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diff --git a/libtommath/tommath.tex b/libtommath/tommath.tex index c79a537..d70b64b 100644 --- a/libtommath/tommath.tex +++ b/libtommath/tommath.tex @@ -66,31 +66,20 @@ QUALCOMM Australia \\ } } \maketitle -This text has been placed in the public domain. This text corresponds to the v0.39 release of the +This text has been placed in the public domain. This text corresponds to the v0.39 release of the LibTomMath project. -\begin{alltt} -Tom St Denis -111 Banning Rd -Ottawa, Ontario -K2L 1C3 -Canada - -Phone: 1-613-836-3160 -Email: tomstdenis@gmail.com -\end{alltt} - -This text is formatted to the international B5 paper size of 176mm wide by 250mm tall using the \LaTeX{} +This text is formatted to the international B5 paper size of 176mm wide by 250mm tall using the \LaTeX{} {\em book} macro package and the Perl {\em booker} package. \tableofcontents \listoffigures \chapter*{Prefaces} -When I tell people about my LibTom projects and that I release them as public domain they are often puzzled. -They ask why I did it and especially why I continue to work on them for free. The best I can explain it is ``Because I can.'' -Which seems odd and perhaps too terse for adult conversation. I often qualify it with ``I am able, I am willing.'' which +When I tell people about my LibTom projects and that I release them as public domain they are often puzzled. +They ask why I did it and especially why I continue to work on them for free. The best I can explain it is ``Because I can.'' +Which seems odd and perhaps too terse for adult conversation. I often qualify it with ``I am able, I am willing.'' which perhaps explains it better. I am the first to admit there is not anything that special with what I have done. Perhaps -others can see that too and then we would have a society to be proud of. My LibTom projects are what I am doing to give +others can see that too and then we would have a society to be proud of. My LibTom projects are what I am doing to give back to society in the form of tools and knowledge that can help others in their endeavours. I started writing this book because it was the most logical task to further my goal of open academia. The LibTomMath source @@ -103,9 +92,9 @@ from relatively straightforward algebra and I hope that this book can be a valua This book and indeed much of the LibTom projects would not exist in their current form if it was not for a plethora of kind people donating their time, resources and kind words to help support my work. Writing a text of significant length (along with the source code) is a tiresome and lengthy process. Currently the LibTom project is four years old, -comprises of literally thousands of users and over 100,000 lines of source code, TeX and other material. People like Mads and Greg -were there at the beginning to encourage me to work well. It is amazing how timely validation from others can boost morale to -continue the project. Definitely my parents were there for me by providing room and board during the many months of work in 2003. +comprises of literally thousands of users and over 100,000 lines of source code, TeX and other material. People like Mads and Greg +were there at the beginning to encourage me to work well. It is amazing how timely validation from others can boost morale to +continue the project. Definitely my parents were there for me by providing room and board during the many months of work in 2003. To my many friends whom I have met through the years I thank you for the good times and the words of encouragement. I hope I honour your kind gestures with this project. @@ -115,22 +104,22 @@ Open Source. Open Academia. Open Minds. \begin{flushright} Tom St Denis \end{flushright} \newpage -I found the opportunity to work with Tom appealing for several reasons, not only could I broaden my own horizons, but also +I found the opportunity to work with Tom appealing for several reasons, not only could I broaden my own horizons, but also contribute to educate others facing the problem of having to handle big number mathematical calculations. -This book is Tom's child and he has been caring and fostering the project ever since the beginning with a clear mind of -how he wanted the project to turn out. I have helped by proofreading the text and we have had several discussions about +This book is Tom's child and he has been caring and fostering the project ever since the beginning with a clear mind of +how he wanted the project to turn out. I have helped by proofreading the text and we have had several discussions about the layout and language used. -I hold a masters degree in cryptography from the University of Southern Denmark and have always been interested in the -practical aspects of cryptography. +I hold a masters degree in cryptography from the University of Southern Denmark and have always been interested in the +practical aspects of cryptography. -Having worked in the security consultancy business for several years in S\~{a}o Paulo, Brazil, I have been in touch with a -great deal of work in which multiple precision mathematics was needed. Understanding the possibilities for speeding up -multiple precision calculations is often very important since we deal with outdated machine architecture where modular +Having worked in the security consultancy business for several years in S\~{a}o Paulo, Brazil, I have been in touch with a +great deal of work in which multiple precision mathematics was needed. Understanding the possibilities for speeding up +multiple precision calculations is often very important since we deal with outdated machine architecture where modular reductions, for example, become painfully slow. -This text is for people who stop and wonder when first examining algorithms such as RSA for the first time and asks +This text is for people who stop and wonder when first examining algorithms such as RSA for the first time and asks themselves, ``You tell me this is only secure for large numbers, fine; but how do you implement these numbers?'' \begin{flushright} @@ -142,22 +131,22 @@ Brazil \end{flushright} \newpage -It's all because I broke my leg. That just happened to be at about the same time that Tom asked for someone to review the section of the book about -Karatsuba multiplication. I was laid up, alone and immobile, and thought ``Why not?'' I vaguely knew what Karatsuba multiplication was, but not +It's all because I broke my leg. That just happened to be at about the same time that Tom asked for someone to review the section of the book about +Karatsuba multiplication. I was laid up, alone and immobile, and thought ``Why not?'' I vaguely knew what Karatsuba multiplication was, but not really, so I thought I could help, learn, and stop myself from watching daytime cable TV, all at once. -At the time of writing this, I've still not met Tom or Mads in meatspace. I've been following Tom's progress since his first splash on the +At the time of writing this, I've still not met Tom or Mads in meatspace. I've been following Tom's progress since his first splash on the sci.crypt Usenet news group. I watched him go from a clueless newbie, to the cryptographic equivalent of a reformed smoker, to a real -contributor to the field, over a period of about two years. I've been impressed with his obvious intelligence, and astounded by his productivity. +contributor to the field, over a period of about two years. I've been impressed with his obvious intelligence, and astounded by his productivity. Of course, he's young enough to be my own child, so he doesn't have my problems with staying awake. -When I reviewed that single section of the book, in its very earliest form, I was very pleasantly surprised. So I decided to collaborate more fully, -and at least review all of it, and perhaps write some bits too. There's still a long way to go with it, and I have watched a number of close -friends go through the mill of publication, so I think that the way to go is longer than Tom thinks it is. Nevertheless, it's a good effort, +When I reviewed that single section of the book, in its very earliest form, I was very pleasantly surprised. So I decided to collaborate more fully, +and at least review all of it, and perhaps write some bits too. There's still a long way to go with it, and I have watched a number of close +friends go through the mill of publication, so I think that the way to go is longer than Tom thinks it is. Nevertheless, it's a good effort, and I'm pleased to be involved with it. \begin{flushright} -Greg Rose, Sydney, Australia, June 2003. +Greg Rose, Sydney, Australia, June 2003. \end{flushright} \mainmatter @@ -167,23 +156,23 @@ Greg Rose, Sydney, Australia, June 2003. \subsection{What is Multiple Precision Arithmetic?} When we think of long-hand arithmetic such as addition or multiplication we rarely consider the fact that we instinctively -raise or lower the precision of the numbers we are dealing with. For example, in decimal we almost immediate can -reason that $7$ times $6$ is $42$. However, $42$ has two digits of precision as opposed to one digit we started with. -Further multiplications of say $3$ result in a larger precision result $126$. In these few examples we have multiple +raise or lower the precision of the numbers we are dealing with. For example, in decimal we almost immediate can +reason that $7$ times $6$ is $42$. However, $42$ has two digits of precision as opposed to one digit we started with. +Further multiplications of say $3$ result in a larger precision result $126$. In these few examples we have multiple precisions for the numbers we are working with. Despite the various levels of precision a single subset\footnote{With the occasional optimization.} - of algorithms can be designed to accomodate them. + of algorithms can be designed to accomodate them. By way of comparison a fixed or single precision operation would lose precision on various operations. For example, in the decimal system with fixed precision $6 \cdot 7 = 2$. Essentially at the heart of computer based multiple precision arithmetic are the same long-hand algorithms taught in -schools to manually add, subtract, multiply and divide. +schools to manually add, subtract, multiply and divide. \subsection{The Need for Multiple Precision Arithmetic} The most prevalent need for multiple precision arithmetic, often referred to as ``bignum'' math, is within the implementation -of public-key cryptography algorithms. Algorithms such as RSA \cite{RSAREF} and Diffie-Hellman \cite{DHREF} require -integers of significant magnitude to resist known cryptanalytic attacks. For example, at the time of this writing a -typical RSA modulus would be at least greater than $10^{309}$. However, modern programming languages such as ISO C \cite{ISOC} and +of public-key cryptography algorithms. Algorithms such as RSA \cite{RSAREF} and Diffie-Hellman \cite{DHREF} require +integers of significant magnitude to resist known cryptanalytic attacks. For example, at the time of this writing a +typical RSA modulus would be at least greater than $10^{309}$. However, modern programming languages such as ISO C \cite{ISOC} and Java \cite{JAVA} only provide instrinsic support for integers which are relatively small and single precision. \begin{figure}[!here] @@ -201,27 +190,27 @@ Java \cite{JAVA} only provide instrinsic support for integers which are relative \label{fig:ISOC} \end{figure} -The largest data type guaranteed to be provided by the ISO C programming -language\footnote{As per the ISO C standard. However, each compiler vendor is allowed to augment the precision as they -see fit.} can only represent values up to $10^{19}$ as shown in figure \ref{fig:ISOC}. On its own the C language is -insufficient to accomodate the magnitude required for the problem at hand. An RSA modulus of magnitude $10^{19}$ could be -trivially factored\footnote{A Pollard-Rho factoring would take only $2^{16}$ time.} on the average desktop computer, -rendering any protocol based on the algorithm insecure. Multiple precision algorithms solve this very problem by +The largest data type guaranteed to be provided by the ISO C programming +language\footnote{As per the ISO C standard. However, each compiler vendor is allowed to augment the precision as they +see fit.} can only represent values up to $10^{19}$ as shown in figure \ref{fig:ISOC}. On its own the C language is +insufficient to accomodate the magnitude required for the problem at hand. An RSA modulus of magnitude $10^{19}$ could be +trivially factored\footnote{A Pollard-Rho factoring would take only $2^{16}$ time.} on the average desktop computer, +rendering any protocol based on the algorithm insecure. Multiple precision algorithms solve this very problem by extending the range of representable integers while using single precision data types. -Most advancements in fast multiple precision arithmetic stem from the need for faster and more efficient cryptographic -primitives. Faster modular reduction and exponentiation algorithms such as Barrett's algorithm, which have appeared in -various cryptographic journals, can render algorithms such as RSA and Diffie-Hellman more efficient. In fact, several -major companies such as RSA Security, Certicom and Entrust have built entire product lines on the implementation and +Most advancements in fast multiple precision arithmetic stem from the need for faster and more efficient cryptographic +primitives. Faster modular reduction and exponentiation algorithms such as Barrett's algorithm, which have appeared in +various cryptographic journals, can render algorithms such as RSA and Diffie-Hellman more efficient. In fact, several +major companies such as RSA Security, Certicom and Entrust have built entire product lines on the implementation and deployment of efficient algorithms. -However, cryptography is not the only field of study that can benefit from fast multiple precision integer routines. -Another auxiliary use of multiple precision integers is high precision floating point data types. -The basic IEEE \cite{IEEE} standard floating point type is made up of an integer mantissa $q$, an exponent $e$ and a sign bit $s$. -Numbers are given in the form $n = q \cdot b^e \cdot -1^s$ where $b = 2$ is the most common base for IEEE. Since IEEE -floating point is meant to be implemented in hardware the precision of the mantissa is often fairly small +However, cryptography is not the only field of study that can benefit from fast multiple precision integer routines. +Another auxiliary use of multiple precision integers is high precision floating point data types. +The basic IEEE \cite{IEEE} standard floating point type is made up of an integer mantissa $q$, an exponent $e$ and a sign bit $s$. +Numbers are given in the form $n = q \cdot b^e \cdot -1^s$ where $b = 2$ is the most common base for IEEE. Since IEEE +floating point is meant to be implemented in hardware the precision of the mantissa is often fairly small (\textit{23, 48 and 64 bits}). The mantissa is merely an integer and a multiple precision integer could be used to create -a mantissa of much larger precision than hardware alone can efficiently support. This approach could be useful where +a mantissa of much larger precision than hardware alone can efficiently support. This approach could be useful where scientific applications must minimize the total output error over long calculations. Yet another use for large integers is within arithmetic on polynomials of large characteristic (i.e. $GF(p)[x]$ for large $p$). @@ -229,152 +218,152 @@ In fact the library discussed within this text has already been used to form a p \subsection{Benefits of Multiple Precision Arithmetic} \index{precision} -The benefit of multiple precision representations over single or fixed precision representations is that -no precision is lost while representing the result of an operation which requires excess precision. For example, -the product of two $n$-bit integers requires at least $2n$ bits of precision to be represented faithfully. A multiple -precision algorithm would augment the precision of the destination to accomodate the result while a single precision system +The benefit of multiple precision representations over single or fixed precision representations is that +no precision is lost while representing the result of an operation which requires excess precision. For example, +the product of two $n$-bit integers requires at least $2n$ bits of precision to be represented faithfully. A multiple +precision algorithm would augment the precision of the destination to accomodate the result while a single precision system would truncate excess bits to maintain a fixed level of precision. It is possible to implement algorithms which require large integers with fixed precision algorithms. For example, elliptic -curve cryptography (\textit{ECC}) is often implemented on smartcards by fixing the precision of the integers to the maximum -size the system will ever need. Such an approach can lead to vastly simpler algorithms which can accomodate the -integers required even if the host platform cannot natively accomodate them\footnote{For example, the average smartcard +curve cryptography (\textit{ECC}) is often implemented on smartcards by fixing the precision of the integers to the maximum +size the system will ever need. Such an approach can lead to vastly simpler algorithms which can accomodate the +integers required even if the host platform cannot natively accomodate them\footnote{For example, the average smartcard processor has an 8 bit accumulator.}. However, as efficient as such an approach may be, the resulting source code is not normally very flexible. It cannot, at runtime, accomodate inputs of higher magnitude than the designer anticipated. -Multiple precision algorithms have the most overhead of any style of arithmetic. For the the most part the +Multiple precision algorithms have the most overhead of any style of arithmetic. For the the most part the overhead can be kept to a minimum with careful planning, but overall, it is not well suited for most memory starved -platforms. However, multiple precision algorithms do offer the most flexibility in terms of the magnitude of the -inputs. That is, the same algorithms based on multiple precision integers can accomodate any reasonable size input -without the designer's explicit forethought. This leads to lower cost of ownership for the code as it only has to +platforms. However, multiple precision algorithms do offer the most flexibility in terms of the magnitude of the +inputs. That is, the same algorithms based on multiple precision integers can accomodate any reasonable size input +without the designer's explicit forethought. This leads to lower cost of ownership for the code as it only has to be written and tested once. \section{Purpose of This Text} -The purpose of this text is to instruct the reader regarding how to implement efficient multiple precision algorithms. -That is to not only explain a limited subset of the core theory behind the algorithms but also the various ``house keeping'' -elements that are neglected by authors of other texts on the subject. Several well reknowned texts \cite{TAOCPV2,HAC} -give considerably detailed explanations of the theoretical aspects of algorithms and often very little information -regarding the practical implementation aspects. - -In most cases how an algorithm is explained and how it is actually implemented are two very different concepts. For -example, the Handbook of Applied Cryptography (\textit{HAC}), algorithm 14.7 on page 594, gives a relatively simple -algorithm for performing multiple precision integer addition. However, the description lacks any discussion concerning +The purpose of this text is to instruct the reader regarding how to implement efficient multiple precision algorithms. +That is to not only explain a limited subset of the core theory behind the algorithms but also the various ``house keeping'' +elements that are neglected by authors of other texts on the subject. Several well reknowned texts \cite{TAOCPV2,HAC} +give considerably detailed explanations of the theoretical aspects of algorithms and often very little information +regarding the practical implementation aspects. + +In most cases how an algorithm is explained and how it is actually implemented are two very different concepts. For +example, the Handbook of Applied Cryptography (\textit{HAC}), algorithm 14.7 on page 594, gives a relatively simple +algorithm for performing multiple precision integer addition. However, the description lacks any discussion concerning the fact that the two integer inputs may be of differing magnitudes. As a result the implementation is not as simple -as the text would lead people to believe. Similarly the division routine (\textit{algorithm 14.20, pp. 598}) does not +as the text would lead people to believe. Similarly the division routine (\textit{algorithm 14.20, pp. 598}) does not discuss how to handle sign or handle the dividend's decreasing magnitude in the main loop (\textit{step \#3}). -Both texts also do not discuss several key optimal algorithms required such as ``Comba'' and Karatsuba multipliers -and fast modular inversion, which we consider practical oversights. These optimal algorithms are vital to achieve -any form of useful performance in non-trivial applications. +Both texts also do not discuss several key optimal algorithms required such as ``Comba'' and Karatsuba multipliers +and fast modular inversion, which we consider practical oversights. These optimal algorithms are vital to achieve +any form of useful performance in non-trivial applications. To solve this problem the focus of this text is on the practical aspects of implementing a multiple precision integer -package. As a case study the ``LibTomMath''\footnote{Available at \url{http://math.libtomcrypt.com}} package is used -to demonstrate algorithms with real implementations\footnote{In the ISO C programming language.} that have been field -tested and work very well. The LibTomMath library is freely available on the Internet for all uses and this text +package. As a case study the ``LibTomMath''\footnote{Available at \url{http://math.libtomcrypt.com}} package is used +to demonstrate algorithms with real implementations\footnote{In the ISO C programming language.} that have been field +tested and work very well. The LibTomMath library is freely available on the Internet for all uses and this text discusses a very large portion of the inner workings of the library. -The algorithms that are presented will always include at least one ``pseudo-code'' description followed -by the actual C source code that implements the algorithm. The pseudo-code can be used to implement the same -algorithm in other programming languages as the reader sees fit. +The algorithms that are presented will always include at least one ``pseudo-code'' description followed +by the actual C source code that implements the algorithm. The pseudo-code can be used to implement the same +algorithm in other programming languages as the reader sees fit. This text shall also serve as a walkthrough of the creation of multiple precision algorithms from scratch. Showing -the reader how the algorithms fit together as well as where to start on various taskings. +the reader how the algorithms fit together as well as where to start on various taskings. \section{Discussion and Notation} \subsection{Notation} A multiple precision integer of $n$-digits shall be denoted as $x = (x_{n-1}, \ldots, x_1, x_0)_{ \beta }$ and represent -the integer $x \equiv \sum_{i=0}^{n-1} x_i\beta^i$. The elements of the array $x$ are said to be the radix $\beta$ digits -of the integer. For example, $x = (1,2,3)_{10}$ would represent the integer -$1\cdot 10^2 + 2\cdot10^1 + 3\cdot10^0 = 123$. +the integer $x \equiv \sum_{i=0}^{n-1} x_i\beta^i$. The elements of the array $x$ are said to be the radix $\beta$ digits +of the integer. For example, $x = (1,2,3)_{10}$ would represent the integer +$1\cdot 10^2 + 2\cdot10^1 + 3\cdot10^0 = 123$. \index{mp\_int} -The term ``mp\_int'' shall refer to a composite structure which contains the digits of the integer it represents, as well -as auxilary data required to manipulate the data. These additional members are discussed further in section -\ref{sec:MPINT}. For the purposes of this text a ``multiple precision integer'' and an ``mp\_int'' are assumed to be -synonymous. When an algorithm is specified to accept an mp\_int variable it is assumed the various auxliary data members -are present as well. An expression of the type \textit{variablename.item} implies that it should evaluate to the -member named ``item'' of the variable. For example, a string of characters may have a member ``length'' which would -evaluate to the number of characters in the string. If the string $a$ equals ``hello'' then it follows that -$a.length = 5$. +The term ``mp\_int'' shall refer to a composite structure which contains the digits of the integer it represents, as well +as auxilary data required to manipulate the data. These additional members are discussed further in section +\ref{sec:MPINT}. For the purposes of this text a ``multiple precision integer'' and an ``mp\_int'' are assumed to be +synonymous. When an algorithm is specified to accept an mp\_int variable it is assumed the various auxliary data members +are present as well. An expression of the type \textit{variablename.item} implies that it should evaluate to the +member named ``item'' of the variable. For example, a string of characters may have a member ``length'' which would +evaluate to the number of characters in the string. If the string $a$ equals ``hello'' then it follows that +$a.length = 5$. For certain discussions more generic algorithms are presented to help the reader understand the final algorithm used -to solve a given problem. When an algorithm is described as accepting an integer input it is assumed the input is -a plain integer with no additional multiple-precision members. That is, algorithms that use integers as opposed to -mp\_ints as inputs do not concern themselves with the housekeeping operations required such as memory management. These +to solve a given problem. When an algorithm is described as accepting an integer input it is assumed the input is +a plain integer with no additional multiple-precision members. That is, algorithms that use integers as opposed to +mp\_ints as inputs do not concern themselves with the housekeeping operations required such as memory management. These algorithms will be used to establish the relevant theory which will subsequently be used to describe a multiple -precision algorithm to solve the same problem. +precision algorithm to solve the same problem. \subsection{Precision Notation} -The variable $\beta$ represents the radix of a single digit of a multiple precision integer and -must be of the form $q^p$ for $q, p \in \Z^+$. A single precision variable must be able to represent integers in -the range $0 \le x < q \beta$ while a double precision variable must be able to represent integers in the range -$0 \le x < q \beta^2$. The extra radix-$q$ factor allows additions and subtractions to proceed without truncation of the +The variable $\beta$ represents the radix of a single digit of a multiple precision integer and +must be of the form $q^p$ for $q, p \in \Z^+$. A single precision variable must be able to represent integers in +the range $0 \le x < q \beta$ while a double precision variable must be able to represent integers in the range +$0 \le x < q \beta^2$. The extra radix-$q$ factor allows additions and subtractions to proceed without truncation of the carry. Since all modern computers are binary, it is assumed that $q$ is two. \index{mp\_digit} \index{mp\_word} -Within the source code that will be presented for each algorithm, the data type \textbf{mp\_digit} will represent -a single precision integer type, while, the data type \textbf{mp\_word} will represent a double precision integer type. In -several algorithms (notably the Comba routines) temporary results will be stored in arrays of double precision mp\_words. -For the purposes of this text $x_j$ will refer to the $j$'th digit of a single precision array and $\hat x_j$ will refer to +Within the source code that will be presented for each algorithm, the data type \textbf{mp\_digit} will represent +a single precision integer type, while, the data type \textbf{mp\_word} will represent a double precision integer type. In +several algorithms (notably the Comba routines) temporary results will be stored in arrays of double precision mp\_words. +For the purposes of this text $x_j$ will refer to the $j$'th digit of a single precision array and $\hat x_j$ will refer to the $j$'th digit of a double precision array. Whenever an expression is to be assigned to a double precision -variable it is assumed that all single precision variables are promoted to double precision during the evaluation. +variable it is assumed that all single precision variables are promoted to double precision during the evaluation. Expressions that are assigned to a single precision variable are truncated to fit within the precision of a single precision data type. -For example, if $\beta = 10^2$ a single precision data type may represent a value in the +For example, if $\beta = 10^2$ a single precision data type may represent a value in the range $0 \le x < 10^3$, while a double precision data type may represent a value in the range $0 \le x < 10^5$. Let $a = 23$ and $b = 49$ represent two single precision variables. The single precision product shall be written as $c \leftarrow a \cdot b$ while the double precision product shall be written as $\hat c \leftarrow a \cdot b$. -In this particular case, $\hat c = 1127$ and $c = 127$. The most significant digit of the product would not fit -in a single precision data type and as a result $c \ne \hat c$. +In this particular case, $\hat c = 1127$ and $c = 127$. The most significant digit of the product would not fit +in a single precision data type and as a result $c \ne \hat c$. \subsection{Algorithm Inputs and Outputs} Within the algorithm descriptions all variables are assumed to be scalars of either single or double precision -as indicated. The only exception to this rule is when variables have been indicated to be of type mp\_int. This -distinction is important as scalars are often used as array indicies and various other counters. +as indicated. The only exception to this rule is when variables have been indicated to be of type mp\_int. This +distinction is important as scalars are often used as array indicies and various other counters. \subsection{Mathematical Expressions} -The $\lfloor \mbox{ } \rfloor$ brackets imply an expression truncated to an integer not greater than the expression +The $\lfloor \mbox{ } \rfloor$ brackets imply an expression truncated to an integer not greater than the expression itself. For example, $\lfloor 5.7 \rfloor = 5$. Similarly the $\lceil \mbox{ } \rceil$ brackets imply an expression -rounded to an integer not less than the expression itself. For example, $\lceil 5.1 \rceil = 6$. Typically when -the $/$ division symbol is used the intention is to perform an integer division with truncation. For example, -$5/2 = 2$ which will often be written as $\lfloor 5/2 \rfloor = 2$ for clarity. When an expression is written as a -fraction a real value division is implied, for example ${5 \over 2} = 2.5$. +rounded to an integer not less than the expression itself. For example, $\lceil 5.1 \rceil = 6$. Typically when +the $/$ division symbol is used the intention is to perform an integer division with truncation. For example, +$5/2 = 2$ which will often be written as $\lfloor 5/2 \rfloor = 2$ for clarity. When an expression is written as a +fraction a real value division is implied, for example ${5 \over 2} = 2.5$. The norm of a multiple precision integer, for example $\vert \vert x \vert \vert$, will be used to represent the number of digits in the representation -of the integer. For example, $\vert \vert 123 \vert \vert = 3$ and $\vert \vert 79452 \vert \vert = 5$. +of the integer. For example, $\vert \vert 123 \vert \vert = 3$ and $\vert \vert 79452 \vert \vert = 5$. \subsection{Work Effort} \index{big-Oh} -To measure the efficiency of the specified algorithms, a modified big-Oh notation is used. In this system all -single precision operations are considered to have the same cost\footnote{Except where explicitly noted.}. -That is a single precision addition, multiplication and division are assumed to take the same time to +To measure the efficiency of the specified algorithms, a modified big-Oh notation is used. In this system all +single precision operations are considered to have the same cost\footnote{Except where explicitly noted.}. +That is a single precision addition, multiplication and division are assumed to take the same time to complete. While this is generally not true in practice, it will simplify the discussions considerably. -Some algorithms have slight advantages over others which is why some constants will not be removed in -the notation. For example, a normal baseline multiplication (section \ref{sec:basemult}) requires $O(n^2)$ work while a -baseline squaring (section \ref{sec:basesquare}) requires $O({{n^2 + n}\over 2})$ work. In standard big-Oh notation these -would both be said to be equivalent to $O(n^2)$. However, -in the context of the this text this is not the case as the magnitude of the inputs will typically be rather small. As a +Some algorithms have slight advantages over others which is why some constants will not be removed in +the notation. For example, a normal baseline multiplication (section \ref{sec:basemult}) requires $O(n^2)$ work while a +baseline squaring (section \ref{sec:basesquare}) requires $O({{n^2 + n}\over 2})$ work. In standard big-Oh notation these +would both be said to be equivalent to $O(n^2)$. However, +in the context of the this text this is not the case as the magnitude of the inputs will typically be rather small. As a result small constant factors in the work effort will make an observable difference in algorithm efficiency. -All of the algorithms presented in this text have a polynomial time work level. That is, of the form -$O(n^k)$ for $n, k \in \Z^{+}$. This will help make useful comparisons in terms of the speed of the algorithms and how +All of the algorithms presented in this text have a polynomial time work level. That is, of the form +$O(n^k)$ for $n, k \in \Z^{+}$. This will help make useful comparisons in terms of the speed of the algorithms and how various optimizations will help pay off in the long run. \section{Exercises} Within the more advanced chapters a section will be set aside to give the reader some challenging exercises related to -the discussion at hand. These exercises are not designed to be prize winning problems, but instead to be thought -provoking. Wherever possible the problems are forward minded, stating problems that will be answered in subsequent -chapters. The reader is encouraged to finish the exercises as they appear to get a better understanding of the -subject material. +the discussion at hand. These exercises are not designed to be prize winning problems, but instead to be thought +provoking. Wherever possible the problems are forward minded, stating problems that will be answered in subsequent +chapters. The reader is encouraged to finish the exercises as they appear to get a better understanding of the +subject material. That being said, the problems are designed to affirm knowledge of a particular subject matter. Students in particular are encouraged to verify they can answer the problems correctly before moving on. Similar to the exercises of \cite[pp. ix]{TAOCPV2} these exercises are given a scoring system based on the difficulty of -the problem. However, unlike \cite{TAOCPV2} the problems do not get nearly as hard. The scoring of these -exercises ranges from one (the easiest) to five (the hardest). The following table sumarizes the +the problem. However, unlike \cite{TAOCPV2} the problems do not get nearly as hard. The scoring of these +exercises ranges from one (the easiest) to five (the hardest). The following table sumarizes the scoring system used. \begin{figure}[here] @@ -404,21 +393,21 @@ scoring system used. \end{figure} Problems at the first level are meant to be simple questions that the reader can answer quickly without programming a solution or -devising new theory. These problems are quick tests to see if the material is understood. Problems at the second level +devising new theory. These problems are quick tests to see if the material is understood. Problems at the second level are also designed to be easy but will require a program or algorithm to be implemented to arrive at the answer. These -two levels are essentially entry level questions. +two levels are essentially entry level questions. -Problems at the third level are meant to be a bit more difficult than the first two levels. The answer is often -fairly obvious but arriving at an exacting solution requires some thought and skill. These problems will almost always +Problems at the third level are meant to be a bit more difficult than the first two levels. The answer is often +fairly obvious but arriving at an exacting solution requires some thought and skill. These problems will almost always involve devising a new algorithm or implementing a variation of another algorithm previously presented. Readers who can answer these questions will feel comfortable with the concepts behind the topic at hand. -Problems at the fourth level are meant to be similar to those of the level three questions except they will require -additional research to be completed. The reader will most likely not know the answer right away, nor will the text provide -the exact details of the answer until a subsequent chapter. +Problems at the fourth level are meant to be similar to those of the level three questions except they will require +additional research to be completed. The reader will most likely not know the answer right away, nor will the text provide +the exact details of the answer until a subsequent chapter. -Problems at the fifth level are meant to be the hardest -problems relative to all the other problems in the chapter. People who can correctly answer fifth level problems have a +Problems at the fifth level are meant to be the hardest +problems relative to all the other problems in the chapter. People who can correctly answer fifth level problems have a mastery of the subject matter at hand. Often problems will be tied together. The purpose of this is to start a chain of thought that will be discussed in future chapters. The reader @@ -427,43 +416,43 @@ is encouraged to answer the follow-up problems and try to draw the relevance of \section{Introduction to LibTomMath} \subsection{What is LibTomMath?} -LibTomMath is a free and open source multiple precision integer library written entirely in portable ISO C. By portable it -is meant that the library does not contain any code that is computer platform dependent or otherwise problematic to use on -any given platform. +LibTomMath is a free and open source multiple precision integer library written entirely in portable ISO C. By portable it +is meant that the library does not contain any code that is computer platform dependent or otherwise problematic to use on +any given platform. The library has been successfully tested under numerous operating systems including Unix\footnote{All of these -trademarks belong to their respective rightful owners.}, MacOS, Windows, Linux, PalmOS and on standalone hardware such -as the Gameboy Advance. The library is designed to contain enough functionality to be able to develop applications such +trademarks belong to their respective rightful owners.}, MacOS, Windows, Linux, PalmOS and on standalone hardware such +as the Gameboy Advance. The library is designed to contain enough functionality to be able to develop applications such as public key cryptosystems and still maintain a relatively small footprint. \subsection{Goals of LibTomMath} -Libraries which obtain the most efficiency are rarely written in a high level programming language such as C. However, -even though this library is written entirely in ISO C, considerable care has been taken to optimize the algorithm implementations within the -library. Specifically the code has been written to work well with the GNU C Compiler (\textit{GCC}) on both x86 and ARM -processors. Wherever possible, highly efficient algorithms, such as Karatsuba multiplication, sliding window -exponentiation and Montgomery reduction have been provided to make the library more efficient. +Libraries which obtain the most efficiency are rarely written in a high level programming language such as C. However, +even though this library is written entirely in ISO C, considerable care has been taken to optimize the algorithm implementations within the +library. Specifically the code has been written to work well with the GNU C Compiler (\textit{GCC}) on both x86 and ARM +processors. Wherever possible, highly efficient algorithms, such as Karatsuba multiplication, sliding window +exponentiation and Montgomery reduction have been provided to make the library more efficient. -Even with the nearly optimal and specialized algorithms that have been included the Application Programing Interface -(\textit{API}) has been kept as simple as possible. Often generic place holder routines will make use of specialized -algorithms automatically without the developer's specific attention. One such example is the generic multiplication -algorithm \textbf{mp\_mul()} which will automatically use Toom--Cook, Karatsuba, Comba or baseline multiplication -based on the magnitude of the inputs and the configuration of the library. +Even with the nearly optimal and specialized algorithms that have been included the Application Programing Interface +(\textit{API}) has been kept as simple as possible. Often generic place holder routines will make use of specialized +algorithms automatically without the developer's specific attention. One such example is the generic multiplication +algorithm \textbf{mp\_mul()} which will automatically use Toom--Cook, Karatsuba, Comba or baseline multiplication +based on the magnitude of the inputs and the configuration of the library. -Making LibTomMath as efficient as possible is not the only goal of the LibTomMath project. Ideally the library should +Making LibTomMath as efficient as possible is not the only goal of the LibTomMath project. Ideally the library should be source compatible with another popular library which makes it more attractive for developers to use. In this case the -MPI library was used as a API template for all the basic functions. MPI was chosen because it is another library that fits -in the same niche as LibTomMath. Even though LibTomMath uses MPI as the template for the function names and argument +MPI library was used as a API template for all the basic functions. MPI was chosen because it is another library that fits +in the same niche as LibTomMath. Even though LibTomMath uses MPI as the template for the function names and argument passing conventions, it has been written from scratch by Tom St Denis. -The project is also meant to act as a learning tool for students, the logic being that no easy-to-follow ``bignum'' -library exists which can be used to teach computer science students how to perform fast and reliable multiple precision -integer arithmetic. To this end the source code has been given quite a few comments and algorithm discussion points. +The project is also meant to act as a learning tool for students, the logic being that no easy-to-follow ``bignum'' +library exists which can be used to teach computer science students how to perform fast and reliable multiple precision +integer arithmetic. To this end the source code has been given quite a few comments and algorithm discussion points. \section{Choice of LibTomMath} LibTomMath was chosen as the case study of this text not only because the author of both projects is one and the same but -for more worthy reasons. Other libraries such as GMP \cite{GMP}, MPI \cite{MPI}, LIP \cite{LIP} and OpenSSL -\cite{OPENSSL} have multiple precision integer arithmetic routines but would not be ideal for this text for +for more worthy reasons. Other libraries such as GMP \cite{GMP}, MPI \cite{MPI}, LIP \cite{LIP} and OpenSSL +\cite{OPENSSL} have multiple precision integer arithmetic routines but would not be ideal for this text for reasons that will be explained in the following sub-sections. \subsection{Code Base} @@ -472,76 +461,76 @@ segments of code littered throughout the source. This clean and uncluttered app developer can more readily discern the true intent of a given section of source code without trying to keep track of what conditional code will be used. -The code base of LibTomMath is well organized. Each function is in its own separate source code file +The code base of LibTomMath is well organized. Each function is in its own separate source code file which allows the reader to find a given function very quickly. On average there are $76$ lines of code per source file which makes the source very easily to follow. By comparison MPI and LIP are single file projects making code tracing -very hard. GMP has many conditional code segments which also hinder tracing. +very hard. GMP has many conditional code segments which also hinder tracing. When compiled with GCC for the x86 processor and optimized for speed the entire library is approximately $100$KiB\footnote{The notation ``KiB'' means $2^{10}$ octets, similarly ``MiB'' means $2^{20}$ octets.} - which is fairly small compared to GMP (over $250$KiB). LibTomMath is slightly larger than MPI (which compiles to about + which is fairly small compared to GMP (over $250$KiB). LibTomMath is slightly larger than MPI (which compiles to about $50$KiB) but LibTomMath is also much faster and more complete than MPI. \subsection{API Simplicity} -LibTomMath is designed after the MPI library and shares the API design. Quite often programs that use MPI will build -with LibTomMath without change. The function names correlate directly to the action they perform. Almost all of the -functions share the same parameter passing convention. The learning curve is fairly shallow with the API provided -which is an extremely valuable benefit for the student and developer alike. +LibTomMath is designed after the MPI library and shares the API design. Quite often programs that use MPI will build +with LibTomMath without change. The function names correlate directly to the action they perform. Almost all of the +functions share the same parameter passing convention. The learning curve is fairly shallow with the API provided +which is an extremely valuable benefit for the student and developer alike. -The LIP library is an example of a library with an API that is awkward to work with. LIP uses function names that are often ``compressed'' to -illegible short hand. LibTomMath does not share this characteristic. +The LIP library is an example of a library with an API that is awkward to work with. LIP uses function names that are often ``compressed'' to +illegible short hand. LibTomMath does not share this characteristic. The GMP library also does not return error codes. Instead it uses a POSIX.1 \cite{POSIX1} signal system where errors are signaled to the host application. This happens to be the fastest approach but definitely not the most versatile. In -effect a math error (i.e. invalid input, heap error, etc) can cause a program to stop functioning which is definitely +effect a math error (i.e. invalid input, heap error, etc) can cause a program to stop functioning which is definitely undersireable in many situations. \subsection{Optimizations} While LibTomMath is certainly not the fastest library (GMP often beats LibTomMath by a factor of two) it does -feature a set of optimal algorithms for tasks such as modular reduction, exponentiation, multiplication and squaring. GMP +feature a set of optimal algorithms for tasks such as modular reduction, exponentiation, multiplication and squaring. GMP and LIP also feature such optimizations while MPI only uses baseline algorithms with no optimizations. GMP lacks a few of the additional modular reduction optimizations that LibTomMath features\footnote{At the time of this writing GMP -only had Barrett and Montgomery modular reduction algorithms.}. +only had Barrett and Montgomery modular reduction algorithms.}. LibTomMath is almost always an order of magnitude faster than the MPI library at computationally expensive tasks such as modular -exponentiation. In the grand scheme of ``bignum'' libraries LibTomMath is faster than the average library and usually +exponentiation. In the grand scheme of ``bignum'' libraries LibTomMath is faster than the average library and usually slower than the best libraries such as GMP and OpenSSL by only a small factor. \subsection{Portability and Stability} -LibTomMath will build ``out of the box'' on any platform equipped with a modern version of the GNU C Compiler -(\textit{GCC}). This means that without changes the library will build without configuration or setting up any -variables. LIP and MPI will build ``out of the box'' as well but have numerous known bugs. Most notably the author of -MPI has recently stopped working on his library and LIP has long since been discontinued. +LibTomMath will build ``out of the box'' on any platform equipped with a modern version of the GNU C Compiler +(\textit{GCC}). This means that without changes the library will build without configuration or setting up any +variables. LIP and MPI will build ``out of the box'' as well but have numerous known bugs. Most notably the author of +MPI has recently stopped working on his library and LIP has long since been discontinued. GMP requires a configuration script to run and will not build out of the box. GMP and LibTomMath are still in active development and are very stable across a variety of platforms. \subsection{Choice} LibTomMath is a relatively compact, well documented, highly optimized and portable library which seems only natural for -the case study of this text. Various source files from the LibTomMath project will be included within the text. However, -the reader is encouraged to download their own copy of the library to actually be able to work with the library. +the case study of this text. Various source files from the LibTomMath project will be included within the text. However, +the reader is encouraged to download their own copy of the library to actually be able to work with the library. \chapter{Getting Started} \section{Library Basics} -The trick to writing any useful library of source code is to build a solid foundation and work outwards from it. First, -a problem along with allowable solution parameters should be identified and analyzed. In this particular case the +The trick to writing any useful library of source code is to build a solid foundation and work outwards from it. First, +a problem along with allowable solution parameters should be identified and analyzed. In this particular case the inability to accomodate multiple precision integers is the problem. Futhermore, the solution must be written as portable source code that is reasonably efficient across several different computer platforms. -After a foundation is formed the remainder of the library can be designed and implemented in a hierarchical fashion. -That is, to implement the lowest level dependencies first and work towards the most abstract functions last. For example, +After a foundation is formed the remainder of the library can be designed and implemented in a hierarchical fashion. +That is, to implement the lowest level dependencies first and work towards the most abstract functions last. For example, before implementing a modular exponentiation algorithm one would implement a modular reduction algorithm. -By building outwards from a base foundation instead of using a parallel design methodology the resulting project is +By building outwards from a base foundation instead of using a parallel design methodology the resulting project is highly modular. Being highly modular is a desirable property of any project as it often means the resulting product -has a small footprint and updates are easy to perform. +has a small footprint and updates are easy to perform. -Usually when I start a project I will begin with the header files. I define the data types I think I will need and -prototype the initial functions that are not dependent on other functions (within the library). After I +Usually when I start a project I will begin with the header files. I define the data types I think I will need and +prototype the initial functions that are not dependent on other functions (within the library). After I implement these base functions I prototype more dependent functions and implement them. The process repeats until -I implement all of the functions I require. For example, in the case of LibTomMath I implemented functions such as -mp\_init() well before I implemented mp\_mul() and even further before I implemented mp\_exptmod(). As an example as to -why this design works note that the Karatsuba and Toom-Cook multipliers were written \textit{after} the -dependent function mp\_exptmod() was written. Adding the new multiplication algorithms did not require changes to the -mp\_exptmod() function itself and lowered the total cost of ownership (\textit{so to speak}) and of development +I implement all of the functions I require. For example, in the case of LibTomMath I implemented functions such as +mp\_init() well before I implemented mp\_mul() and even further before I implemented mp\_exptmod(). As an example as to +why this design works note that the Karatsuba and Toom-Cook multipliers were written \textit{after} the +dependent function mp\_exptmod() was written. Adding the new multiplication algorithms did not require changes to the +mp\_exptmod() function itself and lowered the total cost of ownership (\textit{so to speak}) and of development for new algorithms. This methodology allows new algorithms to be tested in a complete framework with relative ease. \begin{center} @@ -553,40 +542,40 @@ for new algorithms. This methodology allows new algorithms to be tested in a co \end{center} Only after the majority of the functions were in place did I pursue a less hierarchical approach to auditing and optimizing -the source code. For example, one day I may audit the multipliers and the next day the polynomial basis functions. +the source code. For example, one day I may audit the multipliers and the next day the polynomial basis functions. -It only makes sense to begin the text with the preliminary data types and support algorithms required as well. +It only makes sense to begin the text with the preliminary data types and support algorithms required as well. This chapter discusses the core algorithms of the library which are the dependents for every other algorithm. \section{What is a Multiple Precision Integer?} -Recall that most programming languages, in particular ISO C \cite{ISOC}, only have fixed precision data types that on their own cannot -be used to represent values larger than their precision will allow. The purpose of multiple precision algorithms is -to use fixed precision data types to create and manipulate multiple precision integers which may represent values -that are very large. +Recall that most programming languages, in particular ISO C \cite{ISOC}, only have fixed precision data types that on their own cannot +be used to represent values larger than their precision will allow. The purpose of multiple precision algorithms is +to use fixed precision data types to create and manipulate multiple precision integers which may represent values +that are very large. As a well known analogy, school children are taught how to form numbers larger than nine by prepending more radix ten digits. In the decimal system -the largest single digit value is $9$. However, by concatenating digits together larger numbers may be represented. Newly prepended digits -(\textit{to the left}) are said to be in a different power of ten column. That is, the number $123$ can be described as having a $1$ in the hundreds -column, $2$ in the tens column and $3$ in the ones column. Or more formally $123 = 1 \cdot 10^2 + 2 \cdot 10^1 + 3 \cdot 10^0$. Computer based -multiple precision arithmetic is essentially the same concept. Larger integers are represented by adjoining fixed +the largest single digit value is $9$. However, by concatenating digits together larger numbers may be represented. Newly prepended digits +(\textit{to the left}) are said to be in a different power of ten column. That is, the number $123$ can be described as having a $1$ in the hundreds +column, $2$ in the tens column and $3$ in the ones column. Or more formally $123 = 1 \cdot 10^2 + 2 \cdot 10^1 + 3 \cdot 10^0$. Computer based +multiple precision arithmetic is essentially the same concept. Larger integers are represented by adjoining fixed precision computer words with the exception that a different radix is used. -What most people probably do not think about explicitly are the various other attributes that describe a multiple precision -integer. For example, the integer $154_{10}$ has two immediately obvious properties. First, the integer is positive, -that is the sign of this particular integer is positive as opposed to negative. Second, the integer has three digits in -its representation. There is an additional property that the integer posesses that does not concern pencil-and-paper -arithmetic. The third property is how many digits placeholders are available to hold the integer. +What most people probably do not think about explicitly are the various other attributes that describe a multiple precision +integer. For example, the integer $154_{10}$ has two immediately obvious properties. First, the integer is positive, +that is the sign of this particular integer is positive as opposed to negative. Second, the integer has three digits in +its representation. There is an additional property that the integer posesses that does not concern pencil-and-paper +arithmetic. The third property is how many digits placeholders are available to hold the integer. The human analogy of this third property is ensuring there is enough space on the paper to write the integer. For example, -if one starts writing a large number too far to the right on a piece of paper they will have to erase it and move left. +if one starts writing a large number too far to the right on a piece of paper they will have to erase it and move left. Similarly, computer algorithms must maintain strict control over memory usage to ensure that the digits of an integer -will not exceed the allowed boundaries. These three properties make up what is known as a multiple precision -integer or mp\_int for short. +will not exceed the allowed boundaries. These three properties make up what is known as a multiple precision +integer or mp\_int for short. \subsection{The mp\_int Structure} \label{sec:MPINT} -The mp\_int structure is the ISO C based manifestation of what represents a multiple precision integer. The ISO C standard does not provide for -any such data type but it does provide for making composite data types known as structures. The following is the structure definition +The mp\_int structure is the ISO C based manifestation of what represents a multiple precision integer. The ISO C standard does not provide for +any such data type but it does provide for making composite data types known as structures. The following is the structure definition used within LibTomMath. \index{mp\_int} @@ -613,46 +602,46 @@ The mp\_int structure (fig. \ref{fig:mpint}) can be broken down as follows. \begin{enumerate} \item The \textbf{used} parameter denotes how many digits of the array \textbf{dp} contain the digits used to represent -a given integer. The \textbf{used} count must be positive (or zero) and may not exceed the \textbf{alloc} count. +a given integer. The \textbf{used} count must be positive (or zero) and may not exceed the \textbf{alloc} count. -\item The \textbf{alloc} parameter denotes how -many digits are available in the array to use by functions before it has to increase in size. When the \textbf{used} count -of a result would exceed the \textbf{alloc} count all of the algorithms will automatically increase the size of the -array to accommodate the precision of the result. +\item The \textbf{alloc} parameter denotes how +many digits are available in the array to use by functions before it has to increase in size. When the \textbf{used} count +of a result would exceed the \textbf{alloc} count all of the algorithms will automatically increase the size of the +array to accommodate the precision of the result. -\item The pointer \textbf{dp} points to a dynamically allocated array of digits that represent the given multiple -precision integer. It is padded with $(\textbf{alloc} - \textbf{used})$ zero digits. The array is maintained in a least +\item The pointer \textbf{dp} points to a dynamically allocated array of digits that represent the given multiple +precision integer. It is padded with $(\textbf{alloc} - \textbf{used})$ zero digits. The array is maintained in a least significant digit order. As a pencil and paper analogy the array is organized such that the right most digits are stored -first starting at the location indexed by zero\footnote{In C all arrays begin at zero.} in the array. For example, -if \textbf{dp} contains $\lbrace a, b, c, \ldots \rbrace$ where \textbf{dp}$_0 = a$, \textbf{dp}$_1 = b$, \textbf{dp}$_2 = c$, $\ldots$ then -it would represent the integer $a + b\beta + c\beta^2 + \ldots$ +first starting at the location indexed by zero\footnote{In C all arrays begin at zero.} in the array. For example, +if \textbf{dp} contains $\lbrace a, b, c, \ldots \rbrace$ where \textbf{dp}$_0 = a$, \textbf{dp}$_1 = b$, \textbf{dp}$_2 = c$, $\ldots$ then +it would represent the integer $a + b\beta + c\beta^2 + \ldots$ \index{MP\_ZPOS} \index{MP\_NEG} -\item The \textbf{sign} parameter denotes the sign as either zero/positive (\textbf{MP\_ZPOS}) or negative (\textbf{MP\_NEG}). +\item The \textbf{sign} parameter denotes the sign as either zero/positive (\textbf{MP\_ZPOS}) or negative (\textbf{MP\_NEG}). \end{enumerate} \subsubsection{Valid mp\_int Structures} -Several rules are placed on the state of an mp\_int structure and are assumed to be followed for reasons of efficiency. +Several rules are placed on the state of an mp\_int structure and are assumed to be followed for reasons of efficiency. The only exceptions are when the structure is passed to initialization functions such as mp\_init() and mp\_init\_copy(). \begin{enumerate} \item The value of \textbf{alloc} may not be less than one. That is \textbf{dp} always points to a previously allocated array of digits. \item The value of \textbf{used} may not exceed \textbf{alloc} and must be greater than or equal to zero. -\item The value of \textbf{used} implies the digit at index $(used - 1)$ of the \textbf{dp} array is non-zero. That is, +\item The value of \textbf{used} implies the digit at index $(used - 1)$ of the \textbf{dp} array is non-zero. That is, leading zero digits in the most significant positions must be trimmed. \begin{enumerate} \item Digits in the \textbf{dp} array at and above the \textbf{used} location must be zero. \end{enumerate} -\item The value of \textbf{sign} must be \textbf{MP\_ZPOS} if \textbf{used} is zero; +\item The value of \textbf{sign} must be \textbf{MP\_ZPOS} if \textbf{used} is zero; this represents the mp\_int value of zero. \end{enumerate} \section{Argument Passing} -A convention of argument passing must be adopted early on in the development of any library. Making the function -prototypes consistent will help eliminate many headaches in the future as the library grows to significant complexity. -In LibTomMath the multiple precision integer functions accept parameters from left to right as pointers to mp\_int -structures. That means that the source (input) operands are placed on the left and the destination (output) on the right. +A convention of argument passing must be adopted early on in the development of any library. Making the function +prototypes consistent will help eliminate many headaches in the future as the library grows to significant complexity. +In LibTomMath the multiple precision integer functions accept parameters from left to right as pointers to mp\_int +structures. That means that the source (input) operands are placed on the left and the destination (output) on the right. Consider the following examples. \begin{verbatim} @@ -665,25 +654,25 @@ The left to right order is a fairly natural way to implement the functions since functions and make sense of them. For example, the first function would read ``multiply a and b and store in c''. Certain libraries (\textit{LIP by Lenstra for instance}) accept parameters the other way around, to mimic the order -of assignment expressions. That is, the destination (output) is on the left and arguments (inputs) are on the right. In -truth, it is entirely a matter of preference. In the case of LibTomMath the convention from the MPI library has been -adopted. - -Another very useful design consideration, provided for in LibTomMath, is whether to allow argument sources to also be a -destination. For example, the second example (\textit{mp\_add}) adds $a$ to $b$ and stores in $a$. This is an important -feature to implement since it allows the calling functions to cut down on the number of variables it must maintain. -However, to implement this feature specific care has to be given to ensure the destination is not modified before the +of assignment expressions. That is, the destination (output) is on the left and arguments (inputs) are on the right. In +truth, it is entirely a matter of preference. In the case of LibTomMath the convention from the MPI library has been +adopted. + +Another very useful design consideration, provided for in LibTomMath, is whether to allow argument sources to also be a +destination. For example, the second example (\textit{mp\_add}) adds $a$ to $b$ and stores in $a$. This is an important +feature to implement since it allows the calling functions to cut down on the number of variables it must maintain. +However, to implement this feature specific care has to be given to ensure the destination is not modified before the source is fully read. \section{Return Values} -A well implemented application, no matter what its purpose, should trap as many runtime errors as possible and return them -to the caller. By catching runtime errors a library can be guaranteed to prevent undefined behaviour. However, the end +A well implemented application, no matter what its purpose, should trap as many runtime errors as possible and return them +to the caller. By catching runtime errors a library can be guaranteed to prevent undefined behaviour. However, the end developer can still manage to cause a library to crash. For example, by passing an invalid pointer an application may fault by dereferencing memory not owned by the application. -In the case of LibTomMath the only errors that are checked for are related to inappropriate inputs (division by zero for -instance) and memory allocation errors. It will not check that the mp\_int passed to any function is valid nor -will it check pointers for validity. Any function that can cause a runtime error will return an error code as an +In the case of LibTomMath the only errors that are checked for are related to inappropriate inputs (division by zero for +instance) and memory allocation errors. It will not check that the mp\_int passed to any function is valid nor +will it check pointers for validity. Any function that can cause a runtime error will return an error code as an \textbf{int} data type with one of the following values (fig \ref{fig:errcodes}). \index{MP\_OKAY} \index{MP\_VAL} \index{MP\_MEM} @@ -702,7 +691,7 @@ will it check pointers for validity. Any function that can cause a runtime erro \end{figure} When an error is detected within a function it should free any memory it allocated, often during the initialization of -temporary mp\_ints, and return as soon as possible. The goal is to leave the system in the same state it was when the +temporary mp\_ints, and return as soon as possible. The goal is to leave the system in the same state it was when the function was called. Error checking with this style of API is fairly simple. \begin{verbatim} @@ -713,19 +702,19 @@ function was called. Error checking with this style of API is fairly simple. } \end{verbatim} -The GMP \cite{GMP} library uses C style \textit{signals} to flag errors which is of questionable use. Not all errors are fatal +The GMP \cite{GMP} library uses C style \textit{signals} to flag errors which is of questionable use. Not all errors are fatal and it was not deemed ideal by the author of LibTomMath to force developers to have signal handlers for such cases. \section{Initialization and Clearing} -The logical starting point when actually writing multiple precision integer functions is the initialization and +The logical starting point when actually writing multiple precision integer functions is the initialization and clearing of the mp\_int structures. These two algorithms will be used by the majority of the higher level algorithms. Given the basic mp\_int structure an initialization routine must first allocate memory to hold the digits of the integer. Often it is optimal to allocate a sufficiently large pre-set number of digits even though the initial integer will represent zero. If only a single digit were allocated quite a few subsequent re-allocations would occur when operations are performed on the integers. There is a tradeoff between how many default digits to allocate -and how many re-allocations are tolerable. Obviously allocating an excessive amount of digits initially will waste -memory and become unmanageable. +and how many re-allocations are tolerable. Obviously allocating an excessive amount of digits initially will waste +memory and become unmanageable. If the memory for the digits has been successfully allocated then the rest of the members of the structure must be initialized. Since the initial state of an mp\_int is to represent the zero integer, the allocated digits must be set @@ -760,16 +749,16 @@ structure are set to valid values. The mp\_init algorithm will perform such an \textbf{Algorithm mp\_init.} The purpose of this function is to initialize an mp\_int structure so that the rest of the library can properly manipulte it. It is assumed that the input may not have had any of its members previously initialized which is certainly -a valid assumption if the input resides on the stack. +a valid assumption if the input resides on the stack. Before any of the members such as \textbf{sign}, \textbf{used} or \textbf{alloc} are initialized the memory for -the digits is allocated. If this fails the function returns before setting any of the other members. The \textbf{MP\_PREC} -name represents a constant\footnote{Defined in the ``tommath.h'' header file within LibTomMath.} +the digits is allocated. If this fails the function returns before setting any of the other members. The \textbf{MP\_PREC} +name represents a constant\footnote{Defined in the ``tommath.h'' header file within LibTomMath.} used to dictate the minimum precision of newly initialized mp\_int integers. Ideally, it is at least equal to the smallest precision number you'll be working with. Allocating a block of digits at first instead of a single digit has the benefit of lowering the number of usually slow -heap operations later functions will have to perform in the future. If \textbf{MP\_PREC} is set correctly the slack +heap operations later functions will have to perform in the future. If \textbf{MP\_PREC} is set correctly the slack memory and the number of heap operations will be trivial. Once the allocation has been made the digits have to be set to zero as well as the \textbf{used}, \textbf{sign} and @@ -781,38 +770,65 @@ This function introduces the idiosyncrasy that all iterative loops, commonly ini when the ``to'' keyword is placed between two expressions. For example, ``for $a$ from $b$ to $c$ do'' means that a subsequent expression (or body of expressions) are to be evaluated upto $c - b$ times so long as $b \le c$. In each iteration the variable $a$ is substituted for a new integer that lies inclusively between $b$ and $c$. If $b > c$ occured -the loop would not iterate. By contrast if the ``downto'' keyword were used in place of ``to'' the loop would iterate +the loop would not iterate. By contrast if the ``downto'' keyword were used in place of ``to'' the loop would iterate decrementally. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_init.c \vspace{-3mm} \begin{alltt} +016 +017 /* init a new mp_int */ +018 int mp_init (mp_int * a) +019 \{ +020 int i; +021 +022 /* allocate memory required and clear it */ +023 a->dp = OPT_CAST(mp_digit) XMALLOC (sizeof (mp_digit) * MP_PREC); +024 if (a->dp == NULL) \{ +025 return MP_MEM; +026 \} +027 +028 /* set the digits to zero */ +029 for (i = 0; i < MP_PREC; i++) \{ +030 a->dp[i] = 0; +031 \} +032 +033 /* set the used to zero, allocated digits to the default precision +034 * and sign to positive */ +035 a->used = 0; +036 a->alloc = MP_PREC; +037 a->sign = MP_ZPOS; +038 +039 return MP_OKAY; +040 \} +041 #endif +042 \end{alltt} \end{small} -One immediate observation of this initializtion function is that it does not return a pointer to a mp\_int structure. It -is assumed that the caller has already allocated memory for the mp\_int structure, typically on the application stack. The -call to mp\_init() is used only to initialize the members of the structure to a known default state. +One immediate observation of this initializtion function is that it does not return a pointer to a mp\_int structure. It +is assumed that the caller has already allocated memory for the mp\_int structure, typically on the application stack. The +call to mp\_init() is used only to initialize the members of the structure to a known default state. -Here we see (line 24) the memory allocation is performed first. This allows us to exit cleanly and quickly +Here we see (line 23) the memory allocation is performed first. This allows us to exit cleanly and quickly if there is an error. If the allocation fails the routine will return \textbf{MP\_MEM} to the caller to indicate there was a memory error. The function XMALLOC is what actually allocates the memory. Technically XMALLOC is not a function but a macro defined in ``tommath.h``. By default, XMALLOC will evaluate to malloc() which is the C library's built--in memory allocation routine. In order to assure the mp\_int is in a known state the digits must be set to zero. On most platforms this could have been -accomplished by using calloc() instead of malloc(). However, to correctly initialize a integer type to a given value in a -portable fashion you have to actually assign the value. The for loop (line 30) performs this required +accomplished by using calloc() instead of malloc(). However, to correctly initialize a integer type to a given value in a +portable fashion you have to actually assign the value. The for loop (line 29) performs this required operation. -After the memory has been successfully initialized the remainder of the members are initialized -(lines 34 through 35) to their respective default states. At this point the algorithm has succeeded and -a success code is returned to the calling function. If this function returns \textbf{MP\_OKAY} it is safe to assume the -mp\_int structure has been properly initialized and is safe to use with other functions within the library. +After the memory has been successfully initialized the remainder of the members are initialized +(lines 33 through 34) to their respective default states. At this point the algorithm has succeeded and +a success code is returned to the calling function. If this function returns \textbf{MP\_OKAY} it is safe to assume the +mp\_int structure has been properly initialized and is safe to use with other functions within the library. \subsection{Clearing an mp\_int} -When an mp\_int is no longer required by the application, the memory that has been allocated for its digits must be +When an mp\_int is no longer required by the application, the memory that has been allocated for its digits must be returned to the application's memory pool with the mp\_clear algorithm. \begin{figure}[here] @@ -837,12 +853,12 @@ returned to the application's memory pool with the mp\_clear algorithm. \end{figure} \textbf{Algorithm mp\_clear.} -This algorithm accomplishes two goals. First, it clears the digits and the other mp\_int members. This ensures that +This algorithm accomplishes two goals. First, it clears the digits and the other mp\_int members. This ensures that if a developer accidentally re-uses a cleared structure it is less likely to cause problems. The second goal is to free the allocated memory. The logic behind the algorithm is extended by marking cleared mp\_int structures so that subsequent calls to this -algorithm will not try to free the memory multiple times. Cleared mp\_ints are detectable by having a pre-defined invalid +algorithm will not try to free the memory multiple times. Cleared mp\_ints are detectable by having a pre-defined invalid digit pointer \textbf{dp} setting. Once an mp\_int has been cleared the mp\_int structure is no longer in a valid state for any other algorithm @@ -852,36 +868,61 @@ with the exception of algorithms mp\_init, mp\_init\_copy, mp\_init\_size and mp \hspace{-5.1mm}{\bf File}: bn\_mp\_clear.c \vspace{-3mm} \begin{alltt} +016 +017 /* clear one (frees) */ +018 void +019 mp_clear (mp_int * a) +020 \{ +021 int i; +022 +023 /* only do anything if a hasn't been freed previously */ +024 if (a->dp != NULL) \{ +025 /* first zero the digits */ +026 for (i = 0; i < a->used; i++) \{ +027 a->dp[i] = 0; +028 \} +029 +030 /* free ram */ +031 XFREE(a->dp); +032 +033 /* reset members to make debugging easier */ +034 a->dp = NULL; +035 a->alloc = a->used = 0; +036 a->sign = MP_ZPOS; +037 \} +038 \} +039 #endif +040 \end{alltt} \end{small} -The algorithm only operates on the mp\_int if it hasn't been previously cleared. The if statement (line 25) +The algorithm only operates on the mp\_int if it hasn't been previously cleared. The if statement (line 24) checks to see if the \textbf{dp} member is not \textbf{NULL}. If the mp\_int is a valid mp\_int then \textbf{dp} cannot be \textbf{NULL} in which case the if statement will evaluate to true. -The digits of the mp\_int are cleared by the for loop (line 27) which assigns a zero to every digit. Similar to mp\_init() -the digits are assigned zero instead of using block memory operations (such as memset()) since this is more portable. +The digits of the mp\_int are cleared by the for loop (line 26) which assigns a zero to every digit. Similar to mp\_init() +the digits are assigned zero instead of using block memory operations (such as memset()) since this is more portable. The digits are deallocated off the heap via the XFREE macro. Similar to XMALLOC the XFREE macro actually evaluates to a standard C library function. In this case the free() function. Since free() only deallocates the memory the pointer -still has to be reset to \textbf{NULL} manually (line 35). +still has to be reset to \textbf{NULL} manually (line 34). -Now that the digits have been cleared and deallocated the other members are set to their final values (lines 36 and 37). +Now that the digits have been cleared and deallocated the other members are set to their final values (lines 35 and 36). \section{Maintenance Algorithms} The previous sections describes how to initialize and clear an mp\_int structure. To further support operations that are to be performed on mp\_int structures (such as addition and multiplication) the dependent algorithms must be -able to augment the precision of an mp\_int and -initialize mp\_ints with differing initial conditions. +able to augment the precision of an mp\_int and +initialize mp\_ints with differing initial conditions. These algorithms complete the set of low level algorithms required to work with mp\_int structures in the higher level algorithms such as addition, multiplication and modular exponentiation. \subsection{Augmenting an mp\_int's Precision} -When storing a value in an mp\_int structure, a sufficient number of digits must be available to accomodate the entire -result of an operation without loss of precision. Quite often the size of the array given by the \textbf{alloc} member -is large enough to simply increase the \textbf{used} digit count. However, when the size of the array is too small it +When storing a value in an mp\_int structure, a sufficient number of digits must be available to accomodate the entire +result of an operation without loss of precision. Quite often the size of the array given by the \textbf{alloc} member +is large enough to simply increase the \textbf{used} digit count. However, when the size of the array is too small it must be re-sized appropriately to accomodate the result. The mp\_grow algorithm will provide this functionality. \newpage\begin{figure}[here] @@ -907,20 +948,58 @@ must be re-sized appropriately to accomodate the result. The mp\_grow algorithm \end{figure} \textbf{Algorithm mp\_grow.} -It is ideal to prevent re-allocations from being performed if they are not required (step one). This is useful to -prevent mp\_ints from growing excessively in code that erroneously calls mp\_grow. +It is ideal to prevent re-allocations from being performed if they are not required (step one). This is useful to +prevent mp\_ints from growing excessively in code that erroneously calls mp\_grow. -The requested digit count is padded up to next multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} (steps two and three). -This helps prevent many trivial reallocations that would grow an mp\_int by trivially small values. +The requested digit count is padded up to next multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} (steps two and three). +This helps prevent many trivial reallocations that would grow an mp\_int by trivially small values. -It is assumed that the reallocation (step four) leaves the lower $a.alloc$ digits of the mp\_int intact. This is much -akin to how the \textit{realloc} function from the standard C library works. Since the newly allocated digits are +It is assumed that the reallocation (step four) leaves the lower $a.alloc$ digits of the mp\_int intact. This is much +akin to how the \textit{realloc} function from the standard C library works. Since the newly allocated digits are assumed to contain undefined values they are initially set to zero. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_grow.c \vspace{-3mm} \begin{alltt} +016 +017 /* grow as required */ +018 int mp_grow (mp_int * a, int size) +019 \{ +020 int i; +021 mp_digit *tmp; +022 +023 /* if the alloc size is smaller alloc more ram */ +024 if (a->alloc < size) \{ +025 /* ensure there are always at least MP_PREC digits extra on top */ +026 size += (MP_PREC * 2) - (size % MP_PREC); +027 +028 /* reallocate the array a->dp +029 * +030 * We store the return in a temporary variable +031 * in case the operation failed we don't want +032 * to overwrite the dp member of a. +033 */ +034 tmp = OPT_CAST(mp_digit) XREALLOC (a->dp, sizeof (mp_digit) * size); +035 if (tmp == NULL) \{ +036 /* reallocation failed but "a" is still valid [can be freed] */ +037 return MP_MEM; +038 \} +039 +040 /* reallocation succeeded so set a->dp */ +041 a->dp = tmp; +042 +043 /* zero excess digits */ +044 i = a->alloc; +045 a->alloc = size; +046 for (; i < a->alloc; i++) \{ +047 a->dp[i] = 0; +048 \} +049 \} +050 return MP_OKAY; +051 \} +052 #endif +053 \end{alltt} \end{small} @@ -929,19 +1008,19 @@ if the \textbf{alloc} member of the mp\_int is smaller than the requested digit the function skips the re-allocation part thus saving time. When a re-allocation is performed it is turned into an optimal request to save time in the future. The requested digit count is -padded upwards to 2nd multiple of \textbf{MP\_PREC} larger than \textbf{alloc} (line 25). The XREALLOC function is used +padded upwards to 2nd multiple of \textbf{MP\_PREC} larger than \textbf{alloc} (line 26). The XREALLOC function is used to re-allocate the memory. As per the other functions XREALLOC is actually a macro which evaluates to realloc by default. The realloc function leaves the base of the allocation intact which means the first \textbf{alloc} digits of the mp\_int are the same as before the re-allocation. All that is left is to clear the newly allocated digits and return. Note that the re-allocation result is actually stored in a temporary pointer $tmp$. This is to allow this function to return an error with a valid pointer. Earlier releases of the library stored the result of XREALLOC into the mp\_int $a$. That would -result in a memory leak if XREALLOC ever failed. +result in a memory leak if XREALLOC ever failed. \subsection{Initializing Variable Precision mp\_ints} -Occasionally the number of digits required will be known in advance of an initialization, based on, for example, the size -of input mp\_ints to a given algorithm. The purpose of algorithm mp\_init\_size is similar to mp\_init except that it -will allocate \textit{at least} a specified number of digits. +Occasionally the number of digits required will be known in advance of an initialization, based on, for example, the size +of input mp\_ints to a given algorithm. The purpose of algorithm mp\_init\_size is similar to mp\_init except that it +will allocate \textit{at least} a specified number of digits. \begin{figure}[here] \begin{small} @@ -968,12 +1047,12 @@ will allocate \textit{at least} a specified number of digits. \end{figure} \textbf{Algorithm mp\_init\_size.} -This algorithm will initialize an mp\_int structure $a$ like algorithm mp\_init with the exception that the number of -digits allocated can be controlled by the second input argument $b$. The input size is padded upwards so it is a -multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} digits. This padding is used to prevent trivial +This algorithm will initialize an mp\_int structure $a$ like algorithm mp\_init with the exception that the number of +digits allocated can be controlled by the second input argument $b$. The input size is padded upwards so it is a +multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} digits. This padding is used to prevent trivial allocations from becoming a bottleneck in the rest of the algorithms. -Like algorithm mp\_init, the mp\_int structure is initialized to a default state representing the integer zero. This +Like algorithm mp\_init, the mp\_int structure is initialized to a default state representing the integer zero. This particular algorithm is useful if it is known ahead of time the approximate size of the input. If the approximation is correct no further memory re-allocations are required to work with the mp\_int. @@ -981,22 +1060,51 @@ correct no further memory re-allocations are required to work with the mp\_int. \hspace{-5.1mm}{\bf File}: bn\_mp\_init\_size.c \vspace{-3mm} \begin{alltt} +016 +017 /* init an mp_init for a given size */ +018 int mp_init_size (mp_int * a, int size) +019 \{ +020 int x; +021 +022 /* pad size so there are always extra digits */ +023 size += (MP_PREC * 2) - (size % MP_PREC); +024 +025 /* alloc mem */ +026 a->dp = OPT_CAST(mp_digit) XMALLOC (sizeof (mp_digit) * size); +027 if (a->dp == NULL) \{ +028 return MP_MEM; +029 \} +030 +031 /* set the members */ +032 a->used = 0; +033 a->alloc = size; +034 a->sign = MP_ZPOS; +035 +036 /* zero the digits */ +037 for (x = 0; x < size; x++) \{ +038 a->dp[x] = 0; +039 \} +040 +041 return MP_OKAY; +042 \} +043 #endif +044 \end{alltt} \end{small} -The number of digits $b$ requested is padded (line 24) by first augmenting it to the next multiple of -\textbf{MP\_PREC} and then adding \textbf{MP\_PREC} to the result. If the memory can be successfully allocated the -mp\_int is placed in a default state representing the integer zero. Otherwise, the error code \textbf{MP\_MEM} will be -returned (line 29). +The number of digits $b$ requested is padded (line 23) by first augmenting it to the next multiple of +\textbf{MP\_PREC} and then adding \textbf{MP\_PREC} to the result. If the memory can be successfully allocated the +mp\_int is placed in a default state representing the integer zero. Otherwise, the error code \textbf{MP\_MEM} will be +returned (line 28). -The digits are allocated and set to zero at the same time with the calloc() function (line @25,XCALLOC@). The -\textbf{used} count is set to zero, the \textbf{alloc} count set to the padded digit count and the \textbf{sign} flag set -to \textbf{MP\_ZPOS} to achieve a default valid mp\_int state (lines 33, 34 and 35). If the function -returns succesfully then it is correct to assume that the mp\_int structure is in a valid state for the remainder of the +The digits are allocated with the malloc() function (line 26) and set to zero afterwards (line 37). The +\textbf{used} count is set to zero, the \textbf{alloc} count set to the padded digit count and the \textbf{sign} flag set +to \textbf{MP\_ZPOS} to achieve a default valid mp\_int state (lines 32, 33 and 34). If the function +returns succesfully then it is correct to assume that the mp\_int structure is in a valid state for the remainder of the functions to work with. \subsection{Multiple Integer Initializations and Clearings} -Occasionally a function will require a series of mp\_int data types to be made available simultaneously. +Occasionally a function will require a series of mp\_int data types to be made available simultaneously. The purpose of algorithm mp\_init\_multi is to initialize a variable length array of mp\_int structures in a single statement. It is essentially a shortcut to multiple initializations. @@ -1021,47 +1129,87 @@ statement. It is essentially a shortcut to multiple initializations. \end{figure} \textbf{Algorithm mp\_init\_multi.} -The algorithm will initialize the array of mp\_int variables one at a time. If a runtime error has been detected -(\textit{step 1.2}) all of the previously initialized variables are cleared. The goal is an ``all or nothing'' +The algorithm will initialize the array of mp\_int variables one at a time. If a runtime error has been detected +(\textit{step 1.2}) all of the previously initialized variables are cleared. The goal is an ``all or nothing'' initialization which allows for quick recovery from runtime errors. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_init\_multi.c \vspace{-3mm} \begin{alltt} +016 #include <stdarg.h> +017 +018 int mp_init_multi(mp_int *mp, ...) +019 \{ +020 mp_err res = MP_OKAY; /* Assume ok until proven otherwise */ +021 int n = 0; /* Number of ok inits */ +022 mp_int* cur_arg = mp; +023 va_list args; +024 +025 va_start(args, mp); /* init args to next argument from caller */ +026 while (cur_arg != NULL) \{ +027 if (mp_init(cur_arg) != MP_OKAY) \{ +028 /* Oops - error! Back-track and mp_clear what we already +029 succeeded in init-ing, then return error. +030 */ +031 va_list clean_args; +032 +033 /* end the current list */ +034 va_end(args); +035 +036 /* now start cleaning up */ +037 cur_arg = mp; +038 va_start(clean_args, mp); +039 while (n-- != 0) \{ +040 mp_clear(cur_arg); +041 cur_arg = va_arg(clean_args, mp_int*); +042 \} +043 va_end(clean_args); +044 res = MP_MEM; +045 break; +046 \} +047 n++; +048 cur_arg = va_arg(args, mp_int*); +049 \} +050 va_end(args); +051 return res; /* Assumed ok, if error flagged above. */ +052 \} +053 +054 #endif +055 \end{alltt} \end{small} This function intializes a variable length list of mp\_int structure pointers. However, instead of having the mp\_int -structures in an actual C array they are simply passed as arguments to the function. This function makes use of the -``...'' argument syntax of the C programming language. The list is terminated with a final \textbf{NULL} argument -appended on the right. +structures in an actual C array they are simply passed as arguments to the function. This function makes use of the +``...'' argument syntax of the C programming language. The list is terminated with a final \textbf{NULL} argument +appended on the right. The function uses the ``stdarg.h'' \textit{va} functions to step portably through the arguments to the function. A count -$n$ of succesfully initialized mp\_int structures is maintained (line 48) such that if a failure does occur, -the algorithm can backtrack and free the previously initialized structures (lines 28 to 47). +$n$ of succesfully initialized mp\_int structures is maintained (line 47) such that if a failure does occur, +the algorithm can backtrack and free the previously initialized structures (lines 27 to 46). \subsection{Clamping Excess Digits} -When a function anticipates a result will be $n$ digits it is simpler to assume this is true within the body of -the function instead of checking during the computation. For example, a multiplication of a $i$ digit number by a -$j$ digit produces a result of at most $i + j$ digits. It is entirely possible that the result is $i + j - 1$ -though, with no final carry into the last position. However, suppose the destination had to be first expanded -(\textit{via mp\_grow}) to accomodate $i + j - 1$ digits than further expanded to accomodate the final carry. +When a function anticipates a result will be $n$ digits it is simpler to assume this is true within the body of +the function instead of checking during the computation. For example, a multiplication of a $i$ digit number by a +$j$ digit produces a result of at most $i + j$ digits. It is entirely possible that the result is $i + j - 1$ +though, with no final carry into the last position. However, suppose the destination had to be first expanded +(\textit{via mp\_grow}) to accomodate $i + j - 1$ digits than further expanded to accomodate the final carry. That would be a considerable waste of time since heap operations are relatively slow. The ideal solution is to always assume the result is $i + j$ and fix up the \textbf{used} count after the function terminates. This way a single heap operation (\textit{at most}) is required. However, if the result was not checked -there would be an excess high order zero digit. +there would be an excess high order zero digit. -For example, suppose the product of two integers was $x_n = (0x_{n-1}x_{n-2}...x_0)_{\beta}$. The leading zero digit +For example, suppose the product of two integers was $x_n = (0x_{n-1}x_{n-2}...x_0)_{\beta}$. The leading zero digit will not contribute to the precision of the result. In fact, through subsequent operations more leading zero digits would -accumulate to the point the size of the integer would be prohibitive. As a result even though the precision is very -low the representation is excessively large. +accumulate to the point the size of the integer would be prohibitive. As a result even though the precision is very +low the representation is excessively large. -The mp\_clamp algorithm is designed to solve this very problem. It will trim high-order zeros by decrementing the -\textbf{used} count until a non-zero most significant digit is found. Also in this system, zero is considered to be a -positive number which means that if the \textbf{used} count is decremented to zero, the sign must be set to +The mp\_clamp algorithm is designed to solve this very problem. It will trim high-order zeros by decrementing the +\textbf{used} count until a non-zero most significant digit is found. Also in this system, zero is considered to be a +positive number which means that if the \textbf{used} count is decremented to zero, the sign must be set to \textbf{MP\_ZPOS}. \begin{figure}[here] @@ -1083,21 +1231,46 @@ positive number which means that if the \textbf{used} count is decremented to ze \textbf{Algorithm mp\_clamp.} As can be expected this algorithm is very simple. The loop on step one is expected to iterate only once or twice at -the most. For example, this will happen in cases where there is not a carry to fill the last position. Step two fixes the sign for +the most. For example, this will happen in cases where there is not a carry to fill the last position. Step two fixes the sign for when all of the digits are zero to ensure that the mp\_int is valid at all times. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_clamp.c \vspace{-3mm} \begin{alltt} +016 +017 /* trim unused digits +018 * +019 * This is used to ensure that leading zero digits are +020 * trimed and the leading "used" digit will be non-zero +021 * Typically very fast. Also fixes the sign if there +022 * are no more leading digits +023 */ +024 void +025 mp_clamp (mp_int * a) +026 \{ +027 /* decrease used while the most significant digit is +028 * zero. +029 */ +030 while ((a->used > 0) && (a->dp[a->used - 1] == 0)) \{ +031 --(a->used); +032 \} +033 +034 /* reset the sign flag if used == 0 */ +035 if (a->used == 0) \{ +036 a->sign = MP_ZPOS; +037 \} +038 \} +039 #endif +040 \end{alltt} \end{small} -Note on line 28 how to test for the \textbf{used} count is made on the left of the \&\& operator. In the C programming -language the terms to \&\& are evaluated left to right with a boolean short-circuit if any condition fails. This is -important since if the \textbf{used} is zero the test on the right would fetch below the array. That is obviously -undesirable. The parenthesis on line 31 is used to make sure the \textbf{used} count is decremented and not -the pointer ``a''. +Note on line 27 how to test for the \textbf{used} count is made on the left of the \&\& operator. In the C programming +language the terms to \&\& are evaluated left to right with a boolean short-circuit if any condition fails. This is +important since if the \textbf{used} is zero the test on the right would fetch below the array. That is obviously +undesirable. The parenthesis on line 30 is used to make sure the \textbf{used} count is decremented and not +the pointer ``a''. \section*{Exercises} \begin{tabular}{cl} @@ -1123,19 +1296,19 @@ $\left [ 1 \right ]$ & Give an example of when the algorithm mp\_init\_copy mig \section{Introduction} In the previous chapter a series of low level algorithms were established that dealt with initializing and maintaining -mp\_int structures. This chapter will discuss another set of seemingly non-algebraic algorithms which will form the low +mp\_int structures. This chapter will discuss another set of seemingly non-algebraic algorithms which will form the low level basis of the entire library. While these algorithm are relatively trivial it is important to understand how they work before proceeding since these algorithms will be used almost intrinsically in the following chapters. The algorithms in this chapter deal primarily with more ``programmer'' related tasks such as creating copies of mp\_int structures, assigning small values to mp\_int structures and comparisons of the values mp\_int structures -represent. +represent. \section{Assigning Values to mp\_int Structures} \subsection{Copying an mp\_int} Assigning the value that a given mp\_int structure represents to another mp\_int structure shall be known as making a copy for the purposes of this text. The copy of the mp\_int will be a separate entity that represents the same -value as the mp\_int it was copied from. The mp\_copy algorithm provides this functionality. +value as the mp\_int it was copied from. The mp\_copy algorithm provides this functionality. \newpage\begin{figure}[here] \begin{center} @@ -1160,45 +1333,94 @@ value as the mp\_int it was copied from. The mp\_copy algorithm provides this f \textbf{Algorithm mp\_copy.} This algorithm copies the mp\_int $a$ such that upon succesful termination of the algorithm the mp\_int $b$ will -represent the same integer as the mp\_int $a$. The mp\_int $b$ shall be a complete and distinct copy of the +represent the same integer as the mp\_int $a$. The mp\_int $b$ shall be a complete and distinct copy of the mp\_int $a$ meaing that the mp\_int $a$ can be modified and it shall not affect the value of the mp\_int $b$. -If $b$ does not have enough room for the digits of $a$ it must first have its precision augmented via the mp\_grow +If $b$ does not have enough room for the digits of $a$ it must first have its precision augmented via the mp\_grow algorithm. The digits of $a$ are copied over the digits of $b$ and any excess digits of $b$ are set to zero (step two and three). The \textbf{used} and \textbf{sign} members of $a$ are finally copied over the respective members of $b$. \textbf{Remark.} This algorithm also introduces a new idiosyncrasy that will be used throughout the rest of the -text. The error return codes of other algorithms are not explicitly checked in the pseudo-code presented. For example, in -step one of the mp\_copy algorithm the return of mp\_grow is not explicitly checked to ensure it succeeded. Text space is +text. The error return codes of other algorithms are not explicitly checked in the pseudo-code presented. For example, in +step one of the mp\_copy algorithm the return of mp\_grow is not explicitly checked to ensure it succeeded. Text space is limited so it is assumed that if a algorithm fails it will clear all temporarily allocated mp\_ints and return -the error code itself. However, the C code presented will demonstrate all of the error handling logic required to +the error code itself. However, the C code presented will demonstrate all of the error handling logic required to implement the pseudo-code. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_copy.c \vspace{-3mm} \begin{alltt} +016 +017 /* copy, b = a */ +018 int +019 mp_copy (mp_int * a, mp_int * b) +020 \{ +021 int res, n; +022 +023 /* if dst == src do nothing */ +024 if (a == b) \{ +025 return MP_OKAY; +026 \} +027 +028 /* grow dest */ +029 if (b->alloc < a->used) \{ +030 if ((res = mp_grow (b, a->used)) != MP_OKAY) \{ +031 return res; +032 \} +033 \} +034 +035 /* zero b and copy the parameters over */ +036 \{ +037 mp_digit *tmpa, *tmpb; +038 +039 /* pointer aliases */ +040 +041 /* source */ +042 tmpa = a->dp; +043 +044 /* destination */ +045 tmpb = b->dp; +046 +047 /* copy all the digits */ +048 for (n = 0; n < a->used; n++) \{ +049 *tmpb++ = *tmpa++; +050 \} +051 +052 /* clear high digits */ +053 for (; n < b->used; n++) \{ +054 *tmpb++ = 0; +055 \} +056 \} +057 +058 /* copy used count and sign */ +059 b->used = a->used; +060 b->sign = a->sign; +061 return MP_OKAY; +062 \} +063 #endif +064 \end{alltt} \end{small} Occasionally a dependent algorithm may copy an mp\_int effectively into itself such as when the input and output -mp\_int structures passed to a function are one and the same. For this case it is optimal to return immediately without -copying digits (line 25). +mp\_int structures passed to a function are one and the same. For this case it is optimal to return immediately without +copying digits (line 24). The mp\_int $b$ must have enough digits to accomodate the used digits of the mp\_int $a$. If $b.alloc$ is less than -$a.used$ the algorithm mp\_grow is used to augment the precision of $b$ (lines 30 to 33). In order to +$a.used$ the algorithm mp\_grow is used to augment the precision of $b$ (lines 29 to 33). In order to simplify the inner loop that copies the digits from $a$ to $b$, two aliases $tmpa$ and $tmpb$ point directly at the digits -of the mp\_ints $a$ and $b$ respectively. These aliases (lines 43 and 46) allow the compiler to access the digits without first dereferencing the -mp\_int pointers and then subsequently the pointer to the digits. +of the mp\_ints $a$ and $b$ respectively. These aliases (lines 42 and 45) allow the compiler to access the digits without first dereferencing the +mp\_int pointers and then subsequently the pointer to the digits. -After the aliases are established the digits from $a$ are copied into $b$ (lines 49 to 51) and then the excess -digits of $b$ are set to zero (lines 54 to 56). Both ``for'' loops make use of the pointer aliases and in -fact the alias for $b$ is carried through into the second ``for'' loop to clear the excess digits. This optimization +After the aliases are established the digits from $a$ are copied into $b$ (lines 48 to 50) and then the excess +digits of $b$ are set to zero (lines 53 to 55). Both ``for'' loops make use of the pointer aliases and in +fact the alias for $b$ is carried through into the second ``for'' loop to clear the excess digits. This optimization allows the alias to stay in a machine register fairly easy between the two loops. \textbf{Remarks.} The use of pointer aliases is an implementation methodology first introduced in this function that will -be used considerably in other functions. Technically, a pointer alias is simply a short hand alias used to lower the +be used considerably in other functions. Technically, a pointer alias is simply a short hand alias used to lower the number of pointer dereferencing operations required to access data. For example, a for loop may resemble \begin{alltt} @@ -1207,7 +1429,7 @@ for (x = 0; x < 100; x++) \{ \} \end{alltt} -This could be re-written using aliases as +This could be re-written using aliases as \begin{alltt} mp_digit *tmpa; @@ -1217,17 +1439,17 @@ for (x = 0; x < 100; x++) \{ \} \end{alltt} -In this case an alias is used to access the -array of digits within an mp\_int structure directly. It may seem that a pointer alias is strictly not required +In this case an alias is used to access the +array of digits within an mp\_int structure directly. It may seem that a pointer alias is strictly not required as a compiler may optimize out the redundant pointer operations. However, there are two dominant reasons to use aliases. -The first reason is that most compilers will not effectively optimize pointer arithmetic. For example, some optimizations -may work for the Microsoft Visual C++ compiler (MSVC) and not for the GNU C Compiler (GCC). Also some optimizations may -work for GCC and not MSVC. As such it is ideal to find a common ground for as many compilers as possible. Pointer -aliases optimize the code considerably before the compiler even reads the source code which means the end compiled code +The first reason is that most compilers will not effectively optimize pointer arithmetic. For example, some optimizations +may work for the Microsoft Visual C++ compiler (MSVC) and not for the GNU C Compiler (GCC). Also some optimizations may +work for GCC and not MSVC. As such it is ideal to find a common ground for as many compilers as possible. Pointer +aliases optimize the code considerably before the compiler even reads the source code which means the end compiled code stands a better chance of being faster. -The second reason is that pointer aliases often can make an algorithm simpler to read. Consider the first ``for'' +The second reason is that pointer aliases often can make an algorithm simpler to read. Consider the first ``for'' loop of the function mp\_copy() re-written to not use pointer aliases. \begin{alltt} @@ -1237,13 +1459,13 @@ loop of the function mp\_copy() re-written to not use pointer aliases. \} \end{alltt} -Whether this code is harder to read depends strongly on the individual. However, it is quantifiably slightly more +Whether this code is harder to read depends strongly on the individual. However, it is quantifiably slightly more complicated as there are four variables within the statement instead of just two. \subsubsection{Nested Statements} Another commonly used technique in the source routines is that certain sections of code are nested. This is used in particular with the pointer aliases to highlight code phases. For example, a Comba multiplier (discussed in chapter six) -will typically have three different phases. First the temporaries are initialized, then the columns calculated and +will typically have three different phases. First the temporaries are initialized, then the columns calculated and finally the carries are propagated. In this example the middle column production phase will typically be nested as it uses temporary variables and aliases the most. @@ -1252,9 +1474,9 @@ the various temporary variables required do not propagate into other sections of \subsection{Creating a Clone} -Another common operation is to make a local temporary copy of an mp\_int argument. To initialize an mp\_int -and then copy another existing mp\_int into the newly intialized mp\_int will be known as creating a clone. This is -useful within functions that need to modify an argument but do not wish to actually modify the original copy. The +Another common operation is to make a local temporary copy of an mp\_int argument. To initialize an mp\_int +and then copy another existing mp\_int into the newly intialized mp\_int will be known as creating a clone. This is +useful within functions that need to modify an argument but do not wish to actually modify the original copy. The mp\_init\_copy algorithm has been designed to help perform this task. \begin{figure}[here] @@ -1274,19 +1496,32 @@ mp\_init\_copy algorithm has been designed to help perform this task. \end{figure} \textbf{Algorithm mp\_init\_copy.} -This algorithm will initialize an mp\_int variable and copy another previously initialized mp\_int variable into it. As -such this algorithm will perform two operations in one step. +This algorithm will initialize an mp\_int variable and copy another previously initialized mp\_int variable into it. As +such this algorithm will perform two operations in one step. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_init\_copy.c \vspace{-3mm} \begin{alltt} +016 +017 /* creates "a" then copies b into it */ +018 int mp_init_copy (mp_int * a, mp_int * b) +019 \{ +020 int res; +021 +022 if ((res = mp_init_size (a, b->used)) != MP_OKAY) \{ +023 return res; +024 \} +025 return mp_copy (b, a); +026 \} +027 #endif +028 \end{alltt} \end{small} -This will initialize \textbf{a} and make it a verbatim copy of the contents of \textbf{b}. Note that +This will initialize \textbf{a} and make it a verbatim copy of the contents of \textbf{b}. Note that \textbf{a} will have its own memory allocated which means that \textbf{b} may be cleared after the call -and \textbf{a} will be left intact. +and \textbf{a} will be left intact. \section{Zeroing an Integer} Reseting an mp\_int to the default state is a common step in many algorithms. The mp\_zero algorithm will be the algorithm used to @@ -1310,16 +1545,33 @@ perform this task. \end{figure} \textbf{Algorithm mp\_zero.} -This algorithm simply resets a mp\_int to the default state. +This algorithm simply resets a mp\_int to the default state. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_zero.c \vspace{-3mm} \begin{alltt} +016 +017 /* set to zero */ +018 void mp_zero (mp_int * a) +019 \{ +020 int n; +021 mp_digit *tmp; +022 +023 a->sign = MP_ZPOS; +024 a->used = 0; +025 +026 tmp = a->dp; +027 for (n = 0; n < a->alloc; n++) \{ +028 *tmp++ = 0; +029 \} +030 \} +031 #endif +032 \end{alltt} \end{small} -After the function is completed, all of the digits are zeroed, the \textbf{used} count is zeroed and the +After the function is completed, all of the digits are zeroed, the \textbf{used} count is zeroed and the \textbf{sign} variable is set to \textbf{MP\_ZPOS}. \section{Sign Manipulation} @@ -1347,17 +1599,41 @@ the absolute value of an mp\_int. \textbf{Algorithm mp\_abs.} This algorithm computes the absolute of an mp\_int input. First it copies $a$ over $b$. This is an example of an algorithm where the check in mp\_copy that determines if the source and destination are equal proves useful. This allows, -for instance, the developer to pass the same mp\_int as the source and destination to this function without addition +for instance, the developer to pass the same mp\_int as the source and destination to this function without addition logic to handle it. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_abs.c \vspace{-3mm} \begin{alltt} +016 +017 /* b = |a| +018 * +019 * Simple function copies the input and fixes the sign to positive +020 */ +021 int +022 mp_abs (mp_int * a, mp_int * b) +023 \{ +024 int res; +025 +026 /* copy a to b */ +027 if (a != b) \{ +028 if ((res = mp_copy (a, b)) != MP_OKAY) \{ +029 return res; +030 \} +031 \} +032 +033 /* force the sign of b to positive */ +034 b->sign = MP_ZPOS; +035 +036 return MP_OKAY; +037 \} +038 #endif +039 \end{alltt} \end{small} -This fairly trivial algorithm first eliminates non--required duplications (line 28) and then sets the +This fairly trivial algorithm first eliminates non--required duplications (line 27) and then sets the \textbf{sign} flag to \textbf{MP\_ZPOS}. \subsection{Integer Negation} @@ -1387,7 +1663,7 @@ the negative of an mp\_int input. \textbf{Algorithm mp\_neg.} This algorithm computes the negation of an input. First it copies $a$ over $b$. If $a$ has no used digits then -the algorithm returns immediately. Otherwise it flips the sign flag and stores the result in $b$. Note that if +the algorithm returns immediately. Otherwise it flips the sign flag and stores the result in $b$. Note that if $a$ had no digits then it must be positive by definition. Had step three been omitted then the algorithm would return zero as negative. @@ -1395,10 +1671,31 @@ zero as negative. \hspace{-5.1mm}{\bf File}: bn\_mp\_neg.c \vspace{-3mm} \begin{alltt} +016 +017 /* b = -a */ +018 int mp_neg (mp_int * a, mp_int * b) +019 \{ +020 int res; +021 if (a != b) \{ +022 if ((res = mp_copy (a, b)) != MP_OKAY) \{ +023 return res; +024 \} +025 \} +026 +027 if (mp_iszero(b) != MP_YES) \{ +028 b->sign = (a->sign == MP_ZPOS) ? MP_NEG : MP_ZPOS; +029 \} else \{ +030 b->sign = MP_ZPOS; +031 \} +032 +033 return MP_OKAY; +034 \} +035 #endif +036 \end{alltt} \end{small} -Like mp\_abs() this function avoids non--required duplications (line 22) and then sets the sign. We +Like mp\_abs() this function avoids non--required duplications (line 21) and then sets the sign. We have to make sure that only non--zero values get a \textbf{sign} of \textbf{MP\_NEG}. If the mp\_int is zero than the \textbf{sign} is hard--coded to \textbf{MP\_ZPOS}. @@ -1417,9 +1714,9 @@ Often a mp\_int must be set to a relatively small value such as $1$ or $2$. For 2. $a_0 \leftarrow b \mbox{ (mod }\beta\mbox{)}$ \\ 3. $a.used \leftarrow \left \lbrace \begin{array}{ll} 1 & \mbox{if }a_0 > 0 \\ - 0 & \mbox{if }a_0 = 0 + 0 & \mbox{if }a_0 = 0 \end{array} \right .$ \\ -\hline +\hline \end{tabular} \end{center} \caption{Algorithm mp\_set} @@ -1433,19 +1730,29 @@ single digit is set (\textit{modulo $\beta$}) and the \textbf{used} count is adj \hspace{-5.1mm}{\bf File}: bn\_mp\_set.c \vspace{-3mm} \begin{alltt} +016 +017 /* set to a digit */ +018 void mp_set (mp_int * a, mp_digit b) +019 \{ +020 mp_zero (a); +021 a->dp[0] = b & MP_MASK; +022 a->used = (a->dp[0] != 0) ? 1 : 0; +023 \} +024 #endif +025 \end{alltt} \end{small} -First we zero (line 21) the mp\_int to make sure that the other members are initialized for a +First we zero (line 20) the mp\_int to make sure that the other members are initialized for a small positive constant. mp\_zero() ensures that the \textbf{sign} is positive and the \textbf{used} count -is zero. Next we set the digit and reduce it modulo $\beta$ (line 22). After this step we have to +is zero. Next we set the digit and reduce it modulo $\beta$ (line 21). After this step we have to check if the resulting digit is zero or not. If it is not then we set the \textbf{used} count to one, otherwise to zero. -We can quickly reduce modulo $\beta$ since it is of the form $2^k$ and a quick binary AND operation with +We can quickly reduce modulo $\beta$ since it is of the form $2^k$ and a quick binary AND operation with $2^k - 1$ will perform the same operation. -One important limitation of this function is that it will only set one digit. The size of a digit is not fixed, meaning source that uses +One important limitation of this function is that it will only set one digit. The size of a digit is not fixed, meaning source that uses this function should take that into account. Only trivially small constants can be set using this function. \subsection{Setting Large Constants} @@ -1473,9 +1780,9 @@ data type as input and will always treat it as a 32-bit integer. \end{figure} \textbf{Algorithm mp\_set\_int.} -The algorithm performs eight iterations of a simple loop where in each iteration four bits from the source are added to the +The algorithm performs eight iterations of a simple loop where in each iteration four bits from the source are added to the mp\_int. Step 2.1 will multiply the current result by sixteen making room for four more bits in the less significant positions. In step 2.2 the -next four bits from the source are extracted and are added to the mp\_int. The \textbf{used} digit count is +next four bits from the source are extracted and are added to the mp\_int. The \textbf{used} digit count is incremented to reflect the addition. The \textbf{used} digit counter is incremented since if any of the leading digits were zero the mp\_int would have zero digits used and the newly added four bits would be ignored. @@ -1485,27 +1792,56 @@ Excess zero digits are trimmed in steps 2.1 and 3 by using higher level algorith \hspace{-5.1mm}{\bf File}: bn\_mp\_set\_int.c \vspace{-3mm} \begin{alltt} +016 +017 /* set a 32-bit const */ +018 int mp_set_int (mp_int * a, unsigned long b) +019 \{ +020 int x, res; +021 +022 mp_zero (a); +023 +024 /* set four bits at a time */ +025 for (x = 0; x < 8; x++) \{ +026 /* shift the number up four bits */ +027 if ((res = mp_mul_2d (a, 4, a)) != MP_OKAY) \{ +028 return res; +029 \} +030 +031 /* OR in the top four bits of the source */ +032 a->dp[0] |= (b >> 28) & 15; +033 +034 /* shift the source up to the next four bits */ +035 b <<= 4; +036 +037 /* ensure that digits are not clamped off */ +038 a->used += 1; +039 \} +040 mp_clamp (a); +041 return MP_OKAY; +042 \} +043 #endif +044 \end{alltt} \end{small} This function sets four bits of the number at a time to handle all practical \textbf{DIGIT\_BIT} sizes. The weird -addition on line 39 ensures that the newly added in bits are added to the number of digits. While it may not -seem obvious as to why the digit counter does not grow exceedingly large it is because of the shift on line 28 -as well as the call to mp\_clamp() on line 41. Both functions will clamp excess leading digits which keeps +addition on line 38 ensures that the newly added in bits are added to the number of digits. While it may not +seem obvious as to why the digit counter does not grow exceedingly large it is because of the shift on line 27 +as well as the call to mp\_clamp() on line 40. Both functions will clamp excess leading digits which keeps the number of used digits low. \section{Comparisons} \subsection{Unsigned Comparisions} Comparing a multiple precision integer is performed with the exact same algorithm used to compare two decimal numbers. For example, to compare $1,234$ to $1,264$ the digits are extracted by their positions. That is we compare $1 \cdot 10^3 + 2 \cdot 10^2 + 3 \cdot 10^1 + 4 \cdot 10^0$ -to $1 \cdot 10^3 + 2 \cdot 10^2 + 6 \cdot 10^1 + 4 \cdot 10^0$ by comparing single digits at a time starting with the highest magnitude -positions. If any leading digit of one integer is greater than a digit in the same position of another integer then obviously it must be greater. +to $1 \cdot 10^3 + 2 \cdot 10^2 + 6 \cdot 10^1 + 4 \cdot 10^0$ by comparing single digits at a time starting with the highest magnitude +positions. If any leading digit of one integer is greater than a digit in the same position of another integer then obviously it must be greater. The first comparision routine that will be developed is the unsigned magnitude compare which will perform a comparison based on the digits of two -mp\_int variables alone. It will ignore the sign of the two inputs. Such a function is useful when an absolute comparison is required or if the +mp\_int variables alone. It will ignore the sign of the two inputs. Such a function is useful when an absolute comparison is required or if the signs are known to agree in advance. -To facilitate working with the results of the comparison functions three constants are required. +To facilitate working with the results of the comparison functions three constants are required. \begin{figure}[here] \begin{center} @@ -1541,9 +1877,9 @@ To facilitate working with the results of the comparison functions three constan \textbf{Algorithm mp\_cmp\_mag.} By saying ``$a$ to the left of $b$'' it is meant that the comparison is with respect to $a$, that is if $a$ is greater than $b$ it will return -\textbf{MP\_GT} and similar with respect to when $a = b$ and $a < b$. The first two steps compare the number of digits used in both $a$ and $b$. -Obviously if the digit counts differ there would be an imaginary zero digit in the smaller number where the leading digit of the larger number is. -If both have the same number of digits than the actual digits themselves must be compared starting at the leading digit. +\textbf{MP\_GT} and similar with respect to when $a = b$ and $a < b$. The first two steps compare the number of digits used in both $a$ and $b$. +Obviously if the digit counts differ there would be an imaginary zero digit in the smaller number where the leading digit of the larger number is. +If both have the same number of digits than the actual digits themselves must be compared starting at the leading digit. By step three both inputs must have the same number of digits so its safe to start from either $a.used - 1$ or $b.used - 1$ and count down to the zero'th digit. If after all of the digits have been compared, no difference is found, the algorithm returns \textbf{MP\_EQ}. @@ -1552,18 +1888,54 @@ the zero'th digit. If after all of the digits have been compared, no difference \hspace{-5.1mm}{\bf File}: bn\_mp\_cmp\_mag.c \vspace{-3mm} \begin{alltt} +016 +017 /* compare maginitude of two ints (unsigned) */ +018 int mp_cmp_mag (mp_int * a, mp_int * b) +019 \{ +020 int n; +021 mp_digit *tmpa, *tmpb; +022 +023 /* compare based on # of non-zero digits */ +024 if (a->used > b->used) \{ +025 return MP_GT; +026 \} +027 +028 if (a->used < b->used) \{ +029 return MP_LT; +030 \} +031 +032 /* alias for a */ +033 tmpa = a->dp + (a->used - 1); +034 +035 /* alias for b */ +036 tmpb = b->dp + (a->used - 1); +037 +038 /* compare based on digits */ +039 for (n = 0; n < a->used; ++n, --tmpa, --tmpb) \{ +040 if (*tmpa > *tmpb) \{ +041 return MP_GT; +042 \} +043 +044 if (*tmpa < *tmpb) \{ +045 return MP_LT; +046 \} +047 \} +048 return MP_EQ; +049 \} +050 #endif +051 \end{alltt} \end{small} -The two if statements (lines 25 and 29) compare the number of digits in the two inputs. These two are -performed before all of the digits are compared since it is a very cheap test to perform and can potentially save -considerable time. The implementation given is also not valid without those two statements. $b.alloc$ may be +The two if statements (lines 24 and 28) compare the number of digits in the two inputs. These two are +performed before all of the digits are compared since it is a very cheap test to perform and can potentially save +considerable time. The implementation given is also not valid without those two statements. $b.alloc$ may be smaller than $a.used$, meaning that undefined values will be read from $b$ past the end of the array of digits. \subsection{Signed Comparisons} -Comparing with sign considerations is also fairly critical in several routines (\textit{division for example}). Based on an unsigned magnitude +Comparing with sign considerations is also fairly critical in several routines (\textit{division for example}). Based on an unsigned magnitude comparison a trivial signed comparison algorithm can be written. \begin{figure}[here] @@ -1586,21 +1958,45 @@ comparison a trivial signed comparison algorithm can be written. \end{figure} \textbf{Algorithm mp\_cmp.} -The first two steps compare the signs of the two inputs. If the signs do not agree then it can return right away with the appropriate -comparison code. When the signs are equal the digits of the inputs must be compared to determine the correct result. In step -three the unsigned comparision flips the order of the arguments since they are both negative. For instance, if $-a > -b$ then +The first two steps compare the signs of the two inputs. If the signs do not agree then it can return right away with the appropriate +comparison code. When the signs are equal the digits of the inputs must be compared to determine the correct result. In step +three the unsigned comparision flips the order of the arguments since they are both negative. For instance, if $-a > -b$ then $\vert a \vert < \vert b \vert$. Step number four will compare the two when they are both positive. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_cmp.c \vspace{-3mm} \begin{alltt} +016 +017 /* compare two ints (signed)*/ +018 int +019 mp_cmp (mp_int * a, mp_int * b) +020 \{ +021 /* compare based on sign */ +022 if (a->sign != b->sign) \{ +023 if (a->sign == MP_NEG) \{ +024 return MP_LT; +025 \} else \{ +026 return MP_GT; +027 \} +028 \} +029 +030 /* compare digits */ +031 if (a->sign == MP_NEG) \{ +032 /* if negative compare opposite direction */ +033 return mp_cmp_mag(b, a); +034 \} else \{ +035 return mp_cmp_mag(a, b); +036 \} +037 \} +038 #endif +039 \end{alltt} \end{small} -The two if statements (lines 23 and 24) perform the initial sign comparison. If the signs are not the equal then which ever -has the positive sign is larger. The inputs are compared (line 32) based on magnitudes. If the signs were both -negative then the unsigned comparison is performed in the opposite direction (line 34). Otherwise, the signs are assumed to +The two if statements (lines 22 and 23) perform the initial sign comparison. If the signs are not the equal then which ever +has the positive sign is larger. The inputs are compared (line 31) based on magnitudes. If the signs were both +negative then the unsigned comparison is performed in the opposite direction (line 33). Otherwise, the signs are assumed to be both positive and a forward direction unsigned comparison is performed. \section*{Exercises} @@ -1617,37 +2013,37 @@ $\left [ 1 \right ]$ & Suggest a simple method to speed up the implementation of \chapter{Basic Arithmetic} \section{Introduction} -At this point algorithms for initialization, clearing, zeroing, copying, comparing and setting small constants have been -established. The next logical set of algorithms to develop are addition, subtraction and digit shifting algorithms. These -algorithms make use of the lower level algorithms and are the cruicial building block for the multiplication algorithms. It is very important -that these algorithms are highly optimized. On their own they are simple $O(n)$ algorithms but they can be called from higher level algorithms -which easily places them at $O(n^2)$ or even $O(n^3)$ work levels. - -All of the algorithms within this chapter make use of the logical bit shift operations denoted by $<<$ and $>>$ for left and right -logical shifts respectively. A logical shift is analogous to sliding the decimal point of radix-10 representations. For example, the real -number $0.9345$ is equivalent to $93.45\%$ which is found by sliding the the decimal two places to the right (\textit{multiplying by $\beta^2 = 10^2$}). -Algebraically a binary logical shift is equivalent to a division or multiplication by a power of two. +At this point algorithms for initialization, clearing, zeroing, copying, comparing and setting small constants have been +established. The next logical set of algorithms to develop are addition, subtraction and digit shifting algorithms. These +algorithms make use of the lower level algorithms and are the cruicial building block for the multiplication algorithms. It is very important +that these algorithms are highly optimized. On their own they are simple $O(n)$ algorithms but they can be called from higher level algorithms +which easily places them at $O(n^2)$ or even $O(n^3)$ work levels. + +All of the algorithms within this chapter make use of the logical bit shift operations denoted by $<<$ and $>>$ for left and right +logical shifts respectively. A logical shift is analogous to sliding the decimal point of radix-10 representations. For example, the real +number $0.9345$ is equivalent to $93.45\%$ which is found by sliding the the decimal two places to the right (\textit{multiplying by $\beta^2 = 10^2$}). +Algebraically a binary logical shift is equivalent to a division or multiplication by a power of two. For example, $a << k = a \cdot 2^k$ while $a >> k = \lfloor a/2^k \rfloor$. One significant difference between a logical shift and the way decimals are shifted is that digits below the zero'th position are removed -from the number. For example, consider $1101_2 >> 1$ using decimal notation this would produce $110.1_2$. However, with a logical shift the -result is $110_2$. +from the number. For example, consider $1101_2 >> 1$ using decimal notation this would produce $110.1_2$. However, with a logical shift the +result is $110_2$. \section{Addition and Subtraction} In common twos complement fixed precision arithmetic negative numbers are easily represented by subtraction from the modulus. For example, with 32-bit integers -$a - b\mbox{ (mod }2^{32}\mbox{)}$ is the same as $a + (2^{32} - b) \mbox{ (mod }2^{32}\mbox{)}$ since $2^{32} \equiv 0 \mbox{ (mod }2^{32}\mbox{)}$. +$a - b\mbox{ (mod }2^{32}\mbox{)}$ is the same as $a + (2^{32} - b) \mbox{ (mod }2^{32}\mbox{)}$ since $2^{32} \equiv 0 \mbox{ (mod }2^{32}\mbox{)}$. As a result subtraction can be performed with a trivial series of logical operations and an addition. However, in multiple precision arithmetic negative numbers are not represented in the same way. Instead a sign flag is used to keep track of the -sign of the integer. As a result signed addition and subtraction are actually implemented as conditional usage of lower level addition or +sign of the integer. As a result signed addition and subtraction are actually implemented as conditional usage of lower level addition or subtraction algorithms with the sign fixed up appropriately. The lower level algorithms will add or subtract integers without regard to the sign flag. That is they will add or subtract the magnitude of the integers respectively. \subsection{Low Level Addition} -An unsigned addition of multiple precision integers is performed with the same long-hand algorithm used to add decimal numbers. That is to add the -trailing digits first and propagate the resulting carry upwards. Since this is a lower level algorithm the name will have a ``s\_'' prefix. +An unsigned addition of multiple precision integers is performed with the same long-hand algorithm used to add decimal numbers. That is to add the +trailing digits first and propagate the resulting carry upwards. Since this is a lower level algorithm the name will have a ``s\_'' prefix. Historically that convention stems from the MPI library where ``s\_'' stood for static functions that were hidden from the developer entirely. \newpage @@ -1694,18 +2090,18 @@ Historically that convention stems from the MPI library where ``s\_'' stood for \end{figure} \textbf{Algorithm s\_mp\_add.} -This algorithm is loosely based on algorithm 14.7 of HAC \cite[pp. 594]{HAC} but has been extended to allow the inputs to have different magnitudes. -Coincidentally the description of algorithm A in Knuth \cite[pp. 266]{TAOCPV2} shares the same deficiency as the algorithm from \cite{HAC}. Even the +This algorithm is loosely based on algorithm 14.7 of HAC \cite[pp. 594]{HAC} but has been extended to allow the inputs to have different magnitudes. +Coincidentally the description of algorithm A in Knuth \cite[pp. 266]{TAOCPV2} shares the same deficiency as the algorithm from \cite{HAC}. Even the MIX pseudo machine code presented by Knuth \cite[pp. 266-267]{TAOCPV2} is incapable of handling inputs which are of different magnitudes. The first thing that has to be accomplished is to sort out which of the two inputs is the largest. The addition logic will simply add all of the smallest input to the largest input and store that first part of the result in the destination. Then it will apply a simpler addition loop to excess digits of the larger input. -The first two steps will handle sorting the inputs such that $min$ and $max$ hold the digit counts of the two +The first two steps will handle sorting the inputs such that $min$ and $max$ hold the digit counts of the two inputs. The variable $x$ will be an mp\_int alias for the largest input or the second input $b$ if they have the -same number of digits. After the inputs are sorted the destination $c$ is grown as required to accomodate the sum -of the two inputs. The original \textbf{used} count of $c$ is copied and set to the new used count. +same number of digits. After the inputs are sorted the destination $c$ is grown as required to accomodate the sum +of the two inputs. The original \textbf{used} count of $c$ is copied and set to the new used count. At this point the first addition loop will go through as many digit positions that both inputs have. The carry variable $\mu$ is set to zero outside the loop. Inside the loop an ``addition'' step requires three statements to produce @@ -1724,36 +2120,126 @@ The final carry is stored in $c_{max}$ and digits above $max$ upto $oldused$ are \hspace{-5.1mm}{\bf File}: bn\_s\_mp\_add.c \vspace{-3mm} \begin{alltt} +016 +017 /* low level addition, based on HAC pp.594, Algorithm 14.7 */ +018 int +019 s_mp_add (mp_int * a, mp_int * b, mp_int * c) +020 \{ +021 mp_int *x; +022 int olduse, res, min, max; +023 +024 /* find sizes, we let |a| <= |b| which means we have to sort +025 * them. "x" will point to the input with the most digits +026 */ +027 if (a->used > b->used) \{ +028 min = b->used; +029 max = a->used; +030 x = a; +031 \} else \{ +032 min = a->used; +033 max = b->used; +034 x = b; +035 \} +036 +037 /* init result */ +038 if (c->alloc < (max + 1)) \{ +039 if ((res = mp_grow (c, max + 1)) != MP_OKAY) \{ +040 return res; +041 \} +042 \} +043 +044 /* get old used digit count and set new one */ +045 olduse = c->used; +046 c->used = max + 1; +047 +048 \{ +049 mp_digit u, *tmpa, *tmpb, *tmpc; +050 int i; +051 +052 /* alias for digit pointers */ +053 +054 /* first input */ +055 tmpa = a->dp; +056 +057 /* second input */ +058 tmpb = b->dp; +059 +060 /* destination */ +061 tmpc = c->dp; +062 +063 /* zero the carry */ +064 u = 0; +065 for (i = 0; i < min; i++) \{ +066 /* Compute the sum at one digit, T[i] = A[i] + B[i] + U */ +067 *tmpc = *tmpa++ + *tmpb++ + u; +068 +069 /* U = carry bit of T[i] */ +070 u = *tmpc >> ((mp_digit)DIGIT_BIT); +071 +072 /* take away carry bit from T[i] */ +073 *tmpc++ &= MP_MASK; +074 \} +075 +076 /* now copy higher words if any, that is in A+B +077 * if A or B has more digits add those in +078 */ +079 if (min != max) \{ +080 for (; i < max; i++) \{ +081 /* T[i] = X[i] + U */ +082 *tmpc = x->dp[i] + u; +083 +084 /* U = carry bit of T[i] */ +085 u = *tmpc >> ((mp_digit)DIGIT_BIT); +086 +087 /* take away carry bit from T[i] */ +088 *tmpc++ &= MP_MASK; +089 \} +090 \} +091 +092 /* add carry */ +093 *tmpc++ = u; +094 +095 /* clear digits above oldused */ +096 for (i = c->used; i < olduse; i++) \{ +097 *tmpc++ = 0; +098 \} +099 \} +100 +101 mp_clamp (c); +102 return MP_OKAY; +103 \} +104 #endif +105 \end{alltt} \end{small} -We first sort (lines 28 to 36) the inputs based on magnitude and determine the $min$ and $max$ variables. +We first sort (lines 27 to 35) the inputs based on magnitude and determine the $min$ and $max$ variables. Note that $x$ is a pointer to an mp\_int assigned to the largest input, in effect it is a local alias. Next we -grow the destination (38 to 42) ensure that it can accomodate the result of the addition. +grow the destination (37 to 42) ensure that it can accomodate the result of the addition. -Similar to the implementation of mp\_copy this function uses the braced code and local aliases coding style. The three aliases that are on -lines 56, 59 and 62 represent the two inputs and destination variables respectively. These aliases are used to ensure the +Similar to the implementation of mp\_copy this function uses the braced code and local aliases coding style. The three aliases that are on +lines 55, 58 and 61 represent the two inputs and destination variables respectively. These aliases are used to ensure the compiler does not have to dereference $a$, $b$ or $c$ (respectively) to access the digits of the respective mp\_int. -The initial carry $u$ will be cleared (line 65), note that $u$ is of type mp\_digit which ensures type -compatibility within the implementation. The initial addition (line 66 to 75) adds digits from +The initial carry $u$ will be cleared (line 64), note that $u$ is of type mp\_digit which ensures type +compatibility within the implementation. The initial addition (line 65 to 74) adds digits from both inputs until the smallest input runs out of digits. Similarly the conditional addition loop -(line 81 to 90) adds the remaining digits from the larger of the two inputs. The addition is finished -with the final carry being stored in $tmpc$ (line 94). Note the ``++'' operator within the same expression. -After line 94, $tmpc$ will point to the $c.used$'th digit of the mp\_int $c$. This is useful -for the next loop (line 97 to 99) which set any old upper digits to zero. +(line 80 to 90) adds the remaining digits from the larger of the two inputs. The addition is finished +with the final carry being stored in $tmpc$ (line 93). Note the ``++'' operator within the same expression. +After line 93, $tmpc$ will point to the $c.used$'th digit of the mp\_int $c$. This is useful +for the next loop (line 96 to 99) which set any old upper digits to zero. \subsection{Low Level Subtraction} The low level unsigned subtraction algorithm is very similar to the low level unsigned addition algorithm. The principle difference is that the -unsigned subtraction algorithm requires the result to be positive. That is when computing $a - b$ the condition $\vert a \vert \ge \vert b\vert$ must -be met for this algorithm to function properly. Keep in mind this low level algorithm is not meant to be used in higher level algorithms directly. +unsigned subtraction algorithm requires the result to be positive. That is when computing $a - b$ the condition $\vert a \vert \ge \vert b\vert$ must +be met for this algorithm to function properly. Keep in mind this low level algorithm is not meant to be used in higher level algorithms directly. This algorithm as will be shown can be used to create functional signed addition and subtraction algorithms. For this algorithm a new variable is required to make the description simpler. Recall from section 1.3.1 that a mp\_digit must be able to represent -the range $0 \le x < 2\beta$ for the algorithms to work correctly. However, it is allowable that a mp\_digit represent a larger range of values. For -this algorithm we will assume that the variable $\gamma$ represents the number of bits available in a -mp\_digit (\textit{this implies $2^{\gamma} > \beta$}). +the range $0 \le x < 2\beta$ for the algorithms to work correctly. However, it is allowable that a mp\_digit represent a larger range of values. For +this algorithm we will assume that the variable $\gamma$ represents the number of bits available in a +mp\_digit (\textit{this implies $2^{\gamma} > \beta$}). For example, the default for LibTomMath is to use a ``unsigned long'' for the mp\_digit ``type'' while $\beta = 2^{28}$. In ISO C an ``unsigned long'' data type must be able to represent $0 \le x < 2^{32}$ meaning that in this case $\gamma \ge 32$. @@ -1769,7 +2255,7 @@ data type must be able to represent $0 \le x < 2^{32}$ meaning that in this case 1. $min \leftarrow b.used$ \\ 2. $max \leftarrow a.used$ \\ 3. If $c.alloc < max$ then grow $c$ to hold at least $max$ digits. (\textit{mp\_grow}) \\ -4. $oldused \leftarrow c.used$ \\ +4. $oldused \leftarrow c.used$ \\ 5. $c.used \leftarrow max$ \\ 6. $u \leftarrow 0$ \\ 7. for $n$ from $0$ to $min - 1$ do \\ @@ -1799,24 +2285,24 @@ passing variables $a$ and $b$ the condition that $\vert a \vert \ge \vert b \ver algorithm is loosely based on algorithm 14.9 \cite[pp. 595]{HAC} and is similar to algorithm S in \cite[pp. 267]{TAOCPV2} as well. As was the case of the algorithm s\_mp\_add both other references lack discussion concerning various practical details such as when the inputs differ in magnitude. -The initial sorting of the inputs is trivial in this algorithm since $a$ is guaranteed to have at least the same magnitude of $b$. Steps 1 and 2 -set the $min$ and $max$ variables. Unlike the addition routine there is guaranteed to be no carry which means that the final result can be at -most $max$ digits in length as opposed to $max + 1$. Similar to the addition algorithm the \textbf{used} count of $c$ is copied locally and +The initial sorting of the inputs is trivial in this algorithm since $a$ is guaranteed to have at least the same magnitude of $b$. Steps 1 and 2 +set the $min$ and $max$ variables. Unlike the addition routine there is guaranteed to be no carry which means that the final result can be at +most $max$ digits in length as opposed to $max + 1$. Similar to the addition algorithm the \textbf{used} count of $c$ is copied locally and set to the maximal count for the operation. -The subtraction loop that begins on step seven is essentially the same as the addition loop of algorithm s\_mp\_add except single precision -subtraction is used instead. Note the use of the $\gamma$ variable to extract the carry (\textit{also known as the borrow}) within the subtraction -loops. Under the assumption that two's complement single precision arithmetic is used this will successfully extract the desired carry. +The subtraction loop that begins on step seven is essentially the same as the addition loop of algorithm s\_mp\_add except single precision +subtraction is used instead. Note the use of the $\gamma$ variable to extract the carry (\textit{also known as the borrow}) within the subtraction +loops. Under the assumption that two's complement single precision arithmetic is used this will successfully extract the desired carry. -For example, consider subtracting $0101_2$ from $0100_2$ where $\gamma = 4$ and $\beta = 2$. The least significant bit will force a carry upwards to -the third bit which will be set to zero after the borrow. After the very first bit has been subtracted $4 - 1 \equiv 0011_2$ will remain, When the -third bit of $0101_2$ is subtracted from the result it will cause another carry. In this case though the carry will be forced to propagate all the -way to the most significant bit. +For example, consider subtracting $0101_2$ from $0100_2$ where $\gamma = 4$ and $\beta = 2$. The least significant bit will force a carry upwards to +the third bit which will be set to zero after the borrow. After the very first bit has been subtracted $4 - 1 \equiv 0011_2$ will remain, When the +third bit of $0101_2$ is subtracted from the result it will cause another carry. In this case though the carry will be forced to propagate all the +way to the most significant bit. -Recall that $\beta < 2^{\gamma}$. This means that if a carry does occur just before the $lg(\beta)$'th bit it will propagate all the way to the most +Recall that $\beta < 2^{\gamma}$. This means that if a carry does occur just before the $lg(\beta)$'th bit it will propagate all the way to the most significant bit. Thus, the high order bits of the mp\_digit that are not part of the actual digit will either be all zero, or all one. All that -is needed is a single zero or one bit for the carry. Therefore a single logical shift right by $\gamma - 1$ positions is sufficient to extract the -carry. This method of carry extraction may seem awkward but the reason for it becomes apparent when the implementation is discussed. +is needed is a single zero or one bit for the carry. Therefore a single logical shift right by $\gamma - 1$ positions is sufficient to extract the +carry. This method of carry extraction may seem awkward but the reason for it becomes apparent when the implementation is discussed. If $b$ has a smaller magnitude than $a$ then step 9 will force the carry and copy operation to propagate through the larger input $a$ into $c$. Step 10 will ensure that any leading digits of $c$ above the $max$'th position are zeroed. @@ -1825,33 +2311,104 @@ If $b$ has a smaller magnitude than $a$ then step 9 will force the carry and cop \hspace{-5.1mm}{\bf File}: bn\_s\_mp\_sub.c \vspace{-3mm} \begin{alltt} +016 +017 /* low level subtraction (assumes |a| > |b|), HAC pp.595 Algorithm 14.9 */ +018 int +019 s_mp_sub (mp_int * a, mp_int * b, mp_int * c) +020 \{ +021 int olduse, res, min, max; +022 +023 /* find sizes */ +024 min = b->used; +025 max = a->used; +026 +027 /* init result */ +028 if (c->alloc < max) \{ +029 if ((res = mp_grow (c, max)) != MP_OKAY) \{ +030 return res; +031 \} +032 \} +033 olduse = c->used; +034 c->used = max; +035 +036 \{ +037 mp_digit u, *tmpa, *tmpb, *tmpc; +038 int i; +039 +040 /* alias for digit pointers */ +041 tmpa = a->dp; +042 tmpb = b->dp; +043 tmpc = c->dp; +044 +045 /* set carry to zero */ +046 u = 0; +047 for (i = 0; i < min; i++) \{ +048 /* T[i] = A[i] - B[i] - U */ +049 *tmpc = (*tmpa++ - *tmpb++) - u; +050 +051 /* U = carry bit of T[i] +052 * Note this saves performing an AND operation since +053 * if a carry does occur it will propagate all the way to the +054 * MSB. As a result a single shift is enough to get the carry +055 */ +056 u = *tmpc >> ((mp_digit)((CHAR_BIT * sizeof(mp_digit)) - 1)); +057 +058 /* Clear carry from T[i] */ +059 *tmpc++ &= MP_MASK; +060 \} +061 +062 /* now copy higher words if any, e.g. if A has more digits than B */ +063 for (; i < max; i++) \{ +064 /* T[i] = A[i] - U */ +065 *tmpc = *tmpa++ - u; +066 +067 /* U = carry bit of T[i] */ +068 u = *tmpc >> ((mp_digit)((CHAR_BIT * sizeof(mp_digit)) - 1)); +069 +070 /* Clear carry from T[i] */ +071 *tmpc++ &= MP_MASK; +072 \} +073 +074 /* clear digits above used (since we may not have grown result above) */ + +075 for (i = c->used; i < olduse; i++) \{ +076 *tmpc++ = 0; +077 \} +078 \} +079 +080 mp_clamp (c); +081 return MP_OKAY; +082 \} +083 +084 #endif +085 \end{alltt} \end{small} -Like low level addition we ``sort'' the inputs. Except in this case the sorting is hardcoded -(lines 25 and 26). In reality the $min$ and $max$ variables are only aliases and are only -used to make the source code easier to read. Again the pointer alias optimization is used +Like low level addition we ``sort'' the inputs. Except in this case the sorting is hardcoded +(lines 24 and 25). In reality the $min$ and $max$ variables are only aliases and are only +used to make the source code easier to read. Again the pointer alias optimization is used within this algorithm. The aliases $tmpa$, $tmpb$ and $tmpc$ are initialized -(lines 42, 43 and 44) for $a$, $b$ and $c$ respectively. - -The first subtraction loop (lines 47 through 61) subtract digits from both inputs until the smaller of -the two inputs has been exhausted. As remarked earlier there is an implementation reason for using the ``awkward'' -method of extracting the carry (line 57). The traditional method for extracting the carry would be to shift -by $lg(\beta)$ positions and logically AND the least significant bit. The AND operation is required because all of -the bits above the $\lg(\beta)$'th bit will be set to one after a carry occurs from subtraction. This carry -extraction requires two relatively cheap operations to extract the carry. The other method is to simply shift the -most significant bit to the least significant bit thus extracting the carry with a single cheap operation. This +(lines 41, 42 and 43) for $a$, $b$ and $c$ respectively. + +The first subtraction loop (lines 46 through 60) subtract digits from both inputs until the smaller of +the two inputs has been exhausted. As remarked earlier there is an implementation reason for using the ``awkward'' +method of extracting the carry (line 56). The traditional method for extracting the carry would be to shift +by $lg(\beta)$ positions and logically AND the least significant bit. The AND operation is required because all of +the bits above the $\lg(\beta)$'th bit will be set to one after a carry occurs from subtraction. This carry +extraction requires two relatively cheap operations to extract the carry. The other method is to simply shift the +most significant bit to the least significant bit thus extracting the carry with a single cheap operation. This optimization only works on twos compliment machines which is a safe assumption to make. -If $a$ has a larger magnitude than $b$ an additional loop (lines 64 through 73) is required to propagate -the carry through $a$ and copy the result to $c$. +If $a$ has a larger magnitude than $b$ an additional loop (lines 63 through 72) is required to propagate +the carry through $a$ and copy the result to $c$. \subsection{High Level Addition} Now that both lower level addition and subtraction algorithms have been established an effective high level signed addition algorithm can be -established. This high level addition algorithm will be what other algorithms and developers will use to perform addition of mp\_int data -types. +established. This high level addition algorithm will be what other algorithms and developers will use to perform addition of mp\_int data +types. -Recall from section 5.2 that an mp\_int represents an integer with an unsigned mantissa (\textit{the array of digits}) and a \textbf{sign} +Recall from section 5.2 that an mp\_int represents an integer with an unsigned mantissa (\textit{the array of digits}) and a \textbf{sign} flag. A high level addition is actually performed as a series of eight separate cases which can be optimized down to three unique cases. \begin{figure}[!here] @@ -1879,8 +2436,8 @@ flag. A high level addition is actually performed as a series of eight separate \end{figure} \textbf{Algorithm mp\_add.} -This algorithm performs the signed addition of two mp\_int variables. There is no reference algorithm to draw upon from -either \cite{TAOCPV2} or \cite{HAC} since they both only provide unsigned operations. The algorithm is fairly +This algorithm performs the signed addition of two mp\_int variables. There is no reference algorithm to draw upon from +either \cite{TAOCPV2} or \cite{HAC} since they both only provide unsigned operations. The algorithm is fairly straightforward but restricted since subtraction can only produce positive results. \begin{figure}[here] @@ -1910,9 +2467,9 @@ straightforward but restricted since subtraction can only produce positive resul \label{fig:AddChart} \end{figure} -Figure~\ref{fig:AddChart} lists all of the eight possible input combinations and is sorted to show that only three -specific cases need to be handled. The return code of the unsigned operations at step 1.2, 2.1.2 and 2.2.2 are -forwarded to step three to check for errors. This simplifies the description of the algorithm considerably and best +Figure~\ref{fig:AddChart} lists all of the eight possible input combinations and is sorted to show that only three +specific cases need to be handled. The return code of the unsigned operations at step 1.2, 2.1.2 and 2.2.2 are +forwarded to step three to check for errors. This simplifies the description of the algorithm considerably and best follows how the implementation actually was achieved. Also note how the \textbf{sign} is set before the unsigned addition or subtraction is performed. Recall from the descriptions of algorithms @@ -1920,13 +2477,47 @@ s\_mp\_add and s\_mp\_sub that the mp\_clamp function is used at the end to trim to \textbf{MP\_ZPOS} when the \textbf{used} digit count reaches zero. For example, consider performing $-a + a$ with algorithm mp\_add. By the description of the algorithm the sign is set to \textbf{MP\_NEG} which would -produce a result of $-0$. However, since the sign is set first then the unsigned addition is performed the subsequent usage of algorithm mp\_clamp -within algorithm s\_mp\_add will force $-0$ to become $0$. +produce a result of $-0$. However, since the sign is set first then the unsigned addition is performed the subsequent usage of algorithm mp\_clamp +within algorithm s\_mp\_add will force $-0$ to become $0$. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_add.c \vspace{-3mm} \begin{alltt} +016 +017 /* high level addition (handles signs) */ +018 int mp_add (mp_int * a, mp_int * b, mp_int * c) +019 \{ +020 int sa, sb, res; +021 +022 /* get sign of both inputs */ +023 sa = a->sign; +024 sb = b->sign; +025 +026 /* handle two cases, not four */ +027 if (sa == sb) \{ +028 /* both positive or both negative */ +029 /* add their magnitudes, copy the sign */ +030 c->sign = sa; +031 res = s_mp_add (a, b, c); +032 \} else \{ +033 /* one positive, the other negative */ +034 /* subtract the one with the greater magnitude from */ +035 /* the one of the lesser magnitude. The result gets */ +036 /* the sign of the one with the greater magnitude. */ +037 if (mp_cmp_mag (a, b) == MP_LT) \{ +038 c->sign = sb; +039 res = s_mp_sub (b, a, c); +040 \} else \{ +041 c->sign = sa; +042 res = s_mp_sub (a, b, c); +043 \} +044 \} +045 return res; +046 \} +047 +048 #endif +049 \end{alltt} \end{small} @@ -1936,7 +2527,7 @@ explicitly checking it and returning the constant \textbf{MP\_OKAY}. The observ level functions do so. Returning their return code is sufficient. \subsection{High Level Subtraction} -The high level signed subtraction algorithm is essentially the same as the high level signed addition algorithm. +The high level signed subtraction algorithm is essentially the same as the high level signed addition algorithm. \newpage\begin{figure}[!here] \begin{center} @@ -1966,7 +2557,7 @@ The high level signed subtraction algorithm is essentially the same as the high \end{figure} \textbf{Algorithm mp\_sub.} -This algorithm performs the signed subtraction of two inputs. Similar to algorithm mp\_add there is no reference in either \cite{TAOCPV2} or +This algorithm performs the signed subtraction of two inputs. Similar to algorithm mp\_add there is no reference in either \cite{TAOCPV2} or \cite{HAC}. Also this algorithm is restricted by algorithm s\_mp\_sub. Chart \ref{fig:SubChart} lists the eight possible inputs and the operations required. @@ -1993,32 +2584,72 @@ the operations required. \label{fig:SubChart} \end{figure} -Similar to the case of algorithm mp\_add the \textbf{sign} is set first before the unsigned addition or subtraction. That is to prevent the -algorithm from producing $-a - -a = -0$ as a result. +Similar to the case of algorithm mp\_add the \textbf{sign} is set first before the unsigned addition or subtraction. That is to prevent the +algorithm from producing $-a - -a = -0$ as a result. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_sub.c \vspace{-3mm} \begin{alltt} +016 +017 /* high level subtraction (handles signs) */ +018 int +019 mp_sub (mp_int * a, mp_int * b, mp_int * c) +020 \{ +021 int sa, sb, res; +022 +023 sa = a->sign; +024 sb = b->sign; +025 +026 if (sa != sb) \{ +027 /* subtract a negative from a positive, OR */ +028 /* subtract a positive from a negative. */ +029 /* In either case, ADD their magnitudes, */ +030 /* and use the sign of the first number. */ +031 c->sign = sa; +032 res = s_mp_add (a, b, c); +033 \} else \{ +034 /* subtract a positive from a positive, OR */ +035 /* subtract a negative from a negative. */ +036 /* First, take the difference between their */ +037 /* magnitudes, then... */ +038 if (mp_cmp_mag (a, b) != MP_LT) \{ +039 /* Copy the sign from the first */ +040 c->sign = sa; +041 /* The first has a larger or equal magnitude */ +042 res = s_mp_sub (a, b, c); +043 \} else \{ +044 /* The result has the *opposite* sign from */ +045 /* the first number. */ +046 c->sign = (sa == MP_ZPOS) ? MP_NEG : MP_ZPOS; +047 /* The second has a larger magnitude */ +048 res = s_mp_sub (b, a, c); +049 \} +050 \} +051 return res; +052 \} +053 +054 #endif +055 \end{alltt} \end{small} Much like the implementation of algorithm mp\_add the variable $res$ is used to catch the return code of the unsigned addition or subtraction operations -and forward it to the end of the function. On line 39 the ``not equal to'' \textbf{MP\_LT} expression is used to emulate a -``greater than or equal to'' comparison. +and forward it to the end of the function. On line 38 the ``not equal to'' \textbf{MP\_LT} expression is used to emulate a +``greater than or equal to'' comparison. \section{Bit and Digit Shifting} -It is quite common to think of a multiple precision integer as a polynomial in $x$, that is $y = f(\beta)$ where $f(x) = \sum_{i=0}^{n-1} a_i x^i$. -This notation arises within discussion of Montgomery and Diminished Radix Reduction as well as Karatsuba multiplication and squaring. +It is quite common to think of a multiple precision integer as a polynomial in $x$, that is $y = f(\beta)$ where $f(x) = \sum_{i=0}^{n-1} a_i x^i$. +This notation arises within discussion of Montgomery and Diminished Radix Reduction as well as Karatsuba multiplication and squaring. In order to facilitate operations on polynomials in $x$ as above a series of simple ``digit'' algorithms have to be established. That is to shift the digits left or right as well to shift individual bits of the digits left and right. It is important to note that not all ``shift'' operations -are on radix-$\beta$ digits. +are on radix-$\beta$ digits. \subsection{Multiplication by Two} -In a binary system where the radix is a power of two multiplication by two not only arises often in other algorithms it is a fairly efficient -operation to perform. A single precision logical shift left is sufficient to multiply a single digit by two. +In a binary system where the radix is a power of two multiplication by two not only arises often in other algorithms it is a fairly efficient +operation to perform. A single precision logical shift left is sufficient to multiply a single digit by two. \newpage\begin{figure}[!here] \begin{small} @@ -2052,31 +2683,94 @@ operation to perform. A single precision logical shift left is sufficient to mu \end{figure} \textbf{Algorithm mp\_mul\_2.} -This algorithm will quickly multiply a mp\_int by two provided $\beta$ is a power of two. Neither \cite{TAOCPV2} nor \cite{HAC} describe such -an algorithm despite the fact it arises often in other algorithms. The algorithm is setup much like the lower level algorithm s\_mp\_add since -it is for all intents and purposes equivalent to the operation $b = \vert a \vert + \vert a \vert$. +This algorithm will quickly multiply a mp\_int by two provided $\beta$ is a power of two. Neither \cite{TAOCPV2} nor \cite{HAC} describe such +an algorithm despite the fact it arises often in other algorithms. The algorithm is setup much like the lower level algorithm s\_mp\_add since +it is for all intents and purposes equivalent to the operation $b = \vert a \vert + \vert a \vert$. Step 1 and 2 grow the input as required to accomodate the maximum number of \textbf{used} digits in the result. The initial \textbf{used} count is set to $a.used$ at step 4. Only if there is a final carry will the \textbf{used} count require adjustment. -Step 6 is an optimization implementation of the addition loop for this specific case. That is since the two values being added together +Step 6 is an optimization implementation of the addition loop for this specific case. That is since the two values being added together are the same there is no need to perform two reads from the digits of $a$. Step 6.1 performs a single precision shift on the current digit $a_n$ to obtain what will be the carry for the next iteration. Step 6.2 calculates the $n$'th digit of the result as single precision shift of $a_n$ plus -the previous carry. Recall from section 4.1 that $a_n << 1$ is equivalent to $a_n \cdot 2$. An iteration of the addition loop is finished with +the previous carry. Recall from section 4.1 that $a_n << 1$ is equivalent to $a_n \cdot 2$. An iteration of the addition loop is finished with forwarding the carry to the next iteration. -Step 7 takes care of any final carry by setting the $a.used$'th digit of the result to the carry and augmenting the \textbf{used} count of $b$. +Step 7 takes care of any final carry by setting the $a.used$'th digit of the result to the carry and augmenting the \textbf{used} count of $b$. Step 8 clears any leading digits of $b$ in case it originally had a larger magnitude than $a$. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_mul\_2.c \vspace{-3mm} \begin{alltt} +016 +017 /* b = a*2 */ +018 int mp_mul_2(mp_int * a, mp_int * b) +019 \{ +020 int x, res, oldused; +021 +022 /* grow to accomodate result */ +023 if (b->alloc < (a->used + 1)) \{ +024 if ((res = mp_grow (b, a->used + 1)) != MP_OKAY) \{ +025 return res; +026 \} +027 \} +028 +029 oldused = b->used; +030 b->used = a->used; +031 +032 \{ +033 mp_digit r, rr, *tmpa, *tmpb; +034 +035 /* alias for source */ +036 tmpa = a->dp; +037 +038 /* alias for dest */ +039 tmpb = b->dp; +040 +041 /* carry */ +042 r = 0; +043 for (x = 0; x < a->used; x++) \{ +044 +045 /* get what will be the *next* carry bit from the +046 * MSB of the current digit +047 */ +048 rr = *tmpa >> ((mp_digit)(DIGIT_BIT - 1)); +049 +050 /* now shift up this digit, add in the carry [from the previous] */ +051 *tmpb++ = ((*tmpa++ << ((mp_digit)1)) | r) & MP_MASK; +052 +053 /* copy the carry that would be from the source +054 * digit into the next iteration +055 */ +056 r = rr; +057 \} +058 +059 /* new leading digit? */ +060 if (r != 0) \{ +061 /* add a MSB which is always 1 at this point */ +062 *tmpb = 1; +063 ++(b->used); +064 \} +065 +066 /* now zero any excess digits on the destination +067 * that we didn't write to +068 */ +069 tmpb = b->dp + b->used; +070 for (x = b->used; x < oldused; x++) \{ +071 *tmpb++ = 0; +072 \} +073 \} +074 b->sign = a->sign; +075 return MP_OKAY; +076 \} +077 #endif +078 \end{alltt} \end{small} This implementation is essentially an optimized implementation of s\_mp\_add for the case of doubling an input. The only noteworthy difference -is the use of the logical shift operator on line 52 to perform a single precision doubling. +is the use of the logical shift operator on line 51 to perform a single precision doubling. \subsection{Division by Two} A division by two can just as easily be accomplished with a logical shift right as multiplication by two can be with a logical shift left. @@ -2117,31 +2811,80 @@ core as the basis of the algorithm. Unlike mp\_mul\_2 the shift operations work could be written to work from the trailing digit to the leading digit however, it would have to stop one short of $a.used - 1$ digits to prevent reading past the end of the array of digits. -Essentially the loop at step 6 is similar to that of mp\_mul\_2 except the logical shifts go in the opposite direction and the carry is at the -least significant bit not the most significant bit. +Essentially the loop at step 6 is similar to that of mp\_mul\_2 except the logical shifts go in the opposite direction and the carry is at the +least significant bit not the most significant bit. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_div\_2.c \vspace{-3mm} \begin{alltt} +016 +017 /* b = a/2 */ +018 int mp_div_2(mp_int * a, mp_int * b) +019 \{ +020 int x, res, oldused; +021 +022 /* copy */ +023 if (b->alloc < a->used) \{ +024 if ((res = mp_grow (b, a->used)) != MP_OKAY) \{ +025 return res; +026 \} +027 \} +028 +029 oldused = b->used; +030 b->used = a->used; +031 \{ +032 mp_digit r, rr, *tmpa, *tmpb; +033 +034 /* source alias */ +035 tmpa = a->dp + b->used - 1; +036 +037 /* dest alias */ +038 tmpb = b->dp + b->used - 1; +039 +040 /* carry */ +041 r = 0; +042 for (x = b->used - 1; x >= 0; x--) \{ +043 /* get the carry for the next iteration */ +044 rr = *tmpa & 1; +045 +046 /* shift the current digit, add in carry and store */ +047 *tmpb-- = (*tmpa-- >> 1) | (r << (DIGIT_BIT - 1)); +048 +049 /* forward carry to next iteration */ +050 r = rr; +051 \} +052 +053 /* zero excess digits */ +054 tmpb = b->dp + b->used; +055 for (x = b->used; x < oldused; x++) \{ +056 *tmpb++ = 0; +057 \} +058 \} +059 b->sign = a->sign; +060 mp_clamp (b); +061 return MP_OKAY; +062 \} +063 #endif +064 \end{alltt} \end{small} \section{Polynomial Basis Operations} Recall from section 4.3 that any integer can be represented as a polynomial in $x$ as $y = f(\beta)$. Such a representation is also known as -the polynomial basis \cite[pp. 48]{ROSE}. Given such a notation a multiplication or division by $x$ amounts to shifting whole digits a single +the polynomial basis \cite[pp. 48]{ROSE}. Given such a notation a multiplication or division by $x$ amounts to shifting whole digits a single place. The need for such operations arises in several other higher level algorithms such as Barrett and Montgomery reduction, integer -division and Karatsuba multiplication. +division and Karatsuba multiplication. Converting from an array of digits to polynomial basis is very simple. Consider the integer $y \equiv (a_2, a_1, a_0)_{\beta}$ and recall that $y = \sum_{i=0}^{2} a_i \beta^i$. Simply replace $\beta$ with $x$ and the expression is in polynomial basis. For example, $f(x) = 8x + 9$ is the -polynomial basis representation for $89$ using radix ten. That is, $f(10) = 8(10) + 9 = 89$. +polynomial basis representation for $89$ using radix ten. That is, $f(10) = 8(10) + 9 = 89$. \subsection{Multiplication by $x$} -Given a polynomial in $x$ such as $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0$ multiplying by $x$ amounts to shifting the coefficients up one +Given a polynomial in $x$ such as $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0$ multiplying by $x$ amounts to shifting the coefficients up one degree. In this case $f(x) \cdot x = a_n x^{n+1} + a_{n-1} x^n + ... + a_0 x$. From a scalar basis point of view multiplying by $x$ is equivalent to -multiplying by the integer $\beta$. +multiplying by the integer $\beta$. \newpage\begin{figure}[!here] \begin{small} @@ -2172,16 +2915,16 @@ multiplying by the integer $\beta$. \end{figure} \textbf{Algorithm mp\_lshd.} -This algorithm multiplies an mp\_int by the $b$'th power of $x$. This is equivalent to multiplying by $\beta^b$. The algorithm differs +This algorithm multiplies an mp\_int by the $b$'th power of $x$. This is equivalent to multiplying by $\beta^b$. The algorithm differs from the other algorithms presented so far as it performs the operation in place instead storing the result in a separate location. The motivation behind this change is due to the way this function is typically used. Algorithms such as mp\_add store the result in an optionally different third mp\_int because the original inputs are often still required. Algorithm mp\_lshd (\textit{and similarly algorithm mp\_rshd}) is -typically used on values where the original value is no longer required. The algorithm will return success immediately if -$b \le 0$ since the rest of algorithm is only valid when $b > 0$. +typically used on values where the original value is no longer required. The algorithm will return success immediately if +$b \le 0$ since the rest of algorithm is only valid when $b > 0$. First the destination $a$ is grown as required to accomodate the result. The counters $i$ and $j$ are used to form a \textit{sliding window} over -the digits of $a$ of length $b$. The head of the sliding window is at $i$ (\textit{the leading digit}) and the tail at $j$ (\textit{the trailing digit}). -The loop on step 7 copies the digit from the tail to the head. In each iteration the window is moved down one digit. The last loop on +the digits of $a$ of length $b$. The head of the sliding window is at $i$ (\textit{the leading digit}) and the tail at $j$ (\textit{the trailing digit}). +The loop on step 7 copies the digit from the tail to the head. In each iteration the window is moved down one digit. The last loop on step 8 sets the lower $b$ digits to zero. \newpage @@ -2197,18 +2940,66 @@ step 8 sets the lower $b$ digits to zero. \hspace{-5.1mm}{\bf File}: bn\_mp\_lshd.c \vspace{-3mm} \begin{alltt} +016 +017 /* shift left a certain amount of digits */ +018 int mp_lshd (mp_int * a, int b) +019 \{ +020 int x, res; +021 +022 /* if its less than zero return */ +023 if (b <= 0) \{ +024 return MP_OKAY; +025 \} +026 +027 /* grow to fit the new digits */ +028 if (a->alloc < (a->used + b)) \{ +029 if ((res = mp_grow (a, a->used + b)) != MP_OKAY) \{ +030 return res; +031 \} +032 \} +033 +034 \{ +035 mp_digit *top, *bottom; +036 +037 /* increment the used by the shift amount then copy upwards */ +038 a->used += b; +039 +040 /* top */ +041 top = a->dp + a->used - 1; +042 +043 /* base */ +044 bottom = (a->dp + a->used - 1) - b; +045 +046 /* much like mp_rshd this is implemented using a sliding window +047 * except the window goes the otherway around. Copying from +048 * the bottom to the top. see bn_mp_rshd.c for more info. +049 */ +050 for (x = a->used - 1; x >= b; x--) \{ +051 *top-- = *bottom--; +052 \} +053 +054 /* zero the lower digits */ +055 top = a->dp; +056 for (x = 0; x < b; x++) \{ +057 *top++ = 0; +058 \} +059 \} +060 return MP_OKAY; +061 \} +062 #endif +063 \end{alltt} \end{small} -The if statement (line 24) ensures that the $b$ variable is greater than zero since we do not interpret negative -shift counts properly. The \textbf{used} count is incremented by $b$ before the copy loop begins. This elminates -the need for an additional variable in the for loop. The variable $top$ (line 42) is an alias -for the leading digit while $bottom$ (line 45) is an alias for the trailing edge. The aliases form a -window of exactly $b$ digits over the input. +The if statement (line 23) ensures that the $b$ variable is greater than zero since we do not interpret negative +shift counts properly. The \textbf{used} count is incremented by $b$ before the copy loop begins. This elminates +the need for an additional variable in the for loop. The variable $top$ (line 41) is an alias +for the leading digit while $bottom$ (line 44) is an alias for the trailing edge. The aliases form a +window of exactly $b$ digits over the input. \subsection{Division by $x$} -Division by powers of $x$ is easily achieved by shifting the digits right and removing any that will end up to the right of the zero'th digit. +Division by powers of $x$ is easily achieved by shifting the digits right and removing any that will end up to the right of the zero'th digit. \newpage\begin{figure}[!here] \begin{small} @@ -2241,13 +3032,13 @@ Division by powers of $x$ is easily achieved by shifting the digits right and re \textbf{Algorithm mp\_rshd.} This algorithm divides the input in place by the $b$'th power of $x$. It is analogous to dividing by a $\beta^b$ but much quicker since -it does not require single precision division. This algorithm does not actually return an error code as it cannot fail. +it does not require single precision division. This algorithm does not actually return an error code as it cannot fail. If the input $b$ is less than one the algorithm quickly returns without performing any work. If the \textbf{used} count is less than or equal to the shift count $b$ then it will simply zero the input and return. After the trivial cases of inputs have been handled the sliding window is setup. Much like the case of algorithm mp\_lshd a sliding window that -is $b$ digits wide is used to copy the digits. Unlike mp\_lshd the window slides in the opposite direction from the trailing to the leading digit. +is $b$ digits wide is used to copy the digits. Unlike mp\_lshd the window slides in the opposite direction from the trailing to the leading digit. Also the digits are copied from the leading to the trailing edge. Once the window copy is complete the upper digits must be zeroed and the \textbf{used} count decremented. @@ -2256,18 +3047,71 @@ Once the window copy is complete the upper digits must be zeroed and the \textbf \hspace{-5.1mm}{\bf File}: bn\_mp\_rshd.c \vspace{-3mm} \begin{alltt} +016 +017 /* shift right a certain amount of digits */ +018 void mp_rshd (mp_int * a, int b) +019 \{ +020 int x; +021 +022 /* if b <= 0 then ignore it */ +023 if (b <= 0) \{ +024 return; +025 \} +026 +027 /* if b > used then simply zero it and return */ +028 if (a->used <= b) \{ +029 mp_zero (a); +030 return; +031 \} +032 +033 \{ +034 mp_digit *bottom, *top; +035 +036 /* shift the digits down */ +037 +038 /* bottom */ +039 bottom = a->dp; +040 +041 /* top [offset into digits] */ +042 top = a->dp + b; +043 +044 /* this is implemented as a sliding window where +045 * the window is b-digits long and digits from +046 * the top of the window are copied to the bottom +047 * +048 * e.g. +049 +050 b-2 | b-1 | b0 | b1 | b2 | ... | bb | ----> +051 /\symbol{92} | ----> +052 \symbol{92}-------------------/ ----> +053 */ +054 for (x = 0; x < (a->used - b); x++) \{ +055 *bottom++ = *top++; +056 \} +057 +058 /* zero the top digits */ +059 for (; x < a->used; x++) \{ +060 *bottom++ = 0; +061 \} +062 \} +063 +064 /* remove excess digits */ +065 a->used -= b; +066 \} +067 #endif +068 \end{alltt} \end{small} The only noteworthy element of this routine is the lack of a return type since it cannot fail. Like mp\_lshd() we -form a sliding window except we copy in the other direction. After the window (line 60) we then zero +form a sliding window except we copy in the other direction. After the window (line 59) we then zero the upper digits of the input to make sure the result is correct. \section{Powers of Two} -Now that algorithms for moving single bits as well as whole digits exist algorithms for moving the ``in between'' distances are required. For +Now that algorithms for moving single bits as well as whole digits exist algorithms for moving the ``in between'' distances are required. For example, to quickly multiply by $2^k$ for any $k$ without using a full multiplier algorithm would prove useful. Instead of performing single -shifts $k$ times to achieve a multiplication by $2^{\pm k}$ a mixture of whole digit shifting and partial digit shifting is employed. +shifts $k$ times to achieve a multiplication by $2^{\pm k}$ a mixture of whole digit shifting and partial digit shifting is employed. \subsection{Multiplication by Power of Two} @@ -2308,34 +3152,100 @@ shifts $k$ times to achieve a multiplication by $2^{\pm k}$ a mixture of whole d This algorithm multiplies $a$ by $2^b$ and stores the result in $c$. The algorithm uses algorithm mp\_lshd and a derivative of algorithm mp\_mul\_2 to quickly compute the product. -First the algorithm will multiply $a$ by $x^{\lfloor b / lg(\beta) \rfloor}$ which will ensure that the remainder multiplicand is less than -$\beta$. For example, if $b = 37$ and $\beta = 2^{28}$ then this step will multiply by $x$ leaving a multiplication by $2^{37 - 28} = 2^{9}$ +First the algorithm will multiply $a$ by $x^{\lfloor b / lg(\beta) \rfloor}$ which will ensure that the remainder multiplicand is less than +$\beta$. For example, if $b = 37$ and $\beta = 2^{28}$ then this step will multiply by $x$ leaving a multiplication by $2^{37 - 28} = 2^{9}$ left. -After the digits have been shifted appropriately at most $lg(\beta) - 1$ shifts are left to perform. Step 5 calculates the number of remaining shifts -required. If it is non-zero a modified shift loop is used to calculate the remaining product. +After the digits have been shifted appropriately at most $lg(\beta) - 1$ shifts are left to perform. Step 5 calculates the number of remaining shifts +required. If it is non-zero a modified shift loop is used to calculate the remaining product. Essentially the loop is a generic version of algorithm mp\_mul\_2 designed to handle any shift count in the range $1 \le x < lg(\beta)$. The $mask$ -variable is used to extract the upper $d$ bits to form the carry for the next iteration. +variable is used to extract the upper $d$ bits to form the carry for the next iteration. -This algorithm is loosely measured as a $O(2n)$ algorithm which means that if the input is $n$-digits that it takes $2n$ ``time'' to +This algorithm is loosely measured as a $O(2n)$ algorithm which means that if the input is $n$-digits that it takes $2n$ ``time'' to complete. It is possible to optimize this algorithm down to a $O(n)$ algorithm at a cost of making the algorithm slightly harder to follow. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_mul\_2d.c \vspace{-3mm} \begin{alltt} +016 +017 /* shift left by a certain bit count */ +018 int mp_mul_2d (mp_int * a, int b, mp_int * c) +019 \{ +020 mp_digit d; +021 int res; +022 +023 /* copy */ +024 if (a != c) \{ +025 if ((res = mp_copy (a, c)) != MP_OKAY) \{ +026 return res; +027 \} +028 \} +029 +030 if (c->alloc < (int)(c->used + (b / DIGIT_BIT) + 1)) \{ +031 if ((res = mp_grow (c, c->used + (b / DIGIT_BIT) + 1)) != MP_OKAY) \{ +032 return res; +033 \} +034 \} +035 +036 /* shift by as many digits in the bit count */ +037 if (b >= (int)DIGIT_BIT) \{ +038 if ((res = mp_lshd (c, b / DIGIT_BIT)) != MP_OKAY) \{ +039 return res; +040 \} +041 \} +042 +043 /* shift any bit count < DIGIT_BIT */ +044 d = (mp_digit) (b % DIGIT_BIT); +045 if (d != 0) \{ +046 mp_digit *tmpc, shift, mask, r, rr; +047 int x; +048 +049 /* bitmask for carries */ +050 mask = (((mp_digit)1) << d) - 1; +051 +052 /* shift for msbs */ +053 shift = DIGIT_BIT - d; +054 +055 /* alias */ +056 tmpc = c->dp; +057 +058 /* carry */ +059 r = 0; +060 for (x = 0; x < c->used; x++) \{ +061 /* get the higher bits of the current word */ +062 rr = (*tmpc >> shift) & mask; +063 +064 /* shift the current word and OR in the carry */ +065 *tmpc = ((*tmpc << d) | r) & MP_MASK; +066 ++tmpc; +067 +068 /* set the carry to the carry bits of the current word */ +069 r = rr; +070 \} +071 +072 /* set final carry */ +073 if (r != 0) \{ +074 c->dp[(c->used)++] = r; +075 \} +076 \} +077 mp_clamp (c); +078 return MP_OKAY; +079 \} +080 #endif +081 \end{alltt} \end{small} -The shifting is performed in--place which means the first step (line 25) is to copy the input to the +The shifting is performed in--place which means the first step (line 24) is to copy the input to the destination. We avoid calling mp\_copy() by making sure the mp\_ints are different. The destination then -has to be grown (line 32) to accomodate the result. +has to be grown (line 31) to accomodate the result. -If the shift count $b$ is larger than $lg(\beta)$ then a call to mp\_lshd() is used to handle all of the multiples -of $lg(\beta)$. Leaving only a remaining shift of $lg(\beta) - 1$ or fewer bits left. Inside the actual shift -loop (lines 46 to 76) we make use of pre--computed values $shift$ and $mask$. These are used to -extract the carry bit(s) to pass into the next iteration of the loop. The $r$ and $rr$ variables form a -chain between consecutive iterations to propagate the carry. +If the shift count $b$ is larger than $lg(\beta)$ then a call to mp\_lshd() is used to handle all of the multiples +of $lg(\beta)$. Leaving only a remaining shift of $lg(\beta) - 1$ or fewer bits left. Inside the actual shift +loop (lines 45 to 76) we make use of pre--computed values $shift$ and $mask$. These are used to +extract the carry bit(s) to pass into the next iteration of the loop. The $r$ and $rr$ variables form a +chain between consecutive iterations to propagate the carry. \subsection{Division by Power of Two} @@ -2373,7 +3283,7 @@ chain between consecutive iterations to propagate the carry. \end{figure} \textbf{Algorithm mp\_div\_2d.} -This algorithm will divide an input $a$ by $2^b$ and produce the quotient and remainder. The algorithm is designed much like algorithm +This algorithm will divide an input $a$ by $2^b$ and produce the quotient and remainder. The algorithm is designed much like algorithm mp\_mul\_2d by first using whole digit shifts then single precision shifts. This algorithm will also produce the remainder of the division by using algorithm mp\_mod\_2d. @@ -2381,11 +3291,91 @@ by using algorithm mp\_mod\_2d. \hspace{-5.1mm}{\bf File}: bn\_mp\_div\_2d.c \vspace{-3mm} \begin{alltt} +016 +017 /* shift right by a certain bit count (store quotient in c, optional remaind + er in d) */ +018 int mp_div_2d (mp_int * a, int b, mp_int * c, mp_int * d) +019 \{ +020 mp_digit D, r, rr; +021 int x, res; +022 mp_int t; +023 +024 +025 /* if the shift count is <= 0 then we do no work */ +026 if (b <= 0) \{ +027 res = mp_copy (a, c); +028 if (d != NULL) \{ +029 mp_zero (d); +030 \} +031 return res; +032 \} +033 +034 if ((res = mp_init (&t)) != MP_OKAY) \{ +035 return res; +036 \} +037 +038 /* get the remainder */ +039 if (d != NULL) \{ +040 if ((res = mp_mod_2d (a, b, &t)) != MP_OKAY) \{ +041 mp_clear (&t); +042 return res; +043 \} +044 \} +045 +046 /* copy */ +047 if ((res = mp_copy (a, c)) != MP_OKAY) \{ +048 mp_clear (&t); +049 return res; +050 \} +051 +052 /* shift by as many digits in the bit count */ +053 if (b >= (int)DIGIT_BIT) \{ +054 mp_rshd (c, b / DIGIT_BIT); +055 \} +056 +057 /* shift any bit count < DIGIT_BIT */ +058 D = (mp_digit) (b % DIGIT_BIT); +059 if (D != 0) \{ +060 mp_digit *tmpc, mask, shift; +061 +062 /* mask */ +063 mask = (((mp_digit)1) << D) - 1; +064 +065 /* shift for lsb */ +066 shift = DIGIT_BIT - D; +067 +068 /* alias */ +069 tmpc = c->dp + (c->used - 1); +070 +071 /* carry */ +072 r = 0; +073 for (x = c->used - 1; x >= 0; x--) \{ +074 /* get the lower bits of this word in a temp */ +075 rr = *tmpc & mask; +076 +077 /* shift the current word and mix in the carry bits from the previous + word */ +078 *tmpc = (*tmpc >> D) | (r << shift); +079 --tmpc; +080 +081 /* set the carry to the carry bits of the current word found above */ +082 r = rr; +083 \} +084 \} +085 mp_clamp (c); +086 if (d != NULL) \{ +087 mp_exch (&t, d); +088 \} +089 mp_clear (&t); +090 return MP_OKAY; +091 \} +092 #endif +093 \end{alltt} \end{small} -The implementation of algorithm mp\_div\_2d is slightly different than the algorithm specifies. The remainder $d$ may be optionally -ignored by passing \textbf{NULL} as the pointer to the mp\_int variable. The temporary mp\_int variable $t$ is used to hold the +The implementation of algorithm mp\_div\_2d is slightly different than the algorithm specifies. The remainder $d$ may be optionally +ignored by passing \textbf{NULL} as the pointer to the mp\_int variable. The temporary mp\_int variable $t$ is used to hold the result of the remainder operation until the end. This allows $d$ and $a$ to represent the same mp\_int without modifying $a$ before the quotient is obtained. @@ -2395,7 +3385,7 @@ the direction of the shifts. \subsection{Remainder of Division by Power of Two} The last algorithm in the series of polynomial basis power of two algorithms is calculating the remainder of division by $2^b$. This -algorithm benefits from the fact that in twos complement arithmetic $a \mbox{ (mod }2^b\mbox{)}$ is the same as $a$ AND $2^b - 1$. +algorithm benefits from the fact that in twos complement arithmetic $a \mbox{ (mod }2^b\mbox{)}$ is the same as $a$ AND $2^b - 1$. \begin{figure}[!here] \begin{small} @@ -2427,14 +3417,52 @@ algorithm benefits from the fact that in twos complement arithmetic $a \mbox{ (m \end{figure} \textbf{Algorithm mp\_mod\_2d.} -This algorithm will quickly calculate the value of $a \mbox{ (mod }2^b\mbox{)}$. First if $b$ is less than or equal to zero the -result is set to zero. If $b$ is greater than the number of bits in $a$ then it simply copies $a$ to $c$ and returns. Otherwise, $a$ +This algorithm will quickly calculate the value of $a \mbox{ (mod }2^b\mbox{)}$. First if $b$ is less than or equal to zero the +result is set to zero. If $b$ is greater than the number of bits in $a$ then it simply copies $a$ to $c$ and returns. Otherwise, $a$ is copied to $b$, leading digits are removed and the remaining leading digit is trimed to the exact bit count. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_mod\_2d.c \vspace{-3mm} \begin{alltt} +016 +017 /* calc a value mod 2**b */ +018 int +019 mp_mod_2d (mp_int * a, int b, mp_int * c) +020 \{ +021 int x, res; +022 +023 /* if b is <= 0 then zero the int */ +024 if (b <= 0) \{ +025 mp_zero (c); +026 return MP_OKAY; +027 \} +028 +029 /* if the modulus is larger than the value than return */ +030 if (b >= (int) (a->used * DIGIT_BIT)) \{ +031 res = mp_copy (a, c); +032 return res; +033 \} +034 +035 /* copy */ +036 if ((res = mp_copy (a, c)) != MP_OKAY) \{ +037 return res; +038 \} +039 +040 /* zero digits above the last digit of the modulus */ +041 for (x = (b / DIGIT_BIT) + (((b % DIGIT_BIT) == 0) ? 0 : 1); x < c->used; + x++) \{ +042 c->dp[x] = 0; +043 \} +044 /* clear the digit that is not completely outside/inside the modulus */ +045 c->dp[b / DIGIT_BIT] &= +046 (mp_digit) ((((mp_digit) 1) << (((mp_digit) b) % DIGIT_BIT)) - ((mp_digi + t) 1)); +047 mp_clamp (c); +048 return MP_OKAY; +049 \} +050 #endif +051 \end{alltt} \end{small} @@ -2442,9 +3470,9 @@ We first avoid cases of $b \le 0$ by simply mp\_zero()'ing the destination in su than the input we just mp\_copy() the input and return right away. After this point we know we must actually perform some work to produce the remainder. -Recalling that reducing modulo $2^k$ and a binary ``and'' with $2^k - 1$ are numerically equivalent we can quickly reduce -the number. First we zero any digits above the last digit in $2^b$ (line 42). Next we reduce the -leading digit of both (line 46) and then mp\_clamp(). +Recalling that reducing modulo $2^k$ and a binary ``and'' with $2^k - 1$ are numerically equivalent we can quickly reduce +the number. First we zero any digits above the last digit in $2^b$ (line 41). Next we reduce the +leading digit of both (line 45) and then mp\_clamp(). \section*{Exercises} \begin{tabular}{cl} @@ -2477,40 +3505,40 @@ $\left [ 2 \right ] $ & Using only algorithms mp\_abs and mp\_sub devise another \chapter{Multiplication and Squaring} \section{The Multipliers} -For most number theoretic problems including certain public key cryptographic algorithms, the ``multipliers'' form the most important subset of -algorithms of any multiple precision integer package. The set of multiplier algorithms include integer multiplication, squaring and modular reduction -where in each of the algorithms single precision multiplication is the dominant operation performed. This chapter will discuss integer multiplication -and squaring, leaving modular reductions for the subsequent chapter. +For most number theoretic problems including certain public key cryptographic algorithms, the ``multipliers'' form the most important subset of +algorithms of any multiple precision integer package. The set of multiplier algorithms include integer multiplication, squaring and modular reduction +where in each of the algorithms single precision multiplication is the dominant operation performed. This chapter will discuss integer multiplication +and squaring, leaving modular reductions for the subsequent chapter. -The importance of the multiplier algorithms is for the most part driven by the fact that certain popular public key algorithms are based on modular +The importance of the multiplier algorithms is for the most part driven by the fact that certain popular public key algorithms are based on modular exponentiation, that is computing $d \equiv a^b \mbox{ (mod }c\mbox{)}$ for some arbitrary choice of $a$, $b$, $c$ and $d$. During a modular -exponentiation the majority\footnote{Roughly speaking a modular exponentiation will spend about 40\% of the time performing modular reductions, -35\% of the time performing squaring and 25\% of the time performing multiplications.} of the processor time is spent performing single precision +exponentiation the majority\footnote{Roughly speaking a modular exponentiation will spend about 40\% of the time performing modular reductions, +35\% of the time performing squaring and 25\% of the time performing multiplications.} of the processor time is spent performing single precision multiplications. -For centuries general purpose multiplication has required a lengthly $O(n^2)$ process, whereby each digit of one multiplicand has to be multiplied -against every digit of the other multiplicand. Traditional long-hand multiplication is based on this process; while the techniques can differ the -overall algorithm used is essentially the same. Only ``recently'' have faster algorithms been studied. First Karatsuba multiplication was discovered in -1962. This algorithm can multiply two numbers with considerably fewer single precision multiplications when compared to the long-hand approach. -This technique led to the discovery of polynomial basis algorithms (\textit{good reference?}) and subquently Fourier Transform based solutions. +For centuries general purpose multiplication has required a lengthly $O(n^2)$ process, whereby each digit of one multiplicand has to be multiplied +against every digit of the other multiplicand. Traditional long-hand multiplication is based on this process; while the techniques can differ the +overall algorithm used is essentially the same. Only ``recently'' have faster algorithms been studied. First Karatsuba multiplication was discovered in +1962. This algorithm can multiply two numbers with considerably fewer single precision multiplications when compared to the long-hand approach. +This technique led to the discovery of polynomial basis algorithms (\textit{good reference?}) and subquently Fourier Transform based solutions. \section{Multiplication} \subsection{The Baseline Multiplication} \label{sec:basemult} \index{baseline multiplication} Computing the product of two integers in software can be achieved using a trivial adaptation of the standard $O(n^2)$ long-hand multiplication -algorithm that school children are taught. The algorithm is considered an $O(n^2)$ algorithm since for two $n$-digit inputs $n^2$ single precision -multiplications are required. More specifically for a $m$ and $n$ digit input $m \cdot n$ single precision multiplications are required. To -simplify most discussions, it will be assumed that the inputs have comparable number of digits. - -The ``baseline multiplication'' algorithm is designed to act as the ``catch-all'' algorithm, only to be used when the faster algorithms cannot be -used. This algorithm does not use any particularly interesting optimizations and should ideally be avoided if possible. One important -facet of this algorithm, is that it has been modified to only produce a certain amount of output digits as resolution. The importance of this -modification will become evident during the discussion of Barrett modular reduction. Recall that for a $n$ and $m$ digit input the product -will be at most $n + m$ digits. Therefore, this algorithm can be reduced to a full multiplier by having it produce $n + m$ digits of the product. - -Recall from sub-section 4.2.2 the definition of $\gamma$ as the number of bits in the type \textbf{mp\_digit}. We shall now extend the variable set to -include $\alpha$ which shall represent the number of bits in the type \textbf{mp\_word}. This implies that $2^{\alpha} > 2 \cdot \beta^2$. The +algorithm that school children are taught. The algorithm is considered an $O(n^2)$ algorithm since for two $n$-digit inputs $n^2$ single precision +multiplications are required. More specifically for a $m$ and $n$ digit input $m \cdot n$ single precision multiplications are required. To +simplify most discussions, it will be assumed that the inputs have comparable number of digits. + +The ``baseline multiplication'' algorithm is designed to act as the ``catch-all'' algorithm, only to be used when the faster algorithms cannot be +used. This algorithm does not use any particularly interesting optimizations and should ideally be avoided if possible. One important +facet of this algorithm, is that it has been modified to only produce a certain amount of output digits as resolution. The importance of this +modification will become evident during the discussion of Barrett modular reduction. Recall that for a $n$ and $m$ digit input the product +will be at most $n + m$ digits. Therefore, this algorithm can be reduced to a full multiplier by having it produce $n + m$ digits of the product. + +Recall from sub-section 4.2.2 the definition of $\gamma$ as the number of bits in the type \textbf{mp\_digit}. We shall now extend the variable set to +include $\alpha$ which shall represent the number of bits in the type \textbf{mp\_word}. This implies that $2^{\alpha} > 2 \cdot \beta^2$. The constant $\delta = 2^{\alpha - 2lg(\beta)}$ will represent the maximal weight of any column in a product (\textit{see sub-section 5.2.2 for more information}). \newpage\begin{figure}[!here] @@ -2554,20 +3582,20 @@ Compute the product. \\ \textbf{Algorithm s\_mp\_mul\_digs.} This algorithm computes the unsigned product of two inputs $a$ and $b$, limited to an output precision of $digs$ digits. While it may seem -a bit awkward to modify the function from its simple $O(n^2)$ description, the usefulness of partial multipliers will arise in a subsequent -algorithm. The algorithm is loosely based on algorithm 14.12 from \cite[pp. 595]{HAC} and is similar to Algorithm M of Knuth \cite[pp. 268]{TAOCPV2}. -Algorithm s\_mp\_mul\_digs differs from these cited references since it can produce a variable output precision regardless of the precision of the +a bit awkward to modify the function from its simple $O(n^2)$ description, the usefulness of partial multipliers will arise in a subsequent +algorithm. The algorithm is loosely based on algorithm 14.12 from \cite[pp. 595]{HAC} and is similar to Algorithm M of Knuth \cite[pp. 268]{TAOCPV2}. +Algorithm s\_mp\_mul\_digs differs from these cited references since it can produce a variable output precision regardless of the precision of the inputs. The first thing this algorithm checks for is whether a Comba multiplier can be used instead. If the minimum digit count of either -input is less than $\delta$, then the Comba method may be used instead. After the Comba method is ruled out, the baseline algorithm begins. A -temporary mp\_int variable $t$ is used to hold the intermediate result of the product. This allows the algorithm to be used to -compute products when either $a = c$ or $b = c$ without overwriting the inputs. +input is less than $\delta$, then the Comba method may be used instead. After the Comba method is ruled out, the baseline algorithm begins. A +temporary mp\_int variable $t$ is used to hold the intermediate result of the product. This allows the algorithm to be used to +compute products when either $a = c$ or $b = c$ without overwriting the inputs. All of step 5 is the infamous $O(n^2)$ multiplication loop slightly modified to only produce upto $digs$ digits of output. The $pb$ variable is given the count of digits to read from $b$ inside the nested loop. If $pb \le 1$ then no more output digits can be produced and the algorithm -will exit the loop. The best way to think of the loops are as a series of $pb \times 1$ multiplications. That is, in each pass of the -innermost loop $a_{ix}$ is multiplied against $b$ and the result is added (\textit{with an appropriate shift}) to $t$. +will exit the loop. The best way to think of the loops are as a series of $pb \times 1$ multiplications. That is, in each pass of the +innermost loop $a_{ix}$ is multiplied against $b$ and the result is added (\textit{with an appropriate shift}) to $t$. For example, consider multiplying $576$ by $241$. That is equivalent to computing $10^0(1)(576) + 10^1(4)(576) + 10^2(2)(576)$ which is best visualized in the following table. @@ -2581,20 +3609,20 @@ visualized in the following table. && & 5 & 7 & 6 & $10^0(1)(576)$ \\ &2 & 3 & 6 & 1 & 6 & $10^1(4)(576) + 10^0(1)(576)$ \\ 1 & 3 & 8 & 8 & 1 & 6 & $10^2(2)(576) + 10^1(4)(576) + 10^0(1)(576)$ \\ -\hline +\hline \end{tabular} \end{center} \caption{Long-Hand Multiplication Diagram} \end{figure} -Each row of the product is added to the result after being shifted to the left (\textit{multiplied by a power of the radix}) by the appropriate +Each row of the product is added to the result after being shifted to the left (\textit{multiplied by a power of the radix}) by the appropriate count. That is in pass $ix$ of the inner loop the product is added starting at the $ix$'th digit of the reult. Step 5.4.1 introduces the hat symbol (\textit{e.g. $\hat r$}) which represents a double precision variable. The multiplication on that step is assumed to be a double wide output single precision multiplication. That is, two single precision variables are multiplied to produce a double precision result. The step is somewhat optimized from a long-hand multiplication algorithm because the carry from the addition in step -5.4.1 is propagated through the nested loop. If the carry was not propagated immediately it would overflow the single precision digit -$t_{ix+iy}$ and the result would be lost. +5.4.1 is propagated through the nested loop. If the carry was not propagated immediately it would overflow the single precision digit +$t_{ix+iy}$ and the result would be lost. At step 5.5 the nested loop is finished and any carry that was left over should be forwarded. The carry does not have to be added to the $ix+pb$'th digit since that digit is assumed to be zero at this point. However, if $ix + pb \ge digs$ the carry is not set as it would make the result @@ -2604,55 +3632,126 @@ exceed the precision requested. \hspace{-5.1mm}{\bf File}: bn\_s\_mp\_mul\_digs.c \vspace{-3mm} \begin{alltt} +016 +017 /* multiplies |a| * |b| and only computes upto digs digits of result +018 * HAC pp. 595, Algorithm 14.12 Modified so you can control how +019 * many digits of output are created. +020 */ +021 int s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs) +022 \{ +023 mp_int t; +024 int res, pa, pb, ix, iy; +025 mp_digit u; +026 mp_word r; +027 mp_digit tmpx, *tmpt, *tmpy; +028 +029 /* can we use the fast multiplier? */ +030 if (((digs) < MP_WARRAY) && +031 (MIN (a->used, b->used) < +032 (1 << ((CHAR_BIT * sizeof(mp_word)) - (2 * DIGIT_BIT))))) \{ +033 return fast_s_mp_mul_digs (a, b, c, digs); +034 \} +035 +036 if ((res = mp_init_size (&t, digs)) != MP_OKAY) \{ +037 return res; +038 \} +039 t.used = digs; +040 +041 /* compute the digits of the product directly */ +042 pa = a->used; +043 for (ix = 0; ix < pa; ix++) \{ +044 /* set the carry to zero */ +045 u = 0; +046 +047 /* limit ourselves to making digs digits of output */ +048 pb = MIN (b->used, digs - ix); +049 +050 /* setup some aliases */ +051 /* copy of the digit from a used within the nested loop */ +052 tmpx = a->dp[ix]; +053 +054 /* an alias for the destination shifted ix places */ +055 tmpt = t.dp + ix; +056 +057 /* an alias for the digits of b */ +058 tmpy = b->dp; +059 +060 /* compute the columns of the output and propagate the carry */ +061 for (iy = 0; iy < pb; iy++) \{ +062 /* compute the column as a mp_word */ +063 r = (mp_word)*tmpt + +064 ((mp_word)tmpx * (mp_word)*tmpy++) + +065 (mp_word)u; +066 +067 /* the new column is the lower part of the result */ +068 *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK)); +069 +070 /* get the carry word from the result */ +071 u = (mp_digit) (r >> ((mp_word) DIGIT_BIT)); +072 \} +073 /* set carry if it is placed below digs */ +074 if ((ix + iy) < digs) \{ +075 *tmpt = u; +076 \} +077 \} +078 +079 mp_clamp (&t); +080 mp_exch (&t, c); +081 +082 mp_clear (&t); +083 return MP_OKAY; +084 \} +085 #endif +086 \end{alltt} \end{small} -First we determine (line 31) if the Comba method can be used first since it's faster. The conditions for -sing the Comba routine are that min$(a.used, b.used) < \delta$ and the number of digits of output is less than -\textbf{MP\_WARRAY}. This new constant is used to control the stack usage in the Comba routines. By default it is +First we determine (line 30) if the Comba method can be used first since it's faster. The conditions for +sing the Comba routine are that min$(a.used, b.used) < \delta$ and the number of digits of output is less than +\textbf{MP\_WARRAY}. This new constant is used to control the stack usage in the Comba routines. By default it is set to $\delta$ but can be reduced when memory is at a premium. If we cannot use the Comba method we proceed to setup the baseline routine. We allocate the the destination mp\_int -$t$ (line 37) to the exact size of the output to avoid further re--allocations. At this point we now +$t$ (line 36) to the exact size of the output to avoid further re--allocations. At this point we now begin the $O(n^2)$ loop. This implementation of multiplication has the caveat that it can be trimmed to only produce a variable number of -digits as output. In each iteration of the outer loop the $pb$ variable is set (line 49) to the maximum -number of inner loop iterations. +digits as output. In each iteration of the outer loop the $pb$ variable is set (line 48) to the maximum +number of inner loop iterations. Inside the inner loop we calculate $\hat r$ as the mp\_word product of the two mp\_digits and the addition of the -carry from the previous iteration. A particularly important observation is that most modern optimizing -C compilers (GCC for instance) can recognize that a $N \times N \rightarrow 2N$ multiplication is all that +carry from the previous iteration. A particularly important observation is that most modern optimizing +C compilers (GCC for instance) can recognize that a $N \times N \rightarrow 2N$ multiplication is all that is required for the product. In x86 terms for example, this means using the MUL instruction. -Each digit of the product is stored in turn (line 69) and the carry propagated (line 72) to the +Each digit of the product is stored in turn (line 68) and the carry propagated (line 71) to the next iteration. \subsection{Faster Multiplication by the ``Comba'' Method} -One of the huge drawbacks of the ``baseline'' algorithms is that at the $O(n^2)$ level the carry must be -computed and propagated upwards. This makes the nested loop very sequential and hard to unroll and implement -in parallel. The ``Comba'' \cite{COMBA} method is named after little known (\textit{in cryptographic venues}) Paul G. -Comba who described a method of implementing fast multipliers that do not require nested carry fixup operations. As an -interesting aside it seems that Paul Barrett describes a similar technique in his 1986 paper \cite{BARRETT} written +One of the huge drawbacks of the ``baseline'' algorithms is that at the $O(n^2)$ level the carry must be +computed and propagated upwards. This makes the nested loop very sequential and hard to unroll and implement +in parallel. The ``Comba'' \cite{COMBA} method is named after little known (\textit{in cryptographic venues}) Paul G. +Comba who described a method of implementing fast multipliers that do not require nested carry fixup operations. As an +interesting aside it seems that Paul Barrett describes a similar technique in his 1986 paper \cite{BARRETT} written five years before. -At the heart of the Comba technique is once again the long-hand algorithm. Except in this case a slight -twist is placed on how the columns of the result are produced. In the standard long-hand algorithm rows of products -are produced then added together to form the final result. In the baseline algorithm the columns are added together -after each iteration to get the result instantaneously. +At the heart of the Comba technique is once again the long-hand algorithm. Except in this case a slight +twist is placed on how the columns of the result are produced. In the standard long-hand algorithm rows of products +are produced then added together to form the final result. In the baseline algorithm the columns are added together +after each iteration to get the result instantaneously. -In the Comba algorithm the columns of the result are produced entirely independently of each other. That is at -the $O(n^2)$ level a simple multiplication and addition step is performed. The carries of the columns are propagated -after the nested loop to reduce the amount of work requiored. Succintly the first step of the algorithm is to compute -the product vector $\vec x$ as follows. +In the Comba algorithm the columns of the result are produced entirely independently of each other. That is at +the $O(n^2)$ level a simple multiplication and addition step is performed. The carries of the columns are propagated +after the nested loop to reduce the amount of work requiored. Succintly the first step of the algorithm is to compute +the product vector $\vec x$ as follows. \begin{equation} \vec x_n = \sum_{i+j = n} a_ib_j, \forall n \in \lbrace 0, 1, 2, \ldots, i + j \rbrace \end{equation} Where $\vec x_n$ is the $n'th$ column of the output vector. Consider the following example which computes the vector $\vec x$ for the multiplication -of $576$ and $241$. +of $576$ and $241$. \newpage\begin{figure}[here] \begin{small} @@ -2663,15 +3762,15 @@ of $576$ and $241$. \hline & & $1 \cdot 5 = 5$ & $1 \cdot 7 = 7$ & $1 \cdot 6 = 6$ & First pass \\ & $4 \cdot 5 = 20$ & $4 \cdot 7+5=33$ & $4 \cdot 6+7=31$ & 6 & Second pass \\ $2 \cdot 5 = 10$ & $2 \cdot 7 + 20 = 34$ & $2 \cdot 6+33=45$ & 31 & 6 & Third pass \\ -\hline 10 & 34 & 45 & 31 & 6 & Final Result \\ -\hline +\hline 10 & 34 & 45 & 31 & 6 & Final Result \\ +\hline \end{tabular} \end{center} \end{small} \caption{Comba Multiplication Diagram} \end{figure} -At this point the vector $x = \left < 10, 34, 45, 31, 6 \right >$ is the result of the first step of the Comba multipler. +At this point the vector $x = \left < 10, 34, 45, 31, 6 \right >$ is the result of the first step of the Comba multipler. Now the columns must be fixed by propagating the carry upwards. The resultant vector will have one extra dimension over the input vector which is congruent to adding a leading zero digit. @@ -2694,16 +3793,16 @@ congruent to adding a leading zero digit. \caption{Algorithm Comba Fixup} \end{figure} -With that algorithm and $k = 5$ and $\beta = 10$ the following vector is produced $\vec x= \left < 1, 3, 8, 8, 1, 6 \right >$. In this case +With that algorithm and $k = 5$ and $\beta = 10$ the following vector is produced $\vec x= \left < 1, 3, 8, 8, 1, 6 \right >$. In this case $241 \cdot 576$ is in fact $138816$ and the procedure succeeded. If the algorithm is correct and as will be demonstrated shortly more efficient than the baseline algorithm why not simply always use this algorithm? \subsubsection{Column Weight.} -At the nested $O(n^2)$ level the Comba method adds the product of two single precision variables to each column of the output +At the nested $O(n^2)$ level the Comba method adds the product of two single precision variables to each column of the output independently. A serious obstacle is if the carry is lost, due to lack of precision before the algorithm has a chance to fix the carries. For example, in the multiplication of two three-digit numbers the third column of output will be the sum of three single precision multiplications. If the precision of the accumulator for the output digits is less then $3 \cdot (\beta - 1)^2$ then -an overflow can occur and the carry information will be lost. For any $m$ and $n$ digit inputs the maximum weight of any column is +an overflow can occur and the carry information will be lost. For any $m$ and $n$ digit inputs the maximum weight of any column is min$(m, n)$ which is fairly obvious. The maximum number of terms in any column of a product is known as the ``column weight'' and strictly governs when the algorithm can be used. Recall @@ -2714,7 +3813,7 @@ two quantities we must not violate the following k \cdot \left (\beta - 1 \right )^2 < 2^{\alpha} \end{equation} -Which reduces to +Which reduces to \begin{equation} k \cdot \left ( \beta^2 - 2\beta + 1 \right ) < 2^{\alpha} @@ -2727,9 +3826,9 @@ found. k < {{2^{\alpha}} \over {\left (2^{2\rho} - 2^{\rho + 1} + 1 \right )}} \end{equation} -The defaults for LibTomMath are $\beta = 2^{28}$ and $\alpha = 2^{64}$ which means that $k$ is bounded by $k < 257$. In this configuration -the smaller input may not have more than $256$ digits if the Comba method is to be used. This is quite satisfactory for most applications since -$256$ digits would allow for numbers in the range of $0 \le x < 2^{7168}$ which, is much larger than most public key cryptographic algorithms require. +The defaults for LibTomMath are $\beta = 2^{28}$ and $\alpha = 2^{64}$ which means that $k$ is bounded by $k < 257$. In this configuration +the smaller input may not have more than $256$ digits if the Comba method is to be used. This is quite satisfactory for most applications since +$256$ digits would allow for numbers in the range of $0 \le x < 2^{7168}$ which, is much larger than most public key cryptographic algorithms require. \newpage\begin{figure}[!here] \begin{small} @@ -2775,79 +3874,167 @@ Place an array of \textbf{MP\_WARRAY} single precision digits named $W$ on the s \textbf{Algorithm fast\_s\_mp\_mul\_digs.} This algorithm performs the unsigned multiplication of $a$ and $b$ using the Comba method limited to $digs$ digits of precision. -The outer loop of this algorithm is more complicated than that of the baseline multiplier. This is because on the inside of the +The outer loop of this algorithm is more complicated than that of the baseline multiplier. This is because on the inside of the loop we want to produce one column per pass. This allows the accumulator $\_ \hat W$ to be placed in CPU registers and reduce the memory bandwidth to two \textbf{mp\_digit} reads per iteration. The $ty$ variable is set to the minimum count of $ix$ or the number of digits in $b$. That way if $a$ has more digits than $b$ this will be limited to $b.used - 1$. The $tx$ variable is set to the to the distance past $b.used$ the variable -$ix$ is. This is used for the immediately subsequent statement where we find $iy$. +$ix$ is. This is used for the immediately subsequent statement where we find $iy$. The variable $iy$ is the minimum digits we can read from either $a$ or $b$ before running out. Computing one column at a time means we have to scan one integer upwards and the other downwards. $a$ starts at $tx$ and $b$ starts at $ty$. In each -pass we are producing the $ix$'th output column and we note that $tx + ty = ix$. As we move $tx$ upwards we have to -move $ty$ downards so the equality remains valid. The $iy$ variable is the number of iterations until +pass we are producing the $ix$'th output column and we note that $tx + ty = ix$. As we move $tx$ upwards we have to +move $ty$ downards so the equality remains valid. The $iy$ variable is the number of iterations until $tx \ge a.used$ or $ty < 0$ occurs. After every inner pass we store the lower half of the accumulator into $W_{ix}$ and then propagate the carry of the accumulator into the next round by dividing $\_ \hat W$ by $\beta$. -To measure the benefits of the Comba method over the baseline method consider the number of operations that are required. If the -cost in terms of time of a multiply and addition is $p$ and the cost of a carry propagation is $q$ then a baseline multiplication would require -$O \left ((p + q)n^2 \right )$ time to multiply two $n$-digit numbers. The Comba method requires only $O(pn^2 + qn)$ time, however in practice, +To measure the benefits of the Comba method over the baseline method consider the number of operations that are required. If the +cost in terms of time of a multiply and addition is $p$ and the cost of a carry propagation is $q$ then a baseline multiplication would require +$O \left ((p + q)n^2 \right )$ time to multiply two $n$-digit numbers. The Comba method requires only $O(pn^2 + qn)$ time, however in practice, the speed increase is actually much more. With $O(n)$ space the algorithm can be reduced to $O(pn + qn)$ time by implementing the $n$ multiply -and addition operations in the nested loop in parallel. +and addition operations in the nested loop in parallel. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_fast\_s\_mp\_mul\_digs.c \vspace{-3mm} \begin{alltt} +016 +017 /* Fast (comba) multiplier +018 * +019 * This is the fast column-array [comba] multiplier. It is +020 * designed to compute the columns of the product first +021 * then handle the carries afterwards. This has the effect +022 * of making the nested loops that compute the columns very +023 * simple and schedulable on super-scalar processors. +024 * +025 * This has been modified to produce a variable number of +026 * digits of output so if say only a half-product is required +027 * you don't have to compute the upper half (a feature +028 * required for fast Barrett reduction). +029 * +030 * Based on Algorithm 14.12 on pp.595 of HAC. +031 * +032 */ +033 int fast_s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs) +034 \{ +035 int olduse, res, pa, ix, iz; +036 mp_digit W[MP_WARRAY]; +037 mp_word _W; +038 +039 /* grow the destination as required */ +040 if (c->alloc < digs) \{ +041 if ((res = mp_grow (c, digs)) != MP_OKAY) \{ +042 return res; +043 \} +044 \} +045 +046 /* number of output digits to produce */ +047 pa = MIN(digs, a->used + b->used); +048 +049 /* clear the carry */ +050 _W = 0; +051 for (ix = 0; ix < pa; ix++) \{ +052 int tx, ty; +053 int iy; +054 mp_digit *tmpx, *tmpy; +055 +056 /* get offsets into the two bignums */ +057 ty = MIN(b->used-1, ix); +058 tx = ix - ty; +059 +060 /* setup temp aliases */ +061 tmpx = a->dp + tx; +062 tmpy = b->dp + ty; +063 +064 /* this is the number of times the loop will iterrate, essentially +065 while (tx++ < a->used && ty-- >= 0) \{ ... \} +066 */ +067 iy = MIN(a->used-tx, ty+1); +068 +069 /* execute loop */ +070 for (iz = 0; iz < iy; ++iz) \{ +071 _W += ((mp_word)*tmpx++)*((mp_word)*tmpy--); +072 +073 \} +074 +075 /* store term */ +076 W[ix] = ((mp_digit)_W) & MP_MASK; +077 +078 /* make next carry */ +079 _W = _W >> ((mp_word)DIGIT_BIT); +080 \} +081 +082 /* setup dest */ +083 olduse = c->used; +084 c->used = pa; +085 +086 \{ +087 mp_digit *tmpc; +088 tmpc = c->dp; +089 for (ix = 0; ix < (pa + 1); ix++) \{ +090 /* now extract the previous digit [below the carry] */ +091 *tmpc++ = W[ix]; +092 \} +093 +094 /* clear unused digits [that existed in the old copy of c] */ +095 for (; ix < olduse; ix++) \{ +096 *tmpc++ = 0; +097 \} +098 \} +099 mp_clamp (c); +100 return MP_OKAY; +101 \} +102 #endif +103 \end{alltt} \end{small} -As per the pseudo--code we first calculate $pa$ (line 48) as the number of digits to output. Next we begin the outer loop -to produce the individual columns of the product. We use the two aliases $tmpx$ and $tmpy$ (lines 62, 63) to point -inside the two multiplicands quickly. +As per the pseudo--code we first calculate $pa$ (line 47) as the number of digits to output. Next we begin the outer loop +to produce the individual columns of the product. We use the two aliases $tmpx$ and $tmpy$ (lines 61, 62) to point +inside the two multiplicands quickly. -The inner loop (lines 71 to 74) of this implementation is where the tradeoff come into play. Originally this comba -implementation was ``row--major'' which means it adds to each of the columns in each pass. After the outer loop it would then fix -the carries. This was very fast except it had an annoying drawback. You had to read a mp\_word and two mp\_digits and write -one mp\_word per iteration. On processors such as the Athlon XP and P4 this did not matter much since the cache bandwidth -is very high and it can keep the ALU fed with data. It did, however, matter on older and embedded cpus where cache is often -slower and also often doesn't exist. This new algorithm only performs two reads per iteration under the assumption that the +The inner loop (lines 70 to 73) of this implementation is where the tradeoff come into play. Originally this comba +implementation was ``row--major'' which means it adds to each of the columns in each pass. After the outer loop it would then fix +the carries. This was very fast except it had an annoying drawback. You had to read a mp\_word and two mp\_digits and write +one mp\_word per iteration. On processors such as the Athlon XP and P4 this did not matter much since the cache bandwidth +is very high and it can keep the ALU fed with data. It did, however, matter on older and embedded cpus where cache is often +slower and also often doesn't exist. This new algorithm only performs two reads per iteration under the assumption that the compiler has aliased $\_ \hat W$ to a CPU register. -After the inner loop we store the current accumulator in $W$ and shift $\_ \hat W$ (lines 77, 80) to forward it as -a carry for the next pass. After the outer loop we use the final carry (line 77) as the last digit of the product. +After the inner loop we store the current accumulator in $W$ and shift $\_ \hat W$ (lines 76, 79) to forward it as +a carry for the next pass. After the outer loop we use the final carry (line 76) as the last digit of the product. \subsection{Polynomial Basis Multiplication} To break the $O(n^2)$ barrier in multiplication requires a completely different look at integer multiplication. In the following algorithms -the use of polynomial basis representation for two integers $a$ and $b$ as $f(x) = \sum_{i=0}^{n} a_i x^i$ and +the use of polynomial basis representation for two integers $a$ and $b$ as $f(x) = \sum_{i=0}^{n} a_i x^i$ and $g(x) = \sum_{i=0}^{n} b_i x^i$ respectively, is required. In this system both $f(x)$ and $g(x)$ have $n + 1$ terms and are of the $n$'th degree. - + The product $a \cdot b \equiv f(x)g(x)$ is the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$. The coefficients $w_i$ will directly yield the desired product when $\beta$ is substituted for $x$. The direct solution to solve for the $2n + 1$ coefficients requires $O(n^2)$ time and would in practice be slower than the Comba technique. -However, numerical analysis theory indicates that only $2n + 1$ distinct points in $W(x)$ are required to determine the values of the $2n + 1$ unknown -coefficients. This means by finding $\zeta_y = W(y)$ for $2n + 1$ small values of $y$ the coefficients of $W(x)$ can be found with -Gaussian elimination. This technique is also occasionally refered to as the \textit{interpolation technique} (\textit{references please...}) since in -effect an interpolation based on $2n + 1$ points will yield a polynomial equivalent to $W(x)$. +However, numerical analysis theory indicates that only $2n + 1$ distinct points in $W(x)$ are required to determine the values of the $2n + 1$ unknown +coefficients. This means by finding $\zeta_y = W(y)$ for $2n + 1$ small values of $y$ the coefficients of $W(x)$ can be found with +Gaussian elimination. This technique is also occasionally refered to as the \textit{interpolation technique} (\textit{references please...}) since in +effect an interpolation based on $2n + 1$ points will yield a polynomial equivalent to $W(x)$. -The coefficients of the polynomial $W(x)$ are unknown which makes finding $W(y)$ for any value of $y$ impossible. However, since -$W(x) = f(x)g(x)$ the equivalent $\zeta_y = f(y) g(y)$ can be used in its place. The benefit of this technique stems from the -fact that $f(y)$ and $g(y)$ are much smaller than either $a$ or $b$ respectively. As a result finding the $2n + 1$ relations required +The coefficients of the polynomial $W(x)$ are unknown which makes finding $W(y)$ for any value of $y$ impossible. However, since +$W(x) = f(x)g(x)$ the equivalent $\zeta_y = f(y) g(y)$ can be used in its place. The benefit of this technique stems from the +fact that $f(y)$ and $g(y)$ are much smaller than either $a$ or $b$ respectively. As a result finding the $2n + 1$ relations required by multiplying $f(y)g(y)$ involves multiplying integers that are much smaller than either of the inputs. When picking points to gather relations there are always three obvious points to choose, $y = 0, 1$ and $ \infty$. The $\zeta_0$ term -is simply the product $W(0) = w_0 = a_0 \cdot b_0$. The $\zeta_1$ term is the product +is simply the product $W(0) = w_0 = a_0 \cdot b_0$. The $\zeta_1$ term is the product $W(1) = \left (\sum_{i = 0}^{n} a_i \right ) \left (\sum_{i = 0}^{n} b_i \right )$. The third point $\zeta_{\infty}$ is less obvious but rather -simple to explain. The $2n + 1$'th coefficient of $W(x)$ is numerically equivalent to the most significant column in an integer multiplication. -The point at $\infty$ is used symbolically to represent the most significant column, that is $W(\infty) = w_{2n} = a_nb_n$. Note that the +simple to explain. The $2n + 1$'th coefficient of $W(x)$ is numerically equivalent to the most significant column in an integer multiplication. +The point at $\infty$ is used symbolically to represent the most significant column, that is $W(\infty) = w_{2n} = a_nb_n$. Note that the points at $y = 0$ and $\infty$ yield the coefficients $w_0$ and $w_{2n}$ directly. -If more points are required they should be of small values and powers of two such as $2^q$ and the related \textit{mirror points} -$\left (2^q \right )^{2n} \cdot \zeta_{2^{-q}}$ for small values of $q$. The term ``mirror point'' stems from the fact that +If more points are required they should be of small values and powers of two such as $2^q$ and the related \textit{mirror points} +$\left (2^q \right )^{2n} \cdot \zeta_{2^{-q}}$ for small values of $q$. The term ``mirror point'' stems from the fact that $\left (2^q \right )^{2n} \cdot \zeta_{2^{-q}}$ can be calculated in the exact opposite fashion as $\zeta_{2^q}$. For example, when $n = 2$ and $q = 1$ then following two equations are equivalent to the point $\zeta_{2}$ and its mirror. @@ -2857,10 +4044,10 @@ example, when $n = 2$ and $q = 1$ then following two equations are equivalent to \end{eqnarray} Using such points will allow the values of $f(y)$ and $g(y)$ to be independently calculated using only left shifts. For example, when $n = 2$ the -polynomial $f(2^q)$ is equal to $2^q((2^qa_2) + a_1) + a_0$. This technique of polynomial representation is known as Horner's method. +polynomial $f(2^q)$ is equal to $2^q((2^qa_2) + a_1) + a_0$. This technique of polynomial representation is known as Horner's method. -As a general rule of the algorithm when the inputs are split into $n$ parts each there are $2n - 1$ multiplications. Each multiplication is of -multiplicands that have $n$ times fewer digits than the inputs. The asymptotic running time of this algorithm is +As a general rule of the algorithm when the inputs are split into $n$ parts each there are $2n - 1$ multiplications. Each multiplication is of +multiplicands that have $n$ times fewer digits than the inputs. The asymptotic running time of this algorithm is $O \left ( k^{lg_n(2n - 1)} \right )$ for $k$ digit inputs (\textit{assuming they have the same number of digits}). Figure~\ref{fig:exponent} summarizes the exponents for various values of $n$. @@ -2885,23 +4072,23 @@ summarizes the exponents for various values of $n$. At first it may seem like a good idea to choose $n = 1000$ since the exponent is approximately $1.1$. However, the overhead of solving for the 2001 terms of $W(x)$ will certainly consume any savings the algorithm could offer for all but exceedingly large -numbers. +numbers. \subsubsection{Cutoff Point} -The polynomial basis multiplication algorithms all require fewer single precision multiplications than a straight Comba approach. However, +The polynomial basis multiplication algorithms all require fewer single precision multiplications than a straight Comba approach. However, the algorithms incur an overhead (\textit{at the $O(n)$ work level}) since they require a system of equations to be solved. This makes the polynomial basis approach more costly to use with small inputs. -Let $m$ represent the number of digits in the multiplicands (\textit{assume both multiplicands have the same number of digits}). There exists a -point $y$ such that when $m < y$ the polynomial basis algorithms are more costly than Comba, when $m = y$ they are roughly the same cost and -when $m > y$ the Comba methods are slower than the polynomial basis algorithms. +Let $m$ represent the number of digits in the multiplicands (\textit{assume both multiplicands have the same number of digits}). There exists a +point $y$ such that when $m < y$ the polynomial basis algorithms are more costly than Comba, when $m = y$ they are roughly the same cost and +when $m > y$ the Comba methods are slower than the polynomial basis algorithms. The exact location of $y$ depends on several key architectural elements of the computer platform in question. \begin{enumerate} \item The ratio of clock cycles for single precision multiplication versus other simpler operations such as addition, shifting, etc. For example on the AMD Athlon the ratio is roughly $17 : 1$ while on the Intel P4 it is $29 : 1$. The higher the ratio in favour of multiplication the lower -the cutoff point $y$ will be. +the cutoff point $y$ will be. \item The complexity of the linear system of equations (\textit{for the coefficients of $W(x)$}) is. Generally speaking as the number of splits grows the complexity grows substantially. Ideally solving the system will only involve addition, subtraction and shifting of integers. This @@ -2914,11 +4101,11 @@ influence over the cutoff point. A clean cutoff point separation occurs when a point $y$ is found such that all of the cutoff point conditions are met. For example, if the point is too low then there will be values of $m$ such that $m > y$ and the Comba method is still faster. Finding the cutoff points is fairly simple when -a high resolution timer is available. +a high resolution timer is available. \subsection{Karatsuba Multiplication} Karatsuba \cite{KARA} multiplication when originally proposed in 1962 was among the first set of algorithms to break the $O(n^2)$ barrier for -general purpose multiplication. Given two polynomial basis representations $f(x) = ax + b$ and $g(x) = cx + d$, Karatsuba proved with +general purpose multiplication. Given two polynomial basis representations $f(x) = ax + b$ and $g(x) = cx + d$, Karatsuba proved with light algebra \cite{KARAP} that the following polynomial is equivalent to multiplication of the two integers the polynomials represent. \begin{equation} @@ -2926,8 +4113,8 @@ f(x) \cdot g(x) = acx^2 + ((a + b)(c + d) - (ac + bd))x + bd \end{equation} Using the observation that $ac$ and $bd$ could be re-used only three half sized multiplications would be required to produce the product. Applying -this algorithm recursively, the work factor becomes $O(n^{lg(3)})$ which is substantially better than the work factor $O(n^2)$ of the Comba technique. It turns -out what Karatsuba did not know or at least did not publish was that this is simply polynomial basis multiplication with the points +this algorithm recursively, the work factor becomes $O(n^{lg(3)})$ which is substantially better than the work factor $O(n^2)$ of the Comba technique. It turns +out what Karatsuba did not know or at least did not publish was that this is simply polynomial basis multiplication with the points $\zeta_0$, $\zeta_{\infty}$ and $\zeta_{1}$. Consider the resultant system of equations. \begin{center} @@ -2940,7 +4127,7 @@ $\zeta_{\infty}$ & $=$ & $w_2$ & & & & \\ By adding the first and last equation to the equation in the middle the term $w_1$ can be isolated and all three coefficients solved for. The simplicity of this system of equations has made Karatsuba fairly popular. In fact the cutoff point is often fairly low\footnote{With LibTomMath 0.18 it is 70 and 109 digits for the Intel P4 and AMD Athlon respectively.} -making it an ideal algorithm to speed up certain public key cryptosystems such as RSA and Diffie-Hellman. +making it an ideal algorithm to speed up certain public key cryptosystems such as RSA and Diffie-Hellman. \newpage\begin{figure}[!here] \begin{small} @@ -2978,7 +4165,7 @@ Calculate the final product. \\ 18. $c \leftarrow t1 + x1y1$ \\ 19. Clear all of the temporary variables. \\ 20. Return(\textit{MP\_OKAY}).\\ -\hline +\hline \end{tabular} \end{center} \end{small} @@ -2987,13 +4174,13 @@ Calculate the final product. \\ \textbf{Algorithm mp\_karatsuba\_mul.} This algorithm computes the unsigned product of two inputs using the Karatsuba multiplication algorithm. It is loosely based on the description -from Knuth \cite[pp. 294-295]{TAOCPV2}. +from Knuth \cite[pp. 294-295]{TAOCPV2}. \index{radix point} In order to split the two inputs into their respective halves, a suitable \textit{radix point} must be chosen. The radix point chosen must -be used for both of the inputs meaning that it must be smaller than the smallest input. Step 3 chooses the radix point $B$ as half of the -smallest input \textbf{used} count. After the radix point is chosen the inputs are split into lower and upper halves. Step 4 and 5 -compute the lower halves. Step 6 and 7 computer the upper halves. +be used for both of the inputs meaning that it must be smaller than the smallest input. Step 3 chooses the radix point $B$ as half of the +smallest input \textbf{used} count. After the radix point is chosen the inputs are split into lower and upper halves. Step 4 and 5 +compute the lower halves. Step 6 and 7 computer the upper halves. After the halves have been computed the three intermediate half-size products must be computed. Step 8 and 9 compute the trivial products $x0 \cdot y0$ and $x1 \cdot y1$. The mp\_int $x0$ is used as a temporary variable after $x1 + x0$ has been computed. By using $x0$ instead @@ -3005,34 +4192,182 @@ The remaining steps 13 through 18 compute the Karatsuba polynomial through a var \hspace{-5.1mm}{\bf File}: bn\_mp\_karatsuba\_mul.c \vspace{-3mm} \begin{alltt} +016 +017 /* c = |a| * |b| using Karatsuba Multiplication using +018 * three half size multiplications +019 * +020 * Let B represent the radix [e.g. 2**DIGIT_BIT] and +021 * let n represent half of the number of digits in +022 * the min(a,b) +023 * +024 * a = a1 * B**n + a0 +025 * b = b1 * B**n + b0 +026 * +027 * Then, a * b => +028 a1b1 * B**2n + ((a1 + a0)(b1 + b0) - (a0b0 + a1b1)) * B + a0b0 +029 * +030 * Note that a1b1 and a0b0 are used twice and only need to be +031 * computed once. So in total three half size (half # of +032 * digit) multiplications are performed, a0b0, a1b1 and +033 * (a1+b1)(a0+b0) +034 * +035 * Note that a multiplication of half the digits requires +036 * 1/4th the number of single precision multiplications so in +037 * total after one call 25% of the single precision multiplications +038 * are saved. Note also that the call to mp_mul can end up back +039 * in this function if the a0, a1, b0, or b1 are above the threshold. +040 * This is known as divide-and-conquer and leads to the famous +041 * O(N**lg(3)) or O(N**1.584) work which is asymptopically lower than +042 * the standard O(N**2) that the baseline/comba methods use. +043 * Generally though the overhead of this method doesn't pay off +044 * until a certain size (N ~ 80) is reached. +045 */ +046 int mp_karatsuba_mul (mp_int * a, mp_int * b, mp_int * c) +047 \{ +048 mp_int x0, x1, y0, y1, t1, x0y0, x1y1; +049 int B, err; +050 +051 /* default the return code to an error */ +052 err = MP_MEM; +053 +054 /* min # of digits */ +055 B = MIN (a->used, b->used); +056 +057 /* now divide in two */ +058 B = B >> 1; +059 +060 /* init copy all the temps */ +061 if (mp_init_size (&x0, B) != MP_OKAY) +062 goto ERR; +063 if (mp_init_size (&x1, a->used - B) != MP_OKAY) +064 goto X0; +065 if (mp_init_size (&y0, B) != MP_OKAY) +066 goto X1; +067 if (mp_init_size (&y1, b->used - B) != MP_OKAY) +068 goto Y0; +069 +070 /* init temps */ +071 if (mp_init_size (&t1, B * 2) != MP_OKAY) +072 goto Y1; +073 if (mp_init_size (&x0y0, B * 2) != MP_OKAY) +074 goto T1; +075 if (mp_init_size (&x1y1, B * 2) != MP_OKAY) +076 goto X0Y0; +077 +078 /* now shift the digits */ +079 x0.used = y0.used = B; +080 x1.used = a->used - B; +081 y1.used = b->used - B; +082 +083 \{ +084 int x; +085 mp_digit *tmpa, *tmpb, *tmpx, *tmpy; +086 +087 /* we copy the digits directly instead of using higher level functions +088 * since we also need to shift the digits +089 */ +090 tmpa = a->dp; +091 tmpb = b->dp; +092 +093 tmpx = x0.dp; +094 tmpy = y0.dp; +095 for (x = 0; x < B; x++) \{ +096 *tmpx++ = *tmpa++; +097 *tmpy++ = *tmpb++; +098 \} +099 +100 tmpx = x1.dp; +101 for (x = B; x < a->used; x++) \{ +102 *tmpx++ = *tmpa++; +103 \} +104 +105 tmpy = y1.dp; +106 for (x = B; x < b->used; x++) \{ +107 *tmpy++ = *tmpb++; +108 \} +109 \} +110 +111 /* only need to clamp the lower words since by definition the +112 * upper words x1/y1 must have a known number of digits +113 */ +114 mp_clamp (&x0); +115 mp_clamp (&y0); +116 +117 /* now calc the products x0y0 and x1y1 */ +118 /* after this x0 is no longer required, free temp [x0==t2]! */ +119 if (mp_mul (&x0, &y0, &x0y0) != MP_OKAY) +120 goto X1Y1; /* x0y0 = x0*y0 */ +121 if (mp_mul (&x1, &y1, &x1y1) != MP_OKAY) +122 goto X1Y1; /* x1y1 = x1*y1 */ +123 +124 /* now calc x1+x0 and y1+y0 */ +125 if (s_mp_add (&x1, &x0, &t1) != MP_OKAY) +126 goto X1Y1; /* t1 = x1 - x0 */ +127 if (s_mp_add (&y1, &y0, &x0) != MP_OKAY) +128 goto X1Y1; /* t2 = y1 - y0 */ +129 if (mp_mul (&t1, &x0, &t1) != MP_OKAY) +130 goto X1Y1; /* t1 = (x1 + x0) * (y1 + y0) */ +131 +132 /* add x0y0 */ +133 if (mp_add (&x0y0, &x1y1, &x0) != MP_OKAY) +134 goto X1Y1; /* t2 = x0y0 + x1y1 */ +135 if (s_mp_sub (&t1, &x0, &t1) != MP_OKAY) +136 goto X1Y1; /* t1 = (x1+x0)*(y1+y0) - (x1y1 + x0y0) */ +137 +138 /* shift by B */ +139 if (mp_lshd (&t1, B) != MP_OKAY) +140 goto X1Y1; /* t1 = (x0y0 + x1y1 - (x1-x0)*(y1-y0))<<B */ +141 if (mp_lshd (&x1y1, B * 2) != MP_OKAY) +142 goto X1Y1; /* x1y1 = x1y1 << 2*B */ +143 +144 if (mp_add (&x0y0, &t1, &t1) != MP_OKAY) +145 goto X1Y1; /* t1 = x0y0 + t1 */ +146 if (mp_add (&t1, &x1y1, c) != MP_OKAY) +147 goto X1Y1; /* t1 = x0y0 + t1 + x1y1 */ +148 +149 /* Algorithm succeeded set the return code to MP_OKAY */ +150 err = MP_OKAY; +151 +152 X1Y1:mp_clear (&x1y1); +153 X0Y0:mp_clear (&x0y0); +154 T1:mp_clear (&t1); +155 Y1:mp_clear (&y1); +156 Y0:mp_clear (&y0); +157 X1:mp_clear (&x1); +158 X0:mp_clear (&x0); +159 ERR: +160 return err; +161 \} +162 #endif +163 \end{alltt} \end{small} The new coding element in this routine, not seen in previous routines, is the usage of goto statements. The conventional wisdom is that goto statements should be avoided. This is generally true, however when every single function call can fail, it makes sense -to handle error recovery with a single piece of code. Lines 62 to 76 handle initializing all of the temporary variables +to handle error recovery with a single piece of code. Lines 61 to 75 handle initializing all of the temporary variables required. Note how each of the if statements goes to a different label in case of failure. This allows the routine to correctly free only the temporaries that have been successfully allocated so far. -The temporary variables are all initialized using the mp\_init\_size routine since they are expected to be large. This saves the +The temporary variables are all initialized using the mp\_init\_size routine since they are expected to be large. This saves the additional reallocation that would have been necessary. Also $x0$, $x1$, $y0$ and $y1$ have to be able to hold at least their respective number of digits for the next section of code. The first algebraic portion of the algorithm is to split the two inputs into their halves. However, instead of using mp\_mod\_2d and mp\_rshd -to extract the halves, the respective code has been placed inline within the body of the function. To initialize the halves, the \textbf{used} and -\textbf{sign} members are copied first. The first for loop on line 96 copies the lower halves. Since they are both the same magnitude it -is simpler to calculate both lower halves in a single loop. The for loop on lines 102 and 107 calculate the upper halves $x1$ and +to extract the halves, the respective code has been placed inline within the body of the function. To initialize the halves, the \textbf{used} and +\textbf{sign} members are copied first. The first for loop on line 101 copies the lower halves. Since they are both the same magnitude it +is simpler to calculate both lower halves in a single loop. The for loop on lines 106 and 106 calculate the upper halves $x1$ and $y1$ respectively. By inlining the calculation of the halves, the Karatsuba multiplier has a slightly lower overhead and can be used for smaller magnitude inputs. -When line 151 is reached, the algorithm has completed succesfully. The ``error status'' variable $err$ is set to \textbf{MP\_OKAY} so that -the same code that handles errors can be used to clear the temporary variables and return. +When line 150 is reached, the algorithm has completed succesfully. The ``error status'' variable $err$ is set to \textbf{MP\_OKAY} so that +the same code that handles errors can be used to clear the temporary variables and return. \subsection{Toom-Cook $3$-Way Multiplication} -Toom-Cook $3$-Way \cite{TOOM} multiplication is essentially the polynomial basis algorithm for $n = 2$ except that the points are -chosen such that $\zeta$ is easy to compute and the resulting system of equations easy to reduce. Here, the points $\zeta_{0}$, -$16 \cdot \zeta_{1 \over 2}$, $\zeta_1$, $\zeta_2$ and $\zeta_{\infty}$ make up the five required points to solve for the coefficients +Toom-Cook $3$-Way \cite{TOOM} multiplication is essentially the polynomial basis algorithm for $n = 2$ except that the points are +chosen such that $\zeta$ is easy to compute and the resulting system of equations easy to reduce. Here, the points $\zeta_{0}$, +$16 \cdot \zeta_{1 \over 2}$, $\zeta_1$, $\zeta_2$ and $\zeta_{\infty}$ make up the five required points to solve for the coefficients of the $W(x)$. With the five relations that Toom-Cook specifies, the following system of equations is formed. @@ -3050,7 +4385,7 @@ $\zeta_{\infty}$ & $=$ & $1w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ A trivial solution to this matrix requires $12$ subtractions, two multiplications by a small power of two, two divisions by a small power of two, two divisions by three and one multiplication by three. All of these $19$ sub-operations require less than quadratic time, meaning that the algorithm can be faster than a baseline multiplication. However, the greater complexity of this algorithm places the cutoff point -(\textbf{TOOM\_MUL\_CUTOFF}) where Toom-Cook becomes more efficient much higher than the Karatsuba cutoff point. +(\textbf{TOOM\_MUL\_CUTOFF}) where Toom-Cook becomes more efficient much higher than the Karatsuba cutoff point. \begin{figure}[!here] \begin{small} @@ -3120,7 +4455,7 @@ Now substitute $\beta^k$ for $x$ by shifting $w_0, w_1, ..., w_4$. \\ \end{figure} \textbf{Algorithm mp\_toom\_mul.} -This algorithm computes the product of two mp\_int variables $a$ and $b$ using the Toom-Cook approach. Compared to the Karatsuba multiplication, this +This algorithm computes the product of two mp\_int variables $a$ and $b$ using the Toom-Cook approach. Compared to the Karatsuba multiplication, this algorithm has a lower asymptotic running time of approximately $O(n^{1.464})$ but at an obvious cost in overhead. In this description, several statements have been compounded to save space. The intention is that the statements are executed from left to right across any given step. @@ -3132,39 +4467,306 @@ The first two relations $w_0$ and $w_4$ are the points $\zeta_{0}$ and $\zeta_{\ to the points $16 \cdot \zeta_{1 \over 2}, \zeta_{2}$ and $\zeta_{1}$ respectively. These are found using logical shifts to independently find $f(y)$ and $g(y)$ which significantly speeds up the algorithm. -After the five relations $w_0, w_1, \ldots, w_4$ have been computed, the system they represent must be solved in order for the unknown coefficients +After the five relations $w_0, w_1, \ldots, w_4$ have been computed, the system they represent must be solved in order for the unknown coefficients $w_1, w_2$ and $w_3$ to be isolated. The steps 18 through 25 perform the system reduction required as previously described. Each step of the reduction represents the comparable matrix operation that would be performed had this been performed by pencil. For example, step 18 indicates -that row $1$ must be subtracted from row $4$ and simultaneously row $0$ subtracted from row $3$. +that row $1$ must be subtracted from row $4$ and simultaneously row $0$ subtracted from row $3$. -Once the coeffients have been isolated, the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$ is known. By substituting $\beta^{k}$ for $x$, the integer +Once the coeffients have been isolated, the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$ is known. By substituting $\beta^{k}$ for $x$, the integer result $a \cdot b$ is produced. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_toom\_mul.c \vspace{-3mm} \begin{alltt} +016 +017 /* multiplication using the Toom-Cook 3-way algorithm +018 * +019 * Much more complicated than Karatsuba but has a lower +020 * asymptotic running time of O(N**1.464). This algorithm is +021 * only particularly useful on VERY large inputs +022 * (we're talking 1000s of digits here...). +023 */ +024 int mp_toom_mul(mp_int *a, mp_int *b, mp_int *c) +025 \{ +026 mp_int w0, w1, w2, w3, w4, tmp1, tmp2, a0, a1, a2, b0, b1, b2; +027 int res, B; +028 +029 /* init temps */ +030 if ((res = mp_init_multi(&w0, &w1, &w2, &w3, &w4, +031 &a0, &a1, &a2, &b0, &b1, +032 &b2, &tmp1, &tmp2, NULL)) != MP_OKAY) \{ +033 return res; +034 \} +035 +036 /* B */ +037 B = MIN(a->used, b->used) / 3; +038 +039 /* a = a2 * B**2 + a1 * B + a0 */ +040 if ((res = mp_mod_2d(a, DIGIT_BIT * B, &a0)) != MP_OKAY) \{ +041 goto ERR; +042 \} +043 +044 if ((res = mp_copy(a, &a1)) != MP_OKAY) \{ +045 goto ERR; +046 \} +047 mp_rshd(&a1, B); +048 if ((res = mp_mod_2d(&a1, DIGIT_BIT * B, &a1)) != MP_OKAY) \{ +049 goto ERR; +050 \} +051 +052 if ((res = mp_copy(a, &a2)) != MP_OKAY) \{ +053 goto ERR; +054 \} +055 mp_rshd(&a2, B*2); +056 +057 /* b = b2 * B**2 + b1 * B + b0 */ +058 if ((res = mp_mod_2d(b, DIGIT_BIT * B, &b0)) != MP_OKAY) \{ +059 goto ERR; +060 \} +061 +062 if ((res = mp_copy(b, &b1)) != MP_OKAY) \{ +063 goto ERR; +064 \} +065 mp_rshd(&b1, B); +066 (void)mp_mod_2d(&b1, DIGIT_BIT * B, &b1); +067 +068 if ((res = mp_copy(b, &b2)) != MP_OKAY) \{ +069 goto ERR; +070 \} +071 mp_rshd(&b2, B*2); +072 +073 /* w0 = a0*b0 */ +074 if ((res = mp_mul(&a0, &b0, &w0)) != MP_OKAY) \{ +075 goto ERR; +076 \} +077 +078 /* w4 = a2 * b2 */ +079 if ((res = mp_mul(&a2, &b2, &w4)) != MP_OKAY) \{ +080 goto ERR; +081 \} +082 +083 /* w1 = (a2 + 2(a1 + 2a0))(b2 + 2(b1 + 2b0)) */ +084 if ((res = mp_mul_2(&a0, &tmp1)) != MP_OKAY) \{ +085 goto ERR; +086 \} +087 if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) \{ +088 goto ERR; +089 \} +090 if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) \{ +091 goto ERR; +092 \} +093 if ((res = mp_add(&tmp1, &a2, &tmp1)) != MP_OKAY) \{ +094 goto ERR; +095 \} +096 +097 if ((res = mp_mul_2(&b0, &tmp2)) != MP_OKAY) \{ +098 goto ERR; +099 \} +100 if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) \{ +101 goto ERR; +102 \} +103 if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) \{ +104 goto ERR; +105 \} +106 if ((res = mp_add(&tmp2, &b2, &tmp2)) != MP_OKAY) \{ +107 goto ERR; +108 \} +109 +110 if ((res = mp_mul(&tmp1, &tmp2, &w1)) != MP_OKAY) \{ +111 goto ERR; +112 \} +113 +114 /* w3 = (a0 + 2(a1 + 2a2))(b0 + 2(b1 + 2b2)) */ +115 if ((res = mp_mul_2(&a2, &tmp1)) != MP_OKAY) \{ +116 goto ERR; +117 \} +118 if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) \{ +119 goto ERR; +120 \} +121 if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) \{ +122 goto ERR; +123 \} +124 if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) \{ +125 goto ERR; +126 \} +127 +128 if ((res = mp_mul_2(&b2, &tmp2)) != MP_OKAY) \{ +129 goto ERR; +130 \} +131 if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) \{ +132 goto ERR; +133 \} +134 if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) \{ +135 goto ERR; +136 \} +137 if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) \{ +138 goto ERR; +139 \} +140 +141 if ((res = mp_mul(&tmp1, &tmp2, &w3)) != MP_OKAY) \{ +142 goto ERR; +143 \} +144 +145 +146 /* w2 = (a2 + a1 + a0)(b2 + b1 + b0) */ +147 if ((res = mp_add(&a2, &a1, &tmp1)) != MP_OKAY) \{ +148 goto ERR; +149 \} +150 if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) \{ +151 goto ERR; +152 \} +153 if ((res = mp_add(&b2, &b1, &tmp2)) != MP_OKAY) \{ +154 goto ERR; +155 \} +156 if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) \{ +157 goto ERR; +158 \} +159 if ((res = mp_mul(&tmp1, &tmp2, &w2)) != MP_OKAY) \{ +160 goto ERR; +161 \} +162 +163 /* now solve the matrix +164 +165 0 0 0 0 1 +166 1 2 4 8 16 +167 1 1 1 1 1 +168 16 8 4 2 1 +169 1 0 0 0 0 +170 +171 using 12 subtractions, 4 shifts, +172 2 small divisions and 1 small multiplication +173 */ +174 +175 /* r1 - r4 */ +176 if ((res = mp_sub(&w1, &w4, &w1)) != MP_OKAY) \{ +177 goto ERR; +178 \} +179 /* r3 - r0 */ +180 if ((res = mp_sub(&w3, &w0, &w3)) != MP_OKAY) \{ +181 goto ERR; +182 \} +183 /* r1/2 */ +184 if ((res = mp_div_2(&w1, &w1)) != MP_OKAY) \{ +185 goto ERR; +186 \} +187 /* r3/2 */ +188 if ((res = mp_div_2(&w3, &w3)) != MP_OKAY) \{ +189 goto ERR; +190 \} +191 /* r2 - r0 - r4 */ +192 if ((res = mp_sub(&w2, &w0, &w2)) != MP_OKAY) \{ +193 goto ERR; +194 \} +195 if ((res = mp_sub(&w2, &w4, &w2)) != MP_OKAY) \{ +196 goto ERR; +197 \} +198 /* r1 - r2 */ +199 if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) \{ +200 goto ERR; +201 \} +202 /* r3 - r2 */ +203 if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) \{ +204 goto ERR; +205 \} +206 /* r1 - 8r0 */ +207 if ((res = mp_mul_2d(&w0, 3, &tmp1)) != MP_OKAY) \{ +208 goto ERR; +209 \} +210 if ((res = mp_sub(&w1, &tmp1, &w1)) != MP_OKAY) \{ +211 goto ERR; +212 \} +213 /* r3 - 8r4 */ +214 if ((res = mp_mul_2d(&w4, 3, &tmp1)) != MP_OKAY) \{ +215 goto ERR; +216 \} +217 if ((res = mp_sub(&w3, &tmp1, &w3)) != MP_OKAY) \{ +218 goto ERR; +219 \} +220 /* 3r2 - r1 - r3 */ +221 if ((res = mp_mul_d(&w2, 3, &w2)) != MP_OKAY) \{ +222 goto ERR; +223 \} +224 if ((res = mp_sub(&w2, &w1, &w2)) != MP_OKAY) \{ +225 goto ERR; +226 \} +227 if ((res = mp_sub(&w2, &w3, &w2)) != MP_OKAY) \{ +228 goto ERR; +229 \} +230 /* r1 - r2 */ +231 if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) \{ +232 goto ERR; +233 \} +234 /* r3 - r2 */ +235 if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) \{ +236 goto ERR; +237 \} +238 /* r1/3 */ +239 if ((res = mp_div_3(&w1, &w1, NULL)) != MP_OKAY) \{ +240 goto ERR; +241 \} +242 /* r3/3 */ +243 if ((res = mp_div_3(&w3, &w3, NULL)) != MP_OKAY) \{ +244 goto ERR; +245 \} +246 +247 /* at this point shift W[n] by B*n */ +248 if ((res = mp_lshd(&w1, 1*B)) != MP_OKAY) \{ +249 goto ERR; +250 \} +251 if ((res = mp_lshd(&w2, 2*B)) != MP_OKAY) \{ +252 goto ERR; +253 \} +254 if ((res = mp_lshd(&w3, 3*B)) != MP_OKAY) \{ +255 goto ERR; +256 \} +257 if ((res = mp_lshd(&w4, 4*B)) != MP_OKAY) \{ +258 goto ERR; +259 \} +260 +261 if ((res = mp_add(&w0, &w1, c)) != MP_OKAY) \{ +262 goto ERR; +263 \} +264 if ((res = mp_add(&w2, &w3, &tmp1)) != MP_OKAY) \{ +265 goto ERR; +266 \} +267 if ((res = mp_add(&w4, &tmp1, &tmp1)) != MP_OKAY) \{ +268 goto ERR; +269 \} +270 if ((res = mp_add(&tmp1, c, c)) != MP_OKAY) \{ +271 goto ERR; +272 \} +273 +274 ERR: +275 mp_clear_multi(&w0, &w1, &w2, &w3, &w4, +276 &a0, &a1, &a2, &b0, &b1, +277 &b2, &tmp1, &tmp2, NULL); +278 return res; +279 \} +280 +281 #endif +282 \end{alltt} \end{small} -The first obvious thing to note is that this algorithm is complicated. The complexity is worth it if you are multiplying very +The first obvious thing to note is that this algorithm is complicated. The complexity is worth it if you are multiplying very large numbers. For example, a 10,000 digit multiplication takes approximaly 99,282,205 fewer single precision multiplications with Toom--Cook than a Comba or baseline approach (this is a savings of more than 99$\%$). For most ``crypto'' sized numbers this algorithm is not practical as Karatsuba has a much lower cutoff point. -First we split $a$ and $b$ into three roughly equal portions. This has been accomplished (lines 41 to 70) with +First we split $a$ and $b$ into three roughly equal portions. This has been accomplished (lines 40 to 71) with combinations of mp\_rshd() and mp\_mod\_2d() function calls. At this point $a = a2 \cdot \beta^2 + a1 \cdot \beta + a0$ and similiarly -for $b$. +for $b$. Next we compute the five points $w0, w1, w2, w3$ and $w4$. Recall that $w0$ and $w4$ can be computed directly from the portions so -we get those out of the way first (lines 73 and 78). Next we compute $w1, w2$ and $w3$ using Horners method. +we get those out of the way first (lines 74 and 79). Next we compute $w1, w2$ and $w3$ using Horners method. After this point we solve for the actual values of $w1, w2$ and $w3$ by reducing the $5 \times 5$ system which is relatively -straight forward. +straight forward. \subsection{Signed Multiplication} Now that algorithms to handle multiplications of every useful dimensions have been developed, a rather simple finishing touch is required. So far all -of the multiplication algorithms have been unsigned multiplications which leaves only a signed multiplication algorithm to be established. +of the multiplication algorithms have been unsigned multiplications which leaves only a signed multiplication algorithm to be established. \begin{figure}[!here] \begin{small} @@ -3198,29 +4800,77 @@ of the multiplication algorithms have been unsigned multiplications which leaves \end{figure} \textbf{Algorithm mp\_mul.} -This algorithm performs the signed multiplication of two inputs. It will make use of any of the three unsigned multiplication algorithms +This algorithm performs the signed multiplication of two inputs. It will make use of any of the three unsigned multiplication algorithms available when the input is of appropriate size. The \textbf{sign} of the result is not set until the end of the algorithm since algorithm -s\_mp\_mul\_digs will clear it. +s\_mp\_mul\_digs will clear it. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_mul.c \vspace{-3mm} \begin{alltt} +016 +017 /* high level multiplication (handles sign) */ +018 int mp_mul (mp_int * a, mp_int * b, mp_int * c) +019 \{ +020 int res, neg; +021 neg = (a->sign == b->sign) ? MP_ZPOS : MP_NEG; +022 +023 /* use Toom-Cook? */ +024 #ifdef BN_MP_TOOM_MUL_C +025 if (MIN (a->used, b->used) >= TOOM_MUL_CUTOFF) \{ +026 res = mp_toom_mul(a, b, c); +027 \} else +028 #endif +029 #ifdef BN_MP_KARATSUBA_MUL_C +030 /* use Karatsuba? */ +031 if (MIN (a->used, b->used) >= KARATSUBA_MUL_CUTOFF) \{ +032 res = mp_karatsuba_mul (a, b, c); +033 \} else +034 #endif +035 \{ +036 /* can we use the fast multiplier? +037 * +038 * The fast multiplier can be used if the output will +039 * have less than MP_WARRAY digits and the number of +040 * digits won't affect carry propagation +041 */ +042 int digs = a->used + b->used + 1; +043 +044 #ifdef BN_FAST_S_MP_MUL_DIGS_C +045 if ((digs < MP_WARRAY) && +046 (MIN(a->used, b->used) <= +047 (1 << ((CHAR_BIT * sizeof(mp_word)) - (2 * DIGIT_BIT))))) \{ +048 res = fast_s_mp_mul_digs (a, b, c, digs); +049 \} else +050 #endif +051 \{ +052 #ifdef BN_S_MP_MUL_DIGS_C +053 res = s_mp_mul (a, b, c); /* uses s_mp_mul_digs */ +054 #else +055 res = MP_VAL; +056 #endif +057 \} +058 \} +059 c->sign = (c->used > 0) ? neg : MP_ZPOS; +060 return res; +061 \} +062 #endif +063 \end{alltt} \end{small} -The implementation is rather simplistic and is not particularly noteworthy. Line 22 computes the sign of the result using the ``?'' -operator from the C programming language. Line 48 computes $\delta$ using the fact that $1 << k$ is equal to $2^k$. +The implementation is rather simplistic and is not particularly noteworthy. Line 23 computes the sign of the result using the ``?'' +operator from the C programming language. Line 47 computes $\delta$ using the fact that $1 << k$ is equal to $2^k$. \section{Squaring} \label{sec:basesquare} Squaring is a special case of multiplication where both multiplicands are equal. At first it may seem like there is no significant optimization available but in fact there is. Consider the multiplication of $576$ against $241$. In total there will be nine single precision multiplications -performed which are $1\cdot 6$, $1 \cdot 7$, $1 \cdot 5$, $4 \cdot 6$, $4 \cdot 7$, $4 \cdot 5$, $2 \cdot 6$, $2 \cdot 7$ and $2 \cdot 5$. Now consider -the multiplication of $123$ against $123$. The nine products are $3 \cdot 3$, $3 \cdot 2$, $3 \cdot 1$, $2 \cdot 3$, $2 \cdot 2$, $2 \cdot 1$, -$1 \cdot 3$, $1 \cdot 2$ and $1 \cdot 1$. On closer inspection some of the products are equivalent. For example, $3 \cdot 2 = 2 \cdot 3$ -and $3 \cdot 1 = 1 \cdot 3$. +performed which are $1\cdot 6$, $1 \cdot 7$, $1 \cdot 5$, $4 \cdot 6$, $4 \cdot 7$, $4 \cdot 5$, $2 \cdot 6$, $2 \cdot 7$ and $2 \cdot 5$. Now consider +the multiplication of $123$ against $123$. The nine products are $3 \cdot 3$, $3 \cdot 2$, $3 \cdot 1$, $2 \cdot 3$, $2 \cdot 2$, $2 \cdot 1$, +$1 \cdot 3$, $1 \cdot 2$ and $1 \cdot 1$. On closer inspection some of the products are equivalent. For example, $3 \cdot 2 = 2 \cdot 3$ +and $3 \cdot 1 = 1 \cdot 3$. For any $n$-digit input, there are ${{\left (n^2 + n \right)}\over 2}$ possible unique single precision multiplications required compared to the $n^2$ required for multiplication. The following diagram gives an example of the operations required. @@ -3239,19 +4889,19 @@ $\times$ &&1&2&3&\\ \end{figure} Starting from zero and numbering the columns from right to left a very simple pattern becomes obvious. For the purposes of this discussion let $x$ -represent the number being squared. The first observation is that in row $k$ the $2k$'th column of the product has a $\left (x_k \right)^2$ term in it. +represent the number being squared. The first observation is that in row $k$ the $2k$'th column of the product has a $\left (x_k \right)^2$ term in it. The second observation is that every column $j$ in row $k$ where $j \ne 2k$ is part of a double product. Every non-square term of a column will -appear twice hence the name ``double product''. Every odd column is made up entirely of double products. In fact every column is made up of double -products and at most one square (\textit{see the exercise section}). +appear twice hence the name ``double product''. Every odd column is made up entirely of double products. In fact every column is made up of double +products and at most one square (\textit{see the exercise section}). -The third and final observation is that for row $k$ the first unique non-square term, that is, one that hasn't already appeared in an earlier row, -occurs at column $2k + 1$. For example, on row $1$ of the previous squaring, column one is part of the double product with column one from row zero. +The third and final observation is that for row $k$ the first unique non-square term, that is, one that hasn't already appeared in an earlier row, +occurs at column $2k + 1$. For example, on row $1$ of the previous squaring, column one is part of the double product with column one from row zero. Column two of row one is a square and column three is the first unique column. \subsection{The Baseline Squaring Algorithm} The baseline squaring algorithm is meant to be a catch-all squaring algorithm. It will handle any of the input sizes that the faster routines -will not handle. +will not handle. \begin{figure}[!here] \begin{small} @@ -3293,35 +4943,100 @@ will not handle. \textbf{Algorithm s\_mp\_sqr.} This algorithm computes the square of an input using the three observations on squaring. It is based fairly faithfully on algorithm 14.16 of HAC -\cite[pp.596-597]{HAC}. Similar to algorithm s\_mp\_mul\_digs, a temporary mp\_int is allocated to hold the result of the squaring. This allows the +\cite[pp.596-597]{HAC}. Similar to algorithm s\_mp\_mul\_digs, a temporary mp\_int is allocated to hold the result of the squaring. This allows the destination mp\_int to be the same as the source mp\_int. The outer loop of this algorithm begins on step 4. It is best to think of the outer loop as walking down the rows of the partial results, while the inner loop computes the columns of the partial result. Step 4.1 and 4.2 compute the square term for each row, and step 4.3 and 4.4 propagate -the carry and compute the double products. +the carry and compute the double products. The requirement that a mp\_word be able to represent the range $0 \le x < 2 \beta^2$ arises from this very algorithm. The product $a_{ix}a_{iy}$ will lie in the range $0 \le x \le \beta^2 - 2\beta + 1$ which is obviously less than $\beta^2$ meaning that when it is multiplied by two, it can be properly represented by a mp\_word. -Similar to algorithm s\_mp\_mul\_digs, after every pass of the inner loop, the destination is correctly set to the sum of all of the partial -results calculated so far. This involves expensive carry propagation which will be eliminated in the next algorithm. +Similar to algorithm s\_mp\_mul\_digs, after every pass of the inner loop, the destination is correctly set to the sum of all of the partial +results calculated so far. This involves expensive carry propagation which will be eliminated in the next algorithm. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_s\_mp\_sqr.c \vspace{-3mm} \begin{alltt} +016 +017 /* low level squaring, b = a*a, HAC pp.596-597, Algorithm 14.16 */ +018 int s_mp_sqr (mp_int * a, mp_int * b) +019 \{ +020 mp_int t; +021 int res, ix, iy, pa; +022 mp_word r; +023 mp_digit u, tmpx, *tmpt; +024 +025 pa = a->used; +026 if ((res = mp_init_size (&t, (2 * pa) + 1)) != MP_OKAY) \{ +027 return res; +028 \} +029 +030 /* default used is maximum possible size */ +031 t.used = (2 * pa) + 1; +032 +033 for (ix = 0; ix < pa; ix++) \{ +034 /* first calculate the digit at 2*ix */ +035 /* calculate double precision result */ +036 r = (mp_word)t.dp[2*ix] + +037 ((mp_word)a->dp[ix] * (mp_word)a->dp[ix]); +038 +039 /* store lower part in result */ +040 t.dp[ix+ix] = (mp_digit) (r & ((mp_word) MP_MASK)); +041 +042 /* get the carry */ +043 u = (mp_digit)(r >> ((mp_word) DIGIT_BIT)); +044 +045 /* left hand side of A[ix] * A[iy] */ +046 tmpx = a->dp[ix]; +047 +048 /* alias for where to store the results */ +049 tmpt = t.dp + ((2 * ix) + 1); +050 +051 for (iy = ix + 1; iy < pa; iy++) \{ +052 /* first calculate the product */ +053 r = ((mp_word)tmpx) * ((mp_word)a->dp[iy]); +054 +055 /* now calculate the double precision result, note we use +056 * addition instead of *2 since it's easier to optimize +057 */ +058 r = ((mp_word) *tmpt) + r + r + ((mp_word) u); +059 +060 /* store lower part */ +061 *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK)); +062 +063 /* get carry */ +064 u = (mp_digit)(r >> ((mp_word) DIGIT_BIT)); +065 \} +066 /* propagate upwards */ +067 while (u != ((mp_digit) 0)) \{ +068 r = ((mp_word) *tmpt) + ((mp_word) u); +069 *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK)); +070 u = (mp_digit)(r >> ((mp_word) DIGIT_BIT)); +071 \} +072 \} +073 +074 mp_clamp (&t); +075 mp_exch (&t, b); +076 mp_clear (&t); +077 return MP_OKAY; +078 \} +079 #endif +080 \end{alltt} \end{small} -Inside the outer loop (line 34) the square term is calculated on line 37. The carry (line 44) has been -extracted from the mp\_word accumulator using a right shift. Aliases for $a_{ix}$ and $t_{ix+iy}$ are initialized -(lines 47 and 50) to simplify the inner loop. The doubling is performed using two -additions (line 59) since it is usually faster than shifting, if not at least as fast. +Inside the outer loop (line 33) the square term is calculated on line 36. The carry (line 43) has been +extracted from the mp\_word accumulator using a right shift. Aliases for $a_{ix}$ and $t_{ix+iy}$ are initialized +(lines 46 and 49) to simplify the inner loop. The doubling is performed using two +additions (line 58) since it is usually faster than shifting, if not at least as fast. The important observation is that the inner loop does not begin at $iy = 0$ like for multiplication. As such the inner loops get progressively shorter as the algorithm proceeds. This is what leads to the savings compared to using a multiplication to -square a number. +square a number. \subsection{Faster Squaring by the ``Comba'' Method} A major drawback to the baseline method is the requirement for single precision shifting inside the $O(n^2)$ nested loop. Squaring has an additional @@ -3331,10 +5046,10 @@ performance hazards. The first obvious solution is to make an array of mp\_words which will hold all of the columns. This will indeed eliminate all of the carry propagation operations from the inner loop. However, the inner product must still be doubled $O(n^2)$ times. The solution stems from the simple fact that $2a + 2b + 2c = 2(a + b + c)$. That is the sum of all of the double products is equal to double the sum of all the products. For example, -$ab + ba + ac + ca = 2ab + 2ac = 2(ab + ac)$. +$ab + ba + ac + ca = 2ab + 2ac = 2(ab + ac)$. -However, we cannot simply double all of the columns, since the squares appear only once per row. The most practical solution is to have two -mp\_word arrays. One array will hold the squares and the other array will hold the double products. With both arrays the doubling and +However, we cannot simply double all of the columns, since the squares appear only once per row. The most practical solution is to have two +mp\_word arrays. One array will hold the squares and the other array will hold the double products. With both arrays the doubling and carry propagation can be moved to a $O(n)$ work level outside the $O(n^2)$ level. In this case, we have an even simpler solution in mind. \newpage\begin{figure}[!here] @@ -3372,7 +5087,7 @@ Place an array of \textbf{MP\_WARRAY} mp\_digits named $W$ on the stack. \\ 9. for $ix$ from $pa$ to $oldused - 1$ do \\ \hspace{3mm}9.1 $b_{ix} \leftarrow 0$ \\ 10. Clamp excess digits from $b$. (\textit{mp\_clamp}) \\ -11. Return(\textit{MP\_OKAY}). \\ +11. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} @@ -3381,8 +5096,8 @@ Place an array of \textbf{MP\_WARRAY} mp\_digits named $W$ on the stack. \\ \end{figure} \textbf{Algorithm fast\_s\_mp\_sqr.} -This algorithm computes the square of an input using the Comba technique. It is designed to be a replacement for algorithm -s\_mp\_sqr when the number of input digits is less than \textbf{MP\_WARRAY} and less than $\delta \over 2$. +This algorithm computes the square of an input using the Comba technique. It is designed to be a replacement for algorithm +s\_mp\_sqr when the number of input digits is less than \textbf{MP\_WARRAY} and less than $\delta \over 2$. This algorithm is very similar to the Comba multiplier except with a few key differences we shall make note of. First, we have an accumulator and carry variables $\_ \hat W$ and $\hat W1$ respectively. This is because the inner loop @@ -3399,38 +5114,133 @@ only to even outputs and it is the square of the term at the $\lfloor ix / 2 \rf \hspace{-5.1mm}{\bf File}: bn\_fast\_s\_mp\_sqr.c \vspace{-3mm} \begin{alltt} +016 +017 /* the jist of squaring... +018 * you do like mult except the offset of the tmpx [one that +019 * starts closer to zero] can't equal the offset of tmpy. +020 * So basically you set up iy like before then you min it with +021 * (ty-tx) so that it never happens. You double all those +022 * you add in the inner loop +023 +024 After that loop you do the squares and add them in. +025 */ +026 +027 int fast_s_mp_sqr (mp_int * a, mp_int * b) +028 \{ +029 int olduse, res, pa, ix, iz; +030 mp_digit W[MP_WARRAY], *tmpx; +031 mp_word W1; +032 +033 /* grow the destination as required */ +034 pa = a->used + a->used; +035 if (b->alloc < pa) \{ +036 if ((res = mp_grow (b, pa)) != MP_OKAY) \{ +037 return res; +038 \} +039 \} +040 +041 /* number of output digits to produce */ +042 W1 = 0; +043 for (ix = 0; ix < pa; ix++) \{ +044 int tx, ty, iy; +045 mp_word _W; +046 mp_digit *tmpy; +047 +048 /* clear counter */ +049 _W = 0; +050 +051 /* get offsets into the two bignums */ +052 ty = MIN(a->used-1, ix); +053 tx = ix - ty; +054 +055 /* setup temp aliases */ +056 tmpx = a->dp + tx; +057 tmpy = a->dp + ty; +058 +059 /* this is the number of times the loop will iterrate, essentially +060 while (tx++ < a->used && ty-- >= 0) \{ ... \} +061 */ +062 iy = MIN(a->used-tx, ty+1); +063 +064 /* now for squaring tx can never equal ty +065 * we halve the distance since they approach at a rate of 2x +066 * and we have to round because odd cases need to be executed +067 */ +068 iy = MIN(iy, ((ty-tx)+1)>>1); +069 +070 /* execute loop */ +071 for (iz = 0; iz < iy; iz++) \{ +072 _W += ((mp_word)*tmpx++)*((mp_word)*tmpy--); +073 \} +074 +075 /* double the inner product and add carry */ +076 _W = _W + _W + W1; +077 +078 /* even columns have the square term in them */ +079 if ((ix&1) == 0) \{ +080 _W += ((mp_word)a->dp[ix>>1])*((mp_word)a->dp[ix>>1]); +081 \} +082 +083 /* store it */ +084 W[ix] = (mp_digit)(_W & MP_MASK); +085 +086 /* make next carry */ +087 W1 = _W >> ((mp_word)DIGIT_BIT); +088 \} +089 +090 /* setup dest */ +091 olduse = b->used; +092 b->used = a->used+a->used; +093 +094 \{ +095 mp_digit *tmpb; +096 tmpb = b->dp; +097 for (ix = 0; ix < pa; ix++) \{ +098 *tmpb++ = W[ix] & MP_MASK; +099 \} +100 +101 /* clear unused digits [that existed in the old copy of c] */ +102 for (; ix < olduse; ix++) \{ +103 *tmpb++ = 0; +104 \} +105 \} +106 mp_clamp (b); +107 return MP_OKAY; +108 \} +109 #endif +110 \end{alltt} \end{small} -This implementation is essentially a copy of Comba multiplication with the appropriate changes added to make it faster for -the special case of squaring. +This implementation is essentially a copy of Comba multiplication with the appropriate changes added to make it faster for +the special case of squaring. \subsection{Polynomial Basis Squaring} The same algorithm that performs optimal polynomial basis multiplication can be used to perform polynomial basis squaring. The minor exception is that $\zeta_y = f(y)g(y)$ is actually equivalent to $\zeta_y = f(y)^2$ since $f(y) = g(y)$. Instead of performing $2n + 1$ -multiplications to find the $\zeta$ relations, squaring operations are performed instead. +multiplications to find the $\zeta$ relations, squaring operations are performed instead. \subsection{Karatsuba Squaring} -Let $f(x) = ax + b$ represent the polynomial basis representation of a number to square. -Let $h(x) = \left ( f(x) \right )^2$ represent the square of the polynomial. The Karatsuba equation can be modified to square a +Let $f(x) = ax + b$ represent the polynomial basis representation of a number to square. +Let $h(x) = \left ( f(x) \right )^2$ represent the square of the polynomial. The Karatsuba equation can be modified to square a number with the following equation. \begin{equation} h(x) = a^2x^2 + \left ((a + b)^2 - (a^2 + b^2) \right )x + b^2 \end{equation} -Upon closer inspection this equation only requires the calculation of three half-sized squares: $a^2$, $b^2$ and $(a + b)^2$. As in -Karatsuba multiplication, this algorithm can be applied recursively on the input and will achieve an asymptotic running time of +Upon closer inspection this equation only requires the calculation of three half-sized squares: $a^2$, $b^2$ and $(a + b)^2$. As in +Karatsuba multiplication, this algorithm can be applied recursively on the input and will achieve an asymptotic running time of $O \left ( n^{lg(3)} \right )$. -If the asymptotic times of Karatsuba squaring and multiplication are the same, why not simply use the multiplication algorithm -instead? The answer to this arises from the cutoff point for squaring. As in multiplication there exists a cutoff point, at which the -time required for a Comba based squaring and a Karatsuba based squaring meet. Due to the overhead inherent in the Karatsuba method, the cutoff -point is fairly high. For example, on an AMD Athlon XP processor with $\beta = 2^{28}$, the cutoff point is around 127 digits. +If the asymptotic times of Karatsuba squaring and multiplication are the same, why not simply use the multiplication algorithm +instead? The answer to this arises from the cutoff point for squaring. As in multiplication there exists a cutoff point, at which the +time required for a Comba based squaring and a Karatsuba based squaring meet. Due to the overhead inherent in the Karatsuba method, the cutoff +point is fairly high. For example, on an AMD Athlon XP processor with $\beta = 2^{28}$, the cutoff point is around 127 digits. -Consider squaring a 200 digit number with this technique. It will be split into two 100 digit halves which are subsequently squared. +Consider squaring a 200 digit number with this technique. It will be split into two 100 digit halves which are subsequently squared. The 100 digit halves will not be squared using Karatsuba, but instead using the faster Comba based squaring algorithm. If Karatsuba multiplication -were used instead, the 100 digit numbers would be squared with a slower Comba based multiplication. +were used instead, the 100 digit numbers would be squared with a slower Comba based multiplication. \newpage\begin{figure}[!here] \begin{small} @@ -3484,7 +5294,7 @@ Now if $5n$ single precision additions and a squaring of $n$-digits is faster th this method is faster. Assuming no further recursions occur, the difference can be estimated with the following inequality. Let $p$ represent the cost of a single precision addition and $q$ the cost of a single precision multiplication both in terms of time\footnote{Or -machine clock cycles.}. +machine clock cycles.}. \begin{equation} 5pn +{{q(n^2 + n)} \over 2} \le pn + qn^2 @@ -3502,32 +5312,134 @@ ${13 \over 9}$ & $<$ & $n$ \\ This results in a cutoff point around $n = 2$. As a consequence it is actually faster to compute the middle term the ``long way'' on processors where multiplication is substantially slower\footnote{On the Athlon there is a 1:17 ratio between clock cycles for addition and multiplication. On the Intel P4 processor this ratio is 1:29 making this method even more beneficial. The only common exception is the ARMv4 processor which has a -ratio of 1:7. } than simpler operations such as addition. +ratio of 1:7. } than simpler operations such as addition. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_karatsuba\_sqr.c \vspace{-3mm} \begin{alltt} +016 +017 /* Karatsuba squaring, computes b = a*a using three +018 * half size squarings +019 * +020 * See comments of karatsuba_mul for details. It +021 * is essentially the same algorithm but merely +022 * tuned to perform recursive squarings. +023 */ +024 int mp_karatsuba_sqr (mp_int * a, mp_int * b) +025 \{ +026 mp_int x0, x1, t1, t2, x0x0, x1x1; +027 int B, err; +028 +029 err = MP_MEM; +030 +031 /* min # of digits */ +032 B = a->used; +033 +034 /* now divide in two */ +035 B = B >> 1; +036 +037 /* init copy all the temps */ +038 if (mp_init_size (&x0, B) != MP_OKAY) +039 goto ERR; +040 if (mp_init_size (&x1, a->used - B) != MP_OKAY) +041 goto X0; +042 +043 /* init temps */ +044 if (mp_init_size (&t1, a->used * 2) != MP_OKAY) +045 goto X1; +046 if (mp_init_size (&t2, a->used * 2) != MP_OKAY) +047 goto T1; +048 if (mp_init_size (&x0x0, B * 2) != MP_OKAY) +049 goto T2; +050 if (mp_init_size (&x1x1, (a->used - B) * 2) != MP_OKAY) +051 goto X0X0; +052 +053 \{ +054 int x; +055 mp_digit *dst, *src; +056 +057 src = a->dp; +058 +059 /* now shift the digits */ +060 dst = x0.dp; +061 for (x = 0; x < B; x++) \{ +062 *dst++ = *src++; +063 \} +064 +065 dst = x1.dp; +066 for (x = B; x < a->used; x++) \{ +067 *dst++ = *src++; +068 \} +069 \} +070 +071 x0.used = B; +072 x1.used = a->used - B; +073 +074 mp_clamp (&x0); +075 +076 /* now calc the products x0*x0 and x1*x1 */ +077 if (mp_sqr (&x0, &x0x0) != MP_OKAY) +078 goto X1X1; /* x0x0 = x0*x0 */ +079 if (mp_sqr (&x1, &x1x1) != MP_OKAY) +080 goto X1X1; /* x1x1 = x1*x1 */ +081 +082 /* now calc (x1+x0)**2 */ +083 if (s_mp_add (&x1, &x0, &t1) != MP_OKAY) +084 goto X1X1; /* t1 = x1 - x0 */ +085 if (mp_sqr (&t1, &t1) != MP_OKAY) +086 goto X1X1; /* t1 = (x1 - x0) * (x1 - x0) */ +087 +088 /* add x0y0 */ +089 if (s_mp_add (&x0x0, &x1x1, &t2) != MP_OKAY) +090 goto X1X1; /* t2 = x0x0 + x1x1 */ +091 if (s_mp_sub (&t1, &t2, &t1) != MP_OKAY) +092 goto X1X1; /* t1 = (x1+x0)**2 - (x0x0 + x1x1) */ +093 +094 /* shift by B */ +095 if (mp_lshd (&t1, B) != MP_OKAY) +096 goto X1X1; /* t1 = (x0x0 + x1x1 - (x1-x0)*(x1-x0))<<B */ +097 if (mp_lshd (&x1x1, B * 2) != MP_OKAY) +098 goto X1X1; /* x1x1 = x1x1 << 2*B */ +099 +100 if (mp_add (&x0x0, &t1, &t1) != MP_OKAY) +101 goto X1X1; /* t1 = x0x0 + t1 */ +102 if (mp_add (&t1, &x1x1, b) != MP_OKAY) +103 goto X1X1; /* t1 = x0x0 + t1 + x1x1 */ +104 +105 err = MP_OKAY; +106 +107 X1X1:mp_clear (&x1x1); +108 X0X0:mp_clear (&x0x0); +109 T2:mp_clear (&t2); +110 T1:mp_clear (&t1); +111 X1:mp_clear (&x1); +112 X0:mp_clear (&x0); +113 ERR: +114 return err; +115 \} +116 #endif +117 \end{alltt} \end{small} -This implementation is largely based on the implementation of algorithm mp\_karatsuba\_mul. It uses the same inline style to copy and -shift the input into the two halves. The loop from line 54 to line 70 has been modified since only one input exists. The \textbf{used} +This implementation is largely based on the implementation of algorithm mp\_karatsuba\_mul. It uses the same inline style to copy and +shift the input into the two halves. The loop from line 53 to line 69 has been modified since only one input exists. The \textbf{used} count of both $x0$ and $x1$ is fixed up and $x0$ is clamped before the calculations begin. At this point $x1$ and $x0$ are valid equivalents -to the respective halves as if mp\_rshd and mp\_mod\_2d had been used. +to the respective halves as if mp\_rshd and mp\_mod\_2d had been used. By inlining the copy and shift operations the cutoff point for Karatsuba multiplication can be lowered. On the Athlon the cutoff point is exactly at the point where Comba squaring can no longer be used (\textit{128 digits}). On slower processors such as the Intel P4 it is actually below the Comba limit (\textit{at 110 digits}). -This routine uses the same error trap coding style as mp\_karatsuba\_sqr. As the temporary variables are initialized errors are -redirected to the error trap higher up. If the algorithm completes without error the error code is set to \textbf{MP\_OKAY} and +This routine uses the same error trap coding style as mp\_karatsuba\_sqr. As the temporary variables are initialized errors are +redirected to the error trap higher up. If the algorithm completes without error the error code is set to \textbf{MP\_OKAY} and mp\_clears are executed normally. \subsection{Toom-Cook Squaring} The Toom-Cook squaring algorithm mp\_toom\_sqr is heavily based on the algorithm mp\_toom\_mul with the exception that squarings are used -instead of multiplication to find the five relations. The reader is encouraged to read the description of the latter algorithm and try to -derive their own Toom-Cook squaring algorithm. +instead of multiplication to find the five relations. The reader is encouraged to read the description of the latter algorithm and try to +derive their own Toom-Cook squaring algorithm. \subsection{High Level Squaring} \newpage\begin{figure}[!here] @@ -3560,12 +5472,53 @@ derive their own Toom-Cook squaring algorithm. \textbf{Algorithm mp\_sqr.} This algorithm computes the square of the input using one of four different algorithms. If the input is very large and has at least \textbf{TOOM\_SQR\_CUTOFF} or \textbf{KARATSUBA\_SQR\_CUTOFF} digits then either the Toom-Cook or the Karatsuba Squaring algorithm is used. If -neither of the polynomial basis algorithms should be used then either the Comba or baseline algorithm is used. +neither of the polynomial basis algorithms should be used then either the Comba or baseline algorithm is used. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_sqr.c \vspace{-3mm} \begin{alltt} +016 +017 /* computes b = a*a */ +018 int +019 mp_sqr (mp_int * a, mp_int * b) +020 \{ +021 int res; +022 +023 #ifdef BN_MP_TOOM_SQR_C +024 /* use Toom-Cook? */ +025 if (a->used >= TOOM_SQR_CUTOFF) \{ +026 res = mp_toom_sqr(a, b); +027 /* Karatsuba? */ +028 \} else +029 #endif +030 #ifdef BN_MP_KARATSUBA_SQR_C +031 if (a->used >= KARATSUBA_SQR_CUTOFF) \{ +032 res = mp_karatsuba_sqr (a, b); +033 \} else +034 #endif +035 \{ +036 #ifdef BN_FAST_S_MP_SQR_C +037 /* can we use the fast comba multiplier? */ +038 if ((((a->used * 2) + 1) < MP_WARRAY) && +039 (a->used < +040 (1 << (((sizeof(mp_word) * CHAR_BIT) - (2 * DIGIT_BIT)) - 1)))) \{ +041 res = fast_s_mp_sqr (a, b); +042 \} else +043 #endif +044 \{ +045 #ifdef BN_S_MP_SQR_C +046 res = s_mp_sqr (a, b); +047 #else +048 res = MP_VAL; +049 #endif +050 \} +051 \} +052 b->sign = MP_ZPOS; +053 return res; +054 \} +055 #endif +056 \end{alltt} \end{small} @@ -3576,11 +5529,11 @@ $\left [ 3 \right ] $ & Devise an efficient algorithm for selection of the radix & \\ $\left [ 2 \right ] $ & In section 5.3 the fact that every column of a squaring is made up \\ & of double products and at most one square is stated. Prove this statement. \\ - & \\ + & \\ $\left [ 3 \right ] $ & Prove the equation for Karatsuba squaring. \\ & \\ $\left [ 1 \right ] $ & Prove that Karatsuba squaring requires $O \left (n^{lg(3)} \right )$ time. \\ - & \\ + & \\ $\left [ 2 \right ] $ & Determine the minimal ratio between addition and multiplication clock cycles \\ & required for equation $6.7$ to be true. \\ & \\ @@ -3597,61 +5550,61 @@ $\left [ 4 \right ] $ & Same as the previous but also modify the Karatsuba and T \chapter{Modular Reduction} \section{Basics of Modular Reduction} \index{modular residue} -Modular reduction is an operation that arises quite often within public key cryptography algorithms and various number theoretic algorithms, +Modular reduction is an operation that arises quite often within public key cryptography algorithms and various number theoretic algorithms, such as factoring. Modular reduction algorithms are the third class of algorithms of the ``multipliers'' set. A number $a$ is said to be \textit{reduced} -modulo another number $b$ by finding the remainder of the division $a/b$. Full integer division with remainder is a topic to be covered +modulo another number $b$ by finding the remainder of the division $a/b$. Full integer division with remainder is a topic to be covered in~\ref{sec:division}. -Modular reduction is equivalent to solving for $r$ in the following equation. $a = bq + r$ where $q = \lfloor a/b \rfloor$. The result -$r$ is said to be ``congruent to $a$ modulo $b$'' which is also written as $r \equiv a \mbox{ (mod }b\mbox{)}$. In other vernacular $r$ is known as the +Modular reduction is equivalent to solving for $r$ in the following equation. $a = bq + r$ where $q = \lfloor a/b \rfloor$. The result +$r$ is said to be ``congruent to $a$ modulo $b$'' which is also written as $r \equiv a \mbox{ (mod }b\mbox{)}$. In other vernacular $r$ is known as the ``modular residue'' which leads to ``quadratic residue''\footnote{That's fancy talk for $b \equiv a^2 \mbox{ (mod }p\mbox{)}$.} and -other forms of residues. +other forms of residues. -Modular reductions are normally used to create either finite groups, rings or fields. The most common usage for performance driven modular reductions -is in modular exponentiation algorithms. That is to compute $d = a^b \mbox{ (mod }c\mbox{)}$ as fast as possible. This operation is used in the -RSA and Diffie-Hellman public key algorithms, for example. Modular multiplication and squaring also appears as a fundamental operation in -elliptic curve cryptographic algorithms. As will be discussed in the subsequent chapter there exist fast algorithms for computing modular -exponentiations without having to perform (\textit{in this example}) $b - 1$ multiplications. These algorithms will produce partial results in the -range $0 \le x < c^2$ which can be taken advantage of to create several efficient algorithms. They have also been used to create redundancy check -algorithms known as CRCs, error correction codes such as Reed-Solomon and solve a variety of number theoeretic problems. +Modular reductions are normally used to create either finite groups, rings or fields. The most common usage for performance driven modular reductions +is in modular exponentiation algorithms. That is to compute $d = a^b \mbox{ (mod }c\mbox{)}$ as fast as possible. This operation is used in the +RSA and Diffie-Hellman public key algorithms, for example. Modular multiplication and squaring also appears as a fundamental operation in +elliptic curve cryptographic algorithms. As will be discussed in the subsequent chapter there exist fast algorithms for computing modular +exponentiations without having to perform (\textit{in this example}) $b - 1$ multiplications. These algorithms will produce partial results in the +range $0 \le x < c^2$ which can be taken advantage of to create several efficient algorithms. They have also been used to create redundancy check +algorithms known as CRCs, error correction codes such as Reed-Solomon and solve a variety of number theoeretic problems. \section{The Barrett Reduction} The Barrett reduction algorithm \cite{BARRETT} was inspired by fast division algorithms which multiply by the reciprocal to emulate -division. Barretts observation was that the residue $c$ of $a$ modulo $b$ is equal to +division. Barretts observation was that the residue $c$ of $a$ modulo $b$ is equal to \begin{equation} c = a - b \cdot \lfloor a/b \rfloor \end{equation} -Since algorithms such as modular exponentiation would be using the same modulus extensively, typical DSP\footnote{It is worth noting that Barrett's paper -targeted the DSP56K processor.} intuition would indicate the next step would be to replace $a/b$ by a multiplication by the reciprocal. However, -DSP intuition on its own will not work as these numbers are considerably larger than the precision of common DSP floating point data types. +Since algorithms such as modular exponentiation would be using the same modulus extensively, typical DSP\footnote{It is worth noting that Barrett's paper +targeted the DSP56K processor.} intuition would indicate the next step would be to replace $a/b$ by a multiplication by the reciprocal. However, +DSP intuition on its own will not work as these numbers are considerably larger than the precision of common DSP floating point data types. It would take another common optimization to optimize the algorithm. \subsection{Fixed Point Arithmetic} The trick used to optimize the above equation is based on a technique of emulating floating point data types with fixed precision integers. Fixed -point arithmetic would become very popular as it greatly optimize the ``3d-shooter'' genre of games in the mid 1990s when floating point units were -fairly slow if not unavailable. The idea behind fixed point arithmetic is to take a normal $k$-bit integer data type and break it into $p$-bit -integer and a $q$-bit fraction part (\textit{where $p+q = k$}). +point arithmetic would become very popular as it greatly optimize the ``3d-shooter'' genre of games in the mid 1990s when floating point units were +fairly slow if not unavailable. The idea behind fixed point arithmetic is to take a normal $k$-bit integer data type and break it into $p$-bit +integer and a $q$-bit fraction part (\textit{where $p+q = k$}). In this system a $k$-bit integer $n$ would actually represent $n/2^q$. For example, with $q = 4$ the integer $n = 37$ would actually represent the -value $2.3125$. To multiply two fixed point numbers the integers are multiplied using traditional arithmetic and subsequently normalized by -moving the implied decimal point back to where it should be. For example, with $q = 4$ to multiply the integers $9$ and $5$ they must be converted -to fixed point first by multiplying by $2^q$. Let $a = 9(2^q)$ represent the fixed point representation of $9$ and $b = 5(2^q)$ represent the -fixed point representation of $5$. The product $ab$ is equal to $45(2^{2q})$ which when normalized by dividing by $2^q$ produces $45(2^q)$. +value $2.3125$. To multiply two fixed point numbers the integers are multiplied using traditional arithmetic and subsequently normalized by +moving the implied decimal point back to where it should be. For example, with $q = 4$ to multiply the integers $9$ and $5$ they must be converted +to fixed point first by multiplying by $2^q$. Let $a = 9(2^q)$ represent the fixed point representation of $9$ and $b = 5(2^q)$ represent the +fixed point representation of $5$. The product $ab$ is equal to $45(2^{2q})$ which when normalized by dividing by $2^q$ produces $45(2^q)$. This technique became popular since a normal integer multiplication and logical shift right are the only required operations to perform a multiplication -of two fixed point numbers. Using fixed point arithmetic, division can be easily approximated by multiplying by the reciprocal. If $2^q$ is -equivalent to one than $2^q/b$ is equivalent to the fixed point approximation of $1/b$ using real arithmetic. Using this fact dividing an integer +of two fixed point numbers. Using fixed point arithmetic, division can be easily approximated by multiplying by the reciprocal. If $2^q$ is +equivalent to one than $2^q/b$ is equivalent to the fixed point approximation of $1/b$ using real arithmetic. Using this fact dividing an integer $a$ by another integer $b$ can be achieved with the following expression. \begin{equation} \lfloor a / b \rfloor \mbox{ }\approx\mbox{ } \lfloor (a \cdot \lfloor 2^q / b \rfloor)/2^q \rfloor \end{equation} -The precision of the division is proportional to the value of $q$. If the divisor $b$ is used frequently as is the case with +The precision of the division is proportional to the value of $q$. If the divisor $b$ is used frequently as is the case with modular exponentiation pre-computing $2^q/b$ will allow a division to be performed with a multiplication and a right shift. Both operations -are considerably faster than division on most processors. +are considerably faster than division on most processors. Consider dividing $19$ by $5$. The correct result is $\lfloor 19/5 \rfloor = 3$. With $q = 3$ the reciprocal is $\lfloor 2^q/5 \rfloor = 1$ which leads to a product of $19$ which when divided by $2^q$ produces $2$. However, with $q = 4$ the reciprocal is $\lfloor 2^q/5 \rfloor = 3$ and @@ -3672,11 +5625,11 @@ c = a - b \cdot \lfloor (a \cdot \mu)/2^q \rfloor Provided that $2^q \ge a$ this algorithm will produce a quotient that is either exactly correct or off by a value of one. In the context of Barrett reduction the value of $a$ is bound by $0 \le a \le (b - 1)^2$ meaning that $2^q \ge b^2$ is sufficient to ensure the reciprocal will have enough -precision. +precision. -Let $n$ represent the number of digits in $b$. This algorithm requires approximately $2n^2$ single precision multiplications to produce the quotient and -another $n^2$ single precision multiplications to find the residue. In total $3n^2$ single precision multiplications are required to -reduce the number. +Let $n$ represent the number of digits in $b$. This algorithm requires approximately $2n^2$ single precision multiplications to produce the quotient and +another $n^2$ single precision multiplications to find the residue. In total $3n^2$ single precision multiplications are required to +reduce the number. For example, if $b = 1179677$ and $q = 41$ ($2^q > b^2$), then the reciprocal $\mu$ is equal to $\lfloor 2^q / b \rfloor = 1864089$. Consider reducing $a = 180388626447$ modulo $b$ using the above reduction equation. The quotient using the new formula is $\lfloor (a \cdot \mu) / 2^q \rfloor = 152913$. @@ -3684,19 +5637,19 @@ By subtracting $152913b$ from $a$ the correct residue $a \equiv 677346 \mbox{ (m \subsection{Choosing a Radix Point} Using the fixed point representation a modular reduction can be performed with $3n^2$ single precision multiplications. If that were the best -that could be achieved a full division\footnote{A division requires approximately $O(2cn^2)$ single precision multiplications for a small value of $c$. +that could be achieved a full division\footnote{A division requires approximately $O(2cn^2)$ single precision multiplications for a small value of $c$. See~\ref{sec:division} for further details.} might as well be used in its place. The key to optimizing the reduction is to reduce the precision of -the initial multiplication that finds the quotient. +the initial multiplication that finds the quotient. Let $a$ represent the number of which the residue is sought. Let $b$ represent the modulus used to find the residue. Let $m$ represent -the number of digits in $b$. For the purposes of this discussion we will assume that the number of digits in $a$ is $2m$, which is generally true if -two $m$-digit numbers have been multiplied. Dividing $a$ by $b$ is the same as dividing a $2m$ digit integer by a $m$ digit integer. Digits below the +the number of digits in $b$. For the purposes of this discussion we will assume that the number of digits in $a$ is $2m$, which is generally true if +two $m$-digit numbers have been multiplied. Dividing $a$ by $b$ is the same as dividing a $2m$ digit integer by a $m$ digit integer. Digits below the $m - 1$'th digit of $a$ will contribute at most a value of $1$ to the quotient because $\beta^k < b$ for any $0 \le k \le m - 1$. Another way to -express this is by re-writing $a$ as two parts. If $a' \equiv a \mbox{ (mod }b^m\mbox{)}$ and $a'' = a - a'$ then +express this is by re-writing $a$ as two parts. If $a' \equiv a \mbox{ (mod }b^m\mbox{)}$ and $a'' = a - a'$ then ${a \over b} \equiv {{a' + a''} \over b}$ which is equivalent to ${a' \over b} + {a'' \over b}$. Since $a'$ is bound to be less than $b$ the quotient is bound by $0 \le {a' \over b} < 1$. -Since the digits of $a'$ do not contribute much to the quotient the observation is that they might as well be zero. However, if the digits +Since the digits of $a'$ do not contribute much to the quotient the observation is that they might as well be zero. However, if the digits ``might as well be zero'' they might as well not be there in the first place. Let $q_0 = \lfloor a/\beta^{m-1} \rfloor$ represent the input with the irrelevant digits trimmed. Now the modular reduction is trimmed to the almost equivalent equation @@ -3704,52 +5657,52 @@ with the irrelevant digits trimmed. Now the modular reduction is trimmed to the c = a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor \end{equation} -Note that the original divisor $2^q$ has been replaced with $\beta^{m+1}$ where in this case $q$ is a multiple of $lg(\beta)$. Also note that the -exponent on the divisor when added to the amount $q_0$ was shifted by equals $2m$. If the optimization had not been performed the divisor -would have the exponent $2m$ so in the end the exponents do ``add up''. Using the above equation the quotient +Note that the original divisor $2^q$ has been replaced with $\beta^{m+1}$ where in this case $q$ is a multiple of $lg(\beta)$. Also note that the +exponent on the divisor when added to the amount $q_0$ was shifted by equals $2m$. If the optimization had not been performed the divisor +would have the exponent $2m$ so in the end the exponents do ``add up''. Using the above equation the quotient $\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ can be off from the true quotient by at most two. The original fixed point quotient can be off by as much as one (\textit{provided the radix point is chosen suitably}) and now that the lower irrelevent digits have been trimmed the quotient -can be off by an additional value of one for a total of at most two. This implies that -$0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$. By first subtracting $b$ times the quotient and then conditionally subtracting +can be off by an additional value of one for a total of at most two. This implies that +$0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$. By first subtracting $b$ times the quotient and then conditionally subtracting $b$ once or twice the residue is found. The quotient is now found using $(m + 1)(m) = m^2 + m$ single precision multiplications and the residue with an additional $m^2$ single -precision multiplications, ignoring the subtractions required. In total $2m^2 + m$ single precision multiplications are required to find the residue. +precision multiplications, ignoring the subtractions required. In total $2m^2 + m$ single precision multiplications are required to find the residue. This is considerably faster than the original attempt. -For example, let $\beta = 10$ represent the radix of the digits. Let $b = 9999$ represent the modulus which implies $m = 4$. Let $a = 99929878$ -represent the value of which the residue is desired. In this case $q = 8$ since $10^7 < 9999^2$ meaning that $\mu = \lfloor \beta^{q}/b \rfloor = 10001$. -With the new observation the multiplicand for the quotient is equal to $q_0 = \lfloor a / \beta^{m - 1} \rfloor = 99929$. The quotient is then -$\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor = 9993$. Subtracting $9993b$ from $a$ and the correct residue $a \equiv 9871 \mbox{ (mod }b\mbox{)}$ -is found. +For example, let $\beta = 10$ represent the radix of the digits. Let $b = 9999$ represent the modulus which implies $m = 4$. Let $a = 99929878$ +represent the value of which the residue is desired. In this case $q = 8$ since $10^7 < 9999^2$ meaning that $\mu = \lfloor \beta^{q}/b \rfloor = 10001$. +With the new observation the multiplicand for the quotient is equal to $q_0 = \lfloor a / \beta^{m - 1} \rfloor = 99929$. The quotient is then +$\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor = 9993$. Subtracting $9993b$ from $a$ and the correct residue $a \equiv 9871 \mbox{ (mod }b\mbox{)}$ +is found. \subsection{Trimming the Quotient} -So far the reduction algorithm has been optimized from $3m^2$ single precision multiplications down to $2m^2 + m$ single precision multiplications. As +So far the reduction algorithm has been optimized from $3m^2$ single precision multiplications down to $2m^2 + m$ single precision multiplications. As it stands now the algorithm is already fairly fast compared to a full integer division algorithm. However, there is still room for -optimization. +optimization. After the first multiplication inside the quotient ($q_0 \cdot \mu$) the value is shifted right by $m + 1$ places effectively nullifying the lower -half of the product. It would be nice to be able to remove those digits from the product to effectively cut down the number of single precision -multiplications. If the number of digits in the modulus $m$ is far less than $\beta$ a full product is not required for the algorithm to work properly. -In fact the lower $m - 2$ digits will not affect the upper half of the product at all and do not need to be computed. +half of the product. It would be nice to be able to remove those digits from the product to effectively cut down the number of single precision +multiplications. If the number of digits in the modulus $m$ is far less than $\beta$ a full product is not required for the algorithm to work properly. +In fact the lower $m - 2$ digits will not affect the upper half of the product at all and do not need to be computed. The value of $\mu$ is a $m$-digit number and $q_0$ is a $m + 1$ digit number. Using a full multiplier $(m + 1)(m) = m^2 + m$ single precision multiplications would be required. Using a multiplier that will only produce digits at and above the $m - 1$'th digit reduces the number -of single precision multiplications to ${m^2 + m} \over 2$ single precision multiplications. +of single precision multiplications to ${m^2 + m} \over 2$ single precision multiplications. \subsection{Trimming the Residue} After the quotient has been calculated it is used to reduce the input. As previously noted the algorithm is not exact and it can be off by a small -multiple of the modulus, that is $0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$. If $b$ is $m$ digits than the +multiple of the modulus, that is $0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$. If $b$ is $m$ digits than the result of reduction equation is a value of at most $m + 1$ digits (\textit{provided $3 < \beta$}) implying that the upper $m - 1$ digits are -implicitly zero. +implicitly zero. The next optimization arises from this very fact. Instead of computing $b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ using a full $O(m^2)$ multiplication algorithm only the lower $m+1$ digits of the product have to be computed. Similarly the value of $a$ can -be reduced modulo $\beta^{m+1}$ before the multiple of $b$ is subtracted which simplifes the subtraction as well. A multiplication that produces -only the lower $m+1$ digits requires ${m^2 + 3m - 2} \over 2$ single precision multiplications. +be reduced modulo $\beta^{m+1}$ before the multiple of $b$ is subtracted which simplifes the subtraction as well. A multiplication that produces +only the lower $m+1$ digits requires ${m^2 + 3m - 2} \over 2$ single precision multiplications. With both optimizations in place the algorithm is the algorithm Barrett proposed. It requires $m^2 + 2m - 1$ single precision multiplications which -is considerably faster than the straightforward $3m^2$ method. +is considerably faster than the straightforward $3m^2$ method. \subsection{The Barrett Algorithm} \newpage\begin{figure}[!here] @@ -3793,43 +5746,124 @@ Now subtract the modulus if the residue is too large (e.g. quotient too small). \textbf{Algorithm mp\_reduce.} This algorithm will reduce the input $a$ modulo $b$ in place using the Barrett algorithm. It is loosely based on algorithm 14.42 of HAC -\cite[pp. 602]{HAC} which is based on the paper from Paul Barrett \cite{BARRETT}. The algorithm has several restrictions and assumptions which must +\cite[pp. 602]{HAC} which is based on the paper from Paul Barrett \cite{BARRETT}. The algorithm has several restrictions and assumptions which must be adhered to for the algorithm to work. First the modulus $b$ is assumed to be positive and greater than one. If the modulus were less than or equal to one than subtracting a multiple of it would either accomplish nothing or actually enlarge the input. The input $a$ must be in the range $0 \le a < b^2$ in order for the quotient to have enough precision. If $a$ is the product of two numbers that were already reduced modulo $b$, this will not be a problem. -Technically the algorithm will still work if $a \ge b^2$ but it will take much longer to finish. The value of $\mu$ is passed as an argument to this -algorithm and is assumed to be calculated and stored before the algorithm is used. +Technically the algorithm will still work if $a \ge b^2$ but it will take much longer to finish. The value of $\mu$ is passed as an argument to this +algorithm and is assumed to be calculated and stored before the algorithm is used. -Recall that the multiplication for the quotient on step 3 must only produce digits at or above the $m-1$'th position. An algorithm called +Recall that the multiplication for the quotient on step 3 must only produce digits at or above the $m-1$'th position. An algorithm called $s\_mp\_mul\_high\_digs$ which has not been presented is used to accomplish this task. The algorithm is based on $s\_mp\_mul\_digs$ except that instead of stopping at a given level of precision it starts at a given level of precision. This optimal algorithm can only be used if the number -of digits in $b$ is very much smaller than $\beta$. +of digits in $b$ is very much smaller than $\beta$. -While it is known that -$a \ge b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ only the lower $m+1$ digits are being used to compute the residue, so an implied -``borrow'' from the higher digits might leave a negative result. After the multiple of the modulus has been subtracted from $a$ the residue must be -fixed up in case it is negative. The invariant $\beta^{m+1}$ must be added to the residue to make it positive again. +While it is known that +$a \ge b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ only the lower $m+1$ digits are being used to compute the residue, so an implied +``borrow'' from the higher digits might leave a negative result. After the multiple of the modulus has been subtracted from $a$ the residue must be +fixed up in case it is negative. The invariant $\beta^{m+1}$ must be added to the residue to make it positive again. -The while loop at step 9 will subtract $b$ until the residue is less than $b$. If the algorithm is performed correctly this step is +The while loop at step 9 will subtract $b$ until the residue is less than $b$. If the algorithm is performed correctly this step is performed at most twice, and on average once. However, if $a \ge b^2$ than it will iterate substantially more times than it should. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_reduce.c \vspace{-3mm} \begin{alltt} +016 +017 /* reduces x mod m, assumes 0 < x < m**2, mu is +018 * precomputed via mp_reduce_setup. +019 * From HAC pp.604 Algorithm 14.42 +020 */ +021 int mp_reduce (mp_int * x, mp_int * m, mp_int * mu) +022 \{ +023 mp_int q; +024 int res, um = m->used; +025 +026 /* q = x */ +027 if ((res = mp_init_copy (&q, x)) != MP_OKAY) \{ +028 return res; +029 \} +030 +031 /* q1 = x / b**(k-1) */ +032 mp_rshd (&q, um - 1); +033 +034 /* according to HAC this optimization is ok */ +035 if (((mp_digit) um) > (((mp_digit)1) << (DIGIT_BIT - 1))) \{ +036 if ((res = mp_mul (&q, mu, &q)) != MP_OKAY) \{ +037 goto CLEANUP; +038 \} +039 \} else \{ +040 #ifdef BN_S_MP_MUL_HIGH_DIGS_C +041 if ((res = s_mp_mul_high_digs (&q, mu, &q, um)) != MP_OKAY) \{ +042 goto CLEANUP; +043 \} +044 #elif defined(BN_FAST_S_MP_MUL_HIGH_DIGS_C) +045 if ((res = fast_s_mp_mul_high_digs (&q, mu, &q, um)) != MP_OKAY) \{ +046 goto CLEANUP; +047 \} +048 #else +049 \{ +050 res = MP_VAL; +051 goto CLEANUP; +052 \} +053 #endif +054 \} +055 +056 /* q3 = q2 / b**(k+1) */ +057 mp_rshd (&q, um + 1); +058 +059 /* x = x mod b**(k+1), quick (no division) */ +060 if ((res = mp_mod_2d (x, DIGIT_BIT * (um + 1), x)) != MP_OKAY) \{ +061 goto CLEANUP; +062 \} +063 +064 /* q = q * m mod b**(k+1), quick (no division) */ +065 if ((res = s_mp_mul_digs (&q, m, &q, um + 1)) != MP_OKAY) \{ +066 goto CLEANUP; +067 \} +068 +069 /* x = x - q */ +070 if ((res = mp_sub (x, &q, x)) != MP_OKAY) \{ +071 goto CLEANUP; +072 \} +073 +074 /* If x < 0, add b**(k+1) to it */ +075 if (mp_cmp_d (x, 0) == MP_LT) \{ +076 mp_set (&q, 1); +077 if ((res = mp_lshd (&q, um + 1)) != MP_OKAY) +078 goto CLEANUP; +079 if ((res = mp_add (x, &q, x)) != MP_OKAY) +080 goto CLEANUP; +081 \} +082 +083 /* Back off if it's too big */ +084 while (mp_cmp (x, m) != MP_LT) \{ +085 if ((res = s_mp_sub (x, m, x)) != MP_OKAY) \{ +086 goto CLEANUP; +087 \} +088 \} +089 +090 CLEANUP: +091 mp_clear (&q); +092 +093 return res; +094 \} +095 #endif +096 \end{alltt} \end{small} The first multiplication that determines the quotient can be performed by only producing the digits from $m - 1$ and up. This essentially halves the number of single precision multiplications required. However, the optimization is only safe if $\beta$ is much larger than the number of digits -in the modulus. In the source code this is evaluated on lines 36 to 44 where algorithm s\_mp\_mul\_high\_digs is used when it is -safe to do so. +in the modulus. In the source code this is evaluated on lines 36 to 43 where algorithm s\_mp\_mul\_high\_digs is used when it is +safe to do so. \subsection{The Barrett Setup Algorithm} In order to use algorithm mp\_reduce the value of $\mu$ must be calculated in advance. Ideally this value should be computed once and stored for -future use so that the Barrett algorithm can be used without delay. +future use so that the Barrett algorithm can be used without delay. \newpage\begin{figure}[!here] \begin{small} @@ -3857,28 +5891,43 @@ is equivalent and much faster. The final value is computed by taking the intege \hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_setup.c \vspace{-3mm} \begin{alltt} +016 +017 /* pre-calculate the value required for Barrett reduction +018 * For a given modulus "b" it calulates the value required in "a" +019 */ +020 int mp_reduce_setup (mp_int * a, mp_int * b) +021 \{ +022 int res; +023 +024 if ((res = mp_2expt (a, b->used * 2 * DIGIT_BIT)) != MP_OKAY) \{ +025 return res; +026 \} +027 return mp_div (a, b, a, NULL); +028 \} +029 #endif +030 \end{alltt} \end{small} This simple routine calculates the reciprocal $\mu$ required by Barrett reduction. Note the extended usage of algorithm mp\_div where the variable -which would received the remainder is passed as NULL. As will be discussed in~\ref{sec:division} the division routine allows both the quotient and the -remainder to be passed as NULL meaning to ignore the value. +which would received the remainder is passed as NULL. As will be discussed in~\ref{sec:division} the division routine allows both the quotient and the +remainder to be passed as NULL meaning to ignore the value. \section{The Montgomery Reduction} -Montgomery reduction\footnote{Thanks to Niels Ferguson for his insightful explanation of the algorithm.} \cite{MONT} is by far the most interesting -form of reduction in common use. It computes a modular residue which is not actually equal to the residue of the input yet instead equal to a -residue times a constant. However, as perplexing as this may sound the algorithm is relatively simple and very efficient. +Montgomery reduction\footnote{Thanks to Niels Ferguson for his insightful explanation of the algorithm.} \cite{MONT} is by far the most interesting +form of reduction in common use. It computes a modular residue which is not actually equal to the residue of the input yet instead equal to a +residue times a constant. However, as perplexing as this may sound the algorithm is relatively simple and very efficient. Throughout this entire section the variable $n$ will represent the modulus used to form the residue. As will be discussed shortly the value of $n$ must be odd. The variable $x$ will represent the quantity of which the residue is sought. Similar to the Barrett algorithm the input is restricted to $0 \le x < n^2$. To begin the description some simple number theory facts must be established. \textbf{Fact 1.} Adding $n$ to $x$ does not change the residue since in effect it adds one to the quotient $\lfloor x / n \rfloor$. Another way -to explain this is that $n$ is (\textit{or multiples of $n$ are}) congruent to zero modulo $n$. Adding zero will not change the value of the residue. +to explain this is that $n$ is (\textit{or multiples of $n$ are}) congruent to zero modulo $n$. Adding zero will not change the value of the residue. \textbf{Fact 2.} If $x$ is even then performing a division by two in $\Z$ is congruent to $x \cdot 2^{-1} \mbox{ (mod }n\mbox{)}$. Actually -this is an application of the fact that if $x$ is evenly divisible by any $k \in \Z$ then division in $\Z$ will be congruent to -multiplication by $k^{-1}$ modulo $n$. +this is an application of the fact that if $x$ is evenly divisible by any $k \in \Z$ then division in $\Z$ will be congruent to +multiplication by $k^{-1}$ modulo $n$. From these two simple facts the following simple algorithm can be derived. @@ -3904,8 +5953,8 @@ From these two simple facts the following simple algorithm can be derived. The algorithm reduces the input one bit at a time using the two congruencies stated previously. Inside the loop $n$, which is odd, is added to $x$ if $x$ is odd. This forces $x$ to be even which allows the division by two in $\Z$ to be congruent to a modular division by two. Since -$x$ is assumed to be initially much larger than $n$ the addition of $n$ will contribute an insignificant magnitude to $x$. Let $r$ represent the -final result of the Montgomery algorithm. If $k > lg(n)$ and $0 \le x < n^2$ then the final result is limited to +$x$ is assumed to be initially much larger than $n$ the addition of $n$ will contribute an insignificant magnitude to $x$. Let $r$ represent the +final result of the Montgomery algorithm. If $k > lg(n)$ and $0 \le x < n^2$ then the final result is limited to $0 \le r < \lfloor x/2^k \rfloor + n$. As a result at most a single subtraction is required to get the residue desired. \begin{figure}[here] @@ -3930,12 +5979,12 @@ $0 \le r < \lfloor x/2^k \rfloor + n$. As a result at most a single subtraction \label{fig:MONT1} \end{figure} -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 -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 -$r \equiv 158$ is produced. +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 +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 +$r \equiv 158$ is produced. Let $k = \lfloor lg(n) \rfloor + 1$ represent the number of bits in $n$. The current algorithm requires $2k^2$ single precision shifts -and $k^2$ single precision additions. At this rate the algorithm is most certainly slower than Barrett reduction and not terribly useful. +and $k^2$ single precision additions. At this rate the algorithm is most certainly slower than Barrett reduction and not terribly useful. Fortunately there exists an alternative representation of the algorithm. \begin{figure}[!here] @@ -3984,10 +6033,10 @@ precision shifts has now been reduced from $2k^2$ to $k^2 + k$ which is only a s \label{fig:MONT2} \end{figure} -Figure~\ref{fig:MONT2} demonstrates the modified algorithm reducing $x = 5555$ modulo $n = 257$ with $k = 9$. -With this algorithm a single shift right at the end is the only right shift required to reduce the input instead of $k$ right shifts inside the -loop. Note that for the iterations $t = 2, 5, 6$ and $8$ where the result $x$ is not changed. In those iterations the $t$'th bit of $x$ is -zero and the appropriate multiple of $n$ does not need to be added to force the $t$'th bit of the result to zero. +Figure~\ref{fig:MONT2} demonstrates the modified algorithm reducing $x = 5555$ modulo $n = 257$ with $k = 9$. +With this algorithm a single shift right at the end is the only right shift required to reduce the input instead of $k$ right shifts inside the +loop. Note that for the iterations $t = 2, 5, 6$ and $8$ where the result $x$ is not changed. In those iterations the $t$'th bit of $x$ is +zero and the appropriate multiple of $n$ does not need to be added to force the $t$'th bit of the result to zero. \subsection{Digit Based Montgomery Reduction} Instead of computing the reduction on a bit-by-bit basis it is actually much faster to compute it on digit-by-digit basis. Consider the @@ -4011,9 +6060,9 @@ previous algorithm re-written to compute the Montgomery reduction in this new fa \caption{Algorithm Montgomery Reduction (modified II)} \end{figure} -The value $\mu n \beta^t$ is a multiple of the modulus $n$ meaning that it will not change the residue. If the first digit of +The value $\mu n \beta^t$ is a multiple of the modulus $n$ meaning that it will not change the residue. If the first digit of the value $\mu n \beta^t$ equals the negative (modulo $\beta$) of the $t$'th digit of $x$ then the addition will result in a zero digit. This -problem breaks down to solving the following congruency. +problem breaks down to solving the following congruency. \begin{center} \begin{tabular}{rcl} @@ -4023,10 +6072,10 @@ $\mu$ & $\equiv$ & $-x_t/n_0 \mbox{ (mod }\beta\mbox{)}$ \\ \end{tabular} \end{center} -In each iteration of the loop on step 1 a new value of $\mu$ must be calculated. The value of $-1/n_0 \mbox{ (mod }\beta\mbox{)}$ is used -extensively in this algorithm and should be precomputed. Let $\rho$ represent the negative of the modular inverse of $n_0$ modulo $\beta$. +In each iteration of the loop on step 1 a new value of $\mu$ must be calculated. The value of $-1/n_0 \mbox{ (mod }\beta\mbox{)}$ is used +extensively in this algorithm and should be precomputed. Let $\rho$ represent the negative of the modular inverse of $n_0$ modulo $\beta$. -For example, let $\beta = 10$ represent the radix. Let $n = 17$ represent the modulus which implies $k = 2$ and $\rho \equiv 7$. Let $x = 33$ +For example, let $\beta = 10$ represent the radix. Let $n = 17$ represent the modulus which implies $k = 2$ and $\rho \equiv 7$. Let $x = 33$ represent the value to reduce. \newpage\begin{figure} @@ -4042,14 +6091,14 @@ represent the value to reduce. \caption{Example of Montgomery Reduction} \end{figure} -The final result $900$ is then divided by $\beta^k$ to produce the final result $9$. The first observation is that $9 \nequiv x \mbox{ (mod }n\mbox{)}$ +The final result $900$ is then divided by $\beta^k$ to produce the final result $9$. The first observation is that $9 \nequiv x \mbox{ (mod }n\mbox{)}$ which implies the result is not the modular residue of $x$ modulo $n$. However, recall that the residue is actually multiplied by $\beta^{-k}$ in the algorithm. To get the true residue the value must be multiplied by $\beta^k$. In this case $\beta^k \equiv 15 \mbox{ (mod }n\mbox{)}$ and -the correct residue is $9 \cdot 15 \equiv 16 \mbox{ (mod }n\mbox{)}$. +the correct residue is $9 \cdot 15 \equiv 16 \mbox{ (mod }n\mbox{)}$. \subsection{Baseline Montgomery Reduction} -The baseline Montgomery reduction algorithm will produce the residue for any size input. It is designed to be a catch-all algororithm for -Montgomery reductions. +The baseline Montgomery reduction algorithm will produce the residue for any size input. It is designed to be a catch-all algororithm for +Montgomery reductions. \newpage\begin{figure}[!here] \begin{small} @@ -4097,9 +6146,9 @@ Divide by $\beta^k$ and fix up as required. \\ \textbf{Algorithm mp\_montgomery\_reduce.} This algorithm reduces the input $x$ modulo $n$ in place using the Montgomery reduction algorithm. The algorithm is loosely based on algorithm 14.32 of \cite[pp.601]{HAC} except it merges the multiplication of $\mu n \beta^t$ with the addition in the inner loop. The -restrictions on this algorithm are fairly easy to adapt to. First $0 \le x < n^2$ bounds the input to numbers in the same range as +restrictions on this algorithm are fairly easy to adapt to. First $0 \le x < n^2$ bounds the input to numbers in the same range as for the Barrett algorithm. Additionally if $n > 1$ and $n$ is odd there will exist a modular inverse $\rho$. $\rho$ must be calculated in -advance of this algorithm. Finally the variable $k$ is fixed and a pseudonym for $n.used$. +advance of this algorithm. Finally the variable $k$ is fixed and a pseudonym for $n.used$. Step 2 decides whether a faster Montgomery algorithm can be used. It is based on the Comba technique meaning that there are limits on the size of the input. This algorithm is discussed in sub-section 6.3.3. @@ -4108,36 +6157,135 @@ Step 5 is the main reduction loop of the algorithm. The value of $\mu$ is calcu calculates $x + \mu n \beta^{ix}$ by multiplying $\mu n$ and adding the result to $x$ shifted by $ix$ digits. Both the addition and multiplication are performed in the same loop to save time and memory. Step 5.4 will handle any additional carries that escape the inner loop. -Using a quick inspection this algorithm requires $n$ single precision multiplications for the outer loop and $n^2$ single precision multiplications +Using a quick inspection this algorithm requires $n$ single precision multiplications for the outer loop and $n^2$ single precision multiplications in the inner loop. In total $n^2 + n$ single precision multiplications which compares favourably to Barrett at $n^2 + 2n - 1$ single precision -multiplications. +multiplications. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_montgomery\_reduce.c \vspace{-3mm} \begin{alltt} +016 +017 /* computes xR**-1 == x (mod N) via Montgomery Reduction */ +018 int +019 mp_montgomery_reduce (mp_int * x, mp_int * n, mp_digit rho) +020 \{ +021 int ix, res, digs; +022 mp_digit mu; +023 +024 /* can the fast reduction [comba] method be used? +025 * +026 * Note that unlike in mul you're safely allowed *less* +027 * than the available columns [255 per default] since carries +028 * are fixed up in the inner loop. +029 */ +030 digs = (n->used * 2) + 1; +031 if ((digs < MP_WARRAY) && +032 (n->used < +033 (1 << ((CHAR_BIT * sizeof(mp_word)) - (2 * DIGIT_BIT))))) \{ +034 return fast_mp_montgomery_reduce (x, n, rho); +035 \} +036 +037 /* grow the input as required */ +038 if (x->alloc < digs) \{ +039 if ((res = mp_grow (x, digs)) != MP_OKAY) \{ +040 return res; +041 \} +042 \} +043 x->used = digs; +044 +045 for (ix = 0; ix < n->used; ix++) \{ +046 /* mu = ai * rho mod b +047 * +048 * The value of rho must be precalculated via +049 * montgomery_setup() such that +050 * it equals -1/n0 mod b this allows the +051 * following inner loop to reduce the +052 * input one digit at a time +053 */ +054 mu = (mp_digit) (((mp_word)x->dp[ix] * (mp_word)rho) & MP_MASK); +055 +056 /* a = a + mu * m * b**i */ +057 \{ +058 int iy; +059 mp_digit *tmpn, *tmpx, u; +060 mp_word r; +061 +062 /* alias for digits of the modulus */ +063 tmpn = n->dp; +064 +065 /* alias for the digits of x [the input] */ +066 tmpx = x->dp + ix; +067 +068 /* set the carry to zero */ +069 u = 0; +070 +071 /* Multiply and add in place */ +072 for (iy = 0; iy < n->used; iy++) \{ +073 /* compute product and sum */ +074 r = ((mp_word)mu * (mp_word)*tmpn++) + +075 (mp_word) u + (mp_word) *tmpx; +076 +077 /* get carry */ +078 u = (mp_digit)(r >> ((mp_word) DIGIT_BIT)); +079 +080 /* fix digit */ +081 *tmpx++ = (mp_digit)(r & ((mp_word) MP_MASK)); +082 \} +083 /* At this point the ix'th digit of x should be zero */ +084 +085 +086 /* propagate carries upwards as required*/ +087 while (u != 0) \{ +088 *tmpx += u; +089 u = *tmpx >> DIGIT_BIT; +090 *tmpx++ &= MP_MASK; +091 \} +092 \} +093 \} +094 +095 /* at this point the n.used'th least +096 * significant digits of x are all zero +097 * which means we can shift x to the +098 * right by n.used digits and the +099 * residue is unchanged. +100 */ +101 +102 /* x = x/b**n.used */ +103 mp_clamp(x); +104 mp_rshd (x, n->used); +105 +106 /* if x >= n then x = x - n */ +107 if (mp_cmp_mag (x, n) != MP_LT) \{ +108 return s_mp_sub (x, n, x); +109 \} +110 +111 return MP_OKAY; +112 \} +113 #endif +114 \end{alltt} \end{small} -This is the baseline implementation of the Montgomery reduction algorithm. Lines 31 to 36 determine if the Comba based -routine can be used instead. Line 47 computes the value of $\mu$ for that particular iteration of the outer loop. +This is the baseline implementation of the Montgomery reduction algorithm. Lines 30 to 35 determine if the Comba based +routine can be used instead. Line 48 computes the value of $\mu$ for that particular iteration of the outer loop. The multiplication $\mu n \beta^{ix}$ is performed in one step in the inner loop. The alias $tmpx$ refers to the $ix$'th digit of $x$ and -the alias $tmpn$ refers to the modulus $n$. +the alias $tmpn$ refers to the modulus $n$. \subsection{Faster ``Comba'' Montgomery Reduction} The Montgomery reduction requires fewer single precision multiplications than a Barrett reduction, however it is much slower due to the serial nature of the inner loop. The Barrett reduction algorithm requires two slightly modified multipliers which can be implemented with the Comba technique. The Montgomery reduction algorithm cannot directly use the Comba technique to any significant advantage since the inner loop calculates -a $k \times 1$ product $k$ times. +a $k \times 1$ product $k$ times. -The biggest obstacle is that at the $ix$'th iteration of the outer loop the value of $x_{ix}$ is required to calculate $\mu$. This means the -carries from $0$ to $ix - 1$ must have been propagated upwards to form a valid $ix$'th digit. The solution as it turns out is very simple. -Perform a Comba like multiplier and inside the outer loop just after the inner loop fix up the $ix + 1$'th digit by forwarding the carry. +The biggest obstacle is that at the $ix$'th iteration of the outer loop the value of $x_{ix}$ is required to calculate $\mu$. This means the +carries from $0$ to $ix - 1$ must have been propagated upwards to form a valid $ix$'th digit. The solution as it turns out is very simple. +Perform a Comba like multiplier and inside the outer loop just after the inner loop fix up the $ix + 1$'th digit by forwarding the carry. With this change in place the Montgomery reduction algorithm can be performed with a Comba style multiplication loop which substantially increases -the speed of the algorithm. +the speed of the algorithm. \newpage\begin{figure}[!here] \begin{small} @@ -4186,9 +6334,9 @@ Zero excess digits and fixup $x$. \\ \textbf{Algorithm fast\_mp\_montgomery\_reduce.} This algorithm will compute the Montgomery reduction of $x$ modulo $n$ using the Comba technique. It is on most computer platforms significantly faster than algorithm mp\_montgomery\_reduce and algorithm mp\_reduce (\textit{Barrett reduction}). The algorithm has the same restrictions -on the input as the baseline reduction algorithm. An additional two restrictions are imposed on this algorithm. The number of digits $k$ in the +on the input as the baseline reduction algorithm. An additional two restrictions are imposed on this algorithm. The number of digits $k$ in the the modulus $n$ must not violate $MP\_WARRAY > 2k +1$ and $n < \delta$. When $\beta = 2^{28}$ this algorithm can be used to reduce modulo -a modulus of at most $3,556$ bits in length. +a modulus of at most $3,556$ bits in length. As in the other Comba reduction algorithms there is a $\hat W$ array which stores the columns of the product. It is initially filled with the contents of $x$ with the excess digits zeroed. The reduction loop is very similar the to the baseline loop at heart. The multiplication on step @@ -4202,28 +6350,181 @@ how the upper bits of those same words are not reduced modulo $\beta$. This is point. Step 5 will propagate the remainder of the carries upwards. On step 6 the columns are reduced modulo $\beta$ and shifted simultaneously as they are -stored in the destination $x$. +stored in the destination $x$. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_fast\_mp\_montgomery\_reduce.c \vspace{-3mm} \begin{alltt} +016 +017 /* computes xR**-1 == x (mod N) via Montgomery Reduction +018 * +019 * This is an optimized implementation of montgomery_reduce +020 * which uses the comba method to quickly calculate the columns of the +021 * reduction. +022 * +023 * Based on Algorithm 14.32 on pp.601 of HAC. +024 */ +025 int fast_mp_montgomery_reduce (mp_int * x, mp_int * n, mp_digit rho) +026 \{ +027 int ix, res, olduse; +028 mp_word W[MP_WARRAY]; +029 +030 /* get old used count */ +031 olduse = x->used; +032 +033 /* grow a as required */ +034 if (x->alloc < (n->used + 1)) \{ +035 if ((res = mp_grow (x, n->used + 1)) != MP_OKAY) \{ +036 return res; +037 \} +038 \} +039 +040 /* first we have to get the digits of the input into +041 * an array of double precision words W[...] +042 */ +043 \{ +044 mp_word *_W; +045 mp_digit *tmpx; +046 +047 /* alias for the W[] array */ +048 _W = W; +049 +050 /* alias for the digits of x*/ +051 tmpx = x->dp; +052 +053 /* copy the digits of a into W[0..a->used-1] */ +054 for (ix = 0; ix < x->used; ix++) \{ +055 *_W++ = *tmpx++; +056 \} +057 +058 /* zero the high words of W[a->used..m->used*2] */ +059 for (; ix < ((n->used * 2) + 1); ix++) \{ +060 *_W++ = 0; +061 \} +062 \} +063 +064 /* now we proceed to zero successive digits +065 * from the least significant upwards +066 */ +067 for (ix = 0; ix < n->used; ix++) \{ +068 /* mu = ai * m' mod b +069 * +070 * We avoid a double precision multiplication (which isn't required) +071 * by casting the value down to a mp_digit. Note this requires +072 * that W[ix-1] have the carry cleared (see after the inner loop) +073 */ +074 mp_digit mu; +075 mu = (mp_digit) (((W[ix] & MP_MASK) * rho) & MP_MASK); +076 +077 /* a = a + mu * m * b**i +078 * +079 * This is computed in place and on the fly. The multiplication +080 * by b**i is handled by offseting which columns the results +081 * are added to. +082 * +083 * Note the comba method normally doesn't handle carries in the +084 * inner loop In this case we fix the carry from the previous +085 * column since the Montgomery reduction requires digits of the +086 * result (so far) [see above] to work. This is +087 * handled by fixing up one carry after the inner loop. The +088 * carry fixups are done in order so after these loops the +089 * first m->used words of W[] have the carries fixed +090 */ +091 \{ +092 int iy; +093 mp_digit *tmpn; +094 mp_word *_W; +095 +096 /* alias for the digits of the modulus */ +097 tmpn = n->dp; +098 +099 /* Alias for the columns set by an offset of ix */ +100 _W = W + ix; +101 +102 /* inner loop */ +103 for (iy = 0; iy < n->used; iy++) \{ +104 *_W++ += ((mp_word)mu) * ((mp_word)*tmpn++); +105 \} +106 \} +107 +108 /* now fix carry for next digit, W[ix+1] */ +109 W[ix + 1] += W[ix] >> ((mp_word) DIGIT_BIT); +110 \} +111 +112 /* now we have to propagate the carries and +113 * shift the words downward [all those least +114 * significant digits we zeroed]. +115 */ +116 \{ +117 mp_digit *tmpx; +118 mp_word *_W, *_W1; +119 +120 /* nox fix rest of carries */ +121 +122 /* alias for current word */ +123 _W1 = W + ix; +124 +125 /* alias for next word, where the carry goes */ +126 _W = W + ++ix; +127 +128 for (; ix <= ((n->used * 2) + 1); ix++) \{ +129 *_W++ += *_W1++ >> ((mp_word) DIGIT_BIT); +130 \} +131 +132 /* copy out, A = A/b**n +133 * +134 * The result is A/b**n but instead of converting from an +135 * array of mp_word to mp_digit than calling mp_rshd +136 * we just copy them in the right order +137 */ +138 +139 /* alias for destination word */ +140 tmpx = x->dp; +141 +142 /* alias for shifted double precision result */ +143 _W = W + n->used; +144 +145 for (ix = 0; ix < (n->used + 1); ix++) \{ +146 *tmpx++ = (mp_digit)(*_W++ & ((mp_word) MP_MASK)); +147 \} +148 +149 /* zero oldused digits, if the input a was larger than +150 * m->used+1 we'll have to clear the digits +151 */ +152 for (; ix < olduse; ix++) \{ +153 *tmpx++ = 0; +154 \} +155 \} +156 +157 /* set the max used and clamp */ +158 x->used = n->used + 1; +159 mp_clamp (x); +160 +161 /* if A >= m then A = A - m */ +162 if (mp_cmp_mag (x, n) != MP_LT) \{ +163 return s_mp_sub (x, n, x); +164 \} +165 return MP_OKAY; +166 \} +167 #endif +168 \end{alltt} \end{small} -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 -the same alias variables to make the code easier to read. +The $\hat W$ array is first filled with digits of $x$ on line 50 then the rest of the digits are zeroed on line 54. Both loops share +the same alias variables to make the code easier to read. The value of $\mu$ is calculated in an interesting fashion. First the value $\hat W_{ix}$ is reduced modulo $\beta$ and cast to a mp\_digit. This -forces the compiler to use a single precision multiplication and prevents any concerns about loss of precision. Line 110 fixes the carry +forces the compiler to use a single precision multiplication and prevents any concerns about loss of precision. Line 109 fixes the carry for the next iteration of the loop by propagating the carry from $\hat W_{ix}$ to $\hat W_{ix+1}$. -The for loop on line 109 propagates the rest of the carries upwards through the columns. The for loop on line 126 reduces the columns +The for loop on line 108 propagates the rest of the carries upwards through the columns. The for loop on line 125 reduces the columns modulo $\beta$ and shifts them $k$ places at the same time. The alias $\_ \hat W$ actually refers to the array $\hat W$ starting at the $n.used$'th -digit, that is $\_ \hat W_{t} = \hat W_{n.used + t}$. +digit, that is $\_ \hat W_{t} = \hat W_{n.used + t}$. \subsection{Montgomery Setup} -To calculate the variable $\rho$ a relatively simple algorithm will be required. +To calculate the variable $\rho$ a relatively simple algorithm will be required. \begin{figure}[!here] \begin{small} @@ -4244,22 +6545,62 @@ To calculate the variable $\rho$ a relatively simple algorithm will be required. \end{tabular} \end{center} \end{small} -\caption{Algorithm mp\_montgomery\_setup} +\caption{Algorithm mp\_montgomery\_setup} \end{figure} \textbf{Algorithm mp\_montgomery\_setup.} -This algorithm will calculate the value of $\rho$ required within the Montgomery reduction algorithms. It uses a very interesting trick -to calculate $1/n_0$ when $\beta$ is a power of two. +This algorithm will calculate the value of $\rho$ required within the Montgomery reduction algorithms. It uses a very interesting trick +to calculate $1/n_0$ when $\beta$ is a power of two. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_montgomery\_setup.c \vspace{-3mm} \begin{alltt} +016 +017 /* setups the montgomery reduction stuff */ +018 int +019 mp_montgomery_setup (mp_int * n, mp_digit * rho) +020 \{ +021 mp_digit x, b; +022 +023 /* fast inversion mod 2**k +024 * +025 * Based on the fact that +026 * +027 * XA = 1 (mod 2**n) => (X(2-XA)) A = 1 (mod 2**2n) +028 * => 2*X*A - X*X*A*A = 1 +029 * => 2*(1) - (1) = 1 +030 */ +031 b = n->dp[0]; +032 +033 if ((b & 1) == 0) \{ +034 return MP_VAL; +035 \} +036 +037 x = (((b + 2) & 4) << 1) + b; /* here x*a==1 mod 2**4 */ +038 x *= 2 - (b * x); /* here x*a==1 mod 2**8 */ +039 #if !defined(MP_8BIT) +040 x *= 2 - (b * x); /* here x*a==1 mod 2**16 */ +041 #endif +042 #if defined(MP_64BIT) || !(defined(MP_8BIT) || defined(MP_16BIT)) +043 x *= 2 - (b * x); /* here x*a==1 mod 2**32 */ +044 #endif +045 #ifdef MP_64BIT +046 x *= 2 - (b * x); /* here x*a==1 mod 2**64 */ +047 #endif +048 +049 /* rho = -1/m mod b */ +050 *rho = (mp_digit)(((mp_word)1 << ((mp_word) DIGIT_BIT)) - x) & MP_MASK; +051 +052 return MP_OKAY; +053 \} +054 #endif +055 \end{alltt} \end{small} This source code computes the value of $\rho$ required to perform Montgomery reduction. It has been modified to avoid performing excess -multiplications when $\beta$ is not the default 28-bits. +multiplications when $\beta$ is not the default 28-bits. \section{The Diminished Radix Algorithm} The Diminished Radix method of modular reduction \cite{DRMET} is a fairly clever technique which can be more efficient than either the Barrett @@ -4269,7 +6610,7 @@ or Montgomery methods for certain forms of moduli. The technique is based on th (x \mbox{ mod } n) + k \lfloor x / n \rfloor \equiv x \mbox{ (mod }(n - k)\mbox{)} \end{equation} -This observation was used in the MMB \cite{MMB} block cipher to create a diffusion primitive. It used the fact that if $n = 2^{31}$ and $k=1$ that +This observation was used in the MMB \cite{MMB} block cipher to create a diffusion primitive. It used the fact that if $n = 2^{31}$ and $k=1$ that then a x86 multiplier could produce the 62-bit product and use the ``shrd'' instruction to perform a double-precision right shift. The proof of the above equation is very simple. First write $x$ in the product form. @@ -4283,7 +6624,7 @@ Now reduce both sides modulo $(n - k)$. x \equiv qk + r \mbox{ (mod }(n-k)\mbox{)} \end{equation} -The variable $n$ reduces modulo $n - k$ to $k$. By putting $q = \lfloor x/n \rfloor$ and $r = x \mbox{ mod } n$ +The variable $n$ reduces modulo $n - k$ to $k$. By putting $q = \lfloor x/n \rfloor$ and $r = x \mbox{ mod } n$ into the equation the original congruence is reproduced, thus concluding the proof. The following algorithm is based on this observation. \begin{figure}[!here] @@ -4313,7 +6654,7 @@ into the equation the original congruence is reproduced, thus concluding the pro This algorithm will reduce $x$ modulo $n - k$ and return the residue. If $0 \le x < (n - k)^2$ then the algorithm will loop almost always once or twice and occasionally three times. For simplicity sake the value of $x$ is bounded by the following simple polynomial. -\begin{equation} +\begin{equation} 0 \le x < n^2 + k^2 - 2nk \end{equation} @@ -4324,15 +6665,15 @@ q < n - 2k - k^2/n \end{equation} Since $k^2$ is going to be considerably smaller than $n$ that term will always be zero. The value of $x$ after step 3 is bounded trivially as -$0 \le x < n$. By step four the sum $x + q$ is bounded by +$0 \le x < n$. By step four the sum $x + q$ is bounded by \begin{equation} 0 \le q + x < (k + 1)n - 2k^2 - 1 \end{equation} With a second pass $q$ will be loosely bounded by $0 \le q < k^2$ after step 2 while $x$ will still be loosely bounded by $0 \le x < n$ after step 3. After the second pass it is highly unlike that the -sum in step 4 will exceed $n - k$. In practice fewer than three passes of the algorithm are required to reduce virtually every input in the -range $0 \le x < (n - k - 1)^2$. +sum in step 4 will exceed $n - k$. In practice fewer than three passes of the algorithm are required to reduce virtually every input in the +range $0 \le x < (n - k - 1)^2$. \begin{figure} \begin{small} @@ -4345,13 +6686,13 @@ $q \leftarrow q*k = 1446759$ \\ $x \leftarrow x \mbox{ mod } n = 21$ \\ $x \leftarrow x + q = 1446780$ \\ $x \leftarrow x - (n - k) = 1446527$ \\ -\hline +\hline $q \leftarrow \lfloor x/n \rfloor = 5650$ \\ $q \leftarrow q*k = 16950$ \\ $x \leftarrow x \mbox{ mod } n = 127$ \\ $x \leftarrow x + q = 17077$ \\ $x \leftarrow x - (n - k) = 16824$ \\ -\hline +\hline $q \leftarrow \lfloor x/n \rfloor = 65$ \\ $q \leftarrow q*k = 195$ \\ $x \leftarrow x \mbox{ mod } n = 184$ \\ @@ -4374,29 +6715,29 @@ three passes were required to find the residue $x \equiv 126$. On the surface this algorithm looks like a very expensive algorithm. It requires a couple of subtractions followed by multiplication and other modular reductions. The usefulness of this algorithm becomes exceedingly clear when an appropriate modulus is chosen. -Division in general is a very expensive operation to perform. The one exception is when the division is by a power of the radix of representation used. -Division by ten for example is simple for pencil and paper mathematics since it amounts to shifting the decimal place to the right. Similarly division -by two (\textit{or powers of two}) is very simple for binary computers to perform. It would therefore seem logical to choose $n$ of the form $2^p$ -which would imply that $\lfloor x / n \rfloor$ is a simple shift of $x$ right $p$ bits. +Division in general is a very expensive operation to perform. The one exception is when the division is by a power of the radix of representation used. +Division by ten for example is simple for pencil and paper mathematics since it amounts to shifting the decimal place to the right. Similarly division +by two (\textit{or powers of two}) is very simple for binary computers to perform. It would therefore seem logical to choose $n$ of the form $2^p$ +which would imply that $\lfloor x / n \rfloor$ is a simple shift of $x$ right $p$ bits. -However, there is one operation related to division of power of twos that is even faster than this. If $n = \beta^p$ then the division may be -performed by moving whole digits to the right $p$ places. In practice division by $\beta^p$ is much faster than division by $2^p$ for any $p$. -Also with the choice of $n = \beta^p$ reducing $x$ modulo $n$ merely requires zeroing the digits above the $p-1$'th digit of $x$. +However, there is one operation related to division of power of twos that is even faster than this. If $n = \beta^p$ then the division may be +performed by moving whole digits to the right $p$ places. In practice division by $\beta^p$ is much faster than division by $2^p$ for any $p$. +Also with the choice of $n = \beta^p$ reducing $x$ modulo $n$ merely requires zeroing the digits above the $p-1$'th digit of $x$. Throughout the next section the term ``restricted modulus'' will refer to a modulus of the form $\beta^p - k$ whereas the term ``unrestricted -modulus'' will refer to a modulus of the form $2^p - k$. The word ``restricted'' in this case refers to the fact that it is based on the -$2^p$ logic except $p$ must be a multiple of $lg(\beta)$. +modulus'' will refer to a modulus of the form $2^p - k$. The word ``restricted'' in this case refers to the fact that it is based on the +$2^p$ logic except $p$ must be a multiple of $lg(\beta)$. \subsection{Choice of $k$} Now that division and reduction (\textit{step 1 and 3 of figure~\ref{fig:DR}}) have been optimized to simple digit operations the multiplication by $k$ in step 2 is the most expensive operation. Fortunately the choice of $k$ is not terribly limited. For all intents and purposes it might -as well be a single digit. The smaller the value of $k$ is the faster the algorithm will be. +as well be a single digit. The smaller the value of $k$ is the faster the algorithm will be. \subsection{Restricted Diminished Radix Reduction} -The restricted Diminished Radix algorithm can quickly reduce an input modulo a modulus of the form $n = \beta^p - k$. This algorithm can reduce +The restricted Diminished Radix algorithm can quickly reduce an input modulo a modulus of the form $n = \beta^p - k$. This algorithm can reduce an input $x$ within the range $0 \le x < n^2$ using only a couple passes of the algorithm demonstrated in figure~\ref{fig:DR}. The implementation -of this algorithm has been optimized to avoid additional overhead associated with a division by $\beta^p$, the multiplication by $k$ or the addition -of $x$ and $q$. The resulting algorithm is very efficient and can lead to substantial improvements over Barrett and Montgomery reduction when modular +of this algorithm has been optimized to avoid additional overhead associated with a division by $\beta^p$, the multiplication by $k$ or the addition +of $x$ and $q$. The resulting algorithm is very efficient and can lead to substantial improvements over Barrett and Montgomery reduction when modular exponentiations are performed. \newpage\begin{figure}[!here] @@ -4432,37 +6773,114 @@ exponentiations are performed. \textbf{Algorithm mp\_dr\_reduce.} This algorithm will perform the Dimished Radix reduction of $x$ modulo $n$. It has similar restrictions to that of the Barrett reduction -with the addition that $n$ must be of the form $n = \beta^m - k$ where $0 < k <\beta$. +with the addition that $n$ must be of the form $n = \beta^m - k$ where $0 < k <\beta$. This algorithm essentially implements the pseudo-code in figure~\ref{fig:DR} except with a slight optimization. The division by $\beta^m$, multiplication by $k$ and addition of $x \mbox{ mod }\beta^m$ are all performed simultaneously inside the loop on step 4. The division by $\beta^m$ is emulated by accessing the term at the $m+i$'th position which is subsequently multiplied by $k$ and added to the term at the $i$'th position. After the loop the $m$'th -digit is set to the carry and the upper digits are zeroed. Steps 5 and 6 emulate the reduction modulo $\beta^m$ that should have happend to -$x$ before the addition of the multiple of the upper half. +digit is set to the carry and the upper digits are zeroed. Steps 5 and 6 emulate the reduction modulo $\beta^m$ that should have happend to +$x$ before the addition of the multiple of the upper half. At step 8 if $x$ is still larger than $n$ another pass of the algorithm is required. First $n$ is subtracted from $x$ and then the algorithm resumes -at step 3. +at step 3. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_reduce.c \vspace{-3mm} \begin{alltt} +016 +017 /* reduce "x" in place modulo "n" using the Diminished Radix algorithm. +018 * +019 * Based on algorithm from the paper +020 * +021 * "Generating Efficient Primes for Discrete Log Cryptosystems" +022 * Chae Hoon Lim, Pil Joong Lee, +023 * POSTECH Information Research Laboratories +024 * +025 * The modulus must be of a special format [see manual] +026 * +027 * Has been modified to use algorithm 7.10 from the LTM book instead +028 * +029 * Input x must be in the range 0 <= x <= (n-1)**2 +030 */ +031 int +032 mp_dr_reduce (mp_int * x, mp_int * n, mp_digit k) +033 \{ +034 int err, i, m; +035 mp_word r; +036 mp_digit mu, *tmpx1, *tmpx2; +037 +038 /* m = digits in modulus */ +039 m = n->used; +040 +041 /* ensure that "x" has at least 2m digits */ +042 if (x->alloc < (m + m)) \{ +043 if ((err = mp_grow (x, m + m)) != MP_OKAY) \{ +044 return err; +045 \} +046 \} +047 +048 /* top of loop, this is where the code resumes if +049 * another reduction pass is required. +050 */ +051 top: +052 /* aliases for digits */ +053 /* alias for lower half of x */ +054 tmpx1 = x->dp; +055 +056 /* alias for upper half of x, or x/B**m */ +057 tmpx2 = x->dp + m; +058 +059 /* set carry to zero */ +060 mu = 0; +061 +062 /* compute (x mod B**m) + k * [x/B**m] inline and inplace */ +063 for (i = 0; i < m; i++) \{ +064 r = (((mp_word)*tmpx2++) * (mp_word)k) + *tmpx1 + mu; +065 *tmpx1++ = (mp_digit)(r & MP_MASK); +066 mu = (mp_digit)(r >> ((mp_word)DIGIT_BIT)); +067 \} +068 +069 /* set final carry */ +070 *tmpx1++ = mu; +071 +072 /* zero words above m */ +073 for (i = m + 1; i < x->used; i++) \{ +074 *tmpx1++ = 0; +075 \} +076 +077 /* clamp, sub and return */ +078 mp_clamp (x); +079 +080 /* if x >= n then subtract and reduce again +081 * Each successive "recursion" makes the input smaller and smaller. +082 */ +083 if (mp_cmp_mag (x, n) != MP_LT) \{ +084 if ((err = s_mp_sub(x, n, x)) != MP_OKAY) \{ +085 return err; +086 \} +087 goto top; +088 \} +089 return MP_OKAY; +090 \} +091 #endif +092 \end{alltt} \end{small} -The first step is to grow $x$ as required to $2m$ digits since the reduction is performed in place on $x$. The label on line 52 is where +The first step is to grow $x$ as required to $2m$ digits since the reduction is performed in place on $x$. The label on line 51 is where the algorithm will resume if further reduction passes are required. In theory it could be placed at the top of the function however, the size of -the modulus and question of whether $x$ is large enough are invariant after the first pass meaning that it would be a waste of time. +the modulus and question of whether $x$ is large enough are invariant after the first pass meaning that it would be a waste of time. The aliases $tmpx1$ and $tmpx2$ refer to the digits of $x$ where the latter is offset by $m$ digits. By reading digits from $x$ offset by $m$ digits -a division by $\beta^m$ can be simulated virtually for free. The loop on line 64 performs the bulk of the work (\textit{corresponds to step 4 of algorithm 7.11}) +a division by $\beta^m$ can be simulated virtually for free. The loop on line 63 performs the bulk of the work (\textit{corresponds to step 4 of algorithm 7.11}) in this algorithm. -By line 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 -same pointer will point to the $m+1$'th digit where the zeroes will be placed. +By line 70 the pointer $tmpx1$ points to the $m$'th digit of $x$ which is where the final carry will be placed. Similarly by line 73 the +same pointer will point to the $m+1$'th digit where the zeroes will be placed. -Since the algorithm is only valid if both $x$ and $n$ are greater than zero an unsigned comparison suffices to determine if another pass is required. -With the same logic at line 81 the value of $x$ is known to be greater than or equal to $n$ meaning that an unsigned subtraction can be used +Since the algorithm is only valid if both $x$ and $n$ are greater than zero an unsigned comparison suffices to determine if another pass is required. +With the same logic at line 84 the value of $x$ is known to be greater than or equal to $n$ meaning that an unsigned subtraction can be used as well. Since the destination of the subtraction is the larger of the inputs the call to algorithm s\_mp\_sub cannot fail and the return code does not need to be checked. @@ -4490,6 +6908,19 @@ completeness. \hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_setup.c \vspace{-3mm} \begin{alltt} +016 +017 /* determines the setup value */ +018 void mp_dr_setup(mp_int *a, mp_digit *d) +019 \{ +020 /* the casts are required if DIGIT_BIT is one less than +021 * the number of bits in a mp_digit [e.g. DIGIT_BIT==31] +022 */ +023 *d = (mp_digit)((((mp_word)1) << ((mp_word)DIGIT_BIT)) - +024 ((mp_word)a->dp[0])); +025 \} +026 +027 #endif +028 \end{alltt} \end{small} @@ -4525,6 +6956,30 @@ step 3 then $n$ must be of Diminished Radix form. \hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_is\_modulus.c \vspace{-3mm} \begin{alltt} +016 +017 /* determines if a number is a valid DR modulus */ +018 int mp_dr_is_modulus(mp_int *a) +019 \{ +020 int ix; +021 +022 /* must be at least two digits */ +023 if (a->used < 2) \{ +024 return 0; +025 \} +026 +027 /* must be of the form b**k - a [a <= b] so all +028 * but the first digit must be equal to -1 (mod b). +029 */ +030 for (ix = 1; ix < a->used; ix++) \{ +031 if (a->dp[ix] != MP_MASK) \{ +032 return 0; +033 \} +034 \} +035 return 1; +036 \} +037 +038 #endif +039 \end{alltt} \end{small} @@ -4562,25 +7017,69 @@ algorithm is much faster than either Montgomery or Barrett reduction when the mo \textbf{Algorithm mp\_reduce\_2k.} This algorithm quickly reduces an input $a$ modulo an unrestricted Diminished Radix modulus $n$. Division by $2^p$ is emulated with a right -shift which makes the algorithm fairly inexpensive to use. +shift which makes the algorithm fairly inexpensive to use. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_2k.c \vspace{-3mm} \begin{alltt} +016 +017 /* reduces a modulo n where n is of the form 2**p - d */ +018 int mp_reduce_2k(mp_int *a, mp_int *n, mp_digit d) +019 \{ +020 mp_int q; +021 int p, res; +022 +023 if ((res = mp_init(&q)) != MP_OKAY) \{ +024 return res; +025 \} +026 +027 p = mp_count_bits(n); +028 top: +029 /* q = a/2**p, a = a mod 2**p */ +030 if ((res = mp_div_2d(a, p, &q, a)) != MP_OKAY) \{ +031 goto ERR; +032 \} +033 +034 if (d != 1) \{ +035 /* q = q * d */ +036 if ((res = mp_mul_d(&q, d, &q)) != MP_OKAY) \{ +037 goto ERR; +038 \} +039 \} +040 +041 /* a = a + q */ +042 if ((res = s_mp_add(a, &q, a)) != MP_OKAY) \{ +043 goto ERR; +044 \} +045 +046 if (mp_cmp_mag(a, n) != MP_LT) \{ +047 if ((res = s_mp_sub(a, n, a)) != MP_OKAY) \{ +048 goto ERR; +049 \} +050 goto top; +051 \} +052 +053 ERR: +054 mp_clear(&q); +055 return res; +056 \} +057 +058 #endif +059 \end{alltt} \end{small} The algorithm mp\_count\_bits calculates the number of bits in an mp\_int which is used to find the initial value of $p$. The call to mp\_div\_2d -on line 31 calculates both the quotient $q$ and the remainder $a$ required. By doing both in a single function call the code size +on line 30 calculates both the quotient $q$ and the remainder $a$ required. By doing both in a single function call the code size is kept fairly small. The multiplication by $k$ is only performed if $k > 1$. This allows reductions modulo $2^p - 1$ to be performed without -any multiplications. +any multiplications. -The unsigned s\_mp\_add, mp\_cmp\_mag and s\_mp\_sub are used in place of their full sign counterparts since the inputs are only valid if they are -positive. By using the unsigned versions the overhead is kept to a minimum. +The unsigned s\_mp\_add, mp\_cmp\_mag and s\_mp\_sub are used in place of their full sign counterparts since the inputs are only valid if they are +positive. By using the unsigned versions the overhead is kept to a minimum. \subsubsection{Unrestricted Setup} -To setup this reduction algorithm the value of $k = 2^p - n$ is required. +To setup this reduction algorithm the value of $k = 2^p - n$ is required. \begin{figure}[!here] \begin{small} @@ -4604,12 +7103,40 @@ To setup this reduction algorithm the value of $k = 2^p - n$ is required. \textbf{Algorithm mp\_reduce\_2k\_setup.} This algorithm computes the value of $k$ required for the algorithm mp\_reduce\_2k. By making a temporary variable $x$ equal to $2^p$ a subtraction -is sufficient to solve for $k$. Alternatively if $n$ has more than one digit the value of $k$ is simply $\beta - n_0$. +is sufficient to solve for $k$. Alternatively if $n$ has more than one digit the value of $k$ is simply $\beta - n_0$. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_2k\_setup.c \vspace{-3mm} \begin{alltt} +016 +017 /* determines the setup value */ +018 int mp_reduce_2k_setup(mp_int *a, mp_digit *d) +019 \{ +020 int res, p; +021 mp_int tmp; +022 +023 if ((res = mp_init(&tmp)) != MP_OKAY) \{ +024 return res; +025 \} +026 +027 p = mp_count_bits(a); +028 if ((res = mp_2expt(&tmp, p)) != MP_OKAY) \{ +029 mp_clear(&tmp); +030 return res; +031 \} +032 +033 if ((res = s_mp_sub(&tmp, a, &tmp)) != MP_OKAY) \{ +034 mp_clear(&tmp); +035 return res; +036 \} +037 +038 *d = tmp.dp[0]; +039 mp_clear(&tmp); +040 return MP_OKAY; +041 \} +042 #endif +043 \end{alltt} \end{small} @@ -4648,12 +7175,45 @@ significant bit. The resulting sum will be a power of two. \end{figure} \textbf{Algorithm mp\_reduce\_is\_2k.} -This algorithm quickly determines if a modulus is of the form required for algorithm mp\_reduce\_2k to function properly. +This algorithm quickly determines if a modulus is of the form required for algorithm mp\_reduce\_2k to function properly. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_is\_2k.c \vspace{-3mm} \begin{alltt} +016 +017 /* determines if mp_reduce_2k can be used */ +018 int mp_reduce_is_2k(mp_int *a) +019 \{ +020 int ix, iy, iw; +021 mp_digit iz; +022 +023 if (a->used == 0) \{ +024 return MP_NO; +025 \} else if (a->used == 1) \{ +026 return MP_YES; +027 \} else if (a->used > 1) \{ +028 iy = mp_count_bits(a); +029 iz = 1; +030 iw = 1; +031 +032 /* Test every bit from the second digit up, must be 1 */ +033 for (ix = DIGIT_BIT; ix < iy; ix++) \{ +034 if ((a->dp[iw] & iz) == 0) \{ +035 return MP_NO; +036 \} +037 iz <<= 1; +038 if (iz > (mp_digit)MP_MASK) \{ +039 ++iw; +040 iz = 1; +041 \} +042 \} +043 \} +044 return MP_YES; +045 \} +046 +047 #endif +048 \end{alltt} \end{small} @@ -4662,7 +7222,7 @@ This algorithm quickly determines if a modulus is of the form required for algor \section{Algorithm Comparison} So far three very different algorithms for modular reduction have been discussed. Each of the algorithms have their own strengths and weaknesses that makes having such a selection very useful. The following table sumarizes the three algorithms along with comparisons of work factors. Since -all three algorithms have the restriction that $0 \le x < n^2$ and $n > 1$ those limitations are not included in the table. +all three algorithms have the restriction that $0 \le x < n^2$ and $n > 1$ those limitations are not included in the table. \begin{center} \begin{small} @@ -4698,12 +7258,12 @@ $\left [ 4 \right ]$ & Prove that the pseudo-code algorithm ``Diminished Radix R & (\textit{figure~\ref{fig:DR}}) terminates. Also prove the probability that it will \\ & terminate within $1 \le k \le 10$ iterations. \\ & \\ -\end{tabular} +\end{tabular} \chapter{Exponentiation} Exponentiation is the operation of raising one variable to the power of another, for example, $a^b$. A variant of exponentiation, computed -in a finite field or ring, is called modular exponentiation. This latter style of operation is typically used in public key +in a finite field or ring, is called modular exponentiation. This latter style of operation is typically used in public key cryptosystems such as RSA and Diffie-Hellman. The ability to quickly compute modular exponentiations is of great benefit to any such cryptosystem and many methods have been sought to speed it up. @@ -4713,7 +7273,7 @@ the number of multiplications becomes prohibitive. Imagine what would happen if with a $1024$-bit key. Such a calculation could never be completed as it would take simply far too long. Fortunately there is a very simple algorithm based on the laws of exponents. Recall that $lg_a(a^b) = b$ and that $lg_a(a^ba^c) = b + c$ which -are two trivial relationships between the base and the exponent. Let $b_i$ represent the $i$'th bit of $b$ starting from the least +are two trivial relationships between the base and the exponent. Let $b_i$ represent the $i$'th bit of $b$ starting from the least significant bit. If $b$ is a $k$-bit integer than the following equation is true. \begin{equation} @@ -4730,7 +7290,7 @@ The term $a^{2^i}$ can be found from the $i - 1$'th term by squaring the term si $a^{2^{i+1}}$. This observation forms the basis of essentially all fast exponentiation algorithms. It requires $k$ squarings and on average $k \over 2$ multiplications to compute the result. This is indeed quite an improvement over simply multiplying by $a$ a total of $b-1$ times. -While this current method is a considerable speed up there are further improvements to be made. For example, the $a^{2^i}$ term does not need to +While this current method is a considerable speed up there are further improvements to be made. For example, the $a^{2^i}$ term does not need to be computed in an auxilary variable. Consider the following equivalent algorithm. \begin{figure}[!here] @@ -4756,7 +7316,7 @@ be computed in an auxilary variable. Consider the following equivalent algorith This algorithm starts from the most significant bit and works towards the least significant bit. When the $i$'th bit of $b$ is set $a$ is multiplied against the current product. In each iteration the product is squared which doubles the exponent of the individual terms of the -product. +product. For example, let $b = 101100_2 \equiv 44_{10}$. The following chart demonstrates the actions of the algorithm. @@ -4777,13 +7337,13 @@ For example, let $b = 101100_2 \equiv 44_{10}$. The following chart demonstrate \caption{Example of Left to Right Exponentiation} \end{figure} -When the product $a^{32} \cdot a^8 \cdot a^4$ is simplified it is equal $a^{44}$ which is the desired exponentiation. This particular algorithm is -called ``Left to Right'' because it reads the exponent in that order. All of the exponentiation algorithms that will be presented are of this nature. +When the product $a^{32} \cdot a^8 \cdot a^4$ is simplified it is equal $a^{44}$ which is the desired exponentiation. This particular algorithm is +called ``Left to Right'' because it reads the exponent in that order. All of the exponentiation algorithms that will be presented are of this nature. \subsection{Single Digit Exponentiation} -The first algorithm in the series of exponentiation algorithms will be an unbounded algorithm where the exponent is a single digit. It is intended -to be used when a small power of an input is required (\textit{e.g. $a^5$}). It is faster than simply multiplying $b - 1$ times for all values of -$b$ that are greater than three. +The first algorithm in the series of exponentiation algorithms will be an unbounded algorithm where the exponent is a single digit. It is intended +to be used when a small power of an input is required (\textit{e.g. $a^5$}). It is faster than simply multiplying $b - 1$ times for all values of +$b$ that are greater than three. \newpage\begin{figure}[!here] \begin{small} @@ -4811,10 +7371,10 @@ $b$ that are greater than three. \textbf{Algorithm mp\_expt\_d.} This algorithm computes the value of $a$ raised to the power of a single digit $b$. It uses the left to right exponentiation algorithm to -quickly compute the exponentiation. It is loosely based on algorithm 14.79 of HAC \cite[pp. 615]{HAC} with the difference that the -exponent is a fixed width. +quickly compute the exponentiation. It is loosely based on algorithm 14.79 of HAC \cite[pp. 615]{HAC} with the difference that the +exponent is a fixed width. -A copy of $a$ is made first to allow destination variable $c$ be the same as the source variable $a$. The result is set to the initial value of +A copy of $a$ is made first to allow destination variable $c$ be the same as the source variable $a$. The result is set to the initial value of $1$ in the subsequent step. Inside the loop the exponent is read from the most significant bit first down to the least significant bit. First $c$ is invariably squared @@ -4823,16 +7383,80 @@ of $b$ is shifted left one bit to make the next bit down from the most signfican iteration of the loop moves the bits of the exponent $b$ upwards to the most significant location. \vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_expt\_d.c +\hspace{-5.1mm}{\bf File}: bn\_mp\_expt\_d\_ex.c \vspace{-3mm} \begin{alltt} +016 +017 /* calculate c = a**b using a square-multiply algorithm */ +018 int mp_expt_d_ex (mp_int * a, mp_digit b, mp_int * c, int fast) +019 \{ +020 int res; +021 unsigned int x; +022 +023 mp_int g; +024 +025 if ((res = mp_init_copy (&g, a)) != MP_OKAY) \{ +026 return res; +027 \} +028 +029 /* set initial result */ +030 mp_set (c, 1); +031 +032 if (fast != 0) \{ +033 while (b > 0) \{ +034 /* if the bit is set multiply */ +035 if ((b & 1) != 0) \{ +036 if ((res = mp_mul (c, &g, c)) != MP_OKAY) \{ +037 mp_clear (&g); +038 return res; +039 \} +040 \} +041 +042 /* square */ +043 if (b > 1) \{ +044 if ((res = mp_sqr (&g, &g)) != MP_OKAY) \{ +045 mp_clear (&g); +046 return res; +047 \} +048 \} +049 +050 /* shift to next bit */ +051 b >>= 1; +052 \} +053 \} +054 else \{ +055 for (x = 0; x < DIGIT_BIT; x++) \{ +056 /* square */ +057 if ((res = mp_sqr (c, c)) != MP_OKAY) \{ +058 mp_clear (&g); +059 return res; +060 \} +061 +062 /* if the bit is set multiply */ +063 if ((b & (mp_digit) (((mp_digit)1) << (DIGIT_BIT - 1))) != 0) \{ +064 if ((res = mp_mul (c, &g, c)) != MP_OKAY) \{ +065 mp_clear (&g); +066 return res; +067 \} +068 \} +069 +070 /* shift to next bit */ +071 b <<= 1; +072 \} +073 \} /* if ... else */ +074 +075 mp_clear (&g); +076 return MP_OKAY; +077 \} +078 #endif +079 \end{alltt} \end{small} -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 -the most significant down towards the least significant. The invariant squaring operation placed on line 33 is performed first. After +This describes only the algorithm that is used when the parameter $fast$ is $0$. Line 30 sets the initial value of the result to $1$. Next the loop on line 55 steps through each bit of the exponent starting from +the most significant down towards the least significant. The invariant squaring operation placed on line 57 is performed first. After the squaring the result $c$ is multiplied by the base $g$ if and only if the most significant bit of the exponent is set. The shift on line -47 moves all of the bits of the exponent upwards towards the most significant location. +71 moves all of the bits of the exponent upwards towards the most significant location. \section{$k$-ary Exponentiation} When calculating an exponentiation the most time consuming bottleneck is the multiplications which are in general a small factor @@ -4865,7 +7489,7 @@ portion of the entire exponent. Consider the following modification to the basi The squaring on step 2.1 can be calculated by squaring the value $c$ successively $k$ times. If the values of $a^g$ for $0 < g < 2^k$ have been precomputed this algorithm requires only $t$ multiplications and $tk$ squarings. The table can be generated with $2^{k - 1} - 1$ squarings and -$2^{k - 1} + 1$ multiplications. This algorithm assumes that the number of bits in the exponent is evenly divisible by $k$. +$2^{k - 1} + 1$ multiplications. This algorithm assumes that the number of bits in the exponent is evenly divisible by $k$. However, when it is not the remaining $0 < x \le k - 1$ bits can be handled with algorithm~\ref{fig:LTOR}. Suppose $k = 4$ and $t = 100$. This modified algorithm will require $109$ multiplications and $408$ squarings to compute the exponentiation. The @@ -4875,7 +7499,7 @@ has increased slightly but the number of multiplications has nearly halved. \subsection{Optimal Values of $k$} An optimal value of $k$ will minimize $2^{k} + \lceil n / k \rceil + n - 1$ for a fixed number of bits in the exponent $n$. The simplest approach is to brute force search amongst the values $k = 2, 3, \ldots, 8$ for the lowest result. Table~\ref{fig:OPTK} lists optimal values of $k$ -for various exponent sizes and compares the number of multiplication and squarings required against algorithm~\ref{fig:LTOR}. +for various exponent sizes and compares the number of multiplication and squarings required against algorithm~\ref{fig:LTOR}. \begin{figure}[here] \begin{center} @@ -4901,10 +7525,10 @@ for various exponent sizes and compares the number of multiplication and squarin \subsection{Sliding-Window Exponentiation} A simple modification to the previous algorithm is only generate the upper half of the table in the range $2^{k-1} \le g < 2^k$. Essentially -this is a table for all values of $g$ where the most significant bit of $g$ is a one. However, in order for this to be allowed in the -algorithm values of $g$ in the range $0 \le g < 2^{k-1}$ must be avoided. +this is a table for all values of $g$ where the most significant bit of $g$ is a one. However, in order for this to be allowed in the +algorithm values of $g$ in the range $0 \le g < 2^{k-1}$ must be avoided. -Table~\ref{fig:OPTK2} lists optimal values of $k$ for various exponent sizes and compares the work required against algorithm~\ref{fig:KARY}. +Table~\ref{fig:OPTK2} lists optimal values of $k$ for various exponent sizes and compares the work required against algorithm {\ref{fig:KARY}}. \begin{figure}[here] \begin{center} @@ -4955,29 +7579,29 @@ Table~\ref{fig:OPTK2} lists optimal values of $k$ for various exponent sizes and Similar to the previous algorithm this algorithm must have a special handler when fewer than $k$ bits are left in the exponent. While this algorithm requires the same number of squarings it can potentially have fewer multiplications. The pre-computed table $a^g$ is also half -the size as the previous table. +the size as the previous table. -Consider the exponent $b = 111101011001000_2 \equiv 31432_{10}$ with $k = 3$ using both algorithms. The first algorithm will divide the exponent up as -the following five $3$-bit words $b \equiv \left ( 111, 101, 011, 001, 000 \right )_{2}$. The second algorithm will break the +Consider the exponent $b = 111101011001000_2 \equiv 31432_{10}$ with $k = 3$ using both algorithms. The first algorithm will divide the exponent up as +the following five $3$-bit words $b \equiv \left ( 111, 101, 011, 001, 000 \right )_{2}$. The second algorithm will break the exponent as $b \equiv \left ( 111, 101, 0, 110, 0, 100, 0 \right )_{2}$. The single digit $0$ in the second representation are where -a single squaring took place instead of a squaring and multiplication. In total the first method requires $10$ multiplications and $18$ -squarings. The second method requires $8$ multiplications and $18$ squarings. +a single squaring took place instead of a squaring and multiplication. In total the first method requires $10$ multiplications and $18$ +squarings. The second method requires $8$ multiplications and $18$ squarings. -In general the sliding window method is never slower than the generic $k$-ary method and often it is slightly faster. +In general the sliding window method is never slower than the generic $k$-ary method and often it is slightly faster. \section{Modular Exponentiation} -Modular exponentiation is essentially computing the power of a base within a finite field or ring. For example, computing -$d \equiv a^b \mbox{ (mod }c\mbox{)}$ is a modular exponentiation. Instead of first computing $a^b$ and then reducing it -modulo $c$ the intermediate result is reduced modulo $c$ after every squaring or multiplication operation. +Modular exponentiation is essentially computing the power of a base within a finite field or ring. For example, computing +$d \equiv a^b \mbox{ (mod }c\mbox{)}$ is a modular exponentiation. Instead of first computing $a^b$ and then reducing it +modulo $c$ the intermediate result is reduced modulo $c$ after every squaring or multiplication operation. This guarantees that any intermediate result is bounded by $0 \le d \le c^2 - 2c + 1$ and can be reduced modulo $c$ quickly using -one of the algorithms presented in chapter six. +one of the algorithms presented in chapter six. Before the actual modular exponentiation algorithm can be written a wrapper algorithm must be written first. This algorithm will allow the exponent $b$ to be negative which is computed as $c \equiv \left (1 / a \right )^{\vert b \vert} \mbox{(mod }d\mbox{)}$. The value of $(1/a) \mbox{ mod }c$ is computed using the modular inverse (\textit{see \ref{sec;modinv}}). If no inverse exists the algorithm -terminates with an error. +terminates with an error. \begin{figure}[!here] \begin{small} @@ -5004,25 +7628,119 @@ terminates with an error. \end{figure} \textbf{Algorithm mp\_exptmod.} -The first algorithm which actually performs modular exponentiation is algorithm s\_mp\_exptmod. It is a sliding window $k$-ary algorithm -which uses Barrett reduction to reduce the product modulo $p$. The second algorithm mp\_exptmod\_fast performs the same operation +The first algorithm which actually performs modular exponentiation is algorithm s\_mp\_exptmod. It is a sliding window $k$-ary algorithm +which uses Barrett reduction to reduce the product modulo $p$. The second algorithm mp\_exptmod\_fast performs the same operation except it uses either Montgomery or Diminished Radix reduction. The two latter reduction algorithms are clumped in the same exponentiation -algorithm since their arguments are essentially the same (\textit{two mp\_ints and one mp\_digit}). +algorithm since their arguments are essentially the same (\textit{two mp\_ints and one mp\_digit}). \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_exptmod.c \vspace{-3mm} \begin{alltt} +016 +017 +018 /* this is a shell function that calls either the normal or Montgomery +019 * exptmod functions. Originally the call to the montgomery code was +020 * embedded in the normal function but that wasted alot of stack space +021 * for nothing (since 99% of the time the Montgomery code would be called) +022 */ +023 int mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y) +024 \{ +025 int dr; +026 +027 /* modulus P must be positive */ +028 if (P->sign == MP_NEG) \{ +029 return MP_VAL; +030 \} +031 +032 /* if exponent X is negative we have to recurse */ +033 if (X->sign == MP_NEG) \{ +034 #ifdef BN_MP_INVMOD_C +035 mp_int tmpG, tmpX; +036 int err; +037 +038 /* first compute 1/G mod P */ +039 if ((err = mp_init(&tmpG)) != MP_OKAY) \{ +040 return err; +041 \} +042 if ((err = mp_invmod(G, P, &tmpG)) != MP_OKAY) \{ +043 mp_clear(&tmpG); +044 return err; +045 \} +046 +047 /* now get |X| */ +048 if ((err = mp_init(&tmpX)) != MP_OKAY) \{ +049 mp_clear(&tmpG); +050 return err; +051 \} +052 if ((err = mp_abs(X, &tmpX)) != MP_OKAY) \{ +053 mp_clear_multi(&tmpG, &tmpX, NULL); +054 return err; +055 \} +056 +057 /* and now compute (1/G)**|X| instead of G**X [X < 0] */ +058 err = mp_exptmod(&tmpG, &tmpX, P, Y); +059 mp_clear_multi(&tmpG, &tmpX, NULL); +060 return err; +061 #else +062 /* no invmod */ +063 return MP_VAL; +064 #endif +065 \} +066 +067 /* modified diminished radix reduction */ +068 #if defined(BN_MP_REDUCE_IS_2K_L_C) && defined(BN_MP_REDUCE_2K_L_C) && defin + ed(BN_S_MP_EXPTMOD_C) +069 if (mp_reduce_is_2k_l(P) == MP_YES) \{ +070 return s_mp_exptmod(G, X, P, Y, 1); +071 \} +072 #endif +073 +074 #ifdef BN_MP_DR_IS_MODULUS_C +075 /* is it a DR modulus? */ +076 dr = mp_dr_is_modulus(P); +077 #else +078 /* default to no */ +079 dr = 0; +080 #endif +081 +082 #ifdef BN_MP_REDUCE_IS_2K_C +083 /* if not, is it a unrestricted DR modulus? */ +084 if (dr == 0) \{ +085 dr = mp_reduce_is_2k(P) << 1; +086 \} +087 #endif +088 +089 /* if the modulus is odd or dr != 0 use the montgomery method */ +090 #ifdef BN_MP_EXPTMOD_FAST_C +091 if ((mp_isodd (P) == MP_YES) || (dr != 0)) \{ +092 return mp_exptmod_fast (G, X, P, Y, dr); +093 \} else \{ +094 #endif +095 #ifdef BN_S_MP_EXPTMOD_C +096 /* otherwise use the generic Barrett reduction technique */ +097 return s_mp_exptmod (G, X, P, Y, 0); +098 #else +099 /* no exptmod for evens */ +100 return MP_VAL; +101 #endif +102 #ifdef BN_MP_EXPTMOD_FAST_C +103 \} +104 #endif +105 \} +106 +107 #endif +108 \end{alltt} \end{small} -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 +In order to keep the algorithms in a known state the first step on line 28 is to reject any negative modulus as input. If the exponent is negative the algorithm tries to perform a modular exponentiation with the modular inverse of the base $G$. The temporary variable $tmpG$ is assigned the modular inverse of $G$ and $tmpX$ is assigned the absolute value of $X$. The algorithm will recuse with these new values with a positive exponent. -If the exponent is positive the algorithm resumes the exponentiation. Line 77 determines if the modulus is of the restricted Diminished Radix -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 +If the exponent is positive the algorithm resumes the exponentiation. Line 76 determines if the modulus is of the restricted Diminished Radix +form. If it is not line 69 attempts to determine if it is of a unrestricted Diminished Radix form. The integer $dr$ will take on one of three values. \begin{enumerate} @@ -5032,7 +7750,7 @@ of three values. \end{enumerate} Line 69 determines if the fast modular exponentiation algorithm can be used. It is allowed if $dr \ne 0$ or if the modulus is odd. Otherwise, -the slower s\_mp\_exptmod algorithm is used which uses Barrett reduction. +the slower s\_mp\_exptmod algorithm is used which uses Barrett reduction. \subsection{Barrett Modular Exponentiation} @@ -5137,9 +7855,9 @@ No more windows left. Check for residual bits of exponent. \\ This algorithm computes the $x$'th power of $g$ modulo $p$ and stores the result in $y$. It takes advantage of the Barrett reduction algorithm to keep the product small throughout the algorithm. -The first two steps determine the optimal window size based on the number of bits in the exponent. The larger the exponent the +The first two steps determine the optimal window size based on the number of bits in the exponent. The larger the exponent the larger the window size becomes. After a window size $winsize$ has been chosen an array of $2^{winsize}$ mp\_int variables is allocated. This -table will hold the values of $g^x \mbox{ (mod }p\mbox{)}$ for $2^{winsize - 1} \le x < 2^{winsize}$. +table will hold the values of $g^x \mbox{ (mod }p\mbox{)}$ for $2^{winsize - 1} \le x < 2^{winsize}$. After the table is allocated the first power of $g$ is found. Since $g \ge p$ is allowed it must be first reduced modulo $p$ to make the rest of the algorithm more efficient. The first element of the table at $2^{winsize - 1}$ is found by squaring $M_1$ successively $winsize - 2$ @@ -5147,35 +7865,35 @@ times. The rest of the table elements are found by multiplying the previous ele Now that the table is available the sliding window may begin. The following list describes the functions of all the variables in the window. \begin{enumerate} -\item The variable $mode$ dictates how the bits of the exponent are interpreted. +\item The variable $mode$ dictates how the bits of the exponent are interpreted. \begin{enumerate} - \item When $mode = 0$ the bits are ignored since no non-zero bit of the exponent has been seen yet. For example, if the exponent were simply - $1$ then there would be $lg(\beta) - 1$ zero bits before the first non-zero bit. In this case bits are ignored until a non-zero bit is found. - \item When $mode = 1$ a non-zero bit has been seen before and a new $winsize$-bit window has not been formed yet. In this mode leading $0$ bits - are read and a single squaring is performed. If a non-zero bit is read a new window is created. + \item When $mode = 0$ the bits are ignored since no non-zero bit of the exponent has been seen yet. For example, if the exponent were simply + $1$ then there would be $lg(\beta) - 1$ zero bits before the first non-zero bit. In this case bits are ignored until a non-zero bit is found. + \item When $mode = 1$ a non-zero bit has been seen before and a new $winsize$-bit window has not been formed yet. In this mode leading $0$ bits + are read and a single squaring is performed. If a non-zero bit is read a new window is created. \item When $mode = 2$ the algorithm is in the middle of forming a window and new bits are appended to the window from the most significant bit downwards. \end{enumerate} \item The variable $bitcnt$ indicates how many bits are left in the current digit of the exponent left to be read. When it reaches zero a new digit is fetched from the exponent. -\item The variable $buf$ holds the currently read digit of the exponent. +\item The variable $buf$ holds the currently read digit of the exponent. \item The variable $digidx$ is an index into the exponents digits. It starts at the leading digit $x.used - 1$ and moves towards the trailing digit. \item The variable $bitcpy$ indicates how many bits are in the currently formed window. When it reaches $winsize$ the window is flushed and the appropriate operations performed. -\item The variable $bitbuf$ holds the current bits of the window being formed. +\item The variable $bitbuf$ holds the current bits of the window being formed. \end{enumerate} All of step 12 is the window processing loop. It will iterate while there are digits available form the exponent to read. The first step inside this loop is to extract a new digit if no more bits are available in the current digit. If there are no bits left a new digit is -read and if there are no digits left than the loop terminates. +read and if there are no digits left than the loop terminates. After a digit is made available step 12.3 will extract the most significant bit of the current digit and move all other bits in the digit -upwards. In effect the digit is read from most significant bit to least significant bit and since the digits are read from leading to +upwards. In effect the digit is read from most significant bit to least significant bit and since the digits are read from leading to trailing edges the entire exponent is read from most significant bit to least significant bit. -At step 12.5 if the $mode$ and currently extracted bit $y$ are both zero the bit is ignored and the next bit is read. This prevents the +At step 12.5 if the $mode$ and currently extracted bit $y$ are both zero the bit is ignored and the next bit is read. This prevents the algorithm from having to perform trivial squaring and reduction operations before the first non-zero bit is read. Step 12.6 and 12.7-10 handle -the two cases of $mode = 1$ and $mode = 2$ respectively. +the two cases of $mode = 1$ and $mode = 2$ respectively. \begin{center} \begin{figure}[here] @@ -5185,24 +7903,258 @@ the two cases of $mode = 1$ and $mode = 2$ respectively. \end{figure} \end{center} -By step 13 there are no more digits left in the exponent. However, there may be partial bits in the window left. If $mode = 2$ then -a Left-to-Right algorithm is used to process the remaining few bits. +By step 13 there are no more digits left in the exponent. However, there may be partial bits in the window left. If $mode = 2$ then +a Left-to-Right algorithm is used to process the remaining few bits. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_s\_mp\_exptmod.c \vspace{-3mm} \begin{alltt} +016 #ifdef MP_LOW_MEM +017 #define TAB_SIZE 32 +018 #else +019 #define TAB_SIZE 256 +020 #endif +021 +022 int s_mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y, int redmod + e) +023 \{ +024 mp_int M[TAB_SIZE], res, mu; +025 mp_digit buf; +026 int err, bitbuf, bitcpy, bitcnt, mode, digidx, x, y, winsize; +027 int (*redux)(mp_int*,mp_int*,mp_int*); +028 +029 /* find window size */ +030 x = mp_count_bits (X); +031 if (x <= 7) \{ +032 winsize = 2; +033 \} else if (x <= 36) \{ +034 winsize = 3; +035 \} else if (x <= 140) \{ +036 winsize = 4; +037 \} else if (x <= 450) \{ +038 winsize = 5; +039 \} else if (x <= 1303) \{ +040 winsize = 6; +041 \} else if (x <= 3529) \{ +042 winsize = 7; +043 \} else \{ +044 winsize = 8; +045 \} +046 +047 #ifdef MP_LOW_MEM +048 if (winsize > 5) \{ +049 winsize = 5; +050 \} +051 #endif +052 +053 /* init M array */ +054 /* init first cell */ +055 if ((err = mp_init(&M[1])) != MP_OKAY) \{ +056 return err; +057 \} +058 +059 /* now init the second half of the array */ +060 for (x = 1<<(winsize-1); x < (1 << winsize); x++) \{ +061 if ((err = mp_init(&M[x])) != MP_OKAY) \{ +062 for (y = 1<<(winsize-1); y < x; y++) \{ +063 mp_clear (&M[y]); +064 \} +065 mp_clear(&M[1]); +066 return err; +067 \} +068 \} +069 +070 /* create mu, used for Barrett reduction */ +071 if ((err = mp_init (&mu)) != MP_OKAY) \{ +072 goto LBL_M; +073 \} +074 +075 if (redmode == 0) \{ +076 if ((err = mp_reduce_setup (&mu, P)) != MP_OKAY) \{ +077 goto LBL_MU; +078 \} +079 redux = mp_reduce; +080 \} else \{ +081 if ((err = mp_reduce_2k_setup_l (P, &mu)) != MP_OKAY) \{ +082 goto LBL_MU; +083 \} +084 redux = mp_reduce_2k_l; +085 \} +086 +087 /* create M table +088 * +089 * The M table contains powers of the base, +090 * e.g. M[x] = G**x mod P +091 * +092 * The first half of the table is not +093 * computed though accept for M[0] and M[1] +094 */ +095 if ((err = mp_mod (G, P, &M[1])) != MP_OKAY) \{ +096 goto LBL_MU; +097 \} +098 +099 /* compute the value at M[1<<(winsize-1)] by squaring +100 * M[1] (winsize-1) times +101 */ +102 if ((err = mp_copy (&M[1], &M[1 << (winsize - 1)])) != MP_OKAY) \{ +103 goto LBL_MU; +104 \} +105 +106 for (x = 0; x < (winsize - 1); x++) \{ +107 /* square it */ +108 if ((err = mp_sqr (&M[1 << (winsize - 1)], +109 &M[1 << (winsize - 1)])) != MP_OKAY) \{ +110 goto LBL_MU; +111 \} +112 +113 /* reduce modulo P */ +114 if ((err = redux (&M[1 << (winsize - 1)], P, &mu)) != MP_OKAY) \{ +115 goto LBL_MU; +116 \} +117 \} +118 +119 /* create upper table, that is M[x] = M[x-1] * M[1] (mod P) +120 * for x = (2**(winsize - 1) + 1) to (2**winsize - 1) +121 */ +122 for (x = (1 << (winsize - 1)) + 1; x < (1 << winsize); x++) \{ +123 if ((err = mp_mul (&M[x - 1], &M[1], &M[x])) != MP_OKAY) \{ +124 goto LBL_MU; +125 \} +126 if ((err = redux (&M[x], P, &mu)) != MP_OKAY) \{ +127 goto LBL_MU; +128 \} +129 \} +130 +131 /* setup result */ +132 if ((err = mp_init (&res)) != MP_OKAY) \{ +133 goto LBL_MU; +134 \} +135 mp_set (&res, 1); +136 +137 /* set initial mode and bit cnt */ +138 mode = 0; +139 bitcnt = 1; +140 buf = 0; +141 digidx = X->used - 1; +142 bitcpy = 0; +143 bitbuf = 0; +144 +145 for (;;) \{ +146 /* grab next digit as required */ +147 if (--bitcnt == 0) \{ +148 /* if digidx == -1 we are out of digits */ +149 if (digidx == -1) \{ +150 break; +151 \} +152 /* read next digit and reset the bitcnt */ +153 buf = X->dp[digidx--]; +154 bitcnt = (int) DIGIT_BIT; +155 \} +156 +157 /* grab the next msb from the exponent */ +158 y = (buf >> (mp_digit)(DIGIT_BIT - 1)) & 1; +159 buf <<= (mp_digit)1; +160 +161 /* if the bit is zero and mode == 0 then we ignore it +162 * These represent the leading zero bits before the first 1 bit +163 * in the exponent. Technically this opt is not required but it +164 * does lower the # of trivial squaring/reductions used +165 */ +166 if ((mode == 0) && (y == 0)) \{ +167 continue; +168 \} +169 +170 /* if the bit is zero and mode == 1 then we square */ +171 if ((mode == 1) && (y == 0)) \{ +172 if ((err = mp_sqr (&res, &res)) != MP_OKAY) \{ +173 goto LBL_RES; +174 \} +175 if ((err = redux (&res, P, &mu)) != MP_OKAY) \{ +176 goto LBL_RES; +177 \} +178 continue; +179 \} +180 +181 /* else we add it to the window */ +182 bitbuf |= (y << (winsize - ++bitcpy)); +183 mode = 2; +184 +185 if (bitcpy == winsize) \{ +186 /* ok window is filled so square as required and multiply */ +187 /* square first */ +188 for (x = 0; x < winsize; x++) \{ +189 if ((err = mp_sqr (&res, &res)) != MP_OKAY) \{ +190 goto LBL_RES; +191 \} +192 if ((err = redux (&res, P, &mu)) != MP_OKAY) \{ +193 goto LBL_RES; +194 \} +195 \} +196 +197 /* then multiply */ +198 if ((err = mp_mul (&res, &M[bitbuf], &res)) != MP_OKAY) \{ +199 goto LBL_RES; +200 \} +201 if ((err = redux (&res, P, &mu)) != MP_OKAY) \{ +202 goto LBL_RES; +203 \} +204 +205 /* empty window and reset */ +206 bitcpy = 0; +207 bitbuf = 0; +208 mode = 1; +209 \} +210 \} +211 +212 /* if bits remain then square/multiply */ +213 if ((mode == 2) && (bitcpy > 0)) \{ +214 /* square then multiply if the bit is set */ +215 for (x = 0; x < bitcpy; x++) \{ +216 if ((err = mp_sqr (&res, &res)) != MP_OKAY) \{ +217 goto LBL_RES; +218 \} +219 if ((err = redux (&res, P, &mu)) != MP_OKAY) \{ +220 goto LBL_RES; +221 \} +222 +223 bitbuf <<= 1; +224 if ((bitbuf & (1 << winsize)) != 0) \{ +225 /* then multiply */ +226 if ((err = mp_mul (&res, &M[1], &res)) != MP_OKAY) \{ +227 goto LBL_RES; +228 \} +229 if ((err = redux (&res, P, &mu)) != MP_OKAY) \{ +230 goto LBL_RES; +231 \} +232 \} +233 \} +234 \} +235 +236 mp_exch (&res, Y); +237 err = MP_OKAY; +238 LBL_RES:mp_clear (&res); +239 LBL_MU:mp_clear (&mu); +240 LBL_M: +241 mp_clear(&M[1]); +242 for (x = 1<<(winsize-1); x < (1 << winsize); x++) \{ +243 mp_clear (&M[x]); +244 \} +245 return err; +246 \} +247 #endif +248 \end{alltt} \end{small} -Lines 32 through 46 determine the optimal window size based on the length of the exponent in bits. The window divisions are sorted -from smallest to greatest so that in each \textbf{if} statement only one condition must be tested. For example, by the \textbf{if} statement -on line 38 the value of $x$ is already known to be greater than $140$. +Lines 31 through 45 determine the optimal window size based on the length of the exponent in bits. The window divisions are sorted +from smallest to greatest so that in each \textbf{if} statement only one condition must be tested. For example, by the \textbf{if} statement +on line 37 the value of $x$ is already known to be greater than $140$. -The conditional piece of code beginning on line 48 allows the window size to be restricted to five bits. This logic is used to ensure -the table of precomputed powers of $G$ remains relatively small. +The conditional piece of code beginning on line 47 allows the window size to be restricted to five bits. This logic is used to ensure +the table of precomputed powers of $G$ remains relatively small. -The for loop on line 61 initializes the $M$ array while lines 72 and 77 through 86 initialize the reduction +The for loop on line 60 initializes the $M$ array while lines 71 and 76 through 85 initialize the reduction function that will be used for this modulus. -- More later. @@ -5237,17 +8189,46 @@ equivalent to $m \cdot 2^k$. By this logic when $m = 1$ a quick power of two ca \hspace{-5.1mm}{\bf File}: bn\_mp\_2expt.c \vspace{-3mm} \begin{alltt} +016 +017 /* computes a = 2**b +018 * +019 * Simple algorithm which zeroes the int, grows it then just sets one bit +020 * as required. +021 */ +022 int +023 mp_2expt (mp_int * a, int b) +024 \{ +025 int res; +026 +027 /* zero a as per default */ +028 mp_zero (a); +029 +030 /* grow a to accomodate the single bit */ +031 if ((res = mp_grow (a, (b / DIGIT_BIT) + 1)) != MP_OKAY) \{ +032 return res; +033 \} +034 +035 /* set the used count of where the bit will go */ +036 a->used = (b / DIGIT_BIT) + 1; +037 +038 /* put the single bit in its place */ +039 a->dp[b / DIGIT_BIT] = ((mp_digit)1) << (b % DIGIT_BIT); +040 +041 return MP_OKAY; +042 \} +043 #endif +044 \end{alltt} \end{small} \chapter{Higher Level Algorithms} This chapter discusses the various higher level algorithms that are required to complete a well rounded multiple precision integer package. These -routines are less performance oriented than the algorithms of chapters five, six and seven but are no less important. +routines are less performance oriented than the algorithms of chapters five, six and seven but are no less important. The first section describes a method of integer division with remainder that is universally well known. It provides the signed division logic -for the package. The subsequent section discusses a set of algorithms which allow a single digit to be the 2nd operand for a variety of operations. -These algorithms serve mostly to simplify other algorithms where small constants are required. The last two sections discuss how to manipulate +for the package. The subsequent section discusses a set of algorithms which allow a single digit to be the 2nd operand for a variety of operations. +These algorithms serve mostly to simplify other algorithms where small constants are required. The last two sections discuss how to manipulate various representations of integers. For example, converting from an mp\_int to a string of character. \section{Integer Division with Remainder} @@ -5255,7 +8236,7 @@ various representations of integers. For example, converting from an mp\_int to Integer division aside from modular exponentiation is the most intensive algorithm to compute. Like addition, subtraction and multiplication the basis of this algorithm is the long-hand division algorithm taught to school children. Throughout this discussion several common variables -will be used. Let $x$ represent the divisor and $y$ represent the dividend. Let $q$ represent the integer quotient $\lfloor y / x \rfloor$ and +will be used. Let $x$ represent the divisor and $y$ represent the dividend. Let $q$ represent the integer quotient $\lfloor y / x \rfloor$ and let $r$ represent the remainder $r = y - x \lfloor y / x \rfloor$. The following simple algorithm will be used to start the discussion. \newpage\begin{figure}[!here] @@ -5285,42 +8266,42 @@ let $r$ represent the remainder $r = y - x \lfloor y / x \rfloor$. The followin As children we are taught this very simple algorithm for the case of $\beta = 10$. Almost instinctively several optimizations are taught for which their reason of existing are never explained. For this example let $y = 5471$ represent the dividend and $x = 23$ represent the divisor. -To find the first digit of the quotient the value of $k$ must be maximized such that $kx\beta^t$ is less than or equal to $y$ and +To find the first digit of the quotient the value of $k$ must be maximized such that $kx\beta^t$ is less than or equal to $y$ and simultaneously $(k + 1)x\beta^t$ is greater than $y$. Implicitly $k$ is the maximum value the $t$'th digit of the quotient may have. The habitual method used to find the maximum is to ``eyeball'' the two numbers, typically only the leading digits and quickly estimate a quotient. By only using leading -digits a much simpler division may be used to form an educated guess at what the value must be. In this case $k = \lfloor 54/23\rfloor = 2$ quickly -arises as a possible solution. Indeed $2x\beta^2 = 4600$ is less than $y = 5471$ and simultaneously $(k + 1)x\beta^2 = 6900$ is larger than $y$. +digits a much simpler division may be used to form an educated guess at what the value must be. In this case $k = \lfloor 54/23\rfloor = 2$ quickly +arises as a possible solution. Indeed $2x\beta^2 = 4600$ is less than $y = 5471$ and simultaneously $(k + 1)x\beta^2 = 6900$ is larger than $y$. As a result $k\beta^2$ is added to the quotient which now equals $q = 200$ and $4600$ is subtracted from $y$ to give a remainder of $y = 841$. -Again this process is repeated to produce the quotient digit $k = 3$ which makes the quotient $q = 200 + 3\beta = 230$ and the remainder +Again this process is repeated to produce the quotient digit $k = 3$ which makes the quotient $q = 200 + 3\beta = 230$ and the remainder $y = 841 - 3x\beta = 181$. Finally the last iteration of the loop produces $k = 7$ which leads to the quotient $q = 230 + 7 = 237$ and the -remainder $y = 181 - 7x = 20$. The final quotient and remainder found are $q = 237$ and $r = y = 20$ which are indeed correct since -$237 \cdot 23 + 20 = 5471$ is true. +remainder $y = 181 - 7x = 20$. The final quotient and remainder found are $q = 237$ and $r = y = 20$ which are indeed correct since +$237 \cdot 23 + 20 = 5471$ is true. \subsection{Quotient Estimation} \label{sec:divest} As alluded to earlier the quotient digit $k$ can be estimated from only the leading digits of both the divisor and dividend. When $p$ leading digits are used from both the divisor and dividend to form an estimation the accuracy of the estimation rises as $p$ grows. Technically speaking the estimation is based on assuming the lower $\vert \vert y \vert \vert - p$ and $\vert \vert x \vert \vert - p$ lower digits of the -dividend and divisor are zero. +dividend and divisor are zero. The value of the estimation may off by a few values in either direction and in general is fairly correct. A simplification \cite[pp. 271]{TAOCPV2} -of the estimation technique is to use $t + 1$ digits of the dividend and $t$ digits of the divisor, in particularly when $t = 1$. The estimate -using this technique is never too small. For the following proof let $t = \vert \vert y \vert \vert - 1$ and $s = \vert \vert x \vert \vert - 1$ +of the estimation technique is to use $t + 1$ digits of the dividend and $t$ digits of the divisor, in particularly when $t = 1$. The estimate +using this technique is never too small. For the following proof let $t = \vert \vert y \vert \vert - 1$ and $s = \vert \vert x \vert \vert - 1$ represent the most significant digits of the dividend and divisor respectively. -\textbf{Proof.}\textit{ The quotient $\hat k = \lfloor (y_t\beta + y_{t-1}) / x_s \rfloor$ is greater than or equal to +\textbf{Proof.}\textit{ The quotient $\hat k = \lfloor (y_t\beta + y_{t-1}) / x_s \rfloor$ is greater than or equal to $k = \lfloor y / (x \cdot \beta^{\vert \vert y \vert \vert - \vert \vert x \vert \vert - 1}) \rfloor$. } -The first obvious case is when $\hat k = \beta - 1$ in which case the proof is concluded since the real quotient cannot be larger. For all other +The first obvious case is when $\hat k = \beta - 1$ in which case the proof is concluded since the real quotient cannot be larger. For all other cases $\hat k = \lfloor (y_t\beta + y_{t-1}) / x_s \rfloor$ and $\hat k x_s \ge y_t\beta + y_{t-1} - x_s + 1$. The latter portion of the inequalility -$-x_s + 1$ arises from the fact that a truncated integer division will give the same quotient for at most $x_s - 1$ values. Next a series of +$-x_s + 1$ arises from the fact that a truncated integer division will give the same quotient for at most $x_s - 1$ values. Next a series of inequalities will prove the hypothesis. \begin{equation} y - \hat k x \le y - \hat k x_s\beta^s \end{equation} -This is trivially true since $x \ge x_s\beta^s$. Next we replace $\hat kx_s\beta^s$ by the previous inequality for $\hat kx_s$. +This is trivially true since $x \ge x_s\beta^s$. Next we replace $\hat kx_s\beta^s$ by the previous inequality for $\hat kx_s$. \begin{equation} y - \hat k x \le y_t\beta^t + \ldots + y_0 - (y_t\beta^t + y_{t-1}\beta^{t-1} - x_s\beta^t + \beta^s) @@ -5345,13 +8326,13 @@ Which proves that $y - \hat kx \le x$ and by consequence $\hat k \ge k$ which co For the purposes of division a normalized input is when the divisors leading digit $x_n$ is greater than or equal to $\beta / 2$. By multiplying both $x$ and $y$ by $j = \lfloor (\beta / 2) / x_n \rfloor$ the quotient remains unchanged and the remainder is simply $j$ times the original remainder. The purpose of normalization is to ensure the leading digit of the divisor is sufficiently large such that the estimated quotient will -lie in the domain of a single digit. Consider the maximum dividend $(\beta - 1) \cdot \beta + (\beta - 1)$ and the minimum divisor $\beta / 2$. +lie in the domain of a single digit. Consider the maximum dividend $(\beta - 1) \cdot \beta + (\beta - 1)$ and the minimum divisor $\beta / 2$. -\begin{equation} -{{\beta^2 - 1} \over { \beta / 2}} \le 2\beta - {2 \over \beta} +\begin{equation} +{{\beta^2 - 1} \over { \beta / 2}} \le 2\beta - {2 \over \beta} \end{equation} -At most the quotient approaches $2\beta$, however, in practice this will not occur since that would imply the previous quotient digit was too small. +At most the quotient approaches $2\beta$, however, in practice this will not occur since that would imply the previous quotient digit was too small. \subsection{Radix-$\beta$ Division with Remainder} \newpage\begin{figure}[!here] @@ -5449,23 +8430,23 @@ Finalize the result. \\ This algorithm will calculate quotient and remainder from an integer division given a dividend and divisor. The algorithm is a signed division and will produce a fully qualified quotient and remainder. -First the divisor $b$ must be non-zero which is enforced in step one. If the divisor is larger than the dividend than the quotient is implicitly -zero and the remainder is the dividend. +First the divisor $b$ must be non-zero which is enforced in step one. If the divisor is larger than the dividend than the quotient is implicitly +zero and the remainder is the dividend. After the first two trivial cases of inputs are handled the variable $q$ is setup to receive the digits of the quotient. Two unsigned copies of the divisor $y$ and dividend $x$ are made as well. The core of the division algorithm is an unsigned division and will only work if the values are -positive. Now the two values $x$ and $y$ must be normalized such that the leading digit of $y$ is greater than or equal to $\beta / 2$. -This is performed by shifting both to the left by enough bits to get the desired normalization. +positive. Now the two values $x$ and $y$ must be normalized such that the leading digit of $y$ is greater than or equal to $\beta / 2$. +This is performed by shifting both to the left by enough bits to get the desired normalization. -At this point the division algorithm can begin producing digits of the quotient. Recall that maximum value of the estimation used is +At this point the division algorithm can begin producing digits of the quotient. Recall that maximum value of the estimation used is $2\beta - {2 \over \beta}$ which means that a digit of the quotient must be first produced by another means. In this case $y$ is shifted -to the left (\textit{step ten}) so that it has the same number of digits as $x$. The loop on step eleven will subtract multiples of the +to the left (\textit{step ten}) so that it has the same number of digits as $x$. The loop on step eleven will subtract multiples of the shifted copy of $y$ until $x$ is smaller. Since the leading digit of $y$ is greater than or equal to $\beta/2$ this loop will iterate at most two -times to produce the desired leading digit of the quotient. +times to produce the desired leading digit of the quotient. Now the remainder of the digits can be produced. The equation $\hat q = \lfloor {{x_i \beta + x_{i-1}}\over y_t} \rfloor$ is used to fairly accurately approximate the true quotient digit. The estimation can in theory produce an estimation as high as $2\beta - {2 \over \beta}$ but by -induction the upper quotient digit is correct (\textit{as established on step eleven}) and the estimate must be less than $\beta$. +induction the upper quotient digit is correct (\textit{as established on step eleven}) and the estimate must be less than $\beta$. Recall from section~\ref{sec:divest} that the estimation is never too low but may be too high. The next step of the estimation process is to refine the estimation. The loop on step 13.5 uses $x_i\beta^2 + x_{i-1}\beta + x_{i-2}$ and $q_{i - t - 1}(y_t\beta + y_{t-1})$ as a higher @@ -5473,56 +8454,332 @@ order approximation to adjust the quotient digit. After both phases of estimation the quotient digit may still be off by a value of one\footnote{This is similar to the error introduced by optimizing Barrett reduction.}. Steps 13.6 and 13.7 subtract the multiple of the divisor from the dividend (\textit{Similar to step 3.3 of -algorithm~\ref{fig:raddiv}} and then subsequently add a multiple of the divisor if the quotient was too large. +algorithm~\ref{fig:raddiv}} and then subsequently add a multiple of the divisor if the quotient was too large. -Now that the quotient has been determine finializing the result is a matter of clamping the quotient, fixing the sizes and de-normalizing the +Now that the quotient has been determine finializing the result is a matter of clamping the quotient, fixing the sizes and de-normalizing the remainder. An important aspect of this algorithm seemingly overlooked in other descriptions such as that of Algorithm 14.20 HAC \cite[pp. 598]{HAC} -is that when the estimations are being made (\textit{inside the loop on step 13.5}) that the digits $y_{t-1}$, $x_{i-2}$ and $x_{i-1}$ may lie +is that when the estimations are being made (\textit{inside the loop on step 13.5}) that the digits $y_{t-1}$, $x_{i-2}$ and $x_{i-1}$ may lie outside their respective boundaries. For example, if $t = 0$ or $i \le 1$ then the digits would be undefined. In those cases the digits should -respectively be replaced with a zero. +respectively be replaced with a zero. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_div.c \vspace{-3mm} \begin{alltt} +016 +017 #ifdef BN_MP_DIV_SMALL +018 +019 /* slower bit-bang division... also smaller */ +020 int mp_div(mp_int * a, mp_int * b, mp_int * c, mp_int * d) +021 \{ +022 mp_int ta, tb, tq, q; +023 int res, n, n2; +024 +025 /* is divisor zero ? */ +026 if (mp_iszero (b) == MP_YES) \{ +027 return MP_VAL; +028 \} +029 +030 /* if a < b then q=0, r = a */ +031 if (mp_cmp_mag (a, b) == MP_LT) \{ +032 if (d != NULL) \{ +033 res = mp_copy (a, d); +034 \} else \{ +035 res = MP_OKAY; +036 \} +037 if (c != NULL) \{ +038 mp_zero (c); +039 \} +040 return res; +041 \} +042 +043 /* init our temps */ +044 if ((res = mp_init_multi(&ta, &tb, &tq, &q, NULL)) != MP_OKAY) \{ +045 return res; +046 \} +047 +048 +049 mp_set(&tq, 1); +050 n = mp_count_bits(a) - mp_count_bits(b); +051 if (((res = mp_abs(a, &ta)) != MP_OKAY) || +052 ((res = mp_abs(b, &tb)) != MP_OKAY) || +053 ((res = mp_mul_2d(&tb, n, &tb)) != MP_OKAY) || +054 ((res = mp_mul_2d(&tq, n, &tq)) != MP_OKAY)) \{ +055 goto LBL_ERR; +056 \} +057 +058 while (n-- >= 0) \{ +059 if (mp_cmp(&tb, &ta) != MP_GT) \{ +060 if (((res = mp_sub(&ta, &tb, &ta)) != MP_OKAY) || +061 ((res = mp_add(&q, &tq, &q)) != MP_OKAY)) \{ +062 goto LBL_ERR; +063 \} +064 \} +065 if (((res = mp_div_2d(&tb, 1, &tb, NULL)) != MP_OKAY) || +066 ((res = mp_div_2d(&tq, 1, &tq, NULL)) != MP_OKAY)) \{ +067 goto LBL_ERR; +068 \} +069 \} +070 +071 /* now q == quotient and ta == remainder */ +072 n = a->sign; +073 n2 = (a->sign == b->sign) ? MP_ZPOS : MP_NEG; +074 if (c != NULL) \{ +075 mp_exch(c, &q); +076 c->sign = (mp_iszero(c) == MP_YES) ? MP_ZPOS : n2; +077 \} +078 if (d != NULL) \{ +079 mp_exch(d, &ta); +080 d->sign = (mp_iszero(d) == MP_YES) ? MP_ZPOS : n; +081 \} +082 LBL_ERR: +083 mp_clear_multi(&ta, &tb, &tq, &q, NULL); +084 return res; +085 \} +086 +087 #else +088 +089 /* integer signed division. +090 * c*b + d == a [e.g. a/b, c=quotient, d=remainder] +091 * HAC pp.598 Algorithm 14.20 +092 * +093 * Note that the description in HAC is horribly +094 * incomplete. For example, it doesn't consider +095 * the case where digits are removed from 'x' in +096 * the inner loop. It also doesn't consider the +097 * case that y has fewer than three digits, etc.. +098 * +099 * The overall algorithm is as described as +100 * 14.20 from HAC but fixed to treat these cases. +101 */ +102 int mp_div (mp_int * a, mp_int * b, mp_int * c, mp_int * d) +103 \{ +104 mp_int q, x, y, t1, t2; +105 int res, n, t, i, norm, neg; +106 +107 /* is divisor zero ? */ +108 if (mp_iszero (b) == MP_YES) \{ +109 return MP_VAL; +110 \} +111 +112 /* if a < b then q=0, r = a */ +113 if (mp_cmp_mag (a, b) == MP_LT) \{ +114 if (d != NULL) \{ +115 res = mp_copy (a, d); +116 \} else \{ +117 res = MP_OKAY; +118 \} +119 if (c != NULL) \{ +120 mp_zero (c); +121 \} +122 return res; +123 \} +124 +125 if ((res = mp_init_size (&q, a->used + 2)) != MP_OKAY) \{ +126 return res; +127 \} +128 q.used = a->used + 2; +129 +130 if ((res = mp_init (&t1)) != MP_OKAY) \{ +131 goto LBL_Q; +132 \} +133 +134 if ((res = mp_init (&t2)) != MP_OKAY) \{ +135 goto LBL_T1; +136 \} +137 +138 if ((res = mp_init_copy (&x, a)) != MP_OKAY) \{ +139 goto LBL_T2; +140 \} +141 +142 if ((res = mp_init_copy (&y, b)) != MP_OKAY) \{ +143 goto LBL_X; +144 \} +145 +146 /* fix the sign */ +147 neg = (a->sign == b->sign) ? MP_ZPOS : MP_NEG; +148 x.sign = y.sign = MP_ZPOS; +149 +150 /* normalize both x and y, ensure that y >= b/2, [b == 2**DIGIT_BIT] */ +151 norm = mp_count_bits(&y) % DIGIT_BIT; +152 if (norm < (int)(DIGIT_BIT-1)) \{ +153 norm = (DIGIT_BIT-1) - norm; +154 if ((res = mp_mul_2d (&x, norm, &x)) != MP_OKAY) \{ +155 goto LBL_Y; +156 \} +157 if ((res = mp_mul_2d (&y, norm, &y)) != MP_OKAY) \{ +158 goto LBL_Y; +159 \} +160 \} else \{ +161 norm = 0; +162 \} +163 +164 /* note hac does 0 based, so if used==5 then its 0,1,2,3,4, e.g. use 4 */ +165 n = x.used - 1; +166 t = y.used - 1; +167 +168 /* while (x >= y*b**n-t) do \{ q[n-t] += 1; x -= y*b**\{n-t\} \} */ +169 if ((res = mp_lshd (&y, n - t)) != MP_OKAY) \{ /* y = y*b**\{n-t\} */ +170 goto LBL_Y; +171 \} +172 +173 while (mp_cmp (&x, &y) != MP_LT) \{ +174 ++(q.dp[n - t]); +175 if ((res = mp_sub (&x, &y, &x)) != MP_OKAY) \{ +176 goto LBL_Y; +177 \} +178 \} +179 +180 /* reset y by shifting it back down */ +181 mp_rshd (&y, n - t); +182 +183 /* step 3. for i from n down to (t + 1) */ +184 for (i = n; i >= (t + 1); i--) \{ +185 if (i > x.used) \{ +186 continue; +187 \} +188 +189 /* step 3.1 if xi == yt then set q\{i-t-1\} to b-1, +190 * otherwise set q\{i-t-1\} to (xi*b + x\{i-1\})/yt */ +191 if (x.dp[i] == y.dp[t]) \{ +192 q.dp[(i - t) - 1] = ((((mp_digit)1) << DIGIT_BIT) - 1); +193 \} else \{ +194 mp_word tmp; +195 tmp = ((mp_word) x.dp[i]) << ((mp_word) DIGIT_BIT); +196 tmp |= ((mp_word) x.dp[i - 1]); +197 tmp /= ((mp_word) y.dp[t]); +198 if (tmp > (mp_word) MP_MASK) \{ +199 tmp = MP_MASK; +200 \} +201 q.dp[(i - t) - 1] = (mp_digit) (tmp & (mp_word) (MP_MASK)); +202 \} +203 +204 /* while (q\{i-t-1\} * (yt * b + y\{t-1\})) > +205 xi * b**2 + xi-1 * b + xi-2 +206 +207 do q\{i-t-1\} -= 1; +208 */ +209 q.dp[(i - t) - 1] = (q.dp[(i - t) - 1] + 1) & MP_MASK; +210 do \{ +211 q.dp[(i - t) - 1] = (q.dp[(i - t) - 1] - 1) & MP_MASK; +212 +213 /* find left hand */ +214 mp_zero (&t1); +215 t1.dp[0] = ((t - 1) < 0) ? 0 : y.dp[t - 1]; +216 t1.dp[1] = y.dp[t]; +217 t1.used = 2; +218 if ((res = mp_mul_d (&t1, q.dp[(i - t) - 1], &t1)) != MP_OKAY) \{ +219 goto LBL_Y; +220 \} +221 +222 /* find right hand */ +223 t2.dp[0] = ((i - 2) < 0) ? 0 : x.dp[i - 2]; +224 t2.dp[1] = ((i - 1) < 0) ? 0 : x.dp[i - 1]; +225 t2.dp[2] = x.dp[i]; +226 t2.used = 3; +227 \} while (mp_cmp_mag(&t1, &t2) == MP_GT); +228 +229 /* step 3.3 x = x - q\{i-t-1\} * y * b**\{i-t-1\} */ +230 if ((res = mp_mul_d (&y, q.dp[(i - t) - 1], &t1)) != MP_OKAY) \{ +231 goto LBL_Y; +232 \} +233 +234 if ((res = mp_lshd (&t1, (i - t) - 1)) != MP_OKAY) \{ +235 goto LBL_Y; +236 \} +237 +238 if ((res = mp_sub (&x, &t1, &x)) != MP_OKAY) \{ +239 goto LBL_Y; +240 \} +241 +242 /* if x < 0 then \{ x = x + y*b**\{i-t-1\}; q\{i-t-1\} -= 1; \} */ +243 if (x.sign == MP_NEG) \{ +244 if ((res = mp_copy (&y, &t1)) != MP_OKAY) \{ +245 goto LBL_Y; +246 \} +247 if ((res = mp_lshd (&t1, (i - t) - 1)) != MP_OKAY) \{ +248 goto LBL_Y; +249 \} +250 if ((res = mp_add (&x, &t1, &x)) != MP_OKAY) \{ +251 goto LBL_Y; +252 \} +253 +254 q.dp[(i - t) - 1] = (q.dp[(i - t) - 1] - 1UL) & MP_MASK; +255 \} +256 \} +257 +258 /* now q is the quotient and x is the remainder +259 * [which we have to normalize] +260 */ +261 +262 /* get sign before writing to c */ +263 x.sign = (x.used == 0) ? MP_ZPOS : a->sign; +264 +265 if (c != NULL) \{ +266 mp_clamp (&q); +267 mp_exch (&q, c); +268 c->sign = neg; +269 \} +270 +271 if (d != NULL) \{ +272 if ((res = mp_div_2d (&x, norm, &x, NULL)) != MP_OKAY) \{ +273 goto LBL_Y; +274 \} +275 mp_exch (&x, d); +276 \} +277 +278 res = MP_OKAY; +279 +280 LBL_Y:mp_clear (&y); +281 LBL_X:mp_clear (&x); +282 LBL_T2:mp_clear (&t2); +283 LBL_T1:mp_clear (&t1); +284 LBL_Q:mp_clear (&q); +285 return res; +286 \} +287 +288 #endif +289 +290 #endif +291 \end{alltt} \end{small} The implementation of this algorithm differs slightly from the pseudo code presented previously. In this algorithm either of the quotient $c$ or remainder $d$ may be passed as a \textbf{NULL} pointer which indicates their value is not desired. For example, the C code to call the division -algorithm with only the quotient is +algorithm with only the quotient is \begin{verbatim} mp_div(&a, &b, &c, NULL); /* c = [a/b] */ \end{verbatim} -Lines 109 and 113 handle the two trivial cases of inputs which are division by zero and dividend smaller than the divisor -respectively. After the two trivial cases all of the temporary variables are initialized. Line 148 determines the sign of -the quotient and line 148 ensures that both $x$ and $y$ are positive. +Lines 108 and 113 handle the two trivial cases of inputs which are division by zero and dividend smaller than the divisor +respectively. After the two trivial cases all of the temporary variables are initialized. Line 147 determines the sign of +the quotient and line 148 ensures that both $x$ and $y$ are positive. 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 of precision which when reduced modulo $lg(\beta)$ produces the value of $k$. In this case $k$ is the number of bits in the leading digit which is exactly what is required. For the algorithm to operate $k$ must equal $lg(\beta) - 1$ and when it does not the inputs must be normalized by shifting them to the left by $lg(\beta) - 1 - k$ bits. -Throughout the variables $n$ and $t$ will represent the highest digit of $x$ and $y$ respectively. These are first used to produce the +Throughout the variables $n$ and $t$ will represent the highest digit of $x$ and $y$ respectively. These are first used to produce the leading digit of the quotient. The loop beginning on line 184 will produce the remainder of the quotient digits. -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 +The conditional ``continue'' on line 186 is used to prevent the algorithm from reading past the leading edge of $x$ which can occur when the algorithm eliminates multiple non-zero digits in a single iteration. This ensures that $x_i$ is always non-zero since by definition the digits -above the $i$'th position $x$ must be zero in order for the quotient to be precise\footnote{Precise as far as integer division is concerned.}. +above the $i$'th position $x$ must be zero in order for the quotient to be precise\footnote{Precise as far as integer division is concerned.}. -Lines 214, 216 and 223 through 225 manually construct the high accuracy estimations by setting the digits of the two mp\_int -variables directly. +Lines 214, 216 and 223 through 225 manually construct the high accuracy estimations by setting the digits of the two mp\_int +variables directly. \section{Single Digit Helpers} -This section briefly describes a series of single digit helper algorithms which come in handy when working with small constants. All of +This section briefly describes a series of single digit helper algorithms which come in handy when working with small constants. All of the helper functions assume the single digit input is positive and will treat them as such. \subsection{Single Digit Addition and Subtraction} -Both addition and subtraction are performed by ``cheating'' and using mp\_set followed by the higher level addition or subtraction +Both addition and subtraction are performed by ``cheating'' and using mp\_set followed by the higher level addition or subtraction algorithms. As a result these algorithms are subtantially simpler with a slight cost in performance. \newpage\begin{figure}[!here] @@ -5550,6 +8807,99 @@ This algorithm initiates a temporary mp\_int with the value of the single digit \hspace{-5.1mm}{\bf File}: bn\_mp\_add\_d.c \vspace{-3mm} \begin{alltt} +016 +017 /* single digit addition */ +018 int +019 mp_add_d (mp_int * a, mp_digit b, mp_int * c) +020 \{ +021 int res, ix, oldused; +022 mp_digit *tmpa, *tmpc, mu; +023 +024 /* grow c as required */ +025 if (c->alloc < (a->used + 1)) \{ +026 if ((res = mp_grow(c, a->used + 1)) != MP_OKAY) \{ +027 return res; +028 \} +029 \} +030 +031 /* if a is negative and |a| >= b, call c = |a| - b */ +032 if ((a->sign == MP_NEG) && ((a->used > 1) || (a->dp[0] >= b))) \{ +033 /* temporarily fix sign of a */ +034 a->sign = MP_ZPOS; +035 +036 /* c = |a| - b */ +037 res = mp_sub_d(a, b, c); +038 +039 /* fix sign */ +040 a->sign = c->sign = MP_NEG; +041 +042 /* clamp */ +043 mp_clamp(c); +044 +045 return res; +046 \} +047 +048 /* old number of used digits in c */ +049 oldused = c->used; +050 +051 /* sign always positive */ +052 c->sign = MP_ZPOS; +053 +054 /* source alias */ +055 tmpa = a->dp; +056 +057 /* destination alias */ +058 tmpc = c->dp; +059 +060 /* if a is positive */ +061 if (a->sign == MP_ZPOS) \{ +062 /* add digit, after this we're propagating +063 * the carry. +064 */ +065 *tmpc = *tmpa++ + b; +066 mu = *tmpc >> DIGIT_BIT; +067 *tmpc++ &= MP_MASK; +068 +069 /* now handle rest of the digits */ +070 for (ix = 1; ix < a->used; ix++) \{ +071 *tmpc = *tmpa++ + mu; +072 mu = *tmpc >> DIGIT_BIT; +073 *tmpc++ &= MP_MASK; +074 \} +075 /* set final carry */ +076 ix++; +077 *tmpc++ = mu; +078 +079 /* setup size */ +080 c->used = a->used + 1; +081 \} else \{ +082 /* a was negative and |a| < b */ +083 c->used = 1; +084 +085 /* the result is a single digit */ +086 if (a->used == 1) \{ +087 *tmpc++ = b - a->dp[0]; +088 \} else \{ +089 *tmpc++ = b; +090 \} +091 +092 /* setup count so the clearing of oldused +093 * can fall through correctly +094 */ +095 ix = 1; +096 \} +097 +098 /* now zero to oldused */ +099 while (ix++ < oldused) \{ +100 *tmpc++ = 0; +101 \} +102 mp_clamp(c); +103 +104 return MP_OKAY; +105 \} +106 +107 #endif +108 \end{alltt} \end{small} @@ -5593,22 +8943,82 @@ only has one digit. \caption{Algorithm mp\_mul\_d} \end{figure} \textbf{Algorithm mp\_mul\_d.} -This algorithm quickly multiplies an mp\_int by a small single digit value. It is specially tailored to the job and has a minimal of overhead. -Unlike the full multiplication algorithms this algorithm does not require any significnat temporary storage or memory allocations. +This algorithm quickly multiplies an mp\_int by a small single digit value. It is specially tailored to the job and has a minimal of overhead. +Unlike the full multiplication algorithms this algorithm does not require any significnat temporary storage or memory allocations. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_mul\_d.c \vspace{-3mm} \begin{alltt} +016 +017 /* multiply by a digit */ +018 int +019 mp_mul_d (mp_int * a, mp_digit b, mp_int * c) +020 \{ +021 mp_digit u, *tmpa, *tmpc; +022 mp_word r; +023 int ix, res, olduse; +024 +025 /* make sure c is big enough to hold a*b */ +026 if (c->alloc < (a->used + 1)) \{ +027 if ((res = mp_grow (c, a->used + 1)) != MP_OKAY) \{ +028 return res; +029 \} +030 \} +031 +032 /* get the original destinations used count */ +033 olduse = c->used; +034 +035 /* set the sign */ +036 c->sign = a->sign; +037 +038 /* alias for a->dp [source] */ +039 tmpa = a->dp; +040 +041 /* alias for c->dp [dest] */ +042 tmpc = c->dp; +043 +044 /* zero carry */ +045 u = 0; +046 +047 /* compute columns */ +048 for (ix = 0; ix < a->used; ix++) \{ +049 /* compute product and carry sum for this term */ +050 r = (mp_word)u + ((mp_word)*tmpa++ * (mp_word)b); +051 +052 /* mask off higher bits to get a single digit */ +053 *tmpc++ = (mp_digit) (r & ((mp_word) MP_MASK)); +054 +055 /* send carry into next iteration */ +056 u = (mp_digit) (r >> ((mp_word) DIGIT_BIT)); +057 \} +058 +059 /* store final carry [if any] and increment ix offset */ +060 *tmpc++ = u; +061 ++ix; +062 +063 /* now zero digits above the top */ +064 while (ix++ < olduse) \{ +065 *tmpc++ = 0; +066 \} +067 +068 /* set used count */ +069 c->used = a->used + 1; +070 mp_clamp(c); +071 +072 return MP_OKAY; +073 \} +074 #endif +075 \end{alltt} \end{small} -In this implementation the destination $c$ may point to the same mp\_int as the source $a$ since the result is written after the digit is -read from the source. This function uses pointer aliases $tmpa$ and $tmpc$ for the digits of $a$ and $c$ respectively. +In this implementation the destination $c$ may point to the same mp\_int as the source $a$ since the result is written after the digit is +read from the source. This function uses pointer aliases $tmpa$ and $tmpc$ for the digits of $a$ and $c$ respectively. \subsection{Single Digit Division} Like the single digit multiplication algorithm, single digit division is also a fairly common algorithm used in radix conversion. Since the -divisor is only a single digit a specialized variant of the division algorithm can be used to compute the quotient. +divisor is only a single digit a specialized variant of the division algorithm can be used to compute the quotient. \newpage\begin{figure}[!here] \begin{small} @@ -5645,40 +9055,136 @@ divisor is only a single digit a specialized variant of the division algorithm c \textbf{Algorithm mp\_div\_d.} This algorithm divides the mp\_int $a$ by the single mp\_digit $b$ using an optimized approach. Essentially in every iteration of the algorithm another digit of the dividend is reduced and another digit of quotient produced. Provided $b < \beta$ the value of $\hat w$ -after step 7.1 will be limited such that $0 \le \lfloor \hat w / b \rfloor < \beta$. +after step 7.1 will be limited such that $0 \le \lfloor \hat w / b \rfloor < \beta$. If the divisor $b$ is equal to three a variant of this algorithm is used which is called mp\_div\_3. It replaces the division by three with a multiplication by $\lfloor \beta / 3 \rfloor$ and the appropriate shift and residual fixup. In essence it is much like the Barrett reduction -from chapter seven. +from chapter seven. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_div\_d.c \vspace{-3mm} \begin{alltt} +016 +017 static int s_is_power_of_two(mp_digit b, int *p) +018 \{ +019 int x; +020 +021 /* fast return if no power of two */ +022 if ((b == 0) || ((b & (b-1)) != 0)) \{ +023 return 0; +024 \} +025 +026 for (x = 0; x < DIGIT_BIT; x++) \{ +027 if (b == (((mp_digit)1)<<x)) \{ +028 *p = x; +029 return 1; +030 \} +031 \} +032 return 0; +033 \} +034 +035 /* single digit division (based on routine from MPI) */ +036 int mp_div_d (mp_int * a, mp_digit b, mp_int * c, mp_digit * d) +037 \{ +038 mp_int q; +039 mp_word w; +040 mp_digit t; +041 int res, ix; +042 +043 /* cannot divide by zero */ +044 if (b == 0) \{ +045 return MP_VAL; +046 \} +047 +048 /* quick outs */ +049 if ((b == 1) || (mp_iszero(a) == MP_YES)) \{ +050 if (d != NULL) \{ +051 *d = 0; +052 \} +053 if (c != NULL) \{ +054 return mp_copy(a, c); +055 \} +056 return MP_OKAY; +057 \} +058 +059 /* power of two ? */ +060 if (s_is_power_of_two(b, &ix) == 1) \{ +061 if (d != NULL) \{ +062 *d = a->dp[0] & ((((mp_digit)1)<<ix) - 1); +063 \} +064 if (c != NULL) \{ +065 return mp_div_2d(a, ix, c, NULL); +066 \} +067 return MP_OKAY; +068 \} +069 +070 #ifdef BN_MP_DIV_3_C +071 /* three? */ +072 if (b == 3) \{ +073 return mp_div_3(a, c, d); +074 \} +075 #endif +076 +077 /* no easy answer [c'est la vie]. Just division */ +078 if ((res = mp_init_size(&q, a->used)) != MP_OKAY) \{ +079 return res; +080 \} +081 +082 q.used = a->used; +083 q.sign = a->sign; +084 w = 0; +085 for (ix = a->used - 1; ix >= 0; ix--) \{ +086 w = (w << ((mp_word)DIGIT_BIT)) | ((mp_word)a->dp[ix]); +087 +088 if (w >= b) \{ +089 t = (mp_digit)(w / b); +090 w -= ((mp_word)t) * ((mp_word)b); +091 \} else \{ +092 t = 0; +093 \} +094 q.dp[ix] = (mp_digit)t; +095 \} +096 +097 if (d != NULL) \{ +098 *d = (mp_digit)w; +099 \} +100 +101 if (c != NULL) \{ +102 mp_clamp(&q); +103 mp_exch(&q, c); +104 \} +105 mp_clear(&q); +106 +107 return res; +108 \} +109 +110 #endif +111 \end{alltt} \end{small} Like the implementation of algorithm mp\_div this algorithm allows either of the quotient or remainder to be passed as a \textbf{NULL} pointer to indicate the respective value is not required. This allows a trivial single digit modular reduction algorithm, mp\_mod\_d to be created. -The division and remainder on lines 44 and @45,%@ can be replaced often by a single division on most processors. For example, the 32-bit x86 based -processors can divide a 64-bit quantity by a 32-bit quantity and produce the quotient and remainder simultaneously. Unfortunately the GCC -compiler does not recognize that optimization and will actually produce two function calls to find the quotient and remainder respectively. +The division and remainder on lines 89 and 90 can be replaced often by a single division on most processors. For example, the 32-bit x86 based +processors can divide a 64-bit quantity by a 32-bit quantity and produce the quotient and remainder simultaneously. Unfortunately the GCC +compiler does not recognize that optimization and will actually produce two function calls to find the quotient and remainder respectively. \subsection{Single Digit Root Extraction} -Finding the $n$'th root of an integer is fairly easy as far as numerical analysis is concerned. Algorithms such as the Newton-Raphson approximation -(\ref{eqn:newton}) series will converge very quickly to a root for any continuous function $f(x)$. +Finding the $n$'th root of an integer is fairly easy as far as numerical analysis is concerned. Algorithms such as the Newton-Raphson approximation +(\ref{eqn:newton}) series will converge very quickly to a root for any continuous function $f(x)$. \begin{equation} x_{i+1} = x_i - {f(x_i) \over f'(x_i)} \label{eqn:newton} \end{equation} -In this case the $n$'th root is desired and $f(x) = x^n - a$ where $a$ is the integer of which the root is desired. The derivative of $f(x)$ is +In this case the $n$'th root is desired and $f(x) = x^n - a$ where $a$ is the integer of which the root is desired. The derivative of $f(x)$ is simply $f'(x) = nx^{n - 1}$. Of particular importance is that this algorithm will be used over the integers not over the a more continuous domain -such as the real numbers. As a result the root found can be above the true root by few and must be manually adjusted. Ideally at the end of the -algorithm the $n$'th root $b$ of an integer $a$ is desired such that $b^n \le a$. +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 +algorithm the $n$'th root $b$ of an integer $a$ is desired such that $b^n \le a$. \newpage\begin{figure}[!here] \begin{small} @@ -5719,24 +9225,35 @@ algorithm the $n$'th root $b$ of an integer $a$ is desired such that $b^n \le a$ \textbf{Algorithm mp\_n\_root.} This algorithm finds the integer $n$'th root of an input using the Newton-Raphson approach. It is partially optimized based on the observation 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 -$x^{b - 1}$. This value can then be multiplied by $x$ and have $a$ subtracted from it to find the numerator. This saves a total of $b - 1$ -multiplications by t$1$ inside the loop. +$x^{b - 1}$. This value can then be multiplied by $x$ and have $a$ subtracted from it to find the numerator. This saves a total of $b - 1$ +multiplications by t$1$ inside the loop. 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 -root. Ideally this algorithm is meant to find the $n$'th root of an input where $n$ is bounded by $2 \le n \le 5$. +root. Ideally this algorithm is meant to find the $n$'th root of an input where $n$ is bounded by $2 \le n \le 5$. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_n\_root.c \vspace{-3mm} \begin{alltt} +016 +017 /* wrapper function for mp_n_root_ex() +018 * computes c = (a)**(1/b) such that (c)**b <= a and (c+1)**b > a +019 */ +020 int mp_n_root (mp_int * a, mp_digit b, mp_int * c) +021 \{ +022 return mp_n_root_ex(a, b, c, 0); +023 \} +024 +025 #endif +026 \end{alltt} \end{small} \section{Random Number Generation} -Random numbers come up in a variety of activities from public key cryptography to simple simulations and various randomized algorithms. Pollard-Rho +Random numbers come up in a variety of activities from public key cryptography to simple simulations and various randomized algorithms. Pollard-Rho 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 -is solely for simulations and not intended for cryptographic use. +is solely for simulations and not intended for cryptographic use. \newpage\begin{figure}[!here] \begin{small} @@ -5764,12 +9281,49 @@ is solely for simulations and not intended for cryptographic use. \textbf{Algorithm mp\_rand.} 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 final result has at least $b$ digits. It relies heavily on a third-part random number generator which should ideally generate uniformly all of -the integers from $0$ to $\beta - 1$. +the integers from $0$ to $\beta - 1$. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_rand.c \vspace{-3mm} \begin{alltt} +016 +017 /* makes a pseudo-random int of a given size */ +018 int +019 mp_rand (mp_int * a, int digits) +020 \{ +021 int res; +022 mp_digit d; +023 +024 mp_zero (a); +025 if (digits <= 0) \{ +026 return MP_OKAY; +027 \} +028 +029 /* first place a random non-zero digit */ +030 do \{ +031 d = ((mp_digit) abs (MP_GEN_RANDOM())) & MP_MASK; +032 \} while (d == 0); +033 +034 if ((res = mp_add_d (a, d, a)) != MP_OKAY) \{ +035 return res; +036 \} +037 +038 while (--digits > 0) \{ +039 if ((res = mp_lshd (a, 1)) != MP_OKAY) \{ +040 return res; +041 \} +042 +043 if ((res = mp_add_d (a, ((mp_digit) abs (MP_GEN_RANDOM())), a)) != MP_OK + AY) \{ +044 return res; +045 \} +046 \} +047 +048 return MP_OKAY; +049 \} +050 #endif +051 \end{alltt} \end{small} @@ -5779,7 +9333,7 @@ be given a string of characters such as ``114585'' and turn it into the radix-$\ into a program. \subsection{Reading Radix-n Input} -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 +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 printable characters. For example, when the character ``N'' is read it represents the integer $23$. The first $16$ characters of the map are for the common representations up to hexadecimal. After that they match the ``base64'' encoding scheme which are suitable chosen such that they are printable. While outputting as base64 may not be too helpful for human operators it does allow communication via non binary @@ -5789,7 +9343,7 @@ mediums. \begin{center} \begin{tabular}{cc|cc|cc|cc} \hline \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} \\ -\hline +\hline 0 & 0 & 1 & 1 & 2 & 2 & 3 & 3 \\ 4 & 4 & 5 & 5 & 6 & 6 & 7 & 7 \\ 8 & 8 & 9 & 9 & 10 & A & 11 & B \\ @@ -5843,7 +9397,7 @@ mediums. \caption{Algorithm mp\_read\_radix} \end{figure} \textbf{Algorithm mp\_read\_radix.} -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 +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 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 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 as part of larger input without any significant problem. @@ -5852,11 +9406,77 @@ as part of larger input without any significant problem. \hspace{-5.1mm}{\bf File}: bn\_mp\_read\_radix.c \vspace{-3mm} \begin{alltt} +016 +017 /* read a string [ASCII] in a given radix */ +018 int mp_read_radix (mp_int * a, const char *str, int radix) +019 \{ +020 int y, res, neg; +021 char ch; +022 +023 /* zero the digit bignum */ +024 mp_zero(a); +025 +026 /* make sure the radix is ok */ +027 if ((radix < 2) || (radix > 64)) \{ +028 return MP_VAL; +029 \} +030 +031 /* if the leading digit is a +032 * minus set the sign to negative. +033 */ +034 if (*str == '-') \{ +035 ++str; +036 neg = MP_NEG; +037 \} else \{ +038 neg = MP_ZPOS; +039 \} +040 +041 /* set the integer to the default of zero */ +042 mp_zero (a); +043 +044 /* process each digit of the string */ +045 while (*str != '\symbol{92}0') \{ +046 /* if the radix <= 36 the conversion is case insensitive +047 * this allows numbers like 1AB and 1ab to represent the same value +048 * [e.g. in hex] +049 */ +050 ch = (radix <= 36) ? (char)toupper((int)*str) : *str; +051 for (y = 0; y < 64; y++) \{ +052 if (ch == mp_s_rmap[y]) \{ +053 break; +054 \} +055 \} +056 +057 /* if the char was found in the map +058 * and is less than the given radix add it +059 * to the number, otherwise exit the loop. +060 */ +061 if (y < radix) \{ +062 if ((res = mp_mul_d (a, (mp_digit) radix, a)) != MP_OKAY) \{ +063 return res; +064 \} +065 if ((res = mp_add_d (a, (mp_digit) y, a)) != MP_OKAY) \{ +066 return res; +067 \} +068 \} else \{ +069 break; +070 \} +071 ++str; +072 \} +073 +074 /* set the sign only if a != 0 */ +075 if (mp_iszero(a) != MP_YES) \{ +076 a->sign = neg; +077 \} +078 return MP_OKAY; +079 \} +080 #endif +081 \end{alltt} \end{small} \subsection{Generating Radix-$n$ Output} -Generating radix-$n$ output is fairly trivial with a division and remainder algorithm. +Generating radix-$n$ output is fairly trivial with a division and remainder algorithm. \newpage\begin{figure}[!here] \begin{small} @@ -5890,10 +9510,10 @@ Generating radix-$n$ output is fairly trivial with a division and remainder algo \caption{Algorithm mp\_toradix} \end{figure} \textbf{Algorithm mp\_toradix.} -This algorithm computes the radix-$r$ representation of an mp\_int $a$. The ``digits'' of the representation are extracted by reducing +This algorithm computes the radix-$r$ representation of an mp\_int $a$. The ``digits'' of the representation are extracted by reducing 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 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 -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 +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 (see~\ref{fig:mpradix}). To remedy this flaw the digits must be swapped or simply ``reversed''. \begin{figure} @@ -5916,11 +9536,67 @@ are required instead of a series of $n \times k$ divisions. One design flaw of \hspace{-5.1mm}{\bf File}: bn\_mp\_toradix.c \vspace{-3mm} \begin{alltt} +016 +017 /* stores a bignum as a ASCII string in a given radix (2..64) */ +018 int mp_toradix (mp_int * a, char *str, int radix) +019 \{ +020 int res, digs; +021 mp_int t; +022 mp_digit d; +023 char *_s = str; +024 +025 /* check range of the radix */ +026 if ((radix < 2) || (radix > 64)) \{ +027 return MP_VAL; +028 \} +029 +030 /* quick out if its zero */ +031 if (mp_iszero(a) == MP_YES) \{ +032 *str++ = '0'; +033 *str = '\symbol{92}0'; +034 return MP_OKAY; +035 \} +036 +037 if ((res = mp_init_copy (&t, a)) != MP_OKAY) \{ +038 return res; +039 \} +040 +041 /* if it is negative output a - */ +042 if (t.sign == MP_NEG) \{ +043 ++_s; +044 *str++ = '-'; +045 t.sign = MP_ZPOS; +046 \} +047 +048 digs = 0; +049 while (mp_iszero (&t) == MP_NO) \{ +050 if ((res = mp_div_d (&t, (mp_digit) radix, &t, &d)) != MP_OKAY) \{ +051 mp_clear (&t); +052 return res; +053 \} +054 *str++ = mp_s_rmap[d]; +055 ++digs; +056 \} +057 +058 /* reverse the digits of the string. In this case _s points +059 * to the first digit [exluding the sign] of the number] +060 */ +061 bn_reverse ((unsigned char *)_s, digs); +062 +063 /* append a NULL so the string is properly terminated */ +064 *str = '\symbol{92}0'; +065 +066 mp_clear (&t); +067 return MP_OKAY; +068 \} +069 +070 #endif +071 \end{alltt} \end{small} \chapter{Number Theoretic Algorithms} -This chapter discusses several fundamental number theoretic algorithms such as the greatest common divisor, least common multiple and Jacobi +This chapter discusses several fundamental number theoretic algorithms such as the greatest common divisor, least common multiple and Jacobi symbol computation. These algorithms arise as essential components in several key cryptographic algorithms such as the RSA public key algorithm and various Sieve based factoring algorithms. @@ -5930,7 +9606,7 @@ both $a$ and $b$. That is, $k$ is the largest integer such that $0 \equiv a \mb simultaneously. 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 -$r$ is also divisible by $k$. The reduction pattern follows $\left < a , b \right > \rightarrow \left < b, a \mbox{ mod } b \right >$. +$r$ is also divisible by $k$. The reduction pattern follows $\left < a , b \right > \rightarrow \left < b, a \mbox{ mod } b \right >$. \newpage\begin{figure}[!here] \begin{small} @@ -5954,9 +9630,9 @@ $r$ is also divisible by $k$. The reduction pattern follows $\left < a , b \rig \end{figure} This algorithm will quickly converge on the greatest common divisor since the residue $r$ tends diminish rapidly. However, divisions are -relatively expensive operations to perform and should ideally be avoided. There is another approach based on a similar relationship of -greatest common divisors. The faster approach is based on the observation that if $k$ divides both $a$ and $b$ it will also divide $a - b$. -In particular, we would like $a - b$ to decrease in magnitude which implies that $b \ge a$. +relatively expensive operations to perform and should ideally be avoided. There is another approach based on a similar relationship of +greatest common divisors. The faster approach is based on the observation that if $k$ divides both $a$ and $b$ it will also divide $a - b$. +In particular, we would like $a - b$ to decrease in magnitude which implies that $b \ge a$. \begin{figure}[!here] \begin{small} @@ -5980,17 +9656,17 @@ In particular, we would like $a - b$ to decrease in magnitude which implies that \textbf{Proof} \textit{Algorithm~\ref{fig:gcd2} will return the greatest common divisor of $a$ and $b$.} 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 -words in every iteration that tuple $\left < a, b \right >$ decrease in magnitude until eventually $a = b$. Since both $a$ and $b$ are always -divisible by the greatest common divisor (\textit{until the last iteration}) and in the last iteration of the algorithm $b = 0$, therefore, in the +words in every iteration that tuple $\left < a, b \right >$ decrease in magnitude until eventually $a = b$. Since both $a$ and $b$ are always +divisible by the greatest common divisor (\textit{until the last iteration}) and in the last iteration of the algorithm $b = 0$, therefore, in the second to last iteration of the algorithm $b = a$ and clearly $(a, a) = a$ which concludes the proof. \textbf{QED}. -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 +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 $b - a$ 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 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 the greatest common divisor. -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. -Then inside the loop whenever $b - a$ is divisible by some power of $p$ it can be safely removed. +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. +Then inside the loop whenever $b - a$ is divisible by some power of $p$ it can be safely removed. \begin{figure}[!here] \begin{small} @@ -6023,14 +9699,14 @@ Then inside the loop whenever $b - a$ is divisible by some power of $p$ it can b \label{fig:gcd3} \end{figure} -This algorithm is based on the first except it removes powers of $p$ first and inside the main loop to ensure the tuple $\left < a, b \right >$ +This algorithm is based on the first except it removes powers of $p$ first and inside the main loop to ensure the tuple $\left < a, b \right >$ 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 -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 -divided out of the difference $b - a$ so long as the division leaves no remainder. +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 +divided out of the difference $b - a$ so long as the division leaves no remainder. 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 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 -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 +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 largest of the pair. \subsection{Complete Greatest Common Divisor} @@ -6078,15 +9754,15 @@ and will produce the greatest common divisor. \textbf{Algorithm mp\_gcd.} This algorithm will produce the greatest common divisor of two mp\_ints $a$ and $b$. The algorithm was originally based on Algorithm B of Knuth \cite[pp. 338]{TAOCPV2} but has been modified to be simpler to explain. In theory it achieves the same asymptotic working time as -Algorithm B and in practice this appears to be true. +Algorithm B and in practice this appears to be true. -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 -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 +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 +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 $a$ and $b$ respectively and the algorithm will proceed to reduce the pair. 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 -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 -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 +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 +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 they cannot both be even. 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 @@ -6094,29 +9770,115 @@ or greater than $u$. This ensures that the subtraction on step 8.2 will always factors of two from the difference $u$ to ensure that in the next iteration of the loop both are once again odd. 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 -must be adjusted by multiplying by the common factors of two ($2^k$) removed earlier. +must be adjusted by multiplying by the common factors of two ($2^k$) removed earlier. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_gcd.c \vspace{-3mm} \begin{alltt} +016 +017 /* Greatest Common Divisor using the binary method */ +018 int mp_gcd (mp_int * a, mp_int * b, mp_int * c) +019 \{ +020 mp_int u, v; +021 int k, u_lsb, v_lsb, res; +022 +023 /* either zero than gcd is the largest */ +024 if (mp_iszero (a) == MP_YES) \{ +025 return mp_abs (b, c); +026 \} +027 if (mp_iszero (b) == MP_YES) \{ +028 return mp_abs (a, c); +029 \} +030 +031 /* get copies of a and b we can modify */ +032 if ((res = mp_init_copy (&u, a)) != MP_OKAY) \{ +033 return res; +034 \} +035 +036 if ((res = mp_init_copy (&v, b)) != MP_OKAY) \{ +037 goto LBL_U; +038 \} +039 +040 /* must be positive for the remainder of the algorithm */ +041 u.sign = v.sign = MP_ZPOS; +042 +043 /* B1. Find the common power of two for u and v */ +044 u_lsb = mp_cnt_lsb(&u); +045 v_lsb = mp_cnt_lsb(&v); +046 k = MIN(u_lsb, v_lsb); +047 +048 if (k > 0) \{ +049 /* divide the power of two out */ +050 if ((res = mp_div_2d(&u, k, &u, NULL)) != MP_OKAY) \{ +051 goto LBL_V; +052 \} +053 +054 if ((res = mp_div_2d(&v, k, &v, NULL)) != MP_OKAY) \{ +055 goto LBL_V; +056 \} +057 \} +058 +059 /* divide any remaining factors of two out */ +060 if (u_lsb != k) \{ +061 if ((res = mp_div_2d(&u, u_lsb - k, &u, NULL)) != MP_OKAY) \{ +062 goto LBL_V; +063 \} +064 \} +065 +066 if (v_lsb != k) \{ +067 if ((res = mp_div_2d(&v, v_lsb - k, &v, NULL)) != MP_OKAY) \{ +068 goto LBL_V; +069 \} +070 \} +071 +072 while (mp_iszero(&v) == MP_NO) \{ +073 /* make sure v is the largest */ +074 if (mp_cmp_mag(&u, &v) == MP_GT) \{ +075 /* swap u and v to make sure v is >= u */ +076 mp_exch(&u, &v); +077 \} +078 +079 /* subtract smallest from largest */ +080 if ((res = s_mp_sub(&v, &u, &v)) != MP_OKAY) \{ +081 goto LBL_V; +082 \} +083 +084 /* Divide out all factors of two */ +085 if ((res = mp_div_2d(&v, mp_cnt_lsb(&v), &v, NULL)) != MP_OKAY) \{ +086 goto LBL_V; +087 \} +088 \} +089 +090 /* multiply by 2**k which we divided out at the beginning */ +091 if ((res = mp_mul_2d (&u, k, c)) != MP_OKAY) \{ +092 goto LBL_V; +093 \} +094 c->sign = MP_ZPOS; +095 res = MP_OKAY; +096 LBL_V:mp_clear (&u); +097 LBL_U:mp_clear (&v); +098 return res; +099 \} +100 #endif +101 \end{alltt} \end{small} -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 +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 integer zero otherwise it evaluates to $0$. The latter evaluates to $1$ if the input mp\_int represents a non-zero even integer otherwise -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 -trivial cases of inputs are handled on lines 24 through 30. After those lines the inputs are assumed to be non-zero. +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 +trivial cases of inputs are handled on lines 23 through 29. After those lines the inputs are assumed to be non-zero. -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 -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 -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 -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 -entries than are accessible by an ``int'' so this is not a limitation.}. +Lines 32 and 36 make local copies $u$ and $v$ of the inputs $a$ and $b$ respectively. At this point the common factors of two +must be divided out of the two inputs. The block starting at line 43 removes common factors of two by first counting the number of trailing +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 +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 +entries than are accessible by an ``int'' so this is not a limitation.}. -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 +At this point there are no more common factors of two in the two values. The divisions by a power of two on lines 61 and 67 remove 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 -on line 73 performs the reduction of the pair until $v$ is equal to zero. The unsigned comparison and subtraction algorithms are used in +on line 72 performs the reduction of the pair until $v$ is equal to zero. The unsigned comparison and subtraction algorithms are used in 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. \section{Least Common Multiple} @@ -6125,8 +9887,8 @@ least common multiple is normally denoted as $[ a, b ]$ and numerically equivale and $b = 2 \cdot 3 \cdot 3 \cdot 7 = 126$ the least common multiple is ${126 \over {(12, 126)}} = {126 \over 6} = 21$. 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 -collide, that is be in synchronous states, after only $[ a, b ]$ iterations. This is why, for example, random number generators based on -Linear Feedback Shift Registers (LFSR) tend to use registers with periods which are co-prime (\textit{e.g. the greatest common divisor is one.}). +collide, that is be in synchronous states, after only $[ a, b ]$ iterations. This is why, for example, random number generators based on +Linear Feedback Shift Registers (LFSR) tend to use registers with periods which are co-prime (\textit{e.g. the greatest common divisor is one.}). 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] $. \begin{figure}[!here] @@ -6155,11 +9917,52 @@ dividing the product of the two inputs by their greatest common divisor. \hspace{-5.1mm}{\bf File}: bn\_mp\_lcm.c \vspace{-3mm} \begin{alltt} +016 +017 /* computes least common multiple as |a*b|/(a, b) */ +018 int mp_lcm (mp_int * a, mp_int * b, mp_int * c) +019 \{ +020 int res; +021 mp_int t1, t2; +022 +023 +024 if ((res = mp_init_multi (&t1, &t2, NULL)) != MP_OKAY) \{ +025 return res; +026 \} +027 +028 /* t1 = get the GCD of the two inputs */ +029 if ((res = mp_gcd (a, b, &t1)) != MP_OKAY) \{ +030 goto LBL_T; +031 \} +032 +033 /* divide the smallest by the GCD */ +034 if (mp_cmp_mag(a, b) == MP_LT) \{ +035 /* store quotient in t2 such that t2 * b is the LCM */ +036 if ((res = mp_div(a, &t1, &t2, NULL)) != MP_OKAY) \{ +037 goto LBL_T; +038 \} +039 res = mp_mul(b, &t2, c); +040 \} else \{ +041 /* store quotient in t2 such that t2 * a is the LCM */ +042 if ((res = mp_div(b, &t1, &t2, NULL)) != MP_OKAY) \{ +043 goto LBL_T; +044 \} +045 res = mp_mul(a, &t2, c); +046 \} +047 +048 /* fix the sign to positive */ +049 c->sign = MP_ZPOS; +050 +051 LBL_T: +052 mp_clear_multi (&t1, &t2, NULL); +053 return res; +054 \} +055 #endif +056 \end{alltt} \end{small} \section{Jacobi Symbol Computation} -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 +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 defined. The Legendre function computes whether or not an integer $a$ is a quadratic residue modulo an odd prime $p$. Numerically it is equivalent to equation \ref{eqn:legendre}. @@ -6169,9 +9972,9 @@ equivalent to equation \ref{eqn:legendre}. a^{(p-1)/2} \equiv \begin{array}{rl} -1 & \mbox{if }a\mbox{ is a quadratic non-residue.} \\ 0 & \mbox{if }a\mbox{ divides }p\mbox{.} \\ - 1 & \mbox{if }a\mbox{ is a quadratic residue}. + 1 & \mbox{if }a\mbox{ is a quadratic residue}. \end{array} \mbox{ (mod }p\mbox{)} -\label{eqn:legendre} +\label{eqn:legendre} \end{equation} \textbf{Proof.} \textit{Equation \ref{eqn:legendre} correctly identifies the residue status of an integer $a$ modulo a prime $p$.} @@ -6195,7 +9998,7 @@ then the quantity in the braces must be zero. By reduction, \begin{eqnarray} \left (x^2 \right )^{(p-1)/2} - a^{(p-1)/2} \equiv 0 \nonumber \\ \left (x^2 \right )^{(p-1)/2} \equiv a^{(p-1)/2} \nonumber \\ -x^2 \equiv a \mbox{ (mod }p\mbox{)} +x^2 \equiv a \mbox{ (mod }p\mbox{)} \end{eqnarray} 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 $a$ @@ -6215,47 +10018,47 @@ the Jacobi symbol $\left ( { a \over p } \right )$ is equal to the following equ 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 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 -following are true. +following are true. \begin{enumerate} -\item $\left ( { a \over p} \right )$ equals $-1$, $0$ or $1$. +\item $\left ( { a \over p} \right )$ equals $-1$, $0$ or $1$. \item $\left ( { ab \over p} \right ) = \left ( { a \over p} \right )\left ( { b \over p} \right )$. \item If $a \equiv b$ then $\left ( { a \over p} \right ) = \left ( { b \over p} \right )$. \item $\left ( { 2 \over p} \right )$ equals $1$ if $p \equiv 1$ or $7 \mbox{ (mod }8\mbox{)}$. Otherwise, it equals $-1$. -\item $\left ( { a \over p} \right ) \equiv \left ( { p \over a} \right ) \cdot (-1)^{(p-1)(a-1)/4}$. More specifically -$\left ( { a \over p} \right ) = \left ( { p \over a} \right )$ if $p \equiv a \equiv 1 \mbox{ (mod }4\mbox{)}$. +\item $\left ( { a \over p} \right ) \equiv \left ( { p \over a} \right ) \cdot (-1)^{(p-1)(a-1)/4}$. More specifically +$\left ( { a \over p} \right ) = \left ( { p \over a} \right )$ if $p \equiv a \equiv 1 \mbox{ (mod }4\mbox{)}$. \end{enumerate} Using these facts if $a = 2^k \cdot a'$ then \begin{eqnarray} \left ( { a \over p } \right ) = \left ( {{2^k} \over p } \right ) \left ( {a' \over p} \right ) \nonumber \\ - = \left ( {2 \over p } \right )^k \left ( {a' \over p} \right ) + = \left ( {2 \over p } \right )^k \left ( {a' \over p} \right ) \label{eqn:jacobi} \end{eqnarray} -By fact five, +By fact five, \begin{equation} -\left ( { a \over p } \right ) = \left ( { p \over a } \right ) \cdot (-1)^{(p-1)(a-1)/4} +\left ( { a \over p } \right ) = \left ( { p \over a } \right ) \cdot (-1)^{(p-1)(a-1)/4} \end{equation} -Subsequently by fact three since $p \equiv (p \mbox{ mod }a) \mbox{ (mod }a\mbox{)}$ then +Subsequently by fact three since $p \equiv (p \mbox{ mod }a) \mbox{ (mod }a\mbox{)}$ then \begin{equation} -\left ( { a \over p } \right ) = \left ( { {p \mbox{ mod } a} \over a } \right ) \cdot (-1)^{(p-1)(a-1)/4} +\left ( { a \over p } \right ) = \left ( { {p \mbox{ mod } a} \over a } \right ) \cdot (-1)^{(p-1)(a-1)/4} \end{equation} By putting both observations into equation \ref{eqn:jacobi} the following simplified equation is formed. \begin{equation} -\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} +\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} \end{equation} -The value of $\left ( {{p \mbox{ mod }a'} \over a'} \right )$ can be found by using the same equation recursively. The value of -$\left ( {2 \over p } \right )^k$ equals $1$ if $k$ is even otherwise it equals $\left ( {2 \over p } \right )$. Using this approach the -factors of $p$ do not have to be known. Furthermore, if $(a, p) = 1$ then the algorithm will terminate when the recursion requests the -Jacobi symbol computation of $\left ( {1 \over a'} \right )$ which is simply $1$. +The value of $\left ( {{p \mbox{ mod }a'} \over a'} \right )$ can be found by using the same equation recursively. The value of +$\left ( {2 \over p } \right )^k$ equals $1$ if $k$ is even otherwise it equals $\left ( {2 \over p } \right )$. Using this approach the +factors of $p$ do not have to be known. Furthermore, if $(a, p) = 1$ then the algorithm will terminate when the recursion requests the +Jacobi symbol computation of $\left ( {1 \over a'} \right )$ which is simply $1$. \newpage\begin{figure}[!here] \begin{small} @@ -6299,12 +10102,12 @@ Jacobi symbol computation of $\left ( {1 \over a'} \right )$ which is simply $1$ \end{figure} \textbf{Algorithm mp\_jacobi.} This algorithm computes the Jacobi symbol for an arbitrary positive integer $a$ with respect to an odd integer $p$ greater than three. The algorithm -is based on algorithm 2.149 of HAC \cite[pp. 73]{HAC}. +is based on algorithm 2.149 of HAC \cite[pp. 73]{HAC}. 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 -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 -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 -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 $a'$ +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 +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 +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 $a'$ are congruent to one modulo four, otherwise it evaluates to negative one. 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 @@ -6314,25 +10117,123 @@ $\left ( {p' \over a'} \right )$ which is multiplied against the current Jacobi \hspace{-5.1mm}{\bf File}: bn\_mp\_jacobi.c \vspace{-3mm} \begin{alltt} +016 +017 /* computes the jacobi c = (a | n) (or Legendre if n is prime) +018 * HAC pp. 73 Algorithm 2.149 +019 * HAC is wrong here, as the special case of (0 | 1) is not +020 * handled correctly. +021 */ +022 int mp_jacobi (mp_int * a, mp_int * n, int *c) +023 \{ +024 mp_int a1, p1; +025 int k, s, r, res; +026 mp_digit residue; +027 +028 /* if a < 0 return MP_VAL */ +029 if (mp_isneg(a) == MP_YES) \{ +030 return MP_VAL; +031 \} +032 +033 /* if n <= 0 return MP_VAL */ +034 if (mp_cmp_d(n, 0) != MP_GT) \{ +035 return MP_VAL; +036 \} +037 +038 /* step 1. handle case of a == 0 */ +039 if (mp_iszero (a) == MP_YES) \{ +040 /* special case of a == 0 and n == 1 */ +041 if (mp_cmp_d (n, 1) == MP_EQ) \{ +042 *c = 1; +043 \} else \{ +044 *c = 0; +045 \} +046 return MP_OKAY; +047 \} +048 +049 /* step 2. if a == 1, return 1 */ +050 if (mp_cmp_d (a, 1) == MP_EQ) \{ +051 *c = 1; +052 return MP_OKAY; +053 \} +054 +055 /* default */ +056 s = 0; +057 +058 /* step 3. write a = a1 * 2**k */ +059 if ((res = mp_init_copy (&a1, a)) != MP_OKAY) \{ +060 return res; +061 \} +062 +063 if ((res = mp_init (&p1)) != MP_OKAY) \{ +064 goto LBL_A1; +065 \} +066 +067 /* divide out larger power of two */ +068 k = mp_cnt_lsb(&a1); +069 if ((res = mp_div_2d(&a1, k, &a1, NULL)) != MP_OKAY) \{ +070 goto LBL_P1; +071 \} +072 +073 /* step 4. if e is even set s=1 */ +074 if ((k & 1) == 0) \{ +075 s = 1; +076 \} else \{ +077 /* else set s=1 if p = 1/7 (mod 8) or s=-1 if p = 3/5 (mod 8) */ +078 residue = n->dp[0] & 7; +079 +080 if ((residue == 1) || (residue == 7)) \{ +081 s = 1; +082 \} else if ((residue == 3) || (residue == 5)) \{ +083 s = -1; +084 \} +085 \} +086 +087 /* step 5. if p == 3 (mod 4) *and* a1 == 3 (mod 4) then s = -s */ +088 if ( ((n->dp[0] & 3) == 3) && ((a1.dp[0] & 3) == 3)) \{ +089 s = -s; +090 \} +091 +092 /* if a1 == 1 we're done */ +093 if (mp_cmp_d (&a1, 1) == MP_EQ) \{ +094 *c = s; +095 \} else \{ +096 /* n1 = n mod a1 */ +097 if ((res = mp_mod (n, &a1, &p1)) != MP_OKAY) \{ +098 goto LBL_P1; +099 \} +100 if ((res = mp_jacobi (&p1, &a1, &r)) != MP_OKAY) \{ +101 goto LBL_P1; +102 \} +103 *c = s * r; +104 \} +105 +106 /* done */ +107 res = MP_OKAY; +108 LBL_P1:mp_clear (&p1); +109 LBL_A1:mp_clear (&a1); +110 return res; +111 \} +112 #endif +113 \end{alltt} \end{small} -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 -variable name character. +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 +variable name character. 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 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 -the values it may obtain are merely $-1$, $0$ and $1$. +the values it may obtain are merely $-1$, $0$ and $1$. 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 -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 +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 processor requirements and neither is faster than the other. -Line 58 through 71 determines the value of $\left ( { 2 \over p } \right )^k$. If the least significant bit of $k$ is zero than +Line 59 through 71 determines the value of $\left ( { 2 \over p } \right )^k$. If the least significant bit of $k$ is zero than $k$ 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 -$(-1)^{(p-1)(a'-1)/4}$ is compute and multiplied against $s$ on lines 71 through 74. +$(-1)^{(p-1)(a'-1)/4}$ is compute and multiplied against $s$ on lines 73 through 76. -Finally, if $a1$ does not equal one the algorithm must recurse and compute $\left ( {p' \over a'} \right )$. +Finally, if $a1$ does not equal one the algorithm must recurse and compute $\left ( {p' \over a'} \right )$. \textit{-- Comment about default $s$ and such...} @@ -6340,31 +10241,31 @@ Finally, if $a1$ does not equal one the algorithm must recurse and compute $\lef \label{sec:modinv} The modular inverse of a number actually refers to the modular multiplicative inverse. Essentially for any integer $a$ such that $(a, p) = 1$ there 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 -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 +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 fields of integers. However, the former will be the matter of discussion. -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 +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 order of the multiplicative subgroup modulo $p$ then $b$ must be the multiplicative inverse of $a$. The proof of which is trivial. \begin{equation} ab \equiv a \left (a^{\Phi(p) - 1} \right ) \equiv a^{\Phi(p)} \equiv a^0 \equiv 1 \mbox{ (mod }p\mbox{)} \end{equation} -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 -requires all of the prime factors. This approach also is very slow as the size of $p$ grows. +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 +requires all of the prime factors. This approach also is very slow as the size of $p$ grows. -A simpler approach is based on the observation that solving for the multiplicative inverse is equivalent to solving the linear +A simpler approach is based on the observation that solving for the multiplicative inverse is equivalent to solving the linear Diophantine\footnote{See LeVeque \cite[pp. 40-43]{LeVeque} for more information.} equation. \begin{equation} ab + pq = 1 \end{equation} -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 -$a$ modulo $p$. The extended Euclidean algorithm (Knuth \cite[pp. 342]{TAOCPV2}) can be used to solve such equations provided $(a, p) = 1$. +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 +$a$ modulo $p$. The extended Euclidean algorithm (Knuth \cite[pp. 342]{TAOCPV2}) can be used to solve such equations provided $(a, p) = 1$. However, instead of using that algorithm directly a variant known as the binary Extended Euclidean algorithm will be used in its place. The -binary approach is very similar to the binary greatest common divisor algorithm except it will produce a full solution to the Diophantine -equation. +binary approach is very similar to the binary greatest common divisor algorithm except it will produce a full solution to the Diophantine +equation. \subsection{General Case} \newpage\begin{figure}[!here] @@ -6416,12 +10317,12 @@ equation. \end{small} \end{figure} \textbf{Algorithm mp\_invmod.} -This algorithm computes the modular multiplicative inverse of an integer $a$ modulo an integer $b$. This algorithm is a variation of the +This algorithm computes the modular multiplicative inverse of an integer $a$ modulo an integer $b$. This algorithm is a variation of the extended binary Euclidean algorithm from HAC \cite[pp. 608]{HAC}. It has been modified to only compute the modular inverse and not a complete -Diophantine solution. +Diophantine solution. 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 -inverse for $a$ and the error is reported. +inverse for $a$ and the error is reported. 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 the other variables to the Diophantine equation are solved. The algorithm terminates when $u = 0$ in which case the solution is @@ -6431,36 +10332,60 @@ Ca + Db = v \end{equation} 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, $C$ -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 -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 $0 < a < p$ +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 +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 $0 < a < p$ then only a couple of additions or subtractions will be required to adjust the inverse. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_invmod.c \vspace{-3mm} \begin{alltt} +016 +017 /* hac 14.61, pp608 */ +018 int mp_invmod (mp_int * a, mp_int * b, mp_int * c) +019 \{ +020 /* b cannot be negative */ +021 if ((b->sign == MP_NEG) || (mp_iszero(b) == MP_YES)) \{ +022 return MP_VAL; +023 \} +024 +025 #ifdef BN_FAST_MP_INVMOD_C +026 /* if the modulus is odd we can use a faster routine instead */ +027 if (mp_isodd (b) == MP_YES) \{ +028 return fast_mp_invmod (a, b, c); +029 \} +030 #endif +031 +032 #ifdef BN_MP_INVMOD_SLOW_C +033 return mp_invmod_slow(a, b, c); +034 #else +035 return MP_VAL; +036 #endif +037 \} +038 #endif +039 \end{alltt} \end{small} \subsubsection{Odd Moduli} 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 -the Diophantine $Cb + Da = 1$ only $B$ and $D$ are required to find the inverse of $a$. +the Diophantine $Cb + Da = 1$ only $B$ and $D$ are required to find the inverse of $a$. -The algorithm fast\_mp\_invmod is a direct adaptation of algorithm mp\_invmod with all all steps involving either $A$ or $C$ removed. This +The algorithm fast\_mp\_invmod is a direct adaptation of algorithm mp\_invmod with all all steps involving either $A$ or $C$ removed. This optimization will halve the time required to compute the modular inverse. \section{Primality Tests} -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 -since the integers $2 \ldots 6$ do not evenly divide $a$. By contrast, $a = 6$ is not prime since $a = 6 = 2 \cdot 3$. +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 +since the integers $2 \ldots 6$ do not evenly divide $a$. By contrast, $a = 6$ is not prime since $a = 6 = 2 \cdot 3$. Prime numbers arise in cryptography considerably as they allow finite fields to be formed. The ability to determine whether an integer is prime or not quickly has been a viable subject in cryptography and number theory for considerable time. The algorithms that will be presented are all probablistic algorithms in that when they report an integer is composite it must be composite. However, when the algorithms report an integer is -prime the algorithm may be incorrect. +prime the algorithm may be incorrect. -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 +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 well be zero. For the purposes of these discussions let $n$ represent the candidate integer of which the primality is in question. \subsection{Trial Division} @@ -6474,13 +10399,13 @@ of the primes less than $\sqrt{n} + 1$ the algorithm cannot prove if a candidate The benefit of this test is that trial division by small values is fairly efficient. Specially compared to the other algorithms that will be discussed shortly. The probability that this approach correctly identifies a composite candidate when tested with all primes upto $q$ is given by -$1 - {1.12 \over ln(q)}$. The graph (\ref{pic:primality}, will be added later) demonstrates the probability of success for the range -$3 \le q \le 100$. +$1 - {1.12 \over ln(q)}$. The graph (\ref{pic:primality}, will be added later) demonstrates the probability of success for the range +$3 \le q \le 100$. -At approximately $q = 30$ the gain of performing further tests diminishes fairly quickly. At $q = 90$ further testing is generally not going to -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 -approximately $80\%$ of all candidate integers. The constant \textbf{PRIME\_SIZE} is equal to the number of primes in the test base. The -array \_\_prime\_tab is an array of the first \textbf{PRIME\_SIZE} prime numbers. +At approximately $q = 30$ the gain of performing further tests diminishes fairly quickly. At $q = 90$ further testing is generally not going to +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 +approximately $80\%$ of all candidate integers. The constant \textbf{PRIME\_SIZE} is equal to the number of primes in the test base. The +array \_\_prime\_tab is an array of the first \textbf{PRIME\_SIZE} prime numbers. \begin{figure}[!here] \begin{small} @@ -6504,37 +10429,110 @@ array \_\_prime\_tab is an array of the first \textbf{PRIME\_SIZE} prime numbers \caption{Algorithm mp\_prime\_is\_divisible} \end{figure} \textbf{Algorithm mp\_prime\_is\_divisible.} -This algorithm attempts to determine if a candidate integer $n$ is composite by performing trial divisions. +This algorithm attempts to determine if a candidate integer $n$ is composite by performing trial divisions. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_prime\_is\_divisible.c \vspace{-3mm} \begin{alltt} +016 +017 /* determines if an integers is divisible by one +018 * of the first PRIME_SIZE primes or not +019 * +020 * sets result to 0 if not, 1 if yes +021 */ +022 int mp_prime_is_divisible (mp_int * a, int *result) +023 \{ +024 int err, ix; +025 mp_digit res; +026 +027 /* default to not */ +028 *result = MP_NO; +029 +030 for (ix = 0; ix < PRIME_SIZE; ix++) \{ +031 /* what is a mod LBL_prime_tab[ix] */ +032 if ((err = mp_mod_d (a, ltm_prime_tab[ix], &res)) != MP_OKAY) \{ +033 return err; +034 \} +035 +036 /* is the residue zero? */ +037 if (res == 0) \{ +038 *result = MP_YES; +039 return MP_OKAY; +040 \} +041 \} +042 +043 return MP_OKAY; +044 \} +045 #endif +046 \end{alltt} \end{small} -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 +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 mp\_digit. The table \_\_prime\_tab is defined in the following file. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_prime\_tab.c \vspace{-3mm} \begin{alltt} +016 const mp_digit ltm_prime_tab[] = \{ +017 0x0002, 0x0003, 0x0005, 0x0007, 0x000B, 0x000D, 0x0011, 0x0013, +018 0x0017, 0x001D, 0x001F, 0x0025, 0x0029, 0x002B, 0x002F, 0x0035, +019 0x003B, 0x003D, 0x0043, 0x0047, 0x0049, 0x004F, 0x0053, 0x0059, +020 0x0061, 0x0065, 0x0067, 0x006B, 0x006D, 0x0071, 0x007F, +021 #ifndef MP_8BIT +022 0x0083, +023 0x0089, 0x008B, 0x0095, 0x0097, 0x009D, 0x00A3, 0x00A7, 0x00AD, +024 0x00B3, 0x00B5, 0x00BF, 0x00C1, 0x00C5, 0x00C7, 0x00D3, 0x00DF, +025 0x00E3, 0x00E5, 0x00E9, 0x00EF, 0x00F1, 0x00FB, 0x0101, 0x0107, +026 0x010D, 0x010F, 0x0115, 0x0119, 0x011B, 0x0125, 0x0133, 0x0137, +027 +028 0x0139, 0x013D, 0x014B, 0x0151, 0x015B, 0x015D, 0x0161, 0x0167, +029 0x016F, 0x0175, 0x017B, 0x017F, 0x0185, 0x018D, 0x0191, 0x0199, +030 0x01A3, 0x01A5, 0x01AF, 0x01B1, 0x01B7, 0x01BB, 0x01C1, 0x01C9, +031 0x01CD, 0x01CF, 0x01D3, 0x01DF, 0x01E7, 0x01EB, 0x01F3, 0x01F7, +032 0x01FD, 0x0209, 0x020B, 0x021D, 0x0223, 0x022D, 0x0233, 0x0239, +033 0x023B, 0x0241, 0x024B, 0x0251, 0x0257, 0x0259, 0x025F, 0x0265, +034 0x0269, 0x026B, 0x0277, 0x0281, 0x0283, 0x0287, 0x028D, 0x0293, +035 0x0295, 0x02A1, 0x02A5, 0x02AB, 0x02B3, 0x02BD, 0x02C5, 0x02CF, +036 +037 0x02D7, 0x02DD, 0x02E3, 0x02E7, 0x02EF, 0x02F5, 0x02F9, 0x0301, +038 0x0305, 0x0313, 0x031D, 0x0329, 0x032B, 0x0335, 0x0337, 0x033B, +039 0x033D, 0x0347, 0x0355, 0x0359, 0x035B, 0x035F, 0x036D, 0x0371, +040 0x0373, 0x0377, 0x038B, 0x038F, 0x0397, 0x03A1, 0x03A9, 0x03AD, +041 0x03B3, 0x03B9, 0x03C7, 0x03CB, 0x03D1, 0x03D7, 0x03DF, 0x03E5, +042 0x03F1, 0x03F5, 0x03FB, 0x03FD, 0x0407, 0x0409, 0x040F, 0x0419, +043 0x041B, 0x0425, 0x0427, 0x042D, 0x043F, 0x0443, 0x0445, 0x0449, +044 0x044F, 0x0455, 0x045D, 0x0463, 0x0469, 0x047F, 0x0481, 0x048B, +045 +046 0x0493, 0x049D, 0x04A3, 0x04A9, 0x04B1, 0x04BD, 0x04C1, 0x04C7, +047 0x04CD, 0x04CF, 0x04D5, 0x04E1, 0x04EB, 0x04FD, 0x04FF, 0x0503, +048 0x0509, 0x050B, 0x0511, 0x0515, 0x0517, 0x051B, 0x0527, 0x0529, +049 0x052F, 0x0551, 0x0557, 0x055D, 0x0565, 0x0577, 0x0581, 0x058F, +050 0x0593, 0x0595, 0x0599, 0x059F, 0x05A7, 0x05AB, 0x05AD, 0x05B3, +051 0x05BF, 0x05C9, 0x05CB, 0x05CF, 0x05D1, 0x05D5, 0x05DB, 0x05E7, +052 0x05F3, 0x05FB, 0x0607, 0x060D, 0x0611, 0x0617, 0x061F, 0x0623, +053 0x062B, 0x062F, 0x063D, 0x0641, 0x0647, 0x0649, 0x064D, 0x0653 +054 #endif +055 \}; +056 #endif +057 \end{alltt} \end{small} 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 -upto $1619$ are used. Note that the value of \textbf{PRIME\_SIZE} is a constant dependent on the size of a mp\_digit. +upto $1619$ are used. Note that the value of \textbf{PRIME\_SIZE} is a constant dependent on the size of a mp\_digit. \subsection{The Fermat Test} -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 +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 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 -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 -$a^1 = a$. +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 +$a^1 = a$. -If $n$ is composite then any given base $a$ does not have to have a period which divides $n - 1$. In which case +If $n$ is composite then any given base $a$ does not have to have a period which divides $n - 1$. In which case 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 -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 +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 integers known as Carmichael numbers will be a pseudo-prime to all valid bases. Fortunately such numbers are extremely rare as $n$ grows in size. @@ -6560,18 +10558,61 @@ in size. \end{figure} \textbf{Algorithm mp\_prime\_fermat.} 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 -determine the result. +determine the result. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_prime\_fermat.c \vspace{-3mm} \begin{alltt} +016 +017 /* performs one Fermat test. +018 * +019 * If "a" were prime then b**a == b (mod a) since the order of +020 * the multiplicative sub-group would be phi(a) = a-1. That means +021 * it would be the same as b**(a mod (a-1)) == b**1 == b (mod a). +022 * +023 * Sets result to 1 if the congruence holds, or zero otherwise. +024 */ +025 int mp_prime_fermat (mp_int * a, mp_int * b, int *result) +026 \{ +027 mp_int t; +028 int err; +029 +030 /* default to composite */ +031 *result = MP_NO; +032 +033 /* ensure b > 1 */ +034 if (mp_cmp_d(b, 1) != MP_GT) \{ +035 return MP_VAL; +036 \} +037 +038 /* init t */ +039 if ((err = mp_init (&t)) != MP_OKAY) \{ +040 return err; +041 \} +042 +043 /* compute t = b**a mod a */ +044 if ((err = mp_exptmod (b, a, a, &t)) != MP_OKAY) \{ +045 goto LBL_T; +046 \} +047 +048 /* is it equal to b? */ +049 if (mp_cmp (&t, b) == MP_EQ) \{ +050 *result = MP_YES; +051 \} +052 +053 err = MP_OKAY; +054 LBL_T:mp_clear (&t); +055 return err; +056 \} +057 #endif +058 \end{alltt} \end{small} \subsection{The Miller-Rabin Test} -The Miller-Rabin (citation) test is another primality test which has tighter error bounds than the Fermat test specifically with sequentially chosen -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 +The Miller-Rabin (citation) test is another primality test which has tighter error bounds than the Fermat test specifically with sequentially chosen +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 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 some value not congruent to $\pm 1$ when squared equals one which cannot occur if $n$ is prime. @@ -6607,17 +10648,101 @@ some value not congruent to $\pm 1$ when squared equals one which cannot occur i \end{figure} \textbf{Algorithm mp\_prime\_miller\_rabin.} 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 -if $b$ is composite or $c = 0$ if $b$ is provably composite. The values of $s$ and $r$ are computed such that $a' = a - 1 = 2^sr$. +if $b$ is composite or $c = 0$ if $b$ is provably composite. The values of $s$ and $r$ are computed such that $a' = a - 1 = 2^sr$. 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 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 $a$ -is provably composite. If the algorithm performs $s - 1$ squarings and $y \nequiv -1$ then $a$ is provably composite. If $a$ is not provably +is provably composite. If the algorithm performs $s - 1$ squarings and $y \nequiv -1$ then $a$ is provably composite. If $a$ is not provably composite then it is \textit{probably} prime. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_mp\_prime\_miller\_rabin.c \vspace{-3mm} \begin{alltt} +016 +017 /* Miller-Rabin test of "a" to the base of "b" as described in +018 * HAC pp. 139 Algorithm 4.24 +019 * +020 * Sets result to 0 if definitely composite or 1 if probably prime. +021 * Randomly the chance of error is no more than 1/4 and often +022 * very much lower. +023 */ +024 int mp_prime_miller_rabin (mp_int * a, mp_int * b, int *result) +025 \{ +026 mp_int n1, y, r; +027 int s, j, err; +028 +029 /* default */ +030 *result = MP_NO; +031 +032 /* ensure b > 1 */ +033 if (mp_cmp_d(b, 1) != MP_GT) \{ +034 return MP_VAL; +035 \} +036 +037 /* get n1 = a - 1 */ +038 if ((err = mp_init_copy (&n1, a)) != MP_OKAY) \{ +039 return err; +040 \} +041 if ((err = mp_sub_d (&n1, 1, &n1)) != MP_OKAY) \{ +042 goto LBL_N1; +043 \} +044 +045 /* set 2**s * r = n1 */ +046 if ((err = mp_init_copy (&r, &n1)) != MP_OKAY) \{ +047 goto LBL_N1; +048 \} +049 +050 /* count the number of least significant bits +051 * which are zero +052 */ +053 s = mp_cnt_lsb(&r); +054 +055 /* now divide n - 1 by 2**s */ +056 if ((err = mp_div_2d (&r, s, &r, NULL)) != MP_OKAY) \{ +057 goto LBL_R; +058 \} +059 +060 /* compute y = b**r mod a */ +061 if ((err = mp_init (&y)) != MP_OKAY) \{ +062 goto LBL_R; +063 \} +064 if ((err = mp_exptmod (b, &r, a, &y)) != MP_OKAY) \{ +065 goto LBL_Y; +066 \} +067 +068 /* if y != 1 and y != n1 do */ +069 if ((mp_cmp_d (&y, 1) != MP_EQ) && (mp_cmp (&y, &n1) != MP_EQ)) \{ +070 j = 1; +071 /* while j <= s-1 and y != n1 */ +072 while ((j <= (s - 1)) && (mp_cmp (&y, &n1) != MP_EQ)) \{ +073 if ((err = mp_sqrmod (&y, a, &y)) != MP_OKAY) \{ +074 goto LBL_Y; +075 \} +076 +077 /* if y == 1 then composite */ +078 if (mp_cmp_d (&y, 1) == MP_EQ) \{ +079 goto LBL_Y; +080 \} +081 +082 ++j; +083 \} +084 +085 /* if y != n1 then composite */ +086 if (mp_cmp (&y, &n1) != MP_EQ) \{ +087 goto LBL_Y; +088 \} +089 \} +090 +091 /* probably prime now */ +092 *result = MP_YES; +093 LBL_Y:mp_clear (&y); +094 LBL_R:mp_clear (&r); +095 LBL_N1:mp_clear (&n1); +096 return err; +097 \} +098 #endif +099 \end{alltt} \end{small} |