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a86e33af | 1 | /* -*-c-*- |
2 | * | |
12ed8a1f | 3 | * $Id$ |
a86e33af | 4 | * |
5 | * Karatsuba's multiplication algorithm | |
6 | * | |
7 | * (c) 1999 Straylight/Edgeware | |
8 | */ | |
9 | ||
45c0fd36 | 10 | /*----- Licensing notice --------------------------------------------------* |
a86e33af | 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. | |
45c0fd36 | 18 | * |
a86e33af | 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. | |
45c0fd36 | 23 | * |
a86e33af | 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 | ||
a86e33af | 30 | /*----- Header files ------------------------------------------------------*/ |
31 | ||
4468424e | 32 | #include <assert.h> |
a86e33af | 33 | #include <stdio.h> |
34 | ||
35 | #include "mpx.h" | |
52cdaca9 | 36 | #include "karatsuba.h" |
a86e33af | 37 | |
38 | /*----- Tweakables --------------------------------------------------------*/ | |
39 | ||
a86e33af | 40 | #ifdef TEST_RIG |
52cdaca9 | 41 | # undef MPK_THRESH |
dd22938e | 42 | # define MPK_THRESH 4 /* Smallest possible correct value */ |
a86e33af | 43 | #endif |
44 | ||
a86e33af | 45 | /*----- Main code ---------------------------------------------------------*/ |
46 | ||
47 | /* --- @mpx_kmul@ --- * | |
48 | * | |
49 | * Arguments: @mpw *dv, *dvl@ = pointer to destination buffer | |
50 | * @const mpw *av, *avl@ = pointer to first argument | |
51 | * @const mpw *bv, *bvl@ = pointer to second argument | |
52 | * @mpw *sv, *svl@ = pointer to scratch workspace | |
53 | * | |
54 | * Returns: --- | |
55 | * | |
56 | * Use: Multiplies two multiprecision integers using Karatsuba's | |
57 | * algorithm. This is rather faster than traditional long | |
58 | * multiplication (e.g., @mpx_umul@) on large numbers, although | |
59 | * more expensive on small ones. | |
60 | * | |
dd22938e | 61 | * The destination must be three times as large as the larger |
62 | * argument. The scratch space must be five times as large as | |
63 | * the larger argument. | |
a86e33af | 64 | */ |
65 | ||
66 | void mpx_kmul(mpw *dv, mpw *dvl, | |
67 | const mpw *av, const mpw *avl, | |
68 | const mpw *bv, const mpw *bvl, | |
69 | mpw *sv, mpw *svl) | |
70 | { | |
71 | const mpw *avm, *bvm; | |
72 | size_t m; | |
73 | ||
74 | /* --- Dispose of easy cases to @mpx_umul@ --- * | |
75 | * | |
76 | * Karatsuba is only a win on large numbers, because of all the | |
77 | * recursiveness and bookkeeping. The recursive calls make a quick check | |
78 | * to see whether to bottom out to @mpx_umul@ which should help quite a | |
79 | * lot, but sometimes the only way to know is to make sure... | |
80 | */ | |
81 | ||
82 | MPX_SHRINK(av, avl); | |
83 | MPX_SHRINK(bv, bvl); | |
84 | ||
52cdaca9 | 85 | if (avl - av <= MPK_THRESH || bvl - bv <= MPK_THRESH) { |
a86e33af | 86 | mpx_umul(dv, dvl, av, avl, bv, bvl); |
87 | return; | |
88 | } | |
89 | ||
90 | /* --- How the algorithm works --- * | |
91 | * | |
7d5fa32a | 92 | * Let %$A = xb + y$% and %$B = ub + v$%. Then, simply by expanding, |
93 | * %$AB = x u b^2 + b(x v + y u) + y v$%. That's not helped any, because | |
94 | * I've got four multiplications, each four times easier than the one I | |
95 | * started with. However, note that I can rewrite the coefficient of %$b$% | |
96 | * as %$xv + yu = (x + y)(u + v) - xu - yv$%. The terms %$xu$% and %$yv$% | |
a86e33af | 97 | * I've already calculated, and that leaves only one more multiplication to |
98 | * do. So now I have three multiplications, each four times easier, and | |
99 | * that's a win. | |
100 | */ | |
101 | ||
102 | /* --- First things --- * | |
103 | * | |
104 | * Sort out where to break the factors in half. I'll choose the midpoint | |
52cdaca9 | 105 | * of the larger one, since this minimizes the amount of work I have to do |
a86e33af | 106 | * most effectively. |
107 | */ | |
108 | ||
109 | if (avl - av > bvl - bv) { | |
110 | m = (avl - av + 1) >> 1; | |
111 | avm = av + m; | |
112 | if (bvl - bv > m) | |
113 | bvm = bv + m; | |
114 | else | |
115 | bvm = bvl; | |
116 | } else { | |
117 | m = (bvl - bv + 1) >> 1; | |
118 | bvm = bv + m; | |
119 | if (avl - av > m) | |
120 | avm = av + m; | |
121 | else | |
122 | avm = avl; | |
123 | } | |
124 | ||
4468424e | 125 | /* --- Sort out the middle term --- */ |
a86e33af | 126 | |
127 | { | |
4468424e | 128 | mpw *bsv = sv + m + 1, *ssv = bsv + m + 1; |
129 | mpw *rdv = dv + m, *rdvl = rdv + 2 * (m + 2); | |
130 | ||
432c4e18 | 131 | assert(rdvl <= dvl); |
132 | assert(ssv <= svl); | |
4468424e | 133 | UADD2(sv, bsv, av, avm, avm, avl); |
134 | UADD2(bsv, ssv, bv, bvm, bvm, bvl); | |
52cdaca9 | 135 | if (m > MPK_THRESH) |
a86e33af | 136 | mpx_kmul(rdv, rdvl, sv, bsv, bsv, ssv, ssv, svl); |
137 | else | |
138 | mpx_umul(rdv, rdvl, sv, bsv, bsv, ssv); | |
a86e33af | 139 | } |
140 | ||
141 | /* --- Sort out the other two terms --- */ | |
142 | ||
143 | { | |
4468424e | 144 | mpw *svm = sv + m, *svn = svm + m, *ssv = svn + 4; |
a86e33af | 145 | mpw *tdv = dv + m; |
146 | mpw *rdv = tdv + m; | |
147 | ||
4468424e | 148 | if (avl == avm || bvl == bvm) |
149 | MPX_ZERO(rdv + m + 1, dvl); | |
150 | else { | |
52cdaca9 | 151 | if (m > MPK_THRESH) |
4468424e | 152 | mpx_kmul(sv, ssv, avm, avl, bvm, bvl, ssv, svl); |
153 | else | |
154 | mpx_umul(sv, ssv, avm, avl, bvm, bvl); | |
155 | MPX_COPY(rdv + m + 1, dvl, svm + 1, svn); | |
156 | UADD(rdv, sv, svm + 1); | |
157 | USUB(tdv, sv, svn); | |
158 | } | |
159 | ||
52cdaca9 | 160 | if (m > MPK_THRESH) |
a86e33af | 161 | mpx_kmul(sv, ssv, av, avm, bv, bvm, ssv, svl); |
162 | else | |
163 | mpx_umul(sv, ssv, av, avm, bv, bvm); | |
4468424e | 164 | MPX_COPY(dv, tdv, sv, svm); |
165 | USUB(tdv, sv, svn); | |
166 | UADD(tdv, svm, svn); | |
a86e33af | 167 | } |
168 | } | |
169 | ||
170 | /*----- Test rig ----------------------------------------------------------*/ | |
171 | ||
172 | #ifdef TEST_RIG | |
173 | ||
174 | #include <mLib/alloc.h> | |
175 | #include <mLib/testrig.h> | |
176 | ||
45c0fd36 MW |
177 | #define ALLOC(v, vl, sz) do { \ |
178 | size_t _sz = (sz); \ | |
179 | mpw *_vv = xmalloc(MPWS(_sz)); \ | |
180 | mpw *_vvl = _vv + _sz; \ | |
181 | (v) = _vv; \ | |
182 | (vl) = _vvl; \ | |
a86e33af | 183 | } while (0) |
184 | ||
45c0fd36 MW |
185 | #define LOAD(v, vl, d) do { \ |
186 | const dstr *_d = (d); \ | |
187 | mpw *_v, *_vl; \ | |
188 | ALLOC(_v, _vl, MPW_RQ(_d->len)); \ | |
189 | mpx_loadb(_v, _vl, _d->buf, _d->len); \ | |
190 | (v) = _v; \ | |
191 | (vl) = _vl; \ | |
a86e33af | 192 | } while (0) |
193 | ||
194 | #define MAX(x, y) ((x) > (y) ? (x) : (y)) | |
195 | ||
196 | static void dumpmp(const char *msg, const mpw *v, const mpw *vl) | |
197 | { | |
198 | fputs(msg, stderr); | |
199 | MPX_SHRINK(v, vl); | |
200 | while (v < vl) | |
201 | fprintf(stderr, " %08lx", (unsigned long)*--vl); | |
202 | fputc('\n', stderr); | |
203 | } | |
204 | ||
205 | static int umul(dstr *v) | |
206 | { | |
207 | mpw *a, *al; | |
208 | mpw *b, *bl; | |
209 | mpw *c, *cl; | |
210 | mpw *d, *dl; | |
211 | mpw *s, *sl; | |
212 | size_t m; | |
213 | int ok = 1; | |
214 | ||
215 | LOAD(a, al, &v[0]); | |
216 | LOAD(b, bl, &v[1]); | |
217 | LOAD(c, cl, &v[2]); | |
218 | m = MAX(al - a, bl - b) + 1; | |
dd22938e | 219 | ALLOC(d, dl, 3 * m); |
220 | ALLOC(s, sl, 5 * m); | |
a86e33af | 221 | |
222 | mpx_kmul(d, dl, a, al, b, bl, s, sl); | |
c9060100 | 223 | if (!mpx_ueq(d, dl, c, cl)) { |
a86e33af | 224 | fprintf(stderr, "\n*** umul failed\n"); |
45c0fd36 MW |
225 | dumpmp(" a", a, al); |
226 | dumpmp(" b", b, bl); | |
a86e33af | 227 | dumpmp("expected", c, cl); |
228 | dumpmp(" result", d, dl); | |
229 | ok = 0; | |
230 | } | |
231 | ||
12ed8a1f | 232 | xfree(a); xfree(b); xfree(c); xfree(d); xfree(s); |
a86e33af | 233 | return (ok); |
234 | } | |
235 | ||
236 | static test_chunk defs[] = { | |
237 | { "umul", umul, { &type_hex, &type_hex, &type_hex, 0 } }, | |
238 | { 0, 0, { 0 } } | |
239 | }; | |
240 | ||
241 | int main(int argc, char *argv[]) | |
242 | { | |
243 | test_run(argc, argv, defs, SRCDIR"/tests/mpx"); | |
244 | return (0); | |
245 | } | |
246 | ||
247 | #endif | |
248 | ||
249 | /*----- That's all, folks -------------------------------------------------*/ |