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
)
71 if (EC_ATINF(a
) || F_ZEROP(c
->f
, a
->y
))
78 dx
= F_SQR(f
, MP_NEW
, a
->x
); /* %$x^2$% */
79 dy
= F_DBL(f
, MP_NEW
, a
->y
); /* %$2 y$% */
80 dx
= F_TPL(f
, dx
, dx
); /* %$3 x^2$% */
81 dx
= F_ADD(f
, dx
, dx
, c
->a
); /* %$3 x^2 + A$% */
82 dy
= F_INV(f
, dy
, dy
); /* %$(2 y)^{-1}$% */
83 lambda
= F_MUL(f
, MP_NEW
, dx
, dy
); /* %$\lambda = (3 x^2 + A)/(2 y)$% */
85 dx
= F_SQR(f
, dx
, lambda
); /* %$\lambda^2$% */
86 dy
= F_DBL(f
, dy
, a
->x
); /* %$2 x$% */
87 dx
= F_SUB(f
, dx
, dx
, dy
); /* %$x' = \lambda^2 - 2 x */
88 dy
= F_SUB(f
, dy
, a
->x
, dx
); /* %$x - x'$% */
89 dy
= F_MUL(f
, dy
, lambda
, dy
); /* %$\lambda (x - x')$% */
90 dy
= F_SUB(f
, dy
, dy
, a
->y
); /* %$y' = \lambda (x - x') - y$% */
101 static ec
*ecprojdbl(ec_curve
*c
, ec
*d
, const ec
*a
)
103 if (EC_ATINF(a
) || F_ZEROP(c
->f
, a
->y
))
107 mp
*p
, *q
, *m
, *s
, *dx
, *dy
, *dz
;
109 p
= F_SQR(f
, MP_NEW
, a
->z
); /* %$z^2$% */
110 q
= F_SQR(f
, MP_NEW
, p
); /* %$z^4$% */
111 p
= F_MUL(f
, p
, q
, c
->a
); /* %$A z^4$% */
112 m
= F_SQR(f
, MP_NEW
, a
->x
); /* %$x^2$% */
113 m
= F_TPL(f
, m
, m
); /* %$3 x^2$% */
114 m
= F_ADD(f
, m
, m
, p
); /* %$m = 3 x^2 + A z^4$% */
116 q
= F_DBL(f
, q
, a
->y
); /* %$2 y$% */
117 dz
= F_MUL(f
, MP_NEW
, q
, a
->z
); /* %$z' = 2 y z$% */
119 p
= F_SQR(f
, p
, q
); /* %$4 y^2$% */
120 s
= F_MUL(f
, MP_NEW
, p
, a
->x
); /* %$s = 4 x y^2$% */
121 q
= F_SQR(f
, q
, p
); /* %$16 y^4$% */
122 q
= F_HLV(f
, q
, q
); /* %$t = 8 y^4$% */
124 p
= F_DBL(f
, p
, s
); /* %$2 s$% */
125 dx
= F_SQR(f
, MP_NEW
, m
); /* %$m^2$% */
126 dx
= F_SUB(f
, dx
, dx
, p
); /* %$x' = m^2 - 2 s$% */
128 s
= F_SUB(f
, s
, s
, dx
); /* %$s - x'$% */
129 dy
= F_MUL(f
, p
, m
, s
); /* %$m (s - x')$% */
130 dy
= F_SUB(f
, dy
, dy
, q
); /* %$y' = m (s - x') - t$% */
143 static ec
*ecprojxdbl(ec_curve
*c
, ec
*d
, const ec
*a
)
145 if (EC_ATINF(a
) || F_ZEROP(c
->f
, a
->y
))
149 mp
*p
, *q
, *m
, *s
, *dx
, *dy
, *dz
;
151 m
= F_SQR(f
, MP_NEW
, a
->z
); /* %$z^2$% */
152 p
= F_SUB(f
, MP_NEW
, a
->x
, m
); /* %$x - z^2$% */
153 q
= F_ADD(f
, MP_NEW
, a
->x
, m
); /* %$x + z^2$% */
154 m
= F_MUL(f
, m
, p
, q
); /* %$x^2 - z^4$% */
155 m
= F_TPL(f
, m
, m
); /* %$m = 3 x^2 - 3 z^4$% */
157 q
= F_DBL(f
, q
, a
->y
); /* %$2 y$% */
158 dz
= F_MUL(f
, MP_NEW
, q
, a
->z
); /* %$z' = 2 y z$% */
160 p
= F_SQR(f
, p
, q
); /* %$4 y^2$% */
161 s
= F_MUL(f
, MP_NEW
, p
, a
->x
); /* %$s = 4 x y^2$% */
162 q
= F_SQR(f
, q
, p
); /* %$16 y^4$% */
163 q
= F_HLV(f
, q
, q
); /* %$t = 8 y^4$% */
165 p
= F_DBL(f
, p
, s
); /* %$2 s$% */
166 dx
= F_SQR(f
, MP_NEW
, m
); /* %$m^2$% */
167 dx
= F_SUB(f
, dx
, dx
, p
); /* %$x' = m^2 - 2 s$% */
169 s
= F_SUB(f
, s
, s
, dx
); /* %$s - x'$% */
170 dy
= F_MUL(f
, p
, m
, s
); /* %$m (s - x')$% */
171 dy
= F_SUB(f
, dy
, dy
, q
); /* %$y' = m (s - x') - t$% */
184 static ec
*ecadd(ec_curve
*c
, ec
*d
, const ec
*a
, const ec
*b
)
188 else if (EC_ATINF(a
))
190 else if (EC_ATINF(b
))
197 if (!MP_EQ(a
->x
, b
->x
)) {
198 dy
= F_SUB(f
, MP_NEW
, a
->y
, b
->y
); /* %$y_0 - y_1$% */
199 dx
= F_SUB(f
, MP_NEW
, a
->x
, b
->x
); /* %$x_0 - x_1$% */
200 dx
= F_INV(f
, dx
, dx
); /* %$(x_0 - x_1)^{-1}$% */
201 lambda
= F_MUL(f
, MP_NEW
, dy
, dx
);
202 /* %$\lambda = (y_0 - y1)/(x_0 - x_1)$% */
203 } else if (F_ZEROP(c
->f
, a
->y
) || !MP_EQ(a
->y
, b
->y
)) {
207 dx
= F_SQR(f
, MP_NEW
, a
->x
); /* %$x_0^2$% */
208 dx
= F_TPL(f
, dx
, dx
); /* %$3 x_0^2$% */
209 dx
= F_ADD(f
, dx
, dx
, c
->a
); /* %$3 x_0^2 + A$% */
210 dy
= F_DBL(f
, MP_NEW
, a
->y
); /* %$2 y_0$% */
211 dy
= F_INV(f
, dy
, dy
); /* %$(2 y_0)^{-1}$% */
212 lambda
= F_MUL(f
, MP_NEW
, dx
, dy
);
213 /* %$\lambda = (3 x_0^2 + A)/(2 y_0)$% */
216 dx
= F_SQR(f
, dx
, lambda
); /* %$\lambda^2$% */
217 dx
= F_SUB(f
, dx
, dx
, a
->x
); /* %$\lambda^2 - x_0$% */
218 dx
= F_SUB(f
, dx
, dx
, b
->x
); /* %$x' = \lambda^2 - x_0 - x_1$% */
219 dy
= F_SUB(f
, dy
, b
->x
, dx
); /* %$x_1 - x'$% */
220 dy
= F_MUL(f
, dy
, lambda
, dy
); /* %$\lambda (x_1 - x')$% */
221 dy
= F_SUB(f
, dy
, dy
, b
->y
); /* %$y' = \lambda (x_1 - x') - y_1$% */
232 static ec
*ecprojadd(ec_curve
*c
, ec
*d
, const ec
*a
, const ec
*b
)
235 c
->ops
->dbl(c
, d
, a
);
236 else if (EC_ATINF(a
))
238 else if (EC_ATINF(b
))
242 mp
*p
, *q
, *r
, *w
, *u
, *uu
, *s
, *ss
, *dx
, *dy
, *dz
;
244 q
= F_SQR(f
, MP_NEW
, a
->z
); /* %$z_0^2$% */
245 u
= F_MUL(f
, MP_NEW
, q
, b
->x
); /* %$u = x_1 z_0^2$% */
246 p
= F_MUL(f
, MP_NEW
, q
, b
->y
); /* %$y_1 z_0^2$% */
247 s
= F_MUL(f
, q
, p
, a
->z
); /* %$s = y_1 z_0^3$% */
249 q
= F_SQR(f
, MP_NEW
, b
->z
); /* %$z_1^2$% */
250 uu
= F_MUL(f
, MP_NEW
, q
, a
->x
); /* %$uu = x_0 z_1^2$%*/
251 p
= F_MUL(f
, p
, q
, a
->y
); /* %$y_0 z_1^2$% */
252 ss
= F_MUL(f
, q
, p
, b
->z
); /* %$ss = y_0 z_1^3$% */
254 w
= F_SUB(f
, p
, uu
, u
); /* %$w = uu - u$% */
255 r
= F_SUB(f
, MP_NEW
, ss
, s
); /* %$r = ss - s$% */
264 return (c
->ops
->dbl(c
, d
, a
));
271 u
= F_ADD(f
, u
, u
, uu
); /* %$t = uu + u$% */
272 s
= F_ADD(f
, s
, s
, ss
); /* %$m = ss + r$% */
274 uu
= F_MUL(f
, uu
, a
->z
, w
); /* %$z_0 w$% */
275 dz
= F_MUL(f
, ss
, uu
, b
->z
); /* %$z' = z_0 z_1 w$% */
277 p
= F_SQR(f
, uu
, w
); /* %$w^2$% */
278 q
= F_MUL(f
, MP_NEW
, p
, u
); /* %$t w^2$% */
279 u
= F_MUL(f
, u
, p
, w
); /* %$w^3$% */
280 p
= F_MUL(f
, p
, u
, s
); /* %$m w^3$% */
282 dx
= F_SQR(f
, u
, r
); /* %$r^2$% */
283 dx
= F_SUB(f
, dx
, dx
, q
); /* %$x' = r^2 - t w^2$% */
285 s
= F_DBL(f
, s
, dx
); /* %$2 x'$% */
286 q
= F_SUB(f
, q
, q
, s
); /* %$v = t w^2 - 2 x'$% */
287 dy
= F_MUL(f
, s
, q
, r
); /* %$v r$% */
288 dy
= F_SUB(f
, dy
, dy
, p
); /* %$v r - m w^3$% */
289 dy
= F_HLV(f
, dy
, dy
); /* %$y' = (v r - m w^3)/2$% */
303 static int eccheck(ec_curve
*c
, const ec
*p
)
308 if (EC_ATINF(p
)) return (0);
309 l
= F_SQR(f
, MP_NEW
, p
->y
);
310 x
= F_SQR(f
, MP_NEW
, p
->x
);
311 r
= F_MUL(f
, MP_NEW
, x
, p
->x
);
312 x
= F_MUL(f
, x
, c
->a
, p
->x
);
313 r
= F_ADD(f
, r
, r
, x
);
314 r
= F_ADD(f
, r
, r
, c
->b
);
315 rc
= MP_EQ(l
, r
) ?
0 : -1;
322 static int ecprojcheck(ec_curve
*c
, const ec
*p
)
327 c
->ops
->fix(c
, &t
, p
);
333 static void ecdestroy(ec_curve
*c
)
340 /* --- @ec_prime@, @ec_primeproj@ --- *
342 * Arguments: @field *f@ = the underlying field for this elliptic curve
343 * @mp *a, *b@ = the coefficients for this curve
345 * Returns: A pointer to the curve, or null.
347 * Use: Creates a curve structure for an elliptic curve defined over
348 * a prime field. The @primeproj@ variant uses projective
349 * coordinates, which can be a win.
352 extern ec_curve
*ec_prime(field
*f
, mp
*a
, mp
*b
)
354 ec_curve
*c
= CREATE(ec_curve
);
355 c
->ops
= &ec_primeops
;
357 c
->a
= F_IN(f
, MP_NEW
, a
);
358 c
->b
= F_IN(f
, MP_NEW
, b
);
362 extern ec_curve
*ec_primeproj(field
*f
, mp
*a
, mp
*b
)
364 ec_curve
*c
= CREATE(ec_curve
);
367 ax
= mp_add(MP_NEW
, a
, MP_THREE
);
368 ax
= F_IN(f
, ax
, ax
);
370 c
->ops
= &ec_primeprojxops
;
372 c
->ops
= &ec_primeprojops
;
375 c
->a
= F_IN(f
, MP_NEW
, a
);
376 c
->b
= F_IN(f
, MP_NEW
, b
);
380 static const ec_ops ec_primeops
= {
382 ecdestroy
, ec_stdsamep
, ec_idin
, ec_idout
, ec_idfix
,
383 ecfind
, ecneg
, ecadd
, ec_stdsub
, ecdbl
, eccheck
386 static const ec_ops ec_primeprojops
= {
388 ecdestroy
, ec_stdsamep
, ec_projin
, ec_projout
, ec_projfix
,
389 ecfind
, ecneg
, ecprojadd
, ec_stdsub
, ecprojdbl
, ecprojcheck
392 static const ec_ops ec_primeprojxops
= {
394 ecdestroy
, ec_stdsamep
, ec_projin
, ec_projout
, ec_projfix
,
395 ecfind
, ecneg
, ecprojadd
, ec_stdsub
, ecprojxdbl
, ecprojcheck
398 /*----- Test rig ----------------------------------------------------------*/
402 #define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
404 int main(int argc
, char *argv
[])
408 ec g
= EC_INIT
, d
= EC_INIT
;
410 int i
, n
= argc
== 1 ?
1 : atoi(argv
[1]);
412 printf("ec-prime: ");
415 b
= MP(0xb3312fa7e23ee7e4988e056be3f82d19181d9c6efe8141120314088f5013875ac656398d8a2ed19d2a85c8edd3ec2aef);
416 p
= MP(39402006196394479212279040100143613805079739270465446667948293404245721771496870329047266088258938001861606973112319);
417 r
= MP(39402006196394479212279040100143613805079739270465446667946905279627659399113263569398956308152294913554433653942642);
419 f
= field_niceprime(p
);
420 c
= ec_primeproj(f
, a
, b
);
422 g
.x
= MP(0xaa87ca22be8b05378eb1c71ef320ad746e1d3b628ba79b9859f741e082542a385502f25dbf55296c3a545e3872760ab7);
423 g
.y
= MP(0x3617de4a96262c6f5d9e98bf9292dc29f8f41dbd289a147ce9da3113b5f0b8c00a60b1ce1d7e819d7a431d7c90ea0e5f);
425 for (i
= 0; i
< n
; i
++) {
426 ec_mul(c
, &d
, &g
, r
);
428 fprintf(stderr
, "zero too early\n");
431 ec_add(c
, &d
, &d
, &g
);
433 fprintf(stderr
, "didn't reach zero\n");
434 MP_EPRINT("d.x", d
.x
);
435 MP_EPRINT("d.y", d
.y
);
443 MP_DROP(p
); MP_DROP(a
); MP_DROP(b
); MP_DROP(r
);
444 assert(!mparena_count(&mparena_global
));
451 /*----- That's all, folks -------------------------------------------------*/