ceb3f0c0 |
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 | |
46 | typedef struct ecctx { |
47 | ec_curve c; |
48 | mp *a, *b; |
49 | mp *bb; |
50 | } ecctx; |
51 | |
52 | /*----- Main code ---------------------------------------------------------*/ |
53 | |
54 | static const ec_ops ec_binops, ec_binprojops; |
55 | |
56 | static 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 | |
64 | static 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 | |
75 | static ec *ecfind(ec_curve *c, ec *d, mp *x) |
76 | { |
77 | /* write me */ |
78 | return (0); |
79 | } |
80 | |
81 | static 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'$% */ |
103 | |
104 | EC_DESTROY(d); |
105 | d->x = dx; |
106 | d->y = dy; |
107 | d->z = 0; |
108 | MP_DROP(lambda); |
109 | } |
110 | return (d); |
111 | } |
112 | |
113 | static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a) |
114 | { |
115 | if (EC_ATINF(a) || F_ZEROP(c->f, a->x)) |
116 | EC_SETINF(d); |
117 | else { |
118 | field *f = c->f; |
119 | ecctx *cc = (ecctx *)c; |
120 | mp *dx, *dy, *dz, *u, *v; |
121 | |
122 | dy = F_SQR(f, MP_NEW, a->z); /* %$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'$% */ |
139 | |
140 | EC_DESTROY(d); |
141 | d->x = dx; |
142 | d->y = dy; |
143 | d->z = dz; |
144 | MP_DROP(u); |
145 | MP_DROP(v); |
146 | assert(!(d->x->f & MP_DESTROYED)); |
147 | assert(!(d->y->f & MP_DESTROYED)); |
148 | assert(!(d->z->f & MP_DESTROYED)); |
149 | } |
150 | return (d); |
151 | } |
152 | |
153 | static ec *ecadd(ec_curve *c, ec *d, const ec *a, const ec *b) |
154 | { |
155 | if (a == b) |
156 | ecdbl(c, d, a); |
157 | else if (EC_ATINF(a)) |
158 | EC_COPY(d, b); |
159 | else if (EC_ATINF(b)) |
160 | EC_COPY(d, a); |
161 | else { |
162 | field *f = c->f; |
163 | ecctx *cc = (ecctx *)c; |
164 | mp *lambda; |
165 | mp *dx, *dy; |
166 | |
167 | if (!MP_EQ(a->x, b->x)) { |
168 | dx = F_ADD(f, MP_NEW, a->x, b->x); /* %$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'$% */ |
198 | |
199 | EC_DESTROY(d); |
200 | d->x = dx; |
201 | d->y = dy; |
202 | d->z = 0; |
203 | MP_DROP(lambda); |
204 | } |
205 | return (d); |
206 | } |
207 | |
208 | static ec *ecprojadd(ec_curve *c, ec *d, const ec *a, const ec *b) |
209 | { |
210 | if (a == b) |
211 | c->ops->dbl(c, d, a); |
212 | else if (EC_ATINF(a)) |
213 | EC_COPY(d, b); |
214 | else if (EC_ATINF(b)) |
215 | EC_COPY(d, a); |
216 | else { |
217 | field *f = c->f; |
218 | ecctx *cc = (ecctx *)c; |
219 | mp *dx, *dy, *dz, *u, *uu, *v, *t, *s, *ss, *r, *w, *l; |
220 | |
221 | dz = F_SQR(f, MP_NEW, b->z); /* %$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 | |
285 | static 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 | |
307 | static 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 | |
318 | static 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 | |
339 | ec_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 | |
350 | ec_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 | |
362 | static const ec_ops ec_binops = { |
363 | ecdestroy, ec_idin, ec_idout, ec_idfix, |
364 | 0, ecneg, ecadd, ec_stdsub, ecdbl, eccheck |
365 | }; |
366 | |
367 | static 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 | |
378 | int 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 -------------------------------------------------*/ |