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Generate, store and retreive elliptic curve keys.
[u/mdw/catacomb] / ec-bin.c
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1/* -*-c-*-
2 *
3 * $Id: ec-bin.c,v 1.5 2004/03/27 17:54:11 mdw Exp $
4 *
5 * Arithmetic for elliptic curves over binary fields
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: ec-bin.c,v $
33 * Revision 1.5 2004/03/27 17:54:11 mdw
34 * Standard curves and curve checking.
35 *
36 * Revision 1.4 2004/03/23 15:19:32 mdw
37 * Test elliptic curves more thoroughly.
38 *
39 * Revision 1.3 2004/03/22 02:19:09 mdw
40 * Rationalise the sliding-window threshold. Drop guarantee that right
41 * arguments to EC @add@ are canonical, and fix up projective implementations
42 * to cope.
43 *
44 * Revision 1.2 2004/03/21 22:52:06 mdw
45 * Merge and close elliptic curve branch.
46 *
47 * Revision 1.1.2.1 2004/03/21 22:39:46 mdw
48 * Elliptic curves on binary fields work.
49 *
50 */
51
52/*----- Header files ------------------------------------------------------*/
53
54#include <mLib/sub.h>
55
56#include "ec.h"
57
58/*----- Data structures ---------------------------------------------------*/
59
60typedef struct ecctx {
61 ec_curve c;
62 mp *bb;
63} ecctx;
64
65/*----- Main code ---------------------------------------------------------*/
66
67static const ec_ops ec_binops, ec_binprojops;
68
69static ec *ecneg(ec_curve *c, ec *d, const ec *p)
70{
71 EC_COPY(d, p);
72 if (d->x)
73 d->y = F_ADD(c->f, d->y, d->y, d->x);
74 return (d);
75}
76
77static ec *ecprojneg(ec_curve *c, ec *d, const ec *p)
78{
79 EC_COPY(d, p);
80 if (d->x) {
81 mp *t = F_MUL(c->f, MP_NEW, d->x, d->z);
82 d->y = F_ADD(c->f, d->y, d->y, t);
83 MP_DROP(t);
84 }
85 return (d);
86}
87
88static ec *ecfind(ec_curve *c, ec *d, mp *x)
89{
90 field *f = c->f;
91 mp *y, *u, *v;
92
93 if (F_ZEROP(f, x))
94 y = F_SQRT(f, MP_NEW, c->b);
95 else {
96 u = F_SQR(f, MP_NEW, x); /* %$x^2$% */
97 y = F_MUL(f, MP_NEW, u, c->a); /* %$a x^2$% */
98 y = F_ADD(f, y, y, c->b); /* %$a x^2 + b$% */
99 v = F_MUL(f, MP_NEW, u, x); /* %$x^3$% */
100 y = F_ADD(f, y, y, v); /* %$A = x^3 + a x^2 + b$% */
101 if (!F_ZEROP(f, y)) {
102 u = F_INV(f, u, u); /* %$x^{-2}$% */
103 v = F_MUL(f, v, u, y); /* %$B = A x^{-2} = x + a + b x^{-2}$% */
104 y = F_QUADSOLVE(f, y, v); /* %$z^2 + z = B$% */
105 if (y) y = F_MUL(f, y, y, x); /* %$y = z x$% */
106 }
107 MP_DROP(u);
108 MP_DROP(v);
109 }
110 if (!y) return (0);
111 EC_DESTROY(d);
112 d->x = MP_COPY(x);
113 d->y = y;
114 d->z = MP_COPY(f->one);
115 return (d);
116}
117
118static ec *ecdbl(ec_curve *c, ec *d, const ec *a)
119{
120 if (EC_ATINF(a) || F_ZEROP(c->f, a->x))
121 EC_SETINF(d);
122 else {
123 field *f = c->f;
124 mp *lambda;
125 mp *dx, *dy;
126
127 dx = F_INV(f, MP_NEW, a->x); /* %$x^{-1}$% */
128 dy = F_MUL(f, MP_NEW, dx, a->y); /* %$y/x$% */
129 lambda = F_ADD(f, dy, dy, a->x); /* %$\lambda = x + y/x$% */
130
131 dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
132 dx = F_ADD(f, dx, dx, lambda); /* %$\lambda^2 + \lambda$% */
133 dx = F_ADD(f, dx, dx, c->a); /* %$x' = a + \lambda^2 + \lambda$% */
134
135 dy = F_ADD(f, MP_NEW, a->x, dx); /* %$ x + x' $% */
136 dy = F_MUL(f, dy, dy, lambda); /* %$ (x + x') \lambda$% */
137 dy = F_ADD(f, dy, dy, a->y); /* %$ (x + x') \lambda + y$% */
138 dy = F_ADD(f, dy, dy, dx); /* %$ y' = (x + x') \lambda + y + x'$% */
139
140 EC_DESTROY(d);
141 d->x = dx;
142 d->y = dy;
143 d->z = 0;
144 MP_DROP(lambda);
145 }
146 return (d);
147}
148
149static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a)
150{
151 if (EC_ATINF(a) || F_ZEROP(c->f, a->x))
152 EC_SETINF(d);
153 else {
154 field *f = c->f;
155 ecctx *cc = (ecctx *)c;
156 mp *dx, *dy, *dz, *u, *v;
157
158 dy = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */
159 dx = F_MUL(f, MP_NEW, dy, cc->bb); /* %$c z^2$% */
160 dx = F_ADD(f, dx, dx, a->x); /* %$x + c z^2$% */
161 dz = F_SQR(f, MP_NEW, dx); /* %$(x + c z^2)^2$% */
162 dx = F_SQR(f, dx, dz); /* %$x' = (x + c z^2)^4$% */
163
164 dz = F_MUL(f, dz, dy, a->x); /* %$z' = x z^2$% */
165
166 dy = F_SQR(f, dy, a->x); /* %$x^2$% */
167 u = F_MUL(f, MP_NEW, a->y, a->z); /* %$y z$% */
168 u = F_ADD(f, u, u, dz); /* %$z' + y z$% */
169 u = F_ADD(f, u, u, dy); /* %$u = z' + x^2 + y z$% */
170
171 v = F_SQR(f, MP_NEW, dy); /* %$x^4$% */
172 dy = F_MUL(f, dy, v, dz); /* %$x^4 z'$% */
173 v = F_MUL(f, v, u, dx); /* %$u x'$% */
174 dy = F_ADD(f, dy, dy, v); /* %$y' = x^4 z' + u x'$% */
175
176 EC_DESTROY(d);
177 d->x = dx;
178 d->y = dy;
179 d->z = dz;
180 MP_DROP(u);
181 MP_DROP(v);
182 assert(!(d->x->f & MP_DESTROYED));
183 assert(!(d->y->f & MP_DESTROYED));
184 assert(!(d->z->f & MP_DESTROYED));
185 }
186 return (d);
187}
188
189static ec *ecadd(ec_curve *c, ec *d, const ec *a, const ec *b)
190{
191 if (a == b)
192 ecdbl(c, d, a);
193 else if (EC_ATINF(a))
194 EC_COPY(d, b);
195 else if (EC_ATINF(b))
196 EC_COPY(d, a);
197 else {
198 field *f = c->f;
199 mp *lambda;
200 mp *dx, *dy;
201
202 if (!MP_EQ(a->x, b->x)) {
203 dx = F_ADD(f, MP_NEW, a->x, b->x); /* %$x_0 + x_1$% */
204 dy = F_INV(f, MP_NEW, dx); /* %$(x_0 + x_1)^{-1}$% */
205 dx = F_ADD(f, dx, a->y, b->y); /* %$y_0 + y_1$% */
206 lambda = F_MUL(f, MP_NEW, dy, dx);
207 /* %$\lambda = (y_0 + y_1)/(x_0 + x_1)$% */
208
209 dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
210 dx = F_ADD(f, dx, dx, lambda); /* %$\lambda^2 + \lambda$% */
211 dx = F_ADD(f, dx, dx, c->a); /* %$a + \lambda^2 + \lambda$% */
212 dx = F_ADD(f, dx, dx, a->x); /* %$a + \lambda^2 + \lambda + x_0$% */
213 dx = F_ADD(f, dx, dx, b->x);
214 /* %$x' = a + \lambda^2 + \lambda + x_0 + x_1$% */
215 } else if (!MP_EQ(a->y, b->y) || F_ZEROP(f, a->x)) {
216 EC_SETINF(d);
217 return (d);
218 } else {
219 dx = F_INV(f, MP_NEW, a->x); /* %$x^{-1}$% */
220 dy = F_MUL(f, MP_NEW, dx, a->y); /* %$y/x$% */
221 lambda = F_ADD(f, dy, dy, a->x); /* %$\lambda = x + y/x$% */
222
223 dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
224 dx = F_ADD(f, dx, dx, lambda); /* %$\lambda^2 + \lambda$% */
225 dx = F_ADD(f, dx, dx, c->a); /* %$x' = a + \lambda^2 + \lambda$% */
226 dy = MP_NEW;
227 }
228
229 dy = F_ADD(f, dy, a->x, dx); /* %$ x + x' $% */
230 dy = F_MUL(f, dy, dy, lambda); /* %$ (x + x') \lambda$% */
231 dy = F_ADD(f, dy, dy, a->y); /* %$ (x + x') \lambda + y$% */
232 dy = F_ADD(f, dy, dy, dx); /* %$ y' = (x + x') \lambda + y + x'$% */
233
234 EC_DESTROY(d);
235 d->x = dx;
236 d->y = dy;
237 d->z = 0;
238 MP_DROP(lambda);
239 }
240 return (d);
241}
242
243static ec *ecprojadd(ec_curve *c, ec *d, const ec *a, const ec *b)
244{
245 if (a == b)
246 c->ops->dbl(c, d, a);
247 else if (EC_ATINF(a))
248 EC_COPY(d, b);
249 else if (EC_ATINF(b))
250 EC_COPY(d, a);
251 else {
252 field *f = c->f;
253 mp *dx, *dy, *dz, *u, *uu, *v, *t, *s, *ss, *r, *w, *l;
254
255 dz = F_SQR(f, MP_NEW, b->z); /* %$z_1^2$% */
256 u = F_MUL(f, MP_NEW, dz, a->x); /* %$u_0 = x_0 z_1^2$% */
257 t = F_MUL(f, MP_NEW, dz, b->z); /* %$z_1^3$% */
258 s = F_MUL(f, MP_NEW, t, a->y); /* %$s_0 = y_0 z_1^3$% */
259
260 dz = F_SQR(f, dz, a->z); /* %$z_0^2$% */
261 uu = F_MUL(f, MP_NEW, dz, b->x); /* %$u_1 = x_1 z_0^2$% */
262 t = F_MUL(f, t, dz, a->z); /* %$z_0^3$% */
263 ss = F_MUL(f, MP_NEW, t, b->y); /* %$s_1 = y_1 z_0^3$% */
264
265 w = F_ADD(f, u, u, uu); /* %$r = u_0 + u_1$% */
266 r = F_ADD(f, s, s, ss); /* %$w = s_0 + s_1$% */
267 if (F_ZEROP(f, w)) {
268 MP_DROP(w);
269 MP_DROP(uu);
270 MP_DROP(ss);
271 MP_DROP(t);
272 MP_DROP(dz);
273 if (F_ZEROP(f, r)) {
274 MP_DROP(r);
275 return (c->ops->dbl(c, d, a));
276 } else {
277 MP_DROP(r);
278 EC_SETINF(d);
279 return (d);
280 }
281 }
282
283 l = F_MUL(f, t, a->z, w); /* %$l = z_0 w$% */
284
285 dz = F_MUL(f, dz, l, b->z); /* %$z' = l z_1$% */
286
287 ss = F_MUL(f, ss, r, b->x); /* %$r x_1$% */
288 t = F_MUL(f, uu, l, b->y); /* %$l y_1$% */
289 v = F_ADD(f, ss, ss, t); /* %$v = r x_1 + l y_1$% */
290
291 t = F_ADD(f, t, r, dz); /* %$t = r + z'$% */
292
293 uu = F_SQR(f, MP_NEW, dz); /* %$z'^2$% */
294 dx = F_MUL(f, MP_NEW, uu, c->a); /* %$a z'^2$% */
295 uu = F_MUL(f, uu, t, r); /* %$t r$% */
296 dx = F_ADD(f, dx, dx, uu); /* %$a z'^2 + t r$% */
297 r = F_SQR(f, r, w); /* %$w^2$% */
298 uu = F_MUL(f, uu, r, w); /* %$w^3$% */
299 dx = F_ADD(f, dx, dx, uu); /* %$x' = a z'^2 + t r + w^3$% */
300
301 r = F_SQR(f, r, l); /* %$l^2$% */
302 dy = F_MUL(f, uu, v, r); /* %$v l^2$% */
303 l = F_MUL(f, l, t, dx); /* %$t x'$% */
304 dy = F_ADD(f, dy, dy, l); /* %$y' = t x' + v l^2$% */
305
306 EC_DESTROY(d);
307 d->x = dx;
308 d->y = dy;
309 d->z = dz;
310 MP_DROP(l);
311 MP_DROP(r);
312 MP_DROP(w);
313 MP_DROP(t);
314 MP_DROP(v);
315 }
316 return (d);
317}
318
319static int eccheck(ec_curve *c, const ec *p)
320{
321 field *f = c->f;
322 int rc;
323 mp *u, *v;
324
325 v = F_SQR(f, MP_NEW, p->x);
326 u = F_MUL(f, MP_NEW, v, p->x);
327 v = F_MUL(f, v, v, c->a);
328 u = F_ADD(f, u, u, v);
329 u = F_ADD(f, u, u, c->b);
330 v = F_MUL(f, v, p->x, p->y);
331 u = F_ADD(f, u, u, v);
332 v = F_SQR(f, v, p->y);
333 u = F_ADD(f, u, u, v);
334 rc = F_ZEROP(f, u) ? 0 : -1;
335 mp_drop(u);
336 mp_drop(v);
337 return (rc);
338}
339
340static int ecprojcheck(ec_curve *c, const ec *p)
341{
342 ec t = EC_INIT;
343 int rc;
344
345 c->ops->fix(c, &t, p);
346 rc = eccheck(c, &t);
347 EC_DESTROY(&t);
348 return (rc);
349}
350
351static void ecdestroy(ec_curve *c)
352{
353 ecctx *cc = (ecctx *)c;
354 MP_DROP(cc->c.a);
355 MP_DROP(cc->c.b);
356 if (cc->bb) MP_DROP(cc->bb);
357 DESTROY(cc);
358}
359
360/* --- @ec_bin@, @ec_binproj@ --- *
361 *
362 * Arguments: @field *f@ = the underlying field for this elliptic curve
363 * @mp *a, *b@ = the coefficients for this curve
364 *
365 * Returns: A pointer to the curve.
366 *
367 * Use: Creates a curve structure for an elliptic curve defined over
368 * a binary field. The @binproj@ variant uses projective
369 * coordinates, which can be a win.
370 */
371
372ec_curve *ec_bin(field *f, mp *a, mp *b)
373{
374 ecctx *cc = CREATE(ecctx);
375 cc->c.ops = &ec_binops;
376 cc->c.f = f;
377 cc->c.a = F_IN(f, MP_NEW, a);
378 cc->c.b = F_IN(f, MP_NEW, b);
379 cc->bb = 0;
380 return (&cc->c);
381}
382
383ec_curve *ec_binproj(field *f, mp *a, mp *b)
384{
385 ecctx *cc = CREATE(ecctx);
386 cc->c.ops = &ec_binprojops;
387 cc->c.f = f;
388 cc->c.a = F_IN(f, MP_NEW, a);
389 cc->c.b = F_IN(f, MP_NEW, b);
390 cc->bb = F_SQRT(f, MP_NEW, b);
391 cc->bb = F_SQRT(f, cc->bb, cc->bb);
392 return (&cc->c);
393}
394
395static const ec_ops ec_binops = {
396 ecdestroy, ec_idin, ec_idout, ec_idfix,
397 ecfind, ecneg, ecadd, ec_stdsub, ecdbl, eccheck
398};
399
400static const ec_ops ec_binprojops = {
401 ecdestroy, ec_projin, ec_projout, ec_projfix,
402 ecfind, ecprojneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck
403};
404
405/*----- Test rig ----------------------------------------------------------*/
406
407#ifdef TEST_RIG
408
409#define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
410
411int main(int argc, char *argv[])
412{
413 field *f;
414 ec_curve *c;
415 ec g = EC_INIT, d = EC_INIT;
416 mp *p, *a, *b, *r;
417 int i, n = argc == 1 ? 1 : atoi(argv[1]);
418
419 printf("ec-bin: ");
420 fflush(stdout);
421 a = MP(1);
422 b = MP(0x021a5c2c8ee9feb5c4b9a753b7b476b7fd6422ef1f3dd674761fa99d6ac27c8a9a197b272822f6cd57a55aa4f50ae317b13545f);
423 p = MP(0x2000000000000000000000000000000000000000000000000000000000000000000000000000000008000000000000000000001);
424 r =
425 MP(661055968790248598951915308032771039828404682964281219284648798304157774827374805208143723762179110965979867288366567526770);
426
427 f = field_binpoly(p);
428 c = ec_binproj(f, a, b);
429
430 g.x = MP(0x15d4860d088ddb3496b0c6064756260441cde4af1771d4db01ffe5b34e59703dc255a868a1180515603aeab60794e54bb7996a7);
431 g.y = MP(0x061b1cfab6be5f32bbfa78324ed106a7636b9c5a7bd198d0158aa4f5488d08f38514f1fdf4b4f40d2181b3681c364ba0273c706);
432
433 for (i = 0; i < n; i++) {
434 ec_mul(c, &d, &g, r);
435 if (EC_ATINF(&d)) {
436 fprintf(stderr, "zero too early\n");
437 return (1);
438 }
439 ec_add(c, &d, &d, &g);
440 if (!EC_ATINF(&d)) {
441 fprintf(stderr, "didn't reach zero\n");
442 MP_EPRINTX("d.x", d.x);
443 MP_EPRINTX("d.y", d.y);
444 return (1);
445 }
446 ec_destroy(&d);
447 }
448
449 ec_destroy(&g);
450 ec_destroycurve(c);
451 F_DESTROY(f);
452 MP_DROP(p); MP_DROP(a); MP_DROP(b); MP_DROP(r);
453 assert(!mparena_count(&mparena_global));
454 printf("ok\n");
455 return (0);
456}
457
458#endif
459
460/*----- That's all, folks -------------------------------------------------*/
Catacomb cryptographic library (downstream of http://git.distorted.org.uk/~mdw/catacomb/)