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 /*----- Header files ------------------------------------------------------*/
37 /*----- Main code ---------------------------------------------------------*/
39 static const ec_ops ec_binops
, ec_binprojops
;
41 static ec
*ecneg(ec_curve
*c
, ec
*d
, const ec
*p
)
45 d
->y
= F_ADD(c
->f
, d
->y
, d
->y
, d
->x
);
49 static ec
*ecprojneg(ec_curve
*c
, ec
*d
, const ec
*p
)
53 mp
*t
= F_MUL(c
->f
, MP_NEW
, d
->x
, d
->z
);
54 d
->y
= F_ADD(c
->f
, d
->y
, d
->y
, t
);
60 static ec
*ecfind(ec_curve
*c
, ec
*d
, mp
*x
)
66 y
= F_SQRT(f
, MP_NEW
, c
->b
);
68 u
= F_SQR(f
, MP_NEW
, x
); /* %$x^2$% */
69 y
= F_MUL(f
, MP_NEW
, u
, c
->a
); /* %$a x^2$% */
70 y
= F_ADD(f
, y
, y
, c
->b
); /* %$a x^2 + b$% */
71 v
= F_MUL(f
, MP_NEW
, u
, x
); /* %$x^3$% */
72 y
= F_ADD(f
, y
, y
, v
); /* %$A = x^3 + a x^2 + b$% */
74 u
= F_INV(f
, u
, u
); /* %$x^{-2}$% */
75 v
= F_MUL(f
, v
, u
, y
); /* %$B = A x^{-2} = x + a + b x^{-2}$% */
76 y
= F_QUADSOLVE(f
, y
, v
); /* %$z^2 + z = B$% */
77 if (y
) y
= F_MUL(f
, y
, y
, x
); /* %$y = z x$% */
86 d
->z
= MP_COPY(f
->one
);
90 static ec
*ecdbl(ec_curve
*c
, ec
*d
, const ec
*a
)
92 if (EC_ATINF(a
) || F_ZEROP(c
->f
, a
->x
))
99 dx
= F_INV(f
, MP_NEW
, a
->x
); /* %$x^{-1}$% */
100 dy
= F_MUL(f
, MP_NEW
, dx
, a
->y
); /* %$y/x$% */
101 lambda
= F_ADD(f
, dy
, dy
, a
->x
); /* %$\lambda = x + y/x$% */
103 dx
= F_SQR(f
, dx
, lambda
); /* %$\lambda^2$% */
104 dx
= F_ADD(f
, dx
, dx
, lambda
); /* %$\lambda^2 + \lambda$% */
105 dx
= F_ADD(f
, dx
, dx
, c
->a
); /* %$x' = a + \lambda^2 + \lambda$% */
107 dy
= F_ADD(f
, MP_NEW
, a
->x
, dx
); /* %$ x + x' $% */
108 dy
= F_MUL(f
, dy
, dy
, lambda
); /* %$ (x + x') \lambda$% */
109 dy
= F_ADD(f
, dy
, dy
, a
->y
); /* %$ (x + x') \lambda + y$% */
110 dy
= F_ADD(f
, dy
, dy
, dx
); /* %$ y' = (x + x') \lambda + y + x'$% */
121 static ec
*ecprojdbl(ec_curve
*c
, ec
*d
, const ec
*a
)
123 if (EC_ATINF(a
) || F_ZEROP(c
->f
, a
->x
))
127 ecctx_bin
*cc
= (ecctx_bin
*)c
;
128 mp
*dx
, *dy
, *dz
, *u
, *v
;
130 dy
= F_SQR(f
, MP_NEW
, a
->z
); /* %$z^2$% */
131 dx
= F_MUL(f
, MP_NEW
, dy
, cc
->bb
); /* %$c z^2$% */
132 dx
= F_ADD(f
, dx
, dx
, a
->x
); /* %$x + c z^2$% */
133 dz
= F_SQR(f
, MP_NEW
, dx
); /* %$(x + c z^2)^2$% */
134 dx
= F_SQR(f
, dx
, dz
); /* %$x' = (x + c z^2)^4$% */
136 dz
= F_MUL(f
, dz
, dy
, a
->x
); /* %$z' = x z^2$% */
138 dy
= F_SQR(f
, dy
, a
->x
); /* %$x^2$% */
139 u
= F_MUL(f
, MP_NEW
, a
->y
, a
->z
); /* %$y z$% */
140 u
= F_ADD(f
, u
, u
, dz
); /* %$z' + y z$% */
141 u
= F_ADD(f
, u
, u
, dy
); /* %$u = z' + x^2 + y z$% */
143 v
= F_SQR(f
, MP_NEW
, dy
); /* %$x^4$% */
144 dy
= F_MUL(f
, dy
, v
, dz
); /* %$x^4 z'$% */
145 v
= F_MUL(f
, v
, u
, dx
); /* %$u x'$% */
146 dy
= F_ADD(f
, dy
, dy
, v
); /* %$y' = x^4 z' + u x'$% */
158 static ec
*ecadd(ec_curve
*c
, ec
*d
, const ec
*a
, const ec
*b
)
162 else if (EC_ATINF(a
))
164 else if (EC_ATINF(b
))
171 if (!MP_EQ(a
->x
, b
->x
)) {
172 dx
= F_ADD(f
, MP_NEW
, a
->x
, b
->x
); /* %$x_0 + x_1$% */
173 dy
= F_INV(f
, MP_NEW
, dx
); /* %$(x_0 + x_1)^{-1}$% */
174 dx
= F_ADD(f
, dx
, a
->y
, b
->y
); /* %$y_0 + y_1$% */
175 lambda
= F_MUL(f
, MP_NEW
, dy
, dx
);
176 /* %$\lambda = (y_0 + y_1)/(x_0 + x_1)$% */
178 dx
= F_SQR(f
, dx
, lambda
); /* %$\lambda^2$% */
179 dx
= F_ADD(f
, dx
, dx
, lambda
); /* %$\lambda^2 + \lambda$% */
180 dx
= F_ADD(f
, dx
, dx
, c
->a
); /* %$a + \lambda^2 + \lambda$% */
181 dx
= F_ADD(f
, dx
, dx
, a
->x
); /* %$a + \lambda^2 + \lambda + x_0$% */
182 dx
= F_ADD(f
, dx
, dx
, b
->x
);
183 /* %$x' = a + \lambda^2 + \lambda + x_0 + x_1$% */
184 } else if (!MP_EQ(a
->y
, b
->y
) || F_ZEROP(f
, a
->x
)) {
188 dx
= F_INV(f
, MP_NEW
, a
->x
); /* %$x^{-1}$% */
189 dy
= F_MUL(f
, MP_NEW
, dx
, a
->y
); /* %$y/x$% */
190 lambda
= F_ADD(f
, dy
, dy
, a
->x
); /* %$\lambda = x + y/x$% */
192 dx
= F_SQR(f
, dx
, lambda
); /* %$\lambda^2$% */
193 dx
= F_ADD(f
, dx
, dx
, lambda
); /* %$\lambda^2 + \lambda$% */
194 dx
= F_ADD(f
, dx
, dx
, c
->a
); /* %$x' = a + \lambda^2 + \lambda$% */
198 dy
= F_ADD(f
, dy
, a
->x
, dx
); /* %$ x + x' $% */
199 dy
= F_MUL(f
, dy
, dy
, lambda
); /* %$ (x + x') \lambda$% */
200 dy
= F_ADD(f
, dy
, dy
, a
->y
); /* %$ (x + x') \lambda + y$% */
201 dy
= F_ADD(f
, dy
, dy
, dx
); /* %$ y' = (x + x') \lambda + y + x'$% */
212 static ec
*ecprojadd(ec_curve
*c
, ec
*d
, const ec
*a
, const ec
*b
)
215 c
->ops
->dbl(c
, d
, a
);
216 else if (EC_ATINF(a
))
218 else if (EC_ATINF(b
))
222 mp
*dx
, *dy
, *dz
, *u
, *uu
, *v
, *t
, *s
, *ss
, *r
, *w
, *l
;
224 dz
= F_SQR(f
, MP_NEW
, b
->z
); /* %$z_1^2$% */
225 u
= F_MUL(f
, MP_NEW
, dz
, a
->x
); /* %$u_0 = x_0 z_1^2$% */
226 t
= F_MUL(f
, MP_NEW
, dz
, b
->z
); /* %$z_1^3$% */
227 s
= F_MUL(f
, MP_NEW
, t
, a
->y
); /* %$s_0 = y_0 z_1^3$% */
229 dz
= F_SQR(f
, dz
, a
->z
); /* %$z_0^2$% */
230 uu
= F_MUL(f
, MP_NEW
, dz
, b
->x
); /* %$u_1 = x_1 z_0^2$% */
231 t
= F_MUL(f
, t
, dz
, a
->z
); /* %$z_0^3$% */
232 ss
= F_MUL(f
, MP_NEW
, t
, b
->y
); /* %$s_1 = y_1 z_0^3$% */
234 w
= F_ADD(f
, u
, u
, uu
); /* %$r = u_0 + u_1$% */
235 r
= F_ADD(f
, s
, s
, ss
); /* %$w = s_0 + s_1$% */
244 return (c
->ops
->dbl(c
, d
, a
));
252 l
= F_MUL(f
, t
, a
->z
, w
); /* %$l = z_0 w$% */
254 dz
= F_MUL(f
, dz
, l
, b
->z
); /* %$z' = l z_1$% */
256 ss
= F_MUL(f
, ss
, r
, b
->x
); /* %$r x_1$% */
257 t
= F_MUL(f
, uu
, l
, b
->y
); /* %$l y_1$% */
258 v
= F_ADD(f
, ss
, ss
, t
); /* %$v = r x_1 + l y_1$% */
260 t
= F_ADD(f
, t
, r
, dz
); /* %$t = r + z'$% */
262 uu
= F_SQR(f
, MP_NEW
, dz
); /* %$z'^2$% */
263 dx
= F_MUL(f
, MP_NEW
, uu
, c
->a
); /* %$a z'^2$% */
264 uu
= F_MUL(f
, uu
, t
, r
); /* %$t r$% */
265 dx
= F_ADD(f
, dx
, dx
, uu
); /* %$a z'^2 + t r$% */
266 r
= F_SQR(f
, r
, w
); /* %$w^2$% */
267 uu
= F_MUL(f
, uu
, r
, w
); /* %$w^3$% */
268 dx
= F_ADD(f
, dx
, dx
, uu
); /* %$x' = a z'^2 + t r + w^3$% */
270 r
= F_SQR(f
, r
, l
); /* %$l^2$% */
271 dy
= F_MUL(f
, uu
, v
, r
); /* %$v l^2$% */
272 l
= F_MUL(f
, l
, t
, dx
); /* %$t x'$% */
273 dy
= F_ADD(f
, dy
, dy
, l
); /* %$y' = t x' + v l^2$% */
288 static int eccheck(ec_curve
*c
, const ec
*p
)
294 if (EC_ATINF(p
)) return (0);
295 v
= F_SQR(f
, MP_NEW
, p
->x
);
296 u
= F_MUL(f
, MP_NEW
, v
, p
->x
);
297 v
= F_MUL(f
, v
, v
, c
->a
);
298 u
= F_ADD(f
, u
, u
, v
);
299 u
= F_ADD(f
, u
, u
, c
->b
);
300 v
= F_MUL(f
, v
, p
->x
, p
->y
);
301 u
= F_ADD(f
, u
, u
, v
);
302 v
= F_SQR(f
, v
, p
->y
);
303 u
= F_ADD(f
, u
, u
, v
);
304 rc
= F_ZEROP(f
, u
) ?
0 : -1;
310 static int ecprojcheck(ec_curve
*c
, const ec
*p
)
315 c
->ops
->fix(c
, &t
, p
);
321 static void ecdestroy(ec_curve
*c
)
323 ecctx_bin
*cc
= (ecctx_bin
*)c
;
326 if (cc
->bb
) MP_DROP(cc
->bb
);
330 /* --- @ec_bin@, @ec_binproj@ --- *
332 * Arguments: @field *f@ = the underlying field for this elliptic curve
333 * @mp *a, *b@ = the coefficients for this curve
335 * Returns: A pointer to the curve, or null.
337 * Use: Creates a curve structure for an elliptic curve defined over
338 * a binary field. The @binproj@ variant uses projective
339 * coordinates, which can be a win.
342 ec_curve
*ec_bin(field
*f
, mp
*a
, mp
*b
)
344 ecctx_bin
*cc
= CREATE(ecctx_bin
);
345 cc
->c
.ops
= &ec_binops
;
347 cc
->c
.a
= F_IN(f
, MP_NEW
, a
);
348 cc
->c
.b
= F_IN(f
, MP_NEW
, b
);
353 ec_curve
*ec_binproj(field
*f
, mp
*a
, mp
*b
)
355 ecctx_bin
*cc
= CREATE(ecctx_bin
);
356 cc
->c
.ops
= &ec_binprojops
;
358 cc
->c
.a
= F_IN(f
, MP_NEW
, a
);
359 cc
->c
.b
= F_IN(f
, MP_NEW
, b
);
360 cc
->bb
= F_SQRT(f
, MP_NEW
, cc
->c
.b
);
362 cc
->bb
= F_SQRT(f
, cc
->bb
, cc
->bb
);
372 static const ec_ops ec_binops
= {
374 ecdestroy
, ec_stdsamep
, ec_idin
, ec_idout
, ec_idfix
,
375 ecfind
, ecneg
, ecadd
, ec_stdsub
, ecdbl
, eccheck
378 static const ec_ops ec_binprojops
= {
380 ecdestroy
, ec_stdsamep
, ec_projin
, ec_projout
, ec_projfix
,
381 ecfind
, ecprojneg
, ecprojadd
, ec_stdsub
, ecprojdbl
, ecprojcheck
384 /*----- Test rig ----------------------------------------------------------*/
388 #define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
390 int main(int argc
, char *argv
[])
394 ec g
= EC_INIT
, d
= EC_INIT
;
395 mp
*p
, *a
, *b
, *r
, *beta
;
396 int i
, n
= argc
== 1 ?
1 : atoi(argv
[1]);
400 a
= MP(0x7ffffffffffffffffffffffffffffffffffffffff);
401 b
= MP(0x6645f3cacf1638e139c6cd13ef61734fbc9e3d9fb);
402 p
= MP(0x800000000000000000000000000000000000000c9);
403 beta
= MP(0x715169c109c612e390d347c748342bcd3b02a0bef);
404 r
= MP(0x040000000000000000000292fe77e70c12a4234c32);
406 f
= field_binnorm(p
, beta
);
407 c
= ec_binproj(f
, a
, b
);
408 g
.x
= MP(0x0311103c17167564ace77ccb09c681f886ba54ee8);
409 g
.y
= MP(0x333ac13c6447f2e67613bf7009daf98c87bb50c7f);
411 for (i
= 0; i
< n
; i
++) {
412 ec_mul(c
, &d
, &g
, r
);
414 fprintf(stderr
, "zero too early\n");
417 ec_add(c
, &d
, &d
, &g
);
419 fprintf(stderr
, "didn't reach zero\n");
420 MP_EPRINTX("d.x", d
.x
);
421 MP_EPRINTX("d.y", d
.y
);
430 MP_DROP(p
); MP_DROP(a
); MP_DROP(b
); MP_DROP(r
); MP_DROP(beta
);
431 assert(!mparena_count(&mparena_global
));
438 /*----- That's all, folks -------------------------------------------------*/