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diff --git a/libtommath/tommath.tex b/libtommath/tommath.tex index 9c4dc82..b016010 100644 --- a/libtommath/tommath.tex +++ b/libtommath/tommath.tex @@ -49,7 +49,7 @@ \begin{document} \frontmatter \pagestyle{empty} -\title{Implementing Multiple Precision Arithmetic \\ ~ \\ Draft Edition } +\title{Multi--Precision Math} \author{\mbox{ %\begin{small} \begin{tabular}{c} @@ -66,7 +66,7 @@ QUALCOMM Australia \\ } } \maketitle -This text has been placed in the public domain. This text corresponds to the v0.30 release of the +This text has been placed in the public domain. This text corresponds to the v0.35 release of the LibTomMath project. \begin{alltt} @@ -85,66 +85,32 @@ This text is formatted to the international B5 paper size of 176mm wide by 250mm \tableofcontents \listoffigures -\chapter*{Prefaces to the Draft Edition} -I started this text in April 2003 to complement my LibTomMath library. That is, explain how to implement the functions -contained in LibTomMath. The goal is to have a textbook that any Computer Science student can use when implementing their -own multiple precision arithmetic. The plan I wanted to follow was flesh out all the -ideas and concepts I had floating around in my head and then work on it afterwards refining a little bit at a time. Chance -would have it that I ended up with my summer off from Algonquin College and I was given four months solid to work on the -text. - -Choosing to not waste any time I dove right into the project even before my spring semester was finished. I wrote a bit -off and on at first. The moment my exams were finished I jumped into long 12 to 16 hour days. The result after only -a couple of months was a ten chapter, three hundred page draft that I quickly had distributed to anyone who wanted -to read it. I had Jean-Luc Cooke print copies for me and I brought them to Crypto'03 in Santa Barbara. So far I have -managed to grab a certain level of attention having people from around the world ask me for copies of the text was certain -rewarding. - -Now we are past December 2003. By this time I had pictured that I would have at least finished my second draft of the text. -Currently I am far off from this goal. I've done partial re-writes of chapters one, two and three but they are not even -finished yet. I haven't given up on the project, only had some setbacks. First O'Reilly declined to publish the text then -Addison-Wesley and Greg is tried another which I don't know the name of. However, at this point I want to focus my energy -onto finishing the book not securing a contract. - -So why am I writing this text? It seems like a lot of work right? Most certainly it is a lot of work writing a textbook. -Even the simplest introductory material has to be lined with references and figures. A lot of the text has to be re-written -from point form to prose form to ensure an easier read. Why am I doing all this work for free then? Simple. My philosophy -is quite simply ``Open Source. Open Academia. Open Minds'' which means that to achieve a goal of open minds, that is, -people willing to accept new ideas and explore the unknown you have to make available material they can access freely -without hinderance. - -I've been writing free software since I was about sixteen but only recently have I hit upon software that people have come -to depend upon. I started LibTomCrypt in December 2001 and now several major companies use it as integral portions of their -software. Several educational institutions use it as a matter of course and many freelance developers use it as -part of their projects. To further my contributions I started the LibTomMath project in December 2002 aimed at providing -multiple precision arithmetic routines that students could learn from. That is write routines that are not only easy -to understand and follow but provide quite impressive performance considering they are all in standard portable ISO C. - -The second leg of my philosophy is ``Open Academia'' which is where this textbook comes in. In the end, when all is -said and done the text will be useable by educational institutions as a reference on multiple precision arithmetic. - -At this time I feel I should share a little information about myself. The most common question I was asked at -Crypto'03, perhaps just out of professional courtesy, was which school I either taught at or attended. The unfortunate -truth is that I neither teach at or attend a school of academic reputation. I'm currently at Algonquin College which -is what I'd like to call ``somewhat academic but mostly vocational'' college. In otherwords, job training. - -I'm a 21 year old computer science student mostly self-taught in the areas I am aware of (which includes a half-dozen -computer science fields, a few fields of mathematics and some English). I look forward to teaching someday but I am -still far off from that goal. - -Now it would be improper for me to not introduce the rest of the texts co-authors. While they are only contributing -corrections and editorial feedback their support has been tremendously helpful in presenting the concepts laid out -in the text so far. Greg has always been there for me. He has tracked my LibTom projects since their inception and even -sent cheques to help pay tuition from time to time. His background has provided a wonderful source to bounce ideas off -of and improve the quality of my writing. Mads is another fellow who has just ``been there''. I don't even recall what -his interest in the LibTom projects is but I'm definitely glad he has been around. His ability to catch logical errors -in my written English have saved me on several occasions to say the least. - -What to expect next? Well this is still a rough draft. I've only had the chance to update a few chapters. However, I've -been getting the feeling that people are starting to use my text and I owe them some updated material. My current tenative -plan is to edit one chapter every two weeks starting January 4th. It seems insane but my lower course load at college -should provide ample time. By Crypto'04 I plan to have a 2nd draft of the text polished and ready to hand out to as many -people who will take it. +\chapter*{Prefaces} +When I tell people about my LibTom projects and that I release them as public domain they are often puzzled. +They ask why I did it and especially why I continue to work on them for free. The best I can explain it is ``Because I can.'' +Which seems odd and perhaps too terse for adult conversation. I often qualify it with ``I am able, I am willing.'' which +perhaps explains it better. I am the first to admit there is not anything that special with what I have done. Perhaps +others can see that too and then we would have a society to be proud of. My LibTom projects are what I am doing to give +back to society in the form of tools and knowledge that can help others in their endeavours. + +I started writing this book because it was the most logical task to further my goal of open academia. The LibTomMath source +code itself was written to be easy to follow and learn from. There are times, however, where pure C source code does not +explain the algorithms properly. Hence this book. The book literally starts with the foundation of the library and works +itself outwards to the more complicated algorithms. The use of both pseudo--code and verbatim source code provides a duality +of ``theory'' and ``practice'' that the computer science students of the world shall appreciate. I never deviate too far +from relatively straightforward algebra and I hope that this book can be a valuable learning asset. + +This book and indeed much of the LibTom projects would not exist in their current form if it was not for a plethora +of kind people donating their time, resources and kind words to help support my work. Writing a text of significant +length (along with the source code) is a tiresome and lengthy process. Currently the LibTom project is four years old, +comprises of literally thousands of users and over 100,000 lines of source code, TeX and other material. People like Mads and Greg +were there at the beginning to encourage me to work well. It is amazing how timely validation from others can boost morale to +continue the project. Definitely my parents were there for me by providing room and board during the many months of work in 2003. + +To my many friends whom I have met through the years I thank you for the good times and the words of encouragement. I hope I +honour your kind gestures with this project. + +Open Source. Open Academia. Open Minds. \begin{flushright} Tom St Denis \end{flushright} @@ -1045,7 +1011,7 @@ assumed to contain undefined values they are initially set to zero. \end{alltt} \end{small} -A quick optimization is to first determine if a memory re-allocation is required at all. The if statement (line 23) checks +A quick optimization is to first determine if a memory re-allocation is required at all. The if statement (line 24) checks if the \textbf{alloc} member of the mp\_int is smaller than the requested digit count. If the count is not larger than \textbf{alloc} the function skips the re-allocation part thus saving time. @@ -1590,14 +1556,20 @@ This algorithm simply resets a mp\_int to the default state. \begin{alltt} 016 017 /* set to zero */ -018 void -019 mp_zero (mp_int * a) -020 \{ -021 a->sign = MP_ZPOS; -022 a->used = 0; -023 memset (a->dp, 0, sizeof (mp_digit) * a->alloc); -024 \} -025 #endif +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 \end{alltt} \end{small} @@ -1609,7 +1581,7 @@ After the function is completed, all of the digits are zeroed, the \textbf{used} With the mp\_int representation of an integer, calculating the absolute value is trivial. The mp\_abs algorithm will compute the absolute value of an mp\_int. -\newpage\begin{figure}[here] +\begin{figure}[here] \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_abs}. \\ @@ -1662,6 +1634,9 @@ logic to handle it. \end{alltt} \end{small} +This fairly trivial algorithm first eliminates non--required duplications (line 27) and then sets the +\textbf{sign} flag to \textbf{MP\_ZPOS}. + \subsection{Integer Negation} With the mp\_int representation of an integer, calculating the negation is also trivial. The mp\_neg algorithm will compute the negative of an mp\_int input. @@ -1702,23 +1677,33 @@ zero as negative. 018 int mp_neg (mp_int * a, mp_int * b) 019 \{ 020 int res; -021 if ((res = mp_copy (a, b)) != MP_OKAY) \{ -022 return res; -023 \} -024 if (mp_iszero(b) != MP_YES) \{ -025 b->sign = (a->sign == MP_ZPOS) ? MP_NEG : MP_ZPOS; -026 \} -027 return MP_OKAY; -028 \} -029 #endif +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 \end{alltt} \end{small} +Like mp\_abs() this function avoids non--required duplications (line 21) and then sets the sign. We +have to make sure that only non--zero values get a \textbf{sign} of \textbf{MP\_NEG}. If the mp\_int is zero +than the \textbf{sign} is hard--coded to \textbf{MP\_ZPOS}. + \section{Small Constants} \subsection{Setting Small Constants} Often a mp\_int must be set to a relatively small value such as $1$ or $2$. For these cases the mp\_set algorithm is useful. -\begin{figure}[here] +\newpage\begin{figure}[here] \begin{center} \begin{tabular}{l} \hline Algorithm \textbf{mp\_set}. \\ @@ -1757,11 +1742,14 @@ single digit is set (\textit{modulo $\beta$}) and the \textbf{used} count is adj \end{alltt} \end{small} -Line 20 calls mp\_zero() to clear the mp\_int and reset the sign. Line 21 copies the digit -into the least significant location. Note the usage of a new constant \textbf{MP\_MASK}. This constant is used to quickly -reduce an integer modulo $\beta$. Since $\beta$ is of the form $2^k$ for any suitable $k$ it suffices to perform a binary AND with -$MP\_MASK = 2^k - 1$ to perform the reduction. Finally line 22 will set the \textbf{used} member with respect to the -digit actually set. This function will always make the integer positive. +First we zero (line 20) the mp\_int to make sure that the other members are initialized for a +small positive constant. mp\_zero() ensures that the \textbf{sign} is positive and the \textbf{used} count +is zero. Next we set the digit and reduce it modulo $\beta$ (line 21). After this step we have to +check if the resulting digit is zero or not. If it is not then we set the \textbf{used} count to one, otherwise +to zero. + +We can quickly reduce modulo $\beta$ since it is of the form $2^k$ and a quick binary AND operation with +$2^k - 1$ will perform the same operation. One important limitation of this function is that it will only set one digit. The size of a digit is not fixed, meaning source that uses this function should take that into account. Only trivially small constants can be set using this function. @@ -1936,10 +1924,12 @@ the zero'th digit. If after all of the digits have been compared, no difference \end{alltt} \end{small} -The two if statements on lines 24 and 28 compare the number of digits in the two inputs. These two are performed before all of the digits -are compared since it is a very cheap test to perform and can potentially save considerable time. The implementation given is also not valid -without those two statements. $b.alloc$ may be smaller than $a.used$, meaning that undefined values will be read from $b$ past the end of the -array of digits. +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 @@ -2000,9 +1990,9 @@ $\vert a \vert < \vert b \vert$. Step number four will compare the two when the \end{alltt} \end{small} -The two if statements on 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. At line 31, the inputs are compared based on magnitudes. If the signs were both negative then -the unsigned comparison is performed in the opposite direction (\textit{line 33}). Otherwise, the signs are assumed to +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} @@ -2218,19 +2208,21 @@ The final carry is stored in $c_{max}$ and digits above $max$ upto $oldused$ are \end{alltt} \end{small} -Lines 27 to 35 perform the initial sorting of the inputs and determine the $min$ and $max$ variables. Note that $x$ is a pointer to a -mp\_int assigned to the largest input, in effect it is a local alias. Lines 37 to 42 ensure that the destination is grown to -accomodate the result of the addition. +We first sort (lines 27 to 35) the inputs based on magnitude and determine the $min$ and $max$ variables. +Note that $x$ is a pointer to an mp\_int assigned to the largest input, in effect it is a local alias. Next we +grow the destination (37 to 42) ensure that it can accomodate the result of the addition. Similar to the implementation of mp\_copy this function uses the braced code and local aliases coding style. The three aliases that are on lines 55, 58 and 61 represent the two inputs and destination variables respectively. These aliases are used to ensure the compiler does not have to dereference $a$, $b$ or $c$ (respectively) to access the digits of the respective mp\_int. -The initial carry $u$ is cleared on line 64, note that $u$ is of type mp\_digit which ensures type compatibility within the -implementation. The initial addition loop begins on line 65 and ends on line 74. Similarly the conditional addition loop -begins on line 80 and ends on line 90. The addition is finished with the final carry being stored in $tmpc$ on line 93. -Note the ``++'' operator on the same line. After line 93 $tmpc$ will point to the $c.used$'th digit of the mp\_int $c$. This is useful -for the next loop on lines 96 to 99 which set any old upper digits to zero. +The initial carry $u$ will be cleared (line 64), note that $u$ is of type mp\_digit which ensures type +compatibility within the implementation. The initial addition (line 65 to 74) adds digits from +both inputs until the smallest input runs out of digits. Similarly the conditional addition loop +(line 80 to 90) adds the remaining digits from the larger of the two inputs. The addition is finished +with the final carry being stored in $tmpc$ (line 93). Note the ``++'' operator within the same expression. +After line 93, $tmpc$ will point to the $c.used$'th digit of the mp\_int $c$. This is useful +for the next loop (line 96 to 99) which set any old upper digits to zero. \subsection{Low Level Subtraction} The low level unsigned subtraction algorithm is very similar to the low level unsigned addition algorithm. The principle difference is that the @@ -2245,7 +2237,7 @@ this algorithm we will assume that the variable $\gamma$ represents the number o mp\_digit (\textit{this implies $2^{\gamma} > \beta$}). For example, the default for LibTomMath is to use a ``unsigned long'' for the mp\_digit ``type'' while $\beta = 2^{28}$. In ISO C an ``unsigned long'' -data type must be able to represent $0 \le x < 2^{32}$ meaning that in this case $\gamma = 32$. +data type must be able to represent $0 \le x < 2^{32}$ meaning that in this case $\gamma \ge 32$. \newpage\begin{figure}[!here] \begin{center} @@ -2387,20 +2379,23 @@ If $b$ has a smaller magnitude than $a$ then step 9 will force the carry and cop \end{alltt} \end{small} -Line 24 and 25 perform the initial hardcoded sorting of the inputs. In reality the $min$ and $max$ variables are only aliases and are only -used to make the source code easier to read. Again the pointer alias optimization is used within this algorithm. Lines 41, 42 and 43 initialize the aliases for -$a$, $b$ and $c$ respectively. +Like low level addition we ``sort'' the inputs. Except in this case the sorting is hardcoded +(lines 24 and 25). In reality the $min$ and $max$ variables are only aliases and are only +used to make the source code easier to read. Again the pointer alias optimization is used +within this algorithm. The aliases $tmpa$, $tmpb$ and $tmpc$ are initialized +(lines 41, 42 and 43) for $a$, $b$ and $c$ respectively. -The first subtraction loop occurs on lines 46 through 60. The theory behind the subtraction loop is exactly the same as that for -the addition loop. As remarked earlier there is an implementation reason for using the ``awkward'' method of extracting the carry -(\textit{see line 56}). The traditional method for extracting the carry would be to shift by $lg(\beta)$ positions and logically AND -the least significant bit. The AND operation is required because all of the bits above the $\lg(\beta)$'th bit will be set to one after a carry -occurs from subtraction. This carry extraction requires two relatively cheap operations to extract the carry. The other method is to simply -shift the most significant bit to the least significant bit thus extracting the carry with a single cheap operation. This optimization only works on -twos compliment machines which is a safe assumption to make. +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 (\textit{see lines 63 through 72}) 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 @@ -2985,10 +2980,11 @@ step 8 sets the lower $b$ digits to zero. \end{alltt} \end{small} -The if statement on line 23 ensures that the $b$ variable is greater than zero. The \textbf{used} count is incremented by $b$ before -the copy loop begins. This elminates the need for an additional variable in the for loop. The variable $top$ on line 41 is an alias -for the leading digit while $bottom$ on line 44 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 $x$} @@ -3095,9 +3091,9 @@ Once the window copy is complete the upper digits must be zeroed and the \textbf \end{alltt} \end{small} -The only noteworthy element of this routine is the lack of a return type. - --- Will update later to give it a return type...Tom +The only noteworthy element of this routine is the lack of a return type since it cannot fail. Like mp\_lshd() we +form a sliding window except we copy in the other direction. After the window (line 59) we then zero +the upper digits of the input to make sure the result is correct. \section{Powers of Two} @@ -3228,7 +3224,15 @@ complete. It is possible to optimize this algorithm down to a $O(n)$ algorithm \end{alltt} \end{small} -Notes to be revised when code is updated. -- Tom +The shifting is performed in--place which means the first step (line 24) is to copy the input to the +destination. We avoid calling mp\_copy() by making sure the mp\_ints are different. The destination then +has to be grown (line 31) to accomodate the result. + +If the shift count $b$ is larger than $lg(\beta)$ then a call to mp\_lshd() is used to handle all of the multiples +of $lg(\beta)$. Leaving only a remaining shift of $lg(\beta) - 1$ or fewer bits left. Inside the actual shift +loop (lines 45 to 76) we make use of pre--computed values $shift$ and $mask$. These are used to +extract the carry bit(s) to pass into the next iteration of the loop. The $r$ and $rr$ variables form a +chain between consecutive iterations to propagate the carry. \subsection{Division by Power of Two} @@ -3361,7 +3365,8 @@ ignored by passing \textbf{NULL} as the pointer to the mp\_int variable. The result of the remainder operation until the end. This allows $d$ and $a$ to represent the same mp\_int without modifying $a$ before the quotient is obtained. -The remainder of the source code is essentially the same as the source code for mp\_mul\_2d. (-- Fix this paragraph up later, Tom). +The remainder of the source code is essentially the same as the source code for mp\_mul\_2d. The only significant difference is +the direction of the shifts. \subsection{Remainder of Division by Power of Two} @@ -3446,7 +3451,13 @@ is copied to $b$, leading digits are removed and the remaining leading digit is \end{alltt} \end{small} --- Add comments later, Tom. +We first avoid cases of $b \le 0$ by simply mp\_zero()'ing the destination in such cases. Next if $2^b$ is larger +than the input we just mp\_copy() the input and return right away. After this point we know we must actually +perform some work to produce the remainder. + +Recalling that reducing modulo $2^k$ and a binary ``and'' with $2^k - 1$ are numerically equivalent we can quickly reduce +the number. First we zero any digits above the last digit in $2^b$ (line 41). Next we reduce the +leading digit of both (line 45) and then mp\_clamp(). \section*{Exercises} \begin{tabular}{cl} @@ -3611,101 +3622,113 @@ exceed the precision requested. 018 * HAC pp. 595, Algorithm 14.12 Modified so you can control how 019 * many digits of output are created. 020 */ -021 int -022 s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs) -023 \{ -024 mp_int t; -025 int res, pa, pb, ix, iy; -026 mp_digit u; -027 mp_word r; -028 mp_digit tmpx, *tmpt, *tmpy; -029 -030 /* can we use the fast multiplier? */ -031 if (((digs) < MP_WARRAY) && -032 MIN (a->used, b->used) < -033 (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) \{ -034 return fast_s_mp_mul_digs (a, b, c, digs); -035 \} -036 -037 if ((res = mp_init_size (&t, digs)) != MP_OKAY) \{ -038 return res; -039 \} -040 t.used = digs; -041 -042 /* compute the digits of the product directly */ -043 pa = a->used; -044 for (ix = 0; ix < pa; ix++) \{ -045 /* set the carry to zero */ -046 u = 0; -047 -048 /* limit ourselves to making digs digits of output */ -049 pb = MIN (b->used, digs - ix); -050 -051 /* setup some aliases */ -052 /* copy of the digit from a used within the nested loop */ -053 tmpx = a->dp[ix]; -054 -055 /* an alias for the destination shifted ix places */ -056 tmpt = t.dp + ix; -057 -058 /* an alias for the digits of b */ -059 tmpy = b->dp; -060 -061 /* compute the columns of the output and propagate the carry */ -062 for (iy = 0; iy < pb; iy++) \{ -063 /* compute the column as a mp_word */ -064 r = ((mp_word)*tmpt) + -065 ((mp_word)tmpx) * ((mp_word)*tmpy++) + -066 ((mp_word) u); -067 -068 /* the new column is the lower part of the result */ -069 *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK)); -070 -071 /* get the carry word from the result */ -072 u = (mp_digit) (r >> ((mp_word) DIGIT_BIT)); -073 \} -074 /* set carry if it is placed below digs */ -075 if (ix + iy < digs) \{ -076 *tmpt = u; -077 \} -078 \} -079 -080 mp_clamp (&t); -081 mp_exch (&t, c); -082 -083 mp_clear (&t); -084 return MP_OKAY; -085 \} -086 #endif +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 \end{alltt} \end{small} -Lines 31 to 35 determine if the Comba method can be used first. The conditions for using the Comba routine are that min$(a.used, b.used) < \delta$ and -the number of digits of output is less than \textbf{MP\_WARRAY}. This new constant is used to control -the stack usage in the Comba routines. By default it is set to $\delta$ but can be reduced when memory is at a premium. +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 36) to the exact size of the output to avoid further re--allocations. At this point we now +begin the $O(n^2)$ loop. -Of particular importance is the calculation of the $ix+iy$'th column on lines 64, 65 and 66. Note how all of the -variables are cast to the type \textbf{mp\_word}, which is also the type of variable $\hat r$. That is to ensure that double precision operations -are used instead of single precision. The multiplication on line 65 makes use of a specific GCC optimizer behaviour. On the outset it looks like -the compiler will have to use a double precision multiplication to produce the result required. Such an operation would be horribly slow on most -processors and drag this to a crawl. However, GCC is smart enough to realize that double wide output single precision multipliers can be used. For -example, the instruction ``MUL'' on the x86 processor can multiply two 32-bit values and produce a 64-bit result. +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 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 +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 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 five years before. +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 @@ -3799,38 +3822,32 @@ $256$ digits would allow for numbers in the range of $0 \le x < 2^{7168}$ which, \textbf{Input}. mp\_int $a$, mp\_int $b$ and an integer $digs$ \\ \textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert \mbox{ (mod }\beta^{digs}\mbox{)}$. \\ \hline \\ -Place an array of \textbf{MP\_WARRAY} double precision digits named $\hat W$ on the stack. \\ +Place an array of \textbf{MP\_WARRAY} single precision digits named $W$ on the stack. \\ 1. If $c.alloc < digs$ then grow $c$ to $digs$ digits. (\textit{mp\_grow}) \\ 2. If step 1 failed return(\textit{MP\_MEM}).\\ \\ -Zero the temporary array $\hat W$. \\ -3. for $n$ from $0$ to $digs - 1$ do \\ -\hspace{3mm}3.1 $\hat W_n \leftarrow 0$ \\ +3. $pa \leftarrow \mbox{MIN}(digs, a.used + b.used)$ \\ \\ -Compute the columns. \\ -4. for $ix$ from $0$ to $a.used - 1$ do \\ -\hspace{3mm}4.1 $pb \leftarrow \mbox{min}(b.used, digs - ix)$ \\ -\hspace{3mm}4.2 If $pb < 1$ then goto step 5. \\ -\hspace{3mm}4.3 for $iy$ from $0$ to $pb - 1$ do \\ -\hspace{6mm}4.3.1 $\hat W_{ix+iy} \leftarrow \hat W_{ix+iy} + a_{ix}b_{iy}$ \\ +4. $\_ \hat W \leftarrow 0$ \\ +5. for $ix$ from 0 to $pa - 1$ do \\ +\hspace{3mm}5.1 $ty \leftarrow \mbox{MIN}(b.used - 1, ix)$ \\ +\hspace{3mm}5.2 $tx \leftarrow ix - ty$ \\ +\hspace{3mm}5.3 $iy \leftarrow \mbox{MIN}(a.used - tx, ty + 1)$ \\ +\hspace{3mm}5.4 for $iz$ from 0 to $iy - 1$ do \\ +\hspace{6mm}5.4.1 $\_ \hat W \leftarrow \_ \hat W + a_{tx+iy}b_{ty-iy}$ \\ +\hspace{3mm}5.5 $W_{ix} \leftarrow \_ \hat W (\mbox{mod }\beta)$\\ +\hspace{3mm}5.6 $\_ \hat W \leftarrow \lfloor \_ \hat W / \beta \rfloor$ \\ +6. $W_{pa} \leftarrow \_ \hat W (\mbox{mod }\beta)$ \\ \\ -Propagate the carries upwards. \\ -5. $oldused \leftarrow c.used$ \\ -6. $c.used \leftarrow digs$ \\ -7. If $digs > 1$ then do \\ -\hspace{3mm}7.1. for $ix$ from $1$ to $digs - 1$ do \\ -\hspace{6mm}7.1.1 $\hat W_{ix} \leftarrow \hat W_{ix} + \lfloor \hat W_{ix-1} / \beta \rfloor$ \\ -\hspace{6mm}7.1.2 $c_{ix - 1} \leftarrow \hat W_{ix - 1} \mbox{ (mod }\beta\mbox{)}$ \\ -8. else do \\ -\hspace{3mm}8.1 $ix \leftarrow 0$ \\ -9. $c_{ix} \leftarrow \hat W_{ix} \mbox{ (mod }\beta\mbox{)}$ \\ +7. $oldused \leftarrow c.used$ \\ +8. $c.used \leftarrow digs$ \\ +9. for $ix$ from $0$ to $pa$ do \\ +\hspace{3mm}9.1 $c_{ix} \leftarrow W_{ix}$ \\ +10. for $ix$ from $pa + 1$ to $oldused - 1$ do \\ +\hspace{3mm}10.1 $c_{ix} \leftarrow 0$ \\ \\ -Zero excess digits. \\ -10. If $digs < oldused$ then do \\ -\hspace{3mm}10.1 for $n$ from $digs$ to $oldused - 1$ do \\ -\hspace{6mm}10.1.1 $c_n \leftarrow 0$ \\ -11. Clamp excessive digits of $c$. (\textit{mp\_clamp}) \\ -12. Return(\textit{MP\_OKAY}). \\ +11. Clamp $c$. \\ +12. Return MP\_OKAY. \\ \hline \end{tabular} \end{center} @@ -3840,15 +3857,24 @@ Zero excess digits. \\ \end{figure} \textbf{Algorithm fast\_s\_mp\_mul\_digs.} -This algorithm performs the unsigned multiplication of $a$ and $b$ using the Comba method limited to $digs$ digits of precision. The algorithm -essentially peforms the same calculation as algorithm s\_mp\_mul\_digs, just much faster. +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 +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 array $\hat W$ is meant to be on the stack when the algorithm is used. The size of the array does not change which is ideal. Note also that -unlike algorithm s\_mp\_mul\_digs no temporary mp\_int is required since the result is calculated directly in $\hat W$. +The $ty$ 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 $iy$. -The $O(n^2)$ loop on step four is where the Comba method's advantages begin to show through in comparison to the baseline algorithm. The lack of -a carry variable or propagation in this loop allows the loop to be performed with only single precision multiplication and additions. Now that each -iteration of the inner loop can be performed independent of the others the inner loop can be performed with a high level of parallelism. +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 +$tx \ge a.used$ 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 @@ -3877,97 +3903,95 @@ and addition operations in the nested loop in parallel. 030 * Based on Algorithm 14.12 on pp.595 of HAC. 031 * 032 */ -033 int -034 fast_s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs) -035 \{ -036 int olduse, res, pa, ix, iz; -037 mp_digit W[MP_WARRAY]; -038 register mp_word _W; -039 -040 /* grow the destination as required */ -041 if (c->alloc < digs) \{ -042 if ((res = mp_grow (c, digs)) != MP_OKAY) \{ -043 return res; -044 \} -045 \} -046 -047 /* number of output digits to produce */ -048 pa = MIN(digs, a->used + b->used); -049 -050 /* clear the carry */ -051 _W = 0; -052 for (ix = 0; ix < pa; ix++) \{ -053 int tx, ty; -054 int iy; -055 mp_digit *tmpx, *tmpy; -056 -057 /* get offsets into the two bignums */ -058 ty = MIN(b->used-1, ix); -059 tx = ix - ty; -060 -061 /* setup temp aliases */ -062 tmpx = a->dp + tx; -063 tmpy = b->dp + ty; -064 -065 /* this is the number of times the loop will iterrate, essentially its - -066 while (tx++ < a->used && ty-- >= 0) \{ ... \} -067 */ -068 iy = MIN(a->used-tx, ty+1); -069 -070 /* execute loop */ -071 for (iz = 0; iz < iy; ++iz) \{ -072 _W += ((mp_word)*tmpx++)*((mp_word)*tmpy--); -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 /* store final carry */ -083 W[ix] = _W; -084 -085 /* setup dest */ -086 olduse = c->used; -087 c->used = digs; -088 -089 \{ -090 register mp_digit *tmpc; -091 tmpc = c->dp; -092 for (ix = 0; ix < digs; ix++) \{ -093 /* now extract the previous digit [below the carry] */ -094 *tmpc++ = W[ix]; -095 \} -096 -097 /* clear unused digits [that existed in the old copy of c] */ -098 for (; ix < olduse; ix++) \{ -099 *tmpc++ = 0; -100 \} -101 \} -102 mp_clamp (c); -103 return MP_OKAY; -104 \} -105 #endif +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 register 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 /* store term */ +075 W[ix] = ((mp_digit)_W) & MP_MASK; +076 +077 /* make next carry */ +078 _W = _W >> ((mp_word)DIGIT_BIT); +079 \} +080 +081 /* store final carry */ +082 W[ix] = (mp_digit)(_W & MP_MASK); +083 +084 /* setup dest */ +085 olduse = c->used; +086 c->used = pa; +087 +088 \{ +089 register mp_digit *tmpc; +090 tmpc = c->dp; +091 for (ix = 0; ix < pa+1; ix++) \{ +092 /* now extract the previous digit [below the carry] */ +093 *tmpc++ = W[ix]; +094 \} +095 +096 /* clear unused digits [that existed in the old copy of c] */ +097 for (; ix < olduse; ix++) \{ +098 *tmpc++ = 0; +099 \} +100 \} +101 mp_clamp (c); +102 return MP_OKAY; +103 \} +104 #endif \end{alltt} \end{small} -The memset on line @47,memset@ clears the initial $\hat W$ array to zero in a single step. Like the slower baseline multiplication -implementation a series of aliases (\textit{lines 62, 63 and 76}) are used to simplify the inner $O(n^2)$ loop. -In this case a new alias $\_\hat W$ has been added which refers to the double precision columns offset by $ix$ in each pass. +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 on lines 92, 79 and 80 is where the algorithm will spend the majority of the time, which is why it has been -stripped to the bones of any extra baggage\footnote{Hence the pointer aliases.}. On x86 processors the multiplication and additions amount to at the -very least five instructions (\textit{two loads, two additions, one multiply}) while on the ARMv4 processors they amount to only three -(\textit{one load, one store, one multiply-add}). For both of the x86 and ARMv4 processors the GCC compiler performs a good job at unrolling the loop -and scheduling the instructions so there are very few dependency stalls. +The inner loop (lines 70 to 72) 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. -In theory the difference between the baseline and comba algorithms is a mere $O(qn)$ time difference. However, in the $O(n^2)$ nested loop of the -baseline method there are dependency stalls as the algorithm must wait for the multiplier to finish before propagating the carry to the next -digit. As a result fewer of the often multiple execution units\footnote{The AMD Athlon has three execution units and the Intel P4 has four.} can -be simultaneously used. +After the inner loop we store the current accumulator in $W$ and shift $\_ \hat W$ (lines 75, 78) to forward it as +a carry for the next pass. After the outer loop we use the final carry (line 82) 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 @@ -4444,277 +4468,290 @@ result $a \cdot b$ is produced. 016 017 /* multiplication using the Toom-Cook 3-way algorithm 018 * -019 * Much more complicated than Karatsuba but has a lower asymptotic running t - ime of -020 * O(N**1.464). This algorithm is only particularly useful on VERY large -021 * inputs (we're talking 1000s of digits here...). -022 */ -023 int mp_toom_mul(mp_int *a, mp_int *b, mp_int *c) -024 \{ -025 mp_int w0, w1, w2, w3, w4, tmp1, tmp2, a0, a1, a2, b0, b1, b2; -026 int res, B; -027 -028 /* init temps */ -029 if ((res = mp_init_multi(&w0, &w1, &w2, &w3, &w4, -030 &a0, &a1, &a2, &b0, &b1, -031 &b2, &tmp1, &tmp2, NULL)) != MP_OKAY) \{ -032 return res; -033 \} -034 -035 /* B */ -036 B = MIN(a->used, b->used) / 3; -037 -038 /* a = a2 * B**2 + a1 * B + a0 */ -039 if ((res = mp_mod_2d(a, DIGIT_BIT * B, &a0)) != MP_OKAY) \{ -040 goto ERR; -041 \} -042 -043 if ((res = mp_copy(a, &a1)) != MP_OKAY) \{ -044 goto ERR; -045 \} -046 mp_rshd(&a1, B); -047 mp_mod_2d(&a1, DIGIT_BIT * B, &a1); -048 -049 if ((res = mp_copy(a, &a2)) != MP_OKAY) \{ -050 goto ERR; -051 \} -052 mp_rshd(&a2, B*2); -053 -054 /* b = b2 * B**2 + b1 * B + b0 */ -055 if ((res = mp_mod_2d(b, DIGIT_BIT * B, &b0)) != MP_OKAY) \{ -056 goto ERR; -057 \} -058 -059 if ((res = mp_copy(b, &b1)) != MP_OKAY) \{ -060 goto ERR; -061 \} -062 mp_rshd(&b1, B); -063 mp_mod_2d(&b1, DIGIT_BIT * B, &b1); -064 -065 if ((res = mp_copy(b, &b2)) != MP_OKAY) \{ -066 goto ERR; -067 \} -068 mp_rshd(&b2, B*2); -069 -070 /* w0 = a0*b0 */ -071 if ((res = mp_mul(&a0, &b0, &w0)) != MP_OKAY) \{ -072 goto ERR; -073 \} -074 -075 /* w4 = a2 * b2 */ -076 if ((res = mp_mul(&a2, &b2, &w4)) != MP_OKAY) \{ -077 goto ERR; -078 \} -079 -080 /* w1 = (a2 + 2(a1 + 2a0))(b2 + 2(b1 + 2b0)) */ -081 if ((res = mp_mul_2(&a0, &tmp1)) != MP_OKAY) \{ -082 goto ERR; -083 \} -084 if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) \{ -085 goto ERR; -086 \} -087 if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) \{ -088 goto ERR; -089 \} -090 if ((res = mp_add(&tmp1, &a2, &tmp1)) != MP_OKAY) \{ -091 goto ERR; -092 \} -093 -094 if ((res = mp_mul_2(&b0, &tmp2)) != MP_OKAY) \{ -095 goto ERR; -096 \} -097 if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) \{ -098 goto ERR; -099 \} -100 if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) \{ -101 goto ERR; -102 \} -103 if ((res = mp_add(&tmp2, &b2, &tmp2)) != MP_OKAY) \{ -104 goto ERR; -105 \} -106 -107 if ((res = mp_mul(&tmp1, &tmp2, &w1)) != MP_OKAY) \{ -108 goto ERR; -109 \} -110 -111 /* w3 = (a0 + 2(a1 + 2a2))(b0 + 2(b1 + 2b2)) */ -112 if ((res = mp_mul_2(&a2, &tmp1)) != MP_OKAY) \{ -113 goto ERR; -114 \} -115 if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) \{ -116 goto ERR; -117 \} -118 if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) \{ -119 goto ERR; -120 \} -121 if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) \{ -122 goto ERR; -123 \} -124 -125 if ((res = mp_mul_2(&b2, &tmp2)) != MP_OKAY) \{ -126 goto ERR; -127 \} -128 if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) \{ -129 goto ERR; -130 \} -131 if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) \{ -132 goto ERR; -133 \} -134 if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) \{ -135 goto ERR; -136 \} -137 -138 if ((res = mp_mul(&tmp1, &tmp2, &w3)) != MP_OKAY) \{ -139 goto ERR; -140 \} -141 -142 -143 /* w2 = (a2 + a1 + a0)(b2 + b1 + b0) */ -144 if ((res = mp_add(&a2, &a1, &tmp1)) != MP_OKAY) \{ -145 goto ERR; -146 \} -147 if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) \{ -148 goto ERR; -149 \} -150 if ((res = mp_add(&b2, &b1, &tmp2)) != MP_OKAY) \{ -151 goto ERR; -152 \} -153 if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) \{ -154 goto ERR; -155 \} -156 if ((res = mp_mul(&tmp1, &tmp2, &w2)) != MP_OKAY) \{ -157 goto ERR; -158 \} -159 -160 /* now solve the matrix -161 -162 0 0 0 0 1 -163 1 2 4 8 16 -164 1 1 1 1 1 -165 16 8 4 2 1 -166 1 0 0 0 0 -167 -168 using 12 subtractions, 4 shifts, -169 2 small divisions and 1 small multiplication -170 */ -171 -172 /* r1 - r4 */ -173 if ((res = mp_sub(&w1, &w4, &w1)) != MP_OKAY) \{ -174 goto ERR; -175 \} -176 /* r3 - r0 */ -177 if ((res = mp_sub(&w3, &w0, &w3)) != MP_OKAY) \{ -178 goto ERR; -179 \} -180 /* r1/2 */ -181 if ((res = mp_div_2(&w1, &w1)) != MP_OKAY) \{ -182 goto ERR; -183 \} -184 /* r3/2 */ -185 if ((res = mp_div_2(&w3, &w3)) != MP_OKAY) \{ -186 goto ERR; -187 \} -188 /* r2 - r0 - r4 */ -189 if ((res = mp_sub(&w2, &w0, &w2)) != MP_OKAY) \{ -190 goto ERR; -191 \} -192 if ((res = mp_sub(&w2, &w4, &w2)) != MP_OKAY) \{ -193 goto ERR; -194 \} -195 /* r1 - r2 */ -196 if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) \{ -197 goto ERR; -198 \} -199 /* r3 - r2 */ -200 if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) \{ -201 goto ERR; -202 \} -203 /* r1 - 8r0 */ -204 if ((res = mp_mul_2d(&w0, 3, &tmp1)) != MP_OKAY) \{ -205 goto ERR; -206 \} -207 if ((res = mp_sub(&w1, &tmp1, &w1)) != MP_OKAY) \{ -208 goto ERR; -209 \} -210 /* r3 - 8r4 */ -211 if ((res = mp_mul_2d(&w4, 3, &tmp1)) != MP_OKAY) \{ -212 goto ERR; -213 \} -214 if ((res = mp_sub(&w3, &tmp1, &w3)) != MP_OKAY) \{ -215 goto ERR; -216 \} -217 /* 3r2 - r1 - r3 */ -218 if ((res = mp_mul_d(&w2, 3, &w2)) != MP_OKAY) \{ -219 goto ERR; -220 \} -221 if ((res = mp_sub(&w2, &w1, &w2)) != MP_OKAY) \{ -222 goto ERR; -223 \} -224 if ((res = mp_sub(&w2, &w3, &w2)) != MP_OKAY) \{ -225 goto ERR; -226 \} -227 /* r1 - r2 */ -228 if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) \{ -229 goto ERR; -230 \} -231 /* r3 - r2 */ -232 if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) \{ -233 goto ERR; -234 \} -235 /* r1/3 */ -236 if ((res = mp_div_3(&w1, &w1, NULL)) != MP_OKAY) \{ -237 goto ERR; -238 \} -239 /* r3/3 */ -240 if ((res = mp_div_3(&w3, &w3, NULL)) != MP_OKAY) \{ -241 goto ERR; -242 \} -243 -244 /* at this point shift W[n] by B*n */ -245 if ((res = mp_lshd(&w1, 1*B)) != MP_OKAY) \{ -246 goto ERR; -247 \} -248 if ((res = mp_lshd(&w2, 2*B)) != MP_OKAY) \{ -249 goto ERR; -250 \} -251 if ((res = mp_lshd(&w3, 3*B)) != MP_OKAY) \{ -252 goto ERR; -253 \} -254 if ((res = mp_lshd(&w4, 4*B)) != MP_OKAY) \{ -255 goto ERR; -256 \} -257 -258 if ((res = mp_add(&w0, &w1, c)) != MP_OKAY) \{ -259 goto ERR; -260 \} -261 if ((res = mp_add(&w2, &w3, &tmp1)) != MP_OKAY) \{ -262 goto ERR; -263 \} -264 if ((res = mp_add(&w4, &tmp1, &tmp1)) != MP_OKAY) \{ -265 goto ERR; -266 \} -267 if ((res = mp_add(&tmp1, c, c)) != MP_OKAY) \{ -268 goto ERR; -269 \} -270 -271 ERR: -272 mp_clear_multi(&w0, &w1, &w2, &w3, &w4, -273 &a0, &a1, &a2, &b0, &b1, -274 &b2, &tmp1, &tmp2, NULL); -275 return res; -276 \} -277 -278 #endif +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 mp_mod_2d(&a1, DIGIT_BIT * B, &a1); +049 +050 if ((res = mp_copy(a, &a2)) != MP_OKAY) \{ +051 goto ERR; +052 \} +053 mp_rshd(&a2, B*2); +054 +055 /* b = b2 * B**2 + b1 * B + b0 */ +056 if ((res = mp_mod_2d(b, DIGIT_BIT * B, &b0)) != MP_OKAY) \{ +057 goto ERR; +058 \} +059 +060 if ((res = mp_copy(b, &b1)) != MP_OKAY) \{ +061 goto ERR; +062 \} +063 mp_rshd(&b1, B); +064 mp_mod_2d(&b1, DIGIT_BIT * B, &b1); +065 +066 if ((res = mp_copy(b, &b2)) != MP_OKAY) \{ +067 goto ERR; +068 \} +069 mp_rshd(&b2, B*2); +070 +071 /* w0 = a0*b0 */ +072 if ((res = mp_mul(&a0, &b0, &w0)) != MP_OKAY) \{ +073 goto ERR; +074 \} +075 +076 /* w4 = a2 * b2 */ +077 if ((res = mp_mul(&a2, &b2, &w4)) != MP_OKAY) \{ +078 goto ERR; +079 \} +080 +081 /* w1 = (a2 + 2(a1 + 2a0))(b2 + 2(b1 + 2b0)) */ +082 if ((res = mp_mul_2(&a0, &tmp1)) != MP_OKAY) \{ +083 goto ERR; +084 \} +085 if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) \{ +086 goto ERR; +087 \} +088 if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) \{ +089 goto ERR; +090 \} +091 if ((res = mp_add(&tmp1, &a2, &tmp1)) != MP_OKAY) \{ +092 goto ERR; +093 \} +094 +095 if ((res = mp_mul_2(&b0, &tmp2)) != MP_OKAY) \{ +096 goto ERR; +097 \} +098 if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) \{ +099 goto ERR; +100 \} +101 if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) \{ +102 goto ERR; +103 \} +104 if ((res = mp_add(&tmp2, &b2, &tmp2)) != MP_OKAY) \{ +105 goto ERR; +106 \} +107 +108 if ((res = mp_mul(&tmp1, &tmp2, &w1)) != MP_OKAY) \{ +109 goto ERR; +110 \} +111 +112 /* w3 = (a0 + 2(a1 + 2a2))(b0 + 2(b1 + 2b2)) */ +113 if ((res = mp_mul_2(&a2, &tmp1)) != MP_OKAY) \{ +114 goto ERR; +115 \} +116 if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) \{ +117 goto ERR; +118 \} +119 if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) \{ +120 goto ERR; +121 \} +122 if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) \{ +123 goto ERR; +124 \} +125 +126 if ((res = mp_mul_2(&b2, &tmp2)) != MP_OKAY) \{ +127 goto ERR; +128 \} +129 if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) \{ +130 goto ERR; +131 \} +132 if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) \{ +133 goto ERR; +134 \} +135 if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) \{ +136 goto ERR; +137 \} +138 +139 if ((res = mp_mul(&tmp1, &tmp2, &w3)) != MP_OKAY) \{ +140 goto ERR; +141 \} +142 +143 +144 /* w2 = (a2 + a1 + a0)(b2 + b1 + b0) */ +145 if ((res = mp_add(&a2, &a1, &tmp1)) != MP_OKAY) \{ +146 goto ERR; +147 \} +148 if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) \{ +149 goto ERR; +150 \} +151 if ((res = mp_add(&b2, &b1, &tmp2)) != MP_OKAY) \{ +152 goto ERR; +153 \} +154 if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) \{ +155 goto ERR; +156 \} +157 if ((res = mp_mul(&tmp1, &tmp2, &w2)) != MP_OKAY) \{ +158 goto ERR; +159 \} +160 +161 /* now solve the matrix +162 +163 0 0 0 0 1 +164 1 2 4 8 16 +165 1 1 1 1 1 +166 16 8 4 2 1 +167 1 0 0 0 0 +168 +169 using 12 subtractions, 4 shifts, +170 2 small divisions and 1 small multiplication +171 */ +172 +173 /* r1 - r4 */ +174 if ((res = mp_sub(&w1, &w4, &w1)) != MP_OKAY) \{ +175 goto ERR; +176 \} +177 /* r3 - r0 */ +178 if ((res = mp_sub(&w3, &w0, &w3)) != MP_OKAY) \{ +179 goto ERR; +180 \} +181 /* r1/2 */ +182 if ((res = mp_div_2(&w1, &w1)) != MP_OKAY) \{ +183 goto ERR; +184 \} +185 /* r3/2 */ +186 if ((res = mp_div_2(&w3, &w3)) != MP_OKAY) \{ +187 goto ERR; +188 \} +189 /* r2 - r0 - r4 */ +190 if ((res = mp_sub(&w2, &w0, &w2)) != MP_OKAY) \{ +191 goto ERR; +192 \} +193 if ((res = mp_sub(&w2, &w4, &w2)) != MP_OKAY) \{ +194 goto ERR; +195 \} +196 /* r1 - r2 */ +197 if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) \{ +198 goto ERR; +199 \} +200 /* r3 - r2 */ +201 if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) \{ +202 goto ERR; +203 \} +204 /* r1 - 8r0 */ +205 if ((res = mp_mul_2d(&w0, 3, &tmp1)) != MP_OKAY) \{ +206 goto ERR; +207 \} +208 if ((res = mp_sub(&w1, &tmp1, &w1)) != MP_OKAY) \{ +209 goto ERR; +210 \} +211 /* r3 - 8r4 */ +212 if ((res = mp_mul_2d(&w4, 3, &tmp1)) != MP_OKAY) \{ +213 goto ERR; +214 \} +215 if ((res = mp_sub(&w3, &tmp1, &w3)) != MP_OKAY) \{ +216 goto ERR; +217 \} +218 /* 3r2 - r1 - r3 */ +219 if ((res = mp_mul_d(&w2, 3, &w2)) != MP_OKAY) \{ +220 goto ERR; +221 \} +222 if ((res = mp_sub(&w2, &w1, &w2)) != MP_OKAY) \{ +223 goto ERR; +224 \} +225 if ((res = mp_sub(&w2, &w3, &w2)) != MP_OKAY) \{ +226 goto ERR; +227 \} +228 /* r1 - r2 */ +229 if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) \{ +230 goto ERR; +231 \} +232 /* r3 - r2 */ +233 if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) \{ +234 goto ERR; +235 \} +236 /* r1/3 */ +237 if ((res = mp_div_3(&w1, &w1, NULL)) != MP_OKAY) \{ +238 goto ERR; +239 \} +240 /* r3/3 */ +241 if ((res = mp_div_3(&w3, &w3, NULL)) != MP_OKAY) \{ +242 goto ERR; +243 \} +244 +245 /* at this point shift W[n] by B*n */ +246 if ((res = mp_lshd(&w1, 1*B)) != MP_OKAY) \{ +247 goto ERR; +248 \} +249 if ((res = mp_lshd(&w2, 2*B)) != MP_OKAY) \{ +250 goto ERR; +251 \} +252 if ((res = mp_lshd(&w3, 3*B)) != MP_OKAY) \{ +253 goto ERR; +254 \} +255 if ((res = mp_lshd(&w4, 4*B)) != MP_OKAY) \{ +256 goto ERR; +257 \} +258 +259 if ((res = mp_add(&w0, &w1, c)) != MP_OKAY) \{ +260 goto ERR; +261 \} +262 if ((res = mp_add(&w2, &w3, &tmp1)) != MP_OKAY) \{ +263 goto ERR; +264 \} +265 if ((res = mp_add(&w4, &tmp1, &tmp1)) != MP_OKAY) \{ +266 goto ERR; +267 \} +268 if ((res = mp_add(&tmp1, c, c)) != MP_OKAY) \{ +269 goto ERR; +270 \} +271 +272 ERR: +273 mp_clear_multi(&w0, &w1, &w2, &w3, &w4, +274 &a0, &a1, &a2, &b0, &b1, +275 &b2, &tmp1, &tmp2, NULL); +276 return res; +277 \} +278 +279 #endif \end{alltt} \end{small} --- Comments to be added during editing phase. +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 40 to 69) 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 $b$. + +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 72 and 77). 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. \subsection{Signed Multiplication} Now that algorithms to handle multiplications of every useful dimensions have been developed, a rather simple finishing touch is required. So far all of the multiplication algorithms have been unsigned multiplications which leaves only a signed multiplication algorithm to be established. -\newpage\begin{figure}[!here] +\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} @@ -4847,7 +4884,7 @@ Column two of row one is a square and column three is the first unique column. The baseline squaring algorithm is meant to be a catch-all squaring algorithm. It will handle any of the input sizes that the faster routines will not handle. -\newpage\begin{figure}[!here] +\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} @@ -4907,75 +4944,79 @@ results calculated so far. This involves expensive carry propagation which will \begin{alltt} 016 017 /* low level squaring, b = a*a, HAC pp.596-597, Algorithm 14.16 */ -018 int -019 s_mp_sqr (mp_int * a, mp_int * b) -020 \{ -021 mp_int t; -022 int res, ix, iy, pa; -023 mp_word r; -024 mp_digit u, tmpx, *tmpt; -025 -026 pa = a->used; -027 if ((res = mp_init_size (&t, 2*pa + 1)) != MP_OKAY) \{ -028 return res; -029 \} -030 -031 /* default used is maximum possible size */ -032 t.used = 2*pa + 1; -033 -034 for (ix = 0; ix < pa; ix++) \{ -035 /* first calculate the digit at 2*ix */ -036 /* calculate double precision result */ -037 r = ((mp_word) t.dp[2*ix]) + -038 ((mp_word)a->dp[ix])*((mp_word)a->dp[ix]); -039 -040 /* store lower part in result */ -041 t.dp[ix+ix] = (mp_digit) (r & ((mp_word) MP_MASK)); -042 -043 /* get the carry */ -044 u = (mp_digit)(r >> ((mp_word) DIGIT_BIT)); -045 -046 /* left hand side of A[ix] * A[iy] */ -047 tmpx = a->dp[ix]; -048 -049 /* alias for where to store the results */ -050 tmpt = t.dp + (2*ix + 1); -051 -052 for (iy = ix + 1; iy < pa; iy++) \{ -053 /* first calculate the product */ -054 r = ((mp_word)tmpx) * ((mp_word)a->dp[iy]); -055 -056 /* now calculate the double precision result, note we use -057 * addition instead of *2 since it's easier to optimize -058 */ -059 r = ((mp_word) *tmpt) + r + r + ((mp_word) u); -060 -061 /* store lower part */ -062 *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK)); -063 -064 /* get carry */ -065 u = (mp_digit)(r >> ((mp_word) DIGIT_BIT)); -066 \} -067 /* propagate upwards */ -068 while (u != ((mp_digit) 0)) \{ -069 r = ((mp_word) *tmpt) + ((mp_word) u); -070 *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK)); -071 u = (mp_digit)(r >> ((mp_word) DIGIT_BIT)); -072 \} -073 \} -074 -075 mp_clamp (&t); -076 mp_exch (&t, b); -077 mp_clear (&t); -078 return MP_OKAY; -079 \} -080 #endif +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 \end{alltt} \end{small} -Inside the outer loop (\textit{see line 34}) the square term is calculated on line 37. Line 44 extracts the carry from the square -term. Aliases for $a_{ix}$ and $t_{ix+iy}$ are initialized on lines 47 and 50 respectively. The doubling is performed using two -additions (\textit{see 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. \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 @@ -4987,9 +5028,9 @@ propagation operations from the inner loop. However, the inner product must sti that $2a + 2b + 2c = 2(a + b + c)$. That is the sum of all of the double products is equal to double the sum of all the products. For example, $ab + ba + ac + ca = 2ab + 2ac = 2(ab + ac)$. -However, we cannot simply double all of the columns, since the squares appear only once per row. The most practical solution is to have two mp\_word -arrays. One array will hold the squares and the other array will hold the double products. With both arrays the doubling and carry propagation can be -moved to a $O(n)$ work level outside the $O(n^2)$ level. +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] \begin{small} @@ -4999,34 +5040,34 @@ moved to a $O(n)$ work level outside the $O(n^2)$ level. \textbf{Input}. mp\_int $a$ \\ \textbf{Output}. $b \leftarrow a^2$ \\ \hline \\ -Place two arrays of \textbf{MP\_WARRAY} mp\_words named $\hat W$ and $\hat {X}$ on the stack. \\ +Place an array of \textbf{MP\_WARRAY} mp\_digits named $W$ on the stack. \\ 1. If $b.alloc < 2a.used + 1$ then grow $b$ to $2a.used + 1$ digits. (\textit{mp\_grow}). \\ 2. If step 1 failed return(\textit{MP\_MEM}). \\ -3. for $ix$ from $0$ to $2a.used + 1$ do \\ -\hspace{3mm}3.1 $\hat W_{ix} \leftarrow 0$ \\ -\hspace{3mm}3.2 $\hat {X}_{ix} \leftarrow 0$ \\ -4. for $ix$ from $0$ to $a.used - 1$ do \\ -\hspace{3mm}Compute the square.\\ -\hspace{3mm}4.1 $\hat {X}_{ix+ix} \leftarrow \left ( a_{ix} \right )^2$ \\ \\ -\hspace{3mm}Compute the double products.\\ -\hspace{3mm}4.2 for $iy$ from $ix + 1$ to $a.used - 1$ do \\ -\hspace{6mm}4.2.1 $\hat W_{ix+iy} \leftarrow \hat W_{ix+iy} + a_{ix}a_{iy}$ \\ -5. $oldused \leftarrow b.used$ \\ -6. $b.used \leftarrow 2a.used + 1$ \\ +3. $pa \leftarrow 2 \cdot a.used$ \\ +4. $\hat W1 \leftarrow 0$ \\ +5. for $ix$ from $0$ to $pa - 1$ do \\ +\hspace{3mm}5.1 $\_ \hat W \leftarrow 0$ \\ +\hspace{3mm}5.2 $ty \leftarrow \mbox{MIN}(a.used - 1, ix)$ \\ +\hspace{3mm}5.3 $tx \leftarrow ix - ty$ \\ +\hspace{3mm}5.4 $iy \leftarrow \mbox{MIN}(a.used - tx, ty + 1)$ \\ +\hspace{3mm}5.5 $iy \leftarrow \mbox{MIN}(iy, \lfloor \left (ty - tx + 1 \right )/2 \rfloor)$ \\ +\hspace{3mm}5.6 for $iz$ from $0$ to $iz - 1$ do \\ +\hspace{6mm}5.6.1 $\_ \hat W \leftarrow \_ \hat W + a_{tx + iz}a_{ty - iz}$ \\ +\hspace{3mm}5.7 $\_ \hat W \leftarrow 2 \cdot \_ \hat W + \hat W1$ \\ +\hspace{3mm}5.8 if $ix$ is even then \\ +\hspace{6mm}5.8.1 $\_ \hat W \leftarrow \_ \hat W + \left ( a_{\lfloor ix/2 \rfloor}\right )^2$ \\ +\hspace{3mm}5.9 $W_{ix} \leftarrow \_ \hat W (\mbox{mod }\beta)$ \\ +\hspace{3mm}5.10 $\hat W1 \leftarrow \lfloor \_ \hat W / \beta \rfloor$ \\ \\ -Double the products and propagate the carries simultaneously. \\ -7. $\hat W_0 \leftarrow 2 \hat W_0 + \hat {X}_0$ \\ -8. for $ix$ from $1$ to $2a.used$ do \\ -\hspace{3mm}8.1 $\hat W_{ix} \leftarrow 2 \hat W_{ix} + \hat {X}_{ix}$ \\ -\hspace{3mm}8.2 $\hat W_{ix} \leftarrow \hat W_{ix} + \lfloor \hat W_{ix - 1} / \beta \rfloor$ \\ -\hspace{3mm}8.3 $b_{ix-1} \leftarrow W_{ix-1} \mbox{ (mod }\beta\mbox{)}$ \\ -9. $b_{2a.used} \leftarrow \hat W_{2a.used} \mbox{ (mod }\beta\mbox{)}$ \\ -10. if $2a.used + 1 < oldused$ then do \\ -\hspace{3mm}10.1 for $ix$ from $2a.used + 1$ to $oldused$ do \\ -\hspace{6mm}10.1.1 $b_{ix} \leftarrow 0$ \\ -11. Clamp excess digits from $b$. (\textit{mp\_clamp}) \\ -12. Return(\textit{MP\_OKAY}). \\ +6. $oldused \leftarrow b.used$ \\ +7. $b.used \leftarrow 2 \cdot a.used$ \\ +8. for $ix$ from $0$ to $pa - 1$ do \\ +\hspace{3mm}8.1 $b_{ix} \leftarrow W_{ix}$ \\ +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}). \\ \hline \end{tabular} \end{center} @@ -5035,146 +5076,123 @@ Double the products and propagate the carries simultaneously. \\ \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. -This routine requires two arrays of mp\_words to be placed on the stack. The first array $\hat W$ will hold the double products and the second -array $\hat X$ will hold the squares. Though only at most $MP\_WARRAY \over 2$ words of $\hat X$ are used, it has proven faster on most -processors to simply make it a full size array. +First, we have an accumulator and carry variables $\_ \hat W$ and $\hat W1$ respectively. This is because the inner loop +products are to be doubled. If we had added the previous carry in we would be doubling too much. Next we perform an +addition MIN condition on $iy$ (step 5.5) to prevent overlapping digits. For example, $a_3 \cdot a_5$ is equal +$a_5 \cdot a_3$. Whereas in the multiplication case we would have $5 < a.used$ and $3 \ge 0$ is maintained since we double the sum +of the products just outside the inner loop we have to avoid doing this. This is also a good thing since we perform +fewer multiplications and the routine ends up being faster. -The loop on step 3 will zero the two arrays to prepare them for the squaring step. Step 4.1 computes the squares of the product. Note how -it simply assigns the value into the $\hat X$ array. The nested loop on step 4.2 computes the doubles of the products. This loop -computes the sum of the products for each column. They are not doubled until later. - -After the squaring loop, the products stored in $\hat W$ musted be doubled and the carries propagated forwards. It makes sense to do both -operations at the same time. The expression $\hat W_{ix} \leftarrow 2 \hat W_{ix} + \hat {X}_{ix}$ computes the sum of the double product and the -squares in place. +Finally the last difference is the addition of the ``square'' term outside the inner loop (step 5.8). We add in the square +only to even outputs and it is the square of the term at the $\lfloor ix / 2 \rfloor$ position. \vspace{+3mm}\begin{small} \hspace{-5.1mm}{\bf File}: bn\_fast\_s\_mp\_sqr.c \vspace{-3mm} \begin{alltt} 016 -017 /* fast squaring -018 * -019 * This is the comba method where the columns of the product -020 * are computed first then the carries are computed. This -021 * has the effect of making a very simple inner loop that -022 * is executed the most -023 * -024 * W2 represents the outer products and W the inner. -025 * -026 * A further optimizations is made because the inner -027 * products are of the form "A * B * 2". The *2 part does -028 * not need to be computed until the end which is good -029 * because 64-bit shifts are slow! -030 * -031 * Based on Algorithm 14.16 on pp.597 of HAC. -032 * -033 */ -034 /* the jist of squaring... -035 -036 you do like mult except the offset of the tmpx [one that starts closer to ze - ro] -037 can't equal the offset of tmpy. So basically you set up iy like before then - you min it with -038 (ty-tx) so that it never happens. You double all those you add in the inner - loop -039 -040 After that loop you do the squares and add them in. -041 -042 Remove W2 and don't memset W -043 -044 */ -045 -046 int fast_s_mp_sqr (mp_int * a, mp_int * b) -047 \{ -048 int olduse, res, pa, ix, iz; -049 mp_digit W[MP_WARRAY], *tmpx; -050 mp_word W1; -051 -052 /* grow the destination as required */ -053 pa = a->used + a->used; -054 if (b->alloc < pa) \{ -055 if ((res = mp_grow (b, pa)) != MP_OKAY) \{ -056 return res; -057 \} -058 \} -059 -060 /* number of output digits to produce */ -061 W1 = 0; -062 for (ix = 0; ix < pa; ix++) \{ -063 int tx, ty, iy; -064 mp_word _W; -065 mp_digit *tmpy; -066 -067 /* clear counter */ -068 _W = 0; +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 /* get offsets into the two bignums */ -071 ty = MIN(a->used-1, ix); -072 tx = ix - ty; -073 -074 /* setup temp aliases */ -075 tmpx = a->dp + tx; -076 tmpy = a->dp + ty; +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 /* this is the number of times the loop will iterrate, essentially its - -079 while (tx++ < a->used && ty-- >= 0) \{ ... \} -080 */ -081 iy = MIN(a->used-tx, ty+1); +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 /* now for squaring tx can never equal ty -084 * we halve the distance since they approach at a rate of 2x -085 * and we have to round because odd cases need to be executed -086 */ -087 iy = MIN(iy, (ty-tx+1)>>1); -088 -089 /* execute loop */ -090 for (iz = 0; iz < iy; iz++) \{ -091 _W += ((mp_word)*tmpx++)*((mp_word)*tmpy--); -092 \} +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 /* double the inner product and add carry */ -095 _W = _W + _W + W1; -096 -097 /* even columns have the square term in them */ -098 if ((ix&1) == 0) \{ -099 _W += ((mp_word)a->dp[ix>>1])*((mp_word)a->dp[ix>>1]); -100 \} -101 -102 /* store it */ -103 W[ix] = _W; -104 -105 /* make next carry */ -106 W1 = _W >> ((mp_word)DIGIT_BIT); -107 \} -108 -109 /* setup dest */ -110 olduse = b->used; -111 b->used = a->used+a->used; -112 -113 \{ -114 mp_digit *tmpb; -115 tmpb = b->dp; -116 for (ix = 0; ix < pa; ix++) \{ -117 *tmpb++ = W[ix] & MP_MASK; -118 \} -119 -120 /* clear unused digits [that existed in the old copy of c] */ -121 for (; ix < olduse; ix++) \{ -122 *tmpb++ = 0; -123 \} -124 \} -125 mp_clamp (b); -126 return MP_OKAY; -127 \} -128 #endif +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 \end{alltt} \end{small} --- Write something deep and insightful later, Tom. +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 @@ -5392,14 +5410,13 @@ By inlining the copy and shift operations the cutoff point for Karatsuba multipl is exactly at the point where Comba squaring can no longer be used (\textit{128 digits}). On slower processors such as the Intel P4 it is actually below the Comba limit (\textit{at 110 digits}). -This routine uses the same error trap coding style as mp\_karatsuba\_sqr. As the temporary variables are initialized errors are redirected to -the error trap higher up. If the algorithm completes without error the error code is set to \textbf{MP\_OKAY} and mp\_clears are executed normally. - -\textit{Last paragraph sucks. re-write! -- Tom} +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 +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} @@ -5485,12 +5502,9 @@ neither of the polynomial basis algorithms should be used then either the Comba $\left [ 3 \right ] $ & Devise an efficient algorithm for selection of the radix point to handle inputs \\ & that have different number of digits in Karatsuba multiplication. \\ & \\ -$\left [ 3 \right ] $ & In section 5.3 the fact that every column of a squaring is made up \\ +$\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 [ 2 \right ] $ & In the Comba squaring algorithm half of the $\hat X$ variables are not used. \\ - & Revise algorithm fast\_s\_mp\_sqr to shrink the $\hat X$ array. \\ - & \\ $\left [ 3 \right ] $ & Prove the equation for Karatsuba squaring. \\ & \\ $\left [ 1 \right ] $ & Prove that Karatsuba squaring requires $O \left (n^{lg(3)} \right )$ time. \\ @@ -5498,6 +5512,14 @@ $\left [ 1 \right ] $ & Prove that Karatsuba squaring requires $O \left (n^{lg(3 $\left [ 2 \right ] $ & Determine the minimal ratio between addition and multiplication clock cycles \\ & required for equation $6.7$ to be true. \\ & \\ +$\left [ 3 \right ] $ & Implement a threaded version of Comba multiplication (and squaring) where you \\ + & compute subsets of the columns in each thread. Determine a cutoff point where \\ + & it is effective and add the logic to mp\_mul() and mp\_sqr(). \\ + &\\ +$\left [ 4 \right ] $ & Same as the previous but also modify the Karatsuba and Toom-Cook. You must \\ + & increase the throughput of mp\_exptmod() for random odd moduli in the range \\ + & $512 \ldots 4096$ bits significantly ($> 2x$) to complete this challenge. \\ + & \\ \end{tabular} \chapter{Modular Reduction} @@ -5516,7 +5538,7 @@ 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 +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. @@ -5730,95 +5752,94 @@ performed at most twice, and on average once. However, if $a \ge b^2$ than it wi 018 * precomputed via mp_reduce_setup. 019 * From HAC pp.604 Algorithm 14.42 020 */ -021 int -022 mp_reduce (mp_int * x, mp_int * m, mp_int * mu) -023 \{ -024 mp_int q; -025 int res, um = m->used; -026 -027 /* q = x */ -028 if ((res = mp_init_copy (&q, x)) != MP_OKAY) \{ -029 return res; -030 \} -031 -032 /* q1 = x / b**(k-1) */ -033 mp_rshd (&q, um - 1); -034 -035 /* according to HAC this optimization is ok */ -036 if (((unsigned long) um) > (((mp_digit)1) << (DIGIT_BIT - 1))) \{ -037 if ((res = mp_mul (&q, mu, &q)) != MP_OKAY) \{ -038 goto CLEANUP; -039 \} -040 \} else \{ -041 #ifdef BN_S_MP_MUL_HIGH_DIGS_C -042 if ((res = s_mp_mul_high_digs (&q, mu, &q, um - 1)) != MP_OKAY) \{ -043 goto CLEANUP; -044 \} -045 #elif defined(BN_FAST_S_MP_MUL_HIGH_DIGS_C) -046 if ((res = fast_s_mp_mul_high_digs (&q, mu, &q, um - 1)) != MP_OKAY) \{ -047 goto CLEANUP; -048 \} -049 #else -050 \{ -051 res = MP_VAL; -052 goto CLEANUP; -053 \} -054 #endif -055 \} -056 -057 /* q3 = q2 / b**(k+1) */ -058 mp_rshd (&q, um + 1); -059 -060 /* x = x mod b**(k+1), quick (no division) */ -061 if ((res = mp_mod_2d (x, DIGIT_BIT * (um + 1), x)) != MP_OKAY) \{ -062 goto CLEANUP; -063 \} -064 -065 /* q = q * m mod b**(k+1), quick (no division) */ -066 if ((res = s_mp_mul_digs (&q, m, &q, um + 1)) != MP_OKAY) \{ -067 goto CLEANUP; -068 \} -069 -070 /* x = x - q */ -071 if ((res = mp_sub (x, &q, x)) != MP_OKAY) \{ -072 goto CLEANUP; -073 \} -074 -075 /* If x < 0, add b**(k+1) to it */ -076 if (mp_cmp_d (x, 0) == MP_LT) \{ -077 mp_set (&q, 1); -078 if ((res = mp_lshd (&q, um + 1)) != MP_OKAY) -079 goto CLEANUP; -080 if ((res = mp_add (x, &q, x)) != MP_OKAY) -081 goto CLEANUP; -082 \} -083 -084 /* Back off if it's too big */ -085 while (mp_cmp (x, m) != MP_LT) \{ -086 if ((res = s_mp_sub (x, m, x)) != MP_OKAY) \{ -087 goto CLEANUP; -088 \} -089 \} -090 -091 CLEANUP: -092 mp_clear (&q); -093 -094 return res; -095 \} -096 #endif +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 (((unsigned long) 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 \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 +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. -\begin{figure}[!here] +\newpage\begin{figure}[!here] \begin{small} \begin{center} \begin{tabular}{l} @@ -6314,161 +6335,160 @@ stored in the destination $x$. 022 * 023 * Based on Algorithm 14.32 on pp.601 of HAC. 024 */ -025 int -026 fast_mp_montgomery_reduce (mp_int * x, mp_int * n, mp_digit rho) -027 \{ -028 int ix, res, olduse; -029 mp_word W[MP_WARRAY]; -030 -031 /* get old used count */ -032 olduse = x->used; -033 -034 /* grow a as required */ -035 if (x->alloc < n->used + 1) \{ -036 if ((res = mp_grow (x, n->used + 1)) != MP_OKAY) \{ -037 return res; -038 \} -039 \} -040 -041 /* first we have to get the digits of the input into -042 * an array of double precision words W[...] -043 */ -044 \{ -045 register mp_word *_W; -046 register mp_digit *tmpx; -047 -048 /* alias for the W[] array */ -049 _W = W; -050 -051 /* alias for the digits of x*/ -052 tmpx = x->dp; -053 -054 /* copy the digits of a into W[0..a->used-1] */ -055 for (ix = 0; ix < x->used; ix++) \{ -056 *_W++ = *tmpx++; -057 \} -058 -059 /* zero the high words of W[a->used..m->used*2] */ -060 for (; ix < n->used * 2 + 1; ix++) \{ -061 *_W++ = 0; -062 \} -063 \} -064 -065 /* now we proceed to zero successive digits -066 * from the least significant upwards -067 */ -068 for (ix = 0; ix < n->used; ix++) \{ -069 /* mu = ai * m' mod b -070 * -071 * We avoid a double precision multiplication (which isn't required) -072 * by casting the value down to a mp_digit. Note this requires -073 * that W[ix-1] have the carry cleared (see after the inner loop) -074 */ -075 register mp_digit mu; -076 mu = (mp_digit) (((W[ix] & MP_MASK) * rho) & MP_MASK); -077 -078 /* a = a + mu * m * b**i -079 * -080 * This is computed in place and on the fly. The multiplication -081 * by b**i is handled by offseting which columns the results -082 * are added to. -083 * -084 * Note the comba method normally doesn't handle carries in the -085 * inner loop In this case we fix the carry from the previous -086 * column since the Montgomery reduction requires digits of the -087 * result (so far) [see above] to work. This is -088 * handled by fixing up one carry after the inner loop. The -089 * carry fixups are done in order so after these loops the -090 * first m->used words of W[] have the carries fixed -091 */ -092 \{ -093 register int iy; -094 register mp_digit *tmpn; -095 register mp_word *_W; -096 -097 /* alias for the digits of the modulus */ -098 tmpn = n->dp; -099 -100 /* Alias for the columns set by an offset of ix */ -101 _W = W + ix; -102 -103 /* inner loop */ -104 for (iy = 0; iy < n->used; iy++) \{ -105 *_W++ += ((mp_word)mu) * ((mp_word)*tmpn++); -106 \} -107 \} -108 -109 /* now fix carry for next digit, W[ix+1] */ -110 W[ix + 1] += W[ix] >> ((mp_word) DIGIT_BIT); -111 \} -112 -113 /* now we have to propagate the carries and -114 * shift the words downward [all those least -115 * significant digits we zeroed]. -116 */ -117 \{ -118 register mp_digit *tmpx; -119 register mp_word *_W, *_W1; -120 -121 /* nox fix rest of carries */ -122 -123 /* alias for current word */ -124 _W1 = W + ix; -125 -126 /* alias for next word, where the carry goes */ -127 _W = W + ++ix; -128 -129 for (; ix <= n->used * 2 + 1; ix++) \{ -130 *_W++ += *_W1++ >> ((mp_word) DIGIT_BIT); -131 \} -132 -133 /* copy out, A = A/b**n -134 * -135 * The result is A/b**n but instead of converting from an -136 * array of mp_word to mp_digit than calling mp_rshd -137 * we just copy them in the right order -138 */ -139 -140 /* alias for destination word */ -141 tmpx = x->dp; -142 -143 /* alias for shifted double precision result */ -144 _W = W + n->used; -145 -146 for (ix = 0; ix < n->used + 1; ix++) \{ -147 *tmpx++ = (mp_digit)(*_W++ & ((mp_word) MP_MASK)); -148 \} -149 -150 /* zero oldused digits, if the input a was larger than -151 * m->used+1 we'll have to clear the digits -152 */ -153 for (; ix < olduse; ix++) \{ -154 *tmpx++ = 0; -155 \} -156 \} -157 -158 /* set the max used and clamp */ -159 x->used = n->used + 1; -160 mp_clamp (x); -161 -162 /* if A >= m then A = A - m */ -163 if (mp_cmp_mag (x, n) != MP_LT) \{ -164 return s_mp_sub (x, n, x); -165 \} -166 return MP_OKAY; -167 \} -168 #endif +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 register mp_word *_W; +045 register 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 register 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 register int iy; +093 register mp_digit *tmpn; +094 register 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 register mp_digit *tmpx; +118 register 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 \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 $\hat W$ 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 $\_ \hat W_{t} = \hat W_{n.used + t}$. @@ -6968,51 +6988,50 @@ shift which makes the algorithm fairly inexpensive to use. \begin{alltt} 016 017 /* reduces a modulo n where n is of the form 2**p - d */ -018 int -019 mp_reduce_2k(mp_int *a, mp_int *n, mp_digit d) -020 \{ -021 mp_int q; -022 int p, res; -023 -024 if ((res = mp_init(&q)) != MP_OKAY) \{ -025 return res; -026 \} -027 -028 p = mp_count_bits(n); -029 top: -030 /* q = a/2**p, a = a mod 2**p */ -031 if ((res = mp_div_2d(a, p, &q, a)) != MP_OKAY) \{ -032 goto ERR; -033 \} -034 -035 if (d != 1) \{ -036 /* q = q * d */ -037 if ((res = mp_mul_d(&q, d, &q)) != MP_OKAY) \{ -038 goto ERR; -039 \} -040 \} -041 -042 /* a = a + q */ -043 if ((res = s_mp_add(a, &q, a)) != MP_OKAY) \{ -044 goto ERR; -045 \} -046 -047 if (mp_cmp_mag(a, n) != MP_LT) \{ -048 s_mp_sub(a, n, a); -049 goto top; -050 \} -051 -052 ERR: -053 mp_clear(&q); -054 return res; -055 \} -056 -057 #endif +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 s_mp_sub(a, n, a); +048 goto top; +049 \} +050 +051 ERR: +052 mp_clear(&q); +053 return res; +054 \} +055 +056 #endif \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. @@ -7052,32 +7071,31 @@ is sufficient to solve for $k$. Alternatively if $n$ has more than one digit th \begin{alltt} 016 017 /* determines the setup value */ -018 int -019 mp_reduce_2k_setup(mp_int *a, mp_digit *d) -020 \{ -021 int res, p; -022 mp_int tmp; -023 -024 if ((res = mp_init(&tmp)) != MP_OKAY) \{ -025 return res; -026 \} -027 -028 p = mp_count_bits(a); -029 if ((res = mp_2expt(&tmp, p)) != MP_OKAY) \{ -030 mp_clear(&tmp); -031 return res; -032 \} -033 -034 if ((res = s_mp_sub(&tmp, a, &tmp)) != MP_OKAY) \{ -035 mp_clear(&tmp); -036 return res; -037 \} -038 -039 *d = tmp.dp[0]; -040 mp_clear(&tmp); -041 return MP_OKAY; -042 \} -043 #endif +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 \end{alltt} \end{small} @@ -7130,9 +7148,9 @@ This algorithm quickly determines if a modulus is of the form required for algor 021 mp_digit iz; 022 023 if (a->used == 0) \{ -024 return 0; +024 return MP_NO; 025 \} else if (a->used == 1) \{ -026 return 1; +026 return MP_YES; 027 \} else if (a->used > 1) \{ 028 iy = mp_count_bits(a); 029 iz = 1; @@ -7141,7 +7159,7 @@ This algorithm quickly determines if a modulus is of the form required for algor 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 0; +035 return MP_NO; 036 \} 037 iz <<= 1; 038 if (iz > (mp_digit)MP_MASK) \{ @@ -7150,7 +7168,7 @@ This algorithm quickly determines if a modulus is of the form required for algor 041 \} 042 \} 043 \} -044 return 1; +044 return MP_YES; 045 \} 046 047 #endif @@ -7601,39 +7619,47 @@ algorithm since their arguments are essentially the same (\textit{two mp\_ints a 064 #endif 065 \} 066 -067 #ifdef BN_MP_DR_IS_MODULUS_C -068 /* is it a DR modulus? */ -069 dr = mp_dr_is_modulus(P); -070 #else -071 dr = 0; +067 /* modified diminished radix reduction */ +068 #if defined(BN_MP_REDUCE_IS_2K_L_C) && defined(BN_MP_REDUCE_2K_L_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_REDUCE_IS_2K_C -075 /* if not, is it a uDR modulus? */ -076 if (dr == 0) \{ -077 dr = mp_reduce_is_2k(P) << 1; -078 \} -079 #endif -080 -081 /* if the modulus is odd or dr != 0 use the fast method */ -082 #ifdef BN_MP_EXPTMOD_FAST_C -083 if (mp_isodd (P) == 1 || dr != 0) \{ -084 return mp_exptmod_fast (G, X, P, Y, dr); -085 \} else \{ -086 #endif -087 #ifdef BN_S_MP_EXPTMOD_C -088 /* otherwise use the generic Barrett reduction technique */ -089 return s_mp_exptmod (G, X, P, Y); -090 #else -091 /* no exptmod for evens */ -092 return MP_VAL; -093 #endif -094 #ifdef BN_MP_EXPTMOD_FAST_C -095 \} -096 #endif -097 \} -098 -099 #endif +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) == 1 || 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 \end{alltt} \end{small} @@ -7642,8 +7668,8 @@ negative the algorithm tries to perform a modular exponentiation with the modula 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 69 determines if the modulus is of the restricted Diminished Radix -form. If it is not line 77 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} @@ -7652,7 +7678,7 @@ of three values. \item $dr = 2$ means that the modulus is of unrestricted Diminished Radix form. \end{enumerate} -Line 67 determines if the fast modular exponentiation algorithm can be used. It is allowed if $dr \ne 0$ or if the modulus is odd. Otherwise, +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. \subsection{Barrett Modular Exponentiation} @@ -7820,230 +7846,244 @@ a Left-to-Right algorithm is used to process the remaining few bits. 020 #define TAB_SIZE 256 021 #endif 022 -023 int s_mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y) +023 int s_mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y, int redmod + e) 024 \{ 025 mp_int M[TAB_SIZE], res, mu; 026 mp_digit buf; 027 int err, bitbuf, bitcpy, bitcnt, mode, digidx, x, y, winsize; -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 if ((err = mp_reduce_setup (&mu, P)) != MP_OKAY) \{ -075 goto LBL_MU; -076 \} -077 -078 /* create M table -079 * -080 * The M table contains powers of the base, -081 * e.g. M[x] = G**x mod P -082 * -083 * The first half of the table is not -084 * computed though accept for M[0] and M[1] -085 */ -086 if ((err = mp_mod (G, P, &M[1])) != MP_OKAY) \{ -087 goto LBL_MU; -088 \} -089 -090 /* compute the value at M[1<<(winsize-1)] by squaring -091 * M[1] (winsize-1) times -092 */ -093 if ((err = mp_copy (&M[1], &M[1 << (winsize - 1)])) != MP_OKAY) \{ -094 goto LBL_MU; -095 \} -096 -097 for (x = 0; x < (winsize - 1); x++) \{ -098 if ((err = mp_sqr (&M[1 << (winsize - 1)], -099 &M[1 << (winsize - 1)])) != MP_OKAY) \{ -100 goto LBL_MU; -101 \} -102 if ((err = mp_reduce (&M[1 << (winsize - 1)], P, &mu)) != MP_OKAY) \{ -103 goto LBL_MU; -104 \} +028 int (*redux)(mp_int*,mp_int*,mp_int*); +029 +030 /* find window size */ +031 x = mp_count_bits (X); +032 if (x <= 7) \{ +033 winsize = 2; +034 \} else if (x <= 36) \{ +035 winsize = 3; +036 \} else if (x <= 140) \{ +037 winsize = 4; +038 \} else if (x <= 450) \{ +039 winsize = 5; +040 \} else if (x <= 1303) \{ +041 winsize = 6; +042 \} else if (x <= 3529) \{ +043 winsize = 7; +044 \} else \{ +045 winsize = 8; +046 \} +047 +048 #ifdef MP_LOW_MEM +049 if (winsize > 5) \{ +050 winsize = 5; +051 \} +052 #endif +053 +054 /* init M array */ +055 /* init first cell */ +056 if ((err = mp_init(&M[1])) != MP_OKAY) \{ +057 return err; +058 \} +059 +060 /* now init the second half of the array */ +061 for (x = 1<<(winsize-1); x < (1 << winsize); x++) \{ +062 if ((err = mp_init(&M[x])) != MP_OKAY) \{ +063 for (y = 1<<(winsize-1); y < x; y++) \{ +064 mp_clear (&M[y]); +065 \} +066 mp_clear(&M[1]); +067 return err; +068 \} +069 \} +070 +071 /* create mu, used for Barrett reduction */ +072 if ((err = mp_init (&mu)) != MP_OKAY) \{ +073 goto LBL_M; +074 \} +075 +076 if (redmode == 0) \{ +077 if ((err = mp_reduce_setup (&mu, P)) != MP_OKAY) \{ +078 goto LBL_MU; +079 \} +080 redux = mp_reduce; +081 \} else \{ +082 if ((err = mp_reduce_2k_setup_l (P, &mu)) != MP_OKAY) \{ +083 goto LBL_MU; +084 \} +085 redux = mp_reduce_2k_l; +086 \} +087 +088 /* create M table +089 * +090 * The M table contains powers of the base, +091 * e.g. M[x] = G**x mod P +092 * +093 * The first half of the table is not +094 * computed though accept for M[0] and M[1] +095 */ +096 if ((err = mp_mod (G, P, &M[1])) != MP_OKAY) \{ +097 goto LBL_MU; +098 \} +099 +100 /* compute the value at M[1<<(winsize-1)] by squaring +101 * M[1] (winsize-1) times +102 */ +103 if ((err = mp_copy (&M[1], &M[1 << (winsize - 1)])) != MP_OKAY) \{ +104 goto LBL_MU; 105 \} 106 -107 /* create upper table, that is M[x] = M[x-1] * M[1] (mod P) -108 * for x = (2**(winsize - 1) + 1) to (2**winsize - 1) -109 */ -110 for (x = (1 << (winsize - 1)) + 1; x < (1 << winsize); x++) \{ -111 if ((err = mp_mul (&M[x - 1], &M[1], &M[x])) != MP_OKAY) \{ -112 goto LBL_MU; -113 \} -114 if ((err = mp_reduce (&M[x], P, &mu)) != MP_OKAY) \{ -115 goto LBL_MU; -116 \} -117 \} -118 -119 /* setup result */ -120 if ((err = mp_init (&res)) != MP_OKAY) \{ -121 goto LBL_MU; -122 \} -123 mp_set (&res, 1); -124 -125 /* set initial mode and bit cnt */ -126 mode = 0; -127 bitcnt = 1; -128 buf = 0; -129 digidx = X->used - 1; -130 bitcpy = 0; -131 bitbuf = 0; -132 -133 for (;;) \{ -134 /* grab next digit as required */ -135 if (--bitcnt == 0) \{ -136 /* if digidx == -1 we are out of digits */ -137 if (digidx == -1) \{ -138 break; -139 \} -140 /* read next digit and reset the bitcnt */ -141 buf = X->dp[digidx--]; -142 bitcnt = (int) DIGIT_BIT; -143 \} -144 -145 /* grab the next msb from the exponent */ -146 y = (buf >> (mp_digit)(DIGIT_BIT - 1)) & 1; -147 buf <<= (mp_digit)1; -148 -149 /* if the bit is zero and mode == 0 then we ignore it -150 * These represent the leading zero bits before the first 1 bit -151 * in the exponent. Technically this opt is not required but it -152 * does lower the # of trivial squaring/reductions used -153 */ -154 if (mode == 0 && y == 0) \{ -155 continue; +107 for (x = 0; x < (winsize - 1); x++) \{ +108 /* square it */ +109 if ((err = mp_sqr (&M[1 << (winsize - 1)], +110 &M[1 << (winsize - 1)])) != MP_OKAY) \{ +111 goto LBL_MU; +112 \} +113 +114 /* reduce modulo P */ +115 if ((err = redux (&M[1 << (winsize - 1)], P, &mu)) != MP_OKAY) \{ +116 goto LBL_MU; +117 \} +118 \} +119 +120 /* create upper table, that is M[x] = M[x-1] * M[1] (mod P) +121 * for x = (2**(winsize - 1) + 1) to (2**winsize - 1) +122 */ +123 for (x = (1 << (winsize - 1)) + 1; x < (1 << winsize); x++) \{ +124 if ((err = mp_mul (&M[x - 1], &M[1], &M[x])) != MP_OKAY) \{ +125 goto LBL_MU; +126 \} +127 if ((err = redux (&M[x], P, &mu)) != MP_OKAY) \{ +128 goto LBL_MU; +129 \} +130 \} +131 +132 /* setup result */ +133 if ((err = mp_init (&res)) != MP_OKAY) \{ +134 goto LBL_MU; +135 \} +136 mp_set (&res, 1); +137 +138 /* set initial mode and bit cnt */ +139 mode = 0; +140 bitcnt = 1; +141 buf = 0; +142 digidx = X->used - 1; +143 bitcpy = 0; +144 bitbuf = 0; +145 +146 for (;;) \{ +147 /* grab next digit as required */ +148 if (--bitcnt == 0) \{ +149 /* if digidx == -1 we are out of digits */ +150 if (digidx == -1) \{ +151 break; +152 \} +153 /* read next digit and reset the bitcnt */ +154 buf = X->dp[digidx--]; +155 bitcnt = (int) DIGIT_BIT; 156 \} 157 -158 /* if the bit is zero and mode == 1 then we square */ -159 if (mode == 1 && y == 0) \{ -160 if ((err = mp_sqr (&res, &res)) != MP_OKAY) \{ -161 goto LBL_RES; -162 \} -163 if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) \{ -164 goto LBL_RES; -165 \} -166 continue; -167 \} -168 -169 /* else we add it to the window */ -170 bitbuf |= (y << (winsize - ++bitcpy)); -171 mode = 2; -172 -173 if (bitcpy == winsize) \{ -174 /* ok window is filled so square as required and multiply */ -175 /* square first */ -176 for (x = 0; x < winsize; x++) \{ -177 if ((err = mp_sqr (&res, &res)) != MP_OKAY) \{ -178 goto LBL_RES; -179 \} -180 if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) \{ -181 goto LBL_RES; -182 \} -183 \} -184 -185 /* then multiply */ -186 if ((err = mp_mul (&res, &M[bitbuf], &res)) != MP_OKAY) \{ -187 goto LBL_RES; -188 \} -189 if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) \{ -190 goto LBL_RES; -191 \} -192 -193 /* empty window and reset */ -194 bitcpy = 0; -195 bitbuf = 0; -196 mode = 1; -197 \} -198 \} -199 -200 /* if bits remain then square/multiply */ -201 if (mode == 2 && bitcpy > 0) \{ -202 /* square then multiply if the bit is set */ -203 for (x = 0; x < bitcpy; x++) \{ -204 if ((err = mp_sqr (&res, &res)) != MP_OKAY) \{ -205 goto LBL_RES; -206 \} -207 if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) \{ -208 goto LBL_RES; -209 \} -210 -211 bitbuf <<= 1; -212 if ((bitbuf & (1 << winsize)) != 0) \{ -213 /* then multiply */ -214 if ((err = mp_mul (&res, &M[1], &res)) != MP_OKAY) \{ -215 goto LBL_RES; -216 \} -217 if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) \{ -218 goto LBL_RES; -219 \} -220 \} -221 \} -222 \} +158 /* grab the next msb from the exponent */ +159 y = (buf >> (mp_digit)(DIGIT_BIT - 1)) & 1; +160 buf <<= (mp_digit)1; +161 +162 /* if the bit is zero and mode == 0 then we ignore it +163 * These represent the leading zero bits before the first 1 bit +164 * in the exponent. Technically this opt is not required but it +165 * does lower the # of trivial squaring/reductions used +166 */ +167 if (mode == 0 && y == 0) \{ +168 continue; +169 \} +170 +171 /* if the bit is zero and mode == 1 then we square */ +172 if (mode == 1 && y == 0) \{ +173 if ((err = mp_sqr (&res, &res)) != MP_OKAY) \{ +174 goto LBL_RES; +175 \} +176 if ((err = redux (&res, P, &mu)) != MP_OKAY) \{ +177 goto LBL_RES; +178 \} +179 continue; +180 \} +181 +182 /* else we add it to the window */ +183 bitbuf |= (y << (winsize - ++bitcpy)); +184 mode = 2; +185 +186 if (bitcpy == winsize) \{ +187 /* ok window is filled so square as required and multiply */ +188 /* square first */ +189 for (x = 0; x < winsize; x++) \{ +190 if ((err = mp_sqr (&res, &res)) != MP_OKAY) \{ +191 goto LBL_RES; +192 \} +193 if ((err = redux (&res, P, &mu)) != MP_OKAY) \{ +194 goto LBL_RES; +195 \} +196 \} +197 +198 /* then multiply */ +199 if ((err = mp_mul (&res, &M[bitbuf], &res)) != MP_OKAY) \{ +200 goto LBL_RES; +201 \} +202 if ((err = redux (&res, P, &mu)) != MP_OKAY) \{ +203 goto LBL_RES; +204 \} +205 +206 /* empty window and reset */ +207 bitcpy = 0; +208 bitbuf = 0; +209 mode = 1; +210 \} +211 \} +212 +213 /* if bits remain then square/multiply */ +214 if (mode == 2 && bitcpy > 0) \{ +215 /* square then multiply if the bit is set */ +216 for (x = 0; x < bitcpy; x++) \{ +217 if ((err = mp_sqr (&res, &res)) != MP_OKAY) \{ +218 goto LBL_RES; +219 \} +220 if ((err = redux (&res, P, &mu)) != MP_OKAY) \{ +221 goto LBL_RES; +222 \} 223 -224 mp_exch (&res, Y); -225 err = MP_OKAY; -226 LBL_RES:mp_clear (&res); -227 LBL_MU:mp_clear (&mu); -228 LBL_M: -229 mp_clear(&M[1]); -230 for (x = 1<<(winsize-1); x < (1 << winsize); x++) \{ -231 mp_clear (&M[x]); -232 \} -233 return err; -234 \} -235 #endif +224 bitbuf <<= 1; +225 if ((bitbuf & (1 << winsize)) != 0) \{ +226 /* then multiply */ +227 if ((err = mp_mul (&res, &M[1], &res)) != MP_OKAY) \{ +228 goto LBL_RES; +229 \} +230 if ((err = redux (&res, P, &mu)) != MP_OKAY) \{ +231 goto LBL_RES; +232 \} +233 \} +234 \} +235 \} +236 +237 mp_exch (&res, Y); +238 err = MP_OKAY; +239 LBL_RES:mp_clear (&res); +240 LBL_MU:mp_clear (&mu); +241 LBL_M: +242 mp_clear(&M[1]); +243 for (x = 1<<(winsize-1); x < (1 << winsize); x++) \{ +244 mp_clear (&M[x]); +245 \} +246 return err; +247 \} +248 #endif \end{alltt} \end{small} -Lines 31 through 41 determine the optimal window size based on the length of the exponent in bits. The window divisions are sorted +Lines 21 through 40 determine the optimal window size based on the length of the exponent in bits. The window divisions are sorted from smallest to greatest so that in each \textbf{if} statement only one condition must be tested. For example, by the \textbf{if} statement -on line 33 the value of $x$ is already known to be greater than $140$. +on line 32 the value of $x$ is already known to be greater than $140$. -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 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 for loop on line 60 initializes the $M$ array while lines 61 and 74 compute the value of $\mu$ required for +The for loop on line 61 initializes the $M$ array while lines 62 and 77 compute the value of $\mu$ required for Barrett reduction. -- More later. @@ -8873,21 +8913,22 @@ Unlike the full multiplication algorithms this algorithm does not require any si 056 u = (mp_digit) (r >> ((mp_word) DIGIT_BIT)); 057 \} 058 -059 /* store final carry [if any] */ +059 /* store final carry [if any] and increment ix offset */ 060 *tmpc++ = u; -061 -062 /* now zero digits above the top */ -063 while (ix++ < olduse) \{ -064 *tmpc++ = 0; -065 \} -066 -067 /* set used count */ -068 c->used = a->used + 1; -069 mp_clamp(c); -070 -071 return MP_OKAY; -072 \} -073 #endif +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 \end{alltt} \end{small} @@ -9275,14 +9316,14 @@ the integers from $0$ to $\beta - 1$. 028 029 /* first place a random non-zero digit */ 030 do \{ -031 d = ((mp_digit) abs (rand ())); +031 d = ((mp_digit) abs (rand ())) & 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) \{ +038 while (--digits > 0) \{ 039 if ((res = mp_lshd (a, 1)) != MP_OKAY) \{ 040 return res; 041 \} @@ -9379,7 +9420,7 @@ as part of larger input without any significant problem. \begin{alltt} 016 017 /* read a string [ASCII] in a given radix */ -018 int mp_read_radix (mp_int * a, char *str, int radix) +018 int mp_read_radix (mp_int * a, const char *str, int radix) 019 \{ 020 int y, res, neg; 021 char ch; @@ -9941,6 +9982,8 @@ To explain the Jacobi Symbol we shall first discuss the Legendre function\footno 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}. +\textit{-- Tom, don't be an ass, cite your source here...!} + \begin{equation} a^{(p-1)/2} \equiv \begin{array}{rl} -1 & \mbox{if }a\mbox{ is a quadratic non-residue.} \\ |