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/*
* Copyright (C) 2018 Denys Vlasenko
*
* Licensed under GPLv2, see file LICENSE in this source tree.
*/
#include "tls.h"
typedef uint8_t byte;
typedef uint16_t word16;
typedef uint32_t word32;
#define XMEMSET memset
#define F25519_SIZE CURVE25519_KEYSIZE
/* The code below is taken from wolfssl-3.15.3/wolfcrypt/src/fe_low_mem.c
* Header comment is kept intact:
*/
/* fe_low_mem.c
*
* Copyright (C) 2006-2017 wolfSSL Inc.
*
* This file is part of wolfSSL.
*
* wolfSSL is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 2 of the License, or
* (at your option) any later version.
*
* wolfSSL is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1335, USA
*/
/* Based from Daniel Beer's public domain work. */
#if 0 //UNUSED
static void fprime_copy(byte *x, const byte *a)
{
int i;
for (i = 0; i < F25519_SIZE; i++)
x[i] = a[i];
}
#endif
static void lm_copy(byte* x, const byte* a)
{
int i;
for (i = 0; i < F25519_SIZE; i++)
x[i] = a[i];
}
#if 0 //UNUSED
static void fprime_select(byte *dst, const byte *zero, const byte *one, byte condition)
{
const byte mask = -condition;
int i;
for (i = 0; i < F25519_SIZE; i++)
dst[i] = zero[i] ^ (mask & (one[i] ^ zero[i]));
}
#endif
static void fe_select(byte *dst,
const byte *zero, const byte *one,
byte condition)
{
const byte mask = -condition;
int i;
for (i = 0; i < F25519_SIZE; i++)
dst[i] = zero[i] ^ (mask & (one[i] ^ zero[i]));
}
#if 0 //UNUSED
static void raw_add(byte *x, const byte *p)
{
word16 c = 0;
int i;
for (i = 0; i < F25519_SIZE; i++) {
c += ((word16)x[i]) + ((word16)p[i]);
x[i] = (byte)c;
c >>= 8;
}
}
#endif
#if 0 //UNUSED
static void raw_try_sub(byte *x, const byte *p)
{
byte minusp[F25519_SIZE];
word16 c = 0;
int i;
for (i = 0; i < F25519_SIZE; i++) {
c = ((word16)x[i]) - ((word16)p[i]) - c;
minusp[i] = (byte)c;
c = (c >> 8) & 1;
}
fprime_select(x, minusp, x, (byte)c);
}
#endif
#if 0 //UNUSED
static int prime_msb(const byte *p)
{
int i;
byte x;
int shift = 1;
int z = F25519_SIZE - 1;
/*
Test for any hot bits.
As soon as one instance is encountered set shift to 0.
*/
for (i = F25519_SIZE - 1; i >= 0; i--) {
shift &= ((shift ^ ((-p[i] | p[i]) >> 7)) & 1);
z -= shift;
}
x = p[z];
z <<= 3;
shift = 1;
for (i = 0; i < 8; i++) {
shift &= ((-(x >> i) | (x >> i)) >> (7 - i) & 1);
z += shift;
}
return z - 1;
}
#endif
#if 0 //UNUSED
static void fprime_add(byte *r, const byte *a, const byte *modulus)
{
raw_add(r, a);
raw_try_sub(r, modulus);
}
#endif
#if 0 //UNUSED
static void fprime_sub(byte *r, const byte *a, const byte *modulus)
{
raw_add(r, modulus);
raw_try_sub(r, a);
raw_try_sub(r, modulus);
}
#endif
#if 0 //UNUSED
static void fprime_mul(byte *r, const byte *a, const byte *b,
const byte *modulus)
{
word16 c = 0;
int i,j;
XMEMSET(r, 0, F25519_SIZE);
for (i = prime_msb(modulus); i >= 0; i--) {
const byte bit = (b[i >> 3] >> (i & 7)) & 1;
byte plusa[F25519_SIZE];
for (j = 0; j < F25519_SIZE; j++) {
c |= ((word16)r[j]) << 1;
r[j] = (byte)c;
c >>= 8;
}
raw_try_sub(r, modulus);
fprime_copy(plusa, r);
fprime_add(plusa, a, modulus);
fprime_select(r, r, plusa, bit);
}
}
#endif
#if 0 //UNUSED
static void fe_load(byte *x, word32 c)
{
word32 i;
for (i = 0; i < sizeof(c); i++) {
x[i] = c;
c >>= 8;
}
for (; i < F25519_SIZE; i++)
x[i] = 0;
}
#endif
static void fe_normalize(byte *x)
{
byte minusp[F25519_SIZE];
word16 c;
int i;
/* Reduce using 2^255 = 19 mod p */
c = (x[31] >> 7) * 19;
x[31] &= 127;
for (i = 0; i < F25519_SIZE; i++) {
c += x[i];
x[i] = (byte)c;
c >>= 8;
}
/* The number is now less than 2^255 + 18, and therefore less than
* 2p. Try subtracting p, and conditionally load the subtracted
* value if underflow did not occur.
*/
c = 19;
for (i = 0; i + 1 < F25519_SIZE; i++) {
c += x[i];
minusp[i] = (byte)c;
c >>= 8;
}
c += ((word16)x[i]) - 128;
minusp[31] = (byte)c;
/* Load x-p if no underflow */
fe_select(x, minusp, x, (c >> 15) & 1);
}
static void lm_add(byte* r, const byte* a, const byte* b)
{
word16 c = 0;
int i;
/* Add */
for (i = 0; i < F25519_SIZE; i++) {
c >>= 8;
c += ((word16)a[i]) + ((word16)b[i]);
r[i] = (byte)c;
}
/* Reduce with 2^255 = 19 mod p */
r[31] &= 127;
c = (c >> 7) * 19;
for (i = 0; i < F25519_SIZE; i++) {
c += r[i];
r[i] = (byte)c;
c >>= 8;
}
}
static void lm_sub(byte* r, const byte* a, const byte* b)
{
word32 c = 0;
int i;
/* Calculate a + 2p - b, to avoid underflow */
c = 218;
for (i = 0; i + 1 < F25519_SIZE; i++) {
c += 65280 + ((word32)a[i]) - ((word32)b[i]);
r[i] = c;
c >>= 8;
}
c += ((word32)a[31]) - ((word32)b[31]);
r[31] = c & 127;
c = (c >> 7) * 19;
for (i = 0; i < F25519_SIZE; i++) {
c += r[i];
r[i] = c;
c >>= 8;
}
}
#if 0 //UNUSED
static void lm_neg(byte* r, const byte* a)
{
word32 c = 0;
int i;
/* Calculate 2p - a, to avoid underflow */
c = 218;
for (i = 0; i + 1 < F25519_SIZE; i++) {
c += 65280 - ((word32)a[i]);
r[i] = c;
c >>= 8;
}
c -= ((word32)a[31]);
r[31] = c & 127;
c = (c >> 7) * 19;
for (i = 0; i < F25519_SIZE; i++) {
c += r[i];
r[i] = c;
c >>= 8;
}
}
#endif
static void fe_mul__distinct(byte *r, const byte *a, const byte *b)
{
word32 c = 0;
int i;
for (i = 0; i < F25519_SIZE; i++) {
int j;
c >>= 8;
for (j = 0; j <= i; j++)
c += ((word32)a[j]) * ((word32)b[i - j]);
for (; j < F25519_SIZE; j++)
c += ((word32)a[j]) *
((word32)b[i + F25519_SIZE - j]) * 38;
r[i] = c;
}
r[31] &= 127;
c = (c >> 7) * 19;
for (i = 0; i < F25519_SIZE; i++) {
c += r[i];
r[i] = c;
c >>= 8;
}
}
#if 0 //UNUSED
static void lm_mul(byte *r, const byte* a, const byte *b)
{
byte tmp[F25519_SIZE];
fe_mul__distinct(tmp, a, b);
lm_copy(r, tmp);
}
#endif
static void fe_mul_c(byte *r, const byte *a, word32 b)
{
word32 c = 0;
int i;
for (i = 0; i < F25519_SIZE; i++) {
c >>= 8;
c += b * ((word32)a[i]);
r[i] = c;
}
r[31] &= 127;
c >>= 7;
c *= 19;
for (i = 0; i < F25519_SIZE; i++) {
c += r[i];
r[i] = c;
c >>= 8;
}
}
static void fe_inv__distinct(byte *r, const byte *x)
{
byte s[F25519_SIZE];
int i;
/* This is a prime field, so by Fermat's little theorem:
*
* x^(p-1) = 1 mod p
*
* Therefore, raise to (p-2) = 2^255-21 to get a multiplicative
* inverse.
*
* This is a 255-bit binary number with the digits:
*
* 11111111... 01011
*
* We compute the result by the usual binary chain, but
* alternate between keeping the accumulator in r and s, so as
* to avoid copying temporaries.
*/
/* 1 1 */
fe_mul__distinct(s, x, x);
fe_mul__distinct(r, s, x);
/* 1 x 248 */
for (i = 0; i < 248; i++) {
fe_mul__distinct(s, r, r);
fe_mul__distinct(r, s, x);
}
/* 0 */
fe_mul__distinct(s, r, r);
/* 1 */
fe_mul__distinct(r, s, s);
fe_mul__distinct(s, r, x);
/* 0 */
fe_mul__distinct(r, s, s);
/* 1 */
fe_mul__distinct(s, r, r);
fe_mul__distinct(r, s, x);
/* 1 */
fe_mul__distinct(s, r, r);
fe_mul__distinct(r, s, x);
}
#if 0 //UNUSED
static void lm_invert(byte *r, const byte *x)
{
byte tmp[F25519_SIZE];
fe_inv__distinct(tmp, x);
lm_copy(r, tmp);
}
#endif
#if 0 //UNUSED
/* Raise x to the power of (p-5)/8 = 2^252-3, using s for temporary
* storage.
*/
static void exp2523(byte *r, const byte *x, byte *s)
{
int i;
/* This number is a 252-bit number with the binary expansion:
*
* 111111... 01
*/
/* 1 1 */
fe_mul__distinct(r, x, x);
fe_mul__distinct(s, r, x);
/* 1 x 248 */
for (i = 0; i < 248; i++) {
fe_mul__distinct(r, s, s);
fe_mul__distinct(s, r, x);
}
/* 0 */
fe_mul__distinct(r, s, s);
/* 1 */
fe_mul__distinct(s, r, r);
fe_mul__distinct(r, s, x);
}
#endif
#if 0 //UNUSED
static void fe_sqrt(byte *r, const byte *a)
{
byte v[F25519_SIZE];
byte i[F25519_SIZE];
byte x[F25519_SIZE];
byte y[F25519_SIZE];
/* v = (2a)^((p-5)/8) [x = 2a] */
fe_mul_c(x, a, 2);
exp2523(v, x, y);
/* i = 2av^2 - 1 */
fe_mul__distinct(y, v, v);
fe_mul__distinct(i, x, y);
fe_load(y, 1);
lm_sub(i, i, y);
/* r = avi */
fe_mul__distinct(x, v, a);
fe_mul__distinct(r, x, i);
}
#endif
/* Differential addition */
static void xc_diffadd(byte *x5, byte *z5,
const byte *x1, const byte *z1,
const byte *x2, const byte *z2,
const byte *x3, const byte *z3)
{
/* Explicit formulas database: dbl-1987-m3
*
* source 1987 Montgomery "Speeding the Pollard and elliptic curve
* methods of factorization", page 261, fifth display, plus
* common-subexpression elimination
* compute A = X2+Z2
* compute B = X2-Z2
* compute C = X3+Z3
* compute D = X3-Z3
* compute DA = D A
* compute CB = C B
* compute X5 = Z1(DA+CB)^2
* compute Z5 = X1(DA-CB)^2
*/
byte da[F25519_SIZE];
byte cb[F25519_SIZE];
byte a[F25519_SIZE];
byte b[F25519_SIZE];
lm_add(a, x2, z2);
lm_sub(b, x3, z3); /* D */
fe_mul__distinct(da, a, b);
lm_sub(b, x2, z2);
lm_add(a, x3, z3); /* C */
fe_mul__distinct(cb, a, b);
lm_add(a, da, cb);
fe_mul__distinct(b, a, a);
fe_mul__distinct(x5, z1, b);
lm_sub(a, da, cb);
fe_mul__distinct(b, a, a);
fe_mul__distinct(z5, x1, b);
}
/* Double an X-coordinate */
static void xc_double(byte *x3, byte *z3,
const byte *x1, const byte *z1)
{
/* Explicit formulas database: dbl-1987-m
*
* source 1987 Montgomery "Speeding the Pollard and elliptic
* curve methods of factorization", page 261, fourth display
* compute X3 = (X1^2-Z1^2)^2
* compute Z3 = 4 X1 Z1 (X1^2 + a X1 Z1 + Z1^2)
*/
byte x1sq[F25519_SIZE];
byte z1sq[F25519_SIZE];
byte x1z1[F25519_SIZE];
byte a[F25519_SIZE];
fe_mul__distinct(x1sq, x1, x1);
fe_mul__distinct(z1sq, z1, z1);
fe_mul__distinct(x1z1, x1, z1);
lm_sub(a, x1sq, z1sq);
fe_mul__distinct(x3, a, a);
fe_mul_c(a, x1z1, 486662);
lm_add(a, x1sq, a);
lm_add(a, z1sq, a);
fe_mul__distinct(x1sq, x1z1, a);
fe_mul_c(z3, x1sq, 4);
}
void FAST_FUNC curve25519(byte *result, const byte *e, const byte *q)
{
int i;
struct {
/* from wolfssl-3.15.3/wolfssl/wolfcrypt/fe_operations.h */
/*static const*/ byte f25519_one[F25519_SIZE]; // = {1};
/* Current point: P_m */
byte xm[F25519_SIZE];
byte zm[F25519_SIZE]; // = {1};
/* Predecessor: P_(m-1) */
byte xm1[F25519_SIZE]; // = {1};
byte zm1[F25519_SIZE]; // = {0};
} z;
#define f25519_one z.f25519_one
#define xm z.xm
#define zm z.zm
#define xm1 z.xm1
#define zm1 z.zm1
memset(&z, 0, sizeof(z));
f25519_one[0] = 1;
zm[0] = 1;
xm1[0] = 1;
/* Note: bit 254 is assumed to be 1 */
lm_copy(xm, q);
for (i = 253; i >= 0; i--) {
const int bit = (e[i >> 3] >> (i & 7)) & 1;
byte xms[F25519_SIZE];
byte zms[F25519_SIZE];
/* From P_m and P_(m-1), compute P_(2m) and P_(2m-1) */
xc_diffadd(xm1, zm1, q, f25519_one, xm, zm, xm1, zm1);
xc_double(xm, zm, xm, zm);
/* Compute P_(2m+1) */
xc_diffadd(xms, zms, xm1, zm1, xm, zm, q, f25519_one);
/* Select:
* bit = 1 --> (P_(2m+1), P_(2m))
* bit = 0 --> (P_(2m), P_(2m-1))
*/
fe_select(xm1, xm1, xm, bit);
fe_select(zm1, zm1, zm, bit);
fe_select(xm, xm, xms, bit);
fe_select(zm, zm, zms, bit);
}
/* Freeze out of projective coordinates */
fe_inv__distinct(zm1, zm);
fe_mul__distinct(result, zm1, xm);
fe_normalize(result);
}