3 * $Id: ec-prime.c,v 1.3.4.2 2004/03/20 00:13:31 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 1.3.4.2 2004/03/20 00:13:31 mdw
34 * Projective coordinates for prime curves
36 * Revision 1.3.4.1 2003/06/10 13:43:53 mdw
37 * Simple (non-projective) curves over prime fields now seem to work.
39 * Revision 1.3 2003/05/15 23:25:59 mdw
40 * Make elliptic curve stuff build.
42 * Revision 1.2 2002/01/13 13:48:44 mdw
45 * Revision 1.1 2001/04/29 18:12:33 mdw
50 /*----- Header files ------------------------------------------------------*/
56 /*----- Data structures ---------------------------------------------------*/
58 typedef struct ecctx
{
63 /*----- Simple prime curves -----------------------------------------------*/
65 static const ec_ops ec_primeops
, ec_primeprojops
, ec_primeprojxops
;
67 static ec
*ecneg(ec_curve
*c
, ec
*d
, const ec
*p
)
70 d
->y
= F_NEG(c
->f
, d
->y
, d
->y
);
74 static ec
*ecfind(ec_curve
*c
, ec
*d
, mp
*x
)
77 ecctx
*cc
= (ecctx
*)c
;
80 q
= F_SQR(f
, MP_NEW
, x
);
81 p
= F_MUL(f
, MP_NEW
, x
, q
);
82 q
= F_MUL(f
, q
, x
, cc
->a
);
83 p
= F_ADD(f
, p
, p
, q
);
84 p
= F_ADD(f
, p
, p
, cc
->b
);
92 d
->z
= MP_COPY(f
->one
);
96 static ec
*ecdbl(ec_curve
*c
, ec
*d
, const ec
*a
)
100 else if (F_ZEROP(c
->f
, a
->y
))
104 ecctx
*cc
= (ecctx
*)c
;
108 dx
= F_SQR(f
, MP_NEW
, a
->x
); /* %$x^2$% */
109 dy
= F_DBL(f
, MP_NEW
, a
->y
); /* %$2 y$% */
110 dx
= F_TPL(f
, dx
, dx
); /* %$3 x^2$% */
111 dx
= F_ADD(f
, dx
, dx
, cc
->a
); /* %$3 x^2 + A$% */
112 dy
= F_INV(f
, dy
, dy
); /* %$(2 y)^{-1}$% */
113 lambda
= F_MUL(f
, MP_NEW
, dx
, dy
); /* %$\lambda = (3 x^2 + A)/(2 y)$% */
115 dx
= F_SQR(f
, dx
, lambda
); /* %$\lambda^2$% */
116 dy
= F_DBL(f
, dy
, a
->x
); /* %$2 x$% */
117 dx
= F_SUB(f
, dx
, dx
, dy
); /* %$x' = \lambda^2 - 2 x */
118 dy
= F_SUB(f
, dy
, a
->x
, dx
); /* %$x - x'$% */
119 dy
= F_MUL(f
, dy
, lambda
, dy
); /* %$\lambda (x - x')$% */
120 dy
= F_SUB(f
, dy
, dy
, a
->y
); /* %$y' = \lambda (x - x') - y$% */
131 static ec
*ecprojdbl(ec_curve
*c
, ec
*d
, const ec
*a
)
135 else if (F_ZEROP(c
->f
, a
->y
))
139 ecctx
*cc
= (ecctx
*)c
;
140 mp
*p
, *q
, *m
, *s
, *dx
, *dy
, *dz
;
142 p
= F_SQR(f
, MP_NEW
, a
->z
); /* %$z^2$% */
143 q
= F_SQR(f
, MP_NEW
, p
); /* %$z^4$% */
144 p
= F_MUL(f
, p
, q
, cc
->a
); /* %$A z^4$% */
145 m
= F_SQR(f
, MP_NEW
, a
->x
); /* %$x^2$% */
146 m
= F_TPL(f
, m
, m
); /* %$3 x^2$% */
147 m
= F_ADD(f
, m
, m
, p
); /* %$m = 3 x^2 + A z^4$% */
149 q
= F_DBL(f
, q
, a
->y
); /* %$2 y$% */
150 dz
= F_MUL(f
, MP_NEW
, q
, a
->z
); /* %$z' = 2 y z$% */
152 p
= F_SQR(f
, p
, q
); /* %$4 y^2$% */
153 s
= F_MUL(f
, MP_NEW
, p
, a
->x
); /* %$s = 4 x y^2$% */
154 q
= F_SQR(f
, q
, p
); /* %$16 y^4$% */
155 q
= F_HLV(f
, q
, q
); /* %$t = 8 y^4$% */
157 p
= F_DBL(f
, p
, s
); /* %$2 s$% */
158 dx
= F_SQR(f
, MP_NEW
, m
); /* %$m^2$% */
159 dx
= F_SUB(f
, dx
, dx
, p
); /* %$x' = m^2 - 2 s$% */
161 s
= F_SUB(f
, s
, s
, dx
); /* %$s - x'$% */
162 dy
= F_MUL(f
, p
, m
, s
); /* %$m (s - x')$% */
163 dy
= F_SUB(f
, dy
, dy
, q
); /* %$y' = m (s - x') - t$% */
176 static ec
*ecprojxdbl(ec_curve
*c
, ec
*d
, const ec
*a
)
180 else if (F_ZEROP(c
->f
, a
->y
))
184 mp
*p
, *q
, *m
, *s
, *dx
, *dy
, *dz
;
186 m
= F_SQR(f
, MP_NEW
, a
->z
); /* %$z^2$% */
187 p
= F_SUB(f
, MP_NEW
, a
->x
, m
); /* %$x - z^2$% */
188 q
= F_ADD(f
, MP_NEW
, a
->x
, m
); /* %$x + z^2$% */
189 m
= F_MUL(f
, m
, p
, q
); /* %$x^2 - z^4$% */
190 m
= F_TPL(f
, m
, m
); /* %$m = 3 x^2 - 3 z^4$% */
192 q
= F_DBL(f
, q
, a
->y
); /* %$2 y$% */
193 dz
= F_MUL(f
, MP_NEW
, q
, a
->z
); /* %$z' = 2 y z$% */
195 p
= F_SQR(f
, p
, q
); /* %$4 y^2$% */
196 s
= F_MUL(f
, MP_NEW
, p
, a
->x
); /* %$s = 4 x y^2$% */
197 q
= F_SQR(f
, q
, p
); /* %$16 y^4$% */
198 q
= F_HLV(f
, q
, q
); /* %$t = 8 y^4$% */
200 p
= F_DBL(f
, p
, s
); /* %$2 s$% */
201 dx
= F_SQR(f
, MP_NEW
, m
); /* %$m^2$% */
202 dx
= F_SUB(f
, dx
, dx
, p
); /* %$x' = m^2 - 2 s$% */
204 s
= F_SUB(f
, s
, s
, dx
); /* %$s - x'$% */
205 dy
= F_MUL(f
, p
, m
, s
); /* %$m (s - x')$% */
206 dy
= F_SUB(f
, dy
, dy
, q
); /* %$y' = m (s - x') - t$% */
219 static ec
*ecadd(ec_curve
*c
, ec
*d
, const ec
*a
, const ec
*b
)
223 else if (EC_ATINF(a
))
225 else if (EC_ATINF(b
))
232 if (!MP_EQ(a
->x
, b
->x
)) {
233 dy
= F_SUB(f
, MP_NEW
, a
->y
, b
->y
); /* %$y_0 - y_1$% */
234 dx
= F_SUB(f
, MP_NEW
, a
->x
, b
->x
); /* %$x_0 - x_1$% */
235 dx
= F_INV(f
, dx
, dx
); /* %$(x_0 - x_1)^{-1}$% */
236 lambda
= F_MUL(f
, MP_NEW
, dy
, dx
);
237 /* %$\lambda = (y_0 - y1)/(x_0 - x_1)$% */
238 } else if (F_ZEROP(c
->f
, a
->y
) || !MP_EQ(a
->y
, b
->y
)) {
242 ecctx
*cc
= (ecctx
*)c
;
243 dx
= F_SQR(f
, MP_NEW
, a
->x
); /* %$x_0^2$% */
244 dx
= F_TPL(f
, dx
, dx
); /* %$3 x_0^2$% */
245 dx
= F_ADD(f
, dx
, dx
, cc
->a
); /* %$3 x_0^2 + A$% */
246 dy
= F_DBL(f
, MP_NEW
, a
->y
); /* %$2 y_0$% */
247 dy
= F_INV(f
, dy
, dy
); /* %$(2 y_0)^{-1}$% */
248 lambda
= F_MUL(f
, MP_NEW
, dx
, dy
);
249 /* %$\lambda = (3 x_0^2 + A)/(2 y_0)$% */
252 dx
= F_SQR(f
, dx
, lambda
); /* %$\lambda^2$% */
253 dx
= F_SUB(f
, dx
, dx
, a
->x
); /* %$\lambda^2 - x_0$% */
254 dx
= F_SUB(f
, dx
, dx
, b
->x
); /* %$x' = \lambda^2 - x_0 - x_1$% */
255 dy
= F_SUB(f
, dy
, b
->x
, dx
); /* %$x_1 - x'$% */
256 dy
= F_MUL(f
, dy
, lambda
, dy
); /* %$\lambda (x_1 - x')$% */
257 dy
= F_SUB(f
, dy
, dy
, b
->y
);
258 /* %$y' = \lambda (x_1 - x') - y_1$% */
269 static ec
*ecprojadd(ec_curve
*c
, ec
*d
, const ec
*a
, const ec
*b
)
272 c
->ops
->dbl(c
, d
, a
);
273 else if (EC_ATINF(a
))
275 else if (EC_ATINF(b
))
279 mp
*p
, *q
, *r
, *w
, *u
, *s
, *dx
, *dy
, *dz
;
281 q
= F_SQR(f
, MP_NEW
, a
->z
); /* %$z_0^2$% */
282 u
= F_MUL(f
, MP_NEW
, q
, b
->x
); /* %$u = x_1 z_0^2$% */
283 p
= F_MUL(f
, MP_NEW
, q
, b
->y
); /* %$y_1 z_0^2$% */
284 s
= F_MUL(f
, q
, p
, a
->z
); /* %$s = y_1 z_0^3$% */
286 w
= F_SUB(f
, p
, a
->x
, u
); /* %$w = x_0 - u$% */
287 r
= F_SUB(f
, MP_NEW
, a
->y
, s
); /* %$r = y_0 - s$% */
294 return (c
->ops
->dbl(c
, d
, a
));
304 u
= F_ADD(f
, u
, u
, a
->x
); /* %$t = x_0 + u$% */
305 s
= F_ADD(f
, s
, s
, a
->y
); /* %$m = y_0 + r$% */
307 dz
= F_MUL(f
, MP_NEW
, a
->z
, w
); /* %$z' = z_0 w$% */
309 p
= F_SQR(f
, MP_NEW
, w
); /* %$w^2$% */
310 q
= F_MUL(f
, MP_NEW
, p
, u
); /* %$t w^2$% */
311 u
= F_MUL(f
, u
, p
, w
); /* %$w^3$% */
312 p
= F_MUL(f
, p
, u
, s
); /* %$m w^3$% */
314 dx
= F_SQR(f
, u
, r
); /* %$r^2$% */
315 dx
= F_SUB(f
, dx
, dx
, q
); /* %$x' = r^2 - t w^2$% */
317 s
= F_DBL(f
, s
, dx
); /* %$2 x'$% */
318 q
= F_SUB(f
, q
, q
, s
); /* %$v = t w^2 - 2 x'$% */
319 dy
= F_MUL(f
, s
, q
, r
); /* %$v r$% */
320 dy
= F_SUB(f
, dy
, dy
, p
); /* %$v r - m w^3$% */
321 dy
= F_HLV(f
, dy
, dy
); /* %$y' = (v r - m w^3)/2$% */
335 static int eccheck(ec_curve
*c
, const ec
*p
)
337 ecctx
*cc
= (ecctx
*)c
;
340 mp
*l
= F_SQR(f
, MP_NEW
, p
->y
);
341 mp
*x
= F_SQR(f
, MP_NEW
, p
->x
);
342 mp
*r
= F_MUL(f
, MP_NEW
, x
, p
->x
);
343 x
= F_MUL(f
, x
, cc
->a
, p
->x
);
344 r
= F_ADD(f
, r
, r
, x
);
345 r
= F_ADD(f
, r
, r
, cc
->b
);
346 rc
= MP_EQ(l
, r
) ?
0 : -1;
353 static int ecprojcheck(ec_curve
*c
, const ec
*p
)
358 c
->ops
->fix(c
, &t
, p
);
364 static void ecdestroy(ec_curve
*c
)
366 ecctx
*cc
= (ecctx
*)c
;
372 /* --- @ec_prime@, @ec_primeproj@ --- *
374 * Arguments: @field *f@ = the underlying field for this elliptic curve
375 * @mp *a, *b@ = the coefficients for this curve
377 * Returns: A pointer to the curve.
379 * Use: Creates a curve structure for an elliptic curve defined over
380 * a prime field. The @primeproj@ variant uses projective
381 * coordinates, which can be a win.
384 extern ec_curve
*ec_prime(field
*f
, mp
*a
, mp
*b
)
386 ecctx
*cc
= CREATE(ecctx
);
387 cc
->c
.ops
= &ec_primeops
;
389 cc
->a
= F_IN(f
, MP_NEW
, a
);
390 cc
->b
= F_IN(f
, MP_NEW
, b
);
394 extern ec_curve
*ec_primeproj(field
*f
, mp
*a
, mp
*b
)
396 ecctx
*cc
= CREATE(ecctx
);
399 ax
= mp_add(MP_NEW
, a
, MP_THREE
);
400 ax
= F_IN(f
, ax
, ax
);
402 cc
->c
.ops
= &ec_primeprojxops
;
404 cc
->c
.ops
= &ec_primeprojops
;
407 cc
->a
= F_IN(f
, MP_NEW
, a
);
408 cc
->b
= F_IN(f
, MP_NEW
, b
);
412 static const ec_ops ec_primeops
= {
413 ecdestroy
, ec_idin
, ec_idout
, ec_idfix
,
414 0, ecneg
, ecadd
, ec_stdsub
, ecdbl
, eccheck
417 static const ec_ops ec_primeprojops
= {
418 ecdestroy
, ec_projin
, ec_projout
, ec_projfix
,
419 0, ecneg
, ecprojadd
, ec_stdsub
, ecprojdbl
, ecprojcheck
422 static const ec_ops ec_primeprojxops
= {
423 ecdestroy
, ec_projin
, ec_projout
, ec_projfix
,
424 0, ecneg
, ecprojadd
, ec_stdsub
, ecprojxdbl
, ecprojcheck
427 /*----- Test rig ----------------------------------------------------------*/
431 #define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
437 ec g
= EC_INIT
, d
= EC_INIT
;
440 printf("ec-prime: ");
443 b
= MP(0x64210519e59c80e70fa7e9ab72243049feb8deecc146b9b1);
444 p
= MP(6277101735386680763835789423207666416083908700390324961279);
445 r
= MP(6277101735386680763835789423176059013767194773182842284080);
448 c
= ec_prime(f
, a
, b
);
450 g
.x
= MP(0x188da80eb03090f67cbf20eb43a18800f4ff0afd82ff1012);
451 g
.y
= MP(0x07192b95ffc8da78631011ed6b24cdd573f977a11e794811);
453 ec_mul(c
, &d
, &g
, r
);
455 fprintf(stderr
, "zero too early\n");
458 ec_add(c
, &d
, &d
, &g
);
460 fprintf(stderr
, "didn't reach zero\n");
461 MP_EPRINT("d.x", d
.x
);
462 MP_EPRINT("d.y", d
.y
);
470 MP_DROP(p
); MP_DROP(a
); MP_DROP(b
); MP_DROP(r
);
471 assert(!mparena_count(&mparena_global
));
478 /*----- That's all, folks -------------------------------------------------*/