ceb3f0c0 |
1 | /* -*-c-*- |
2 | * |
b817bfc6 |
3 | * $Id: ec-bin.c,v 1.9 2004/04/08 01:36:15 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 | |
ceb3f0c0 |
30 | /*----- Header files ------------------------------------------------------*/ |
31 | |
32 | #include <mLib/sub.h> |
33 | |
34 | #include "ec.h" |
35 | |
36 | /*----- Data structures ---------------------------------------------------*/ |
37 | |
38 | typedef struct ecctx { |
39 | ec_curve c; |
ceb3f0c0 |
40 | mp *bb; |
41 | } ecctx; |
42 | |
43 | /*----- Main code ---------------------------------------------------------*/ |
44 | |
45 | static const ec_ops ec_binops, ec_binprojops; |
46 | |
47 | static ec *ecneg(ec_curve *c, ec *d, const ec *p) |
48 | { |
49 | EC_COPY(d, p); |
50 | if (d->x) |
51 | d->y = F_ADD(c->f, d->y, d->y, d->x); |
52 | return (d); |
53 | } |
54 | |
55 | static ec *ecprojneg(ec_curve *c, ec *d, const ec *p) |
56 | { |
57 | EC_COPY(d, p); |
58 | if (d->x) { |
59 | mp *t = F_MUL(c->f, MP_NEW, d->x, d->z); |
60 | d->y = F_ADD(c->f, d->y, d->y, t); |
61 | MP_DROP(t); |
62 | } |
63 | return (d); |
64 | } |
65 | |
66 | static ec *ecfind(ec_curve *c, ec *d, mp *x) |
67 | { |
bc985cef |
68 | field *f = c->f; |
bc985cef |
69 | mp *y, *u, *v; |
70 | |
71 | if (F_ZEROP(f, x)) |
432c4e18 |
72 | y = F_SQRT(f, MP_NEW, c->b); |
bc985cef |
73 | else { |
74 | u = F_SQR(f, MP_NEW, x); /* %$x^2$% */ |
432c4e18 |
75 | y = F_MUL(f, MP_NEW, u, c->a); /* %$a x^2$% */ |
76 | y = F_ADD(f, y, y, c->b); /* %$a x^2 + b$% */ |
bc985cef |
77 | v = F_MUL(f, MP_NEW, u, x); /* %$x^3$% */ |
78 | y = F_ADD(f, y, y, v); /* %$A = x^3 + a x^2 + b$% */ |
79 | if (!F_ZEROP(f, y)) { |
80 | u = F_INV(f, u, u); /* %$x^{-2}$% */ |
81 | v = F_MUL(f, v, u, y); /* %$B = A x^{-2} = x + a + b x^{-2}$% */ |
82 | y = F_QUADSOLVE(f, y, v); /* %$z^2 + z = B$% */ |
83 | if (y) y = F_MUL(f, y, y, x); /* %$y = z x$% */ |
84 | } |
85 | MP_DROP(u); |
86 | MP_DROP(v); |
87 | } |
88 | if (!y) return (0); |
89 | EC_DESTROY(d); |
90 | d->x = MP_COPY(x); |
91 | d->y = y; |
92 | d->z = MP_COPY(f->one); |
93 | return (d); |
ceb3f0c0 |
94 | } |
95 | |
96 | static ec *ecdbl(ec_curve *c, ec *d, const ec *a) |
97 | { |
98 | if (EC_ATINF(a) || F_ZEROP(c->f, a->x)) |
99 | EC_SETINF(d); |
100 | else { |
101 | field *f = c->f; |
ceb3f0c0 |
102 | mp *lambda; |
103 | mp *dx, *dy; |
104 | |
105 | dx = F_INV(f, MP_NEW, a->x); /* %$x^{-1}$% */ |
106 | dy = F_MUL(f, MP_NEW, dx, a->y); /* %$y/x$% */ |
107 | lambda = F_ADD(f, dy, dy, a->x); /* %$\lambda = x + y/x$% */ |
108 | |
109 | dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */ |
110 | dx = F_ADD(f, dx, dx, lambda); /* %$\lambda^2 + \lambda$% */ |
432c4e18 |
111 | dx = F_ADD(f, dx, dx, c->a); /* %$x' = a + \lambda^2 + \lambda$% */ |
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112 | |
113 | dy = F_ADD(f, MP_NEW, a->x, dx); /* %$ x + x' $% */ |
114 | dy = F_MUL(f, dy, dy, lambda); /* %$ (x + x') \lambda$% */ |
115 | dy = F_ADD(f, dy, dy, a->y); /* %$ (x + x') \lambda + y$% */ |
116 | dy = F_ADD(f, dy, dy, dx); /* %$ y' = (x + x') \lambda + y + x'$% */ |
117 | |
118 | EC_DESTROY(d); |
119 | d->x = dx; |
120 | d->y = dy; |
121 | d->z = 0; |
122 | MP_DROP(lambda); |
123 | } |
124 | return (d); |
125 | } |
126 | |
127 | static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a) |
128 | { |
129 | if (EC_ATINF(a) || F_ZEROP(c->f, a->x)) |
130 | EC_SETINF(d); |
131 | else { |
132 | field *f = c->f; |
133 | ecctx *cc = (ecctx *)c; |
134 | mp *dx, *dy, *dz, *u, *v; |
135 | |
136 | dy = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */ |
137 | dx = F_MUL(f, MP_NEW, dy, cc->bb); /* %$c z^2$% */ |
138 | dx = F_ADD(f, dx, dx, a->x); /* %$x + c z^2$% */ |
139 | dz = F_SQR(f, MP_NEW, dx); /* %$(x + c z^2)^2$% */ |
140 | dx = F_SQR(f, dx, dz); /* %$x' = (x + c z^2)^4$% */ |
141 | |
142 | dz = F_MUL(f, dz, dy, a->x); /* %$z' = x z^2$% */ |
143 | |
144 | dy = F_SQR(f, dy, a->x); /* %$x^2$% */ |
145 | u = F_MUL(f, MP_NEW, a->y, a->z); /* %$y z$% */ |
146 | u = F_ADD(f, u, u, dz); /* %$z' + y z$% */ |
147 | u = F_ADD(f, u, u, dy); /* %$u = z' + x^2 + y z$% */ |
148 | |
149 | v = F_SQR(f, MP_NEW, dy); /* %$x^4$% */ |
150 | dy = F_MUL(f, dy, v, dz); /* %$x^4 z'$% */ |
151 | v = F_MUL(f, v, u, dx); /* %$u x'$% */ |
152 | dy = F_ADD(f, dy, dy, v); /* %$y' = x^4 z' + u x'$% */ |
153 | |
154 | EC_DESTROY(d); |
155 | d->x = dx; |
156 | d->y = dy; |
157 | d->z = dz; |
158 | MP_DROP(u); |
159 | MP_DROP(v); |
ceb3f0c0 |
160 | } |
161 | return (d); |
162 | } |
163 | |
164 | static ec *ecadd(ec_curve *c, ec *d, const ec *a, const ec *b) |
165 | { |
166 | if (a == b) |
167 | ecdbl(c, d, a); |
168 | else if (EC_ATINF(a)) |
169 | EC_COPY(d, b); |
170 | else if (EC_ATINF(b)) |
171 | EC_COPY(d, a); |
172 | else { |
173 | field *f = c->f; |
ceb3f0c0 |
174 | mp *lambda; |
175 | mp *dx, *dy; |
176 | |
177 | if (!MP_EQ(a->x, b->x)) { |
178 | dx = F_ADD(f, MP_NEW, a->x, b->x); /* %$x_0 + x_1$% */ |
179 | dy = F_INV(f, MP_NEW, dx); /* %$(x_0 + x_1)^{-1}$% */ |
180 | dx = F_ADD(f, dx, a->y, b->y); /* %$y_0 + y_1$% */ |
181 | lambda = F_MUL(f, MP_NEW, dy, dx); |
182 | /* %$\lambda = (y_0 + y_1)/(x_0 + x_1)$% */ |
183 | |
184 | dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */ |
185 | dx = F_ADD(f, dx, dx, lambda); /* %$\lambda^2 + \lambda$% */ |
432c4e18 |
186 | dx = F_ADD(f, dx, dx, c->a); /* %$a + \lambda^2 + \lambda$% */ |
ceb3f0c0 |
187 | dx = F_ADD(f, dx, dx, a->x); /* %$a + \lambda^2 + \lambda + x_0$% */ |
188 | dx = F_ADD(f, dx, dx, b->x); |
189 | /* %$x' = a + \lambda^2 + \lambda + x_0 + x_1$% */ |
190 | } else if (!MP_EQ(a->y, b->y) || F_ZEROP(f, a->x)) { |
191 | EC_SETINF(d); |
192 | return (d); |
193 | } else { |
194 | dx = F_INV(f, MP_NEW, a->x); /* %$x^{-1}$% */ |
195 | dy = F_MUL(f, MP_NEW, dx, a->y); /* %$y/x$% */ |
196 | lambda = F_ADD(f, dy, dy, a->x); /* %$\lambda = x + y/x$% */ |
197 | |
198 | dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */ |
199 | dx = F_ADD(f, dx, dx, lambda); /* %$\lambda^2 + \lambda$% */ |
432c4e18 |
200 | dx = F_ADD(f, dx, dx, c->a); /* %$x' = a + \lambda^2 + \lambda$% */ |
ceb3f0c0 |
201 | dy = MP_NEW; |
202 | } |
203 | |
204 | dy = F_ADD(f, dy, a->x, dx); /* %$ x + x' $% */ |
205 | dy = F_MUL(f, dy, dy, lambda); /* %$ (x + x') \lambda$% */ |
206 | dy = F_ADD(f, dy, dy, a->y); /* %$ (x + x') \lambda + y$% */ |
207 | dy = F_ADD(f, dy, dy, dx); /* %$ y' = (x + x') \lambda + y + x'$% */ |
208 | |
209 | EC_DESTROY(d); |
210 | d->x = dx; |
211 | d->y = dy; |
212 | d->z = 0; |
213 | MP_DROP(lambda); |
214 | } |
215 | return (d); |
216 | } |
217 | |
218 | static ec *ecprojadd(ec_curve *c, ec *d, const ec *a, const ec *b) |
219 | { |
220 | if (a == b) |
221 | c->ops->dbl(c, d, a); |
222 | else if (EC_ATINF(a)) |
223 | EC_COPY(d, b); |
224 | else if (EC_ATINF(b)) |
225 | EC_COPY(d, a); |
226 | else { |
227 | field *f = c->f; |
ceb3f0c0 |
228 | mp *dx, *dy, *dz, *u, *uu, *v, *t, *s, *ss, *r, *w, *l; |
229 | |
230 | dz = F_SQR(f, MP_NEW, b->z); /* %$z_1^2$% */ |
231 | u = F_MUL(f, MP_NEW, dz, a->x); /* %$u_0 = x_0 z_1^2$% */ |
232 | t = F_MUL(f, MP_NEW, dz, b->z); /* %$z_1^3$% */ |
233 | s = F_MUL(f, MP_NEW, t, a->y); /* %$s_0 = y_0 z_1^3$% */ |
234 | |
235 | dz = F_SQR(f, dz, a->z); /* %$z_0^2$% */ |
236 | uu = F_MUL(f, MP_NEW, dz, b->x); /* %$u_1 = x_1 z_0^2$% */ |
237 | t = F_MUL(f, t, dz, a->z); /* %$z_0^3$% */ |
238 | ss = F_MUL(f, MP_NEW, t, b->y); /* %$s_1 = y_1 z_0^3$% */ |
239 | |
240 | w = F_ADD(f, u, u, uu); /* %$r = u_0 + u_1$% */ |
241 | r = F_ADD(f, s, s, ss); /* %$w = s_0 + s_1$% */ |
242 | if (F_ZEROP(f, w)) { |
243 | MP_DROP(w); |
244 | MP_DROP(uu); |
245 | MP_DROP(ss); |
246 | MP_DROP(t); |
247 | MP_DROP(dz); |
248 | if (F_ZEROP(f, r)) { |
249 | MP_DROP(r); |
250 | return (c->ops->dbl(c, d, a)); |
251 | } else { |
252 | MP_DROP(r); |
253 | EC_SETINF(d); |
254 | return (d); |
255 | } |
256 | } |
257 | |
258 | l = F_MUL(f, t, a->z, w); /* %$l = z_0 w$% */ |
259 | |
260 | dz = F_MUL(f, dz, l, b->z); /* %$z' = l z_1$% */ |
261 | |
262 | ss = F_MUL(f, ss, r, b->x); /* %$r x_1$% */ |
263 | t = F_MUL(f, uu, l, b->y); /* %$l y_1$% */ |
264 | v = F_ADD(f, ss, ss, t); /* %$v = r x_1 + l y_1$% */ |
265 | |
266 | t = F_ADD(f, t, r, dz); /* %$t = r + z'$% */ |
267 | |
268 | uu = F_SQR(f, MP_NEW, dz); /* %$z'^2$% */ |
432c4e18 |
269 | dx = F_MUL(f, MP_NEW, uu, c->a); /* %$a z'^2$% */ |
ceb3f0c0 |
270 | uu = F_MUL(f, uu, t, r); /* %$t r$% */ |
271 | dx = F_ADD(f, dx, dx, uu); /* %$a z'^2 + t r$% */ |
272 | r = F_SQR(f, r, w); /* %$w^2$% */ |
273 | uu = F_MUL(f, uu, r, w); /* %$w^3$% */ |
274 | dx = F_ADD(f, dx, dx, uu); /* %$x' = a z'^2 + t r + w^3$% */ |
275 | |
276 | r = F_SQR(f, r, l); /* %$l^2$% */ |
277 | dy = F_MUL(f, uu, v, r); /* %$v l^2$% */ |
278 | l = F_MUL(f, l, t, dx); /* %$t x'$% */ |
279 | dy = F_ADD(f, dy, dy, l); /* %$y' = t x' + v l^2$% */ |
280 | |
281 | EC_DESTROY(d); |
282 | d->x = dx; |
283 | d->y = dy; |
284 | d->z = dz; |
285 | MP_DROP(l); |
286 | MP_DROP(r); |
287 | MP_DROP(w); |
288 | MP_DROP(t); |
289 | MP_DROP(v); |
290 | } |
291 | return (d); |
292 | } |
293 | |
294 | static int eccheck(ec_curve *c, const ec *p) |
295 | { |
ceb3f0c0 |
296 | field *f = c->f; |
297 | int rc; |
298 | mp *u, *v; |
299 | |
34e4f738 |
300 | if (EC_ATINF(p)) return (0); |
ceb3f0c0 |
301 | v = F_SQR(f, MP_NEW, p->x); |
302 | u = F_MUL(f, MP_NEW, v, p->x); |
432c4e18 |
303 | v = F_MUL(f, v, v, c->a); |
ceb3f0c0 |
304 | u = F_ADD(f, u, u, v); |
432c4e18 |
305 | u = F_ADD(f, u, u, c->b); |
ceb3f0c0 |
306 | v = F_MUL(f, v, p->x, p->y); |
307 | u = F_ADD(f, u, u, v); |
308 | v = F_SQR(f, v, p->y); |
309 | u = F_ADD(f, u, u, v); |
bc985cef |
310 | rc = F_ZEROP(f, u) ? 0 : -1; |
ceb3f0c0 |
311 | mp_drop(u); |
312 | mp_drop(v); |
313 | return (rc); |
314 | } |
315 | |
316 | static int ecprojcheck(ec_curve *c, const ec *p) |
317 | { |
318 | ec t = EC_INIT; |
319 | int rc; |
320 | |
321 | c->ops->fix(c, &t, p); |
322 | rc = eccheck(c, &t); |
323 | EC_DESTROY(&t); |
324 | return (rc); |
325 | } |
326 | |
327 | static void ecdestroy(ec_curve *c) |
328 | { |
329 | ecctx *cc = (ecctx *)c; |
432c4e18 |
330 | MP_DROP(cc->c.a); |
331 | MP_DROP(cc->c.b); |
ceb3f0c0 |
332 | if (cc->bb) MP_DROP(cc->bb); |
333 | DESTROY(cc); |
334 | } |
335 | |
336 | /* --- @ec_bin@, @ec_binproj@ --- * |
337 | * |
338 | * Arguments: @field *f@ = the underlying field for this elliptic curve |
339 | * @mp *a, *b@ = the coefficients for this curve |
340 | * |
02d7884d |
341 | * Returns: A pointer to the curve, or null. |
ceb3f0c0 |
342 | * |
343 | * Use: Creates a curve structure for an elliptic curve defined over |
344 | * a binary field. The @binproj@ variant uses projective |
345 | * coordinates, which can be a win. |
346 | */ |
347 | |
348 | ec_curve *ec_bin(field *f, mp *a, mp *b) |
349 | { |
350 | ecctx *cc = CREATE(ecctx); |
351 | cc->c.ops = &ec_binops; |
352 | cc->c.f = f; |
432c4e18 |
353 | cc->c.a = F_IN(f, MP_NEW, a); |
354 | cc->c.b = F_IN(f, MP_NEW, b); |
ceb3f0c0 |
355 | cc->bb = 0; |
356 | return (&cc->c); |
357 | } |
358 | |
359 | ec_curve *ec_binproj(field *f, mp *a, mp *b) |
360 | { |
361 | ecctx *cc = CREATE(ecctx); |
362 | cc->c.ops = &ec_binprojops; |
363 | cc->c.f = f; |
432c4e18 |
364 | cc->c.a = F_IN(f, MP_NEW, a); |
365 | cc->c.b = F_IN(f, MP_NEW, b); |
4edc47b8 |
366 | cc->bb = F_SQRT(f, MP_NEW, cc->c.b); |
02d7884d |
367 | if (cc->bb) |
368 | cc->bb = F_SQRT(f, cc->bb, cc->bb); |
369 | if (!cc->bb) { |
370 | MP_DROP(cc->c.a); |
371 | MP_DROP(cc->c.b); |
372 | DESTROY(cc); |
373 | return (0); |
374 | } |
ceb3f0c0 |
375 | return (&cc->c); |
376 | } |
377 | |
378 | static const ec_ops ec_binops = { |
34e4f738 |
379 | ecdestroy, ec_stdsamep, ec_idin, ec_idout, ec_idfix, |
bc985cef |
380 | ecfind, ecneg, ecadd, ec_stdsub, ecdbl, eccheck |
ceb3f0c0 |
381 | }; |
382 | |
383 | static const ec_ops ec_binprojops = { |
34e4f738 |
384 | ecdestroy, ec_stdsamep, ec_projin, ec_projout, ec_projfix, |
bc985cef |
385 | ecfind, ecprojneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck |
ceb3f0c0 |
386 | }; |
387 | |
388 | /*----- Test rig ----------------------------------------------------------*/ |
389 | |
390 | #ifdef TEST_RIG |
391 | |
392 | #define MP(x) mp_readstring(MP_NEW, #x, 0, 0) |
393 | |
394 | int main(int argc, char *argv[]) |
395 | { |
396 | field *f; |
397 | ec_curve *c; |
398 | ec g = EC_INIT, d = EC_INIT; |
4edc47b8 |
399 | mp *p, *a, *b, *r, *beta; |
ceb3f0c0 |
400 | int i, n = argc == 1 ? 1 : atoi(argv[1]); |
401 | |
402 | printf("ec-bin: "); |
403 | fflush(stdout); |
4edc47b8 |
404 | a = MP(0x7ffffffffffffffffffffffffffffffffffffffff); |
405 | b = MP(0x6645f3cacf1638e139c6cd13ef61734fbc9e3d9fb); |
406 | p = MP(0x800000000000000000000000000000000000000c9); |
407 | beta = MP(0x715169c109c612e390d347c748342bcd3b02a0bef); |
408 | r = MP(0x040000000000000000000292fe77e70c12a4234c32); |
ceb3f0c0 |
409 | |
4edc47b8 |
410 | f = field_binnorm(p, beta); |
ceb3f0c0 |
411 | c = ec_binproj(f, a, b); |
4edc47b8 |
412 | g.x = MP(0x0311103c17167564ace77ccb09c681f886ba54ee8); |
413 | g.y = MP(0x333ac13c6447f2e67613bf7009daf98c87bb50c7f); |
ceb3f0c0 |
414 | |
415 | for (i = 0; i < n; i++) { |
416 | ec_mul(c, &d, &g, r); |
417 | if (EC_ATINF(&d)) { |
418 | fprintf(stderr, "zero too early\n"); |
419 | return (1); |
420 | } |
421 | ec_add(c, &d, &d, &g); |
422 | if (!EC_ATINF(&d)) { |
423 | fprintf(stderr, "didn't reach zero\n"); |
424 | MP_EPRINTX("d.x", d.x); |
425 | MP_EPRINTX("d.y", d.y); |
ceb3f0c0 |
426 | return (1); |
427 | } |
428 | ec_destroy(&d); |
429 | } |
430 | |
431 | ec_destroy(&g); |
432 | ec_destroycurve(c); |
433 | F_DESTROY(f); |
4edc47b8 |
434 | MP_DROP(p); MP_DROP(a); MP_DROP(b); MP_DROP(r); MP_DROP(beta); |
ceb3f0c0 |
435 | assert(!mparena_count(&mparena_global)); |
436 | printf("ok\n"); |
437 | return (0); |
438 | } |
439 | |
440 | #endif |
441 | |
442 | /*----- That's all, folks -------------------------------------------------*/ |