| 1 | /* -*-c-*- |
| 2 | * |
| 3 | * $Id: mpx-kmul.c,v 1.8 2002/10/09 00:36:03 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 1.8 2002/10/09 00:36:03 mdw |
| 34 | * Fix bounds on workspace for Karatsuba operations. |
| 35 | * |
| 36 | * Revision 1.7 2000/10/08 15:48:35 mdw |
| 37 | * Rename Karatsuba constants now that we have @gfx_kmul@ too. |
| 38 | * |
| 39 | * Revision 1.6 2000/10/08 12:11:01 mdw |
| 40 | * Use @mpx_ueq@ instead of @MPX_UCMP@. |
| 41 | * |
| 42 | * Revision 1.5 2000/07/29 17:04:02 mdw |
| 43 | * Remove useless header `mpscan.h'. |
| 44 | * |
| 45 | * Revision 1.4 2000/06/17 11:42:11 mdw |
| 46 | * Moved the Karatsuba macros into a separate file for better sharing. |
| 47 | * Fixed some comments. |
| 48 | * |
| 49 | * Revision 1.3 1999/12/13 15:35:01 mdw |
| 50 | * Simplify and improve. |
| 51 | * |
| 52 | * Revision 1.2 1999/12/11 10:58:02 mdw |
| 53 | * Remove tweakable comments. |
| 54 | * |
| 55 | * Revision 1.1 1999/12/10 23:23:51 mdw |
| 56 | * Karatsuba-Ofman multiplication algorithm. |
| 57 | * |
| 58 | */ |
| 59 | |
| 60 | /*----- Header files ------------------------------------------------------*/ |
| 61 | |
| 62 | #include <assert.h> |
| 63 | #include <stdio.h> |
| 64 | |
| 65 | #include "mpx.h" |
| 66 | #include "karatsuba.h" |
| 67 | |
| 68 | /*----- Tweakables --------------------------------------------------------*/ |
| 69 | |
| 70 | #ifdef TEST_RIG |
| 71 | # undef MPK_THRESH |
| 72 | # define MPK_THRESH 4 /* Smallest possible correct value */ |
| 73 | #endif |
| 74 | |
| 75 | /*----- Main code ---------------------------------------------------------*/ |
| 76 | |
| 77 | /* --- @mpx_kmul@ --- * |
| 78 | * |
| 79 | * Arguments: @mpw *dv, *dvl@ = pointer to destination buffer |
| 80 | * @const mpw *av, *avl@ = pointer to first argument |
| 81 | * @const mpw *bv, *bvl@ = pointer to second argument |
| 82 | * @mpw *sv, *svl@ = pointer to scratch workspace |
| 83 | * |
| 84 | * Returns: --- |
| 85 | * |
| 86 | * Use: Multiplies two multiprecision integers using Karatsuba's |
| 87 | * algorithm. This is rather faster than traditional long |
| 88 | * multiplication (e.g., @mpx_umul@) on large numbers, although |
| 89 | * more expensive on small ones. |
| 90 | * |
| 91 | * The destination must be three times as large as the larger |
| 92 | * argument. The scratch space must be five times as large as |
| 93 | * the larger argument. |
| 94 | */ |
| 95 | |
| 96 | void mpx_kmul(mpw *dv, mpw *dvl, |
| 97 | const mpw *av, const mpw *avl, |
| 98 | const mpw *bv, const mpw *bvl, |
| 99 | mpw *sv, mpw *svl) |
| 100 | { |
| 101 | const mpw *avm, *bvm; |
| 102 | size_t m; |
| 103 | |
| 104 | /* --- Dispose of easy cases to @mpx_umul@ --- * |
| 105 | * |
| 106 | * Karatsuba is only a win on large numbers, because of all the |
| 107 | * recursiveness and bookkeeping. The recursive calls make a quick check |
| 108 | * to see whether to bottom out to @mpx_umul@ which should help quite a |
| 109 | * lot, but sometimes the only way to know is to make sure... |
| 110 | */ |
| 111 | |
| 112 | MPX_SHRINK(av, avl); |
| 113 | MPX_SHRINK(bv, bvl); |
| 114 | |
| 115 | if (avl - av <= MPK_THRESH || bvl - bv <= MPK_THRESH) { |
| 116 | mpx_umul(dv, dvl, av, avl, bv, bvl); |
| 117 | return; |
| 118 | } |
| 119 | |
| 120 | /* --- How the algorithm works --- * |
| 121 | * |
| 122 | * Let %$A = xb + y$% and %$B = ub + v$%. Then, simply by expanding, |
| 123 | * %$AB = x u b^2 + b(x v + y u) + y v$%. That's not helped any, because |
| 124 | * I've got four multiplications, each four times easier than the one I |
| 125 | * started with. However, note that I can rewrite the coefficient of %$b$% |
| 126 | * as %$xv + yu = (x + y)(u + v) - xu - yv$%. The terms %$xu$% and %$yv$% |
| 127 | * I've already calculated, and that leaves only one more multiplication to |
| 128 | * do. So now I have three multiplications, each four times easier, and |
| 129 | * that's a win. |
| 130 | */ |
| 131 | |
| 132 | /* --- First things --- * |
| 133 | * |
| 134 | * Sort out where to break the factors in half. I'll choose the midpoint |
| 135 | * of the larger one, since this minimizes the amount of work I have to do |
| 136 | * most effectively. |
| 137 | */ |
| 138 | |
| 139 | if (avl - av > bvl - bv) { |
| 140 | m = (avl - av + 1) >> 1; |
| 141 | avm = av + m; |
| 142 | if (bvl - bv > m) |
| 143 | bvm = bv + m; |
| 144 | else |
| 145 | bvm = bvl; |
| 146 | } else { |
| 147 | m = (bvl - bv + 1) >> 1; |
| 148 | bvm = bv + m; |
| 149 | if (avl - av > m) |
| 150 | avm = av + m; |
| 151 | else |
| 152 | avm = avl; |
| 153 | } |
| 154 | |
| 155 | /* --- Sort out the middle term --- */ |
| 156 | |
| 157 | { |
| 158 | mpw *bsv = sv + m + 1, *ssv = bsv + m + 1; |
| 159 | mpw *rdv = dv + m, *rdvl = rdv + 2 * (m + 2); |
| 160 | |
| 161 | assert(rdvl < dvl); |
| 162 | assert(ssv < svl); |
| 163 | UADD2(sv, bsv, av, avm, avm, avl); |
| 164 | UADD2(bsv, ssv, bv, bvm, bvm, bvl); |
| 165 | if (m > MPK_THRESH) |
| 166 | mpx_kmul(rdv, rdvl, sv, bsv, bsv, ssv, ssv, svl); |
| 167 | else |
| 168 | mpx_umul(rdv, rdvl, sv, bsv, bsv, ssv); |
| 169 | } |
| 170 | |
| 171 | /* --- Sort out the other two terms --- */ |
| 172 | |
| 173 | { |
| 174 | mpw *svm = sv + m, *svn = svm + m, *ssv = svn + 4; |
| 175 | mpw *tdv = dv + m; |
| 176 | mpw *rdv = tdv + m; |
| 177 | |
| 178 | if (avl == avm || bvl == bvm) |
| 179 | MPX_ZERO(rdv + m + 1, dvl); |
| 180 | else { |
| 181 | if (m > MPK_THRESH) |
| 182 | mpx_kmul(sv, ssv, avm, avl, bvm, bvl, ssv, svl); |
| 183 | else |
| 184 | mpx_umul(sv, ssv, avm, avl, bvm, bvl); |
| 185 | MPX_COPY(rdv + m + 1, dvl, svm + 1, svn); |
| 186 | UADD(rdv, sv, svm + 1); |
| 187 | USUB(tdv, sv, svn); |
| 188 | } |
| 189 | |
| 190 | if (m > MPK_THRESH) |
| 191 | mpx_kmul(sv, ssv, av, avm, bv, bvm, ssv, svl); |
| 192 | else |
| 193 | mpx_umul(sv, ssv, av, avm, bv, bvm); |
| 194 | MPX_COPY(dv, tdv, sv, svm); |
| 195 | USUB(tdv, sv, svn); |
| 196 | UADD(tdv, svm, svn); |
| 197 | } |
| 198 | } |
| 199 | |
| 200 | /*----- Test rig ----------------------------------------------------------*/ |
| 201 | |
| 202 | #ifdef TEST_RIG |
| 203 | |
| 204 | #include <mLib/alloc.h> |
| 205 | #include <mLib/testrig.h> |
| 206 | |
| 207 | #define ALLOC(v, vl, sz) do { \ |
| 208 | size_t _sz = (sz); \ |
| 209 | mpw *_vv = xmalloc(MPWS(_sz)); \ |
| 210 | mpw *_vvl = _vv + _sz; \ |
| 211 | (v) = _vv; \ |
| 212 | (vl) = _vvl; \ |
| 213 | } while (0) |
| 214 | |
| 215 | #define LOAD(v, vl, d) do { \ |
| 216 | const dstr *_d = (d); \ |
| 217 | mpw *_v, *_vl; \ |
| 218 | ALLOC(_v, _vl, MPW_RQ(_d->len)); \ |
| 219 | mpx_loadb(_v, _vl, _d->buf, _d->len); \ |
| 220 | (v) = _v; \ |
| 221 | (vl) = _vl; \ |
| 222 | } while (0) |
| 223 | |
| 224 | #define MAX(x, y) ((x) > (y) ? (x) : (y)) |
| 225 | |
| 226 | static void dumpmp(const char *msg, const mpw *v, const mpw *vl) |
| 227 | { |
| 228 | fputs(msg, stderr); |
| 229 | MPX_SHRINK(v, vl); |
| 230 | while (v < vl) |
| 231 | fprintf(stderr, " %08lx", (unsigned long)*--vl); |
| 232 | fputc('\n', stderr); |
| 233 | } |
| 234 | |
| 235 | static int umul(dstr *v) |
| 236 | { |
| 237 | mpw *a, *al; |
| 238 | mpw *b, *bl; |
| 239 | mpw *c, *cl; |
| 240 | mpw *d, *dl; |
| 241 | mpw *s, *sl; |
| 242 | size_t m; |
| 243 | int ok = 1; |
| 244 | |
| 245 | LOAD(a, al, &v[0]); |
| 246 | LOAD(b, bl, &v[1]); |
| 247 | LOAD(c, cl, &v[2]); |
| 248 | m = MAX(al - a, bl - b) + 1; |
| 249 | ALLOC(d, dl, 3 * m); |
| 250 | ALLOC(s, sl, 5 * m); |
| 251 | |
| 252 | mpx_kmul(d, dl, a, al, b, bl, s, sl); |
| 253 | if (!mpx_ueq(d, dl, c, cl)) { |
| 254 | fprintf(stderr, "\n*** umul failed\n"); |
| 255 | dumpmp(" a", a, al); |
| 256 | dumpmp(" b", b, bl); |
| 257 | dumpmp("expected", c, cl); |
| 258 | dumpmp(" result", d, dl); |
| 259 | ok = 0; |
| 260 | } |
| 261 | |
| 262 | free(a); free(b); free(c); free(d); free(s); |
| 263 | return (ok); |
| 264 | } |
| 265 | |
| 266 | static test_chunk defs[] = { |
| 267 | { "umul", umul, { &type_hex, &type_hex, &type_hex, 0 } }, |
| 268 | { 0, 0, { 0 } } |
| 269 | }; |
| 270 | |
| 271 | int main(int argc, char *argv[]) |
| 272 | { |
| 273 | test_run(argc, argv, defs, SRCDIR"/tests/mpx"); |
| 274 | return (0); |
| 275 | } |
| 276 | |
| 277 | #endif |
| 278 | |
| 279 | /*----- That's all, folks -------------------------------------------------*/ |