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Elliptic curves on binary fields work.
[u/mdw/catacomb] / ec-bin.c
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1/* -*-c-*-
2 *
3 * $Id: ec-bin.c,v 1.1.2.1 2004/03/21 22:39:46 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 1.1.2.1 2004/03/21 22:39:46 mdw
34 * Elliptic curves on binary fields work.
35 *
36 */
37
38/*----- Header files ------------------------------------------------------*/
39
40#include <mLib/sub.h>
41
42#include "ec.h"
43
44/*----- Data structures ---------------------------------------------------*/
45
46typedef struct ecctx {
47 ec_curve c;
48 mp *a, *b;
49 mp *bb;
50} ecctx;
51
52/*----- Main code ---------------------------------------------------------*/
53
54static const ec_ops ec_binops, ec_binprojops;
55
56static ec *ecneg(ec_curve *c, ec *d, const ec *p)
57{
58 EC_COPY(d, p);
59 if (d->x)
60 d->y = F_ADD(c->f, d->y, d->y, d->x);
61 return (d);
62}
63
64static ec *ecprojneg(ec_curve *c, ec *d, const ec *p)
65{
66 EC_COPY(d, p);
67 if (d->x) {
68 mp *t = F_MUL(c->f, MP_NEW, d->x, d->z);
69 d->y = F_ADD(c->f, d->y, d->y, t);
70 MP_DROP(t);
71 }
72 return (d);
73}
74
75static ec *ecfind(ec_curve *c, ec *d, mp *x)
76{
77 /* write me */
78 return (0);
79}
80
81static ec *ecdbl(ec_curve *c, ec *d, const ec *a)
82{
83 if (EC_ATINF(a) || F_ZEROP(c->f, a->x))
84 EC_SETINF(d);
85 else {
86 field *f = c->f;
87 ecctx *cc = (ecctx *)c;
88 mp *lambda;
89 mp *dx, *dy;
90
91 dx = F_INV(f, MP_NEW, a->x); /* %$x^{-1}$% */
92 dy = F_MUL(f, MP_NEW, dx, a->y); /* %$y/x$% */
93 lambda = F_ADD(f, dy, dy, a->x); /* %$\lambda = x + y/x$% */
94
95 dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
96 dx = F_ADD(f, dx, dx, lambda); /* %$\lambda^2 + \lambda$% */
97 dx = F_ADD(f, dx, dx, cc->a); /* %$x' = a + \lambda^2 + \lambda$% */
98
99 dy = F_ADD(f, MP_NEW, a->x, dx); /* %$ x + x' $% */
100 dy = F_MUL(f, dy, dy, lambda); /* %$ (x + x') \lambda$% */
101 dy = F_ADD(f, dy, dy, a->y); /* %$ (x + x') \lambda + y$% */
102 dy = F_ADD(f, dy, dy, dx); /* %$ y' = (x + x') \lambda + y + x'$% */
103
104 EC_DESTROY(d);
105 d->x = dx;
106 d->y = dy;
107 d->z = 0;
108 MP_DROP(lambda);
109 }
110 return (d);
111}
112
113static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a)
114{
115 if (EC_ATINF(a) || F_ZEROP(c->f, a->x))
116 EC_SETINF(d);
117 else {
118 field *f = c->f;
119 ecctx *cc = (ecctx *)c;
120 mp *dx, *dy, *dz, *u, *v;
121
122 dy = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */
123 dx = F_MUL(f, MP_NEW, dy, cc->bb); /* %$c z^2$% */
124 dx = F_ADD(f, dx, dx, a->x); /* %$x + c z^2$% */
125 dz = F_SQR(f, MP_NEW, dx); /* %$(x + c z^2)^2$% */
126 dx = F_SQR(f, dx, dz); /* %$x' = (x + c z^2)^4$% */
127
128 dz = F_MUL(f, dz, dy, a->x); /* %$z' = x z^2$% */
129
130 dy = F_SQR(f, dy, a->x); /* %$x^2$% */
131 u = F_MUL(f, MP_NEW, a->y, a->z); /* %$y z$% */
132 u = F_ADD(f, u, u, dz); /* %$z' + y z$% */
133 u = F_ADD(f, u, u, dy); /* %$u = z' + x^2 + y z$% */
134
135 v = F_SQR(f, MP_NEW, dy); /* %$x^4$% */
136 dy = F_MUL(f, dy, v, dz); /* %$x^4 z'$% */
137 v = F_MUL(f, v, u, dx); /* %$u x'$% */
138 dy = F_ADD(f, dy, dy, v); /* %$y' = x^4 z' + u x'$% */
139
140 EC_DESTROY(d);
141 d->x = dx;
142 d->y = dy;
143 d->z = dz;
144 MP_DROP(u);
145 MP_DROP(v);
146 assert(!(d->x->f & MP_DESTROYED));
147 assert(!(d->y->f & MP_DESTROYED));
148 assert(!(d->z->f & MP_DESTROYED));
149 }
150 return (d);
151}
152
153static ec *ecadd(ec_curve *c, ec *d, const ec *a, const ec *b)
154{
155 if (a == b)
156 ecdbl(c, d, a);
157 else if (EC_ATINF(a))
158 EC_COPY(d, b);
159 else if (EC_ATINF(b))
160 EC_COPY(d, a);
161 else {
162 field *f = c->f;
163 ecctx *cc = (ecctx *)c;
164 mp *lambda;
165 mp *dx, *dy;
166
167 if (!MP_EQ(a->x, b->x)) {
168 dx = F_ADD(f, MP_NEW, a->x, b->x); /* %$x_0 + x_1$% */
169 dy = F_INV(f, MP_NEW, dx); /* %$(x_0 + x_1)^{-1}$% */
170 dx = F_ADD(f, dx, a->y, b->y); /* %$y_0 + y_1$% */
171 lambda = F_MUL(f, MP_NEW, dy, dx);
172 /* %$\lambda = (y_0 + y_1)/(x_0 + x_1)$% */
173
174 dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
175 dx = F_ADD(f, dx, dx, lambda); /* %$\lambda^2 + \lambda$% */
176 dx = F_ADD(f, dx, dx, cc->a); /* %$a + \lambda^2 + \lambda$% */
177 dx = F_ADD(f, dx, dx, a->x); /* %$a + \lambda^2 + \lambda + x_0$% */
178 dx = F_ADD(f, dx, dx, b->x);
179 /* %$x' = a + \lambda^2 + \lambda + x_0 + x_1$% */
180 } else if (!MP_EQ(a->y, b->y) || F_ZEROP(f, a->x)) {
181 EC_SETINF(d);
182 return (d);
183 } else {
184 dx = F_INV(f, MP_NEW, a->x); /* %$x^{-1}$% */
185 dy = F_MUL(f, MP_NEW, dx, a->y); /* %$y/x$% */
186 lambda = F_ADD(f, dy, dy, a->x); /* %$\lambda = x + y/x$% */
187
188 dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
189 dx = F_ADD(f, dx, dx, lambda); /* %$\lambda^2 + \lambda$% */
190 dx = F_ADD(f, dx, dx, cc->a); /* %$x' = a + \lambda^2 + \lambda$% */
191 dy = MP_NEW;
192 }
193
194 dy = F_ADD(f, dy, a->x, dx); /* %$ x + x' $% */
195 dy = F_MUL(f, dy, dy, lambda); /* %$ (x + x') \lambda$% */
196 dy = F_ADD(f, dy, dy, a->y); /* %$ (x + x') \lambda + y$% */
197 dy = F_ADD(f, dy, dy, dx); /* %$ y' = (x + x') \lambda + y + x'$% */
198
199 EC_DESTROY(d);
200 d->x = dx;
201 d->y = dy;
202 d->z = 0;
203 MP_DROP(lambda);
204 }
205 return (d);
206}
207
208static ec *ecprojadd(ec_curve *c, ec *d, const ec *a, const ec *b)
209{
210 if (a == b)
211 c->ops->dbl(c, d, a);
212 else if (EC_ATINF(a))
213 EC_COPY(d, b);
214 else if (EC_ATINF(b))
215 EC_COPY(d, a);
216 else {
217 field *f = c->f;
218 ecctx *cc = (ecctx *)c;
219 mp *dx, *dy, *dz, *u, *uu, *v, *t, *s, *ss, *r, *w, *l;
220
221 dz = F_SQR(f, MP_NEW, b->z); /* %$z_1^2$% */
222 u = F_MUL(f, MP_NEW, dz, a->x); /* %$u_0 = x_0 z_1^2$% */
223 t = F_MUL(f, MP_NEW, dz, b->z); /* %$z_1^3$% */
224 s = F_MUL(f, MP_NEW, t, a->y); /* %$s_0 = y_0 z_1^3$% */
225
226 dz = F_SQR(f, dz, a->z); /* %$z_0^2$% */
227 uu = F_MUL(f, MP_NEW, dz, b->x); /* %$u_1 = x_1 z_0^2$% */
228 t = F_MUL(f, t, dz, a->z); /* %$z_0^3$% */
229 ss = F_MUL(f, MP_NEW, t, b->y); /* %$s_1 = y_1 z_0^3$% */
230
231 w = F_ADD(f, u, u, uu); /* %$r = u_0 + u_1$% */
232 r = F_ADD(f, s, s, ss); /* %$w = s_0 + s_1$% */
233 if (F_ZEROP(f, w)) {
234 MP_DROP(w);
235 MP_DROP(uu);
236 MP_DROP(ss);
237 MP_DROP(t);
238 MP_DROP(dz);
239 if (F_ZEROP(f, r)) {
240 MP_DROP(r);
241 return (c->ops->dbl(c, d, a));
242 } else {
243 MP_DROP(r);
244 EC_SETINF(d);
245 return (d);
246 }
247 }
248
249 l = F_MUL(f, t, a->z, w); /* %$l = z_0 w$% */
250
251 dz = F_MUL(f, dz, l, b->z); /* %$z' = l z_1$% */
252
253 ss = F_MUL(f, ss, r, b->x); /* %$r x_1$% */
254 t = F_MUL(f, uu, l, b->y); /* %$l y_1$% */
255 v = F_ADD(f, ss, ss, t); /* %$v = r x_1 + l y_1$% */
256
257 t = F_ADD(f, t, r, dz); /* %$t = r + z'$% */
258
259 uu = F_SQR(f, MP_NEW, dz); /* %$z'^2$% */
260 dx = F_MUL(f, MP_NEW, uu, cc->a); /* %$a z'^2$% */
261 uu = F_MUL(f, uu, t, r); /* %$t r$% */
262 dx = F_ADD(f, dx, dx, uu); /* %$a z'^2 + t r$% */
263 r = F_SQR(f, r, w); /* %$w^2$% */
264 uu = F_MUL(f, uu, r, w); /* %$w^3$% */
265 dx = F_ADD(f, dx, dx, uu); /* %$x' = a z'^2 + t r + w^3$% */
266
267 r = F_SQR(f, r, l); /* %$l^2$% */
268 dy = F_MUL(f, uu, v, r); /* %$v l^2$% */
269 l = F_MUL(f, l, t, dx); /* %$t x'$% */
270 dy = F_ADD(f, dy, dy, l); /* %$y' = t x' + v l^2$% */
271
272 EC_DESTROY(d);
273 d->x = dx;
274 d->y = dy;
275 d->z = dz;
276 MP_DROP(l);
277 MP_DROP(r);
278 MP_DROP(w);
279 MP_DROP(t);
280 MP_DROP(v);
281 }
282 return (d);
283}
284
285static int eccheck(ec_curve *c, const ec *p)
286{
287 ecctx *cc = (ecctx *)c;
288 field *f = c->f;
289 int rc;
290 mp *u, *v;
291
292 v = F_SQR(f, MP_NEW, p->x);
293 u = F_MUL(f, MP_NEW, v, p->x);
294 v = F_MUL(f, v, v, cc->a);
295 u = F_ADD(f, u, u, v);
296 u = F_ADD(f, u, u, cc->b);
297 v = F_MUL(f, v, p->x, p->y);
298 u = F_ADD(f, u, u, v);
299 v = F_SQR(f, v, p->y);
300 u = F_ADD(f, u, u, v);
301 rc = F_ZEROP(f, u);
302 mp_drop(u);
303 mp_drop(v);
304 return (rc);
305}
306
307static int ecprojcheck(ec_curve *c, const ec *p)
308{
309 ec t = EC_INIT;
310 int rc;
311
312 c->ops->fix(c, &t, p);
313 rc = eccheck(c, &t);
314 EC_DESTROY(&t);
315 return (rc);
316}
317
318static void ecdestroy(ec_curve *c)
319{
320 ecctx *cc = (ecctx *)c;
321 MP_DROP(cc->a);
322 MP_DROP(cc->b);
323 if (cc->bb) MP_DROP(cc->bb);
324 DESTROY(cc);
325}
326
327/* --- @ec_bin@, @ec_binproj@ --- *
328 *
329 * Arguments: @field *f@ = the underlying field for this elliptic curve
330 * @mp *a, *b@ = the coefficients for this curve
331 *
332 * Returns: A pointer to the curve.
333 *
334 * Use: Creates a curve structure for an elliptic curve defined over
335 * a binary field. The @binproj@ variant uses projective
336 * coordinates, which can be a win.
337 */
338
339ec_curve *ec_bin(field *f, mp *a, mp *b)
340{
341 ecctx *cc = CREATE(ecctx);
342 cc->c.ops = &ec_binops;
343 cc->c.f = f;
344 cc->a = F_IN(f, MP_NEW, a);
345 cc->b = F_IN(f, MP_NEW, b);
346 cc->bb = 0;
347 return (&cc->c);
348}
349
350ec_curve *ec_binproj(field *f, mp *a, mp *b)
351{
352 ecctx *cc = CREATE(ecctx);
353 cc->c.ops = &ec_binprojops;
354 cc->c.f = f;
355 cc->a = F_IN(f, MP_NEW, a);
356 cc->b = F_IN(f, MP_NEW, b);
357 cc->bb = F_SQRT(f, MP_NEW, b);
358 cc->bb = F_SQRT(f, cc->bb, cc->bb);
359 return (&cc->c);
360}
361
362static const ec_ops ec_binops = {
363 ecdestroy, ec_idin, ec_idout, ec_idfix,
364 0, ecneg, ecadd, ec_stdsub, ecdbl, eccheck
365};
366
367static const ec_ops ec_binprojops = {
368 ecdestroy, ec_projin, ec_projout, ec_projfix,
369 0, ecprojneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck
370};
371
372/*----- Test rig ----------------------------------------------------------*/
373
374#ifdef TEST_RIG
375
376#define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
377
378int main(int argc, char *argv[])
379{
380 field *f;
381 ec_curve *c;
382 ec g = EC_INIT, d = EC_INIT;
383 mp *p, *a, *b, *r;
384 int i, n = argc == 1 ? 1 : atoi(argv[1]);
385
386 printf("ec-bin: ");
387 fflush(stdout);
388 a = MP(1);
389 b = MP(0x066647ede6c332c7f8c0923bb58213b333b20e9ce4281fe115f7d8f90ad);
390 p = MP(0x20000000000000000000000000000000000000004000000000000000001);
391 r =
392 MP(6901746346790563787434755862277025555839812737345013555379383634485462);
393
394 f = field_binpoly(p);
395 c = ec_binproj(f, a, b);
396
397 g.x = MP(0x0fac9dfcbac8313bb2139f1bb755fef65bc391f8b36f8f8eb7371fd558b);
398 g.y = MP(0x1006a08a41903350678e58528bebf8a0beff867a7ca36716f7e01f81052);
399
400 for (i = 0; i < n; i++) {
401 ec_mul(c, &d, &g, r);
402 if (EC_ATINF(&d)) {
403 fprintf(stderr, "zero too early\n");
404 return (1);
405 }
406 ec_add(c, &d, &d, &g);
407 if (!EC_ATINF(&d)) {
408 fprintf(stderr, "didn't reach zero\n");
409 MP_EPRINTX("d.x", d.x);
410 MP_EPRINTX("d.y", d.y);
411 MP_EPRINTX("d.z", d.y);
412 return (1);
413 }
414 ec_destroy(&d);
415 }
416
417 ec_destroy(&g);
418 ec_destroycurve(c);
419 F_DESTROY(f);
420 MP_DROP(p); MP_DROP(a); MP_DROP(b); MP_DROP(r);
421 assert(!mparena_count(&mparena_global));
422 printf("ok\n");
423 return (0);
424}
425
426#endif
427
428/*----- That's all, folks -------------------------------------------------*/
Catacomb cryptographic library (downstream of http://git.distorted.org.uk/~mdw/catacomb/)