3 * $Id: ec-prime.c,v 1.11 2004/04/08 01:36:15 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 /*----- Header files ------------------------------------------------------*/
36 /*----- Simple prime curves -----------------------------------------------*/
38 static const ec_ops ec_primeops
, ec_primeprojops
, ec_primeprojxops
;
40 static ec
*ecneg(ec_curve
*c
, ec
*d
, const ec
*p
)
44 d
->y
= F_NEG(c
->f
, d
->y
, d
->y
);
48 static ec
*ecfind(ec_curve
*c
, ec
*d
, mp
*x
)
53 q
= F_SQR(f
, MP_NEW
, x
);
54 p
= F_MUL(f
, MP_NEW
, x
, q
);
55 q
= F_MUL(f
, q
, x
, c
->a
);
56 p
= F_ADD(f
, p
, p
, q
);
57 p
= F_ADD(f
, p
, p
, c
->b
);
65 d
->z
= MP_COPY(f
->one
);
69 static ec
*ecdbl(ec_curve
*c
, ec
*d
, const ec
*a
)
73 else if (F_ZEROP(c
->f
, a
->y
))
80 dx
= F_SQR(f
, MP_NEW
, a
->x
); /* %$x^2$% */
81 dy
= F_DBL(f
, MP_NEW
, a
->y
); /* %$2 y$% */
82 dx
= F_TPL(f
, dx
, dx
); /* %$3 x^2$% */
83 dx
= F_ADD(f
, dx
, dx
, c
->a
); /* %$3 x^2 + A$% */
84 dy
= F_INV(f
, dy
, dy
); /* %$(2 y)^{-1}$% */
85 lambda
= F_MUL(f
, MP_NEW
, dx
, dy
); /* %$\lambda = (3 x^2 + A)/(2 y)$% */
87 dx
= F_SQR(f
, dx
, lambda
); /* %$\lambda^2$% */
88 dy
= F_DBL(f
, dy
, a
->x
); /* %$2 x$% */
89 dx
= F_SUB(f
, dx
, dx
, dy
); /* %$x' = \lambda^2 - 2 x */
90 dy
= F_SUB(f
, dy
, a
->x
, dx
); /* %$x - x'$% */
91 dy
= F_MUL(f
, dy
, lambda
, dy
); /* %$\lambda (x - x')$% */
92 dy
= F_SUB(f
, dy
, dy
, a
->y
); /* %$y' = \lambda (x - x') - y$% */
103 static ec
*ecprojdbl(ec_curve
*c
, ec
*d
, const ec
*a
)
107 else if (F_ZEROP(c
->f
, a
->y
))
111 mp
*p
, *q
, *m
, *s
, *dx
, *dy
, *dz
;
113 p
= F_SQR(f
, MP_NEW
, a
->z
); /* %$z^2$% */
114 q
= F_SQR(f
, MP_NEW
, p
); /* %$z^4$% */
115 p
= F_MUL(f
, p
, q
, c
->a
); /* %$A z^4$% */
116 m
= F_SQR(f
, MP_NEW
, a
->x
); /* %$x^2$% */
117 m
= F_TPL(f
, m
, m
); /* %$3 x^2$% */
118 m
= F_ADD(f
, m
, m
, p
); /* %$m = 3 x^2 + A z^4$% */
120 q
= F_DBL(f
, q
, a
->y
); /* %$2 y$% */
121 dz
= F_MUL(f
, MP_NEW
, q
, a
->z
); /* %$z' = 2 y z$% */
123 p
= F_SQR(f
, p
, q
); /* %$4 y^2$% */
124 s
= F_MUL(f
, MP_NEW
, p
, a
->x
); /* %$s = 4 x y^2$% */
125 q
= F_SQR(f
, q
, p
); /* %$16 y^4$% */
126 q
= F_HLV(f
, q
, q
); /* %$t = 8 y^4$% */
128 p
= F_DBL(f
, p
, s
); /* %$2 s$% */
129 dx
= F_SQR(f
, MP_NEW
, m
); /* %$m^2$% */
130 dx
= F_SUB(f
, dx
, dx
, p
); /* %$x' = m^2 - 2 s$% */
132 s
= F_SUB(f
, s
, s
, dx
); /* %$s - x'$% */
133 dy
= F_MUL(f
, p
, m
, s
); /* %$m (s - x')$% */
134 dy
= F_SUB(f
, dy
, dy
, q
); /* %$y' = m (s - x') - t$% */
147 static ec
*ecprojxdbl(ec_curve
*c
, ec
*d
, const ec
*a
)
151 else if (F_ZEROP(c
->f
, a
->y
))
155 mp
*p
, *q
, *m
, *s
, *dx
, *dy
, *dz
;
157 m
= F_SQR(f
, MP_NEW
, a
->z
); /* %$z^2$% */
158 p
= F_SUB(f
, MP_NEW
, a
->x
, m
); /* %$x - z^2$% */
159 q
= F_ADD(f
, MP_NEW
, a
->x
, m
); /* %$x + z^2$% */
160 m
= F_MUL(f
, m
, p
, q
); /* %$x^2 - z^4$% */
161 m
= F_TPL(f
, m
, m
); /* %$m = 3 x^2 - 3 z^4$% */
163 q
= F_DBL(f
, q
, a
->y
); /* %$2 y$% */
164 dz
= F_MUL(f
, MP_NEW
, q
, a
->z
); /* %$z' = 2 y z$% */
166 p
= F_SQR(f
, p
, q
); /* %$4 y^2$% */
167 s
= F_MUL(f
, MP_NEW
, p
, a
->x
); /* %$s = 4 x y^2$% */
168 q
= F_SQR(f
, q
, p
); /* %$16 y^4$% */
169 q
= F_HLV(f
, q
, q
); /* %$t = 8 y^4$% */
171 p
= F_DBL(f
, p
, s
); /* %$2 s$% */
172 dx
= F_SQR(f
, MP_NEW
, m
); /* %$m^2$% */
173 dx
= F_SUB(f
, dx
, dx
, p
); /* %$x' = m^2 - 2 s$% */
175 s
= F_SUB(f
, s
, s
, dx
); /* %$s - x'$% */
176 dy
= F_MUL(f
, p
, m
, s
); /* %$m (s - x')$% */
177 dy
= F_SUB(f
, dy
, dy
, q
); /* %$y' = m (s - x') - t$% */
190 static ec
*ecadd(ec_curve
*c
, ec
*d
, const ec
*a
, const ec
*b
)
194 else if (EC_ATINF(a
))
196 else if (EC_ATINF(b
))
203 if (!MP_EQ(a
->x
, b
->x
)) {
204 dy
= F_SUB(f
, MP_NEW
, a
->y
, b
->y
); /* %$y_0 - y_1$% */
205 dx
= F_SUB(f
, MP_NEW
, a
->x
, b
->x
); /* %$x_0 - x_1$% */
206 dx
= F_INV(f
, dx
, dx
); /* %$(x_0 - x_1)^{-1}$% */
207 lambda
= F_MUL(f
, MP_NEW
, dy
, dx
);
208 /* %$\lambda = (y_0 - y1)/(x_0 - x_1)$% */
209 } else if (F_ZEROP(c
->f
, a
->y
) || !MP_EQ(a
->y
, b
->y
)) {
213 dx
= F_SQR(f
, MP_NEW
, a
->x
); /* %$x_0^2$% */
214 dx
= F_TPL(f
, dx
, dx
); /* %$3 x_0^2$% */
215 dx
= F_ADD(f
, dx
, dx
, c
->a
); /* %$3 x_0^2 + A$% */
216 dy
= F_DBL(f
, MP_NEW
, a
->y
); /* %$2 y_0$% */
217 dy
= F_INV(f
, dy
, dy
); /* %$(2 y_0)^{-1}$% */
218 lambda
= F_MUL(f
, MP_NEW
, dx
, dy
);
219 /* %$\lambda = (3 x_0^2 + A)/(2 y_0)$% */
222 dx
= F_SQR(f
, dx
, lambda
); /* %$\lambda^2$% */
223 dx
= F_SUB(f
, dx
, dx
, a
->x
); /* %$\lambda^2 - x_0$% */
224 dx
= F_SUB(f
, dx
, dx
, b
->x
); /* %$x' = \lambda^2 - x_0 - x_1$% */
225 dy
= F_SUB(f
, dy
, b
->x
, dx
); /* %$x_1 - x'$% */
226 dy
= F_MUL(f
, dy
, lambda
, dy
); /* %$\lambda (x_1 - x')$% */
227 dy
= F_SUB(f
, dy
, dy
, b
->y
); /* %$y' = \lambda (x_1 - x') - y_1$% */
238 static ec
*ecprojadd(ec_curve
*c
, ec
*d
, const ec
*a
, const ec
*b
)
241 c
->ops
->dbl(c
, d
, a
);
242 else if (EC_ATINF(a
))
244 else if (EC_ATINF(b
))
248 mp
*p
, *q
, *r
, *w
, *u
, *uu
, *s
, *ss
, *dx
, *dy
, *dz
;
250 q
= F_SQR(f
, MP_NEW
, a
->z
); /* %$z_0^2$% */
251 u
= F_MUL(f
, MP_NEW
, q
, b
->x
); /* %$u = x_1 z_0^2$% */
252 p
= F_MUL(f
, MP_NEW
, q
, b
->y
); /* %$y_1 z_0^2$% */
253 s
= F_MUL(f
, q
, p
, a
->z
); /* %$s = y_1 z_0^3$% */
255 q
= F_SQR(f
, MP_NEW
, b
->z
); /* %$z_1^2$% */
256 uu
= F_MUL(f
, MP_NEW
, q
, a
->x
); /* %$uu = x_0 z_1^2$%*/
257 p
= F_MUL(f
, p
, q
, a
->y
); /* %$y_0 z_1^2$% */
258 ss
= F_MUL(f
, q
, p
, b
->z
); /* %$ss = y_0 z_1^3$% */
260 w
= F_SUB(f
, p
, uu
, u
); /* %$w = uu - u$% */
261 r
= F_SUB(f
, MP_NEW
, ss
, s
); /* %$r = ss - s$% */
270 return (c
->ops
->dbl(c
, d
, a
));
277 u
= F_ADD(f
, u
, u
, uu
); /* %$t = uu + u$% */
278 s
= F_ADD(f
, s
, s
, ss
); /* %$m = ss + r$% */
280 uu
= F_MUL(f
, uu
, a
->z
, w
); /* %$z_0 w$% */
281 dz
= F_MUL(f
, ss
, uu
, b
->z
); /* %$z' = z_0 z_1 w$% */
283 p
= F_SQR(f
, uu
, w
); /* %$w^2$% */
284 q
= F_MUL(f
, MP_NEW
, p
, u
); /* %$t w^2$% */
285 u
= F_MUL(f
, u
, p
, w
); /* %$w^3$% */
286 p
= F_MUL(f
, p
, u
, s
); /* %$m w^3$% */
288 dx
= F_SQR(f
, u
, r
); /* %$r^2$% */
289 dx
= F_SUB(f
, dx
, dx
, q
); /* %$x' = r^2 - t w^2$% */
291 s
= F_DBL(f
, s
, dx
); /* %$2 x'$% */
292 q
= F_SUB(f
, q
, q
, s
); /* %$v = t w^2 - 2 x'$% */
293 dy
= F_MUL(f
, s
, q
, r
); /* %$v r$% */
294 dy
= F_SUB(f
, dy
, dy
, p
); /* %$v r - m w^3$% */
295 dy
= F_HLV(f
, dy
, dy
); /* %$y' = (v r - m w^3)/2$% */
309 static int eccheck(ec_curve
*c
, const ec
*p
)
314 if (EC_ATINF(p
)) return (0);
315 l
= F_SQR(f
, MP_NEW
, p
->y
);
316 x
= F_SQR(f
, MP_NEW
, p
->x
);
317 r
= F_MUL(f
, MP_NEW
, x
, p
->x
);
318 x
= F_MUL(f
, x
, c
->a
, p
->x
);
319 r
= F_ADD(f
, r
, r
, x
);
320 r
= F_ADD(f
, r
, r
, c
->b
);
321 rc
= MP_EQ(l
, r
) ?
0 : -1;
328 static int ecprojcheck(ec_curve
*c
, const ec
*p
)
333 c
->ops
->fix(c
, &t
, p
);
339 static void ecdestroy(ec_curve
*c
)
346 /* --- @ec_prime@, @ec_primeproj@ --- *
348 * Arguments: @field *f@ = the underlying field for this elliptic curve
349 * @mp *a, *b@ = the coefficients for this curve
351 * Returns: A pointer to the curve, or null.
353 * Use: Creates a curve structure for an elliptic curve defined over
354 * a prime field. The @primeproj@ variant uses projective
355 * coordinates, which can be a win.
358 extern ec_curve
*ec_prime(field
*f
, mp
*a
, mp
*b
)
360 ec_curve
*c
= CREATE(ec_curve
);
361 c
->ops
= &ec_primeops
;
363 c
->a
= F_IN(f
, MP_NEW
, a
);
364 c
->b
= F_IN(f
, MP_NEW
, b
);
368 extern ec_curve
*ec_primeproj(field
*f
, mp
*a
, mp
*b
)
370 ec_curve
*c
= CREATE(ec_curve
);
373 ax
= mp_add(MP_NEW
, a
, MP_THREE
);
374 ax
= F_IN(f
, ax
, ax
);
376 c
->ops
= &ec_primeprojxops
;
378 c
->ops
= &ec_primeprojops
;
381 c
->a
= F_IN(f
, MP_NEW
, a
);
382 c
->b
= F_IN(f
, MP_NEW
, b
);
386 static const ec_ops ec_primeops
= {
387 ecdestroy
, ec_stdsamep
, ec_idin
, ec_idout
, ec_idfix
,
388 ecfind
, ecneg
, ecadd
, ec_stdsub
, ecdbl
, eccheck
391 static const ec_ops ec_primeprojops
= {
392 ecdestroy
, ec_stdsamep
, ec_projin
, ec_projout
, ec_projfix
,
393 ecfind
, ecneg
, ecprojadd
, ec_stdsub
, ecprojdbl
, ecprojcheck
396 static const ec_ops ec_primeprojxops
= {
397 ecdestroy
, ec_stdsamep
, ec_projin
, ec_projout
, ec_projfix
,
398 ecfind
, ecneg
, ecprojadd
, ec_stdsub
, ecprojxdbl
, ecprojcheck
401 /*----- Test rig ----------------------------------------------------------*/
405 #define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
407 int main(int argc
, char *argv
[])
411 ec g
= EC_INIT
, d
= EC_INIT
;
413 int i
, n
= argc
== 1 ?
1 : atoi(argv
[1]);
415 printf("ec-prime: ");
418 b
= MP(0xb3312fa7e23ee7e4988e056be3f82d19181d9c6efe8141120314088f5013875ac656398d8a2ed19d2a85c8edd3ec2aef);
419 p
= MP(39402006196394479212279040100143613805079739270465446667948293404245721771496870329047266088258938001861606973112319);
420 r
= MP(39402006196394479212279040100143613805079739270465446667946905279627659399113263569398956308152294913554433653942642);
422 f
= field_niceprime(p
);
423 c
= ec_primeproj(f
, a
, b
);
425 g
.x
= MP(0xaa87ca22be8b05378eb1c71ef320ad746e1d3b628ba79b9859f741e082542a385502f25dbf55296c3a545e3872760ab7);
426 g
.y
= MP(0x3617de4a96262c6f5d9e98bf9292dc29f8f41dbd289a147ce9da3113b5f0b8c00a60b1ce1d7e819d7a431d7c90ea0e5f);
428 for (i
= 0; i
< n
; i
++) {
429 ec_mul(c
, &d
, &g
, r
);
431 fprintf(stderr
, "zero too early\n");
434 ec_add(c
, &d
, &d
, &g
);
436 fprintf(stderr
, "didn't reach zero\n");
437 MP_EPRINT("d.x", d
.x
);
438 MP_EPRINT("d.y", d
.y
);
446 MP_DROP(p
); MP_DROP(a
); MP_DROP(b
); MP_DROP(r
);
447 assert(!mparena_count(&mparena_global
));
454 /*----- That's all, folks -------------------------------------------------*/