7712bbf037614e935d2c7d5765168fa75c96814a
3 * $Id: ec-prime.c,v 1.7 2004/03/27 00:04:46 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 1.7 2004/03/27 00:04:46 mdw
34 * Implement efficient reduction for pleasant-looking primes.
36 * Revision 1.6 2004/03/23 15:19:32 mdw
37 * Test elliptic curves more thoroughly.
39 * Revision 1.5 2004/03/22 02:19:10 mdw
40 * Rationalise the sliding-window threshold. Drop guarantee that right
41 * arguments to EC @add@ are canonical, and fix up projective implementations
44 * Revision 1.4 2004/03/21 22:52:06 mdw
45 * Merge and close elliptic curve branch.
47 * Revision 1.3.4.3 2004/03/21 22:39:46 mdw
48 * Elliptic curves on binary fields work.
50 * Revision 1.3.4.2 2004/03/20 00:13:31 mdw
51 * Projective coordinates for prime curves
53 * Revision 1.3.4.1 2003/06/10 13:43:53 mdw
54 * Simple (non-projective) curves over prime fields now seem to work.
56 * Revision 1.3 2003/05/15 23:25:59 mdw
57 * Make elliptic curve stuff build.
59 * Revision 1.2 2002/01/13 13:48:44 mdw
62 * Revision 1.1 2001/04/29 18:12:33 mdw
67 /*----- Header files ------------------------------------------------------*/
73 /*----- Data structures ---------------------------------------------------*/
75 typedef struct ecctx
{
80 /*----- Simple prime curves -----------------------------------------------*/
82 static const ec_ops ec_primeops
, ec_primeprojops
, ec_primeprojxops
;
84 static ec
*ecneg(ec_curve
*c
, ec
*d
, const ec
*p
)
88 d
->y
= F_NEG(c
->f
, d
->y
, d
->y
);
92 static ec
*ecfind(ec_curve
*c
, ec
*d
, mp
*x
)
95 ecctx
*cc
= (ecctx
*)c
;
98 q
= F_SQR(f
, MP_NEW
, x
);
99 p
= F_MUL(f
, MP_NEW
, x
, q
);
100 q
= F_MUL(f
, q
, x
, cc
->a
);
101 p
= F_ADD(f
, p
, p
, q
);
102 p
= F_ADD(f
, p
, p
, cc
->b
);
110 d
->z
= MP_COPY(f
->one
);
114 static ec
*ecdbl(ec_curve
*c
, ec
*d
, const ec
*a
)
118 else if (F_ZEROP(c
->f
, a
->y
))
122 ecctx
*cc
= (ecctx
*)c
;
126 dx
= F_SQR(f
, MP_NEW
, a
->x
); /* %$x^2$% */
127 dy
= F_DBL(f
, MP_NEW
, a
->y
); /* %$2 y$% */
128 dx
= F_TPL(f
, dx
, dx
); /* %$3 x^2$% */
129 dx
= F_ADD(f
, dx
, dx
, cc
->a
); /* %$3 x^2 + A$% */
130 dy
= F_INV(f
, dy
, dy
); /* %$(2 y)^{-1}$% */
131 lambda
= F_MUL(f
, MP_NEW
, dx
, dy
); /* %$\lambda = (3 x^2 + A)/(2 y)$% */
133 dx
= F_SQR(f
, dx
, lambda
); /* %$\lambda^2$% */
134 dy
= F_DBL(f
, dy
, a
->x
); /* %$2 x$% */
135 dx
= F_SUB(f
, dx
, dx
, dy
); /* %$x' = \lambda^2 - 2 x */
136 dy
= F_SUB(f
, dy
, a
->x
, dx
); /* %$x - x'$% */
137 dy
= F_MUL(f
, dy
, lambda
, dy
); /* %$\lambda (x - x')$% */
138 dy
= F_SUB(f
, dy
, dy
, a
->y
); /* %$y' = \lambda (x - x') - y$% */
149 static ec
*ecprojdbl(ec_curve
*c
, ec
*d
, const ec
*a
)
153 else if (F_ZEROP(c
->f
, a
->y
))
157 ecctx
*cc
= (ecctx
*)c
;
158 mp
*p
, *q
, *m
, *s
, *dx
, *dy
, *dz
;
160 p
= F_SQR(f
, MP_NEW
, a
->z
); /* %$z^2$% */
161 q
= F_SQR(f
, MP_NEW
, p
); /* %$z^4$% */
162 p
= F_MUL(f
, p
, q
, cc
->a
); /* %$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 ecctx
*cc
= (ecctx
*)c
;
261 dx
= F_SQR(f
, MP_NEW
, a
->x
); /* %$x_0^2$% */
262 dx
= F_TPL(f
, dx
, dx
); /* %$3 x_0^2$% */
263 dx
= F_ADD(f
, dx
, dx
, cc
->a
); /* %$3 x_0^2 + A$% */
264 dy
= F_DBL(f
, MP_NEW
, a
->y
); /* %$2 y_0$% */
265 dy
= F_INV(f
, dy
, dy
); /* %$(2 y_0)^{-1}$% */
266 lambda
= F_MUL(f
, MP_NEW
, dx
, dy
);
267 /* %$\lambda = (3 x_0^2 + A)/(2 y_0)$% */
270 dx
= F_SQR(f
, dx
, lambda
); /* %$\lambda^2$% */
271 dx
= F_SUB(f
, dx
, dx
, a
->x
); /* %$\lambda^2 - x_0$% */
272 dx
= F_SUB(f
, dx
, dx
, b
->x
); /* %$x' = \lambda^2 - x_0 - x_1$% */
273 dy
= F_SUB(f
, dy
, b
->x
, dx
); /* %$x_1 - x'$% */
274 dy
= F_MUL(f
, dy
, lambda
, dy
); /* %$\lambda (x_1 - x')$% */
275 dy
= F_SUB(f
, dy
, dy
, b
->y
); /* %$y' = \lambda (x_1 - x') - y_1$% */
286 static ec
*ecprojadd(ec_curve
*c
, ec
*d
, const ec
*a
, const ec
*b
)
289 c
->ops
->dbl(c
, d
, a
);
290 else if (EC_ATINF(a
))
292 else if (EC_ATINF(b
))
296 mp
*p
, *q
, *r
, *w
, *u
, *uu
, *s
, *ss
, *dx
, *dy
, *dz
;
298 q
= F_SQR(f
, MP_NEW
, a
->z
); /* %$z_0^2$% */
299 u
= F_MUL(f
, MP_NEW
, q
, b
->x
); /* %$u = x_1 z_0^2$% */
300 p
= F_MUL(f
, MP_NEW
, q
, b
->y
); /* %$y_1 z_0^2$% */
301 s
= F_MUL(f
, q
, p
, a
->z
); /* %$s = y_1 z_0^3$% */
303 q
= F_SQR(f
, MP_NEW
, b
->z
); /* %$z_1^2$% */
304 uu
= F_MUL(f
, MP_NEW
, q
, a
->x
); /* %$uu = x_0 z_1^2$%*/
305 p
= F_MUL(f
, p
, q
, a
->y
); /* %$y_0 z_1^2$% */
306 ss
= F_MUL(f
, q
, p
, b
->z
); /* %$ss = y_0 z_1^3$% */
308 w
= F_SUB(f
, p
, uu
, u
); /* %$w = uu - u$% */
309 r
= F_SUB(f
, MP_NEW
, ss
, s
); /* %$r = ss - s$% */
318 return (c
->ops
->dbl(c
, d
, a
));
325 u
= F_ADD(f
, u
, u
, uu
); /* %$t = uu + u$% */
326 s
= F_ADD(f
, s
, s
, ss
); /* %$m = ss + r$% */
328 uu
= F_MUL(f
, uu
, a
->z
, w
); /* %$z_0 w$% */
329 dz
= F_MUL(f
, ss
, uu
, b
->z
); /* %$z' = z_0 z_1 w$% */
331 p
= F_SQR(f
, uu
, w
); /* %$w^2$% */
332 q
= F_MUL(f
, MP_NEW
, p
, u
); /* %$t w^2$% */
333 u
= F_MUL(f
, u
, p
, w
); /* %$w^3$% */
334 p
= F_MUL(f
, p
, u
, s
); /* %$m w^3$% */
336 dx
= F_SQR(f
, u
, r
); /* %$r^2$% */
337 dx
= F_SUB(f
, dx
, dx
, q
); /* %$x' = r^2 - t w^2$% */
339 s
= F_DBL(f
, s
, dx
); /* %$2 x'$% */
340 q
= F_SUB(f
, q
, q
, s
); /* %$v = t w^2 - 2 x'$% */
341 dy
= F_MUL(f
, s
, q
, r
); /* %$v r$% */
342 dy
= F_SUB(f
, dy
, dy
, p
); /* %$v r - m w^3$% */
343 dy
= F_HLV(f
, dy
, dy
); /* %$y' = (v r - m w^3)/2$% */
357 static int eccheck(ec_curve
*c
, const ec
*p
)
359 ecctx
*cc
= (ecctx
*)c
;
362 mp
*l
= F_SQR(f
, MP_NEW
, p
->y
);
363 mp
*x
= F_SQR(f
, MP_NEW
, p
->x
);
364 mp
*r
= F_MUL(f
, MP_NEW
, x
, p
->x
);
365 x
= F_MUL(f
, x
, cc
->a
, p
->x
);
366 r
= F_ADD(f
, r
, r
, x
);
367 r
= F_ADD(f
, r
, r
, cc
->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
)
388 ecctx
*cc
= (ecctx
*)c
;
394 /* --- @ec_prime@, @ec_primeproj@ --- *
396 * Arguments: @field *f@ = the underlying field for this elliptic curve
397 * @mp *a, *b@ = the coefficients for this curve
399 * Returns: A pointer to the curve.
401 * Use: Creates a curve structure for an elliptic curve defined over
402 * a prime field. The @primeproj@ variant uses projective
403 * coordinates, which can be a win.
406 extern ec_curve
*ec_prime(field
*f
, mp
*a
, mp
*b
)
408 ecctx
*cc
= CREATE(ecctx
);
409 cc
->c
.ops
= &ec_primeops
;
411 cc
->a
= F_IN(f
, MP_NEW
, a
);
412 cc
->b
= F_IN(f
, MP_NEW
, b
);
416 extern ec_curve
*ec_primeproj(field
*f
, mp
*a
, mp
*b
)
418 ecctx
*cc
= CREATE(ecctx
);
421 ax
= mp_add(MP_NEW
, a
, MP_THREE
);
422 ax
= F_IN(f
, ax
, ax
);
424 cc
->c
.ops
= &ec_primeprojxops
;
426 cc
->c
.ops
= &ec_primeprojops
;
429 cc
->a
= F_IN(f
, MP_NEW
, a
);
430 cc
->b
= F_IN(f
, MP_NEW
, b
);
434 static const ec_ops ec_primeops
= {
435 ecdestroy
, ec_idin
, ec_idout
, ec_idfix
,
436 ecfind
, ecneg
, ecadd
, ec_stdsub
, ecdbl
, eccheck
439 static const ec_ops ec_primeprojops
= {
440 ecdestroy
, ec_projin
, ec_projout
, ec_projfix
,
441 ecfind
, ecneg
, ecprojadd
, ec_stdsub
, ecprojdbl
, ecprojcheck
444 static const ec_ops ec_primeprojxops
= {
445 ecdestroy
, ec_projin
, ec_projout
, ec_projfix
,
446 ecfind
, ecneg
, ecprojadd
, ec_stdsub
, ecprojxdbl
, ecprojcheck
449 /*----- Test rig ----------------------------------------------------------*/
453 #define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
455 int main(int argc
, char *argv
[])
459 ec g
= EC_INIT
, d
= EC_INIT
;
461 int i
, n
= argc
== 1 ?
1 : atoi(argv
[1]);
463 printf("ec-prime: ");
466 b
= MP(0x64210519e59c80e70fa7e9ab72243049feb8deecc146b9b1);
467 p
= MP(6277101735386680763835789423207666416083908700390324961279);
468 r
= MP(6277101735386680763835789423176059013767194773182842284080);
470 f
= field_niceprime(p
);
471 c
= ec_primeproj(f
, a
, b
);
473 g
.x
= MP(0x188da80eb03090f67cbf20eb43a18800f4ff0afd82ff1012);
474 g
.y
= MP(0x07192b95ffc8da78631011ed6b24cdd573f977a11e794811);
476 for (i
= 0; i
< n
; i
++) {
477 ec_mul(c
, &d
, &g
, r
);
479 fprintf(stderr
, "zero too early\n");
482 ec_add(c
, &d
, &d
, &g
);
484 fprintf(stderr
, "didn't reach zero\n");
485 MP_EPRINT("d.x", d
.x
);
486 MP_EPRINT("d.y", d
.y
);
494 MP_DROP(p
); MP_DROP(a
); MP_DROP(b
); MP_DROP(r
);
495 assert(!mparena_count(&mparena_global
));
502 /*----- That's all, folks -------------------------------------------------*/