b0ab12e6 |
1 | /* -*-c-*- |
2 | * |
b817bfc6 |
3 | * $Id: ec-prime.c,v 1.11 2004/04/08 01:36:15 mdw Exp $ |
b0ab12e6 |
4 | * |
5 | * Elliptic curves over prime fields |
6 | * |
7 | * (c) 2001 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 | |
b0ab12e6 |
30 | /*----- Header files ------------------------------------------------------*/ |
31 | |
41cb1beb |
32 | #include <mLib/sub.h> |
33 | |
b0ab12e6 |
34 | #include "ec.h" |
35 | |
dbfee00a |
36 | /*----- Simple prime curves -----------------------------------------------*/ |
b0ab12e6 |
37 | |
8823192f |
38 | static const ec_ops ec_primeops, ec_primeprojops, ec_primeprojxops; |
41cb1beb |
39 | |
40 | static ec *ecneg(ec_curve *c, ec *d, const ec *p) |
b085fd91 |
41 | { |
42 | EC_COPY(d, p); |
ceb3f0c0 |
43 | if (d->y) |
44 | d->y = F_NEG(c->f, d->y, d->y); |
b085fd91 |
45 | return (d); |
46 | } |
47 | |
8823192f |
48 | static ec *ecfind(ec_curve *c, ec *d, mp *x) |
49 | { |
50 | mp *p, *q; |
8823192f |
51 | field *f = c->f; |
52 | |
53 | q = F_SQR(f, MP_NEW, x); |
54 | p = F_MUL(f, MP_NEW, x, q); |
432c4e18 |
55 | q = F_MUL(f, q, x, c->a); |
8823192f |
56 | p = F_ADD(f, p, p, q); |
432c4e18 |
57 | p = F_ADD(f, p, p, c->b); |
8823192f |
58 | MP_DROP(q); |
59 | p = F_SQRT(f, p, p); |
60 | if (!p) |
61 | return (0); |
62 | EC_DESTROY(d); |
63 | d->x = MP_COPY(x); |
64 | d->y = p; |
65 | d->z = MP_COPY(f->one); |
b085fd91 |
66 | return (d); |
67 | } |
68 | |
69 | static ec *ecdbl(ec_curve *c, ec *d, const ec *a) |
b0ab12e6 |
70 | { |
b085fd91 |
71 | if (EC_ATINF(a)) |
72 | EC_SETINF(d); |
8823192f |
73 | else if (F_ZEROP(c->f, a->y)) |
b085fd91 |
74 | EC_COPY(d, a); |
75 | else { |
76 | field *f = c->f; |
b085fd91 |
77 | mp *lambda; |
78 | mp *dy, *dx; |
79 | |
8823192f |
80 | dx = F_SQR(f, MP_NEW, a->x); /* %$x^2$% */ |
81 | dy = F_DBL(f, MP_NEW, a->y); /* %$2 y$% */ |
82 | dx = F_TPL(f, dx, dx); /* %$3 x^2$% */ |
432c4e18 |
83 | dx = F_ADD(f, dx, dx, c->a); /* %$3 x^2 + A$% */ |
8823192f |
84 | dy = F_INV(f, dy, dy); /* %$(2 y)^{-1}$% */ |
85 | lambda = F_MUL(f, MP_NEW, dx, dy); /* %$\lambda = (3 x^2 + A)/(2 y)$% */ |
b085fd91 |
86 | |
8823192f |
87 | dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */ |
88 | dy = F_DBL(f, dy, a->x); /* %$2 x$% */ |
89 | dx = F_SUB(f, dx, dx, dy); /* %$x' = \lambda^2 - 2 x */ |
90 | dy = F_SUB(f, dy, a->x, dx); /* %$x - x'$% */ |
91 | dy = F_MUL(f, dy, lambda, dy); /* %$\lambda (x - x')$% */ |
92 | dy = F_SUB(f, dy, dy, a->y); /* %$y' = \lambda (x - x') - y$% */ |
b0ab12e6 |
93 | |
b085fd91 |
94 | EC_DESTROY(d); |
95 | d->x = dx; |
96 | d->y = dy; |
97 | d->z = 0; |
98 | MP_DROP(lambda); |
99 | } |
100 | return (d); |
101 | } |
102 | |
8823192f |
103 | static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a) |
104 | { |
105 | if (EC_ATINF(a)) |
106 | EC_SETINF(d); |
107 | else if (F_ZEROP(c->f, a->y)) |
108 | EC_COPY(d, a); |
109 | else { |
110 | field *f = c->f; |
8823192f |
111 | mp *p, *q, *m, *s, *dx, *dy, *dz; |
112 | |
113 | p = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */ |
114 | q = F_SQR(f, MP_NEW, p); /* %$z^4$% */ |
432c4e18 |
115 | p = F_MUL(f, p, q, c->a); /* %$A z^4$% */ |
8823192f |
116 | m = F_SQR(f, MP_NEW, a->x); /* %$x^2$% */ |
117 | m = F_TPL(f, m, m); /* %$3 x^2$% */ |
118 | m = F_ADD(f, m, m, p); /* %$m = 3 x^2 + A z^4$% */ |
119 | |
120 | q = F_DBL(f, q, a->y); /* %$2 y$% */ |
121 | dz = F_MUL(f, MP_NEW, q, a->z); /* %$z' = 2 y z$% */ |
122 | |
123 | p = F_SQR(f, p, q); /* %$4 y^2$% */ |
124 | s = F_MUL(f, MP_NEW, p, a->x); /* %$s = 4 x y^2$% */ |
125 | q = F_SQR(f, q, p); /* %$16 y^4$% */ |
126 | q = F_HLV(f, q, q); /* %$t = 8 y^4$% */ |
127 | |
128 | p = F_DBL(f, p, s); /* %$2 s$% */ |
129 | dx = F_SQR(f, MP_NEW, m); /* %$m^2$% */ |
130 | dx = F_SUB(f, dx, dx, p); /* %$x' = m^2 - 2 s$% */ |
131 | |
132 | s = F_SUB(f, s, s, dx); /* %$s - x'$% */ |
133 | dy = F_MUL(f, p, m, s); /* %$m (s - x')$% */ |
134 | dy = F_SUB(f, dy, dy, q); /* %$y' = m (s - x') - t$% */ |
135 | |
136 | EC_DESTROY(d); |
137 | d->x = dx; |
138 | d->y = dy; |
139 | d->z = dz; |
140 | MP_DROP(m); |
141 | MP_DROP(q); |
142 | MP_DROP(s); |
143 | } |
144 | return (d); |
145 | } |
146 | |
147 | static ec *ecprojxdbl(ec_curve *c, ec *d, const ec *a) |
148 | { |
149 | if (EC_ATINF(a)) |
150 | EC_SETINF(d); |
151 | else if (F_ZEROP(c->f, a->y)) |
152 | EC_COPY(d, a); |
153 | else { |
154 | field *f = c->f; |
155 | mp *p, *q, *m, *s, *dx, *dy, *dz; |
156 | |
157 | m = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */ |
158 | p = F_SUB(f, MP_NEW, a->x, m); /* %$x - z^2$% */ |
159 | q = F_ADD(f, MP_NEW, a->x, m); /* %$x + z^2$% */ |
160 | m = F_MUL(f, m, p, q); /* %$x^2 - z^4$% */ |
161 | m = F_TPL(f, m, m); /* %$m = 3 x^2 - 3 z^4$% */ |
162 | |
163 | q = F_DBL(f, q, a->y); /* %$2 y$% */ |
164 | dz = F_MUL(f, MP_NEW, q, a->z); /* %$z' = 2 y z$% */ |
165 | |
166 | p = F_SQR(f, p, q); /* %$4 y^2$% */ |
167 | s = F_MUL(f, MP_NEW, p, a->x); /* %$s = 4 x y^2$% */ |
168 | q = F_SQR(f, q, p); /* %$16 y^4$% */ |
169 | q = F_HLV(f, q, q); /* %$t = 8 y^4$% */ |
170 | |
171 | p = F_DBL(f, p, s); /* %$2 s$% */ |
172 | dx = F_SQR(f, MP_NEW, m); /* %$m^2$% */ |
173 | dx = F_SUB(f, dx, dx, p); /* %$x' = m^2 - 2 s$% */ |
174 | |
175 | s = F_SUB(f, s, s, dx); /* %$s - x'$% */ |
176 | dy = F_MUL(f, p, m, s); /* %$m (s - x')$% */ |
177 | dy = F_SUB(f, dy, dy, q); /* %$y' = m (s - x') - t$% */ |
178 | |
179 | EC_DESTROY(d); |
180 | d->x = dx; |
181 | d->y = dy; |
182 | d->z = dz; |
183 | MP_DROP(m); |
184 | MP_DROP(q); |
185 | MP_DROP(s); |
186 | } |
187 | return (d); |
188 | } |
189 | |
b085fd91 |
190 | static ec *ecadd(ec_curve *c, ec *d, const ec *a, const ec *b) |
191 | { |
b0ab12e6 |
192 | if (a == b) |
193 | ecdbl(c, d, a); |
194 | else if (EC_ATINF(a)) |
195 | EC_COPY(d, b); |
196 | else if (EC_ATINF(b)) |
197 | EC_COPY(d, a); |
b085fd91 |
198 | else { |
199 | field *f = c->f; |
200 | mp *lambda; |
201 | mp *dy, *dx; |
202 | |
203 | if (!MP_EQ(a->x, b->x)) { |
8823192f |
204 | dy = F_SUB(f, MP_NEW, a->y, b->y); /* %$y_0 - y_1$% */ |
205 | dx = F_SUB(f, MP_NEW, a->x, b->x); /* %$x_0 - x_1$% */ |
206 | dx = F_INV(f, dx, dx); /* %$(x_0 - x_1)^{-1}$% */ |
b085fd91 |
207 | lambda = F_MUL(f, MP_NEW, dy, dx); |
8823192f |
208 | /* %$\lambda = (y_0 - y1)/(x_0 - x_1)$% */ |
209 | } else if (F_ZEROP(c->f, a->y) || !MP_EQ(a->y, b->y)) { |
b0ab12e6 |
210 | EC_SETINF(d); |
b085fd91 |
211 | return (d); |
212 | } else { |
8823192f |
213 | dx = F_SQR(f, MP_NEW, a->x); /* %$x_0^2$% */ |
214 | dx = F_TPL(f, dx, dx); /* %$3 x_0^2$% */ |
432c4e18 |
215 | dx = F_ADD(f, dx, dx, c->a); /* %$3 x_0^2 + A$% */ |
8823192f |
216 | dy = F_DBL(f, MP_NEW, a->y); /* %$2 y_0$% */ |
217 | dy = F_INV(f, dy, dy); /* %$(2 y_0)^{-1}$% */ |
41cb1beb |
218 | lambda = F_MUL(f, MP_NEW, dx, dy); |
8823192f |
219 | /* %$\lambda = (3 x_0^2 + A)/(2 y_0)$% */ |
b085fd91 |
220 | } |
221 | |
8823192f |
222 | dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */ |
223 | dx = F_SUB(f, dx, dx, a->x); /* %$\lambda^2 - x_0$% */ |
224 | dx = F_SUB(f, dx, dx, b->x); /* %$x' = \lambda^2 - x_0 - x_1$% */ |
225 | dy = F_SUB(f, dy, b->x, dx); /* %$x_1 - x'$% */ |
226 | dy = F_MUL(f, dy, lambda, dy); /* %$\lambda (x_1 - x')$% */ |
ceb3f0c0 |
227 | dy = F_SUB(f, dy, dy, b->y); /* %$y' = \lambda (x_1 - x') - y_1$% */ |
b0ab12e6 |
228 | |
b085fd91 |
229 | EC_DESTROY(d); |
230 | d->x = dx; |
231 | d->y = dy; |
232 | d->z = 0; |
233 | MP_DROP(lambda); |
b0ab12e6 |
234 | } |
b085fd91 |
235 | return (d); |
b0ab12e6 |
236 | } |
237 | |
8823192f |
238 | static ec *ecprojadd(ec_curve *c, ec *d, const ec *a, const ec *b) |
239 | { |
240 | if (a == b) |
241 | c->ops->dbl(c, d, a); |
242 | else if (EC_ATINF(a)) |
243 | EC_COPY(d, b); |
244 | else if (EC_ATINF(b)) |
245 | EC_COPY(d, a); |
246 | else { |
247 | field *f = c->f; |
391faf42 |
248 | mp *p, *q, *r, *w, *u, *uu, *s, *ss, *dx, *dy, *dz; |
8823192f |
249 | |
250 | q = F_SQR(f, MP_NEW, a->z); /* %$z_0^2$% */ |
251 | u = F_MUL(f, MP_NEW, q, b->x); /* %$u = x_1 z_0^2$% */ |
252 | p = F_MUL(f, MP_NEW, q, b->y); /* %$y_1 z_0^2$% */ |
253 | s = F_MUL(f, q, p, a->z); /* %$s = y_1 z_0^3$% */ |
254 | |
391faf42 |
255 | q = F_SQR(f, MP_NEW, b->z); /* %$z_1^2$% */ |
256 | uu = F_MUL(f, MP_NEW, q, a->x); /* %$uu = x_0 z_1^2$%*/ |
257 | p = F_MUL(f, p, q, a->y); /* %$y_0 z_1^2$% */ |
258 | ss = F_MUL(f, q, p, b->z); /* %$ss = y_0 z_1^3$% */ |
259 | |
260 | w = F_SUB(f, p, uu, u); /* %$w = uu - u$% */ |
261 | r = F_SUB(f, MP_NEW, ss, s); /* %$r = ss - s$% */ |
8823192f |
262 | if (F_ZEROP(f, w)) { |
ceb3f0c0 |
263 | MP_DROP(w); |
264 | MP_DROP(u); |
265 | MP_DROP(s); |
391faf42 |
266 | MP_DROP(uu); |
267 | MP_DROP(ss); |
8823192f |
268 | if (F_ZEROP(f, r)) { |
8823192f |
269 | MP_DROP(r); |
8823192f |
270 | return (c->ops->dbl(c, d, a)); |
271 | } else { |
8823192f |
272 | MP_DROP(r); |
8823192f |
273 | EC_SETINF(d); |
274 | return (d); |
275 | } |
276 | } |
391faf42 |
277 | u = F_ADD(f, u, u, uu); /* %$t = uu + u$% */ |
278 | s = F_ADD(f, s, s, ss); /* %$m = ss + r$% */ |
8823192f |
279 | |
391faf42 |
280 | uu = F_MUL(f, uu, a->z, w); /* %$z_0 w$% */ |
281 | dz = F_MUL(f, ss, uu, b->z); /* %$z' = z_0 z_1 w$% */ |
8823192f |
282 | |
391faf42 |
283 | p = F_SQR(f, uu, w); /* %$w^2$% */ |
8823192f |
284 | q = F_MUL(f, MP_NEW, p, u); /* %$t w^2$% */ |
285 | u = F_MUL(f, u, p, w); /* %$w^3$% */ |
286 | p = F_MUL(f, p, u, s); /* %$m w^3$% */ |
287 | |
288 | dx = F_SQR(f, u, r); /* %$r^2$% */ |
289 | dx = F_SUB(f, dx, dx, q); /* %$x' = r^2 - t w^2$% */ |
290 | |
291 | s = F_DBL(f, s, dx); /* %$2 x'$% */ |
292 | q = F_SUB(f, q, q, s); /* %$v = t w^2 - 2 x'$% */ |
293 | dy = F_MUL(f, s, q, r); /* %$v r$% */ |
294 | dy = F_SUB(f, dy, dy, p); /* %$v r - m w^3$% */ |
295 | dy = F_HLV(f, dy, dy); /* %$y' = (v r - m w^3)/2$% */ |
296 | |
297 | EC_DESTROY(d); |
298 | d->x = dx; |
299 | d->y = dy; |
300 | d->z = dz; |
301 | MP_DROP(p); |
302 | MP_DROP(q); |
303 | MP_DROP(r); |
304 | MP_DROP(w); |
305 | } |
306 | return (d); |
307 | } |
308 | |
309 | static int eccheck(ec_curve *c, const ec *p) |
310 | { |
8823192f |
311 | field *f = c->f; |
34e4f738 |
312 | mp *l, *x, *r; |
8823192f |
313 | int rc; |
34e4f738 |
314 | if (EC_ATINF(p)) return (0); |
315 | l = F_SQR(f, MP_NEW, p->y); |
316 | x = F_SQR(f, MP_NEW, p->x); |
317 | r = F_MUL(f, MP_NEW, x, p->x); |
432c4e18 |
318 | x = F_MUL(f, x, c->a, p->x); |
8823192f |
319 | r = F_ADD(f, r, r, x); |
432c4e18 |
320 | r = F_ADD(f, r, r, c->b); |
8823192f |
321 | rc = MP_EQ(l, r) ? 0 : -1; |
322 | mp_drop(l); |
323 | mp_drop(x); |
324 | mp_drop(r); |
325 | return (rc); |
326 | } |
327 | |
328 | static int ecprojcheck(ec_curve *c, const ec *p) |
329 | { |
330 | ec t = EC_INIT; |
331 | int rc; |
332 | |
333 | c->ops->fix(c, &t, p); |
334 | rc = eccheck(c, &t); |
335 | EC_DESTROY(&t); |
336 | return (rc); |
337 | } |
338 | |
41cb1beb |
339 | static void ecdestroy(ec_curve *c) |
340 | { |
432c4e18 |
341 | MP_DROP(c->a); |
342 | MP_DROP(c->b); |
343 | DESTROY(c); |
41cb1beb |
344 | } |
345 | |
346 | /* --- @ec_prime@, @ec_primeproj@ --- * |
347 | * |
dbfee00a |
348 | * Arguments: @field *f@ = the underlying field for this elliptic curve |
41cb1beb |
349 | * @mp *a, *b@ = the coefficients for this curve |
350 | * |
02d7884d |
351 | * Returns: A pointer to the curve, or null. |
41cb1beb |
352 | * |
353 | * Use: Creates a curve structure for an elliptic curve defined over |
354 | * a prime field. The @primeproj@ variant uses projective |
355 | * coordinates, which can be a win. |
356 | */ |
357 | |
358 | extern ec_curve *ec_prime(field *f, mp *a, mp *b) |
359 | { |
432c4e18 |
360 | ec_curve *c = CREATE(ec_curve); |
361 | c->ops = &ec_primeops; |
362 | c->f = f; |
363 | c->a = F_IN(f, MP_NEW, a); |
364 | c->b = F_IN(f, MP_NEW, b); |
365 | return (c); |
41cb1beb |
366 | } |
367 | |
8823192f |
368 | extern ec_curve *ec_primeproj(field *f, mp *a, mp *b) |
369 | { |
432c4e18 |
370 | ec_curve *c = CREATE(ec_curve); |
8823192f |
371 | mp *ax; |
372 | |
373 | ax = mp_add(MP_NEW, a, MP_THREE); |
374 | ax = F_IN(f, ax, ax); |
375 | if (F_ZEROP(f, ax)) |
432c4e18 |
376 | c->ops = &ec_primeprojxops; |
8823192f |
377 | else |
432c4e18 |
378 | c->ops = &ec_primeprojops; |
8823192f |
379 | MP_DROP(ax); |
432c4e18 |
380 | c->f = f; |
381 | c->a = F_IN(f, MP_NEW, a); |
382 | c->b = F_IN(f, MP_NEW, b); |
383 | return (c); |
41cb1beb |
384 | } |
385 | |
386 | static const ec_ops ec_primeops = { |
34e4f738 |
387 | ecdestroy, ec_stdsamep, ec_idin, ec_idout, ec_idfix, |
bc985cef |
388 | ecfind, ecneg, ecadd, ec_stdsub, ecdbl, eccheck |
8823192f |
389 | }; |
390 | |
391 | static const ec_ops ec_primeprojops = { |
34e4f738 |
392 | ecdestroy, ec_stdsamep, ec_projin, ec_projout, ec_projfix, |
bc985cef |
393 | ecfind, ecneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck |
8823192f |
394 | }; |
395 | |
396 | static const ec_ops ec_primeprojxops = { |
34e4f738 |
397 | ecdestroy, ec_stdsamep, ec_projin, ec_projout, ec_projfix, |
bc985cef |
398 | ecfind, ecneg, ecprojadd, ec_stdsub, ecprojxdbl, ecprojcheck |
41cb1beb |
399 | }; |
400 | |
401 | /*----- Test rig ----------------------------------------------------------*/ |
402 | |
403 | #ifdef TEST_RIG |
404 | |
405 | #define MP(x) mp_readstring(MP_NEW, #x, 0, 0) |
406 | |
ceb3f0c0 |
407 | int main(int argc, char *argv[]) |
41cb1beb |
408 | { |
409 | field *f; |
410 | ec_curve *c; |
411 | ec g = EC_INIT, d = EC_INIT; |
412 | mp *p, *a, *b, *r; |
ceb3f0c0 |
413 | int i, n = argc == 1 ? 1 : atoi(argv[1]); |
41cb1beb |
414 | |
dbfee00a |
415 | printf("ec-prime: "); |
416 | fflush(stdout); |
41cb1beb |
417 | a = MP(-3); |
432c4e18 |
418 | b = MP(0xb3312fa7e23ee7e4988e056be3f82d19181d9c6efe8141120314088f5013875ac656398d8a2ed19d2a85c8edd3ec2aef); |
419 | p = MP(39402006196394479212279040100143613805079739270465446667948293404245721771496870329047266088258938001861606973112319); |
420 | r = MP(39402006196394479212279040100143613805079739270465446667946905279627659399113263569398956308152294913554433653942642); |
41cb1beb |
421 | |
f46efa79 |
422 | f = field_niceprime(p); |
ceb3f0c0 |
423 | c = ec_primeproj(f, a, b); |
41cb1beb |
424 | |
432c4e18 |
425 | g.x = MP(0xaa87ca22be8b05378eb1c71ef320ad746e1d3b628ba79b9859f741e082542a385502f25dbf55296c3a545e3872760ab7); |
426 | g.y = MP(0x3617de4a96262c6f5d9e98bf9292dc29f8f41dbd289a147ce9da3113b5f0b8c00a60b1ce1d7e819d7a431d7c90ea0e5f); |
41cb1beb |
427 | |
ceb3f0c0 |
428 | for (i = 0; i < n; i++) { |
429 | ec_mul(c, &d, &g, r); |
430 | if (EC_ATINF(&d)) { |
431 | fprintf(stderr, "zero too early\n"); |
432 | return (1); |
433 | } |
434 | ec_add(c, &d, &d, &g); |
435 | if (!EC_ATINF(&d)) { |
436 | fprintf(stderr, "didn't reach zero\n"); |
437 | MP_EPRINT("d.x", d.x); |
438 | MP_EPRINT("d.y", d.y); |
439 | return (1); |
440 | } |
441 | ec_destroy(&d); |
dbfee00a |
442 | } |
41cb1beb |
443 | ec_destroy(&g); |
444 | ec_destroycurve(c); |
445 | F_DESTROY(f); |
dbfee00a |
446 | MP_DROP(p); MP_DROP(a); MP_DROP(b); MP_DROP(r); |
447 | assert(!mparena_count(&mparena_global)); |
448 | printf("ok\n"); |
41cb1beb |
449 | return (0); |
450 | } |
451 | |
452 | #endif |
453 | |
b0ab12e6 |
454 | /*----- That's all, folks -------------------------------------------------*/ |