/* ecc-25519.c
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
#include <assert.h>
#include "ecc.h"
#include "ecc-internal.h"
#define USE_REDC 0
#include "ecc-25519.h"
#define PHIGH_BITS (GMP_NUMB_BITS * ECC_LIMB_SIZE - 255)
#if HAVE_NATIVE_ecc_25519_modp
#define ecc_25519_modp nettle_ecc_25519_modp
void
ecc_25519_modp (const struct ecc_modulo *m, mp_limb_t *rp);
#else
#if PHIGH_BITS == 0
#error Unsupported limb size */
#endif
static void
ecc_25519_modp(const struct ecc_modulo *m UNUSED, mp_limb_t *rp)
{
mp_limb_t hi, cy;
cy = mpn_addmul_1 (rp, rp + ECC_LIMB_SIZE, ECC_LIMB_SIZE,
(mp_limb_t) 19 << PHIGH_BITS);
hi = rp[ECC_LIMB_SIZE-1];
cy = (cy << PHIGH_BITS) + (hi >> (GMP_NUMB_BITS - PHIGH_BITS));
rp[ECC_LIMB_SIZE-1] = (hi & (GMP_NUMB_MASK >> PHIGH_BITS))
+ sec_add_1 (rp, rp, ECC_LIMB_SIZE - 1, 19 * cy);
}
#endif /* HAVE_NATIVE_ecc_25519_modp */
#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_modulo *q, 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;)
{
cy = mpn_submul_1 (rp + n,
q->B_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, q->m, ECC_LIMB_SIZE);
}
cy = mpn_submul_1 (rp, q->m, ECC_LIMB_SIZE,
rp[ECC_LIMB_SIZE-1] >> (GMP_NUMB_BITS - QHIGH_BITS));
assert (cy < 2);
cnd_add_n (cy, rp, q->m, ECC_LIMB_SIZE);
}
/* Needs 2*ecc->size limbs at rp, and 2*ecc->size additional limbs of
scratch space. No overlap allowed. */
static void
ecc_mod_pow_2kp1 (const struct ecc_modulo *m,
mp_limb_t *rp, const mp_limb_t *xp,
unsigned k, mp_limb_t *tp)
{
if (k & 1)
{
ecc_mod_sqr (m, tp, xp);
k--;
}
else
{
ecc_mod_sqr (m, rp, xp);
ecc_mod_sqr (m, tp, rp);
k -= 2;
}
while (k > 0)
{
ecc_mod_sqr (m, rp, tp);
ecc_mod_sqr (m, tp, rp);
k -= 2;
}
ecc_mod_mul (m, rp, tp, xp);
}
/* Computes a^{(p-5)/8} = a^{2^{252-3}} mod m. Needs 5 * n scratch
space. */
static void
ecc_mod_pow_252m3 (const struct ecc_modulo *m,
mp_limb_t *rp, const mp_limb_t *ap, mp_limb_t *scratch)
{
#define a7 scratch
#define t0 (scratch + ECC_LIMB_SIZE)
#define t1 (scratch + 3*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_mod_pow_2kp1 (m, t0, ap, 1, t1); /* a^3 */
ecc_mod_sqr (m, rp, t0); /* a^6 */
ecc_mod_mul (m, a7, rp, ap); /* a^7 */
ecc_mod_pow_2kp1 (m, rp, a7, 3, t0); /* a^63 = a^{2^6-1} */
ecc_mod_sqr (m, t0, rp); /* a^{2^7-2} */
ecc_mod_mul (m, rp, t0, ap); /* a^{2^7-1} */
ecc_mod_pow_2kp1 (m, t0, rp, 7, t1); /* a^{2^14-1}*/
ecc_mod_pow_2kp1 (m, rp, t0, 14, t1); /* a^{2^28-1} */
ecc_mod_sqr (m, t0, rp); /* a^{2^29-2} */
ecc_mod_sqr (m, t1, t0); /* a^{2^30-4} */
ecc_mod_sqr (m, t0, t1); /* a^{2^31-8} */
ecc_mod_mul (m, rp, t0, a7); /* a^{2^31-1} */
ecc_mod_pow_2kp1 (m, t0, rp, 31, t1); /* a^{2^62-1} */
ecc_mod_pow_2kp1 (m, rp, t0, 62, t1); /* a^{2^124-1}*/
ecc_mod_sqr (m, t0, rp); /* a^{2^125-2} */
ecc_mod_mul (m, rp, t0, ap); /* a^{2^125-1} */
ecc_mod_pow_2kp1 (m, t0, rp, 125, t1);/* a^{2^250-1} */
ecc_mod_sqr (m, rp, t0); /* a^{2^251-2} */
ecc_mod_sqr (m, t0, rp); /* a^{2^252-4} */
ecc_mod_mul (m, rp, t0, ap); /* a^{2^252-3} */
#undef t0
#undef t1
#undef a7
}
/* Needs 5*ECC_LIMB_SIZE scratch space. */
#define ECC_25519_INV_ITCH (5*ECC_LIMB_SIZE)
static void ecc_25519_inv (const struct ecc_modulo *p,
mp_limb_t *rp, const mp_limb_t *ap,
mp_limb_t *scratch)
{
#define t0 scratch
/* Addition chain
p - 2 = 2^{255} - 21
= 1 + 2 (1 + 4 (2^{252}-3))
*/
ecc_mod_pow_252m3 (p, rp, ap, t0);
ecc_mod_sqr (p, t0, rp);
ecc_mod_sqr (p, rp, t0);
ecc_mod_mul (p, t0, ap, rp);
ecc_mod_sqr (p, rp, t0);
ecc_mod_mul (p, t0, ap, rp);
mpn_copyi (rp, t0, ECC_LIMB_SIZE); /* FIXME: Eliminate copy? */
#undef t0
}
/* First, do a canonical reduction, then check if zero */
static int
ecc_25519_zero_p (const struct ecc_modulo *p, mp_limb_t *xp)
{
mp_limb_t cy;
mp_limb_t w;
mp_size_t i;
#if PHIGH_BITS > 0
mp_limb_t hi = xp[ECC_LIMB_SIZE-1];
xp[ECC_LIMB_SIZE-1] = (hi & (GMP_NUMB_MASK >> PHIGH_BITS))
+ sec_add_1 (xp, xp, ECC_LIMB_SIZE - 1, 19 * (hi >> (GMP_NUMB_BITS - PHIGH_BITS)));
#endif
cy = mpn_sub_n (xp, xp, p->m, ECC_LIMB_SIZE);
cnd_add_n (cy, xp, p->m, ECC_LIMB_SIZE);
for (i = 0, w = 0; i < ECC_LIMB_SIZE; i++)
w |= xp[i];
return w == 0;
}
/* Compute x such that x^2 = u/v (mod p). Returns one on success, zero
on failure. We use 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).
To avoid a separate inversion, we also use a trick of djb's, to
compute the candidate root as
x = (u/v)^{(p+3)/8} = u v^3 (u v^7)^{(p-5)/8}.
*/
#if ECC_SQRT_E != 2
#error Broken curve25519 parameters
#endif
/* Needs 4*n space + scratch for ecc_mod_pow_252m3. */
#define ECC_25519_SQRT_ITCH (9*ECC_LIMB_SIZE)
static int
ecc_25519_sqrt(const struct ecc_modulo *p, mp_limb_t *rp,
const mp_limb_t *up, const mp_limb_t *vp,
mp_limb_t *scratch)
{
int pos, neg;
#define uv3 scratch
#define uv7 (scratch + ECC_LIMB_SIZE)
#define uv7p (scratch + 2*ECC_LIMB_SIZE)
#define v2 (scratch + 2*ECC_LIMB_SIZE)
#define uv (scratch + 3*ECC_LIMB_SIZE)
#define v4 (scratch + 3*ECC_LIMB_SIZE)
#define scratch_out (scratch + 4 * ECC_LIMB_SIZE)
#define x2 scratch
#define vx2 (scratch + ECC_LIMB_SIZE)
#define t0 (scratch + 2*ECC_LIMB_SIZE)
/* Live values */
ecc_mod_sqr (p, v2, vp); /* v2 */
ecc_mod_mul (p, uv, up, vp); /* uv, v2 */
ecc_mod_mul (p, uv3, uv, v2); /* uv3, v2 */
ecc_mod_sqr (p, v4, v2); /* uv3, v4 */
ecc_mod_mul (p, uv7, uv3, v4); /* uv3, uv7 */
ecc_mod_pow_252m3 (p, uv7p, uv7, scratch_out); /* uv3, uv7p */
ecc_mod_mul (p, rp, uv7p, uv3); /* none */
/* Check sign. If square root exists, have v x^2 = ±u */
ecc_mod_sqr (p, x2, rp);
ecc_mod_mul (p, vx2, x2, vp);
ecc_mod_add (p, t0, vx2, up);
neg = ecc_25519_zero_p (p, t0);
ecc_mod_sub (p, t0, up, vx2);
pos = ecc_25519_zero_p (p, t0);
ecc_mod_mul (p, t0, rp, ecc_sqrt_z);
cnd_copy (neg, rp, t0, ECC_LIMB_SIZE);
return pos | neg;
#undef uv3
#undef uv7
#undef uv7p
#undef v2
#undef v4
#undef scratch_out
#undef x2
#undef vx2
#undef t0
}
const struct ecc_curve _nettle_curve25519 =
{
{
255,
ECC_LIMB_SIZE,
ECC_BMODP_SIZE,
0,
ECC_25519_INV_ITCH,
ECC_25519_SQRT_ITCH,
ecc_p,
ecc_Bmodp,
ecc_Bmodp_shifted,
NULL,
ecc_pp1h,
ecc_25519_modp,
ecc_25519_modp,
ecc_25519_inv,
ecc_25519_sqrt,
},
{
253,
ECC_LIMB_SIZE,
ECC_BMODQ_SIZE,
0,
ECC_MOD_INV_ITCH (ECC_LIMB_SIZE),
0,
ecc_q,
ecc_Bmodq,
ecc_mBmodq_shifted, /* Use q - 2^{252} instead. */
NULL,
ecc_qp1h,
ecc_25519_modq,
ecc_25519_modq,
ecc_mod_inv,
NULL,
},
0, /* No redc */
ECC_PIPPENGER_K,
ECC_PIPPENGER_C,
ECC_ADD_EHH_ITCH (ECC_LIMB_SIZE),
ECC_MUL_A_EH_ITCH (ECC_LIMB_SIZE),
ECC_MUL_G_EH_ITCH (ECC_LIMB_SIZE),
ECC_EH_TO_A_ITCH (ECC_LIMB_SIZE, ECC_25519_INV_ITCH),
ecc_add_ehh,
ecc_mul_a_eh,
ecc_mul_g_eh,
ecc_eh_to_a,
ecc_d, /* Use the Edwards curve constant. */
ecc_g,
ecc_edwards,
ecc_unit,
ecc_table
};