Squaring by the ``Comba'' Method}
3327
A major drawback to the baseline method is the requirement for single precision shifting inside the $O(n^2)$ nested loop. Squaring has an additional
3328
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
3329
performance hazards.
3331
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
3332
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
3333
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,
3334
$ab + ba + ac + ca = 2ab + 2ac = 2(ab + ac)$.
3336
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
3337
mp\_word arrays. One array will hold the squares and the other array will hold the double products. With both arrays the doubling and
3338
carry propagation can be moved to a $O(n)$ work level outside the $O(n^2)$ level. In this case, we have an even simpler solution in mind.
3340
\newpage\begin{figure}[!here]
3344
\hline Algorithm \textbf{fast\_s\_mp\_sqr}. \\
3345
\textbf{Input}. mp\_int $a$ \\
3346
\textbf{Output}. $b \leftarrow a^2$ \\
3348
Place an array of \textbf{MP\_WARRAY} mp\_digits named $W$ on the stack. \\
3349
1. If $b.alloc < 2a.used + 1$ then grow $b$ to $2a.used + 1$ digits. (\textit{mp\_grow}). \\
3350
2. If step 1 failed return(\textit{MP\_MEM}). \\
3352
3. $pa \leftarrow 2 \cdot a.used$ \\
3353
4. $\hat W1 \leftarrow 0$ \\
3354
5. for $ix$ from $0$ to $pa - 1$ do \\
3355
\hspace{3mm}5.1 $\_ \hat W \leftarrow 0$ \\
3356
\hspace{3mm}5.2 $ty \leftarrow \mbox{MIN}(a.used - 1, ix)$ \\
3357
\hspace{3mm}5.3 $tx \leftarrow ix - ty$ \\
3358
\hspace{3mm}5.4 $iy \leftarrow \mbox{MIN}(a.used - tx, ty + 1)$ \\
3359
\hspace{3mm}5.5 $iy \leftarrow \mbox{MIN}(iy, \lfloor \left (ty - tx + 1 \right )/2 \rfloor)$ \\
3360
\hspace{3mm}5.6 for $iz$ from $0$ to $iz - 1$ do \\
3361
\hspace{6mm}5.6.1 $\_ \hat W \leftarrow \_ \hat W + a_{tx + iz}a_{ty - iz}$ \\
3362
\hspace{3mm}5.7 $\_ \hat W \leftarrow 2 \cdot \_ \hat W + \hat W1$ \\
3363
\hspace{3mm}5.8 if $ix$ is even then \\
3364
\hspace{6mm}5.8.1 $\_ \hat W \leftarrow \_ \hat W + \left ( a_{\lfloor ix/2 \rfloor}\right )^2$ \\
3365
\hspace{3mm}5.9 $W_{ix} \leftarrow \_ \hat W (\mbox{mod }\beta)$ \\
3366
\hspace{3mm}5.10 $\hat W1 \leftarrow \lfloor \_ \hat W / \beta \rfloor$ \\
3368
6. $oldused \leftarrow b.used$ \\
3369
7. $b.used \leftarrow 2 \cdot a.used$ \\
3370
8. for $ix$ from $0$ to $pa - 1$ do \\
3371
\hspace{3mm}8.1 $b_{ix} \leftarrow W_{ix}$ \\
3372
9. for $ix$ from $pa$ to $oldused - 1$ do \\
3373
\hspace{3mm}9.1 $b_{ix} \leftarrow 0$ \\
3374
10. Clamp excess digits from $b$. (\textit{mp\_clamp}) \\
3375
11. Return(\textit{MP\_OKAY}). \\
3380
\caption{Algorithm fast\_s\_mp\_sqr}
3383
\textbf{Algorithm fast\_s\_mp\_sqr.}
3384
This algorithm computes the square of an input using the Comba technique. It is designed to be a replacement for algorithm
3385
s\_mp\_sqr when the number of input digits is less than \textbf{MP\_WARRAY} and less than $\delta \over 2$.
3386
This algorithm is very similar to the Comba multiplier except with a few key differences we shall make note of.
3388
First, we have an accumulator and carry variables $\_ \hat W$ and $\hat W1$ respectively. This is because the inner loop
3389
products are to be doubled. If we had added the previous carry in we would be doubling too much. Next we perform an
3390
addition MIN condition on $iy$ (step 5.5) to prevent overlapping digits. For example, $a_3 \cdot a_5$ is equal
3391
$a_5 \cdot a_3$. Whereas in the multiplication case we would have $5 < a.used$ and $3 \ge 0$ is maintained since we double the sum
3392
of the products just outside the inner loop we have to avoid doing this. This is also a good thing since we perform
3393
fewer multiplications and the routine ends up being faster.
3395
Finally the last difference is the addition of the ``square'' term outside the inner loop (step 5.8). We add in the square
3396
only to even outputs and it is the square of the term at the $\lfloor ix / 2 \rfloor$ position.
3398
\vspace{+3mm}\begin{small}
3399
\hspace{-5.1mm}{\bf File}: bn\_fast\_s\_mp\_sqr.c
3405
This implementation is essentially a copy of Comba multiplication with the appropriate changes added to make it faster for
3406
the special case of squaring.
3408
\subsection{Polynomial Basis Squaring}
3409
The same algorithm that performs optimal polynomial basis multiplication can be used to perform polynomial basis squaring. The minor exception
3410
is that $\zeta_y = f(y)g(y)$ is actually equivalent to $\zeta_y = f(y)^2$ since $f(y) = g(y)$. Instead of performing $2n + 1$
3411
multiplications to find the $\zeta$ relations, squaring operations are performed instead.
3413
\subsection{Karatsuba Squaring}
3414
Let $f(x) = ax + b$ represent the polynomial basis representation of a number to square.
3415
Let $h(x) = \left ( f(x) \right )^2$ represent the square of the polynomial. The Karatsuba equation can be modified to square a
3416
number with the following equation.
3419
h(x) = a^2x^2 + \left ((a + b)^2 - (a^2 + b^2) \right )x + b^2
3422
Upon closer inspection this equation only requires the calculation of three half-sized squares: $a^2$, $b^2$ and $(a + b)^2$. As in
3423
Karatsuba multiplication, this algorithm can be applied recursively on the input and will achieve an asymptotic running time of
3424
$O \left ( n^{lg(3)} \right )$.
3426
If the asymptotic times of Karatsuba squaring and multiplication are the same, why not simply use the multiplication algorithm
3427
instead? The answer to this arises from the cutoff point for squaring. As in multiplication there exists a cutoff point, at which the
3428
time required for a Comba based squaring and a Karatsuba based squaring meet. Due to the overhead inherent in the Karatsuba method, the cutoff
3429
point is fairly high. For example, on an AMD Athlon XP processor with $\beta = 2^{28}$, the cutoff point is around 127 digits.
3431
Consider squaring a 200 digit number with this technique. It will be split into two 100 digit halves which are subsequently squared.
3432
The 100 digit halves will not be squared using Karatsuba, but instead using the faster Comba based squaring algorithm. If Karatsuba multiplication
3433
were used instead, the 100 digit numbers would be squared with a slower Comba based multiplication.
3435
\newpage\begin{figure}[!here]
3439
\hline Algorithm \textbf{mp\_karatsuba\_sqr}. \\
3440
\textbf{Input}. mp\_int $a$ \\
3441
\textbf{Output}. $b \leftarrow a^2$ \\
3443
1. Initialize the following temporary mp\_ints: $x0$, $x1$, $t1$, $t2$, $x0x0$ and $x1x1$. \\
3444
2. If any of the initializations on step 1 failed return(\textit{MP\_MEM}). \\
3446
Split the input. e.g. $a = x1\beta^B + x0$ \\
3447
3. $B \leftarrow \lfloor a.used / 2 \rfloor$ \\
3448
4. $x0 \leftarrow a \mbox{ (mod }\beta^B\mbox{)}$ (\textit{mp\_mod\_2d}) \\
3449
5. $x1 \leftarrow \lfloor a / \beta^B \rfloor$ (\textit{mp\_lshd}) \\
3451
Calculate the three squares. \\
3452
6. $x0x0 \leftarrow x0^2$ (\textit{mp\_sqr}) \\
3453
7. $x1x1 \leftarrow x1^2$ \\
3454
8. $t1 \leftarrow x1 + x0$ (\textit{s\_mp\_add}) \\
3455
9. $t1 \leftarrow t1^2$ \\
3457
Compute the middle term. \\
3458
10. $t2 \leftarrow x0x0 + x1x1$ (\textit{s\_mp\_add}) \\
3459
11. $t1 \leftarrow t1 - t2$ \\
3461
Compute final product. \\
3462
12. $t1 \leftarrow t1\beta^B$ (\textit{mp\_lshd}) \\
3463
13. $x1x1 \leftarrow x1x1\beta^{2B}$ \\
3464
14. $t1 \leftarrow t1 + x0x0$ \\
3465
15. $b \leftarrow t1 + x1x1$ \\
3466
16. Return(\textit{MP\_OKAY}). \\
3471
\caption{Algorithm mp\_karatsuba\_sqr}
3474
\textbf{Algorithm mp\_karatsuba\_sqr.}
3475
This algorithm computes the square of an input $a$ using the Karatsuba technique. This algorithm is very similar to the Karatsuba based
3476
multiplication algorithm with the exception that the three half-size multiplications have been replaced with three half-size squarings.
3478
The radix point for squaring is simply placed exactly in the middle of the digits when the input has an odd number of digits, otherwise it is
3479
placed just below the middle. Step 3, 4 and 5 compute the two halves required using $B$
3480
as the radix point. The first two squares in steps 6 and 7 are rather straightforward while the last square is of a more compact form.
3482
By expanding $\left (x1 + x0 \right )^2$, the $x1^2$ and $x0^2$ terms in the middle disappear, that is $(x0 - x1)^2 - (x1^2 + x0^2) = 2 \cdot x0 \cdot x1$.
3483
Now if $5n$ single precision additions and a squaring of $n$-digits is faster than multiplying two $n$-digit numbers and doubling then
3484
this method is faster. Assuming no further recursions occur, the difference can be estimated with the following inequality.
3486
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
3487
machine clock cycles.}.
3490
5pn +{{q(n^2 + n)} \over 2} \le pn + qn^2
3493
For example, on an AMD Athlon XP processor $p = {1 \over 3}$ and $q = 6$. This implies that the following inequality should hold.
3495
\begin{tabular}{rcl}
3496
${5n \over 3} + 3n^2 + 3n$ & $<$ & ${n \over 3} + 6n^2$ \\
3497
${5 \over 3} + 3n + 3$ & $<$ & ${1 \over 3} + 6n$ \\
3498
${13 \over 9}$ & $<$ & $n$ \\
3502
This results in a cutoff point around $n = 2$. As a consequence it is actually faster to compute the middle term the ``long way'' on processors
3503
where multiplication is substantially slower\footnote{On the Athlon there is a 1:17 ratio between clock cycles for addition and multiplication. On
3504
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
3505
ratio of 1:7. } than simpler operations such as addition.
3507
\vspace{+3mm}\begin{small}
3508
\hspace{-5.1mm}{\bf File}: bn\_mp\_karatsuba\_sqr.c
3514
This implementation is largely based on the implementation of algorithm mp\_karatsuba\_mul. It uses the same inline style to copy and
3515
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}
3516
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
3517
to the respective halves as if mp\_rshd and mp\_mod\_2d had been used.
3519
By inlining the copy and shift operations the cutoff point for Karatsuba multiplication can be lowered. On the Athlon the cutoff point
3520
is exactly at the point where Comba squaring can no longer be used (\textit{128 digits}). On slower processors such as the Intel P4
3521
it is actually below the Comba limit (\textit{at 110 digits}).
3523
This routine uses the same error trap coding style as mp\_karatsuba\_sqr. As the temporary variables are initialized errors are
3524
redirected to the error trap higher up. If the algorithm completes without error the error code is set to \textbf{MP\_OKAY} and
3525
mp\_clears are executed normally.
3527
\subsection{Toom-Cook Squaring}
3528
The Toom-Cook squaring algorithm mp\_toom\_sqr is heavily based on the algorithm mp\_toom\_mul with the exception that squarings are used
3529
instead of multiplication to find the five relations. The reader is encouraged to read the description of the latter algorithm and try to
3530
derive their own Toom-Cook squaring algorithm.
3532
\subsection{High Level Squaring}
3533
\newpage\begin{figure}[!here]
3537
\hline Algorithm \textbf{mp\_sqr}. \\
3538
\textbf{Input}. mp\_int $a$ \\
3539
\textbf{Output}. $b \leftarrow a^2$ \\
3541
1. If $a.used \ge TOOM\_SQR\_CUTOFF$ then \\
3542
\hspace{3mm}1.1 $b \leftarrow a^2$ using algorithm mp\_toom\_sqr \\
3543
2. else if $a.used \ge KARATSUBA\_SQR\_CUTOFF$ then \\
3544
\hspace{3mm}2.1 $b \leftarrow a^2$ using algorithm mp\_karatsuba\_sqr \\
3546
\hspace{3mm}3.1 $digs \leftarrow a.used + b.used + 1$ \\
3547
\hspace{3mm}3.2 If $digs < MP\_ARRAY$ and $a.used \le \delta$ then \\
3548
\hspace{6mm}3.2.1 $b \leftarrow a^2$ using algorithm fast\_s\_mp\_sqr. \\
3549
\hspace{3mm}3.3 else \\
3550
\hspace{6mm}3.3.1 $b \leftarrow a^2$ using algorithm s\_mp\_sqr. \\
3551
4. $b.sign \leftarrow MP\_ZPOS$ \\
3552
5. Return the result of the unsigned squaring performed. \\
3557
\caption{Algorithm mp\_sqr}
3560
\textbf{Algorithm mp\_sqr.}
3561
This algorithm computes the square of the input using one of four different algorithms. If the input is very large and has at least
3562
\textbf{TOOM\_SQR\_CUTOFF} or \textbf{KARATSUBA\_SQR\_CUTOFF} digits then either the Toom-Cook or the Karatsuba Squaring algorithm is used. If
3563
neither of the polynomial basis algorithms should be used then either the Comba or baseline algorithm is used.
3565
\vspace{+3mm}\begin{small}
3566
\hspace{-5.1mm}{\bf File}: bn\_mp\_sqr.c
3572
\section*{Exercises}
3574
$\left [ 3 \right ] $ & Devise an efficient algorithm for selection of the radix point to handle inputs \\
3575
& that have different number of digits in Karatsuba multiplication. \\
3577
$\left [ 2 \right ] $ & In section 5.3 the fact that every column of a squaring is made up \\
3578
& of double products and at most one square is stated. Prove this statement. \\
3580
$\left [ 3 \right ] $ & Prove the equation for Karatsuba squaring. \\
3582
$\left [ 1 \right ] $ & Prove that Karatsuba squaring requires $O \left (n^{lg(3)} \right )$ time. \\
3584
$\left [ 2 \right ] $ & Determine the minimal ratio between addition and multiplication clock cycles \\
3585
& required for equation $6.7$ to be true. \\
3587
$\left [ 3 \right ] $ & Implement a threaded version of Comba multiplication (and squaring) where you \\
3588
& compute subsets of the columns in each thread. Determine a cutoff point where \\
3589
& it is effective and add the logic to mp\_mul() and mp\_sqr(). \\
3591
$\left [ 4 \right ] $ & Same as the previous but also modify the Karatsuba and Toom-Cook. You must \\
3592
& increase the throughput of mp\_exptmod() for random odd moduli in the range \\
3593
& $512 \ldots 4096$ bits significantly ($> 2x$) to complete this challenge. \\
3597
\chapter{Modular Reduction}
3598
\section{Basics of Modular Reduction}
3599
\index{modular residue}
3600
Modular reduction is an operation that arises quite often within public key cryptography algorithms and various number theoretic algorithms,
3601
such as factoring. Modular reduction algorithms are the third class of algorithms of the ``multipliers'' set. A number $a$ is said to be \textit{reduced}
3602
modulo another number $b$ by finding the remainder of the division $a/b$. Full integer division with remainder is a topic to be covered
3603
in~\ref{sec:division}.
3605
Modular reduction is equivalent to solving for $r$ in the following equation. $a = bq + r$ where $q = \lfloor a/b \rfloor$. The result
3606
$r$ is said to be ``congruent to $a$ modulo $b$'' which is also written as $r \equiv a \mbox{ (mod }b\mbox{)}$. In other vernacular $r$ is known as the
3607
``modular residue'' which leads to ``quadratic residue''\footnote{That's fancy talk for $b \equiv a^2 \mbox{ (mod }p\mbox{)}$.} and
3608
other forms of residues.
3610
Modular reductions are normally used to create either finite groups, rings or fields. The most common usage for performance driven modular reductions
3611
is in modular exponentiation algorithms. That is to compute $d = a^b \mbox{ (mod }c\mbox{)}$ as fast as possible. This operation is used in the
3612
RSA and Diffie-Hellman public key algorithms, for example. Modular multiplication and squaring also appears as a fundamental operation in
3613
elliptic curve cryptographic algorithms. As will be discussed in the subsequent chapter there exist fast algorithms for computing modular
3614
exponentiations without having to perform (\textit{in this example}) $b - 1$ multiplications. These algorithms will produce partial results in the
3615
range $0 \le x < c^2$ which can be taken advantage of to create several efficient algorithms. They have also been used to create redundancy check
3616
algorithms known as CRCs, error correction codes such as Reed-Solomon and solve a variety of number theoeretic problems.
3618
\section{The Barrett Reduction}
3619
The Barrett reduction algorithm \cite{BARRETT} was inspired by fast division algorithms which multiply by the reciprocal to emulate
3620
division. Barretts observation was that the residue $c$ of $a$ modulo $b$ is equal to
3623
c = a - b \cdot \lfloor a/b \rfloor
3626
Since algorithms such as modular exponentiation would be using the same modulus extensively, typical DSP\footnote{It is worth noting that Barrett's paper
3627
targeted the DSP56K processor.} intuition would indicate the next step would be to replace $a/b$ by a multiplication by the reciprocal. However,
3628
DSP intuition on its own will not work as these numbers are considerably larger than the precision of common DSP floating point data types.
3629
It would take another common optimization to optimize the algorithm.
3631
\subsection{Fixed Point Arithmetic}
3632
The trick used to optimize the above equation is based on a technique of emulating floating point data types with fixed precision integers. Fixed
3633
point arithmetic would become very popular as it greatly optimize the ``3d-shooter'' genre of games in the mid 1990s when floating point units were
3634
fairly slow if not unavailable. The idea behind fixed point arithmetic is to take a normal $k$-bit integer data type and break it into $p$-bit
3635
integer and a $q$-bit fraction part (\textit{where $p+q = k$}).
3637
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
3638
value $2.3125$. To multiply two fixed point numbers the integers are multiplied using traditional arithmetic and subsequently normalized by
3639
moving the implied decimal point back to where it should be. For example, with $q = 4$ to multiply the integers $9$ and $5$ they must be converted
3640
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
3641
fixed point representation of $5$. The product $ab$ is equal to $45(2^{2q})$ which when normalized by dividing by $2^q$ produces $45(2^q)$.
3643
This technique became popular since a normal integer multiplication and logical shift right are the only required operations to perform a multiplication
3644
of two fixed point numbers. Using fixed point arithmetic, division can be easily approximated by multiplying by the reciprocal. If $2^q$ is
3645
equivalent to one than $2^q/b$ is equivalent to the fixed point approximation of $1/b$ using real arithmetic. Using this fact dividing an integer
3646
$a$ by another integer $b$ can be achieved with the following expression.
3649
\lfloor a / b \rfloor \mbox{ }\approx\mbox{ } \lfloor (a \cdot \lfloor 2^q / b \rfloor)/2^q \rfloor
3652
The precision of the division is proportional to the value of $q$. If the divisor $b$ is used frequently as is the case with
3653
modular exponentiation pre-computing $2^q/b$ will allow a division to be performed with a multiplication and a right shift. Both operations
3654
are considerably faster than division on most processors.
3656
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
3657
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
3658
the result of the emulated division is $\lfloor 3 \cdot 19 / 2^q \rfloor = 3$ which is correct. The value of $2^q$ must be close to or ideally
3659
larger than the dividend. In effect if $a$ is the dividend then $q$ should allow $0 \le \lfloor a/2^q \rfloor \le 1$ in order for this approach
3660
to work correctly. Plugging this form of divison into the original equation the following modular residue equation arises.
3663
c = a - b \cdot \lfloor (a \cdot \lfloor 2^q / b \rfloor)/2^q \rfloor
3666
Using the notation from \cite{BARRETT} the value of $\lfloor 2^q / b \rfloor$ will be represented by the $\mu$ symbol. Using the $\mu$
3667
variable also helps re-inforce the idea that it is meant to be computed once and re-used.
3670
c = a - b \cdot \lfloor (a \cdot \mu)/2^q \rfloor
3673
Provided that $2^q \ge a$ this algorithm will produce a quotient that is either exactly correct or off by a value of one. In the context of Barrett
3674
reduction the value of $a$ is bound by $0 \le a \le (b - 1)^2$ meaning that $2^q \ge b^2$ is sufficient to ensure the reciprocal will have enough
3677
Let $n$ represent the number of digits in $b$. This algorithm requires approximately $2n^2$ single precision multiplications to produce the quotient and
3678
another $n^2$ single precision multiplications to find the residue. In total $3n^2$ single precision multiplications are required to
3681
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
3682
$a = 180388626447$ modulo $b$ using the above reduction equation. The quotient using the new formula is $\lfloor (a \cdot \mu) / 2^q \rfloor = 152913$.
3683
By subtracting $152913b$ from $a$ the correct residue $a \equiv 677346 \mbox{ (mod }b\mbox{)}$ is found.
3685
\subsection{Choosing a Radix Point}
3686
Using the fixed point representation a modular reduction can be performed with $3n^2$ single precision multiplications. If that were the best
3687
that could be achieved a full division\footnote{A division requires approximately $O(2cn^2)$ single precision multiplications for a small value of $c$.
3688
See~\ref{sec:division} for further details.} might as well be used in its place. The key to optimizing the reduction is to reduce the precision of
3689
the initial multiplication that finds the quotient.
3691
Let $a$ represent the number of which the residue is sought. Let $b$ represent the modulus used to find the residue. Let $m$ represent
3692
the number of digits in $b$. For the purposes of this discussion we will assume that the number of digits in $a$ is $2m$, which is generally true if
3693
two $m$-digit numbers have been multiplied. Dividing $a$ by $b$ is the same as dividing a $2m$ digit integer by a $m$ digit integer. Digits below the
3694
$m - 1$'th digit of $a$ will contribute at most a value of $1$ to the quotient because $\beta^k < b$ for any $0 \le k \le m - 1$. Another way to
3695
express this is by re-writing $a$ as two parts. If $a' \equiv a \mbox{ (mod }b^m\mbox{)}$ and $a'' = a - a'$ then
3696
${a \over b} \equiv {{a' + a''} \over b}$ which is equivalent to ${a' \over b} + {a'' \over b}$. Since $a'$ is bound to be less than $b$ the quotient
3697
is bound by $0 \le {a' \over b} < 1$.
3699
Since the digits of $a'$ do not contribute much to the quotient the observation is that they might as well be zero. However, if the digits
3700
``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
3701
with the irrelevant digits trimmed. Now the modular reduction is trimmed to the almost equivalent equation
3704
c = a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor
3707
Note that the original divisor $2^q$ has been replaced with $\beta^{m+1}$ where in this case $q$ is a multiple of $lg(\beta)$. Also note that the
3708
exponent on the divisor when added to the amount $q_0$ was shifted by equals $2m$. If the optimization had not been performed the divisor
3709
would have the exponent $2m$ so in the end the exponents do ``add up''. Using the above equation the quotient
3710
$\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ can be off from the true quotient by at most two. The original fixed point quotient can be off
3711
by as much as one (\textit{provided the radix point is chosen suitably}) and now that the lower irrelevent digits have been trimmed the quotient
3712
can be off by an additional value of one for a total of at most two. This implies that
3713
$0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$. By first subtracting $b$ times the quotient and then conditionally subtracting
3714
$b$ once or twice the residue is found.
3716
The quotient is now found using $(m + 1)(m) = m^2 + m$ single precision multiplications and the residue with an additional $m^2$ single
3717
precision multiplications, ignoring the subtractions required. In total $2m^2 + m$ single precision multiplications are required to find the residue.
3718
This is considerably faster than the original attempt.
3720
For example, let $\beta = 10$ represent the radix of the digits. Let $b = 9999$ represent the modulus which implies $m = 4$. Let $a = 99929878$
3721
represent the value of which the residue is desired. In this case $q = 8$ since $10^7 < 9999^2$ meaning that $\mu = \lfloor \beta^{q}/b \rfloor = 10001$.
3722
With the new observation the multiplicand for the quotient is equal to $q_0 = \lfloor a / \beta^{m - 1} \rfloor = 99929$. The quotient is then
3723
$\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor = 9993$. Subtracting $9993b$ from $a$ and the correct residue $a \equiv 9871 \mbox{ (mod }b\mbox{)}$
3726
\subsection{Trimming the Quotient}
3727
So far the reduction algorithm has been optimized from $3m^2$ single precision multiplications down to $2m^2 + m$ single precision multiplications. As
3728
it stands now the algorithm is already fairly fast compared to a full integer division algorithm. However, there is still room for
3731
After the first multiplication inside the quotient ($q_0 \cdot \mu$) the value is shifted right by $m + 1$ places effectively nullifying the lower
3732
half of the product. It would be nice to be able to remove those digits from the product to effectively cut down the number of single precision
3733
multiplications. If the number of digits in the modulus $m$ is far less than $\beta$ a full product is not required for the algorithm to work properly.
3734
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.
3736
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
3737
multiplications would be required. Using a multiplier that will only produce digits at and above the $m - 1$'th digit reduces the number
3738
of single precision multiplications to ${m^2 + m} \over 2$ single precision multiplications.
3740
\subsection{Trimming the Residue}
3741
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
3742
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
3743
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
3746
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
3747
$O(m^2)$ multiplication algorithm only the lower $m+1$ digits of the product have to be computed. Similarly the value of $a$ can
3748
be reduced modulo $\beta^{m+1}$ before the multiple of $b$ is subtracted which simplifes the subtraction as well. A multiplication that produces
3749
only the lower $m+1$ digits requires ${m^2 + 3m - 2} \over 2$ single precision multiplications.
3751
With both optimizations in place the algorithm is the algorithm Barrett proposed. It requires $m^2 + 2m - 1$ single precision multiplications which
3752
is considerably faster than the straightforward $3m^2$ method.
3754
\subsection{The Barrett Algorithm}
3755
\newpage\begin{figure}[!here]
3759
\hline Algorithm \textbf{mp\_reduce}. \\
3760
\textbf{Input}. mp\_int $a$, mp\_int $b$ and $\mu = \lfloor \beta^{2m}/b \rfloor, m = \lceil lg_{\beta}(b) \rceil, (0 \le a < b^2, b > 1)$ \\
3761
\textbf{Output}. $a \mbox{ (mod }b\mbox{)}$ \\
3763
Let $m$ represent the number of digits in $b$. \\
3764
1. Make a copy of $a$ and store it in $q$. (\textit{mp\_init\_copy}) \\
3765
2. $q \leftarrow \lfloor q / \beta^{m - 1} \rfloor$ (\textit{mp\_rshd}) \\
3767
Produce the quotient. \\
3768
3. $q \leftarrow q \cdot \mu$ (\textit{note: only produce digits at or above $m-1$}) \\
3769
4. $q \leftarrow \lfloor q / \beta^{m + 1} \rfloor$ \\
3771
Subtract the multiple of modulus from the input. \\
3772
5. $a \leftarrow a \mbox{ (mod }\beta^{m+1}\mbox{)}$ (\textit{mp\_mod\_2d}) \\
3773
6. $q \leftarrow q \cdot b \mbox{ (mod }\beta^{m+1}\mbox{)}$ (\textit{s\_mp\_mul\_digs}) \\
3774
7. $a \leftarrow a - q$ (\textit{mp\_sub}) \\
3776
Add $\beta^{m+1}$ if a carry occured. \\
3777
8. If $a < 0$ then (\textit{mp\_cmp\_d}) \\
3778
\hspace{3mm}8.1 $q \leftarrow 1$ (\textit{mp\_set}) \\
3779
\hspace{3mm}8.2 $q \leftarrow q \cdot \beta^{m+1}$ (\textit{mp\_lshd}) \\
3780
\hspace{3mm}8.3 $a \leftarrow a + q$ \\
3782
Now subtract the modulus if the residue is too large (e.g. quotient too small). \\
3783
9. While $a \ge b$ do (\textit{mp\_cmp}) \\
3784
\hspace{3mm}9.1 $c \leftarrow a - b$ \\
3786
11. Return(\textit{MP\_OKAY}) \\
3791
\caption{Algorithm mp\_reduce}
3794
\textbf{Algorithm mp\_reduce.}
3795
This algorithm will reduce the input $a$ modulo $b$ in place using the Barrett algorithm. It is loosely based on algorithm 14.42 of HAC
3796
\cite[pp. 602]{HAC} which is based on the paper from Paul Barrett \cite{BARRETT}. The algorithm has several restrictions and assumptions which must
3797
be adhered to for the algorithm to work.
3799
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
3800
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
3801
for the quotient to have enough precision. If $a$ is the product of two numbers that were already reduced modulo $b$, this will not be a problem.
3802
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
3803
algorithm and is assumed to be calculated and stored before the algorithm is used.
3805
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
3806
$s\_mp\_mul\_high\_digs$ which has not been presented is used to accomplish this task. The algorithm is based on $s\_mp\_mul\_digs$ except that
3807
instead of stopping at a given level of precision it starts at a given level of precision. This optimal algorithm can only be used if the number
3808
of digits in $b$ is very much smaller than $\beta$.
3810
While it is known that
3811
$a \ge b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ only the lower $m+1$ digits are being used to compute the residue, so an implied
3812
``borrow'' from the higher digits might leave a negative result. After the multiple of the modulus has been subtracted from $a$ the residue must be
3813
fixed up in case it is negative. The invariant $\beta^{m+1}$ must be added to the residue to make it positive again.
3815
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
3816
performed at most twice, and on average once. However, if $a \ge b^2$ than it will iterate substantially more times than it should.
3818
\vspace{+3mm}\begin{small}
3819
\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce.c
3825
The first multiplication that determines the quotient can be performed by only producing the digits from $m - 1$ and up. This essentially halves
3826
the number of single precision multiplications required. However, the optimization is only safe if $\beta$ is much larger than the number of digits
3827
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
3830
\subsection{The Barrett Setup Algorithm}
3831
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
3832
future use so that the Barrett algorithm can be used without delay.
3834
\newpage\begin{figure}[!here]
3838
\hline Algorithm \textbf{mp\_reduce\_setup}. \\
3839
\textbf{Input}. mp\_int $a$ ($a > 1$) \\
3840
\textbf{Output}. $\mu \leftarrow \lfloor \beta^{2m}/a \rfloor$ \\
3842
1. $\mu \leftarrow 2^{2 \cdot lg(\beta) \cdot m}$ (\textit{mp\_2expt}) \\
3843
2. $\mu \leftarrow \lfloor \mu / b \rfloor$ (\textit{mp\_div}) \\
3844
3. Return(\textit{MP\_OKAY}) \\
3849
\caption{Algorithm mp\_reduce\_setup}
3852
\textbf{Algorithm mp\_reduce\_setup.}
3853
This algorithm computes the reciprocal $\mu$ required for Barrett reduction. First $\beta^{2m}$ is calculated as $2^{2 \cdot lg(\beta) \cdot m}$ which
3854
is equivalent and much faster. The final value is computed by taking the integer quotient of $\lfloor \mu / b \rfloor$.
3856
\vspace{+3mm}\begin{small}
3857
\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_setup.c
3863
This simple routine calculates the reciprocal $\mu$ required by Barrett reduction. Note the extended usage of algorithm mp\_div where the variable
3864
which would received the remainder is passed as NULL. As will be discussed in~\ref{sec:division} the division routine allows both the quotient and the
3865
remainder to be passed as NULL meaning to ignore the value.
3867
\section{The Montgomery Reduction}
3868
Montgomery reduction\footnote{Thanks to Niels Ferguson for his insightful explanation of the algorithm.} \cite{MONT} is by far the most interesting
3869
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
3870
residue times a constant. However, as perplexing as this may sound the algorithm is relatively simple and very efficient.
3872
Throughout this entire section the variable $n$ will represent the modulus used to form the residue. As will be discussed shortly the value of
3873
$n$ must be odd. The variable $x$ will represent the quantity of which the residue is sought. Similar to the Barrett algorithm the input
3874
is restricted to $0 \le x < n^2$. To begin the description some simple number theory facts must be established.
3876
\textbf{Fact 1.} Adding $n$ to $x$ does not change the residue since in effect it adds one to the quotient $\lfloor x / n \rfloor$. Another way
3877
to explain this is that $n$ is (\textit{or multiples of $n$ are}) congruent to zero modulo $n$. Adding zero will not change the value of the residue.
3879
\textbf{Fact 2.} If $x$ is even then performing a division by two in $\Z$ is congruent to $x \cdot 2^{-1} \mbox{ (mod }n\mbox{)}$. Actually
3880
this is an application of the fact that if $x$ is evenly divisible by any $k \in \Z$ then division in $\Z$ will be congruent to
3881
multiplication by $k^{-1}$ modulo $n$.
3883
From these two simple facts the following simple algorithm can be derived.
3885
\newpage\begin{figure}[!here]
3889
\hline Algorithm \textbf{Montgomery Reduction}. \\
3890
\textbf{Input}. Integer $x$, $n$ and $k$ \\
3891
\textbf{Output}. $2^{-k}x \mbox{ (mod }n\mbox{)}$ \\
3893
1. for $t$ from $1$ to $k$ do \\
3894
\hspace{3mm}1.1 If $x$ is odd then \\
3895
\hspace{6mm}1.1.1 $x \leftarrow x + n$ \\
3896
\hspace{3mm}1.2 $x \leftarrow x/2$ \\
3902
\caption{Algorithm Montgomery Reduction}
3905
The algorithm reduces the input one bit at a time using the two congruencies stated previously. Inside the loop $n$, which is odd, is
3906
added to $x$ if $x$ is odd. This forces $x$ to be even which allows the division by two in $\Z$ to be congruent to a modular division by two. Since
3907
$x$ is assumed to be initially much larger than $n$ the addition of $n$ will contribute an insignificant magnitude to $x$. Let $r$ represent the
3908
final result of the Montgomery algorithm. If $k > lg(n)$ and $0 \le x < n^2$ then the final result is limited to
3909
$0 \le r < \lfloor x/2^k \rfloor + n$. As a result at most a single subtraction is required to get the residue desired.
3911
\begin{figure}[here]
3914
\begin{tabular}{|c|l|}
3915
\hline \textbf{Step number ($t$)} & \textbf{Result ($x$)} \\
3916
\hline $1$ & $x + n = 5812$, $x/2 = 2906$ \\
3917
\hline $2$ & $x/2 = 1453$ \\
3918
\hline $3$ & $x + n = 1710$, $x/2 = 855$ \\
3919
\hline $4$ & $x + n = 1112$, $x/2 = 556$ \\
3920
\hline $5$ & $x/2 = 278$ \\
3921
\hline $6$ & $x/2 = 139$ \\
3922
\hline $7$ & $x + n = 396$, $x/2 = 198$ \\
3923
\hline $8$ & $x/2 = 99$ \\
3924
\hline $9$ & $x + n = 356$, $x/2 = 178$ \\
3929
\caption{Example of Montgomery Reduction (I)}
3933
Consider the example in figure~\ref{fig:MONT1} which reduces $x = 5555$ modulo $n = 257$ when $k = 9$ (note $\beta^k = 512$ which is larger than $n$). The result of
3934
the algorithm $r = 178$ is congruent to the value of $2^{-9} \cdot 5555 \mbox{ (mod }257\mbox{)}$. When $r$ is multiplied by $2^9$ modulo $257$ the correct residue
3935
$r \equiv 158$ is produced.
3937
Let $k = \lfloor lg(n) \rfloor + 1$ represent the number of bits in $n$. The current algorithm requires $2k^2$ single precision shifts
3938
and $k^2$ single precision additions. At this rate the algorithm is most certainly slower than Barrett reduction and not terribly useful.
3939
Fortunately there exists an alternative representation of the algorithm.
3941
\begin{figure}[!here]
3945
\hline Algorithm \textbf{Montgomery Reduction} (modified I). \\
3946
\textbf{Input}. Integer $x$, $n$ and $k$ ($2^k > n$) \\
3947
\textbf{Output}. $2^{-k}x \mbox{ (mod }n\mbox{)}$ \\
3949
1. for $t$ from $1$ to $k$ do \\
3950
\hspace{3mm}1.1 If the $t$'th bit of $x$ is one then \\
3951
\hspace{6mm}1.1.1 $x \leftarrow x + 2^tn$ \\
3952
2. Return $x/2^k$. \\
3957
\caption{Algorithm Montgomery Reduction (modified I)}
3960
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
3961
precision shifts has now been reduced from $2k^2$ to $k^2 + k$ which is only a small improvement.
3963
\begin{figure}[here]
3966
\begin{tabular}{|c|l|r|}
3967
\hline \textbf{Step number ($t$)} & \textbf{Result ($x$)} & \textbf{Result ($x$) in Binary} \\
3968
\hline -- & $5555$ & $1010110110011$ \\
3969
\hline $1$ & $x + 2^{0}n = 5812$ & $1011010110100$ \\
3970
\hline $2$ & $5812$ & $1011010110100$ \\
3971
\hline $3$ & $x + 2^{2}n = 6840$ & $1101010111000$ \\
3972
\hline $4$ & $x + 2^{3}n = 8896$ & $10001011000000$ \\
3973
\hline $5$ & $8896$ & $10001011000000$ \\
3974
\hline $6$ & $8896$ & $10001011000000$ \\
3975
\hline $7$ & $x + 2^{6}n = 25344$ & $110001100000000$ \\
3976
\hline $8$ & $25344$ & $110001100000000$ \\
3977
\hline $9$ & $x + 2^{7}n = 91136$ & $10110010000000000$ \\
3978
\hline -- & $x/2^k = 178$ & \\
3983
\caption{Example of Montgomery Reduction (II)}
3987
Figure~\ref{fig:MONT2} demonstrates the modified algorithm reducing $x = 5555$ modulo $n = 257$ with $k = 9$.
3988
With this algorithm a single shift right at the end is the only right shift required to reduce the input instead of $k$ right shifts inside the
3989
loop. Note that for the iterations $t = 2, 5, 6$ and $8$ where the result $x$ is not changed. In those iterations the $t$'th bit of $x$ is
3990
zero and the appropriate multiple of $n$ does not need to be added to force the $t$'th bit of the result to zero.
3992
\subsection{Digit Based Montgomery Reduction}
3993
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
3994
previous algorithm re-written to compute the Montgomery reduction in this new fashion.
3996
\begin{figure}[!here]
4000
\hline Algorithm \textbf{Montgomery Reduction} (modified II). \\
4001
\textbf{Input}. Integer $x$, $n$ and $k$ ($\beta^k > n$) \\
4002
\textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\
4004
1. for $t$ from $0$ to $k - 1$ do \\
4005
\hspace{3mm}1.1 $x \leftarrow x + \mu n \beta^t$ \\
4006
2. Return $x/\beta^k$. \\
4011
\caption{Algorithm Montgomery Reduction (modified II)}
4014
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
4015
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
4016
problem breaks down to solving the following congruency.
4019
\begin{tabular}{rcl}
4020
$x_t + \mu n_0$ & $\equiv$ & $0 \mbox{ (mod }\beta\mbox{)}$ \\
4021
$\mu n_0$ & $\equiv$ & $-x_t \mbox{ (mod }\beta\mbox{)}$ \\
4022
$\mu$ & $\equiv$ & $-x_t/n_0 \mbox{ (mod }\beta\mbox{)}$ \\
4026
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
4027
extensively in this algorithm and should be precomputed. Let $\rho$ represent the negative of the modular inverse of $n_0$ modulo $\beta$.
4029
For example, let $\beta = 10$ represent the radix. Let $n = 17$ represent the modulus which implies $k = 2$ and $\rho \equiv 7$. Let $x = 33$
4030
represent the value to reduce.
4032
\newpage\begin{figure}
4034
\begin{tabular}{|c|c|c|}
4035
\hline \textbf{Step ($t$)} & \textbf{Value of $x$} & \textbf{Value of $\mu$} \\
4036
\hline -- & $33$ & --\\
4037
\hline $0$ & $33 + \mu n = 50$ & $1$ \\
4038
\hline $1$ & $50 + \mu n \beta = 900$ & $5$ \\
4042
\caption{Example of Montgomery Reduction}
4045
The final result $900$ is then divided by $\beta^k$ to produce the final result $9$. The first observation is that $9 \nequiv x \mbox{ (mod }n\mbox{)}$
4046
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
4047
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
4048
the correct residue is $9 \cdot 15 \equiv 16 \mbox{ (mod }n\mbox{)}$.
4050
\subsection{Baseline Montgomery Reduction}
4051
The baseline Montgomery reduction algorithm will produce the residue for any size input. It is designed to be a catch-all algororithm for
4052
Montgomery reductions.
4054
\newpage\begin{figure}[!here]
4058
\hline Algorithm \textbf{mp\_montgomery\_reduce}. \\
4059
\textbf{Input}. mp\_int $x$, mp\_int $n$ and a digit $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\
4060
\hspace{11.5mm}($0 \le x < n^2, n > 1, (n, \beta) = 1, \beta^k > n$) \\
4061
\textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\
4063
1. $digs \leftarrow 2n.used + 1$ \\
4064
2. If $digs < MP\_ARRAY$ and $m.used < \delta$ then \\
4065
\hspace{3mm}2.1 Use algorithm fast\_mp\_montgomery\_reduce instead. \\
4067
Setup $x$ for the reduction. \\
4068
3. If $x.alloc < digs$ then grow $x$ to $digs$ digits. \\
4069
4. $x.used \leftarrow digs$ \\
4071
Eliminate the lower $k$ digits. \\
4072
5. For $ix$ from $0$ to $k - 1$ do \\
4073
\hspace{3mm}5.1 $\mu \leftarrow x_{ix} \cdot \rho \mbox{ (mod }\beta\mbox{)}$ \\
4074
\hspace{3mm}5.2 $u \leftarrow 0$ \\
4075
\hspace{3mm}5.3 For $iy$ from $0$ to $k - 1$ do \\
4076
\hspace{6mm}5.3.1 $\hat r \leftarrow \mu n_{iy} + x_{ix + iy} + u$ \\
4077
\hspace{6mm}5.3.2 $x_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\
4078
\hspace{6mm}5.3.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\
4079
\hspace{3mm}5.4 While $u > 0$ do \\
4080
\hspace{6mm}5.4.1 $iy \leftarrow iy + 1$ \\
4081
\hspace{6mm}5.4.2 $x_{ix + iy} \leftarrow x_{ix + iy} + u$ \\
4082
\hspace{6mm}5.4.3 $u \leftarrow \lfloor x_{ix+iy} / \beta \rfloor$ \\
4083
\hspace{6mm}5.4.4 $x_{ix + iy} \leftarrow x_{ix+iy} \mbox{ (mod }\beta\mbox{)}$ \\
4085
Divide by $\beta^k$ and fix up as required. \\
4086
6. $x \leftarrow \lfloor x / \beta^k \rfloor$ \\
4087
7. If $x \ge n$ then \\
4088
\hspace{3mm}7.1 $x \leftarrow x - n$ \\
4089
8. Return(\textit{MP\_OKAY}). \\
4094
\caption{Algorithm mp\_montgomery\_reduce}
4097
\textbf{Algorithm mp\_montgomery\_reduce.}
4098
This algorithm reduces the input $x$ modulo $n$ in place using the Montgomery reduction algorithm. The algorithm is loosely based
4099
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
4100
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
4101
for the Barrett algorithm. Additionally if $n > 1$ and $n$ is odd there will exist a modular inverse $\rho$. $\rho$ must be calculated in
4102
advance of this algorithm. Finally the variable $k$ is fixed and a pseudonym for $n.used$.
4104
Step 2 decides whether a faster Montgomery algorithm can be used. It is based on the Comba technique meaning that there are limits on
4105
the size of the input. This algorithm is discussed in sub-section 6.3.3.
4107
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
4108
calculates $x + \mu n \beta^{ix}$ by multiplying $\mu n$ and adding the result to $x$ shifted by $ix$ digits. Both the addition and
4109
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.
4111
Using a quick inspection this algorithm requires $n$ single precision multiplications for the outer loop and $n^2$ single precision multiplications
4112
in the inner loop. In total $n^2 + n$ single precision multiplications which compares favourably to Barrett at $n^2 + 2n - 1$ single precision
4115
\vspace{+3mm}\begin{small}
4116
\hspace{-5.1mm}{\bf File}: bn\_mp\_montgomery\_reduce.c
4122
This is the baseline implementation of the Montgomery reduction algorithm. Lines 31 to 36 determine if the Comba based
4123
routine can be used instead. Line 47 computes the value of $\mu$ for that particular iteration of the outer loop.
4125
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
4126
the alias $tmpn$ refers to the modulus $n$.
4128
\subsection{Faster ``Comba'' Montgomery Reduction}
4130
The Montgomery reduction requires fewer single precision multiplications than a Barrett reduction, however it is much slower due to the serial
4131
nature of the inner loop. The Barrett reduction algorithm requires two slightly modified multipliers which can be implemented with the Comba
4132
technique. The Montgomery reduction algorithm cannot directly use the Comba technique to any significant advantage since the inner loop calculates
4133
a $k \times 1$ product $k$ times.
4135
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
4136
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.
4137
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.
4139
With this change in place the Montgomery reduction algorithm can be performed with a Comba style multiplication loop which substantially increases
4140
the speed of the algorithm.
4142
\newpage\begin{figure}[!here]
4146
\hline Algorithm \textbf{fast\_mp\_montgomery\_reduce}. \\
4147
\textbf{Input}. mp\_int $x$, mp\_int $n$ and a digit $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\
4148
\hspace{11.5mm}($0 \le x < n^2, n > 1, (n, \beta) = 1, \beta^k > n$) \\
4149
\textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\
4151
Place an array of \textbf{MP\_WARRAY} mp\_word variables called $\hat W$ on the stack. \\
4152
1. if $x.alloc < n.used + 1$ then grow $x$ to $n.used + 1$ digits. \\
4153
Copy the digits of $x$ into the array $\hat W$ \\
4154
2. For $ix$ from $0$ to $x.used - 1$ do \\
4155
\hspace{3mm}2.1 $\hat W_{ix} \leftarrow x_{ix}$ \\
4156
3. For $ix$ from $x.used$ to $2n.used - 1$ do \\
4157
\hspace{3mm}3.1 $\hat W_{ix} \leftarrow 0$ \\
4158
Elimiate the lower $k$ digits. \\
4159
4. for $ix$ from $0$ to $n.used - 1$ do \\
4160
\hspace{3mm}4.1 $\mu \leftarrow \hat W_{ix} \cdot \rho \mbox{ (mod }\beta\mbox{)}$ \\
4161
\hspace{3mm}4.2 For $iy$ from $0$ to $n.used - 1$ do \\
4162
\hspace{6mm}4.2.1 $\hat W_{iy + ix} \leftarrow \hat W_{iy + ix} + \mu \cdot n_{iy}$ \\
4163
\hspace{3mm}4.3 $\hat W_{ix + 1} \leftarrow \hat W_{ix + 1} + \lfloor \hat W_{ix} / \beta \rfloor$ \\
4164
Propagate carries upwards. \\
4165
5. for $ix$ from $n.used$ to $2n.used + 1$ do \\
4166
\hspace{3mm}5.1 $\hat W_{ix + 1} \leftarrow \hat W_{ix + 1} + \lfloor \hat W_{ix} / \beta \rfloor$ \\
4167
Shift right and reduce modulo $\beta$ simultaneously. \\
4168
6. for $ix$ from $0$ to $n.used + 1$ do \\
4169
\hspace{3mm}6.1 $x_{ix} \leftarrow \hat W_{ix + n.used} \mbox{ (mod }\beta\mbox{)}$ \\
4170
Zero excess digits and fixup $x$. \\
4171
7. if $x.used > n.used + 1$ then do \\
4172
\hspace{3mm}7.1 for $ix$ from $n.used + 1$ to $x.used - 1$ do \\
4173
\hspace{6mm}7.1.1 $x_{ix} \leftarrow 0$ \\
4174
8. $x.used \leftarrow n.used + 1$ \\
4175
9. Clamp excessive digits of $x$. \\
4176
10. If $x \ge n$ then \\
4177
\hspace{3mm}10.1 $x \leftarrow x - n$ \\
4178
11. Return(\textit{MP\_OKAY}). \\
4183
\caption{Algorithm fast\_mp\_montgomery\_reduce}
4186
\textbf{Algorithm fast\_mp\_montgomery\_reduce.}
4187
This algorithm will compute the Montgomery reduction of $x$ modulo $n$ using the Comba technique. It is on most computer platforms significantly
4188
faster than algorithm mp\_montgomery\_reduce and algorithm mp\_reduce (\textit{Barrett reduction}). The algorithm has the same restrictions
4189
on the input as the baseline reduction algorithm. An additional two restrictions are imposed on this algorithm. The number of digits $k$ in the
4190
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
4191
a modulus of at most $3,556$ bits in length.
4193
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
4194
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
4195
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
4196
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
4197
a single precision multiplication instead half the amount of time is spent.
4199
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
4200
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
4201
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
4204
Step 5 will propagate the remainder of the carries upwards. On step 6 the columns are reduced modulo $\beta$ and shifted simultaneously as they are
4205
stored in the destination $x$.
4207
\vspace{+3mm}\begin{small}
4208
\hspace{-5.1mm}{\bf File}: bn\_fast\_mp\_montgomery\_reduce.c
4214
The $\hat W$ array is first filled with digits of $x$ on line 48 then the rest of the digits are zeroed on line 55. Both loops share
4215
the same alias variables to make the code easier to read.
4217
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
4218
forces the compiler to use a single precision multiplication and prevents any concerns about loss of precision. Line 110 fixes the carry
4219
for the next iteration of the loop by propagating the carry from $\hat W_{ix}$ to $\hat W_{ix+1}$.
4221
The for loop on line 109 propagates the rest of the carries upwards through the columns. The for loop on line 126 reduces the columns
4222
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
4223
digit, that is $\_ \hat W_{t} = \hat W_{n.used + t}$.
4225
\subsection{Montgomery Setup}
4226
To calculate the variable $\rho$ a relatively simple algorithm will be required.
4228
\begin{figure}[!here]
4232
\hline Algorithm \textbf{mp\_montgomery\_setup}. \\
4233
\textbf{Input}. mp\_int $n$ ($n > 1$ and $(n, 2) = 1$) \\
4234
\textbf{Output}. $\rho \equiv -1/n_0 \mbox{ (mod }\beta\mbox{)}$ \\
4236
1. $b \leftarrow n_0$ \\
4237
2. If $b$ is even return(\textit{MP\_VAL}) \\
4238
3. $x \leftarrow (((b + 2) \mbox{ AND } 4) << 1) + b$ \\
4239
4. for $k$ from 0 to $\lceil lg(lg(\beta)) \rceil - 2$ do \\
4240
\hspace{3mm}4.1 $x \leftarrow x \cdot (2 - bx)$ \\
4241
5. $\rho \leftarrow \beta - x \mbox{ (mod }\beta\mbox{)}$ \\
4242
6. Return(\textit{MP\_OKAY}). \\
4247
\caption{Algorithm mp\_montgomery\_setup}
4250
\textbf{Algorithm mp\_montgomery\_setup.}
4251
This algorithm will calculate the value of $\rho$ required within the Montgomery reduction algorithms. It uses a very interesting trick
4252
to calculate $1/n_0$ when $\beta$ is a power of two.
4254
\vspace{+3mm}\begin{small}
4255
\hspace{-5.1mm}{\bf File}: bn\_mp\_montgomery\_setup.c
4261
This source code computes the value of $\rho$ required to perform Montgomery reduction. It has been modified to avoid performing excess
4262
multiplications when $\beta$ is not the default 28-bits.
4264
\section{The Diminished Radix Algorithm}
4265
The Diminished Radix method of modular reduction \cite{DRMET} is a fairly clever technique which can be more efficient than either the Barrett
4266
or Montgomery methods for certain forms of moduli. The technique is based on the following simple congruence.
4269
(x \mbox{ mod } n) + k \lfloor x / n \rfloor \equiv x \mbox{ (mod }(n - k)\mbox{)}
4272
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
4273
then a x86 multiplier could produce the 62-bit product and use the ``shrd'' instruction to perform a double-precision right shift. The proof
4274
of the above equation is very simple. First write $x$ in the product form.
4280
Now reduce both sides modulo $(n - k)$.
4283
x \equiv qk + r \mbox{ (mod }(n-k)\mbox{)}
4286
The variable $n$ reduces modulo $n - k$ to $k$. By putting $q = \lfloor x/n \rfloor$ and $r = x \mbox{ mod } n$
4287
into the equation the original congruence is reproduced, thus concluding the proof. The following algorithm is based on this observation.
4289
\begin{figure}[!here]
4293
\hline Algorithm \textbf{Diminished Radix Reduction}. \\
4294
\textbf{Input}. Integer $x$, $n$, $k$ \\
4295
\textbf{Output}. $x \mbox{ mod } (n - k)$ \\
4297
1. $q \leftarrow \lfloor x / n \rfloor$ \\
4298
2. $q \leftarrow k \cdot q$ \\
4299
3. $x \leftarrow x \mbox{ (mod }n\mbox{)}$ \\
4300
4. $x \leftarrow x + q$ \\
4301
5. If $x \ge (n - k)$ then \\
4302
\hspace{3mm}5.1 $x \leftarrow x - (n - k)$ \\
4303
\hspace{3mm}5.2 Goto step 1. \\
4309
\caption{Algorithm Diminished Radix Reduction}
4313
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
4314
once or twice and occasionally three times. For simplicity sake the value of $x$ is bounded by the following simple polynomial.
4317
0 \le x < n^2 + k^2 - 2nk
4320
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.
4326
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
4327
$0 \le x < n$. By step four the sum $x + q$ is bounded by
4330
0 \le q + x < (k + 1)n - 2k^2 - 1
4333
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
4334
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
4335
range $0 \le x < (n - k - 1)^2$.
4340
\begin{tabular}{|l|}
4342
$x = 123456789, n = 256, k = 3$ \\
4343
\hline $q \leftarrow \lfloor x/n \rfloor = 482253$ \\
4344
$q \leftarrow q*k = 1446759$ \\
4345
$x \leftarrow x \mbox{ mod } n = 21$ \\
4346
$x \leftarrow x + q = 1446780$ \\
4347
$x \leftarrow x - (n - k) = 1446527$ \\
4349
$q \leftarrow \lfloor x/n \rfloor = 5650$ \\
4350
$q \leftarrow q*k = 16950$ \\
4351
$x \leftarrow x \mbox{ mod } n = 127$ \\
4352
$x \leftarrow x + q = 17077$ \\
4353
$x \leftarrow x - (n - k) = 16824$ \\
4355
$q \leftarrow \lfloor x/n \rfloor = 65$ \\
4356
$q \leftarrow q*k = 195$ \\
4357
$x \leftarrow x \mbox{ mod } n = 184$ \\
4358
$x \leftarrow x + q = 379$ \\
4359
$x \leftarrow x - (n - k) = 126$ \\
4364
\caption{Example Diminished Radix Reduction}
4368
Figure~\ref{fig:EXDR} demonstrates the reduction of $x = 123456789$ modulo $n - k = 253$ when $n = 256$ and $k = 3$. Note that even while $x$
4369
is considerably larger than $(n - k - 1)^2 = 63504$ the algorithm still converges on the modular residue exceedingly fast. In this case only
4370
three passes were required to find the residue $x \equiv 126$.
4373
\subsection{Choice of Moduli}
4374
On the surface this algorithm looks like a very expensive algorithm. It requires a couple of subtractions followed by multiplication and other
4375
modular reductions. The usefulness of this algorithm becomes exceedingly clear when an appropriate modulus is chosen.
4377
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.
4378
Division by ten for example is simple for pencil and paper mathematics since it amounts to shifting the decimal place to the right. Similarly division
4379
by two (\textit{or powers of two}) is very simple for binary computers to perform. It would therefore seem logical to choose $n$ of the form $2^p$
4380
which would imply that $\lfloor x / n \rfloor$ is a simple shift of $x$ right $p$ bits.
4382
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
4383
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 $p$.
4384
Also with the choice of $n = \beta^p$ reducing $x$ modulo $n$ merely requires zeroing the digits above the $p-1$'th digit of $x$.
4386
Throughout the next section the term ``restricted modulus'' will refer to a modulus of the form $\beta^p - k$ whereas the term ``unrestricted
4387
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
4388
$2^p$ logic except $p$ must be a multiple of $lg(\beta)$.
4390
\subsection{Choice of $k$}
4391
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 $k$
4392
in step 2 is the most expensive operation. Fortunately the choice of $k$ is not terribly limited. For all intents and purposes it might
4393
as well be a single digit. The smaller the value of $k$ is the faster the algorithm will be.
4395
\subsection{Restricted Diminished Radix Reduction}
4396
The restricted Diminished Radix algorithm can quickly reduce an input modulo a modulus of the form $n = \beta^p - k$. This algorithm can reduce
4397
an input $x$ within the range $0 \le x < n^2$ using only a couple passes of the algorithm demonstrated in figure~\ref{fig:DR}. The implementation
4398
of this algorithm has been optimized to avoid additional overhead associated with a division by $\beta^p$, the multiplication by $k$ or the addition
4399
of $x$ and $q$. The resulting algorithm is very efficient and can lead to substantial improvements over Barrett and Montgomery reduction when modular
4400
exponentiations are performed.
4402
\newpage\begin{figure}[!here]
4406
\hline Algorithm \textbf{mp\_dr\_reduce}. \\
4407
\textbf{Input}. mp\_int $x$, $n$ and a mp\_digit $k = \beta - n_0$ \\
4408
\hspace{11.5mm}($0 \le x < n^2$, $n > 1$, $0 < k < \beta$) \\
4409
\textbf{Output}. $x \mbox{ mod } n$ \\
4411
1. $m \leftarrow n.used$ \\
4412
2. If $x.alloc < 2m$ then grow $x$ to $2m$ digits. \\
4413
3. $\mu \leftarrow 0$ \\
4414
4. for $i$ from $0$ to $m - 1$ do \\
4415
\hspace{3mm}4.1 $\hat r \leftarrow k \cdot x_{m+i} + x_{i} + \mu$ \\
4416
\hspace{3mm}4.2 $x_{i} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\
4417
\hspace{3mm}4.3 $\mu \leftarrow \lfloor \hat r / \beta \rfloor$ \\
4418
5. $x_{m} \leftarrow \mu$ \\
4419
6. for $i$ from $m + 1$ to $x.used - 1$ do \\
4420
\hspace{3mm}6.1 $x_{i} \leftarrow 0$ \\
4421
7. Clamp excess digits of $x$. \\
4422
8. If $x \ge n$ then \\
4423
\hspace{3mm}8.1 $x \leftarrow x - n$ \\
4424
\hspace{3mm}8.2 Goto step 3. \\
4425
9. Return(\textit{MP\_OKAY}). \\
4430
\caption{Algorithm mp\_dr\_reduce}
4433
\textbf{Algorithm mp\_dr\_reduce.}
4434
This algorithm will perform the Dimished Radix reduction of $x$ modulo $n$. It has similar restrictions to that of the Barrett reduction
4435
with the addition that $n$ must be of the form $n = \beta^m - k$ where $0 < k <\beta$.
4437
This algorithm essentially implements the pseudo-code in figure~\ref{fig:DR} except with a slight optimization. The division by $\beta^m$, multiplication by $k$
4438
and addition of $x \mbox{ mod }\beta^m$ are all performed simultaneously inside the loop on step 4. The division by $\beta^m$ is emulated by accessing
4439
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
4440
digit is set to the carry and the upper digits are zeroed. Steps 5 and 6 emulate the reduction modulo $\beta^m$ that should have happend to
4441
$x$ before the addition of the multiple of the upper half.
4443
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
4446
\vspace{+3mm}\begin{small}
4447
\hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_reduce.c
4453
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 52 is where
4454
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
4455
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.
4457
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
4458
a division by $\beta^m$ can be simulated virtually for free. The loop on line 64 performs the bulk of the work (\textit{corresponds to step 4 of algorithm 7.11})
4461
By line 67 the pointer $tmpx1$ points to the $m$'th digit of $x$ which is where the final carry will be placed. Similarly by line 74 the
4462
same pointer will point to the $m+1$'th digit where the zeroes will be placed.
4464
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.
4465
With the same logic at line 81 the value of $x$ is known to be greater than or equal to $n$ meaning that an unsigned subtraction can be used
4466
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
4467
does not need to be checked.
4469
\subsubsection{Setup}
4470
To setup the restricted Diminished Radix algorithm the value $k = \beta - n_0$ is required. This algorithm is not really complicated but provided for
4473
\begin{figure}[!here]
4477
\hline Algorithm \textbf{mp\_dr\_setup}. \\
4478
\textbf{Input}. mp\_int $n$ \\
4479
\textbf{Output}. $k = \beta - n_0$ \\
4481
1. $k \leftarrow \beta - n_0$ \\
4486
\caption{Algorithm mp\_dr\_setup}
4489
\vspace{+3mm}\begin{small}
4490
\hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_setup.c
4496
\subsubsection{Modulus Detection}
4497
Another algorithm which will be useful is the ability to detect a restricted Diminished Radix modulus. An integer is said to be
4498
of restricted Diminished Radix form if all of the digits are equal to $\beta - 1$ except the trailing digit which may be any value.
4500
\begin{figure}[!here]
4504
\hline Algorithm \textbf{mp\_dr\_is\_modulus}. \\
4505
\textbf{Input}. mp\_int $n$ \\
4506
\textbf{Output}. $1$ if $n$ is in D.R form, $0$ otherwise \\
4508
1. If $n.used < 2$ then return($0$). \\
4509
2. for $ix$ from $1$ to $n.used - 1$ do \\
4510
\hspace{3mm}2.1 If $n_{ix} \ne \beta - 1$ return($0$). \\
4516
\caption{Algorithm mp\_dr\_is\_modulus}
4519
\textbf{Algorithm mp\_dr\_is\_modulus.}
4520
This algorithm determines if a value is in Diminished Radix form. Step 1 rejects obvious cases where fewer than two digits are
4521
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
4522
step 3 then $n$ must be of Diminished Radix form.
4524
\vspace{+3mm}\begin{small}
4525
\hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_is\_modulus.c
4531
\subsection{Unrestricted Diminished Radix Reduction}
4532
The unrestricted Diminished Radix algorithm allows modular reductions to be performed when the modulus is of the form $2^p - k$. This algorithm
4533
is a straightforward adaptation of algorithm~\ref{fig:DR}.
4535
In general the restricted Diminished Radix reduction algorithm is much faster since it has considerably lower overhead. However, this new
4536
algorithm is much faster than either Montgomery or Barrett reduction when the moduli are of the appropriate form.
4538
\begin{figure}[!here]
4542
\hline Algorithm \textbf{mp\_reduce\_2k}. \\
4543
\textbf{Input}. mp\_int $a$ and $n$. mp\_digit $k$ \\
4544
\hspace{11.5mm}($a \ge 0$, $n > 1$, $0 < k < \beta$, $n + k$ is a power of two) \\
4545
\textbf{Output}. $a \mbox{ (mod }n\mbox{)}$ \\
4547
1. $p \leftarrow \lceil lg(n) \rceil$ (\textit{mp\_count\_bits}) \\
4548
2. While $a \ge n$ do \\
4549
\hspace{3mm}2.1 $q \leftarrow \lfloor a / 2^p \rfloor$ (\textit{mp\_div\_2d}) \\
4550
\hspace{3mm}2.2 $a \leftarrow a \mbox{ (mod }2^p\mbox{)}$ (\textit{mp\_mod\_2d}) \\
4551
\hspace{3mm}2.3 $q \leftarrow q \cdot k$ (\textit{mp\_mul\_d}) \\
4552
\hspace{3mm}2.4 $a \leftarrow a - q$ (\textit{s\_mp\_sub}) \\
4553
\hspace{3mm}2.5 If $a \ge n$ then do \\
4554
\hspace{6mm}2.5.1 $a \leftarrow a - n$ \\
4555
3. Return(\textit{MP\_OKAY}). \\
4560
\caption{Algorithm mp\_reduce\_2k}
4563
\textbf{Algorithm mp\_reduce\_2k.}
4564
This algorithm quickly reduces an input $a$ modulo an unrestricted Diminished Radix modulus $n$. Division by $2^p$ is emulated with a right
4565
shift which makes the algorithm fairly inexpensive to use.
4567
\vspace{+3mm}\begin{small}
4568
\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_2k.c
4574
The algorithm mp\_count\_bits calculates the number of bits in an mp\_int which is used to find the initial value of $p$. The call to mp\_div\_2d
4575
on line 31 calculates both the quotient $q$ and the remainder $a$ required. By doing both in a single function call the code size
4576
is kept fairly small. The multiplication by $k$ is only performed if $k > 1$. This allows reductions modulo $2^p - 1$ to be performed without
4577
any multiplications.
4579
The unsigned s\_mp\_add, mp\_cmp\_mag and s\_mp\_sub are used in place of their full sign counterparts since the inputs are only valid if they are
4580
positive. By using the unsigned versions the overhead is kept to a minimum.
4582
\subsubsection{Unrestricted Setup}
4583
To setup this reduction algorithm the value of $k = 2^p - n$ is required.
4585
\begin{figure}[!here]
4589
\hline Algorithm \textbf{mp\_reduce\_2k\_setup}. \\
4590
\textbf{Input}. mp\_int $n$ \\
4591
\textbf{Output}. $k = 2^p - n$ \\
4593
1. $p \leftarrow \lceil lg(n) \rceil$ (\textit{mp\_count\_bits}) \\
4594
2. $x \leftarrow 2^p$ (\textit{mp\_2expt}) \\
4595
3. $x \leftarrow x - n$ (\textit{mp\_sub}) \\
4596
4. $k \leftarrow x_0$ \\
4597
5. Return(\textit{MP\_OKAY}). \\
4602
\caption{Algorithm mp\_reduce\_2k\_setup}
4605
\textbf{Algorithm mp\_reduce\_2k\_setup.}
4606
This algorithm computes the value of $k$ required for the algorithm mp\_reduce\_2k. By making a temporary variable $x$ equal to $2^p$ a subtraction
4607
is sufficient to solve for $k$. Alternatively if $n$ has more than one digit the value of $k$ is simply $\beta - n_0$.
4609
\vspace{+3mm}\begin{small}
4610
\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_2k\_setup.c
4616
\subsubsection{Unrestricted Detection}
4617
An integer $n$ is a valid unrestricted Diminished Radix modulus if either of the following are true.
4620
\item The number has only one digit.
4621
\item The number has more than one digit and every bit from the $\beta$'th to the most significant is one.
4624
If either condition is true than there is a power of two $2^p$ such that $0 < 2^p - n < \beta$. If the input is only
4625
one digit than it will always be of the correct form. Otherwise all of the bits above the first digit must be one. This arises from the fact
4626
that there will be value of $k$ that when added to the modulus causes a carry in the first digit which propagates all the way to the most
4627
significant bit. The resulting sum will be a power of two.
4629
\begin{figure}[!here]
4633
\hline Algorithm \textbf{mp\_reduce\_is\_2k}. \\
4634
\textbf{Input}. mp\_int $n$ \\
4635
\textbf{Output}. $1$ if of proper form, $0$ otherwise \\
4637
1. If $n.used = 0$ then return($0$). \\
4638
2. If $n.used = 1$ then return($1$). \\
4639
3. $p \leftarrow \lceil lg(n) \rceil$ (\textit{mp\_count\_bits}) \\
4640
4. for $x$ from $lg(\beta)$ to $p$ do \\
4641
\hspace{3mm}4.1 If the ($x \mbox{ mod }lg(\beta)$)'th bit of the $\lfloor x / lg(\beta) \rfloor$ of $n$ is zero then return($0$). \\
4647
\caption{Algorithm mp\_reduce\_is\_2k}
4650
\textbf{Algorithm mp\_reduce\_is\_2k.}
4651
This algorithm quickly determines if a modulus is of the form required for algorithm mp\_reduce\_2k to function properly.
4653
\vspace{+3mm}\begin{small}
4654
\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_is\_2k.c
4662
\section{Algorithm Comparison}
4663
So far three very different algorithms for modular reduction have been discussed. Each of the algorithms have their own strengths and weaknesses
4664
that makes having such a selection very useful. The following table sumarizes the three algorithms along with comparisons of work factors. Since
4665
all three algorithms have the restriction that $0 \le x < n^2$ and $n > 1$ those limitations are not included in the table.
4669
\begin{tabular}{|c|c|c|c|c|c|}
4670
\hline \textbf{Method} & \textbf{Work Required} & \textbf{Limitations} & \textbf{$m = 8$} & \textbf{$m = 32$} & \textbf{$m = 64$} \\
4671
\hline Barrett & $m^2 + 2m - 1$ & None & $79$ & $1087$ & $4223$ \\
4672
\hline Montgomery & $m^2 + m$ & $n$ must be odd & $72$ & $1056$ & $4160$ \\
4673
\hline D.R. & $2m$ & $n = \beta^m - k$ & $16$ & $64$ & $128$ \\
4679
In theory Montgomery and Barrett reductions would require roughly the same amount of time to complete. However, in practice since Montgomery
4680
reduction can be written as a single function with the Comba technique it is much faster. Barrett reduction suffers from the overhead of
4681
calling the half precision multipliers, addition and division by $\beta$ algorithms.
4683
For almost every cryptographic algorithm Montgomery reduction is the algorithm of choice. The one set of algorithms where Diminished Radix reduction truly
4684
shines are based on the discrete logarithm problem such as Diffie-Hellman \cite{DH} and ElGamal \cite{ELGAMAL}. In these algorithms
4685
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
4686
modular exponentiation to greatly speed up the operation.
4690
\section*{Exercises}
4692
$\left [ 3 \right ]$ & Prove that the ``trick'' in algorithm mp\_montgomery\_setup actually \\
4693
& calculates the correct value of $\rho$. \\
4695
$\left [ 2 \right ]$ & Devise an algorithm to reduce modulo $n + k$ for small $k$ quickly. \\
4697
$\left [ 4 \right ]$ & Prove that the pseudo-code algorithm ``Diminished Radix Reduction'' \\
4698
& (\textit{figure~\ref{fig:DR}}) terminates. Also prove the probability that it will \\
4699
& terminate within $1 \le k \le 10$ iterations. \\
4704
\chapter{Exponentiation}
4705
Exponentiation is the operation of raising one variable to the power of another, for example, $a^b$. A variant of exponentiation, computed
4706
in a finite field or ring, is called modular exponentiation. This latter style of operation is typically used in public key
4707
cryptosystems such as RSA and Diffie-Hellman. The ability to quickly compute modular exponentiations is of great benefit to any
4708
such cryptosystem and many methods have been sought to speed it up.
4710
\section{Exponentiation Basics}
4711
A trivial algorithm would simply multiply $a$ against itself $b - 1$ times to compute the exponentiation desired. However, as $b$ grows in size
4712
the number of multiplications becomes prohibitive. Imagine what would happen if $b$ $\approx$ $2^{1024}$ as is the case when computing an RSA signature
4713
with a $1024$-bit key. Such a calculation could never be completed as it would take simply far too long.
4715
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
4716
are two trivial relationships between the base and the exponent. Let $b_i$ represent the $i$'th bit of $b$ starting from the least
4717
significant bit. If $b$ is a $k$-bit integer than the following equation is true.
4720
a^b = \prod_{i=0}^{k-1} a^{2^i \cdot b_i}
4723
By taking the base $a$ logarithm of both sides of the equation the following equation is the result.
4726
b = \sum_{i=0}^{k-1}2^i \cdot b_i
4729
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
4730
$a^{2^{i+1}}$. This observation forms the basis of essentially all fast exponentiation algorithms. It requires $k$ squarings and on average
4731
$k \over 2$ multiplications to compute the result. This is indeed quite an improvement over simply multiplying by $a$ a total of $b-1$ times.
4733
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
4734
be computed in an auxilary variable. Consider the following equivalent algorithm.
4736
\begin{figure}[!here]
4740
\hline Algorithm \textbf{Left to Right Exponentiation}. \\
4741
\textbf{Input}. Integer $a$, $b$ and $k$ \\
4742
\textbf{Output}. $c = a^b$ \\
4744
1. $c \leftarrow 1$ \\
4745
2. for $i$ from $k - 1$ to $0$ do \\
4746
\hspace{3mm}2.1 $c \leftarrow c^2$ \\
4747
\hspace{3mm}2.2 $c \leftarrow c \cdot a^{b_i}$ \\
4753
\caption{Left to Right Exponentiation}
4757
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
4758
multiplied against the current product. In each iteration the product is squared which doubles the exponent of the individual terms of the
4761
For example, let $b = 101100_2 \equiv 44_{10}$. The following chart demonstrates the actions of the algorithm.
4763
\newpage\begin{figure}
4765
\begin{tabular}{|c|c|}
4766
\hline \textbf{Value of $i$} & \textbf{Value of $c$} \\
4769
\hline $4$ & $a^2$ \\
4770
\hline $3$ & $a^4 \cdot a$ \\
4771
\hline $2$ & $a^8 \cdot a^2 \cdot a$ \\
4772
\hline $1$ & $a^{16} \cdot a^4 \cdot a^2$ \\
4773
\hline $0$ & $a^{32} \cdot a^8 \cdot a^4$ \\
4777
\caption{Example of Left to Right Exponentiation}
4780
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
4781
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.
4783
\subsection{Single Digit Exponentiation}
4784
The first algorithm in the series of exponentiation algorithms will be an unbounded algorithm where the exponent is a single digit. It is intended
4785
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
4786
$b$ that are greater than three.
4788
\newpage\begin{figure}[!here]
4792
\hline Algorithm \textbf{mp\_expt\_d}. \\
4793
\textbf{Input}. mp\_int $a$ and mp\_digit $b$ \\
4794
\textbf{Output}. $c = a^b$ \\
4796
1. $g \leftarrow a$ (\textit{mp\_init\_copy}) \\
4797
2. $c \leftarrow 1$ (\textit{mp\_set}) \\
4798
3. for $x$ from 1 to $lg(\beta)$ do \\
4799
\hspace{3mm}3.1 $c \leftarrow c^2$ (\textit{mp\_sqr}) \\
4800
\hspace{3mm}3.2 If $b$ AND $2^{lg(\beta) - 1} \ne 0$ then \\
4801
\hspace{6mm}3.2.1 $c \leftarrow c \cdot g$ (\textit{mp\_mul}) \\
4802
\hspace{3mm}3.3 $b \leftarrow b << 1$ \\
4804
5. Return(\textit{MP\_OKAY}). \\
4809
\caption{Algorithm mp\_expt\_d}
4812
\textbf{Algorithm mp\_expt\_d.}
4813
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
4814
quickly compute the exponentiation. It is loosely based on algorithm 14.79 of HAC \cite[pp. 615]{HAC} with the difference that the
4815
exponent is a fixed width.
4817
A copy of $a$ is made first to allow destination variable $c$ be the same as the source variable $a$. The result is set to the initial value of
4818
$1$ in the subsequent step.
4820
Inside the loop the exponent is read from the most significant bit first down to the least significant bit. First $c$ is invariably squared
4821
on step 3.1. In the following step if the most significant bit of $b$ is one the copy of $a$ is multiplied against $c$. The value
4822
of $b$ is shifted left one bit to make the next bit down from the most signficant bit the new most significant bit. In effect each
4823
iteration of the loop moves the bits of the exponent $b$ upwards to the most significant location.
4825
\vspace{+3mm}\begin{small}
4826
\hspace{-5.1mm}{\bf File}: bn\_mp\_expt\_d.c
4832
Line 29 sets the initial value of the result to $1$. Next the loop on line 31 steps through each bit of the exponent starting from
4833
the most significant down towards the least significant. The invariant squaring operation placed on line 33 is performed first. After
4834
the squaring the result $c$ is multiplied by the base $g$ if and only if the most significant bit of the exponent is set. The shift on line
4835
47 moves all of the bits of the exponent upwards towards the most significant location.
4837
\section{$k$-ary Exponentiation}
4838
When calculating an exponentiation the most time consuming bottleneck is the multiplications which are in general a small factor
4839
slower than squaring. Recall from the previous algorithm that $b_{i}$ refers to the $i$'th bit of the exponent $b$. Suppose instead it referred to
4840
the $i$'th $k$-bit digit of the exponent of $b$. For $k = 1$ the definitions are synonymous and for $k > 1$ algorithm~\ref{fig:KARY}
4841
computes the same exponentiation. A group of $k$ bits from the exponent is called a \textit{window}. That is it is a small window on only a
4842
portion of the entire exponent. Consider the following modification to the basic left to right exponentiation algorithm.
4844
\begin{figure}[!here]
4848
\hline Algorithm \textbf{$k$-ary Exponentiation}. \\
4849
\textbf{Input}. Integer $a$, $b$, $k$ and $t$ \\
4850
\textbf{Output}. $c = a^b$ \\
4852
1. $c \leftarrow 1$ \\
4853
2. for $i$ from $t - 1$ to $0$ do \\
4854
\hspace{3mm}2.1 $c \leftarrow c^{2^k} $ \\
4855
\hspace{3mm}2.2 Extract the $i$'th $k$-bit word from $b$ and store it in $g$. \\
4856
\hspace{3mm}2.3 $c \leftarrow c \cdot a^g$ \\
4862
\caption{$k$-ary Exponentiation}
4866
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
4867
precomputed this algorithm requires only $t$ multiplications and $tk$ squarings. The table can be generated with $2^{k - 1} - 1$ squarings and
4868
$2^{k - 1} + 1$ multiplications. This algorithm assumes that the number of bits in the exponent is evenly divisible by $k$.
4869
However, when it is not the remaining $0 < x \le k - 1$ bits can be handled with algorithm~\ref{fig:LTOR}.
4871
Suppose $k = 4$ and $t = 100$. This modified algorithm will require $109$ multiplications and $408$ squarings to compute the exponentiation. The
4872
original algorithm would on average have required $200$ multiplications and $400$ squrings to compute the same value. The total number of squarings
4873
has increased slightly but the number of multiplications has nearly halved.
4875
\subsection{Optimal Values of $k$}
4876
An optimal value of $k$ will minimize $2^{k} + \lceil n / k \rceil + n - 1$ for a fixed number of bits in the exponent $n$. The simplest
4877
approach is to brute force search amongst the values $k = 2, 3, \ldots, 8$ for the lowest result. Table~\ref{fig:OPTK} lists optimal values of $k$
4878
for various exponent sizes and compares the number of multiplication and squarings required against algorithm~\ref{fig:LTOR}.
4880
\begin{figure}[here]
4883
\begin{tabular}{|c|c|c|c|c|c|}
4884
\hline \textbf{Exponent (bits)} & \textbf{Optimal $k$} & \textbf{Work at $k$} & \textbf{Work with ~\ref{fig:LTOR}} \\
4885
\hline $16$ & $2$ & $27$ & $24$ \\
4886
\hline $32$ & $3$ & $49$ & $48$ \\
4887
\hline $64$ & $3$ & $92$ & $96$ \\
4888
\hline $128$ & $4$ & $175$ & $192$ \\
4889
\hline $256$ & $4$ & $335$ & $384$ \\
4890
\hline $512$ & $5$ & $645$ & $768$ \\
4891
\hline $1024$ & $6$ & $1257$ & $1536$ \\
4892
\hline $2048$ & $6$ & $2452$ & $3072$ \\
4893
\hline $4096$ & $7$ & $4808$ & $6144$ \\
4898
\caption{Optimal Values of $k$ for $k$-ary Exponentiation}
4902
\subsection{Sliding-Window Exponentiation}
4903
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
4904
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
4905
algorithm values of $g$ in the range $0 \le g < 2^{k-1}$ must be avoided.
4907
Table~\ref{fig:OPTK2} lists optimal values of $k$ for various exponent sizes and compares the work required against algorithm~\ref{fig:KARY}.
4909
\begin{figure}[here]
4912
\begin{tabular}{|c|c|c|c|c|c|}
4913
\hline \textbf{Exponent (bits)} & \textbf{Optimal $k$} & \textbf{Work at $k$} & \textbf{Work with ~\ref{fig:KARY}} \\
4914
\hline $16$ & $3$ & $24$ & $27$ \\
4915
\hline $32$ & $3$ & $45$ & $49$ \\
4916
\hline $64$ & $4$ & $87$ & $92$ \\
4917
\hline $128$ & $4$ & $167$ & $175$ \\
4918
\hline $256$ & $5$ & $322$ & $335$ \\
4919
\hline $512$ & $6$ & $628$ & $645$ \\
4920
\hline $1024$ & $6$ & $1225$ & $1257$ \\
4921
\hline $2048$ & $7$ & $2403$ & $2452$ \\
4922
\hline $4096$ & $8$ & $4735$ & $4808$ \\
4927
\caption{Optimal Values of $k$ for Sliding Window Exponentiation}
4931
\newpage\begin{figure}[!here]
4935
\hline Algorithm \textbf{Sliding Window $k$-ary Exponentiation}. \\
4936
\textbf{Input}. Integer $a$, $b$, $k$ and $t$ \\
4937
\textbf{Output}. $c = a^b$ \\
4939
1. $c \leftarrow 1$ \\
4940
2. for $i$ from $t - 1$ to $0$ do \\
4941
\hspace{3mm}2.1 If the $i$'th bit of $b$ is a zero then \\
4942
\hspace{6mm}2.1.1 $c \leftarrow c^2$ \\
4943
\hspace{3mm}2.2 else do \\
4944
\hspace{6mm}2.2.1 $c \leftarrow c^{2^k}$ \\
4945
\hspace{6mm}2.2.2 Extract the $k$ bits from $(b_{i}b_{i-1}\ldots b_{i-(k-1)})$ and store it in $g$. \\
4946
\hspace{6mm}2.2.3 $c \leftarrow c \cdot a^g$ \\
4947
\hspace{6mm}2.2.4 $i \leftarrow i - k$ \\
4953
\caption{Sliding Window $k$-ary Exponentiation}
4956
Similar to the previous algorithm this algorithm must have a special handler when fewer than $k$ bits are left in the exponent. While this
4957
algorithm requires the same number of squarings it can potentially have fewer multiplications. The pre-computed table $a^g$ is also half
4958
the size as the previous table.
4960
Consider the exponent $b = 111101011001000_2 \equiv 31432_{10}$ with $k = 3$ using both algorithms. The first algorithm will divide the exponent up as
4961
the following five $3$-bit words $b \equiv \left ( 111, 101, 011, 001, 000 \right )_{2}$. The second algorithm will break the
4962
exponent as $b \equiv \left ( 111, 101, 0, 110, 0, 100, 0 \right )_{2}$. The single digit $0$ in the second representation are where
4963
a single squaring took place instead of a squaring and multiplication. In total the first method requires $10$ multiplications and $18$
4964
squarings. The second method requires $8$ multiplications and $18$ squarings.
4966
In general the sliding window method is never slower than the generic $k$-ary method and often it is slightly faster.
4968
\section{Modular Exponentiation}
4970
Modular exponentiation is essentially computing the power of a base within a finite field or ring. For example, computing
4971
$d \equiv a^b \mbox{ (mod }c\mbox{)}$ is a modular exponentiation. Instead of first computing $a^b$ and then reducing it
4972
modulo $c$ the intermediate result is reduced modulo $c$ after every squaring or multiplication operation.
4974
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
4975
one of the algorithms presented in chapter six.
4977
Before the actual modular exponentiation algorithm can be written a wrapper algorithm must be written first. This algorithm
4978
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
4979
value of $(1/a) \mbox{ mod }c$ is computed using the modular inverse (\textit{see \ref{sec;modinv}}). If no inverse exists the algorithm
4980
terminates with an error.
4982
\begin{figure}[!here]
4986
\hline Algorithm \textbf{mp\_exptmod}. \\
4987
\textbf{Input}. mp\_int $a$, $b$ and $c$ \\
4988
\textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\
4990
1. If $c.sign = MP\_NEG$ return(\textit{MP\_VAL}). \\
4991
2. If $b.sign = MP\_NEG$ then \\
4992
\hspace{3mm}2.1 $g' \leftarrow g^{-1} \mbox{ (mod }c\mbox{)}$ \\
4993
\hspace{3mm}2.2 $x' \leftarrow \vert x \vert$ \\
4994
\hspace{3mm}2.3 Compute $d \equiv g'^{x'} \mbox{ (mod }c\mbox{)}$ via recursion. \\
4995
3. if $p$ is odd \textbf{OR} $p$ is a D.R. modulus then \\
4996
\hspace{3mm}3.1 Compute $y \equiv g^{x} \mbox{ (mod }p\mbox{)}$ via algorithm mp\_exptmod\_fast. \\
4998
\hspace{3mm}4.1 Compute $y \equiv g^{x} \mbox{ (mod }p\mbox{)}$ via algorithm s\_mp\_exptmod. \\
5003
\caption{Algorithm mp\_exptmod}
5006
\textbf{Algorithm mp\_exptmod.}
5007
The first algorithm which actually performs modular exponentiation is algorithm s\_mp\_exptmod. It is a sliding window $k$-ary algorithm
5008
which uses Barrett reduction to reduce the product modulo $p$. The second algorithm mp\_exptmod\_fast performs the same operation
5009
except it uses either Montgomery or Diminished Radix reduction. The two latter reduction algorithms are clumped in the same exponentiation
5010
algorithm since their arguments are essentially the same (\textit{two mp\_ints and one mp\_digit}).
5012
\vspace{+3mm}\begin{small}
5013
\hspace{-5.1mm}{\bf File}: bn\_mp\_exptmod.c
5019
In order to keep the algorithms in a known state the first step on line 29 is to reject any negative modulus as input. If the exponent is
5020
negative the algorithm tries to perform a modular exponentiation with the modular inverse of the base $G$. The temporary variable $tmpG$ is assigned
5021
the modular inverse of $G$ and $tmpX$ is assigned the absolute value of $X$. The algorithm will recuse with these new values with a positive
5024
If the exponent is positive the algorithm resumes the exponentiation. Line 77 determines if the modulus is of the restricted Diminished Radix
5025
form. If it is not line 70 attempts to determine if it is of a unrestricted Diminished Radix form. The integer $dr$ will take on one
5029
\item $dr = 0$ means that the modulus is not of either restricted or unrestricted Diminished Radix form.
5030
\item $dr = 1$ means that the modulus is of restricted Diminished Radix form.
5031
\item $dr = 2$ means that the modulus is of unrestricted Diminished Radix form.
5034
Line 69 determines if the fast modular exponentiation algorithm can be used. It is allowed if $dr \ne 0$ or if the modulus is odd. Otherwise,
5035
the slower s\_mp\_exptmod algorithm is used which uses Barrett reduction.
5037
\subsection{Barrett Modular Exponentiation}
5039
\newpage\begin{figure}[!here]
5043
\hline Algorithm \textbf{s\_mp\_exptmod}. \\
5044
\textbf{Input}. mp\_int $a$, $b$ and $c$ \\
5045
\textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\
5047
1. $k \leftarrow lg(x)$ \\
5048
2. $winsize \leftarrow \left \lbrace \begin{array}{ll}
5049
2 & \mbox{if }k \le 7 \\
5050
3 & \mbox{if }7 < k \le 36 \\
5051
4 & \mbox{if }36 < k \le 140 \\
5052
5 & \mbox{if }140 < k \le 450 \\
5053
6 & \mbox{if }450 < k \le 1303 \\
5054
7 & \mbox{if }1303 < k \le 3529 \\
5055
8 & \mbox{if }3529 < k \\
5056
\end{array} \right .$ \\
5057
3. Initialize $2^{winsize}$ mp\_ints in an array named $M$ and one mp\_int named $\mu$ \\
5058
4. Calculate the $\mu$ required for Barrett Reduction (\textit{mp\_reduce\_setup}). \\
5059
5. $M_1 \leftarrow g \mbox{ (mod }p\mbox{)}$ \\
5061
Setup the table of small powers of $g$. First find $g^{2^{winsize}}$ and then all multiples of it. \\
5062
6. $k \leftarrow 2^{winsize - 1}$ \\
5063
7. $M_{k} \leftarrow M_1$ \\
5064
8. for $ix$ from 0 to $winsize - 2$ do \\
5065
\hspace{3mm}8.1 $M_k \leftarrow \left ( M_k \right )^2$ (\textit{mp\_sqr}) \\
5066
\hspace{3mm}8.2 $M_k \leftarrow M_k \mbox{ (mod }p\mbox{)}$ (\textit{mp\_reduce}) \\
5067
9. for $ix$ from $2^{winsize - 1} + 1$ to $2^{winsize} - 1$ do \\
5068
\hspace{3mm}9.1 $M_{ix} \leftarrow M_{ix - 1} \cdot M_{1}$ (\textit{mp\_mul}) \\
5069
\hspace{3mm}9.2 $M_{ix} \leftarrow M_{ix} \mbox{ (mod }p\mbox{)}$ (\textit{mp\_reduce}) \\
5070
10. $res \leftarrow 1$ \\
5072
Start Sliding Window. \\
5073
11. $mode \leftarrow 0, bitcnt \leftarrow 1, buf \leftarrow 0, digidx \leftarrow x.used - 1, bitcpy \leftarrow 0, bitbuf \leftarrow 0$ \\
5075
\hspace{3mm}12.1 $bitcnt \leftarrow bitcnt - 1$ \\
5076
\hspace{3mm}12.2 If $bitcnt = 0$ then do \\
5077
\hspace{6mm}12.2.1 If $digidx = -1$ goto step 13. \\
5078
\hspace{6mm}12.2.2 $buf \leftarrow x_{digidx}$ \\
5079
\hspace{6mm}12.2.3 $digidx \leftarrow digidx - 1$ \\
5080
\hspace{6mm}12.2.4 $bitcnt \leftarrow lg(\beta)$ \\
5081
Continued on next page. \\
5086
\caption{Algorithm s\_mp\_exptmod}
5089
\newpage\begin{figure}[!here]
5093
\hline Algorithm \textbf{s\_mp\_exptmod} (\textit{continued}). \\
5094
\textbf{Input}. mp\_int $a$, $b$ and $c$ \\
5095
\textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\
5097
\hspace{3mm}12.3 $y \leftarrow (buf >> (lg(\beta) - 1))$ AND $1$ \\
5098
\hspace{3mm}12.4 $buf \leftarrow buf << 1$ \\
5099
\hspace{3mm}12.5 if $mode = 0$ and $y = 0$ then goto step 12. \\
5100
\hspace{3mm}12.6 if $mode = 1$ and $y = 0$ then do \\
5101
\hspace{6mm}12.6.1 $res \leftarrow res^2$ \\
5102
\hspace{6mm}12.6.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\
5103
\hspace{6mm}12.6.3 Goto step 12. \\
5104
\hspace{3mm}12.7 $bitcpy \leftarrow bitcpy + 1$ \\
5105
\hspace{3mm}12.8 $bitbuf \leftarrow bitbuf + (y << (winsize - bitcpy))$ \\
5106
\hspace{3mm}12.9 $mode \leftarrow 2$ \\
5107
\hspace{3mm}12.10 If $bitcpy = winsize$ then do \\
5108
\hspace{6mm}Window is full so perform the squarings and single multiplication. \\
5109
\hspace{6mm}12.10.1 for $ix$ from $0$ to $winsize -1$ do \\
5110
\hspace{9mm}12.10.1.1 $res \leftarrow res^2$ \\
5111
\hspace{9mm}12.10.1.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\
5112
\hspace{6mm}12.10.2 $res \leftarrow res \cdot M_{bitbuf}$ \\
5113
\hspace{6mm}12.10.3 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\
5114
\hspace{6mm}Reset the window. \\
5115
\hspace{6mm}12.10.4 $bitcpy \leftarrow 0, bitbuf \leftarrow 0, mode \leftarrow 1$ \\
5117
No more windows left. Check for residual bits of exponent. \\
5118
13. If $mode = 2$ and $bitcpy > 0$ then do \\
5119
\hspace{3mm}13.1 for $ix$ form $0$ to $bitcpy - 1$ do \\
5120
\hspace{6mm}13.1.1 $res \leftarrow res^2$ \\
5121
\hspace{6mm}13.1.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\
5122
\hspace{6mm}13.1.3 $bitbuf \leftarrow bitbuf << 1$ \\
5123
\hspace{6mm}13.1.4 If $bitbuf$ AND $2^{winsize} \ne 0$ then do \\
5124
\hspace{9mm}13.1.4.1 $res \leftarrow res \cdot M_{1}$ \\
5125
\hspace{9mm}13.1.4.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\
5126
14. $y \leftarrow res$ \\
5127
15. Clear $res$, $mu$ and the $M$ array. \\
5128
16. Return(\textit{MP\_OKAY}). \\
5133
\caption{Algorithm s\_mp\_exptmod (continued)}
5136
\textbf{Algorithm s\_mp\_exptmod.}
5137
This algorithm computes the $x$'th power of $g$ modulo $p$ and stores the result in $y$. It takes advantage of the Barrett reduction
5138
algorithm to keep the product small throughout the algorithm.
5140
The first two steps determine the optimal window size based on the number of bits in the exponent. The larger the exponent the
5141
larger the window size becomes. After a window size $winsize$ has been chosen an array of $2^{winsize}$ mp\_int variables is allocated. This
5142
table will hold the values of $g^x \mbox{ (mod }p\mbox{)}$ for $2^{winsize - 1} \le x < 2^{winsize}$.
5144
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
5145
the rest of the algorithm more efficient. The first element of the table at $2^{winsize - 1}$ is found by squaring $M_1$ successively $winsize - 2$
5146
times. The rest of the table elements are found by multiplying the previous element by $M_1$ modulo $p$.
5148
Now that the table is available the sliding window may begin. The following list describes the functions of all the variables in the window.
5150
\item The variable $mode$ dictates how the bits of the exponent are interpreted.
5152
\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
5153
$1$ 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.
5154
\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
5155
are read and a single squaring is performed. If a non-zero bit is read a new window is created.
5156
\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
5159
\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
5160
is fetched from the exponent.
5161
\item The variable $buf$ holds the currently read digit of the exponent.
5162
\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.
5163
\item The variable $bitcpy$ indicates how many bits are in the currently formed window. When it reaches $winsize$ the window is flushed and
5164
the appropriate operations performed.
5165
\item The variable $bitbuf$ holds the current bits of the window being formed.
5168
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
5169
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
5170
read and if there are no digits left than the loop terminates.
5172
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
5173
upwards. In effect the digit is read from most significant bit to least significant bit and since the digits are read from leading to
5174
trailing edges the entire exponent is read from most significant bit to least significant bit.
5176
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
5177
algorithm from having to perform trivial squaring and reduction operations before the first non-zero bit is read. Step 12.6 and 12.7-10 handle
5178
the two cases of $mode = 1$ and $mode = 2$ respectively.
5181
\begin{figure}[here]
5182
\includegraphics{pics/expt_state.ps}
5183
\caption{Sliding Window State Diagram}
5184
\label{pic:expt_state}
5188
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
5189
a Left-to-Right algorithm is used to process the remaining few bits.
5191
\vspace{+3mm}\begin{small}
5192
\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_exptmod.c
5198
Lines 32 through 46 determine the optimal window size based on the length of the exponent in bits. The window divisions are sorted
5199
from smallest to greatest so that in each \textbf{if} statement only one condition must be tested. For example, by the \textbf{if} statement
5200
on line 38 the value of $x$ is already known to be greater than $140$.
5202
The conditional piece of code beginning on line 48 allows the window size to be restricted to five bits. This logic is used to ensure
5203
the table of precomputed powers of $G$ remains relatively small.
5205
The for loop on line 61 initializes the $M$ array while lines 72 and 77 through 86 initialize the reduction
5206
function that will be used for this modulus.
5209
\section{Quick Power of Two}
5210
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
5211
equivalent to $m \cdot 2^k$. By this logic when $m = 1$ a quick power of two can be achieved.
5213
\begin{figure}[!here]
5217
\hline Algorithm \textbf{mp\_2expt}. \\
5218
\textbf{Input}. integer $b$ \\
5219
\textbf{Output}. $a \leftarrow 2^b$ \\
5221
1. $a \leftarrow 0$ \\
5222
2. If $a.alloc < \lfloor b / lg(\beta) \rfloor + 1$ then grow $a$ appropriately. \\
5223
3. $a.used \leftarrow \lfloor b / lg(\beta) \rfloor + 1$ \\
5224
4. $a_{\lfloor b / lg(\beta) \rfloor} \leftarrow 1 << (b \mbox{ mod } lg(\beta))$ \\
5225
5. Return(\textit{MP\_OKAY}). \\
5230
\caption{Algorithm mp\_2expt}
5233
\textbf{Algorithm mp\_2expt.}
5235
\vspace{+3mm}\begin{small}
5236
\hspace{-5.1mm}{\bf File}: bn\_mp\_2expt.c
5242
\chapter{Higher Level Algorithms}
5244
This chapter discusses the various higher level algorithms that are required to complete a well rounded multiple precision integer package. These
5245
routines are less performance oriented than the algorithms of chapters five, six and seven but are no less important.
5247
The first section describes a method of integer division with remainder that is universally well known. It provides the signed division logic
5248
for the package. The subsequent section discusses a set of algorithms which allow a single digit to be the 2nd operand for a variety of operations.
5249
These algorithms serve mostly to simplify other algorithms where small constants are required. The last two sections discuss how to manipulate
5250
various representations of integers. For example, converting from an mp\_int to a string of character.
5252
\section{Integer Division with Remainder}
5253
\label{sec:division}
5255
Integer division aside from modular exponentiation is the most intensive algorithm to compute. Like addition, subtraction and multiplication
5256
the basis of this algorithm is the long-hand division algorithm taught to school children. Throughout this discussion several common variables
5257
will be used. Let $x$ represent the divisor and $y$ represent the dividend. Let $q$ represent the integer quotient $\lfloor y / x \rfloor$ and
5258
let $r$ represent the remainder $r = y - x \lfloor y / x \rfloor$. The following simple algorithm will be used to start the discussion.
5260
\newpage\begin{figure}[!here]
5264
\hline Algorithm \textbf{Radix-$\beta$ Integer Division}. \\
5265
\textbf{Input}. integer $x$ and $y$ \\
5266
\textbf{Output}. $q = \lfloor y/x\rfloor, r = y - xq$ \\
5268
1. $q \leftarrow 0$ \\
5269
2. $n \leftarrow \vert \vert y \vert \vert - \vert \vert x \vert \vert$ \\
5270
3. for $t$ from $n$ down to $0$ do \\
5271
\hspace{3mm}3.1 Maximize $k$ such that $kx\beta^t$ is less than or equal to $y$ and $(k + 1)x\beta^t$ is greater. \\
5272
\hspace{3mm}3.2 $q \leftarrow q + k\beta^t$ \\
5273
\hspace{3mm}3.3 $y \leftarrow y - kx\beta^t$ \\
5274
4. $r \leftarrow y$ \\
5275
5. Return($q, r$) \\
5280
\caption{Algorithm Radix-$\beta$ Integer Division}
5284
As children we are taught this very simple algorithm for the case of $\beta = 10$. Almost instinctively several optimizations are taught for which
5285
their reason of existing are never explained. For this example let $y = 5471$ represent the dividend and $x = 23$ represent the divisor.
5287
To find the first digit of the quotient the value of $k$ must be maximized such that $kx\beta^t$ is less than or equal to $y$ and
5288
simultaneously $(k + 1)x\beta^t$ is greater than $y$. Implicitly $k$ is the maximum value the $t$'th digit of the quotient may have. The habitual method
5289
used to find the maximum is to ``eyeball'' the two numbers, typically only the leading digits and quickly estimate a quotient. By only using leading
5290
digits a much simpler division may be used to form an educated guess at what the value must be. In this case $k = \lfloor 54/23\rfloor = 2$ quickly
5291
arises as a possible solution. Indeed $2x\beta^2 = 4600$ is less than $y = 5471$ and simultaneously $(k + 1)x\beta^2 = 6900$ is larger than $y$.
5292
As a result $k\beta^2$ is added to the quotient which now equals $q = 200$ and $4600$ is subtracted from $y$ to give a remainder of $y = 841$.
5294
Again this process is repeated to produce the quotient digit $k = 3$ which makes the quotient $q = 200 + 3\beta = 230$ and the remainder
5295
$y = 841 - 3x\beta = 181$. Finally the last iteration of the loop produces $k = 7$ which leads to the quotient $q = 230 + 7 = 237$ and the
5296
remainder $y = 181 - 7x = 20$. The final quotient and remainder found are $q = 237$ and $r = y = 20$ which are indeed correct since
5297
$237 \cdot 23 + 20 = 5471$ is true.
5299
\subsection{Quotient Estimation}
5301
As alluded to earlier the quotient digit $k$ can be estimated from only the leading digits of both the divisor and dividend. When $p$ leading
5302
digits are used from both the divisor and dividend to form an estimation the accuracy of the estimation rises as $p$ grows. Technically
5303
speaking the estimation is based on assuming the lower $\vert \vert y \vert \vert - p$ and $\vert \vert x \vert \vert - p$ lower digits of the
5304
dividend and divisor are zero.
5306
The value of the estimation may off by a few values in either direction and in general is fairly correct. A simplification \cite[pp. 271]{TAOCPV2}
5307
of the estimation technique is to use $t + 1$ digits of the dividend and $t$ digits of the divisor, in particularly when $t = 1$. The estimate
5308
using this technique is never too small. For the following proof let $t = \vert \vert y \vert \vert - 1$ and $s = \vert \vert x \vert \vert - 1$
5309
represent the most significant digits of the dividend and divisor respectively.
5311
\textbf{Proof.}\textit{ The quotient $\hat k = \lfloor (y_t\beta + y_{t-1}) / x_s \rfloor$ is greater than or equal to
5312
$k = \lfloor y / (x \cdot \beta^{\vert \vert y \vert \vert - \vert \vert x \vert \vert - 1}) \rfloor$. }
5313
The first obvious case is when $\hat k = \beta - 1$ in which case the proof is concluded since the real quotient cannot be larger. For all other
5314
cases $\hat k = \lfloor (y_t\beta + y_{t-1}) / x_s \rfloor$ and $\hat k x_s \ge y_t\beta + y_{t-1} - x_s + 1$. The latter portion of the inequalility
5315
$-x_s + 1$ arises from the fact that a truncated integer division will give the same quotient for at most $x_s - 1$ values. Next a series of
5316
inequalities will prove the hypothesis.
5319
y - \hat k x \le y - \hat k x_s\beta^s
5322
This is trivially true since $x \ge x_s\beta^s$. Next we replace $\hat kx_s\beta^s$ by the previous inequality for $\hat kx_s$.
5325
y - \hat k x \le y_t\beta^t + \ldots + y_0 - (y_t\beta^t + y_{t-1}\beta^{t-1} - x_s\beta^t + \beta^s)
5328
By simplifying the previous inequality the following inequality is formed.
5331
y - \hat k x \le y_{t-2}\beta^{t-2} + \ldots + y_0 + x_s\beta^s - \beta^s
5337
y_{t-2}\beta^{t-2} + \ldots + y_0 + x_s\beta^s - \beta^s < x_s\beta^s \le x
5340
Which proves that $y - \hat kx \le x$ and by consequence $\hat k \ge k$ which concludes the proof. \textbf{QED}
5343
\subsection{Normalized Integers}
5344
For the purposes of division a normalized input is when the divisors leading digit $x_n$ is greater than or equal to $\beta / 2$. By multiplying both
5345
$x$ and $y$ by $j = \lfloor (\beta / 2) / x_n \rfloor$ the quotient remains unchanged and the remainder is simply $j$ times the original
5346
remainder. The purpose of normalization is to ensure the leading digit of the divisor is sufficiently large such that the estimated quotient will
5347
lie in the domain of a single digit. Consider the maximum dividend $(\beta - 1) \cdot \beta + (\beta - 1)$ and the minimum divisor $\beta / 2$.
5350
{{\beta^2 - 1} \over { \beta / 2}} \le 2\beta - {2 \over \beta}
5353
At most the quotient approaches $2\beta$, however, in practice this will not occur since that would imply the previous quotient digit was too small.
5355
\subsection{Radix-$\beta$ Division with Remainder}
5356
\newpage\begin{figure}[!here]
5360
\hline Algorithm \textbf{mp\_div}. \\
5361
\textbf{Input}. mp\_int $a, b$ \\
5362
\textbf{Output}. $c = \lfloor a/b \rfloor$, $d = a - bc$ \\
5364
1. If $b = 0$ return(\textit{MP\_VAL}). \\
5365
2. If $\vert a \vert < \vert b \vert$ then do \\
5366
\hspace{3mm}2.1 $d \leftarrow a$ \\
5367
\hspace{3mm}2.2 $c \leftarrow 0$ \\
5368
\hspace{3mm}2.3 Return(\textit{MP\_OKAY}). \\
5370
Setup the quotient to receive the digits. \\
5371
3. Grow $q$ to $a.used + 2$ digits. \\
5372
4. $q \leftarrow 0$ \\
5373
5. $x \leftarrow \vert a \vert , y \leftarrow \vert b \vert$ \\
5374
6. $sign \leftarrow \left \lbrace \begin{array}{ll}
5375
MP\_ZPOS & \mbox{if }a.sign = b.sign \\
5376
MP\_NEG & \mbox{otherwise} \\
5377
\end{array} \right .$ \\
5379
Normalize the inputs such that the leading digit of $y$ is greater than or equal to $\beta / 2$. \\
5380
7. $norm \leftarrow (lg(\beta) - 1) - (\lceil lg(y) \rceil \mbox{ (mod }lg(\beta)\mbox{)})$ \\
5381
8. $x \leftarrow x \cdot 2^{norm}, y \leftarrow y \cdot 2^{norm}$ \\
5383
Find the leading digit of the quotient. \\
5384
9. $n \leftarrow x.used - 1, t \leftarrow y.used - 1$ \\
5385
10. $y \leftarrow y \cdot \beta^{n - t}$ \\
5386
11. While ($x \ge y$) do \\
5387
\hspace{3mm}11.1 $q_{n - t} \leftarrow q_{n - t} + 1$ \\
5388
\hspace{3mm}11.2 $x \leftarrow x - y$ \\
5389
12. $y \leftarrow \lfloor y / \beta^{n-t} \rfloor$ \\
5391
Continued on the next page. \\
5396
\caption{Algorithm mp\_div}
5399
\newpage\begin{figure}[!here]
5403
\hline Algorithm \textbf{mp\_div} (continued). \\
5404
\textbf{Input}. mp\_int $a, b$ \\
5405
\textbf{Output}. $c = \lfloor a/b \rfloor$, $d = a - bc$ \\
5407
Now find the remainder fo the digits. \\
5408
13. for $i$ from $n$ down to $(t + 1)$ do \\
5409
\hspace{3mm}13.1 If $i > x.used$ then jump to the next iteration of this loop. \\
5410
\hspace{3mm}13.2 If $x_{i} = y_{t}$ then \\
5411
\hspace{6mm}13.2.1 $q_{i - t - 1} \leftarrow \beta - 1$ \\
5412
\hspace{3mm}13.3 else \\
5413
\hspace{6mm}13.3.1 $\hat r \leftarrow x_{i} \cdot \beta + x_{i - 1}$ \\
5414
\hspace{6mm}13.3.2 $\hat r \leftarrow \lfloor \hat r / y_{t} \rfloor$ \\
5415
\hspace{6mm}13.3.3 $q_{i - t - 1} \leftarrow \hat r$ \\
5416
\hspace{3mm}13.4 $q_{i - t - 1} \leftarrow q_{i - t - 1} + 1$ \\
5418
Fixup quotient estimation. \\
5419
\hspace{3mm}13.5 Loop \\
5420
\hspace{6mm}13.5.1 $q_{i - t - 1} \leftarrow q_{i - t - 1} - 1$ \\
5421
\hspace{6mm}13.5.2 t$1 \leftarrow 0$ \\
5422
\hspace{6mm}13.5.3 t$1_0 \leftarrow y_{t - 1}, $ t$1_1 \leftarrow y_t,$ t$1.used \leftarrow 2$ \\
5423
\hspace{6mm}13.5.4 $t1 \leftarrow t1 \cdot q_{i - t - 1}$ \\
5424
\hspace{6mm}13.5.5 t$2_0 \leftarrow x_{i - 2}, $ t$2_1 \leftarrow x_{i - 1}, $ t$2_2 \leftarrow x_i, $ t$2.used \leftarrow 3$ \\
5425
\hspace{6mm}13.5.6 If $\vert t1 \vert > \vert t2 \vert$ then goto step 13.5. \\
5426
\hspace{3mm}13.6 t$1 \leftarrow y \cdot q_{i - t - 1}$ \\
5427
\hspace{3mm}13.7 t$1 \leftarrow $ t$1 \cdot \beta^{i - t - 1}$ \\
5428
\hspace{3mm}13.8 $x \leftarrow x - $ t$1$ \\
5429
\hspace{3mm}13.9 If $x.sign = MP\_NEG$ then \\
5430
\hspace{6mm}13.10 t$1 \leftarrow y$ \\
5431
\hspace{6mm}13.11 t$1 \leftarrow $ t$1 \cdot \beta^{i - t - 1}$ \\
5432
\hspace{6mm}13.12 $x \leftarrow x + $ t$1$ \\
5433
\hspace{6mm}13.13 $q_{i - t - 1} \leftarrow q_{i - t - 1} - 1$ \\
5435
Finalize the result. \\
5436
14. Clamp excess digits of $q$ \\
5437
15. $c \leftarrow q, c.sign \leftarrow sign$ \\
5438
16. $x.sign \leftarrow a.sign$ \\
5439
17. $d \leftarrow \lfloor x / 2^{norm} \rfloor$ \\
5440
18. Return(\textit{MP\_OKAY}). \\
5445
\caption{Algorithm mp\_div (continued)}
5447
\textbf{Algorithm mp\_div.}
5448
This algorithm will calculate quotient and remainder from an integer division given a dividend and divisor. The algorithm is a signed
5449
division and will produce a fully qualified quotient and remainder.
5451
First the divisor $b$ must be non-zero which is enforced in step one. If the divisor is larger than the dividend than the quotient is implicitly
5452
zero and the remainder is the dividend.
5454
After the first two trivial cases of inputs are handled the variable $q$ is setup to receive the digits of the quotient. Two unsigned copies of the
5455
divisor $y$ and dividend $x$ are made as well. The core of the division algorithm is an unsigned division and will only work if the values are
5456
positive. Now the two values $x$ and $y$ must be normalized such that the leading digit of $y$ is greater than or equal to $\beta / 2$.
5457
This is performed by shifting both to the left by enough bits to get the desired normalization.
5459
At this point the division algorithm can begin producing digits of the quotient. Recall that maximum value of the estimation used is
5460
$2\beta - {2 \over \beta}$ which means that a digit of the quotient must be first produced by another means. In this case $y$ is shifted
5461
to the left (\textit{step ten}) so that it has the same number of digits as $x$. The loop on step eleven will subtract multiples of the
5462
shifted copy of $y$ until $x$ is smaller. Since the leading digit of $y$ is greater than or equal to $\beta/2$ this loop will iterate at most two
5463
times to produce the desired leading digit of the quotient.
5465
Now the remainder of the digits can be produced. The equation $\hat q = \lfloor {{x_i \beta + x_{i-1}}\over y_t} \rfloor$ is used to fairly
5466
accurately approximate the true quotient digit. The estimation can in theory produce an estimation as high as $2\beta - {2 \over \beta}$ but by
5467
induction the upper quotient digit is correct (\textit{as established on step eleven}) and the estimate must be less than $\beta$.
5469
Recall from section~\ref{sec:divest} that the estimation is never too low but may be too high. The next step of the estimation process is
5470
to refine the estimation. The loop on step 13.5 uses $x_i\beta^2 + x_{i-1}\beta + x_{i-2}$ and $q_{i - t - 1}(y_t\beta + y_{t-1})$ as a higher
5471
order approximation to adjust the quotient digit.
5473
After both phases of estimation the quotient digit may still be off by a value of one\footnote{This is similar to the error introduced
5474
by optimizing Barrett reduction.}. Steps 13.6 and 13.7 subtract the multiple of the divisor from the dividend (\textit{Similar to step 3.3 of
5475
algorithm~\ref{fig:raddiv}} and then subsequently add a multiple of the divisor if the quotient was too large.
5477
Now that the quotient has been determine finializing the result is a matter of clamping the quotient, fixing the sizes and de-normalizing the
5478
remainder. An important aspect of this algorithm seemingly overlooked in other descriptions such as that of Algorithm 14.20 HAC \cite[pp. 598]{HAC}
5479
is that when the estimations are being made (\textit{inside the loop on step 13.5}) that the digits $y_{t-1}$, $x_{i-2}$ and $x_{i-1}$ may lie
5480
outside their respective boundaries. For example, if $t = 0$ or $i \le 1$ then the digits would be undefined. In those cases the digits should
5481
respectively be replaced with a zero.
5483
\vspace{+3mm}\begin{small}
5484
\hspace{-5.1mm}{\bf File}: bn\_mp\_div.c
5490
The implementation of this algorithm differs slightly from the pseudo code presented previously. In this algorithm either of the quotient $c$ or
5491
remainder $d$ may be passed as a \textbf{NULL} pointer which indicates their value is not desired. For example, the C code to call the division
5492
algorithm with only the quotient is
5495
mp_div(&a, &b, &c, NULL); /* c = [a/b] */
5498
Lines 109 and 113 handle the two trivial cases of inputs which are division by zero and dividend smaller than the divisor
5499
respectively. After the two trivial cases all of the temporary variables are initialized. Line 148 determines the sign of
5500
the quotient and line 148 ensures that both $x$ and $y$ are positive.
5502
The number of bits in the leading digit is calculated on line 151. Implictly an mp\_int with $r$ digits will require $lg(\beta)(r-1) + k$ bits
5503
of precision which when reduced modulo $lg(\beta)$ produces the value of $k$. In this case $k$ is the number of bits in the leading digit which is
5504
exactly what is required. For the algorithm to operate $k$ must equal $lg(\beta) - 1$ and when it does not the inputs must be normalized by shifting
5505
them to the left by $lg(\beta) - 1 - k$ bits.
5507
Throughout the variables $n$ and $t$ will represent the highest digit of $x$ and $y$ respectively. These are first used to produce the
5508
leading digit of the quotient. The loop beginning on line 184 will produce the remainder of the quotient digits.
5510
The conditional ``continue'' on line 187 is used to prevent the algorithm from reading past the leading edge of $x$ which can occur when the
5511
algorithm eliminates multiple non-zero digits in a single iteration. This ensures that $x_i$ is always non-zero since by definition the digits
5512
above the $i$'th position $x$ must be zero in order for the quotient to be precise\footnote{Precise as far as integer division is concerned.}.
5514
Lines 214, 216 and 223 through 225 manually construct the high accuracy estimations by setting the digits of the two mp\_int
5517
\section{Single Digit Helpers}
5519
This section briefly describes a series of single digit helper algorithms which come in handy when working with small constants. All of
5520
the helper functions assume the single digit input is positive and will treat them as such.
5522
\subsection{Single Digit Addition and Subtraction}
5524
Both addition and subtraction are performed by ``cheating'' and using mp\_set followed by the higher level addition or subtraction
5525
algorithms. As a result these algorithms are subtantially simpler with a slight cost in performance.
5527
\newpage\begin{figure}[!here]
5531
\hline Algorithm \textbf{mp\_add\_d}. \\
5532
\textbf{Input}. mp\_int $a$ and a mp\_digit $b$ \\
5533
\textbf{Output}. $c = a + b$ \\
5535
1. $t \leftarrow b$ (\textit{mp\_set}) \\
5536
2. $c \leftarrow a + t$ \\
5537
3. Return(\textit{MP\_OKAY}) \\
5542
\caption{Algorithm mp\_add\_d}
5545
\textbf{Algorithm mp\_add\_d.}
5546
This algorithm initiates a temporary mp\_int with the value of the single digit and uses algorithm mp\_add to add the two values together.
5548
\vspace{+3mm}\begin{small}
5549
\hspace{-5.1mm}{\bf File}: bn\_mp\_add\_d.c
5555
Clever use of the letter 't'.
5557
\subsubsection{Subtraction}
5558
The single digit subtraction algorithm mp\_sub\_d is essentially the same except it uses mp\_sub to subtract the digit from the mp\_int.
5560
\subsection{Single Digit Multiplication}
5561
Single digit multiplication arises enough in division and radix conversion that it ought to be implement as a special case of the baseline
5562
multiplication algorithm. Essentially this algorithm is a modified version of algorithm s\_mp\_mul\_digs where one of the multiplicands
5565
\begin{figure}[!here]
5569
\hline Algorithm \textbf{mp\_mul\_d}. \\
5570
\textbf{Input}. mp\_int $a$ and a mp\_digit $b$ \\
5571
\textbf{Output}. $c = ab$ \\
5573
1. $pa \leftarrow a.used$ \\
5574
2. Grow $c$ to at least $pa + 1$ digits. \\
5575
3. $oldused \leftarrow c.used$ \\
5576
4. $c.used \leftarrow pa + 1$ \\
5577
5. $c.sign \leftarrow a.sign$ \\
5578
6. $\mu \leftarrow 0$ \\
5579
7. for $ix$ from $0$ to $pa - 1$ do \\
5580
\hspace{3mm}7.1 $\hat r \leftarrow \mu + a_{ix}b$ \\
5581
\hspace{3mm}7.2 $c_{ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\
5582
\hspace{3mm}7.3 $\mu \leftarrow \lfloor \hat r / \beta \rfloor$ \\
5583
8. $c_{pa} \leftarrow \mu$ \\
5584
9. for $ix$ from $pa + 1$ to $oldused$ do \\
5585
\hspace{3mm}9.1 $c_{ix} \leftarrow 0$ \\
5586
10. Clamp excess digits of $c$. \\
5587
11. Return(\textit{MP\_OKAY}). \\
5592
\caption{Algorithm mp\_mul\_d}
5594
\textbf{Algorithm mp\_mul\_d.}
5595
This algorithm quickly multiplies an mp\_int by a small single digit value. It is specially tailored to the job and has a minimal of overhead.
5596
Unlike the full multiplication algorithms this algorithm does not require any significnat temporary storage or memory allocations.
5598
\vspace{+3mm}\begin{small}
5599
\hspace{-5.1mm}{\bf File}: bn\_mp\_mul\_d.c
5605
In this implementation the destination $c$ may point to the same mp\_int as the source $a$ since the result is written after the digit is
5606
read from the source. This function uses pointer aliases $tmpa$ and $tmpc$ for the digits of $a$ and $c$ respectively.
5608
\subsection{Single Digit Division}
5609
Like the single digit multiplication algorithm, single digit division is also a fairly common algorithm used in radix conversion. Since the
5610
divisor is only a single digit a specialized variant of the division algorithm can be used to compute the quotient.
5612
\newpage\begin{figure}[!here]
5616
\hline Algorithm \textbf{mp\_div\_d}. \\
5617
\textbf{Input}. mp\_int $a$ and a mp\_digit $b$ \\
5618
\textbf{Output}. $c = \lfloor a / b \rfloor, d = a - cb$ \\
5620
1. If $b = 0$ then return(\textit{MP\_VAL}).\\
5621
2. If $b = 3$ then use algorithm mp\_div\_3 instead. \\
5622
3. Init $q$ to $a.used$ digits. \\
5623
4. $q.used \leftarrow a.used$ \\
5624
5. $q.sign \leftarrow a.sign$ \\
5625
6. $\hat w \leftarrow 0$ \\
5626
7. for $ix$ from $a.used - 1$ down to $0$ do \\
5627
\hspace{3mm}7.1 $\hat w \leftarrow \hat w \beta + a_{ix}$ \\
5628
\hspace{3mm}7.2 If $\hat w \ge b$ then \\
5629
\hspace{6mm}7.2.1 $t \leftarrow \lfloor \hat w / b \rfloor$ \\
5630
\hspace{6mm}7.2.2 $\hat w \leftarrow \hat w \mbox{ (mod }b\mbox{)}$ \\
5631
\hspace{3mm}7.3 else\\
5632
\hspace{6mm}7.3.1 $t \leftarrow 0$ \\
5633
\hspace{3mm}7.4 $q_{ix} \leftarrow t$ \\
5634
8. $d \leftarrow \hat w$ \\
5635
9. Clamp excess digits of $q$. \\
5636
10. $c \leftarrow q$ \\
5637
11. Return(\textit{MP\_OKAY}). \\
5642
\caption{Algorithm mp\_div\_d}
5644
\textbf{Algorithm mp\_div\_d.}
5645
This algorithm divides the mp\_int $a$ by the single mp\_digit $b$ using an optimized approach. Essentially in every iteration of the
5646
algorithm another digit of the dividend is reduced and another digit of quotient produced. Provided $b < \beta$ the value of $\hat w$
5647
after step 7.1 will be limited such that $0 \le \lfloor \hat w / b \rfloor < \beta$.
5649
If the divisor $b$ is equal to three a variant of this algorithm is used which is called mp\_div\_3. It replaces the division by three with
5650
a multiplication by $\lfloor \beta / 3 \rfloor$ and the appropriate shift and residual fixup. In essence it is much like the Barrett reduction
5653
\vspace{+3mm}\begin{small}
5654
\hspace{-5.1mm}{\bf File}: bn\_mp\_div\_d.c
5660
Like the implementation of algorithm mp\_div this algorithm allows either of the quotient or remainder to be passed as a \textbf{NULL} pointer to
5661
indicate the respective value is not required. This allows a trivial single digit modular reduction algorithm, mp\_mod\_d to be created.
5663
The division and remainder on lines 44 and @45,%@ can be replaced often by a single division on most processors. For example, the 32-bit x86 based
5664
processors can divide a 64-bit quantity by a 32-bit quantity and produce the quotient and remainder simultaneously. Unfortunately the GCC
5665
compiler does not recognize that optimization and will actually produce two function calls to find the quotient and remainder respectively.
5667
\subsection{Single Digit Root Extraction}
5669
Finding the $n$'th root of an integer is fairly easy as far as numerical analysis is concerned. Algorithms such as the Newton-Raphson approximation
5670
(\ref{eqn:newton}) series will converge very quickly to a root for any continuous function $f(x)$.
5673
x_{i+1} = x_i - {f(x_i) \over f'(x_i)}
5677
In this case the $n$'th root is desired and $f(x) = x^n - a$ where $a$ is the integer of which the root is desired. The derivative of $f(x)$ is
5678
simply $f'(x) = nx^{n - 1}$. Of particular importance is that this algorithm will be used over the integers not over the a more continuous domain
5679
such as the real numbers. As a result the root found can be above the true root by few and must be manually adjusted. Ideally at the end of the
5680
algorithm the $n$'th root $b$ of an integer $a$ is desired such that $b^n \le a$.
5682
\newpage\begin{figure}[!here]
5686
\hline Algorithm \textbf{mp\_n\_root}. \\
5687
\textbf{Input}. mp\_int $a$ and a mp\_digit $b$ \\
5688
\textbf{Output}. $c^b \le a$ \\
5690
1. If $b$ is even and $a.sign = MP\_NEG$ return(\textit{MP\_VAL}). \\
5691
2. $sign \leftarrow a.sign$ \\
5692
3. $a.sign \leftarrow MP\_ZPOS$ \\
5693
4. t$2 \leftarrow 2$ \\
5695
\hspace{3mm}5.1 t$1 \leftarrow $ t$2$ \\
5696
\hspace{3mm}5.2 t$3 \leftarrow $ t$1^{b - 1}$ \\
5697
\hspace{3mm}5.3 t$2 \leftarrow $ t$3 $ $\cdot$ t$1$ \\
5698
\hspace{3mm}5.4 t$2 \leftarrow $ t$2 - a$ \\
5699
\hspace{3mm}5.5 t$3 \leftarrow $ t$3 \cdot b$ \\
5700
\hspace{3mm}5.6 t$3 \leftarrow \lfloor $t$2 / $t$3 \rfloor$ \\
5701
\hspace{3mm}5.7 t$2 \leftarrow $ t$1 - $ t$3$ \\
5702
\hspace{3mm}5.8 If t$1 \ne $ t$2$ then goto step 5. \\
5704
\hspace{3mm}6.1 t$2 \leftarrow $ t$1^b$ \\
5705
\hspace{3mm}6.2 If t$2 > a$ then \\
5706
\hspace{6mm}6.2.1 t$1 \leftarrow $ t$1 - 1$ \\
5707
\hspace{6mm}6.2.2 Goto step 6. \\
5708
7. $a.sign \leftarrow sign$ \\
5709
8. $c \leftarrow $ t$1$ \\
5710
9. $c.sign \leftarrow sign$ \\
5711
10. Return(\textit{MP\_OKAY}). \\
5716
\caption{Algorithm mp\_n\_root}
5718
\textbf{Algorithm mp\_n\_root.}
5719
This algorithm finds the integer $n$'th root of an input using the Newton-Raphson approach. It is partially optimized based on the observation
5720
that the numerator of ${f(x) \over f'(x)}$ can be derived from a partial denominator. That is at first the denominator is calculated by finding
5721
$x^{b - 1}$. This value can then be multiplied by $x$ and have $a$ subtracted from it to find the numerator. This saves a total of $b - 1$
5722
multiplications by t$1$ inside the loop.
5724
The initial value of the approximation is t$2 = 2$ which allows the algorithm to start with very small values and quickly converge on the
5725
root. Ideally this algorithm is meant to find the $n$'th root of an input where $n$ is bounded by $2 \le n \le 5$.
5727
\vspace{+3mm}\begin{small}
5728
\hspace{-5.1mm}{\bf File}: bn\_mp\_n\_root.c
5734
\section{Random Number Generation}
5736
Random numbers come up in a variety of activities from public key cryptography to simple simulations and various randomized algorithms. Pollard-Rho
5737
factoring for example, can make use of random values as starting points to find factors of a composite integer. In this case the algorithm presented
5738
is solely for simulations and not intended for cryptographic use.
5740
\newpage\begin{figure}[!here]
5744
\hline Algorithm \textbf{mp\_rand}. \\
5745
\textbf{Input}. An integer $b$ \\
5746
\textbf{Output}. A pseudo-random number of $b$ digits \\
5748
1. $a \leftarrow 0$ \\
5749
2. If $b \le 0$ return(\textit{MP\_OKAY}) \\
5750
3. Pick a non-zero random digit $d$. \\
5751
4. $a \leftarrow a + d$ \\
5752
5. for $ix$ from 1 to $d - 1$ do \\
5753
\hspace{3mm}5.1 $a \leftarrow a \cdot \beta$ \\
5754
\hspace{3mm}5.2 Pick a random digit $d$. \\
5755
\hspace{3mm}5.3 $a \leftarrow a + d$ \\
5756
6. Return(\textit{MP\_OKAY}). \\
5761
\caption{Algorithm mp\_rand}
5763
\textbf{Algorithm mp\_rand.}
5764
This algorithm produces a pseudo-random integer of $b$ digits. By ensuring that the first digit is non-zero the algorithm also guarantees that the
5765
final result has at least $b$ digits. It relies heavily on a third-part random number generator which should ideally generate uniformly all of
5766
the integers from $0$ to $\beta - 1$.
5768
\vspace{+3mm}\begin{small}
5769
\hspace{-5.1mm}{\bf File}: bn\_mp\_rand.c
5775
\section{Formatted Representations}
5776
The ability to emit a radix-$n$ textual representation of an integer is useful for interacting with human parties. For example, the ability to
5777
be given a string of characters such as ``114585'' and turn it into the radix-$\beta$ equivalent would make it easier to enter numbers
5780
\subsection{Reading Radix-n Input}
5781
For the purposes of this text we will assume that a simple lower ASCII map (\ref{fig:ASC}) is used for the values of from $0$ to $63$ to
5782
printable characters. For example, when the character ``N'' is read it represents the integer $23$. The first $16$ characters of the
5783
map are for the common representations up to hexadecimal. After that they match the ``base64'' encoding scheme which are suitable chosen
5784
such that they are printable. While outputting as base64 may not be too helpful for human operators it does allow communication via non binary
5787
\newpage\begin{figure}[here]
5789
\begin{tabular}{cc|cc|cc|cc}
5790
\hline \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} \\
5792
0 & 0 & 1 & 1 & 2 & 2 & 3 & 3 \\
5793
4 & 4 & 5 & 5 & 6 & 6 & 7 & 7 \\
5794
8 & 8 & 9 & 9 & 10 & A & 11 & B \\
5795
12 & C & 13 & D & 14 & E & 15 & F \\
5796
16 & G & 17 & H & 18 & I & 19 & J \\
5797
20 & K & 21 & L & 22 & M & 23 & N \\
5798
24 & O & 25 & P & 26 & Q & 27 & R \\
5799
28 & S & 29 & T & 30 & U & 31 & V \\
5800
32 & W & 33 & X & 34 & Y & 35 & Z \\
5801
36 & a & 37 & b & 38 & c & 39 & d \\
5802
40 & e & 41 & f & 42 & g & 43 & h \\
5803
44 & i & 45 & j & 46 & k & 47 & l \\
5804
48 & m & 49 & n & 50 & o & 51 & p \\
5805
52 & q & 53 & r & 54 & s & 55 & t \\
5806
56 & u & 57 & v & 58 & w & 59 & x \\
5807
60 & y & 61 & z & 62 & $+$ & 63 & $/$ \\
5811
\caption{Lower ASCII Map}
5815
\newpage\begin{figure}[!here]
5819
\hline Algorithm \textbf{mp\_read\_radix}. \\
5820
\textbf{Input}. A string $str$ of length $sn$ and radix $r$. \\
5821
\textbf{Output}. The radix-$\beta$ equivalent mp\_int. \\
5823
1. If $r < 2$ or $r > 64$ return(\textit{MP\_VAL}). \\
5824
2. $ix \leftarrow 0$ \\
5825
3. If $str_0 =$ ``-'' then do \\
5826
\hspace{3mm}3.1 $ix \leftarrow ix + 1$ \\
5827
\hspace{3mm}3.2 $sign \leftarrow MP\_NEG$ \\
5829
\hspace{3mm}4.1 $sign \leftarrow MP\_ZPOS$ \\
5830
5. $a \leftarrow 0$ \\
5831
6. for $iy$ from $ix$ to $sn - 1$ do \\
5832
\hspace{3mm}6.1 Let $y$ denote the position in the map of $str_{iy}$. \\
5833
\hspace{3mm}6.2 If $str_{iy}$ is not in the map or $y \ge r$ then goto step 7. \\
5834
\hspace{3mm}6.3 $a \leftarrow a \cdot r$ \\
5835
\hspace{3mm}6.4 $a \leftarrow a + y$ \\
5836
7. If $a \ne 0$ then $a.sign \leftarrow sign$ \\
5837
8. Return(\textit{MP\_OKAY}). \\
5842
\caption{Algorithm mp\_read\_radix}
5844
\textbf{Algorithm mp\_read\_radix.}
5845
This algorithm will read an ASCII string and produce the radix-$\beta$ mp\_int representation of the same integer. A minus symbol ``-'' may precede the
5846
string to indicate the value is negative, otherwise it is assumed to be positive. The algorithm will read up to $sn$ characters from the input
5847
and will stop when it reads a character it cannot map the algorithm stops reading characters from the string. This allows numbers to be embedded
5848
as part of larger input without any significant problem.
5850
\vspace{+3mm}\begin{small}
5851
\hspace{-5.1mm}{\bf File}: bn\_mp\_read\_radix.c
5857
\subsection{Generating Radix-$n$ Output}
5858
Generating radix-$n$ output is fairly trivial with a division and remainder algorithm.
5860
\newpage\begin{figure}[!here]
5864
\hline Algorithm \textbf{mp\_toradix}. \\
5865
\textbf{Input}. A mp\_int $a$ and an integer $r$\\
5866
\textbf{Output}. The radix-$r$ representation of $a$ \\
5868
1. If $r < 2$ or $r > 64$ return(\textit{MP\_VAL}). \\
5869
2. If $a = 0$ then $str = $ ``$0$'' and return(\textit{MP\_OKAY}). \\
5870
3. $t \leftarrow a$ \\
5871
4. $str \leftarrow$ ``'' \\
5872
5. if $t.sign = MP\_NEG$ then \\
5873
\hspace{3mm}5.1 $str \leftarrow str + $ ``-'' \\
5874
\hspace{3mm}5.2 $t.sign = MP\_ZPOS$ \\
5875
6. While ($t \ne 0$) do \\
5876
\hspace{3mm}6.1 $d \leftarrow t \mbox{ (mod }r\mbox{)}$ \\
5877
\hspace{3mm}6.2 $t \leftarrow \lfloor t / r \rfloor$ \\
5878
\hspace{3mm}6.3 Look up $d$ in the map and store the equivalent character in $y$. \\
5879
\hspace{3mm}6.4 $str \leftarrow str + y$ \\
5880
7. If $str_0 = $``$-$'' then \\
5881
\hspace{3mm}7.1 Reverse the digits $str_1, str_2, \ldots str_n$. \\
5883
\hspace{3mm}8.1 Reverse the digits $str_0, str_1, \ldots str_n$. \\
5884
9. Return(\textit{MP\_OKAY}).\\
5889
\caption{Algorithm mp\_toradix}
5891
\textbf{Algorithm mp\_toradix.}
5892
This algorithm computes the radix-$r$ representation of an mp\_int $a$. The ``digits'' of the representation are extracted by reducing
5893
successive powers of $\lfloor a / r^k \rfloor$ the input modulo $r$ until $r^k > a$. Note that instead of actually dividing by $r^k$ in
5894
each iteration the quotient $\lfloor a / r \rfloor$ is saved for the next iteration. As a result a series of trivial $n \times 1$ divisions
5895
are required instead of a series of $n \times k$ divisions. One design flaw of this approach is that the digits are produced in the reverse order
5896
(see~\ref{fig:mpradix}). To remedy this flaw the digits must be swapped or simply ``reversed''.
5900
\begin{tabular}{|c|c|c|}
5901
\hline \textbf{Value of $a$} & \textbf{Value of $d$} & \textbf{Value of $str$} \\
5902
\hline $1234$ & -- & -- \\
5903
\hline $123$ & $4$ & ``4'' \\
5904
\hline $12$ & $3$ & ``43'' \\
5905
\hline $1$ & $2$ & ``432'' \\
5906
\hline $0$ & $1$ & ``4321'' \\
5910
\caption{Example of Algorithm mp\_toradix.}
5914
\vspace{+3mm}\begin{small}
5915
\hspace{-5.1mm}{\bf File}: bn\_mp\_toradix.c
5921
\chapter{Number Theoretic Algorithms}
5922
This chapter discusses several fundamental number theoretic algorithms such as the greatest common divisor, least common multiple and Jacobi
5923
symbol computation. These algorithms arise as essential components in several key cryptographic algorithms such as the RSA public key algorithm and
5924
various Sieve based factoring algorithms.
5926
\section{Greatest Common Divisor}
5927
The greatest common divisor of two integers $a$ and $b$, often denoted as $(a, b)$ is the largest integer $k$ that is a proper divisor of
5928
both $a$ and $b$. That is, $k$ is the largest integer such that $0 \equiv a \mbox{ (mod }k\mbox{)}$ and $0 \equiv b \mbox{ (mod }k\mbox{)}$ occur
5931
The most common approach (cite) is to reduce one input modulo another. That is if $a$ and $b$ are divisible by some integer $k$ and if $qa + r = b$ then
5932
$r$ is also divisible by $k$. The reduction pattern follows $\left < a , b \right > \rightarrow \left < b, a \mbox{ mod } b \right >$.
5934
\newpage\begin{figure}[!here]
5938
\hline Algorithm \textbf{Greatest Common Divisor (I)}. \\
5939
\textbf{Input}. Two positive integers $a$ and $b$ greater than zero. \\
5940
\textbf{Output}. The greatest common divisor $(a, b)$. \\
5942
1. While ($b > 0$) do \\
5943
\hspace{3mm}1.1 $r \leftarrow a \mbox{ (mod }b\mbox{)}$ \\
5944
\hspace{3mm}1.2 $a \leftarrow b$ \\
5945
\hspace{3mm}1.3 $b \leftarrow r$ \\
5951
\caption{Algorithm Greatest Common Divisor (I)}
5955
This algorithm will quickly converge on the greatest common divisor since the residue $r$ tends diminish rapidly. However, divisions are
5956
relatively expensive operations to perform and should ideally be avoided. There is another approach based on a similar relationship of
5957
greatest common divisors. The faster approach is based on the observation that if $k$ divides both $a$ and $b$ it will also divide $a - b$.
5958
In particular, we would like $a - b$ to decrease in magnitude which implies that $b \ge a$.
5960
\begin{figure}[!here]
5964
\hline Algorithm \textbf{Greatest Common Divisor (II)}. \\
5965
\textbf{Input}. Two positive integers $a$ and $b$ greater than zero. \\
5966
\textbf{Output}. The greatest common divisor $(a, b)$. \\
5968
1. While ($b > 0$) do \\
5969
\hspace{3mm}1.1 Swap $a$ and $b$ such that $a$ is the smallest of the two. \\
5970
\hspace{3mm}1.2 $b \leftarrow b - a$ \\
5976
\caption{Algorithm Greatest Common Divisor (II)}
5980
\textbf{Proof} \textit{Algorithm~\ref{fig:gcd2} will return the greatest common divisor of $a$ and $b$.}
5981
The algorithm in figure~\ref{fig:gcd2} will eventually terminate since $b \ge a$ the subtraction in step 1.2 will be a value less than $b$. In other
5982
words in every iteration that tuple $\left < a, b \right >$ decrease in magnitude until eventually $a = b$. Since both $a$ and $b$ are always
5983
divisible by the greatest common divisor (\textit{until the last iteration}) and in the last iteration of the algorithm $b = 0$, therefore, in the
5984
second to last iteration of the algorithm $b = a$ and clearly $(a, a) = a$ which concludes the proof. \textbf{QED}.
5986
As a matter of practicality algorithm \ref{fig:gcd1} decreases far too slowly to be useful. Specially if $b$ is much larger than $a$ such that
5987
$b - a$ is still very much larger than $a$. A simple addition to the algorithm is to divide $b - a$ by a power of some integer $p$ which does
5988
not divide the greatest common divisor but will divide $b - a$. In this case ${b - a} \over p$ is also an integer and still divisible by
5989
the greatest common divisor.
5991
However, instead of factoring $b - a$ to find a suitable value of $p$ the powers of $p$ can be removed from $a$ and $b$ that are in common first.
5992
Then inside the loop whenever $b - a$ is divisible by some power of $p$ it can be safely removed.
5994
\begin{figure}[!here]
5998
\hline Algorithm \textbf{Greatest Common Divisor (III)}. \\
5999
\textbf{Input}. Two positive integers $a$ and $b$ greater than zero. \\
6000
\textbf{Output}. The greatest common divisor $(a, b)$. \\
6002
1. $k \leftarrow 0$ \\
6003
2. While $a$ and $b$ are both divisible by $p$ do \\
6004
\hspace{3mm}2.1 $a \leftarrow \lfloor a / p \rfloor$ \\
6005
\hspace{3mm}2.2 $b \leftarrow \lfloor b / p \rfloor$ \\
6006
\hspace{3mm}2.3 $k \leftarrow k + 1$ \\
6007
3. While $a$ is divisible by $p$ do \\
6008
\hspace{3mm}3.1 $a \leftarrow \lfloor a / p \rfloor$ \\
6009
4. While $b$ is divisible by $p$ do \\
6010
\hspace{3mm}4.1 $b \leftarrow \lfloor b / p \rfloor$ \\
6011
5. While ($b > 0$) do \\
6012
\hspace{3mm}5.1 Swap $a$ and $b$ such that $a$ is the smallest of the two. \\
6013
\hspace{3mm}5.2 $b \leftarrow b - a$ \\
6014
\hspace{3mm}5.3 While $b$ is divisible by $p$ do \\
6015
\hspace{6mm}5.3.1 $b \leftarrow \lfloor b / p \rfloor$ \\
6016
6. Return($a \cdot p^k$). \\
6021
\caption{Algorithm Greatest Common Divisor (III)}
6025
This algorithm is based on the first except it removes powers of $p$ first and inside the main loop to ensure the tuple $\left < a, b \right >$
6026
decreases more rapidly. The first loop on step two removes powers of $p$ that are in common. A count, $k$, is kept which will present a common
6027
divisor of $p^k$. After step two the remaining common divisor of $a$ and $b$ cannot be divisible by $p$. This means that $p$ can be safely
6028
divided out of the difference $b - a$ so long as the division leaves no remainder.
6030
In particular the value of $p$ should be chosen such that the division on step 5.3.1 occur often. It also helps that division by $p$ be easy
6031
to compute. The ideal choice of $p$ is two since division by two amounts to a right logical shift. Another important observation is that by
6032
step five both $a$ and $b$ are odd. Therefore, the diffrence $b - a$ must be even which means that each iteration removes one bit from the
6033
largest of the pair.
6035
\subsection{Complete Greatest Common Divisor}
6036
The algorithms presented so far cannot handle inputs which are zero or negative. The following algorithm can handle all input cases properly
6037
and will produce the greatest common divisor.
6039
\newpage\begin{figure}[!here]
6043
\hline Algorithm \textbf{mp\_gcd}. \\
6044
\textbf{Input}. mp\_int $a$ and $b$ \\
6045
\textbf{Output}. The greatest common divisor $c = (a, b)$. \\
6047
1. If $a = 0$ then \\
6048
\hspace{3mm}1.1 $c \leftarrow \vert b \vert $ \\
6049
\hspace{3mm}1.2 Return(\textit{MP\_OKAY}). \\
6050
2. If $b = 0$ then \\
6051
\hspace{3mm}2.1 $c \leftarrow \vert a \vert $ \\
6052
\hspace{3mm}2.2 Return(\textit{MP\_OKAY}). \\
6053
3. $u \leftarrow \vert a \vert, v \leftarrow \vert b \vert$ \\
6054
4. $k \leftarrow 0$ \\
6055
5. While $u.used > 0$ and $v.used > 0$ and $u_0 \equiv v_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\
6056
\hspace{3mm}5.1 $k \leftarrow k + 1$ \\
6057
\hspace{3mm}5.2 $u \leftarrow \lfloor u / 2 \rfloor$ \\
6058
\hspace{3mm}5.3 $v \leftarrow \lfloor v / 2 \rfloor$ \\
6059
6. While $u.used > 0$ and $u_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\
6060
\hspace{3mm}6.1 $u \leftarrow \lfloor u / 2 \rfloor$ \\
6061
7. While $v.used > 0$ and $v_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\
6062
\hspace{3mm}7.1 $v \leftarrow \lfloor v / 2 \rfloor$ \\
6063
8. While $v.used > 0$ \\
6064
\hspace{3mm}8.1 If $\vert u \vert > \vert v \vert$ then \\
6065
\hspace{6mm}8.1.1 Swap $u$ and $v$. \\
6066
\hspace{3mm}8.2 $v \leftarrow \vert v \vert - \vert u \vert$ \\
6067
\hspace{3mm}8.3 While $v.used > 0$ and $v_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\
6068
\hspace{6mm}8.3.1 $v \leftarrow \lfloor v / 2 \rfloor$ \\
6069
9. $c \leftarrow u \cdot 2^k$ \\
6070
10. Return(\textit{MP\_OKAY}). \\
6075
\caption{Algorithm mp\_gcd}
6077
\textbf{Algorithm mp\_gcd.}
6078
This algorithm will produce the greatest common divisor of two mp\_ints $a$ and $b$. The algorithm was originally based on Algorithm B of
6079
Knuth \cite[pp. 338]{TAOCPV2} but has been modified to be simpler to explain. In theory it achieves the same asymptotic working time as
6080
Algorithm B and in practice this appears to be true.
6082
The first two steps handle the cases where either one of or both inputs are zero. If either input is zero the greatest common divisor is the
6083
largest input or zero if they are both zero. If the inputs are not trivial than $u$ and $v$ are assigned the absolute values of
6084
$a$ and $b$ respectively and the algorithm will proceed to reduce the pair.
6086
Step five will divide out any common factors of two and keep track of the count in the variable $k$. After this step, two is no longer a
6087
factor of the remaining greatest common divisor between $u$ and $v$ and can be safely evenly divided out of either whenever they are even. Step
6088
six and seven ensure that the $u$ and $v$ respectively have no more factors of two. At most only one of the while--loops will iterate since
6089
they cannot both be even.
6091
By step eight both of $u$ and $v$ are odd which is required for the inner logic. First the pair are swapped such that $v$ is equal to
6092
or greater than $u$. This ensures that the subtraction on step 8.2 will always produce a positive and even result. Step 8.3 removes any
6093
factors of two from the difference $u$ to ensure that in the next iteration of the loop both are once again odd.
6095
After $v = 0$ occurs the variable $u$ has the greatest common divisor of the pair $\left < u, v \right >$ just after step six. The result
6096
must be adjusted by multiplying by the common factors of two ($2^k$) removed earlier.
6098
\vspace{+3mm}\begin{small}
6099
\hspace{-5.1mm}{\bf File}: bn\_mp\_gcd.c
6105
This function makes use of the macros mp\_iszero and mp\_iseven. The former evaluates to $1$ if the input mp\_int is equivalent to the
6106
integer zero otherwise it evaluates to $0$. The latter evaluates to $1$ if the input mp\_int represents a non-zero even integer otherwise
6107
it evaluates to $0$. Note that just because mp\_iseven may evaluate to $0$ does not mean the input is odd, it could also be zero. The three
6108
trivial cases of inputs are handled on lines 24 through 30. After those lines the inputs are assumed to be non-zero.
6110
Lines 32 and 37 make local copies $u$ and $v$ of the inputs $a$ and $b$ respectively. At this point the common factors of two
6111
must be divided out of the two inputs. The block starting at line 44 removes common factors of two by first counting the number of trailing
6112
zero bits in both. The local integer $k$ is used to keep track of how many factors of $2$ are pulled out of both values. It is assumed that
6113
the number of factors will not exceed the maximum value of a C ``int'' data type\footnote{Strictly speaking no array in C may have more than
6114
entries than are accessible by an ``int'' so this is not a limitation.}.
6116
At this point there are no more common factors of two in the two values. The divisions by a power of two on lines 62 and 68 remove
6117
any independent factors of two such that both $u$ and $v$ are guaranteed to be an odd integer before hitting the main body of the algorithm. The while loop
6118
on line 73 performs the reduction of the pair until $v$ is equal to zero. The unsigned comparison and subtraction algorithms are used in
6119
place of the full signed routines since both values are guaranteed to be positive and the result of the subtraction is guaranteed to be non-negative.
6121
\section{Least Common Multiple}
6122
The least common multiple of a pair of integers is their product divided by their greatest common divisor. For two integers $a$ and $b$ the
6123
least common multiple is normally denoted as $[ a, b ]$ and numerically equivalent to ${ab} \over {(a, b)}$. For example, if $a = 2 \cdot 2 \cdot 3 = 12$
6124
and $b = 2 \cdot 3 \cdot 3 \cdot 7 = 126$ the least common multiple is ${126 \over {(12, 126)}} = {126 \over 6} = 21$.
6126
The least common multiple arises often in coding theory as well as number theory. If two functions have periods of $a$ and $b$ respectively they will
6127
collide, that is be in synchronous states, after only $[ a, b ]$ iterations. This is why, for example, random number generators based on
6128
Linear Feedback Shift Registers (LFSR) tend to use registers with periods which are co-prime (\textit{e.g. the greatest common divisor is one.}).
6129
Similarly in number theory if a composite $n$ has two prime factors $p$ and $q$ then maximal order of any unit of $\Z/n\Z$ will be $[ p - 1, q - 1] $.
6131
\begin{figure}[!here]
6135
\hline Algorithm \textbf{mp\_lcm}. \\
6136
\textbf{Input}. mp\_int $a$ and $b$ \\
6137
\textbf{Output}. The least common multiple $c = [a, b]$. \\
6139
1. $c \leftarrow (a, b)$ \\
6140
2. $t \leftarrow a \cdot b$ \\
6141
3. $c \leftarrow \lfloor t / c \rfloor$ \\
6142
4. Return(\textit{MP\_OKAY}). \\
6147
\caption{Algorithm mp\_lcm}
6149
\textbf{Algorithm mp\_lcm.}
6150
This algorithm computes the least common multiple of two mp\_int inputs $a$ and $b$. It computes the least common multiple directly by
6151
dividing the product of the two inputs by their greatest common divisor.
6153
\vspace{+3mm}\begin{small}
6154
\hspace{-5.1mm}{\bf File}: bn\_mp\_lcm.c
6160
\section{Jacobi Symbol Computation}
6161
To explain the Jacobi Symbol we shall first discuss the Legendre function\footnote{Arrg. What is the name of this?} off which the Jacobi symbol is
6162
defined. The Legendre function computes whether or not an integer $a$ is a quadratic residue modulo an odd prime $p$. Numerically it is
6163
equivalent to equation \ref{eqn:legendre}.
6165
\textit{-- Tom, don't be an ass, cite your source here...!}
6168
a^{(p-1)/2} \equiv \begin{array}{rl}
6169
-1 & \mbox{if }a\mbox{ is a quadratic non-residue.} \\
6170
0 & \mbox{if }a\mbox{ divides }p\mbox{.} \\
6171
1 & \mbox{if }a\mbox{ is a quadratic residue}.
6172
\end{array} \mbox{ (mod }p\mbox{)}
6173
\label{eqn:legendre}
6176
\textbf{Proof.} \textit{Equation \ref{eqn:legendre} correctly identifies the residue status of an integer $a$ modulo a prime $p$.}
6177
An integer $a$ is a quadratic residue if the following equation has a solution.
6180
x^2 \equiv a \mbox{ (mod }p\mbox{)}
6184
Consider the following equation.
6187
0 \equiv x^{p-1} - 1 \equiv \left \lbrace \left (x^2 \right )^{(p-1)/2} - a^{(p-1)/2} \right \rbrace + \left ( a^{(p-1)/2} - 1 \right ) \mbox{ (mod }p\mbox{)}
6191
Whether equation \ref{eqn:root} has a solution or not equation \ref{eqn:rooti} is always true. If $a^{(p-1)/2} - 1 \equiv 0 \mbox{ (mod }p\mbox{)}$
6192
then the quantity in the braces must be zero. By reduction,
6195
\left (x^2 \right )^{(p-1)/2} - a^{(p-1)/2} \equiv 0 \nonumber \\
6196
\left (x^2 \right )^{(p-1)/2} \equiv a^{(p-1)/2} \nonumber \\
6197
x^2 \equiv a \mbox{ (mod }p\mbox{)}
6200
As a result there must be a solution to the quadratic equation and in turn $a$ must be a quadratic residue. If $a$ does not divide $p$ and $a$
6201
is not a quadratic residue then the only other value $a^{(p-1)/2}$ may be congruent to is $-1$ since
6203
0 \equiv a^{p - 1} - 1 \equiv (a^{(p-1)/2} + 1)(a^{(p-1)/2} - 1) \mbox{ (mod }p\mbox{)}
6205
One of the terms on the right hand side must be zero. \textbf{QED}
6207
\subsection{Jacobi Symbol}
6208
The Jacobi symbol is a generalization of the Legendre function for any odd non prime moduli $p$ greater than 2. If $p = \prod_{i=0}^n p_i$ then
6209
the Jacobi symbol $\left ( { a \over p } \right )$ is equal to the following equation.
6212
\left ( { a \over p } \right ) = \left ( { a \over p_0} \right ) \left ( { a \over p_1} \right ) \ldots \left ( { a \over p_n} \right )
6215
By inspection if $p$ is prime the Jacobi symbol is equivalent to the Legendre function. The following facts\footnote{See HAC \cite[pp. 72-74]{HAC} for
6216
further details.} will be used to derive an efficient Jacobi symbol algorithm. Where $p$ is an odd integer greater than two and $a, b \in \Z$ the
6220
\item $\left ( { a \over p} \right )$ equals $-1$, $0$ or $1$.
6221
\item $\left ( { ab \over p} \right ) = \left ( { a \over p} \right )\left ( { b \over p} \right )$.
6222
\item If $a \equiv b$ then $\left ( { a \over p} \right ) = \left ( { b \over p} \right )$.
6223
\item $\left ( { 2 \over p} \right )$ equals $1$ if $p \equiv 1$ or $7 \mbox{ (mod }8\mbox{)}$. Otherwise, it equals $-1$.
6224
\item $\left ( { a \over p} \right ) \equiv \left ( { p \over a} \right ) \cdot (-1)^{(p-1)(a-1)/4}$. More specifically
6225
$\left ( { a \over p} \right ) = \left ( { p \over a} \right )$ if $p \equiv a \equiv 1 \mbox{ (mod }4\mbox{)}$.
6228
Using these facts if $a = 2^k \cdot a'$ then
6231
\left ( { a \over p } \right ) = \left ( {{2^k} \over p } \right ) \left ( {a' \over p} \right ) \nonumber \\
6232
= \left ( {2 \over p } \right )^k \left ( {a' \over p} \right )
6239
\left ( { a \over p } \right ) = \left ( { p \over a } \right ) \cdot (-1)^{(p-1)(a-1)/4}
6242
Subsequently by fact three since $p \equiv (p \mbox{ mod }a) \mbox{ (mod }a\mbox{)}$ then
6245
\left ( { a \over p } \right ) = \left ( { {p \mbox{ mod } a} \over a } \right ) \cdot (-1)^{(p-1)(a-1)/4}
6248
By putting both observations into equation \ref{eqn:jacobi} the following simplified equation is formed.
6251
\left ( { a \over p } \right ) = \left ( {2 \over p } \right )^k \left ( {{p\mbox{ mod }a'} \over a'} \right ) \cdot (-1)^{(p-1)(a'-1)/4}
6254
The value of $\left ( {{p \mbox{ mod }a'} \over a'} \right )$ can be found by using the same equation recursively. The value of
6255
$\left ( {2 \over p } \right )^k$ equals $1$ if $k$ is even otherwise it equals $\left ( {2 \over p } \right )$. Using this approach the
6256
factors of $p$ do not have to be known. Furthermore, if $(a, p) = 1$ then the algorithm will terminate when the recursion requests the
6257
Jacobi symbol computation of $\left ( {1 \over a'} \right )$ which is simply $1$.
6259
\newpage\begin{figure}[!here]
6263
\hline Algorithm \textbf{mp\_jacobi}. \\
6264
\textbf{Input}. mp\_int $a$ and $p$, $a \ge 0$, $p \ge 3$, $p \equiv 1 \mbox{ (mod }2\mbox{)}$ \\
6265
\textbf{Output}. The Jacobi symbol $c = \left ( {a \over p } \right )$. \\
6267
1. If $a = 0$ then \\
6268
\hspace{3mm}1.1 $c \leftarrow 0$ \\
6269
\hspace{3mm}1.2 Return(\textit{MP\_OKAY}). \\
6270
2. If $a = 1$ then \\
6271
\hspace{3mm}2.1 $c \leftarrow 1$ \\
6272
\hspace{3mm}2.2 Return(\textit{MP\_OKAY}). \\
6273
3. $a' \leftarrow a$ \\
6274
4. $k \leftarrow 0$ \\
6275
5. While $a'.used > 0$ and $a'_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\
6276
\hspace{3mm}5.1 $k \leftarrow k + 1$ \\
6277
\hspace{3mm}5.2 $a' \leftarrow \lfloor a' / 2 \rfloor$ \\
6278
6. If $k \equiv 0 \mbox{ (mod }2\mbox{)}$ then \\
6279
\hspace{3mm}6.1 $s \leftarrow 1$ \\
6281
\hspace{3mm}7.1 $r \leftarrow p_0 \mbox{ (mod }8\mbox{)}$ \\
6282
\hspace{3mm}7.2 If $r = 1$ or $r = 7$ then \\
6283
\hspace{6mm}7.2.1 $s \leftarrow 1$ \\
6284
\hspace{3mm}7.3 else \\
6285
\hspace{6mm}7.3.1 $s \leftarrow -1$ \\
6286
8. If $p_0 \equiv a'_0 \equiv 3 \mbox{ (mod }4\mbox{)}$ then \\
6287
\hspace{3mm}8.1 $s \leftarrow -s$ \\
6288
9. If $a' \ne 1$ then \\
6289
\hspace{3mm}9.1 $p' \leftarrow p \mbox{ (mod }a'\mbox{)}$ \\
6290
\hspace{3mm}9.2 $s \leftarrow s \cdot \mbox{mp\_jacobi}(p', a')$ \\
6291
10. $c \leftarrow s$ \\
6292
11. Return(\textit{MP\_OKAY}). \\
6297
\caption{Algorithm mp\_jacobi}
6299
\textbf{Algorithm mp\_jacobi.}
6300
This algorithm computes the Jacobi symbol for an arbitrary positive integer $a$ with respect to an odd integer $p$ greater than three. The algorithm
6301
is based on algorithm 2.149 of HAC \cite[pp. 73]{HAC}.
6303
Step numbers one and two handle the trivial cases of $a = 0$ and $a = 1$ respectively. Step five determines the number of two factors in the
6304
input $a$. If $k$ is even than the term $\left ( { 2 \over p } \right )^k$ must always evaluate to one. If $k$ is odd than the term evaluates to one
6305
if $p_0$ is congruent to one or seven modulo eight, otherwise it evaluates to $-1$. After the the $\left ( { 2 \over p } \right )^k$ term is handled
6306
the $(-1)^{(p-1)(a'-1)/4}$ is computed and multiplied against the current product $s$. The latter term evaluates to one if both $p$ and $a'$
6307
are congruent to one modulo four, otherwise it evaluates to negative one.
6309
By step nine if $a'$ does not equal one a recursion is required. Step 9.1 computes $p' \equiv p \mbox{ (mod }a'\mbox{)}$ and will recurse to compute
6310
$\left ( {p' \over a'} \right )$ which is multiplied against the current Jacobi product.
6312
\vspace{+3mm}\begin{small}
6313
\hspace{-5.1mm}{\bf File}: bn\_mp\_jacobi.c
6319
As a matter of practicality the variable $a'$ as per the pseudo-code is reprensented by the variable $a1$ since the $'$ symbol is not valid for a C
6320
variable name character.
6322
The two simple cases of $a = 0$ and $a = 1$ are handled at the very beginning to simplify the algorithm. If the input is non-trivial the algorithm
6323
has to proceed compute the Jacobi. The variable $s$ is used to hold the current Jacobi product. Note that $s$ is merely a C ``int'' data type since
6324
the values it may obtain are merely $-1$, $0$ and $1$.
6326
After a local copy of $a$ is made all of the factors of two are divided out and the total stored in $k$. Technically only the least significant
6327
bit of $k$ is required, however, it makes the algorithm simpler to follow to perform an addition. In practice an exclusive-or and addition have the same
6328
processor requirements and neither is faster than the other.
6330
Line 58 through 71 determines the value of $\left ( { 2 \over p } \right )^k$. If the least significant bit of $k$ is zero than
6331
$k$ is even and the value is one. Otherwise, the value of $s$ depends on which residue class $p$ belongs to modulo eight. The value of
6332
$(-1)^{(p-1)(a'-1)/4}$ is compute and multiplied against $s$ on lines 71 through 74.
6334
Finally, if $a1$ does not equal one the algorithm must recurse and compute $\left ( {p' \over a'} \right )$.
6336
\textit{-- Comment about default $s$ and such...}
6338
\section{Modular Inverse}
6340
The modular inverse of a number actually refers to the modular multiplicative inverse. Essentially for any integer $a$ such that $(a, p) = 1$ there
6341
exist another integer $b$ such that $ab \equiv 1 \mbox{ (mod }p\mbox{)}$. The integer $b$ is called the multiplicative inverse of $a$ which is
6342
denoted as $b = a^{-1}$. Technically speaking modular inversion is a well defined operation for any finite ring or field not just for rings and
6343
fields of integers. However, the former will be the matter of discussion.
6345
The simplest approach is to compute the algebraic inverse of the input. That is to compute $b \equiv a^{\Phi(p) - 1}$. If $\Phi(p)$ is the
6346
order of the multiplicative subgroup modulo $p$ then $b$ must be the multiplicative inverse of $a$. The proof of which is trivial.
6349
ab \equiv a \left (a^{\Phi(p) - 1} \right ) \equiv a^{\Phi(p)} \equiv a^0 \equiv 1 \mbox{ (mod }p\mbox{)}
6352
However, as simple as this approach may be it has two serious flaws. It requires that the value of $\Phi(p)$ be known which if $p$ is composite
6353
requires all of the prime factors. This approach also is very slow as the size of $p$ grows.
6355
A simpler approach is based on the observation that solving for the multiplicative inverse is equivalent to solving the linear
6356
Diophantine\footnote{See LeVeque \cite[pp. 40-43]{LeVeque} for more information.} equation.
6362
Where $a$, $b$, $p$ and $q$ are all integers. If such a pair of integers $ \left < b, q \right >$ exist than $b$ is the multiplicative inverse of
6363
$a$ modulo $p$. The extended Euclidean algorithm (Knuth \cite[pp. 342]{TAOCPV2}) can be used to solve such equations provided $(a, p) = 1$.
6364
However, instead of using that algorithm directly a variant known as the binary Extended Euclidean algorithm will be used in its place. The
6365
binary approach is very similar to the binary greatest common divisor algorithm except it will produce a full solution to the Diophantine
6368
\subsection{General Case}
6369
\newpage\begin{figure}[!here]
6373
\hline Algorithm \textbf{mp\_invmod}. \\
6374
\textbf{Input}. mp\_int $a$ and $b$, $(a, b) = 1$, $p \ge 2$, $0 < a < p$. \\
6375
\textbf{Output}. The modular inverse $c \equiv a^{-1} \mbox{ (mod }b\mbox{)}$. \\
6377
1. If $b \le 0$ then return(\textit{MP\_VAL}). \\
6378
2. If $b_0 \equiv 1 \mbox{ (mod }2\mbox{)}$ then use algorithm fast\_mp\_invmod. \\
6379
3. $x \leftarrow \vert a \vert, y \leftarrow b$ \\
6380
4. If $x_0 \equiv y_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ then return(\textit{MP\_VAL}). \\
6381
5. $B \leftarrow 0, C \leftarrow 0, A \leftarrow 1, D \leftarrow 1$ \\
6382
6. While $u.used > 0$ and $u_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\
6383
\hspace{3mm}6.1 $u \leftarrow \lfloor u / 2 \rfloor$ \\
6384
\hspace{3mm}6.2 If ($A.used > 0$ and $A_0 \equiv 1 \mbox{ (mod }2\mbox{)}$) or ($B.used > 0$ and $B_0 \equiv 1 \mbox{ (mod }2\mbox{)}$) then \\
6385
\hspace{6mm}6.2.1 $A \leftarrow A + y$ \\
6386
\hspace{6mm}6.2.2 $B \leftarrow B - x$ \\
6387
\hspace{3mm}6.3 $A \leftarrow \lfloor A / 2 \rfloor$ \\
6388
\hspace{3mm}6.4 $B \leftarrow \lfloor B / 2 \rfloor$ \\
6389
7. While $v.used > 0$ and $v_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\
6390
\hspace{3mm}7.1 $v \leftarrow \lfloor v / 2 \rfloor$ \\
6391
\hspace{3mm}7.2 If ($C.used > 0$ and $C_0 \equiv 1 \mbox{ (mod }2\mbox{)}$) or ($D.used > 0$ and $D_0 \equiv 1 \mbox{ (mod }2\mbox{)}$) then \\
6392
\hspace{6mm}7.2.1 $C \leftarrow C + y$ \\
6393
\hspace{6mm}7.2.2 $D \leftarrow D - x$ \\
6394
\hspace{3mm}7.3 $C \leftarrow \lfloor C / 2 \rfloor$ \\
6395
\hspace{3mm}7.4 $D \leftarrow \lfloor D / 2 \rfloor$ \\
6396
8. If $u \ge v$ then \\
6397
\hspace{3mm}8.1 $u \leftarrow u - v$ \\
6398
\hspace{3mm}8.2 $A \leftarrow A - C$ \\
6399
\hspace{3mm}8.3 $B \leftarrow B - D$ \\
6401
\hspace{3mm}9.1 $v \leftarrow v - u$ \\
6402
\hspace{3mm}9.2 $C \leftarrow C - A$ \\
6403
\hspace{3mm}9.3 $D \leftarrow D - B$ \\
6404
10. If $u \ne 0$ goto step 6. \\
6405
11. If $v \ne 1$ return(\textit{MP\_VAL}). \\
6406
12. While $C \le 0$ do \\
6407
\hspace{3mm}12.1 $C \leftarrow C + b$ \\
6408
13. While $C \ge b$ do \\
6409
\hspace{3mm}13.1 $C \leftarrow C - b$ \\
6410
14. $c \leftarrow C$ \\
6411
15. Return(\textit{MP\_OKAY}). \\
6417
\textbf{Algorithm mp\_invmod.}
6418
This algorithm computes the modular multiplicative inverse of an integer $a$ modulo an integer $b$. This algorithm is a variation of the
6419
extended binary Euclidean algorithm from HAC \cite[pp. 608]{HAC}. It has been modified to only compute the modular inverse and not a complete
6420
Diophantine solution.
6422
If $b \le 0$ than the modulus is invalid and MP\_VAL is returned. Similarly if both $a$ and $b$ are even then there cannot be a multiplicative
6423
inverse for $a$ and the error is reported.
6425
The astute reader will observe that steps seven through nine are very similar to the binary greatest common divisor algorithm mp\_gcd. In this case
6426
the other variables to the Diophantine equation are solved. The algorithm terminates when $u = 0$ in which case the solution is
6432
If $v$, the greatest common divisor of $a$ and $b$ is not equal to one then the algorithm will report an error as no inverse exists. Otherwise, $C$
6433
is the modular inverse of $a$. The actual value of $C$ is congruent to, but not necessarily equal to, the ideal modular inverse which should lie
6434
within $1 \le a^{-1} < b$. Step numbers twelve and thirteen adjust the inverse until it is in range. If the original input $a$ is within $0 < a < p$
6435
then only a couple of additions or subtractions will be required to adjust the inverse.
6437
\vspace{+3mm}\begin{small}
6438
\hspace{-5.1mm}{\bf File}: bn\_mp\_invmod.c
6444
\subsubsection{Odd Moduli}
6446
When the modulus $b$ is odd the variables $A$ and $C$ are fixed and are not required to compute the inverse. In particular by attempting to solve
6447
the Diophantine $Cb + Da = 1$ only $B$ and $D$ are required to find the inverse of $a$.
6449
The algorithm fast\_mp\_invmod is a direct adaptation of algorithm mp\_invmod with all all steps involving either $A$ or $C$ removed. This
6450
optimization will halve the time required to compute the modular inverse.
6452
\section{Primality Tests}
6454
A non-zero integer $a$ is said to be prime if it is not divisible by any other integer excluding one and itself. For example, $a = 7$ is prime
6455
since the integers $2 \ldots 6$ do not evenly divide $a$. By contrast, $a = 6$ is not prime since $a = 6 = 2 \cdot 3$.
6457
Prime numbers arise in cryptography considerably as they allow finite fields to be formed. The ability to determine whether an integer is prime or
6458
not quickly has been a viable subject in cryptography and number theory for considerable time. The algorithms that will be presented are all
6459
probablistic algorithms in that when they report an integer is composite it must be composite. However, when the algorithms report an integer is
6460
prime the algorithm may be incorrect.
6462
As will be discussed it is possible to limit the probability of error so well that for practical purposes the probablity of error might as
6463
well be zero. For the purposes of these discussions let $n$ represent the candidate integer of which the primality is in question.
6465
\subsection{Trial Division}
6467
Trial division means to attempt to evenly divide a candidate integer by small prime integers. If the candidate can be evenly divided it obviously
6468
cannot be prime. By dividing by all primes $1 < p \le \sqrt{n}$ this test can actually prove whether an integer is prime. However, such a test
6469
would require a prohibitive amount of time as $n$ grows.
6471
Instead of dividing by every prime, a smaller, more mangeable set of primes may be used instead. By performing trial division with only a subset
6472
of the primes less than $\sqrt{n} + 1$ the algorithm cannot prove if a candidate is prime. However, often it can prove a candidate is not prime.
6474
The benefit of this test is that trial division by small values is fairly efficient. Specially compared to the other algorithms that will be
6475
discussed shortly. The probability that this approach correctly identifies a composite candidate when tested with all primes upto $q$ is given by
6476
$1 - {1.12 \over ln(q)}$. The graph (\ref{pic:primality}, will be added later) demonstrates the probability of success for the range
6479
At approximately $q = 30$ the gain of performing further tests diminishes fairly quickly. At $q = 90$ further testing is generally not going to
6480
be of any practical use. In the case of LibTomMath the default limit $q = 256$ was chosen since it is not too high and will eliminate
6481
approximately $80\%$ of all candidate integers. The constant \textbf{PRIME\_SIZE} is equal to the number of primes in the test base. The
6482
array \_\_prime\_tab is an array of the first \textbf{PRIME\_SIZE} prime numbers.
6484
\begin{figure}[!here]
6488
\hline Algorithm \textbf{mp\_prime\_is\_divisible}. \\
6489
\textbf{Input}. mp\_int $a$ \\
6490
\textbf{Output}. $c = 1$ if $n$ is divisible by a small prime, otherwise $c = 0$. \\
6492
1. for $ix$ from $0$ to $PRIME\_SIZE$ do \\
6493
\hspace{3mm}1.1 $d \leftarrow n \mbox{ (mod }\_\_prime\_tab_{ix}\mbox{)}$ \\
6494
\hspace{3mm}1.2 If $d = 0$ then \\
6495
\hspace{6mm}1.2.1 $c \leftarrow 1$ \\
6496
\hspace{6mm}1.2.2 Return(\textit{MP\_OKAY}). \\
6497
2. $c \leftarrow 0$ \\
6498
3. Return(\textit{MP\_OKAY}). \\
6503
\caption{Algorithm mp\_prime\_is\_divisible}
6505
\textbf{Algorithm mp\_prime\_is\_divisible.}
6506
This algorithm attempts to determine if a candidate integer $n$ is composite by performing trial divisions.
6508
\vspace{+3mm}\begin{small}
6509
\hspace{-5.1mm}{\bf File}: bn\_mp\_prime\_is\_divisible.c
6515
The algorithm defaults to a return of $0$ in case an error occurs. The values in the prime table are all specified to be in the range of a
6516
mp\_digit. The table \_\_prime\_tab is defined in the following file.
6518
\vspace{+3mm}\begin{small}
6519
\hspace{-5.1mm}{\bf File}: bn\_prime\_tab.c
6525
Note that there are two possible tables. When an mp\_digit is 7-bits long only the primes upto $127$ may be included, otherwise the primes
6526
upto $1619$ are used. Note that the value of \textbf{PRIME\_SIZE} is a constant dependent on the size of a mp\_digit.
6528
\subsection{The Fermat Test}
6529
The Fermat test is probably one the oldest tests to have a non-trivial probability of success. It is based on the fact that if $n$ is in
6530
fact prime then $a^{n} \equiv a \mbox{ (mod }n\mbox{)}$ for all $0 < a < n$. The reason being that if $n$ is prime than the order of
6531
the multiplicative sub group is $n - 1$. Any base $a$ must have an order which divides $n - 1$ and as such $a^n$ is equivalent to
6534
If $n$ is composite then any given base $a$ does not have to have a period which divides $n - 1$. In which case
6535
it is possible that $a^n \nequiv a \mbox{ (mod }n\mbox{)}$. However, this test is not absolute as it is possible that the order
6536
of a base will divide $n - 1$ which would then be reported as prime. Such a base yields what is known as a Fermat pseudo-prime. Several
6537
integers known as Carmichael numbers will be a pseudo-prime to all valid bases. Fortunately such numbers are extremely rare as $n$ grows
6540
\begin{figure}[!here]
6544
\hline Algorithm \textbf{mp\_prime\_fermat}. \\
6545
\textbf{Input}. mp\_int $a$ and $b$, $a \ge 2$, $0 < b < a$. \\
6546
\textbf{Output}. $c = 1$ if $b^a \equiv b \mbox{ (mod }a\mbox{)}$, otherwise $c = 0$. \\
6548
1. $t \leftarrow b^a \mbox{ (mod }a\mbox{)}$ \\
6549
2. If $t = b$ then \\
6550
\hspace{3mm}2.1 $c = 1$ \\
6552
\hspace{3mm}3.1 $c = 0$ \\
6553
4. Return(\textit{MP\_OKAY}). \\
6558
\caption{Algorithm mp\_prime\_fermat}
6560
\textbf{Algorithm mp\_prime\_fermat.}
6561
This algorithm determines whether an mp\_int $a$ is a Fermat prime to the base $b$ or not. It uses a single modular exponentiation to
6562
determine the result.
6564
\vspace{+3mm}\begin{small}
6565
\hspace{-5.1mm}{\bf File}: bn\_mp\_prime\_fermat.c
6571
\subsection{The Miller-Rabin Test}
6572
The Miller-Rabin (citation) test is another primality test which has tighter error bounds than the Fermat test specifically with sequentially chosen
6573
candidate integers. The algorithm is based on the observation that if $n - 1 = 2^kr$ and if $b^r \nequiv \pm 1$ then after upto $k - 1$ squarings the
6574
value must be equal to $-1$. The squarings are stopped as soon as $-1$ is observed. If the value of $1$ is observed first it means that
6575
some value not congruent to $\pm 1$ when squared equals one which cannot occur if $n$ is prime.
6577
\begin{figure}[!here]
6581
\hline Algorithm \textbf{mp\_prime\_miller\_rabin}. \\
6582
\textbf{Input}. mp\_int $a$ and $b$, $a \ge 2$, $0 < b < a$. \\
6583
\textbf{Output}. $c = 1$ if $a$ is a Miller-Rabin prime to the base $a$, otherwise $c = 0$. \\
6585
1. $a' \leftarrow a - 1$ \\
6586
2. $r \leftarrow n1$ \\
6587
3. $c \leftarrow 0, s \leftarrow 0$ \\
6588
4. While $r.used > 0$ and $r_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\
6589
\hspace{3mm}4.1 $s \leftarrow s + 1$ \\
6590
\hspace{3mm}4.2 $r \leftarrow \lfloor r / 2 \rfloor$ \\
6591
5. $y \leftarrow b^r \mbox{ (mod }a\mbox{)}$ \\
6592
6. If $y \nequiv \pm 1$ then \\
6593
\hspace{3mm}6.1 $j \leftarrow 1$ \\
6594
\hspace{3mm}6.2 While $j \le (s - 1)$ and $y \nequiv a'$ \\
6595
\hspace{6mm}6.2.1 $y \leftarrow y^2 \mbox{ (mod }a\mbox{)}$ \\
6596
\hspace{6mm}6.2.2 If $y = 1$ then goto step 8. \\
6597
\hspace{6mm}6.2.3 $j \leftarrow j + 1$ \\
6598
\hspace{3mm}6.3 If $y \nequiv a'$ goto step 8. \\
6599
7. $c \leftarrow 1$\\
6600
8. Return(\textit{MP\_OKAY}). \\
6605
\caption{Algorithm mp\_prime\_miller\_rabin}
6607
\textbf{Algorithm mp\_prime\_miller\_rabin.}
6608
This algorithm performs one trial round of the Miller-Rabin algorithm to the base $b$. It will set $c = 1$ if the algorithm cannot determine
6609
if $b$ is composite or $c = 0$ if $b$ is provably composite. The values of $s$ and $r$ are computed such that $a' = a - 1 = 2^sr$.
6611
If the value $y \equiv b^r$ is congruent to $\pm 1$ then the algorithm cannot prove if $a$ is composite or not. Otherwise, the algorithm will
6612
square $y$ upto $s - 1$ times stopping only when $y \equiv -1$. If $y^2 \equiv 1$ and $y \nequiv \pm 1$ then the algorithm can report that $a$
6613
is provably composite. If the algorithm performs $s - 1$ squarings and $y \nequiv -1$ then $a$ is provably composite. If $a$ is not provably
6614
composite then it is \textit{probably} prime.
6616
\vspace{+3mm}\begin{small}
6617
\hspace{-5.1mm}{\bf File}: bn\_mp\_prime\_miller\_rabin.c
6628
\begin{thebibliography}{ABCDEF}
6629
\bibitem[1]{TAOCPV2}
6630
Donald Knuth, \textit{The Art of Computer Programming}, Third Edition, Volume Two, Seminumerical Algorithms, Addison-Wesley, 1998
6633
A. Menezes, P. van Oorschot, S. Vanstone, \textit{Handbook of Applied Cryptography}, CRC Press, 1996
6636
Michael Rosing, \textit{Implementing Elliptic Curve Cryptography}, Manning Publications, 1999
6639
Paul G. Comba, \textit{Exponentiation Cryptosystems on the IBM PC}. IBM Systems Journal 29(4): 526-538 (1990)
6642
A. Karatsuba, Doklay Akad. Nauk SSSR 145 (1962), pp.293-294
6645
Andre Weimerskirch and Christof Paar, \textit{Generalizations of the Karatsuba Algorithm for Polynomial Multiplication}, Submitted to Design, Codes and Cryptography, March 2002
6647
\bibitem[7]{BARRETT}
6648
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.
6651
P.L.Montgomery. \textit{Modular multiplication without trial division}. Mathematics of Computation, 44(170):519-521, April 1985.
6654
Chae Hoon Lim and Pil Joong Lee, \textit{Generating Efficient Primes for Discrete Log Cryptosystems}, POSTECH Information Research Laboratories
6657
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
6659
\bibitem[11]{RSAREF}
6660
R.L. Rivest, A. Shamir, L. Adleman, \textit{A Method for Obtaining Digital Signatures and Public-Key Cryptosystems}
6663
Whitfield Diffie, Martin E. Hellman, \textit{New Directions in Cryptography}, IEEE Transactions on Information Theory, 1976
6666
IEEE Standard for Binary Floating-Point Arithmetic (ANSI/IEEE Std 754-1985)
6669
GNU Multiple Precision (GMP), \url{http://www.swox.com/gmp/}
6672
Multiple Precision Integer Library (MPI), Michael Fromberger, \url{http://thayer.dartmouth.edu/~sting/mpi/}
6674
\bibitem[16]{OPENSSL}
6675
OpenSSL Cryptographic Toolkit, \url{http://openssl.org}
6678
Large Integer Package, \url{http://home.hetnet.nl/~ecstr/LIP.zip}
6681
JTC1/SC22/WG14, ISO/IEC 9899:1999, ``A draft rationale for the C99 standard.''
6684
The Sun Java Website, \url{http://java.sun.com/}
6686
\end{thebibliography}
added
removed
lib/hcrypto/libtommath/tommath.tex
