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1 | /* -*-c-*- |
2 | * |
432c4e18 |
3 | * $Id: ec-bin.c,v 1.5 2004/03/27 17:54:11 mdw Exp $ |
ceb3f0c0 |
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 $ |
432c4e18 |
33 | * Revision 1.5 2004/03/27 17:54:11 mdw |
34 | * Standard curves and curve checking. |
35 | * |
bc985cef |
36 | * Revision 1.4 2004/03/23 15:19:32 mdw |
37 | * Test elliptic curves more thoroughly. |
38 | * |
391faf42 |
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 | * |
c3caa2fa |
44 | * Revision 1.2 2004/03/21 22:52:06 mdw |
45 | * Merge and close elliptic curve branch. |
46 | * |
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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 | |
60 | typedef struct ecctx { |
61 | ec_curve c; |
ceb3f0c0 |
62 | mp *bb; |
63 | } ecctx; |
64 | |
65 | /*----- Main code ---------------------------------------------------------*/ |
66 | |
67 | static const ec_ops ec_binops, ec_binprojops; |
68 | |
69 | static 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 | |
77 | static 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 | |
88 | static ec *ecfind(ec_curve *c, ec *d, mp *x) |
89 | { |
bc985cef |
90 | field *f = c->f; |
bc985cef |
91 | mp *y, *u, *v; |
92 | |
93 | if (F_ZEROP(f, x)) |
432c4e18 |
94 | y = F_SQRT(f, MP_NEW, c->b); |
bc985cef |
95 | else { |
96 | u = F_SQR(f, MP_NEW, x); /* %$x^2$% */ |
432c4e18 |
97 | y = F_MUL(f, MP_NEW, u, c->a); /* %$a x^2$% */ |
98 | y = F_ADD(f, y, y, c->b); /* %$a x^2 + b$% */ |
bc985cef |
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); |
ceb3f0c0 |
116 | } |
117 | |
118 | static 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; |
ceb3f0c0 |
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); /* %$\lambda^2 + \lambda$% */ |
432c4e18 |
133 | dx = F_ADD(f, dx, dx, c->a); /* %$x' = a + \lambda^2 + \lambda$% */ |
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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'$% */ |
139 | |
140 | EC_DESTROY(d); |
141 | d->x = dx; |
142 | d->y = dy; |
143 | d->z = 0; |
144 | MP_DROP(lambda); |
145 | } |
146 | return (d); |
147 | } |
148 | |
149 | static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a) |
150 | { |
151 | if (EC_ATINF(a) || F_ZEROP(c->f, a->x)) |
152 | EC_SETINF(d); |
153 | else { |
154 | field *f = c->f; |
155 | ecctx *cc = (ecctx *)c; |
156 | mp *dx, *dy, *dz, *u, *v; |
157 | |
158 | dy = F_SQR(f, MP_NEW, a->z); /* %$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'$% */ |
175 | |
176 | EC_DESTROY(d); |
177 | d->x = dx; |
178 | d->y = dy; |
179 | d->z = dz; |
180 | MP_DROP(u); |
181 | MP_DROP(v); |
182 | assert(!(d->x->f & MP_DESTROYED)); |
183 | assert(!(d->y->f & MP_DESTROYED)); |
184 | assert(!(d->z->f & MP_DESTROYED)); |
185 | } |
186 | return (d); |
187 | } |
188 | |
189 | static ec *ecadd(ec_curve *c, ec *d, const ec *a, const ec *b) |
190 | { |
191 | if (a == b) |
192 | ecdbl(c, d, a); |
193 | else if (EC_ATINF(a)) |
194 | EC_COPY(d, b); |
195 | else if (EC_ATINF(b)) |
196 | EC_COPY(d, a); |
197 | else { |
198 | field *f = c->f; |
ceb3f0c0 |
199 | mp *lambda; |
200 | mp *dx, *dy; |
201 | |
202 | if (!MP_EQ(a->x, b->x)) { |
203 | dx = F_ADD(f, MP_NEW, a->x, b->x); /* %$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$% */ |
432c4e18 |
211 | dx = F_ADD(f, dx, dx, c->a); /* %$a + \lambda^2 + \lambda$% */ |
ceb3f0c0 |
212 | dx = F_ADD(f, dx, dx, a->x); /* %$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$% */ |
432c4e18 |
225 | dx = F_ADD(f, dx, dx, c->a); /* %$x' = a + \lambda^2 + \lambda$% */ |
ceb3f0c0 |
226 | dy = MP_NEW; |
227 | } |
228 | |
229 | dy = F_ADD(f, dy, a->x, dx); /* %$ 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'$% */ |
233 | |
234 | EC_DESTROY(d); |
235 | d->x = dx; |
236 | d->y = dy; |
237 | d->z = 0; |
238 | MP_DROP(lambda); |
239 | } |
240 | return (d); |
241 | } |
242 | |
243 | static ec *ecprojadd(ec_curve *c, ec *d, const ec *a, const ec *b) |
244 | { |
245 | if (a == b) |
246 | c->ops->dbl(c, d, a); |
247 | else if (EC_ATINF(a)) |
248 | EC_COPY(d, b); |
249 | else if (EC_ATINF(b)) |
250 | EC_COPY(d, a); |
251 | else { |
252 | field *f = c->f; |
ceb3f0c0 |
253 | mp *dx, *dy, *dz, *u, *uu, *v, *t, *s, *ss, *r, *w, *l; |
254 | |
255 | dz = F_SQR(f, MP_NEW, b->z); /* %$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$% */ |
432c4e18 |
294 | dx = F_MUL(f, MP_NEW, uu, c->a); /* %$a z'^2$% */ |
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295 | uu = F_MUL(f, uu, t, r); /* %$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 | |
319 | static int eccheck(ec_curve *c, const ec *p) |
320 | { |
ceb3f0c0 |
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); |
432c4e18 |
327 | v = F_MUL(f, v, v, c->a); |
ceb3f0c0 |
328 | u = F_ADD(f, u, u, v); |
432c4e18 |
329 | u = F_ADD(f, u, u, c->b); |
ceb3f0c0 |
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); |
bc985cef |
334 | rc = F_ZEROP(f, u) ? 0 : -1; |
ceb3f0c0 |
335 | mp_drop(u); |
336 | mp_drop(v); |
337 | return (rc); |
338 | } |
339 | |
340 | static 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 | |
351 | static void ecdestroy(ec_curve *c) |
352 | { |
353 | ecctx *cc = (ecctx *)c; |
432c4e18 |
354 | MP_DROP(cc->c.a); |
355 | MP_DROP(cc->c.b); |
ceb3f0c0 |
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 | |
372 | ec_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; |
432c4e18 |
377 | cc->c.a = F_IN(f, MP_NEW, a); |
378 | cc->c.b = F_IN(f, MP_NEW, b); |
ceb3f0c0 |
379 | cc->bb = 0; |
380 | return (&cc->c); |
381 | } |
382 | |
383 | ec_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; |
432c4e18 |
388 | cc->c.a = F_IN(f, MP_NEW, a); |
389 | cc->c.b = F_IN(f, MP_NEW, b); |
ceb3f0c0 |
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 | |
395 | static const ec_ops ec_binops = { |
396 | ecdestroy, ec_idin, ec_idout, ec_idfix, |
bc985cef |
397 | ecfind, ecneg, ecadd, ec_stdsub, ecdbl, eccheck |
ceb3f0c0 |
398 | }; |
399 | |
400 | static const ec_ops ec_binprojops = { |
401 | ecdestroy, ec_projin, ec_projout, ec_projfix, |
bc985cef |
402 | ecfind, ecprojneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck |
ceb3f0c0 |
403 | }; |
404 | |
405 | /*----- Test rig ----------------------------------------------------------*/ |
406 | |
407 | #ifdef TEST_RIG |
408 | |
409 | #define MP(x) mp_readstring(MP_NEW, #x, 0, 0) |
410 | |
411 | int 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); |
bc985cef |
422 | b = MP(0x021a5c2c8ee9feb5c4b9a753b7b476b7fd6422ef1f3dd674761fa99d6ac27c8a9a197b272822f6cd57a55aa4f50ae317b13545f); |
423 | p = MP(0x2000000000000000000000000000000000000000000000000000000000000000000000000000000008000000000000000000001); |
ceb3f0c0 |
424 | r = |
bc985cef |
425 | MP(661055968790248598951915308032771039828404682964281219284648798304157774827374805208143723762179110965979867288366567526770); |
ceb3f0c0 |
426 | |
427 | f = field_binpoly(p); |
428 | c = ec_binproj(f, a, b); |
429 | |
bc985cef |
430 | g.x = MP(0x15d4860d088ddb3496b0c6064756260441cde4af1771d4db01ffe5b34e59703dc255a868a1180515603aeab60794e54bb7996a7); |
431 | g.y = MP(0x061b1cfab6be5f32bbfa78324ed106a7636b9c5a7bd198d0158aa4f5488d08f38514f1fdf4b4f40d2181b3681c364ba0273c706); |
ceb3f0c0 |
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); |
ceb3f0c0 |
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 -------------------------------------------------*/ |