2017-12-01 18:12:25 +00:00
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/*
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2018-01-18 13:12:46 +00:00
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* Copyright 2017-2018 The OpenSSL Project Authors. All Rights Reserved.
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2017-12-01 18:12:25 +00:00
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* Copyright 2015-2016 Cryptography Research, Inc.
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2017-11-15 15:27:21 +00:00
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*
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2017-12-01 18:12:25 +00:00
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* Licensed under the OpenSSL license (the "License"). You may not use
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* this file except in compliance with the License. You can obtain a copy
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* in the file LICENSE in the source distribution or at
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* https://www.openssl.org/source/license.html
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2017-11-15 15:27:21 +00:00
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*
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2017-12-01 18:12:25 +00:00
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* Originally written by Mike Hamburg
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2017-11-15 15:27:21 +00:00
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*/
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2017-11-15 16:39:33 +00:00
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#include <openssl/crypto.h>
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2017-11-15 15:27:21 +00:00
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#include "word.h"
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#include "field.h"
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2017-11-15 16:15:18 +00:00
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#include "point_448.h"
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#include "ed448.h"
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2017-11-16 11:13:53 +00:00
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#include "curve448_lcl.h"
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2017-11-15 15:27:21 +00:00
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#define COFACTOR 4
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/* Comb config: number of combs, n, t, s. */
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#define COMBS_N 5
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#define COMBS_T 5
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#define COMBS_S 18
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2017-12-05 16:37:57 +00:00
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#define C448_WNAF_FIXED_TABLE_BITS 5
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#define C448_WNAF_VAR_TABLE_BITS 3
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2017-11-15 15:27:21 +00:00
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static const int EDWARDS_D = -39081;
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2017-12-04 13:30:53 +00:00
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static const curve448_scalar_t precomputed_scalarmul_adjustment = {
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{
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{
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SC_LIMB(0xc873d6d54a7bb0cf), SC_LIMB(0xe933d8d723a70aad),
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SC_LIMB(0xbb124b65129c96fd), SC_LIMB(0x00000008335dc163)
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}
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}
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2017-12-04 11:38:58 +00:00
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};
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2017-11-15 15:27:21 +00:00
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#define TWISTED_D ((EDWARDS_D)-1)
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2017-12-05 16:37:57 +00:00
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#define WBITS C448_WORD_BITS /* NB this may be different from ARCH_WORD_BITS */
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2017-11-15 15:27:21 +00:00
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/* Projective Niels coordinates */
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2017-12-04 11:38:58 +00:00
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typedef struct {
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gf a, b, c;
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} niels_s, niels_t[1];
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typedef struct {
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niels_t n;
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gf z;
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2017-12-05 16:14:11 +00:00
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} VECTOR_ALIGNED pniels_t[1];
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2017-11-15 15:27:21 +00:00
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/* Precomputed base */
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2017-12-04 11:38:58 +00:00
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struct curve448_precomputed_s {
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niels_t table[COMBS_N << (COMBS_T - 1)];
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};
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2017-11-15 15:27:21 +00:00
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2017-11-16 16:27:48 +00:00
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extern const gf curve448_precomputed_base_as_fe[];
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2017-11-16 16:58:20 +00:00
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const curve448_precomputed_s *curve448_precomputed_base =
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2017-12-04 11:38:58 +00:00
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(const curve448_precomputed_s *)&curve448_precomputed_base_as_fe;
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2017-11-15 15:27:21 +00:00
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2017-12-04 13:30:53 +00:00
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/* Inverse. */
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2017-12-04 11:38:58 +00:00
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static void gf_invert(gf y, const gf x, int assert_nonzero)
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{
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2018-01-18 12:55:23 +00:00
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mask_t ret;
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2017-11-15 15:27:21 +00:00
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gf t1, t2;
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2017-12-04 11:38:58 +00:00
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gf_sqr(t1, x); /* o^2 */
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ret = gf_isr(t2, t1); /* +-1/sqrt(o^2) = +-1/o */
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2017-11-15 15:27:21 +00:00
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(void)ret;
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2017-12-04 11:38:58 +00:00
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if (assert_nonzero)
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assert(ret);
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2017-11-15 15:27:21 +00:00
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gf_sqr(t1, t2);
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2017-12-04 11:38:58 +00:00
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gf_mul(t2, t1, x); /* not direct to y in case of alias. */
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2017-11-15 15:27:21 +00:00
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gf_copy(y, t2);
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}
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/** identity = (0,1) */
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2017-12-04 11:38:58 +00:00
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const curve448_point_t curve448_point_identity =
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{ {{{{0}}}, {{{1}}}, {{{1}}}, {{{0}}}} };
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2017-11-15 15:27:21 +00:00
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2017-12-04 13:30:53 +00:00
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static void point_double_internal(curve448_point_t p, const curve448_point_t q,
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int before_double)
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2017-12-04 11:38:58 +00:00
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{
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2017-11-15 15:27:21 +00:00
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gf a, b, c, d;
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2017-12-04 13:30:53 +00:00
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2017-12-04 11:38:58 +00:00
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gf_sqr(c, q->x);
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gf_sqr(a, q->y);
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gf_add_nr(d, c, a); /* 2+e */
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gf_add_nr(p->t, q->y, q->x); /* 2+e */
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gf_sqr(b, p->t);
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gf_subx_nr(b, b, d, 3); /* 4+e */
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gf_sub_nr(p->t, a, c); /* 3+e */
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gf_sqr(p->x, q->z);
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gf_add_nr(p->z, p->x, p->x); /* 2+e */
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gf_subx_nr(a, p->z, p->t, 4); /* 6+e */
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if (GF_HEADROOM == 5)
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gf_weak_reduce(a); /* or 1+e */
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gf_mul(p->x, a, b);
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gf_mul(p->z, p->t, a);
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gf_mul(p->y, p->t, d);
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if (!before_double)
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gf_mul(p->t, b, d);
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2017-11-15 15:27:21 +00:00
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}
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2017-12-04 11:38:58 +00:00
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void curve448_point_double(curve448_point_t p, const curve448_point_t q)
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{
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point_double_internal(p, q, 0);
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2017-11-15 15:27:21 +00:00
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}
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/* Operations on [p]niels */
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2017-12-04 11:38:58 +00:00
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static ossl_inline void cond_neg_niels(niels_t n, mask_t neg)
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{
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2017-11-15 15:27:21 +00:00
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gf_cond_swap(n->a, n->b, neg);
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gf_cond_neg(n->c, neg);
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}
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2017-12-04 11:38:58 +00:00
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static void pt_to_pniels(pniels_t b, const curve448_point_t a)
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{
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gf_sub(b->n->a, a->y, a->x);
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gf_add(b->n->b, a->x, a->y);
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gf_mulw(b->n->c, a->t, 2 * TWISTED_D);
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gf_add(b->z, a->z, a->z);
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2017-11-15 15:27:21 +00:00
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}
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2017-12-04 11:38:58 +00:00
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static void pniels_to_pt(curve448_point_t e, const pniels_t d)
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{
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2017-11-15 15:27:21 +00:00
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gf eu;
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2017-12-04 13:30:53 +00:00
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2017-12-04 11:38:58 +00:00
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gf_add(eu, d->n->b, d->n->a);
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gf_sub(e->y, d->n->b, d->n->a);
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gf_mul(e->t, e->y, eu);
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gf_mul(e->x, d->z, e->y);
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gf_mul(e->y, d->z, eu);
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gf_sqr(e->z, d->z);
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2017-11-15 15:27:21 +00:00
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}
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2017-12-04 11:38:58 +00:00
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static void niels_to_pt(curve448_point_t e, const niels_t n)
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{
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gf_add(e->y, n->b, n->a);
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gf_sub(e->x, n->b, n->a);
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gf_mul(e->t, e->y, e->x);
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gf_copy(e->z, ONE);
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2017-11-15 15:27:21 +00:00
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}
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2017-12-04 13:30:53 +00:00
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static void add_niels_to_pt(curve448_point_t d, const niels_t e,
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int before_double)
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2017-12-04 11:38:58 +00:00
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{
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2017-11-15 15:27:21 +00:00
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gf a, b, c;
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2017-12-04 13:30:53 +00:00
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2017-12-04 11:38:58 +00:00
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gf_sub_nr(b, d->y, d->x); /* 3+e */
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gf_mul(a, e->a, b);
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gf_add_nr(b, d->x, d->y); /* 2+e */
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gf_mul(d->y, e->b, b);
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gf_mul(d->x, e->c, d->t);
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gf_add_nr(c, a, d->y); /* 2+e */
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gf_sub_nr(b, d->y, a); /* 3+e */
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gf_sub_nr(d->y, d->z, d->x); /* 3+e */
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gf_add_nr(a, d->x, d->z); /* 2+e */
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gf_mul(d->z, a, d->y);
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gf_mul(d->x, d->y, b);
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gf_mul(d->y, a, c);
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if (!before_double)
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gf_mul(d->t, b, c);
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2017-11-15 15:27:21 +00:00
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}
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2017-12-04 13:30:53 +00:00
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static void sub_niels_from_pt(curve448_point_t d, const niels_t e,
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int before_double)
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2017-12-04 11:38:58 +00:00
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{
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2017-11-15 15:27:21 +00:00
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gf a, b, c;
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2018-02-12 14:27:02 +00:00
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2017-12-04 11:38:58 +00:00
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gf_sub_nr(b, d->y, d->x); /* 3+e */
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gf_mul(a, e->b, b);
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gf_add_nr(b, d->x, d->y); /* 2+e */
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gf_mul(d->y, e->a, b);
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gf_mul(d->x, e->c, d->t);
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gf_add_nr(c, a, d->y); /* 2+e */
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gf_sub_nr(b, d->y, a); /* 3+e */
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gf_add_nr(d->y, d->z, d->x); /* 2+e */
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gf_sub_nr(a, d->z, d->x); /* 3+e */
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gf_mul(d->z, a, d->y);
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gf_mul(d->x, d->y, b);
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gf_mul(d->y, a, c);
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if (!before_double)
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gf_mul(d->t, b, c);
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2017-11-15 15:27:21 +00:00
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}
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2017-12-04 13:30:53 +00:00
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static void add_pniels_to_pt(curve448_point_t p, const pniels_t pn,
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int before_double)
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2017-12-04 11:38:58 +00:00
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{
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2017-11-15 15:27:21 +00:00
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gf L0;
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2017-12-04 13:30:53 +00:00
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2017-12-04 11:38:58 +00:00
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gf_mul(L0, p->z, pn->z);
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gf_copy(p->z, L0);
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add_niels_to_pt(p, pn->n, before_double);
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2017-11-15 15:27:21 +00:00
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}
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2017-12-04 13:30:53 +00:00
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static void sub_pniels_from_pt(curve448_point_t p, const pniels_t pn,
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int before_double)
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2017-12-04 11:38:58 +00:00
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{
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2017-11-15 15:27:21 +00:00
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gf L0;
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2017-12-04 13:30:53 +00:00
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2017-12-04 11:38:58 +00:00
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gf_mul(L0, p->z, pn->z);
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gf_copy(p->z, L0);
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sub_niels_from_pt(p, pn->n, before_double);
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2017-11-15 15:27:21 +00:00
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}
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2017-12-05 16:37:57 +00:00
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c448_bool_t curve448_point_eq(const curve448_point_t p,
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const curve448_point_t q)
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2017-12-04 11:38:58 +00:00
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{
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2018-01-18 12:55:23 +00:00
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mask_t succ;
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2018-02-12 14:27:02 +00:00
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gf a, b;
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2018-01-18 12:55:23 +00:00
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2017-11-15 15:27:21 +00:00
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/* equality mod 2-torsion compares x/y */
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2017-12-04 11:38:58 +00:00
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gf_mul(a, p->y, q->x);
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gf_mul(b, q->y, p->x);
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succ = gf_eq(a, b);
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2017-11-16 16:22:00 +00:00
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2017-11-15 15:27:21 +00:00
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return mask_to_bool(succ);
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}
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2017-12-05 16:37:57 +00:00
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c448_bool_t curve448_point_valid(const curve448_point_t p)
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2017-12-04 11:38:58 +00:00
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{
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2018-01-18 12:55:23 +00:00
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mask_t out;
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2017-12-04 11:38:58 +00:00
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gf a, b, c;
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2018-02-12 14:27:02 +00:00
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2017-12-04 11:38:58 +00:00
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gf_mul(a, p->x, p->y);
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gf_mul(b, p->z, p->t);
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out = gf_eq(a, b);
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gf_sqr(a, p->x);
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gf_sqr(b, p->y);
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gf_sub(a, b, a);
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gf_sqr(b, p->t);
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gf_mulw(c, b, TWISTED_D);
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gf_sqr(b, p->z);
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gf_add(b, b, c);
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out &= gf_eq(a, b);
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out &= ~gf_eq(p->z, ZERO);
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2017-11-15 15:27:21 +00:00
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return mask_to_bool(out);
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}
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2017-12-12 14:17:40 +00:00
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static ossl_inline void constant_time_lookup_niels(niels_s * RESTRICT ni,
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2017-12-04 13:30:53 +00:00
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const niels_t * table,
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int nelts, int idx)
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2017-12-04 11:38:58 +00:00
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{
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2017-11-15 15:27:21 +00:00
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constant_time_lookup(ni, table, sizeof(niels_s), nelts, idx);
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}
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2017-12-04 11:38:58 +00:00
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void curve448_precomputed_scalarmul(curve448_point_t out,
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const curve448_precomputed_s * table,
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const curve448_scalar_t scalar)
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{
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2018-02-12 14:27:02 +00:00
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unsigned int i, j, k;
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2017-11-15 15:27:21 +00:00
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const unsigned int n = COMBS_N, t = COMBS_T, s = COMBS_S;
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2018-01-18 12:55:23 +00:00
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niels_t ni;
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2017-11-16 16:58:20 +00:00
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curve448_scalar_t scalar1x;
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2018-02-12 14:27:02 +00:00
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2017-11-16 16:27:48 +00:00
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curve448_scalar_add(scalar1x, scalar, precomputed_scalarmul_adjustment);
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2017-12-04 11:38:58 +00:00
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curve448_scalar_halve(scalar1x, scalar1x);
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2018-02-12 14:27:02 +00:00
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for (i = s; i > 0; i--) {
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if (i != s)
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2017-12-04 11:38:58 +00:00
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point_double_internal(out, out, 0);
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for (j = 0; j < n; j++) {
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2017-11-15 15:27:21 +00:00
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int tab = 0;
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2018-01-18 12:55:23 +00:00
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mask_t invert;
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2017-12-04 11:38:58 +00:00
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|
|
|
|
for (k = 0; k < t; k++) {
|
2018-02-12 14:27:02 +00:00
|
|
|
unsigned int bit = (i - 1) + s * (k + j * t);
|
|
|
|
|
2017-12-06 13:06:13 +00:00
|
|
|
if (bit < C448_SCALAR_BITS) {
|
2017-12-04 11:38:58 +00:00
|
|
|
tab |=
|
|
|
|
(scalar1x->limb[bit / WBITS] >> (bit % WBITS) & 1) << k;
|
2017-11-15 15:27:21 +00:00
|
|
|
}
|
|
|
|
}
|
2017-12-04 11:38:58 +00:00
|
|
|
|
|
|
|
invert = (tab >> (t - 1)) - 1;
|
2017-11-15 15:27:21 +00:00
|
|
|
tab ^= invert;
|
2017-12-04 11:38:58 +00:00
|
|
|
tab &= (1 << (t - 1)) - 1;
|
2017-11-15 15:27:21 +00:00
|
|
|
|
2017-12-04 11:38:58 +00:00
|
|
|
constant_time_lookup_niels(ni, &table->table[j << (t - 1)],
|
|
|
|
1 << (t - 1), tab);
|
2017-11-15 15:27:21 +00:00
|
|
|
|
|
|
|
cond_neg_niels(ni, invert);
|
2018-02-12 14:27:02 +00:00
|
|
|
if ((i != s) || j != 0) {
|
|
|
|
add_niels_to_pt(out, ni, j == n - 1 && i != 1);
|
2017-11-15 15:27:21 +00:00
|
|
|
} else {
|
|
|
|
niels_to_pt(out, ni);
|
|
|
|
}
|
|
|
|
}
|
|
|
|
}
|
2017-12-04 11:38:58 +00:00
|
|
|
|
|
|
|
OPENSSL_cleanse(ni, sizeof(ni));
|
|
|
|
OPENSSL_cleanse(scalar1x, sizeof(scalar1x));
|
2017-11-15 15:27:21 +00:00
|
|
|
}
|
|
|
|
|
2017-12-04 13:30:53 +00:00
|
|
|
void curve448_point_mul_by_ratio_and_encode_like_eddsa(
|
2017-12-06 13:06:13 +00:00
|
|
|
uint8_t enc[EDDSA_448_PUBLIC_BYTES],
|
2017-12-04 13:30:53 +00:00
|
|
|
const curve448_point_t p)
|
2017-12-04 11:38:58 +00:00
|
|
|
{
|
2017-11-15 15:27:21 +00:00
|
|
|
gf x, y, z, t;
|
2017-11-16 16:58:20 +00:00
|
|
|
curve448_point_t q;
|
2018-02-12 14:27:02 +00:00
|
|
|
|
|
|
|
/* The point is now on the twisted curve. Move it to untwisted. */
|
2017-12-04 11:38:58 +00:00
|
|
|
curve448_point_copy(q, p);
|
2017-11-16 16:22:00 +00:00
|
|
|
|
2017-11-15 15:27:21 +00:00
|
|
|
{
|
|
|
|
/* 4-isogeny: 2xy/(y^+x^2), (y^2-x^2)/(2z^2-y^2+x^2) */
|
|
|
|
gf u;
|
2017-12-04 13:30:53 +00:00
|
|
|
|
2017-12-04 11:38:58 +00:00
|
|
|
gf_sqr(x, q->x);
|
|
|
|
gf_sqr(t, q->y);
|
|
|
|
gf_add(u, x, t);
|
|
|
|
gf_add(z, q->y, q->x);
|
|
|
|
gf_sqr(y, z);
|
|
|
|
gf_sub(y, y, u);
|
|
|
|
gf_sub(z, t, x);
|
|
|
|
gf_sqr(x, q->z);
|
|
|
|
gf_add(t, x, x);
|
|
|
|
gf_sub(t, t, z);
|
|
|
|
gf_mul(x, t, y);
|
|
|
|
gf_mul(y, z, u);
|
|
|
|
gf_mul(z, u, t);
|
|
|
|
OPENSSL_cleanse(u, sizeof(u));
|
2017-11-15 15:27:21 +00:00
|
|
|
}
|
2017-11-16 16:22:00 +00:00
|
|
|
|
2017-11-15 15:27:21 +00:00
|
|
|
/* Affinize */
|
2017-12-04 11:38:58 +00:00
|
|
|
gf_invert(z, z, 1);
|
|
|
|
gf_mul(t, x, z);
|
|
|
|
gf_mul(x, y, z);
|
|
|
|
|
2017-11-15 15:27:21 +00:00
|
|
|
/* Encode */
|
2017-12-06 13:06:13 +00:00
|
|
|
enc[EDDSA_448_PRIVATE_BYTES - 1] = 0;
|
2017-11-15 15:27:21 +00:00
|
|
|
gf_serialize(enc, x, 1);
|
2017-12-06 13:06:13 +00:00
|
|
|
enc[EDDSA_448_PRIVATE_BYTES - 1] |= 0x80 & gf_lobit(t);
|
2017-11-15 15:27:21 +00:00
|
|
|
|
2017-12-04 11:38:58 +00:00
|
|
|
OPENSSL_cleanse(x, sizeof(x));
|
|
|
|
OPENSSL_cleanse(y, sizeof(y));
|
|
|
|
OPENSSL_cleanse(z, sizeof(z));
|
|
|
|
OPENSSL_cleanse(t, sizeof(t));
|
2017-11-16 16:27:48 +00:00
|
|
|
curve448_point_destroy(q);
|
2017-11-15 15:27:21 +00:00
|
|
|
}
|
|
|
|
|
2017-12-05 16:37:57 +00:00
|
|
|
c448_error_t curve448_point_decode_like_eddsa_and_mul_by_ratio(
|
2017-12-04 13:30:53 +00:00
|
|
|
curve448_point_t p,
|
2017-12-06 13:06:13 +00:00
|
|
|
const uint8_t enc[EDDSA_448_PUBLIC_BYTES])
|
2017-12-04 11:38:58 +00:00
|
|
|
{
|
2017-12-06 13:06:13 +00:00
|
|
|
uint8_t enc2[EDDSA_448_PUBLIC_BYTES];
|
2018-01-18 12:55:23 +00:00
|
|
|
mask_t low;
|
|
|
|
mask_t succ;
|
|
|
|
|
2017-12-04 11:38:58 +00:00
|
|
|
memcpy(enc2, enc, sizeof(enc2));
|
|
|
|
|
2017-12-06 13:06:13 +00:00
|
|
|
low = ~word_is_zero(enc2[EDDSA_448_PRIVATE_BYTES - 1] & 0x80);
|
|
|
|
enc2[EDDSA_448_PRIVATE_BYTES - 1] &= ~0x80;
|
2017-11-15 15:27:21 +00:00
|
|
|
|
2018-01-18 12:55:23 +00:00
|
|
|
succ = gf_deserialize(p->y, enc2, 1, 0);
|
2017-12-06 13:06:13 +00:00
|
|
|
succ &= word_is_zero(enc2[EDDSA_448_PRIVATE_BYTES - 1]);
|
2017-11-15 15:27:21 +00:00
|
|
|
|
2017-12-04 11:38:58 +00:00
|
|
|
gf_sqr(p->x, p->y);
|
|
|
|
gf_sub(p->z, ONE, p->x); /* num = 1-y^2 */
|
|
|
|
gf_mulw(p->t, p->x, EDWARDS_D); /* dy^2 */
|
|
|
|
gf_sub(p->t, ONE, p->t); /* denom = 1-dy^2 or 1-d + dy^2 */
|
|
|
|
|
|
|
|
gf_mul(p->x, p->z, p->t);
|
|
|
|
succ &= gf_isr(p->t, p->x); /* 1/sqrt(num * denom) */
|
|
|
|
|
|
|
|
gf_mul(p->x, p->t, p->z); /* sqrt(num / denom) */
|
|
|
|
gf_cond_neg(p->x, gf_lobit(p->x) ^ low);
|
|
|
|
gf_copy(p->z, ONE);
|
|
|
|
|
2017-11-15 15:27:21 +00:00
|
|
|
{
|
|
|
|
gf a, b, c, d;
|
2018-02-12 14:27:02 +00:00
|
|
|
|
|
|
|
/* 4-isogeny 2xy/(y^2-ax^2), (y^2+ax^2)/(2-y^2-ax^2) */
|
2017-12-04 11:38:58 +00:00
|
|
|
gf_sqr(c, p->x);
|
|
|
|
gf_sqr(a, p->y);
|
|
|
|
gf_add(d, c, a);
|
|
|
|
gf_add(p->t, p->y, p->x);
|
|
|
|
gf_sqr(b, p->t);
|
|
|
|
gf_sub(b, b, d);
|
|
|
|
gf_sub(p->t, a, c);
|
|
|
|
gf_sqr(p->x, p->z);
|
|
|
|
gf_add(p->z, p->x, p->x);
|
|
|
|
gf_sub(a, p->z, d);
|
|
|
|
gf_mul(p->x, a, b);
|
|
|
|
gf_mul(p->z, p->t, a);
|
|
|
|
gf_mul(p->y, p->t, d);
|
|
|
|
gf_mul(p->t, b, d);
|
|
|
|
OPENSSL_cleanse(a, sizeof(a));
|
|
|
|
OPENSSL_cleanse(b, sizeof(b));
|
|
|
|
OPENSSL_cleanse(c, sizeof(c));
|
|
|
|
OPENSSL_cleanse(d, sizeof(d));
|
2017-11-15 15:27:21 +00:00
|
|
|
}
|
2017-12-04 11:38:58 +00:00
|
|
|
|
|
|
|
OPENSSL_cleanse(enc2, sizeof(enc2));
|
2017-11-16 16:27:48 +00:00
|
|
|
assert(curve448_point_valid(p) || ~succ);
|
2017-12-04 11:38:58 +00:00
|
|
|
|
2017-12-05 16:37:57 +00:00
|
|
|
return c448_succeed_if(mask_to_bool(succ));
|
2017-11-15 15:27:21 +00:00
|
|
|
}
|
|
|
|
|
2017-12-06 13:06:13 +00:00
|
|
|
c448_error_t x448_int(uint8_t out[X_PUBLIC_BYTES],
|
|
|
|
const uint8_t base[X_PUBLIC_BYTES],
|
|
|
|
const uint8_t scalar[X_PRIVATE_BYTES])
|
2017-12-04 11:38:58 +00:00
|
|
|
{
|
2017-11-15 15:27:21 +00:00
|
|
|
gf x1, x2, z2, x3, z3, t1, t2;
|
2018-01-18 12:55:23 +00:00
|
|
|
int t;
|
|
|
|
mask_t swap = 0;
|
|
|
|
mask_t nz;
|
|
|
|
|
2017-12-04 11:38:58 +00:00
|
|
|
ignore_result(gf_deserialize(x1, base, 1, 0));
|
|
|
|
gf_copy(x2, ONE);
|
|
|
|
gf_copy(z2, ZERO);
|
|
|
|
gf_copy(x3, x1);
|
|
|
|
gf_copy(z3, ONE);
|
|
|
|
|
|
|
|
for (t = X_PRIVATE_BITS - 1; t >= 0; t--) {
|
|
|
|
uint8_t sb = scalar[t / 8];
|
2018-01-18 12:55:23 +00:00
|
|
|
mask_t k_t;
|
2017-12-04 11:38:58 +00:00
|
|
|
|
2017-11-15 15:27:21 +00:00
|
|
|
/* Scalar conditioning */
|
2017-12-04 11:38:58 +00:00
|
|
|
if (t / 8 == 0)
|
|
|
|
sb &= -(uint8_t)COFACTOR;
|
|
|
|
else if (t == X_PRIVATE_BITS - 1)
|
|
|
|
sb = -1;
|
|
|
|
|
|
|
|
k_t = (sb >> (t % 8)) & 1;
|
2017-12-12 14:17:40 +00:00
|
|
|
k_t = 0 - k_t; /* set to all 0s or all 1s */
|
2017-12-04 11:38:58 +00:00
|
|
|
|
2017-11-15 15:27:21 +00:00
|
|
|
swap ^= k_t;
|
2017-12-04 11:38:58 +00:00
|
|
|
gf_cond_swap(x2, x3, swap);
|
|
|
|
gf_cond_swap(z2, z3, swap);
|
2017-11-15 15:27:21 +00:00
|
|
|
swap = k_t;
|
2017-12-04 11:38:58 +00:00
|
|
|
|
|
|
|
gf_add_nr(t1, x2, z2); /* A = x2 + z2 *//* 2+e */
|
|
|
|
gf_sub_nr(t2, x2, z2); /* B = x2 - z2 *//* 3+e */
|
|
|
|
gf_sub_nr(z2, x3, z3); /* D = x3 - z3 *//* 3+e */
|
|
|
|
gf_mul(x2, t1, z2); /* DA */
|
|
|
|
gf_add_nr(z2, z3, x3); /* C = x3 + z3 *//* 2+e */
|
|
|
|
gf_mul(x3, t2, z2); /* CB */
|
|
|
|
gf_sub_nr(z3, x2, x3); /* DA-CB *//* 3+e */
|
|
|
|
gf_sqr(z2, z3); /* (DA-CB)^2 */
|
|
|
|
gf_mul(z3, x1, z2); /* z3 = x1(DA-CB)^2 */
|
|
|
|
gf_add_nr(z2, x2, x3); /* (DA+CB) *//* 2+e */
|
|
|
|
gf_sqr(x3, z2); /* x3 = (DA+CB)^2 */
|
|
|
|
|
|
|
|
gf_sqr(z2, t1); /* AA = A^2 */
|
|
|
|
gf_sqr(t1, t2); /* BB = B^2 */
|
|
|
|
gf_mul(x2, z2, t1); /* x2 = AA*BB */
|
|
|
|
gf_sub_nr(t2, z2, t1); /* E = AA-BB *//* 3+e */
|
|
|
|
|
|
|
|
gf_mulw(t1, t2, -EDWARDS_D); /* E*-d = a24*E */
|
|
|
|
gf_add_nr(t1, t1, z2); /* AA + a24*E *//* 2+e */
|
|
|
|
gf_mul(z2, t2, t1); /* z2 = E(AA+a24*E) */
|
2017-11-15 15:27:21 +00:00
|
|
|
}
|
2017-12-04 11:38:58 +00:00
|
|
|
|
2017-11-15 15:27:21 +00:00
|
|
|
/* Finish */
|
2017-12-04 11:38:58 +00:00
|
|
|
gf_cond_swap(x2, x3, swap);
|
|
|
|
gf_cond_swap(z2, z3, swap);
|
|
|
|
gf_invert(z2, z2, 0);
|
|
|
|
gf_mul(x1, x2, z2);
|
|
|
|
gf_serialize(out, x1, 1);
|
|
|
|
nz = ~gf_eq(x1, ZERO);
|
|
|
|
|
|
|
|
OPENSSL_cleanse(x1, sizeof(x1));
|
|
|
|
OPENSSL_cleanse(x2, sizeof(x2));
|
|
|
|
OPENSSL_cleanse(z2, sizeof(z2));
|
|
|
|
OPENSSL_cleanse(x3, sizeof(x3));
|
|
|
|
OPENSSL_cleanse(z3, sizeof(z3));
|
|
|
|
OPENSSL_cleanse(t1, sizeof(t1));
|
|
|
|
OPENSSL_cleanse(t2, sizeof(t2));
|
|
|
|
|
2017-12-05 16:37:57 +00:00
|
|
|
return c448_succeed_if(mask_to_bool(nz));
|
2017-11-15 15:27:21 +00:00
|
|
|
}
|
|
|
|
|
2017-12-04 11:38:58 +00:00
|
|
|
void curve448_point_mul_by_ratio_and_encode_like_x448(uint8_t
|
|
|
|
out[X_PUBLIC_BYTES],
|
|
|
|
const curve448_point_t p)
|
|
|
|
{
|
2017-11-16 16:58:20 +00:00
|
|
|
curve448_point_t q;
|
2018-02-12 14:27:02 +00:00
|
|
|
|
2017-12-04 11:38:58 +00:00
|
|
|
curve448_point_copy(q, p);
|
|
|
|
gf_invert(q->t, q->x, 0); /* 1/x */
|
|
|
|
gf_mul(q->z, q->t, q->y); /* y/x */
|
|
|
|
gf_sqr(q->y, q->z); /* (y/x)^2 */
|
|
|
|
gf_serialize(out, q->y, 1);
|
2017-11-16 16:27:48 +00:00
|
|
|
curve448_point_destroy(q);
|
2017-11-15 15:27:21 +00:00
|
|
|
}
|
|
|
|
|
2017-12-06 13:06:13 +00:00
|
|
|
void x448_derive_public_key(uint8_t out[X_PUBLIC_BYTES],
|
|
|
|
const uint8_t scalar[X_PRIVATE_BYTES])
|
2017-12-04 11:38:58 +00:00
|
|
|
{
|
2017-11-15 15:27:21 +00:00
|
|
|
/* Scalar conditioning */
|
|
|
|
uint8_t scalar2[X_PRIVATE_BYTES];
|
2018-01-18 12:55:23 +00:00
|
|
|
curve448_scalar_t the_scalar;
|
|
|
|
curve448_point_t p;
|
|
|
|
unsigned int i;
|
|
|
|
|
2017-12-04 11:38:58 +00:00
|
|
|
memcpy(scalar2, scalar, sizeof(scalar2));
|
2017-11-15 15:27:21 +00:00
|
|
|
scalar2[0] &= -(uint8_t)COFACTOR;
|
2017-12-04 11:38:58 +00:00
|
|
|
|
2017-12-12 14:17:40 +00:00
|
|
|
scalar2[X_PRIVATE_BYTES - 1] &= ~((0u - 1u) << ((X_PRIVATE_BITS + 7) % 8));
|
2017-12-04 11:38:58 +00:00
|
|
|
scalar2[X_PRIVATE_BYTES - 1] |= 1 << ((X_PRIVATE_BITS + 7) % 8);
|
|
|
|
|
|
|
|
curve448_scalar_decode_long(the_scalar, scalar2, sizeof(scalar2));
|
|
|
|
|
2017-11-15 15:27:21 +00:00
|
|
|
/* Compensate for the encoding ratio */
|
2017-12-06 13:06:13 +00:00
|
|
|
for (i = 1; i < X448_ENCODE_RATIO; i <<= 1) {
|
2017-12-04 11:38:58 +00:00
|
|
|
curve448_scalar_halve(the_scalar, the_scalar);
|
2017-11-15 15:27:21 +00:00
|
|
|
}
|
2017-12-04 11:38:58 +00:00
|
|
|
curve448_precomputed_scalarmul(p, curve448_precomputed_base, the_scalar);
|
|
|
|
curve448_point_mul_by_ratio_and_encode_like_x448(out, p);
|
2017-11-16 16:27:48 +00:00
|
|
|
curve448_point_destroy(p);
|
2017-11-15 15:27:21 +00:00
|
|
|
}
|
|
|
|
|
2017-12-04 13:30:53 +00:00
|
|
|
/* Control for variable-time scalar multiply algorithms. */
|
2017-11-15 15:27:21 +00:00
|
|
|
struct smvt_control {
|
2017-12-04 11:38:58 +00:00
|
|
|
int power, addend;
|
2017-11-15 15:27:21 +00:00
|
|
|
};
|
|
|
|
|
2017-12-12 14:17:40 +00:00
|
|
|
#if defined(__GNUC__) || defined(__clang__)
|
|
|
|
# define NUMTRAILINGZEROS __builtin_ctz
|
|
|
|
#else
|
|
|
|
# define NUMTRAILINGZEROS numtrailingzeros
|
|
|
|
static uint32_t numtrailingzeros(uint32_t i)
|
|
|
|
{
|
2018-02-12 14:27:02 +00:00
|
|
|
uint32_t tmp;
|
2017-12-12 14:17:40 +00:00
|
|
|
uint32_t num = 31;
|
|
|
|
|
|
|
|
if (i == 0)
|
|
|
|
return 32;
|
|
|
|
|
|
|
|
tmp = i << 16;
|
|
|
|
if (tmp != 0) {
|
|
|
|
i = tmp;
|
|
|
|
num -= 16;
|
|
|
|
}
|
|
|
|
tmp = i << 8;
|
|
|
|
if (tmp != 0) {
|
|
|
|
i = tmp;
|
|
|
|
num -= 8;
|
|
|
|
}
|
|
|
|
tmp = i << 4;
|
|
|
|
if (tmp != 0) {
|
|
|
|
i = tmp;
|
|
|
|
num -= 4;
|
|
|
|
}
|
|
|
|
tmp = i << 2;
|
|
|
|
if (tmp != 0) {
|
|
|
|
i = tmp;
|
|
|
|
num -= 2;
|
|
|
|
}
|
2018-02-12 14:27:02 +00:00
|
|
|
tmp = i << 1;
|
|
|
|
if (tmp != 0)
|
2017-12-12 14:17:40 +00:00
|
|
|
num--;
|
|
|
|
|
|
|
|
return num;
|
|
|
|
}
|
|
|
|
#endif
|
|
|
|
|
2017-12-04 13:30:53 +00:00
|
|
|
static int recode_wnaf(struct smvt_control *control,
|
|
|
|
/* [nbits/(table_bits + 1) + 3] */
|
|
|
|
const curve448_scalar_t scalar,
|
|
|
|
unsigned int table_bits)
|
2017-12-04 11:38:58 +00:00
|
|
|
{
|
2017-12-06 13:06:13 +00:00
|
|
|
unsigned int table_size = C448_SCALAR_BITS / (table_bits + 1) + 3;
|
2017-11-15 15:27:21 +00:00
|
|
|
int position = table_size - 1; /* at the end */
|
2018-01-18 12:55:23 +00:00
|
|
|
uint64_t current = scalar->limb[0] & 0xFFFF;
|
2017-12-04 11:38:58 +00:00
|
|
|
uint32_t mask = (1 << (table_bits + 1)) - 1;
|
2018-01-18 12:55:23 +00:00
|
|
|
unsigned int w;
|
|
|
|
const unsigned int B_OVER_16 = sizeof(scalar->limb[0]) / 2;
|
|
|
|
unsigned int n, i;
|
|
|
|
|
2017-11-15 15:27:21 +00:00
|
|
|
/* place the end marker */
|
|
|
|
control[position].power = -1;
|
|
|
|
control[position].addend = 0;
|
|
|
|
position--;
|
|
|
|
|
2017-12-04 11:38:58 +00:00
|
|
|
/*
|
|
|
|
* PERF: Could negate scalar if it's large. But then would need more cases
|
|
|
|
* in the actual code that uses it, all for an expected reduction of like
|
|
|
|
* 1/5 op. Probably not worth it.
|
2017-11-15 15:27:21 +00:00
|
|
|
*/
|
|
|
|
|
2017-12-06 13:06:13 +00:00
|
|
|
for (w = 1; w < (C448_SCALAR_BITS - 1) / 16 + 3; w++) {
|
|
|
|
if (w < (C448_SCALAR_BITS - 1) / 16 + 1) {
|
2017-11-15 15:27:21 +00:00
|
|
|
/* Refill the 16 high bits of current */
|
2017-12-04 13:30:53 +00:00
|
|
|
current += (uint32_t)((scalar->limb[w / B_OVER_16]
|
|
|
|
>> (16 * (w % B_OVER_16))) << 16);
|
2017-11-15 15:27:21 +00:00
|
|
|
}
|
2017-12-04 11:38:58 +00:00
|
|
|
|
2017-11-15 15:27:21 +00:00
|
|
|
while (current & 0xFFFF) {
|
2017-12-12 14:17:40 +00:00
|
|
|
uint32_t pos = NUMTRAILINGZEROS((uint32_t)current);
|
2017-12-04 13:30:53 +00:00
|
|
|
uint32_t odd = (uint32_t)current >> pos;
|
2017-11-15 15:27:21 +00:00
|
|
|
int32_t delta = odd & mask;
|
2018-01-18 12:55:23 +00:00
|
|
|
|
|
|
|
assert(position >= 0);
|
2018-02-12 14:27:02 +00:00
|
|
|
if (odd & (1 << (table_bits + 1)))
|
2017-12-04 11:38:58 +00:00
|
|
|
delta -= (1 << (table_bits + 1));
|
2017-11-15 15:27:21 +00:00
|
|
|
current -= delta << pos;
|
2017-12-04 11:38:58 +00:00
|
|
|
control[position].power = pos + 16 * (w - 1);
|
2017-11-15 15:27:21 +00:00
|
|
|
control[position].addend = delta;
|
|
|
|
position--;
|
|
|
|
}
|
|
|
|
current >>= 16;
|
|
|
|
}
|
2017-12-04 11:38:58 +00:00
|
|
|
assert(current == 0);
|
|
|
|
|
2017-11-15 15:27:21 +00:00
|
|
|
position++;
|
2018-01-18 12:55:23 +00:00
|
|
|
n = table_size - position;
|
2018-02-12 14:27:02 +00:00
|
|
|
for (i = 0; i < n; i++)
|
2017-12-04 11:38:58 +00:00
|
|
|
control[i] = control[i + position];
|
2018-02-12 14:27:02 +00:00
|
|
|
|
2017-12-04 11:38:58 +00:00
|
|
|
return n - 1;
|
2017-11-15 15:27:21 +00:00
|
|
|
}
|
|
|
|
|
2017-12-04 13:30:53 +00:00
|
|
|
static void prepare_wnaf_table(pniels_t * output,
|
|
|
|
const curve448_point_t working,
|
|
|
|
unsigned int tbits)
|
2017-12-04 11:38:58 +00:00
|
|
|
{
|
2017-11-16 16:58:20 +00:00
|
|
|
curve448_point_t tmp;
|
2017-11-15 15:27:21 +00:00
|
|
|
int i;
|
2018-01-18 12:55:23 +00:00
|
|
|
pniels_t twop;
|
|
|
|
|
2017-11-15 15:27:21 +00:00
|
|
|
pt_to_pniels(output[0], working);
|
|
|
|
|
2017-12-04 11:38:58 +00:00
|
|
|
if (tbits == 0)
|
|
|
|
return;
|
2017-11-15 15:27:21 +00:00
|
|
|
|
2017-12-04 11:38:58 +00:00
|
|
|
curve448_point_double(tmp, working);
|
2017-11-15 15:27:21 +00:00
|
|
|
pt_to_pniels(twop, tmp);
|
|
|
|
|
2017-12-04 11:38:58 +00:00
|
|
|
add_pniels_to_pt(tmp, output[0], 0);
|
2017-11-15 15:27:21 +00:00
|
|
|
pt_to_pniels(output[1], tmp);
|
|
|
|
|
2017-12-04 11:38:58 +00:00
|
|
|
for (i = 2; i < 1 << tbits; i++) {
|
|
|
|
add_pniels_to_pt(tmp, twop, 0);
|
2017-11-15 15:27:21 +00:00
|
|
|
pt_to_pniels(output[i], tmp);
|
|
|
|
}
|
2017-12-04 11:38:58 +00:00
|
|
|
|
2017-11-16 16:27:48 +00:00
|
|
|
curve448_point_destroy(tmp);
|
2017-12-04 11:38:58 +00:00
|
|
|
OPENSSL_cleanse(twop, sizeof(twop));
|
2017-11-15 15:27:21 +00:00
|
|
|
}
|
|
|
|
|
2017-11-16 16:27:48 +00:00
|
|
|
extern const gf curve448_precomputed_wnaf_as_fe[];
|
2017-12-04 11:38:58 +00:00
|
|
|
static const niels_t *curve448_wnaf_base =
|
|
|
|
(const niels_t *)curve448_precomputed_wnaf_as_fe;
|
|
|
|
|
|
|
|
void curve448_base_double_scalarmul_non_secret(curve448_point_t combo,
|
|
|
|
const curve448_scalar_t scalar1,
|
|
|
|
const curve448_point_t base2,
|
|
|
|
const curve448_scalar_t scalar2)
|
|
|
|
{
|
2018-02-12 14:27:02 +00:00
|
|
|
const int table_bits_var = C448_WNAF_VAR_TABLE_BITS;
|
|
|
|
const int table_bits_pre = C448_WNAF_FIXED_TABLE_BITS;
|
2017-12-06 13:06:13 +00:00
|
|
|
struct smvt_control control_var[C448_SCALAR_BITS /
|
2017-12-05 16:37:57 +00:00
|
|
|
(C448_WNAF_VAR_TABLE_BITS + 1) + 3];
|
2017-12-06 13:06:13 +00:00
|
|
|
struct smvt_control control_pre[C448_SCALAR_BITS /
|
2017-12-05 16:37:57 +00:00
|
|
|
(C448_WNAF_FIXED_TABLE_BITS + 1) + 3];
|
2017-11-15 15:27:21 +00:00
|
|
|
int ncb_pre = recode_wnaf(control_pre, scalar1, table_bits_pre);
|
|
|
|
int ncb_var = recode_wnaf(control_var, scalar2, table_bits_var);
|
2017-12-05 16:37:57 +00:00
|
|
|
pniels_t precmp_var[1 << C448_WNAF_VAR_TABLE_BITS];
|
2017-12-04 11:38:58 +00:00
|
|
|
int contp = 0, contv = 0, i;
|
2018-01-18 12:55:23 +00:00
|
|
|
|
2017-11-15 15:27:21 +00:00
|
|
|
prepare_wnaf_table(precmp_var, base2, table_bits_var);
|
2018-01-18 12:55:23 +00:00
|
|
|
i = control_var[0].power;
|
2017-11-15 15:27:21 +00:00
|
|
|
|
|
|
|
if (i < 0) {
|
2017-11-16 16:27:48 +00:00
|
|
|
curve448_point_copy(combo, curve448_point_identity);
|
2017-11-15 15:27:21 +00:00
|
|
|
return;
|
|
|
|
} else if (i > control_pre[0].power) {
|
|
|
|
pniels_to_pt(combo, precmp_var[control_var[0].addend >> 1]);
|
|
|
|
contv++;
|
2017-12-04 11:38:58 +00:00
|
|
|
} else if (i == control_pre[0].power && i >= 0) {
|
2017-11-15 15:27:21 +00:00
|
|
|
pniels_to_pt(combo, precmp_var[control_var[0].addend >> 1]);
|
2017-12-04 11:38:58 +00:00
|
|
|
add_niels_to_pt(combo, curve448_wnaf_base[control_pre[0].addend >> 1],
|
|
|
|
i);
|
|
|
|
contv++;
|
|
|
|
contp++;
|
2017-11-15 15:27:21 +00:00
|
|
|
} else {
|
|
|
|
i = control_pre[0].power;
|
2017-11-16 16:27:48 +00:00
|
|
|
niels_to_pt(combo, curve448_wnaf_base[control_pre[0].addend >> 1]);
|
2017-11-15 15:27:21 +00:00
|
|
|
contp++;
|
|
|
|
}
|
2017-12-04 11:38:58 +00:00
|
|
|
|
2017-11-15 15:27:21 +00:00
|
|
|
for (i--; i >= 0; i--) {
|
2018-02-12 14:27:02 +00:00
|
|
|
int cv = (i == control_var[contv].power);
|
|
|
|
int cp = (i == control_pre[contp].power);
|
|
|
|
|
2017-12-04 11:38:58 +00:00
|
|
|
point_double_internal(combo, combo, i && !(cv || cp));
|
2017-11-15 15:27:21 +00:00
|
|
|
|
|
|
|
if (cv) {
|
|
|
|
assert(control_var[contv].addend);
|
|
|
|
|
2018-02-12 14:27:02 +00:00
|
|
|
if (control_var[contv].addend > 0)
|
2017-12-04 11:38:58 +00:00
|
|
|
add_pniels_to_pt(combo,
|
2017-12-04 13:30:53 +00:00
|
|
|
precmp_var[control_var[contv].addend >> 1],
|
|
|
|
i && !cp);
|
2018-02-12 14:27:02 +00:00
|
|
|
else
|
2017-12-04 11:38:58 +00:00
|
|
|
sub_pniels_from_pt(combo,
|
2017-12-04 13:30:53 +00:00
|
|
|
precmp_var[(-control_var[contv].addend)
|
|
|
|
>> 1], i && !cp);
|
2017-11-15 15:27:21 +00:00
|
|
|
contv++;
|
|
|
|
}
|
|
|
|
|
|
|
|
if (cp) {
|
|
|
|
assert(control_pre[contp].addend);
|
|
|
|
|
2018-02-12 14:27:02 +00:00
|
|
|
if (control_pre[contp].addend > 0)
|
2017-12-04 11:38:58 +00:00
|
|
|
add_niels_to_pt(combo,
|
2017-12-04 13:30:53 +00:00
|
|
|
curve448_wnaf_base[control_pre[contp].addend
|
|
|
|
>> 1], i);
|
2018-02-12 14:27:02 +00:00
|
|
|
else
|
2017-12-04 11:38:58 +00:00
|
|
|
sub_niels_from_pt(combo,
|
|
|
|
curve448_wnaf_base[(-control_pre
|
|
|
|
[contp].addend) >> 1], i);
|
2017-11-15 15:27:21 +00:00
|
|
|
contp++;
|
|
|
|
}
|
|
|
|
}
|
|
|
|
|
2017-12-04 11:38:58 +00:00
|
|
|
/* This function is non-secret, but whatever this is cheap. */
|
|
|
|
OPENSSL_cleanse(control_var, sizeof(control_var));
|
|
|
|
OPENSSL_cleanse(control_pre, sizeof(control_pre));
|
|
|
|
OPENSSL_cleanse(precmp_var, sizeof(precmp_var));
|
|
|
|
|
|
|
|
assert(contv == ncb_var);
|
|
|
|
(void)ncb_var;
|
|
|
|
assert(contp == ncb_pre);
|
|
|
|
(void)ncb_pre;
|
2017-11-15 15:27:21 +00:00
|
|
|
}
|
|
|
|
|
2017-12-04 11:38:58 +00:00
|
|
|
void curve448_point_destroy(curve448_point_t point)
|
|
|
|
{
|
2017-11-16 16:58:20 +00:00
|
|
|
OPENSSL_cleanse(point, sizeof(curve448_point_t));
|
2017-11-15 15:27:21 +00:00
|
|
|
}
|
|
|
|
|
2017-11-16 11:13:53 +00:00
|
|
|
int X448(uint8_t out_shared_key[56], const uint8_t private_key[56],
|
|
|
|
const uint8_t peer_public_value[56])
|
|
|
|
{
|
2017-12-06 13:06:13 +00:00
|
|
|
return x448_int(out_shared_key, peer_public_value, private_key)
|
|
|
|
== C448_SUCCESS;
|
2017-11-16 11:13:53 +00:00
|
|
|
}
|
|
|
|
|
|
|
|
void X448_public_from_private(uint8_t out_public_value[56],
|
|
|
|
const uint8_t private_key[56])
|
|
|
|
{
|
2017-12-06 13:06:13 +00:00
|
|
|
x448_derive_public_key(out_public_value, private_key);
|
2017-11-16 11:13:53 +00:00
|
|
|
}
|