/* Copyright (c) 2020, Google Inc.
*
* Permission to use, copy, modify, and/or distribute this software for any
* purpose with or without fee is hereby granted, provided that the above
* copyright notice and this permission notice appear in all copies.
*
* THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
* WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
* MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY
* SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
* WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION
* OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN
* CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. */
// Some of this code is taken from the ref10 version of Ed25519 in SUPERCOP
// 20141124 (http://bench.cr.yp.to/supercop.html). That code is released as
// public domain. Other parts have been replaced to call into code generated by
// Fiat (https://github.com/mit-plv/fiat-crypto) in //third_party/fiat.
//
// The field functions are shared by Ed25519 and X25519 where possible.
#include <openssl_grpc/curve25519.h>
#include <assert.h>
#include <string.h>
#include <openssl_grpc/cpu.h>
#include <openssl_grpc/mem.h>
#include <openssl_grpc/rand.h>
#include <openssl_grpc/sha.h>
#include <openssl_grpc/type_check.h>
#include "internal.h"
#include "../internal.h"
// Various pre-computed constants.
#include "./curve25519_tables.h"
#if defined(BORINGSSL_CURVE25519_64BIT)
#include "../../third_party/fiat/curve25519_64.h"
#else
#include "../../third_party/fiat/curve25519_32.h"
#endif // BORINGSSL_CURVE25519_64BIT
// Low-level intrinsic operations
static uint64_t load_3(const uint8_t *in) {
uint64_t result;
result = (uint64_t)in[0];
result |= ((uint64_t)in[1]) << 8;
result |= ((uint64_t)in[2]) << 16;
return result;
}
static uint64_t load_4(const uint8_t *in) {
uint64_t result;
result = (uint64_t)in[0];
result |= ((uint64_t)in[1]) << 8;
result |= ((uint64_t)in[2]) << 16;
result |= ((uint64_t)in[3]) << 24;
return result;
}
// Field operations.
#if defined(BORINGSSL_CURVE25519_64BIT)
typedef uint64_t fe_limb_t;
#define FE_NUM_LIMBS 5
// assert_fe asserts that |f| satisfies bounds:
//
// [[0x0 ~> 0x8cccccccccccc],
// [0x0 ~> 0x8cccccccccccc],
// [0x0 ~> 0x8cccccccccccc],
// [0x0 ~> 0x8cccccccccccc],
// [0x0 ~> 0x8cccccccccccc]]
//
// See comments in curve25519_64.h for which functions use these bounds for
// inputs or outputs.
#define assert_fe(f) \
do { \
for (unsigned _assert_fe_i = 0; _assert_fe_i < 5; _assert_fe_i++) { \
assert(f[_assert_fe_i] <= UINT64_C(0x8cccccccccccc)); \
} \
} while (0)
// assert_fe_loose asserts that |f| satisfies bounds:
//
// [[0x0 ~> 0x1a666666666664],
// [0x0 ~> 0x1a666666666664],
// [0x0 ~> 0x1a666666666664],
// [0x0 ~> 0x1a666666666664],
// [0x0 ~> 0x1a666666666664]]
//
// See comments in curve25519_64.h for which functions use these bounds for
// inputs or outputs.
#define assert_fe_loose(f) \
do { \
for (unsigned _assert_fe_i = 0; _assert_fe_i < 5; _assert_fe_i++) { \
assert(f[_assert_fe_i] <= UINT64_C(0x1a666666666664)); \
} \
} while (0)
#else
typedef uint32_t fe_limb_t;
#define FE_NUM_LIMBS 10
// assert_fe asserts that |f| satisfies bounds:
//
// [[0x0 ~> 0x4666666], [0x0 ~> 0x2333333],
// [0x0 ~> 0x4666666], [0x0 ~> 0x2333333],
// [0x0 ~> 0x4666666], [0x0 ~> 0x2333333],
// [0x0 ~> 0x4666666], [0x0 ~> 0x2333333],
// [0x0 ~> 0x4666666], [0x0 ~> 0x2333333]]
//
// See comments in curve25519_32.h for which functions use these bounds for
// inputs or outputs.
#define assert_fe(f) \
do { \
for (unsigned _assert_fe_i = 0; _assert_fe_i < 10; _assert_fe_i++) { \
assert(f[_assert_fe_i] <= \
((_assert_fe_i & 1) ? 0x2333333u : 0x4666666u)); \
} \
} while (0)
// assert_fe_loose asserts that |f| satisfies bounds:
//
// [[0x0 ~> 0xd333332], [0x0 ~> 0x6999999],
// [0x0 ~> 0xd333332], [0x0 ~> 0x6999999],
// [0x0 ~> 0xd333332], [0x0 ~> 0x6999999],
// [0x0 ~> 0xd333332], [0x0 ~> 0x6999999],
// [0x0 ~> 0xd333332], [0x0 ~> 0x6999999]]
//
// See comments in curve25519_32.h for which functions use these bounds for
// inputs or outputs.
#define assert_fe_loose(f) \
do { \
for (unsigned _assert_fe_i = 0; _assert_fe_i < 10; _assert_fe_i++) { \
assert(f[_assert_fe_i] <= \
((_assert_fe_i & 1) ? 0x6999999u : 0xd333332u)); \
} \
} while (0)
#endif // BORINGSSL_CURVE25519_64BIT
OPENSSL_STATIC_ASSERT(sizeof(fe) == sizeof(fe_limb_t) * FE_NUM_LIMBS,
"fe_limb_t[FE_NUM_LIMBS] is inconsistent with fe");
static void fe_frombytes_strict(fe *h, const uint8_t s[32]) {
// |fiat_25519_from_bytes| requires the top-most bit be clear.
assert((s[31] & 0x80) == 0);
fiat_25519_from_bytes(h->v, s);
assert_fe(h->v);
}
static void fe_frombytes(fe *h, const uint8_t s[32]) {
uint8_t s_copy[32];
OPENSSL_memcpy(s_copy, s, 32);
s_copy[31] &= 0x7f;
fe_frombytes_strict(h, s_copy);
}
static void fe_tobytes(uint8_t s[32], const fe *f) {
assert_fe(f->v);
fiat_25519_to_bytes(s, f->v);
}
// h = 0
static void fe_0(fe *h) {
OPENSSL_memset(h, 0, sizeof(fe));
}
static void fe_loose_0(fe_loose *h) {
OPENSSL_memset(h, 0, sizeof(fe_loose));
}
// h = 1
static void fe_1(fe *h) {
OPENSSL_memset(h, 0, sizeof(fe));
h->v[0] = 1;
}
static void fe_loose_1(fe_loose *h) {
OPENSSL_memset(h, 0, sizeof(fe_loose));
h->v[0] = 1;
}
// h = f + g
// Can overlap h with f or g.
static void fe_add(fe_loose *h, const fe *f, const fe *g) {
assert_fe(f->v);
assert_fe(g->v);
fiat_25519_add(h->v, f->v, g->v);
assert_fe_loose(h->v);
}
// h = f - g
// Can overlap h with f or g.
static void fe_sub(fe_loose *h, const fe *f, const fe *g) {
assert_fe(f->v);
assert_fe(g->v);
fiat_25519_sub(h->v, f->v, g->v);
assert_fe_loose(h->v);
}
static void fe_carry(fe *h, const fe_loose* f) {
assert_fe_loose(f->v);
fiat_25519_carry(h->v, f->v);
assert_fe(h->v);
}
static void fe_mul_impl(fe_limb_t out[FE_NUM_LIMBS],
const fe_limb_t in1[FE_NUM_LIMBS],
const fe_limb_t in2[FE_NUM_LIMBS]) {
assert_fe_loose(in1);
assert_fe_loose(in2);
fiat_25519_carry_mul(out, in1, in2);
assert_fe(out);
}
static void fe_mul_ltt(fe_loose *h, const fe *f, const fe *g) {
fe_mul_impl(h->v, f->v, g->v);
}
static void fe_mul_llt(fe_loose *h, const fe_loose *f, const fe *g) {
fe_mul_impl(h->v, f->v, g->v);
}
static void fe_mul_ttt(fe *h, const fe *f, const fe *g) {
fe_mul_impl(h->v, f->v, g->v);
}
static void fe_mul_tlt(fe *h, const fe_loose *f, const fe *g) {
fe_mul_impl(h->v, f->v, g->v);
}
static void fe_mul_ttl(fe *h, const fe *f, const fe_loose *g) {
fe_mul_impl(h->v, f->v, g->v);
}
static void fe_mul_tll(fe *h, const fe_loose *f, const fe_loose *g) {
fe_mul_impl(h->v, f->v, g->v);
}
static void fe_sq_tl(fe *h, const fe_loose *f) {
assert_fe_loose(f->v);
fiat_25519_carry_square(h->v, f->v);
assert_fe(h->v);
}
static void fe_sq_tt(fe *h, const fe *f) {
assert_fe_loose(f->v);
fiat_25519_carry_square(h->v, f->v);
assert_fe(h->v);
}
// Replace (f,g) with (g,f) if b == 1;
// replace (f,g) with (f,g) if b == 0.
//
// Preconditions: b in {0,1}.
static void fe_cswap(fe *f, fe *g, fe_limb_t b) {
b = 0-b;
for (unsigned i = 0; i < FE_NUM_LIMBS; i++) {
fe_limb_t x = f->v[i] ^ g->v[i];
x &= b;
f->v[i] ^= x;
g->v[i] ^= x;
}
}
static void fe_mul121666(fe *h, const fe_loose *f) {
assert_fe_loose(f->v);
fiat_25519_carry_scmul_121666(h->v, f->v);
assert_fe(h->v);
}
// h = -f
static void fe_neg(fe_loose *h, const fe *f) {
assert_fe(f->v);
fiat_25519_opp(h->v, f->v);
assert_fe_loose(h->v);
}
// Replace (f,g) with (g,g) if b == 1;
// replace (f,g) with (f,g) if b == 0.
//
// Preconditions: b in {0,1}.
static void fe_cmov(fe_loose *f, const fe_loose *g, fe_limb_t b) {
// Silence an unused function warning. |fiat_25519_selectznz| isn't quite the
// calling convention the rest of this code wants, so implement it by hand.
//
// TODO(davidben): Switch to fiat's calling convention, or ask fiat to emit a
// different one.
(void)fiat_25519_selectznz;
b = 0-b;
for (unsigned i = 0; i < FE_NUM_LIMBS; i++) {
fe_limb_t x = f->v[i] ^ g->v[i];
x &= b;
f->v[i] ^= x;
}
}
// h = f
static void fe_copy(fe *h, const fe *f) {
OPENSSL_memmove(h, f, sizeof(fe));
}
static void fe_copy_lt(fe_loose *h, const fe *f) {
OPENSSL_STATIC_ASSERT(sizeof(fe_loose) == sizeof(fe),
"fe and fe_loose mismatch");
OPENSSL_memmove(h, f, sizeof(fe));
}
#if !defined(OPENSSL_SMALL)
static void fe_copy_ll(fe_loose *h, const fe_loose *f) {
OPENSSL_memmove(h, f, sizeof(fe_loose));
}
#endif // !defined(OPENSSL_SMALL)
static void fe_loose_invert(fe *out, const fe_loose *z) {
fe t0;
fe t1;
fe t2;
fe t3;
int i;
fe_sq_tl(&t0, z);
fe_sq_tt(&t1, &t0);
for (i = 1; i < 2; ++i) {
fe_sq_tt(&t1, &t1);
}
fe_mul_tlt(&t1, z, &t1);
fe_mul_ttt(&t0, &t0, &t1);
fe_sq_tt(&t2, &t0);
fe_mul_ttt(&t1, &t1, &t2);
fe_sq_tt(&t2, &t1);
for (i = 1; i < 5; ++i) {
fe_sq_tt(&t2, &t2);
}
fe_mul_ttt(&t1, &t2, &t1);
fe_sq_tt(&t2, &t1);
for (i = 1; i < 10; ++i) {
fe_sq_tt(&t2, &t2);
}
fe_mul_ttt(&t2, &t2, &t1);
fe_sq_tt(&t3, &t2);
for (i = 1; i < 20; ++i) {
fe_sq_tt(&t3, &t3);
}
fe_mul_ttt(&t2, &t3, &t2);
fe_sq_tt(&t2, &t2);
for (i = 1; i < 10; ++i) {
fe_sq_tt(&t2, &t2);
}
fe_mul_ttt(&t1, &t2, &t1);
fe_sq_tt(&t2, &t1);
for (i = 1; i < 50; ++i) {
fe_sq_tt(&t2, &t2);
}
fe_mul_ttt(&t2, &t2, &t1);
fe_sq_tt(&t3, &t2);
for (i = 1; i < 100; ++i) {
fe_sq_tt(&t3, &t3);
}
fe_mul_ttt(&t2, &t3, &t2);
fe_sq_tt(&t2, &t2);
for (i = 1; i < 50; ++i) {
fe_sq_tt(&t2, &t2);
}
fe_mul_ttt(&t1, &t2, &t1);
fe_sq_tt(&t1, &t1);
for (i = 1; i < 5; ++i) {
fe_sq_tt(&t1, &t1);
}
fe_mul_ttt(out, &t1, &t0);
}
static void fe_invert(fe *out, const fe *z) {
fe_loose l;
fe_copy_lt(&l, z);
fe_loose_invert(out, &l);
}
// return 0 if f == 0
// return 1 if f != 0
static int fe_isnonzero(const fe_loose *f) {
fe tight;
fe_carry(&tight, f);
uint8_t s[32];
fe_tobytes(s, &tight);
static const uint8_t zero[32] = {0};
return CRYPTO_memcmp(s, zero, sizeof(zero)) != 0;
}
// return 1 if f is in {1,3,5,...,q-2}
// return 0 if f is in {0,2,4,...,q-1}
static int fe_isnegative(const fe *f) {
uint8_t s[32];
fe_tobytes(s, f);
return s[0] & 1;
}
static void fe_sq2_tt(fe *h, const fe *f) {
// h = f^2
fe_sq_tt(h, f);
// h = h + h
fe_loose tmp;
fe_add(&tmp, h, h);
fe_carry(h, &tmp);
}
static void fe_pow22523(fe *out, const fe *z) {
fe t0;
fe t1;
fe t2;
int i;
fe_sq_tt(&t0, z);
fe_sq_tt(&t1, &t0);
for (i = 1; i < 2; ++i) {
fe_sq_tt(&t1, &t1);
}
fe_mul_ttt(&t1, z, &t1);
fe_mul_ttt(&t0, &t0, &t1);
fe_sq_tt(&t0, &t0);
fe_mul_ttt(&t0, &t1, &t0);
fe_sq_tt(&t1, &t0);
for (i = 1; i < 5; ++i) {
fe_sq_tt(&t1, &t1);
}
fe_mul_ttt(&t0, &t1, &t0);
fe_sq_tt(&t1, &t0);
for (i = 1; i < 10; ++i) {
fe_sq_tt(&t1, &t1);
}
fe_mul_ttt(&t1, &t1, &t0);
fe_sq_tt(&t2, &t1);
for (i = 1; i < 20; ++i) {
fe_sq_tt(&t2, &t2);
}
fe_mul_ttt(&t1, &t2, &t1);
fe_sq_tt(&t1, &t1);
for (i = 1; i < 10; ++i) {
fe_sq_tt(&t1, &t1);
}
fe_mul_ttt(&t0, &t1, &t0);
fe_sq_tt(&t1, &t0);
for (i = 1; i < 50; ++i) {
fe_sq_tt(&t1, &t1);
}
fe_mul_ttt(&t1, &t1, &t0);
fe_sq_tt(&t2, &t1);
for (i = 1; i < 100; ++i) {
fe_sq_tt(&t2, &t2);
}
fe_mul_ttt(&t1, &t2, &t1);
fe_sq_tt(&t1, &t1);
for (i = 1; i < 50; ++i) {
fe_sq_tt(&t1, &t1);
}
fe_mul_ttt(&t0, &t1, &t0);
fe_sq_tt(&t0, &t0);
for (i = 1; i < 2; ++i) {
fe_sq_tt(&t0, &t0);
}
fe_mul_ttt(out, &t0, z);
}
// Group operations.
void x25519_ge_tobytes(uint8_t s[32], const ge_p2 *h) {
fe recip;
fe x;
fe y;
fe_invert(&recip, &h->Z);
fe_mul_ttt(&x, &h->X, &recip);
fe_mul_ttt(&y, &h->Y, &recip);
fe_tobytes(s, &y);
s[31] ^= fe_isnegative(&x) << 7;
}
static void ge_p3_tobytes(uint8_t s[32], const ge_p3 *h) {
fe recip;
fe x;
fe y;
fe_invert(&recip, &h->Z);
fe_mul_ttt(&x, &h->X, &recip);
fe_mul_ttt(&y, &h->Y, &recip);
fe_tobytes(s, &y);
s[31] ^= fe_isnegative(&x) << 7;
}
int x25519_ge_frombytes_vartime(ge_p3 *h, const uint8_t s[32]) {
fe u;
fe_loose v;
fe v3;
fe vxx;
fe_loose check;
fe_frombytes(&h->Y, s);
fe_1(&h->Z);
fe_sq_tt(&v3, &h->Y);
fe_mul_ttt(&vxx, &v3, &d);
fe_sub(&v, &v3, &h->Z); // u = y^2-1
fe_carry(&u, &v);
fe_add(&v, &vxx, &h->Z); // v = dy^2+1
fe_sq_tl(&v3, &v);
fe_mul_ttl(&v3, &v3, &v); // v3 = v^3
fe_sq_tt(&h->X, &v3);
fe_mul_ttl(&h->X, &h->X, &v);
fe_mul_ttt(&h->X, &h->X, &u); // x = uv^7
fe_pow22523(&h->X, &h->X); // x = (uv^7)^((q-5)/8)
fe_mul_ttt(&h->X, &h->X, &v3);
fe_mul_ttt(&h->X, &h->X, &u); // x = uv^3(uv^7)^((q-5)/8)
fe_sq_tt(&vxx, &h->X);
fe_mul_ttl(&vxx, &vxx, &v);
fe_sub(&check, &vxx, &u);
if (fe_isnonzero(&check)) {
fe_add(&check, &vxx, &u);
if (fe_isnonzero(&check)) {
return 0;
}
fe_mul_ttt(&h->X, &h->X, &sqrtm1);
}
if (fe_isnegative(&h->X) != (s[31] >> 7)) {
fe_loose t;
fe_neg(&t, &h->X);
fe_carry(&h->X, &t);
}
fe_mul_ttt(&h->T, &h->X, &h->Y);
return 1;
}
static void ge_p2_0(ge_p2 *h) {
fe_0(&h->X);
fe_1(&h->Y);
fe_1(&h->Z);
}
static void ge_p3_0(ge_p3 *h) {
fe_0(&h->X);
fe_1(&h->Y);
fe_1(&h->Z);
fe_0(&h->T);
}
static void ge_cached_0(ge_cached *h) {
fe_loose_1(&h->YplusX);
fe_loose_1(&h->YminusX);
fe_loose_1(&h->Z);
fe_loose_0(&h->T2d);
}
static void ge_precomp_0(ge_precomp *h) {
fe_loose_1(&h->yplusx);
fe_loose_1(&h->yminusx);
fe_loose_0(&h->xy2d);
}
// r = p
static void ge_p3_to_p2(ge_p2 *r, const ge_p3 *p) {
fe_copy(&r->X, &p->X);
fe_copy(&r->Y, &p->Y);
fe_copy(&r->Z, &p->Z);
}
// r = p
void x25519_ge_p3_to_cached(ge_cached *r, const ge_p3 *p) {
fe_add(&r->YplusX, &p->Y, &p->X);
fe_sub(&r->YminusX, &p->Y, &p->X);
fe_copy_lt(&r->Z, &p->Z);
fe_mul_ltt(&r->T2d, &p->T, &d2);
}
// r = p
void x25519_ge_p1p1_to_p2(ge_p2 *r, const ge_p1p1 *p) {
fe_mul_tll(&r->X, &p->X, &p->T);
fe_mul_tll(&r->Y, &p->Y, &p->Z);
fe_mul_tll(&r->Z, &p->Z, &p->T);
}
// r = p
void x25519_ge_p1p1_to_p3(ge_p3 *r, const ge_p1p1 *p) {
fe_mul_tll(&r->X, &p->X, &p->T);
fe_mul_tll(&r->Y, &p->Y, &p->Z);
fe_mul_tll(&r->Z, &p->Z, &p->T);
fe_mul_tll(&r->T, &p->X, &p->Y);
}
// r = p
static void ge_p1p1_to_cached(ge_cached *r, const ge_p1p1 *p) {
ge_p3 t;
x25519_ge_p1p1_to_p3(&t, p);
x25519_ge_p3_to_cached(r, &t);
}
// r = 2 * p
static void ge_p2_dbl(ge_p1p1 *r, const ge_p2 *p) {
fe trX, trZ, trT;
fe t0;
fe_sq_tt(&trX, &p->X);
fe_sq_tt(&trZ, &p->Y);
fe_sq2_tt(&trT, &p->Z);
fe_add(&r->Y, &p->X, &p->Y);
fe_sq_tl(&t0, &r->Y);
fe_add(&r->Y, &trZ, &trX);
fe_sub(&r->Z, &trZ, &trX);
fe_carry(&trZ, &r->Y);
fe_sub(&r->X, &t0, &trZ);
fe_carry(&trZ, &r->Z);
fe_sub(&r->T, &trT, &trZ);
}
// r = 2 * p
static void ge_p3_dbl(ge_p1p1 *r, const ge_p3 *p) {
ge_p2 q;
ge_p3_to_p2(&q, p);
ge_p2_dbl(r, &q);
}
// r = p + q
static void ge_madd(ge_p1p1 *r, const ge_p3 *p, const ge_precomp *q) {
fe trY, trZ, trT;
fe_add(&r->X, &p->Y, &p->X);
fe_sub(&r->Y, &p->Y, &p->X);
fe_mul_tll(&trZ, &r->X, &q->yplusx);
fe_mul_tll(&trY, &r->Y, &q->yminusx);
fe_mul_tlt(&trT, &q->xy2d, &p->T);
fe_add(&r->T, &p->Z, &p->Z);
fe_sub(&r->X, &trZ, &trY);
fe_add(&r->Y, &trZ, &trY);
fe_carry(&trZ, &r->T);
fe_add(&r->Z, &trZ, &trT);
fe_sub(&r->T, &trZ, &trT);
}
// r = p - q
static void ge_msub(ge_p1p1 *r, const ge_p3 *p, const ge_precomp *q) {
fe trY, trZ, trT;
fe_add(&r->X, &p->Y, &p->X);
fe_sub(&r->Y, &p->Y, &p->X);
fe_mul_tll(&trZ, &r->X, &q->yminusx);
fe_mul_tll(&trY, &r->Y, &q->yplusx);
fe_mul_tlt(&trT, &q->xy2d, &p->T);
fe_add(&r->T, &p->Z, &p->Z);
fe_sub(&r->X, &trZ, &trY);
fe_add(&r->Y, &trZ, &trY);
fe_carry(&trZ, &r->T);
fe_sub(&r->Z, &trZ, &trT);
fe_add(&r->T, &trZ, &trT);
}
// r = p + q
void x25519_ge_add(ge_p1p1 *r, const ge_p3 *p, const ge_cached *q) {
fe trX, trY, trZ, trT;
fe_add(&r->X, &p->Y, &p->X);
fe_sub(&r->Y, &p->Y, &p->X);
fe_mul_tll(&trZ, &r->X, &q->YplusX);
fe_mul_tll(&trY, &r->Y, &q->YminusX);
fe_mul_tlt(&trT, &q->T2d, &p->T);
fe_mul_ttl(&trX, &p->Z, &q->Z);
fe_add(&r->T, &trX, &trX);
fe_sub(&r->X, &trZ, &trY);
fe_add(&r->Y, &trZ, &trY);
fe_carry(&trZ, &r->T);
fe_add(&r->Z, &trZ, &trT);
fe_sub(&r->T, &trZ, &trT);
}
// r = p - q
void x25519_ge_sub(ge_p1p1 *r, const ge_p3 *p, const ge_cached *q) {
fe trX, trY, trZ, trT;
fe_add(&r->X, &p->Y, &p->X);
fe_sub(&r->Y, &p->Y, &p->X);
fe_mul_tll(&trZ, &r->X, &q->YminusX);
fe_mul_tll(&trY, &r->Y, &q->YplusX);
fe_mul_tlt(&trT, &q->T2d, &p->T);
fe_mul_ttl(&trX, &p->Z, &q->Z);
fe_add(&r->T, &trX, &trX);
fe_sub(&r->X, &trZ, &trY);
fe_add(&r->Y, &trZ, &trY);
fe_carry(&trZ, &r->T);
fe_sub(&r->Z, &trZ, &trT);
fe_add(&r->T, &trZ, &trT);
}
static uint8_t equal(signed char b, signed char c) {
uint8_t ub = b;
uint8_t uc = c;
uint8_t x = ub ^ uc; // 0: yes; 1..255: no
uint32_t y = x; // 0: yes; 1..255: no
y -= 1; // 4294967295: yes; 0..254: no
y >>= 31; // 1: yes; 0: no
return y;
}
static void cmov(ge_precomp *t, const ge_precomp *u, uint8_t b) {
fe_cmov(&t->yplusx, &u->yplusx, b);
fe_cmov(&t->yminusx, &u->yminusx, b);
fe_cmov(&t->xy2d, &u->xy2d, b);
}
void x25519_ge_scalarmult_small_precomp(
ge_p3 *h, const uint8_t a[32], const uint8_t precomp_table[15 * 2 * 32]) {
// precomp_table is first expanded into matching |ge_precomp|
// elements.
ge_precomp multiples[15];
unsigned i;
for (i = 0; i < 15; i++) {
// The precomputed table is assumed to already clear the top bit, so
// |fe_frombytes_strict| may be used directly.
const uint8_t *bytes = &precomp_table[i*(2 * 32)];
fe x, y;
fe_frombytes_strict(&x, bytes);
fe_frombytes_strict(&y, bytes + 32);
ge_precomp *out = &multiples[i];
fe_add(&out->yplusx, &y, &x);
fe_sub(&out->yminusx, &y, &x);
fe_mul_ltt(&out->xy2d, &x, &y);
fe_mul_llt(&out->xy2d, &out->xy2d, &d2);
}
// See the comment above |k25519SmallPrecomp| about the structure of the
// precomputed elements. This loop does 64 additions and 64 doublings to
// calculate the result.
ge_p3_0(h);
for (i = 63; i < 64; i--) {
unsigned j;
signed char index = 0;
for (j = 0; j < 4; j++) {
const uint8_t bit = 1 & (a[(8 * j) + (i / 8)] >> (i & 7));
index |= (bit << j);
}
ge_precomp e;
ge_precomp_0(&e);
for (j = 1; j < 16; j++) {
cmov(&e, &multiples[j-1], equal(index, j));
}
ge_cached cached;
ge_p1p1 r;
x25519_ge_p3_to_cached(&cached, h);
x25519_ge_add(&r, h, &cached);
x25519_ge_p1p1_to_p3(h, &r);
ge_madd(&r, h, &e);
x25519_ge_p1p1_to_p3(h, &r);
}
}
#if defined(OPENSSL_SMALL)
void x25519_ge_scalarmult_base(ge_p3 *h, const uint8_t a[32]) {
x25519_ge_scalarmult_small_precomp(h, a, k25519SmallPrecomp);
}
#else
static uint8_t negative(signed char b) {
uint32_t x = b;
x >>= 31; // 1: yes; 0: no
return x;
}
static void table_select(ge_precomp *t, int pos, signed char b) {
ge_precomp minust;
uint8_t bnegative = negative(b);
uint8_t babs = b - ((uint8_t)((-bnegative) & b) << 1);
ge_precomp_0(t);
cmov(t, &k25519Precomp[pos][0], equal(babs, 1));
cmov(t, &k25519Precomp[pos][1], equal(babs, 2));
cmov(t, &k25519Precomp[pos][2], equal(babs, 3));
cmov(t, &k25519Precomp[pos][3], equal(babs, 4));
cmov(t, &k25519Precomp[pos][4], equal(babs, 5));
cmov(t, &k25519Precomp[pos][5], equal(babs, 6));
cmov(t, &k25519Precomp[pos][6], equal(babs, 7));
cmov(t, &k25519Precomp[pos][7], equal(babs, 8));
fe_copy_ll(&minust.yplusx, &t->yminusx);
fe_copy_ll(&minust.yminusx, &t->yplusx);
// NOTE: the input table is canonical, but types don't encode it
fe tmp;
fe_carry(&tmp, &t->xy2d);
fe_neg(&minust.xy2d, &tmp);
cmov(t, &minust, bnegative);
}
// h = a * B
// where a = a[0]+256*a[1]+...+256^31 a[31]
// B is the Ed25519 base point (x,4/5) with x positive.
//
// Preconditions:
// a[31] <= 127
void x25519_ge_scalarmult_base(ge_p3 *h, const uint8_t a[32]) {
signed char e[64];
signed char carry;
ge_p1p1 r;
ge_p2 s;
ge_precomp t;
int i;
for (i = 0; i < 32; ++i) {
e[2 * i + 0] = (a[i] >> 0) & 15;
e[2 * i + 1] = (a[i] >> 4) & 15;
}
// each e[i] is between 0 and 15
// e[63] is between 0 and 7
carry = 0;
for (i = 0; i < 63; ++i) {
e[i] += carry;
carry = e[i] + 8;
carry >>= 4;
e[i] -= carry << 4;
}
e[63] += carry;
// each e[i] is between -8 and 8
ge_p3_0(h);
for (i = 1; i < 64; i += 2) {
table_select(&t, i / 2, e[i]);
ge_madd(&r, h, &t);
x25519_ge_p1p1_to_p3(h, &r);
}
ge_p3_dbl(&r, h);
x25519_ge_p1p1_to_p2(&s, &r);
ge_p2_dbl(&r, &s);
x25519_ge_p1p1_to_p2(&s, &r);
ge_p2_dbl(&r, &s);
x25519_ge_p1p1_to_p2(&s, &r);
ge_p2_dbl(&r, &s);
x25519_ge_p1p1_to_p3(h, &r);
for (i = 0; i < 64; i += 2) {
table_select(&t, i / 2, e[i]);
ge_madd(&r, h, &t);
x25519_ge_p1p1_to_p3(h, &r);
}
}
#endif
static void cmov_cached(ge_cached *t, ge_cached *u, uint8_t b) {
fe_cmov(&t->YplusX, &u->YplusX, b);
fe_cmov(&t->YminusX, &u->YminusX, b);
fe_cmov(&t->Z, &u->Z, b);
fe_cmov(&t->T2d, &u->T2d, b);
}
// r = scalar * A.
// where a = a[0]+256*a[1]+...+256^31 a[31].
void x25519_ge_scalarmult(ge_p2 *r, const uint8_t *scalar, const ge_p3 *A) {
ge_p2 Ai_p2[8];
ge_cached Ai[16];
ge_p1p1 t;
ge_cached_0(&Ai[0]);
x25519_ge_p3_to_cached(&Ai[1], A);
ge_p3_to_p2(&Ai_p2[1], A);
unsigned i;
for (i = 2; i < 16; i += 2) {
ge_p2_dbl(&t, &Ai_p2[i / 2]);
ge_p1p1_to_cached(&Ai[i], &t);
if (i < 8) {
x25519_ge_p1p1_to_p2(&Ai_p2[i], &t);
}
x25519_ge_add(&t, A, &Ai[i]);
ge_p1p1_to_cached(&Ai[i + 1], &t);
if (i < 7) {
x25519_ge_p1p1_to_p2(&Ai_p2[i + 1], &t);
}
}
ge_p2_0(r);
ge_p3 u;
for (i = 0; i < 256; i += 4) {
ge_p2_dbl(&t, r);
x25519_ge_p1p1_to_p2(r, &t);
ge_p2_dbl(&t, r);
x25519_ge_p1p1_to_p2(r, &t);
ge_p2_dbl(&t, r);
x25519_ge_p1p1_to_p2(r, &t);
ge_p2_dbl(&t, r);
x25519_ge_p1p1_to_p3(&u, &t);
uint8_t index = scalar[31 - i/8];
index >>= 4 - (i & 4);
index &= 0xf;
unsigned j;
ge_cached selected;
ge_cached_0(&selected);
for (j = 0; j < 16; j++) {
cmov_cached(&selected, &Ai[j], equal(j, index));
}
x25519_ge_add(&t, &u, &selected);
x25519_ge_p1p1_to_p2(r, &t);
}
}
static void slide(signed char *r, const uint8_t *a) {
int i;
int b;
int k;
for (i = 0; i < 256; ++i) {
r[i] = 1 & (a[i >> 3] >> (i & 7));
}
for (i = 0; i < 256; ++i) {
if (r[i]) {
for (b = 1; b <= 6 && i + b < 256; ++b) {
if (r[i + b]) {
if (r[i] + (r[i + b] << b) <= 15) {
r[i] += r[i + b] << b;
r[i + b] = 0;
} else if (r[i] - (r[i + b] << b) >= -15) {
r[i] -= r[i + b] << b;
for (k = i + b; k < 256; ++k) {
if (!r[k]) {
r[k] = 1;
break;
}
r[k] = 0;
}
} else {
break;
}
}
}
}
}
}
// r = a * A + b * B
// where a = a[0]+256*a[1]+...+256^31 a[31].
// and b = b[0]+256*b[1]+...+256^31 b[31].
// B is the Ed25519 base point (x,4/5) with x positive.
static void ge_double_scalarmult_vartime(ge_p2 *r, const uint8_t *a,
const ge_p3 *A, const uint8_t *b) {
signed char aslide[256];
signed char bslide[256];
ge_cached Ai[8]; // A,3A,5A,7A,9A,11A,13A,15A
ge_p1p1 t;
ge_p3 u;
ge_p3 A2;
int i;
slide(aslide, a);
slide(bslide, b);
x25519_ge_p3_to_cached(&Ai[0], A);
ge_p3_dbl(&t, A);
x25519_ge_p1p1_to_p3(&A2, &t);
x25519_ge_add(&t, &A2, &Ai[0]);
x25519_ge_p1p1_to_p3(&u, &t);
x25519_ge_p3_to_cached(&Ai[1], &u);
x25519_ge_add(&t, &A2, &Ai[1]);
x25519_ge_p1p1_to_p3(&u, &t);
x25519_ge_p3_to_cached(&Ai[2], &u);
x25519_ge_add(&t, &A2, &Ai[2]);
x25519_ge_p1p1_to_p3(&u, &t);
x25519_ge_p3_to_cached(&Ai[3], &u);
x25519_ge_add(&t, &A2, &Ai[3]);
x25519_ge_p1p1_to_p3(&u, &t);
x25519_ge_p3_to_cached(&Ai[4], &u);
x25519_ge_add(&t, &A2, &Ai[4]);
x25519_ge_p1p1_to_p3(&u, &t);
x25519_ge_p3_to_cached(&Ai[5], &u);
x25519_ge_add(&t, &A2, &Ai[5]);
x25519_ge_p1p1_to_p3(&u, &t);
x25519_ge_p3_to_cached(&Ai[6], &u);
x25519_ge_add(&t, &A2, &Ai[6]);
x25519_ge_p1p1_to_p3(&u, &t);
x25519_ge_p3_to_cached(&Ai[7], &u);
ge_p2_0(r);
for (i = 255; i >= 0; --i) {
if (aslide[i] || bslide[i]) {
break;
}
}
for (; i >= 0; --i) {
ge_p2_dbl(&t, r);
if (aslide[i] > 0) {
x25519_ge_p1p1_to_p3(&u, &t);
x25519_ge_add(&t, &u, &Ai[aslide[i] / 2]);
} else if (aslide[i] < 0) {
x25519_ge_p1p1_to_p3(&u, &t);
x25519_ge_sub(&t, &u, &Ai[(-aslide[i]) / 2]);
}
if (bslide[i] > 0) {
x25519_ge_p1p1_to_p3(&u, &t);
ge_madd(&t, &u, &Bi[bslide[i] / 2]);
} else if (bslide[i] < 0) {
x25519_ge_p1p1_to_p3(&u, &t);
ge_msub(&t, &u, &Bi[(-bslide[i]) / 2]);
}
x25519_ge_p1p1_to_p2(r, &t);
}
}
// int64_lshift21 returns |a << 21| but is defined when shifting bits into the
// sign bit. This works around a language flaw in C.
static inline int64_t int64_lshift21(int64_t a) {
return (int64_t)((uint64_t)a << 21);
}
// The set of scalars is \Z/l
// where l = 2^252 + 27742317777372353535851937790883648493.
// Input:
// s[0]+256*s[1]+...+256^63*s[63] = s
//
// Output:
// s[0]+256*s[1]+...+256^31*s[31] = s mod l
// where l = 2^252 + 27742317777372353535851937790883648493.
// Overwrites s in place.
void x25519_sc_reduce(uint8_t s[64]) {
int64_t s0 = 2097151 & load_3(s);
int64_t s1 = 2097151 & (load_4(s + 2) >> 5);
int64_t s2 = 2097151 & (load_3(s + 5) >> 2);
int64_t s3 = 2097151 & (load_4(s + 7) >> 7);
int64_t s4 = 2097151 & (load_4(s + 10) >> 4);
int64_t s5 = 2097151 & (load_3(s + 13) >> 1);
int64_t s6 = 2097151 & (load_4(s + 15) >> 6);
int64_t s7 = 2097151 & (load_3(s + 18) >> 3);
int64_t s8 = 2097151 & load_3(s + 21);
int64_t s9 = 2097151 & (load_4(s + 23) >> 5);
int64_t s10 = 2097151 & (load_3(s + 26) >> 2);
int64_t s11 = 2097151 & (load_4(s + 28) >> 7);
int64_t s12 = 2097151 & (load_4(s + 31) >> 4);
int64_t s13 = 2097151 & (load_3(s + 34) >> 1);
int64_t s14 = 2097151 & (load_4(s + 36) >> 6);
int64_t s15 = 2097151 & (load_3(s + 39) >> 3);
int64_t s16 = 2097151 & load_3(s + 42);
int64_t s17 = 2097151 & (load_4(s + 44) >> 5);
int64_t s18 = 2097151 & (load_3(s + 47) >> 2);
int64_t s19 = 2097151 & (load_4(s + 49) >> 7);
int64_t s20 = 2097151 & (load_4(s + 52) >> 4);
int64_t s21 = 2097151 & (load_3(s + 55) >> 1);
int64_t s22 = 2097151 & (load_4(s + 57) >> 6);
int64_t s23 = (load_4(s + 60) >> 3);
int64_t carry0;
int64_t carry1;
int64_t carry2;
int64_t carry3;
int64_t carry4;
int64_t carry5;
int64_t carry6;
int64_t carry7;
int64_t carry8;
int64_t carry9;
int64_t carry10;
int64_t carry11;
int64_t carry12;
int64_t carry13;
int64_t carry14;
int64_t carry15;
int64_t carry16;
s11 += s23 * 666643;
s12 += s23 * 470296;
s13 += s23 * 654183;
s14 -= s23 * 997805;
s15 += s23 * 136657;
s16 -= s23 * 683901;
s23 = 0;
s10 += s22 * 666643;
s11 += s22 * 470296;
s12 += s22 * 654183;
s13 -= s22 * 997805;
s14 += s22 * 136657;
s15 -= s22 * 683901;
s22 = 0;
s9 += s21 * 666643;
s10 += s21 * 470296;
s11 += s21 * 654183;
s12 -= s21 * 997805;
s13 += s21 * 136657;
s14 -= s21 * 683901;
s21 = 0;
s8 += s20 * 666643;
s9 += s20 * 470296;
s10 += s20 * 654183;
s11 -= s20 * 997805;
s12 += s20 * 136657;
s13 -= s20 * 683901;
s20 = 0;
s7 += s19 * 666643;
s8 += s19 * 470296;
s9 += s19 * 654183;
s10 -= s19 * 997805;
s11 += s19 * 136657;
s12 -= s19 * 683901;
s19 = 0;
s6 += s18 * 666643;
s7 += s18 * 470296;
s8 += s18 * 654183;
s9 -= s18 * 997805;
s10 += s18 * 136657;
s11 -= s18 * 683901;
s18 = 0;
carry6 = (s6 + (1 << 20)) >> 21;
s7 += carry6;
s6 -= int64_lshift21(carry6);
carry8 = (s8 + (1 << 20)) >> 21;
s9 += carry8;
s8 -= int64_lshift21(carry8);
carry10 = (s10 + (1 << 20)) >> 21;
s11 += carry10;
s10 -= int64_lshift21(carry10);
carry12 = (s12 + (1 << 20)) >> 21;
s13 += carry12;
s12 -= int64_lshift21(carry12);
carry14 = (s14 + (1 << 20)) >> 21;
s15 += carry14;
s14 -= int64_lshift21(carry14);
carry16 = (s16 + (1 << 20)) >> 21;
s17 += carry16;
s16 -= int64_lshift21(carry16);
carry7 = (s7 + (1 << 20)) >> 21;
s8 += carry7;
s7 -= int64_lshift21(carry7);
carry9 = (s9 + (1 << 20)) >> 21;
s10 += carry9;
s9 -= int64_lshift21(carry9);
carry11 = (s11 + (1 << 20)) >> 21;
s12 += carry11;
s11 -= int64_lshift21(carry11);
carry13 = (s13 + (1 << 20)) >> 21;
s14 += carry13;
s13 -= int64_lshift21(carry13);
carry15 = (s15 + (1 << 20)) >> 21;
s16 += carry15;
s15 -= int64_lshift21(carry15);
s5 += s17 * 666643;
s6 += s17 * 470296;
s7 += s17 * 654183;
s8 -= s17 * 997805;
s9 += s17 * 136657;
s10 -= s17 * 683901;
s17 = 0;
s4 += s16 * 666643;
s5 += s16 * 470296;
s6 += s16 * 654183;
s7 -= s16 * 997805;
s8 += s16 * 136657;
s9 -= s16 * 683901;
s16 = 0;
s3 += s15 * 666643;
s4 += s15 * 470296;
s5 += s15 * 654183;
s6 -= s15 * 997805;
s7 += s15 * 136657;
s8 -= s15 * 683901;
s15 = 0;
s2 += s14 * 666643;
s3 += s14 * 470296;
s4 += s14 * 654183;
s5 -= s14 * 997805;
s6 += s14 * 136657;
s7 -= s14 * 683901;
s14 = 0;
s1 += s13 * 666643;
s2 += s13 * 470296;
s3 += s13 * 654183;
s4 -= s13 * 997805;
s5 += s13 * 136657;
s6 -= s13 * 683901;
s13 = 0;
s0 += s12 * 666643;
s1 += s12 * 470296;
s2 += s12 * 654183;
s3 -= s12 * 997805;
s4 += s12 * 136657;
s5 -= s12 * 683901;
s12 = 0;
carry0 = (s0 + (1 << 20)) >> 21;
s1 += carry0;
s0 -= int64_lshift21(carry0);
carry2 = (s2 + (1 << 20)) >> 21;
s3 += carry2;
s2 -= int64_lshift21(carry2);
carry4 = (s4 + (1 << 20)) >> 21;
s5 += carry4;
s4 -= int64_lshift21(carry4);
carry6 = (s6 + (1 << 20)) >> 21;
s7 += carry6;
s6 -= int64_lshift21(carry6);
carry8 = (s8 + (1 << 20)) >> 21;
s9 += carry8;
s8 -= int64_lshift21(carry8);
carry10 = (s10 + (1 << 20)) >> 21;
s11 += carry10;
s10 -= int64_lshift21(carry10);
carry1 = (s1 + (1 << 20)) >> 21;
s2 += carry1;
s1 -= int64_lshift21(carry1);
carry3 = (s3 + (1 << 20)) >> 21;
s4 += carry3;
s3 -= int64_lshift21(carry3);
carry5 = (s5 + (1 << 20)) >> 21;
s6 += carry5;
s5 -= int64_lshift21(carry5);
carry7 = (s7 + (1 << 20)) >> 21;
s8 += carry7;
s7 -= int64_lshift21(carry7);
carry9 = (s9 + (1 << 20)) >> 21;
s10 += carry9;
s9 -= int64_lshift21(carry9);
carry11 = (s11 + (1 << 20)) >> 21;
s12 += carry11;
s11 -= int64_lshift21(carry11);
s0 += s12 * 666643;
s1 += s12 * 470296;
s2 += s12 * 654183;
s3 -= s12 * 997805;
s4 += s12 * 136657;
s5 -= s12 * 683901;
s12 = 0;
carry0 = s0 >> 21;
s1 += carry0;
s0 -= int64_lshift21(carry0);
carry1 = s1 >> 21;
s2 += carry1;
s1 -= int64_lshift21(carry1);
carry2 = s2 >> 21;
s3 += carry2;
s2 -= int64_lshift21(carry2);
carry3 = s3 >> 21;
s4 += carry3;
s3 -= int64_lshift21(carry3);
carry4 = s4 >> 21;
s5 += carry4;
s4 -= int64_lshift21(carry4);
carry5 = s5 >> 21;
s6 += carry5;
s5 -= int64_lshift21(carry5);
carry6 = s6 >> 21;
s7 += carry6;
s6 -= int64_lshift21(carry6);
carry7 = s7 >> 21;
s8 += carry7;
s7 -= int64_lshift21(carry7);
carry8 = s8 >> 21;
s9 += carry8;
s8 -= int64_lshift21(carry8);
carry9 = s9 >> 21;
s10 += carry9;
s9 -= int64_lshift21(carry9);
carry10 = s10 >> 21;
s11 += carry10;
s10 -= int64_lshift21(carry10);
carry11 = s11 >> 21;
s12 += carry11;
s11 -= int64_lshift21(carry11);
s0 += s12 * 666643;
s1 += s12 * 470296;
s2 += s12 * 654183;
s3 -= s12 * 997805;
s4 += s12 * 136657;
s5 -= s12 * 683901;
s12 = 0;
carry0 = s0 >> 21;
s1 += carry0;
s0 -= int64_lshift21(carry0);
carry1 = s1 >> 21;
s2 += carry1;
s1 -= int64_lshift21(carry1);
carry2 = s2 >> 21;
s3 += carry2;
s2 -= int64_lshift21(carry2);
carry3 = s3 >> 21;
s4 += carry3;
s3 -= int64_lshift21(carry3);
carry4 = s4 >> 21;
s5 += carry4;
s4 -= int64_lshift21(carry4);
carry5 = s5 >> 21;
s6 += carry5;
s5 -= int64_lshift21(carry5);
carry6 = s6 >> 21;
s7 += carry6;
s6 -= int64_lshift21(carry6);
carry7 = s7 >> 21;
s8 += carry7;
s7 -= int64_lshift21(carry7);
carry8 = s8 >> 21;
s9 += carry8;
s8 -= int64_lshift21(carry8);
carry9 = s9 >> 21;
s10 += carry9;
s9 -= int64_lshift21(carry9);
carry10 = s10 >> 21;
s11 += carry10;
s10 -= int64_lshift21(carry10);
s[0] = s0 >> 0;
s[1] = s0 >> 8;
s[2] = (s0 >> 16) | (s1 << 5);
s[3] = s1 >> 3;
s[4] = s1 >> 11;
s[5] = (s1 >> 19) | (s2 << 2);
s[6] = s2 >> 6;
s[7] = (s2 >> 14) | (s3 << 7);
s[8] = s3 >> 1;
s[9] = s3 >> 9;
s[10] = (s3 >> 17) | (s4 << 4);
s[11] = s4 >> 4;
s[12] = s4 >> 12;
s[13] = (s4 >> 20) | (s5 << 1);
s[14] = s5 >> 7;
s[15] = (s5 >> 15) | (s6 << 6);
s[16] = s6 >> 2;
s[17] = s6 >> 10;
s[18] = (s6 >> 18) | (s7 << 3);
s[19] = s7 >> 5;
s[20] = s7 >> 13;
s[21] = s8 >> 0;
s[22] = s8 >> 8;
s[23] = (s8 >> 16) | (s9 << 5);
s[24] = s9 >> 3;
s[25] = s9 >> 11;
s[26] = (s9 >> 19) | (s10 << 2);
s[27] = s10 >> 6;
s[28] = (s10 >> 14) | (s11 << 7);
s[29] = s11 >> 1;
s[30] = s11 >> 9;
s[31] = s11 >> 17;
}
// Input:
// a[0]+256*a[1]+...+256^31*a[31] = a
// b[0]+256*b[1]+...+256^31*b[31] = b
// c[0]+256*c[1]+...+256^31*c[31] = c
//
// Output:
// s[0]+256*s[1]+...+256^31*s[31] = (ab+c) mod l
// where l = 2^252 + 27742317777372353535851937790883648493.
static void sc_muladd(uint8_t *s, const uint8_t *a, const uint8_t *b,
const uint8_t *c) {
int64_t a0 = 2097151 & load_3(a);
int64_t a1 = 2097151 & (load_4(a + 2) >> 5);
int64_t a2 = 2097151 & (load_3(a + 5) >> 2);
int64_t a3 = 2097151 & (load_4(a + 7) >> 7);
int64_t a4 = 2097151 & (load_4(a + 10) >> 4);
int64_t a5 = 2097151 & (load_3(a + 13) >> 1);
int64_t a6 = 2097151 & (load_4(a + 15) >> 6);
int64_t a7 = 2097151 & (load_3(a + 18) >> 3);
int64_t a8 = 2097151 & load_3(a + 21);
int64_t a9 = 2097151 & (load_4(a + 23) >> 5);
int64_t a10 = 2097151 & (load_3(a + 26) >> 2);
int64_t a11 = (load_4(a + 28) >> 7);
int64_t b0 = 2097151 & load_3(b);
int64_t b1 = 2097151 & (load_4(b + 2) >> 5);
int64_t b2 = 2097151 & (load_3(b + 5) >> 2);
int64_t b3 = 2097151 & (load_4(b + 7) >> 7);
int64_t b4 = 2097151 & (load_4(b + 10) >> 4);
int64_t b5 = 2097151 & (load_3(b + 13) >> 1);
int64_t b6 = 2097151 & (load_4(b + 15) >> 6);
int64_t b7 = 2097151 & (load_3(b + 18) >> 3);
int64_t b8 = 2097151 & load_3(b + 21);
int64_t b9 = 2097151 & (load_4(b + 23) >> 5);
int64_t b10 = 2097151 & (load_3(b + 26) >> 2);
int64_t b11 = (load_4(b + 28) >> 7);
int64_t c0 = 2097151 & load_3(c);
int64_t c1 = 2097151 & (load_4(c + 2) >> 5);
int64_t c2 = 2097151 & (load_3(c + 5) >> 2);
int64_t c3 = 2097151 & (load_4(c + 7) >> 7);
int64_t c4 = 2097151 & (load_4(c + 10) >> 4);
int64_t c5 = 2097151 & (load_3(c + 13) >> 1);
int64_t c6 = 2097151 & (load_4(c + 15) >> 6);
int64_t c7 = 2097151 & (load_3(c + 18) >> 3);
int64_t c8 = 2097151 & load_3(c + 21);
int64_t c9 = 2097151 & (load_4(c + 23) >> 5);
int64_t c10 = 2097151 & (load_3(c + 26) >> 2);
int64_t c11 = (load_4(c + 28) >> 7);
int64_t s0;
int64_t s1;
int64_t s2;
int64_t s3;
int64_t s4;
int64_t s5;
int64_t s6;
int64_t s7;
int64_t s8;
int64_t s9;
int64_t s10;
int64_t s11;
int64_t s12;
int64_t s13;
int64_t s14;
int64_t s15;
int64_t s16;
int64_t s17;
int64_t s18;
int64_t s19;
int64_t s20;
int64_t s21;
int64_t s22;
int64_t s23;
int64_t carry0;
int64_t carry1;
int64_t carry2;
int64_t carry3;
int64_t carry4;
int64_t carry5;
int64_t carry6;
int64_t carry7;
int64_t carry8;
int64_t carry9;
int64_t carry10;
int64_t carry11;
int64_t carry12;
int64_t carry13;
int64_t carry14;
int64_t carry15;
int64_t carry16;
int64_t carry17;
int64_t carry18;
int64_t carry19;
int64_t carry20;
int64_t carry21;
int64_t carry22;
s0 = c0 + a0 * b0;
s1 = c1 + a0 * b1 + a1 * b0;
s2 = c2 + a0 * b2 + a1 * b1 + a2 * b0;
s3 = c3 + a0 * b3 + a1 * b2 + a2 * b1 + a3 * b0;
s4 = c4 + a0 * b4 + a1 * b3 + a2 * b2 + a3 * b1 + a4 * b0;
s5 = c5 + a0 * b5 + a1 * b4 + a2 * b3 + a3 * b2 + a4 * b1 + a5 * b0;
s6 = c6 + a0 * b6 + a1 * b5 + a2 * b4 + a3 * b3 + a4 * b2 + a5 * b1 + a6 * b0;
s7 = c7 + a0 * b7 + a1 * b6 + a2 * b5 + a3 * b4 + a4 * b3 + a5 * b2 +
a6 * b1 + a7 * b0;
s8 = c8 + a0 * b8 + a1 * b7 + a2 * b6 + a3 * b5 + a4 * b4 + a5 * b3 +
a6 * b2 + a7 * b1 + a8 * b0;
s9 = c9 + a0 * b9 + a1 * b8 + a2 * b7 + a3 * b6 + a4 * b5 + a5 * b4 +
a6 * b3 + a7 * b2 + a8 * b1 + a9 * b0;
s10 = c10 + a0 * b10 + a1 * b9 + a2 * b8 + a3 * b7 + a4 * b6 + a5 * b5 +
a6 * b4 + a7 * b3 + a8 * b2 + a9 * b1 + a10 * b0;
s11 = c11 + a0 * b11 + a1 * b10 + a2 * b9 + a3 * b8 + a4 * b7 + a5 * b6 +
a6 * b5 + a7 * b4 + a8 * b3 + a9 * b2 + a10 * b1 + a11 * b0;
s12 = a1 * b11 + a2 * b10 + a3 * b9 + a4 * b8 + a5 * b7 + a6 * b6 + a7 * b5 +
a8 * b4 + a9 * b3 + a10 * b2 + a11 * b1;
s13 = a2 * b11 + a3 * b10 + a4 * b9 + a5 * b8 + a6 * b7 + a7 * b6 + a8 * b5 +
a9 * b4 + a10 * b3 + a11 * b2;
s14 = a3 * b11 + a4 * b10 + a5 * b9 + a6 * b8 + a7 * b7 + a8 * b6 + a9 * b5 +
a10 * b4 + a11 * b3;
s15 = a4 * b11 + a5 * b10 + a6 * b9 + a7 * b8 + a8 * b7 + a9 * b6 + a10 * b5 +
a11 * b4;
s16 = a5 * b11 + a6 * b10 + a7 * b9 + a8 * b8 + a9 * b7 + a10 * b6 + a11 * b5;
s17 = a6 * b11 + a7 * b10 + a8 * b9 + a9 * b8 + a10 * b7 + a11 * b6;
s18 = a7 * b11 + a8 * b10 + a9 * b9 + a10 * b8 + a11 * b7;
s19 = a8 * b11 + a9 * b10 + a10 * b9 + a11 * b8;
s20 = a9 * b11 + a10 * b10 + a11 * b9;
s21 = a10 * b11 + a11 * b10;
s22 = a11 * b11;
s23 = 0;
carry0 = (s0 + (1 << 20)) >> 21;
s1 += carry0;
s0 -= int64_lshift21(carry0);
carry2 = (s2 + (1 << 20)) >> 21;
s3 += carry2;
s2 -= int64_lshift21(carry2);
carry4 = (s4 + (1 << 20)) >> 21;
s5 += carry4;
s4 -= int64_lshift21(carry4);
carry6 = (s6 + (1 << 20)) >> 21;
s7 += carry6;
s6 -= int64_lshift21(carry6);
carry8 = (s8 + (1 << 20)) >> 21;
s9 += carry8;
s8 -= int64_lshift21(carry8);
carry10 = (s10 + (1 << 20)) >> 21;
s11 += carry10;
s10 -= int64_lshift21(carry10);
carry12 = (s12 + (1 << 20)) >> 21;
s13 += carry12;
s12 -= int64_lshift21(carry12);
carry14 = (s14 + (1 << 20)) >> 21;
s15 += carry14;
s14 -= int64_lshift21(carry14);
carry16 = (s16 + (1 << 20)) >> 21;
s17 += carry16;
s16 -= int64_lshift21(carry16);
carry18 = (s18 + (1 << 20)) >> 21;
s19 += carry18;
s18 -= int64_lshift21(carry18);
carry20 = (s20 + (1 << 20)) >> 21;
s21 += carry20;
s20 -= int64_lshift21(carry20);
carry22 = (s22 + (1 << 20)) >> 21;
s23 += carry22;
s22 -= int64_lshift21(carry22);
carry1 = (s1 + (1 << 20)) >> 21;
s2 += carry1;
s1 -= int64_lshift21(carry1);
carry3 = (s3 + (1 << 20)) >> 21;
s4 += carry3;
s3 -= int64_lshift21(carry3);
carry5 = (s5 + (1 << 20)) >> 21;
s6 += carry5;
s5 -= int64_lshift21(carry5);
carry7 = (s7 + (1 << 20)) >> 21;
s8 += carry7;
s7 -= int64_lshift21(carry7);
carry9 = (s9 + (1 << 20)) >> 21;
s10 += carry9;
s9 -= int64_lshift21(carry9);
carry11 = (s11 + (1 << 20)) >> 21;
s12 += carry11;
s11 -= int64_lshift21(carry11);
carry13 = (s13 + (1 << 20)) >> 21;
s14 += carry13;
s13 -= int64_lshift21(carry13);
carry15 = (s15 + (1 << 20)) >> 21;
s16 += carry15;
s15 -= int64_lshift21(carry15);
carry17 = (s17 + (1 << 20)) >> 21;
s18 += carry17;
s17 -= int64_lshift21(carry17);
carry19 = (s19 + (1 << 20)) >> 21;
s20 += carry19;
s19 -= int64_lshift21(carry19);
carry21 = (s21 + (1 << 20)) >> 21;
s22 += carry21;
s21 -= int64_lshift21(carry21);
s11 += s23 * 666643;
s12 += s23 * 470296;
s13 += s23 * 654183;
s14 -= s23 * 997805;
s15 += s23 * 136657;
s16 -= s23 * 683901;
s23 = 0;
s10 += s22 * 666643;
s11 += s22 * 470296;
s12 += s22 * 654183;
s13 -= s22 * 997805;
s14 += s22 * 136657;
s15 -= s22 * 683901;
s22 = 0;
s9 += s21 * 666643;
s10 += s21 * 470296;
s11 += s21 * 654183;
s12 -= s21 * 997805;
s13 += s21 * 136657;
s14 -= s21 * 683901;
s21 = 0;
s8 += s20 * 666643;
s9 += s20 * 470296;
s10 += s20 * 654183;
s11 -= s20 * 997805;
s12 += s20 * 136657;
s13 -= s20 * 683901;
s20 = 0;
s7 += s19 * 666643;
s8 += s19 * 470296;
s9 += s19 * 654183;
s10 -= s19 * 997805;
s11 += s19 * 136657;
s12 -= s19 * 683901;
s19 = 0;
s6 += s18 * 666643;
s7 += s18 * 470296;
s8 += s18 * 654183;
s9 -= s18 * 997805;
s10 += s18 * 136657;
s11 -= s18 * 683901;
s18 = 0;
carry6 = (s6 + (1 << 20)) >> 21;
s7 += carry6;
s6 -= int64_lshift21(carry6);
carry8 = (s8 + (1 << 20)) >> 21;
s9 += carry8;
s8 -= int64_lshift21(carry8);
carry10 = (s10 + (1 << 20)) >> 21;
s11 += carry10;
s10 -= int64_lshift21(carry10);
carry12 = (s12 + (1 << 20)) >> 21;
s13 += carry12;
s12 -= int64_lshift21(carry12);
carry14 = (s14 + (1 << 20)) >> 21;
s15 += carry14;
s14 -= int64_lshift21(carry14);
carry16 = (s16 + (1 << 20)) >> 21;
s17 += carry16;
s16 -= int64_lshift21(carry16);
carry7 = (s7 + (1 << 20)) >> 21;
s8 += carry7;
s7 -= int64_lshift21(carry7);
carry9 = (s9 + (1 << 20)) >> 21;
s10 += carry9;
s9 -= int64_lshift21(carry9);
carry11 = (s11 + (1 << 20)) >> 21;
s12 += carry11;
s11 -= int64_lshift21(carry11);
carry13 = (s13 + (1 << 20)) >> 21;
s14 += carry13;
s13 -= int64_lshift21(carry13);
carry15 = (s15 + (1 << 20)) >> 21;
s16 += carry15;
s15 -= int64_lshift21(carry15);
s5 += s17 * 666643;
s6 += s17 * 470296;
s7 += s17 * 654183;
s8 -= s17 * 997805;
s9 += s17 * 136657;
s10 -= s17 * 683901;
s17 = 0;
s4 += s16 * 666643;
s5 += s16 * 470296;
s6 += s16 * 654183;
s7 -= s16 * 997805;
s8 += s16 * 136657;
s9 -= s16 * 683901;
s16 = 0;
s3 += s15 * 666643;
s4 += s15 * 470296;
s5 += s15 * 654183;
s6 -= s15 * 997805;
s7 += s15 * 136657;
s8 -= s15 * 683901;
s15 = 0;
s2 += s14 * 666643;
s3 += s14 * 470296;
s4 += s14 * 654183;
s5 -= s14 * 997805;
s6 += s14 * 136657;
s7 -= s14 * 683901;
s14 = 0;
s1 += s13 * 666643;
s2 += s13 * 470296;
s3 += s13 * 654183;
s4 -= s13 * 997805;
s5 += s13 * 136657;
s6 -= s13 * 683901;
s13 = 0;
s0 += s12 * 666643;
s1 += s12 * 470296;
s2 += s12 * 654183;
s3 -= s12 * 997805;
s4 += s12 * 136657;
s5 -= s12 * 683901;
s12 = 0;
carry0 = (s0 + (1 << 20)) >> 21;
s1 += carry0;
s0 -= int64_lshift21(carry0);
carry2 = (s2 + (1 << 20)) >> 21;
s3 += carry2;
s2 -= int64_lshift21(carry2);
carry4 = (s4 + (1 << 20)) >> 21;
s5 += carry4;
s4 -= int64_lshift21(carry4);
carry6 = (s6 + (1 << 20)) >> 21;
s7 += carry6;
s6 -= int64_lshift21(carry6);
carry8 = (s8 + (1 << 20)) >> 21;
s9 += carry8;
s8 -= int64_lshift21(carry8);
carry10 = (s10 + (1 << 20)) >> 21;
s11 += carry10;
s10 -= int64_lshift21(carry10);
carry1 = (s1 + (1 << 20)) >> 21;
s2 += carry1;
s1 -= int64_lshift21(carry1);
carry3 = (s3 + (1 << 20)) >> 21;
s4 += carry3;
s3 -= int64_lshift21(carry3);
carry5 = (s5 + (1 << 20)) >> 21;
s6 += carry5;
s5 -= int64_lshift21(carry5);
carry7 = (s7 + (1 << 20)) >> 21;
s8 += carry7;
s7 -= int64_lshift21(carry7);
carry9 = (s9 + (1 << 20)) >> 21;
s10 += carry9;
s9 -= int64_lshift21(carry9);
carry11 = (s11 + (1 << 20)) >> 21;
s12 += carry11;
s11 -= int64_lshift21(carry11);
s0 += s12 * 666643;
s1 += s12 * 470296;
s2 += s12 * 654183;
s3 -= s12 * 997805;
s4 += s12 * 136657;
s5 -= s12 * 683901;
s12 = 0;
carry0 = s0 >> 21;
s1 += carry0;
s0 -= int64_lshift21(carry0);
carry1 = s1 >> 21;
s2 += carry1;
s1 -= int64_lshift21(carry1);
carry2 = s2 >> 21;
s3 += carry2;
s2 -= int64_lshift21(carry2);
carry3 = s3 >> 21;
s4 += carry3;
s3 -= int64_lshift21(carry3);
carry4 = s4 >> 21;
s5 += carry4;
s4 -= int64_lshift21(carry4);
carry5 = s5 >> 21;
s6 += carry5;
s5 -= int64_lshift21(carry5);
carry6 = s6 >> 21;
s7 += carry6;
s6 -= int64_lshift21(carry6);
carry7 = s7 >> 21;
s8 += carry7;
s7 -= int64_lshift21(carry7);
carry8 = s8 >> 21;
s9 += carry8;
s8 -= int64_lshift21(carry8);
carry9 = s9 >> 21;
s10 += carry9;
s9 -= int64_lshift21(carry9);
carry10 = s10 >> 21;
s11 += carry10;
s10 -= int64_lshift21(carry10);
carry11 = s11 >> 21;
s12 += carry11;
s11 -= int64_lshift21(carry11);
s0 += s12 * 666643;
s1 += s12 * 470296;
s2 += s12 * 654183;
s3 -= s12 * 997805;
s4 += s12 * 136657;
s5 -= s12 * 683901;
s12 = 0;
carry0 = s0 >> 21;
s1 += carry0;
s0 -= int64_lshift21(carry0);
carry1 = s1 >> 21;
s2 += carry1;
s1 -= int64_lshift21(carry1);
carry2 = s2 >> 21;
s3 += carry2;
s2 -= int64_lshift21(carry2);
carry3 = s3 >> 21;
s4 += carry3;
s3 -= int64_lshift21(carry3);
carry4 = s4 >> 21;
s5 += carry4;
s4 -= int64_lshift21(carry4);
carry5 = s5 >> 21;
s6 += carry5;
s5 -= int64_lshift21(carry5);
carry6 = s6 >> 21;
s7 += carry6;
s6 -= int64_lshift21(carry6);
carry7 = s7 >> 21;
s8 += carry7;
s7 -= int64_lshift21(carry7);
carry8 = s8 >> 21;
s9 += carry8;
s8 -= int64_lshift21(carry8);
carry9 = s9 >> 21;
s10 += carry9;
s9 -= int64_lshift21(carry9);
carry10 = s10 >> 21;
s11 += carry10;
s10 -= int64_lshift21(carry10);
s[0] = s0 >> 0;
s[1] = s0 >> 8;
s[2] = (s0 >> 16) | (s1 << 5);
s[3] = s1 >> 3;
s[4] = s1 >> 11;
s[5] = (s1 >> 19) | (s2 << 2);
s[6] = s2 >> 6;
s[7] = (s2 >> 14) | (s3 << 7);
s[8] = s3 >> 1;
s[9] = s3 >> 9;
s[10] = (s3 >> 17) | (s4 << 4);
s[11] = s4 >> 4;
s[12] = s4 >> 12;
s[13] = (s4 >> 20) | (s5 << 1);
s[14] = s5 >> 7;
s[15] = (s5 >> 15) | (s6 << 6);
s[16] = s6 >> 2;
s[17] = s6 >> 10;
s[18] = (s6 >> 18) | (s7 << 3);
s[19] = s7 >> 5;
s[20] = s7 >> 13;
s[21] = s8 >> 0;
s[22] = s8 >> 8;
s[23] = (s8 >> 16) | (s9 << 5);
s[24] = s9 >> 3;
s[25] = s9 >> 11;
s[26] = (s9 >> 19) | (s10 << 2);
s[27] = s10 >> 6;
s[28] = (s10 >> 14) | (s11 << 7);
s[29] = s11 >> 1;
s[30] = s11 >> 9;
s[31] = s11 >> 17;
}
void ED25519_keypair(uint8_t out_public_key[32], uint8_t out_private_key[64]) {
uint8_t seed[32];
RAND_bytes(seed, 32);
ED25519_keypair_from_seed(out_public_key, out_private_key, seed);
}
int ED25519_sign(uint8_t out_sig[64], const uint8_t *message,
size_t message_len, const uint8_t private_key[64]) {
// NOTE: The documentation on this function says that it returns zero on
// allocation failure. While that can't happen with the current
// implementation, we want to reserve the ability to allocate in this
// implementation in the future.
uint8_t az[SHA512_DIGEST_LENGTH];
SHA512(private_key, 32, az);
az[0] &= 248;
az[31] &= 63;
az[31] |= 64;
SHA512_CTX hash_ctx;
SHA512_Init(&hash_ctx);
SHA512_Update(&hash_ctx, az + 32, 32);
SHA512_Update(&hash_ctx, message, message_len);
uint8_t nonce[SHA512_DIGEST_LENGTH];
SHA512_Final(nonce, &hash_ctx);
x25519_sc_reduce(nonce);
ge_p3 R;
x25519_ge_scalarmult_base(&R, nonce);
ge_p3_tobytes(out_sig, &R);
SHA512_Init(&hash_ctx);
SHA512_Update(&hash_ctx, out_sig, 32);
SHA512_Update(&hash_ctx, private_key + 32, 32);
SHA512_Update(&hash_ctx, message, message_len);
uint8_t hram[SHA512_DIGEST_LENGTH];
SHA512_Final(hram, &hash_ctx);
x25519_sc_reduce(hram);
sc_muladd(out_sig + 32, hram, az, nonce);
return 1;
}
int ED25519_verify(const uint8_t *message, size_t message_len,
const uint8_t signature[64], const uint8_t public_key[32]) {
ge_p3 A;
if ((signature[63] & 224) != 0 ||
!x25519_ge_frombytes_vartime(&A, public_key)) {
return 0;
}
fe_loose t;
fe_neg(&t, &A.X);
fe_carry(&A.X, &t);
fe_neg(&t, &A.T);
fe_carry(&A.T, &t);
uint8_t pkcopy[32];
OPENSSL_memcpy(pkcopy, public_key, 32);
uint8_t rcopy[32];
OPENSSL_memcpy(rcopy, signature, 32);
union {
uint64_t u64[4];
uint8_t u8[32];
} scopy;
OPENSSL_memcpy(&scopy.u8[0], signature + 32, 32);
// https://tools.ietf.org/html/rfc8032#section-5.1.7 requires that s be in
// the range [0, order) in order to prevent signature malleability.
// kOrder is the order of Curve25519 in little-endian form.
static const uint64_t kOrder[4] = {
UINT64_C(0x5812631a5cf5d3ed),
UINT64_C(0x14def9dea2f79cd6),
0,
UINT64_C(0x1000000000000000),
};
for (size_t i = 3;; i--) {
if (scopy.u64[i] > kOrder[i]) {
return 0;
} else if (scopy.u64[i] < kOrder[i]) {
break;
} else if (i == 0) {
return 0;
}
}
SHA512_CTX hash_ctx;
SHA512_Init(&hash_ctx);
SHA512_Update(&hash_ctx, signature, 32);
SHA512_Update(&hash_ctx, public_key, 32);
SHA512_Update(&hash_ctx, message, message_len);
uint8_t h[SHA512_DIGEST_LENGTH];
SHA512_Final(h, &hash_ctx);
x25519_sc_reduce(h);
ge_p2 R;
ge_double_scalarmult_vartime(&R, h, &A, scopy.u8);
uint8_t rcheck[32];
x25519_ge_tobytes(rcheck, &R);
return CRYPTO_memcmp(rcheck, rcopy, sizeof(rcheck)) == 0;
}
void ED25519_keypair_from_seed(uint8_t out_public_key[32],
uint8_t out_private_key[64],
const uint8_t seed[32]) {
uint8_t az[SHA512_DIGEST_LENGTH];
SHA512(seed, 32, az);
az[0] &= 248;
az[31] &= 127;
az[31] |= 64;
ge_p3 A;
x25519_ge_scalarmult_base(&A, az);
ge_p3_tobytes(out_public_key, &A);
OPENSSL_memcpy(out_private_key, seed, 32);
OPENSSL_memcpy(out_private_key + 32, out_public_key, 32);
}
static void x25519_scalar_mult_generic(uint8_t out[32],
const uint8_t scalar[32],
const uint8_t point[32]) {
fe x1, x2, z2, x3, z3, tmp0, tmp1;
fe_loose x2l, z2l, x3l, tmp0l, tmp1l;
uint8_t e[32];
OPENSSL_memcpy(e, scalar, 32);
e[0] &= 248;
e[31] &= 127;
e[31] |= 64;
// The following implementation was transcribed to Coq and proven to
// correspond to unary scalar multiplication in affine coordinates given that
// x1 != 0 is the x coordinate of some point on the curve. It was also checked
// in Coq that doing a ladderstep with x1 = x3 = 0 gives z2' = z3' = 0, and z2
// = z3 = 0 gives z2' = z3' = 0. The statement was quantified over the
// underlying field, so it applies to Curve25519 itself and the quadratic
// twist of Curve25519. It was not proven in Coq that prime-field arithmetic
// correctly simulates extension-field arithmetic on prime-field values.
// The decoding of the byte array representation of e was not considered.
// Specification of Montgomery curves in affine coordinates:
// <https://github.com/mit-plv/fiat-crypto/blob/2456d821825521f7e03e65882cc3521795b0320f/src/Spec/MontgomeryCurve.v#L27>
// Proof that these form a group that is isomorphic to a Weierstrass curve:
// <https://github.com/mit-plv/fiat-crypto/blob/2456d821825521f7e03e65882cc3521795b0320f/src/Curves/Montgomery/AffineProofs.v#L35>
// Coq transcription and correctness proof of the loop (where scalarbits=255):
// <https://github.com/mit-plv/fiat-crypto/blob/2456d821825521f7e03e65882cc3521795b0320f/src/Curves/Montgomery/XZ.v#L118>
// <https://github.com/mit-plv/fiat-crypto/blob/2456d821825521f7e03e65882cc3521795b0320f/src/Curves/Montgomery/XZProofs.v#L278>
// preconditions: 0 <= e < 2^255 (not necessarily e < order), fe_invert(0) = 0
fe_frombytes(&x1, point);
fe_1(&x2);
fe_0(&z2);
fe_copy(&x3, &x1);
fe_1(&z3);
unsigned swap = 0;
int pos;
for (pos = 254; pos >= 0; --pos) {
// loop invariant as of right before the test, for the case where x1 != 0:
// pos >= -1; if z2 = 0 then x2 is nonzero; if z3 = 0 then x3 is nonzero
// let r := e >> (pos+1) in the following equalities of projective points:
// to_xz (r*P) === if swap then (x3, z3) else (x2, z2)
// to_xz ((r+1)*P) === if swap then (x2, z2) else (x3, z3)
// x1 is the nonzero x coordinate of the nonzero point (r*P-(r+1)*P)
unsigned b = 1 & (e[pos / 8] >> (pos & 7));
swap ^= b;
fe_cswap(&x2, &x3, swap);
fe_cswap(&z2, &z3, swap);
swap = b;
// Coq transcription of ladderstep formula (called from transcribed loop):
// <https://github.com/mit-plv/fiat-crypto/blob/2456d821825521f7e03e65882cc3521795b0320f/src/Curves/Montgomery/XZ.v#L89>
// <https://github.com/mit-plv/fiat-crypto/blob/2456d821825521f7e03e65882cc3521795b0320f/src/Curves/Montgomery/XZProofs.v#L131>
// x1 != 0 <https://github.com/mit-plv/fiat-crypto/blob/2456d821825521f7e03e65882cc3521795b0320f/src/Curves/Montgomery/XZProofs.v#L217>
// x1 = 0 <https://github.com/mit-plv/fiat-crypto/blob/2456d821825521f7e03e65882cc3521795b0320f/src/Curves/Montgomery/XZProofs.v#L147>
fe_sub(&tmp0l, &x3, &z3);
fe_sub(&tmp1l, &x2, &z2);
fe_add(&x2l, &x2, &z2);
fe_add(&z2l, &x3, &z3);
fe_mul_tll(&z3, &tmp0l, &x2l);
fe_mul_tll(&z2, &z2l, &tmp1l);
fe_sq_tl(&tmp0, &tmp1l);
fe_sq_tl(&tmp1, &x2l);
fe_add(&x3l, &z3, &z2);
fe_sub(&z2l, &z3, &z2);
fe_mul_ttt(&x2, &tmp1, &tmp0);
fe_sub(&tmp1l, &tmp1, &tmp0);
fe_sq_tl(&z2, &z2l);
fe_mul121666(&z3, &tmp1l);
fe_sq_tl(&x3, &x3l);
fe_add(&tmp0l, &tmp0, &z3);
fe_mul_ttt(&z3, &x1, &z2);
fe_mul_tll(&z2, &tmp1l, &tmp0l);
}
// here pos=-1, so r=e, so to_xz (e*P) === if swap then (x3, z3) else (x2, z2)
fe_cswap(&x2, &x3, swap);
fe_cswap(&z2, &z3, swap);
fe_invert(&z2, &z2);
fe_mul_ttt(&x2, &x2, &z2);
fe_tobytes(out, &x2);
}
static void x25519_scalar_mult(uint8_t out[32], const uint8_t scalar[32],
const uint8_t point[32]) {
#if defined(BORINGSSL_X25519_NEON)
if (CRYPTO_is_NEON_capable()) {
x25519_NEON(out, scalar, point);
return;
}
#endif
x25519_scalar_mult_generic(out, scalar, point);
}
void X25519_keypair(uint8_t out_public_value[32], uint8_t out_private_key[32]) {
RAND_bytes(out_private_key, 32);
// All X25519 implementations should decode scalars correctly (see
// https://tools.ietf.org/html/rfc7748#section-5). However, if an
// implementation doesn't then it might interoperate with random keys a
// fraction of the time because they'll, randomly, happen to be correctly
// formed.
//
// Thus we do the opposite of the masking here to make sure that our private
// keys are never correctly masked and so, hopefully, any incorrect
// implementations are deterministically broken.
//
// This does not affect security because, although we're throwing away
// entropy, a valid implementation of scalarmult should throw away the exact
// same bits anyway.
out_private_key[0] |= ~248;
out_private_key[31] &= ~64;
out_private_key[31] |= ~127;
X25519_public_from_private(out_public_value, out_private_key);
}
int X25519(uint8_t out_shared_key[32], const uint8_t private_key[32],
const uint8_t peer_public_value[32]) {
static const uint8_t kZeros[32] = {0};
x25519_scalar_mult(out_shared_key, private_key, peer_public_value);
// The all-zero output results when the input is a point of small order.
return CRYPTO_memcmp(kZeros, out_shared_key, 32) != 0;
}
void X25519_public_from_private(uint8_t out_public_value[32],
const uint8_t private_key[32]) {
#if defined(BORINGSSL_X25519_NEON)
if (CRYPTO_is_NEON_capable()) {
static const uint8_t kMongomeryBasePoint[32] = {9};
x25519_NEON(out_public_value, private_key, kMongomeryBasePoint);
return;
}
#endif
uint8_t e[32];
OPENSSL_memcpy(e, private_key, 32);
e[0] &= 248;
e[31] &= 127;
e[31] |= 64;
ge_p3 A;
x25519_ge_scalarmult_base(&A, e);
// We only need the u-coordinate of the curve25519 point. The map is
// u=(y+1)/(1-y). Since y=Y/Z, this gives u=(Z+Y)/(Z-Y).
fe_loose zplusy, zminusy;
fe zminusy_inv;
fe_add(&zplusy, &A.Z, &A.Y);
fe_sub(&zminusy, &A.Z, &A.Y);
fe_loose_invert(&zminusy_inv, &zminusy);
fe_mul_tlt(&zminusy_inv, &zplusy, &zminusy_inv);
fe_tobytes(out_public_value, &zminusy_inv);
}