git.distorted.org.uk Git - u/mdw/catacomb/blob - ec-bin.c

   1 /* -*-c-*-
   2  *
   3  * $Id$
   4  *
   5  * Arithmetic for elliptic curves over binary fields
   6  *
   7  * (c) 2004 Straylight/Edgeware
   8  */
   9
  10 /*----- Licensing notice --------------------------------------------------*
  11  *
  12  * This file is part of Catacomb.
  13  *
  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.
  18  *
  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.
  23  *
  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,
  27  * MA 02111-1307, USA.
  28  */
  29
  30 /*----- Header files ------------------------------------------------------*/
  31
  32 #include <mLib/sub.h>
  33
  34 #include "ec.h"
  35 #include "ec-guts.h"
  36
  37 /*----- Main code ---------------------------------------------------------*/
  38
  39 static const ec_ops ec_binops, ec_binprojops;
  40
  41 static ec *ecneg(ec_curve *c, ec *d, const ec *p)
  42 {
  43   EC_COPY(d, p);
  44   if (d->x)
  45     d->y = F_ADD(c->f, d->y, d->y, d->x);
  46   return (d);
  47 }
  48
  49 static ec *ecprojneg(ec_curve *c, ec *d, const ec *p)
  50 {
  51   EC_COPY(d, p);
  52   if (d->x) {
  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);
  55     MP_DROP(t);
  56   }
  57   return (d);
  58 }
  59
  60 static ec *ecfind(ec_curve *c, ec *d, mp *x)
  61 {
  62   field *f = c->f;
  63   mp *y, *u, *v;
  64
  65   if (F_ZEROP(f, x))
  66     y = F_SQRT(f, MP_NEW, c->b);
  67   else {
  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);              /* %$A = x^3 + a x^2 + b$% */
  73     if (!F_ZEROP(f, y)) {
  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);     /* %$y = z x$% */
  78     }
  79     MP_DROP(u);
  80     MP_DROP(v);
  81   }
  82   if (!y) return (0);
  83   EC_DESTROY(d);
  84   d->x = MP_COPY(x);
  85   d->y = y;
  86   d->z = MP_COPY(f->one);
  87   return (d);
  88 }
  89
  90 static ec *ecdbl(ec_curve *c, ec *d, const ec *a)
  91 {
  92   if (EC_ATINF(a) || F_ZEROP(c->f, a->x))
  93     EC_SETINF(d);
  94   else {
  95     field *f = c->f;
  96     mp *lambda;
  97     mp *dx, *dy;
  98
  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);    /* %$\lambda = x + y/x$% */
 102
 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);       /* %$x' = a + \lambda^2 + \lambda$% */
 106
 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'$% */
 111
 112     EC_DESTROY(d);
 113     d->x = dx;
 114     d->y = dy;
 115     d->z = 0;
 116     MP_DROP(lambda);
 117   }
 118   return (d);
 119 }
 120
 121 static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a)
 122 {
 123   if (EC_ATINF(a) || F_ZEROP(c->f, a->x))
 124     EC_SETINF(d);
 125   else {
 126     field *f = c->f;
 127     ecctx_bin *cc = (ecctx_bin *)c;
 128     mp *dx, *dy, *dz, *u, *v;
 129
 130     dy = F_SQR(f, MP_NEW, a->z);        /* %$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$% */
 135
 136     dz = F_MUL(f, dz, dy, a->x);        /* %$z' = x z^2$% */
 137
 138     dy = F_SQR(f, dy, a->x);            /* %$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$% */
 142
 143     v = F_SQR(f, MP_NEW, dy);           /* %$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'$% */
 147
 148     EC_DESTROY(d);
 149     d->x = dx;
 150     d->y = dy;
 151     d->z = dz;
 152     MP_DROP(u);
 153     MP_DROP(v);
 154   }
 155   return (d);
 156 }
 157
 158 static ec *ecadd(ec_curve *c, ec *d, const ec *a, const ec *b)
 159 {
 160   if (a == b)
 161     ecdbl(c, d, a);
 162   else if (EC_ATINF(a))
 163     EC_COPY(d, b);
 164   else if (EC_ATINF(b))
 165     EC_COPY(d, a);
 166   else {
 167     field *f = c->f;
 168     mp *lambda;
 169     mp *dx, *dy;
 170
 171     if (!MP_EQ(a->x, b->x)) {
 172       dx = F_ADD(f, MP_NEW, a->x, b->x); /* %$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)$% */
 177
 178       dx = F_SQR(f, dx, lambda);        /* %$\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$% */
 184     } else if (!MP_EQ(a->y, b->y) || F_ZEROP(f, a->x)) {
 185       EC_SETINF(d);
 186       return (d);
 187     } else {
 188       dx = F_INV(f, MP_NEW, a->x);      /* %$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$% */
 191
 192       dx = F_SQR(f, dx, lambda);        /* %$\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$% */
 195       dy = MP_NEW;
 196     }
 197
 198     dy = F_ADD(f, dy, a->x, dx);        /* %$ 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'$% */
 202
 203     EC_DESTROY(d);
 204     d->x = dx;
 205     d->y = dy;
 206     d->z = 0;
 207     MP_DROP(lambda);
 208   }
 209   return (d);
 210 }
 211
 212 static ec *ecprojadd(ec_curve *c, ec *d, const ec *a, const ec *b)
 213 {
 214   if (a == b)
 215     c->ops->dbl(c, d, a);
 216   else if (EC_ATINF(a))
 217     EC_COPY(d, b);
 218   else if (EC_ATINF(b))
 219     EC_COPY(d, a);
 220   else {
 221     field *f = c->f;
 222     mp *dx, *dy, *dz, *u, *uu, *v, *t, *s, *ss, *r, *w, *l;
 223
 224     dz = F_SQR(f, MP_NEW, b->z);        /* %$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$% */
 228
 229     dz = F_SQR(f, dz, a->z);            /* %$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$% */
 233
 234     w = F_ADD(f, u, u, uu);             /* %$r = u_0 + u_1$% */
 235     r = F_ADD(f, s, s, ss);             /* %$w = s_0 + s_1$% */
 236     if (F_ZEROP(f, w)) {
 237       MP_DROP(w);
 238       MP_DROP(uu);
 239       MP_DROP(ss);
 240       MP_DROP(t);
 241       MP_DROP(dz);
 242       if (F_ZEROP(f, r)) {
 243         MP_DROP(r);
 244         return (c->ops->dbl(c, d, a));
 245       } else {
 246         MP_DROP(r);
 247         EC_SETINF(d);
 248         return (d);
 249       }
 250     }
 251
 252     l = F_MUL(f, t, a->z, w);           /* %$l = z_0 w$% */
 253
 254     dz = F_MUL(f, dz, l, b->z);         /* %$z' = l z_1$% */
 255
 256     ss = F_MUL(f, ss, r, b->x);         /* %$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$% */
 259
 260     t = F_ADD(f, t, r, dz);             /* %$t = r + z'$% */
 261
 262     uu = F_SQR(f, MP_NEW, dz);          /* %$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$% */
 269
 270     r = F_SQR(f, r, l);                 /* %$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$% */
 274
 275     EC_DESTROY(d);
 276     d->x = dx;
 277     d->y = dy;
 278     d->z = dz;
 279     MP_DROP(l);
 280     MP_DROP(r);
 281     MP_DROP(w);
 282     MP_DROP(t);
 283     MP_DROP(v);
 284   }
 285   return (d);
 286 }
 287
 288 static int eccheck(ec_curve *c, const ec *p)
 289 {
 290   field *f = c->f;
 291   int rc;
 292   mp *u, *v;
 293
 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;
 305   mp_drop(u);
 306   mp_drop(v);
 307   return (rc);
 308 }
 309
 310 static int ecprojcheck(ec_curve *c, const ec *p)
 311 {
 312   ec t = EC_INIT;
 313   int rc;
 314
 315   c->ops->fix(c, &t, p);
 316   rc = eccheck(c, &t);
 317   EC_DESTROY(&t);
 318   return (rc);
 319 }
 320
 321 static void ecdestroy(ec_curve *c)
 322 {
 323   ecctx_bin *cc = (ecctx_bin *)c;
 324   MP_DROP(cc->c.a);
 325   MP_DROP(cc->c.b);
 326   if (cc->bb) MP_DROP(cc->bb);
 327   DESTROY(cc);
 328 }
 329
 330 /* --- @ec_bin@, @ec_binproj@ --- *
 331  *
 332  * Arguments:   @field *f@ = the underlying field for this elliptic curve
 333  *              @mp *a, *b@ = the coefficients for this curve
 334  *
 335  * Returns:     A pointer to the curve, or null.
 336  *
 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.
 340  */
 341
 342 ec_curve *ec_bin(field *f, mp *a, mp *b)
 343 {
 344   ecctx_bin *cc = CREATE(ecctx_bin);
 345   cc->c.ops = &ec_binops;
 346   cc->c.f = f;
 347   cc->c.a = F_IN(f, MP_NEW, a);
 348   cc->c.b = F_IN(f, MP_NEW, b);
 349   cc->bb = 0;
 350   return (&cc->c);
 351 }
 352
 353 ec_curve *ec_binproj(field *f, mp *a, mp *b)
 354 {
 355   ecctx_bin *cc = CREATE(ecctx_bin);
 356   cc->c.ops = &ec_binprojops;
 357   cc->c.f = f;
 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);
 361   if (cc->bb)
 362     cc->bb = F_SQRT(f, cc->bb, cc->bb);
 363   if (!cc->bb) {
 364     MP_DROP(cc->c.a);
 365     MP_DROP(cc->c.b);
 366     DESTROY(cc);
 367     return (0);
 368   }
 369   return (&cc->c);
 370 }
 371
 372 static const ec_ops ec_binops = {
 373   "bin",
 374   ecdestroy, ec_stdsamep, ec_idin, ec_idout, ec_idfix,
 375   ecfind, ecneg, ecadd, ec_stdsub, ecdbl, eccheck
 376 };
 377
 378 static const ec_ops ec_binprojops = {
 379   "binproj",
 380   ecdestroy, ec_stdsamep, ec_projin, ec_projout, ec_projfix,
 381   ecfind, ecprojneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck
 382 };
 383
 384 /*----- Test rig ----------------------------------------------------------*/
 385
 386 #ifdef TEST_RIG
 387
 388 #define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
 389
 390 int main(int argc, char *argv[])
 391 {
 392   field *f;
 393   ec_curve *c;
 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]);
 397
 398   printf("ec-bin: ");
 399   fflush(stdout);
 400   a = MP(0x7ffffffffffffffffffffffffffffffffffffffff);
 401   b = MP(0x6645f3cacf1638e139c6cd13ef61734fbc9e3d9fb);
 402   p = MP(0x800000000000000000000000000000000000000c9);
 403   beta = MP(0x715169c109c612e390d347c748342bcd3b02a0bef);
 404   r = MP(0x040000000000000000000292fe77e70c12a4234c32);
 405
 406   f = field_binnorm(p, beta);
 407   c = ec_binproj(f, a, b);
 408   g.x = MP(0x0311103c17167564ace77ccb09c681f886ba54ee8);
 409   g.y = MP(0x333ac13c6447f2e67613bf7009daf98c87bb50c7f);
 410
 411   for (i = 0; i < n; i++) {
 412     ec_mul(c, &d, &g, r);
 413     if (EC_ATINF(&d)) {
 414       fprintf(stderr, "zero too early\n");
 415       return (1);
 416     }
 417     ec_add(c, &d, &d, &g);
 418     if (!EC_ATINF(&d)) {
 419       fprintf(stderr, "didn't reach zero\n");
 420       MP_EPRINTX("d.x", d.x);
 421       MP_EPRINTX("d.y", d.y);
 422       return (1);
 423     }
 424     ec_destroy(&d);
 425   }
 426
 427   ec_destroy(&g);
 428   ec_destroycurve(c);
 429   F_DESTROY(f);
 430   MP_DROP(p); MP_DROP(a); MP_DROP(b); MP_DROP(r); MP_DROP(beta);
 431   assert(!mparena_count(&mparena_global));
 432   printf("ok\n");
 433   return (0);
 434 }
 435
 436 #endif
 437
 438 /*----- That's all, folks -------------------------------------------------*/