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wireguard-tools/src/curve25519.c

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/* Original author: Adam Langley <agl@imperialviolet.org>
*
* Copyright 2008 Google Inc. All Rights Reserved.
* Copyright 2015-2016 Jason A. Donenfeld <Jason@zx2c4.com>. All Rights Reserved.
*
* Redistribution and use in source and binary forms of this file, with or
* without modification, are permitted provided that the following conditions
* are met:
*
* * Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* * Redistributions in binary form must reproduce the above
* copyright notice, this list of conditions and the following disclaimer
* in the documentation and/or other materials provided with the
* distribution.
* * Neither the name of Google Inc nor the names of its contributors
* may be used to endorse or promote products derived from this software
* without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
* "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
* LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
* A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
* OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
* SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
* LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
* DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
* THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
* OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*/
#include "curve25519.h"
#include <stdint.h>
#include <string.h>
#ifndef __always_inline
#define __always_inline __inline __attribute__((__always_inline__))
#endif
#ifdef __SIZEOF_INT128__
typedef uint64_t limb;
typedef limb felem[5];
typedef __uint128_t uint128_t;
/* Sum two numbers: output += in */
static __always_inline void fsum(limb *output, const limb *in)
{
output[0] += in[0];
output[1] += in[1];
output[2] += in[2];
output[3] += in[3];
output[4] += in[4];
}
/* Find the difference of two numbers: output = in - output
* (note the order of the arguments!)
*
* Assumes that out[i] < 2**52
* On return, out[i] < 2**55
*/
static __always_inline void fdifference_backwards(felem out, const felem in)
{
/* 152 is 19 << 3 */
static const limb two54m152 = (((limb)1) << 54) - 152;
static const limb two54m8 = (((limb)1) << 54) - 8;
out[0] = in[0] + two54m152 - out[0];
out[1] = in[1] + two54m8 - out[1];
out[2] = in[2] + two54m8 - out[2];
out[3] = in[3] + two54m8 - out[3];
out[4] = in[4] + two54m8 - out[4];
}
/* Multiply a number by a scalar: output = in * scalar */
static __always_inline void fscalar_product(felem output, const felem in, const limb scalar)
{
uint128_t a;
a = ((uint128_t) in[0]) * scalar;
output[0] = ((limb)a) & 0x7ffffffffffffUL;
a = ((uint128_t) in[1]) * scalar + ((limb) (a >> 51));
output[1] = ((limb)a) & 0x7ffffffffffffUL;
a = ((uint128_t) in[2]) * scalar + ((limb) (a >> 51));
output[2] = ((limb)a) & 0x7ffffffffffffUL;
a = ((uint128_t) in[3]) * scalar + ((limb) (a >> 51));
output[3] = ((limb)a) & 0x7ffffffffffffUL;
a = ((uint128_t) in[4]) * scalar + ((limb) (a >> 51));
output[4] = ((limb)a) & 0x7ffffffffffffUL;
output[0] += (a >> 51) * 19;
}
/* Multiply two numbers: output = in2 * in
*
* output must be distinct to both inputs. The inputs are reduced coefficient
* form, the output is not.
*
* Assumes that in[i] < 2**55 and likewise for in2.
* On return, output[i] < 2**52
*/
static __always_inline void fmul(felem output, const felem in2, const felem in)
{
uint128_t t[5];
limb r0,r1,r2,r3,r4,s0,s1,s2,s3,s4,c;
r0 = in[0];
r1 = in[1];
r2 = in[2];
r3 = in[3];
r4 = in[4];
s0 = in2[0];
s1 = in2[1];
s2 = in2[2];
s3 = in2[3];
s4 = in2[4];
t[0] = ((uint128_t) r0) * s0;
t[1] = ((uint128_t) r0) * s1 + ((uint128_t) r1) * s0;
t[2] = ((uint128_t) r0) * s2 + ((uint128_t) r2) * s0 + ((uint128_t) r1) * s1;
t[3] = ((uint128_t) r0) * s3 + ((uint128_t) r3) * s0 + ((uint128_t) r1) * s2 + ((uint128_t) r2) * s1;
t[4] = ((uint128_t) r0) * s4 + ((uint128_t) r4) * s0 + ((uint128_t) r3) * s1 + ((uint128_t) r1) * s3 + ((uint128_t) r2) * s2;
r4 *= 19;
r1 *= 19;
r2 *= 19;
r3 *= 19;
t[0] += ((uint128_t) r4) * s1 + ((uint128_t) r1) * s4 + ((uint128_t) r2) * s3 + ((uint128_t) r3) * s2;
t[1] += ((uint128_t) r4) * s2 + ((uint128_t) r2) * s4 + ((uint128_t) r3) * s3;
t[2] += ((uint128_t) r4) * s3 + ((uint128_t) r3) * s4;
t[3] += ((uint128_t) r4) * s4;
r0 = (limb)t[0] & 0x7ffffffffffffUL; c = (limb)(t[0] >> 51);
t[1] += c; r1 = (limb)t[1] & 0x7ffffffffffffUL; c = (limb)(t[1] >> 51);
t[2] += c; r2 = (limb)t[2] & 0x7ffffffffffffUL; c = (limb)(t[2] >> 51);
t[3] += c; r3 = (limb)t[3] & 0x7ffffffffffffUL; c = (limb)(t[3] >> 51);
t[4] += c; r4 = (limb)t[4] & 0x7ffffffffffffUL; c = (limb)(t[4] >> 51);
r0 += c * 19; c = r0 >> 51; r0 = r0 & 0x7ffffffffffffUL;
r1 += c; c = r1 >> 51; r1 = r1 & 0x7ffffffffffffUL;
r2 += c;
output[0] = r0;
output[1] = r1;
output[2] = r2;
output[3] = r3;
output[4] = r4;
}
static __always_inline void fsquare_times(felem output, const felem in, limb count)
{
uint128_t t[5];
limb r0,r1,r2,r3,r4,c;
limb d0,d1,d2,d4,d419;
r0 = in[0];
r1 = in[1];
r2 = in[2];
r3 = in[3];
r4 = in[4];
do {
d0 = r0 * 2;
d1 = r1 * 2;
d2 = r2 * 2 * 19;
d419 = r4 * 19;
d4 = d419 * 2;
t[0] = ((uint128_t) r0) * r0 + ((uint128_t) d4) * r1 + (((uint128_t) d2) * (r3 ));
t[1] = ((uint128_t) d0) * r1 + ((uint128_t) d4) * r2 + (((uint128_t) r3) * (r3 * 19));
t[2] = ((uint128_t) d0) * r2 + ((uint128_t) r1) * r1 + (((uint128_t) d4) * (r3 ));
t[3] = ((uint128_t) d0) * r3 + ((uint128_t) d1) * r2 + (((uint128_t) r4) * (d419 ));
t[4] = ((uint128_t) d0) * r4 + ((uint128_t) d1) * r3 + (((uint128_t) r2) * (r2 ));
r0 = (limb)t[0] & 0x7ffffffffffffUL; c = (limb)(t[0] >> 51);
t[1] += c; r1 = (limb)t[1] & 0x7ffffffffffffUL; c = (limb)(t[1] >> 51);
t[2] += c; r2 = (limb)t[2] & 0x7ffffffffffffUL; c = (limb)(t[2] >> 51);
t[3] += c; r3 = (limb)t[3] & 0x7ffffffffffffUL; c = (limb)(t[3] >> 51);
t[4] += c; r4 = (limb)t[4] & 0x7ffffffffffffUL; c = (limb)(t[4] >> 51);
r0 += c * 19; c = r0 >> 51; r0 = r0 & 0x7ffffffffffffUL;
r1 += c; c = r1 >> 51; r1 = r1 & 0x7ffffffffffffUL;
r2 += c;
} while(--count);
output[0] = r0;
output[1] = r1;
output[2] = r2;
output[3] = r3;
output[4] = r4;
}
/* Load a little-endian 64-bit number */
static limb load_limb(const uint8_t *in)
{
return
((limb)in[0]) |
(((limb)in[1]) << 8) |
(((limb)in[2]) << 16) |
(((limb)in[3]) << 24) |
(((limb)in[4]) << 32) |
(((limb)in[5]) << 40) |
(((limb)in[6]) << 48) |
(((limb)in[7]) << 56);
}
static void store_limb(uint8_t *out, limb in)
{
out[0] = in & 0xff;
out[1] = (in >> 8) & 0xff;
out[2] = (in >> 16) & 0xff;
out[3] = (in >> 24) & 0xff;
out[4] = (in >> 32) & 0xff;
out[5] = (in >> 40) & 0xff;
out[6] = (in >> 48) & 0xff;
out[7] = (in >> 56) & 0xff;
}
/* Take a little-endian, 32-byte number and expand it into polynomial form */
static void fexpand(limb *output, const uint8_t *in)
{
output[0] = load_limb(in) & 0x7ffffffffffffUL;
output[1] = (load_limb(in+6) >> 3) & 0x7ffffffffffffUL;
output[2] = (load_limb(in+12) >> 6) & 0x7ffffffffffffUL;
output[3] = (load_limb(in+19) >> 1) & 0x7ffffffffffffUL;
output[4] = (load_limb(in+24) >> 12) & 0x7ffffffffffffUL;
}
/* Take a fully reduced polynomial form number and contract it into a
* little-endian, 32-byte array
*/
static void fcontract(uint8_t *output, const felem input)
{
uint128_t t[5];
t[0] = input[0];
t[1] = input[1];
t[2] = input[2];
t[3] = input[3];
t[4] = input[4];
t[1] += t[0] >> 51; t[0] &= 0x7ffffffffffffUL;
t[2] += t[1] >> 51; t[1] &= 0x7ffffffffffffUL;
t[3] += t[2] >> 51; t[2] &= 0x7ffffffffffffUL;
t[4] += t[3] >> 51; t[3] &= 0x7ffffffffffffUL;
t[0] += 19 * (t[4] >> 51); t[4] &= 0x7ffffffffffffUL;
t[1] += t[0] >> 51; t[0] &= 0x7ffffffffffffUL;
t[2] += t[1] >> 51; t[1] &= 0x7ffffffffffffUL;
t[3] += t[2] >> 51; t[2] &= 0x7ffffffffffffUL;
t[4] += t[3] >> 51; t[3] &= 0x7ffffffffffffUL;
t[0] += 19 * (t[4] >> 51); t[4] &= 0x7ffffffffffffUL;
/* now t is between 0 and 2^255-1, properly carried. */
/* case 1: between 0 and 2^255-20. case 2: between 2^255-19 and 2^255-1. */
t[0] += 19;
t[1] += t[0] >> 51; t[0] &= 0x7ffffffffffffUL;
t[2] += t[1] >> 51; t[1] &= 0x7ffffffffffffUL;
t[3] += t[2] >> 51; t[2] &= 0x7ffffffffffffUL;
t[4] += t[3] >> 51; t[3] &= 0x7ffffffffffffUL;
t[0] += 19 * (t[4] >> 51); t[4] &= 0x7ffffffffffffUL;
/* now between 19 and 2^255-1 in both cases, and offset by 19. */
t[0] += 0x8000000000000UL - 19;
t[1] += 0x8000000000000UL - 1;
t[2] += 0x8000000000000UL - 1;
t[3] += 0x8000000000000UL - 1;
t[4] += 0x8000000000000UL - 1;
/* now between 2^255 and 2^256-20, and offset by 2^255. */
t[1] += t[0] >> 51; t[0] &= 0x7ffffffffffffUL;
t[2] += t[1] >> 51; t[1] &= 0x7ffffffffffffUL;
t[3] += t[2] >> 51; t[2] &= 0x7ffffffffffffUL;
t[4] += t[3] >> 51; t[3] &= 0x7ffffffffffffUL;
t[4] &= 0x7ffffffffffffUL;
store_limb(output, t[0] | (t[1] << 51));
store_limb(output+8, (t[1] >> 13) | (t[2] << 38));
store_limb(output+16, (t[2] >> 26) | (t[3] << 25));
store_limb(output+24, (t[3] >> 39) | (t[4] << 12));
}
/* Input: Q, Q', Q-Q'
* Output: 2Q, Q+Q'
*
* x2 z3: long form
* x3 z3: long form
* x z: short form, destroyed
* xprime zprime: short form, destroyed
* qmqp: short form, preserved
*/
static void fmonty(limb *x2, limb *z2, /* output 2Q */
limb *x3, limb *z3, /* output Q + Q' */
limb *x, limb *z, /* input Q */
limb *xprime, limb *zprime, /* input Q' */
const limb *qmqp /* input Q - Q' */)
{
limb origx[5], origxprime[5], zzz[5], xx[5], zz[5], xxprime[5], zzprime[5], zzzprime[5];
memcpy(origx, x, 5 * sizeof(limb));
fsum(x, z);
fdifference_backwards(z, origx); // does x - z
memcpy(origxprime, xprime, sizeof(limb) * 5);
fsum(xprime, zprime);
fdifference_backwards(zprime, origxprime);
fmul(xxprime, xprime, z);
fmul(zzprime, x, zprime);
memcpy(origxprime, xxprime, sizeof(limb) * 5);
fsum(xxprime, zzprime);
fdifference_backwards(zzprime, origxprime);
fsquare_times(x3, xxprime, 1);
fsquare_times(zzzprime, zzprime, 1);
fmul(z3, zzzprime, qmqp);
fsquare_times(xx, x, 1);
fsquare_times(zz, z, 1);
fmul(x2, xx, zz);
fdifference_backwards(zz, xx); // does zz = xx - zz
fscalar_product(zzz, zz, 121665);
fsum(zzz, xx);
fmul(z2, zz, zzz);
}
/* Maybe swap the contents of two limb arrays (@a and @b), each @len elements
* long. Perform the swap iff @swap is non-zero.
*
* This function performs the swap without leaking any side-channel
* information.
*/
static void swap_conditional(limb a[static 5], limb b[static 5], limb iswap)
{
unsigned i;
const limb swap = -iswap;
for (i = 0; i < 5; ++i) {
const limb x = swap & (a[i] ^ b[i]);
a[i] ^= x;
b[i] ^= x;
}
}
/* Calculates nQ where Q is the x-coordinate of a point on the curve
*
* resultx/resultz: the x coordinate of the resulting curve point (short form)
* n: a little endian, 32-byte number
* q: a point of the curve (short form)
*/
static void cmult(limb *resultx, limb *resultz, const uint8_t *n, const limb *q)
{
limb a[5] = {0}, b[5] = {1}, c[5] = {1}, d[5] = {0};
limb *nqpqx = a, *nqpqz = b, *nqx = c, *nqz = d, *t;
limb e[5] = {0}, f[5] = {1}, g[5] = {0}, h[5] = {1};
limb *nqpqx2 = e, *nqpqz2 = f, *nqx2 = g, *nqz2 = h;
unsigned i, j;
memcpy(nqpqx, q, sizeof(limb) * 5);
for (i = 0; i < 32; ++i) {
uint8_t byte = n[31 - i];
for (j = 0; j < 8; ++j) {
const limb bit = byte >> 7;
swap_conditional(nqx, nqpqx, bit);
swap_conditional(nqz, nqpqz, bit);
fmonty(nqx2, nqz2,
nqpqx2, nqpqz2,
nqx, nqz,
nqpqx, nqpqz,
q);
swap_conditional(nqx2, nqpqx2, bit);
swap_conditional(nqz2, nqpqz2, bit);
t = nqx;
nqx = nqx2;
nqx2 = t;
t = nqz;
nqz = nqz2;
nqz2 = t;
t = nqpqx;
nqpqx = nqpqx2;
nqpqx2 = t;
t = nqpqz;
nqpqz = nqpqz2;
nqpqz2 = t;
byte <<= 1;
}
}
memcpy(resultx, nqx, sizeof(limb) * 5);
memcpy(resultz, nqz, sizeof(limb) * 5);
}
static void crecip(felem out, const felem z)
{
felem a,t0,b,c;
/* 2 */ fsquare_times(a, z, 1); // a = 2
/* 8 */ fsquare_times(t0, a, 2);
/* 9 */ fmul(b, t0, z); // b = 9
/* 11 */ fmul(a, b, a); // a = 11
/* 22 */ fsquare_times(t0, a, 1);
/* 2^5 - 2^0 = 31 */ fmul(b, t0, b);
/* 2^10 - 2^5 */ fsquare_times(t0, b, 5);
/* 2^10 - 2^0 */ fmul(b, t0, b);
/* 2^20 - 2^10 */ fsquare_times(t0, b, 10);
/* 2^20 - 2^0 */ fmul(c, t0, b);
/* 2^40 - 2^20 */ fsquare_times(t0, c, 20);
/* 2^40 - 2^0 */ fmul(t0, t0, c);
/* 2^50 - 2^10 */ fsquare_times(t0, t0, 10);
/* 2^50 - 2^0 */ fmul(b, t0, b);
/* 2^100 - 2^50 */ fsquare_times(t0, b, 50);
/* 2^100 - 2^0 */ fmul(c, t0, b);
/* 2^200 - 2^100 */ fsquare_times(t0, c, 100);
/* 2^200 - 2^0 */ fmul(t0, t0, c);
/* 2^250 - 2^50 */ fsquare_times(t0, t0, 50);
/* 2^250 - 2^0 */ fmul(t0, t0, b);
/* 2^255 - 2^5 */ fsquare_times(t0, t0, 5);
/* 2^255 - 21 */ fmul(out, t0, a);
}
void curve25519(uint8_t mypublic[static CURVE25519_POINT_SIZE], const uint8_t secret[static CURVE25519_POINT_SIZE], const uint8_t basepoint[static CURVE25519_POINT_SIZE])
{
limb bp[5], x[5], z[5], zmone[5];
uint8_t e[32];
memcpy(e, secret, 32);
curve25519_normalize_secret(e);
fexpand(bp, basepoint);
cmult(x, z, e, bp);
crecip(zmone, z);
fmul(z, x, zmone);
fcontract(mypublic, z);
}
#else
typedef int64_t limb;
/* Field element representation:
*
* Field elements are written as an array of signed, 64-bit limbs, least
* significant first. The value of the field element is:
* x[0] + 2^26·x[1] + x^51·x[2] + 2^102·x[3] + ...
*
* i.e. the limbs are 26, 25, 26, 25, ... bits wide. */
/* Sum two numbers: output += in */
static void fsum(limb *output, const limb *in)
{
unsigned i;
for (i = 0; i < 10; i += 2) {
output[0 + i] = output[0 + i] + in[0 + i];
output[1 + i] = output[1 + i] + in[1 + i];
}
}
/* Find the difference of two numbers: output = in - output
* (note the order of the arguments!). */
static void fdifference(limb *output, const limb *in)
{
unsigned i;
for (i = 0; i < 10; ++i) {
output[i] = in[i] - output[i];
}
}
/* Multiply a number by a scalar: output = in * scalar */
static void fscalar_product(limb *output, const limb *in, const limb scalar)
{
unsigned i;
for (i = 0; i < 10; ++i) {
output[i] = in[i] * scalar;
}
}
/* Multiply two numbers: output = in2 * in
*
* output must be distinct to both inputs. The inputs are reduced coefficient
* form, the output is not.
*
* output[x] <= 14 * the largest product of the input limbs. */
static void fproduct(limb *output, const limb *in2, const limb *in)
{
output[0] = ((limb) ((int32_t) in2[0])) * ((int32_t) in[0]);
output[1] = ((limb) ((int32_t) in2[0])) * ((int32_t) in[1]) +
((limb) ((int32_t) in2[1])) * ((int32_t) in[0]);
output[2] = 2 * ((limb) ((int32_t) in2[1])) * ((int32_t) in[1]) +
((limb) ((int32_t) in2[0])) * ((int32_t) in[2]) +
((limb) ((int32_t) in2[2])) * ((int32_t) in[0]);
output[3] = ((limb) ((int32_t) in2[1])) * ((int32_t) in[2]) +
((limb) ((int32_t) in2[2])) * ((int32_t) in[1]) +
((limb) ((int32_t) in2[0])) * ((int32_t) in[3]) +
((limb) ((int32_t) in2[3])) * ((int32_t) in[0]);
output[4] = ((limb) ((int32_t) in2[2])) * ((int32_t) in[2]) +
2 * (((limb) ((int32_t) in2[1])) * ((int32_t) in[3]) +
((limb) ((int32_t) in2[3])) * ((int32_t) in[1])) +
((limb) ((int32_t) in2[0])) * ((int32_t) in[4]) +
((limb) ((int32_t) in2[4])) * ((int32_t) in[0]);
output[5] = ((limb) ((int32_t) in2[2])) * ((int32_t) in[3]) +
((limb) ((int32_t) in2[3])) * ((int32_t) in[2]) +
((limb) ((int32_t) in2[1])) * ((int32_t) in[4]) +
((limb) ((int32_t) in2[4])) * ((int32_t) in[1]) +
((limb) ((int32_t) in2[0])) * ((int32_t) in[5]) +
((limb) ((int32_t) in2[5])) * ((int32_t) in[0]);
output[6] = 2 * (((limb) ((int32_t) in2[3])) * ((int32_t) in[3]) +
((limb) ((int32_t) in2[1])) * ((int32_t) in[5]) +
((limb) ((int32_t) in2[5])) * ((int32_t) in[1])) +
((limb) ((int32_t) in2[2])) * ((int32_t) in[4]) +
((limb) ((int32_t) in2[4])) * ((int32_t) in[2]) +
((limb) ((int32_t) in2[0])) * ((int32_t) in[6]) +
((limb) ((int32_t) in2[6])) * ((int32_t) in[0]);
output[7] = ((limb) ((int32_t) in2[3])) * ((int32_t) in[4]) +
((limb) ((int32_t) in2[4])) * ((int32_t) in[3]) +
((limb) ((int32_t) in2[2])) * ((int32_t) in[5]) +
((limb) ((int32_t) in2[5])) * ((int32_t) in[2]) +
((limb) ((int32_t) in2[1])) * ((int32_t) in[6]) +
((limb) ((int32_t) in2[6])) * ((int32_t) in[1]) +
((limb) ((int32_t) in2[0])) * ((int32_t) in[7]) +
((limb) ((int32_t) in2[7])) * ((int32_t) in[0]);
output[8] = ((limb) ((int32_t) in2[4])) * ((int32_t) in[4]) +
2 * (((limb) ((int32_t) in2[3])) * ((int32_t) in[5]) +
((limb) ((int32_t) in2[5])) * ((int32_t) in[3]) +
((limb) ((int32_t) in2[1])) * ((int32_t) in[7]) +
((limb) ((int32_t) in2[7])) * ((int32_t) in[1])) +
((limb) ((int32_t) in2[2])) * ((int32_t) in[6]) +
((limb) ((int32_t) in2[6])) * ((int32_t) in[2]) +
((limb) ((int32_t) in2[0])) * ((int32_t) in[8]) +
((limb) ((int32_t) in2[8])) * ((int32_t) in[0]);
output[9] = ((limb) ((int32_t) in2[4])) * ((int32_t) in[5]) +
((limb) ((int32_t) in2[5])) * ((int32_t) in[4]) +
((limb) ((int32_t) in2[3])) * ((int32_t) in[6]) +
((limb) ((int32_t) in2[6])) * ((int32_t) in[3]) +
((limb) ((int32_t) in2[2])) * ((int32_t) in[7]) +
((limb) ((int32_t) in2[7])) * ((int32_t) in[2]) +
((limb) ((int32_t) in2[1])) * ((int32_t) in[8]) +
((limb) ((int32_t) in2[8])) * ((int32_t) in[1]) +
((limb) ((int32_t) in2[0])) * ((int32_t) in[9]) +
((limb) ((int32_t) in2[9])) * ((int32_t) in[0]);
output[10] = 2 * (((limb) ((int32_t) in2[5])) * ((int32_t) in[5]) +
((limb) ((int32_t) in2[3])) * ((int32_t) in[7]) +
((limb) ((int32_t) in2[7])) * ((int32_t) in[3]) +
((limb) ((int32_t) in2[1])) * ((int32_t) in[9]) +
((limb) ((int32_t) in2[9])) * ((int32_t) in[1])) +
((limb) ((int32_t) in2[4])) * ((int32_t) in[6]) +
((limb) ((int32_t) in2[6])) * ((int32_t) in[4]) +
((limb) ((int32_t) in2[2])) * ((int32_t) in[8]) +
((limb) ((int32_t) in2[8])) * ((int32_t) in[2]);
output[11] = ((limb) ((int32_t) in2[5])) * ((int32_t) in[6]) +
((limb) ((int32_t) in2[6])) * ((int32_t) in[5]) +
((limb) ((int32_t) in2[4])) * ((int32_t) in[7]) +
((limb) ((int32_t) in2[7])) * ((int32_t) in[4]) +
((limb) ((int32_t) in2[3])) * ((int32_t) in[8]) +
((limb) ((int32_t) in2[8])) * ((int32_t) in[3]) +
((limb) ((int32_t) in2[2])) * ((int32_t) in[9]) +
((limb) ((int32_t) in2[9])) * ((int32_t) in[2]);
output[12] = ((limb) ((int32_t) in2[6])) * ((int32_t) in[6]) +
2 * (((limb) ((int32_t) in2[5])) * ((int32_t) in[7]) +
((limb) ((int32_t) in2[7])) * ((int32_t) in[5]) +
((limb) ((int32_t) in2[3])) * ((int32_t) in[9]) +
((limb) ((int32_t) in2[9])) * ((int32_t) in[3])) +
((limb) ((int32_t) in2[4])) * ((int32_t) in[8]) +
((limb) ((int32_t) in2[8])) * ((int32_t) in[4]);
output[13] = ((limb) ((int32_t) in2[6])) * ((int32_t) in[7]) +
((limb) ((int32_t) in2[7])) * ((int32_t) in[6]) +
((limb) ((int32_t) in2[5])) * ((int32_t) in[8]) +
((limb) ((int32_t) in2[8])) * ((int32_t) in[5]) +
((limb) ((int32_t) in2[4])) * ((int32_t) in[9]) +
((limb) ((int32_t) in2[9])) * ((int32_t) in[4]);
output[14] = 2 * (((limb) ((int32_t) in2[7])) * ((int32_t) in[7]) +
((limb) ((int32_t) in2[5])) * ((int32_t) in[9]) +
((limb) ((int32_t) in2[9])) * ((int32_t) in[5])) +
((limb) ((int32_t) in2[6])) * ((int32_t) in[8]) +
((limb) ((int32_t) in2[8])) * ((int32_t) in[6]);
output[15] = ((limb) ((int32_t) in2[7])) * ((int32_t) in[8]) +
((limb) ((int32_t) in2[8])) * ((int32_t) in[7]) +
((limb) ((int32_t) in2[6])) * ((int32_t) in[9]) +
((limb) ((int32_t) in2[9])) * ((int32_t) in[6]);
output[16] = ((limb) ((int32_t) in2[8])) * ((int32_t) in[8]) +
2 * (((limb) ((int32_t) in2[7])) * ((int32_t) in[9]) +
((limb) ((int32_t) in2[9])) * ((int32_t) in[7]));
output[17] = ((limb) ((int32_t) in2[8])) * ((int32_t) in[9]) +
((limb) ((int32_t) in2[9])) * ((int32_t) in[8]);
output[18] = 2 * ((limb) ((int32_t) in2[9])) * ((int32_t) in[9]);
}
/* Reduce a long form to a short form by taking the input mod 2^255 - 19.
*
* On entry: |output[i]| < 14*2^54
* On exit: |output[0..8]| < 280*2^54 */
static void freduce_degree(limb *output)
{
/* Each of these shifts and adds ends up multiplying the value by 19.
*
* For output[0..8], the absolute entry value is < 14*2^54 and we add, at
* most, 19*14*2^54 thus, on exit, |output[0..8]| < 280*2^54. */
output[8] += output[18] << 4;
output[8] += output[18] << 1;
output[8] += output[18];
output[7] += output[17] << 4;
output[7] += output[17] << 1;
output[7] += output[17];
output[6] += output[16] << 4;
output[6] += output[16] << 1;
output[6] += output[16];
output[5] += output[15] << 4;
output[5] += output[15] << 1;
output[5] += output[15];
output[4] += output[14] << 4;
output[4] += output[14] << 1;
output[4] += output[14];
output[3] += output[13] << 4;
output[3] += output[13] << 1;
output[3] += output[13];
output[2] += output[12] << 4;
output[2] += output[12] << 1;
output[2] += output[12];
output[1] += output[11] << 4;
output[1] += output[11] << 1;
output[1] += output[11];
output[0] += output[10] << 4;
output[0] += output[10] << 1;
output[0] += output[10];
}
#if (-1 & 3) != 3
#error "This code only works on a two's complement system"
#endif
/* return v / 2^26, using only shifts and adds.
*
* On entry: v can take any value. */
static inline limb div_by_2_26(const limb v)
{
/* High word of v; no shift needed. */
const uint32_t highword = (uint32_t) (((uint64_t) v) >> 32);
/* Set to all 1s if v was negative; else set to 0s. */
const int32_t sign = ((int32_t) highword) >> 31;
/* Set to 0x3ffffff if v was negative; else set to 0. */
const int32_t roundoff = ((uint32_t) sign) >> 6;
/* Should return v / (1<<26) */
return (v + roundoff) >> 26;
}
/* return v / (2^25), using only shifts and adds.
*
* On entry: v can take any value. */
static inline limb div_by_2_25(const limb v)
{
/* High word of v; no shift needed*/
const uint32_t highword = (uint32_t) (((uint64_t) v) >> 32);
/* Set to all 1s if v was negative; else set to 0s. */
const int32_t sign = ((int32_t) highword) >> 31;
/* Set to 0x1ffffff if v was negative; else set to 0. */
const int32_t roundoff = ((uint32_t) sign) >> 7;
/* Should return v / (1<<25) */
return (v + roundoff) >> 25;
}
/* Reduce all coefficients of the short form input so that |x| < 2^26.
*
* On entry: |output[i]| < 280*2^54 */
static void freduce_coefficients(limb *output)
{
unsigned i;
output[10] = 0;
for (i = 0; i < 10; i += 2) {
limb over = div_by_2_26(output[i]);
/* The entry condition (that |output[i]| < 280*2^54) means that over is, at
* most, 280*2^28 in the first iteration of this loop. This is added to the
* next limb and we can approximate the resulting bound of that limb by
* 281*2^54. */
output[i] -= over << 26;
output[i+1] += over;
/* For the first iteration, |output[i+1]| < 281*2^54, thus |over| <
* 281*2^29. When this is added to the next limb, the resulting bound can
* be approximated as 281*2^54.
*
* For subsequent iterations of the loop, 281*2^54 remains a conservative
* bound and no overflow occurs. */
over = div_by_2_25(output[i+1]);
output[i+1] -= over << 25;
output[i+2] += over;
}
/* Now |output[10]| < 281*2^29 and all other coefficients are reduced. */
output[0] += output[10] << 4;
output[0] += output[10] << 1;
output[0] += output[10];
output[10] = 0;
/* Now output[1..9] are reduced, and |output[0]| < 2^26 + 19*281*2^29
* So |over| will be no more than 2^16. */
{
limb over = div_by_2_26(output[0]);
output[0] -= over << 26;
output[1] += over;
}
/* Now output[0,2..9] are reduced, and |output[1]| < 2^25 + 2^16 < 2^26. The
* bound on |output[1]| is sufficient to meet our needs. */
}
/* A helpful wrapper around fproduct: output = in * in2.
*
* On entry: |in[i]| < 2^27 and |in2[i]| < 2^27.
*
* output must be distinct to both inputs. The output is reduced degree
* (indeed, one need only provide storage for 10 limbs) and |output[i]| < 2^26. */
static void fmul(limb *output, const limb *in, const limb *in2)
{
limb t[19];
fproduct(t, in, in2);
/* |t[i]| < 14*2^54 */
freduce_degree(t);
freduce_coefficients(t);
/* |t[i]| < 2^26 */
memcpy(output, t, sizeof(limb) * 10);
}
/* Square a number: output = in**2
*
* output must be distinct from the input. The inputs are reduced coefficient
* form, the output is not.
*
* output[x] <= 14 * the largest product of the input limbs. */
static void fsquare_inner(limb *output, const limb *in)
{
output[0] = ((limb) ((int32_t) in[0])) * ((int32_t) in[0]);
output[1] = 2 * ((limb) ((int32_t) in[0])) * ((int32_t) in[1]);
output[2] = 2 * (((limb) ((int32_t) in[1])) * ((int32_t) in[1]) +
((limb) ((int32_t) in[0])) * ((int32_t) in[2]));
output[3] = 2 * (((limb) ((int32_t) in[1])) * ((int32_t) in[2]) +
((limb) ((int32_t) in[0])) * ((int32_t) in[3]));
output[4] = ((limb) ((int32_t) in[2])) * ((int32_t) in[2]) +
4 * ((limb) ((int32_t) in[1])) * ((int32_t) in[3]) +
2 * ((limb) ((int32_t) in[0])) * ((int32_t) in[4]);
output[5] = 2 * (((limb) ((int32_t) in[2])) * ((int32_t) in[3]) +
((limb) ((int32_t) in[1])) * ((int32_t) in[4]) +
((limb) ((int32_t) in[0])) * ((int32_t) in[5]));
output[6] = 2 * (((limb) ((int32_t) in[3])) * ((int32_t) in[3]) +
((limb) ((int32_t) in[2])) * ((int32_t) in[4]) +
((limb) ((int32_t) in[0])) * ((int32_t) in[6]) +
2 * ((limb) ((int32_t) in[1])) * ((int32_t) in[5]));
output[7] = 2 * (((limb) ((int32_t) in[3])) * ((int32_t) in[4]) +
((limb) ((int32_t) in[2])) * ((int32_t) in[5]) +
((limb) ((int32_t) in[1])) * ((int32_t) in[6]) +
((limb) ((int32_t) in[0])) * ((int32_t) in[7]));
output[8] = ((limb) ((int32_t) in[4])) * ((int32_t) in[4]) +
2 * (((limb) ((int32_t) in[2])) * ((int32_t) in[6]) +
((limb) ((int32_t) in[0])) * ((int32_t) in[8]) +
2 * (((limb) ((int32_t) in[1])) * ((int32_t) in[7]) +
((limb) ((int32_t) in[3])) * ((int32_t) in[5])));
output[9] = 2 * (((limb) ((int32_t) in[4])) * ((int32_t) in[5]) +
((limb) ((int32_t) in[3])) * ((int32_t) in[6]) +
((limb) ((int32_t) in[2])) * ((int32_t) in[7]) +
((limb) ((int32_t) in[1])) * ((int32_t) in[8]) +
((limb) ((int32_t) in[0])) * ((int32_t) in[9]));
output[10] = 2 * (((limb) ((int32_t) in[5])) * ((int32_t) in[5]) +
((limb) ((int32_t) in[4])) * ((int32_t) in[6]) +
((limb) ((int32_t) in[2])) * ((int32_t) in[8]) +
2 * (((limb) ((int32_t) in[3])) * ((int32_t) in[7]) +
((limb) ((int32_t) in[1])) * ((int32_t) in[9])));
output[11] = 2 * (((limb) ((int32_t) in[5])) * ((int32_t) in[6]) +
((limb) ((int32_t) in[4])) * ((int32_t) in[7]) +
((limb) ((int32_t) in[3])) * ((int32_t) in[8]) +
((limb) ((int32_t) in[2])) * ((int32_t) in[9]));
output[12] = ((limb) ((int32_t) in[6])) * ((int32_t) in[6]) +
2 * (((limb) ((int32_t) in[4])) * ((int32_t) in[8]) +
2 * (((limb) ((int32_t) in[5])) * ((int32_t) in[7]) +
((limb) ((int32_t) in[3])) * ((int32_t) in[9])));
output[13] = 2 * (((limb) ((int32_t) in[6])) * ((int32_t) in[7]) +
((limb) ((int32_t) in[5])) * ((int32_t) in[8]) +
((limb) ((int32_t) in[4])) * ((int32_t) in[9]));
output[14] = 2 * (((limb) ((int32_t) in[7])) * ((int32_t) in[7]) +
((limb) ((int32_t) in[6])) * ((int32_t) in[8]) +
2 * ((limb) ((int32_t) in[5])) * ((int32_t) in[9]));
output[15] = 2 * (((limb) ((int32_t) in[7])) * ((int32_t) in[8]) +
((limb) ((int32_t) in[6])) * ((int32_t) in[9]));
output[16] = ((limb) ((int32_t) in[8])) * ((int32_t) in[8]) +
4 * ((limb) ((int32_t) in[7])) * ((int32_t) in[9]);
output[17] = 2 * ((limb) ((int32_t) in[8])) * ((int32_t) in[9]);
output[18] = 2 * ((limb) ((int32_t) in[9])) * ((int32_t) in[9]);
}
/* fsquare sets output = in^2.
*
* On entry: The |in| argument is in reduced coefficients form and |in[i]| <
* 2^27.
*
* On exit: The |output| argument is in reduced coefficients form (indeed, one
* need only provide storage for 10 limbs) and |out[i]| < 2^26. */
static void fsquare(limb *output, const limb *in)
{
limb t[19];
fsquare_inner(t, in);
/* |t[i]| < 14*2^54 because the largest product of two limbs will be <
* 2^(27+27) and fsquare_inner adds together, at most, 14 of those
* products. */
freduce_degree(t);
freduce_coefficients(t);
/* |t[i]| < 2^26 */
memcpy(output, t, sizeof(limb) * 10);
}
/* Take a little-endian, 32-byte number and expand it into polynomial form */
static void fexpand(limb *output, const uint8_t *input)
{
#define F(n,start,shift,mask) \
output[n] = ((((limb) input[start + 0]) | \
((limb) input[start + 1]) << 8 | \
((limb) input[start + 2]) << 16 | \
((limb) input[start + 3]) << 24) >> shift) & mask;
F(0, 0, 0, 0x3ffffff);
F(1, 3, 2, 0x1ffffff);
F(2, 6, 3, 0x3ffffff);
F(3, 9, 5, 0x1ffffff);
F(4, 12, 6, 0x3ffffff);
F(5, 16, 0, 0x1ffffff);
F(6, 19, 1, 0x3ffffff);
F(7, 22, 3, 0x1ffffff);
F(8, 25, 4, 0x3ffffff);
F(9, 28, 6, 0x1ffffff);
#undef F
}
#if (-32 >> 1) != -16
#error "This code only works when >> does sign-extension on negative numbers"
#endif
/* int32_t_eq returns 0xffffffff iff a == b and zero otherwise. */
static int32_t int32_t_eq(int32_t a, int32_t b)
{
a = ~(a ^ b);
a &= a << 16;
a &= a << 8;
a &= a << 4;
a &= a << 2;
a &= a << 1;
return a >> 31;
}
/* int32_t_gte returns 0xffffffff if a >= b and zero otherwise, where a and b are
* both non-negative. */
static int32_t int32_t_gte(int32_t a, int32_t b)
{
a -= b;
/* a >= 0 iff a >= b. */
return ~(a >> 31);
}
/* Take a fully reduced polynomial form number and contract it into a
* little-endian, 32-byte array.
*
* On entry: |input_limbs[i]| < 2^26 */
static void fcontract(uint8_t *output, limb *input_limbs)
{
int i;
int j;
int32_t input[10];
int32_t mask;
/* |input_limbs[i]| < 2^26, so it's valid to convert to an int32_t. */
for (i = 0; i < 10; i++) {
input[i] = input_limbs[i];
}
for (j = 0; j < 2; ++j) {
for (i = 0; i < 9; ++i) {
if ((i & 1) == 1) {
/* This calculation is a time-invariant way to make input[i]
* non-negative by borrowing from the next-larger limb. */
const int32_t mask = input[i] >> 31;
const int32_t carry = -((input[i] & mask) >> 25);
input[i] = input[i] + (carry << 25);
input[i+1] = input[i+1] - carry;
} else {
const int32_t mask = input[i] >> 31;
const int32_t carry = -((input[i] & mask) >> 26);
input[i] = input[i] + (carry << 26);
input[i+1] = input[i+1] - carry;
}
}
/* There's no greater limb for input[9] to borrow from, but we can multiply
* by 19 and borrow from input[0], which is valid mod 2^255-19. */
{
const int32_t mask = input[9] >> 31;
const int32_t carry = -((input[9] & mask) >> 25);
input[9] = input[9] + (carry << 25);
input[0] = input[0] - (carry * 19);
}
/* After the first iteration, input[1..9] are non-negative and fit within
* 25 or 26 bits, depending on position. However, input[0] may be
* negative. */
}
/* The first borrow-propagation pass above ended with every limb
except (possibly) input[0] non-negative.
If input[0] was negative after the first pass, then it was because of a
carry from input[9]. On entry, input[9] < 2^26 so the carry was, at most,
one, since (2**26-1) >> 25 = 1. Thus input[0] >= -19.
In the second pass, each limb is decreased by at most one. Thus the second
borrow-propagation pass could only have wrapped around to decrease
input[0] again if the first pass left input[0] negative *and* input[1]
through input[9] were all zero. In that case, input[1] is now 2^25 - 1,
and this last borrow-propagation step will leave input[1] non-negative. */
{
const int32_t mask = input[0] >> 31;
const int32_t carry = -((input[0] & mask) >> 26);
input[0] = input[0] + (carry << 26);
input[1] = input[1] - carry;
}
/* All input[i] are now non-negative. However, there might be values between
* 2^25 and 2^26 in a limb which is, nominally, 25 bits wide. */
for (j = 0; j < 2; j++) {
for (i = 0; i < 9; i++) {
if ((i & 1) == 1) {
const int32_t carry = input[i] >> 25;
input[i] &= 0x1ffffff;
input[i+1] += carry;
} else {
const int32_t carry = input[i] >> 26;
input[i] &= 0x3ffffff;
input[i+1] += carry;
}
}
{
const int32_t carry = input[9] >> 25;
input[9] &= 0x1ffffff;
input[0] += 19*carry;
}
}
/* If the first carry-chain pass, just above, ended up with a carry from
* input[9], and that caused input[0] to be out-of-bounds, then input[0] was
* < 2^26 + 2*19, because the carry was, at most, two.
*
* If the second pass carried from input[9] again then input[0] is < 2*19 and
* the input[9] -> input[0] carry didn't push input[0] out of bounds. */
/* It still remains the case that input might be between 2^255-19 and 2^255.
* In this case, input[1..9] must take their maximum value and input[0] must
* be >= (2^255-19) & 0x3ffffff, which is 0x3ffffed. */
mask = int32_t_gte(input[0], 0x3ffffed);
for (i = 1; i < 10; i++) {
if ((i & 1) == 1) {
mask &= int32_t_eq(input[i], 0x1ffffff);
} else {
mask &= int32_t_eq(input[i], 0x3ffffff);
}
}
/* mask is either 0xffffffff (if input >= 2^255-19) and zero otherwise. Thus
* this conditionally subtracts 2^255-19. */
input[0] -= mask & 0x3ffffed;
for (i = 1; i < 10; i++) {
if ((i & 1) == 1) {
input[i] -= mask & 0x1ffffff;
} else {
input[i] -= mask & 0x3ffffff;
}
}
input[1] <<= 2;
input[2] <<= 3;
input[3] <<= 5;
input[4] <<= 6;
input[6] <<= 1;
input[7] <<= 3;
input[8] <<= 4;
input[9] <<= 6;
#define F(i, s) \
output[s+0] |= input[i] & 0xff; \
output[s+1] = (input[i] >> 8) & 0xff; \
output[s+2] = (input[i] >> 16) & 0xff; \
output[s+3] = (input[i] >> 24) & 0xff;
output[0] = 0;
output[16] = 0;
F(0,0);
F(1,3);
F(2,6);
F(3,9);
F(4,12);
F(5,16);
F(6,19);
F(7,22);
F(8,25);
F(9,28);
#undef F
}
/* Input: Q, Q', Q-Q'
* Output: 2Q, Q+Q'
*
* x2 z3: long form
* x3 z3: long form
* x z: short form, destroyed
* xprime zprime: short form, destroyed
* qmqp: short form, preserved
*
* On entry and exit, the absolute value of the limbs of all inputs and outputs
* are < 2^26. */
static void fmonty(limb *x2, limb *z2, /* output 2Q */
limb *x3, limb *z3, /* output Q + Q' */
limb *x, limb *z, /* input Q */
limb *xprime, limb *zprime, /* input Q' */
const limb *qmqp /* input Q - Q' */)
{
limb origx[10], origxprime[10], zzz[19], xx[19], zz[19], xxprime[19],
zzprime[19], zzzprime[19], xxxprime[19];
memcpy(origx, x, 10 * sizeof(limb));
fsum(x, z);
/* |x[i]| < 2^27 */
fdifference(z, origx); /* does x - z */
/* |z[i]| < 2^27 */
memcpy(origxprime, xprime, sizeof(limb) * 10);
fsum(xprime, zprime);
/* |xprime[i]| < 2^27 */
fdifference(zprime, origxprime);
/* |zprime[i]| < 2^27 */
fproduct(xxprime, xprime, z);
/* |xxprime[i]| < 14*2^54: the largest product of two limbs will be <
* 2^(27+27) and fproduct adds together, at most, 14 of those products.
* (Approximating that to 2^58 doesn't work out.) */
fproduct(zzprime, x, zprime);
/* |zzprime[i]| < 14*2^54 */
freduce_degree(xxprime);
freduce_coefficients(xxprime);
/* |xxprime[i]| < 2^26 */
freduce_degree(zzprime);
freduce_coefficients(zzprime);
/* |zzprime[i]| < 2^26 */
memcpy(origxprime, xxprime, sizeof(limb) * 10);
fsum(xxprime, zzprime);
/* |xxprime[i]| < 2^27 */
fdifference(zzprime, origxprime);
/* |zzprime[i]| < 2^27 */
fsquare(xxxprime, xxprime);
/* |xxxprime[i]| < 2^26 */
fsquare(zzzprime, zzprime);
/* |zzzprime[i]| < 2^26 */
fproduct(zzprime, zzzprime, qmqp);
/* |zzprime[i]| < 14*2^52 */
freduce_degree(zzprime);
freduce_coefficients(zzprime);
/* |zzprime[i]| < 2^26 */
memcpy(x3, xxxprime, sizeof(limb) * 10);
memcpy(z3, zzprime, sizeof(limb) * 10);
fsquare(xx, x);
/* |xx[i]| < 2^26 */
fsquare(zz, z);
/* |zz[i]| < 2^26 */
fproduct(x2, xx, zz);
/* |x2[i]| < 14*2^52 */
freduce_degree(x2);
freduce_coefficients(x2);
/* |x2[i]| < 2^26 */
fdifference(zz, xx); // does zz = xx - zz
/* |zz[i]| < 2^27 */
memset(zzz + 10, 0, sizeof(limb) * 9);
fscalar_product(zzz, zz, 121665);
/* |zzz[i]| < 2^(27+17) */
/* No need to call freduce_degree here:
fscalar_product doesn't increase the degree of its input. */
freduce_coefficients(zzz);
/* |zzz[i]| < 2^26 */
fsum(zzz, xx);
/* |zzz[i]| < 2^27 */
fproduct(z2, zz, zzz);
/* |z2[i]| < 14*2^(26+27) */
freduce_degree(z2);
freduce_coefficients(z2);
/* |z2|i| < 2^26 */
}
/* Conditionally swap two reduced-form limb arrays if 'iswap' is 1, but leave
* them unchanged if 'iswap' is 0. Runs in data-invariant time to avoid
* side-channel attacks.
*
* NOTE that this function requires that 'iswap' be 1 or 0; other values give
* wrong results. Also, the two limb arrays must be in reduced-coefficient,
* reduced-degree form: the values in a[10..19] or b[10..19] aren't swapped,
* and all all values in a[0..9],b[0..9] must have magnitude less than
* INT32_MAX. */
static void swap_conditional(limb a[static 19], limb b[static 19], limb iswap)
{
unsigned i;
const int32_t swap = (int32_t) -iswap;
for (i = 0; i < 10; ++i) {
const int32_t x = swap & ( ((int32_t)a[i]) ^ ((int32_t)b[i]) );
a[i] = ((int32_t)a[i]) ^ x;
b[i] = ((int32_t)b[i]) ^ x;
}
}
/* Calculates nQ where Q is the x-coordinate of a point on the curve
*
* resultx/resultz: the x coordinate of the resulting curve point (short form)
* n: a little endian, 32-byte number
* q: a point of the curve (short form) */
static void cmult(limb *resultx, limb *resultz, const uint8_t *n, const limb *q)
{
limb a[19] = {0}, b[19] = {1}, c[19] = {1}, d[19] = {0};
limb *nqpqx = a, *nqpqz = b, *nqx = c, *nqz = d, *t;
limb e[19] = {0}, f[19] = {1}, g[19] = {0}, h[19] = {1};
limb *nqpqx2 = e, *nqpqz2 = f, *nqx2 = g, *nqz2 = h;
unsigned i, j;
memcpy(nqpqx, q, sizeof(limb) * 10);
for (i = 0; i < 32; ++i) {
uint8_t byte = n[31 - i];
for (j = 0; j < 8; ++j) {
const limb bit = byte >> 7;
swap_conditional(nqx, nqpqx, bit);
swap_conditional(nqz, nqpqz, bit);
fmonty(nqx2, nqz2,
nqpqx2, nqpqz2,
nqx, nqz,
nqpqx, nqpqz,
q);
swap_conditional(nqx2, nqpqx2, bit);
swap_conditional(nqz2, nqpqz2, bit);
t = nqx;
nqx = nqx2;
nqx2 = t;
t = nqz;
nqz = nqz2;
nqz2 = t;
t = nqpqx;
nqpqx = nqpqx2;
nqpqx2 = t;
t = nqpqz;
nqpqz = nqpqz2;
nqpqz2 = t;
byte <<= 1;
}
}
memcpy(resultx, nqx, sizeof(limb) * 10);
memcpy(resultz, nqz, sizeof(limb) * 10);
}
static void crecip(limb *out, const limb *z)
{
limb z2[10];
limb z9[10];
limb z11[10];
limb z2_5_0[10];
limb z2_10_0[10];
limb z2_20_0[10];
limb z2_50_0[10];
limb z2_100_0[10];
limb t0[10];
limb t1[10];
int i;
/* 2 */ fsquare(z2,z);
/* 4 */ fsquare(t1,z2);
/* 8 */ fsquare(t0,t1);
/* 9 */ fmul(z9,t0,z);
/* 11 */ fmul(z11,z9,z2);
/* 22 */ fsquare(t0,z11);
/* 2^5 - 2^0 = 31 */ fmul(z2_5_0,t0,z9);
/* 2^6 - 2^1 */ fsquare(t0,z2_5_0);
/* 2^7 - 2^2 */ fsquare(t1,t0);
/* 2^8 - 2^3 */ fsquare(t0,t1);
/* 2^9 - 2^4 */ fsquare(t1,t0);
/* 2^10 - 2^5 */ fsquare(t0,t1);
/* 2^10 - 2^0 */ fmul(z2_10_0,t0,z2_5_0);
/* 2^11 - 2^1 */ fsquare(t0,z2_10_0);
/* 2^12 - 2^2 */ fsquare(t1,t0);
/* 2^20 - 2^10 */ for (i = 2; i < 10; i += 2) { fsquare(t0,t1); fsquare(t1,t0); }
/* 2^20 - 2^0 */ fmul(z2_20_0,t1,z2_10_0);
/* 2^21 - 2^1 */ fsquare(t0,z2_20_0);
/* 2^22 - 2^2 */ fsquare(t1,t0);
/* 2^40 - 2^20 */ for (i = 2; i < 20; i += 2) { fsquare(t0,t1); fsquare(t1,t0); }
/* 2^40 - 2^0 */ fmul(t0,t1,z2_20_0);
/* 2^41 - 2^1 */ fsquare(t1,t0);
/* 2^42 - 2^2 */ fsquare(t0,t1);
/* 2^50 - 2^10 */ for (i = 2; i < 10; i += 2) { fsquare(t1,t0); fsquare(t0,t1); }
/* 2^50 - 2^0 */ fmul(z2_50_0,t0,z2_10_0);
/* 2^51 - 2^1 */ fsquare(t0,z2_50_0);
/* 2^52 - 2^2 */ fsquare(t1,t0);
/* 2^100 - 2^50 */ for (i = 2; i < 50; i += 2) { fsquare(t0,t1); fsquare(t1,t0); }
/* 2^100 - 2^0 */ fmul(z2_100_0,t1,z2_50_0);
/* 2^101 - 2^1 */ fsquare(t1,z2_100_0);
/* 2^102 - 2^2 */ fsquare(t0,t1);
/* 2^200 - 2^100 */ for (i = 2; i < 100; i += 2) { fsquare(t1,t0); fsquare(t0,t1); }
/* 2^200 - 2^0 */ fmul(t1,t0,z2_100_0);
/* 2^201 - 2^1 */ fsquare(t0,t1);
/* 2^202 - 2^2 */ fsquare(t1,t0);
/* 2^250 - 2^50 */ for (i = 2; i < 50; i += 2) { fsquare(t0,t1); fsquare(t1,t0); }
/* 2^250 - 2^0 */ fmul(t0,t1,z2_50_0);
/* 2^251 - 2^1 */ fsquare(t1,t0);
/* 2^252 - 2^2 */ fsquare(t0,t1);
/* 2^253 - 2^3 */ fsquare(t1,t0);
/* 2^254 - 2^4 */ fsquare(t0,t1);
/* 2^255 - 2^5 */ fsquare(t1,t0);
/* 2^255 - 21 */ fmul(out,t1,z11);
}
void curve25519(uint8_t mypublic[static CURVE25519_POINT_SIZE], const uint8_t secret[static CURVE25519_POINT_SIZE], const uint8_t basepoint[static CURVE25519_POINT_SIZE])
{
limb bp[10], x[10], z[11], zmone[10];
uint8_t e[32];
memcpy(e, secret, 32);
curve25519_normalize_secret(e);
fexpand(bp, basepoint);
cmult(x, z, e, bp);
crecip(zmone, z);
fmul(z, x, zmone);
fcontract(mypublic, z);
}
#endif
void curve25519_generate_public(uint8_t pub[static CURVE25519_POINT_SIZE], const uint8_t secret[static CURVE25519_POINT_SIZE])
{
static const uint8_t basepoint[CURVE25519_POINT_SIZE] = { 9 };
curve25519(pub, secret, basepoint);
}
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