ecc-25519.c 7.34 KB
Newer Older
Niels Möller's avatar
Niels Möller committed
1
/* ecc-25519.c
2 3 4 5 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 33 34 35 36 37

   Arithmetic and tables for curve25519,

   Copyright (C) 2014 Niels Möller

   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/.
*/

#if HAVE_CONFIG_H
# include "config.h"
#endif

Niels Möller's avatar
Niels Möller committed
38 39
#include <assert.h>

40
#include "ecc.h"
41 42 43 44 45 46
#include "ecc-internal.h"

#define USE_REDC 0

#include "ecc-25519.h"

47 48 49 50 51 52 53
#if HAVE_NATIVE_ecc_25519_modp

#define ecc_25519_modp nettle_ecc_25519_modp
void
ecc_25519_modp (const struct ecc_curve *ecc, mp_limb_t *rp);
#else

Niels Möller's avatar
Niels Möller committed
54
#define PHIGH_BITS (GMP_NUMB_BITS * ECC_LIMB_SIZE - 255)
55

Niels Möller's avatar
Niels Möller committed
56
#if PHIGH_BITS == 0
57 58 59 60 61 62 63 64 65
#error Unsupported limb size */
#endif

static void
ecc_25519_modp(const struct ecc_curve *ecc UNUSED, mp_limb_t *rp)
{
  mp_limb_t hi, cy;

  cy = mpn_addmul_1 (rp, rp + ECC_LIMB_SIZE, ECC_LIMB_SIZE,
Niels Möller's avatar
Niels Möller committed
66
		     (mp_limb_t) 19 << PHIGH_BITS);
67
  hi = rp[ECC_LIMB_SIZE-1];
Niels Möller's avatar
Niels Möller committed
68 69
  cy = (cy << PHIGH_BITS) + (hi >> (GMP_NUMB_BITS - PHIGH_BITS));
  rp[ECC_LIMB_SIZE-1] = (hi & (GMP_NUMB_MASK >> PHIGH_BITS))
70 71
    + sec_add_1 (rp, rp, ECC_LIMB_SIZE - 1, 19 * cy);
}
72 73
#endif /* HAVE_NATIVE_ecc_25519_modp */

Niels Möller's avatar
Niels Möller committed
74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104
#define QHIGH_BITS (GMP_NUMB_BITS * ECC_LIMB_SIZE - 252)

#if QHIGH_BITS == 0
#error Unsupported limb size */
#endif

static void
ecc_25519_modq (const struct ecc_curve *ecc, mp_limb_t *rp)
{
  mp_size_t n;
  mp_limb_t cy;

  /* n is the offset where we add in the next term */
  for (n = ECC_LIMB_SIZE; n-- > 0;)
    {
      mp_limb_t cy;

      cy = mpn_submul_1 (rp + n,
			 ecc->Bmodq_shifted, ECC_LIMB_SIZE,
			 rp[n + ECC_LIMB_SIZE]);
      /* Top limb of mBmodq_shifted is zero, so we get cy == 0 or 1 */
      assert (cy < 2);
      cnd_add_n (cy, rp+n, ecc_q, ECC_LIMB_SIZE);
    }

  cy = mpn_submul_1 (rp, ecc_q, ECC_LIMB_SIZE,
		     rp[ECC_LIMB_SIZE-1] >> (GMP_NUMB_BITS - QHIGH_BITS));
  assert (cy < 2);
  cnd_add_n (cy, rp, ecc_q, ECC_LIMB_SIZE);
}

105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233
/* Needs 2*ecc->size limbs at rp, and 2*ecc->size additional limbs of
   scratch space. No overlap allowed. */
static void
ecc_modp_powm_2kp1 (const struct ecc_curve *ecc,
		    mp_limb_t *rp, const mp_limb_t *xp,
		    unsigned k, mp_limb_t *tp)
{
  if (k & 1)
    {
      ecc_modp_sqr (ecc, tp, xp);
      k--;
    }
  else
    {
      ecc_modp_sqr (ecc, rp, xp);
      ecc_modp_sqr (ecc, tp, rp);
      k -= 2;
    }
  while (k > 0)
    {
      ecc_modp_sqr (ecc, rp, tp);
      ecc_modp_sqr (ecc, tp, rp);
      k -= 2;
    }
  ecc_modp_mul (ecc, rp, tp, xp);
#undef t1
#undef t2
}

/* Compute x such that x^2 = a (mod p). Returns one on success, zero
   on failure. using the e == 2 special case of the Shanks-Tonelli
   algorithm (see http://www.math.vt.edu/people/brown/doc/sqrts.pdf,
   or Henri Cohen, Computational Algebraic Number Theory, 1.5.1.

   NOTE: Not side-channel silent. FIXME: Compute square root in the
   extended field if a isn't a square (mod p)? FIXME: Accept scratch
   space from caller (could allow scratch == rp). */
#if ECC_SQRT_E != 2
#error Broken curve25519 parameters
#endif
int
ecc_25519_sqrt(mp_limb_t *rp, const mp_limb_t *ap)
{
  mp_size_t itch;
  mp_limb_t *scratch;
  int res;
  const struct ecc_curve *ecc = &nettle_curve25519;

  itch = 7*ECC_LIMB_SIZE;
  scratch = gmp_alloc_limbs (itch);

#define t0 scratch
#define a7 (scratch + 2*ECC_LIMB_SIZE)
#define t1 (scratch + 3*ECC_LIMB_SIZE)
#define t2 (scratch + 5*ECC_LIMB_SIZE)
#define scratch_out (scratch + 3*ECC_LIMB_SIZE) /* overlap t1, t2 */

#define xp (scratch + ECC_LIMB_SIZE)
#define bp (scratch + 2*ECC_LIMB_SIZE)

  /* a^{2^252 - 3} = a^{(p-5)/8}, using the addition chain
     2^252 - 3
     = 1 + (2^252-4)
     = 1 + 4 (2^250-1)
     = 1 + 4 (2^125+1)(2^125-1)
     = 1 + 4 (2^125+1)(1+2(2^124-1))
     = 1 + 4 (2^125+1)(1+2(2^62+1)(2^62-1))
     = 1 + 4 (2^125+1)(1+2(2^62+1)(2^31+1)(2^31-1))
     = 1 + 4 (2^125+1)(1+2(2^62+1)(2^31+1)(7+8(2^28-1)))
     = 1 + 4 (2^125+1)(1+2(2^62+1)(2^31+1)(7+8(2^14+1)(2^14-1)))
     = 1 + 4 (2^125+1)(1+2(2^62+1)(2^31+1)(7+8(2^14+1)(2^7+1)(2^7-1)))
     = 1 + 4 (2^125+1)(1+2(2^62+1)(2^31+1)(7+8(2^14+1)(2^7+1)(1+2(2^6-1))))
     = 1 + 4 (2^125+1)(1+2(2^62+1)(2^31+1)(7+8(2^14+1)(2^7+1)(1+2(2^3+1)*7)))
  */ 
     
  ecc_modp_powm_2kp1 (ecc, t1, ap, 1, t2);  /* a^3 */
  ecc_modp_sqr (ecc, t0, t1);		    /* a^6 */
  ecc_modp_mul (ecc, a7, t0, ap);	    /* a^7 */
  ecc_modp_powm_2kp1 (ecc, t0, a7, 3, t1);  /* a^63 = a^{2^6-1} */
  ecc_modp_sqr (ecc, t1, t0);		    /* a^{2^7-2} */
  ecc_modp_mul (ecc, t0, t1, ap);	    /* a^{2^7-1} */
  ecc_modp_powm_2kp1 (ecc, t1, t0, 7, t2);  /* a^{2^14-1}*/
  ecc_modp_powm_2kp1 (ecc, t0, t1, 14, t2); /* a^{2^28-1} */
  ecc_modp_sqr (ecc, t1, t0);		    /* a^{2^29-2} */
  ecc_modp_sqr (ecc, t2, t1);		    /* a^{2^30-4} */
  ecc_modp_sqr (ecc, t1, t2);		    /* a^{2^31-8} */
  ecc_modp_mul (ecc, t0, t1, a7);	    /* a^{2^31-1} */
  ecc_modp_powm_2kp1 (ecc, t1, t0, 31, t2); /* a^{2^62-1} */  
  ecc_modp_powm_2kp1 (ecc, t0, t1, 62, t2); /* a^{2^124-1}*/
  ecc_modp_sqr (ecc, t1, t0);		    /* a^{2^125-2} */
  ecc_modp_mul (ecc, t0, t1, ap);	    /* a^{2^125-1} */
  ecc_modp_powm_2kp1 (ecc, t1, t0, 125, t2); /* a^{2^250-1} */
  ecc_modp_sqr (ecc, t0, t1);		    /* a^{2^251-2} */
  ecc_modp_sqr (ecc, t1, t0);		    /* a^{2^252-4} */
  ecc_modp_mul (ecc, t0, t1, ap);	    /* a^{2^252-3} */

  /* Compute candidate root x and fudgefactor b. */
  ecc_modp_mul (ecc, xp, t0, ap); /* a^{(p+3)/8 */
  ecc_modp_mul (ecc, bp, t0, xp); /* a^{(p-1)/4} */
  /* Check if b == 1 (mod p) */
  if (mpn_cmp (bp, ecc->p, ECC_LIMB_SIZE) >= 0)
    mpn_sub_n (bp, bp, ecc->p, ECC_LIMB_SIZE);
  if (mpn_cmp (bp, ecc->unit, ECC_LIMB_SIZE) == 0)
    {
      mpn_copyi (rp, xp, ECC_LIMB_SIZE);
      res = 1;
    }
  else
    {
      mpn_add_1 (bp, bp, ECC_LIMB_SIZE, 1);
      if (mpn_cmp (bp, ecc->p, ECC_LIMB_SIZE) == 0)
	{
	  ecc_modp_mul (&nettle_curve25519, bp, xp, ecc_sqrt_z);
	  mpn_copyi (rp, bp, ECC_LIMB_SIZE);
	  res = 1;
	}
      else
	res = 0;
    }
  gmp_free_limbs (scratch, itch);
  return res;
#undef t0
#undef t1
#undef t2
#undef a7
#undef xp
#undef bp
#undef scratch_out
}
234

235 236 237 238 239 240 241 242 243 244
const struct ecc_curve nettle_curve25519 =
{
  255,
  ECC_LIMB_SIZE,
  ECC_BMODP_SIZE,
  ECC_BMODQ_SIZE,
  0, /* No redc */
  0,
  ECC_PIPPENGER_K,
  ECC_PIPPENGER_C,
245

246 247 248 249
  ECC_MUL_A_EH_ITCH (ECC_LIMB_SIZE),
  ECC_MUL_G_EH_ITCH (ECC_LIMB_SIZE),
  ECC_EH_TO_A_ITCH (ECC_LIMB_SIZE),

250 251 252
  ecc_25519_modp,
  NULL,
  ecc_25519_modp,
Niels Möller's avatar
Niels Möller committed
253 254
  ecc_25519_modq,

255

256 257 258 259
  ecc_mul_a_eh,
  ecc_mul_g_eh,
  ecc_eh_to_a,

260
  ecc_p,
261
  ecc_d, /* Use the Edwards curve constant. */
262 263 264
  ecc_q,
  ecc_g,
  ecc_redc_g,
265
  ecc_edwards,
266 267 268 269 270
  ecc_Bmodp,
  ecc_Bmodp_shifted,
  ecc_pp1h,
  ecc_redc_ppm1,
  ecc_unit,
Niels Möller's avatar
Niels Möller committed
271 272
  ecc_Bmodq,  
  ecc_mBmodq_shifted, /* Use q - 2^{252} instead. */ 
273 274 275
  ecc_qp1h,
  ecc_table
};