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1 | /* -*-c-*- |
2 | * |
c3caa2fa |
3 | * $Id: ec-bin.c,v 1.2 2004/03/21 22:52:06 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 $ |
c3caa2fa |
33 | * Revision 1.2 2004/03/21 22:52:06 mdw |
34 | * Merge and close elliptic curve branch. |
35 | * |
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36 | * Revision 1.1.2.1 2004/03/21 22:39:46 mdw |
37 | * Elliptic curves on binary fields work. |
38 | * |
39 | */ |
40 | |
41 | /*----- Header files ------------------------------------------------------*/ |
42 | |
43 | #include <mLib/sub.h> |
44 | |
45 | #include "ec.h" |
46 | |
47 | /*----- Data structures ---------------------------------------------------*/ |
48 | |
49 | typedef struct ecctx { |
50 | ec_curve c; |
51 | mp *a, *b; |
52 | mp *bb; |
53 | } ecctx; |
54 | |
55 | /*----- Main code ---------------------------------------------------------*/ |
56 | |
57 | static const ec_ops ec_binops, ec_binprojops; |
58 | |
59 | static ec *ecneg(ec_curve *c, ec *d, const ec *p) |
60 | { |
61 | EC_COPY(d, p); |
62 | if (d->x) |
63 | d->y = F_ADD(c->f, d->y, d->y, d->x); |
64 | return (d); |
65 | } |
66 | |
67 | static ec *ecprojneg(ec_curve *c, ec *d, const ec *p) |
68 | { |
69 | EC_COPY(d, p); |
70 | if (d->x) { |
71 | mp *t = F_MUL(c->f, MP_NEW, d->x, d->z); |
72 | d->y = F_ADD(c->f, d->y, d->y, t); |
73 | MP_DROP(t); |
74 | } |
75 | return (d); |
76 | } |
77 | |
78 | static ec *ecfind(ec_curve *c, ec *d, mp *x) |
79 | { |
80 | /* write me */ |
81 | return (0); |
82 | } |
83 | |
84 | static ec *ecdbl(ec_curve *c, ec *d, const ec *a) |
85 | { |
86 | if (EC_ATINF(a) || F_ZEROP(c->f, a->x)) |
87 | EC_SETINF(d); |
88 | else { |
89 | field *f = c->f; |
90 | ecctx *cc = (ecctx *)c; |
91 | mp *lambda; |
92 | mp *dx, *dy; |
93 | |
94 | dx = F_INV(f, MP_NEW, a->x); /* %$x^{-1}$% */ |
95 | dy = F_MUL(f, MP_NEW, dx, a->y); /* %$y/x$% */ |
96 | lambda = F_ADD(f, dy, dy, a->x); /* %$\lambda = x + y/x$% */ |
97 | |
98 | dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */ |
99 | dx = F_ADD(f, dx, dx, lambda); /* %$\lambda^2 + \lambda$% */ |
100 | dx = F_ADD(f, dx, dx, cc->a); /* %$x' = a + \lambda^2 + \lambda$% */ |
101 | |
102 | dy = F_ADD(f, MP_NEW, a->x, dx); /* %$ x + x' $% */ |
103 | dy = F_MUL(f, dy, dy, lambda); /* %$ (x + x') \lambda$% */ |
104 | dy = F_ADD(f, dy, dy, a->y); /* %$ (x + x') \lambda + y$% */ |
105 | dy = F_ADD(f, dy, dy, dx); /* %$ y' = (x + x') \lambda + y + x'$% */ |
106 | |
107 | EC_DESTROY(d); |
108 | d->x = dx; |
109 | d->y = dy; |
110 | d->z = 0; |
111 | MP_DROP(lambda); |
112 | } |
113 | return (d); |
114 | } |
115 | |
116 | static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a) |
117 | { |
118 | if (EC_ATINF(a) || F_ZEROP(c->f, a->x)) |
119 | EC_SETINF(d); |
120 | else { |
121 | field *f = c->f; |
122 | ecctx *cc = (ecctx *)c; |
123 | mp *dx, *dy, *dz, *u, *v; |
124 | |
125 | dy = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */ |
126 | dx = F_MUL(f, MP_NEW, dy, cc->bb); /* %$c z^2$% */ |
127 | dx = F_ADD(f, dx, dx, a->x); /* %$x + c z^2$% */ |
128 | dz = F_SQR(f, MP_NEW, dx); /* %$(x + c z^2)^2$% */ |
129 | dx = F_SQR(f, dx, dz); /* %$x' = (x + c z^2)^4$% */ |
130 | |
131 | dz = F_MUL(f, dz, dy, a->x); /* %$z' = x z^2$% */ |
132 | |
133 | dy = F_SQR(f, dy, a->x); /* %$x^2$% */ |
134 | u = F_MUL(f, MP_NEW, a->y, a->z); /* %$y z$% */ |
135 | u = F_ADD(f, u, u, dz); /* %$z' + y z$% */ |
136 | u = F_ADD(f, u, u, dy); /* %$u = z' + x^2 + y z$% */ |
137 | |
138 | v = F_SQR(f, MP_NEW, dy); /* %$x^4$% */ |
139 | dy = F_MUL(f, dy, v, dz); /* %$x^4 z'$% */ |
140 | v = F_MUL(f, v, u, dx); /* %$u x'$% */ |
141 | dy = F_ADD(f, dy, dy, v); /* %$y' = x^4 z' + u x'$% */ |
142 | |
143 | EC_DESTROY(d); |
144 | d->x = dx; |
145 | d->y = dy; |
146 | d->z = dz; |
147 | MP_DROP(u); |
148 | MP_DROP(v); |
149 | assert(!(d->x->f & MP_DESTROYED)); |
150 | assert(!(d->y->f & MP_DESTROYED)); |
151 | assert(!(d->z->f & MP_DESTROYED)); |
152 | } |
153 | return (d); |
154 | } |
155 | |
156 | static ec *ecadd(ec_curve *c, ec *d, const ec *a, const ec *b) |
157 | { |
158 | if (a == b) |
159 | ecdbl(c, d, a); |
160 | else if (EC_ATINF(a)) |
161 | EC_COPY(d, b); |
162 | else if (EC_ATINF(b)) |
163 | EC_COPY(d, a); |
164 | else { |
165 | field *f = c->f; |
166 | ecctx *cc = (ecctx *)c; |
167 | mp *lambda; |
168 | mp *dx, *dy; |
169 | |
170 | if (!MP_EQ(a->x, b->x)) { |
171 | dx = F_ADD(f, MP_NEW, a->x, b->x); /* %$x_0 + x_1$% */ |
172 | dy = F_INV(f, MP_NEW, dx); /* %$(x_0 + x_1)^{-1}$% */ |
173 | dx = F_ADD(f, dx, a->y, b->y); /* %$y_0 + y_1$% */ |
174 | lambda = F_MUL(f, MP_NEW, dy, dx); |
175 | /* %$\lambda = (y_0 + y_1)/(x_0 + x_1)$% */ |
176 | |
177 | dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */ |
178 | dx = F_ADD(f, dx, dx, lambda); /* %$\lambda^2 + \lambda$% */ |
179 | dx = F_ADD(f, dx, dx, cc->a); /* %$a + \lambda^2 + \lambda$% */ |
180 | dx = F_ADD(f, dx, dx, a->x); /* %$a + \lambda^2 + \lambda + x_0$% */ |
181 | dx = F_ADD(f, dx, dx, b->x); |
182 | /* %$x' = a + \lambda^2 + \lambda + x_0 + x_1$% */ |
183 | } else if (!MP_EQ(a->y, b->y) || F_ZEROP(f, a->x)) { |
184 | EC_SETINF(d); |
185 | return (d); |
186 | } else { |
187 | dx = F_INV(f, MP_NEW, a->x); /* %$x^{-1}$% */ |
188 | dy = F_MUL(f, MP_NEW, dx, a->y); /* %$y/x$% */ |
189 | lambda = F_ADD(f, dy, dy, a->x); /* %$\lambda = x + y/x$% */ |
190 | |
191 | dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */ |
192 | dx = F_ADD(f, dx, dx, lambda); /* %$\lambda^2 + \lambda$% */ |
193 | dx = F_ADD(f, dx, dx, cc->a); /* %$x' = a + \lambda^2 + \lambda$% */ |
194 | dy = MP_NEW; |
195 | } |
196 | |
197 | dy = F_ADD(f, dy, a->x, dx); /* %$ x + x' $% */ |
198 | dy = F_MUL(f, dy, dy, lambda); /* %$ (x + x') \lambda$% */ |
199 | dy = F_ADD(f, dy, dy, a->y); /* %$ (x + x') \lambda + y$% */ |
200 | dy = F_ADD(f, dy, dy, dx); /* %$ y' = (x + x') \lambda + y + x'$% */ |
201 | |
202 | EC_DESTROY(d); |
203 | d->x = dx; |
204 | d->y = dy; |
205 | d->z = 0; |
206 | MP_DROP(lambda); |
207 | } |
208 | return (d); |
209 | } |
210 | |
211 | static ec *ecprojadd(ec_curve *c, ec *d, const ec *a, const ec *b) |
212 | { |
213 | if (a == b) |
214 | c->ops->dbl(c, d, a); |
215 | else if (EC_ATINF(a)) |
216 | EC_COPY(d, b); |
217 | else if (EC_ATINF(b)) |
218 | EC_COPY(d, a); |
219 | else { |
220 | field *f = c->f; |
221 | ecctx *cc = (ecctx *)c; |
222 | mp *dx, *dy, *dz, *u, *uu, *v, *t, *s, *ss, *r, *w, *l; |
223 | |
224 | dz = F_SQR(f, MP_NEW, b->z); /* %$z_1^2$% */ |
225 | u = F_MUL(f, MP_NEW, dz, a->x); /* %$u_0 = x_0 z_1^2$% */ |
226 | t = F_MUL(f, MP_NEW, dz, b->z); /* %$z_1^3$% */ |
227 | s = F_MUL(f, MP_NEW, t, a->y); /* %$s_0 = y_0 z_1^3$% */ |
228 | |
229 | dz = F_SQR(f, dz, a->z); /* %$z_0^2$% */ |
230 | uu = F_MUL(f, MP_NEW, dz, b->x); /* %$u_1 = x_1 z_0^2$% */ |
231 | t = F_MUL(f, t, dz, a->z); /* %$z_0^3$% */ |
232 | ss = F_MUL(f, MP_NEW, t, b->y); /* %$s_1 = y_1 z_0^3$% */ |
233 | |
234 | w = F_ADD(f, u, u, uu); /* %$r = u_0 + u_1$% */ |
235 | r = F_ADD(f, s, s, ss); /* %$w = s_0 + s_1$% */ |
236 | if (F_ZEROP(f, w)) { |
237 | MP_DROP(w); |
238 | MP_DROP(uu); |
239 | MP_DROP(ss); |
240 | MP_DROP(t); |
241 | MP_DROP(dz); |
242 | if (F_ZEROP(f, r)) { |
243 | MP_DROP(r); |
244 | return (c->ops->dbl(c, d, a)); |
245 | } else { |
246 | MP_DROP(r); |
247 | EC_SETINF(d); |
248 | return (d); |
249 | } |
250 | } |
251 | |
252 | l = F_MUL(f, t, a->z, w); /* %$l = z_0 w$% */ |
253 | |
254 | dz = F_MUL(f, dz, l, b->z); /* %$z' = l z_1$% */ |
255 | |
256 | ss = F_MUL(f, ss, r, b->x); /* %$r x_1$% */ |
257 | t = F_MUL(f, uu, l, b->y); /* %$l y_1$% */ |
258 | v = F_ADD(f, ss, ss, t); /* %$v = r x_1 + l y_1$% */ |
259 | |
260 | t = F_ADD(f, t, r, dz); /* %$t = r + z'$% */ |
261 | |
262 | uu = F_SQR(f, MP_NEW, dz); /* %$z'^2$% */ |
263 | dx = F_MUL(f, MP_NEW, uu, cc->a); /* %$a z'^2$% */ |
264 | uu = F_MUL(f, uu, t, r); /* %$t r$% */ |
265 | dx = F_ADD(f, dx, dx, uu); /* %$a z'^2 + t r$% */ |
266 | r = F_SQR(f, r, w); /* %$w^2$% */ |
267 | uu = F_MUL(f, uu, r, w); /* %$w^3$% */ |
268 | dx = F_ADD(f, dx, dx, uu); /* %$x' = a z'^2 + t r + w^3$% */ |
269 | |
270 | r = F_SQR(f, r, l); /* %$l^2$% */ |
271 | dy = F_MUL(f, uu, v, r); /* %$v l^2$% */ |
272 | l = F_MUL(f, l, t, dx); /* %$t x'$% */ |
273 | dy = F_ADD(f, dy, dy, l); /* %$y' = t x' + v l^2$% */ |
274 | |
275 | EC_DESTROY(d); |
276 | d->x = dx; |
277 | d->y = dy; |
278 | d->z = dz; |
279 | MP_DROP(l); |
280 | MP_DROP(r); |
281 | MP_DROP(w); |
282 | MP_DROP(t); |
283 | MP_DROP(v); |
284 | } |
285 | return (d); |
286 | } |
287 | |
288 | static int eccheck(ec_curve *c, const ec *p) |
289 | { |
290 | ecctx *cc = (ecctx *)c; |
291 | field *f = c->f; |
292 | int rc; |
293 | mp *u, *v; |
294 | |
295 | v = F_SQR(f, MP_NEW, p->x); |
296 | u = F_MUL(f, MP_NEW, v, p->x); |
297 | v = F_MUL(f, v, v, cc->a); |
298 | u = F_ADD(f, u, u, v); |
299 | u = F_ADD(f, u, u, cc->b); |
300 | v = F_MUL(f, v, p->x, p->y); |
301 | u = F_ADD(f, u, u, v); |
302 | v = F_SQR(f, v, p->y); |
303 | u = F_ADD(f, u, u, v); |
304 | rc = F_ZEROP(f, u); |
305 | mp_drop(u); |
306 | mp_drop(v); |
307 | return (rc); |
308 | } |
309 | |
310 | static int ecprojcheck(ec_curve *c, const ec *p) |
311 | { |
312 | ec t = EC_INIT; |
313 | int rc; |
314 | |
315 | c->ops->fix(c, &t, p); |
316 | rc = eccheck(c, &t); |
317 | EC_DESTROY(&t); |
318 | return (rc); |
319 | } |
320 | |
321 | static void ecdestroy(ec_curve *c) |
322 | { |
323 | ecctx *cc = (ecctx *)c; |
324 | MP_DROP(cc->a); |
325 | MP_DROP(cc->b); |
326 | if (cc->bb) MP_DROP(cc->bb); |
327 | DESTROY(cc); |
328 | } |
329 | |
330 | /* --- @ec_bin@, @ec_binproj@ --- * |
331 | * |
332 | * Arguments: @field *f@ = the underlying field for this elliptic curve |
333 | * @mp *a, *b@ = the coefficients for this curve |
334 | * |
335 | * Returns: A pointer to the curve. |
336 | * |
337 | * Use: Creates a curve structure for an elliptic curve defined over |
338 | * a binary field. The @binproj@ variant uses projective |
339 | * coordinates, which can be a win. |
340 | */ |
341 | |
342 | ec_curve *ec_bin(field *f, mp *a, mp *b) |
343 | { |
344 | ecctx *cc = CREATE(ecctx); |
345 | cc->c.ops = &ec_binops; |
346 | cc->c.f = f; |
347 | cc->a = F_IN(f, MP_NEW, a); |
348 | cc->b = F_IN(f, MP_NEW, b); |
349 | cc->bb = 0; |
350 | return (&cc->c); |
351 | } |
352 | |
353 | ec_curve *ec_binproj(field *f, mp *a, mp *b) |
354 | { |
355 | ecctx *cc = CREATE(ecctx); |
356 | cc->c.ops = &ec_binprojops; |
357 | cc->c.f = f; |
358 | cc->a = F_IN(f, MP_NEW, a); |
359 | cc->b = F_IN(f, MP_NEW, b); |
360 | cc->bb = F_SQRT(f, MP_NEW, b); |
361 | cc->bb = F_SQRT(f, cc->bb, cc->bb); |
362 | return (&cc->c); |
363 | } |
364 | |
365 | static const ec_ops ec_binops = { |
366 | ecdestroy, ec_idin, ec_idout, ec_idfix, |
367 | 0, ecneg, ecadd, ec_stdsub, ecdbl, eccheck |
368 | }; |
369 | |
370 | static const ec_ops ec_binprojops = { |
371 | ecdestroy, ec_projin, ec_projout, ec_projfix, |
372 | 0, ecprojneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck |
373 | }; |
374 | |
375 | /*----- Test rig ----------------------------------------------------------*/ |
376 | |
377 | #ifdef TEST_RIG |
378 | |
379 | #define MP(x) mp_readstring(MP_NEW, #x, 0, 0) |
380 | |
381 | int main(int argc, char *argv[]) |
382 | { |
383 | field *f; |
384 | ec_curve *c; |
385 | ec g = EC_INIT, d = EC_INIT; |
386 | mp *p, *a, *b, *r; |
387 | int i, n = argc == 1 ? 1 : atoi(argv[1]); |
388 | |
389 | printf("ec-bin: "); |
390 | fflush(stdout); |
391 | a = MP(1); |
392 | b = MP(0x066647ede6c332c7f8c0923bb58213b333b20e9ce4281fe115f7d8f90ad); |
393 | p = MP(0x20000000000000000000000000000000000000004000000000000000001); |
394 | r = |
395 | MP(6901746346790563787434755862277025555839812737345013555379383634485462); |
396 | |
397 | f = field_binpoly(p); |
398 | c = ec_binproj(f, a, b); |
399 | |
400 | g.x = MP(0x0fac9dfcbac8313bb2139f1bb755fef65bc391f8b36f8f8eb7371fd558b); |
401 | g.y = MP(0x1006a08a41903350678e58528bebf8a0beff867a7ca36716f7e01f81052); |
402 | |
403 | for (i = 0; i < n; i++) { |
404 | ec_mul(c, &d, &g, r); |
405 | if (EC_ATINF(&d)) { |
406 | fprintf(stderr, "zero too early\n"); |
407 | return (1); |
408 | } |
409 | ec_add(c, &d, &d, &g); |
410 | if (!EC_ATINF(&d)) { |
411 | fprintf(stderr, "didn't reach zero\n"); |
412 | MP_EPRINTX("d.x", d.x); |
413 | MP_EPRINTX("d.y", d.y); |
414 | MP_EPRINTX("d.z", d.y); |
415 | return (1); |
416 | } |
417 | ec_destroy(&d); |
418 | } |
419 | |
420 | ec_destroy(&g); |
421 | ec_destroycurve(c); |
422 | F_DESTROY(f); |
423 | MP_DROP(p); MP_DROP(a); MP_DROP(b); MP_DROP(r); |
424 | assert(!mparena_count(&mparena_global)); |
425 | printf("ok\n"); |
426 | return (0); |
427 | } |
428 | |
429 | #endif |
430 | |
431 | /*----- That's all, folks -------------------------------------------------*/ |