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

   1 /* -*-c-*-
   2  *
   3  * Arithmetic for elliptic curves over binary fields
   4  *
   5  * (c) 2004 Straylight/Edgeware
   6  */
   7
   8 /*----- Licensing notice --------------------------------------------------*
   9  *
  10  * This file is part of Catacomb.
  11  *
  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.
  16  *
  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.
  21  *
  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,
  25  * MA 02111-1307, USA.
  26  */
  27
  28 /*----- Header files ------------------------------------------------------*/
  29
  30 #include <mLib/sub.h>
  31
  32 #include "ec.h"
  33 #include "ec-guts.h"
  34
  35 /*----- Main code ---------------------------------------------------------*/
  36
  37 static const ec_ops ec_binops, ec_binprojops;
  38
  39 static ec *ecneg(ec_curve *c, ec *d, const ec *p)
  40 {
  41   EC_COPY(d, p);
  42   if (d->x)
  43     d->y = F_ADD(c->f, d->y, d->y, d->x);
  44   return (d);
  45 }
  46
  47 static ec *ecprojneg(ec_curve *c, ec *d, const ec *p)
  48 {
  49   EC_COPY(d, p);
  50   if (d->x) {
  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);
  53     MP_DROP(t);
  54   }
  55   return (d);
  56 }
  57
  58 static ec *ecfind(ec_curve *c, ec *d, mp *x)
  59 {
  60   field *f = c->f;
  61   mp *y, *u, *v;
  62
  63   if (F_ZEROP(f, x))
  64     y = F_SQRT(f, MP_NEW, c->b);
  65   else {
  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);              /* %$A = x^3 + a x^2 + b$% */
  71     if (!F_ZEROP(f, y)) {
  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);     /* %$y = z x$% */
  76     }
  77     MP_DROP(u);
  78     MP_DROP(v);
  79   }
  80   if (!y) return (0);
  81   EC_DESTROY(d);
  82   d->x = MP_COPY(x);
  83   d->y = y;
  84   d->z = MP_COPY(f->one);
  85   return (d);
  86 }
  87
  88 static ec *ecdbl(ec_curve *c, ec *d, const ec *a)
  89 {
  90   if (EC_ATINF(a) || F_ZEROP(c->f, a->x))
  91     EC_SETINF(d);
  92   else {
  93     field *f = c->f;
  94     mp *lambda;
  95     mp *dx, *dy;
  96
  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);    /* %$\lambda = x + y/x$% */
 100
 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);       /* %$x' = a + \lambda^2 + \lambda$% */
 104
 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'$% */
 109
 110     EC_DESTROY(d);
 111     d->x = dx;
 112     d->y = dy;
 113     d->z = 0;
 114     MP_DROP(lambda);
 115   }
 116   return (d);
 117 }
 118
 119 static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a)
 120 {
 121   if (EC_ATINF(a) || F_ZEROP(c->f, a->x))
 122     EC_SETINF(d);
 123   else {
 124     field *f = c->f;
 125     ecctx_bin *cc = (ecctx_bin *)c;
 126     mp *dx, *dy, *dz, *u, *v;
 127
 128     dy = F_SQR(f, MP_NEW, a->z);        /* %$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$% */
 133
 134     dz = F_MUL(f, dz, dy, a->x);        /* %$z' = x z^2$% */
 135
 136     dy = F_SQR(f, dy, a->x);            /* %$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$% */
 140
 141     v = F_SQR(f, MP_NEW, dy);           /* %$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'$% */
 145
 146     EC_DESTROY(d);
 147     d->x = dx;
 148     d->y = dy;
 149     d->z = dz;
 150     MP_DROP(u);
 151     MP_DROP(v);
 152   }
 153   return (d);
 154 }
 155
 156 static ec *ecadd(ec_curve *c, ec *d, const ec *a, const ec *b)
 157 {
 158   if (a == b)
 159     ecdbl(c, d, a);
 160   else if (EC_ATINF(a))
 161     EC_COPY(d, b);
 162   else if (EC_ATINF(b))
 163     EC_COPY(d, a);
 164   else {
 165     field *f = c->f;
 166     mp *lambda;
 167     mp *dx, *dy;
 168
 169     if (!MP_EQ(a->x, b->x)) {
 170       dx = F_ADD(f, MP_NEW, a->x, b->x); /* %$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)$% */
 175
 176       dx = F_SQR(f, dx, lambda);        /* %$\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$% */
 182     } else if (!MP_EQ(a->y, b->y) || F_ZEROP(f, a->x)) {
 183       EC_SETINF(d);
 184       return (d);
 185     } else {
 186       dx = F_INV(f, MP_NEW, a->x);      /* %$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$% */
 189
 190       dx = F_SQR(f, dx, lambda);        /* %$\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$% */
 193       dy = MP_NEW;
 194     }
 195
 196     dy = F_ADD(f, dy, a->x, dx);        /* %$ 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'$% */
 200
 201     EC_DESTROY(d);
 202     d->x = dx;
 203     d->y = dy;
 204     d->z = 0;
 205     MP_DROP(lambda);
 206   }
 207   return (d);
 208 }
 209
 210 static ec *ecprojadd(ec_curve *c, ec *d, const ec *a, const ec *b)
 211 {
 212   if (a == b)
 213     c->ops->dbl(c, d, a);
 214   else if (EC_ATINF(a))
 215     EC_COPY(d, b);
 216   else if (EC_ATINF(b))
 217     EC_COPY(d, a);
 218   else {
 219     field *f = c->f;
 220     mp *dx, *dy, *dz, *u, *uu, *v, *t, *s, *ss, *r, *w, *l;
 221
 222     dz = F_SQR(f, MP_NEW, b->z);        /* %$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$% */
 226
 227     dz = F_SQR(f, dz, a->z);            /* %$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$% */
 231
 232     w = F_ADD(f, u, u, uu);             /* %$r = u_0 + u_1$% */
 233     r = F_ADD(f, s, s, ss);             /* %$w = s_0 + s_1$% */
 234     if (F_ZEROP(f, w)) {
 235       MP_DROP(w);
 236       MP_DROP(uu);
 237       MP_DROP(ss);
 238       MP_DROP(t);
 239       MP_DROP(dz);
 240       if (F_ZEROP(f, r)) {
 241         MP_DROP(r);
 242         return (c->ops->dbl(c, d, a));
 243       } else {
 244         MP_DROP(r);
 245         EC_SETINF(d);
 246         return (d);
 247       }
 248     }
 249
 250     l = F_MUL(f, t, a->z, w);           /* %$l = z_0 w$% */
 251
 252     dz = F_MUL(f, dz, l, b->z);         /* %$z' = l z_1$% */
 253
 254     ss = F_MUL(f, ss, r, b->x);         /* %$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$% */
 257
 258     t = F_ADD(f, t, r, dz);             /* %$t = r + z'$% */
 259
 260     uu = F_SQR(f, MP_NEW, dz);          /* %$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$% */
 267
 268     r = F_SQR(f, r, l);                 /* %$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$% */
 272
 273     EC_DESTROY(d);
 274     d->x = dx;
 275     d->y = dy;
 276     d->z = dz;
 277     MP_DROP(l);
 278     MP_DROP(r);
 279     MP_DROP(w);
 280     MP_DROP(t);
 281     MP_DROP(v);
 282   }
 283   return (d);
 284 }
 285
 286 static int eccheck(ec_curve *c, const ec *p)
 287 {
 288   field *f = c->f;
 289   int rc;
 290   mp *u, *v;
 291
 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;
 303   mp_drop(u);
 304   mp_drop(v);
 305   return (rc);
 306 }
 307
 308 static int ecprojcheck(ec_curve *c, const ec *p)
 309 {
 310   ec t = EC_INIT;
 311   int rc;
 312
 313   c->ops->fix(c, &t, p);
 314   rc = eccheck(c, &t);
 315   EC_DESTROY(&t);
 316   return (rc);
 317 }
 318
 319 static void ecdestroy(ec_curve *c)
 320 {
 321   ecctx_bin *cc = (ecctx_bin *)c;
 322   MP_DROP(cc->c.a);
 323   MP_DROP(cc->c.b);
 324   if (cc->bb) MP_DROP(cc->bb);
 325   DESTROY(cc);
 326 }
 327
 328 /* --- @ec_bin@, @ec_binproj@ --- *
 329  *
 330  * Arguments:   @field *f@ = the underlying field for this elliptic curve
 331  *              @mp *a, *b@ = the coefficients for this curve
 332  *
 333  * Returns:     A pointer to the curve, or null.
 334  *
 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.
 338  */
 339
 340 ec_curve *ec_bin(field *f, mp *a, mp *b)
 341 {
 342   ecctx_bin *cc = CREATE(ecctx_bin);
 343   cc->c.ops = &ec_binops;
 344   cc->c.f = f;
 345   cc->c.a = F_IN(f, MP_NEW, a);
 346   cc->c.b = F_IN(f, MP_NEW, b);
 347   cc->bb = 0;
 348   return (&cc->c);
 349 }
 350
 351 ec_curve *ec_binproj(field *f, mp *a, mp *b)
 352 {
 353   ecctx_bin *cc = CREATE(ecctx_bin);
 354   int i;
 355   mp *c, *d;
 356
 357   cc->c.ops = &ec_binprojops;
 358   cc->c.f = f;
 359   cc->c.a = F_IN(f, MP_NEW, a);
 360   cc->c.b = F_IN(f, MP_NEW, b);
 361
 362   c = MP_COPY(cc->c.b);
 363   for (i = 0; i < f->nbits - 2; i++)
 364     c = F_SQR(f, c, c);
 365   d = F_SQR(f, MP_NEW, c); d = F_SQR(f, d, d);
 366   if (!MP_EQ(d, cc->c.b)) {
 367     MP_DROP(c);
 368     MP_DROP(d);
 369     MP_DROP(cc->c.a);
 370     MP_DROP(cc->c.b);
 371     DESTROY(cc);
 372     return (0);
 373   }
 374   cc->bb = c;
 375   MP_DROP(d);
 376   return (&cc->c);
 377 }
 378
 379 static const ec_ops ec_binops = {
 380   "bin",
 381   ecdestroy, ec_stdsamep, ec_idin, ec_idout, ec_idfix,
 382   ecfind, ecneg, ecadd, ec_stdsub, ecdbl, eccheck
 383 };
 384
 385 static const ec_ops ec_binprojops = {
 386   "binproj",
 387   ecdestroy, ec_stdsamep, ec_projin, ec_projout, ec_projfix,
 388   ecfind, ecprojneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck
 389 };
 390
 391 /*----- Test rig ----------------------------------------------------------*/
 392
 393 #ifdef TEST_RIG
 394
 395 #define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
 396
 397 int main(int argc, char *argv[])
 398 {
 399   field *f;
 400   ec_curve *c;
 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]);
 404
 405   printf("ec-bin: ");
 406   fflush(stdout);
 407   a = MP(0x7ffffffffffffffffffffffffffffffffffffffff);
 408   b = MP(0x6645f3cacf1638e139c6cd13ef61734fbc9e3d9fb);
 409   p = MP(0x800000000000000000000000000000000000000c9);
 410   beta = MP(0x715169c109c612e390d347c748342bcd3b02a0bef);
 411   r = MP(0x040000000000000000000292fe77e70c12a4234c32);
 412
 413   f = field_binnorm(p, beta);
 414   c = ec_binproj(f, a, b);
 415   g.x = MP(0x0311103c17167564ace77ccb09c681f886ba54ee8);
 416   g.y = MP(0x333ac13c6447f2e67613bf7009daf98c87bb50c7f);
 417
 418   for (i = 0; i < n; i++) {
 419     ec_mul(c, &d, &g, r);
 420     if (EC_ATINF(&d)) {
 421       fprintf(stderr, "zero too early\n");
 422       return (1);
 423     }
 424     ec_add(c, &d, &d, &g);
 425     if (!EC_ATINF(&d)) {
 426       fprintf(stderr, "didn't reach zero\n");
 427       MP_EPRINTX("d.x", d.x);
 428       MP_EPRINTX("d.y", d.y);
 429       return (1);
 430     }
 431     ec_destroy(&d);
 432   }
 433
 434   ec_destroy(&g);
 435   ec_destroycurve(c);
 436   F_DESTROY(f);
 437   MP_DROP(p); MP_DROP(a); MP_DROP(b); MP_DROP(r); MP_DROP(beta);
 438   assert(!mparena_count(&mparena_global));
 439   printf("ok\n");
 440   return (0);
 441 }
 442
 443 #endif
 444
 445 /*----- That's all, folks -------------------------------------------------*/