3 * Arithmetic for elliptic curves over binary fields
5 * (c) 2004 Straylight/Edgeware
8 /*----- Licensing notice --------------------------------------------------*
10 * This file is part of Catacomb.
12 * Catacomb is free software; you can redistribute it and/or modify
13 * it under the terms of the GNU Library General Public License as
14 * published by the Free Software Foundation; either version 2 of the
15 * License, or (at your option) any later version.
17 * Catacomb is distributed in the hope that it will be useful,
18 * but WITHOUT ANY WARRANTY; without even the implied warranty of
19 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
20 * GNU Library General Public License for more details.
22 * You should have received a copy of the GNU Library General Public
23 * License along with Catacomb; if not, write to the Free
24 * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
28 /*----- Header files ------------------------------------------------------*/
35 /*----- Main code ---------------------------------------------------------*/
37 static const ec_ops ec_binops
, ec_binprojops
;
39 static ec
*ecneg(ec_curve
*c
, ec
*d
, const ec
*p
)
43 d
->y
= F_ADD(c
->f
, d
->y
, d
->y
, d
->x
);
47 static ec
*ecprojneg(ec_curve
*c
, ec
*d
, const ec
*p
)
51 mp
*t
= F_MUL(c
->f
, MP_NEW
, d
->x
, d
->z
);
52 d
->y
= F_ADD(c
->f
, d
->y
, d
->y
, t
);
58 static ec
*ecfind(ec_curve
*c
, ec
*d
, mp
*x
)
64 y
= F_SQRT(f
, MP_NEW
, c
->b
);
66 u
= F_SQR(f
, MP_NEW
, x
); /* %$x^2$% */
67 y
= F_MUL(f
, MP_NEW
, u
, c
->a
); /* %$a x^2$% */
68 y
= F_ADD(f
, y
, y
, c
->b
); /* %$a x^2 + b$% */
69 v
= F_MUL(f
, MP_NEW
, u
, x
); /* %$x^3$% */
70 y
= F_ADD(f
, y
, y
, v
); /* %$A = x^3 + a x^2 + b$% */
72 u
= F_INV(f
, u
, u
); /* %$x^{-2}$% */
73 v
= F_MUL(f
, v
, u
, y
); /* %$B = A x^{-2} = x + a + b x^{-2}$% */
74 y
= F_QUADSOLVE(f
, y
, v
); /* %$z^2 + z = B$% */
75 if (y
) y
= F_MUL(f
, y
, y
, x
); /* %$y = z x$% */
84 d
->z
= MP_COPY(f
->one
);
88 static ec
*ecdbl(ec_curve
*c
, ec
*d
, const ec
*a
)
90 if (EC_ATINF(a
) || F_ZEROP(c
->f
, a
->x
))
97 dx
= F_INV(f
, MP_NEW
, a
->x
); /* %$x^{-1}$% */
98 dy
= F_MUL(f
, MP_NEW
, dx
, a
->y
); /* %$y/x$% */
99 lambda
= F_ADD(f
, dy
, dy
, a
->x
); /* %$\lambda = x + y/x$% */
101 dx
= F_SQR(f
, dx
, lambda
); /* %$\lambda^2$% */
102 dx
= F_ADD(f
, dx
, dx
, lambda
); /* %$\lambda^2 + \lambda$% */
103 dx
= F_ADD(f
, dx
, dx
, c
->a
); /* %$x' = a + \lambda^2 + \lambda$% */
105 dy
= F_ADD(f
, MP_NEW
, a
->x
, dx
); /* %$ x + x' $% */
106 dy
= F_MUL(f
, dy
, dy
, lambda
); /* %$ (x + x') \lambda$% */
107 dy
= F_ADD(f
, dy
, dy
, a
->y
); /* %$ (x + x') \lambda + y$% */
108 dy
= F_ADD(f
, dy
, dy
, dx
); /* %$ y' = (x + x') \lambda + y + x'$% */
119 static ec
*ecprojdbl(ec_curve
*c
, ec
*d
, const ec
*a
)
121 if (EC_ATINF(a
) || F_ZEROP(c
->f
, a
->x
))
125 ecctx_bin
*cc
= (ecctx_bin
*)c
;
126 mp
*dx
, *dy
, *dz
, *u
, *v
;
128 dy
= F_SQR(f
, MP_NEW
, a
->z
); /* %$z^2$% */
129 dx
= F_MUL(f
, MP_NEW
, dy
, cc
->bb
); /* %$c z^2$% */
130 dx
= F_ADD(f
, dx
, dx
, a
->x
); /* %$x + c z^2$% */
131 dz
= F_SQR(f
, MP_NEW
, dx
); /* %$(x + c z^2)^2$% */
132 dx
= F_SQR(f
, dx
, dz
); /* %$x' = (x + c z^2)^4$% */
134 dz
= F_MUL(f
, dz
, dy
, a
->x
); /* %$z' = x z^2$% */
136 dy
= F_SQR(f
, dy
, a
->x
); /* %$x^2$% */
137 u
= F_MUL(f
, MP_NEW
, a
->y
, a
->z
); /* %$y z$% */
138 u
= F_ADD(f
, u
, u
, dz
); /* %$z' + y z$% */
139 u
= F_ADD(f
, u
, u
, dy
); /* %$u = z' + x^2 + y z$% */
141 v
= F_SQR(f
, MP_NEW
, dy
); /* %$x^4$% */
142 dy
= F_MUL(f
, dy
, v
, dz
); /* %$x^4 z'$% */
143 v
= F_MUL(f
, v
, u
, dx
); /* %$u x'$% */
144 dy
= F_ADD(f
, dy
, dy
, v
); /* %$y' = x^4 z' + u x'$% */
156 static ec
*ecadd(ec_curve
*c
, ec
*d
, const ec
*a
, const ec
*b
)
160 else if (EC_ATINF(a
))
162 else if (EC_ATINF(b
))
169 if (!MP_EQ(a
->x
, b
->x
)) {
170 dx
= F_ADD(f
, MP_NEW
, a
->x
, b
->x
); /* %$x_0 + x_1$% */
171 dy
= F_INV(f
, MP_NEW
, dx
); /* %$(x_0 + x_1)^{-1}$% */
172 dx
= F_ADD(f
, dx
, a
->y
, b
->y
); /* %$y_0 + y_1$% */
173 lambda
= F_MUL(f
, MP_NEW
, dy
, dx
);
174 /* %$\lambda = (y_0 + y_1)/(x_0 + x_1)$% */
176 dx
= F_SQR(f
, dx
, lambda
); /* %$\lambda^2$% */
177 dx
= F_ADD(f
, dx
, dx
, lambda
); /* %$\lambda^2 + \lambda$% */
178 dx
= F_ADD(f
, dx
, dx
, c
->a
); /* %$a + \lambda^2 + \lambda$% */
179 dx
= F_ADD(f
, dx
, dx
, a
->x
); /* %$a + \lambda^2 + \lambda + x_0$% */
180 dx
= F_ADD(f
, dx
, dx
, b
->x
);
181 /* %$x' = a + \lambda^2 + \lambda + x_0 + x_1$% */
182 } else if (!MP_EQ(a
->y
, b
->y
) || F_ZEROP(f
, a
->x
)) {
186 dx
= F_INV(f
, MP_NEW
, a
->x
); /* %$x^{-1}$% */
187 dy
= F_MUL(f
, MP_NEW
, dx
, a
->y
); /* %$y/x$% */
188 lambda
= F_ADD(f
, dy
, dy
, a
->x
); /* %$\lambda = x + y/x$% */
190 dx
= F_SQR(f
, dx
, lambda
); /* %$\lambda^2$% */
191 dx
= F_ADD(f
, dx
, dx
, lambda
); /* %$\lambda^2 + \lambda$% */
192 dx
= F_ADD(f
, dx
, dx
, c
->a
); /* %$x' = a + \lambda^2 + \lambda$% */
196 dy
= F_ADD(f
, dy
, a
->x
, dx
); /* %$ x + x' $% */
197 dy
= F_MUL(f
, dy
, dy
, lambda
); /* %$ (x + x') \lambda$% */
198 dy
= F_ADD(f
, dy
, dy
, a
->y
); /* %$ (x + x') \lambda + y$% */
199 dy
= F_ADD(f
, dy
, dy
, dx
); /* %$ y' = (x + x') \lambda + y + x'$% */
210 static ec
*ecprojadd(ec_curve
*c
, ec
*d
, const ec
*a
, const ec
*b
)
213 c
->ops
->dbl(c
, d
, a
);
214 else if (EC_ATINF(a
))
216 else if (EC_ATINF(b
))
220 mp
*dx
, *dy
, *dz
, *u
, *uu
, *v
, *t
, *s
, *ss
, *r
, *w
, *l
;
222 dz
= F_SQR(f
, MP_NEW
, b
->z
); /* %$z_1^2$% */
223 u
= F_MUL(f
, MP_NEW
, dz
, a
->x
); /* %$u_0 = x_0 z_1^2$% */
224 t
= F_MUL(f
, MP_NEW
, dz
, b
->z
); /* %$z_1^3$% */
225 s
= F_MUL(f
, MP_NEW
, t
, a
->y
); /* %$s_0 = y_0 z_1^3$% */
227 dz
= F_SQR(f
, dz
, a
->z
); /* %$z_0^2$% */
228 uu
= F_MUL(f
, MP_NEW
, dz
, b
->x
); /* %$u_1 = x_1 z_0^2$% */
229 t
= F_MUL(f
, t
, dz
, a
->z
); /* %$z_0^3$% */
230 ss
= F_MUL(f
, MP_NEW
, t
, b
->y
); /* %$s_1 = y_1 z_0^3$% */
232 w
= F_ADD(f
, u
, u
, uu
); /* %$r = u_0 + u_1$% */
233 r
= F_ADD(f
, s
, s
, ss
); /* %$w = s_0 + s_1$% */
242 return (c
->ops
->dbl(c
, d
, a
));
250 l
= F_MUL(f
, t
, a
->z
, w
); /* %$l = z_0 w$% */
252 dz
= F_MUL(f
, dz
, l
, b
->z
); /* %$z' = l z_1$% */
254 ss
= F_MUL(f
, ss
, r
, b
->x
); /* %$r x_1$% */
255 t
= F_MUL(f
, uu
, l
, b
->y
); /* %$l y_1$% */
256 v
= F_ADD(f
, ss
, ss
, t
); /* %$v = r x_1 + l y_1$% */
258 t
= F_ADD(f
, t
, r
, dz
); /* %$t = r + z'$% */
260 uu
= F_SQR(f
, MP_NEW
, dz
); /* %$z'^2$% */
261 dx
= F_MUL(f
, MP_NEW
, uu
, c
->a
); /* %$a z'^2$% */
262 uu
= F_MUL(f
, uu
, t
, r
); /* %$t r$% */
263 dx
= F_ADD(f
, dx
, dx
, uu
); /* %$a z'^2 + t r$% */
264 r
= F_SQR(f
, r
, w
); /* %$w^2$% */
265 uu
= F_MUL(f
, uu
, r
, w
); /* %$w^3$% */
266 dx
= F_ADD(f
, dx
, dx
, uu
); /* %$x' = a z'^2 + t r + w^3$% */
268 r
= F_SQR(f
, r
, l
); /* %$l^2$% */
269 dy
= F_MUL(f
, uu
, v
, r
); /* %$v l^2$% */
270 l
= F_MUL(f
, l
, t
, dx
); /* %$t x'$% */
271 dy
= F_ADD(f
, dy
, dy
, l
); /* %$y' = t x' + v l^2$% */
286 static int eccheck(ec_curve
*c
, const ec
*p
)
292 if (EC_ATINF(p
)) return (0);
293 v
= F_SQR(f
, MP_NEW
, p
->x
);
294 u
= F_MUL(f
, MP_NEW
, v
, p
->x
);
295 v
= F_MUL(f
, v
, v
, c
->a
);
296 u
= F_ADD(f
, u
, u
, v
);
297 u
= F_ADD(f
, u
, u
, c
->b
);
298 v
= F_MUL(f
, v
, p
->x
, p
->y
);
299 u
= F_ADD(f
, u
, u
, v
);
300 v
= F_SQR(f
, v
, p
->y
);
301 u
= F_ADD(f
, u
, u
, v
);
302 rc
= F_ZEROP(f
, u
) ?
0 : -1;
308 static int ecprojcheck(ec_curve
*c
, const ec
*p
)
313 c
->ops
->fix(c
, &t
, p
);
319 static void ecdestroy(ec_curve
*c
)
321 ecctx_bin
*cc
= (ecctx_bin
*)c
;
324 if (cc
->bb
) MP_DROP(cc
->bb
);
328 /* --- @ec_bin@, @ec_binproj@ --- *
330 * Arguments: @field *f@ = the underlying field for this elliptic curve
331 * @mp *a, *b@ = the coefficients for this curve
333 * Returns: A pointer to the curve, or null.
335 * Use: Creates a curve structure for an elliptic curve defined over
336 * a binary field. The @binproj@ variant uses projective
337 * coordinates, which can be a win.
340 ec_curve
*ec_bin(field
*f
, mp
*a
, mp
*b
)
342 ecctx_bin
*cc
= CREATE(ecctx_bin
);
343 cc
->c
.ops
= &ec_binops
;
345 cc
->c
.a
= F_IN(f
, MP_NEW
, a
);
346 cc
->c
.b
= F_IN(f
, MP_NEW
, b
);
351 ec_curve
*ec_binproj(field
*f
, mp
*a
, mp
*b
)
353 ecctx_bin
*cc
= CREATE(ecctx_bin
);
357 cc
->c
.ops
= &ec_binprojops
;
359 cc
->c
.a
= F_IN(f
, MP_NEW
, a
);
360 cc
->c
.b
= F_IN(f
, MP_NEW
, b
);
362 c
= MP_COPY(cc
->c
.b
);
363 for (i
= 0; i
< f
->nbits
- 2; i
++)
365 d
= F_SQR(f
, MP_NEW
, c
); d
= F_SQR(f
, d
, d
);
366 if (!MP_EQ(d
, cc
->c
.b
)) {
379 static const ec_ops ec_binops
= {
381 ecdestroy
, ec_stdsamep
, ec_idin
, ec_idout
, ec_idfix
,
382 ecfind
, ecneg
, ecadd
, ec_stdsub
, ecdbl
, eccheck
385 static const ec_ops ec_binprojops
= {
387 ecdestroy
, ec_stdsamep
, ec_projin
, ec_projout
, ec_projfix
,
388 ecfind
, ecprojneg
, ecprojadd
, ec_stdsub
, ecprojdbl
, ecprojcheck
391 /*----- Test rig ----------------------------------------------------------*/
395 #define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
397 int main(int argc
, char *argv
[])
401 ec g
= EC_INIT
, d
= EC_INIT
;
402 mp
*p
, *a
, *b
, *r
, *beta
;
403 int i
, n
= argc
== 1 ?
1 : atoi(argv
[1]);
407 a
= MP(0x7ffffffffffffffffffffffffffffffffffffffff);
408 b
= MP(0x6645f3cacf1638e139c6cd13ef61734fbc9e3d9fb);
409 p
= MP(0x800000000000000000000000000000000000000c9);
410 beta
= MP(0x715169c109c612e390d347c748342bcd3b02a0bef);
411 r
= MP(0x040000000000000000000292fe77e70c12a4234c32);
413 f
= field_binnorm(p
, beta
);
414 c
= ec_binproj(f
, a
, b
);
415 g
.x
= MP(0x0311103c17167564ace77ccb09c681f886ba54ee8);
416 g
.y
= MP(0x333ac13c6447f2e67613bf7009daf98c87bb50c7f);
418 for (i
= 0; i
< n
; i
++) {
419 ec_mul(c
, &d
, &g
, r
);
421 fprintf(stderr
, "zero too early\n");
424 ec_add(c
, &d
, &d
, &g
);
426 fprintf(stderr
, "didn't reach zero\n");
427 MP_EPRINTX("d.x", d
.x
);
428 MP_EPRINTX("d.y", d
.y
);
437 MP_DROP(p
); MP_DROP(a
); MP_DROP(b
); MP_DROP(r
); MP_DROP(beta
);
438 assert(!mparena_count(&mparena_global
));
445 /*----- That's all, folks -------------------------------------------------*/