ce81ba17b55ccd461ab121f338fe24d434e1182e
3 * $Id: ec-prime.c,v 1.8 2004/03/27 17:54:11 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 1.8 2004/03/27 17:54:11 mdw
34 * Standard curves and curve checking.
36 * Revision 1.7 2004/03/27 00:04:46 mdw
37 * Implement efficient reduction for pleasant-looking primes.
39 * Revision 1.6 2004/03/23 15:19:32 mdw
40 * Test elliptic curves more thoroughly.
42 * Revision 1.5 2004/03/22 02:19:10 mdw
43 * Rationalise the sliding-window threshold. Drop guarantee that right
44 * arguments to EC @add@ are canonical, and fix up projective implementations
47 * Revision 1.4 2004/03/21 22:52:06 mdw
48 * Merge and close elliptic curve branch.
50 * Revision 1.3.4.3 2004/03/21 22:39:46 mdw
51 * Elliptic curves on binary fields work.
53 * Revision 1.3.4.2 2004/03/20 00:13:31 mdw
54 * Projective coordinates for prime curves
56 * Revision 1.3.4.1 2003/06/10 13:43:53 mdw
57 * Simple (non-projective) curves over prime fields now seem to work.
59 * Revision 1.3 2003/05/15 23:25:59 mdw
60 * Make elliptic curve stuff build.
62 * Revision 1.2 2002/01/13 13:48:44 mdw
65 * Revision 1.1 2001/04/29 18:12:33 mdw
70 /*----- Header files ------------------------------------------------------*/
76 /*----- Simple prime curves -----------------------------------------------*/
78 static const ec_ops ec_primeops
, ec_primeprojops
, ec_primeprojxops
;
80 static ec
*ecneg(ec_curve
*c
, ec
*d
, const ec
*p
)
84 d
->y
= F_NEG(c
->f
, d
->y
, d
->y
);
88 static ec
*ecfind(ec_curve
*c
, ec
*d
, mp
*x
)
93 q
= F_SQR(f
, MP_NEW
, x
);
94 p
= F_MUL(f
, MP_NEW
, x
, q
);
95 q
= F_MUL(f
, q
, x
, c
->a
);
96 p
= F_ADD(f
, p
, p
, q
);
97 p
= F_ADD(f
, p
, p
, c
->b
);
105 d
->z
= MP_COPY(f
->one
);
109 static ec
*ecdbl(ec_curve
*c
, ec
*d
, const ec
*a
)
113 else if (F_ZEROP(c
->f
, a
->y
))
120 dx
= F_SQR(f
, MP_NEW
, a
->x
); /* %$x^2$% */
121 dy
= F_DBL(f
, MP_NEW
, a
->y
); /* %$2 y$% */
122 dx
= F_TPL(f
, dx
, dx
); /* %$3 x^2$% */
123 dx
= F_ADD(f
, dx
, dx
, c
->a
); /* %$3 x^2 + A$% */
124 dy
= F_INV(f
, dy
, dy
); /* %$(2 y)^{-1}$% */
125 lambda
= F_MUL(f
, MP_NEW
, dx
, dy
); /* %$\lambda = (3 x^2 + A)/(2 y)$% */
127 dx
= F_SQR(f
, dx
, lambda
); /* %$\lambda^2$% */
128 dy
= F_DBL(f
, dy
, a
->x
); /* %$2 x$% */
129 dx
= F_SUB(f
, dx
, dx
, dy
); /* %$x' = \lambda^2 - 2 x */
130 dy
= F_SUB(f
, dy
, a
->x
, dx
); /* %$x - x'$% */
131 dy
= F_MUL(f
, dy
, lambda
, dy
); /* %$\lambda (x - x')$% */
132 dy
= F_SUB(f
, dy
, dy
, a
->y
); /* %$y' = \lambda (x - x') - y$% */
143 static ec
*ecprojdbl(ec_curve
*c
, ec
*d
, const ec
*a
)
147 else if (F_ZEROP(c
->f
, a
->y
))
151 mp
*p
, *q
, *m
, *s
, *dx
, *dy
, *dz
;
153 p
= F_SQR(f
, MP_NEW
, a
->z
); /* %$z^2$% */
154 q
= F_SQR(f
, MP_NEW
, p
); /* %$z^4$% */
155 p
= F_MUL(f
, p
, q
, c
->a
); /* %$A z^4$% */
156 m
= F_SQR(f
, MP_NEW
, a
->x
); /* %$x^2$% */
157 m
= F_TPL(f
, m
, m
); /* %$3 x^2$% */
158 m
= F_ADD(f
, m
, m
, p
); /* %$m = 3 x^2 + A z^4$% */
160 q
= F_DBL(f
, q
, a
->y
); /* %$2 y$% */
161 dz
= F_MUL(f
, MP_NEW
, q
, a
->z
); /* %$z' = 2 y z$% */
163 p
= F_SQR(f
, p
, q
); /* %$4 y^2$% */
164 s
= F_MUL(f
, MP_NEW
, p
, a
->x
); /* %$s = 4 x y^2$% */
165 q
= F_SQR(f
, q
, p
); /* %$16 y^4$% */
166 q
= F_HLV(f
, q
, q
); /* %$t = 8 y^4$% */
168 p
= F_DBL(f
, p
, s
); /* %$2 s$% */
169 dx
= F_SQR(f
, MP_NEW
, m
); /* %$m^2$% */
170 dx
= F_SUB(f
, dx
, dx
, p
); /* %$x' = m^2 - 2 s$% */
172 s
= F_SUB(f
, s
, s
, dx
); /* %$s - x'$% */
173 dy
= F_MUL(f
, p
, m
, s
); /* %$m (s - x')$% */
174 dy
= F_SUB(f
, dy
, dy
, q
); /* %$y' = m (s - x') - t$% */
187 static ec
*ecprojxdbl(ec_curve
*c
, ec
*d
, const ec
*a
)
191 else if (F_ZEROP(c
->f
, a
->y
))
195 mp
*p
, *q
, *m
, *s
, *dx
, *dy
, *dz
;
197 m
= F_SQR(f
, MP_NEW
, a
->z
); /* %$z^2$% */
198 p
= F_SUB(f
, MP_NEW
, a
->x
, m
); /* %$x - z^2$% */
199 q
= F_ADD(f
, MP_NEW
, a
->x
, m
); /* %$x + z^2$% */
200 m
= F_MUL(f
, m
, p
, q
); /* %$x^2 - z^4$% */
201 m
= F_TPL(f
, m
, m
); /* %$m = 3 x^2 - 3 z^4$% */
203 q
= F_DBL(f
, q
, a
->y
); /* %$2 y$% */
204 dz
= F_MUL(f
, MP_NEW
, q
, a
->z
); /* %$z' = 2 y z$% */
206 p
= F_SQR(f
, p
, q
); /* %$4 y^2$% */
207 s
= F_MUL(f
, MP_NEW
, p
, a
->x
); /* %$s = 4 x y^2$% */
208 q
= F_SQR(f
, q
, p
); /* %$16 y^4$% */
209 q
= F_HLV(f
, q
, q
); /* %$t = 8 y^4$% */
211 p
= F_DBL(f
, p
, s
); /* %$2 s$% */
212 dx
= F_SQR(f
, MP_NEW
, m
); /* %$m^2$% */
213 dx
= F_SUB(f
, dx
, dx
, p
); /* %$x' = m^2 - 2 s$% */
215 s
= F_SUB(f
, s
, s
, dx
); /* %$s - x'$% */
216 dy
= F_MUL(f
, p
, m
, s
); /* %$m (s - x')$% */
217 dy
= F_SUB(f
, dy
, dy
, q
); /* %$y' = m (s - x') - t$% */
230 static ec
*ecadd(ec_curve
*c
, ec
*d
, const ec
*a
, const ec
*b
)
234 else if (EC_ATINF(a
))
236 else if (EC_ATINF(b
))
243 if (!MP_EQ(a
->x
, b
->x
)) {
244 dy
= F_SUB(f
, MP_NEW
, a
->y
, b
->y
); /* %$y_0 - y_1$% */
245 dx
= F_SUB(f
, MP_NEW
, a
->x
, b
->x
); /* %$x_0 - x_1$% */
246 dx
= F_INV(f
, dx
, dx
); /* %$(x_0 - x_1)^{-1}$% */
247 lambda
= F_MUL(f
, MP_NEW
, dy
, dx
);
248 /* %$\lambda = (y_0 - y1)/(x_0 - x_1)$% */
249 } else if (F_ZEROP(c
->f
, a
->y
) || !MP_EQ(a
->y
, b
->y
)) {
253 dx
= F_SQR(f
, MP_NEW
, a
->x
); /* %$x_0^2$% */
254 dx
= F_TPL(f
, dx
, dx
); /* %$3 x_0^2$% */
255 dx
= F_ADD(f
, dx
, dx
, c
->a
); /* %$3 x_0^2 + A$% */
256 dy
= F_DBL(f
, MP_NEW
, a
->y
); /* %$2 y_0$% */
257 dy
= F_INV(f
, dy
, dy
); /* %$(2 y_0)^{-1}$% */
258 lambda
= F_MUL(f
, MP_NEW
, dx
, dy
);
259 /* %$\lambda = (3 x_0^2 + A)/(2 y_0)$% */
262 dx
= F_SQR(f
, dx
, lambda
); /* %$\lambda^2$% */
263 dx
= F_SUB(f
, dx
, dx
, a
->x
); /* %$\lambda^2 - x_0$% */
264 dx
= F_SUB(f
, dx
, dx
, b
->x
); /* %$x' = \lambda^2 - x_0 - x_1$% */
265 dy
= F_SUB(f
, dy
, b
->x
, dx
); /* %$x_1 - x'$% */
266 dy
= F_MUL(f
, dy
, lambda
, dy
); /* %$\lambda (x_1 - x')$% */
267 dy
= F_SUB(f
, dy
, dy
, b
->y
); /* %$y' = \lambda (x_1 - x') - y_1$% */
278 static ec
*ecprojadd(ec_curve
*c
, ec
*d
, const ec
*a
, const ec
*b
)
281 c
->ops
->dbl(c
, d
, a
);
282 else if (EC_ATINF(a
))
284 else if (EC_ATINF(b
))
288 mp
*p
, *q
, *r
, *w
, *u
, *uu
, *s
, *ss
, *dx
, *dy
, *dz
;
290 q
= F_SQR(f
, MP_NEW
, a
->z
); /* %$z_0^2$% */
291 u
= F_MUL(f
, MP_NEW
, q
, b
->x
); /* %$u = x_1 z_0^2$% */
292 p
= F_MUL(f
, MP_NEW
, q
, b
->y
); /* %$y_1 z_0^2$% */
293 s
= F_MUL(f
, q
, p
, a
->z
); /* %$s = y_1 z_0^3$% */
295 q
= F_SQR(f
, MP_NEW
, b
->z
); /* %$z_1^2$% */
296 uu
= F_MUL(f
, MP_NEW
, q
, a
->x
); /* %$uu = x_0 z_1^2$%*/
297 p
= F_MUL(f
, p
, q
, a
->y
); /* %$y_0 z_1^2$% */
298 ss
= F_MUL(f
, q
, p
, b
->z
); /* %$ss = y_0 z_1^3$% */
300 w
= F_SUB(f
, p
, uu
, u
); /* %$w = uu - u$% */
301 r
= F_SUB(f
, MP_NEW
, ss
, s
); /* %$r = ss - s$% */
310 return (c
->ops
->dbl(c
, d
, a
));
317 u
= F_ADD(f
, u
, u
, uu
); /* %$t = uu + u$% */
318 s
= F_ADD(f
, s
, s
, ss
); /* %$m = ss + r$% */
320 uu
= F_MUL(f
, uu
, a
->z
, w
); /* %$z_0 w$% */
321 dz
= F_MUL(f
, ss
, uu
, b
->z
); /* %$z' = z_0 z_1 w$% */
323 p
= F_SQR(f
, uu
, w
); /* %$w^2$% */
324 q
= F_MUL(f
, MP_NEW
, p
, u
); /* %$t w^2$% */
325 u
= F_MUL(f
, u
, p
, w
); /* %$w^3$% */
326 p
= F_MUL(f
, p
, u
, s
); /* %$m w^3$% */
328 dx
= F_SQR(f
, u
, r
); /* %$r^2$% */
329 dx
= F_SUB(f
, dx
, dx
, q
); /* %$x' = r^2 - t w^2$% */
331 s
= F_DBL(f
, s
, dx
); /* %$2 x'$% */
332 q
= F_SUB(f
, q
, q
, s
); /* %$v = t w^2 - 2 x'$% */
333 dy
= F_MUL(f
, s
, q
, r
); /* %$v r$% */
334 dy
= F_SUB(f
, dy
, dy
, p
); /* %$v r - m w^3$% */
335 dy
= F_HLV(f
, dy
, dy
); /* %$y' = (v r - m w^3)/2$% */
349 static int eccheck(ec_curve
*c
, const ec
*p
)
353 mp
*l
= F_SQR(f
, MP_NEW
, p
->y
);
354 mp
*x
= F_SQR(f
, MP_NEW
, p
->x
);
355 mp
*r
= F_MUL(f
, MP_NEW
, x
, p
->x
);
356 x
= F_MUL(f
, x
, c
->a
, p
->x
);
357 r
= F_ADD(f
, r
, r
, x
);
358 r
= F_ADD(f
, r
, r
, c
->b
);
359 rc
= MP_EQ(l
, r
) ?
0 : -1;
366 static int ecprojcheck(ec_curve
*c
, const ec
*p
)
371 c
->ops
->fix(c
, &t
, p
);
377 static void ecdestroy(ec_curve
*c
)
384 /* --- @ec_prime@, @ec_primeproj@ --- *
386 * Arguments: @field *f@ = the underlying field for this elliptic curve
387 * @mp *a, *b@ = the coefficients for this curve
389 * Returns: A pointer to the curve.
391 * Use: Creates a curve structure for an elliptic curve defined over
392 * a prime field. The @primeproj@ variant uses projective
393 * coordinates, which can be a win.
396 extern ec_curve
*ec_prime(field
*f
, mp
*a
, mp
*b
)
398 ec_curve
*c
= CREATE(ec_curve
);
399 c
->ops
= &ec_primeops
;
401 c
->a
= F_IN(f
, MP_NEW
, a
);
402 c
->b
= F_IN(f
, MP_NEW
, b
);
406 extern ec_curve
*ec_primeproj(field
*f
, mp
*a
, mp
*b
)
408 ec_curve
*c
= CREATE(ec_curve
);
411 ax
= mp_add(MP_NEW
, a
, MP_THREE
);
412 ax
= F_IN(f
, ax
, ax
);
414 c
->ops
= &ec_primeprojxops
;
416 c
->ops
= &ec_primeprojops
;
419 c
->a
= F_IN(f
, MP_NEW
, a
);
420 c
->b
= F_IN(f
, MP_NEW
, b
);
424 static const ec_ops ec_primeops
= {
425 ecdestroy
, ec_idin
, ec_idout
, ec_idfix
,
426 ecfind
, ecneg
, ecadd
, ec_stdsub
, ecdbl
, eccheck
429 static const ec_ops ec_primeprojops
= {
430 ecdestroy
, ec_projin
, ec_projout
, ec_projfix
,
431 ecfind
, ecneg
, ecprojadd
, ec_stdsub
, ecprojdbl
, ecprojcheck
434 static const ec_ops ec_primeprojxops
= {
435 ecdestroy
, ec_projin
, ec_projout
, ec_projfix
,
436 ecfind
, ecneg
, ecprojadd
, ec_stdsub
, ecprojxdbl
, ecprojcheck
439 /*----- Test rig ----------------------------------------------------------*/
443 #define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
445 int main(int argc
, char *argv
[])
449 ec g
= EC_INIT
, d
= EC_INIT
;
451 int i
, n
= argc
== 1 ?
1 : atoi(argv
[1]);
453 printf("ec-prime: ");
456 b
= MP(0xb3312fa7e23ee7e4988e056be3f82d19181d9c6efe8141120314088f5013875ac656398d8a2ed19d2a85c8edd3ec2aef);
457 p
= MP(39402006196394479212279040100143613805079739270465446667948293404245721771496870329047266088258938001861606973112319);
458 r
= MP(39402006196394479212279040100143613805079739270465446667946905279627659399113263569398956308152294913554433653942642);
460 f
= field_niceprime(p
);
461 c
= ec_primeproj(f
, a
, b
);
463 g
.x
= MP(0xaa87ca22be8b05378eb1c71ef320ad746e1d3b628ba79b9859f741e082542a385502f25dbf55296c3a545e3872760ab7);
464 g
.y
= MP(0x3617de4a96262c6f5d9e98bf9292dc29f8f41dbd289a147ce9da3113b5f0b8c00a60b1ce1d7e819d7a431d7c90ea0e5f);
466 for (i
= 0; i
< n
; i
++) {
467 ec_mul(c
, &d
, &g
, r
);
469 fprintf(stderr
, "zero too early\n");
472 ec_add(c
, &d
, &d
, &g
);
474 fprintf(stderr
, "didn't reach zero\n");
475 MP_EPRINT("d.x", d
.x
);
476 MP_EPRINT("d.y", d
.y
);
484 MP_DROP(p
); MP_DROP(a
); MP_DROP(b
); MP_DROP(r
);
485 assert(!mparena_count(&mparena_global
));
492 /*----- That's all, folks -------------------------------------------------*/