| 1 | /* -*-c-*- |
| 2 | * |
| 3 | * $Id: mpx-kmul.c,v 1.1 1999/12/10 23:23:51 mdw Exp $ |
| 4 | * |
| 5 | * Karatsuba's multiplication algorithm |
| 6 | * |
| 7 | * (c) 1999 Straylight/Edgeware |
| 8 | */ |
| 9 | |
| 10 | /*----- Licensing notice --------------------------------------------------* |
| 11 | * |
| 12 | * This file is part of Catacomb. |
| 13 | * |
| 14 | * Catacomb is free software; you can redistribute it and/or modify |
| 15 | * it under the terms of the GNU Library General Public License as |
| 16 | * published by the Free Software Foundation; either version 2 of the |
| 17 | * License, or (at your option) any later version. |
| 18 | * |
| 19 | * Catacomb is distributed in the hope that it will be useful, |
| 20 | * but WITHOUT ANY WARRANTY; without even the implied warranty of |
| 21 | * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
| 22 | * GNU Library General Public License for more details. |
| 23 | * |
| 24 | * You should have received a copy of the GNU Library General Public |
| 25 | * License along with Catacomb; if not, write to the Free |
| 26 | * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, |
| 27 | * MA 02111-1307, USA. |
| 28 | */ |
| 29 | |
| 30 | /*----- Revision history --------------------------------------------------* |
| 31 | * |
| 32 | * $Log: mpx-kmul.c,v $ |
| 33 | * Revision 1.1 1999/12/10 23:23:51 mdw |
| 34 | * Karatsuba-Ofman multiplication algorithm. |
| 35 | * |
| 36 | */ |
| 37 | |
| 38 | /*----- Header files ------------------------------------------------------*/ |
| 39 | |
| 40 | #include <stdio.h> |
| 41 | |
| 42 | #include "mpx.h" |
| 43 | |
| 44 | /*----- Tweakables --------------------------------------------------------*/ |
| 45 | |
| 46 | /* --- @KARATSUBA_CUTOFF@ --- * |
| 47 | * |
| 48 | * If either of the arguments to @mpx_kmul@ contains this number of words or |
| 49 | * fewer, the job is dumped out to @mpx_umul@ instead. Reduce the size when |
| 50 | * testing, to ensure better coverage. |
| 51 | */ |
| 52 | |
| 53 | #ifdef TEST_RIG |
| 54 | # undef KARATSUBA_CUTOFF |
| 55 | # define KARATSUBA_CUTOFF 2 |
| 56 | #endif |
| 57 | |
| 58 | /*----- Addition macros ---------------------------------------------------*/ |
| 59 | |
| 60 | #define UADD(dv, av, avl) do { \ |
| 61 | mpw *_dv = (dv); \ |
| 62 | const mpw *_av = (av), *_avl = (avl); \ |
| 63 | mpw _c = 0; \ |
| 64 | \ |
| 65 | while (_av < _avl) { \ |
| 66 | mpw _a, _b; \ |
| 67 | mpd _x; \ |
| 68 | _a = *_av++; \ |
| 69 | _b = *_dv; \ |
| 70 | _x = (mpd)_a + (mpd)_b + _c; \ |
| 71 | *_dv++ = MPW(_x); \ |
| 72 | _c = _x >> MPW_BITS; \ |
| 73 | } \ |
| 74 | while (_c) { \ |
| 75 | mpd _x = (mpd)*_dv + (mpd)_c; \ |
| 76 | *_dv++ = MPW(_x); \ |
| 77 | _c = _x >> MPW_BITS; \ |
| 78 | } \ |
| 79 | } while (0) |
| 80 | |
| 81 | #define UADD2(dv, dvl, av, avl, bv, bvl) do { \ |
| 82 | mpw *_dv = (dv), *_dvl = (dvl); \ |
| 83 | const mpw *_av = (av), *_avl = (avl); \ |
| 84 | const mpw *_bv = (bv), *_bvl = (bvl); \ |
| 85 | mpw _c = 0; \ |
| 86 | \ |
| 87 | while (_av < _avl || _bv < _bvl) { \ |
| 88 | mpw _a, _b; \ |
| 89 | mpd _x; \ |
| 90 | _a = (_av < _avl) ? *_av++ : 0; \ |
| 91 | _b = (_bv < _bvl) ? *_bv++ : 0; \ |
| 92 | _x = (mpd)_a + (mpd)_b + _c; \ |
| 93 | *_dv++ = MPW(_x); \ |
| 94 | _c = _x >> MPW_BITS; \ |
| 95 | } \ |
| 96 | *_dv++ = _c; \ |
| 97 | while (_dv < _dvl) \ |
| 98 | *_dv++ = 0; \ |
| 99 | } while (0) |
| 100 | |
| 101 | #define USUB(dv, av, avl) do { \ |
| 102 | mpw *_dv = (dv); \ |
| 103 | const mpw *_av = (av), *_avl = (avl); \ |
| 104 | mpw _c = 0; \ |
| 105 | \ |
| 106 | while (_av < _avl) { \ |
| 107 | mpw _a, _b; \ |
| 108 | mpd _x; \ |
| 109 | _a = *_av++; \ |
| 110 | _b = *_dv; \ |
| 111 | _x = (mpd)_b - (mpd)_a - _c; \ |
| 112 | *_dv++ = MPW(_x); \ |
| 113 | if (_x >> MPW_BITS) \ |
| 114 | _c = 1; \ |
| 115 | else \ |
| 116 | _c = 0; \ |
| 117 | } \ |
| 118 | while (_c) { \ |
| 119 | mpd _x = (mpd)*_dv - (mpd)_c; \ |
| 120 | *_dv++ = MPW(_x); \ |
| 121 | if (_x >> MPW_BITS) \ |
| 122 | _c = 1; \ |
| 123 | else \ |
| 124 | _c = 0; \ |
| 125 | } \ |
| 126 | } while (0) |
| 127 | |
| 128 | /*----- Main code ---------------------------------------------------------*/ |
| 129 | |
| 130 | /* --- @mpx_kmul@ --- * |
| 131 | * |
| 132 | * Arguments: @mpw *dv, *dvl@ = pointer to destination buffer |
| 133 | * @const mpw *av, *avl@ = pointer to first argument |
| 134 | * @const mpw *bv, *bvl@ = pointer to second argument |
| 135 | * @mpw *sv, *svl@ = pointer to scratch workspace |
| 136 | * |
| 137 | * Returns: --- |
| 138 | * |
| 139 | * Use: Multiplies two multiprecision integers using Karatsuba's |
| 140 | * algorithm. This is rather faster than traditional long |
| 141 | * multiplication (e.g., @mpx_umul@) on large numbers, although |
| 142 | * more expensive on small ones. |
| 143 | * |
| 144 | * The destination must be twice as large as the larger |
| 145 | * argument. The scratch space must be twice as large as the |
| 146 | * larger argument, plus the magic number @KARATSUBA_SLOP@. |
| 147 | * (Actually, a number of words proportional to the depth of |
| 148 | * recursion, but since recusion is strongly bounded by memory, |
| 149 | * I can replace it with a constant as long as it's `big |
| 150 | * enough'.) |
| 151 | */ |
| 152 | |
| 153 | void mpx_kmul(mpw *dv, mpw *dvl, |
| 154 | const mpw *av, const mpw *avl, |
| 155 | const mpw *bv, const mpw *bvl, |
| 156 | mpw *sv, mpw *svl) |
| 157 | { |
| 158 | const mpw *avm, *bvm; |
| 159 | size_t m; |
| 160 | |
| 161 | /* --- Dispose of easy cases to @mpx_umul@ --- * |
| 162 | * |
| 163 | * Karatsuba is only a win on large numbers, because of all the |
| 164 | * recursiveness and bookkeeping. The recursive calls make a quick check |
| 165 | * to see whether to bottom out to @mpx_umul@ which should help quite a |
| 166 | * lot, but sometimes the only way to know is to make sure... |
| 167 | */ |
| 168 | |
| 169 | MPX_SHRINK(av, avl); |
| 170 | MPX_SHRINK(bv, bvl); |
| 171 | |
| 172 | if (avl - av <= KARATSUBA_CUTOFF || bvl - bv <= KARATSUBA_CUTOFF) { |
| 173 | mpx_umul(dv, dvl, av, avl, bv, bvl); |
| 174 | return; |
| 175 | } |
| 176 | |
| 177 | /* --- How the algorithm works --- * |
| 178 | * |
| 179 | * Let %$A = xb + y$% and %$B = ub + v$%. Then, simply by expanding, %$AB |
| 180 | * = x u b^2 + b(x v + y u) + y v$%. That's not helped any, because I've |
| 181 | * got four multiplications, each four times easier than the one I started |
| 182 | * with. However, note that I can rewrite the coefficient of %$b$% as |
| 183 | * %$xv + yu = (x + y)(u + v) - xu - yv$%. The terms %$xu$% and %$yv$% |
| 184 | * I've already calculated, and that leaves only one more multiplication to |
| 185 | * do. So now I have three multiplications, each four times easier, and |
| 186 | * that's a win. |
| 187 | */ |
| 188 | |
| 189 | /* --- First things --- * |
| 190 | * |
| 191 | * Sort out where to break the factors in half. I'll choose the midpoint |
| 192 | * of the largest one, since this minimizes the amount of work I have to do |
| 193 | * most effectively. |
| 194 | */ |
| 195 | |
| 196 | if (avl - av > bvl - bv) { |
| 197 | m = (avl - av + 1) >> 1; |
| 198 | avm = av + m; |
| 199 | if (bvl - bv > m) |
| 200 | bvm = bv + m; |
| 201 | else |
| 202 | bvm = bvl; |
| 203 | } else { |
| 204 | m = (bvl - bv + 1) >> 1; |
| 205 | bvm = bv + m; |
| 206 | if (avl - av > m) |
| 207 | avm = av + m; |
| 208 | else |
| 209 | avm = avl; |
| 210 | } |
| 211 | |
| 212 | /* --- Sort out the middle term --- * |
| 213 | * |
| 214 | * I'm going to keep track of the carry by hand rather than pass it down to |
| 215 | * the next level, because it means multiplication by one or zero, which I |
| 216 | * can do easily myself. |
| 217 | */ |
| 218 | |
| 219 | { |
| 220 | unsigned f = 0; |
| 221 | enum { |
| 222 | carry_a = 1, |
| 223 | carry_b = 2 |
| 224 | }; |
| 225 | |
| 226 | mpw *bsv = sv + m, *ssv = bsv + m; |
| 227 | mpw *rdv = dv + m, *rdvl = rdv + 2 * m; |
| 228 | |
| 229 | UADD2(sv, bsv + 1, av, avm, avm, avl); |
| 230 | if (*bsv) |
| 231 | f |= carry_a; |
| 232 | UADD2(bsv, ssv + 1, bv, bvm, bvm, bvl); |
| 233 | if (*ssv) |
| 234 | f |= carry_b; |
| 235 | MPX_ZERO(dv, rdv); |
| 236 | if (m > KARATSUBA_CUTOFF) |
| 237 | mpx_kmul(rdv, rdvl, sv, bsv, bsv, ssv, ssv, svl); |
| 238 | else |
| 239 | mpx_umul(rdv, rdvl, sv, bsv, bsv, ssv); |
| 240 | MPX_ZERO(rdvl, dvl); |
| 241 | rdv += m; rdvl += m; |
| 242 | if (f & carry_b) |
| 243 | UADD(rdv, sv, bsv); |
| 244 | if (f & carry_a) |
| 245 | UADD(rdv, bsv, ssv); |
| 246 | if (!(~f & (carry_a | carry_b))) |
| 247 | MPX_UADDN(rdv + m, rdvl, 1); |
| 248 | } |
| 249 | |
| 250 | /* --- Sort out the other two terms --- */ |
| 251 | |
| 252 | { |
| 253 | mpw *ssv = sv + 2 * m; |
| 254 | mpw *tdv = dv + m; |
| 255 | mpw *rdv = tdv + m; |
| 256 | |
| 257 | if (m > KARATSUBA_CUTOFF) |
| 258 | mpx_kmul(sv, ssv, avm, avl, bvm, bvl, ssv, svl); |
| 259 | else |
| 260 | mpx_umul(sv, ssv, avm, avl, bvm, bvl); |
| 261 | UADD(rdv, sv, ssv); |
| 262 | USUB(tdv, sv, ssv); |
| 263 | |
| 264 | if (m > KARATSUBA_CUTOFF) |
| 265 | mpx_kmul(sv, ssv, av, avm, bv, bvm, ssv, svl); |
| 266 | else |
| 267 | mpx_umul(sv, ssv, av, avm, bv, bvm); |
| 268 | USUB(tdv, sv, ssv); |
| 269 | UADD(dv, sv, ssv); |
| 270 | } |
| 271 | } |
| 272 | |
| 273 | /*----- Test rig ----------------------------------------------------------*/ |
| 274 | |
| 275 | #ifdef TEST_RIG |
| 276 | |
| 277 | #include <mLib/alloc.h> |
| 278 | #include <mLib/testrig.h> |
| 279 | |
| 280 | #include "mpscan.h" |
| 281 | |
| 282 | #define ALLOC(v, vl, sz) do { \ |
| 283 | size_t _sz = (sz); \ |
| 284 | mpw *_vv = xmalloc(MPWS(_sz)); \ |
| 285 | mpw *_vvl = _vv + _sz; \ |
| 286 | (v) = _vv; \ |
| 287 | (vl) = _vvl; \ |
| 288 | } while (0) |
| 289 | |
| 290 | #define LOAD(v, vl, d) do { \ |
| 291 | const dstr *_d = (d); \ |
| 292 | mpw *_v, *_vl; \ |
| 293 | ALLOC(_v, _vl, MPW_RQ(_d->len)); \ |
| 294 | mpx_loadb(_v, _vl, _d->buf, _d->len); \ |
| 295 | (v) = _v; \ |
| 296 | (vl) = _vl; \ |
| 297 | } while (0) |
| 298 | |
| 299 | #define MAX(x, y) ((x) > (y) ? (x) : (y)) |
| 300 | |
| 301 | static void dumpmp(const char *msg, const mpw *v, const mpw *vl) |
| 302 | { |
| 303 | fputs(msg, stderr); |
| 304 | MPX_SHRINK(v, vl); |
| 305 | while (v < vl) |
| 306 | fprintf(stderr, " %08lx", (unsigned long)*--vl); |
| 307 | fputc('\n', stderr); |
| 308 | } |
| 309 | |
| 310 | static int umul(dstr *v) |
| 311 | { |
| 312 | mpw *a, *al; |
| 313 | mpw *b, *bl; |
| 314 | mpw *c, *cl; |
| 315 | mpw *d, *dl; |
| 316 | mpw *s, *sl; |
| 317 | size_t m; |
| 318 | int ok = 1; |
| 319 | |
| 320 | LOAD(a, al, &v[0]); |
| 321 | LOAD(b, bl, &v[1]); |
| 322 | LOAD(c, cl, &v[2]); |
| 323 | m = MAX(al - a, bl - b) + 1; |
| 324 | ALLOC(d, dl, 2 * m); |
| 325 | ALLOC(s, sl, 2 * m + 32); |
| 326 | |
| 327 | mpx_kmul(d, dl, a, al, b, bl, s, sl); |
| 328 | if (MPX_UCMP(d, dl, !=, c, cl)) { |
| 329 | fprintf(stderr, "\n*** umul failed\n"); |
| 330 | dumpmp(" a", a, al); |
| 331 | dumpmp(" b", b, bl); |
| 332 | dumpmp("expected", c, cl); |
| 333 | dumpmp(" result", d, dl); |
| 334 | ok = 0; |
| 335 | } |
| 336 | |
| 337 | free(a); free(b); free(c); free(d); free(s); |
| 338 | return (ok); |
| 339 | } |
| 340 | |
| 341 | static test_chunk defs[] = { |
| 342 | { "umul", umul, { &type_hex, &type_hex, &type_hex, 0 } }, |
| 343 | { 0, 0, { 0 } } |
| 344 | }; |
| 345 | |
| 346 | int main(int argc, char *argv[]) |
| 347 | { |
| 348 | test_run(argc, argv, defs, SRCDIR"/tests/mpx"); |
| 349 | return (0); |
| 350 | } |
| 351 | |
| 352 | #endif |
| 353 | |
| 354 | /*----- That's all, folks -------------------------------------------------*/ |