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1 | /* -*-c-*- |
2 | * |
391faf42 |
3 | * $Id: ec-bin.c,v 1.3 2004/03/22 02:19:09 mdw Exp $ |
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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 $ |
391faf42 |
33 | * Revision 1.3 2004/03/22 02:19:09 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 1.2 2004/03/21 22:52:06 mdw |
39 | * Merge and close elliptic curve branch. |
40 | * |
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41 | * Revision 1.1.2.1 2004/03/21 22:39:46 mdw |
42 | * Elliptic curves on binary fields work. |
43 | * |
44 | */ |
45 | |
46 | /*----- Header files ------------------------------------------------------*/ |
47 | |
48 | #include <mLib/sub.h> |
49 | |
50 | #include "ec.h" |
51 | |
52 | /*----- Data structures ---------------------------------------------------*/ |
53 | |
54 | typedef struct ecctx { |
55 | ec_curve c; |
56 | mp *a, *b; |
57 | mp *bb; |
58 | } ecctx; |
59 | |
60 | /*----- Main code ---------------------------------------------------------*/ |
61 | |
62 | static const ec_ops ec_binops, ec_binprojops; |
63 | |
64 | static ec *ecneg(ec_curve *c, ec *d, const ec *p) |
65 | { |
66 | EC_COPY(d, p); |
67 | if (d->x) |
68 | d->y = F_ADD(c->f, d->y, d->y, d->x); |
69 | return (d); |
70 | } |
71 | |
72 | static ec *ecprojneg(ec_curve *c, ec *d, const ec *p) |
73 | { |
74 | EC_COPY(d, p); |
75 | if (d->x) { |
76 | mp *t = F_MUL(c->f, MP_NEW, d->x, d->z); |
77 | d->y = F_ADD(c->f, d->y, d->y, t); |
78 | MP_DROP(t); |
79 | } |
80 | return (d); |
81 | } |
82 | |
83 | static ec *ecfind(ec_curve *c, ec *d, mp *x) |
84 | { |
85 | /* write me */ |
86 | return (0); |
87 | } |
88 | |
89 | static ec *ecdbl(ec_curve *c, ec *d, const ec *a) |
90 | { |
91 | if (EC_ATINF(a) || F_ZEROP(c->f, a->x)) |
92 | EC_SETINF(d); |
93 | else { |
94 | field *f = c->f; |
95 | ecctx *cc = (ecctx *)c; |
96 | mp *lambda; |
97 | mp *dx, *dy; |
98 | |
99 | dx = F_INV(f, MP_NEW, a->x); /* %$x^{-1}$% */ |
100 | dy = F_MUL(f, MP_NEW, dx, a->y); /* %$y/x$% */ |
101 | lambda = F_ADD(f, dy, dy, a->x); /* %$\lambda = x + y/x$% */ |
102 | |
103 | dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */ |
104 | dx = F_ADD(f, dx, dx, lambda); /* %$\lambda^2 + \lambda$% */ |
105 | dx = F_ADD(f, dx, dx, cc->a); /* %$x' = a + \lambda^2 + \lambda$% */ |
106 | |
107 | dy = F_ADD(f, MP_NEW, a->x, dx); /* %$ x + x' $% */ |
108 | dy = F_MUL(f, dy, dy, lambda); /* %$ (x + x') \lambda$% */ |
109 | dy = F_ADD(f, dy, dy, a->y); /* %$ (x + x') \lambda + y$% */ |
110 | dy = F_ADD(f, dy, dy, dx); /* %$ y' = (x + x') \lambda + y + x'$% */ |
111 | |
112 | EC_DESTROY(d); |
113 | d->x = dx; |
114 | d->y = dy; |
115 | d->z = 0; |
116 | MP_DROP(lambda); |
117 | } |
118 | return (d); |
119 | } |
120 | |
121 | static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a) |
122 | { |
123 | if (EC_ATINF(a) || F_ZEROP(c->f, a->x)) |
124 | EC_SETINF(d); |
125 | else { |
126 | field *f = c->f; |
127 | ecctx *cc = (ecctx *)c; |
128 | mp *dx, *dy, *dz, *u, *v; |
129 | |
130 | dy = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */ |
131 | dx = F_MUL(f, MP_NEW, dy, cc->bb); /* %$c z^2$% */ |
132 | dx = F_ADD(f, dx, dx, a->x); /* %$x + c z^2$% */ |
133 | dz = F_SQR(f, MP_NEW, dx); /* %$(x + c z^2)^2$% */ |
134 | dx = F_SQR(f, dx, dz); /* %$x' = (x + c z^2)^4$% */ |
135 | |
136 | dz = F_MUL(f, dz, dy, a->x); /* %$z' = x z^2$% */ |
137 | |
138 | dy = F_SQR(f, dy, a->x); /* %$x^2$% */ |
139 | u = F_MUL(f, MP_NEW, a->y, a->z); /* %$y z$% */ |
140 | u = F_ADD(f, u, u, dz); /* %$z' + y z$% */ |
141 | u = F_ADD(f, u, u, dy); /* %$u = z' + x^2 + y z$% */ |
142 | |
143 | v = F_SQR(f, MP_NEW, dy); /* %$x^4$% */ |
144 | dy = F_MUL(f, dy, v, dz); /* %$x^4 z'$% */ |
145 | v = F_MUL(f, v, u, dx); /* %$u x'$% */ |
146 | dy = F_ADD(f, dy, dy, v); /* %$y' = x^4 z' + u x'$% */ |
147 | |
148 | EC_DESTROY(d); |
149 | d->x = dx; |
150 | d->y = dy; |
151 | d->z = dz; |
152 | MP_DROP(u); |
153 | MP_DROP(v); |
154 | assert(!(d->x->f & MP_DESTROYED)); |
155 | assert(!(d->y->f & MP_DESTROYED)); |
156 | assert(!(d->z->f & MP_DESTROYED)); |
157 | } |
158 | return (d); |
159 | } |
160 | |
161 | static ec *ecadd(ec_curve *c, ec *d, const ec *a, const ec *b) |
162 | { |
163 | if (a == b) |
164 | ecdbl(c, d, a); |
165 | else if (EC_ATINF(a)) |
166 | EC_COPY(d, b); |
167 | else if (EC_ATINF(b)) |
168 | EC_COPY(d, a); |
169 | else { |
170 | field *f = c->f; |
171 | ecctx *cc = (ecctx *)c; |
172 | mp *lambda; |
173 | mp *dx, *dy; |
174 | |
175 | if (!MP_EQ(a->x, b->x)) { |
176 | dx = F_ADD(f, MP_NEW, a->x, b->x); /* %$x_0 + x_1$% */ |
177 | dy = F_INV(f, MP_NEW, dx); /* %$(x_0 + x_1)^{-1}$% */ |
178 | dx = F_ADD(f, dx, a->y, b->y); /* %$y_0 + y_1$% */ |
179 | lambda = F_MUL(f, MP_NEW, dy, dx); |
180 | /* %$\lambda = (y_0 + y_1)/(x_0 + x_1)$% */ |
181 | |
182 | dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */ |
183 | dx = F_ADD(f, dx, dx, lambda); /* %$\lambda^2 + \lambda$% */ |
184 | dx = F_ADD(f, dx, dx, cc->a); /* %$a + \lambda^2 + \lambda$% */ |
185 | dx = F_ADD(f, dx, dx, a->x); /* %$a + \lambda^2 + \lambda + x_0$% */ |
186 | dx = F_ADD(f, dx, dx, b->x); |
187 | /* %$x' = a + \lambda^2 + \lambda + x_0 + x_1$% */ |
188 | } else if (!MP_EQ(a->y, b->y) || F_ZEROP(f, a->x)) { |
189 | EC_SETINF(d); |
190 | return (d); |
191 | } else { |
192 | dx = F_INV(f, MP_NEW, a->x); /* %$x^{-1}$% */ |
193 | dy = F_MUL(f, MP_NEW, dx, a->y); /* %$y/x$% */ |
194 | lambda = F_ADD(f, dy, dy, a->x); /* %$\lambda = x + y/x$% */ |
195 | |
196 | dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */ |
197 | dx = F_ADD(f, dx, dx, lambda); /* %$\lambda^2 + \lambda$% */ |
198 | dx = F_ADD(f, dx, dx, cc->a); /* %$x' = a + \lambda^2 + \lambda$% */ |
199 | dy = MP_NEW; |
200 | } |
201 | |
202 | dy = F_ADD(f, dy, a->x, dx); /* %$ x + x' $% */ |
203 | dy = F_MUL(f, dy, dy, lambda); /* %$ (x + x') \lambda$% */ |
204 | dy = F_ADD(f, dy, dy, a->y); /* %$ (x + x') \lambda + y$% */ |
205 | dy = F_ADD(f, dy, dy, dx); /* %$ y' = (x + x') \lambda + y + x'$% */ |
206 | |
207 | EC_DESTROY(d); |
208 | d->x = dx; |
209 | d->y = dy; |
210 | d->z = 0; |
211 | MP_DROP(lambda); |
212 | } |
213 | return (d); |
214 | } |
215 | |
216 | static ec *ecprojadd(ec_curve *c, ec *d, const ec *a, const ec *b) |
217 | { |
218 | if (a == b) |
219 | c->ops->dbl(c, d, a); |
220 | else if (EC_ATINF(a)) |
221 | EC_COPY(d, b); |
222 | else if (EC_ATINF(b)) |
223 | EC_COPY(d, a); |
224 | else { |
225 | field *f = c->f; |
226 | ecctx *cc = (ecctx *)c; |
227 | mp *dx, *dy, *dz, *u, *uu, *v, *t, *s, *ss, *r, *w, *l; |
228 | |
229 | dz = F_SQR(f, MP_NEW, b->z); /* %$z_1^2$% */ |
230 | u = F_MUL(f, MP_NEW, dz, a->x); /* %$u_0 = x_0 z_1^2$% */ |
231 | t = F_MUL(f, MP_NEW, dz, b->z); /* %$z_1^3$% */ |
232 | s = F_MUL(f, MP_NEW, t, a->y); /* %$s_0 = y_0 z_1^3$% */ |
233 | |
234 | dz = F_SQR(f, dz, a->z); /* %$z_0^2$% */ |
235 | uu = F_MUL(f, MP_NEW, dz, b->x); /* %$u_1 = x_1 z_0^2$% */ |
236 | t = F_MUL(f, t, dz, a->z); /* %$z_0^3$% */ |
237 | ss = F_MUL(f, MP_NEW, t, b->y); /* %$s_1 = y_1 z_0^3$% */ |
238 | |
239 | w = F_ADD(f, u, u, uu); /* %$r = u_0 + u_1$% */ |
240 | r = F_ADD(f, s, s, ss); /* %$w = s_0 + s_1$% */ |
241 | if (F_ZEROP(f, w)) { |
242 | MP_DROP(w); |
243 | MP_DROP(uu); |
244 | MP_DROP(ss); |
245 | MP_DROP(t); |
246 | MP_DROP(dz); |
247 | if (F_ZEROP(f, r)) { |
248 | MP_DROP(r); |
249 | return (c->ops->dbl(c, d, a)); |
250 | } else { |
251 | MP_DROP(r); |
252 | EC_SETINF(d); |
253 | return (d); |
254 | } |
255 | } |
256 | |
257 | l = F_MUL(f, t, a->z, w); /* %$l = z_0 w$% */ |
258 | |
259 | dz = F_MUL(f, dz, l, b->z); /* %$z' = l z_1$% */ |
260 | |
261 | ss = F_MUL(f, ss, r, b->x); /* %$r x_1$% */ |
262 | t = F_MUL(f, uu, l, b->y); /* %$l y_1$% */ |
263 | v = F_ADD(f, ss, ss, t); /* %$v = r x_1 + l y_1$% */ |
264 | |
265 | t = F_ADD(f, t, r, dz); /* %$t = r + z'$% */ |
266 | |
267 | uu = F_SQR(f, MP_NEW, dz); /* %$z'^2$% */ |
268 | dx = F_MUL(f, MP_NEW, uu, cc->a); /* %$a z'^2$% */ |
269 | uu = F_MUL(f, uu, t, r); /* %$t r$% */ |
270 | dx = F_ADD(f, dx, dx, uu); /* %$a z'^2 + t r$% */ |
271 | r = F_SQR(f, r, w); /* %$w^2$% */ |
272 | uu = F_MUL(f, uu, r, w); /* %$w^3$% */ |
273 | dx = F_ADD(f, dx, dx, uu); /* %$x' = a z'^2 + t r + w^3$% */ |
274 | |
275 | r = F_SQR(f, r, l); /* %$l^2$% */ |
276 | dy = F_MUL(f, uu, v, r); /* %$v l^2$% */ |
277 | l = F_MUL(f, l, t, dx); /* %$t x'$% */ |
278 | dy = F_ADD(f, dy, dy, l); /* %$y' = t x' + v l^2$% */ |
279 | |
280 | EC_DESTROY(d); |
281 | d->x = dx; |
282 | d->y = dy; |
283 | d->z = dz; |
284 | MP_DROP(l); |
285 | MP_DROP(r); |
286 | MP_DROP(w); |
287 | MP_DROP(t); |
288 | MP_DROP(v); |
289 | } |
290 | return (d); |
291 | } |
292 | |
293 | static int eccheck(ec_curve *c, const ec *p) |
294 | { |
295 | ecctx *cc = (ecctx *)c; |
296 | field *f = c->f; |
297 | int rc; |
298 | mp *u, *v; |
299 | |
300 | v = F_SQR(f, MP_NEW, p->x); |
301 | u = F_MUL(f, MP_NEW, v, p->x); |
302 | v = F_MUL(f, v, v, cc->a); |
303 | u = F_ADD(f, u, u, v); |
304 | u = F_ADD(f, u, u, cc->b); |
305 | v = F_MUL(f, v, p->x, p->y); |
306 | u = F_ADD(f, u, u, v); |
307 | v = F_SQR(f, v, p->y); |
308 | u = F_ADD(f, u, u, v); |
309 | rc = F_ZEROP(f, u); |
310 | mp_drop(u); |
311 | mp_drop(v); |
312 | return (rc); |
313 | } |
314 | |
315 | static int ecprojcheck(ec_curve *c, const ec *p) |
316 | { |
317 | ec t = EC_INIT; |
318 | int rc; |
319 | |
320 | c->ops->fix(c, &t, p); |
321 | rc = eccheck(c, &t); |
322 | EC_DESTROY(&t); |
323 | return (rc); |
324 | } |
325 | |
326 | static void ecdestroy(ec_curve *c) |
327 | { |
328 | ecctx *cc = (ecctx *)c; |
329 | MP_DROP(cc->a); |
330 | MP_DROP(cc->b); |
331 | if (cc->bb) MP_DROP(cc->bb); |
332 | DESTROY(cc); |
333 | } |
334 | |
335 | /* --- @ec_bin@, @ec_binproj@ --- * |
336 | * |
337 | * Arguments: @field *f@ = the underlying field for this elliptic curve |
338 | * @mp *a, *b@ = the coefficients for this curve |
339 | * |
340 | * Returns: A pointer to the curve. |
341 | * |
342 | * Use: Creates a curve structure for an elliptic curve defined over |
343 | * a binary field. The @binproj@ variant uses projective |
344 | * coordinates, which can be a win. |
345 | */ |
346 | |
347 | ec_curve *ec_bin(field *f, mp *a, mp *b) |
348 | { |
349 | ecctx *cc = CREATE(ecctx); |
350 | cc->c.ops = &ec_binops; |
351 | cc->c.f = f; |
352 | cc->a = F_IN(f, MP_NEW, a); |
353 | cc->b = F_IN(f, MP_NEW, b); |
354 | cc->bb = 0; |
355 | return (&cc->c); |
356 | } |
357 | |
358 | ec_curve *ec_binproj(field *f, mp *a, mp *b) |
359 | { |
360 | ecctx *cc = CREATE(ecctx); |
361 | cc->c.ops = &ec_binprojops; |
362 | cc->c.f = f; |
363 | cc->a = F_IN(f, MP_NEW, a); |
364 | cc->b = F_IN(f, MP_NEW, b); |
365 | cc->bb = F_SQRT(f, MP_NEW, b); |
366 | cc->bb = F_SQRT(f, cc->bb, cc->bb); |
367 | return (&cc->c); |
368 | } |
369 | |
370 | static const ec_ops ec_binops = { |
371 | ecdestroy, ec_idin, ec_idout, ec_idfix, |
372 | 0, ecneg, ecadd, ec_stdsub, ecdbl, eccheck |
373 | }; |
374 | |
375 | static const ec_ops ec_binprojops = { |
376 | ecdestroy, ec_projin, ec_projout, ec_projfix, |
377 | 0, ecprojneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck |
378 | }; |
379 | |
380 | /*----- Test rig ----------------------------------------------------------*/ |
381 | |
382 | #ifdef TEST_RIG |
383 | |
384 | #define MP(x) mp_readstring(MP_NEW, #x, 0, 0) |
385 | |
386 | int main(int argc, char *argv[]) |
387 | { |
388 | field *f; |
389 | ec_curve *c; |
390 | ec g = EC_INIT, d = EC_INIT; |
391 | mp *p, *a, *b, *r; |
392 | int i, n = argc == 1 ? 1 : atoi(argv[1]); |
393 | |
394 | printf("ec-bin: "); |
395 | fflush(stdout); |
396 | a = MP(1); |
397 | b = MP(0x066647ede6c332c7f8c0923bb58213b333b20e9ce4281fe115f7d8f90ad); |
398 | p = MP(0x20000000000000000000000000000000000000004000000000000000001); |
399 | r = |
400 | MP(6901746346790563787434755862277025555839812737345013555379383634485462); |
401 | |
402 | f = field_binpoly(p); |
403 | c = ec_binproj(f, a, b); |
404 | |
405 | g.x = MP(0x0fac9dfcbac8313bb2139f1bb755fef65bc391f8b36f8f8eb7371fd558b); |
406 | g.y = MP(0x1006a08a41903350678e58528bebf8a0beff867a7ca36716f7e01f81052); |
407 | |
408 | for (i = 0; i < n; i++) { |
409 | ec_mul(c, &d, &g, r); |
410 | if (EC_ATINF(&d)) { |
411 | fprintf(stderr, "zero too early\n"); |
412 | return (1); |
413 | } |
414 | ec_add(c, &d, &d, &g); |
415 | if (!EC_ATINF(&d)) { |
416 | fprintf(stderr, "didn't reach zero\n"); |
417 | MP_EPRINTX("d.x", d.x); |
418 | MP_EPRINTX("d.y", d.y); |
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419 | return (1); |
420 | } |
421 | ec_destroy(&d); |
422 | } |
423 | |
424 | ec_destroy(&g); |
425 | ec_destroycurve(c); |
426 | F_DESTROY(f); |
427 | MP_DROP(p); MP_DROP(a); MP_DROP(b); MP_DROP(r); |
428 | assert(!mparena_count(&mparena_global)); |
429 | printf("ok\n"); |
430 | return (0); |
431 | } |
432 | |
433 | #endif |
434 | |
435 | /*----- That's all, folks -------------------------------------------------*/ |