3 * $Id: ec-prime.c,v 1.9 2004/04/01 12:50:09 mdw Exp $
5 * Elliptic curves over prime fields
7 * (c) 2001 Straylight/Edgeware
10 /*----- Licensing notice --------------------------------------------------*
12 * This file is part of Catacomb.
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.
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.
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,
30 /*----- Revision history --------------------------------------------------*
32 * $Log: ec-prime.c,v $
33 * Revision 1.9 2004/04/01 12:50:09 mdw
34 * Add cyclic group abstraction, with test code. Separate off exponentation
35 * functions for better static linking. Fix a buttload of bugs on the way.
36 * Generally ensure that negative exponents do inversion correctly. Add
37 * table of standard prime-field subgroups. (Binary field subgroups are
38 * currently unimplemented but easy to add if anyone ever finds a good one.)
40 * Revision 1.8 2004/03/27 17:54:11 mdw
41 * Standard curves and curve checking.
43 * Revision 1.7 2004/03/27 00:04:46 mdw
44 * Implement efficient reduction for pleasant-looking primes.
46 * Revision 1.6 2004/03/23 15:19:32 mdw
47 * Test elliptic curves more thoroughly.
49 * Revision 1.5 2004/03/22 02:19:10 mdw
50 * Rationalise the sliding-window threshold. Drop guarantee that right
51 * arguments to EC @add@ are canonical, and fix up projective implementations
54 * Revision 1.4 2004/03/21 22:52:06 mdw
55 * Merge and close elliptic curve branch.
57 * Revision 1.3.4.3 2004/03/21 22:39:46 mdw
58 * Elliptic curves on binary fields work.
60 * Revision 1.3.4.2 2004/03/20 00:13:31 mdw
61 * Projective coordinates for prime curves
63 * Revision 1.3.4.1 2003/06/10 13:43:53 mdw
64 * Simple (non-projective) curves over prime fields now seem to work.
66 * Revision 1.3 2003/05/15 23:25:59 mdw
67 * Make elliptic curve stuff build.
69 * Revision 1.2 2002/01/13 13:48:44 mdw
72 * Revision 1.1 2001/04/29 18:12:33 mdw
77 /*----- Header files ------------------------------------------------------*/
83 /*----- Simple prime curves -----------------------------------------------*/
85 static const ec_ops ec_primeops
, ec_primeprojops
, ec_primeprojxops
;
87 static ec
*ecneg(ec_curve
*c
, ec
*d
, const ec
*p
)
91 d
->y
= F_NEG(c
->f
, d
->y
, d
->y
);
95 static ec
*ecfind(ec_curve
*c
, ec
*d
, mp
*x
)
100 q
= F_SQR(f
, MP_NEW
, x
);
101 p
= F_MUL(f
, MP_NEW
, x
, q
);
102 q
= F_MUL(f
, q
, x
, c
->a
);
103 p
= F_ADD(f
, p
, p
, q
);
104 p
= F_ADD(f
, p
, p
, c
->b
);
112 d
->z
= MP_COPY(f
->one
);
116 static ec
*ecdbl(ec_curve
*c
, ec
*d
, const ec
*a
)
120 else if (F_ZEROP(c
->f
, a
->y
))
127 dx
= F_SQR(f
, MP_NEW
, a
->x
); /* %$x^2$% */
128 dy
= F_DBL(f
, MP_NEW
, a
->y
); /* %$2 y$% */
129 dx
= F_TPL(f
, dx
, dx
); /* %$3 x^2$% */
130 dx
= F_ADD(f
, dx
, dx
, c
->a
); /* %$3 x^2 + A$% */
131 dy
= F_INV(f
, dy
, dy
); /* %$(2 y)^{-1}$% */
132 lambda
= F_MUL(f
, MP_NEW
, dx
, dy
); /* %$\lambda = (3 x^2 + A)/(2 y)$% */
134 dx
= F_SQR(f
, dx
, lambda
); /* %$\lambda^2$% */
135 dy
= F_DBL(f
, dy
, a
->x
); /* %$2 x$% */
136 dx
= F_SUB(f
, dx
, dx
, dy
); /* %$x' = \lambda^2 - 2 x */
137 dy
= F_SUB(f
, dy
, a
->x
, dx
); /* %$x - x'$% */
138 dy
= F_MUL(f
, dy
, lambda
, dy
); /* %$\lambda (x - x')$% */
139 dy
= F_SUB(f
, dy
, dy
, a
->y
); /* %$y' = \lambda (x - x') - y$% */
150 static ec
*ecprojdbl(ec_curve
*c
, ec
*d
, const ec
*a
)
154 else if (F_ZEROP(c
->f
, a
->y
))
158 mp
*p
, *q
, *m
, *s
, *dx
, *dy
, *dz
;
160 p
= F_SQR(f
, MP_NEW
, a
->z
); /* %$z^2$% */
161 q
= F_SQR(f
, MP_NEW
, p
); /* %$z^4$% */
162 p
= F_MUL(f
, p
, q
, c
->a
); /* %$A z^4$% */
163 m
= F_SQR(f
, MP_NEW
, a
->x
); /* %$x^2$% */
164 m
= F_TPL(f
, m
, m
); /* %$3 x^2$% */
165 m
= F_ADD(f
, m
, m
, p
); /* %$m = 3 x^2 + A z^4$% */
167 q
= F_DBL(f
, q
, a
->y
); /* %$2 y$% */
168 dz
= F_MUL(f
, MP_NEW
, q
, a
->z
); /* %$z' = 2 y z$% */
170 p
= F_SQR(f
, p
, q
); /* %$4 y^2$% */
171 s
= F_MUL(f
, MP_NEW
, p
, a
->x
); /* %$s = 4 x y^2$% */
172 q
= F_SQR(f
, q
, p
); /* %$16 y^4$% */
173 q
= F_HLV(f
, q
, q
); /* %$t = 8 y^4$% */
175 p
= F_DBL(f
, p
, s
); /* %$2 s$% */
176 dx
= F_SQR(f
, MP_NEW
, m
); /* %$m^2$% */
177 dx
= F_SUB(f
, dx
, dx
, p
); /* %$x' = m^2 - 2 s$% */
179 s
= F_SUB(f
, s
, s
, dx
); /* %$s - x'$% */
180 dy
= F_MUL(f
, p
, m
, s
); /* %$m (s - x')$% */
181 dy
= F_SUB(f
, dy
, dy
, q
); /* %$y' = m (s - x') - t$% */
194 static ec
*ecprojxdbl(ec_curve
*c
, ec
*d
, const ec
*a
)
198 else if (F_ZEROP(c
->f
, a
->y
))
202 mp
*p
, *q
, *m
, *s
, *dx
, *dy
, *dz
;
204 m
= F_SQR(f
, MP_NEW
, a
->z
); /* %$z^2$% */
205 p
= F_SUB(f
, MP_NEW
, a
->x
, m
); /* %$x - z^2$% */
206 q
= F_ADD(f
, MP_NEW
, a
->x
, m
); /* %$x + z^2$% */
207 m
= F_MUL(f
, m
, p
, q
); /* %$x^2 - z^4$% */
208 m
= F_TPL(f
, m
, m
); /* %$m = 3 x^2 - 3 z^4$% */
210 q
= F_DBL(f
, q
, a
->y
); /* %$2 y$% */
211 dz
= F_MUL(f
, MP_NEW
, q
, a
->z
); /* %$z' = 2 y z$% */
213 p
= F_SQR(f
, p
, q
); /* %$4 y^2$% */
214 s
= F_MUL(f
, MP_NEW
, p
, a
->x
); /* %$s = 4 x y^2$% */
215 q
= F_SQR(f
, q
, p
); /* %$16 y^4$% */
216 q
= F_HLV(f
, q
, q
); /* %$t = 8 y^4$% */
218 p
= F_DBL(f
, p
, s
); /* %$2 s$% */
219 dx
= F_SQR(f
, MP_NEW
, m
); /* %$m^2$% */
220 dx
= F_SUB(f
, dx
, dx
, p
); /* %$x' = m^2 - 2 s$% */
222 s
= F_SUB(f
, s
, s
, dx
); /* %$s - x'$% */
223 dy
= F_MUL(f
, p
, m
, s
); /* %$m (s - x')$% */
224 dy
= F_SUB(f
, dy
, dy
, q
); /* %$y' = m (s - x') - t$% */
237 static ec
*ecadd(ec_curve
*c
, ec
*d
, const ec
*a
, const ec
*b
)
241 else if (EC_ATINF(a
))
243 else if (EC_ATINF(b
))
250 if (!MP_EQ(a
->x
, b
->x
)) {
251 dy
= F_SUB(f
, MP_NEW
, a
->y
, b
->y
); /* %$y_0 - y_1$% */
252 dx
= F_SUB(f
, MP_NEW
, a
->x
, b
->x
); /* %$x_0 - x_1$% */
253 dx
= F_INV(f
, dx
, dx
); /* %$(x_0 - x_1)^{-1}$% */
254 lambda
= F_MUL(f
, MP_NEW
, dy
, dx
);
255 /* %$\lambda = (y_0 - y1)/(x_0 - x_1)$% */
256 } else if (F_ZEROP(c
->f
, a
->y
) || !MP_EQ(a
->y
, b
->y
)) {
260 dx
= F_SQR(f
, MP_NEW
, a
->x
); /* %$x_0^2$% */
261 dx
= F_TPL(f
, dx
, dx
); /* %$3 x_0^2$% */
262 dx
= F_ADD(f
, dx
, dx
, c
->a
); /* %$3 x_0^2 + A$% */
263 dy
= F_DBL(f
, MP_NEW
, a
->y
); /* %$2 y_0$% */
264 dy
= F_INV(f
, dy
, dy
); /* %$(2 y_0)^{-1}$% */
265 lambda
= F_MUL(f
, MP_NEW
, dx
, dy
);
266 /* %$\lambda = (3 x_0^2 + A)/(2 y_0)$% */
269 dx
= F_SQR(f
, dx
, lambda
); /* %$\lambda^2$% */
270 dx
= F_SUB(f
, dx
, dx
, a
->x
); /* %$\lambda^2 - x_0$% */
271 dx
= F_SUB(f
, dx
, dx
, b
->x
); /* %$x' = \lambda^2 - x_0 - x_1$% */
272 dy
= F_SUB(f
, dy
, b
->x
, dx
); /* %$x_1 - x'$% */
273 dy
= F_MUL(f
, dy
, lambda
, dy
); /* %$\lambda (x_1 - x')$% */
274 dy
= F_SUB(f
, dy
, dy
, b
->y
); /* %$y' = \lambda (x_1 - x') - y_1$% */
285 static ec
*ecprojadd(ec_curve
*c
, ec
*d
, const ec
*a
, const ec
*b
)
288 c
->ops
->dbl(c
, d
, a
);
289 else if (EC_ATINF(a
))
291 else if (EC_ATINF(b
))
295 mp
*p
, *q
, *r
, *w
, *u
, *uu
, *s
, *ss
, *dx
, *dy
, *dz
;
297 q
= F_SQR(f
, MP_NEW
, a
->z
); /* %$z_0^2$% */
298 u
= F_MUL(f
, MP_NEW
, q
, b
->x
); /* %$u = x_1 z_0^2$% */
299 p
= F_MUL(f
, MP_NEW
, q
, b
->y
); /* %$y_1 z_0^2$% */
300 s
= F_MUL(f
, q
, p
, a
->z
); /* %$s = y_1 z_0^3$% */
302 q
= F_SQR(f
, MP_NEW
, b
->z
); /* %$z_1^2$% */
303 uu
= F_MUL(f
, MP_NEW
, q
, a
->x
); /* %$uu = x_0 z_1^2$%*/
304 p
= F_MUL(f
, p
, q
, a
->y
); /* %$y_0 z_1^2$% */
305 ss
= F_MUL(f
, q
, p
, b
->z
); /* %$ss = y_0 z_1^3$% */
307 w
= F_SUB(f
, p
, uu
, u
); /* %$w = uu - u$% */
308 r
= F_SUB(f
, MP_NEW
, ss
, s
); /* %$r = ss - s$% */
317 return (c
->ops
->dbl(c
, d
, a
));
324 u
= F_ADD(f
, u
, u
, uu
); /* %$t = uu + u$% */
325 s
= F_ADD(f
, s
, s
, ss
); /* %$m = ss + r$% */
327 uu
= F_MUL(f
, uu
, a
->z
, w
); /* %$z_0 w$% */
328 dz
= F_MUL(f
, ss
, uu
, b
->z
); /* %$z' = z_0 z_1 w$% */
330 p
= F_SQR(f
, uu
, w
); /* %$w^2$% */
331 q
= F_MUL(f
, MP_NEW
, p
, u
); /* %$t w^2$% */
332 u
= F_MUL(f
, u
, p
, w
); /* %$w^3$% */
333 p
= F_MUL(f
, p
, u
, s
); /* %$m w^3$% */
335 dx
= F_SQR(f
, u
, r
); /* %$r^2$% */
336 dx
= F_SUB(f
, dx
, dx
, q
); /* %$x' = r^2 - t w^2$% */
338 s
= F_DBL(f
, s
, dx
); /* %$2 x'$% */
339 q
= F_SUB(f
, q
, q
, s
); /* %$v = t w^2 - 2 x'$% */
340 dy
= F_MUL(f
, s
, q
, r
); /* %$v r$% */
341 dy
= F_SUB(f
, dy
, dy
, p
); /* %$v r - m w^3$% */
342 dy
= F_HLV(f
, dy
, dy
); /* %$y' = (v r - m w^3)/2$% */
356 static int eccheck(ec_curve
*c
, const ec
*p
)
361 if (EC_ATINF(p
)) return (0);
362 l
= F_SQR(f
, MP_NEW
, p
->y
);
363 x
= F_SQR(f
, MP_NEW
, p
->x
);
364 r
= F_MUL(f
, MP_NEW
, x
, p
->x
);
365 x
= F_MUL(f
, x
, c
->a
, p
->x
);
366 r
= F_ADD(f
, r
, r
, x
);
367 r
= F_ADD(f
, r
, r
, c
->b
);
368 rc
= MP_EQ(l
, r
) ?
0 : -1;
375 static int ecprojcheck(ec_curve
*c
, const ec
*p
)
380 c
->ops
->fix(c
, &t
, p
);
386 static void ecdestroy(ec_curve
*c
)
393 /* --- @ec_prime@, @ec_primeproj@ --- *
395 * Arguments: @field *f@ = the underlying field for this elliptic curve
396 * @mp *a, *b@ = the coefficients for this curve
398 * Returns: A pointer to the curve.
400 * Use: Creates a curve structure for an elliptic curve defined over
401 * a prime field. The @primeproj@ variant uses projective
402 * coordinates, which can be a win.
405 extern ec_curve
*ec_prime(field
*f
, mp
*a
, mp
*b
)
407 ec_curve
*c
= CREATE(ec_curve
);
408 c
->ops
= &ec_primeops
;
410 c
->a
= F_IN(f
, MP_NEW
, a
);
411 c
->b
= F_IN(f
, MP_NEW
, b
);
415 extern ec_curve
*ec_primeproj(field
*f
, mp
*a
, mp
*b
)
417 ec_curve
*c
= CREATE(ec_curve
);
420 ax
= mp_add(MP_NEW
, a
, MP_THREE
);
421 ax
= F_IN(f
, ax
, ax
);
423 c
->ops
= &ec_primeprojxops
;
425 c
->ops
= &ec_primeprojops
;
428 c
->a
= F_IN(f
, MP_NEW
, a
);
429 c
->b
= F_IN(f
, MP_NEW
, b
);
433 static const ec_ops ec_primeops
= {
434 ecdestroy
, ec_stdsamep
, ec_idin
, ec_idout
, ec_idfix
,
435 ecfind
, ecneg
, ecadd
, ec_stdsub
, ecdbl
, eccheck
438 static const ec_ops ec_primeprojops
= {
439 ecdestroy
, ec_stdsamep
, ec_projin
, ec_projout
, ec_projfix
,
440 ecfind
, ecneg
, ecprojadd
, ec_stdsub
, ecprojdbl
, ecprojcheck
443 static const ec_ops ec_primeprojxops
= {
444 ecdestroy
, ec_stdsamep
, ec_projin
, ec_projout
, ec_projfix
,
445 ecfind
, ecneg
, ecprojadd
, ec_stdsub
, ecprojxdbl
, ecprojcheck
448 /*----- Test rig ----------------------------------------------------------*/
452 #define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
454 int main(int argc
, char *argv
[])
458 ec g
= EC_INIT
, d
= EC_INIT
;
460 int i
, n
= argc
== 1 ?
1 : atoi(argv
[1]);
462 printf("ec-prime: ");
465 b
= MP(0xb3312fa7e23ee7e4988e056be3f82d19181d9c6efe8141120314088f5013875ac656398d8a2ed19d2a85c8edd3ec2aef);
466 p
= MP(39402006196394479212279040100143613805079739270465446667948293404245721771496870329047266088258938001861606973112319);
467 r
= MP(39402006196394479212279040100143613805079739270465446667946905279627659399113263569398956308152294913554433653942642);
469 f
= field_niceprime(p
);
470 c
= ec_primeproj(f
, a
, b
);
472 g
.x
= MP(0xaa87ca22be8b05378eb1c71ef320ad746e1d3b628ba79b9859f741e082542a385502f25dbf55296c3a545e3872760ab7);
473 g
.y
= MP(0x3617de4a96262c6f5d9e98bf9292dc29f8f41dbd289a147ce9da3113b5f0b8c00a60b1ce1d7e819d7a431d7c90ea0e5f);
475 for (i
= 0; i
< n
; i
++) {
476 ec_mul(c
, &d
, &g
, r
);
478 fprintf(stderr
, "zero too early\n");
481 ec_add(c
, &d
, &d
, &g
);
483 fprintf(stderr
, "didn't reach zero\n");
484 MP_EPRINT("d.x", d
.x
);
485 MP_EPRINT("d.y", d
.y
);
493 MP_DROP(p
); MP_DROP(a
); MP_DROP(b
); MP_DROP(r
);
494 assert(!mparena_count(&mparena_global
));
501 /*----- That's all, folks -------------------------------------------------*/