3 * $Id: ec-bin.c,v 1.6 2004/04/01 12:50:09 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 1.6 2004/04/01 12:50:09 mdw
34 * Add cyclic group abstraction, with test code. Separate off exponentation
35 * functions for better static linking. Fix a buttload of bugs on the way.
36 * Generally ensure that negative exponents do inversion correctly. Add
37 * table of standard prime-field subgroups. (Binary field subgroups are
38 * currently unimplemented but easy to add if anyone ever finds a good one.)
40 * Revision 1.5 2004/03/27 17:54:11 mdw
41 * Standard curves and curve checking.
43 * Revision 1.4 2004/03/23 15:19:32 mdw
44 * Test elliptic curves more thoroughly.
46 * Revision 1.3 2004/03/22 02:19:09 mdw
47 * Rationalise the sliding-window threshold. Drop guarantee that right
48 * arguments to EC @add@ are canonical, and fix up projective implementations
51 * Revision 1.2 2004/03/21 22:52:06 mdw
52 * Merge and close elliptic curve branch.
54 * Revision 1.1.2.1 2004/03/21 22:39:46 mdw
55 * Elliptic curves on binary fields work.
59 /*----- Header files ------------------------------------------------------*/
65 /*----- Data structures ---------------------------------------------------*/
67 typedef struct ecctx
{
72 /*----- Main code ---------------------------------------------------------*/
74 static const ec_ops ec_binops
, ec_binprojops
;
76 static ec
*ecneg(ec_curve
*c
, ec
*d
, const ec
*p
)
80 d
->y
= F_ADD(c
->f
, d
->y
, d
->y
, d
->x
);
84 static ec
*ecprojneg(ec_curve
*c
, ec
*d
, const ec
*p
)
88 mp
*t
= F_MUL(c
->f
, MP_NEW
, d
->x
, d
->z
);
89 d
->y
= F_ADD(c
->f
, d
->y
, d
->y
, t
);
95 static ec
*ecfind(ec_curve
*c
, ec
*d
, mp
*x
)
101 y
= F_SQRT(f
, MP_NEW
, c
->b
);
103 u
= F_SQR(f
, MP_NEW
, x
); /* %$x^2$% */
104 y
= F_MUL(f
, MP_NEW
, u
, c
->a
); /* %$a x^2$% */
105 y
= F_ADD(f
, y
, y
, c
->b
); /* %$a x^2 + b$% */
106 v
= F_MUL(f
, MP_NEW
, u
, x
); /* %$x^3$% */
107 y
= F_ADD(f
, y
, y
, v
); /* %$A = x^3 + a x^2 + b$% */
108 if (!F_ZEROP(f
, y
)) {
109 u
= F_INV(f
, u
, u
); /* %$x^{-2}$% */
110 v
= F_MUL(f
, v
, u
, y
); /* %$B = A x^{-2} = x + a + b x^{-2}$% */
111 y
= F_QUADSOLVE(f
, y
, v
); /* %$z^2 + z = B$% */
112 if (y
) y
= F_MUL(f
, y
, y
, x
); /* %$y = z x$% */
121 d
->z
= MP_COPY(f
->one
);
125 static ec
*ecdbl(ec_curve
*c
, ec
*d
, const ec
*a
)
127 if (EC_ATINF(a
) || F_ZEROP(c
->f
, a
->x
))
134 dx
= F_INV(f
, MP_NEW
, a
->x
); /* %$x^{-1}$% */
135 dy
= F_MUL(f
, MP_NEW
, dx
, a
->y
); /* %$y/x$% */
136 lambda
= F_ADD(f
, dy
, dy
, a
->x
); /* %$\lambda = x + y/x$% */
138 dx
= F_SQR(f
, dx
, lambda
); /* %$\lambda^2$% */
139 dx
= F_ADD(f
, dx
, dx
, lambda
); /* %$\lambda^2 + \lambda$% */
140 dx
= F_ADD(f
, dx
, dx
, c
->a
); /* %$x' = a + \lambda^2 + \lambda$% */
142 dy
= F_ADD(f
, MP_NEW
, a
->x
, dx
); /* %$ x + x' $% */
143 dy
= F_MUL(f
, dy
, dy
, lambda
); /* %$ (x + x') \lambda$% */
144 dy
= F_ADD(f
, dy
, dy
, a
->y
); /* %$ (x + x') \lambda + y$% */
145 dy
= F_ADD(f
, dy
, dy
, dx
); /* %$ y' = (x + x') \lambda + y + x'$% */
156 static ec
*ecprojdbl(ec_curve
*c
, ec
*d
, const ec
*a
)
158 if (EC_ATINF(a
) || F_ZEROP(c
->f
, a
->x
))
162 ecctx
*cc
= (ecctx
*)c
;
163 mp
*dx
, *dy
, *dz
, *u
, *v
;
165 dy
= F_SQR(f
, MP_NEW
, a
->z
); /* %$z^2$% */
166 dx
= F_MUL(f
, MP_NEW
, dy
, cc
->bb
); /* %$c z^2$% */
167 dx
= F_ADD(f
, dx
, dx
, a
->x
); /* %$x + c z^2$% */
168 dz
= F_SQR(f
, MP_NEW
, dx
); /* %$(x + c z^2)^2$% */
169 dx
= F_SQR(f
, dx
, dz
); /* %$x' = (x + c z^2)^4$% */
171 dz
= F_MUL(f
, dz
, dy
, a
->x
); /* %$z' = x z^2$% */
173 dy
= F_SQR(f
, dy
, a
->x
); /* %$x^2$% */
174 u
= F_MUL(f
, MP_NEW
, a
->y
, a
->z
); /* %$y z$% */
175 u
= F_ADD(f
, u
, u
, dz
); /* %$z' + y z$% */
176 u
= F_ADD(f
, u
, u
, dy
); /* %$u = z' + x^2 + y z$% */
178 v
= F_SQR(f
, MP_NEW
, dy
); /* %$x^4$% */
179 dy
= F_MUL(f
, dy
, v
, dz
); /* %$x^4 z'$% */
180 v
= F_MUL(f
, v
, u
, dx
); /* %$u x'$% */
181 dy
= F_ADD(f
, dy
, dy
, v
); /* %$y' = x^4 z' + u x'$% */
189 assert(!(d
->x
->f
& MP_DESTROYED
));
190 assert(!(d
->y
->f
& MP_DESTROYED
));
191 assert(!(d
->z
->f
& MP_DESTROYED
));
196 static ec
*ecadd(ec_curve
*c
, ec
*d
, const ec
*a
, const ec
*b
)
200 else if (EC_ATINF(a
))
202 else if (EC_ATINF(b
))
209 if (!MP_EQ(a
->x
, b
->x
)) {
210 dx
= F_ADD(f
, MP_NEW
, a
->x
, b
->x
); /* %$x_0 + x_1$% */
211 dy
= F_INV(f
, MP_NEW
, dx
); /* %$(x_0 + x_1)^{-1}$% */
212 dx
= F_ADD(f
, dx
, a
->y
, b
->y
); /* %$y_0 + y_1$% */
213 lambda
= F_MUL(f
, MP_NEW
, dy
, dx
);
214 /* %$\lambda = (y_0 + y_1)/(x_0 + x_1)$% */
216 dx
= F_SQR(f
, dx
, lambda
); /* %$\lambda^2$% */
217 dx
= F_ADD(f
, dx
, dx
, lambda
); /* %$\lambda^2 + \lambda$% */
218 dx
= F_ADD(f
, dx
, dx
, c
->a
); /* %$a + \lambda^2 + \lambda$% */
219 dx
= F_ADD(f
, dx
, dx
, a
->x
); /* %$a + \lambda^2 + \lambda + x_0$% */
220 dx
= F_ADD(f
, dx
, dx
, b
->x
);
221 /* %$x' = a + \lambda^2 + \lambda + x_0 + x_1$% */
222 } else if (!MP_EQ(a
->y
, b
->y
) || F_ZEROP(f
, a
->x
)) {
226 dx
= F_INV(f
, MP_NEW
, a
->x
); /* %$x^{-1}$% */
227 dy
= F_MUL(f
, MP_NEW
, dx
, a
->y
); /* %$y/x$% */
228 lambda
= F_ADD(f
, dy
, dy
, a
->x
); /* %$\lambda = x + y/x$% */
230 dx
= F_SQR(f
, dx
, lambda
); /* %$\lambda^2$% */
231 dx
= F_ADD(f
, dx
, dx
, lambda
); /* %$\lambda^2 + \lambda$% */
232 dx
= F_ADD(f
, dx
, dx
, c
->a
); /* %$x' = a + \lambda^2 + \lambda$% */
236 dy
= F_ADD(f
, dy
, a
->x
, dx
); /* %$ x + x' $% */
237 dy
= F_MUL(f
, dy
, dy
, lambda
); /* %$ (x + x') \lambda$% */
238 dy
= F_ADD(f
, dy
, dy
, a
->y
); /* %$ (x + x') \lambda + y$% */
239 dy
= F_ADD(f
, dy
, dy
, dx
); /* %$ y' = (x + x') \lambda + y + x'$% */
250 static ec
*ecprojadd(ec_curve
*c
, ec
*d
, const ec
*a
, const ec
*b
)
253 c
->ops
->dbl(c
, d
, a
);
254 else if (EC_ATINF(a
))
256 else if (EC_ATINF(b
))
260 mp
*dx
, *dy
, *dz
, *u
, *uu
, *v
, *t
, *s
, *ss
, *r
, *w
, *l
;
262 dz
= F_SQR(f
, MP_NEW
, b
->z
); /* %$z_1^2$% */
263 u
= F_MUL(f
, MP_NEW
, dz
, a
->x
); /* %$u_0 = x_0 z_1^2$% */
264 t
= F_MUL(f
, MP_NEW
, dz
, b
->z
); /* %$z_1^3$% */
265 s
= F_MUL(f
, MP_NEW
, t
, a
->y
); /* %$s_0 = y_0 z_1^3$% */
267 dz
= F_SQR(f
, dz
, a
->z
); /* %$z_0^2$% */
268 uu
= F_MUL(f
, MP_NEW
, dz
, b
->x
); /* %$u_1 = x_1 z_0^2$% */
269 t
= F_MUL(f
, t
, dz
, a
->z
); /* %$z_0^3$% */
270 ss
= F_MUL(f
, MP_NEW
, t
, b
->y
); /* %$s_1 = y_1 z_0^3$% */
272 w
= F_ADD(f
, u
, u
, uu
); /* %$r = u_0 + u_1$% */
273 r
= F_ADD(f
, s
, s
, ss
); /* %$w = s_0 + s_1$% */
282 return (c
->ops
->dbl(c
, d
, a
));
290 l
= F_MUL(f
, t
, a
->z
, w
); /* %$l = z_0 w$% */
292 dz
= F_MUL(f
, dz
, l
, b
->z
); /* %$z' = l z_1$% */
294 ss
= F_MUL(f
, ss
, r
, b
->x
); /* %$r x_1$% */
295 t
= F_MUL(f
, uu
, l
, b
->y
); /* %$l y_1$% */
296 v
= F_ADD(f
, ss
, ss
, t
); /* %$v = r x_1 + l y_1$% */
298 t
= F_ADD(f
, t
, r
, dz
); /* %$t = r + z'$% */
300 uu
= F_SQR(f
, MP_NEW
, dz
); /* %$z'^2$% */
301 dx
= F_MUL(f
, MP_NEW
, uu
, c
->a
); /* %$a z'^2$% */
302 uu
= F_MUL(f
, uu
, t
, r
); /* %$t r$% */
303 dx
= F_ADD(f
, dx
, dx
, uu
); /* %$a z'^2 + t r$% */
304 r
= F_SQR(f
, r
, w
); /* %$w^2$% */
305 uu
= F_MUL(f
, uu
, r
, w
); /* %$w^3$% */
306 dx
= F_ADD(f
, dx
, dx
, uu
); /* %$x' = a z'^2 + t r + w^3$% */
308 r
= F_SQR(f
, r
, l
); /* %$l^2$% */
309 dy
= F_MUL(f
, uu
, v
, r
); /* %$v l^2$% */
310 l
= F_MUL(f
, l
, t
, dx
); /* %$t x'$% */
311 dy
= F_ADD(f
, dy
, dy
, l
); /* %$y' = t x' + v l^2$% */
326 static int eccheck(ec_curve
*c
, const ec
*p
)
332 if (EC_ATINF(p
)) return (0);
333 v
= F_SQR(f
, MP_NEW
, p
->x
);
334 u
= F_MUL(f
, MP_NEW
, v
, p
->x
);
335 v
= F_MUL(f
, v
, v
, c
->a
);
336 u
= F_ADD(f
, u
, u
, v
);
337 u
= F_ADD(f
, u
, u
, c
->b
);
338 v
= F_MUL(f
, v
, p
->x
, p
->y
);
339 u
= F_ADD(f
, u
, u
, v
);
340 v
= F_SQR(f
, v
, p
->y
);
341 u
= F_ADD(f
, u
, u
, v
);
342 rc
= F_ZEROP(f
, u
) ?
0 : -1;
348 static int ecprojcheck(ec_curve
*c
, const ec
*p
)
353 c
->ops
->fix(c
, &t
, p
);
359 static void ecdestroy(ec_curve
*c
)
361 ecctx
*cc
= (ecctx
*)c
;
364 if (cc
->bb
) MP_DROP(cc
->bb
);
368 /* --- @ec_bin@, @ec_binproj@ --- *
370 * Arguments: @field *f@ = the underlying field for this elliptic curve
371 * @mp *a, *b@ = the coefficients for this curve
373 * Returns: A pointer to the curve.
375 * Use: Creates a curve structure for an elliptic curve defined over
376 * a binary field. The @binproj@ variant uses projective
377 * coordinates, which can be a win.
380 ec_curve
*ec_bin(field
*f
, mp
*a
, mp
*b
)
382 ecctx
*cc
= CREATE(ecctx
);
383 cc
->c
.ops
= &ec_binops
;
385 cc
->c
.a
= F_IN(f
, MP_NEW
, a
);
386 cc
->c
.b
= F_IN(f
, MP_NEW
, b
);
391 ec_curve
*ec_binproj(field
*f
, mp
*a
, mp
*b
)
393 ecctx
*cc
= CREATE(ecctx
);
394 cc
->c
.ops
= &ec_binprojops
;
396 cc
->c
.a
= F_IN(f
, MP_NEW
, a
);
397 cc
->c
.b
= F_IN(f
, MP_NEW
, b
);
398 cc
->bb
= F_SQRT(f
, MP_NEW
, b
);
399 cc
->bb
= F_SQRT(f
, cc
->bb
, cc
->bb
);
403 static const ec_ops ec_binops
= {
404 ecdestroy
, ec_stdsamep
, ec_idin
, ec_idout
, ec_idfix
,
405 ecfind
, ecneg
, ecadd
, ec_stdsub
, ecdbl
, eccheck
408 static const ec_ops ec_binprojops
= {
409 ecdestroy
, ec_stdsamep
, ec_projin
, ec_projout
, ec_projfix
,
410 ecfind
, ecprojneg
, ecprojadd
, ec_stdsub
, ecprojdbl
, ecprojcheck
413 /*----- Test rig ----------------------------------------------------------*/
417 #define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
419 int main(int argc
, char *argv
[])
423 ec g
= EC_INIT
, d
= EC_INIT
;
425 int i
, n
= argc
== 1 ?
1 : atoi(argv
[1]);
430 b
= MP(0x021a5c2c8ee9feb5c4b9a753b7b476b7fd6422ef1f3dd674761fa99d6ac27c8a9a197b272822f6cd57a55aa4f50ae317b13545f);
431 p
= MP(0x2000000000000000000000000000000000000000000000000000000000000000000000000000000008000000000000000000001);
433 MP(661055968790248598951915308032771039828404682964281219284648798304157774827374805208143723762179110965979867288366567526770);
435 f
= field_binpoly(p
);
436 c
= ec_binproj(f
, a
, b
);
438 g
.x
= MP(0x15d4860d088ddb3496b0c6064756260441cde4af1771d4db01ffe5b34e59703dc255a868a1180515603aeab60794e54bb7996a7);
439 g
.y
= MP(0x061b1cfab6be5f32bbfa78324ed106a7636b9c5a7bd198d0158aa4f5488d08f38514f1fdf4b4f40d2181b3681c364ba0273c706);
441 for (i
= 0; i
< n
; i
++) {
442 ec_mul(c
, &d
, &g
, r
);
444 fprintf(stderr
, "zero too early\n");
447 ec_add(c
, &d
, &d
, &g
);
449 fprintf(stderr
, "didn't reach zero\n");
450 MP_EPRINTX("d.x", d
.x
);
451 MP_EPRINTX("d.y", d
.y
);
460 MP_DROP(p
); MP_DROP(a
); MP_DROP(b
); MP_DROP(r
);
461 assert(!mparena_count(&mparena_global
));
468 /*----- That's all, folks -------------------------------------------------*/