3 * $Id: ec-bin.c,v 1.8 2004/04/03 03:32:05 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 1.8 2004/04/03 03:32:05 mdw
34 * General robustification.
36 * Revision 1.7 2004/04/01 21:28:41 mdw
37 * Normal basis support (translates to poly basis internally). Rewrite
38 * EC and prime group table generators in awk, so that they can reuse data
39 * for repeated constants.
41 * Revision 1.6 2004/04/01 12:50:09 mdw
42 * Add cyclic group abstraction, with test code. Separate off exponentation
43 * functions for better static linking. Fix a buttload of bugs on the way.
44 * Generally ensure that negative exponents do inversion correctly. Add
45 * table of standard prime-field subgroups. (Binary field subgroups are
46 * currently unimplemented but easy to add if anyone ever finds a good one.)
48 * Revision 1.5 2004/03/27 17:54:11 mdw
49 * Standard curves and curve checking.
51 * Revision 1.4 2004/03/23 15:19:32 mdw
52 * Test elliptic curves more thoroughly.
54 * Revision 1.3 2004/03/22 02:19:09 mdw
55 * Rationalise the sliding-window threshold. Drop guarantee that right
56 * arguments to EC @add@ are canonical, and fix up projective implementations
59 * Revision 1.2 2004/03/21 22:52:06 mdw
60 * Merge and close elliptic curve branch.
62 * Revision 1.1.2.1 2004/03/21 22:39:46 mdw
63 * Elliptic curves on binary fields work.
67 /*----- Header files ------------------------------------------------------*/
73 /*----- Data structures ---------------------------------------------------*/
75 typedef struct ecctx
{
80 /*----- Main code ---------------------------------------------------------*/
82 static const ec_ops ec_binops
, ec_binprojops
;
84 static ec
*ecneg(ec_curve
*c
, ec
*d
, const ec
*p
)
88 d
->y
= F_ADD(c
->f
, d
->y
, d
->y
, d
->x
);
92 static ec
*ecprojneg(ec_curve
*c
, ec
*d
, const ec
*p
)
96 mp
*t
= F_MUL(c
->f
, MP_NEW
, d
->x
, d
->z
);
97 d
->y
= F_ADD(c
->f
, d
->y
, d
->y
, t
);
103 static ec
*ecfind(ec_curve
*c
, ec
*d
, mp
*x
)
109 y
= F_SQRT(f
, MP_NEW
, c
->b
);
111 u
= F_SQR(f
, MP_NEW
, x
); /* %$x^2$% */
112 y
= F_MUL(f
, MP_NEW
, u
, c
->a
); /* %$a x^2$% */
113 y
= F_ADD(f
, y
, y
, c
->b
); /* %$a x^2 + b$% */
114 v
= F_MUL(f
, MP_NEW
, u
, x
); /* %$x^3$% */
115 y
= F_ADD(f
, y
, y
, v
); /* %$A = x^3 + a x^2 + b$% */
116 if (!F_ZEROP(f
, y
)) {
117 u
= F_INV(f
, u
, u
); /* %$x^{-2}$% */
118 v
= F_MUL(f
, v
, u
, y
); /* %$B = A x^{-2} = x + a + b x^{-2}$% */
119 y
= F_QUADSOLVE(f
, y
, v
); /* %$z^2 + z = B$% */
120 if (y
) y
= F_MUL(f
, y
, y
, x
); /* %$y = z x$% */
129 d
->z
= MP_COPY(f
->one
);
133 static ec
*ecdbl(ec_curve
*c
, ec
*d
, const ec
*a
)
135 if (EC_ATINF(a
) || F_ZEROP(c
->f
, a
->x
))
142 dx
= F_INV(f
, MP_NEW
, a
->x
); /* %$x^{-1}$% */
143 dy
= F_MUL(f
, MP_NEW
, dx
, a
->y
); /* %$y/x$% */
144 lambda
= F_ADD(f
, dy
, dy
, a
->x
); /* %$\lambda = x + y/x$% */
146 dx
= F_SQR(f
, dx
, lambda
); /* %$\lambda^2$% */
147 dx
= F_ADD(f
, dx
, dx
, lambda
); /* %$\lambda^2 + \lambda$% */
148 dx
= F_ADD(f
, dx
, dx
, c
->a
); /* %$x' = a + \lambda^2 + \lambda$% */
150 dy
= F_ADD(f
, MP_NEW
, a
->x
, dx
); /* %$ x + x' $% */
151 dy
= F_MUL(f
, dy
, dy
, lambda
); /* %$ (x + x') \lambda$% */
152 dy
= F_ADD(f
, dy
, dy
, a
->y
); /* %$ (x + x') \lambda + y$% */
153 dy
= F_ADD(f
, dy
, dy
, dx
); /* %$ y' = (x + x') \lambda + y + x'$% */
164 static ec
*ecprojdbl(ec_curve
*c
, ec
*d
, const ec
*a
)
166 if (EC_ATINF(a
) || F_ZEROP(c
->f
, a
->x
))
170 ecctx
*cc
= (ecctx
*)c
;
171 mp
*dx
, *dy
, *dz
, *u
, *v
;
173 dy
= F_SQR(f
, MP_NEW
, a
->z
); /* %$z^2$% */
174 dx
= F_MUL(f
, MP_NEW
, dy
, cc
->bb
); /* %$c z^2$% */
175 dx
= F_ADD(f
, dx
, dx
, a
->x
); /* %$x + c z^2$% */
176 dz
= F_SQR(f
, MP_NEW
, dx
); /* %$(x + c z^2)^2$% */
177 dx
= F_SQR(f
, dx
, dz
); /* %$x' = (x + c z^2)^4$% */
179 dz
= F_MUL(f
, dz
, dy
, a
->x
); /* %$z' = x z^2$% */
181 dy
= F_SQR(f
, dy
, a
->x
); /* %$x^2$% */
182 u
= F_MUL(f
, MP_NEW
, a
->y
, a
->z
); /* %$y z$% */
183 u
= F_ADD(f
, u
, u
, dz
); /* %$z' + y z$% */
184 u
= F_ADD(f
, u
, u
, dy
); /* %$u = z' + x^2 + y z$% */
186 v
= F_SQR(f
, MP_NEW
, dy
); /* %$x^4$% */
187 dy
= F_MUL(f
, dy
, v
, dz
); /* %$x^4 z'$% */
188 v
= F_MUL(f
, v
, u
, dx
); /* %$u x'$% */
189 dy
= F_ADD(f
, dy
, dy
, v
); /* %$y' = x^4 z' + u x'$% */
201 static ec
*ecadd(ec_curve
*c
, ec
*d
, const ec
*a
, const ec
*b
)
205 else if (EC_ATINF(a
))
207 else if (EC_ATINF(b
))
214 if (!MP_EQ(a
->x
, b
->x
)) {
215 dx
= F_ADD(f
, MP_NEW
, a
->x
, b
->x
); /* %$x_0 + x_1$% */
216 dy
= F_INV(f
, MP_NEW
, dx
); /* %$(x_0 + x_1)^{-1}$% */
217 dx
= F_ADD(f
, dx
, a
->y
, b
->y
); /* %$y_0 + y_1$% */
218 lambda
= F_MUL(f
, MP_NEW
, dy
, dx
);
219 /* %$\lambda = (y_0 + y_1)/(x_0 + x_1)$% */
221 dx
= F_SQR(f
, dx
, lambda
); /* %$\lambda^2$% */
222 dx
= F_ADD(f
, dx
, dx
, lambda
); /* %$\lambda^2 + \lambda$% */
223 dx
= F_ADD(f
, dx
, dx
, c
->a
); /* %$a + \lambda^2 + \lambda$% */
224 dx
= F_ADD(f
, dx
, dx
, a
->x
); /* %$a + \lambda^2 + \lambda + x_0$% */
225 dx
= F_ADD(f
, dx
, dx
, b
->x
);
226 /* %$x' = a + \lambda^2 + \lambda + x_0 + x_1$% */
227 } else if (!MP_EQ(a
->y
, b
->y
) || F_ZEROP(f
, a
->x
)) {
231 dx
= F_INV(f
, MP_NEW
, a
->x
); /* %$x^{-1}$% */
232 dy
= F_MUL(f
, MP_NEW
, dx
, a
->y
); /* %$y/x$% */
233 lambda
= F_ADD(f
, dy
, dy
, a
->x
); /* %$\lambda = x + y/x$% */
235 dx
= F_SQR(f
, dx
, lambda
); /* %$\lambda^2$% */
236 dx
= F_ADD(f
, dx
, dx
, lambda
); /* %$\lambda^2 + \lambda$% */
237 dx
= F_ADD(f
, dx
, dx
, c
->a
); /* %$x' = a + \lambda^2 + \lambda$% */
241 dy
= F_ADD(f
, dy
, a
->x
, dx
); /* %$ x + x' $% */
242 dy
= F_MUL(f
, dy
, dy
, lambda
); /* %$ (x + x') \lambda$% */
243 dy
= F_ADD(f
, dy
, dy
, a
->y
); /* %$ (x + x') \lambda + y$% */
244 dy
= F_ADD(f
, dy
, dy
, dx
); /* %$ y' = (x + x') \lambda + y + x'$% */
255 static ec
*ecprojadd(ec_curve
*c
, ec
*d
, const ec
*a
, const ec
*b
)
258 c
->ops
->dbl(c
, d
, a
);
259 else if (EC_ATINF(a
))
261 else if (EC_ATINF(b
))
265 mp
*dx
, *dy
, *dz
, *u
, *uu
, *v
, *t
, *s
, *ss
, *r
, *w
, *l
;
267 dz
= F_SQR(f
, MP_NEW
, b
->z
); /* %$z_1^2$% */
268 u
= F_MUL(f
, MP_NEW
, dz
, a
->x
); /* %$u_0 = x_0 z_1^2$% */
269 t
= F_MUL(f
, MP_NEW
, dz
, b
->z
); /* %$z_1^3$% */
270 s
= F_MUL(f
, MP_NEW
, t
, a
->y
); /* %$s_0 = y_0 z_1^3$% */
272 dz
= F_SQR(f
, dz
, a
->z
); /* %$z_0^2$% */
273 uu
= F_MUL(f
, MP_NEW
, dz
, b
->x
); /* %$u_1 = x_1 z_0^2$% */
274 t
= F_MUL(f
, t
, dz
, a
->z
); /* %$z_0^3$% */
275 ss
= F_MUL(f
, MP_NEW
, t
, b
->y
); /* %$s_1 = y_1 z_0^3$% */
277 w
= F_ADD(f
, u
, u
, uu
); /* %$r = u_0 + u_1$% */
278 r
= F_ADD(f
, s
, s
, ss
); /* %$w = s_0 + s_1$% */
287 return (c
->ops
->dbl(c
, d
, a
));
295 l
= F_MUL(f
, t
, a
->z
, w
); /* %$l = z_0 w$% */
297 dz
= F_MUL(f
, dz
, l
, b
->z
); /* %$z' = l z_1$% */
299 ss
= F_MUL(f
, ss
, r
, b
->x
); /* %$r x_1$% */
300 t
= F_MUL(f
, uu
, l
, b
->y
); /* %$l y_1$% */
301 v
= F_ADD(f
, ss
, ss
, t
); /* %$v = r x_1 + l y_1$% */
303 t
= F_ADD(f
, t
, r
, dz
); /* %$t = r + z'$% */
305 uu
= F_SQR(f
, MP_NEW
, dz
); /* %$z'^2$% */
306 dx
= F_MUL(f
, MP_NEW
, uu
, c
->a
); /* %$a z'^2$% */
307 uu
= F_MUL(f
, uu
, t
, r
); /* %$t r$% */
308 dx
= F_ADD(f
, dx
, dx
, uu
); /* %$a z'^2 + t r$% */
309 r
= F_SQR(f
, r
, w
); /* %$w^2$% */
310 uu
= F_MUL(f
, uu
, r
, w
); /* %$w^3$% */
311 dx
= F_ADD(f
, dx
, dx
, uu
); /* %$x' = a z'^2 + t r + w^3$% */
313 r
= F_SQR(f
, r
, l
); /* %$l^2$% */
314 dy
= F_MUL(f
, uu
, v
, r
); /* %$v l^2$% */
315 l
= F_MUL(f
, l
, t
, dx
); /* %$t x'$% */
316 dy
= F_ADD(f
, dy
, dy
, l
); /* %$y' = t x' + v l^2$% */
331 static int eccheck(ec_curve
*c
, const ec
*p
)
337 if (EC_ATINF(p
)) return (0);
338 v
= F_SQR(f
, MP_NEW
, p
->x
);
339 u
= F_MUL(f
, MP_NEW
, v
, p
->x
);
340 v
= F_MUL(f
, v
, v
, c
->a
);
341 u
= F_ADD(f
, u
, u
, v
);
342 u
= F_ADD(f
, u
, u
, c
->b
);
343 v
= F_MUL(f
, v
, p
->x
, p
->y
);
344 u
= F_ADD(f
, u
, u
, v
);
345 v
= F_SQR(f
, v
, p
->y
);
346 u
= F_ADD(f
, u
, u
, v
);
347 rc
= F_ZEROP(f
, u
) ?
0 : -1;
353 static int ecprojcheck(ec_curve
*c
, const ec
*p
)
358 c
->ops
->fix(c
, &t
, p
);
364 static void ecdestroy(ec_curve
*c
)
366 ecctx
*cc
= (ecctx
*)c
;
369 if (cc
->bb
) MP_DROP(cc
->bb
);
373 /* --- @ec_bin@, @ec_binproj@ --- *
375 * Arguments: @field *f@ = the underlying field for this elliptic curve
376 * @mp *a, *b@ = the coefficients for this curve
378 * Returns: A pointer to the curve, or null.
380 * Use: Creates a curve structure for an elliptic curve defined over
381 * a binary field. The @binproj@ variant uses projective
382 * coordinates, which can be a win.
385 ec_curve
*ec_bin(field
*f
, mp
*a
, mp
*b
)
387 ecctx
*cc
= CREATE(ecctx
);
388 cc
->c
.ops
= &ec_binops
;
390 cc
->c
.a
= F_IN(f
, MP_NEW
, a
);
391 cc
->c
.b
= F_IN(f
, MP_NEW
, b
);
396 ec_curve
*ec_binproj(field
*f
, mp
*a
, mp
*b
)
398 ecctx
*cc
= CREATE(ecctx
);
399 cc
->c
.ops
= &ec_binprojops
;
401 cc
->c
.a
= F_IN(f
, MP_NEW
, a
);
402 cc
->c
.b
= F_IN(f
, MP_NEW
, b
);
403 cc
->bb
= F_SQRT(f
, MP_NEW
, cc
->c
.b
);
405 cc
->bb
= F_SQRT(f
, cc
->bb
, cc
->bb
);
415 static const ec_ops ec_binops
= {
416 ecdestroy
, ec_stdsamep
, ec_idin
, ec_idout
, ec_idfix
,
417 ecfind
, ecneg
, ecadd
, ec_stdsub
, ecdbl
, eccheck
420 static const ec_ops ec_binprojops
= {
421 ecdestroy
, ec_stdsamep
, ec_projin
, ec_projout
, ec_projfix
,
422 ecfind
, ecprojneg
, ecprojadd
, ec_stdsub
, ecprojdbl
, ecprojcheck
425 /*----- Test rig ----------------------------------------------------------*/
429 #define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
431 int main(int argc
, char *argv
[])
435 ec g
= EC_INIT
, d
= EC_INIT
;
436 mp
*p
, *a
, *b
, *r
, *beta
;
437 int i
, n
= argc
== 1 ?
1 : atoi(argv
[1]);
441 a
= MP(0x7ffffffffffffffffffffffffffffffffffffffff);
442 b
= MP(0x6645f3cacf1638e139c6cd13ef61734fbc9e3d9fb);
443 p
= MP(0x800000000000000000000000000000000000000c9);
444 beta
= MP(0x715169c109c612e390d347c748342bcd3b02a0bef);
445 r
= MP(0x040000000000000000000292fe77e70c12a4234c32);
447 f
= field_binnorm(p
, beta
);
448 c
= ec_binproj(f
, a
, b
);
449 g
.x
= MP(0x0311103c17167564ace77ccb09c681f886ba54ee8);
450 g
.y
= MP(0x333ac13c6447f2e67613bf7009daf98c87bb50c7f);
452 for (i
= 0; i
< n
; i
++) {
453 ec_mul(c
, &d
, &g
, r
);
455 fprintf(stderr
, "zero too early\n");
458 ec_add(c
, &d
, &d
, &g
);
460 fprintf(stderr
, "didn't reach zero\n");
461 MP_EPRINTX("d.x", d
.x
);
462 MP_EPRINTX("d.y", d
.y
);
471 MP_DROP(p
); MP_DROP(a
); MP_DROP(b
); MP_DROP(r
); MP_DROP(beta
);
472 assert(!mparena_count(&mparena_global
));
479 /*----- That's all, folks -------------------------------------------------*/