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 | |
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" |
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; |
64 | |
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$% */ |
182 | dx = F_ADD(f, dx, dx, b->x); |
183 | /* %$x' = a + \lambda^2 + \lambda + x_0 + x_1$% */ |
184 | } else if (!MP_EQ(a->y, b->y) || F_ZEROP(f, a->x)) { |
185 | EC_SETINF(d); |
186 | return (d); |
187 | } else { |
188 | dx = F_INV(f, MP_NEW, a->x); /* %$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$% */ |
ceb3f0c0 |
195 | dy = MP_NEW; |
196 | } |
197 | |
198 | dy = F_ADD(f, dy, a->x, dx); /* %$ 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; |
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 | { |
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); |
ceb3f0c0 |
356 | cc->c.ops = &ec_binprojops; |
357 | cc->c.f = f; |
432c4e18 |
358 | cc->c.a = F_IN(f, MP_NEW, a); |
359 | cc->c.b = F_IN(f, MP_NEW, b); |
4edc47b8 |
360 | cc->bb = F_SQRT(f, MP_NEW, cc->c.b); |
02d7884d |
361 | if (cc->bb) |
362 | cc->bb = F_SQRT(f, cc->bb, cc->bb); |
363 | if (!cc->bb) { |
364 | MP_DROP(cc->c.a); |
365 | MP_DROP(cc->c.b); |
366 | DESTROY(cc); |
367 | return (0); |
368 | } |
ceb3f0c0 |
369 | return (&cc->c); |
370 | } |
371 | |
372 | static const ec_ops ec_binops = { |
f94b972d |
373 | "bin", |
34e4f738 |
374 | ecdestroy, ec_stdsamep, ec_idin, ec_idout, ec_idfix, |
bc985cef |
375 | ecfind, ecneg, ecadd, ec_stdsub, ecdbl, eccheck |
ceb3f0c0 |
376 | }; |
377 | |
378 | static const ec_ops ec_binprojops = { |
f94b972d |
379 | "binproj", |
34e4f738 |
380 | ecdestroy, ec_stdsamep, ec_projin, ec_projout, ec_projfix, |
bc985cef |
381 | ecfind, ecprojneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck |
ceb3f0c0 |
382 | }; |
383 | |
384 | /*----- Test rig ----------------------------------------------------------*/ |
385 | |
386 | #ifdef TEST_RIG |
387 | |
388 | #define MP(x) mp_readstring(MP_NEW, #x, 0, 0) |
389 | |
390 | int main(int argc, char *argv[]) |
391 | { |
392 | field *f; |
393 | ec_curve *c; |
394 | ec g = EC_INIT, d = EC_INIT; |
4edc47b8 |
395 | mp *p, *a, *b, *r, *beta; |
ceb3f0c0 |
396 | int i, n = argc == 1 ? 1 : atoi(argv[1]); |
397 | |
398 | printf("ec-bin: "); |
399 | fflush(stdout); |
4edc47b8 |
400 | a = MP(0x7ffffffffffffffffffffffffffffffffffffffff); |
401 | b = MP(0x6645f3cacf1638e139c6cd13ef61734fbc9e3d9fb); |
402 | p = MP(0x800000000000000000000000000000000000000c9); |
403 | beta = MP(0x715169c109c612e390d347c748342bcd3b02a0bef); |
404 | r = MP(0x040000000000000000000292fe77e70c12a4234c32); |
ceb3f0c0 |
405 | |
4edc47b8 |
406 | f = field_binnorm(p, beta); |
ceb3f0c0 |
407 | c = ec_binproj(f, a, b); |
4edc47b8 |
408 | g.x = MP(0x0311103c17167564ace77ccb09c681f886ba54ee8); |
409 | g.y = MP(0x333ac13c6447f2e67613bf7009daf98c87bb50c7f); |
ceb3f0c0 |
410 | |
411 | for (i = 0; i < n; i++) { |
412 | ec_mul(c, &d, &g, r); |
413 | if (EC_ATINF(&d)) { |
414 | fprintf(stderr, "zero too early\n"); |
415 | return (1); |
416 | } |
417 | ec_add(c, &d, &d, &g); |
418 | if (!EC_ATINF(&d)) { |
419 | fprintf(stderr, "didn't reach zero\n"); |
420 | MP_EPRINTX("d.x", d.x); |
421 | MP_EPRINTX("d.y", d.y); |
ceb3f0c0 |
422 | return (1); |
423 | } |
424 | ec_destroy(&d); |
425 | } |
426 | |
427 | ec_destroy(&g); |
428 | ec_destroycurve(c); |
429 | F_DESTROY(f); |
4edc47b8 |
430 | MP_DROP(p); MP_DROP(a); MP_DROP(b); MP_DROP(r); MP_DROP(beta); |
ceb3f0c0 |
431 | assert(!mparena_count(&mparena_global)); |
432 | printf("ok\n"); |
433 | return (0); |
434 | } |
435 | |
436 | #endif |
437 | |
438 | /*----- That's all, folks -------------------------------------------------*/ |