eccdata.c 29.4 KB
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/* eccdata.c

   Generate compile time constant (but machine dependent) tables.

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   Copyright (C) 2013, 2014 Niels Möller
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   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/.
*/
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/* Development of Nettle's ECC support was funded by the .SE Internet Fund. */
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#include <assert.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>

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#include "mini-gmp.c"
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/* Affine coordinates, for simplicity. Infinity point, i.e., te
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   neutral group element, is represented using the is_zero flag. */
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struct ecc_point
{
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  int is_zero;
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  mpz_t x;
  mpz_t y;
};

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

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  enum ecc_type type;

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  /* Prime */
  mpz_t p;
  mpz_t b;

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

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  /* 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;
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  mpz_t d;
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  mpz_t t;

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  /* 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)
{
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  return p->is_zero;
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}

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

static void
ecc_set_zero (struct ecc_point *r)
{
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  r->is_zero = 1;
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}

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

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/* Needs to support in-place operation. */
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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;
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      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);

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      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;
	}
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      mpz_mul (t, t, m);
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      mpz_mod (t, t, ecc->p);
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      /* x' = t^2 - 2 x */
      mpz_mul (x, t, t);
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      mpz_submul_ui (x, p->x, 2);
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      if (ecc->type == ECC_TYPE_MONTGOMERY)
	mpz_sub (x, x, ecc->b);

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      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);

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      r->is_zero = 0;
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      mpz_swap (x, r->x);
      mpz_swap (y, r->y);
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      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);
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      /* This appears to be the only difference between formulas. */
      if (ecc->type == ECC_TYPE_MONTGOMERY)
	mpz_sub (x, x, ecc->b);
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      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);

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      r->is_zero = 0;
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      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)
{
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  /* Avoid the mp_bitcnt_t type for compatibility with older GMP
     versions. */
  unsigned k;
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  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)
{
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  p->is_zero = 0;
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  mpz_set_str (p->x, x, 16);
  mpz_set_str (p->y, y, 16);  
}

static void
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ecc_curve_init_str (struct ecc_curve *ecc, enum ecc_type type,
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		    const char *p, const char *b, const char *q,
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		    const char *gx, const char *gy,
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		    const char *d, const char *t)
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{
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  ecc->type = type;

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  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;
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  mpz_init (ecc->d);
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  mpz_init (ecc->t);

  ecc->use_edwards = (t != NULL);
  if (ecc->use_edwards)
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    {
      mpz_set_str (ecc->t, t, 16);
      mpz_set_str (ecc->d, d, 16);
    }
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}

static void
ecc_curve_init (struct ecc_curve *ecc, unsigned bit_size)
{
  switch (bit_size)
    {
    case 192:      
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      ecc_curve_init_str (ecc, ECC_TYPE_WEIERSTRASS,
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			  /* p = 2^{192} - 2^{64} - 1 */
			  "FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE"
			  "FFFFFFFFFFFFFFFF",

			  "64210519e59c80e70fa7e9ab72243049"
			  "feb8deecc146b9b1", 

			  "ffffffffffffffffffffffff99def836"
			  "146bc9b1b4d22831",

			  "188da80eb03090f67cbf20eb43a18800"
			  "f4ff0afd82ff1012",

			  "07192b95ffc8da78631011ed6b24cdd5"
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			  "73f977a11e794811",
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			  NULL, NULL);
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      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:
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      ecc_curve_init_str (ecc, ECC_TYPE_WEIERSTRASS,
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			  /* p = 2^{224} - 2^{96} + 1 */
			  "ffffffffffffffffffffffffffffffff"
			  "000000000000000000000001",

			  "b4050a850c04b3abf54132565044b0b7"
			  "d7bfd8ba270b39432355ffb4",

			  "ffffffffffffffffffffffffffff16a2"
			  "e0b8f03e13dd29455c5c2a3d",

			  "b70e0cbd6bb4bf7f321390b94a03c1d3"
			  "56c21122343280d6115c1d21",

			  "bd376388b5f723fb4c22dfe6cd4375a0"
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			  "5a07476444d5819985007e34",
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			  NULL, NULL);
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      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:
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      ecc_curve_init_str (ecc, ECC_TYPE_WEIERSTRASS,
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			  /* p = 2^{256} - 2^{224} + 2^{192} + 2^{96} - 1 */
			  "FFFFFFFF000000010000000000000000"
			  "00000000FFFFFFFFFFFFFFFFFFFFFFFF",

			  "5AC635D8AA3A93E7B3EBBD55769886BC"
			  "651D06B0CC53B0F63BCE3C3E27D2604B",

			  "FFFFFFFF00000000FFFFFFFFFFFFFFFF"
			  "BCE6FAADA7179E84F3B9CAC2FC632551",

			  "6B17D1F2E12C4247F8BCE6E563A440F2"
			  "77037D812DEB33A0F4A13945D898C296",

			  "4FE342E2FE1A7F9B8EE7EB4A7C0F9E16"
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			  "2BCE33576B315ECECBB6406837BF51F5",
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			  NULL, NULL);
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      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:
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      ecc_curve_init_str (ecc, ECC_TYPE_WEIERSTRASS,
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			  /* p = 2^{384} - 2^{128} - 2^{96} + 2^{32} - 1 */
			  "ffffffffffffffffffffffffffffffff"
			  "fffffffffffffffffffffffffffffffe"
			  "ffffffff0000000000000000ffffffff",
			  
			  "b3312fa7e23ee7e4988e056be3f82d19"
			  "181d9c6efe8141120314088f5013875a"
			  "c656398d8a2ed19d2a85c8edd3ec2aef",
			  
			  "ffffffffffffffffffffffffffffffff"
			  "ffffffffffffffffc7634d81f4372ddf"
			  "581a0db248b0a77aecec196accc52973",
			  
			  "aa87ca22be8b05378eb1c71ef320ad74"
			  "6e1d3b628ba79b9859f741e082542a38"
			  "5502f25dbf55296c3a545e3872760ab7",
			  
			  "3617de4a96262c6f5d9e98bf9292dc29"
			  "f8f41dbd289a147ce9da3113b5f0b8c0"
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			  "0a60b1ce1d7e819d7a431d7c90ea0e5f",
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			  NULL, NULL);
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      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:
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      ecc_curve_init_str (ecc, ECC_TYPE_WEIERSTRASS,
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			  "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",
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			  NULL, NULL);
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      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;
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    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
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	 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",
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			  /* Order of the subgroup is 2^252 + q_0, where
			     q_0 = 27742317777372353535851937790883648493,
			     125 bits.
			  */
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			  "10000000000000000000000000000000"
			  "14def9dea2f79cd65812631a5cf5d3ed",
			  "9",
			  /* y coordinate from PARI/GP
			     x = Mod(9, 2^255-19); sqrt(x^3 + 486662*x^2 + x)
			  */
			  "20ae19a1b8a086b4e01edd2c7748d14c"
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			  "923d4d7e6d7c61b229e9c5a27eced3d9",
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			  /* (121665/121666) mod p, from PARI/GP
			     c = Mod(121665, p); c / (c+1)
			  */
			  "2dfc9311d490018c7338bf8688861767"
			  "ff8ff5b2bebe27548a14b235eca6874a",
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			  /* A square root of -486664 mod p, PARI/GP
			     -sqrt(Mod(-486664, p)) in PARI/GP.

			     Sign is important to map to the right
			     generator on the twisted edwards curve
			     used for EdDSA. */
			  "70d9120b9f5ff9442d84f723fc03b081"
			  "3a5e2c2eb482e57d3391fb5500ba81e7"
			  );
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      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;

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    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);
}

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static void
ecc_point_out (FILE *f, const struct ecc_point *p)
{
  if (p->is_zero)
    fprintf (f, "zero");
  else
    {
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	fprintf (f, "(");
	mpz_out_str (f, 16, p->x);
	fprintf (f, ",\n     ");
	mpz_out_str (f, 16, (p)->y);
	fprintf (f, ")");
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    }
}
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#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);				\
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	fprintf (stderr, "p = ");					\
	ecc_point_out (stderr, (p));					\
	fprintf (stderr, "\nq = ");					\
	ecc_point_out (stderr, (q));					\
	fprintf (stderr, "\n");						\
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	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);				\
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	fprintf (stderr, "p = ");					\
	ecc_point_out (stderr, (p));					\
	fprintf (stderr, "\n");						\
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	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
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    {
      fprintf (stderr, "g2 = ");
      mpz_out_str (stderr, 16, p.x);
      fprintf (stderr, "\n     ");
      mpz_out_str (stderr, 16, p.y);
      fprintf (stderr, "\n");
    }
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  ecc_add (ecc, &q, &p, &ecc->g);
  if (ecc->ref)
    ASSERT_EQUAL (&q, &ecc->ref[1]);
  else
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    {
      fprintf (stderr, "g3 = ");
      mpz_out_str (stderr, 16, q.x);
      fprintf (stderr, "\n     ");
      mpz_out_str (stderr, 16, q.y);
      fprintf (stderr, "\n");
    }
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  ecc_add (ecc, &q, &q, &ecc->g);
  if (ecc->ref)
    ASSERT_EQUAL (&q, &ecc->ref[2]);
  else
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    {
      fprintf (stderr, "g4 = ");
      mpz_out_str (stderr, 16, q.x);
      fprintf (stderr, "\n     ");
      mpz_out_str (stderr, 16, q.y);
      fprintf (stderr, "\n");
    }
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  ecc_dup (ecc, &q, &p);
  if (ecc->ref)
    ASSERT_EQUAL (&q, &ecc->ref[2]);
  else
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    {
      fprintf (stderr, "g4 = ");
      mpz_out_str (stderr, 16, q.x);
      fprintf (stderr, "\n     ");
      mpz_out_str (stderr, 16, q.y);
      fprintf (stderr, "\n");
    }
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  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);
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      printf (" 0x");
      mpz_out_str (stdout, 16, limb);
      printf ("%s,", suffix);
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      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
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output_point (const char *name, const struct ecc_curve *ecc,
	      const struct ecc_point *p, int use_redc,
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	      unsigned size, unsigned bits_per_limb)
{
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  mpz_t x, y, t;
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  mpz_init (x);
  mpz_init (y);
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  mpz_init (t);
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  if (name)
    printf("static const mp_limb_t %s[%u] = {", name, 2*size);

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  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);
    }
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  output_digits (x, size, bits_per_limb);
  output_digits (y, size, bits_per_limb);
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  if (name)
    printf("\n};\n");

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  mpz_clear (x);
  mpz_clear (y);
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  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;
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  unsigned bits, e;
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  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);
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  if (ecc->use_edwards)
    output_bignum ("ecc_d", ecc->d, limb_size, bits_per_limb);
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  output_bignum ("ecc_q", ecc->q, limb_size, bits_per_limb);
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  output_point ("ecc_g", ecc, &ecc->g, 0, limb_size, bits_per_limb);
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  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);
Niels Möller's avatar
Niels Möller committed
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  bits = mpz_sizeinbase (ecc->q, 2);
  if (bits < ecc->bit_size)
    {
      /* for curve25519, with q = 2^k + q', with a much smaller q' */
      unsigned mbits;
      unsigned shift;

      /* Shift to align the one bit at B */
      shift = bits_per_limb * limb_size + 1 - bits;
      
      mpz_set (t, ecc->q);
      mpz_clrbit (t, bits-1);
      mbits = mpz_sizeinbase (t, 2);

      /* The shifted value must be a limb smaller than q. */
      if (mbits + shift + bits_per_limb <= bits)
	{
	  /* q of the form 2^k + q', with q' a limb smaller */
	  mpz_mul_2exp (t, t, shift);
	  output_bignum ("ecc_mBmodq_shifted", t, limb_size, bits_per_limb);
	}
    }
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  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);
	    }
	}
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    }
  else
    printf ("#define ecc_Bmodp_shifted ecc_Bmodp\n");

  if (bits < limb_size * bits_per_limb)
    {
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      mpz_set_ui (t, 0);
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      mpz_setbit (t, bits);
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      mpz_sub (t, t, ecc->q);      
      output_bignum ("ecc_Bmodq_shifted", t, limb_size, bits_per_limb);      
    }
  else
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    printf ("#define ecc_Bmodq_shifted ecc_Bmodq\n");
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  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);  
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  if (ecc->use_edwards)
    output_bignum ("ecc_edwards", ecc->t, limb_size, bits_per_limb);

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  /* 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);

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  /* For mod p square root computation. */
  if (mpz_fdiv_ui (ecc->p, 4) == 3)
    {
      /* x = a^{(p+1)/4} gives square root of a (if it exists,
	 otherwise the square root of -a). */
      e = 1;
      mpz_add_ui (t, ecc->p, 1);
      mpz_fdiv_q_2exp (t, t, 2); 
    }
  else
    {
      /* p-1 = 2^e s, s odd, t = (s-1)/2*/
      unsigned g, i;
      mpz_t s;
      mpz_t z;

      mpz_init (s);
      mpz_init (z);

      mpz_sub_ui (s, ecc->p, 1);
      e = mpz_scan1 (s, 0);
      assert (e > 1);

      mpz_fdiv_q_2exp (s, s, e);

      /* Find a non-square g, g^{(p-1)/2} = -1,
	 and z = g^{(p-1)/4 */
      for (g = 2; ; g++)
	{
	  mpz_set_ui (z, g);
	  mpz_powm (z, z, s, ecc->p);
	  mpz_mul (t, z, z);
	  mpz_mod (t, t, ecc->p);

	  for (i = 2; i < e; i++)
	    {
	      mpz_mul (t, t, t);
	      mpz_mod (t, t, ecc->p);
	    }
	  if (mpz_cmp_ui (t, 1) != 0)
	    break;
	}
      mpz_add_ui (t, t, 1);
      assert (mpz_cmp (t, ecc->p) == 0);
      output_bignum ("ecc_sqrt_z", z, limb_size, bits_per_limb);

      mpz_fdiv_q_2exp (t, s, 1);

      mpz_clear (s);
      mpz_clear (z);
    }
  printf ("#define ECC_SQRT_E %u\n", e);
  printf ("#define ECC_SQRT_T_BITS %u\n",
	  (unsigned) mpz_sizeinbase (t, 2));
  output_bignum ("ecc_sqrt_t", t, limb_size, bits_per_limb);      

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  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++)
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    output_point (NULL, ecc, &ecc->table[i], 1, limb_size, bits_per_limb);
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  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++)
1140
    output_point (NULL, ecc, &ecc->table[i], 0, limb_size, bits_per_limb);
1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170 1171 1172

  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;
}