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Standard curves and curve checking.
[u/mdw/catacomb] / ec-prime.c
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b0ab12e6 1/* -*-c-*-
2 *
432c4e18 3 * $Id: ec-prime.c,v 1.8 2004/03/27 17:54:11 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 $
432c4e18 33 * Revision 1.8 2004/03/27 17:54:11 mdw
34 * Standard curves and curve checking.
35 *
f46efa79 36 * Revision 1.7 2004/03/27 00:04:46 mdw
37 * Implement efficient reduction for pleasant-looking primes.
38 *
bc985cef 39 * Revision 1.6 2004/03/23 15:19:32 mdw
40 * Test elliptic curves more thoroughly.
41 *
391faf42 42 * Revision 1.5 2004/03/22 02:19:10 mdw
43 * Rationalise the sliding-window threshold. Drop guarantee that right
44 * arguments to EC @add@ are canonical, and fix up projective implementations
45 * to cope.
46 *
c3caa2fa 47 * Revision 1.4 2004/03/21 22:52:06 mdw
48 * Merge and close elliptic curve branch.
49 *
ceb3f0c0 50 * Revision 1.3.4.3 2004/03/21 22:39:46 mdw
51 * Elliptic curves on binary fields work.
52 *
8823192f 53 * Revision 1.3.4.2 2004/03/20 00:13:31 mdw
54 * Projective coordinates for prime curves
55 *
dbfee00a 56 * Revision 1.3.4.1 2003/06/10 13:43:53 mdw
57 * Simple (non-projective) curves over prime fields now seem to work.
58 *
41cb1beb 59 * Revision 1.3 2003/05/15 23:25:59 mdw
60 * Make elliptic curve stuff build.
61 *
b085fd91 62 * Revision 1.2 2002/01/13 13:48:44 mdw
63 * Further progress.
64 *
b0ab12e6 65 * Revision 1.1 2001/04/29 18:12:33 mdw
66 * Prototype version.
67 *
68 */
69
70/*----- Header files ------------------------------------------------------*/
71
41cb1beb 72#include <mLib/sub.h>
73
b0ab12e6 74#include "ec.h"
75
dbfee00a 76/*----- Simple prime curves -----------------------------------------------*/
b0ab12e6 77
8823192f 78static const ec_ops ec_primeops, ec_primeprojops, ec_primeprojxops;
41cb1beb 79
80static ec *ecneg(ec_curve *c, ec *d, const ec *p)
b085fd91 81{
82 EC_COPY(d, p);
ceb3f0c0 83 if (d->y)
84 d->y = F_NEG(c->f, d->y, d->y);
b085fd91 85 return (d);
86}
87
8823192f 88static ec *ecfind(ec_curve *c, ec *d, mp *x)
89{
90 mp *p, *q;
8823192f 91 field *f = c->f;
92
93 q = F_SQR(f, MP_NEW, x);
94 p = F_MUL(f, MP_NEW, x, q);
432c4e18 95 q = F_MUL(f, q, x, c->a);
8823192f 96 p = F_ADD(f, p, p, q);
432c4e18 97 p = F_ADD(f, p, p, c->b);
8823192f 98 MP_DROP(q);
99 p = F_SQRT(f, p, p);
100 if (!p)
101 return (0);
102 EC_DESTROY(d);
103 d->x = MP_COPY(x);
104 d->y = p;
105 d->z = MP_COPY(f->one);
b085fd91 106 return (d);
107}
108
109static ec *ecdbl(ec_curve *c, ec *d, const ec *a)
b0ab12e6 110{
b085fd91 111 if (EC_ATINF(a))
112 EC_SETINF(d);
8823192f 113 else if (F_ZEROP(c->f, a->y))
b085fd91 114 EC_COPY(d, a);
115 else {
116 field *f = c->f;
b085fd91 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); /* %$3 x^2$% */
432c4e18 123 dx = F_ADD(f, dx, dx, c->a); /* %$3 x^2 + A$% */
8823192f 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$% */
b0ab12e6 133
b085fd91 134 EC_DESTROY(d);
135 d->x = dx;
136 d->y = dy;
137 d->z = 0;
138 MP_DROP(lambda);
139 }
140 return (d);
141}
142
8823192f 143static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a)
144{
145 if (EC_ATINF(a))
146 EC_SETINF(d);
147 else if (F_ZEROP(c->f, a->y))
148 EC_COPY(d, a);
149 else {
150 field *f = c->f;
8823192f 151 mp *p, *q, *m, *s, *dx, *dy, *dz;
152
153 p = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */
154 q = F_SQR(f, MP_NEW, p); /* %$z^4$% */
432c4e18 155 p = F_MUL(f, p, q, c->a); /* %$A z^4$% */
8823192f 156 m = F_SQR(f, MP_NEW, a->x); /* %$x^2$% */
157 m = F_TPL(f, m, m); /* %$3 x^2$% */
158 m = F_ADD(f, m, m, p); /* %$m = 3 x^2 + A z^4$% */
159
160 q = F_DBL(f, q, a->y); /* %$2 y$% */
161 dz = F_MUL(f, MP_NEW, q, a->z); /* %$z' = 2 y z$% */
162
163 p = F_SQR(f, p, q); /* %$4 y^2$% */
164 s = F_MUL(f, MP_NEW, p, a->x); /* %$s = 4 x y^2$% */
165 q = F_SQR(f, q, p); /* %$16 y^4$% */
166 q = F_HLV(f, q, q); /* %$t = 8 y^4$% */
167
168 p = F_DBL(f, p, s); /* %$2 s$% */
169 dx = F_SQR(f, MP_NEW, m); /* %$m^2$% */
170 dx = F_SUB(f, dx, dx, p); /* %$x' = m^2 - 2 s$% */
171
172 s = F_SUB(f, s, s, dx); /* %$s - x'$% */
173 dy = F_MUL(f, p, m, s); /* %$m (s - x')$% */
174 dy = F_SUB(f, dy, dy, q); /* %$y' = m (s - x') - t$% */
175
176 EC_DESTROY(d);
177 d->x = dx;
178 d->y = dy;
179 d->z = dz;
180 MP_DROP(m);
181 MP_DROP(q);
182 MP_DROP(s);
183 }
184 return (d);
185}
186
187static ec *ecprojxdbl(ec_curve *c, ec *d, const ec *a)
188{
189 if (EC_ATINF(a))
190 EC_SETINF(d);
191 else if (F_ZEROP(c->f, a->y))
192 EC_COPY(d, a);
193 else {
194 field *f = c->f;
195 mp *p, *q, *m, *s, *dx, *dy, *dz;
196
197 m = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */
198 p = F_SUB(f, MP_NEW, a->x, m); /* %$x - z^2$% */
199 q = F_ADD(f, MP_NEW, a->x, m); /* %$x + z^2$% */
200 m = F_MUL(f, m, p, q); /* %$x^2 - z^4$% */
201 m = F_TPL(f, m, m); /* %$m = 3 x^2 - 3 z^4$% */
202
203 q = F_DBL(f, q, a->y); /* %$2 y$% */
204 dz = F_MUL(f, MP_NEW, q, a->z); /* %$z' = 2 y z$% */
205
206 p = F_SQR(f, p, q); /* %$4 y^2$% */
207 s = F_MUL(f, MP_NEW, p, a->x); /* %$s = 4 x y^2$% */
208 q = F_SQR(f, q, p); /* %$16 y^4$% */
209 q = F_HLV(f, q, q); /* %$t = 8 y^4$% */
210
211 p = F_DBL(f, p, s); /* %$2 s$% */
212 dx = F_SQR(f, MP_NEW, m); /* %$m^2$% */
213 dx = F_SUB(f, dx, dx, p); /* %$x' = m^2 - 2 s$% */
214
215 s = F_SUB(f, s, s, dx); /* %$s - x'$% */
216 dy = F_MUL(f, p, m, s); /* %$m (s - x')$% */
217 dy = F_SUB(f, dy, dy, q); /* %$y' = m (s - x') - t$% */
218
219 EC_DESTROY(d);
220 d->x = dx;
221 d->y = dy;
222 d->z = dz;
223 MP_DROP(m);
224 MP_DROP(q);
225 MP_DROP(s);
226 }
227 return (d);
228}
229
b085fd91 230static ec *ecadd(ec_curve *c, ec *d, const ec *a, const ec *b)
231{
b0ab12e6 232 if (a == b)
233 ecdbl(c, d, a);
234 else if (EC_ATINF(a))
235 EC_COPY(d, b);
236 else if (EC_ATINF(b))
237 EC_COPY(d, a);
b085fd91 238 else {
239 field *f = c->f;
240 mp *lambda;
241 mp *dy, *dx;
242
243 if (!MP_EQ(a->x, b->x)) {
8823192f 244 dy = F_SUB(f, MP_NEW, a->y, b->y); /* %$y_0 - y_1$% */
245 dx = F_SUB(f, MP_NEW, a->x, b->x); /* %$x_0 - x_1$% */
246 dx = F_INV(f, dx, dx); /* %$(x_0 - x_1)^{-1}$% */
b085fd91 247 lambda = F_MUL(f, MP_NEW, dy, dx);
8823192f 248 /* %$\lambda = (y_0 - y1)/(x_0 - x_1)$% */
249 } else if (F_ZEROP(c->f, a->y) || !MP_EQ(a->y, b->y)) {
b0ab12e6 250 EC_SETINF(d);
b085fd91 251 return (d);
252 } else {
8823192f 253 dx = F_SQR(f, MP_NEW, a->x); /* %$x_0^2$% */
254 dx = F_TPL(f, dx, dx); /* %$3 x_0^2$% */
432c4e18 255 dx = F_ADD(f, dx, dx, c->a); /* %$3 x_0^2 + A$% */
8823192f 256 dy = F_DBL(f, MP_NEW, a->y); /* %$2 y_0$% */
257 dy = F_INV(f, dy, dy); /* %$(2 y_0)^{-1}$% */
41cb1beb 258 lambda = F_MUL(f, MP_NEW, dx, dy);
8823192f 259 /* %$\lambda = (3 x_0^2 + A)/(2 y_0)$% */
b085fd91 260 }
261
8823192f 262 dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
263 dx = F_SUB(f, dx, dx, a->x); /* %$\lambda^2 - x_0$% */
264 dx = F_SUB(f, dx, dx, b->x); /* %$x' = \lambda^2 - x_0 - x_1$% */
265 dy = F_SUB(f, dy, b->x, dx); /* %$x_1 - x'$% */
266 dy = F_MUL(f, dy, lambda, dy); /* %$\lambda (x_1 - x')$% */
ceb3f0c0 267 dy = F_SUB(f, dy, dy, b->y); /* %$y' = \lambda (x_1 - x') - y_1$% */
b0ab12e6 268
b085fd91 269 EC_DESTROY(d);
270 d->x = dx;
271 d->y = dy;
272 d->z = 0;
273 MP_DROP(lambda);
b0ab12e6 274 }
b085fd91 275 return (d);
b0ab12e6 276}
277
8823192f 278static ec *ecprojadd(ec_curve *c, ec *d, const ec *a, const ec *b)
279{
280 if (a == b)
281 c->ops->dbl(c, d, a);
282 else if (EC_ATINF(a))
283 EC_COPY(d, b);
284 else if (EC_ATINF(b))
285 EC_COPY(d, a);
286 else {
287 field *f = c->f;
391faf42 288 mp *p, *q, *r, *w, *u, *uu, *s, *ss, *dx, *dy, *dz;
8823192f 289
290 q = F_SQR(f, MP_NEW, a->z); /* %$z_0^2$% */
291 u = F_MUL(f, MP_NEW, q, b->x); /* %$u = x_1 z_0^2$% */
292 p = F_MUL(f, MP_NEW, q, b->y); /* %$y_1 z_0^2$% */
293 s = F_MUL(f, q, p, a->z); /* %$s = y_1 z_0^3$% */
294
391faf42 295 q = F_SQR(f, MP_NEW, b->z); /* %$z_1^2$% */
296 uu = F_MUL(f, MP_NEW, q, a->x); /* %$uu = x_0 z_1^2$%*/
297 p = F_MUL(f, p, q, a->y); /* %$y_0 z_1^2$% */
298 ss = F_MUL(f, q, p, b->z); /* %$ss = y_0 z_1^3$% */
299
300 w = F_SUB(f, p, uu, u); /* %$w = uu - u$% */
301 r = F_SUB(f, MP_NEW, ss, s); /* %$r = ss - s$% */
8823192f 302 if (F_ZEROP(f, w)) {
ceb3f0c0 303 MP_DROP(w);
304 MP_DROP(u);
305 MP_DROP(s);
391faf42 306 MP_DROP(uu);
307 MP_DROP(ss);
8823192f 308 if (F_ZEROP(f, r)) {
8823192f 309 MP_DROP(r);
8823192f 310 return (c->ops->dbl(c, d, a));
311 } else {
8823192f 312 MP_DROP(r);
8823192f 313 EC_SETINF(d);
314 return (d);
315 }
316 }
391faf42 317 u = F_ADD(f, u, u, uu); /* %$t = uu + u$% */
318 s = F_ADD(f, s, s, ss); /* %$m = ss + r$% */
8823192f 319
391faf42 320 uu = F_MUL(f, uu, a->z, w); /* %$z_0 w$% */
321 dz = F_MUL(f, ss, uu, b->z); /* %$z' = z_0 z_1 w$% */
8823192f 322
391faf42 323 p = F_SQR(f, uu, w); /* %$w^2$% */
8823192f 324 q = F_MUL(f, MP_NEW, p, u); /* %$t w^2$% */
325 u = F_MUL(f, u, p, w); /* %$w^3$% */
326 p = F_MUL(f, p, u, s); /* %$m w^3$% */
327
328 dx = F_SQR(f, u, r); /* %$r^2$% */
329 dx = F_SUB(f, dx, dx, q); /* %$x' = r^2 - t w^2$% */
330
331 s = F_DBL(f, s, dx); /* %$2 x'$% */
332 q = F_SUB(f, q, q, s); /* %$v = t w^2 - 2 x'$% */
333 dy = F_MUL(f, s, q, r); /* %$v r$% */
334 dy = F_SUB(f, dy, dy, p); /* %$v r - m w^3$% */
335 dy = F_HLV(f, dy, dy); /* %$y' = (v r - m w^3)/2$% */
336
337 EC_DESTROY(d);
338 d->x = dx;
339 d->y = dy;
340 d->z = dz;
341 MP_DROP(p);
342 MP_DROP(q);
343 MP_DROP(r);
344 MP_DROP(w);
345 }
346 return (d);
347}
348
349static int eccheck(ec_curve *c, const ec *p)
350{
8823192f 351 field *f = c->f;
352 int rc;
353 mp *l = F_SQR(f, MP_NEW, p->y);
354 mp *x = F_SQR(f, MP_NEW, p->x);
355 mp *r = F_MUL(f, MP_NEW, x, p->x);
432c4e18 356 x = F_MUL(f, x, c->a, p->x);
8823192f 357 r = F_ADD(f, r, r, x);
432c4e18 358 r = F_ADD(f, r, r, c->b);
8823192f 359 rc = MP_EQ(l, r) ? 0 : -1;
360 mp_drop(l);
361 mp_drop(x);
362 mp_drop(r);
363 return (rc);
364}
365
366static int ecprojcheck(ec_curve *c, const ec *p)
367{
368 ec t = EC_INIT;
369 int rc;
370
371 c->ops->fix(c, &t, p);
372 rc = eccheck(c, &t);
373 EC_DESTROY(&t);
374 return (rc);
375}
376
41cb1beb 377static void ecdestroy(ec_curve *c)
378{
432c4e18 379 MP_DROP(c->a);
380 MP_DROP(c->b);
381 DESTROY(c);
41cb1beb 382}
383
384/* --- @ec_prime@, @ec_primeproj@ --- *
385 *
dbfee00a 386 * Arguments: @field *f@ = the underlying field for this elliptic curve
41cb1beb 387 * @mp *a, *b@ = the coefficients for this curve
388 *
389 * Returns: A pointer to the curve.
390 *
391 * Use: Creates a curve structure for an elliptic curve defined over
392 * a prime field. The @primeproj@ variant uses projective
393 * coordinates, which can be a win.
394 */
395
396extern ec_curve *ec_prime(field *f, mp *a, mp *b)
397{
432c4e18 398 ec_curve *c = CREATE(ec_curve);
399 c->ops = &ec_primeops;
400 c->f = f;
401 c->a = F_IN(f, MP_NEW, a);
402 c->b = F_IN(f, MP_NEW, b);
403 return (c);
41cb1beb 404}
405
8823192f 406extern ec_curve *ec_primeproj(field *f, mp *a, mp *b)
407{
432c4e18 408 ec_curve *c = CREATE(ec_curve);
8823192f 409 mp *ax;
410
411 ax = mp_add(MP_NEW, a, MP_THREE);
412 ax = F_IN(f, ax, ax);
413 if (F_ZEROP(f, ax))
432c4e18 414 c->ops = &ec_primeprojxops;
8823192f 415 else
432c4e18 416 c->ops = &ec_primeprojops;
8823192f 417 MP_DROP(ax);
432c4e18 418 c->f = f;
419 c->a = F_IN(f, MP_NEW, a);
420 c->b = F_IN(f, MP_NEW, b);
421 return (c);
41cb1beb 422}
423
424static const ec_ops ec_primeops = {
8823192f 425 ecdestroy, ec_idin, ec_idout, ec_idfix,
bc985cef 426 ecfind, ecneg, ecadd, ec_stdsub, ecdbl, eccheck
8823192f 427};
428
429static const ec_ops ec_primeprojops = {
430 ecdestroy, ec_projin, ec_projout, ec_projfix,
bc985cef 431 ecfind, ecneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck
8823192f 432};
433
434static const ec_ops ec_primeprojxops = {
435 ecdestroy, ec_projin, ec_projout, ec_projfix,
bc985cef 436 ecfind, ecneg, ecprojadd, ec_stdsub, ecprojxdbl, ecprojcheck
41cb1beb 437};
438
439/*----- Test rig ----------------------------------------------------------*/
440
441#ifdef TEST_RIG
442
443#define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
444
ceb3f0c0 445int main(int argc, char *argv[])
41cb1beb 446{
447 field *f;
448 ec_curve *c;
449 ec g = EC_INIT, d = EC_INIT;
450 mp *p, *a, *b, *r;
ceb3f0c0 451 int i, n = argc == 1 ? 1 : atoi(argv[1]);
41cb1beb 452
dbfee00a 453 printf("ec-prime: ");
454 fflush(stdout);
41cb1beb 455 a = MP(-3);
432c4e18 456 b = MP(0xb3312fa7e23ee7e4988e056be3f82d19181d9c6efe8141120314088f5013875ac656398d8a2ed19d2a85c8edd3ec2aef);
457 p = MP(39402006196394479212279040100143613805079739270465446667948293404245721771496870329047266088258938001861606973112319);
458 r = MP(39402006196394479212279040100143613805079739270465446667946905279627659399113263569398956308152294913554433653942642);
41cb1beb 459
f46efa79 460 f = field_niceprime(p);
ceb3f0c0 461 c = ec_primeproj(f, a, b);
41cb1beb 462
432c4e18 463 g.x = MP(0xaa87ca22be8b05378eb1c71ef320ad746e1d3b628ba79b9859f741e082542a385502f25dbf55296c3a545e3872760ab7);
464 g.y = MP(0x3617de4a96262c6f5d9e98bf9292dc29f8f41dbd289a147ce9da3113b5f0b8c00a60b1ce1d7e819d7a431d7c90ea0e5f);
41cb1beb 465
ceb3f0c0 466 for (i = 0; i < n; i++) {
467 ec_mul(c, &d, &g, r);
468 if (EC_ATINF(&d)) {
469 fprintf(stderr, "zero too early\n");
470 return (1);
471 }
472 ec_add(c, &d, &d, &g);
473 if (!EC_ATINF(&d)) {
474 fprintf(stderr, "didn't reach zero\n");
475 MP_EPRINT("d.x", d.x);
476 MP_EPRINT("d.y", d.y);
477 return (1);
478 }
479 ec_destroy(&d);
dbfee00a 480 }
41cb1beb 481 ec_destroy(&g);
482 ec_destroycurve(c);
483 F_DESTROY(f);
dbfee00a 484 MP_DROP(p); MP_DROP(a); MP_DROP(b); MP_DROP(r);
485 assert(!mparena_count(&mparena_global));
486 printf("ok\n");
41cb1beb 487 return (0);
488}
489
490#endif
491
b0ab12e6 492/*----- That's all, folks -------------------------------------------------*/
Catacomb cryptographic library (downstream of http://git.distorted.org.uk/~mdw/catacomb/)