3 * $Id: ec-prime.c,v 1.10 2004/04/03 03:32:05 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 1.10 2004/04/03 03:32:05 mdw
34 * General robustification.
36 * Revision 1.9 2004/04/01 12:50:09 mdw
37 * Add cyclic group abstraction, with test code. Separate off exponentation
38 * functions for better static linking. Fix a buttload of bugs on the way.
39 * Generally ensure that negative exponents do inversion correctly. Add
40 * table of standard prime-field subgroups. (Binary field subgroups are
41 * currently unimplemented but easy to add if anyone ever finds a good one.)
43 * Revision 1.8 2004/03/27 17:54:11 mdw
44 * Standard curves and curve checking.
46 * Revision 1.7 2004/03/27 00:04:46 mdw
47 * Implement efficient reduction for pleasant-looking primes.
49 * Revision 1.6 2004/03/23 15:19:32 mdw
50 * Test elliptic curves more thoroughly.
52 * Revision 1.5 2004/03/22 02:19:10 mdw
53 * Rationalise the sliding-window threshold. Drop guarantee that right
54 * arguments to EC @add@ are canonical, and fix up projective implementations
57 * Revision 1.4 2004/03/21 22:52:06 mdw
58 * Merge and close elliptic curve branch.
60 * Revision 1.3.4.3 2004/03/21 22:39:46 mdw
61 * Elliptic curves on binary fields work.
63 * Revision 1.3.4.2 2004/03/20 00:13:31 mdw
64 * Projective coordinates for prime curves
66 * Revision 1.3.4.1 2003/06/10 13:43:53 mdw
67 * Simple (non-projective) curves over prime fields now seem to work.
69 * Revision 1.3 2003/05/15 23:25:59 mdw
70 * Make elliptic curve stuff build.
72 * Revision 1.2 2002/01/13 13:48:44 mdw
75 * Revision 1.1 2001/04/29 18:12:33 mdw
80 /*----- Header files ------------------------------------------------------*/
86 /*----- Simple prime curves -----------------------------------------------*/
88 static const ec_ops ec_primeops
, ec_primeprojops
, ec_primeprojxops
;
90 static ec
*ecneg(ec_curve
*c
, ec
*d
, const ec
*p
)
94 d
->y
= F_NEG(c
->f
, d
->y
, d
->y
);
98 static ec
*ecfind(ec_curve
*c
, ec
*d
, mp
*x
)
103 q
= F_SQR(f
, MP_NEW
, x
);
104 p
= F_MUL(f
, MP_NEW
, x
, q
);
105 q
= F_MUL(f
, q
, x
, c
->a
);
106 p
= F_ADD(f
, p
, p
, q
);
107 p
= F_ADD(f
, p
, p
, c
->b
);
115 d
->z
= MP_COPY(f
->one
);
119 static ec
*ecdbl(ec_curve
*c
, ec
*d
, const ec
*a
)
123 else if (F_ZEROP(c
->f
, a
->y
))
130 dx
= F_SQR(f
, MP_NEW
, a
->x
); /* %$x^2$% */
131 dy
= F_DBL(f
, MP_NEW
, a
->y
); /* %$2 y$% */
132 dx
= F_TPL(f
, dx
, dx
); /* %$3 x^2$% */
133 dx
= F_ADD(f
, dx
, dx
, c
->a
); /* %$3 x^2 + A$% */
134 dy
= F_INV(f
, dy
, dy
); /* %$(2 y)^{-1}$% */
135 lambda
= F_MUL(f
, MP_NEW
, dx
, dy
); /* %$\lambda = (3 x^2 + A)/(2 y)$% */
137 dx
= F_SQR(f
, dx
, lambda
); /* %$\lambda^2$% */
138 dy
= F_DBL(f
, dy
, a
->x
); /* %$2 x$% */
139 dx
= F_SUB(f
, dx
, dx
, dy
); /* %$x' = \lambda^2 - 2 x */
140 dy
= F_SUB(f
, dy
, a
->x
, dx
); /* %$x - x'$% */
141 dy
= F_MUL(f
, dy
, lambda
, dy
); /* %$\lambda (x - x')$% */
142 dy
= F_SUB(f
, dy
, dy
, a
->y
); /* %$y' = \lambda (x - x') - y$% */
153 static ec
*ecprojdbl(ec_curve
*c
, ec
*d
, const ec
*a
)
157 else if (F_ZEROP(c
->f
, a
->y
))
161 mp
*p
, *q
, *m
, *s
, *dx
, *dy
, *dz
;
163 p
= F_SQR(f
, MP_NEW
, a
->z
); /* %$z^2$% */
164 q
= F_SQR(f
, MP_NEW
, p
); /* %$z^4$% */
165 p
= F_MUL(f
, p
, q
, c
->a
); /* %$A z^4$% */
166 m
= F_SQR(f
, MP_NEW
, a
->x
); /* %$x^2$% */
167 m
= F_TPL(f
, m
, m
); /* %$3 x^2$% */
168 m
= F_ADD(f
, m
, m
, p
); /* %$m = 3 x^2 + A z^4$% */
170 q
= F_DBL(f
, q
, a
->y
); /* %$2 y$% */
171 dz
= F_MUL(f
, MP_NEW
, q
, a
->z
); /* %$z' = 2 y z$% */
173 p
= F_SQR(f
, p
, q
); /* %$4 y^2$% */
174 s
= F_MUL(f
, MP_NEW
, p
, a
->x
); /* %$s = 4 x y^2$% */
175 q
= F_SQR(f
, q
, p
); /* %$16 y^4$% */
176 q
= F_HLV(f
, q
, q
); /* %$t = 8 y^4$% */
178 p
= F_DBL(f
, p
, s
); /* %$2 s$% */
179 dx
= F_SQR(f
, MP_NEW
, m
); /* %$m^2$% */
180 dx
= F_SUB(f
, dx
, dx
, p
); /* %$x' = m^2 - 2 s$% */
182 s
= F_SUB(f
, s
, s
, dx
); /* %$s - x'$% */
183 dy
= F_MUL(f
, p
, m
, s
); /* %$m (s - x')$% */
184 dy
= F_SUB(f
, dy
, dy
, q
); /* %$y' = m (s - x') - t$% */
197 static ec
*ecprojxdbl(ec_curve
*c
, ec
*d
, const ec
*a
)
201 else if (F_ZEROP(c
->f
, a
->y
))
205 mp
*p
, *q
, *m
, *s
, *dx
, *dy
, *dz
;
207 m
= F_SQR(f
, MP_NEW
, a
->z
); /* %$z^2$% */
208 p
= F_SUB(f
, MP_NEW
, a
->x
, m
); /* %$x - z^2$% */
209 q
= F_ADD(f
, MP_NEW
, a
->x
, m
); /* %$x + z^2$% */
210 m
= F_MUL(f
, m
, p
, q
); /* %$x^2 - z^4$% */
211 m
= F_TPL(f
, m
, m
); /* %$m = 3 x^2 - 3 z^4$% */
213 q
= F_DBL(f
, q
, a
->y
); /* %$2 y$% */
214 dz
= F_MUL(f
, MP_NEW
, q
, a
->z
); /* %$z' = 2 y z$% */
216 p
= F_SQR(f
, p
, q
); /* %$4 y^2$% */
217 s
= F_MUL(f
, MP_NEW
, p
, a
->x
); /* %$s = 4 x y^2$% */
218 q
= F_SQR(f
, q
, p
); /* %$16 y^4$% */
219 q
= F_HLV(f
, q
, q
); /* %$t = 8 y^4$% */
221 p
= F_DBL(f
, p
, s
); /* %$2 s$% */
222 dx
= F_SQR(f
, MP_NEW
, m
); /* %$m^2$% */
223 dx
= F_SUB(f
, dx
, dx
, p
); /* %$x' = m^2 - 2 s$% */
225 s
= F_SUB(f
, s
, s
, dx
); /* %$s - x'$% */
226 dy
= F_MUL(f
, p
, m
, s
); /* %$m (s - x')$% */
227 dy
= F_SUB(f
, dy
, dy
, q
); /* %$y' = m (s - x') - t$% */
240 static ec
*ecadd(ec_curve
*c
, ec
*d
, const ec
*a
, const ec
*b
)
244 else if (EC_ATINF(a
))
246 else if (EC_ATINF(b
))
253 if (!MP_EQ(a
->x
, b
->x
)) {
254 dy
= F_SUB(f
, MP_NEW
, a
->y
, b
->y
); /* %$y_0 - y_1$% */
255 dx
= F_SUB(f
, MP_NEW
, a
->x
, b
->x
); /* %$x_0 - x_1$% */
256 dx
= F_INV(f
, dx
, dx
); /* %$(x_0 - x_1)^{-1}$% */
257 lambda
= F_MUL(f
, MP_NEW
, dy
, dx
);
258 /* %$\lambda = (y_0 - y1)/(x_0 - x_1)$% */
259 } else if (F_ZEROP(c
->f
, a
->y
) || !MP_EQ(a
->y
, b
->y
)) {
263 dx
= F_SQR(f
, MP_NEW
, a
->x
); /* %$x_0^2$% */
264 dx
= F_TPL(f
, dx
, dx
); /* %$3 x_0^2$% */
265 dx
= F_ADD(f
, dx
, dx
, c
->a
); /* %$3 x_0^2 + A$% */
266 dy
= F_DBL(f
, MP_NEW
, a
->y
); /* %$2 y_0$% */
267 dy
= F_INV(f
, dy
, dy
); /* %$(2 y_0)^{-1}$% */
268 lambda
= F_MUL(f
, MP_NEW
, dx
, dy
);
269 /* %$\lambda = (3 x_0^2 + A)/(2 y_0)$% */
272 dx
= F_SQR(f
, dx
, lambda
); /* %$\lambda^2$% */
273 dx
= F_SUB(f
, dx
, dx
, a
->x
); /* %$\lambda^2 - x_0$% */
274 dx
= F_SUB(f
, dx
, dx
, b
->x
); /* %$x' = \lambda^2 - x_0 - x_1$% */
275 dy
= F_SUB(f
, dy
, b
->x
, dx
); /* %$x_1 - x'$% */
276 dy
= F_MUL(f
, dy
, lambda
, dy
); /* %$\lambda (x_1 - x')$% */
277 dy
= F_SUB(f
, dy
, dy
, b
->y
); /* %$y' = \lambda (x_1 - x') - y_1$% */
288 static ec
*ecprojadd(ec_curve
*c
, ec
*d
, const ec
*a
, const ec
*b
)
291 c
->ops
->dbl(c
, d
, a
);
292 else if (EC_ATINF(a
))
294 else if (EC_ATINF(b
))
298 mp
*p
, *q
, *r
, *w
, *u
, *uu
, *s
, *ss
, *dx
, *dy
, *dz
;
300 q
= F_SQR(f
, MP_NEW
, a
->z
); /* %$z_0^2$% */
301 u
= F_MUL(f
, MP_NEW
, q
, b
->x
); /* %$u = x_1 z_0^2$% */
302 p
= F_MUL(f
, MP_NEW
, q
, b
->y
); /* %$y_1 z_0^2$% */
303 s
= F_MUL(f
, q
, p
, a
->z
); /* %$s = y_1 z_0^3$% */
305 q
= F_SQR(f
, MP_NEW
, b
->z
); /* %$z_1^2$% */
306 uu
= F_MUL(f
, MP_NEW
, q
, a
->x
); /* %$uu = x_0 z_1^2$%*/
307 p
= F_MUL(f
, p
, q
, a
->y
); /* %$y_0 z_1^2$% */
308 ss
= F_MUL(f
, q
, p
, b
->z
); /* %$ss = y_0 z_1^3$% */
310 w
= F_SUB(f
, p
, uu
, u
); /* %$w = uu - u$% */
311 r
= F_SUB(f
, MP_NEW
, ss
, s
); /* %$r = ss - s$% */
320 return (c
->ops
->dbl(c
, d
, a
));
327 u
= F_ADD(f
, u
, u
, uu
); /* %$t = uu + u$% */
328 s
= F_ADD(f
, s
, s
, ss
); /* %$m = ss + r$% */
330 uu
= F_MUL(f
, uu
, a
->z
, w
); /* %$z_0 w$% */
331 dz
= F_MUL(f
, ss
, uu
, b
->z
); /* %$z' = z_0 z_1 w$% */
333 p
= F_SQR(f
, uu
, w
); /* %$w^2$% */
334 q
= F_MUL(f
, MP_NEW
, p
, u
); /* %$t w^2$% */
335 u
= F_MUL(f
, u
, p
, w
); /* %$w^3$% */
336 p
= F_MUL(f
, p
, u
, s
); /* %$m w^3$% */
338 dx
= F_SQR(f
, u
, r
); /* %$r^2$% */
339 dx
= F_SUB(f
, dx
, dx
, q
); /* %$x' = r^2 - t w^2$% */
341 s
= F_DBL(f
, s
, dx
); /* %$2 x'$% */
342 q
= F_SUB(f
, q
, q
, s
); /* %$v = t w^2 - 2 x'$% */
343 dy
= F_MUL(f
, s
, q
, r
); /* %$v r$% */
344 dy
= F_SUB(f
, dy
, dy
, p
); /* %$v r - m w^3$% */
345 dy
= F_HLV(f
, dy
, dy
); /* %$y' = (v r - m w^3)/2$% */
359 static int eccheck(ec_curve
*c
, const ec
*p
)
364 if (EC_ATINF(p
)) return (0);
365 l
= F_SQR(f
, MP_NEW
, p
->y
);
366 x
= F_SQR(f
, MP_NEW
, p
->x
);
367 r
= F_MUL(f
, MP_NEW
, x
, p
->x
);
368 x
= F_MUL(f
, x
, c
->a
, p
->x
);
369 r
= F_ADD(f
, r
, r
, x
);
370 r
= F_ADD(f
, r
, r
, c
->b
);
371 rc
= MP_EQ(l
, r
) ?
0 : -1;
378 static int ecprojcheck(ec_curve
*c
, const ec
*p
)
383 c
->ops
->fix(c
, &t
, p
);
389 static void ecdestroy(ec_curve
*c
)
396 /* --- @ec_prime@, @ec_primeproj@ --- *
398 * Arguments: @field *f@ = the underlying field for this elliptic curve
399 * @mp *a, *b@ = the coefficients for this curve
401 * Returns: A pointer to the curve, or null.
403 * Use: Creates a curve structure for an elliptic curve defined over
404 * a prime field. The @primeproj@ variant uses projective
405 * coordinates, which can be a win.
408 extern ec_curve
*ec_prime(field
*f
, mp
*a
, mp
*b
)
410 ec_curve
*c
= CREATE(ec_curve
);
411 c
->ops
= &ec_primeops
;
413 c
->a
= F_IN(f
, MP_NEW
, a
);
414 c
->b
= F_IN(f
, MP_NEW
, b
);
418 extern ec_curve
*ec_primeproj(field
*f
, mp
*a
, mp
*b
)
420 ec_curve
*c
= CREATE(ec_curve
);
423 ax
= mp_add(MP_NEW
, a
, MP_THREE
);
424 ax
= F_IN(f
, ax
, ax
);
426 c
->ops
= &ec_primeprojxops
;
428 c
->ops
= &ec_primeprojops
;
431 c
->a
= F_IN(f
, MP_NEW
, a
);
432 c
->b
= F_IN(f
, MP_NEW
, b
);
436 static const ec_ops ec_primeops
= {
437 ecdestroy
, ec_stdsamep
, ec_idin
, ec_idout
, ec_idfix
,
438 ecfind
, ecneg
, ecadd
, ec_stdsub
, ecdbl
, eccheck
441 static const ec_ops ec_primeprojops
= {
442 ecdestroy
, ec_stdsamep
, ec_projin
, ec_projout
, ec_projfix
,
443 ecfind
, ecneg
, ecprojadd
, ec_stdsub
, ecprojdbl
, ecprojcheck
446 static const ec_ops ec_primeprojxops
= {
447 ecdestroy
, ec_stdsamep
, ec_projin
, ec_projout
, ec_projfix
,
448 ecfind
, ecneg
, ecprojadd
, ec_stdsub
, ecprojxdbl
, ecprojcheck
451 /*----- Test rig ----------------------------------------------------------*/
455 #define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
457 int main(int argc
, char *argv
[])
461 ec g
= EC_INIT
, d
= EC_INIT
;
463 int i
, n
= argc
== 1 ?
1 : atoi(argv
[1]);
465 printf("ec-prime: ");
468 b
= MP(0xb3312fa7e23ee7e4988e056be3f82d19181d9c6efe8141120314088f5013875ac656398d8a2ed19d2a85c8edd3ec2aef);
469 p
= MP(39402006196394479212279040100143613805079739270465446667948293404245721771496870329047266088258938001861606973112319);
470 r
= MP(39402006196394479212279040100143613805079739270465446667946905279627659399113263569398956308152294913554433653942642);
472 f
= field_niceprime(p
);
473 c
= ec_primeproj(f
, a
, b
);
475 g
.x
= MP(0xaa87ca22be8b05378eb1c71ef320ad746e1d3b628ba79b9859f741e082542a385502f25dbf55296c3a545e3872760ab7);
476 g
.y
= MP(0x3617de4a96262c6f5d9e98bf9292dc29f8f41dbd289a147ce9da3113b5f0b8c00a60b1ce1d7e819d7a431d7c90ea0e5f);
478 for (i
= 0; i
< n
; i
++) {
479 ec_mul(c
, &d
, &g
, r
);
481 fprintf(stderr
, "zero too early\n");
484 ec_add(c
, &d
, &d
, &g
);
486 fprintf(stderr
, "didn't reach zero\n");
487 MP_EPRINT("d.x", d
.x
);
488 MP_EPRINT("d.y", d
.y
);
496 MP_DROP(p
); MP_DROP(a
); MP_DROP(b
); MP_DROP(r
);
497 assert(!mparena_count(&mparena_global
));
504 /*----- That's all, folks -------------------------------------------------*/