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[u/mdw/catacomb] / gfreduce.c
1 /* -*-c-*-
2 *
3 * $Id: gfreduce.c,v 1.4 2004/03/27 00:04:46 mdw Exp $
4 *
5 * Efficient reduction modulo sparse binary polynomials
6 *
7 * (c) 2004 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: gfreduce.c,v $
33 * Revision 1.4 2004/03/27 00:04:46 mdw
34 * Implement efficient reduction for pleasant-looking primes.
35 *
36 * Revision 1.3 2004/03/23 15:19:32 mdw
37 * Test elliptic curves more thoroughly.
38 *
39 * Revision 1.2 2004/03/21 22:52:06 mdw
40 * Merge and close elliptic curve branch.
41 *
42 * Revision 1.1.2.1 2004/03/21 22:39:46 mdw
43 * Elliptic curves on binary fields work.
44 *
45 */
46
47 /*----- Header files ------------------------------------------------------*/
48
49 #include <mLib/alloc.h>
50 #include <mLib/darray.h>
51 #include <mLib/macros.h>
52
53 #include "gf.h"
54 #include "gfreduce.h"
55 #include "gfreduce-exp.h"
56 #include "fibrand.h"
57 #include "mprand.h"
58
59 /*----- Data structures ---------------------------------------------------*/
60
61 DA_DECL(instr_v, gfreduce_instr);
62
63 /*----- Main code ---------------------------------------------------------*/
64
65 /* --- What's going on here? --- *
66 *
67 * Let's face it, @gfx_div@ sucks. It works (I hope), but it's not in any
68 * sense fast. Here, we do efficient reduction modulo sparse polynomials.
69 *
70 * Suppose we have a polynomial @X@ we're trying to reduce mod @P@. If we
71 * take the topmost nonzero word of @X@, call it @w@, then we can eliminate
72 * it by subtracting off @w P x^{k}@ for an appropriate value of @k@. The
73 * trick is in observing that if @P@ is sparse we can do this multiplication
74 * and subtraction efficiently, just by XORing appropriate shifts of @w@ into
75 * @X@.
76 *
77 * The first tricky bit is in working out when to stop. I'll use eight-bit
78 * words to demonstrate what I'm talking about.
79 *
80 * xxxxxx xxxxxxxx xxxxxxxx xxxxxxxx xxxxxxxx xxxxxxxx
81 * 001ppppp pppppppp pppppppp pppppppp
82 * |<rp>|
83 * |<------------ bp ------------->|
84 * |<------------ nw --------------->|
85 *
86 * The trick of taking whole words off of @X@ stops working when there are
87 * only @nw@ words left. Then we have to mask off the bottom bits of @w@
88 * before continuing.
89 */
90
91 /* --- @gfreduce_create@ --- *
92 *
93 * Arguments: @gfreduce *r@ = structure to fill in
94 * @mp *x@ = a (hopefully sparse) polynomial
95 *
96 * Returns: ---
97 *
98 * Use: Initializes a context structure for reduction.
99 */
100
101 void gfreduce_create(gfreduce *r, mp *p)
102 {
103 instr_v iv = DA_INIT;
104 unsigned long d;
105 unsigned dw;
106 mpscan sc;
107 unsigned long i;
108 gfreduce_instr *ip;
109 unsigned f = 0;
110 size_t w, ww, wi, wl, ll;
111
112 /* --- Sort out the easy stuff --- */
113
114 d = mp_bits(p); assert(d); d--;
115 r->lim = d/MPW_BITS;
116 dw = d%MPW_BITS;
117 if (!dw)
118 r->mask = 0;
119 else {
120 r->mask = MPW(((mpw)-1) << dw);
121 r->lim++;
122 }
123 r->p = mp_copy(p);
124
125 /* --- Stash a new instruction --- */
126
127 #define INSTR(op_, arg_) do { \
128 DA_ENSURE(&iv, 1); \
129 DA(&iv)[DA_LEN(&iv)].op = (op_); \
130 DA(&iv)[DA_LEN(&iv)].arg = (arg_); \
131 DA_EXTEND(&iv, 1); \
132 } while (0)
133
134 #define f_lsr 1u
135
136 w = (d + MPW_BITS - 1)/MPW_BITS;
137 INSTR(GFRI_LOAD, w);
138 wi = DA_LEN(&iv);
139 f = 0;
140 ll = 0;
141 for (i = 0, mp_scan(&sc, p); mp_step(&sc) && i < d; i++) {
142 if (!mp_bit(&sc))
143 continue;
144 ww = (d - i + MPW_BITS - 1)/MPW_BITS;
145 if (ww != w) {
146 wl = DA_LEN(&iv);
147 INSTR(GFRI_STORE, w);
148 if (!ll)
149 ll = DA_LEN(&iv);
150 if (!(f & f_lsr))
151 INSTR(GFRI_LOAD, ww);
152 else {
153 INSTR(GFRI_LOAD, w - 1);
154 for (; wi < wl; wi++) {
155 ip = &DA(&iv)[wi];
156 assert(ip->op == GFRI_LSL);
157 if (ip->arg)
158 INSTR(GFRI_LSR, MPW_BITS - ip->arg);
159 }
160 if (w - 1 != ww) {
161 INSTR(GFRI_STORE, w - 1);
162 INSTR(GFRI_LOAD, ww);
163 }
164 f &= ~f_lsr;
165 }
166 w = ww;
167 wi = DA_LEN(&iv);
168 }
169 INSTR(GFRI_LSL, (MPW_BITS + i - d)%MPW_BITS);
170 if ((MPW_BITS + i - d)%MPW_BITS)
171 f |= f_lsr;
172 }
173 wl = DA_LEN(&iv);
174 INSTR(GFRI_STORE, w);
175 if (!ll)
176 ll = DA_LEN(&iv);
177 if (f & f_lsr) {
178 INSTR(GFRI_LOAD, w - 1);
179 for (; wi < wl; wi++) {
180 ip = &DA(&iv)[wi];
181 assert(ip->op == GFRI_LSL);
182 if (ip->arg)
183 INSTR(GFRI_LSR, MPW_BITS - ip->arg);
184 }
185 INSTR(GFRI_STORE, w - 1);
186 }
187
188 #undef INSTR
189
190 r->in = DA_LEN(&iv);
191 r->iv = xmalloc(r->in * sizeof(gfreduce_instr));
192 r->liv = r->iv + ll;
193 memcpy(r->iv, DA(&iv), r->in * sizeof(gfreduce_instr));
194 DA_DESTROY(&iv);
195 }
196
197 /* --- @gfreduce_destroy@ --- *
198 *
199 * Arguments: @gfreduce *r@ = structure to free
200 *
201 * Returns: ---
202 *
203 * Use: Reclaims the resources from a reduction context.
204 */
205
206 void gfreduce_destroy(gfreduce *r)
207 {
208 mp_drop(r->p);
209 xfree(r->iv);
210 }
211
212 /* --- @gfreduce_dump@ --- *
213 *
214 * Arguments: @gfreduce *r@ = structure to dump
215 * @FILE *fp@ = file to dump on
216 *
217 * Returns: ---
218 *
219 * Use: Dumps a reduction context.
220 */
221
222 void gfreduce_dump(gfreduce *r, FILE *fp)
223 {
224 size_t i;
225
226 fprintf(fp, "poly = "); mp_writefile(r->p, fp, 16);
227 fprintf(fp, "\n lim = %lu; mask = %lx\n",
228 (unsigned long)r->lim, (unsigned long)r->mask);
229 for (i = 0; i < r->in; i++) {
230 static const char *opname[] = { "load", "lsl", "lsr", "store" };
231 assert(r->iv[i].op < N(opname));
232 fprintf(fp, " %s %lu\n",
233 opname[r->iv[i].op],
234 (unsigned long)r->iv[i].arg);
235 }
236 }
237
238 /* --- @gfreduce_do@ --- *
239 *
240 * Arguments: @gfreduce *r@ = reduction context
241 * @mp *d@ = destination
242 * @mp *x@ = source
243 *
244 * Returns: Destination, @x@ reduced modulo the reduction poly.
245 */
246
247 static void run(const gfreduce_instr *i, const gfreduce_instr *il,
248 mpw *v, mpw z)
249 {
250 mpw w = 0;
251
252 for (; i < il; i++) {
253 switch (i->op) {
254 case GFRI_LOAD: w = *(v - i->arg); break;
255 case GFRI_LSL: w ^= z << i->arg; break;
256 case GFRI_LSR: w ^= z >> i->arg; break;
257 case GFRI_STORE: *(v - i->arg) = MPW(w); break;
258 default: abort();
259 }
260 }
261 }
262
263 mp *gfreduce_do(gfreduce *r, mp *d, mp *x)
264 {
265 mpw *v, *vl;
266 const gfreduce_instr *il;
267 mpw z;
268
269 /* --- Try to reuse the source's space --- */
270
271 MP_COPY(x);
272 if (d) MP_DROP(d);
273 MP_DEST(x, MP_LEN(x), x->f);
274
275 /* --- Do the reduction --- */
276
277 il = r->iv + r->in;
278 if (MP_LEN(x) >= r->lim) {
279 v = x->v + r->lim;
280 vl = x->vl;
281 while (vl-- > v) {
282 while (*vl) {
283 z = *vl;
284 *vl = 0;
285 run(r->iv, il, vl, z);
286 }
287 }
288 if (r->mask) {
289 while (*vl & r->mask) {
290 z = *vl & r->mask;
291 *vl &= ~r->mask;
292 run(r->iv, il, vl, z);
293 }
294 }
295 }
296
297 /* --- Done --- */
298
299 MP_SHRINK(x);
300 return (x);
301 }
302
303 /* --- @gfreduce_sqrt@ --- *
304 *
305 * Arguments: @gfreduce *r@ = pointer to reduction context
306 * @mp *d@ = destination
307 * @mp *x@ = some polynomial
308 *
309 * Returns: The square root of @x@ modulo @r->p@, or null.
310 */
311
312 mp *gfreduce_sqrt(gfreduce *r, mp *d, mp *x)
313 {
314 mp *y = MP_COPY(x);
315 mp *z, *spare = MP_NEW;
316 unsigned long m = mp_bits(r->p) - 1;
317 unsigned long i;
318
319 for (i = 0; i < m - 1; i++) {
320 mp *t = gf_sqr(spare, y);
321 spare = y;
322 y = gfreduce_do(r, t, t);
323 }
324 z = gf_sqr(spare, y);
325 z = gfreduce_do(r, z, z);
326 if (!MP_EQ(x, z)) {
327 mp_drop(y);
328 y = 0;
329 }
330 mp_drop(z);
331 mp_drop(d);
332 return (y);
333 }
334
335 /* --- @gfreduce_trace@ --- *
336 *
337 * Arguments: @gfreduce *r@ = pointer to reduction context
338 * @mp *x@ = some polynomial
339 *
340 * Returns: The trace of @x@. (%$\Tr(x)=x + x^2 + \cdots + x^{2^{m-1}}$%
341 * if %$x \in \gf{2^m}$%).
342 */
343
344 int gfreduce_trace(gfreduce *r, mp *x)
345 {
346 mp *y = MP_COPY(x);
347 mp *spare = MP_NEW;
348 unsigned long m = mp_bits(r->p) - 1;
349 unsigned long i;
350 int rc;
351
352 for (i = 0; i < m - 1; i++) {
353 mp *t = gf_sqr(spare, y);
354 spare = y;
355 y = gfreduce_do(r, t, t);
356 y = gf_add(y, y, x);
357 }
358 rc = !MP_ISZERO(y);
359 mp_drop(spare);
360 mp_drop(y);
361 return (rc);
362 }
363
364 /* --- @gfreduce_halftrace@ --- *
365 *
366 * Arguments: @gfreduce *r@ = pointer to reduction context
367 * @mp *d@ = destination
368 * @mp *x@ = some polynomial
369 *
370 * Returns: The half-trace of @x@.
371 * (%$\HfTr(x)= x + x^{2^2} + \cdots + x^{2^{m-1}}$%
372 * if %$x \in \gf{2^m}$% with %$m$% odd).
373 */
374
375 mp *gfreduce_halftrace(gfreduce *r, mp *d, mp *x)
376 {
377 mp *y = MP_COPY(x);
378 mp *spare = MP_NEW;
379 unsigned long m = mp_bits(r->p) - 1;
380 unsigned long i;
381
382 mp_drop(d);
383 for (i = 0; i < m - 1; i += 2) {
384 mp *t = gf_sqr(spare, y);
385 spare = y;
386 y = gfreduce_do(r, t, t);
387 t = gf_sqr(spare, y);
388 spare = y;
389 y = gfreduce_do(r, t, t);
390 y = gf_add(y, y, x);
391 }
392 mp_drop(spare);
393 return (y);
394 }
395
396 /* --- @gfreduce_quadsolve@ --- *
397 *
398 * Arguments: @gfreduce *r@ = pointer to reduction context
399 * @mp *d@ = destination
400 * @mp *x@ = some polynomial
401 *
402 * Returns: A polynomial @y@ such that %$y^2 + y = x$%, or null.
403 */
404
405 mp *gfreduce_quadsolve(gfreduce *r, mp *d, mp *x)
406 {
407 unsigned long m = mp_bits(r->p) - 1;
408 mp *t;
409
410 MP_COPY(x);
411 if (m & 1)
412 d = gfreduce_halftrace(r, d, x);
413 else {
414 mp *z, *w, *rho = MP_NEW;
415 mp *spare = MP_NEW;
416 grand *fr = fibrand_create(0);
417 unsigned long i;
418
419 for (;;) {
420 rho = mprand(rho, m, fr, 0);
421 z = MP_ZERO;
422 w = MP_COPY(rho);
423 for (i = 0; i < m - 1; i++) {
424 t = gf_sqr(spare, z); spare = z; z = gfreduce_do(r, t, t);
425 t = gf_sqr(spare, w); spare = w; w = gfreduce_do(r, t, t);
426 t = gf_mul(spare, w, x); t = gfreduce_do(r, t, t); spare = t;
427 z = gf_add(z, z, t);
428 w = gf_add(w, w, rho);
429 }
430 if (!MP_ISZERO(w))
431 break;
432 MP_DROP(z);
433 MP_DROP(w);
434 }
435 if (d) MP_DROP(d);
436 MP_DROP(w);
437 MP_DROP(spare);
438 MP_DROP(rho);
439 fr->ops->destroy(fr);
440 d = z;
441 }
442
443 t = gf_sqr(MP_NEW, d); t = gfreduce_do(r, t, t); t = gf_add(t, t, d);
444 if (!MP_EQ(t, x)) {
445 MP_DROP(d);
446 d = 0;
447 }
448 MP_DROP(t);
449 MP_DROP(x);
450 if (d) d->v[0] &= ~(mpw)1;
451 return (d);
452 }
453
454 /* --- @gfreduce_exp@ --- *
455 *
456 * Arguments: @gfreduce *gr@ = pointer to reduction context
457 * @mp *d@ = fake destination
458 * @mp *a@ = base
459 * @mp *e@ = exponent
460 *
461 * Returns: Result, %$a^e \bmod m$%.
462 */
463
464 mp *gfreduce_exp(gfreduce *gr, mp *d, mp *a, mp *e)
465 {
466 mp *x = MP_ONE;
467 mp *spare = (e->f & MP_BURN) ? MP_NEWSEC : MP_NEW;
468
469 MP_SHRINK(e);
470 if (!MP_LEN(e))
471 ;
472 else if (MP_LEN(e) < EXP_THRESH)
473 EXP_SIMPLE(x, a, e);
474 else
475 EXP_WINDOW(x, a, e);
476 mp_drop(d);
477 mp_drop(spare);
478 return (x);
479 }
480
481 /*----- Test rig ----------------------------------------------------------*/
482
483 #ifdef TEST_RIG
484
485 #define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
486
487 static int vreduce(dstr *v)
488 {
489 mp *d = *(mp **)v[0].buf;
490 mp *n = *(mp **)v[1].buf;
491 mp *r = *(mp **)v[2].buf;
492 mp *c;
493 int ok = 1;
494 gfreduce rr;
495
496 gfreduce_create(&rr, d);
497 c = gfreduce_do(&rr, MP_NEW, n);
498 if (!MP_EQ(c, r)) {
499 fprintf(stderr, "\n*** reduction failed\n*** ");
500 gfreduce_dump(&rr, stderr);
501 fprintf(stderr, "\n*** n = "); mp_writefile(n, stderr, 16);
502 fprintf(stderr, "\n*** r = "); mp_writefile(r, stderr, 16);
503 fprintf(stderr, "\n*** c = "); mp_writefile(c, stderr, 16);
504 fprintf(stderr, "\n");
505 ok = 0;
506 }
507 gfreduce_destroy(&rr);
508 mp_drop(n); mp_drop(d); mp_drop(r); mp_drop(c);
509 assert(mparena_count(MPARENA_GLOBAL) == 0);
510 return (ok);
511 }
512
513 static int vmodexp(dstr *v)
514 {
515 mp *p = *(mp **)v[0].buf;
516 mp *g = *(mp **)v[1].buf;
517 mp *x = *(mp **)v[2].buf;
518 mp *r = *(mp **)v[3].buf;
519 mp *c;
520 int ok = 1;
521 gfreduce rr;
522
523 gfreduce_create(&rr, p);
524 c = gfreduce_exp(&rr, MP_NEW, g, x);
525 if (!MP_EQ(c, r)) {
526 fprintf(stderr, "\n*** modexp failed\n*** ");
527 fprintf(stderr, "\n*** p = "); mp_writefile(p, stderr, 16);
528 fprintf(stderr, "\n*** g = "); mp_writefile(g, stderr, 16);
529 fprintf(stderr, "\n*** x = "); mp_writefile(x, stderr, 16);
530 fprintf(stderr, "\n*** c = "); mp_writefile(c, stderr, 16);
531 fprintf(stderr, "\n*** r = "); mp_writefile(r, stderr, 16);
532 fprintf(stderr, "\n");
533 ok = 0;
534 }
535 gfreduce_destroy(&rr);
536 mp_drop(p); mp_drop(g); mp_drop(r); mp_drop(x); mp_drop(c);
537 assert(mparena_count(MPARENA_GLOBAL) == 0);
538 return (ok);
539 }
540
541 static int vsqrt(dstr *v)
542 {
543 mp *p = *(mp **)v[0].buf;
544 mp *x = *(mp **)v[1].buf;
545 mp *r = *(mp **)v[2].buf;
546 mp *c;
547 int ok = 1;
548 gfreduce rr;
549
550 gfreduce_create(&rr, p);
551 c = gfreduce_sqrt(&rr, MP_NEW, x);
552 if (!MP_EQ(c, r)) {
553 fprintf(stderr, "\n*** sqrt failed\n*** ");
554 fprintf(stderr, "\n*** p = "); mp_writefile(p, stderr, 16);
555 fprintf(stderr, "\n*** x = "); mp_writefile(x, stderr, 16);
556 fprintf(stderr, "\n*** c = "); mp_writefile(c, stderr, 16);
557 fprintf(stderr, "\n*** r = "); mp_writefile(r, stderr, 16);
558 fprintf(stderr, "\n");
559 ok = 0;
560 }
561 gfreduce_destroy(&rr);
562 mp_drop(p); mp_drop(r); mp_drop(x); mp_drop(c);
563 assert(mparena_count(MPARENA_GLOBAL) == 0);
564 return (ok);
565 }
566
567 static int vtr(dstr *v)
568 {
569 mp *p = *(mp **)v[0].buf;
570 mp *x = *(mp **)v[1].buf;
571 int r = *(int *)v[2].buf, c;
572 int ok = 1;
573 gfreduce rr;
574
575 gfreduce_create(&rr, p);
576 c = gfreduce_trace(&rr, x);
577 if (c != r) {
578 fprintf(stderr, "\n*** trace failed\n*** ");
579 fprintf(stderr, "\n*** p = "); mp_writefile(p, stderr, 16);
580 fprintf(stderr, "\n*** x = "); mp_writefile(x, stderr, 16);
581 fprintf(stderr, "\n*** c = %d", c);
582 fprintf(stderr, "\n*** r = %d", r);
583 fprintf(stderr, "\n");
584 ok = 0;
585 }
586 gfreduce_destroy(&rr);
587 mp_drop(p); mp_drop(x);
588 assert(mparena_count(MPARENA_GLOBAL) == 0);
589 return (ok);
590 }
591
592 static int vhftr(dstr *v)
593 {
594 mp *p = *(mp **)v[0].buf;
595 mp *x = *(mp **)v[1].buf;
596 mp *r = *(mp **)v[2].buf;
597 mp *c;
598 int ok = 1;
599 gfreduce rr;
600
601 gfreduce_create(&rr, p);
602 c = gfreduce_halftrace(&rr, MP_NEW, x);
603 if (!MP_EQ(c, r)) {
604 fprintf(stderr, "\n*** halftrace failed\n*** ");
605 fprintf(stderr, "\n*** p = "); mp_writefile(p, stderr, 16);
606 fprintf(stderr, "\n*** x = "); mp_writefile(x, stderr, 16);
607 fprintf(stderr, "\n*** c = "); mp_writefile(c, stderr, 16);
608 fprintf(stderr, "\n*** r = "); mp_writefile(r, stderr, 16);
609 fprintf(stderr, "\n");
610 ok = 0;
611 }
612 gfreduce_destroy(&rr);
613 mp_drop(p); mp_drop(r); mp_drop(x); mp_drop(c);
614 assert(mparena_count(MPARENA_GLOBAL) == 0);
615 return (ok);
616 }
617
618 static int vquad(dstr *v)
619 {
620 mp *p = *(mp **)v[0].buf;
621 mp *x = *(mp **)v[1].buf;
622 mp *r = *(mp **)v[2].buf;
623 mp *c;
624 int ok = 1;
625 gfreduce rr;
626
627 gfreduce_create(&rr, p);
628 c = gfreduce_quadsolve(&rr, MP_NEW, x);
629 if (!MP_EQ(c, r)) {
630 fprintf(stderr, "\n*** quadsolve failed\n*** ");
631 fprintf(stderr, "\n*** p = "); mp_writefile(p, stderr, 16);
632 fprintf(stderr, "\n*** x = "); mp_writefile(x, stderr, 16);
633 fprintf(stderr, "\n*** c = "); mp_writefile(c, stderr, 16);
634 fprintf(stderr, "\n*** r = "); mp_writefile(r, stderr, 16);
635 fprintf(stderr, "\n");
636 ok = 0;
637 }
638 gfreduce_destroy(&rr);
639 mp_drop(p); mp_drop(r); mp_drop(x); mp_drop(c);
640 assert(mparena_count(MPARENA_GLOBAL) == 0);
641 return (ok);
642 }
643
644 static test_chunk defs[] = {
645 { "reduce", vreduce, { &type_mp, &type_mp, &type_mp, 0 } },
646 { "modexp", vmodexp, { &type_mp, &type_mp, &type_mp, &type_mp, 0 } },
647 { "sqrt", vsqrt, { &type_mp, &type_mp, &type_mp, 0 } },
648 { "trace", vtr, { &type_mp, &type_mp, &type_int, 0 } },
649 { "halftrace", vhftr, { &type_mp, &type_mp, &type_mp, 0 } },
650 { "quadsolve", vquad, { &type_mp, &type_mp, &type_mp, 0 } },
651 { 0, 0, { 0 } }
652 };
653
654 int main(int argc, char *argv[])
655 {
656 test_run(argc, argv, defs, SRCDIR"/tests/gfreduce");
657 return (0);
658 }
659
660 #endif
661
662 /*----- That's all, folks -------------------------------------------------*/
Catacomb cryptographic library (downstream of http://git.distorted.org.uk/~mdw/catacomb/)