ceb3f0c0 |
1 | /* -*-c-*- |
2 | * |
4edc47b8 |
3 | * $Id: ec-bin.c,v 1.7 2004/04/01 21:28:41 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 | |
30 | /*----- Revision history --------------------------------------------------* |
31 | * |
32 | * $Log: ec-bin.c,v $ |
4edc47b8 |
33 | * Revision 1.7 2004/04/01 21:28:41 mdw |
34 | * Normal basis support (translates to poly basis internally). Rewrite |
35 | * EC and prime group table generators in awk, so that they can reuse data |
36 | * for repeated constants. |
37 | * |
34e4f738 |
38 | * Revision 1.6 2004/04/01 12:50:09 mdw |
39 | * Add cyclic group abstraction, with test code. Separate off exponentation |
40 | * functions for better static linking. Fix a buttload of bugs on the way. |
41 | * Generally ensure that negative exponents do inversion correctly. Add |
42 | * table of standard prime-field subgroups. (Binary field subgroups are |
43 | * currently unimplemented but easy to add if anyone ever finds a good one.) |
44 | * |
432c4e18 |
45 | * Revision 1.5 2004/03/27 17:54:11 mdw |
46 | * Standard curves and curve checking. |
47 | * |
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48 | * Revision 1.4 2004/03/23 15:19:32 mdw |
49 | * Test elliptic curves more thoroughly. |
50 | * |
391faf42 |
51 | * Revision 1.3 2004/03/22 02:19:09 mdw |
52 | * Rationalise the sliding-window threshold. Drop guarantee that right |
53 | * arguments to EC @add@ are canonical, and fix up projective implementations |
54 | * to cope. |
55 | * |
c3caa2fa |
56 | * Revision 1.2 2004/03/21 22:52:06 mdw |
57 | * Merge and close elliptic curve branch. |
58 | * |
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59 | * Revision 1.1.2.1 2004/03/21 22:39:46 mdw |
60 | * Elliptic curves on binary fields work. |
61 | * |
62 | */ |
63 | |
64 | /*----- Header files ------------------------------------------------------*/ |
65 | |
66 | #include <mLib/sub.h> |
67 | |
68 | #include "ec.h" |
69 | |
70 | /*----- Data structures ---------------------------------------------------*/ |
71 | |
72 | typedef struct ecctx { |
73 | ec_curve c; |
ceb3f0c0 |
74 | mp *bb; |
75 | } ecctx; |
76 | |
77 | /*----- Main code ---------------------------------------------------------*/ |
78 | |
79 | static const ec_ops ec_binops, ec_binprojops; |
80 | |
81 | static ec *ecneg(ec_curve *c, ec *d, const ec *p) |
82 | { |
83 | EC_COPY(d, p); |
84 | if (d->x) |
85 | d->y = F_ADD(c->f, d->y, d->y, d->x); |
86 | return (d); |
87 | } |
88 | |
89 | static ec *ecprojneg(ec_curve *c, ec *d, const ec *p) |
90 | { |
91 | EC_COPY(d, p); |
92 | if (d->x) { |
93 | mp *t = F_MUL(c->f, MP_NEW, d->x, d->z); |
94 | d->y = F_ADD(c->f, d->y, d->y, t); |
95 | MP_DROP(t); |
96 | } |
97 | return (d); |
98 | } |
99 | |
100 | static ec *ecfind(ec_curve *c, ec *d, mp *x) |
101 | { |
bc985cef |
102 | field *f = c->f; |
bc985cef |
103 | mp *y, *u, *v; |
104 | |
105 | if (F_ZEROP(f, x)) |
432c4e18 |
106 | y = F_SQRT(f, MP_NEW, c->b); |
bc985cef |
107 | else { |
108 | u = F_SQR(f, MP_NEW, x); /* %$x^2$% */ |
432c4e18 |
109 | y = F_MUL(f, MP_NEW, u, c->a); /* %$a x^2$% */ |
110 | y = F_ADD(f, y, y, c->b); /* %$a x^2 + b$% */ |
bc985cef |
111 | v = F_MUL(f, MP_NEW, u, x); /* %$x^3$% */ |
112 | y = F_ADD(f, y, y, v); /* %$A = x^3 + a x^2 + b$% */ |
113 | if (!F_ZEROP(f, y)) { |
114 | u = F_INV(f, u, u); /* %$x^{-2}$% */ |
115 | v = F_MUL(f, v, u, y); /* %$B = A x^{-2} = x + a + b x^{-2}$% */ |
116 | y = F_QUADSOLVE(f, y, v); /* %$z^2 + z = B$% */ |
117 | if (y) y = F_MUL(f, y, y, x); /* %$y = z x$% */ |
118 | } |
119 | MP_DROP(u); |
120 | MP_DROP(v); |
121 | } |
122 | if (!y) return (0); |
123 | EC_DESTROY(d); |
124 | d->x = MP_COPY(x); |
125 | d->y = y; |
126 | d->z = MP_COPY(f->one); |
127 | return (d); |
ceb3f0c0 |
128 | } |
129 | |
130 | static ec *ecdbl(ec_curve *c, ec *d, const ec *a) |
131 | { |
132 | if (EC_ATINF(a) || F_ZEROP(c->f, a->x)) |
133 | EC_SETINF(d); |
134 | else { |
135 | field *f = c->f; |
ceb3f0c0 |
136 | mp *lambda; |
137 | mp *dx, *dy; |
138 | |
139 | dx = F_INV(f, MP_NEW, a->x); /* %$x^{-1}$% */ |
140 | dy = F_MUL(f, MP_NEW, dx, a->y); /* %$y/x$% */ |
141 | lambda = F_ADD(f, dy, dy, a->x); /* %$\lambda = x + y/x$% */ |
142 | |
143 | dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */ |
144 | dx = F_ADD(f, dx, dx, lambda); /* %$\lambda^2 + \lambda$% */ |
432c4e18 |
145 | dx = F_ADD(f, dx, dx, c->a); /* %$x' = a + \lambda^2 + \lambda$% */ |
ceb3f0c0 |
146 | |
147 | dy = F_ADD(f, MP_NEW, a->x, dx); /* %$ x + x' $% */ |
148 | dy = F_MUL(f, dy, dy, lambda); /* %$ (x + x') \lambda$% */ |
149 | dy = F_ADD(f, dy, dy, a->y); /* %$ (x + x') \lambda + y$% */ |
150 | dy = F_ADD(f, dy, dy, dx); /* %$ y' = (x + x') \lambda + y + x'$% */ |
151 | |
152 | EC_DESTROY(d); |
153 | d->x = dx; |
154 | d->y = dy; |
155 | d->z = 0; |
156 | MP_DROP(lambda); |
157 | } |
158 | return (d); |
159 | } |
160 | |
161 | static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a) |
162 | { |
163 | if (EC_ATINF(a) || F_ZEROP(c->f, a->x)) |
164 | EC_SETINF(d); |
165 | else { |
166 | field *f = c->f; |
167 | ecctx *cc = (ecctx *)c; |
168 | mp *dx, *dy, *dz, *u, *v; |
169 | |
170 | dy = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */ |
171 | dx = F_MUL(f, MP_NEW, dy, cc->bb); /* %$c z^2$% */ |
172 | dx = F_ADD(f, dx, dx, a->x); /* %$x + c z^2$% */ |
173 | dz = F_SQR(f, MP_NEW, dx); /* %$(x + c z^2)^2$% */ |
174 | dx = F_SQR(f, dx, dz); /* %$x' = (x + c z^2)^4$% */ |
175 | |
176 | dz = F_MUL(f, dz, dy, a->x); /* %$z' = x z^2$% */ |
177 | |
178 | dy = F_SQR(f, dy, a->x); /* %$x^2$% */ |
179 | u = F_MUL(f, MP_NEW, a->y, a->z); /* %$y z$% */ |
180 | u = F_ADD(f, u, u, dz); /* %$z' + y z$% */ |
181 | u = F_ADD(f, u, u, dy); /* %$u = z' + x^2 + y z$% */ |
182 | |
183 | v = F_SQR(f, MP_NEW, dy); /* %$x^4$% */ |
184 | dy = F_MUL(f, dy, v, dz); /* %$x^4 z'$% */ |
185 | v = F_MUL(f, v, u, dx); /* %$u x'$% */ |
186 | dy = F_ADD(f, dy, dy, v); /* %$y' = x^4 z' + u x'$% */ |
187 | |
188 | EC_DESTROY(d); |
189 | d->x = dx; |
190 | d->y = dy; |
191 | d->z = dz; |
192 | MP_DROP(u); |
193 | MP_DROP(v); |
194 | assert(!(d->x->f & MP_DESTROYED)); |
195 | assert(!(d->y->f & MP_DESTROYED)); |
196 | assert(!(d->z->f & MP_DESTROYED)); |
197 | } |
198 | return (d); |
199 | } |
200 | |
201 | static ec *ecadd(ec_curve *c, ec *d, const ec *a, const ec *b) |
202 | { |
203 | if (a == b) |
204 | ecdbl(c, d, a); |
205 | else if (EC_ATINF(a)) |
206 | EC_COPY(d, b); |
207 | else if (EC_ATINF(b)) |
208 | EC_COPY(d, a); |
209 | else { |
210 | field *f = c->f; |
ceb3f0c0 |
211 | mp *lambda; |
212 | mp *dx, *dy; |
213 | |
214 | if (!MP_EQ(a->x, b->x)) { |
215 | dx = F_ADD(f, MP_NEW, a->x, b->x); /* %$x_0 + x_1$% */ |
216 | dy = F_INV(f, MP_NEW, dx); /* %$(x_0 + x_1)^{-1}$% */ |
217 | dx = F_ADD(f, dx, a->y, b->y); /* %$y_0 + y_1$% */ |
218 | lambda = F_MUL(f, MP_NEW, dy, dx); |
219 | /* %$\lambda = (y_0 + y_1)/(x_0 + x_1)$% */ |
220 | |
221 | dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */ |
222 | dx = F_ADD(f, dx, dx, lambda); /* %$\lambda^2 + \lambda$% */ |
432c4e18 |
223 | dx = F_ADD(f, dx, dx, c->a); /* %$a + \lambda^2 + \lambda$% */ |
ceb3f0c0 |
224 | dx = F_ADD(f, dx, dx, a->x); /* %$a + \lambda^2 + \lambda + x_0$% */ |
225 | dx = F_ADD(f, dx, dx, b->x); |
226 | /* %$x' = a + \lambda^2 + \lambda + x_0 + x_1$% */ |
227 | } else if (!MP_EQ(a->y, b->y) || F_ZEROP(f, a->x)) { |
228 | EC_SETINF(d); |
229 | return (d); |
230 | } else { |
231 | dx = F_INV(f, MP_NEW, a->x); /* %$x^{-1}$% */ |
232 | dy = F_MUL(f, MP_NEW, dx, a->y); /* %$y/x$% */ |
233 | lambda = F_ADD(f, dy, dy, a->x); /* %$\lambda = x + y/x$% */ |
234 | |
235 | dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */ |
236 | dx = F_ADD(f, dx, dx, lambda); /* %$\lambda^2 + \lambda$% */ |
432c4e18 |
237 | dx = F_ADD(f, dx, dx, c->a); /* %$x' = a + \lambda^2 + \lambda$% */ |
ceb3f0c0 |
238 | dy = MP_NEW; |
239 | } |
240 | |
241 | dy = F_ADD(f, dy, a->x, dx); /* %$ x + x' $% */ |
242 | dy = F_MUL(f, dy, dy, lambda); /* %$ (x + x') \lambda$% */ |
243 | dy = F_ADD(f, dy, dy, a->y); /* %$ (x + x') \lambda + y$% */ |
244 | dy = F_ADD(f, dy, dy, dx); /* %$ y' = (x + x') \lambda + y + x'$% */ |
245 | |
246 | EC_DESTROY(d); |
247 | d->x = dx; |
248 | d->y = dy; |
249 | d->z = 0; |
250 | MP_DROP(lambda); |
251 | } |
252 | return (d); |
253 | } |
254 | |
255 | static ec *ecprojadd(ec_curve *c, ec *d, const ec *a, const ec *b) |
256 | { |
257 | if (a == b) |
258 | c->ops->dbl(c, d, a); |
259 | else if (EC_ATINF(a)) |
260 | EC_COPY(d, b); |
261 | else if (EC_ATINF(b)) |
262 | EC_COPY(d, a); |
263 | else { |
264 | field *f = c->f; |
ceb3f0c0 |
265 | mp *dx, *dy, *dz, *u, *uu, *v, *t, *s, *ss, *r, *w, *l; |
266 | |
267 | dz = F_SQR(f, MP_NEW, b->z); /* %$z_1^2$% */ |
268 | u = F_MUL(f, MP_NEW, dz, a->x); /* %$u_0 = x_0 z_1^2$% */ |
269 | t = F_MUL(f, MP_NEW, dz, b->z); /* %$z_1^3$% */ |
270 | s = F_MUL(f, MP_NEW, t, a->y); /* %$s_0 = y_0 z_1^3$% */ |
271 | |
272 | dz = F_SQR(f, dz, a->z); /* %$z_0^2$% */ |
273 | uu = F_MUL(f, MP_NEW, dz, b->x); /* %$u_1 = x_1 z_0^2$% */ |
274 | t = F_MUL(f, t, dz, a->z); /* %$z_0^3$% */ |
275 | ss = F_MUL(f, MP_NEW, t, b->y); /* %$s_1 = y_1 z_0^3$% */ |
276 | |
277 | w = F_ADD(f, u, u, uu); /* %$r = u_0 + u_1$% */ |
278 | r = F_ADD(f, s, s, ss); /* %$w = s_0 + s_1$% */ |
279 | if (F_ZEROP(f, w)) { |
280 | MP_DROP(w); |
281 | MP_DROP(uu); |
282 | MP_DROP(ss); |
283 | MP_DROP(t); |
284 | MP_DROP(dz); |
285 | if (F_ZEROP(f, r)) { |
286 | MP_DROP(r); |
287 | return (c->ops->dbl(c, d, a)); |
288 | } else { |
289 | MP_DROP(r); |
290 | EC_SETINF(d); |
291 | return (d); |
292 | } |
293 | } |
294 | |
295 | l = F_MUL(f, t, a->z, w); /* %$l = z_0 w$% */ |
296 | |
297 | dz = F_MUL(f, dz, l, b->z); /* %$z' = l z_1$% */ |
298 | |
299 | ss = F_MUL(f, ss, r, b->x); /* %$r x_1$% */ |
300 | t = F_MUL(f, uu, l, b->y); /* %$l y_1$% */ |
301 | v = F_ADD(f, ss, ss, t); /* %$v = r x_1 + l y_1$% */ |
302 | |
303 | t = F_ADD(f, t, r, dz); /* %$t = r + z'$% */ |
304 | |
305 | uu = F_SQR(f, MP_NEW, dz); /* %$z'^2$% */ |
432c4e18 |
306 | dx = F_MUL(f, MP_NEW, uu, c->a); /* %$a z'^2$% */ |
ceb3f0c0 |
307 | uu = F_MUL(f, uu, t, r); /* %$t r$% */ |
308 | dx = F_ADD(f, dx, dx, uu); /* %$a z'^2 + t r$% */ |
309 | r = F_SQR(f, r, w); /* %$w^2$% */ |
310 | uu = F_MUL(f, uu, r, w); /* %$w^3$% */ |
311 | dx = F_ADD(f, dx, dx, uu); /* %$x' = a z'^2 + t r + w^3$% */ |
312 | |
313 | r = F_SQR(f, r, l); /* %$l^2$% */ |
314 | dy = F_MUL(f, uu, v, r); /* %$v l^2$% */ |
315 | l = F_MUL(f, l, t, dx); /* %$t x'$% */ |
316 | dy = F_ADD(f, dy, dy, l); /* %$y' = t x' + v l^2$% */ |
317 | |
318 | EC_DESTROY(d); |
319 | d->x = dx; |
320 | d->y = dy; |
321 | d->z = dz; |
322 | MP_DROP(l); |
323 | MP_DROP(r); |
324 | MP_DROP(w); |
325 | MP_DROP(t); |
326 | MP_DROP(v); |
327 | } |
328 | return (d); |
329 | } |
330 | |
331 | static int eccheck(ec_curve *c, const ec *p) |
332 | { |
ceb3f0c0 |
333 | field *f = c->f; |
334 | int rc; |
335 | mp *u, *v; |
336 | |
34e4f738 |
337 | if (EC_ATINF(p)) return (0); |
ceb3f0c0 |
338 | v = F_SQR(f, MP_NEW, p->x); |
339 | u = F_MUL(f, MP_NEW, v, p->x); |
432c4e18 |
340 | v = F_MUL(f, v, v, c->a); |
ceb3f0c0 |
341 | u = F_ADD(f, u, u, v); |
432c4e18 |
342 | u = F_ADD(f, u, u, c->b); |
ceb3f0c0 |
343 | v = F_MUL(f, v, p->x, p->y); |
344 | u = F_ADD(f, u, u, v); |
345 | v = F_SQR(f, v, p->y); |
346 | u = F_ADD(f, u, u, v); |
bc985cef |
347 | rc = F_ZEROP(f, u) ? 0 : -1; |
ceb3f0c0 |
348 | mp_drop(u); |
349 | mp_drop(v); |
350 | return (rc); |
351 | } |
352 | |
353 | static int ecprojcheck(ec_curve *c, const ec *p) |
354 | { |
355 | ec t = EC_INIT; |
356 | int rc; |
357 | |
358 | c->ops->fix(c, &t, p); |
359 | rc = eccheck(c, &t); |
360 | EC_DESTROY(&t); |
361 | return (rc); |
362 | } |
363 | |
364 | static void ecdestroy(ec_curve *c) |
365 | { |
366 | ecctx *cc = (ecctx *)c; |
432c4e18 |
367 | MP_DROP(cc->c.a); |
368 | MP_DROP(cc->c.b); |
ceb3f0c0 |
369 | if (cc->bb) MP_DROP(cc->bb); |
370 | DESTROY(cc); |
371 | } |
372 | |
373 | /* --- @ec_bin@, @ec_binproj@ --- * |
374 | * |
375 | * Arguments: @field *f@ = the underlying field for this elliptic curve |
376 | * @mp *a, *b@ = the coefficients for this curve |
377 | * |
378 | * Returns: A pointer to the curve. |
379 | * |
380 | * Use: Creates a curve structure for an elliptic curve defined over |
381 | * a binary field. The @binproj@ variant uses projective |
382 | * coordinates, which can be a win. |
383 | */ |
384 | |
385 | ec_curve *ec_bin(field *f, mp *a, mp *b) |
386 | { |
387 | ecctx *cc = CREATE(ecctx); |
388 | cc->c.ops = &ec_binops; |
389 | cc->c.f = f; |
432c4e18 |
390 | cc->c.a = F_IN(f, MP_NEW, a); |
391 | cc->c.b = F_IN(f, MP_NEW, b); |
ceb3f0c0 |
392 | cc->bb = 0; |
393 | return (&cc->c); |
394 | } |
395 | |
396 | ec_curve *ec_binproj(field *f, mp *a, mp *b) |
397 | { |
398 | ecctx *cc = CREATE(ecctx); |
399 | cc->c.ops = &ec_binprojops; |
400 | cc->c.f = f; |
432c4e18 |
401 | cc->c.a = F_IN(f, MP_NEW, a); |
402 | cc->c.b = F_IN(f, MP_NEW, b); |
4edc47b8 |
403 | cc->bb = F_SQRT(f, MP_NEW, cc->c.b); |
ceb3f0c0 |
404 | cc->bb = F_SQRT(f, cc->bb, cc->bb); |
405 | return (&cc->c); |
406 | } |
407 | |
408 | static const ec_ops ec_binops = { |
34e4f738 |
409 | ecdestroy, ec_stdsamep, ec_idin, ec_idout, ec_idfix, |
bc985cef |
410 | ecfind, ecneg, ecadd, ec_stdsub, ecdbl, eccheck |
ceb3f0c0 |
411 | }; |
412 | |
413 | static const ec_ops ec_binprojops = { |
34e4f738 |
414 | ecdestroy, ec_stdsamep, ec_projin, ec_projout, ec_projfix, |
bc985cef |
415 | ecfind, ecprojneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck |
ceb3f0c0 |
416 | }; |
417 | |
418 | /*----- Test rig ----------------------------------------------------------*/ |
419 | |
420 | #ifdef TEST_RIG |
421 | |
422 | #define MP(x) mp_readstring(MP_NEW, #x, 0, 0) |
423 | |
424 | int main(int argc, char *argv[]) |
425 | { |
426 | field *f; |
427 | ec_curve *c; |
428 | ec g = EC_INIT, d = EC_INIT; |
4edc47b8 |
429 | mp *p, *a, *b, *r, *beta; |
ceb3f0c0 |
430 | int i, n = argc == 1 ? 1 : atoi(argv[1]); |
431 | |
432 | printf("ec-bin: "); |
433 | fflush(stdout); |
4edc47b8 |
434 | a = MP(0x7ffffffffffffffffffffffffffffffffffffffff); |
435 | b = MP(0x6645f3cacf1638e139c6cd13ef61734fbc9e3d9fb); |
436 | p = MP(0x800000000000000000000000000000000000000c9); |
437 | beta = MP(0x715169c109c612e390d347c748342bcd3b02a0bef); |
438 | r = MP(0x040000000000000000000292fe77e70c12a4234c32); |
ceb3f0c0 |
439 | |
4edc47b8 |
440 | f = field_binnorm(p, beta); |
ceb3f0c0 |
441 | c = ec_binproj(f, a, b); |
4edc47b8 |
442 | g.x = MP(0x0311103c17167564ace77ccb09c681f886ba54ee8); |
443 | g.y = MP(0x333ac13c6447f2e67613bf7009daf98c87bb50c7f); |
ceb3f0c0 |
444 | |
445 | for (i = 0; i < n; i++) { |
446 | ec_mul(c, &d, &g, r); |
447 | if (EC_ATINF(&d)) { |
448 | fprintf(stderr, "zero too early\n"); |
449 | return (1); |
450 | } |
451 | ec_add(c, &d, &d, &g); |
452 | if (!EC_ATINF(&d)) { |
453 | fprintf(stderr, "didn't reach zero\n"); |
454 | MP_EPRINTX("d.x", d.x); |
455 | MP_EPRINTX("d.y", d.y); |
ceb3f0c0 |
456 | return (1); |
457 | } |
458 | ec_destroy(&d); |
459 | } |
460 | |
461 | ec_destroy(&g); |
462 | ec_destroycurve(c); |
463 | F_DESTROY(f); |
4edc47b8 |
464 | MP_DROP(p); MP_DROP(a); MP_DROP(b); MP_DROP(r); MP_DROP(beta); |
ceb3f0c0 |
465 | assert(!mparena_count(&mparena_global)); |
466 | printf("ok\n"); |
467 | return (0); |
468 | } |
469 | |
470 | #endif |
471 | |
472 | /*----- That's all, folks -------------------------------------------------*/ |