eccdata.c 27.4 KB
Newer Older
1 2 3 4
/* eccdata.c

   Generate compile time constant (but machine dependent) tables.

5
   Copyright (C) 2013, 2014 Niels Möller
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

   This file is part of GNU Nettle.

   GNU Nettle is free software: you can redistribute it and/or
   modify it under the terms of either:

     * the GNU Lesser General Public License as published by the Free
       Software Foundation; either version 3 of the License, or (at your
       option) any later version.

   or

     * the GNU General Public License as published by the Free
       Software Foundation; either version 2 of the License, or (at your
       option) any later version.

   or both in parallel, as here.

   GNU Nettle is distributed in the hope that it will be useful,
   but WITHOUT ANY WARRANTY; without even the implied warranty of
   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
   General Public License for more details.

   You should have received copies of the GNU General Public License and
   the GNU Lesser General Public License along with this program.  If
   not, see http://www.gnu.org/licenses/.
*/
33

34
/* Development of Nettle's ECC support was funded by the .SE Internet Fund. */
35 36 37 38 39 40

#include <assert.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>

41
#include "mini-gmp.c"
42

43
/* Affine coordinates, for simplicity. Infinity point, i.e., te
44
   neutral group element, is represented using the is_zero flag. */
45 46
struct ecc_point
{
47
  int is_zero;
48 49 50 51
  mpz_t x;
  mpz_t y;
};

52 53 54 55 56 57 58
enum ecc_type
  {
    /* y^2 = x^3 - 3x + b (mod p) */
    ECC_TYPE_WEIERSTRASS,
    /* y^2 = x^3 + b x^2 + x */
    ECC_TYPE_MONTGOMERY
  };
59 60 61 62 63 64 65

struct ecc_curve
{
  unsigned bit_size;
  unsigned pippenger_k;
  unsigned pippenger_c;

66 67
  enum ecc_type type;

68 69 70 71 72 73 74 75
  /* Prime */
  mpz_t p;
  mpz_t b;

  /* Curve order */
  mpz_t q;
  struct ecc_point g;

76 77 78 79 80 81 82
  /* Non-zero if we want elements represented as point s(u, v) on an
     equivalent Edwards curve, using

      u = t x / y
      v = (x-1) / (x+1)
  */
  int use_edwards;
83
  mpz_t d;
84 85
  mpz_t t;

86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118
  /* Table for pippenger's algorithm.
     Element

       i 2^c + j_0 + j_1 2 + j_2 2^2 + ... + j_{c-1} 2^{c-1}

     holds

       2^{ikc} ( j_0 + j_1 2^k + j_2 2^{2k} + ... + j_{c-1} 2^{(c-1)k}) g
   */
  mp_size_t table_size;
  struct ecc_point *table;

  /* If non-NULL, holds 2g, 3g, 4g */
  struct ecc_point *ref;
};

static void
ecc_init (struct ecc_point *p)
{
  mpz_init (p->x);
  mpz_init (p->y);
}

static void
ecc_clear (struct ecc_point *p)
{
  mpz_clear (p->x);
  mpz_clear (p->y);
}

static int
ecc_zero_p (const struct ecc_point *p)
{
119
  return p->is_zero;
120 121 122 123 124
}

static int
ecc_equal_p (const struct ecc_point *p, const struct ecc_point *q)
{
125 126
  return p->is_zero ? q->is_zero
    : !q->is_zero && mpz_cmp (p->x, q->x) == 0 && mpz_cmp (p->y, q->y) == 0;
127 128 129 130 131
}

static void
ecc_set_zero (struct ecc_point *r)
{
132
  r->is_zero = 1;
133 134 135 136 137
}

static void
ecc_set (struct ecc_point *r, const struct ecc_point *p)
{
138
  r->is_zero = p->is_zero;
139 140 141 142
  mpz_set (r->x, p->x);
  mpz_set (r->y, p->y);
}

143
/* Needs to support in-place operation. */
144 145 146 147 148 149 150 151 152 153
static void
ecc_dup (const struct ecc_curve *ecc,
	 struct ecc_point *r, const struct ecc_point *p)
{
  if (ecc_zero_p (p))
    ecc_set_zero (r);

  else
    {
      mpz_t m, t, x, y;
154

155 156 157 158 159 160 161 162 163
      mpz_init (m);
      mpz_init (t);
      mpz_init (x);
      mpz_init (y);

      /* m = (2 y)^-1 */
      mpz_mul_ui (m, p->y, 2);
      mpz_invert (m, m, ecc->p);

164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181
      switch (ecc->type)
	{
	case ECC_TYPE_WEIERSTRASS:
	  /* t = 3 (x^2 - 1) * m */
	  mpz_mul (t, p->x, p->x);
	  mpz_mod (t, t, ecc->p);
	  mpz_sub_ui (t, t, 1);
	  mpz_mul_ui (t, t, 3);
	  break;
	case ECC_TYPE_MONTGOMERY:
	  /* t = (3 x^2 + 2 b x + 1) m = [x(3x+2b)+1] m */
	  mpz_mul_ui (t, ecc->b, 2);
	  mpz_addmul_ui (t, p->x, 3);
	  mpz_mul (t, t, p->x);
	  mpz_mod (t, t, ecc->p);
	  mpz_add_ui (t, t, 1);
	  break;
	}
182
      mpz_mul (t, t, m);
183
      mpz_mod (t, t, ecc->p);
184 185 186

      /* x' = t^2 - 2 x */
      mpz_mul (x, t, t);
Niels Möller's avatar
Niels Möller committed
187
      mpz_submul_ui (x, p->x, 2);
188 189 190
      if (ecc->type == ECC_TYPE_MONTGOMERY)
	mpz_sub (x, x, ecc->b);

191 192 193 194 195 196 197 198
      mpz_mod (x, x, ecc->p);

      /* y' = (x - x') * t - y */
      mpz_sub (y, p->x, x);
      mpz_mul (y, y, t);
      mpz_sub (y, y, p->y);
      mpz_mod (y, y, ecc->p);

199
      r->is_zero = 0;
200 201
      mpz_swap (x, r->x);
      mpz_swap (y, r->y);
202

203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245
      mpz_clear (m);
      mpz_clear (t);
      mpz_clear (x);
      mpz_clear (y);
    }
}

static void
ecc_add (const struct ecc_curve *ecc,
	 struct ecc_point *r, const struct ecc_point *p, const struct ecc_point *q)
{
  if (ecc_zero_p (p))
    ecc_set (r, q);

  else if (ecc_zero_p (q))
    ecc_set (r, p);

  else if (mpz_cmp (p->x, q->x) == 0)
    {
      if (mpz_cmp (p->y, q->y) == 0)
	ecc_dup (ecc, r, p);
      else
	ecc_set_zero (r);
    }
  else
    {
      mpz_t s, t, x, y;
      mpz_init (s);
      mpz_init (t);
      mpz_init (x);
      mpz_init (y);

      /* t = (q_y - p_y) / (q_x - p_x) */
      mpz_sub (t, q->x, p->x);
      mpz_invert (t, t, ecc->p);
      mpz_sub (s, q->y, p->y);
      mpz_mul (t, t, s);
      mpz_mod (t, t, ecc->p);

      /* x' = t^2 - p_x - q_x */
      mpz_mul (x, t, t);
      mpz_sub (x, x, p->x);
      mpz_sub (x, x, q->x);
246 247 248
      /* This appears to be the only difference between formulas. */
      if (ecc->type == ECC_TYPE_MONTGOMERY)
	mpz_sub (x, x, ecc->b);
249 250 251 252 253 254 255 256
      mpz_mod (x, x, ecc->p);

      /* y' = (x - x') * t - y */
      mpz_sub (y, p->x, x);
      mpz_mul (y, y, t);
      mpz_sub (y, y, p->y);
      mpz_mod (y, y, ecc->p);

257
      r->is_zero = 0;
258 259 260 261 262 263 264 265 266 267 268 269 270 271
      mpz_swap (x, r->x);
      mpz_swap (y, r->y);

      mpz_clear (s);
      mpz_clear (t);
      mpz_clear (x);
      mpz_clear (y);
    }
}

static void 
ecc_mul_binary (const struct ecc_curve *ecc,
		struct ecc_point *r, const mpz_t n, const struct ecc_point *p)
{
272 273 274
  /* Avoid the mp_bitcnt_t type for compatibility with older GMP
     versions. */
  unsigned k;
275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310

  assert (r != p);
  assert (mpz_sgn (n) > 0);

  ecc_set (r, p);

  /* Index of highest one bit */
  for (k = mpz_sizeinbase (n, 2) - 1; k-- > 0; )
    {
      ecc_dup (ecc, r, r);
      if (mpz_tstbit (n, k))
	ecc_add (ecc, r, r, p);
    }  
}

static struct ecc_point *
ecc_alloc (size_t n)
{
  struct ecc_point *p = malloc (n * sizeof(*p));
  size_t i;

  if (!p)
    {
      fprintf (stderr, "Virtual memory exhausted.\n");
      exit (EXIT_FAILURE);
    }
  for (i = 0; i < n; i++)
    ecc_init (&p[i]);

  return p;
}

static void
ecc_set_str (struct ecc_point *p,
	     const char *x, const char *y)
{
311
  p->is_zero = 0;
312 313 314 315 316
  mpz_set_str (p->x, x, 16);
  mpz_set_str (p->y, y, 16);  
}

static void
317
ecc_curve_init_str (struct ecc_curve *ecc, enum ecc_type type,
318
		    const char *p, const char *b, const char *q,
319
		    const char *gx, const char *gy,
320
		    const char *d, const char *t)
321
{
322 323
  ecc->type = type;

324 325 326 327 328 329 330 331 332 333 334
  mpz_init_set_str (ecc->p, p, 16);
  mpz_init_set_str (ecc->b, b, 16);
  mpz_init_set_str (ecc->q, q, 16);
  ecc_init (&ecc->g);
  ecc_set_str (&ecc->g, gx, gy);

  ecc->pippenger_k = 0;
  ecc->pippenger_c = 0;
  ecc->table = NULL;

  ecc->ref = NULL;
335

336
  mpz_init (ecc->d);
337 338 339 340
  mpz_init (ecc->t);

  ecc->use_edwards = (t != NULL);
  if (ecc->use_edwards)
341 342 343 344
    {
      mpz_set_str (ecc->t, t, 16);
      mpz_set_str (ecc->d, d, 16);
    }
345 346 347 348 349 350 351 352
}

static void
ecc_curve_init (struct ecc_curve *ecc, unsigned bit_size)
{
  switch (bit_size)
    {
    case 192:      
353
      ecc_curve_init_str (ecc, ECC_TYPE_WEIERSTRASS,
354 355 356 357 358 359 360 361 362 363 364 365 366 367
			  /* p = 2^{192} - 2^{64} - 1 */
			  "FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE"
			  "FFFFFFFFFFFFFFFF",

			  "64210519e59c80e70fa7e9ab72243049"
			  "feb8deecc146b9b1", 

			  "ffffffffffffffffffffffff99def836"
			  "146bc9b1b4d22831",

			  "188da80eb03090f67cbf20eb43a18800"
			  "f4ff0afd82ff1012",

			  "07192b95ffc8da78631011ed6b24cdd5"
368
			  "73f977a11e794811",
369
			  NULL, NULL);
370 371 372 373 374 375 376 377 378 379 380 381 382 383 384
      ecc->ref = ecc_alloc (3);
      ecc_set_str (&ecc->ref[0], /* 2 g */
		   "dafebf5828783f2ad35534631588a3f629a70fb16982a888",
		   "dd6bda0d993da0fa46b27bbc141b868f59331afa5c7e93ab");
      
      ecc_set_str (&ecc->ref[1], /* 3 g */
		   "76e32a2557599e6edcd283201fb2b9aadfd0d359cbb263da",
		   "782c37e372ba4520aa62e0fed121d49ef3b543660cfd05fd");

      ecc_set_str (&ecc->ref[2], /* 4 g */
		   "35433907297cc378b0015703374729d7a4fe46647084e4ba",
		   "a2649984f2135c301ea3acb0776cd4f125389b311db3be32");

      break;
    case 224:
385
      ecc_curve_init_str (ecc, ECC_TYPE_WEIERSTRASS,
386 387 388 389 390 391 392 393 394 395 396 397 398 399
			  /* p = 2^{224} - 2^{96} + 1 */
			  "ffffffffffffffffffffffffffffffff"
			  "000000000000000000000001",

			  "b4050a850c04b3abf54132565044b0b7"
			  "d7bfd8ba270b39432355ffb4",

			  "ffffffffffffffffffffffffffff16a2"
			  "e0b8f03e13dd29455c5c2a3d",

			  "b70e0cbd6bb4bf7f321390b94a03c1d3"
			  "56c21122343280d6115c1d21",

			  "bd376388b5f723fb4c22dfe6cd4375a0"
400
			  "5a07476444d5819985007e34",
401
			  NULL, NULL);
402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417

      ecc->ref = ecc_alloc (3);
      ecc_set_str (&ecc->ref[0], /* 2 g */
		   "706a46dc76dcb76798e60e6d89474788d16dc18032d268fd1a704fa6",
		   "1c2b76a7bc25e7702a704fa986892849fca629487acf3709d2e4e8bb");
      
      ecc_set_str (&ecc->ref[1], /* 3 g */
		   "df1b1d66a551d0d31eff822558b9d2cc75c2180279fe0d08fd896d04",
		   "a3f7f03cadd0be444c0aa56830130ddf77d317344e1af3591981a925");

      ecc_set_str (&ecc->ref[2], /* 4 g */
		   "ae99feebb5d26945b54892092a8aee02912930fa41cd114e40447301",
		   "482580a0ec5bc47e88bc8c378632cd196cb3fa058a7114eb03054c9");

      break;
    case 256:
418
      ecc_curve_init_str (ecc, ECC_TYPE_WEIERSTRASS,
419 420 421 422 423 424 425 426 427 428 429 430 431 432
			  /* p = 2^{256} - 2^{224} + 2^{192} + 2^{96} - 1 */
			  "FFFFFFFF000000010000000000000000"
			  "00000000FFFFFFFFFFFFFFFFFFFFFFFF",

			  "5AC635D8AA3A93E7B3EBBD55769886BC"
			  "651D06B0CC53B0F63BCE3C3E27D2604B",

			  "FFFFFFFF00000000FFFFFFFFFFFFFFFF"
			  "BCE6FAADA7179E84F3B9CAC2FC632551",

			  "6B17D1F2E12C4247F8BCE6E563A440F2"
			  "77037D812DEB33A0F4A13945D898C296",

			  "4FE342E2FE1A7F9B8EE7EB4A7C0F9E16"
433
			  "2BCE33576B315ECECBB6406837BF51F5",
434
			  NULL, NULL);
435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450

      ecc->ref = ecc_alloc (3);
      ecc_set_str (&ecc->ref[0], /* 2 g */
		   "7cf27b188d034f7e8a52380304b51ac3c08969e277f21b35a60b48fc47669978",
		   "7775510db8ed040293d9ac69f7430dbba7dade63ce982299e04b79d227873d1");
      
      ecc_set_str (&ecc->ref[1], /* 3 g */
		   "5ecbe4d1a6330a44c8f7ef951d4bf165e6c6b721efada985fb41661bc6e7fd6c",
		   "8734640c4998ff7e374b06ce1a64a2ecd82ab036384fb83d9a79b127a27d5032");

      ecc_set_str (&ecc->ref[2], /* 4 g */
		   "e2534a3532d08fbba02dde659ee62bd0031fe2db785596ef509302446b030852",
		   "e0f1575a4c633cc719dfee5fda862d764efc96c3f30ee0055c42c23f184ed8c6");

      break;
    case 384:
451
      ecc_curve_init_str (ecc, ECC_TYPE_WEIERSTRASS,
452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470
			  /* p = 2^{384} - 2^{128} - 2^{96} + 2^{32} - 1 */
			  "ffffffffffffffffffffffffffffffff"
			  "fffffffffffffffffffffffffffffffe"
			  "ffffffff0000000000000000ffffffff",
			  
			  "b3312fa7e23ee7e4988e056be3f82d19"
			  "181d9c6efe8141120314088f5013875a"
			  "c656398d8a2ed19d2a85c8edd3ec2aef",
			  
			  "ffffffffffffffffffffffffffffffff"
			  "ffffffffffffffffc7634d81f4372ddf"
			  "581a0db248b0a77aecec196accc52973",
			  
			  "aa87ca22be8b05378eb1c71ef320ad74"
			  "6e1d3b628ba79b9859f741e082542a38"
			  "5502f25dbf55296c3a545e3872760ab7",
			  
			  "3617de4a96262c6f5d9e98bf9292dc29"
			  "f8f41dbd289a147ce9da3113b5f0b8c0"
471
			  "0a60b1ce1d7e819d7a431d7c90ea0e5f",
472
			  NULL, NULL);
473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488

      ecc->ref = ecc_alloc (3);
      ecc_set_str (&ecc->ref[0], /* 2 g */
		   "8d999057ba3d2d969260045c55b97f089025959a6f434d651d207d19fb96e9e4fe0e86ebe0e64f85b96a9c75295df61",
		   "8e80f1fa5b1b3cedb7bfe8dffd6dba74b275d875bc6cc43e904e505f256ab4255ffd43e94d39e22d61501e700a940e80");

      ecc_set_str (&ecc->ref[1], /* 3 g */
		   "77a41d4606ffa1464793c7e5fdc7d98cb9d3910202dcd06bea4f240d3566da6b408bbae5026580d02d7e5c70500c831",
		   "c995f7ca0b0c42837d0bbe9602a9fc998520b41c85115aa5f7684c0edc111eacc24abd6be4b5d298b65f28600a2f1df1");

      ecc_set_str (&ecc->ref[2], /* 4 g */
		   "138251cd52ac9298c1c8aad977321deb97e709bd0b4ca0aca55dc8ad51dcfc9d1589a1597e3a5120e1efd631c63e1835",
		   "cacae29869a62e1631e8a28181ab56616dc45d918abc09f3ab0e63cf792aa4dced7387be37bba569549f1c02b270ed67");

      break;
    case 521:
489
      ecc_curve_init_str (ecc, ECC_TYPE_WEIERSTRASS,
490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517
			  "1ff" /* p = 2^{521} - 1 */
			  "ffffffffffffffffffffffffffffffff"
			  "ffffffffffffffffffffffffffffffff"
			  "ffffffffffffffffffffffffffffffff"
			  "ffffffffffffffffffffffffffffffff",

			  "051"
			  "953eb9618e1c9a1f929a21a0b68540ee"
			  "a2da725b99b315f3b8b489918ef109e1"
			  "56193951ec7e937b1652c0bd3bb1bf07"
			  "3573df883d2c34f1ef451fd46b503f00",

			  "1ff"
			  "ffffffffffffffffffffffffffffffff"
			  "fffffffffffffffffffffffffffffffa"
			  "51868783bf2f966b7fcc0148f709a5d0"
			  "3bb5c9b8899c47aebb6fb71e91386409",

			  "c6"
			  "858e06b70404e9cd9e3ecb662395b442"
			  "9c648139053fb521f828af606b4d3dba"
			  "a14b5e77efe75928fe1dc127a2ffa8de"
			  "3348b3c1856a429bf97e7e31c2e5bd66",

			  "118"
			  "39296a789a3bc0045c8a5fb42c7d1bd9"
			  "98f54449579b446817afbd17273e662c"
			  "97ee72995ef42640c550b9013fad0761"
518
			  "353c7086a272c24088be94769fd16650",
519
			  NULL, NULL);
520 521 522 523 524 525 526 527 528 529 530 531 532 533 534

      ecc->ref = ecc_alloc (3);
      ecc_set_str (&ecc->ref[0], /* 2 g */
		   "433c219024277e7e682fcb288148c282747403279b1ccc06352c6e5505d769be97b3b204da6ef55507aa104a3a35c5af41cf2fa364d60fd967f43e3933ba6d783d",
		   "f4bb8cc7f86db26700a7f3eceeeed3f0b5c6b5107c4da97740ab21a29906c42dbbb3e377de9f251f6b93937fa99a3248f4eafcbe95edc0f4f71be356d661f41b02");
      
      ecc_set_str (&ecc->ref[1], /* 3 g */
		   "1a73d352443de29195dd91d6a64b5959479b52a6e5b123d9ab9e5ad7a112d7a8dd1ad3f164a3a4832051da6bd16b59fe21baeb490862c32ea05a5919d2ede37ad7d",
		   "13e9b03b97dfa62ddd9979f86c6cab814f2f1557fa82a9d0317d2f8ab1fa355ceec2e2dd4cf8dc575b02d5aced1dec3c70cf105c9bc93a590425f588ca1ee86c0e5");

      ecc_set_str (&ecc->ref[2], /* 4 g */
		   "35b5df64ae2ac204c354b483487c9070cdc61c891c5ff39afc06c5d55541d3ceac8659e24afe3d0750e8b88e9f078af066a1d5025b08e5a5e2fbc87412871902f3",
		   "82096f84261279d2b673e0178eb0b4abb65521aef6e6e32e1b5ae63fe2f19907f279f283e54ba385405224f750a95b85eebb7faef04699d1d9e21f47fc346e4d0d");

      break;
535 536 537
    case 255:
      /* curve25519, y^2 = x^3 + 486662 x^2 + x (mod p), with p = 2^{255} - 19.

538
	 According to http://cr.yp.to/papers.html#newelliptic, this
539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560
	 is birationally equivalent to the Edwards curve

	   x^2 + y^2 = 1 + (121665/121666) x^2 y^2 (mod p).

	 And since the constant is not a square, the Edwards formulas
	 should be "complete", with no special cases needed for
	 doubling, neutral element, negatives, etc.

	 Generator is x = 9, with y coordinate
	 14781619447589544791020593568409986887264606134616475288964881837755586237401,
	 according to

	   x = Mod(9, 2^255-19); sqrt(x^3 + 486662*x^2 + x)

	 in PARI/GP. Also, in PARI notation,

	   curve25519 = Mod([0, 486662, 0, 1, 0], 2^255-19)
       */
      ecc_curve_init_str (ecc, ECC_TYPE_MONTGOMERY,
			  "7fffffffffffffffffffffffffffffff"
			  "ffffffffffffffffffffffffffffffed",
			  "76d06",
561 562 563 564
			  /* Order of the subgroup is 2^252 + q_0, where
			     q_0 = 27742317777372353535851937790883648493,
			     125 bits.
			  */
565 566 567 568 569 570 571
			  "10000000000000000000000000000000"
			  "14def9dea2f79cd65812631a5cf5d3ed",
			  "9",
			  /* y coordinate from PARI/GP
			     x = Mod(9, 2^255-19); sqrt(x^3 + 486662*x^2 + x)
			  */
			  "20ae19a1b8a086b4e01edd2c7748d14c"
572
			  "923d4d7e6d7c61b229e9c5a27eced3d9",
573 574 575 576 577
			  /* (121665/121666) mod p, from PARI/GP
			     c = Mod(121665, p); c / (c+1)
			  */
			  "2dfc9311d490018c7338bf8688861767"
			  "ff8ff5b2bebe27548a14b235eca6874a",
578 579 580 581
			  /* sqrt(486664) mod p, from PARI/GP
			     sqrt(Mod(486664, p)) */
			  "141b0b6806563d503de05885280b5910"
			  "9ca5ee38d7b56c9c165db7106377bbd8");
582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601
      ecc->ref = ecc_alloc (3);
      ecc_set_str (&ecc->ref[0], /* 2 g */
		   "20d342d51873f1b7d9750c687d157114"
		   "8f3f5ced1e350b5c5cae469cdd684efb",
		   "13b57e011700e8ae050a00945d2ba2f3"
		   "77659eb28d8d391ebcd70465c72df563");
      ecc_set_str (&ecc->ref[1], /* 3 g */
		   "1c12bc1a6d57abe645534d91c21bba64"
		   "f8824e67621c0859c00a03affb713c12",
		   "2986855cbe387eaeaceea446532c338c"
		   "536af570f71ef7cf75c665019c41222b");

      ecc_set_str (&ecc->ref[2], /* 4 g */
		   "79ce98b7e0689d7de7d1d074a15b315f"
		   "fe1805dfcd5d2a230fee85e4550013ef",
		   "75af5bf4ebdc75c8fe26873427d275d7"
		   "3c0fb13da361077a565539f46de1c30");

      break;

602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684
    default:
      fprintf (stderr, "No known curve for size %d\n", bit_size);
      exit(EXIT_FAILURE);     
    }
  ecc->bit_size = bit_size;
}

static void
ecc_pippenger_precompute (struct ecc_curve *ecc, unsigned k, unsigned c)
{
  unsigned p = (ecc->bit_size + k-1) / k;
  unsigned M = (p + c-1)/c;
  unsigned i, j;

  ecc->pippenger_k = k;
  ecc->pippenger_c = c;
  ecc->table_size = M << c;
  ecc->table = ecc_alloc (ecc->table_size);
  
  /* Compute the first 2^c entries */
  ecc_set_zero (&ecc->table[0]);
  ecc_set (&ecc->table[1], &ecc->g);

  for (j = 2; j < (1U<<c); j <<= 1)
    {
      /* T[j] = 2^k T[j/2] */
      ecc_dup (ecc, &ecc->table[j], &ecc->table[j/2]);
      for (i = 1; i < k; i++)
	ecc_dup (ecc, &ecc->table[j], &ecc->table[j]);

      for (i = 1; i < j; i++)
	ecc_add (ecc, &ecc->table[j + i], &ecc->table[j], &ecc->table[i]);
    }
  for (j = 1<<c; j < ecc->table_size; j++)
    {
      /* T[j] = 2^{kc} T[j-2^c] */
      ecc_dup (ecc, &ecc->table[j], &ecc->table[j - (1<<c)]);
      for (i = 1; i < k*c; i++)
	ecc_dup (ecc, &ecc->table[j], &ecc->table[j]);
    }
}

static void
ecc_mul_pippenger (const struct ecc_curve *ecc,
		   struct ecc_point *r, const mpz_t n_input)
{
  mpz_t n;
  unsigned k, c;
  unsigned i, j;
  unsigned bit_rows;

  mpz_init (n);
  
  mpz_mod (n, n_input, ecc->q);
  ecc_set_zero (r);

  k = ecc->pippenger_k;
  c = ecc->pippenger_c;

  bit_rows = (ecc->bit_size + k - 1) / k;

  for (i = k; i-- > 0; )
    {
      ecc_dup (ecc, r, r);
      for (j = 0; j * c < bit_rows; j++)
	{
	  unsigned bits;
	  mp_size_t bit_index;
	  
	  /* Extract c bits of the exponent, stride k, starting at i + kcj, ending at
	    i + k (cj + c - 1)*/
	  for (bits = 0, bit_index = i + k*(c*j+c); bit_index > i + k*c*j; )
	    {
	      bit_index -= k;
	      bits = (bits << 1) | mpz_tstbit (n, bit_index);
	    }

	  ecc_add (ecc, r, r, &ecc->table[(j << c) | bits]);
	}
    }
  mpz_clear (n);
}

685 686 687 688 689 690 691 692 693 694 695 696 697 698
static void
ecc_point_out (FILE *f, const struct ecc_point *p)
{
  if (p->is_zero)
    fprintf (f, "zero");
  else
    {
	fprintf (stderr, "(");
	mpz_out_str (stderr, 16, p->x);
	fprintf (stderr, ",\n     ");
	mpz_out_str (stderr, 16, (p)->y);
	fprintf (stderr, ")");
    }
}
699 700 701 702 703
#define ASSERT_EQUAL(p, q) do {						\
    if (!ecc_equal_p (p, q))						\
      {									\
	fprintf (stderr, "%s:%d: ASSERT_EQUAL (%s, %s) failed.\n",	\
		 __FILE__, __LINE__, #p, #q);				\
704 705 706 707 708
	fprintf (stderr, "p = ");					\
	ecc_point_out (stderr, (p));					\
	fprintf (stderr, "\nq = ");					\
	ecc_point_out (stderr, (q));					\
	fprintf (stderr, "\n");						\
709 710 711 712 713 714 715 716 717
	abort();							\
      }									\
  } while (0)

#define ASSERT_ZERO(p) do {						\
    if (!ecc_zero_p (p))						\
      {									\
	fprintf (stderr, "%s:%d: ASSERT_ZERO (%s) failed.\n",		\
		 __FILE__, __LINE__, #p);				\
718 719 720
	fprintf (stderr, "p = ");					\
	ecc_point_out (stderr, (p));					\
	fprintf (stderr, "\n");						\
721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738
	abort();							\
      }									\
  } while (0)

static void
ecc_curve_check (const struct ecc_curve *ecc)
{
  struct ecc_point p, q;
  mpz_t n;

  ecc_init (&p);
  ecc_init (&q);
  mpz_init (n);

  ecc_dup (ecc, &p, &ecc->g);
  if (ecc->ref)
    ASSERT_EQUAL (&p, &ecc->ref[0]);
  else
739 740 741 742 743 744 745
    {
      fprintf (stderr, "g2 = ");
      mpz_out_str (stderr, 16, p.x);
      fprintf (stderr, "\n     ");
      mpz_out_str (stderr, 16, p.y);
      fprintf (stderr, "\n");
    }
746 747 748 749
  ecc_add (ecc, &q, &p, &ecc->g);
  if (ecc->ref)
    ASSERT_EQUAL (&q, &ecc->ref[1]);
  else
750 751 752 753 754 755 756
    {
      fprintf (stderr, "g3 = ");
      mpz_out_str (stderr, 16, q.x);
      fprintf (stderr, "\n     ");
      mpz_out_str (stderr, 16, q.y);
      fprintf (stderr, "\n");
    }
757 758 759 760 761

  ecc_add (ecc, &q, &q, &ecc->g);
  if (ecc->ref)
    ASSERT_EQUAL (&q, &ecc->ref[2]);
  else
762 763 764 765 766 767 768
    {
      fprintf (stderr, "g4 = ");
      mpz_out_str (stderr, 16, q.x);
      fprintf (stderr, "\n     ");
      mpz_out_str (stderr, 16, q.y);
      fprintf (stderr, "\n");
    }
769 770 771 772 773

  ecc_dup (ecc, &q, &p);
  if (ecc->ref)
    ASSERT_EQUAL (&q, &ecc->ref[2]);
  else
774 775 776 777 778 779 780
    {
      fprintf (stderr, "g4 = ");
      mpz_out_str (stderr, 16, q.x);
      fprintf (stderr, "\n     ");
      mpz_out_str (stderr, 16, q.y);
      fprintf (stderr, "\n");
    }
781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819

  ecc_mul_binary (ecc, &p, ecc->q, &ecc->g);
  ASSERT_ZERO (&p);

  ecc_mul_pippenger (ecc, &q, ecc->q);
  ASSERT_ZERO (&q);

  ecc_clear (&p);
  ecc_clear (&q);
  mpz_clear (n);
}

static void
output_digits (const mpz_t x,
	       unsigned size, unsigned bits_per_limb)
{  
  mpz_t t;
  mpz_t mask;
  mpz_t limb;
  unsigned i;
  const char *suffix;

  mpz_init (t);
  mpz_init (mask);
  mpz_init (limb);

  mpz_setbit (mask, bits_per_limb);
  mpz_sub_ui (mask, mask, 1);

  suffix = bits_per_limb > 32 ? "ULL" : "UL";

  mpz_init_set (t, x);

  for (i = 0; i < size; i++)
    {
      if ( (i % 8) == 0)
	printf("\n ");
      
      mpz_and (limb, mask, t);
820 821 822
      printf (" 0x");
      mpz_out_str (stdout, 16, limb);
      printf ("%s,", suffix);
823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840
      mpz_tdiv_q_2exp (t, t, bits_per_limb);
    }

  mpz_clear (t);
  mpz_clear (mask);
  mpz_clear (limb);
}

static void
output_bignum (const char *name, const mpz_t x,
	       unsigned size, unsigned bits_per_limb)
{  
  printf ("static const mp_limb_t %s[%d] = {", name, size);
  output_digits (x, size, bits_per_limb);
  printf("\n};\n");
}

static void
841 842
output_point (const char *name, const struct ecc_curve *ecc,
	      const struct ecc_point *p, int use_redc,
843 844
	      unsigned size, unsigned bits_per_limb)
{
845
  mpz_t x, y, t;
846

847 848
  mpz_init (x);
  mpz_init (y);
849
  mpz_init (t);
850
 
851 852 853
  if (name)
    printf("static const mp_limb_t %s[%u] = {", name, 2*size);

854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892
  if (ecc->use_edwards)
    {
      if (ecc_zero_p (p))
	{
	  mpz_set_si (x, 0);
	  mpz_set_si (y, 1);
	}
      else if (!mpz_sgn (p->y))
	{
	  assert (!mpz_sgn (p->x));
	  mpz_set_si (x, 0);
	  mpz_set_si (y, -1);
	}
      else
	{
	  mpz_invert (x, p->y, ecc->p);
	  mpz_mul (x, x, p->x);
	  mpz_mul (x, x, ecc->t);	 
	  mpz_mod (x, x, ecc->p);

	  mpz_sub_ui (y, p->x, 1);
	  mpz_add_ui (t, p->x, 1);
	  mpz_invert (t, t, ecc->p);
	  mpz_mul (y, y, t);
	  mpz_mod (y, y, ecc->p);
	}
    }
  else
    {
      mpz_set (x, p->x);
      mpz_set (y, p->y);
    }
  if (use_redc)
    {
      mpz_mul_2exp (x, x, size * bits_per_limb);
      mpz_mod (x, x, ecc->p);
      mpz_mul_2exp (y, y, size * bits_per_limb);
      mpz_mod (y, y, ecc->p);
    }
893
      
894 895
  output_digits (x, size, bits_per_limb);
  output_digits (y, size, bits_per_limb);
896 897 898 899

  if (name)
    printf("\n};\n");

900 901
  mpz_clear (x);
  mpz_clear (y);
902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942
  mpz_clear (t);
}

static unsigned
output_modulo (const char *name, const mpz_t x,
	       unsigned size, unsigned bits_per_limb)
{
  mpz_t mod;
  unsigned bits;

  mpz_init (mod);

  mpz_setbit (mod, bits_per_limb * size);
  mpz_mod (mod, mod, x);

  bits = mpz_sizeinbase (mod, 2);
  output_bignum (name, mod, size, bits_per_limb);
  
  mpz_clear (mod);
  return bits;
}

static void
output_curve (const struct ecc_curve *ecc, unsigned bits_per_limb)
{
  unsigned limb_size = (ecc->bit_size + bits_per_limb - 1)/bits_per_limb;
  unsigned i;
  unsigned bits;
  int redc_limbs;
  mpz_t t;

  mpz_init (t);

  printf ("/* For NULL. */\n#include <stddef.h>\n");

  printf ("#define ECC_LIMB_SIZE %u\n", limb_size);
  printf ("#define ECC_PIPPENGER_K %u\n", ecc->pippenger_k);
  printf ("#define ECC_PIPPENGER_C %u\n", ecc->pippenger_c);

  output_bignum ("ecc_p", ecc->p, limb_size, bits_per_limb);
  output_bignum ("ecc_b", ecc->b, limb_size, bits_per_limb);
943 944
  if (ecc->use_edwards)
    output_bignum ("ecc_d", ecc->d, limb_size, bits_per_limb);
945
  output_bignum ("ecc_q", ecc->q, limb_size, bits_per_limb);
946 947
  output_point ("ecc_g", ecc, &ecc->g, 0, limb_size, bits_per_limb);
  output_point ("ecc_redc_g", ecc, &ecc->g, 1, limb_size, bits_per_limb);
948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008
  
  bits = output_modulo ("ecc_Bmodp", ecc->p, limb_size, bits_per_limb);
  printf ("#define ECC_BMODP_SIZE %u\n",
	  (bits + bits_per_limb - 1) / bits_per_limb);
  bits = output_modulo ("ecc_Bmodq", ecc->q, limb_size, bits_per_limb);
  printf ("#define ECC_BMODQ_SIZE %u\n",
	  (bits + bits_per_limb - 1) / bits_per_limb);

  if (ecc->bit_size < limb_size * bits_per_limb)
    {
      int shift;

      mpz_set_ui (t, 0);
      mpz_setbit (t, ecc->bit_size);
      mpz_sub (t, t, ecc->p);      
      output_bignum ("ecc_Bmodp_shifted", t, limb_size, bits_per_limb);

      shift = limb_size * bits_per_limb - ecc->bit_size;
      if (shift > 0)
	{
	  /* Check condition for reducing hi limbs. If s is the
	     normalization shift and n is the bit size (so that s + n
	     = limb_size * bite_per_limb), then we need

	       (2^n - 1) + (2^s - 1) (2^n - p) < 2p

	     or equivalently,

	       2^s (2^n - p) <= p

	     To a allow a carry limb to be added in at the same time,
	     substitute s+1 for s.
	  */
	  /* FIXME: For ecdsa verify, we actually need the stricter
	     inequality < 2 q. */
	  mpz_mul_2exp (t, t, shift + 1);
	  if (mpz_cmp (t, ecc->p) > 0)
	    {
	      fprintf (stderr, "Reduction condition failed for %u-bit curve.\n",
		       ecc->bit_size);
	      exit (EXIT_FAILURE);
	    }
	}
      mpz_set_ui (t, 0);
      mpz_setbit (t, ecc->bit_size);
      mpz_sub (t, t, ecc->q);      
      output_bignum ("ecc_Bmodq_shifted", t, limb_size, bits_per_limb);      
    }
  else
    {
      printf ("#define ecc_Bmodp_shifted ecc_Bmodp\n");
      printf ("#define ecc_Bmodq_shifted ecc_Bmodq\n");
    }

  mpz_add_ui (t, ecc->p, 1);
  mpz_fdiv_q_2exp (t, t, 1);
  output_bignum ("ecc_pp1h", t, limb_size, bits_per_limb);      

  mpz_add_ui (t, ecc->q, 1);
  mpz_fdiv_q_2exp (t, t, 1);
  output_bignum ("ecc_qp1h", t, limb_size, bits_per_limb);  
1009 1010 1011 1012

  if (ecc->use_edwards)
    output_bignum ("ecc_edwards", ecc->t, limb_size, bits_per_limb);

1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042
  /* Trailing zeros in p+1 correspond to trailing ones in p. */
  redc_limbs = mpz_scan0 (ecc->p, 0) / bits_per_limb;
  if (redc_limbs > 0)
    {
      mpz_add_ui (t, ecc->p, 1);
      mpz_fdiv_q_2exp (t, t, redc_limbs * bits_per_limb);
      output_bignum ("ecc_redc_ppm1", t, limb_size - redc_limbs, bits_per_limb);
    }
  else
    {    
      /* Trailing zeros in p-1 correspond to zeros just above the low
	 bit of p */
      redc_limbs = mpz_scan1 (ecc->p, 1) / bits_per_limb;
      if (redc_limbs > 0)
	{
	  printf ("#define ecc_redc_ppm1 (ecc_p + %d)\n",
		  redc_limbs);
	  redc_limbs = -redc_limbs;
	}
      else
	printf ("#define ecc_redc_ppm1 NULL\n");
    }
  printf ("#define ECC_REDC_SIZE %d\n", redc_limbs);

  printf ("#if USE_REDC\n");
  printf ("#define ecc_unit ecc_Bmodp\n");

  printf ("static const mp_limb_t ecc_table[%lu] = {",
	 (unsigned long) (2*ecc->table_size * limb_size));
  for (i = 0; i < ecc->table_size; i++)
1043
    output_point (NULL, ecc, &ecc->table[i], 1, limb_size, bits_per_limb);
1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054

  printf("\n};\n");

  printf ("#else\n");

  mpz_init_set_ui (t, 1);
  output_bignum ("ecc_unit", t, limb_size, bits_per_limb);
  
  printf ("static const mp_limb_t ecc_table[%lu] = {",
	 (unsigned long) (2*ecc->table_size * limb_size));
  for (i = 0; i < ecc->table_size; i++)
1055
    output_point (NULL, ecc, &ecc->table[i], 0, limb_size, bits_per_limb);
1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087

  printf("\n};\n");
  printf ("#endif\n");
  
  mpz_clear (t);
}

int
main (int argc, char **argv)
{
  struct ecc_curve ecc;

  if (argc < 4)
    {
      fprintf (stderr, "Usage: %s CURVE-BITS K C [BITS-PER-LIMB]\n", argv[0]);
      return EXIT_FAILURE;
    }

  ecc_curve_init (&ecc, atoi(argv[1]));

  ecc_pippenger_precompute (&ecc, atoi(argv[2]), atoi(argv[3]));

  fprintf (stderr, "Table size: %lu entries\n",
	   (unsigned long) ecc.table_size);

  ecc_curve_check (&ecc);

  if (argc > 4)
    output_curve (&ecc, atoi(argv[4]));

  return EXIT_SUCCESS;
}