| 1 | %%% -*-latex-*- |
| 2 | |
| 3 | \section{Low-level details: MPX} |
| 4 | \label{sec:mpx} |
| 5 | |
| 6 | |
| 7 | \subsection{Data types for low-level arithmetic} |
| 8 | |
| 9 | The header file \hdr{<catacomb/mpw.h>} defines two types, \code{mpw} and |
| 10 | \code{mpd}, and a collection of macros describing them. |
| 11 | |
| 12 | The `multiprecision word' type, \code{mpw}, is an unsigned integer type whose |
| 13 | representation uses \code{MPW_BITS}. The largest value representable in an |
| 14 | \code{mpw} is $\code{MPW_MAX} = 2^{\code{MPW_BITS}} - 1$. |
| 15 | The value of \code{MPW_BITS} is at least 16. Note that, on some |
| 16 | architectures, an \code{mpw} may be capable of representing values larger |
| 17 | than \code{MPW_MAX}. The expression \code{MPW($x$)} returns the value of $x |
| 18 | \bmod \code{MPW_MAX}$ cast to type \code{mpw}. |
| 19 | |
| 20 | The `multiprecision doubleword' type, \code{mpd}, is an unsigned integer type |
| 21 | with at least double the width of \code{mpw}. Hence, an \code{mpd} is |
| 22 | capable of representing the product of two \code{mpw}s, and is useful when |
| 23 | performing multiplications and divisions. The constants \code{MPD_BITS} and |
| 24 | \code{MPD_MAX} are defined to be the number of bits used in the |
| 25 | representation of an \code{mpd} and the largest value representable in an |
| 26 | \code{mpd} respectively. |
| 27 | |
| 28 | A few macros useful for manipulating \code{mpw}s are defined. |
| 29 | |
| 30 | \subsubsection{The \code{MPW} macro} |
| 31 | \label{fn:MPW} |
| 32 | |
| 33 | \fsec{Synopsis} |
| 34 | |
| 35 | \begin{listinglist} |
| 36 | |#include <catacomb/mpw.h>| \\ |
| 37 | |mpw MPW(|$x$|);| |
| 38 | \end{listinglist} |
| 39 | |
| 40 | \fsec{Returns} |
| 41 | |
| 42 | The value of $x \bmod 2^{\code{MPW_BITS}}$, as an \code{mpw}. This is a |
| 43 | frequently used abbreviation for |(mpw)(|$x$| & MPW_MAX)|. |
| 44 | |
| 45 | \subsubsection{The \code{MPWS} macro} |
| 46 | \label{fn:MPWS} |
| 47 | |
| 48 | \fsec{Synopsis} |
| 49 | |
| 50 | \begin{listinglist} |
| 51 | |#include <catacomb/mpw.h>| \\ |
| 52 | |size_t MPWS(size_t |$n$|);| |
| 53 | \end{listinglist} |
| 54 | |
| 55 | \fsec{Returns} |
| 56 | |
| 57 | The number of bytes occupied by an array of $n$~\code{mpw}s (i.e., $n \cdot |
| 58 | |sizeof(mpw)|$). |
| 59 | |
| 60 | \subsubsection{The \code{MPW_RQ} macro} |
| 61 | \label{fn:MPW-RQ} |
| 62 | |
| 63 | \fsec{Synopsis} |
| 64 | |
| 65 | \begin{listinglist} |
| 66 | |#include <catacomb/mpw.h>| \\ |
| 67 | |size_t MPW_RQ(size_t |$\sz$|);| |
| 68 | \end{listinglist} |
| 69 | |
| 70 | \fsec{Returns} |
| 71 | |
| 72 | The number of \code{mpw}s required to represent a multiprecision integer |
| 73 | occupying $\sz$~octets in an external representation. |
| 74 | |
| 75 | |
| 76 | \subsection{Low-level multiprecision integer representation} |
| 77 | |
| 78 | The low-level multiprecision definitions are exported in the header file |
| 79 | \hdr{<catacomb/mpx.h>}. This header includes \hdr{<catacomb/mpw.h>}, so |
| 80 | there's usually no need to include it explicitly. |
| 81 | |
| 82 | A multiprecision integer is represented as an array of \code{mpw}s, |
| 83 | least-significant first. The array's bounds are described by a a pair of |
| 84 | pointers to the array's \emph{base} (i.e., its first element) and |
| 85 | \emph{limit} (i.e., the location immediately following its last element). |
| 86 | |
| 87 | Let $v$ be the base and $\vl$ the limit of a multiprecision integer array. |
| 88 | The array itself is notated as~$v .. \vl$. The array's size in words may be |
| 89 | computed as $\vl - v$ in the normal C fashion. The integer represented by |
| 90 | the array, denoted $\mp(v .. \vl)$, is defined to be |
| 91 | \[ |
| 92 | \mp(v .. \vl) = |
| 93 | \sum_{0 \le i < \vl - v} |
| 94 | 2^{\code{MPW_BITS} \cdot i} v[i] |
| 95 | \] |
| 96 | If the array is empty (i.e., $v = \vl$) then the number is zero. If the |
| 97 | array is empty, or the final word is nonzero, then the representation is said |
| 98 | to be \emph{shrunken}. Shrunken representations are more efficient, since |
| 99 | the arithmetic algorithms don't need to consider high-order words which make |
| 100 | no difference to the final result anyway. |
| 101 | |
| 102 | Whenever a result is too large to be represented in the memory allocated for |
| 103 | it, high-order bits are discarded. Thus, a result written to an array of |
| 104 | $k$~words is reduced modulo $2^{\code{MPW_BITS} \cdot k}$. |
| 105 | |
| 106 | |
| 107 | \subsection{Low-level macros} |
| 108 | \label{sec:mpx-macro} |
| 109 | |
| 110 | The following macros perform various simple operations useful for |
| 111 | manipulating multiprecision integers at the MPX level. |
| 112 | |
| 113 | \subsubsection{The \code{MPX_SHRINK} macro} |
| 114 | \label{fn:MPX-SHRINK} |
| 115 | |
| 116 | \fsec{Synopsis} |
| 117 | |
| 118 | \begin{listinglist} |
| 119 | |#include <catacomb/mpx.h>| \\ |
| 120 | |MPX_SHRINK(const mpw *|$v$|, const mpw *|$\vl$|);| |
| 121 | \end{listinglist} |
| 122 | |
| 123 | \fsec{Description} |
| 124 | |
| 125 | The \code{MPX_SHRINK} macro reduces the limit~$\vl$ of a multiprecision |
| 126 | integer array so that either $v = \vl$ or~$\vl[-1] \ne 0$. The argument~$\vl$ must be an lvalue, since the macro writes the result back when it |
| 127 | finishes. |
| 128 | |
| 129 | \subsubsection{The \code{MPX_BITS} macro} |
| 130 | \label{fn:MPX-BITS} |
| 131 | |
| 132 | \fsec{Synopsis} |
| 133 | |
| 134 | \begin{listinglist} |
| 135 | |#include <catacomb/mpx.h>| \\ |
| 136 | |MPX_BITS(unsigned long |$b$|, const mpw *|$v$|, const mpw *|$\vl$|);| |
| 137 | \end{listinglist} |
| 138 | |
| 139 | \fsec{Description} |
| 140 | |
| 141 | Determines the smallest number of bits which could represent the number |
| 142 | $\mp(v .. \vl)$, and writes the answer to~$b$, which must therefore be an |
| 143 | lvalue. The result is zero if the number is zero; otherwise $b$ is the |
| 144 | largest integer such that |
| 145 | \[ \mp(v .. \vl) \ge 2^{(b - 1) \bmod \code{MPW_BITS}}.\] |
| 146 | |
| 147 | \subsubsection{The \code{MPX_OCTETS} macro} |
| 148 | \label{fn:MPX-OCTETS} |
| 149 | |
| 150 | \fsec{Synopsis} |
| 151 | |
| 152 | \begin{listinglist} |
| 153 | |#include <catacomb/mpx.h>| \\ |
| 154 | |MPX_OCTETS(size_t |$o$|,| |
| 155 | |const mpw *|$v$|, const mpw *|$\vl$|);| |
| 156 | \end{listinglist} |
| 157 | |
| 158 | \fsec{Description} |
| 159 | |
| 160 | Determines the smallest number of octets which could represent the number |
| 161 | $\mp(v .. \vl)$, and writes the answer to~$o$, which must therefore be an |
| 162 | lvalue. This is useful for determining appropriate buffer sizes for the |
| 163 | results of \code{mpx_storel} and \code{mpx_storeb}. |
| 164 | |
| 165 | The result $o$ can be calculated from the number of bits~$b$ reported by |
| 166 | \code{MPX_BITS} as $o = \lceil b / 8 \rceil$; the algorithm used by |
| 167 | \code{MPX_OCTETS} is more efficient than this, however. |
| 168 | |
| 169 | \subsubsection{The \code{MPX_COPY} macro} |
| 170 | \label{fn:MPX-COPY} |
| 171 | |
| 172 | \fsec{Synopsis} |
| 173 | |
| 174 | \begin{listinglist} |
| 175 | |#include <catacomb/mpx.h>| \\ |
| 176 | |MPX_COPY(mpw *|$\dv$|, mpw *|$\dvl$|,| |
| 177 | |const mpw *|$\av$|, const mpw *|$\avl$|);| |
| 178 | \end{listinglist} |
| 179 | |
| 180 | \fsec{Description} |
| 181 | |
| 182 | Copies a multiprecision integer from the source array $\av .. \avl$ to the |
| 183 | destination array $\dv .. \dvl$. If the destination array is large enough, |
| 184 | the result is equal to the source; otherwise, high-order bits are discarded |
| 185 | as usual. |
| 186 | |
| 187 | \subsubsection{The \code{MPX_ZERO} macro} |
| 188 | \label{fn:MPX-ZERO} |
| 189 | |
| 190 | \fsec{Synopsis} |
| 191 | |
| 192 | \begin{listinglist} |
| 193 | |#include <catacomb/mpx.h>| \\ |
| 194 | |MPX_ZERO(mpw *|$v$|, mpw *|$\vl$|);| |
| 195 | \end{listinglist} |
| 196 | |
| 197 | \fsec{Description} |
| 198 | |
| 199 | Sets the integer $\mp(v .. \vl)$ to zero. |
| 200 | |
| 201 | |
| 202 | \subsection{Transfer formats: loading and storing} |
| 203 | \label{sec:mpx-io} |
| 204 | |
| 205 | The MPX layer can translate between internal representations of integers as |
| 206 | arrays of \code{mpw}s and external formats where integers are stored as |
| 207 | arrays of octets. Both little- and big-endian orders are supported. |
| 208 | |
| 209 | If $a_0, a_1, \ldots, a_{k - 1}$ is an array of $k$ octets, with |
| 210 | $0 \le a_i < 256$ for all $0 \le i < k$, then the number $n$ represented by |
| 211 | $a$ in little-endian byte order is |
| 212 | \[ n = \sum_{0 \le i < k} 256^i a_i .\] |
| 213 | Similarly, the number $n'$ represented by $a$ in big-endian byte order is |
| 214 | \[ n' = \sum_{0 \le i < k} 256^{k - i - 1} a_i .\] |
| 215 | |
| 216 | If, in a store operation, the destination octet array is not large enough to |
| 217 | represent the number to be stored, high-order octets are discarded; hence, |
| 218 | the number is reduced modulo $2^{8k}$, where $k$ is the size of the |
| 219 | destination array in octets. |
| 220 | |
| 221 | Two useful macros help when working out how much memory is needed for |
| 222 | translation between internal and transfer formats for multiprecision |
| 223 | integers. The macro \code{MPX_OCTETS} (page~\pageref{fn:MPX-OCTETS}) |
| 224 | calculates the smallest octet array size which can represent a multiprecision |
| 225 | integer. The macro \code{MPW_RQ} (page~\pageref{fn:MPW-RQ}) calculates the |
| 226 | smallest \code{mpw} array which can represent a multiprecision integer held |
| 227 | in an octet array of a particular size. |
| 228 | |
| 229 | \subsubsection{The \code{mpx_storel} function} |
| 230 | \label{fn:mpx-storel} |
| 231 | |
| 232 | \fsec{Synopsis} |
| 233 | |
| 234 | \begin{listinglist} |
| 235 | |#include <catacomb/mpx.h>| \\ |
| 236 | |void mpx_storel(const mpw *|$v$|, const mpw *|$\vl$|,| |
| 237 | |void *|$p$|, size_t |$\sz$|);| |
| 238 | \end{listinglist} |
| 239 | |
| 240 | \fsec{Description} |
| 241 | |
| 242 | Stores the number held in the array $v .. \vl$ to the array of $\sz$~octets |
| 243 | starting at address~$p$ in little-endian byte order (i.e., least significant |
| 244 | byte first). |
| 245 | |
| 246 | \subsubsection{The \code{mpx_loadl} function} |
| 247 | \label{fn:mpx-loadl} |
| 248 | |
| 249 | \fsec{Synopsis} |
| 250 | |
| 251 | \begin{listinglist} |
| 252 | |#include <catacomb/mpx.h>| \\ |
| 253 | |void mpx_loadl(mpw *|$v$|, mpw *|$\vl$|,| |
| 254 | |const void *|$p$|, size_t |$\sz$|);| |
| 255 | \end{listinglist} |
| 256 | |
| 257 | \fsec{Description} |
| 258 | |
| 259 | Loads into the array $v .. \vl$ the number represented in the array of |
| 260 | $\sz$~octets starting at address~$p$ in little-endian byte order (i.e., least |
| 261 | significant byte first). |
| 262 | |
| 263 | \subsubsection{The \code{mpx_storeb} function} |
| 264 | \label{fn:mpx-storeb} |
| 265 | |
| 266 | \fsec{Synopsis} |
| 267 | |
| 268 | \begin{listinglist} |
| 269 | |#include <catacomb/mpx.h>| \\ |
| 270 | |void mpx_storeb(const mpw *|$v$|, const mpw *|$\vl$|,| |
| 271 | |void *|$p$|, size_t |$\sz$|);| |
| 272 | \end{listinglist} |
| 273 | |
| 274 | \fsec{Description} |
| 275 | |
| 276 | Stores the number held in the array $v .. \vl$ to the array of $\sz$~octets |
| 277 | starting at address~$p$ in big-endian byte order (i.e., least significant |
| 278 | byte last). |
| 279 | |
| 280 | \subsubsection{The \code{mpx_loadb} function} |
| 281 | \label{fn:mpx-loadb} |
| 282 | |
| 283 | \fsec{Synopsis} |
| 284 | |
| 285 | \begin{listinglist} |
| 286 | |#include <catacomb/mpx.h>| \\ |
| 287 | |void mpx_loadb(mpw *|$v$|, mpw *|$\vl$|,| |
| 288 | |const void *|$p$|, size_t |$\sz$|);| |
| 289 | \end{listinglist} |
| 290 | |
| 291 | \fsec{Description} |
| 292 | |
| 293 | Loads into the array $v .. \vl$ the number represented in the array of |
| 294 | $\sz$~octets starting at address $p$ in big-endian byte order (i.e., least |
| 295 | significant byte last). |
| 296 | |
| 297 | |
| 298 | \subsection{Bit shifting operations} |
| 299 | \label{sec:mpx-shift} |
| 300 | |
| 301 | The MPX layer provides functions for efficient multiplication and division of |
| 302 | multiprecision integers by powers of two, performed simply by shifting the |
| 303 | binary representation left or right by a number of bits. Shifts by one bit |
| 304 | and by a multiple of \code{MPW_BITS} are particularly fast. |
| 305 | |
| 306 | There are two shift functions, one for left shifts (multiplications) and one |
| 307 | for right shifts (divisions). Each function is passed an array containing |
| 308 | the number to be shifted, a destination array for the result (which may be |
| 309 | the source array), and the number of bits to be shifted as an unsigned |
| 310 | integer. |
| 311 | |
| 312 | |
| 313 | \subsubsection{The \code{mpx_lsl} function} |
| 314 | \label{fn:mpx-lsl} |
| 315 | |
| 316 | \fsec{Synopsis} |
| 317 | |
| 318 | \begin{listinglist} |
| 319 | \begin{tabbing} |
| 320 | |#include <catacomb/mpx.h>| \\ |
| 321 | |void mpx_lsl(|\=|mpw *|$\dv$|, mpw *|$\dvl$|,| \\ |
| 322 | \>|const mpw *|$\av$|, const mpw *|$\avl$|, size_t |$n$|);| |
| 323 | \end{tabbing} |
| 324 | \end{listinglist} |
| 325 | |
| 326 | \fsec{Description} |
| 327 | |
| 328 | Stores in $\dv .. \dvl$ the result of shifting $\mp(\av .. \avl)$ left by |
| 329 | $n$~bits (i.e., multiplying it by~$2^n$). |
| 330 | |
| 331 | \subsubsection{The \code{mpx_lsr} function} |
| 332 | \label{fn:mpx-lsr} |
| 333 | |
| 334 | \fsec{Synopsis} |
| 335 | |
| 336 | \begin{listinglist} |
| 337 | \begin{tabbing} |
| 338 | |#include <catacomb/mpx.h>| \\ |
| 339 | |void mpx_lsr(|\=|mpw *|$\dv$|, mpw *|$\dvl$|,| \\ |
| 340 | \>|const mpw *|$\av$|, const mpw *|$\avl$|, size_t |$n$|);| |
| 341 | \end{tabbing} |
| 342 | \end{listinglist} |
| 343 | |
| 344 | \fsec{Description} |
| 345 | |
| 346 | Stores in $\dv .. \dvl$ the result of shifting $\mp(\av .. \avl)$ right by |
| 347 | $n$~bits (i.e., dividing it by~$2^n$, rounding towards zero). |
| 348 | |
| 349 | |
| 350 | \subsection{Low-level arithmetic} |
| 351 | |
| 352 | The remaining functions perform standard arithmetic operations on large |
| 353 | integers. The form for the arguments is fairly well-standardized: |
| 354 | destination arrays are given first, followed by the source arrays in the |
| 355 | conventional order. (This ordering reflects the usual order in a C |
| 356 | assignment expression, the \code{strcpy} and \code{memcpy} functions, and the |
| 357 | order of operands in many assembly languages.) |
| 358 | |
| 359 | Some functions allow the destination array to be the same as one (or both) |
| 360 | source arrays; others forbid this. Under no circumstances may the |
| 361 | destination partially overlap the sources. |
| 362 | |
| 363 | |
| 364 | \subsubsection{The \code{mpx_2c} function} |
| 365 | \label{fn:mpx-2c} |
| 366 | |
| 367 | \fsec{Synopsis} |
| 368 | |
| 369 | \begin{listinglist} |
| 370 | |#include <catacomb/mpx.h>| \\ |
| 371 | |void mpx_2c(mpw *|$\dv$|, mpw *|$\dvl$|,| |
| 372 | |const mpw *|$v$|, const mpw *|$\vl$|);| |
| 373 | \end{listinglist} |
| 374 | |
| 375 | \fsec{Description} |
| 376 | |
| 377 | Computes the two's complement of the number $\mp(v .. \vl)$ and stores it in |
| 378 | $\dv .. \dvl$. The two arrays $v .. \vl$ and $\dv .. \dvl$ may be the same. |
| 379 | |
| 380 | \subsubsection{The \code{mpx_ucmp} function and \code{MPX_UCMP} macro} |
| 381 | \label{fn:mpx-ucmp} |
| 382 | |
| 383 | \fsec{Synopsis} |
| 384 | \begin{listinglist} |
| 385 | \begin{tabbing} |
| 386 | |#include <catacomb/mpx.h>| \\ |
| 387 | |int mpx_ucmp(|\=|const mpw *|$\av$|, const mpw *|$\avl$|,| \\ |
| 388 | \>|const mpw *|$\bv$|, const mpw *|$\bvl$|);| \\ |
| 389 | |int MPX_UCMP(|\=|const mpw *|$\av$|, const mpw *|$\avl$|, |\synt{rel-op}|,| |
| 390 | \\ \>|const mpw *|$\bv$|, const mpw *|$\bvl$|);| |
| 391 | \end{tabbing} |
| 392 | \end{listinglist} |
| 393 | |
| 394 | \fsec{Description} |
| 395 | |
| 396 | The function \code{mpx_ucmp} performs an unsigned comparison of two unsigned |
| 397 | multiprecision integers $a$ and~$b$, passed in the arrays $\av .. \avl$ and |
| 398 | $\bv .. \bvl$ respectively. |
| 399 | |
| 400 | The macro \code{MPX_UCMP} provides a slightly more readable interface for |
| 401 | comparing integers. The \synt{rel-op} may be any C relational operator |
| 402 | (e.g., |!=|, or |<=|). The macro tests whether |
| 403 | $a \mathrel{\synt{rel-op}} b$. |
| 404 | |
| 405 | \fsec{Returns} |
| 406 | |
| 407 | The function \code{mpx_ucmp} returns a value less then, equal to, or |
| 408 | greater than zero depending on whether $a$ is less than, equal to or greater |
| 409 | than~$b$. |
| 410 | |
| 411 | The macro \code{MPX_UCMP} returns a nonzero result if $a |
| 412 | \mathrel{\synt{rel-op}} b$ is true, and zero if false. |
| 413 | |
| 414 | \subsubsection{The \code{mpx_uadd} function} |
| 415 | \label{fn:mpx-uadd} |
| 416 | |
| 417 | \fsec{Synopsis} |
| 418 | |
| 419 | \begin{listinglist} |
| 420 | \begin{tabbing} |
| 421 | |#include <catacomb/mpx.h>| \\ |
| 422 | |void mpx_uadd(|\=|mpw *|$\dv$|, mpw *|$\dvl$|,| |
| 423 | |const mpw *|$\av$|, const mpw *|$\avl$|,| \\ |
| 424 | \>|const mpw *|$\bv$|, const mpw *|$\bvl$|);| |
| 425 | \end{tabbing} |
| 426 | \end{listinglist} |
| 427 | |
| 428 | \fsec{Description} |
| 429 | |
| 430 | Adds two multiprecision integers. The sum of the two arguments |
| 431 | $\mp(\av .. \avl) + \mp(\bv .. \bvl)$ is stored in $\dv .. \dvl$. The |
| 432 | destination array may be equal to either or both source arrays.\footnote{% |
| 433 | Adding a number to itself is a poor way of doubling. A call to |
| 434 | \code{mpx_lsl} (page~\pageref{fn:mpx-lsl}) is much more efficient.} % |
| 435 | |
| 436 | \subsubsection{The \code{mpx_uaddn} function and \code{MPX_UADDN} macro} |
| 437 | \label{fn:mpx-uaddn} |
| 438 | |
| 439 | \fsec{Synopsis} |
| 440 | |
| 441 | \begin{listinglist} |
| 442 | |#include <catacomb/mpx.h>| \\ |
| 443 | |void mpx_uaddn(mpw *|$\dv$|, mpw *|$\dvl$|, mpw |$n$|);| \\ |
| 444 | |void MPX_UADDN(mpw *|$\dv$|, mpw *|$\dvl$|, mpw |$n$|);| |
| 445 | \end{listinglist} |
| 446 | |
| 447 | \fsec{Description} |
| 448 | |
| 449 | The function \code{mpx_uaddn} adds a small integer~$n$ (expressed as a single |
| 450 | \code{mpw}) to the multiprecision integer held in $\dv .. \dvl$. |
| 451 | |
| 452 | The macro \code{MPX_UADDN} performs exactly the same operation, but uses |
| 453 | inline code rather than calling a function. |
| 454 | |
| 455 | \subsubsection{The \code{mpx_usub} function} |
| 456 | \label{fn:mpx-usub} |
| 457 | |
| 458 | \fsec{Synopsis} |
| 459 | |
| 460 | \begin{listinglist} |
| 461 | \begin{tabbing} |
| 462 | |#include <catacomb/mpx.h>| \\ |
| 463 | |void mpx_usub(|\=|mpw *|$\dv$|, mpw *|$\dvl$|,| |
| 464 | |const mpw *|$\av$|, const mpw *|$\avl$|,| \\ |
| 465 | \>|const mpw *|$\bv$|, const mpw *|$\bvl$|);| |
| 466 | \end{tabbing} |
| 467 | \end{listinglist} |
| 468 | |
| 469 | \fsec{Description} |
| 470 | |
| 471 | Subtracts one multiprecision integer from another. The difference of the two |
| 472 | arguments $\mp(\av .. \avl) - \mp(\bv .. \bvl)$ is stored in $\dv .. \dvl$. |
| 473 | The destination array may be equal to either or both source |
| 474 | arrays.\footnote{% |
| 475 | Subtracting a number from itself is a particularly poor way of clearing an |
| 476 | integer to zero. A call to \code{MPX_ZERO} (page~\pageref{fn:MPX-ZERO}) is |
| 477 | much more efficient.} % |
| 478 | |
| 479 | Because overly large results are truncated to fit the destination array, if |
| 480 | the second argument is larger than the first, a two's-complement result is |
| 481 | obtained. |
| 482 | |
| 483 | \subsubsection{The \code{mpx_usubn} function and \code{MPX_USUBN} macro} |
| 484 | \label{fn:mpx-usubn} |
| 485 | |
| 486 | \fsec{Synopsis} |
| 487 | |
| 488 | \begin{listinglist} |
| 489 | |#include <catacomb/mpx.h>| \\ |
| 490 | |void mpx_usubn(mpw *|$\dv$|, mpw *|$\dvl$|, mpw |$n$|);| \\ |
| 491 | |void MPX_USUBN(mpw *|$\dv$|, mpw *|$\dvl$|, mpw |$n$|);| |
| 492 | \end{listinglist} |
| 493 | |
| 494 | \fsec{Description} |
| 495 | |
| 496 | The function \code{mpx_usubn} subtracts a small integer~$n$ (expressed as a |
| 497 | single \code{mpw}) from the multiprecision integer held in $\dv .. \dvl$. |
| 498 | |
| 499 | The macro \code{MPX_USUBN} performs exactly the same operation, but uses |
| 500 | inline code rather than calling a function. |
| 501 | |
| 502 | \subsubsection{The \code{mpx_umul} function} |
| 503 | \label{fn:mpx-umul} |
| 504 | |
| 505 | \fsec{Synopsis} |
| 506 | |
| 507 | \begin{listinglist} |
| 508 | \begin{tabbing} |
| 509 | |#include <catacomb/mpx.h>| \\ |
| 510 | |void mpx_umul(|\=|mpw *|$\dv$|, mpw *|$\dvl$|,| |
| 511 | |const mpw *|$\av$|, const mpw *|$\avl$|,| \\ |
| 512 | \>|const mpw *|$\bv$|, const mpw *|$\bvl$|);| |
| 513 | \end{tabbing} |
| 514 | \end{listinglist} |
| 515 | |
| 516 | \fsec{Description} |
| 517 | |
| 518 | Multiplies two multiprecision integers. The product of the two arguments |
| 519 | $\mp(\av .. \avl) \times \mp(\bv .. \bvl)$ is stored in $\dv .. \dvl$. The |
| 520 | destination array may not be equal to either source array. |
| 521 | |
| 522 | \subsubsection{The \code{mpx_umuln} function and \code{MPX_UMULN} macro} |
| 523 | \label{fn:mpx-umuln} |
| 524 | |
| 525 | \fsec{Synopsis} |
| 526 | |
| 527 | \begin{listinglist} |
| 528 | \begin{tabbing} |
| 529 | |#include <catacomb/mpx.h>| \\ |
| 530 | |void mpx_umuln(|\=|mpw *|$\dv$|, mpw *|$\dvl$|,| \\ |
| 531 | \> |const mpw *|$\av$|, const mpw *|$\avl$|, mpw |$n$|);| \\ |
| 532 | |void MPX_UMULN(|\=|mpw *|$\dv$|, mpw *|$\dvl$|,| \\ |
| 533 | \> |const mpw *|$\av$|, const mpw *|$\avl$|, mpw |$n$|);| |
| 534 | \end{tabbing} |
| 535 | \end{listinglist} |
| 536 | |
| 537 | \fsec{Description} |
| 538 | |
| 539 | The function \code{mpx_umuln} multiplies the multiprecision integer passed in |
| 540 | $\av .. \avl$ by a small integer~$n$ (expressed as a single \code{mpw}), |
| 541 | writing the product $n \cdot \mp(\av .. \avl)$ to the destination array $\dv |
| 542 | .. \dvl$. The destination array may be equal to the source array $\av |
| 543 | .. \avl$. |
| 544 | |
| 545 | The macro \code{MPX_UMULN} performs exactly the same operation, but uses |
| 546 | inline code rather than calling a function. |
| 547 | |
| 548 | \subsubsection{The \code{mpx_umlan} function and \code{MPX_UMLAN} macro} |
| 549 | \label{fn:mpx-umlan} |
| 550 | |
| 551 | \fsec{Synopsis} |
| 552 | |
| 553 | \begin{listinglist} |
| 554 | \begin{tabbing} |
| 555 | |#include <catacomb/mpx.h>| \\* |
| 556 | |void mpx_umlan(|\=|mpw *|$\dv$|, mpw *|$\dvl$|,| \\* |
| 557 | \> |const mpw *|$\av$|, const mpw *|$\avl$|, mpw |$n$|);| \\ |
| 558 | |void MPX_UMLAN(|\=|mpw *|$\dv$|, mpw *|$\dvl$|,| \\ |
| 559 | \> |const mpw *|$\av$|, const mpw *|$\avl$|, mpw |$n$|);| |
| 560 | \end{tabbing} |
| 561 | \end{listinglist} |
| 562 | |
| 563 | \fsec{Description} |
| 564 | |
| 565 | The function \code{mpx_umlan} multiplies the multiprecision integer passed in |
| 566 | $\av .. \avl$ by a small integer~$n$ (expressed as a single \code{mpw}), and |
| 567 | adds it to the value already stored in the destination array $\dv .. \dvl$. |
| 568 | The destination array may be equal to the source array $\av .. \avl$, |
| 569 | although this isn't very useful. |
| 570 | |
| 571 | The macro \code{MPX_UMLAN} performs exactly the same operation, but uses |
| 572 | inline code rather than calling a function. |
| 573 | |
| 574 | \subsubsection{The \code{mpx_usqr} function} |
| 575 | \label{fn:mpx-usqr} |
| 576 | |
| 577 | \fsec{Synopsis} |
| 578 | |
| 579 | \begin{listinglist} |
| 580 | |#include <catacomb/mpx.h>| \\ |
| 581 | |void mpx_usqr(mpw *|$\dv$|, mpw *|$\dvl$|,| |
| 582 | |const mpw *|$\av$|, const mpw *|$\avl$|);| |
| 583 | \end{listinglist} |
| 584 | |
| 585 | \fsec{Description} |
| 586 | |
| 587 | Squares a multiprecision integer. The result $\mp(\av .. \avl)^2$ is stored |
| 588 | in $\dv .. \dvl$. The destination array may not be equal to the source |
| 589 | array. |
| 590 | |
| 591 | Squaring a number is approximately twice as fast as multiplying a number by |
| 592 | itself. |
| 593 | |
| 594 | \subsubsection{The \code{mpx_udiv} function} |
| 595 | \label{fn:mpx-udiv} |
| 596 | |
| 597 | \fsec{Synopsis} |
| 598 | |
| 599 | \begin{listinglist} |
| 600 | \begin{tabbing} |
| 601 | |#include <catacomb/mpx.h>| \\ |
| 602 | |void mpx_udiv(|\=|mpw *|$\qv$|, mpw *|$\qvl$|, mpw *|$\rv$|, mpw *|$\rvl$|,| |
| 603 | \\ \>|const mpw *|$\dv$|, const mpw *|$\dvl$|,| |
| 604 | |mpw *|$\mathit{sv}$|, mpw *|$\mathit{svl}$|);| |
| 605 | \end{tabbing} |
| 606 | \end{listinglist} |
| 607 | |
| 608 | \fsec{Description} |
| 609 | |
| 610 | Calculates the quotient and remainder when one multiprecision integer is |
| 611 | divided by another. |
| 612 | |
| 613 | The calling convention is slightly strange. The dividend and divisor are |
| 614 | passed in the arrays $\rv .. \rvl$ and $\dv .. \dvl$ respectively. The |
| 615 | quotient is written to the array $\qv .. \qvl$; the remainder is left in $\rv |
| 616 | .. \rvl$. |
| 617 | |
| 618 | The division function requires some workspace. The `scratch' array |
| 619 | $\mathit{sv} .. \mathit{svl}$ must be at least one word longer than the |
| 620 | divisor array $\dv .. \dvl$. The scratch array'sfinal contents are not |
| 621 | useful. Also, the remainder array $\rv .. \rvl$ must have at least two |
| 622 | words' headroom. |
| 623 | |
| 624 | \unverb\| |
| 625 | Given a dividend $x$ and a divisor $y$, the function calculates a quotient |
| 626 | $q$ and remainder $r$ such that $q = \lfloor x / y \rfloor$ and $x = qy + r$. |
| 627 | In particular, this definition implies that $r$ has the same sign as $y$, |
| 628 | which is a useful property when performing modular reductions. |
| 629 | \shortverb\| |
| 630 | |
| 631 | %%% Local Variables: |
| 632 | %%% mode: latex |
| 633 | %%% TeX-master: "catacomb" |
| 634 | %%% End: |