b0ab12e6 |
1 | /* -*-c-*- |
2 | * |
f94b972d |
3 | * $Id$ |
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 | { |
1ccdb13a |
71 | if (EC_ATINF(a) || F_ZEROP(c->f, a->y)) |
b085fd91 |
72 | EC_SETINF(d); |
b085fd91 |
73 | else { |
74 | field *f = c->f; |
b085fd91 |
75 | mp *lambda; |
76 | mp *dy, *dx; |
77 | |
8823192f |
78 | dx = F_SQR(f, MP_NEW, a->x); /* %$x^2$% */ |
79 | dy = F_DBL(f, MP_NEW, a->y); /* %$2 y$% */ |
80 | dx = F_TPL(f, dx, dx); /* %$3 x^2$% */ |
432c4e18 |
81 | dx = F_ADD(f, dx, dx, c->a); /* %$3 x^2 + A$% */ |
8823192f |
82 | dy = F_INV(f, dy, dy); /* %$(2 y)^{-1}$% */ |
83 | lambda = F_MUL(f, MP_NEW, dx, dy); /* %$\lambda = (3 x^2 + A)/(2 y)$% */ |
b085fd91 |
84 | |
8823192f |
85 | dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */ |
86 | dy = F_DBL(f, dy, a->x); /* %$2 x$% */ |
87 | dx = F_SUB(f, dx, dx, dy); /* %$x' = \lambda^2 - 2 x */ |
88 | dy = F_SUB(f, dy, a->x, dx); /* %$x - x'$% */ |
89 | dy = F_MUL(f, dy, lambda, dy); /* %$\lambda (x - x')$% */ |
90 | dy = F_SUB(f, dy, dy, a->y); /* %$y' = \lambda (x - x') - y$% */ |
b0ab12e6 |
91 | |
b085fd91 |
92 | EC_DESTROY(d); |
93 | d->x = dx; |
94 | d->y = dy; |
95 | d->z = 0; |
96 | MP_DROP(lambda); |
97 | } |
98 | return (d); |
99 | } |
100 | |
8823192f |
101 | static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a) |
102 | { |
1ccdb13a |
103 | if (EC_ATINF(a) || F_ZEROP(c->f, a->y)) |
8823192f |
104 | EC_SETINF(d); |
8823192f |
105 | else { |
106 | field *f = c->f; |
8823192f |
107 | mp *p, *q, *m, *s, *dx, *dy, *dz; |
108 | |
109 | p = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */ |
110 | q = F_SQR(f, MP_NEW, p); /* %$z^4$% */ |
432c4e18 |
111 | p = F_MUL(f, p, q, c->a); /* %$A z^4$% */ |
8823192f |
112 | m = F_SQR(f, MP_NEW, a->x); /* %$x^2$% */ |
113 | m = F_TPL(f, m, m); /* %$3 x^2$% */ |
114 | m = F_ADD(f, m, m, p); /* %$m = 3 x^2 + A z^4$% */ |
115 | |
116 | q = F_DBL(f, q, a->y); /* %$2 y$% */ |
117 | dz = F_MUL(f, MP_NEW, q, a->z); /* %$z' = 2 y z$% */ |
118 | |
119 | p = F_SQR(f, p, q); /* %$4 y^2$% */ |
120 | s = F_MUL(f, MP_NEW, p, a->x); /* %$s = 4 x y^2$% */ |
121 | q = F_SQR(f, q, p); /* %$16 y^4$% */ |
122 | q = F_HLV(f, q, q); /* %$t = 8 y^4$% */ |
123 | |
124 | p = F_DBL(f, p, s); /* %$2 s$% */ |
125 | dx = F_SQR(f, MP_NEW, m); /* %$m^2$% */ |
126 | dx = F_SUB(f, dx, dx, p); /* %$x' = m^2 - 2 s$% */ |
127 | |
128 | s = F_SUB(f, s, s, dx); /* %$s - x'$% */ |
129 | dy = F_MUL(f, p, m, s); /* %$m (s - x')$% */ |
130 | dy = F_SUB(f, dy, dy, q); /* %$y' = m (s - x') - t$% */ |
131 | |
132 | EC_DESTROY(d); |
133 | d->x = dx; |
134 | d->y = dy; |
135 | d->z = dz; |
136 | MP_DROP(m); |
137 | MP_DROP(q); |
138 | MP_DROP(s); |
139 | } |
140 | return (d); |
141 | } |
142 | |
143 | static ec *ecprojxdbl(ec_curve *c, ec *d, const ec *a) |
144 | { |
1ccdb13a |
145 | if (EC_ATINF(a) || F_ZEROP(c->f, a->y)) |
8823192f |
146 | EC_SETINF(d); |
8823192f |
147 | else { |
148 | field *f = c->f; |
149 | mp *p, *q, *m, *s, *dx, *dy, *dz; |
150 | |
151 | m = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */ |
152 | p = F_SUB(f, MP_NEW, a->x, m); /* %$x - z^2$% */ |
153 | q = F_ADD(f, MP_NEW, a->x, m); /* %$x + z^2$% */ |
154 | m = F_MUL(f, m, p, q); /* %$x^2 - z^4$% */ |
155 | m = F_TPL(f, m, m); /* %$m = 3 x^2 - 3 z^4$% */ |
156 | |
157 | q = F_DBL(f, q, a->y); /* %$2 y$% */ |
158 | dz = F_MUL(f, MP_NEW, q, a->z); /* %$z' = 2 y z$% */ |
159 | |
160 | p = F_SQR(f, p, q); /* %$4 y^2$% */ |
161 | s = F_MUL(f, MP_NEW, p, a->x); /* %$s = 4 x y^2$% */ |
162 | q = F_SQR(f, q, p); /* %$16 y^4$% */ |
163 | q = F_HLV(f, q, q); /* %$t = 8 y^4$% */ |
164 | |
165 | p = F_DBL(f, p, s); /* %$2 s$% */ |
166 | dx = F_SQR(f, MP_NEW, m); /* %$m^2$% */ |
167 | dx = F_SUB(f, dx, dx, p); /* %$x' = m^2 - 2 s$% */ |
168 | |
169 | s = F_SUB(f, s, s, dx); /* %$s - x'$% */ |
170 | dy = F_MUL(f, p, m, s); /* %$m (s - x')$% */ |
171 | dy = F_SUB(f, dy, dy, q); /* %$y' = m (s - x') - t$% */ |
172 | |
173 | EC_DESTROY(d); |
174 | d->x = dx; |
175 | d->y = dy; |
176 | d->z = dz; |
177 | MP_DROP(m); |
178 | MP_DROP(q); |
179 | MP_DROP(s); |
180 | } |
181 | return (d); |
182 | } |
183 | |
b085fd91 |
184 | static ec *ecadd(ec_curve *c, ec *d, const ec *a, const ec *b) |
185 | { |
b0ab12e6 |
186 | if (a == b) |
187 | ecdbl(c, d, a); |
188 | else if (EC_ATINF(a)) |
189 | EC_COPY(d, b); |
190 | else if (EC_ATINF(b)) |
191 | EC_COPY(d, a); |
b085fd91 |
192 | else { |
193 | field *f = c->f; |
194 | mp *lambda; |
195 | mp *dy, *dx; |
196 | |
197 | if (!MP_EQ(a->x, b->x)) { |
8823192f |
198 | dy = F_SUB(f, MP_NEW, a->y, b->y); /* %$y_0 - y_1$% */ |
199 | dx = F_SUB(f, MP_NEW, a->x, b->x); /* %$x_0 - x_1$% */ |
200 | dx = F_INV(f, dx, dx); /* %$(x_0 - x_1)^{-1}$% */ |
b085fd91 |
201 | lambda = F_MUL(f, MP_NEW, dy, dx); |
8823192f |
202 | /* %$\lambda = (y_0 - y1)/(x_0 - x_1)$% */ |
203 | } else if (F_ZEROP(c->f, a->y) || !MP_EQ(a->y, b->y)) { |
b0ab12e6 |
204 | EC_SETINF(d); |
b085fd91 |
205 | return (d); |
206 | } else { |
8823192f |
207 | dx = F_SQR(f, MP_NEW, a->x); /* %$x_0^2$% */ |
208 | dx = F_TPL(f, dx, dx); /* %$3 x_0^2$% */ |
432c4e18 |
209 | dx = F_ADD(f, dx, dx, c->a); /* %$3 x_0^2 + A$% */ |
8823192f |
210 | dy = F_DBL(f, MP_NEW, a->y); /* %$2 y_0$% */ |
211 | dy = F_INV(f, dy, dy); /* %$(2 y_0)^{-1}$% */ |
41cb1beb |
212 | lambda = F_MUL(f, MP_NEW, dx, dy); |
8823192f |
213 | /* %$\lambda = (3 x_0^2 + A)/(2 y_0)$% */ |
b085fd91 |
214 | } |
215 | |
8823192f |
216 | dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */ |
217 | dx = F_SUB(f, dx, dx, a->x); /* %$\lambda^2 - x_0$% */ |
218 | dx = F_SUB(f, dx, dx, b->x); /* %$x' = \lambda^2 - x_0 - x_1$% */ |
219 | dy = F_SUB(f, dy, b->x, dx); /* %$x_1 - x'$% */ |
220 | dy = F_MUL(f, dy, lambda, dy); /* %$\lambda (x_1 - x')$% */ |
ceb3f0c0 |
221 | dy = F_SUB(f, dy, dy, b->y); /* %$y' = \lambda (x_1 - x') - y_1$% */ |
b0ab12e6 |
222 | |
b085fd91 |
223 | EC_DESTROY(d); |
224 | d->x = dx; |
225 | d->y = dy; |
226 | d->z = 0; |
227 | MP_DROP(lambda); |
b0ab12e6 |
228 | } |
b085fd91 |
229 | return (d); |
b0ab12e6 |
230 | } |
231 | |
8823192f |
232 | static ec *ecprojadd(ec_curve *c, ec *d, const ec *a, const ec *b) |
233 | { |
234 | if (a == b) |
235 | c->ops->dbl(c, d, a); |
236 | else if (EC_ATINF(a)) |
237 | EC_COPY(d, b); |
238 | else if (EC_ATINF(b)) |
239 | EC_COPY(d, a); |
240 | else { |
241 | field *f = c->f; |
391faf42 |
242 | mp *p, *q, *r, *w, *u, *uu, *s, *ss, *dx, *dy, *dz; |
8823192f |
243 | |
244 | q = F_SQR(f, MP_NEW, a->z); /* %$z_0^2$% */ |
245 | u = F_MUL(f, MP_NEW, q, b->x); /* %$u = x_1 z_0^2$% */ |
246 | p = F_MUL(f, MP_NEW, q, b->y); /* %$y_1 z_0^2$% */ |
247 | s = F_MUL(f, q, p, a->z); /* %$s = y_1 z_0^3$% */ |
248 | |
391faf42 |
249 | q = F_SQR(f, MP_NEW, b->z); /* %$z_1^2$% */ |
250 | uu = F_MUL(f, MP_NEW, q, a->x); /* %$uu = x_0 z_1^2$%*/ |
251 | p = F_MUL(f, p, q, a->y); /* %$y_0 z_1^2$% */ |
252 | ss = F_MUL(f, q, p, b->z); /* %$ss = y_0 z_1^3$% */ |
253 | |
254 | w = F_SUB(f, p, uu, u); /* %$w = uu - u$% */ |
255 | r = F_SUB(f, MP_NEW, ss, s); /* %$r = ss - s$% */ |
8823192f |
256 | if (F_ZEROP(f, w)) { |
ceb3f0c0 |
257 | MP_DROP(w); |
258 | MP_DROP(u); |
259 | MP_DROP(s); |
391faf42 |
260 | MP_DROP(uu); |
261 | MP_DROP(ss); |
8823192f |
262 | if (F_ZEROP(f, r)) { |
8823192f |
263 | MP_DROP(r); |
8823192f |
264 | return (c->ops->dbl(c, d, a)); |
265 | } else { |
8823192f |
266 | MP_DROP(r); |
8823192f |
267 | EC_SETINF(d); |
268 | return (d); |
269 | } |
270 | } |
391faf42 |
271 | u = F_ADD(f, u, u, uu); /* %$t = uu + u$% */ |
272 | s = F_ADD(f, s, s, ss); /* %$m = ss + r$% */ |
8823192f |
273 | |
391faf42 |
274 | uu = F_MUL(f, uu, a->z, w); /* %$z_0 w$% */ |
275 | dz = F_MUL(f, ss, uu, b->z); /* %$z' = z_0 z_1 w$% */ |
8823192f |
276 | |
391faf42 |
277 | p = F_SQR(f, uu, w); /* %$w^2$% */ |
8823192f |
278 | q = F_MUL(f, MP_NEW, p, u); /* %$t w^2$% */ |
279 | u = F_MUL(f, u, p, w); /* %$w^3$% */ |
280 | p = F_MUL(f, p, u, s); /* %$m w^3$% */ |
281 | |
282 | dx = F_SQR(f, u, r); /* %$r^2$% */ |
283 | dx = F_SUB(f, dx, dx, q); /* %$x' = r^2 - t w^2$% */ |
284 | |
285 | s = F_DBL(f, s, dx); /* %$2 x'$% */ |
286 | q = F_SUB(f, q, q, s); /* %$v = t w^2 - 2 x'$% */ |
287 | dy = F_MUL(f, s, q, r); /* %$v r$% */ |
288 | dy = F_SUB(f, dy, dy, p); /* %$v r - m w^3$% */ |
289 | dy = F_HLV(f, dy, dy); /* %$y' = (v r - m w^3)/2$% */ |
290 | |
291 | EC_DESTROY(d); |
292 | d->x = dx; |
293 | d->y = dy; |
294 | d->z = dz; |
295 | MP_DROP(p); |
296 | MP_DROP(q); |
297 | MP_DROP(r); |
298 | MP_DROP(w); |
299 | } |
300 | return (d); |
301 | } |
302 | |
303 | static int eccheck(ec_curve *c, const ec *p) |
304 | { |
8823192f |
305 | field *f = c->f; |
34e4f738 |
306 | mp *l, *x, *r; |
8823192f |
307 | int rc; |
34e4f738 |
308 | if (EC_ATINF(p)) return (0); |
309 | l = F_SQR(f, MP_NEW, p->y); |
310 | x = F_SQR(f, MP_NEW, p->x); |
311 | r = F_MUL(f, MP_NEW, x, p->x); |
432c4e18 |
312 | x = F_MUL(f, x, c->a, p->x); |
8823192f |
313 | r = F_ADD(f, r, r, x); |
432c4e18 |
314 | r = F_ADD(f, r, r, c->b); |
8823192f |
315 | rc = MP_EQ(l, r) ? 0 : -1; |
316 | mp_drop(l); |
317 | mp_drop(x); |
318 | mp_drop(r); |
319 | return (rc); |
320 | } |
321 | |
322 | static int ecprojcheck(ec_curve *c, const ec *p) |
323 | { |
324 | ec t = EC_INIT; |
325 | int rc; |
326 | |
327 | c->ops->fix(c, &t, p); |
328 | rc = eccheck(c, &t); |
329 | EC_DESTROY(&t); |
330 | return (rc); |
331 | } |
332 | |
41cb1beb |
333 | static void ecdestroy(ec_curve *c) |
334 | { |
432c4e18 |
335 | MP_DROP(c->a); |
336 | MP_DROP(c->b); |
337 | DESTROY(c); |
41cb1beb |
338 | } |
339 | |
340 | /* --- @ec_prime@, @ec_primeproj@ --- * |
341 | * |
dbfee00a |
342 | * Arguments: @field *f@ = the underlying field for this elliptic curve |
41cb1beb |
343 | * @mp *a, *b@ = the coefficients for this curve |
344 | * |
02d7884d |
345 | * Returns: A pointer to the curve, or null. |
41cb1beb |
346 | * |
347 | * Use: Creates a curve structure for an elliptic curve defined over |
348 | * a prime field. The @primeproj@ variant uses projective |
349 | * coordinates, which can be a win. |
350 | */ |
351 | |
352 | extern ec_curve *ec_prime(field *f, mp *a, mp *b) |
353 | { |
432c4e18 |
354 | ec_curve *c = CREATE(ec_curve); |
355 | c->ops = &ec_primeops; |
356 | c->f = f; |
357 | c->a = F_IN(f, MP_NEW, a); |
358 | c->b = F_IN(f, MP_NEW, b); |
359 | return (c); |
41cb1beb |
360 | } |
361 | |
8823192f |
362 | extern ec_curve *ec_primeproj(field *f, mp *a, mp *b) |
363 | { |
432c4e18 |
364 | ec_curve *c = CREATE(ec_curve); |
8823192f |
365 | mp *ax; |
366 | |
367 | ax = mp_add(MP_NEW, a, MP_THREE); |
368 | ax = F_IN(f, ax, ax); |
369 | if (F_ZEROP(f, ax)) |
432c4e18 |
370 | c->ops = &ec_primeprojxops; |
8823192f |
371 | else |
432c4e18 |
372 | c->ops = &ec_primeprojops; |
8823192f |
373 | MP_DROP(ax); |
432c4e18 |
374 | c->f = f; |
375 | c->a = F_IN(f, MP_NEW, a); |
376 | c->b = F_IN(f, MP_NEW, b); |
377 | return (c); |
41cb1beb |
378 | } |
379 | |
380 | static const ec_ops ec_primeops = { |
f94b972d |
381 | "prime", |
34e4f738 |
382 | ecdestroy, ec_stdsamep, ec_idin, ec_idout, ec_idfix, |
bc985cef |
383 | ecfind, ecneg, ecadd, ec_stdsub, ecdbl, eccheck |
8823192f |
384 | }; |
385 | |
386 | static const ec_ops ec_primeprojops = { |
f94b972d |
387 | "primeproj", |
34e4f738 |
388 | ecdestroy, ec_stdsamep, ec_projin, ec_projout, ec_projfix, |
bc985cef |
389 | ecfind, ecneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck |
8823192f |
390 | }; |
391 | |
392 | static const ec_ops ec_primeprojxops = { |
f94b972d |
393 | "primeproj", |
34e4f738 |
394 | ecdestroy, ec_stdsamep, ec_projin, ec_projout, ec_projfix, |
bc985cef |
395 | ecfind, ecneg, ecprojadd, ec_stdsub, ecprojxdbl, ecprojcheck |
41cb1beb |
396 | }; |
397 | |
398 | /*----- Test rig ----------------------------------------------------------*/ |
399 | |
400 | #ifdef TEST_RIG |
401 | |
402 | #define MP(x) mp_readstring(MP_NEW, #x, 0, 0) |
403 | |
ceb3f0c0 |
404 | int main(int argc, char *argv[]) |
41cb1beb |
405 | { |
406 | field *f; |
407 | ec_curve *c; |
408 | ec g = EC_INIT, d = EC_INIT; |
409 | mp *p, *a, *b, *r; |
ceb3f0c0 |
410 | int i, n = argc == 1 ? 1 : atoi(argv[1]); |
41cb1beb |
411 | |
dbfee00a |
412 | printf("ec-prime: "); |
413 | fflush(stdout); |
41cb1beb |
414 | a = MP(-3); |
432c4e18 |
415 | b = MP(0xb3312fa7e23ee7e4988e056be3f82d19181d9c6efe8141120314088f5013875ac656398d8a2ed19d2a85c8edd3ec2aef); |
416 | p = MP(39402006196394479212279040100143613805079739270465446667948293404245721771496870329047266088258938001861606973112319); |
417 | r = MP(39402006196394479212279040100143613805079739270465446667946905279627659399113263569398956308152294913554433653942642); |
41cb1beb |
418 | |
f46efa79 |
419 | f = field_niceprime(p); |
ceb3f0c0 |
420 | c = ec_primeproj(f, a, b); |
41cb1beb |
421 | |
432c4e18 |
422 | g.x = MP(0xaa87ca22be8b05378eb1c71ef320ad746e1d3b628ba79b9859f741e082542a385502f25dbf55296c3a545e3872760ab7); |
423 | g.y = MP(0x3617de4a96262c6f5d9e98bf9292dc29f8f41dbd289a147ce9da3113b5f0b8c00a60b1ce1d7e819d7a431d7c90ea0e5f); |
41cb1beb |
424 | |
ceb3f0c0 |
425 | for (i = 0; i < n; i++) { |
426 | ec_mul(c, &d, &g, r); |
427 | if (EC_ATINF(&d)) { |
428 | fprintf(stderr, "zero too early\n"); |
429 | return (1); |
430 | } |
431 | ec_add(c, &d, &d, &g); |
432 | if (!EC_ATINF(&d)) { |
433 | fprintf(stderr, "didn't reach zero\n"); |
434 | MP_EPRINT("d.x", d.x); |
435 | MP_EPRINT("d.y", d.y); |
436 | return (1); |
437 | } |
438 | ec_destroy(&d); |
dbfee00a |
439 | } |
41cb1beb |
440 | ec_destroy(&g); |
441 | ec_destroycurve(c); |
442 | F_DESTROY(f); |
dbfee00a |
443 | MP_DROP(p); MP_DROP(a); MP_DROP(b); MP_DROP(r); |
444 | assert(!mparena_count(&mparena_global)); |
445 | printf("ok\n"); |
41cb1beb |
446 | return (0); |
447 | } |
448 | |
449 | #endif |
450 | |
b0ab12e6 |
451 | /*----- That's all, folks -------------------------------------------------*/ |