a86e33af |
1 | /* -*-c-*- |
2 | * |
12ed8a1f |
3 | * $Id$ |
a86e33af |
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 | |
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 | |
a86e33af |
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; \ |
183 | } while (0) |
184 | |
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; \ |
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"); |
225 | dumpmp(" a", a, al); |
226 | dumpmp(" b", b, bl); |
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 -------------------------------------------------*/ |