Commit | Line | Data |
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ceb3f0c0 | 1 | /* -*-c-*- |
2 | * | |
ceb3f0c0 | 3 | * Arithmetic for elliptic curves over binary fields |
4 | * | |
5 | * (c) 2004 Straylight/Edgeware | |
6 | */ | |
7 | ||
45c0fd36 | 8 | /*----- Licensing notice --------------------------------------------------* |
ceb3f0c0 | 9 | * |
10 | * This file is part of Catacomb. | |
11 | * | |
12 | * Catacomb is free software; you can redistribute it and/or modify | |
13 | * it under the terms of the GNU Library General Public License as | |
14 | * published by the Free Software Foundation; either version 2 of the | |
15 | * License, or (at your option) any later version. | |
45c0fd36 | 16 | * |
ceb3f0c0 | 17 | * Catacomb is distributed in the hope that it will be useful, |
18 | * but WITHOUT ANY WARRANTY; without even the implied warranty of | |
19 | * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the | |
20 | * GNU Library General Public License for more details. | |
45c0fd36 | 21 | * |
ceb3f0c0 | 22 | * You should have received a copy of the GNU Library General Public |
23 | * License along with Catacomb; if not, write to the Free | |
24 | * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, | |
25 | * MA 02111-1307, USA. | |
26 | */ | |
27 | ||
ceb3f0c0 | 28 | /*----- Header files ------------------------------------------------------*/ |
29 | ||
30 | #include <mLib/sub.h> | |
31 | ||
32 | #include "ec.h" | |
f94b972d | 33 | #include "ec-guts.h" |
ceb3f0c0 | 34 | |
35 | /*----- Main code ---------------------------------------------------------*/ | |
36 | ||
37 | static const ec_ops ec_binops, ec_binprojops; | |
38 | ||
39 | static ec *ecneg(ec_curve *c, ec *d, const ec *p) | |
40 | { | |
41 | EC_COPY(d, p); | |
42 | if (d->x) | |
43 | d->y = F_ADD(c->f, d->y, d->y, d->x); | |
44 | return (d); | |
45 | } | |
46 | ||
47 | static ec *ecprojneg(ec_curve *c, ec *d, const ec *p) | |
48 | { | |
49 | EC_COPY(d, p); | |
50 | if (d->x) { | |
51 | mp *t = F_MUL(c->f, MP_NEW, d->x, d->z); | |
52 | d->y = F_ADD(c->f, d->y, d->y, t); | |
53 | MP_DROP(t); | |
54 | } | |
55 | return (d); | |
56 | } | |
57 | ||
58 | static ec *ecfind(ec_curve *c, ec *d, mp *x) | |
59 | { | |
bc985cef | 60 | field *f = c->f; |
bc985cef | 61 | mp *y, *u, *v; |
45c0fd36 | 62 | |
bc985cef | 63 | if (F_ZEROP(f, x)) |
432c4e18 | 64 | y = F_SQRT(f, MP_NEW, c->b); |
bc985cef | 65 | else { |
66 | u = F_SQR(f, MP_NEW, x); /* %$x^2$% */ | |
432c4e18 | 67 | y = F_MUL(f, MP_NEW, u, c->a); /* %$a x^2$% */ |
68 | y = F_ADD(f, y, y, c->b); /* %$a x^2 + b$% */ | |
bc985cef | 69 | v = F_MUL(f, MP_NEW, u, x); /* %$x^3$% */ |
70 | y = F_ADD(f, y, y, v); /* %$A = x^3 + a x^2 + b$% */ | |
71 | if (!F_ZEROP(f, y)) { | |
72 | u = F_INV(f, u, u); /* %$x^{-2}$% */ | |
73 | v = F_MUL(f, v, u, y); /* %$B = A x^{-2} = x + a + b x^{-2}$% */ | |
74 | y = F_QUADSOLVE(f, y, v); /* %$z^2 + z = B$% */ | |
75 | if (y) y = F_MUL(f, y, y, x); /* %$y = z x$% */ | |
76 | } | |
77 | MP_DROP(u); | |
78 | MP_DROP(v); | |
79 | } | |
80 | if (!y) return (0); | |
81 | EC_DESTROY(d); | |
82 | d->x = MP_COPY(x); | |
83 | d->y = y; | |
84 | d->z = MP_COPY(f->one); | |
85 | return (d); | |
ceb3f0c0 | 86 | } |
87 | ||
88 | static ec *ecdbl(ec_curve *c, ec *d, const ec *a) | |
89 | { | |
90 | if (EC_ATINF(a) || F_ZEROP(c->f, a->x)) | |
91 | EC_SETINF(d); | |
92 | else { | |
93 | field *f = c->f; | |
ceb3f0c0 | 94 | mp *lambda; |
95 | mp *dx, *dy; | |
96 | ||
97 | dx = F_INV(f, MP_NEW, a->x); /* %$x^{-1}$% */ | |
98 | dy = F_MUL(f, MP_NEW, dx, a->y); /* %$y/x$% */ | |
99 | lambda = F_ADD(f, dy, dy, a->x); /* %$\lambda = x + y/x$% */ | |
100 | ||
101 | dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */ | |
102 | dx = F_ADD(f, dx, dx, lambda); /* %$\lambda^2 + \lambda$% */ | |
432c4e18 | 103 | dx = F_ADD(f, dx, dx, c->a); /* %$x' = a + \lambda^2 + \lambda$% */ |
ceb3f0c0 | 104 | |
105 | dy = F_ADD(f, MP_NEW, a->x, dx); /* %$ x + x' $% */ | |
106 | dy = F_MUL(f, dy, dy, lambda); /* %$ (x + x') \lambda$% */ | |
107 | dy = F_ADD(f, dy, dy, a->y); /* %$ (x + x') \lambda + y$% */ | |
108 | dy = F_ADD(f, dy, dy, dx); /* %$ y' = (x + x') \lambda + y + x'$% */ | |
109 | ||
110 | EC_DESTROY(d); | |
111 | d->x = dx; | |
112 | d->y = dy; | |
113 | d->z = 0; | |
114 | MP_DROP(lambda); | |
115 | } | |
116 | return (d); | |
117 | } | |
118 | ||
119 | static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a) | |
120 | { | |
121 | if (EC_ATINF(a) || F_ZEROP(c->f, a->x)) | |
122 | EC_SETINF(d); | |
123 | else { | |
124 | field *f = c->f; | |
f94b972d | 125 | ecctx_bin *cc = (ecctx_bin *)c; |
ceb3f0c0 | 126 | mp *dx, *dy, *dz, *u, *v; |
127 | ||
128 | dy = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */ | |
129 | dx = F_MUL(f, MP_NEW, dy, cc->bb); /* %$c z^2$% */ | |
130 | dx = F_ADD(f, dx, dx, a->x); /* %$x + c z^2$% */ | |
131 | dz = F_SQR(f, MP_NEW, dx); /* %$(x + c z^2)^2$% */ | |
132 | dx = F_SQR(f, dx, dz); /* %$x' = (x + c z^2)^4$% */ | |
133 | ||
134 | dz = F_MUL(f, dz, dy, a->x); /* %$z' = x z^2$% */ | |
135 | ||
136 | dy = F_SQR(f, dy, a->x); /* %$x^2$% */ | |
137 | u = F_MUL(f, MP_NEW, a->y, a->z); /* %$y z$% */ | |
138 | u = F_ADD(f, u, u, dz); /* %$z' + y z$% */ | |
139 | u = F_ADD(f, u, u, dy); /* %$u = z' + x^2 + y z$% */ | |
140 | ||
141 | v = F_SQR(f, MP_NEW, dy); /* %$x^4$% */ | |
142 | dy = F_MUL(f, dy, v, dz); /* %$x^4 z'$% */ | |
143 | v = F_MUL(f, v, u, dx); /* %$u x'$% */ | |
144 | dy = F_ADD(f, dy, dy, v); /* %$y' = x^4 z' + u x'$% */ | |
145 | ||
146 | EC_DESTROY(d); | |
147 | d->x = dx; | |
148 | d->y = dy; | |
149 | d->z = dz; | |
150 | MP_DROP(u); | |
151 | MP_DROP(v); | |
ceb3f0c0 | 152 | } |
153 | return (d); | |
154 | } | |
155 | ||
156 | static ec *ecadd(ec_curve *c, ec *d, const ec *a, const ec *b) | |
157 | { | |
158 | if (a == b) | |
159 | ecdbl(c, d, a); | |
160 | else if (EC_ATINF(a)) | |
161 | EC_COPY(d, b); | |
162 | else if (EC_ATINF(b)) | |
163 | EC_COPY(d, a); | |
164 | else { | |
165 | field *f = c->f; | |
ceb3f0c0 | 166 | mp *lambda; |
167 | mp *dx, *dy; | |
168 | ||
169 | if (!MP_EQ(a->x, b->x)) { | |
170 | dx = F_ADD(f, MP_NEW, a->x, b->x); /* %$x_0 + x_1$% */ | |
171 | dy = F_INV(f, MP_NEW, dx); /* %$(x_0 + x_1)^{-1}$% */ | |
172 | dx = F_ADD(f, dx, a->y, b->y); /* %$y_0 + y_1$% */ | |
173 | lambda = F_MUL(f, MP_NEW, dy, dx); | |
174 | /* %$\lambda = (y_0 + y_1)/(x_0 + x_1)$% */ | |
175 | ||
176 | dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */ | |
177 | dx = F_ADD(f, dx, dx, lambda); /* %$\lambda^2 + \lambda$% */ | |
432c4e18 | 178 | dx = F_ADD(f, dx, dx, c->a); /* %$a + \lambda^2 + \lambda$% */ |
ceb3f0c0 | 179 | dx = F_ADD(f, dx, dx, a->x); /* %$a + \lambda^2 + \lambda + x_0$% */ |
180 | dx = F_ADD(f, dx, dx, b->x); | |
45c0fd36 | 181 | /* %$x' = a + \lambda^2 + \lambda + x_0 + x_1$% */ |
ceb3f0c0 | 182 | } else if (!MP_EQ(a->y, b->y) || F_ZEROP(f, a->x)) { |
183 | EC_SETINF(d); | |
184 | return (d); | |
185 | } else { | |
186 | dx = F_INV(f, MP_NEW, a->x); /* %$x^{-1}$% */ | |
187 | dy = F_MUL(f, MP_NEW, dx, a->y); /* %$y/x$% */ | |
188 | lambda = F_ADD(f, dy, dy, a->x); /* %$\lambda = x + y/x$% */ | |
189 | ||
190 | dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */ | |
191 | dx = F_ADD(f, dx, dx, lambda); /* %$\lambda^2 + \lambda$% */ | |
432c4e18 | 192 | dx = F_ADD(f, dx, dx, c->a); /* %$x' = a + \lambda^2 + \lambda$% */ |
ceb3f0c0 | 193 | dy = MP_NEW; |
194 | } | |
45c0fd36 | 195 | |
ceb3f0c0 | 196 | dy = F_ADD(f, dy, a->x, dx); /* %$ x + x' $% */ |
197 | dy = F_MUL(f, dy, dy, lambda); /* %$ (x + x') \lambda$% */ | |
198 | dy = F_ADD(f, dy, dy, a->y); /* %$ (x + x') \lambda + y$% */ | |
199 | dy = F_ADD(f, dy, dy, dx); /* %$ y' = (x + x') \lambda + y + x'$% */ | |
200 | ||
201 | EC_DESTROY(d); | |
202 | d->x = dx; | |
203 | d->y = dy; | |
204 | d->z = 0; | |
205 | MP_DROP(lambda); | |
206 | } | |
207 | return (d); | |
208 | } | |
209 | ||
210 | static ec *ecprojadd(ec_curve *c, ec *d, const ec *a, const ec *b) | |
211 | { | |
212 | if (a == b) | |
213 | c->ops->dbl(c, d, a); | |
214 | else if (EC_ATINF(a)) | |
215 | EC_COPY(d, b); | |
216 | else if (EC_ATINF(b)) | |
217 | EC_COPY(d, a); | |
218 | else { | |
219 | field *f = c->f; | |
ceb3f0c0 | 220 | mp *dx, *dy, *dz, *u, *uu, *v, *t, *s, *ss, *r, *w, *l; |
221 | ||
222 | dz = F_SQR(f, MP_NEW, b->z); /* %$z_1^2$% */ | |
223 | u = F_MUL(f, MP_NEW, dz, a->x); /* %$u_0 = x_0 z_1^2$% */ | |
224 | t = F_MUL(f, MP_NEW, dz, b->z); /* %$z_1^3$% */ | |
225 | s = F_MUL(f, MP_NEW, t, a->y); /* %$s_0 = y_0 z_1^3$% */ | |
226 | ||
227 | dz = F_SQR(f, dz, a->z); /* %$z_0^2$% */ | |
228 | uu = F_MUL(f, MP_NEW, dz, b->x); /* %$u_1 = x_1 z_0^2$% */ | |
229 | t = F_MUL(f, t, dz, a->z); /* %$z_0^3$% */ | |
230 | ss = F_MUL(f, MP_NEW, t, b->y); /* %$s_1 = y_1 z_0^3$% */ | |
231 | ||
232 | w = F_ADD(f, u, u, uu); /* %$r = u_0 + u_1$% */ | |
233 | r = F_ADD(f, s, s, ss); /* %$w = s_0 + s_1$% */ | |
234 | if (F_ZEROP(f, w)) { | |
235 | MP_DROP(w); | |
236 | MP_DROP(uu); | |
237 | MP_DROP(ss); | |
238 | MP_DROP(t); | |
239 | MP_DROP(dz); | |
240 | if (F_ZEROP(f, r)) { | |
241 | MP_DROP(r); | |
242 | return (c->ops->dbl(c, d, a)); | |
243 | } else { | |
244 | MP_DROP(r); | |
245 | EC_SETINF(d); | |
246 | return (d); | |
247 | } | |
248 | } | |
249 | ||
250 | l = F_MUL(f, t, a->z, w); /* %$l = z_0 w$% */ | |
251 | ||
252 | dz = F_MUL(f, dz, l, b->z); /* %$z' = l z_1$% */ | |
253 | ||
254 | ss = F_MUL(f, ss, r, b->x); /* %$r x_1$% */ | |
255 | t = F_MUL(f, uu, l, b->y); /* %$l y_1$% */ | |
256 | v = F_ADD(f, ss, ss, t); /* %$v = r x_1 + l y_1$% */ | |
257 | ||
258 | t = F_ADD(f, t, r, dz); /* %$t = r + z'$% */ | |
259 | ||
260 | uu = F_SQR(f, MP_NEW, dz); /* %$z'^2$% */ | |
432c4e18 | 261 | dx = F_MUL(f, MP_NEW, uu, c->a); /* %$a z'^2$% */ |
ceb3f0c0 | 262 | uu = F_MUL(f, uu, t, r); /* %$t r$% */ |
263 | dx = F_ADD(f, dx, dx, uu); /* %$a z'^2 + t r$% */ | |
264 | r = F_SQR(f, r, w); /* %$w^2$% */ | |
265 | uu = F_MUL(f, uu, r, w); /* %$w^3$% */ | |
266 | dx = F_ADD(f, dx, dx, uu); /* %$x' = a z'^2 + t r + w^3$% */ | |
267 | ||
268 | r = F_SQR(f, r, l); /* %$l^2$% */ | |
269 | dy = F_MUL(f, uu, v, r); /* %$v l^2$% */ | |
270 | l = F_MUL(f, l, t, dx); /* %$t x'$% */ | |
271 | dy = F_ADD(f, dy, dy, l); /* %$y' = t x' + v l^2$% */ | |
272 | ||
273 | EC_DESTROY(d); | |
274 | d->x = dx; | |
275 | d->y = dy; | |
276 | d->z = dz; | |
277 | MP_DROP(l); | |
278 | MP_DROP(r); | |
279 | MP_DROP(w); | |
280 | MP_DROP(t); | |
281 | MP_DROP(v); | |
282 | } | |
283 | return (d); | |
284 | } | |
285 | ||
286 | static int eccheck(ec_curve *c, const ec *p) | |
287 | { | |
ceb3f0c0 | 288 | field *f = c->f; |
289 | int rc; | |
290 | mp *u, *v; | |
291 | ||
34e4f738 | 292 | if (EC_ATINF(p)) return (0); |
ceb3f0c0 | 293 | v = F_SQR(f, MP_NEW, p->x); |
294 | u = F_MUL(f, MP_NEW, v, p->x); | |
432c4e18 | 295 | v = F_MUL(f, v, v, c->a); |
ceb3f0c0 | 296 | u = F_ADD(f, u, u, v); |
432c4e18 | 297 | u = F_ADD(f, u, u, c->b); |
ceb3f0c0 | 298 | v = F_MUL(f, v, p->x, p->y); |
299 | u = F_ADD(f, u, u, v); | |
300 | v = F_SQR(f, v, p->y); | |
301 | u = F_ADD(f, u, u, v); | |
bc985cef | 302 | rc = F_ZEROP(f, u) ? 0 : -1; |
ceb3f0c0 | 303 | mp_drop(u); |
304 | mp_drop(v); | |
305 | return (rc); | |
306 | } | |
307 | ||
308 | static int ecprojcheck(ec_curve *c, const ec *p) | |
309 | { | |
310 | ec t = EC_INIT; | |
311 | int rc; | |
45c0fd36 | 312 | |
ceb3f0c0 | 313 | c->ops->fix(c, &t, p); |
314 | rc = eccheck(c, &t); | |
315 | EC_DESTROY(&t); | |
316 | return (rc); | |
317 | } | |
318 | ||
319 | static void ecdestroy(ec_curve *c) | |
320 | { | |
f94b972d | 321 | ecctx_bin *cc = (ecctx_bin *)c; |
432c4e18 | 322 | MP_DROP(cc->c.a); |
323 | MP_DROP(cc->c.b); | |
ceb3f0c0 | 324 | if (cc->bb) MP_DROP(cc->bb); |
325 | DESTROY(cc); | |
326 | } | |
327 | ||
328 | /* --- @ec_bin@, @ec_binproj@ --- * | |
329 | * | |
330 | * Arguments: @field *f@ = the underlying field for this elliptic curve | |
331 | * @mp *a, *b@ = the coefficients for this curve | |
332 | * | |
02d7884d | 333 | * Returns: A pointer to the curve, or null. |
ceb3f0c0 | 334 | * |
335 | * Use: Creates a curve structure for an elliptic curve defined over | |
336 | * a binary field. The @binproj@ variant uses projective | |
337 | * coordinates, which can be a win. | |
338 | */ | |
339 | ||
340 | ec_curve *ec_bin(field *f, mp *a, mp *b) | |
341 | { | |
f94b972d | 342 | ecctx_bin *cc = CREATE(ecctx_bin); |
ceb3f0c0 | 343 | cc->c.ops = &ec_binops; |
344 | cc->c.f = f; | |
432c4e18 | 345 | cc->c.a = F_IN(f, MP_NEW, a); |
346 | cc->c.b = F_IN(f, MP_NEW, b); | |
ceb3f0c0 | 347 | cc->bb = 0; |
348 | return (&cc->c); | |
349 | } | |
350 | ||
351 | ec_curve *ec_binproj(field *f, mp *a, mp *b) | |
352 | { | |
f94b972d | 353 | ecctx_bin *cc = CREATE(ecctx_bin); |
fe6657c9 MW |
354 | int i; |
355 | mp *c, *d; | |
356 | ||
ceb3f0c0 | 357 | cc->c.ops = &ec_binprojops; |
358 | cc->c.f = f; | |
432c4e18 | 359 | cc->c.a = F_IN(f, MP_NEW, a); |
360 | cc->c.b = F_IN(f, MP_NEW, b); | |
fe6657c9 MW |
361 | |
362 | c = MP_COPY(cc->c.b); | |
363 | for (i = 0; i < f->nbits - 2; i++) | |
364 | c = F_SQR(f, c, c); | |
365 | d = F_SQR(f, MP_NEW, c); d = F_SQR(f, d, d); | |
366 | if (!MP_EQ(d, cc->c.b)) { | |
367 | MP_DROP(c); | |
368 | MP_DROP(d); | |
02d7884d | 369 | MP_DROP(cc->c.a); |
370 | MP_DROP(cc->c.b); | |
371 | DESTROY(cc); | |
372 | return (0); | |
373 | } | |
fe6657c9 MW |
374 | cc->bb = c; |
375 | MP_DROP(d); | |
ceb3f0c0 | 376 | return (&cc->c); |
377 | } | |
378 | ||
379 | static const ec_ops ec_binops = { | |
f94b972d | 380 | "bin", |
34e4f738 | 381 | ecdestroy, ec_stdsamep, ec_idin, ec_idout, ec_idfix, |
bc985cef | 382 | ecfind, ecneg, ecadd, ec_stdsub, ecdbl, eccheck |
ceb3f0c0 | 383 | }; |
384 | ||
385 | static const ec_ops ec_binprojops = { | |
f94b972d | 386 | "binproj", |
34e4f738 | 387 | ecdestroy, ec_stdsamep, ec_projin, ec_projout, ec_projfix, |
bc985cef | 388 | ecfind, ecprojneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck |
ceb3f0c0 | 389 | }; |
390 | ||
391 | /*----- Test rig ----------------------------------------------------------*/ | |
392 | ||
393 | #ifdef TEST_RIG | |
394 | ||
395 | #define MP(x) mp_readstring(MP_NEW, #x, 0, 0) | |
396 | ||
397 | int main(int argc, char *argv[]) | |
398 | { | |
399 | field *f; | |
400 | ec_curve *c; | |
401 | ec g = EC_INIT, d = EC_INIT; | |
4edc47b8 | 402 | mp *p, *a, *b, *r, *beta; |
ceb3f0c0 | 403 | int i, n = argc == 1 ? 1 : atoi(argv[1]); |
404 | ||
405 | printf("ec-bin: "); | |
406 | fflush(stdout); | |
4edc47b8 | 407 | a = MP(0x7ffffffffffffffffffffffffffffffffffffffff); |
408 | b = MP(0x6645f3cacf1638e139c6cd13ef61734fbc9e3d9fb); | |
409 | p = MP(0x800000000000000000000000000000000000000c9); | |
410 | beta = MP(0x715169c109c612e390d347c748342bcd3b02a0bef); | |
411 | r = MP(0x040000000000000000000292fe77e70c12a4234c32); | |
ceb3f0c0 | 412 | |
4edc47b8 | 413 | f = field_binnorm(p, beta); |
ceb3f0c0 | 414 | c = ec_binproj(f, a, b); |
4edc47b8 | 415 | g.x = MP(0x0311103c17167564ace77ccb09c681f886ba54ee8); |
416 | g.y = MP(0x333ac13c6447f2e67613bf7009daf98c87bb50c7f); | |
ceb3f0c0 | 417 | |
45c0fd36 | 418 | for (i = 0; i < n; i++) { |
ceb3f0c0 | 419 | ec_mul(c, &d, &g, r); |
420 | if (EC_ATINF(&d)) { | |
421 | fprintf(stderr, "zero too early\n"); | |
422 | return (1); | |
423 | } | |
424 | ec_add(c, &d, &d, &g); | |
425 | if (!EC_ATINF(&d)) { | |
426 | fprintf(stderr, "didn't reach zero\n"); | |
427 | MP_EPRINTX("d.x", d.x); | |
428 | MP_EPRINTX("d.y", d.y); | |
ceb3f0c0 | 429 | return (1); |
430 | } | |
431 | ec_destroy(&d); | |
432 | } | |
433 | ||
434 | ec_destroy(&g); | |
435 | ec_destroycurve(c); | |
436 | F_DESTROY(f); | |
4edc47b8 | 437 | MP_DROP(p); MP_DROP(a); MP_DROP(b); MP_DROP(r); MP_DROP(beta); |
ceb3f0c0 | 438 | assert(!mparena_count(&mparena_global)); |
439 | printf("ok\n"); | |
440 | return (0); | |
441 | } | |
442 | ||
443 | #endif | |
444 | ||
445 | /*----- That's all, folks -------------------------------------------------*/ |