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ceb3f0c0 | 1 | /* -*-c-*- |
2 | * | |
f94b972d | 3 | * $Id$ |
ceb3f0c0 | 4 | * |
5 | * Arithmetic for elliptic curves over binary fields | |
6 | * | |
7 | * (c) 2004 Straylight/Edgeware | |
8 | */ | |
9 | ||
45c0fd36 | 10 | /*----- Licensing notice --------------------------------------------------* |
ceb3f0c0 | 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. | |
45c0fd36 | 18 | * |
ceb3f0c0 | 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. | |
45c0fd36 | 23 | * |
ceb3f0c0 | 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 | ||
ceb3f0c0 | 30 | /*----- Header files ------------------------------------------------------*/ |
31 | ||
32 | #include <mLib/sub.h> | |
33 | ||
34 | #include "ec.h" | |
f94b972d | 35 | #include "ec-guts.h" |
ceb3f0c0 | 36 | |
37 | /*----- Main code ---------------------------------------------------------*/ | |
38 | ||
39 | static const ec_ops ec_binops, ec_binprojops; | |
40 | ||
41 | static ec *ecneg(ec_curve *c, ec *d, const ec *p) | |
42 | { | |
43 | EC_COPY(d, p); | |
44 | if (d->x) | |
45 | d->y = F_ADD(c->f, d->y, d->y, d->x); | |
46 | return (d); | |
47 | } | |
48 | ||
49 | static ec *ecprojneg(ec_curve *c, ec *d, const ec *p) | |
50 | { | |
51 | EC_COPY(d, p); | |
52 | if (d->x) { | |
53 | mp *t = F_MUL(c->f, MP_NEW, d->x, d->z); | |
54 | d->y = F_ADD(c->f, d->y, d->y, t); | |
55 | MP_DROP(t); | |
56 | } | |
57 | return (d); | |
58 | } | |
59 | ||
60 | static ec *ecfind(ec_curve *c, ec *d, mp *x) | |
61 | { | |
bc985cef | 62 | field *f = c->f; |
bc985cef | 63 | mp *y, *u, *v; |
45c0fd36 | 64 | |
bc985cef | 65 | if (F_ZEROP(f, x)) |
432c4e18 | 66 | y = F_SQRT(f, MP_NEW, c->b); |
bc985cef | 67 | else { |
68 | u = F_SQR(f, MP_NEW, x); /* %$x^2$% */ | |
432c4e18 | 69 | y = F_MUL(f, MP_NEW, u, c->a); /* %$a x^2$% */ |
70 | y = F_ADD(f, y, y, c->b); /* %$a x^2 + b$% */ | |
bc985cef | 71 | v = F_MUL(f, MP_NEW, u, x); /* %$x^3$% */ |
72 | y = F_ADD(f, y, y, v); /* %$A = x^3 + a x^2 + b$% */ | |
73 | if (!F_ZEROP(f, y)) { | |
74 | u = F_INV(f, u, u); /* %$x^{-2}$% */ | |
75 | v = F_MUL(f, v, u, y); /* %$B = A x^{-2} = x + a + b x^{-2}$% */ | |
76 | y = F_QUADSOLVE(f, y, v); /* %$z^2 + z = B$% */ | |
77 | if (y) y = F_MUL(f, y, y, x); /* %$y = z x$% */ | |
78 | } | |
79 | MP_DROP(u); | |
80 | MP_DROP(v); | |
81 | } | |
82 | if (!y) return (0); | |
83 | EC_DESTROY(d); | |
84 | d->x = MP_COPY(x); | |
85 | d->y = y; | |
86 | d->z = MP_COPY(f->one); | |
87 | return (d); | |
ceb3f0c0 | 88 | } |
89 | ||
90 | static ec *ecdbl(ec_curve *c, ec *d, const ec *a) | |
91 | { | |
92 | if (EC_ATINF(a) || F_ZEROP(c->f, a->x)) | |
93 | EC_SETINF(d); | |
94 | else { | |
95 | field *f = c->f; | |
ceb3f0c0 | 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); /* %$\lambda^2 + \lambda$% */ | |
432c4e18 | 105 | dx = F_ADD(f, dx, dx, c->a); /* %$x' = a + \lambda^2 + \lambda$% */ |
ceb3f0c0 | 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'$% */ | |
111 | ||
112 | EC_DESTROY(d); | |
113 | d->x = dx; | |
114 | d->y = dy; | |
115 | d->z = 0; | |
116 | MP_DROP(lambda); | |
117 | } | |
118 | return (d); | |
119 | } | |
120 | ||
121 | static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a) | |
122 | { | |
123 | if (EC_ATINF(a) || F_ZEROP(c->f, a->x)) | |
124 | EC_SETINF(d); | |
125 | else { | |
126 | field *f = c->f; | |
f94b972d | 127 | ecctx_bin *cc = (ecctx_bin *)c; |
ceb3f0c0 | 128 | mp *dx, *dy, *dz, *u, *v; |
129 | ||
130 | dy = F_SQR(f, MP_NEW, a->z); /* %$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'$% */ | |
147 | ||
148 | EC_DESTROY(d); | |
149 | d->x = dx; | |
150 | d->y = dy; | |
151 | d->z = dz; | |
152 | MP_DROP(u); | |
153 | MP_DROP(v); | |
ceb3f0c0 | 154 | } |
155 | return (d); | |
156 | } | |
157 | ||
158 | static ec *ecadd(ec_curve *c, ec *d, const ec *a, const ec *b) | |
159 | { | |
160 | if (a == b) | |
161 | ecdbl(c, d, a); | |
162 | else if (EC_ATINF(a)) | |
163 | EC_COPY(d, b); | |
164 | else if (EC_ATINF(b)) | |
165 | EC_COPY(d, a); | |
166 | else { | |
167 | field *f = c->f; | |
ceb3f0c0 | 168 | mp *lambda; |
169 | mp *dx, *dy; | |
170 | ||
171 | if (!MP_EQ(a->x, b->x)) { | |
172 | dx = F_ADD(f, MP_NEW, a->x, b->x); /* %$x_0 + x_1$% */ | |
173 | dy = F_INV(f, MP_NEW, dx); /* %$(x_0 + x_1)^{-1}$% */ | |
174 | dx = F_ADD(f, dx, a->y, b->y); /* %$y_0 + y_1$% */ | |
175 | lambda = F_MUL(f, MP_NEW, dy, dx); | |
176 | /* %$\lambda = (y_0 + y_1)/(x_0 + x_1)$% */ | |
177 | ||
178 | dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */ | |
179 | dx = F_ADD(f, dx, dx, lambda); /* %$\lambda^2 + \lambda$% */ | |
432c4e18 | 180 | dx = F_ADD(f, dx, dx, c->a); /* %$a + \lambda^2 + \lambda$% */ |
ceb3f0c0 | 181 | dx = F_ADD(f, dx, dx, a->x); /* %$a + \lambda^2 + \lambda + x_0$% */ |
182 | dx = F_ADD(f, dx, dx, b->x); | |
45c0fd36 | 183 | /* %$x' = a + \lambda^2 + \lambda + x_0 + x_1$% */ |
ceb3f0c0 | 184 | } else if (!MP_EQ(a->y, b->y) || F_ZEROP(f, a->x)) { |
185 | EC_SETINF(d); | |
186 | return (d); | |
187 | } else { | |
188 | dx = F_INV(f, MP_NEW, a->x); /* %$x^{-1}$% */ | |
189 | dy = F_MUL(f, MP_NEW, dx, a->y); /* %$y/x$% */ | |
190 | lambda = F_ADD(f, dy, dy, a->x); /* %$\lambda = x + y/x$% */ | |
191 | ||
192 | dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */ | |
193 | dx = F_ADD(f, dx, dx, lambda); /* %$\lambda^2 + \lambda$% */ | |
432c4e18 | 194 | dx = F_ADD(f, dx, dx, c->a); /* %$x' = a + \lambda^2 + \lambda$% */ |
ceb3f0c0 | 195 | dy = MP_NEW; |
196 | } | |
45c0fd36 | 197 | |
ceb3f0c0 | 198 | dy = F_ADD(f, dy, a->x, dx); /* %$ x + x' $% */ |
199 | dy = F_MUL(f, dy, dy, lambda); /* %$ (x + x') \lambda$% */ | |
200 | dy = F_ADD(f, dy, dy, a->y); /* %$ (x + x') \lambda + y$% */ | |
201 | dy = F_ADD(f, dy, dy, dx); /* %$ y' = (x + x') \lambda + y + x'$% */ | |
202 | ||
203 | EC_DESTROY(d); | |
204 | d->x = dx; | |
205 | d->y = dy; | |
206 | d->z = 0; | |
207 | MP_DROP(lambda); | |
208 | } | |
209 | return (d); | |
210 | } | |
211 | ||
212 | static ec *ecprojadd(ec_curve *c, ec *d, const ec *a, const ec *b) | |
213 | { | |
214 | if (a == b) | |
215 | c->ops->dbl(c, d, a); | |
216 | else if (EC_ATINF(a)) | |
217 | EC_COPY(d, b); | |
218 | else if (EC_ATINF(b)) | |
219 | EC_COPY(d, a); | |
220 | else { | |
221 | field *f = c->f; | |
ceb3f0c0 | 222 | mp *dx, *dy, *dz, *u, *uu, *v, *t, *s, *ss, *r, *w, *l; |
223 | ||
224 | dz = F_SQR(f, MP_NEW, b->z); /* %$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$% */ | |
432c4e18 | 263 | dx = F_MUL(f, MP_NEW, uu, c->a); /* %$a z'^2$% */ |
ceb3f0c0 | 264 | uu = F_MUL(f, uu, t, r); /* %$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 | { | |
ceb3f0c0 | 290 | field *f = c->f; |
291 | int rc; | |
292 | mp *u, *v; | |
293 | ||
34e4f738 | 294 | if (EC_ATINF(p)) return (0); |
ceb3f0c0 | 295 | v = F_SQR(f, MP_NEW, p->x); |
296 | u = F_MUL(f, MP_NEW, v, p->x); | |
432c4e18 | 297 | v = F_MUL(f, v, v, c->a); |
ceb3f0c0 | 298 | u = F_ADD(f, u, u, v); |
432c4e18 | 299 | u = F_ADD(f, u, u, c->b); |
ceb3f0c0 | 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); | |
bc985cef | 304 | rc = F_ZEROP(f, u) ? 0 : -1; |
ceb3f0c0 | 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; | |
45c0fd36 | 314 | |
ceb3f0c0 | 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 | { | |
f94b972d | 323 | ecctx_bin *cc = (ecctx_bin *)c; |
432c4e18 | 324 | MP_DROP(cc->c.a); |
325 | MP_DROP(cc->c.b); | |
ceb3f0c0 | 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 | * | |
02d7884d | 335 | * Returns: A pointer to the curve, or null. |
ceb3f0c0 | 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 | { | |
f94b972d | 344 | ecctx_bin *cc = CREATE(ecctx_bin); |
ceb3f0c0 | 345 | cc->c.ops = &ec_binops; |
346 | cc->c.f = f; | |
432c4e18 | 347 | cc->c.a = F_IN(f, MP_NEW, a); |
348 | cc->c.b = F_IN(f, MP_NEW, b); | |
ceb3f0c0 | 349 | cc->bb = 0; |
350 | return (&cc->c); | |
351 | } | |
352 | ||
353 | ec_curve *ec_binproj(field *f, mp *a, mp *b) | |
354 | { | |
f94b972d | 355 | ecctx_bin *cc = CREATE(ecctx_bin); |
fe6657c9 MW |
356 | int i; |
357 | mp *c, *d; | |
358 | ||
ceb3f0c0 | 359 | cc->c.ops = &ec_binprojops; |
360 | cc->c.f = f; | |
432c4e18 | 361 | cc->c.a = F_IN(f, MP_NEW, a); |
362 | cc->c.b = F_IN(f, MP_NEW, b); | |
fe6657c9 MW |
363 | |
364 | c = MP_COPY(cc->c.b); | |
365 | for (i = 0; i < f->nbits - 2; i++) | |
366 | c = F_SQR(f, c, c); | |
367 | d = F_SQR(f, MP_NEW, c); d = F_SQR(f, d, d); | |
368 | if (!MP_EQ(d, cc->c.b)) { | |
369 | MP_DROP(c); | |
370 | MP_DROP(d); | |
02d7884d | 371 | MP_DROP(cc->c.a); |
372 | MP_DROP(cc->c.b); | |
373 | DESTROY(cc); | |
374 | return (0); | |
375 | } | |
fe6657c9 MW |
376 | cc->bb = c; |
377 | MP_DROP(d); | |
ceb3f0c0 | 378 | return (&cc->c); |
379 | } | |
380 | ||
381 | static const ec_ops ec_binops = { | |
f94b972d | 382 | "bin", |
34e4f738 | 383 | ecdestroy, ec_stdsamep, ec_idin, ec_idout, ec_idfix, |
bc985cef | 384 | ecfind, ecneg, ecadd, ec_stdsub, ecdbl, eccheck |
ceb3f0c0 | 385 | }; |
386 | ||
387 | static const ec_ops ec_binprojops = { | |
f94b972d | 388 | "binproj", |
34e4f738 | 389 | ecdestroy, ec_stdsamep, ec_projin, ec_projout, ec_projfix, |
bc985cef | 390 | ecfind, ecprojneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck |
ceb3f0c0 | 391 | }; |
392 | ||
393 | /*----- Test rig ----------------------------------------------------------*/ | |
394 | ||
395 | #ifdef TEST_RIG | |
396 | ||
397 | #define MP(x) mp_readstring(MP_NEW, #x, 0, 0) | |
398 | ||
399 | int main(int argc, char *argv[]) | |
400 | { | |
401 | field *f; | |
402 | ec_curve *c; | |
403 | ec g = EC_INIT, d = EC_INIT; | |
4edc47b8 | 404 | mp *p, *a, *b, *r, *beta; |
ceb3f0c0 | 405 | int i, n = argc == 1 ? 1 : atoi(argv[1]); |
406 | ||
407 | printf("ec-bin: "); | |
408 | fflush(stdout); | |
4edc47b8 | 409 | a = MP(0x7ffffffffffffffffffffffffffffffffffffffff); |
410 | b = MP(0x6645f3cacf1638e139c6cd13ef61734fbc9e3d9fb); | |
411 | p = MP(0x800000000000000000000000000000000000000c9); | |
412 | beta = MP(0x715169c109c612e390d347c748342bcd3b02a0bef); | |
413 | r = MP(0x040000000000000000000292fe77e70c12a4234c32); | |
ceb3f0c0 | 414 | |
4edc47b8 | 415 | f = field_binnorm(p, beta); |
ceb3f0c0 | 416 | c = ec_binproj(f, a, b); |
4edc47b8 | 417 | g.x = MP(0x0311103c17167564ace77ccb09c681f886ba54ee8); |
418 | g.y = MP(0x333ac13c6447f2e67613bf7009daf98c87bb50c7f); | |
ceb3f0c0 | 419 | |
45c0fd36 | 420 | for (i = 0; i < n; i++) { |
ceb3f0c0 | 421 | ec_mul(c, &d, &g, r); |
422 | if (EC_ATINF(&d)) { | |
423 | fprintf(stderr, "zero too early\n"); | |
424 | return (1); | |
425 | } | |
426 | ec_add(c, &d, &d, &g); | |
427 | if (!EC_ATINF(&d)) { | |
428 | fprintf(stderr, "didn't reach zero\n"); | |
429 | MP_EPRINTX("d.x", d.x); | |
430 | MP_EPRINTX("d.y", d.y); | |
ceb3f0c0 | 431 | return (1); |
432 | } | |
433 | ec_destroy(&d); | |
434 | } | |
435 | ||
436 | ec_destroy(&g); | |
437 | ec_destroycurve(c); | |
438 | F_DESTROY(f); | |
4edc47b8 | 439 | MP_DROP(p); MP_DROP(a); MP_DROP(b); MP_DROP(r); MP_DROP(beta); |
ceb3f0c0 | 440 | assert(!mparena_count(&mparena_global)); |
441 | printf("ok\n"); | |
442 | return (0); | |
443 | } | |
444 | ||
445 | #endif | |
446 | ||
447 | /*----- That's all, folks -------------------------------------------------*/ |