/* ecc-curve448.c
Arithmetic and tables for curve448,
Copyright (C) 2017 Daiki Ueno
Copyright (C) 2017 Red Hat, Inc.
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 <nettle/ecc.h>
#include "ecc-internal.h"
#define USE_REDC 0
#if defined __clang__ || __GNUC__ > 4 || (__GNUC__ == 4 && __GNUC_MINOR__ >= 5)
# pragma GCC diagnostic ignored "-Wunused-const-variable"
#endif
#if GMP_NUMB_BITS == 32
#include "ecc/ecc-curve448-32.h"
#elif GMP_NUMB_BITS == 64
#include "ecc/ecc-curve448-64.h"
#else
#error unsupported configuration
#endif
#if HAVE_NATIVE_ecc_curve448_modp
#define ecc_curve448_modp _nettle_ecc_curve448_modp
void
ecc_curve448_modp (const struct ecc_modulo *m, mp_limb_t *rp);
#elif GMP_NUMB_BITS == 64
static void
ecc_curve448_modp(const struct ecc_modulo *m, mp_limb_t *rp)
{
/* Let B = 2^64, b = 2^32 = sqrt(B).
p = B^7 - b B^3 - 1 ==> B^7 = b B^3 + 1
We use this to reduce
{r_{13}, ..., r_0} =
{r_6,...,r_0}
+ {r_{10},...,r_7}
+ 2 {r_{13},r_{12}, r_{11}} B^4
+ b {r_{10},...,r_7,r_{13},r_{12},r_{11} (mod p)
or
+----+----+----+----+----+----+----+
|r_6 |r_5 |r_4 |r_3 |r_2 |r_1 |r_0 |
+----+----+----+----+----+----+----+
|r_10|r_9 |r_8 |r_7 |
+----+----+----+----+----+----+----+
2 * |r_13|r_12|r_11|
+----+----+----+----+----+----+----+
+ b * |r_10|r_9 |r_8 |r_7 |r_13|r_12|r_11|
-------+----+----+----+----+----+----+----+
c_7 |r_6 |r_5 |r_4 |r_3 |r_2 |r_1 |r_0 |
+----+----+----+----+----+----+----+
*/
mp_limb_t c3, c4, c7;
mp_limb_t *tp = rp + 7;
c4 = mpn_add_n (rp, rp, rp + 7, 4);
c7 = mpn_addmul_1 (rp + 4, rp + 11, 3, 2);
c3 = mpn_addmul_1 (rp, rp + 11, 3, (mp_limb_t) 1 << 32);
c7 += mpn_addmul_1 (rp + 3, rp + 7, 4, (mp_limb_t) 1 << 32);
tp[0] = c7;
tp[1] = tp[2] = 0;
tp[3] = c3 + (c7 << 32);
tp[4] = c4 + (c7 >> 32) + (tp[3] < c3);
tp[5] = tp[6] = 0;
c7 = mpn_add_n (rp, rp, tp, 7);
c7 = cnd_add_n (c7, rp, m->B, 7);
assert (c7 == 0);
}
#else
#define ecc_curve448_modp ecc_mod
#endif
/* Needs 2*ecc->size limbs at rp, and 2*ecc->size additional limbs of
scratch space. No overlap allowed. */
static void
ecc_mod_pow_2k (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, rp, xp);
k--;
}
else
{
ecc_mod_sqr (m, tp, xp);
ecc_mod_sqr (m, rp, tp);
k -= 2;
}
while (k > 0)
{
ecc_mod_sqr (m, tp, rp);
ecc_mod_sqr (m, rp, tp);
k -= 2;
}
}
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)
{
ecc_mod_pow_2k (m, tp, xp, k, rp);
ecc_mod_mul (m, rp, tp, xp);
}
/* Computes a^{(p-3)/4} = a^{2^446-2^222-1} mod m. Needs 5 * n scratch
space. */
static void
ecc_mod_pow_446m224m1 (const struct ecc_modulo *p,
mp_limb_t *rp, const mp_limb_t *ap,
mp_limb_t *scratch)
{
/* Note overlap: operations writing to t0 clobber t1. */
#define t0 scratch
#define t1 (scratch + 1*ECC_LIMB_SIZE)
#define t2 (scratch + 3*ECC_LIMB_SIZE)
ecc_mod_sqr (p, rp, ap); /* a^2 */
ecc_mod_mul (p, t0, ap, rp); /* a^3 */
ecc_mod_sqr (p, rp, t0); /* a^6 */
ecc_mod_mul (p, t0, ap, rp); /* a^{2^3-1} */
ecc_mod_pow_2kp1 (p, t1, t0, 3, rp); /* a^{2^6-1} */
ecc_mod_pow_2k (p, rp, t1, 3, t2); /* a^{2^9-2^3} */
ecc_mod_mul (p, t2, t0, rp); /* a^{2^9-1} */
ecc_mod_pow_2kp1 (p, t0, t2, 9, rp); /* a^{2^18-1} */
ecc_mod_sqr (p, t1, t0); /* a^{2^19-2} */
ecc_mod_mul (p, rp, ap, t1); /* a^{2^19-1} */
ecc_mod_pow_2k (p, t1, rp, 18, t2); /* a^{2^37-2^18} */
ecc_mod_mul (p, rp, t0, t1); /* a^{2^37-1} */
mpn_copyi (t0, rp, p->size);
ecc_mod_pow_2kp1 (p, rp, t0, 37, t2); /* a^{2^74-1} */
ecc_mod_pow_2k (p, t1, rp, 37, t2); /* a^{2^111-2^37} */
ecc_mod_mul (p, rp, t0, t1); /* a^{2^111-1} */
ecc_mod_pow_2kp1 (p, t0, rp, 111, t2);/* a^{2^222-1} */
ecc_mod_sqr (p, t1, t0); /* a^{2^223-2} */
ecc_mod_mul (p, rp, ap, t1); /* a^{2^223-1} */
ecc_mod_pow_2k (p, t1, rp, 223, t2); /* a^{2^446-2^223} */
ecc_mod_mul (p, rp, t0, t1); /* a^{2^446-2^222-1} */
#undef t0
#undef t1
#undef t2
}
#define ECC_CURVE448_INV_ITCH (5*ECC_LIMB_SIZE)
static void ecc_curve448_inv (const struct ecc_modulo *p,
mp_limb_t *rp, const mp_limb_t *ap,
mp_limb_t *scratch)
{
#define t0 scratch
ecc_mod_pow_446m224m1 (p, rp, ap, scratch); /* a^{2^446-2^222-1} */
ecc_mod_sqr (p, t0, rp); /* a^{2^447-2^223-2} */
ecc_mod_sqr (p, rp, t0); /* a^{2^448-2^224-4} */
ecc_mod_mul (p, t0, ap, rp); /* a^{2^448-2^224-3} */
mpn_copyi (rp, t0, ECC_LIMB_SIZE); /* FIXME: Eliminate copy? */
#undef t0
}
/* First, do a canonical reduction, then check if zero */
static int
ecc_curve448_zero_p (const struct ecc_modulo *p, mp_limb_t *xp)
{
mp_limb_t cy;
mp_limb_t w;
mp_size_t i;
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.
To avoid a separate inversion, we use a trick of djb's, to
compute the candidate root as
x = (u/v)^{(p+1)/4} = u^3 v (u^5 v^3)^{(p-3)/4}.
*/
/* Needs 4*n space + scratch for ecc_mod_pow_446m224m1. */
#define ECC_CURVE448_SQRT_ITCH (9*ECC_LIMB_SIZE)
static int
ecc_curve448_sqrt(const struct ecc_modulo *p, mp_limb_t *rp,
const mp_limb_t *up, const mp_limb_t *vp,
mp_limb_t *scratch)
{
#define u3v scratch
#define u5v3 (scratch + ECC_LIMB_SIZE)
#define u5v3p (scratch + 2*ECC_LIMB_SIZE)
#define u2 (scratch + 2*ECC_LIMB_SIZE)
#define u3 (scratch + 3*ECC_LIMB_SIZE)
#define uv (scratch + 2*ECC_LIMB_SIZE)
#define u2v2 (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, u2, up); /* u2 */
ecc_mod_mul (p, u3, u2, up); /* u3 */
ecc_mod_mul (p, u3v, u3, vp); /* u3v */
ecc_mod_mul (p, uv, up, vp); /* u3v, uv */
ecc_mod_sqr (p, u2v2, uv); /* u3v, u2v2 */
ecc_mod_mul (p, u5v3, u3v, u2v2); /* u3v, u5v3 */
ecc_mod_pow_446m224m1 (p, u5v3p, u5v3, scratch_out); /* u3v, u5v3p */
ecc_mod_mul (p, rp, u5v3p, u3v); /* none */
/* If square root exists, have v x^2 = u */
ecc_mod_sqr (p, x2, rp);
ecc_mod_mul (p, vx2, x2, vp);
ecc_mod_sub (p, t0, vx2, up);
return ecc_curve448_zero_p (p, t0);
#undef u3v
#undef u5v3
#undef u5v3p
#undef u2
#undef u3
#undef uv
#undef u2v2
#undef scratch_out
#undef x2
#undef vx2
#undef t0
}
const struct ecc_curve _nettle_curve448 =
{
{
448,
ECC_LIMB_SIZE,
ECC_BMODP_SIZE,
0,
ECC_CURVE448_INV_ITCH,
ECC_CURVE448_SQRT_ITCH,
ecc_p,
ecc_Bmodp,
ecc_Bmodp_shifted,
NULL,
ecc_pp1h,
ecc_curve448_modp,
ecc_curve448_modp,
ecc_curve448_inv,
ecc_curve448_sqrt,
},
{
446,
ECC_LIMB_SIZE,
ECC_BMODQ_SIZE,
0,
ECC_MOD_INV_ITCH (ECC_LIMB_SIZE),
0,
ecc_q,
ecc_Bmodq,
ecc_Bmodq_shifted,
NULL,
ecc_qp1h,
ecc_mod, /* FIXME: Implement optimized mod function */
ecc_mod, /* FIXME: Implement optimized reduce function */
ecc_mod_inv,
NULL,
},
0, /* No redc */
ECC_PIPPENGER_K,
ECC_PIPPENGER_C,
ECC_ADD_EH_ITCH (ECC_LIMB_SIZE),
ECC_ADD_EHH_ITCH (ECC_LIMB_SIZE),
ECC_DUP_EH_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_CURVE448_INV_ITCH),
ecc_add_eh,
ecc_add_ehh,
ecc_dup_eh,
ecc_mul_a_eh,
ecc_mul_g_eh,
ecc_eh_to_a,
ecc_b,
ecc_unit,
ecc_table
};