0efb72ffba2bdf32f4faefeb60484e7b6740926d
3 * $Id: ec-bin.c,v 1.5 2004/03/27 17:54:11 mdw Exp $
5 * Arithmetic for elliptic curves over binary fields
7 * (c) 2004 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 --------------------------------------------------*
33 * Revision 1.5 2004/03/27 17:54:11 mdw
34 * Standard curves and curve checking.
36 * Revision 1.4 2004/03/23 15:19:32 mdw
37 * Test elliptic curves more thoroughly.
39 * Revision 1.3 2004/03/22 02:19:09 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 1.2 2004/03/21 22:52:06 mdw
45 * Merge and close elliptic curve branch.
47 * Revision 1.1.2.1 2004/03/21 22:39:46 mdw
48 * Elliptic curves on binary fields work.
52 /*----- Header files ------------------------------------------------------*/
58 /*----- Data structures ---------------------------------------------------*/
60 typedef struct ecctx
{
65 /*----- Main code ---------------------------------------------------------*/
67 static const ec_ops ec_binops
, ec_binprojops
;
69 static ec
*ecneg(ec_curve
*c
, ec
*d
, const ec
*p
)
73 d
->y
= F_ADD(c
->f
, d
->y
, d
->y
, d
->x
);
77 static ec
*ecprojneg(ec_curve
*c
, ec
*d
, const ec
*p
)
81 mp
*t
= F_MUL(c
->f
, MP_NEW
, d
->x
, d
->z
);
82 d
->y
= F_ADD(c
->f
, d
->y
, d
->y
, t
);
88 static ec
*ecfind(ec_curve
*c
, ec
*d
, mp
*x
)
94 y
= F_SQRT(f
, MP_NEW
, c
->b
);
96 u
= F_SQR(f
, MP_NEW
, x
); /* %$x^2$% */
97 y
= F_MUL(f
, MP_NEW
, u
, c
->a
); /* %$a x^2$% */
98 y
= F_ADD(f
, y
, y
, c
->b
); /* %$a x^2 + b$% */
99 v
= F_MUL(f
, MP_NEW
, u
, x
); /* %$x^3$% */
100 y
= F_ADD(f
, y
, y
, v
); /* %$A = x^3 + a x^2 + b$% */
101 if (!F_ZEROP(f
, y
)) {
102 u
= F_INV(f
, u
, u
); /* %$x^{-2}$% */
103 v
= F_MUL(f
, v
, u
, y
); /* %$B = A x^{-2} = x + a + b x^{-2}$% */
104 y
= F_QUADSOLVE(f
, y
, v
); /* %$z^2 + z = B$% */
105 if (y
) y
= F_MUL(f
, y
, y
, x
); /* %$y = z x$% */
114 d
->z
= MP_COPY(f
->one
);
118 static ec
*ecdbl(ec_curve
*c
, ec
*d
, const ec
*a
)
120 if (EC_ATINF(a
) || F_ZEROP(c
->f
, a
->x
))
127 dx
= F_INV(f
, MP_NEW
, a
->x
); /* %$x^{-1}$% */
128 dy
= F_MUL(f
, MP_NEW
, dx
, a
->y
); /* %$y/x$% */
129 lambda
= F_ADD(f
, dy
, dy
, a
->x
); /* %$\lambda = x + y/x$% */
131 dx
= F_SQR(f
, dx
, lambda
); /* %$\lambda^2$% */
132 dx
= F_ADD(f
, dx
, dx
, lambda
); /* %$\lambda^2 + \lambda$% */
133 dx
= F_ADD(f
, dx
, dx
, c
->a
); /* %$x' = a + \lambda^2 + \lambda$% */
135 dy
= F_ADD(f
, MP_NEW
, a
->x
, dx
); /* %$ x + x' $% */
136 dy
= F_MUL(f
, dy
, dy
, lambda
); /* %$ (x + x') \lambda$% */
137 dy
= F_ADD(f
, dy
, dy
, a
->y
); /* %$ (x + x') \lambda + y$% */
138 dy
= F_ADD(f
, dy
, dy
, dx
); /* %$ y' = (x + x') \lambda + y + x'$% */
149 static ec
*ecprojdbl(ec_curve
*c
, ec
*d
, const ec
*a
)
151 if (EC_ATINF(a
) || F_ZEROP(c
->f
, a
->x
))
155 ecctx
*cc
= (ecctx
*)c
;
156 mp
*dx
, *dy
, *dz
, *u
, *v
;
158 dy
= F_SQR(f
, MP_NEW
, a
->z
); /* %$z^2$% */
159 dx
= F_MUL(f
, MP_NEW
, dy
, cc
->bb
); /* %$c z^2$% */
160 dx
= F_ADD(f
, dx
, dx
, a
->x
); /* %$x + c z^2$% */
161 dz
= F_SQR(f
, MP_NEW
, dx
); /* %$(x + c z^2)^2$% */
162 dx
= F_SQR(f
, dx
, dz
); /* %$x' = (x + c z^2)^4$% */
164 dz
= F_MUL(f
, dz
, dy
, a
->x
); /* %$z' = x z^2$% */
166 dy
= F_SQR(f
, dy
, a
->x
); /* %$x^2$% */
167 u
= F_MUL(f
, MP_NEW
, a
->y
, a
->z
); /* %$y z$% */
168 u
= F_ADD(f
, u
, u
, dz
); /* %$z' + y z$% */
169 u
= F_ADD(f
, u
, u
, dy
); /* %$u = z' + x^2 + y z$% */
171 v
= F_SQR(f
, MP_NEW
, dy
); /* %$x^4$% */
172 dy
= F_MUL(f
, dy
, v
, dz
); /* %$x^4 z'$% */
173 v
= F_MUL(f
, v
, u
, dx
); /* %$u x'$% */
174 dy
= F_ADD(f
, dy
, dy
, v
); /* %$y' = x^4 z' + u x'$% */
182 assert(!(d
->x
->f
& MP_DESTROYED
));
183 assert(!(d
->y
->f
& MP_DESTROYED
));
184 assert(!(d
->z
->f
& MP_DESTROYED
));
189 static ec
*ecadd(ec_curve
*c
, ec
*d
, const ec
*a
, const ec
*b
)
193 else if (EC_ATINF(a
))
195 else if (EC_ATINF(b
))
202 if (!MP_EQ(a
->x
, b
->x
)) {
203 dx
= F_ADD(f
, MP_NEW
, a
->x
, b
->x
); /* %$x_0 + x_1$% */
204 dy
= F_INV(f
, MP_NEW
, dx
); /* %$(x_0 + x_1)^{-1}$% */
205 dx
= F_ADD(f
, dx
, a
->y
, b
->y
); /* %$y_0 + y_1$% */
206 lambda
= F_MUL(f
, MP_NEW
, dy
, dx
);
207 /* %$\lambda = (y_0 + y_1)/(x_0 + x_1)$% */
209 dx
= F_SQR(f
, dx
, lambda
); /* %$\lambda^2$% */
210 dx
= F_ADD(f
, dx
, dx
, lambda
); /* %$\lambda^2 + \lambda$% */
211 dx
= F_ADD(f
, dx
, dx
, c
->a
); /* %$a + \lambda^2 + \lambda$% */
212 dx
= F_ADD(f
, dx
, dx
, a
->x
); /* %$a + \lambda^2 + \lambda + x_0$% */
213 dx
= F_ADD(f
, dx
, dx
, b
->x
);
214 /* %$x' = a + \lambda^2 + \lambda + x_0 + x_1$% */
215 } else if (!MP_EQ(a
->y
, b
->y
) || F_ZEROP(f
, a
->x
)) {
219 dx
= F_INV(f
, MP_NEW
, a
->x
); /* %$x^{-1}$% */
220 dy
= F_MUL(f
, MP_NEW
, dx
, a
->y
); /* %$y/x$% */
221 lambda
= F_ADD(f
, dy
, dy
, a
->x
); /* %$\lambda = x + y/x$% */
223 dx
= F_SQR(f
, dx
, lambda
); /* %$\lambda^2$% */
224 dx
= F_ADD(f
, dx
, dx
, lambda
); /* %$\lambda^2 + \lambda$% */
225 dx
= F_ADD(f
, dx
, dx
, c
->a
); /* %$x' = a + \lambda^2 + \lambda$% */
229 dy
= F_ADD(f
, dy
, a
->x
, dx
); /* %$ x + x' $% */
230 dy
= F_MUL(f
, dy
, dy
, lambda
); /* %$ (x + x') \lambda$% */
231 dy
= F_ADD(f
, dy
, dy
, a
->y
); /* %$ (x + x') \lambda + y$% */
232 dy
= F_ADD(f
, dy
, dy
, dx
); /* %$ y' = (x + x') \lambda + y + x'$% */
243 static ec
*ecprojadd(ec_curve
*c
, ec
*d
, const ec
*a
, const ec
*b
)
246 c
->ops
->dbl(c
, d
, a
);
247 else if (EC_ATINF(a
))
249 else if (EC_ATINF(b
))
253 mp
*dx
, *dy
, *dz
, *u
, *uu
, *v
, *t
, *s
, *ss
, *r
, *w
, *l
;
255 dz
= F_SQR(f
, MP_NEW
, b
->z
); /* %$z_1^2$% */
256 u
= F_MUL(f
, MP_NEW
, dz
, a
->x
); /* %$u_0 = x_0 z_1^2$% */
257 t
= F_MUL(f
, MP_NEW
, dz
, b
->z
); /* %$z_1^3$% */
258 s
= F_MUL(f
, MP_NEW
, t
, a
->y
); /* %$s_0 = y_0 z_1^3$% */
260 dz
= F_SQR(f
, dz
, a
->z
); /* %$z_0^2$% */
261 uu
= F_MUL(f
, MP_NEW
, dz
, b
->x
); /* %$u_1 = x_1 z_0^2$% */
262 t
= F_MUL(f
, t
, dz
, a
->z
); /* %$z_0^3$% */
263 ss
= F_MUL(f
, MP_NEW
, t
, b
->y
); /* %$s_1 = y_1 z_0^3$% */
265 w
= F_ADD(f
, u
, u
, uu
); /* %$r = u_0 + u_1$% */
266 r
= F_ADD(f
, s
, s
, ss
); /* %$w = s_0 + s_1$% */
275 return (c
->ops
->dbl(c
, d
, a
));
283 l
= F_MUL(f
, t
, a
->z
, w
); /* %$l = z_0 w$% */
285 dz
= F_MUL(f
, dz
, l
, b
->z
); /* %$z' = l z_1$% */
287 ss
= F_MUL(f
, ss
, r
, b
->x
); /* %$r x_1$% */
288 t
= F_MUL(f
, uu
, l
, b
->y
); /* %$l y_1$% */
289 v
= F_ADD(f
, ss
, ss
, t
); /* %$v = r x_1 + l y_1$% */
291 t
= F_ADD(f
, t
, r
, dz
); /* %$t = r + z'$% */
293 uu
= F_SQR(f
, MP_NEW
, dz
); /* %$z'^2$% */
294 dx
= F_MUL(f
, MP_NEW
, uu
, c
->a
); /* %$a z'^2$% */
295 uu
= F_MUL(f
, uu
, t
, r
); /* %$t r$% */
296 dx
= F_ADD(f
, dx
, dx
, uu
); /* %$a z'^2 + t r$% */
297 r
= F_SQR(f
, r
, w
); /* %$w^2$% */
298 uu
= F_MUL(f
, uu
, r
, w
); /* %$w^3$% */
299 dx
= F_ADD(f
, dx
, dx
, uu
); /* %$x' = a z'^2 + t r + w^3$% */
301 r
= F_SQR(f
, r
, l
); /* %$l^2$% */
302 dy
= F_MUL(f
, uu
, v
, r
); /* %$v l^2$% */
303 l
= F_MUL(f
, l
, t
, dx
); /* %$t x'$% */
304 dy
= F_ADD(f
, dy
, dy
, l
); /* %$y' = t x' + v l^2$% */
319 static int eccheck(ec_curve
*c
, const ec
*p
)
325 v
= F_SQR(f
, MP_NEW
, p
->x
);
326 u
= F_MUL(f
, MP_NEW
, v
, p
->x
);
327 v
= F_MUL(f
, v
, v
, c
->a
);
328 u
= F_ADD(f
, u
, u
, v
);
329 u
= F_ADD(f
, u
, u
, c
->b
);
330 v
= F_MUL(f
, v
, p
->x
, p
->y
);
331 u
= F_ADD(f
, u
, u
, v
);
332 v
= F_SQR(f
, v
, p
->y
);
333 u
= F_ADD(f
, u
, u
, v
);
334 rc
= F_ZEROP(f
, u
) ?
0 : -1;
340 static int ecprojcheck(ec_curve
*c
, const ec
*p
)
345 c
->ops
->fix(c
, &t
, p
);
351 static void ecdestroy(ec_curve
*c
)
353 ecctx
*cc
= (ecctx
*)c
;
356 if (cc
->bb
) MP_DROP(cc
->bb
);
360 /* --- @ec_bin@, @ec_binproj@ --- *
362 * Arguments: @field *f@ = the underlying field for this elliptic curve
363 * @mp *a, *b@ = the coefficients for this curve
365 * Returns: A pointer to the curve.
367 * Use: Creates a curve structure for an elliptic curve defined over
368 * a binary field. The @binproj@ variant uses projective
369 * coordinates, which can be a win.
372 ec_curve
*ec_bin(field
*f
, mp
*a
, mp
*b
)
374 ecctx
*cc
= CREATE(ecctx
);
375 cc
->c
.ops
= &ec_binops
;
377 cc
->c
.a
= F_IN(f
, MP_NEW
, a
);
378 cc
->c
.b
= F_IN(f
, MP_NEW
, b
);
383 ec_curve
*ec_binproj(field
*f
, mp
*a
, mp
*b
)
385 ecctx
*cc
= CREATE(ecctx
);
386 cc
->c
.ops
= &ec_binprojops
;
388 cc
->c
.a
= F_IN(f
, MP_NEW
, a
);
389 cc
->c
.b
= F_IN(f
, MP_NEW
, b
);
390 cc
->bb
= F_SQRT(f
, MP_NEW
, b
);
391 cc
->bb
= F_SQRT(f
, cc
->bb
, cc
->bb
);
395 static const ec_ops ec_binops
= {
396 ecdestroy
, ec_idin
, ec_idout
, ec_idfix
,
397 ecfind
, ecneg
, ecadd
, ec_stdsub
, ecdbl
, eccheck
400 static const ec_ops ec_binprojops
= {
401 ecdestroy
, ec_projin
, ec_projout
, ec_projfix
,
402 ecfind
, ecprojneg
, ecprojadd
, ec_stdsub
, ecprojdbl
, ecprojcheck
405 /*----- Test rig ----------------------------------------------------------*/
409 #define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
411 int main(int argc
, char *argv
[])
415 ec g
= EC_INIT
, d
= EC_INIT
;
417 int i
, n
= argc
== 1 ?
1 : atoi(argv
[1]);
422 b
= MP(0x021a5c2c8ee9feb5c4b9a753b7b476b7fd6422ef1f3dd674761fa99d6ac27c8a9a197b272822f6cd57a55aa4f50ae317b13545f);
423 p
= MP(0x2000000000000000000000000000000000000000000000000000000000000000000000000000000008000000000000000000001);
425 MP(661055968790248598951915308032771039828404682964281219284648798304157774827374805208143723762179110965979867288366567526770);
427 f
= field_binpoly(p
);
428 c
= ec_binproj(f
, a
, b
);
430 g
.x
= MP(0x15d4860d088ddb3496b0c6064756260441cde4af1771d4db01ffe5b34e59703dc255a868a1180515603aeab60794e54bb7996a7);
431 g
.y
= MP(0x061b1cfab6be5f32bbfa78324ed106a7636b9c5a7bd198d0158aa4f5488d08f38514f1fdf4b4f40d2181b3681c364ba0273c706);
433 for (i
= 0; i
< n
; i
++) {
434 ec_mul(c
, &d
, &g
, r
);
436 fprintf(stderr
, "zero too early\n");
439 ec_add(c
, &d
, &d
, &g
);
441 fprintf(stderr
, "didn't reach zero\n");
442 MP_EPRINTX("d.x", d
.x
);
443 MP_EPRINTX("d.y", d
.y
);
452 MP_DROP(p
); MP_DROP(a
); MP_DROP(b
); MP_DROP(r
);
453 assert(!mparena_count(&mparena_global
));
460 /*----- That's all, folks -------------------------------------------------*/