3 * $Id: ec-bin.c,v 1.1.2.1 2004/03/21 22:39:46 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 1.1.2.1 2004/03/21 22:39:46 mdw
34 * Elliptic curves on binary fields work.
38 /*----- Header files ------------------------------------------------------*/
44 /*----- Data structures ---------------------------------------------------*/
46 typedef struct ecctx
{
52 /*----- Main code ---------------------------------------------------------*/
54 static const ec_ops ec_binops
, ec_binprojops
;
56 static ec
*ecneg(ec_curve
*c
, ec
*d
, const ec
*p
)
60 d
->y
= F_ADD(c
->f
, d
->y
, d
->y
, d
->x
);
64 static ec
*ecprojneg(ec_curve
*c
, ec
*d
, const ec
*p
)
68 mp
*t
= F_MUL(c
->f
, MP_NEW
, d
->x
, d
->z
);
69 d
->y
= F_ADD(c
->f
, d
->y
, d
->y
, t
);
75 static ec
*ecfind(ec_curve
*c
, ec
*d
, mp
*x
)
81 static ec
*ecdbl(ec_curve
*c
, ec
*d
, const ec
*a
)
83 if (EC_ATINF(a
) || F_ZEROP(c
->f
, a
->x
))
87 ecctx
*cc
= (ecctx
*)c
;
91 dx
= F_INV(f
, MP_NEW
, a
->x
); /* %$x^{-1}$% */
92 dy
= F_MUL(f
, MP_NEW
, dx
, a
->y
); /* %$y/x$% */
93 lambda
= F_ADD(f
, dy
, dy
, a
->x
); /* %$\lambda = x + y/x$% */
95 dx
= F_SQR(f
, dx
, lambda
); /* %$\lambda^2$% */
96 dx
= F_ADD(f
, dx
, dx
, lambda
); /* %$\lambda^2 + \lambda$% */
97 dx
= F_ADD(f
, dx
, dx
, cc
->a
); /* %$x' = a + \lambda^2 + \lambda$% */
99 dy
= F_ADD(f
, MP_NEW
, a
->x
, dx
); /* %$ x + x' $% */
100 dy
= F_MUL(f
, dy
, dy
, lambda
); /* %$ (x + x') \lambda$% */
101 dy
= F_ADD(f
, dy
, dy
, a
->y
); /* %$ (x + x') \lambda + y$% */
102 dy
= F_ADD(f
, dy
, dy
, dx
); /* %$ y' = (x + x') \lambda + y + x'$% */
113 static ec
*ecprojdbl(ec_curve
*c
, ec
*d
, const ec
*a
)
115 if (EC_ATINF(a
) || F_ZEROP(c
->f
, a
->x
))
119 ecctx
*cc
= (ecctx
*)c
;
120 mp
*dx
, *dy
, *dz
, *u
, *v
;
122 dy
= F_SQR(f
, MP_NEW
, a
->z
); /* %$z^2$% */
123 dx
= F_MUL(f
, MP_NEW
, dy
, cc
->bb
); /* %$c z^2$% */
124 dx
= F_ADD(f
, dx
, dx
, a
->x
); /* %$x + c z^2$% */
125 dz
= F_SQR(f
, MP_NEW
, dx
); /* %$(x + c z^2)^2$% */
126 dx
= F_SQR(f
, dx
, dz
); /* %$x' = (x + c z^2)^4$% */
128 dz
= F_MUL(f
, dz
, dy
, a
->x
); /* %$z' = x z^2$% */
130 dy
= F_SQR(f
, dy
, a
->x
); /* %$x^2$% */
131 u
= F_MUL(f
, MP_NEW
, a
->y
, a
->z
); /* %$y z$% */
132 u
= F_ADD(f
, u
, u
, dz
); /* %$z' + y z$% */
133 u
= F_ADD(f
, u
, u
, dy
); /* %$u = z' + x^2 + y z$% */
135 v
= F_SQR(f
, MP_NEW
, dy
); /* %$x^4$% */
136 dy
= F_MUL(f
, dy
, v
, dz
); /* %$x^4 z'$% */
137 v
= F_MUL(f
, v
, u
, dx
); /* %$u x'$% */
138 dy
= F_ADD(f
, dy
, dy
, v
); /* %$y' = x^4 z' + u x'$% */
146 assert(!(d
->x
->f
& MP_DESTROYED
));
147 assert(!(d
->y
->f
& MP_DESTROYED
));
148 assert(!(d
->z
->f
& MP_DESTROYED
));
153 static ec
*ecadd(ec_curve
*c
, ec
*d
, const ec
*a
, const ec
*b
)
157 else if (EC_ATINF(a
))
159 else if (EC_ATINF(b
))
163 ecctx
*cc
= (ecctx
*)c
;
167 if (!MP_EQ(a
->x
, b
->x
)) {
168 dx
= F_ADD(f
, MP_NEW
, a
->x
, b
->x
); /* %$x_0 + x_1$% */
169 dy
= F_INV(f
, MP_NEW
, dx
); /* %$(x_0 + x_1)^{-1}$% */
170 dx
= F_ADD(f
, dx
, a
->y
, b
->y
); /* %$y_0 + y_1$% */
171 lambda
= F_MUL(f
, MP_NEW
, dy
, dx
);
172 /* %$\lambda = (y_0 + y_1)/(x_0 + x_1)$% */
174 dx
= F_SQR(f
, dx
, lambda
); /* %$\lambda^2$% */
175 dx
= F_ADD(f
, dx
, dx
, lambda
); /* %$\lambda^2 + \lambda$% */
176 dx
= F_ADD(f
, dx
, dx
, cc
->a
); /* %$a + \lambda^2 + \lambda$% */
177 dx
= F_ADD(f
, dx
, dx
, a
->x
); /* %$a + \lambda^2 + \lambda + x_0$% */
178 dx
= F_ADD(f
, dx
, dx
, b
->x
);
179 /* %$x' = a + \lambda^2 + \lambda + x_0 + x_1$% */
180 } else if (!MP_EQ(a
->y
, b
->y
) || F_ZEROP(f
, a
->x
)) {
184 dx
= F_INV(f
, MP_NEW
, a
->x
); /* %$x^{-1}$% */
185 dy
= F_MUL(f
, MP_NEW
, dx
, a
->y
); /* %$y/x$% */
186 lambda
= F_ADD(f
, dy
, dy
, a
->x
); /* %$\lambda = x + y/x$% */
188 dx
= F_SQR(f
, dx
, lambda
); /* %$\lambda^2$% */
189 dx
= F_ADD(f
, dx
, dx
, lambda
); /* %$\lambda^2 + \lambda$% */
190 dx
= F_ADD(f
, dx
, dx
, cc
->a
); /* %$x' = a + \lambda^2 + \lambda$% */
194 dy
= F_ADD(f
, dy
, a
->x
, dx
); /* %$ x + x' $% */
195 dy
= F_MUL(f
, dy
, dy
, lambda
); /* %$ (x + x') \lambda$% */
196 dy
= F_ADD(f
, dy
, dy
, a
->y
); /* %$ (x + x') \lambda + y$% */
197 dy
= F_ADD(f
, dy
, dy
, dx
); /* %$ y' = (x + x') \lambda + y + x'$% */
208 static ec
*ecprojadd(ec_curve
*c
, ec
*d
, const ec
*a
, const ec
*b
)
211 c
->ops
->dbl(c
, d
, a
);
212 else if (EC_ATINF(a
))
214 else if (EC_ATINF(b
))
218 ecctx
*cc
= (ecctx
*)c
;
219 mp
*dx
, *dy
, *dz
, *u
, *uu
, *v
, *t
, *s
, *ss
, *r
, *w
, *l
;
221 dz
= F_SQR(f
, MP_NEW
, b
->z
); /* %$z_1^2$% */
222 u
= F_MUL(f
, MP_NEW
, dz
, a
->x
); /* %$u_0 = x_0 z_1^2$% */
223 t
= F_MUL(f
, MP_NEW
, dz
, b
->z
); /* %$z_1^3$% */
224 s
= F_MUL(f
, MP_NEW
, t
, a
->y
); /* %$s_0 = y_0 z_1^3$% */
226 dz
= F_SQR(f
, dz
, a
->z
); /* %$z_0^2$% */
227 uu
= F_MUL(f
, MP_NEW
, dz
, b
->x
); /* %$u_1 = x_1 z_0^2$% */
228 t
= F_MUL(f
, t
, dz
, a
->z
); /* %$z_0^3$% */
229 ss
= F_MUL(f
, MP_NEW
, t
, b
->y
); /* %$s_1 = y_1 z_0^3$% */
231 w
= F_ADD(f
, u
, u
, uu
); /* %$r = u_0 + u_1$% */
232 r
= F_ADD(f
, s
, s
, ss
); /* %$w = s_0 + s_1$% */
241 return (c
->ops
->dbl(c
, d
, a
));
249 l
= F_MUL(f
, t
, a
->z
, w
); /* %$l = z_0 w$% */
251 dz
= F_MUL(f
, dz
, l
, b
->z
); /* %$z' = l z_1$% */
253 ss
= F_MUL(f
, ss
, r
, b
->x
); /* %$r x_1$% */
254 t
= F_MUL(f
, uu
, l
, b
->y
); /* %$l y_1$% */
255 v
= F_ADD(f
, ss
, ss
, t
); /* %$v = r x_1 + l y_1$% */
257 t
= F_ADD(f
, t
, r
, dz
); /* %$t = r + z'$% */
259 uu
= F_SQR(f
, MP_NEW
, dz
); /* %$z'^2$% */
260 dx
= F_MUL(f
, MP_NEW
, uu
, cc
->a
); /* %$a z'^2$% */
261 uu
= F_MUL(f
, uu
, t
, r
); /* %$t r$% */
262 dx
= F_ADD(f
, dx
, dx
, uu
); /* %$a z'^2 + t r$% */
263 r
= F_SQR(f
, r
, w
); /* %$w^2$% */
264 uu
= F_MUL(f
, uu
, r
, w
); /* %$w^3$% */
265 dx
= F_ADD(f
, dx
, dx
, uu
); /* %$x' = a z'^2 + t r + w^3$% */
267 r
= F_SQR(f
, r
, l
); /* %$l^2$% */
268 dy
= F_MUL(f
, uu
, v
, r
); /* %$v l^2$% */
269 l
= F_MUL(f
, l
, t
, dx
); /* %$t x'$% */
270 dy
= F_ADD(f
, dy
, dy
, l
); /* %$y' = t x' + v l^2$% */
285 static int eccheck(ec_curve
*c
, const ec
*p
)
287 ecctx
*cc
= (ecctx
*)c
;
292 v
= F_SQR(f
, MP_NEW
, p
->x
);
293 u
= F_MUL(f
, MP_NEW
, v
, p
->x
);
294 v
= F_MUL(f
, v
, v
, cc
->a
);
295 u
= F_ADD(f
, u
, u
, v
);
296 u
= F_ADD(f
, u
, u
, cc
->b
);
297 v
= F_MUL(f
, v
, p
->x
, p
->y
);
298 u
= F_ADD(f
, u
, u
, v
);
299 v
= F_SQR(f
, v
, p
->y
);
300 u
= F_ADD(f
, u
, u
, v
);
307 static int ecprojcheck(ec_curve
*c
, const ec
*p
)
312 c
->ops
->fix(c
, &t
, p
);
318 static void ecdestroy(ec_curve
*c
)
320 ecctx
*cc
= (ecctx
*)c
;
323 if (cc
->bb
) MP_DROP(cc
->bb
);
327 /* --- @ec_bin@, @ec_binproj@ --- *
329 * Arguments: @field *f@ = the underlying field for this elliptic curve
330 * @mp *a, *b@ = the coefficients for this curve
332 * Returns: A pointer to the curve.
334 * Use: Creates a curve structure for an elliptic curve defined over
335 * a binary field. The @binproj@ variant uses projective
336 * coordinates, which can be a win.
339 ec_curve
*ec_bin(field
*f
, mp
*a
, mp
*b
)
341 ecctx
*cc
= CREATE(ecctx
);
342 cc
->c
.ops
= &ec_binops
;
344 cc
->a
= F_IN(f
, MP_NEW
, a
);
345 cc
->b
= F_IN(f
, MP_NEW
, b
);
350 ec_curve
*ec_binproj(field
*f
, mp
*a
, mp
*b
)
352 ecctx
*cc
= CREATE(ecctx
);
353 cc
->c
.ops
= &ec_binprojops
;
355 cc
->a
= F_IN(f
, MP_NEW
, a
);
356 cc
->b
= F_IN(f
, MP_NEW
, b
);
357 cc
->bb
= F_SQRT(f
, MP_NEW
, b
);
358 cc
->bb
= F_SQRT(f
, cc
->bb
, cc
->bb
);
362 static const ec_ops ec_binops
= {
363 ecdestroy
, ec_idin
, ec_idout
, ec_idfix
,
364 0, ecneg
, ecadd
, ec_stdsub
, ecdbl
, eccheck
367 static const ec_ops ec_binprojops
= {
368 ecdestroy
, ec_projin
, ec_projout
, ec_projfix
,
369 0, ecprojneg
, ecprojadd
, ec_stdsub
, ecprojdbl
, ecprojcheck
372 /*----- Test rig ----------------------------------------------------------*/
376 #define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
378 int main(int argc
, char *argv
[])
382 ec g
= EC_INIT
, d
= EC_INIT
;
384 int i
, n
= argc
== 1 ?
1 : atoi(argv
[1]);
389 b
= MP(0x066647ede6c332c7f8c0923bb58213b333b20e9ce4281fe115f7d8f90ad);
390 p
= MP(0x20000000000000000000000000000000000000004000000000000000001);
392 MP(6901746346790563787434755862277025555839812737345013555379383634485462);
394 f
= field_binpoly(p
);
395 c
= ec_binproj(f
, a
, b
);
397 g
.x
= MP(0x0fac9dfcbac8313bb2139f1bb755fef65bc391f8b36f8f8eb7371fd558b);
398 g
.y
= MP(0x1006a08a41903350678e58528bebf8a0beff867a7ca36716f7e01f81052);
400 for (i
= 0; i
< n
; i
++) {
401 ec_mul(c
, &d
, &g
, r
);
403 fprintf(stderr
, "zero too early\n");
406 ec_add(c
, &d
, &d
, &g
);
408 fprintf(stderr
, "didn't reach zero\n");
409 MP_EPRINTX("d.x", d
.x
);
410 MP_EPRINTX("d.y", d
.y
);
411 MP_EPRINTX("d.z", d
.y
);
420 MP_DROP(p
); MP_DROP(a
); MP_DROP(b
); MP_DROP(r
);
421 assert(!mparena_count(&mparena_global
));
428 /*----- That's all, folks -------------------------------------------------*/