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Random field-element selection.
[u/mdw/catacomb] / ec-prime.c
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b0ab12e6 1/* -*-c-*-
2 *
391faf42 3 * $Id: ec-prime.c,v 1.5 2004/03/22 02:19:10 mdw Exp $
b0ab12e6 4 *
5 * Elliptic curves over prime fields
6 *
7 * (c) 2001 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: ec-prime.c,v $
391faf42 33 * Revision 1.5 2004/03/22 02:19:10 mdw
34 * Rationalise the sliding-window threshold. Drop guarantee that right
35 * arguments to EC @add@ are canonical, and fix up projective implementations
36 * to cope.
37 *
c3caa2fa 38 * Revision 1.4 2004/03/21 22:52:06 mdw
39 * Merge and close elliptic curve branch.
40 *
ceb3f0c0 41 * Revision 1.3.4.3 2004/03/21 22:39:46 mdw
42 * Elliptic curves on binary fields work.
43 *
8823192f 44 * Revision 1.3.4.2 2004/03/20 00:13:31 mdw
45 * Projective coordinates for prime curves
46 *
dbfee00a 47 * Revision 1.3.4.1 2003/06/10 13:43:53 mdw
48 * Simple (non-projective) curves over prime fields now seem to work.
49 *
41cb1beb 50 * Revision 1.3 2003/05/15 23:25:59 mdw
51 * Make elliptic curve stuff build.
52 *
b085fd91 53 * Revision 1.2 2002/01/13 13:48:44 mdw
54 * Further progress.
55 *
b0ab12e6 56 * Revision 1.1 2001/04/29 18:12:33 mdw
57 * Prototype version.
58 *
59 */
60
61/*----- Header files ------------------------------------------------------*/
62
41cb1beb 63#include <mLib/sub.h>
64
b0ab12e6 65#include "ec.h"
66
67/*----- Data structures ---------------------------------------------------*/
68
69typedef struct ecctx {
70 ec_curve c;
71 mp *a, *b;
72} ecctx;
73
dbfee00a 74/*----- Simple prime curves -----------------------------------------------*/
b0ab12e6 75
8823192f 76static const ec_ops ec_primeops, ec_primeprojops, ec_primeprojxops;
41cb1beb 77
78static ec *ecneg(ec_curve *c, ec *d, const ec *p)
b085fd91 79{
80 EC_COPY(d, p);
ceb3f0c0 81 if (d->y)
82 d->y = F_NEG(c->f, d->y, d->y);
b085fd91 83 return (d);
84}
85
8823192f 86static ec *ecfind(ec_curve *c, ec *d, mp *x)
87{
88 mp *p, *q;
89 ecctx *cc = (ecctx *)c;
90 field *f = c->f;
91
92 q = F_SQR(f, MP_NEW, x);
93 p = F_MUL(f, MP_NEW, x, q);
94 q = F_MUL(f, q, x, cc->a);
95 p = F_ADD(f, p, p, q);
96 p = F_ADD(f, p, p, cc->b);
97 MP_DROP(q);
98 p = F_SQRT(f, p, p);
99 if (!p)
100 return (0);
101 EC_DESTROY(d);
102 d->x = MP_COPY(x);
103 d->y = p;
104 d->z = MP_COPY(f->one);
b085fd91 105 return (d);
106}
107
108static ec *ecdbl(ec_curve *c, ec *d, const ec *a)
b0ab12e6 109{
b085fd91 110 if (EC_ATINF(a))
111 EC_SETINF(d);
8823192f 112 else if (F_ZEROP(c->f, a->y))
b085fd91 113 EC_COPY(d, a);
114 else {
115 field *f = c->f;
116 ecctx *cc = (ecctx *)c;
117 mp *lambda;
118 mp *dy, *dx;
119
8823192f 120 dx = F_SQR(f, MP_NEW, a->x); /* %$x^2$% */
121 dy = F_DBL(f, MP_NEW, a->y); /* %$2 y$% */
122 dx = F_TPL(f, dx, dx); /* %$3 x^2$% */
123 dx = F_ADD(f, dx, dx, cc->a); /* %$3 x^2 + A$% */
124 dy = F_INV(f, dy, dy); /* %$(2 y)^{-1}$% */
125 lambda = F_MUL(f, MP_NEW, dx, dy); /* %$\lambda = (3 x^2 + A)/(2 y)$% */
b085fd91 126
8823192f 127 dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
128 dy = F_DBL(f, dy, a->x); /* %$2 x$% */
129 dx = F_SUB(f, dx, dx, dy); /* %$x' = \lambda^2 - 2 x */
130 dy = F_SUB(f, dy, a->x, dx); /* %$x - x'$% */
131 dy = F_MUL(f, dy, lambda, dy); /* %$\lambda (x - x')$% */
132 dy = F_SUB(f, dy, dy, a->y); /* %$y' = \lambda (x - x') - y$% */
b0ab12e6 133
b085fd91 134 EC_DESTROY(d);
135 d->x = dx;
136 d->y = dy;
137 d->z = 0;
138 MP_DROP(lambda);
139 }
140 return (d);
141}
142
8823192f 143static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a)
144{
145 if (EC_ATINF(a))
146 EC_SETINF(d);
147 else if (F_ZEROP(c->f, a->y))
148 EC_COPY(d, a);
149 else {
150 field *f = c->f;
151 ecctx *cc = (ecctx *)c;
152 mp *p, *q, *m, *s, *dx, *dy, *dz;
153
154 p = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */
155 q = F_SQR(f, MP_NEW, p); /* %$z^4$% */
156 p = F_MUL(f, p, q, cc->a); /* %$A z^4$% */
157 m = F_SQR(f, MP_NEW, a->x); /* %$x^2$% */
158 m = F_TPL(f, m, m); /* %$3 x^2$% */
159 m = F_ADD(f, m, m, p); /* %$m = 3 x^2 + A z^4$% */
160
161 q = F_DBL(f, q, a->y); /* %$2 y$% */
162 dz = F_MUL(f, MP_NEW, q, a->z); /* %$z' = 2 y z$% */
163
164 p = F_SQR(f, p, q); /* %$4 y^2$% */
165 s = F_MUL(f, MP_NEW, p, a->x); /* %$s = 4 x y^2$% */
166 q = F_SQR(f, q, p); /* %$16 y^4$% */
167 q = F_HLV(f, q, q); /* %$t = 8 y^4$% */
168
169 p = F_DBL(f, p, s); /* %$2 s$% */
170 dx = F_SQR(f, MP_NEW, m); /* %$m^2$% */
171 dx = F_SUB(f, dx, dx, p); /* %$x' = m^2 - 2 s$% */
172
173 s = F_SUB(f, s, s, dx); /* %$s - x'$% */
174 dy = F_MUL(f, p, m, s); /* %$m (s - x')$% */
175 dy = F_SUB(f, dy, dy, q); /* %$y' = m (s - x') - t$% */
176
177 EC_DESTROY(d);
178 d->x = dx;
179 d->y = dy;
180 d->z = dz;
181 MP_DROP(m);
182 MP_DROP(q);
183 MP_DROP(s);
184 }
185 return (d);
186}
187
188static ec *ecprojxdbl(ec_curve *c, ec *d, const ec *a)
189{
190 if (EC_ATINF(a))
191 EC_SETINF(d);
192 else if (F_ZEROP(c->f, a->y))
193 EC_COPY(d, a);
194 else {
195 field *f = c->f;
196 mp *p, *q, *m, *s, *dx, *dy, *dz;
197
198 m = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */
199 p = F_SUB(f, MP_NEW, a->x, m); /* %$x - z^2$% */
200 q = F_ADD(f, MP_NEW, a->x, m); /* %$x + z^2$% */
201 m = F_MUL(f, m, p, q); /* %$x^2 - z^4$% */
202 m = F_TPL(f, m, m); /* %$m = 3 x^2 - 3 z^4$% */
203
204 q = F_DBL(f, q, a->y); /* %$2 y$% */
205 dz = F_MUL(f, MP_NEW, q, a->z); /* %$z' = 2 y z$% */
206
207 p = F_SQR(f, p, q); /* %$4 y^2$% */
208 s = F_MUL(f, MP_NEW, p, a->x); /* %$s = 4 x y^2$% */
209 q = F_SQR(f, q, p); /* %$16 y^4$% */
210 q = F_HLV(f, q, q); /* %$t = 8 y^4$% */
211
212 p = F_DBL(f, p, s); /* %$2 s$% */
213 dx = F_SQR(f, MP_NEW, m); /* %$m^2$% */
214 dx = F_SUB(f, dx, dx, p); /* %$x' = m^2 - 2 s$% */
215
216 s = F_SUB(f, s, s, dx); /* %$s - x'$% */
217 dy = F_MUL(f, p, m, s); /* %$m (s - x')$% */
218 dy = F_SUB(f, dy, dy, q); /* %$y' = m (s - x') - t$% */
219
220 EC_DESTROY(d);
221 d->x = dx;
222 d->y = dy;
223 d->z = dz;
224 MP_DROP(m);
225 MP_DROP(q);
226 MP_DROP(s);
227 }
228 return (d);
229}
230
b085fd91 231static ec *ecadd(ec_curve *c, ec *d, const ec *a, const ec *b)
232{
b0ab12e6 233 if (a == b)
234 ecdbl(c, d, a);
235 else if (EC_ATINF(a))
236 EC_COPY(d, b);
237 else if (EC_ATINF(b))
238 EC_COPY(d, a);
b085fd91 239 else {
240 field *f = c->f;
241 mp *lambda;
242 mp *dy, *dx;
243
244 if (!MP_EQ(a->x, b->x)) {
8823192f 245 dy = F_SUB(f, MP_NEW, a->y, b->y); /* %$y_0 - y_1$% */
246 dx = F_SUB(f, MP_NEW, a->x, b->x); /* %$x_0 - x_1$% */
247 dx = F_INV(f, dx, dx); /* %$(x_0 - x_1)^{-1}$% */
b085fd91 248 lambda = F_MUL(f, MP_NEW, dy, dx);
8823192f 249 /* %$\lambda = (y_0 - y1)/(x_0 - x_1)$% */
250 } else if (F_ZEROP(c->f, a->y) || !MP_EQ(a->y, b->y)) {
b0ab12e6 251 EC_SETINF(d);
b085fd91 252 return (d);
253 } else {
254 ecctx *cc = (ecctx *)c;
8823192f 255 dx = F_SQR(f, MP_NEW, a->x); /* %$x_0^2$% */
256 dx = F_TPL(f, dx, dx); /* %$3 x_0^2$% */
257 dx = F_ADD(f, dx, dx, cc->a); /* %$3 x_0^2 + A$% */
258 dy = F_DBL(f, MP_NEW, a->y); /* %$2 y_0$% */
259 dy = F_INV(f, dy, dy); /* %$(2 y_0)^{-1}$% */
41cb1beb 260 lambda = F_MUL(f, MP_NEW, dx, dy);
8823192f 261 /* %$\lambda = (3 x_0^2 + A)/(2 y_0)$% */
b085fd91 262 }
263
8823192f 264 dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
265 dx = F_SUB(f, dx, dx, a->x); /* %$\lambda^2 - x_0$% */
266 dx = F_SUB(f, dx, dx, b->x); /* %$x' = \lambda^2 - x_0 - x_1$% */
267 dy = F_SUB(f, dy, b->x, dx); /* %$x_1 - x'$% */
268 dy = F_MUL(f, dy, lambda, dy); /* %$\lambda (x_1 - x')$% */
ceb3f0c0 269 dy = F_SUB(f, dy, dy, b->y); /* %$y' = \lambda (x_1 - x') - y_1$% */
b0ab12e6 270
b085fd91 271 EC_DESTROY(d);
272 d->x = dx;
273 d->y = dy;
274 d->z = 0;
275 MP_DROP(lambda);
b0ab12e6 276 }
b085fd91 277 return (d);
b0ab12e6 278}
279
8823192f 280static ec *ecprojadd(ec_curve *c, ec *d, const ec *a, const ec *b)
281{
282 if (a == b)
283 c->ops->dbl(c, d, a);
284 else if (EC_ATINF(a))
285 EC_COPY(d, b);
286 else if (EC_ATINF(b))
287 EC_COPY(d, a);
288 else {
289 field *f = c->f;
391faf42 290 mp *p, *q, *r, *w, *u, *uu, *s, *ss, *dx, *dy, *dz;
8823192f 291
292 q = F_SQR(f, MP_NEW, a->z); /* %$z_0^2$% */
293 u = F_MUL(f, MP_NEW, q, b->x); /* %$u = x_1 z_0^2$% */
294 p = F_MUL(f, MP_NEW, q, b->y); /* %$y_1 z_0^2$% */
295 s = F_MUL(f, q, p, a->z); /* %$s = y_1 z_0^3$% */
296
391faf42 297 q = F_SQR(f, MP_NEW, b->z); /* %$z_1^2$% */
298 uu = F_MUL(f, MP_NEW, q, a->x); /* %$uu = x_0 z_1^2$%*/
299 p = F_MUL(f, p, q, a->y); /* %$y_0 z_1^2$% */
300 ss = F_MUL(f, q, p, b->z); /* %$ss = y_0 z_1^3$% */
301
302 w = F_SUB(f, p, uu, u); /* %$w = uu - u$% */
303 r = F_SUB(f, MP_NEW, ss, s); /* %$r = ss - s$% */
8823192f 304 if (F_ZEROP(f, w)) {
ceb3f0c0 305 MP_DROP(w);
306 MP_DROP(u);
307 MP_DROP(s);
391faf42 308 MP_DROP(uu);
309 MP_DROP(ss);
8823192f 310 if (F_ZEROP(f, r)) {
8823192f 311 MP_DROP(r);
8823192f 312 return (c->ops->dbl(c, d, a));
313 } else {
8823192f 314 MP_DROP(r);
8823192f 315 EC_SETINF(d);
316 return (d);
317 }
318 }
391faf42 319 u = F_ADD(f, u, u, uu); /* %$t = uu + u$% */
320 s = F_ADD(f, s, s, ss); /* %$m = ss + r$% */
8823192f 321
391faf42 322 uu = F_MUL(f, uu, a->z, w); /* %$z_0 w$% */
323 dz = F_MUL(f, ss, uu, b->z); /* %$z' = z_0 z_1 w$% */
8823192f 324
391faf42 325 p = F_SQR(f, uu, w); /* %$w^2$% */
8823192f 326 q = F_MUL(f, MP_NEW, p, u); /* %$t w^2$% */
327 u = F_MUL(f, u, p, w); /* %$w^3$% */
328 p = F_MUL(f, p, u, s); /* %$m w^3$% */
329
330 dx = F_SQR(f, u, r); /* %$r^2$% */
331 dx = F_SUB(f, dx, dx, q); /* %$x' = r^2 - t w^2$% */
332
333 s = F_DBL(f, s, dx); /* %$2 x'$% */
334 q = F_SUB(f, q, q, s); /* %$v = t w^2 - 2 x'$% */
335 dy = F_MUL(f, s, q, r); /* %$v r$% */
336 dy = F_SUB(f, dy, dy, p); /* %$v r - m w^3$% */
337 dy = F_HLV(f, dy, dy); /* %$y' = (v r - m w^3)/2$% */
338
339 EC_DESTROY(d);
340 d->x = dx;
341 d->y = dy;
342 d->z = dz;
343 MP_DROP(p);
344 MP_DROP(q);
345 MP_DROP(r);
346 MP_DROP(w);
347 }
348 return (d);
349}
350
351static int eccheck(ec_curve *c, const ec *p)
352{
353 ecctx *cc = (ecctx *)c;
354 field *f = c->f;
355 int rc;
356 mp *l = F_SQR(f, MP_NEW, p->y);
357 mp *x = F_SQR(f, MP_NEW, p->x);
358 mp *r = F_MUL(f, MP_NEW, x, p->x);
359 x = F_MUL(f, x, cc->a, p->x);
360 r = F_ADD(f, r, r, x);
361 r = F_ADD(f, r, r, cc->b);
362 rc = MP_EQ(l, r) ? 0 : -1;
363 mp_drop(l);
364 mp_drop(x);
365 mp_drop(r);
366 return (rc);
367}
368
369static int ecprojcheck(ec_curve *c, const ec *p)
370{
371 ec t = EC_INIT;
372 int rc;
373
374 c->ops->fix(c, &t, p);
375 rc = eccheck(c, &t);
376 EC_DESTROY(&t);
377 return (rc);
378}
379
41cb1beb 380static void ecdestroy(ec_curve *c)
381{
382 ecctx *cc = (ecctx *)c;
383 MP_DROP(cc->a);
384 MP_DROP(cc->b);
385 DESTROY(cc);
386}
387
388/* --- @ec_prime@, @ec_primeproj@ --- *
389 *
dbfee00a 390 * Arguments: @field *f@ = the underlying field for this elliptic curve
41cb1beb 391 * @mp *a, *b@ = the coefficients for this curve
392 *
393 * Returns: A pointer to the curve.
394 *
395 * Use: Creates a curve structure for an elliptic curve defined over
396 * a prime field. The @primeproj@ variant uses projective
397 * coordinates, which can be a win.
398 */
399
400extern ec_curve *ec_prime(field *f, mp *a, mp *b)
401{
402 ecctx *cc = CREATE(ecctx);
403 cc->c.ops = &ec_primeops;
404 cc->c.f = f;
dbfee00a 405 cc->a = F_IN(f, MP_NEW, a);
406 cc->b = F_IN(f, MP_NEW, b);
41cb1beb 407 return (&cc->c);
408}
409
8823192f 410extern ec_curve *ec_primeproj(field *f, mp *a, mp *b)
411{
412 ecctx *cc = CREATE(ecctx);
413 mp *ax;
414
415 ax = mp_add(MP_NEW, a, MP_THREE);
416 ax = F_IN(f, ax, ax);
417 if (F_ZEROP(f, ax))
418 cc->c.ops = &ec_primeprojxops;
419 else
420 cc->c.ops = &ec_primeprojops;
421 MP_DROP(ax);
422 cc->c.f = f;
423 cc->a = F_IN(f, MP_NEW, a);
424 cc->b = F_IN(f, MP_NEW, b);
41cb1beb 425 return (&cc->c);
426}
427
428static const ec_ops ec_primeops = {
8823192f 429 ecdestroy, ec_idin, ec_idout, ec_idfix,
430 0, ecneg, ecadd, ec_stdsub, ecdbl, eccheck
431};
432
433static const ec_ops ec_primeprojops = {
434 ecdestroy, ec_projin, ec_projout, ec_projfix,
435 0, ecneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck
436};
437
438static const ec_ops ec_primeprojxops = {
439 ecdestroy, ec_projin, ec_projout, ec_projfix,
440 0, ecneg, ecprojadd, ec_stdsub, ecprojxdbl, ecprojcheck
41cb1beb 441};
442
443/*----- Test rig ----------------------------------------------------------*/
444
445#ifdef TEST_RIG
446
447#define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
448
ceb3f0c0 449int main(int argc, char *argv[])
41cb1beb 450{
451 field *f;
452 ec_curve *c;
453 ec g = EC_INIT, d = EC_INIT;
454 mp *p, *a, *b, *r;
ceb3f0c0 455 int i, n = argc == 1 ? 1 : atoi(argv[1]);
41cb1beb 456
dbfee00a 457 printf("ec-prime: ");
458 fflush(stdout);
41cb1beb 459 a = MP(-3);
460 b = MP(0x64210519e59c80e70fa7e9ab72243049feb8deecc146b9b1);
461 p = MP(6277101735386680763835789423207666416083908700390324961279);
dbfee00a 462 r = MP(6277101735386680763835789423176059013767194773182842284080);
41cb1beb 463
464 f = field_prime(p);
ceb3f0c0 465 c = ec_primeproj(f, a, b);
41cb1beb 466
467 g.x = MP(0x188da80eb03090f67cbf20eb43a18800f4ff0afd82ff1012);
468 g.y = MP(0x07192b95ffc8da78631011ed6b24cdd573f977a11e794811);
469
ceb3f0c0 470 for (i = 0; i < n; i++) {
471 ec_mul(c, &d, &g, r);
472 if (EC_ATINF(&d)) {
473 fprintf(stderr, "zero too early\n");
474 return (1);
475 }
476 ec_add(c, &d, &d, &g);
477 if (!EC_ATINF(&d)) {
478 fprintf(stderr, "didn't reach zero\n");
479 MP_EPRINT("d.x", d.x);
480 MP_EPRINT("d.y", d.y);
481 return (1);
482 }
483 ec_destroy(&d);
dbfee00a 484 }
41cb1beb 485 ec_destroy(&g);
486 ec_destroycurve(c);
487 F_DESTROY(f);
dbfee00a 488 MP_DROP(p); MP_DROP(a); MP_DROP(b); MP_DROP(r);
489 assert(!mparena_count(&mparena_global));
490 printf("ok\n");
41cb1beb 491 return (0);
492}
493
494#endif
495
b0ab12e6 496/*----- That's all, folks -------------------------------------------------*/
Catacomb cryptographic library (downstream of http://git.distorted.org.uk/~mdw/catacomb/)