a86e33af |
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 -------------------------------------------------*/ |