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wolfssl/wolfcrypt/src/fe_448.c

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/* fe_448.c
*
* Copyright (C) 2006-2025 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 On Daniel J Bernstein's curve25519 Public Domain ref10 work.
* Small implementation based on Daniel Beer's curve25519 public domain work.
* Reworked for curve448 by Sean Parkinson.
*/
#include <wolfssl/wolfcrypt/libwolfssl_sources.h>
#if defined(HAVE_CURVE448) || defined(HAVE_ED448)
#include <wolfssl/wolfcrypt/fe_448.h>
#ifdef NO_INLINE
#include <wolfssl/wolfcrypt/misc.h>
#else
#define WOLFSSL_MISC_INCLUDED
#include <wolfcrypt/src/misc.c>
#endif
#if defined(CURVE448_SMALL) || defined(ED448_SMALL)
/* Initialize the field element operations.
*/
void fe448_init(void)
{
}
/* Normalize the field element.
* Ensure result is in range: 0..2^448-2^224-2
*
* a [in] Field element in range 0..2^448-1.
*/
void fe448_norm(word8* a)
{
int i;
sword16 c = 0;
sword16 o = 0;
for (i = 0; i < 56; i++) {
c += a[i];
if ((i == 0) || (i == 28))
c += 1;
c >>= 8;
}
for (i = 0; i < 56; i++) {
if ((i == 0) || (i == 28)) o += c;
o += a[i];
a[i] = (word8)o;
o >>= 8;
}
}
/* Copy one field element into another: d = a.
*
* d [in] Destination field element.
* a [in] Source field element.
*/
void fe448_copy(word8* d, const word8* a)
{
int i;
for (i = 0; i < 56; i++) {
d[i] = a[i];
}
}
/* Conditionally swap the elements.
* Constant time implementation.
*
* a [in] First field element.
* b [in] Second field element.
* c [in] Swap when 1. Valid values: 0, 1.
*/
static void fe448_cswap(word8* a, word8* b, int c)
{
int i;
word8 mask = -(word8)c;
word8 t[56];
for (i = 0; i < 56; i++)
t[i] = (a[i] ^ b[i]) & mask;
for (i = 0; i < 56; i++)
a[i] ^= t[i];
for (i = 0; i < 56; i++)
b[i] ^= t[i];
}
/* Add two field elements. r = (a + b) mod (2^448 - 2^224 - 1)
*
* r [in] Field element to hold sum.
* a [in] Field element to add.
* b [in] Field element to add.
*/
void fe448_add(word8* r, const word8* a, const word8* b)
{
int i;
sword16 c = 0;
sword16 o = 0;
for (i = 0; i < 56; i++) {
c += a[i];
c += b[i];
r[i] = (word8)c;
c >>= 8;
}
for (i = 0; i < 56; i++) {
if ((i == 0) || (i == 28)) o += c;
o += r[i];
r[i] = (word8)o;
o >>= 8;
}
}
/* Subtract a field element from another. r = (a - b) mod (2^448 - 2^224 - 1)
*
* r [in] Field element to hold difference.
* a [in] Field element to subtract from.
* b [in] Field element to subtract.
*/
void fe448_sub(word8* r, const word8* a, const word8* b)
{
int i;
sword16 c = 0;
sword16 o = 0;
for (i = 0; i < 56; i++) {
if (i == 28)
c += 0x1fc;
else
c += 0x1fe;
c += a[i];
c -= b[i];
r[i] = (word8)c;
c >>= 8;
}
for (i = 0; i < 56; i++) {
if ((i == 0) || (i == 28)) o += c;
o += r[i];
r[i] = (word8)o;
o >>= 8;
}
}
/* Multiply a field element by 39081. r = (39081 * a) mod (2^448 - 2^224 - 1)
*
* r [in] Field element to hold result.
* a [in] Field element to multiply.
*/
void fe448_mul39081(word8* r, const word8* a)
{
int i;
sword32 c = 0;
sword32 o = 0;
for (i = 0; i < 56; i++) {
c += a[i] * (sword32)39081;
r[i] = (word8)c;
c >>= 8;
}
for (i = 0; i < 56; i++) {
if ((i == 0) || (i == 28)) o += c;
o += r[i];
r[i] = (word8)o;
o >>= 8;
}
}
/* Multiply two field elements. r = (a * b) mod (2^448 - 2^224 - 1)
*
* r [in] Field element to hold result.
* a [in] Field element to multiply.
* b [in] Field element to multiply.
*/
void fe448_mul(word8* r, const word8* a, const word8* b)
{
int i, k;
sword32 c = 0;
sword16 o = 0, cc = 0;
word8 t[112];
for (k = 0; k < 56; k++) {
i = 0;
for (; i <= k; i++) {
c += (sword32)a[i] * b[k - i];
}
t[k] = (word8)c;
c >>= 8;
}
for (; k < 111; k++) {
i = k - 55;
for (; i < 56; i++) {
c += (sword32)a[i] * b[k - i];
}
t[k] = (word8)c;
c >>= 8;
}
t[k] = (word8)c;
for (i = 0; i < 28; i++) {
o += t[i];
o += t[i + 56];
o += t[i + 84];
r[i] = (word8)o;
o >>= 8;
}
for (i = 28; i < 56; i++) {
o += t[i];
o += t[i + 56];
o += t[i + 28];
o += t[i + 56];
r[i] = (word8)o;
o >>= 8;
}
for (i = 0; i < 56; i++) {
if ((i == 0) || (i == 28)) cc += o;
cc += r[i];
r[i] = (word8)cc;
cc >>= 8;
}
}
/* Square a field element. r = (a * a) mod (2^448 - 2^224 - 1)
*
* r [in] Field element to hold result.
* a [in] Field element to square.
*/
void fe448_sqr(word8* r, const word8* a)
{
int i, k;
sword32 c = 0;
sword32 p;
sword16 o = 0, cc = 0;
word8 t[112];
for (k = 0; k < 56; k++) {
i = 0;
for (; i <= k; i++) {
if (k - i < i)
break;
p = (sword32)a[i] * a[k - i];
if (k - i != i)
p *= 2;
c += p;
}
t[k] = (word8)c;
c >>= 8;
}
for (; k < 111; k++) {
i = k - 55;
for (; i < 56; i++) {
if (k - i < i)
break;
p = (sword32)a[i] * a[k - i];
if (k - i != i)
p *= 2;
c += p;
}
t[k] = (word8)c;
c >>= 8;
}
t[k] = (word8)c;
for (i = 0; i < 28; i++) {
o += t[i];
o += t[i + 56];
o += t[i + 84];
r[i] = (word8)o;
o >>= 8;
}
for (i = 28; i < 56; i++) {
o += t[i];
o += t[i + 56];
o += t[i + 28];
o += t[i + 56];
r[i] = (word8)o;
o >>= 8;
}
for (i = 0; i < 56; i++) {
if ((i == 0) || (i == 28)) cc += o;
cc += r[i];
r[i] = (word8)cc;
cc >>= 8;
}
fe448_norm(r);
}
/* Invert the field element. (r * a) mod (2^448 - 2^224 - 1) = 1
* Constant time implementation - using Fermat's little theorem:
* a^(p-1) mod p = 1 => a^(p-2) mod p = 1/a
* For curve448: p - 2 = 2^448 - 2^224 - 3
*
* r [in] Field element to hold result.
* a [in] Field element to invert.
*/
void fe448_invert(word8* r, const word8* a)
{
int i;
word8 t[56];
fe448_sqr(t, a);
fe448_mul(t, t, a);
for (i = 0; i < 221; i++) {
fe448_sqr(t, t);
fe448_mul(t, t, a);
}
fe448_sqr(t, t);
for (i = 0; i < 222; i++) {
fe448_sqr(t, t);
fe448_mul(t, t, a);
}
fe448_sqr(t, t);
fe448_sqr(t, t);
fe448_mul(r, t, a);
}
/* Scalar multiply the point by a number. r = n.a
* Uses Montgomery ladder and only requires the x-ordinate.
*
* r [in] Field element to hold result.
* n [in] Scalar as an array of bytes.
* a [in] Point to multiply - x-ordinate only.
*/
int curve448(byte* r, const byte* n, const byte* a)
{
word8 x1[56];
word8 x2[56] = {1};
word8 z2[56] = {0};
word8 x3[56];
word8 z3[56] = {1};
word8 t0[56];
word8 t1[56];
int i;
unsigned int swap;
fe448_copy(x1, a);
fe448_copy(x3, a);
swap = 0;
for (i = 447; i >= 0; --i) {
unsigned int b = (n[i >> 3] >> (i & 7)) & 1;
swap ^= b;
fe448_cswap(x2, x3, swap);
fe448_cswap(z2, z3, swap);
swap = b;
/* Montgomery Ladder - double and add */
fe448_add(t0, x2, z2);
fe448_add(t1, x3, z3);
fe448_sub(x2, x2, z2);
fe448_sub(x3, x3, z3);
fe448_mul(t1, t1, x2);
fe448_mul(z3, x3, t0);
fe448_sqr(t0, t0);
fe448_sqr(x2, x2);
fe448_add(x3, z3, t1);
fe448_sqr(x3, x3);
fe448_sub(z3, z3, t1);
fe448_sqr(z3, z3);
fe448_mul(z3, z3, x1);
fe448_sub(t1, t0, x2);
fe448_mul(x2, t0, x2);
fe448_mul39081(z2, t1);
fe448_add(z2, t0, z2);
fe448_mul(z2, z2, t1);
}
fe448_cswap(x2, x3, swap);
fe448_cswap(z2, z3, swap);
fe448_invert(z2, z2);
fe448_mul(r, x2, z2);
fe448_norm(r);
return 0;
}
#ifdef HAVE_ED448
/* Check whether field element is not 0.
* Field element must have been normalized before call.
*
* a [in] Field element.
* returns 0 when zero, and any other value otherwise.
*/
int fe448_isnonzero(const word8* a)
{
int i;
byte c = 0;
for (i = 0; i < 56; i++)
c |= a[i];
return c;
}
/* Negates the field element. r = -a mod (2^448 - 2^224 - 1)
* Add 0x200 to each element and subtract 2 from next.
* Top element overflow handled by subtracting 2 from index 0 and 28.
*
* r [in] Field element to hold result.
* a [in] Field element.
*/
void fe448_neg(word8* r, const word8* a)
{
int i;
sword16 c = 0;
sword16 o = 0;
for (i = 0; i < 56; i++) {
if (i == 28)
c += 0x1fc;
else
c += 0x1fe;
c -= a[i];
r[i] = (word8)c;
c >>= 8;
}
for (i = 0; i < 56; i++) {
if ((i == 0) || (i == 28)) o += c;
o += r[i];
r[i] = (word8)o;
o >>= 8;
}
}
/* Raise field element to (p-3) / 4: 2^446 - 2^222 - 1
* Used for calculating y-ordinate from x-ordinate for Ed448.
*
* r [in] Field element to hold result.
* a [in] Field element to exponentiate.
*/
void fe448_pow_2_446_222_1(word8* r, const word8* a)
{
int i;
word8 t[56];
fe448_sqr(t, a);
fe448_mul(t, t, a);
for (i = 0; i < 221; i++) {
fe448_sqr(t, t);
fe448_mul(t, t, a);
}
fe448_sqr(t, t);
for (i = 0; i < 221; i++) {
fe448_sqr(t, t);
fe448_mul(t, t, a);
}
fe448_sqr(t, t);
fe448_mul(r, t, a);
}
/* Constant time, conditional move of b into a.
* a is not changed if the condition is 0.
*
* a A field element.
* b A field element.
* c If 1 then copy and if 0 then don't copy.
*/
void fe448_cmov(word8* a, const word8* b, int c)
{
int i;
word8 m = -(word8)c;
word8 t[56];
for (i = 0; i < 56; i++)
t[i] = m & (a[i] ^ b[i]);
for (i = 0; i < 56; i++)
a[i] ^= t[i];
}
#endif /* HAVE_ED448 */
#elif defined(CURVED448_128BIT)
/* Initialize the field element operations.
*/
void fe448_init(void)
{
}
/* Convert the field element from a byte array to an array of 56-bits.
*
* r [in] Array to encode into.
* b [in] Byte array.
*/
void fe448_from_bytes(sword64* r, const unsigned char* b)
{
r[ 0] = ((sword64) (b[ 0]) << 0)
| ((sword64) (b[ 1]) << 8)
| ((sword64) (b[ 2]) << 16)
| ((sword64) (b[ 3]) << 24)
| ((sword64) (b[ 4]) << 32)
| ((sword64) (b[ 5]) << 40)
| ((sword64) (b[ 6]) << 48);
r[ 1] = ((sword64) (b[ 7]) << 0)
| ((sword64) (b[ 8]) << 8)
| ((sword64) (b[ 9]) << 16)
| ((sword64) (b[10]) << 24)
| ((sword64) (b[11]) << 32)
| ((sword64) (b[12]) << 40)
| ((sword64) (b[13]) << 48);
r[ 2] = ((sword64) (b[14]) << 0)
| ((sword64) (b[15]) << 8)
| ((sword64) (b[16]) << 16)
| ((sword64) (b[17]) << 24)
| ((sword64) (b[18]) << 32)
| ((sword64) (b[19]) << 40)
| ((sword64) (b[20]) << 48);
r[ 3] = ((sword64) (b[21]) << 0)
| ((sword64) (b[22]) << 8)
| ((sword64) (b[23]) << 16)
| ((sword64) (b[24]) << 24)
| ((sword64) (b[25]) << 32)
| ((sword64) (b[26]) << 40)
| ((sword64) (b[27]) << 48);
r[ 4] = ((sword64) (b[28]) << 0)
| ((sword64) (b[29]) << 8)
| ((sword64) (b[30]) << 16)
| ((sword64) (b[31]) << 24)
| ((sword64) (b[32]) << 32)
| ((sword64) (b[33]) << 40)
| ((sword64) (b[34]) << 48);
r[ 5] = ((sword64) (b[35]) << 0)
| ((sword64) (b[36]) << 8)
| ((sword64) (b[37]) << 16)
| ((sword64) (b[38]) << 24)
| ((sword64) (b[39]) << 32)
| ((sword64) (b[40]) << 40)
| ((sword64) (b[41]) << 48);
r[ 6] = ((sword64) (b[42]) << 0)
| ((sword64) (b[43]) << 8)
| ((sword64) (b[44]) << 16)
| ((sword64) (b[45]) << 24)
| ((sword64) (b[46]) << 32)
| ((sword64) (b[47]) << 40)
| ((sword64) (b[48]) << 48);
r[ 7] = ((sword64) (b[49]) << 0)
| ((sword64) (b[50]) << 8)
| ((sword64) (b[51]) << 16)
| ((sword64) (b[52]) << 24)
| ((sword64) (b[53]) << 32)
| ((sword64) (b[54]) << 40)
| ((sword64) (b[55]) << 48);
}
/* Convert the field element to a byte array from an array of 56-bits.
*
* b [in] Byte array.
* a [in] Array to encode into.
*/
void fe448_to_bytes(unsigned char* b, const sword64* a)
{
sword128 t;
/* Mod */
sword64 in0 = a[0];
sword64 in1 = a[1];
sword64 in2 = a[2];
sword64 in3 = a[3];
sword64 in4 = a[4];
sword64 in5 = a[5];
sword64 in6 = a[6];
sword64 in7 = a[7];
sword64 o = in7 >> 56;
in7 -= o << 56;
in0 += o;
in4 += o;
o = (in0 + 1) >> 56;
o = (o + in1) >> 56;
o = (o + in2) >> 56;
o = (o + in3) >> 56;
o = (o + in4 + 1) >> 56;
o = (o + in5) >> 56;
o = (o + in6) >> 56;
o = (o + in7) >> 56;
in0 += o;
in4 += o;
in7 -= o << 56;
o = (in0 >> 56); in1 += o; t = o << 56; in0 -= (sword64)t;
o = (in1 >> 56); in2 += o; t = o << 56; in1 -= (sword64)t;
o = (in2 >> 56); in3 += o; t = o << 56; in2 -= (sword64)t;
o = (in3 >> 56); in4 += o; t = o << 56; in3 -= (sword64)t;
o = (in4 >> 56); in5 += o; t = o << 56; in4 -= (sword64)t;
o = (in5 >> 56); in6 += o; t = o << 56; in5 -= (sword64)t;
o = (in6 >> 56); in7 += o; t = o << 56; in6 -= (sword64)t;
o = (in7 >> 56); in0 += o;
in4 += o; t = o << 56; in7 -= (sword64)t;
/* Output as bytes */
b[ 0] = (byte)(in0 >> 0);
b[ 1] = (byte)(in0 >> 8);
b[ 2] = (byte)(in0 >> 16);
b[ 3] = (byte)(in0 >> 24);
b[ 4] = (byte)(in0 >> 32);
b[ 5] = (byte)(in0 >> 40);
b[ 6] = (byte)(in0 >> 48);
b[ 7] = (byte)(in1 >> 0);
b[ 8] = (byte)(in1 >> 8);
b[ 9] = (byte)(in1 >> 16);
b[10] = (byte)(in1 >> 24);
b[11] = (byte)(in1 >> 32);
b[12] = (byte)(in1 >> 40);
b[13] = (byte)(in1 >> 48);
b[14] = (byte)(in2 >> 0);
b[15] = (byte)(in2 >> 8);
b[16] = (byte)(in2 >> 16);
b[17] = (byte)(in2 >> 24);
b[18] = (byte)(in2 >> 32);
b[19] = (byte)(in2 >> 40);
b[20] = (byte)(in2 >> 48);
b[21] = (byte)(in3 >> 0);
b[22] = (byte)(in3 >> 8);
b[23] = (byte)(in3 >> 16);
b[24] = (byte)(in3 >> 24);
b[25] = (byte)(in3 >> 32);
b[26] = (byte)(in3 >> 40);
b[27] = (byte)(in3 >> 48);
b[28] = (byte)(in4 >> 0);
b[29] = (byte)(in4 >> 8);
b[30] = (byte)(in4 >> 16);
b[31] = (byte)(in4 >> 24);
b[32] = (byte)(in4 >> 32);
b[33] = (byte)(in4 >> 40);
b[34] = (byte)(in4 >> 48);
b[35] = (byte)(in5 >> 0);
b[36] = (byte)(in5 >> 8);
b[37] = (byte)(in5 >> 16);
b[38] = (byte)(in5 >> 24);
b[39] = (byte)(in5 >> 32);
b[40] = (byte)(in5 >> 40);
b[41] = (byte)(in5 >> 48);
b[42] = (byte)(in6 >> 0);
b[43] = (byte)(in6 >> 8);
b[44] = (byte)(in6 >> 16);
b[45] = (byte)(in6 >> 24);
b[46] = (byte)(in6 >> 32);
b[47] = (byte)(in6 >> 40);
b[48] = (byte)(in6 >> 48);
b[49] = (byte)(in7 >> 0);
b[50] = (byte)(in7 >> 8);
b[51] = (byte)(in7 >> 16);
b[52] = (byte)(in7 >> 24);
b[53] = (byte)(in7 >> 32);
b[54] = (byte)(in7 >> 40);
b[55] = (byte)(in7 >> 48);
}
/* Set the field element to 0.
*
* a [in] Field element.
*/
void fe448_1(sword64* a)
{
a[0] = 1;
a[1] = 0;
a[2] = 0;
a[3] = 0;
a[4] = 0;
a[5] = 0;
a[6] = 0;
a[7] = 0;
}
/* Set the field element to 0.
*
* a [in] Field element.
*/
void fe448_0(sword64* a)
{
a[0] = 0;
a[1] = 0;
a[2] = 0;
a[3] = 0;
a[4] = 0;
a[5] = 0;
a[6] = 0;
a[7] = 0;
}
/* Copy one field element into another: d = a.
*
* d [in] Destination field element.
* a [in] Source field element.
*/
void fe448_copy(sword64* d, const sword64* a)
{
d[0] = a[0];
d[1] = a[1];
d[2] = a[2];
d[3] = a[3];
d[4] = a[4];
d[5] = a[5];
d[6] = a[6];
d[7] = a[7];
}
/* Conditionally swap the elements.
* Constant time implementation.
*
* a [in] First field element.
* b [in] Second field element.
* c [in] Swap when 1. Valid values: 0, 1.
*/
static void fe448_cswap(sword64* a, sword64* b, int c)
{
sword64 mask = -(sword64)c;
sword64 t0 = (a[0] ^ b[0]) & mask;
sword64 t1 = (a[1] ^ b[1]) & mask;
sword64 t2 = (a[2] ^ b[2]) & mask;
sword64 t3 = (a[3] ^ b[3]) & mask;
sword64 t4 = (a[4] ^ b[4]) & mask;
sword64 t5 = (a[5] ^ b[5]) & mask;
sword64 t6 = (a[6] ^ b[6]) & mask;
sword64 t7 = (a[7] ^ b[7]) & mask;
a[0] ^= t0;
a[1] ^= t1;
a[2] ^= t2;
a[3] ^= t3;
a[4] ^= t4;
a[5] ^= t5;
a[6] ^= t6;
a[7] ^= t7;
b[0] ^= t0;
b[1] ^= t1;
b[2] ^= t2;
b[3] ^= t3;
b[4] ^= t4;
b[5] ^= t5;
b[6] ^= t6;
b[7] ^= t7;
}
/* Add two field elements. r = (a + b) mod (2^448 - 2^224 - 1)
*
* r [in] Field element to hold sum.
* a [in] Field element to add.
* b [in] Field element to add.
*/
void fe448_add(sword64* r, const sword64* a, const sword64* b)
{
r[0] = a[0] + b[0];
r[1] = a[1] + b[1];
r[2] = a[2] + b[2];
r[3] = a[3] + b[3];
r[4] = a[4] + b[4];
r[5] = a[5] + b[5];
r[6] = a[6] + b[6];
r[7] = a[7] + b[7];
}
/* Subtract a field element from another. r = (a - b) mod (2^448 - 2^224 - 1)
*
* r [in] Field element to hold difference.
* a [in] Field element to subtract from.
* b [in] Field element to subtract.
*/
void fe448_sub(sword64* r, const sword64* a, const sword64* b)
{
r[0] = a[0] - b[0];
r[1] = a[1] - b[1];
r[2] = a[2] - b[2];
r[3] = a[3] - b[3];
r[4] = a[4] - b[4];
r[5] = a[5] - b[5];
r[6] = a[6] - b[6];
r[7] = a[7] - b[7];
}
/* Multiply a field element by 39081. r = (39081 * a) mod (2^448 - 2^224 - 1)
*
* r [in] Field element to hold result.
* a [in] Field element to multiply.
*/
void fe448_mul39081(sword64* r, const sword64* a)
{
sword128 t;
sword64 o;
sword128 t0 = a[0] * (sword128)39081;
sword128 t1 = a[1] * (sword128)39081;
sword128 t2 = a[2] * (sword128)39081;
sword128 t3 = a[3] * (sword128)39081;
sword128 t4 = a[4] * (sword128)39081;
sword128 t5 = a[5] * (sword128)39081;
sword128 t6 = a[6] * (sword128)39081;
sword128 t7 = a[7] * (sword128)39081;
o = (sword64)(t0 >> 56); t1 += o; t = (sword128)o << 56; t0 -= t;
o = (sword64)(t1 >> 56); t2 += o; t = (sword128)o << 56; t1 -= t;
o = (sword64)(t2 >> 56); t3 += o; t = (sword128)o << 56; t2 -= t;
o = (sword64)(t3 >> 56); t4 += o; t = (sword128)o << 56; t3 -= t;
o = (sword64)(t4 >> 56); t5 += o; t = (sword128)o << 56; t4 -= t;
o = (sword64)(t5 >> 56); t6 += o; t = (sword128)o << 56; t5 -= t;
o = (sword64)(t6 >> 56); t7 += o; t = (sword128)o << 56; t6 -= t;
o = (sword64)(t7 >> 56); t0 += o;
t4 += o; t = (sword128)o << 56; t7 -= t;
/* Store */
r[0] = (sword64)t0;
r[1] = (sword64)t1;
r[2] = (sword64)t2;
r[3] = (sword64)t3;
r[4] = (sword64)t4;
r[5] = (sword64)t5;
r[6] = (sword64)t6;
r[7] = (sword64)t7;
}
/* Multiply two field elements. r = (a * b) mod (2^448 - 2^224 - 1)
*
* r [in] Field element to hold result.
* a [in] Field element to multiply.
* b [in] Field element to multiply.
*/
void fe448_mul(sword64* r, const sword64* a, const sword64* b)
{
sword64 o;
sword64 a1[4];
sword64 b1[4];
sword128 t0;
sword128 t1;
sword128 t2;
sword128 t3;
sword128 t4;
sword128 t5;
sword128 t6;
sword128 t7;
sword128 t03;
sword128 t04;
sword128 t13;
sword128 t14;
a1[0] = a[0] + a[4];
a1[1] = a[1] + a[5];
a1[2] = a[2] + a[6];
a1[3] = a[3] + a[7];
b1[0] = b[0] + b[4];
b1[1] = b[1] + b[5];
b1[2] = b[2] + b[6];
b1[3] = b[3] + b[7];
t03 = ((sword128)a[0] * b[3]) + ((sword128)a[1] * b[2])
+ ((sword128)a[2] * b[1]) + ((sword128)a[3] * b[0]);
t04 = ((sword128)a[1] * b[3]) + ((sword128)a[2] * b[2])
+ ((sword128)a[3] * b[1]);
t04 += t03 >> 56;
t03 &= 0xffffffffffffffL;
t13 = ((sword128)a1[0] * b1[3]) + ((sword128)a1[1] * b1[2])
+ ((sword128)a1[2] * b1[1]) + ((sword128)a1[3] * b1[0]);
t14 = ((sword128)a1[1] * b1[3]) + ((sword128)a1[2] * b1[2])
+ ((sword128)a1[3] * b1[1]);
t14 += t13 >> 56;
t13 &= 0xffffffffffffffL;
t0 = ((sword128)a[0] * b[0]) + ((sword128)a[4] * b[4]) + t14 + -t04;
t1 = ((sword128)a[0] * b[1]) + ((sword128)a[1] * b[0])
+ ((sword128)a[4] * b[5]) + ((sword128)a[5] * b[4])
+ ((sword128)a1[2] * b1[3]) + ((sword128)a1[3] * b1[2])
- ((sword128)a[2] * b[3]) - ((sword128)a[3] * b[2]);
o = (sword64)(t0 >> 56); t1 += o; t0 &= 0xffffffffffffffL;
t2 = ((sword128)a[0] * b[2]) + ((sword128)a[1] * b[1])
+ ((sword128)a[2] * b[0]) + ((sword128)a[4] * b[6])
+ ((sword128)a[5] * b[5]) + ((sword128)a[6] * b[4])
+ ((sword128)a1[3] * b1[3]) - ((sword128)a[3] * b[3]);
o = (sword64)(t1 >> 56); t2 += o; t1 &= 0xffffffffffffffL;
t3 = t03 + ((sword128)a[4] * b[7]) + ((sword128)a[5] * b[6])
+ ((sword128)a[6] * b[5]) + ((sword128)a[7] * b[4]);
o = (sword64)(t2 >> 56); t3 += o; t2 &= 0xffffffffffffffL;
t4 = ((sword128)a[5] * b[7]) + ((sword128)a[6] * b[6])
+ ((sword128)a[7] * b[5]) + ((sword128)a1[0] * b1[0])
- ((sword128)a[0] * b[0]) + t14;
o = (sword64)(t3 >> 56); t4 += o; t3 &= 0xffffffffffffffL;
t5 = ((sword128)a[6] * b[7]) + ((sword128)a[7] * b[6])
+ ((sword128)a1[0] * b1[1]) + ((sword128)a1[1] * b1[0])
- ((sword128)a[0] * b[1]) - ((sword128)a[1] * b[0])
+ ((sword128)a1[2] * b1[3]) + ((sword128)a1[3] * b1[2]);
o = (sword64)(t4 >> 56); t5 += o; t4 &= 0xffffffffffffffL;
t6 = ((sword128)a[7] * b[7]) + ((sword128)a1[0] * b1[2])
+ ((sword128)a1[1] * b1[1]) + ((sword128)a1[2] * b1[0])
- ((sword128)a[0] * b[2]) - ((sword128)a[1] * b[1])
- ((sword128)a[2] * b[0]) + ((sword128)a1[3] * b1[3]);
o = (sword64)(t5 >> 56); t6 += o; t5 &= 0xffffffffffffffL;
t7 = t13 + -t03;
o = (sword64)(t6 >> 56); t7 += o; t6 &= 0xffffffffffffffL;
o = (sword64)(t7 >> 56); t0 += o;
t4 += o; t7 &= 0xffffffffffffffL;
/* Store */
r[0] = (sword64)t0;
r[1] = (sword64)t1;
r[2] = (sword64)t2;
r[3] = (sword64)t3;
r[4] = (sword64)t4;
r[5] = (sword64)t5;
r[6] = (sword64)t6;
r[7] = (sword64)t7;
}
/* Square a field element. r = (a * a) mod (2^448 - 2^224 - 1)
*
* r [in] Field element to hold result.
* a [in] Field element to square.
*/
void fe448_sqr(sword64* r, const sword64* a)
{
sword64 o;
sword64 a1[4];
sword128 t0;
sword128 t1;
sword128 t2;
sword128 t3;
sword128 t4;
sword128 t5;
sword128 t6;
sword128 t7;
sword128 t03;
sword128 t04;
sword128 t13;
sword128 t14;
a1[0] = a[0] + a[4];
a1[1] = a[1] + a[5];
a1[2] = a[2] + a[6];
a1[3] = a[3] + a[7];
t03 = ((sword128)a[0] * (2 * a[3])) + ((sword128)a[1] * (2 * a[2]));
t04 = ((sword128)a[1] * (2 * a[3])) + ((sword128)a[2] * a[2]);
t04 += t03 >> 56;
t03 &= 0xffffffffffffffL;
t13 = ((sword128)a1[0] * (2 * a1[3])) + ((sword128)a1[1] * (2 * a1[2]));
t14 = ((sword128)a1[1] * (2 * a1[3])) + ((sword128)a1[2] * a1[2]);
t14 += t13 >> 56;
t13 &= 0xffffffffffffffL;
t0 = ((sword128)a[0] * a[0]) + ((sword128)a[4] * a[4]) + t14 + -t04;
t1 = ((sword128)a[0] * (2 * a[1])) + ((sword128)a[4] * (2 * a[5]))
+ ((sword128)a1[2] * (2 * a1[3])) - ((sword128)a[2] * (2 * a[3]));
o = (sword64)(t0 >> 56); t1 += o; t0 &= 0xffffffffffffffL;
t2 = ((sword128)a[0] * (2 * a[2])) + ((sword128)a[1] * a[1])
+ ((sword128)a[4] * (2 * a[6])) + ((sword128)a[5] * a[5])
+ ((sword128)a1[3] * a1[3]) - ((sword128)a[3] * a[3]);
o = (sword64)(t1 >> 56); t2 += o; t1 &= 0xffffffffffffffL;
t3 = t03 + ((sword128)a[4] * (2 * a[7])) + ((sword128)a[5] * (2 * a[6]));
o = (sword64)(t2 >> 56); t3 += o; t2 &= 0xffffffffffffffL;
t4 = ((sword128)a[5] * (2 * a[7])) + ((sword128)a[6] * a[6])
+ ((sword128)a1[0] * a1[0]) - ((sword128)a[0] * a[0]) + t14;
o = (sword64)(t3 >> 56); t4 += o; t3 &= 0xffffffffffffffL;
t5 = ((sword128)a[6] * (2 * a[7])) + ((sword128)a1[0] * (2 * a1[1]))
- ((sword128)a[0] * (2 * a[1])) + ((sword128)a1[2] * (2 * a1[3]));
o = (sword64)(t4 >> 56); t5 += o; t4 &= 0xffffffffffffffL;
t6 = ((sword128)a[7] * a[7]) + ((sword128)a1[0] * (2 * a1[2]))
+ ((sword128)a1[1] * a1[1]) - ((sword128)a[0] * (2 * a[2]))
- ((sword128)a[1] * a[1]) + ((sword128)a1[3] * a1[3]);
o = (sword64)(t5 >> 56); t6 += o; t5 &= 0xffffffffffffffL;
t7 = t13 + -t03;
o = (sword64)(t6 >> 56); t7 += o; t6 &= 0xffffffffffffffL;
o = (sword64)(t7 >> 56); t0 += o;
t4 += o; t7 &= 0xffffffffffffffL;
/* Store */
r[0] = (sword64)t0;
r[1] = (sword64)t1;
r[2] = (sword64)t2;
r[3] = (sword64)t3;
r[4] = (sword64)t4;
r[5] = (sword64)t5;
r[6] = (sword64)t6;
r[7] = (sword64)t7;
}
/* Invert the field element. (r * a) mod (2^448 - 2^224 - 1) = 1
* Constant time implementation - using Fermat's little theorem:
* a^(p-1) mod p = 1 => a^(p-2) mod p = 1/a
* For curve448: p - 2 = 2^448 - 2^224 - 3
*
* r [in] Field element to hold result.
* a [in] Field element to invert.
*/
void fe448_invert(sword64* r, const sword64* a)
{
sword64 t1[8];
sword64 t2[8];
sword64 t3[8];
sword64 t4[8];
int i;
fe448_sqr(t1, a);
/* t1 = 2 */
fe448_mul(t1, t1, a);
/* t1 = 3 */
fe448_sqr(t2, t1); for (i = 1; i < 2; ++i) fe448_sqr(t2, t2);
/* t2 = c */
fe448_mul(t3, t2, a);
/* t3 = d */
fe448_mul(t1, t2, t1);
/* t1 = f */
fe448_sqr(t2, t1);
/* t2 = 1e */
fe448_mul(t4, t2, a);
/* t4 = 1f */
fe448_sqr(t2, t4); for (i = 1; i < 5; ++i) fe448_sqr(t2, t2);
/* t2 = 3e0 */
fe448_mul(t1, t2, t4);
/* t1 = 3ff */
fe448_sqr(t2, t1); for (i = 1; i < 10; ++i) fe448_sqr(t2, t2);
/* t2 = ffc00 */
fe448_mul(t1, t2, t1);
/* t1 = fffff */
fe448_sqr(t2, t1); for (i = 1; i < 5; ++i) fe448_sqr(t2, t2);
/* t2 = 1ffffe0 */
fe448_mul(t1, t2, t4);
/* t1 = 1ffffff */
fe448_sqr(t2, t1); for (i = 1; i < 25; ++i) fe448_sqr(t2, t2);
/* t2 = 3fffffe000000 */
fe448_mul(t1, t2, t1);
/* t1 = 3ffffffffffff */
fe448_sqr(t2, t1); for (i = 1; i < 5; ++i) fe448_sqr(t2, t2);
/* t2 = 7fffffffffffe0 */
fe448_mul(t1, t2, t4);
/* t1 = 7fffffffffffff */
fe448_sqr(t2, t1); for (i = 1; i < 55; ++i) fe448_sqr(t2, t2);
/* t2 = 3fffffffffffff80000000000000 */
fe448_mul(t1, t2, t1);
/* t1 = 3fffffffffffffffffffffffffff */
fe448_sqr(t2, t1); for (i = 1; i < 110; ++i) fe448_sqr(t2, t2);
/* t2 = fffffffffffffffffffffffffffc000000000000000000000000000 */
fe448_mul(t1, t2, t1);
/* t1 = fffffffffffffffffffffffffffffffffffffffffffffffffffffff */
fe448_sqr(t2, t1); for (i = 1; i < 4; ++i) fe448_sqr(t2, t2);
/* t2 = fffffffffffffffffffffffffffffffffffffffffffffffffffffff0 */
fe448_mul(t3, t3, t2);
/* t3 = fffffffffffffffffffffffffffffffffffffffffffffffffffffffd */
fe448_mul(t1, t3, a);
/* t1 = fffffffffffffffffffffffffffffffffffffffffffffffffffffffe */
fe448_sqr(t1, t1); for (i = 1; i < 224; ++i) fe448_sqr(t1, t1);
/* t1 = fffffffffffffffffffffffffffffffffffffffffffffffffffffffe00000000000000000000000000000000000000000000000000000000 */
fe448_mul(r, t3, t1);
/* r = fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffffffffffffffffffffffffffffffffffffffffffffffffffffd */
}
/* Scalar multiply the point by a number. r = n.a
* Uses Montgomery ladder and only requires the x-ordinate.
*
* r [in] Field element to hold result.
* n [in] Scalar as an array of bytes.
* a [in] Point to multiply - x-ordinate only.
*/
int curve448(byte* r, const byte* n, const byte* a)
{
sword64 x1[8];
sword64 x2[8];
sword64 z2[8];
sword64 x3[8];
sword64 z3[8];
sword64 t0[8];
sword64 t1[8];
int i;
unsigned int swap;
fe448_from_bytes(x1, a);
fe448_1(x2);
fe448_0(z2);
fe448_copy(x3, x1);
fe448_1(z3);
swap = 0;
for (i = 447; i >= 0; --i) {
unsigned int b = (n[i >> 3] >> (i & 7)) & 1;
swap ^= b;
fe448_cswap(x2, x3, (int)swap);
fe448_cswap(z2, z3, (int)swap);
swap = b;
/* Montgomery Ladder - double and add */
fe448_add(t0, x2, z2);
fe448_reduce(t0);
fe448_add(t1, x3, z3);
fe448_reduce(t1);
fe448_sub(x2, x2, z2);
fe448_sub(x3, x3, z3);
fe448_mul(t1, t1, x2);
fe448_mul(z3, x3, t0);
fe448_sqr(t0, t0);
fe448_sqr(x2, x2);
fe448_add(x3, z3, t1);
fe448_reduce(x3);
fe448_sqr(x3, x3);
fe448_sub(z3, z3, t1);
fe448_sqr(z3, z3);
fe448_mul(z3, z3, x1);
fe448_sub(t1, t0, x2);
fe448_mul(x2, t0, x2);
fe448_mul39081(z2, t1);
fe448_add(z2, t0, z2);
fe448_mul(z2, z2, t1);
}
/* Last two bits are 0 - no final swap check required. */
fe448_invert(z2, z2);
fe448_mul(x2, x2, z2);
fe448_to_bytes(r, x2);
return 0;
}
#ifdef HAVE_ED448
/* Check whether field element is not 0.
* Must convert to a normalized form before checking.
*
* a [in] Field element.
* returns 0 when zero, and any other value otherwise.
*/
int fe448_isnonzero(const sword64* a)
{
byte b[56];
int i;
byte c = 0;
fe448_to_bytes(b, a);
for (i = 0; i < 56; i++)
c |= b[i];
return c;
}
/* Check whether field element is negative.
* Must convert to a normalized form before checking.
*
* a [in] Field element.
* returns 1 when negative, and 0 otherwise.
*/
int fe448_isnegative(const sword64* a)
{
byte b[56];
fe448_to_bytes(b, a);
return b[0] & 1;
}
/* Negates the field element. r = -a
*
* r [in] Field element to hold result.
* a [in] Field element.
*/
void fe448_neg(sword64* r, const sword64* a)
{
r[0] = -a[0];
r[1] = -a[1];
r[2] = -a[2];
r[3] = -a[3];
r[4] = -a[4];
r[5] = -a[5];
r[6] = -a[6];
r[7] = -a[7];
}
/* Raise field element to (p-3) / 4: 2^446 - 2^222 - 1
* Used for calculating y-ordinate from x-ordinate for Ed448.
*
* r [in] Field element to hold result.
* a [in] Field element to exponentiate.
*/
void fe448_pow_2_446_222_1(sword64* r, const sword64* a)
{
sword64 t1[8];
sword64 t2[8];
sword64 t3[8];
sword64 t4[8];
sword64 t5[8];
int i;
fe448_sqr(t3, a);
/* t3 = 2 */
fe448_mul(t1, t3, a);
/* t1 = 3 */
fe448_sqr(t5, t1);
/* t5 = 6 */
fe448_mul(t5, t5, a);
/* t5 = 7 */
fe448_sqr(t2, t1); for (i = 1; i < 2; ++i) fe448_sqr(t2, t2);
/* t2 = c */
fe448_mul(t3, t2, t3);
/* t3 = e */
fe448_mul(t1, t2, t1);
/* t1 = f */
fe448_sqr(t2, t1); for (i = 1; i < 3; ++i) fe448_sqr(t2, t2);
/* t2 = 78 */
fe448_mul(t5, t2, t5);
/* t5 = 7f */
fe448_sqr(t2, t1); for (i = 1; i < 4; ++i) fe448_sqr(t2, t2);
/* t2 = f0 */
fe448_mul(t1, t2, t1);
/* t1 = ff */
fe448_mul(t3, t3, t2);
/* t3 = fe */
fe448_sqr(t2, t1); for (i = 1; i < 7; ++i) fe448_sqr(t2, t2);
/* t2 = 7f80 */
fe448_mul(t5, t2, t5);
/* t5 = 7fff */
fe448_sqr(t2, t1); for (i = 1; i < 8; ++i) fe448_sqr(t2, t2);
/* t2 = ff00 */
fe448_mul(t1, t2, t1);
/* t1 = ffff */
fe448_mul(t3, t3, t2);
/* t3 = fffe */
fe448_sqr(t2, t5); for (i = 1; i < 15; ++i) fe448_sqr(t2, t2);
/* t2 = 3fff8000 */
fe448_mul(t5, t2, t5);
/* t5 = 3fffffff */
fe448_sqr(t2, t1); for (i = 1; i < 16; ++i) fe448_sqr(t2, t2);
/* t2 = ffff0000 */
fe448_mul(t1, t2, t1);
/* t1 = ffffffff */
fe448_mul(t3, t3, t2);
/* t3 = fffffffe */
fe448_sqr(t2, t1); for (i = 1; i < 32; ++i) fe448_sqr(t2, t2);
/* t2 = ffffffff00000000 */
fe448_mul(t2, t2, t1);
/* t2 = ffffffffffffffff */
fe448_sqr(t1, t2); for (i = 1; i < 64; ++i) fe448_sqr(t1, t1);
/* t1 = ffffffffffffffff0000000000000000 */
fe448_mul(t1, t1, t2);
/* t1 = ffffffffffffffffffffffffffffffff */
fe448_sqr(t1, t1); for (i = 1; i < 64; ++i) fe448_sqr(t1, t1);
/* t1 = ffffffffffffffffffffffffffffffff0000000000000000 */
fe448_mul(t4, t1, t2);
/* t4 = ffffffffffffffffffffffffffffffffffffffffffffffff */
fe448_sqr(t2, t4); for (i = 1; i < 32; ++i) fe448_sqr(t2, t2);
/* t2 = ffffffffffffffffffffffffffffffffffffffffffffffff00000000 */
fe448_mul(t3, t3, t2);
/* t3 = fffffffffffffffffffffffffffffffffffffffffffffffffffffffe */
fe448_sqr(t1, t3); for (i = 1; i < 192; ++i) fe448_sqr(t1, t1);
/* t1 = fffffffffffffffffffffffffffffffffffffffffffffffffffffffe000000000000000000000000000000000000000000000000 */
fe448_mul(t1, t1, t4);
/* t1 = fffffffffffffffffffffffffffffffffffffffffffffffffffffffeffffffffffffffffffffffffffffffffffffffffffffffff */
fe448_sqr(t1, t1); for (i = 1; i < 30; ++i) fe448_sqr(t1, t1);
/* t1 = 3fffffffffffffffffffffffffffffffffffffffffffffffffffffffbfffffffffffffffffffffffffffffffffffffffffffffffc0000000 */
fe448_mul(r, t5, t1);
/* r = 3fffffffffffffffffffffffffffffffffffffffffffffffffffffffbfffffffffffffffffffffffffffffffffffffffffffffffffffffff */
}
/* Constant time, conditional move of b into a.
* a is not changed if the condition is 0.
*
* a A field element.
* b A field element.
* c If 1 then copy and if 0 then don't copy.
*/
void fe448_cmov(sword64* a, const sword64* b, int c)
{
sword64 m = -(sword64)c;
sword64 t0 = m & (a[0] ^ b[0]);
sword64 t1 = m & (a[1] ^ b[1]);
sword64 t2 = m & (a[2] ^ b[2]);
sword64 t3 = m & (a[3] ^ b[3]);
sword64 t4 = m & (a[4] ^ b[4]);
sword64 t5 = m & (a[5] ^ b[5]);
sword64 t6 = m & (a[6] ^ b[6]);
sword64 t7 = m & (a[7] ^ b[7]);
a[0] ^= t0;
a[1] ^= t1;
a[2] ^= t2;
a[3] ^= t3;
a[4] ^= t4;
a[5] ^= t5;
a[6] ^= t6;
a[7] ^= t7;
}
#endif /* HAVE_ED448 */
#else
/* Initialize the field element operations.
*/
void fe448_init(void)
{
}
/* Convert the field element from a byte array to an array of 28-bits.
*
* r [in] Array to encode into.
* b [in] Byte array.
*/
void fe448_from_bytes(sword32* r, const unsigned char* b)
{
r[ 0] = (((sword32)((b[ 0] ) >> 0)) << 0)
| (((sword32)((b[ 1] ) >> 0)) << 8)
| (((sword32)((b[ 2] ) >> 0)) << 16)
| ((((sword32)((b[ 3] & 0xf )) >> 0)) << 24);
r[ 1] = (((sword32)((b[ 3] ) >> 4)) << 0)
| (((sword32)((b[ 4] ) >> 0)) << 4)
| (((sword32)((b[ 5] ) >> 0)) << 12)
| (((sword32)((b[ 6] ) >> 0)) << 20);
r[ 2] = (((sword32)((b[ 7] ) >> 0)) << 0)
| (((sword32)((b[ 8] ) >> 0)) << 8)
| (((sword32)((b[ 9] ) >> 0)) << 16)
| ((((sword32)((b[10] & 0xf )) >> 0)) << 24);
r[ 3] = (((sword32)((b[10] ) >> 4)) << 0)
| (((sword32)((b[11] ) >> 0)) << 4)
| (((sword32)((b[12] ) >> 0)) << 12)
| (((sword32)((b[13] ) >> 0)) << 20);
r[ 4] = (((sword32)((b[14] ) >> 0)) << 0)
| (((sword32)((b[15] ) >> 0)) << 8)
| (((sword32)((b[16] ) >> 0)) << 16)
| ((((sword32)((b[17] & 0xf )) >> 0)) << 24);
r[ 5] = (((sword32)((b[17] ) >> 4)) << 0)
| (((sword32)((b[18] ) >> 0)) << 4)
| (((sword32)((b[19] ) >> 0)) << 12)
| (((sword32)((b[20] ) >> 0)) << 20);
r[ 6] = (((sword32)((b[21] ) >> 0)) << 0)
| (((sword32)((b[22] ) >> 0)) << 8)
| (((sword32)((b[23] ) >> 0)) << 16)
| ((((sword32)((b[24] & 0xf )) >> 0)) << 24);
r[ 7] = (((sword32)((b[24] ) >> 4)) << 0)
| (((sword32)((b[25] ) >> 0)) << 4)
| (((sword32)((b[26] ) >> 0)) << 12)
| (((sword32)((b[27] ) >> 0)) << 20);
r[ 8] = (((sword32)((b[28] ) >> 0)) << 0)
| (((sword32)((b[29] ) >> 0)) << 8)
| (((sword32)((b[30] ) >> 0)) << 16)
| ((((sword32)((b[31] & 0xf )) >> 0)) << 24);
r[ 9] = (((sword32)((b[31] ) >> 4)) << 0)
| (((sword32)((b[32] ) >> 0)) << 4)
| (((sword32)((b[33] ) >> 0)) << 12)
| (((sword32)((b[34] ) >> 0)) << 20);
r[10] = (((sword32)((b[35] ) >> 0)) << 0)
| (((sword32)((b[36] ) >> 0)) << 8)
| (((sword32)((b[37] ) >> 0)) << 16)
| ((((sword32)((b[38] & 0xf )) >> 0)) << 24);
r[11] = (((sword32)((b[38] ) >> 4)) << 0)
| (((sword32)((b[39] ) >> 0)) << 4)
| (((sword32)((b[40] ) >> 0)) << 12)
| (((sword32)((b[41] ) >> 0)) << 20);
r[12] = (((sword32)((b[42] ) >> 0)) << 0)
| (((sword32)((b[43] ) >> 0)) << 8)
| (((sword32)((b[44] ) >> 0)) << 16)
| ((((sword32)((b[45] & 0xf )) >> 0)) << 24);
r[13] = (((sword32)((b[45] ) >> 4)) << 0)
| (((sword32)((b[46] ) >> 0)) << 4)
| (((sword32)((b[47] ) >> 0)) << 12)
| (((sword32)((b[48] ) >> 0)) << 20);
r[14] = (((sword32)((b[49] ) >> 0)) << 0)
| (((sword32)((b[50] ) >> 0)) << 8)
| (((sword32)((b[51] ) >> 0)) << 16)
| ((((sword32)((b[52] & 0xf )) >> 0)) << 24);
r[15] = (((sword32)((b[52] ) >> 4)) << 0)
| (((sword32)((b[53] ) >> 0)) << 4)
| (((sword32)((b[54] ) >> 0)) << 12)
| (((sword32)((b[55] ) >> 0)) << 20);
}
/* Convert the field element to a byte array from an array of 28-bits.
*
* b [in] Byte array.
* a [in] Array to encode into.
*/
void fe448_to_bytes(unsigned char* b, const sword32* a)
{
sword64 t;
/* Mod */
sword32 in0 = a[0];
sword32 in1 = a[1];
sword32 in2 = a[2];
sword32 in3 = a[3];
sword32 in4 = a[4];
sword32 in5 = a[5];
sword32 in6 = a[6];
sword32 in7 = a[7];
sword32 in8 = a[8];
sword32 in9 = a[9];
sword32 in10 = a[10];
sword32 in11 = a[11];
sword32 in12 = a[12];
sword32 in13 = a[13];
sword32 in14 = a[14];
sword32 in15 = a[15];
sword32 o = in15 >> 28;
in15 -= o << 28;
in0 += o;
in8 += o;
o = (in0 + 1) >> 28;
o = (o + in1) >> 28;
o = (o + in2) >> 28;
o = (o + in3) >> 28;
o = (o + in4) >> 28;
o = (o + in5) >> 28;
o = (o + in6) >> 28;
o = (o + in7) >> 28;
o = (o + in8 + 1) >> 28;
o = (o + in9) >> 28;
o = (o + in10) >> 28;
o = (o + in11) >> 28;
o = (o + in12) >> 28;
o = (o + in13) >> 28;
o = (o + in14) >> 28;
o = (o + in15) >> 28;
in0 += o;
in8 += o;
in15 -= o << 28;
o = (in0 >> 28); in1 += o; t = o << 28; in0 -= (sword32)t;
o = (in1 >> 28); in2 += o; t = o << 28; in1 -= (sword32)t;
o = (in2 >> 28); in3 += o; t = o << 28; in2 -= (sword32)t;
o = (in3 >> 28); in4 += o; t = o << 28; in3 -= (sword32)t;
o = (in4 >> 28); in5 += o; t = o << 28; in4 -= (sword32)t;
o = (in5 >> 28); in6 += o; t = o << 28; in5 -= (sword32)t;
o = (in6 >> 28); in7 += o; t = o << 28; in6 -= (sword32)t;
o = (in7 >> 28); in8 += o; t = o << 28; in7 -= (sword32)t;
o = (in8 >> 28); in9 += o; t = o << 28; in8 -= (sword32)t;
o = (in9 >> 28); in10 += o; t = o << 28; in9 -= (sword32)t;
o = (in10 >> 28); in11 += o; t = o << 28; in10 -= (sword32)t;
o = (in11 >> 28); in12 += o; t = o << 28; in11 -= (sword32)t;
o = (in12 >> 28); in13 += o; t = o << 28; in12 -= (sword32)t;
o = (in13 >> 28); in14 += o; t = o << 28; in13 -= (sword32)t;
o = (in14 >> 28); in15 += o; t = o << 28; in14 -= (sword32)t;
o = (in15 >> 28); in0 += o;
in8 += o; t = o << 28; in15 -= (sword32)t;
/* Output as bytes */
b[ 0] = (byte)(in0 >> 0);
b[ 1] = (byte)(in0 >> 8);
b[ 2] = (byte)(in0 >> 16);
b[ 3] = (byte)((byte)(in0 >> 24) + (byte)((in1 >> 0) << 4));
b[ 4] = (byte)(in1 >> 4);
b[ 5] = (byte)(in1 >> 12);
b[ 6] = (byte)(in1 >> 20);
b[ 7] = (byte)(in2 >> 0);
b[ 8] = (byte)(in2 >> 8);
b[ 9] = (byte)(in2 >> 16);
b[10] = (byte)((byte)(in2 >> 24) + (byte)((in3 >> 0) << 4));
b[11] = (byte)(in3 >> 4);
b[12] = (byte)(in3 >> 12);
b[13] = (byte)(in3 >> 20);
b[14] = (byte)(in4 >> 0);
b[15] = (byte)(in4 >> 8);
b[16] = (byte)(in4 >> 16);
b[17] = (byte)((byte)(in4 >> 24) + (byte)((in5 >> 0) << 4));
b[18] = (byte)(in5 >> 4);
b[19] = (byte)(in5 >> 12);
b[20] = (byte)(in5 >> 20);
b[21] = (byte)(in6 >> 0);
b[22] = (byte)(in6 >> 8);
b[23] = (byte)(in6 >> 16);
b[24] = (byte)((byte)(in6 >> 24) + (byte)((in7 >> 0) << 4));
b[25] = (byte)(in7 >> 4);
b[26] = (byte)(in7 >> 12);
b[27] = (byte)(in7 >> 20);
b[28] = (byte)(in8 >> 0);
b[29] = (byte)(in8 >> 8);
b[30] = (byte)(in8 >> 16);
b[31] = (byte)((byte)(in8 >> 24) + (byte)((in9 >> 0) << 4));
b[32] = (byte)(in9 >> 4);
b[33] = (byte)(in9 >> 12);
b[34] = (byte)(in9 >> 20);
b[35] = (byte)(in10 >> 0);
b[36] = (byte)(in10 >> 8);
b[37] = (byte)(in10 >> 16);
b[38] = (byte)((byte)(in10 >> 24) + (byte)((in11 >> 0) << 4));
b[39] = (byte)(in11 >> 4);
b[40] = (byte)(in11 >> 12);
b[41] = (byte)(in11 >> 20);
b[42] = (byte)(in12 >> 0);
b[43] = (byte)(in12 >> 8);
b[44] = (byte)(in12 >> 16);
b[45] = (byte)((byte)(in12 >> 24) + (byte)((in13 >> 0) << 4));
b[46] = (byte)(in13 >> 4);
b[47] = (byte)(in13 >> 12);
b[48] = (byte)(in13 >> 20);
b[49] = (byte)(in14 >> 0);
b[50] = (byte)(in14 >> 8);
b[51] = (byte)(in14 >> 16);
b[52] = (byte)((byte)(in14 >> 24) + (byte)((in15 >> 0) << 4));
b[53] = (byte)(in15 >> 4);
b[54] = (byte)(in15 >> 12);
b[55] = (byte)(in15 >> 20);
}
/* Set the field element to 0.
*
* a [in] Field element.
*/
void fe448_1(sword32* a)
{
a[0] = 1;
a[1] = 0;
a[2] = 0;
a[3] = 0;
a[4] = 0;
a[5] = 0;
a[6] = 0;
a[7] = 0;
a[8] = 0;
a[9] = 0;
a[10] = 0;
a[11] = 0;
a[12] = 0;
a[13] = 0;
a[14] = 0;
a[15] = 0;
}
/* Set the field element to 0.
*
* a [in] Field element.
*/
void fe448_0(sword32* a)
{
a[0] = 0;
a[1] = 0;
a[2] = 0;
a[3] = 0;
a[4] = 0;
a[5] = 0;
a[6] = 0;
a[7] = 0;
a[8] = 0;
a[9] = 0;
a[10] = 0;
a[11] = 0;
a[12] = 0;
a[13] = 0;
a[14] = 0;
a[15] = 0;
}
/* Copy one field element into another: d = a.
*
* d [in] Destination field element.
* a [in] Source field element.
*/
void fe448_copy(sword32* d, const sword32* a)
{
d[0] = a[0];
d[1] = a[1];
d[2] = a[2];
d[3] = a[3];
d[4] = a[4];
d[5] = a[5];
d[6] = a[6];
d[7] = a[7];
d[8] = a[8];
d[9] = a[9];
d[10] = a[10];
d[11] = a[11];
d[12] = a[12];
d[13] = a[13];
d[14] = a[14];
d[15] = a[15];
}
/* Conditionally swap the elements.
* Constant time implementation.
*
* a [in] First field element.
* b [in] Second field element.
* c [in] Swap when 1. Valid values: 0, 1.
*/
static void fe448_cswap(sword32* a, sword32* b, int c)
{
sword32 mask = -(sword32)c;
sword32 t0 = (a[0] ^ b[0]) & mask;
sword32 t1 = (a[1] ^ b[1]) & mask;
sword32 t2 = (a[2] ^ b[2]) & mask;
sword32 t3 = (a[3] ^ b[3]) & mask;
sword32 t4 = (a[4] ^ b[4]) & mask;
sword32 t5 = (a[5] ^ b[5]) & mask;
sword32 t6 = (a[6] ^ b[6]) & mask;
sword32 t7 = (a[7] ^ b[7]) & mask;
sword32 t8 = (a[8] ^ b[8]) & mask;
sword32 t9 = (a[9] ^ b[9]) & mask;
sword32 t10 = (a[10] ^ b[10]) & mask;
sword32 t11 = (a[11] ^ b[11]) & mask;
sword32 t12 = (a[12] ^ b[12]) & mask;
sword32 t13 = (a[13] ^ b[13]) & mask;
sword32 t14 = (a[14] ^ b[14]) & mask;
sword32 t15 = (a[15] ^ b[15]) & mask;
a[0] ^= t0;
a[1] ^= t1;
a[2] ^= t2;
a[3] ^= t3;
a[4] ^= t4;
a[5] ^= t5;
a[6] ^= t6;
a[7] ^= t7;
a[8] ^= t8;
a[9] ^= t9;
a[10] ^= t10;
a[11] ^= t11;
a[12] ^= t12;
a[13] ^= t13;
a[14] ^= t14;
a[15] ^= t15;
b[0] ^= t0;
b[1] ^= t1;
b[2] ^= t2;
b[3] ^= t3;
b[4] ^= t4;
b[5] ^= t5;
b[6] ^= t6;
b[7] ^= t7;
b[8] ^= t8;
b[9] ^= t9;
b[10] ^= t10;
b[11] ^= t11;
b[12] ^= t12;
b[13] ^= t13;
b[14] ^= t14;
b[15] ^= t15;
}
/* Add two field elements. r = (a + b) mod (2^448 - 2^224 - 1)
*
* r [in] Field element to hold sum.
* a [in] Field element to add.
* b [in] Field element to add.
*/
void fe448_add(sword32* r, const sword32* a, const sword32* b)
{
r[0] = a[0] + b[0];
r[1] = a[1] + b[1];
r[2] = a[2] + b[2];
r[3] = a[3] + b[3];
r[4] = a[4] + b[4];
r[5] = a[5] + b[5];
r[6] = a[6] + b[6];
r[7] = a[7] + b[7];
r[8] = a[8] + b[8];
r[9] = a[9] + b[9];
r[10] = a[10] + b[10];
r[11] = a[11] + b[11];
r[12] = a[12] + b[12];
r[13] = a[13] + b[13];
r[14] = a[14] + b[14];
r[15] = a[15] + b[15];
}
/* Subtract a field element from another. r = (a - b) mod (2^448 - 2^224 - 1)
*
* r [in] Field element to hold difference.
* a [in] Field element to subtract from.
* b [in] Field element to subtract.
*/
void fe448_sub(sword32* r, const sword32* a, const sword32* b)
{
r[0] = a[0] - b[0];
r[1] = a[1] - b[1];
r[2] = a[2] - b[2];
r[3] = a[3] - b[3];
r[4] = a[4] - b[4];
r[5] = a[5] - b[5];
r[6] = a[6] - b[6];
r[7] = a[7] - b[7];
r[8] = a[8] - b[8];
r[9] = a[9] - b[9];
r[10] = a[10] - b[10];
r[11] = a[11] - b[11];
r[12] = a[12] - b[12];
r[13] = a[13] - b[13];
r[14] = a[14] - b[14];
r[15] = a[15] - b[15];
}
void fe448_reduce(sword32* a)
{
sword64 o;
o = a[0 ] >> 28; a[1 ] += (sword32)o; a[0 ] -= (sword32)(o << 28);
o = a[1 ] >> 28; a[2 ] += (sword32)o; a[1 ] -= (sword32)(o << 28);
o = a[2 ] >> 28; a[3 ] += (sword32)o; a[2 ] -= (sword32)(o << 28);
o = a[3 ] >> 28; a[4 ] += (sword32)o; a[3 ] -= (sword32)(o << 28);
o = a[4 ] >> 28; a[5 ] += (sword32)o; a[4 ] -= (sword32)(o << 28);
o = a[5 ] >> 28; a[6 ] += (sword32)o; a[5 ] -= (sword32)(o << 28);
o = a[6 ] >> 28; a[7 ] += (sword32)o; a[6 ] -= (sword32)(o << 28);
o = a[7 ] >> 28; a[8 ] += (sword32)o; a[7 ] -= (sword32)(o << 28);
o = a[8 ] >> 28; a[9 ] += (sword32)o; a[8 ] -= (sword32)(o << 28);
o = a[9 ] >> 28; a[10] += (sword32)o; a[9 ] -= (sword32)(o << 28);
o = a[10] >> 28; a[11] += (sword32)o; a[10] -= (sword32)(o << 28);
o = a[11] >> 28; a[12] += (sword32)o; a[11] -= (sword32)(o << 28);
o = a[12] >> 28; a[13] += (sword32)o; a[12] -= (sword32)(o << 28);
o = a[13] >> 28; a[14] += (sword32)o; a[13] -= (sword32)(o << 28);
o = a[14] >> 28; a[15] += (sword32)o; a[14] -= (sword32)(o << 28);
o = a[15] >> 28; a[0] += (sword32)o;
a[8] += (sword32)o; a[15] -= (sword32)(o << 28);
}
/* Multiply a field element by 39081. r = (39081 * a) mod (2^448 - 2^224 - 1)
*
* r [in] Field element to hold result.
* a [in] Field element to multiply.
*/
void fe448_mul39081(sword32* r, const sword32* a)
{
sword64 t;
sword32 o;
sword64 t0 = a[0] * (sword64)39081;
sword64 t1 = a[1] * (sword64)39081;
sword64 t2 = a[2] * (sword64)39081;
sword64 t3 = a[3] * (sword64)39081;
sword64 t4 = a[4] * (sword64)39081;
sword64 t5 = a[5] * (sword64)39081;
sword64 t6 = a[6] * (sword64)39081;
sword64 t7 = a[7] * (sword64)39081;
sword64 t8 = a[8] * (sword64)39081;
sword64 t9 = a[9] * (sword64)39081;
sword64 t10 = a[10] * (sword64)39081;
sword64 t11 = a[11] * (sword64)39081;
sword64 t12 = a[12] * (sword64)39081;
sword64 t13 = a[13] * (sword64)39081;
sword64 t14 = a[14] * (sword64)39081;
sword64 t15 = a[15] * (sword64)39081;
o = (sword32)(t0 >> 28); t1 += o; t = (sword64)o << 28; t0 -= t;
o = (sword32)(t1 >> 28); t2 += o; t = (sword64)o << 28; t1 -= t;
o = (sword32)(t2 >> 28); t3 += o; t = (sword64)o << 28; t2 -= t;
o = (sword32)(t3 >> 28); t4 += o; t = (sword64)o << 28; t3 -= t;
o = (sword32)(t4 >> 28); t5 += o; t = (sword64)o << 28; t4 -= t;
o = (sword32)(t5 >> 28); t6 += o; t = (sword64)o << 28; t5 -= t;
o = (sword32)(t6 >> 28); t7 += o; t = (sword64)o << 28; t6 -= t;
o = (sword32)(t7 >> 28); t8 += o; t = (sword64)o << 28; t7 -= t;
o = (sword32)(t8 >> 28); t9 += o; t = (sword64)o << 28; t8 -= t;
o = (sword32)(t9 >> 28); t10 += o; t = (sword64)o << 28; t9 -= t;
o = (sword32)(t10 >> 28); t11 += o; t = (sword64)o << 28; t10 -= t;
o = (sword32)(t11 >> 28); t12 += o; t = (sword64)o << 28; t11 -= t;
o = (sword32)(t12 >> 28); t13 += o; t = (sword64)o << 28; t12 -= t;
o = (sword32)(t13 >> 28); t14 += o; t = (sword64)o << 28; t13 -= t;
o = (sword32)(t14 >> 28); t15 += o; t = (sword64)o << 28; t14 -= t;
o = (sword32)(t15 >> 28); t0 += o;
t8 += o; t = (sword64)o << 28; t15 -= t;
/* Store */
r[0] = (sword32)t0;
r[1] = (sword32)t1;
r[2] = (sword32)t2;
r[3] = (sword32)t3;
r[4] = (sword32)t4;
r[5] = (sword32)t5;
r[6] = (sword32)t6;
r[7] = (sword32)t7;
r[8] = (sword32)t8;
r[9] = (sword32)t9;
r[10] = (sword32)t10;
r[11] = (sword32)t11;
r[12] = (sword32)t12;
r[13] = (sword32)t13;
r[14] = (sword32)t14;
r[15] = (sword32)t15;
}
/* Multiply two field elements. r = a * b
*
* r [in] Field element to hold result.
* a [in] Field element to multiply.
* b [in] Field element to multiply.
*/
static WC_INLINE void fe448_mul_8(sword32* r, const sword32* a, const sword32* b)
{
sword64 t;
sword64 o;
sword64 t15;
sword64 t0 = (sword64)a[ 0] * b[ 0];
sword64 t1 = (sword64)a[ 0] * b[ 1];
sword64 t101 = (sword64)a[ 1] * b[ 0];
sword64 t2 = (sword64)a[ 0] * b[ 2];
sword64 t102 = (sword64)a[ 1] * b[ 1];
sword64 t202 = (sword64)a[ 2] * b[ 0];
sword64 t3 = (sword64)a[ 0] * b[ 3];
sword64 t103 = (sword64)a[ 1] * b[ 2];
sword64 t203 = (sword64)a[ 2] * b[ 1];
sword64 t303 = (sword64)a[ 3] * b[ 0];
sword64 t4 = (sword64)a[ 0] * b[ 4];
sword64 t104 = (sword64)a[ 1] * b[ 3];
sword64 t204 = (sword64)a[ 2] * b[ 2];
sword64 t304 = (sword64)a[ 3] * b[ 1];
sword64 t404 = (sword64)a[ 4] * b[ 0];
sword64 t5 = (sword64)a[ 0] * b[ 5];
sword64 t105 = (sword64)a[ 1] * b[ 4];
sword64 t205 = (sword64)a[ 2] * b[ 3];
sword64 t305 = (sword64)a[ 3] * b[ 2];
sword64 t405 = (sword64)a[ 4] * b[ 1];
sword64 t505 = (sword64)a[ 5] * b[ 0];
sword64 t6 = (sword64)a[ 0] * b[ 6];
sword64 t106 = (sword64)a[ 1] * b[ 5];
sword64 t206 = (sword64)a[ 2] * b[ 4];
sword64 t306 = (sword64)a[ 3] * b[ 3];
sword64 t406 = (sword64)a[ 4] * b[ 2];
sword64 t506 = (sword64)a[ 5] * b[ 1];
sword64 t606 = (sword64)a[ 6] * b[ 0];
sword64 t7 = (sword64)a[ 0] * b[ 7];
sword64 t107 = (sword64)a[ 1] * b[ 6];
sword64 t207 = (sword64)a[ 2] * b[ 5];
sword64 t307 = (sword64)a[ 3] * b[ 4];
sword64 t407 = (sword64)a[ 4] * b[ 3];
sword64 t507 = (sword64)a[ 5] * b[ 2];
sword64 t607 = (sword64)a[ 6] * b[ 1];
sword64 t707 = (sword64)a[ 7] * b[ 0];
sword64 t8 = (sword64)a[ 1] * b[ 7];
sword64 t108 = (sword64)a[ 2] * b[ 6];
sword64 t208 = (sword64)a[ 3] * b[ 5];
sword64 t308 = (sword64)a[ 4] * b[ 4];
sword64 t408 = (sword64)a[ 5] * b[ 3];
sword64 t508 = (sword64)a[ 6] * b[ 2];
sword64 t608 = (sword64)a[ 7] * b[ 1];
sword64 t9 = (sword64)a[ 2] * b[ 7];
sword64 t109 = (sword64)a[ 3] * b[ 6];
sword64 t209 = (sword64)a[ 4] * b[ 5];
sword64 t309 = (sword64)a[ 5] * b[ 4];
sword64 t409 = (sword64)a[ 6] * b[ 3];
sword64 t509 = (sword64)a[ 7] * b[ 2];
sword64 t10 = (sword64)a[ 3] * b[ 7];
sword64 t110 = (sword64)a[ 4] * b[ 6];
sword64 t210 = (sword64)a[ 5] * b[ 5];
sword64 t310 = (sword64)a[ 6] * b[ 4];
sword64 t410 = (sword64)a[ 7] * b[ 3];
sword64 t11 = (sword64)a[ 4] * b[ 7];
sword64 t111 = (sword64)a[ 5] * b[ 6];
sword64 t211 = (sword64)a[ 6] * b[ 5];
sword64 t311 = (sword64)a[ 7] * b[ 4];
sword64 t12 = (sword64)a[ 5] * b[ 7];
sword64 t112 = (sword64)a[ 6] * b[ 6];
sword64 t212 = (sword64)a[ 7] * b[ 5];
sword64 t13 = (sword64)a[ 6] * b[ 7];
sword64 t113 = (sword64)a[ 7] * b[ 6];
sword64 t14 = (sword64)a[ 7] * b[ 7];
t1 += t101;
t2 += t102; t2 += t202;
t3 += t103; t3 += t203; t3 += t303;
t4 += t104; t4 += t204; t4 += t304; t4 += t404;
t5 += t105; t5 += t205; t5 += t305; t5 += t405; t5 += t505;
t6 += t106; t6 += t206; t6 += t306; t6 += t406; t6 += t506;
t6 += t606;
t7 += t107; t7 += t207; t7 += t307; t7 += t407; t7 += t507;
t7 += t607;
t8 += t108; t8 += t208; t8 += t308; t8 += t408; t8 += t508;
t8 += t608;
t9 += t109; t9 += t209; t9 += t309; t9 += t409; t9 += t509;
t10 += t110; t10 += t210; t10 += t310; t10 += t410;
t11 += t111; t11 += t211; t11 += t311;
t12 += t112; t12 += t212;
t13 += t113;
o = t14 >> 28;
t15 = o;
t14 -= o << 28;
o = (t0 >> 28); t1 += o; t = o << 28; t0 -= t;
o = (t1 >> 28); t2 += o; t = o << 28; t1 -= t;
o = (t2 >> 28); t3 += o; t = o << 28; t2 -= t;
o = (t3 >> 28); t4 += o; t = o << 28; t3 -= t;
o = (t4 >> 28); t5 += o; t = o << 28; t4 -= t;
o = (t5 >> 28); t6 += o; t = o << 28; t5 -= t;
o = (t6 >> 28); t7 += o; t = o << 28; t6 -= t;
o = (t707>> 28); t8 += o; t = o << 28; t707-= t; t7 += t707;
o = (t7 >> 28); t8 += o; t = o << 28; t7 -= t;
o = (t8 >> 28); t9 += o; t = o << 28; t8 -= t;
o = (t9 >> 28); t10 += o; t = o << 28; t9 -= t;
o = (t10 >> 28); t11 += o; t = o << 28; t10 -= t;
o = (t11 >> 28); t12 += o; t = o << 28; t11 -= t;
o = (t12 >> 28); t13 += o; t = o << 28; t12 -= t;
o = (t13 >> 28); t14 += o; t = o << 28; t13 -= t;
o = (t14 >> 28); t15 += o; t = o << 28; t14 -= t;
o = (t15 >> 28); t0 += o;
t8 += o; t = o << 28; t15 -= t;
/* Store */
r[0] = (sword32)t0;
r[1] = (sword32)t1;
r[2] = (sword32)t2;
r[3] = (sword32)t3;
r[4] = (sword32)t4;
r[5] = (sword32)t5;
r[6] = (sword32)t6;
r[7] = (sword32)t7;
r[8] = (sword32)t8;
r[9] = (sword32)t9;
r[10] = (sword32)t10;
r[11] = (sword32)t11;
r[12] = (sword32)t12;
r[13] = (sword32)t13;
r[14] = (sword32)t14;
r[15] = (sword32)t15;
}
/* Multiply two field elements. r = (a * b) mod (2^448 - 2^224 - 1)
*
* r [in] Field element to hold result.
* a [in] Field element to multiply.
* b [in] Field element to multiply.
*/
void fe448_mul(sword32* r, const sword32* a, const sword32* b)
{
sword32 r0[16];
sword32 r1[16];
sword32* a1 = r1;
sword32 b1[8];
sword32 r2[16];
a1[0] = a[0] + a[8];
a1[1] = a[1] + a[9];
a1[2] = a[2] + a[10];
a1[3] = a[3] + a[11];
a1[4] = a[4] + a[12];
a1[5] = a[5] + a[13];
a1[6] = a[6] + a[14];
a1[7] = a[7] + a[15];
b1[0] = b[0] + b[8];
b1[1] = b[1] + b[9];
b1[2] = b[2] + b[10];
b1[3] = b[3] + b[11];
b1[4] = b[4] + b[12];
b1[5] = b[5] + b[13];
b1[6] = b[6] + b[14];
b1[7] = b[7] + b[15];
fe448_mul_8(r2, a + 8, b + 8);
fe448_mul_8(r0, a, b);
fe448_mul_8(r1, a1, b1);
r[ 0] = r0[ 0] + r2[ 0] + r1[ 8] - r0[ 8];
r[ 1] = r0[ 1] + r2[ 1] + r1[ 9] - r0[ 9];
r[ 2] = r0[ 2] + r2[ 2] + r1[10] - r0[10];
r[ 3] = r0[ 3] + r2[ 3] + r1[11] - r0[11];
r[ 4] = r0[ 4] + r2[ 4] + r1[12] - r0[12];
r[ 5] = r0[ 5] + r2[ 5] + r1[13] - r0[13];
r[ 6] = r0[ 6] + r2[ 6] + r1[14] - r0[14];
r[ 7] = r0[ 7] + r2[ 7] + r1[15] - r0[15];
r[ 8] = r2[ 8] + r1[ 0] - r0[ 0] + r1[ 8];
r[ 9] = r2[ 9] + r1[ 1] - r0[ 1] + r1[ 9];
r[10] = r2[10] + r1[ 2] - r0[ 2] + r1[10];
r[11] = r2[11] + r1[ 3] - r0[ 3] + r1[11];
r[12] = r2[12] + r1[ 4] - r0[ 4] + r1[12];
r[13] = r2[13] + r1[ 5] - r0[ 5] + r1[13];
r[14] = r2[14] + r1[ 6] - r0[ 6] + r1[14];
r[15] = r2[15] + r1[ 7] - r0[ 7] + r1[15];
}
/* Square a field element. r = a * a
*
* r [in] Field element to hold result.
* a [in] Field element to square.
*/
static WC_INLINE void fe448_sqr_8(sword32* r, const sword32* a)
{
sword64 o;
sword64 t15;
sword64 t;
sword64 t0 = (sword64)a[ 0] * a[ 0];
sword64 t1 = 2 * (sword64)a[ 0] * a[ 1];
sword64 t2 = 2 * (sword64)a[ 0] * a[ 2];
sword64 t102 = (sword64)a[ 1] * a[ 1];
sword64 t3 = 2 * (sword64)a[ 0] * a[ 3];
sword64 t103 = 2 * (sword64)a[ 1] * a[ 2];
sword64 t4 = 2 * (sword64)a[ 0] * a[ 4];
sword64 t104 = 2 * (sword64)a[ 1] * a[ 3];
sword64 t204 = (sword64)a[ 2] * a[ 2];
sword64 t5 = 2 * (sword64)a[ 0] * a[ 5];
sword64 t105 = 2 * (sword64)a[ 1] * a[ 4];
sword64 t205 = 2 * (sword64)a[ 2] * a[ 3];
sword64 t6 = 2 * (sword64)a[ 0] * a[ 6];
sword64 t106 = 2 * (sword64)a[ 1] * a[ 5];
sword64 t206 = 2 * (sword64)a[ 2] * a[ 4];
sword64 t306 = (sword64)a[ 3] * a[ 3];
sword64 t7 = 2 * (sword64)a[ 0] * a[ 7];
sword64 t107 = 2 * (sword64)a[ 1] * a[ 6];
sword64 t207 = 2 * (sword64)a[ 2] * a[ 5];
sword64 t307 = 2 * (sword64)a[ 3] * a[ 4];
sword64 t8 = 2 * (sword64)a[ 1] * a[ 7];
sword64 t108 = 2 * (sword64)a[ 2] * a[ 6];
sword64 t208 = 2 * (sword64)a[ 3] * a[ 5];
sword64 t308 = (sword64)a[ 4] * a[ 4];
sword64 t9 = 2 * (sword64)a[ 2] * a[ 7];
sword64 t109 = 2 * (sword64)a[ 3] * a[ 6];
sword64 t209 = 2 * (sword64)a[ 4] * a[ 5];
sword64 t10 = 2 * (sword64)a[ 3] * a[ 7];
sword64 t110 = 2 * (sword64)a[ 4] * a[ 6];
sword64 t210 = (sword64)a[ 5] * a[ 5];
sword64 t11 = 2 * (sword64)a[ 4] * a[ 7];
sword64 t111 = 2 * (sword64)a[ 5] * a[ 6];
sword64 t12 = 2 * (sword64)a[ 5] * a[ 7];
sword64 t112 = (sword64)a[ 6] * a[ 6];
sword64 t13 = 2 * (sword64)a[ 6] * a[ 7];
sword64 t14 = (sword64)a[ 7] * a[ 7];
t2 += t102;
t3 += t103;
t4 += t104; t4 += t204;
t5 += t105; t5 += t205;
t6 += t106; t6 += t206; t6 += t306;
t7 += t107; t7 += t207;
t8 += t108; t8 += t208; t8 += t308;
t9 += t109; t9 += t209;
t10 += t110; t10 += t210;
t11 += t111;
t12 += t112;
o = t14 >> 28;
t15 = o;
t14 -= o << 28;
o = (t0 >> 28); t1 += o; t = o << 28; t0 -= t;
o = (t1 >> 28); t2 += o; t = o << 28; t1 -= t;
o = (t2 >> 28); t3 += o; t = o << 28; t2 -= t;
o = (t3 >> 28); t4 += o; t = o << 28; t3 -= t;
o = (t4 >> 28); t5 += o; t = o << 28; t4 -= t;
o = (t5 >> 28); t6 += o; t = o << 28; t5 -= t;
o = (t6 >> 28); t7 += o; t = o << 28; t6 -= t;
o = (t307>> 28); t8 += o; t = o << 28; t307-= t; t7 += t307;
o = (t7 >> 28); t8 += o; t = o << 28; t7 -= t;
o = (t8 >> 28); t9 += o; t = o << 28; t8 -= t;
o = (t9 >> 28); t10 += o; t = o << 28; t9 -= t;
o = (t10 >> 28); t11 += o; t = o << 28; t10 -= t;
o = (t11 >> 28); t12 += o; t = o << 28; t11 -= t;
o = (t12 >> 28); t13 += o; t = o << 28; t12 -= t;
o = (t13 >> 28); t14 += o; t = o << 28; t13 -= t;
o = (t14 >> 28); t15 += o; t = o << 28; t14 -= t;
o = (t15 >> 28); t0 += o;
t8 += o; t = o << 28; t15 -= t;
/* Store */
r[0] = (sword32)t0;
r[1] = (sword32)t1;
r[2] = (sword32)t2;
r[3] = (sword32)t3;
r[4] = (sword32)t4;
r[5] = (sword32)t5;
r[6] = (sword32)t6;
r[7] = (sword32)t7;
r[8] = (sword32)t8;
r[9] = (sword32)t9;
r[10] = (sword32)t10;
r[11] = (sword32)t11;
r[12] = (sword32)t12;
r[13] = (sword32)t13;
r[14] = (sword32)t14;
r[15] = (sword32)t15;
}
/* Square a field element. r = (a * a) mod (2^448 - 2^224 - 1)
*
* r [in] Field element to hold result.
* a [in] Field element to square.
*/
void fe448_sqr(sword32* r, const sword32* a)
{
sword32 r0[16];
sword32 r1[16];
sword32* a1 = r1;
sword32 r2[16];
a1[0] = a[0] + a[8];
a1[1] = a[1] + a[9];
a1[2] = a[2] + a[10];
a1[3] = a[3] + a[11];
a1[4] = a[4] + a[12];
a1[5] = a[5] + a[13];
a1[6] = a[6] + a[14];
a1[7] = a[7] + a[15];
fe448_sqr_8(r2, a + 8);
fe448_sqr_8(r0, a);
fe448_sqr_8(r1, a1);
r[ 0] = r0[ 0] + r2[ 0] + r1[ 8] - r0[ 8];
r[ 1] = r0[ 1] + r2[ 1] + r1[ 9] - r0[ 9];
r[ 2] = r0[ 2] + r2[ 2] + r1[10] - r0[10];
r[ 3] = r0[ 3] + r2[ 3] + r1[11] - r0[11];
r[ 4] = r0[ 4] + r2[ 4] + r1[12] - r0[12];
r[ 5] = r0[ 5] + r2[ 5] + r1[13] - r0[13];
r[ 6] = r0[ 6] + r2[ 6] + r1[14] - r0[14];
r[ 7] = r0[ 7] + r2[ 7] + r1[15] - r0[15];
r[ 8] = r2[ 8] + r1[ 0] - r0[ 0] + r1[ 8];
r[ 9] = r2[ 9] + r1[ 1] - r0[ 1] + r1[ 9];
r[10] = r2[10] + r1[ 2] - r0[ 2] + r1[10];
r[11] = r2[11] + r1[ 3] - r0[ 3] + r1[11];
r[12] = r2[12] + r1[ 4] - r0[ 4] + r1[12];
r[13] = r2[13] + r1[ 5] - r0[ 5] + r1[13];
r[14] = r2[14] + r1[ 6] - r0[ 6] + r1[14];
r[15] = r2[15] + r1[ 7] - r0[ 7] + r1[15];
}
/* Invert the field element. (r * a) mod (2^448 - 2^224 - 1) = 1
* Constant time implementation - using Fermat's little theorem:
* a^(p-1) mod p = 1 => a^(p-2) mod p = 1/a
* For curve448: p - 2 = 2^448 - 2^224 - 3
*
* r [in] Field element to hold result.
* a [in] Field element to invert.
*/
void fe448_invert(sword32* r, const sword32* a)
{
sword32 t1[16];
sword32 t2[16];
sword32 t3[16];
sword32 t4[16];
int i;
fe448_sqr(t1, a);
/* t1 = 2 */
fe448_mul(t1, t1, a);
/* t1 = 3 */
fe448_sqr(t2, t1); for (i = 1; i < 2; ++i) fe448_sqr(t2, t2);
/* t2 = c */
fe448_mul(t3, t2, a);
/* t3 = d */
fe448_mul(t1, t2, t1);
/* t1 = f */
fe448_sqr(t2, t1);
/* t2 = 1e */
fe448_mul(t4, t2, a);
/* t4 = 1f */
fe448_sqr(t2, t4); for (i = 1; i < 5; ++i) fe448_sqr(t2, t2);
/* t2 = 3e0 */
fe448_mul(t1, t2, t4);
/* t1 = 3ff */
fe448_sqr(t2, t1); for (i = 1; i < 10; ++i) fe448_sqr(t2, t2);
/* t2 = ffc00 */
fe448_mul(t1, t2, t1);
/* t1 = fffff */
fe448_sqr(t2, t1); for (i = 1; i < 5; ++i) fe448_sqr(t2, t2);
/* t2 = 1ffffe0 */
fe448_mul(t1, t2, t4);
/* t1 = 1ffffff */
fe448_sqr(t2, t1); for (i = 1; i < 25; ++i) fe448_sqr(t2, t2);
/* t2 = 3fffffe000000 */
fe448_mul(t1, t2, t1);
/* t1 = 3ffffffffffff */
fe448_sqr(t2, t1); for (i = 1; i < 5; ++i) fe448_sqr(t2, t2);
/* t2 = 7fffffffffffe0 */
fe448_mul(t1, t2, t4);
/* t1 = 7fffffffffffff */
fe448_sqr(t2, t1); for (i = 1; i < 55; ++i) fe448_sqr(t2, t2);
/* t2 = 3fffffffffffff80000000000000 */
fe448_mul(t1, t2, t1);
/* t1 = 3fffffffffffffffffffffffffff */
fe448_sqr(t2, t1); for (i = 1; i < 110; ++i) fe448_sqr(t2, t2);
/* t2 = fffffffffffffffffffffffffffc000000000000000000000000000 */
fe448_mul(t1, t2, t1);
/* t1 = fffffffffffffffffffffffffffffffffffffffffffffffffffffff */
fe448_sqr(t2, t1); for (i = 1; i < 4; ++i) fe448_sqr(t2, t2);
/* t2 = fffffffffffffffffffffffffffffffffffffffffffffffffffffff0 */
fe448_mul(t3, t3, t2);
/* t3 = fffffffffffffffffffffffffffffffffffffffffffffffffffffffd */
fe448_mul(t1, t3, a);
/* t1 = fffffffffffffffffffffffffffffffffffffffffffffffffffffffe */
fe448_sqr(t1, t1); for (i = 1; i < 224; ++i) fe448_sqr(t1, t1);
/* t1 = fffffffffffffffffffffffffffffffffffffffffffffffffffffffe00000000000000000000000000000000000000000000000000000000 */
fe448_mul(r, t3, t1);
/* r = fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffffffffffffffffffffffffffffffffffffffffffffffffffffd */
}
/* Scalar multiply the point by a number. r = n.a
* Uses Montgomery ladder and only requires the x-ordinate.
*
* r [in] Field element to hold result.
* n [in] Scalar as an array of bytes.
* a [in] Point to multiply - x-ordinate only.
*/
int curve448(byte* r, const byte* n, const byte* a)
{
sword32 x1[16];
sword32 x2[16];
sword32 z2[16];
sword32 x3[16];
sword32 z3[16];
sword32 t0[16];
sword32 t1[16];
int i;
unsigned int swap;
fe448_from_bytes(x1, a);
fe448_1(x2);
fe448_0(z2);
fe448_copy(x3, x1);
fe448_1(z3);
swap = 0;
for (i = 447; i >= 0; --i) {
unsigned int b = (n[i >> 3] >> (i & 7)) & 1;
swap ^= b;
fe448_cswap(x2, x3, (int)swap);
fe448_cswap(z2, z3, (int)swap);
swap = b;
/* Montgomery Ladder - double and add */
fe448_add(t0, x2, z2);
fe448_reduce(t0);
fe448_add(t1, x3, z3);
fe448_reduce(t1);
fe448_sub(x2, x2, z2);
fe448_sub(x3, x3, z3);
fe448_mul(t1, t1, x2);
fe448_mul(z3, x3, t0);
fe448_sqr(t0, t0);
fe448_sqr(x2, x2);
fe448_add(x3, z3, t1);
fe448_reduce(x3);
fe448_sqr(x3, x3);
fe448_sub(z3, z3, t1);
fe448_sqr(z3, z3);
fe448_mul(z3, z3, x1);
fe448_sub(t1, t0, x2);
fe448_mul(x2, t0, x2);
fe448_mul39081(z2, t1);
fe448_add(z2, t0, z2);
fe448_mul(z2, z2, t1);
}
/* Last two bits are 0 - no final swap check required. */
fe448_invert(z2, z2);
fe448_mul(x2, x2, z2);
fe448_to_bytes(r, x2);
return 0;
}
#ifdef HAVE_ED448
/* Check whether field element is not 0.
* Must convert to a normalized form before checking.
*
* a [in] Field element.
* returns 0 when zero, and any other value otherwise.
*/
int fe448_isnonzero(const sword32* a)
{
byte b[56];
int i;
byte c = 0;
fe448_to_bytes(b, a);
for (i = 0; i < 56; i++)
c |= b[i];
return c;
}
/* Check whether field element is negative.
* Must convert to a normalized form before checking.
*
* a [in] Field element.
* returns 1 when negative, and 0 otherwise.
*/
int fe448_isnegative(const sword32* a)
{
byte b[56];
fe448_to_bytes(b, a);
return b[0] & 1;
}
/* Negates the field element. r = -a
*
* r [in] Field element to hold result.
* a [in] Field element.
*/
void fe448_neg(sword32* r, const sword32* a)
{
r[0] = -a[0];
r[1] = -a[1];
r[2] = -a[2];
r[3] = -a[3];
r[4] = -a[4];
r[5] = -a[5];
r[6] = -a[6];
r[7] = -a[7];
r[8] = -a[8];
r[9] = -a[9];
r[10] = -a[10];
r[11] = -a[11];
r[12] = -a[12];
r[13] = -a[13];
r[14] = -a[14];
r[15] = -a[15];
}
/* Raise field element to (p-3) / 4: 2^446 - 2^222 - 1
* Used for calculating y-ordinate from x-ordinate for Ed448.
*
* r [in] Field element to hold result.
* a [in] Field element to exponentiate.
*/
void fe448_pow_2_446_222_1(sword32* r, const sword32* a)
{
sword32 t1[16];
sword32 t2[16];
sword32 t3[16];
sword32 t4[16];
sword32 t5[16];
int i;
fe448_sqr(t3, a);
/* t3 = 2 */
fe448_mul(t1, t3, a);
/* t1 = 3 */
fe448_sqr(t5, t1);
/* t5 = 6 */
fe448_mul(t5, t5, a);
/* t5 = 7 */
fe448_sqr(t2, t1); for (i = 1; i < 2; ++i) fe448_sqr(t2, t2);
/* t2 = c */
fe448_mul(t3, t2, t3);
/* t3 = e */
fe448_mul(t1, t2, t1);
/* t1 = f */
fe448_sqr(t2, t1); for (i = 1; i < 3; ++i) fe448_sqr(t2, t2);
/* t2 = 78 */
fe448_mul(t5, t2, t5);
/* t5 = 7f */
fe448_sqr(t2, t1); for (i = 1; i < 4; ++i) fe448_sqr(t2, t2);
/* t2 = f0 */
fe448_mul(t1, t2, t1);
/* t1 = ff */
fe448_mul(t3, t3, t2);
/* t3 = fe */
fe448_sqr(t2, t1); for (i = 1; i < 7; ++i) fe448_sqr(t2, t2);
/* t2 = 7f80 */
fe448_mul(t5, t2, t5);
/* t5 = 7fff */
fe448_sqr(t2, t1); for (i = 1; i < 8; ++i) fe448_sqr(t2, t2);
/* t2 = ff00 */
fe448_mul(t1, t2, t1);
/* t1 = ffff */
fe448_mul(t3, t3, t2);
/* t3 = fffe */
fe448_sqr(t2, t5); for (i = 1; i < 15; ++i) fe448_sqr(t2, t2);
/* t2 = 3fff8000 */
fe448_mul(t5, t2, t5);
/* t5 = 3fffffff */
fe448_sqr(t2, t1); for (i = 1; i < 16; ++i) fe448_sqr(t2, t2);
/* t2 = ffff0000 */
fe448_mul(t1, t2, t1);
/* t1 = ffffffff */
fe448_mul(t3, t3, t2);
/* t3 = fffffffe */
fe448_sqr(t2, t1); for (i = 1; i < 32; ++i) fe448_sqr(t2, t2);
/* t2 = ffffffff00000000 */
fe448_mul(t2, t2, t1);
/* t2 = ffffffffffffffff */
fe448_sqr(t1, t2); for (i = 1; i < 64; ++i) fe448_sqr(t1, t1);
/* t1 = ffffffffffffffff0000000000000000 */
fe448_mul(t1, t1, t2);
/* t1 = ffffffffffffffffffffffffffffffff */
fe448_sqr(t1, t1); for (i = 1; i < 64; ++i) fe448_sqr(t1, t1);
/* t1 = ffffffffffffffffffffffffffffffff0000000000000000 */
fe448_mul(t4, t1, t2);
/* t4 = ffffffffffffffffffffffffffffffffffffffffffffffff */
fe448_sqr(t2, t4); for (i = 1; i < 32; ++i) fe448_sqr(t2, t2);
/* t2 = ffffffffffffffffffffffffffffffffffffffffffffffff00000000 */
fe448_mul(t3, t3, t2);
/* t3 = fffffffffffffffffffffffffffffffffffffffffffffffffffffffe */
fe448_sqr(t1, t3); for (i = 1; i < 192; ++i) fe448_sqr(t1, t1);
/* t1 = fffffffffffffffffffffffffffffffffffffffffffffffffffffffe000000000000000000000000000000000000000000000000 */
fe448_mul(t1, t1, t4);
/* t1 = fffffffffffffffffffffffffffffffffffffffffffffffffffffffeffffffffffffffffffffffffffffffffffffffffffffffff */
fe448_sqr(t1, t1); for (i = 1; i < 30; ++i) fe448_sqr(t1, t1);
/* t1 = 3fffffffffffffffffffffffffffffffffffffffffffffffffffffffbfffffffffffffffffffffffffffffffffffffffffffffffc0000000 */
fe448_mul(r, t5, t1);
/* r = 3fffffffffffffffffffffffffffffffffffffffffffffffffffffffbfffffffffffffffffffffffffffffffffffffffffffffffffffffff */
}
/* Constant time, conditional move of b into a.
* a is not changed if the condition is 0.
*
* a A field element.
* b A field element.
* c If 1 then copy and if 0 then don't copy.
*/
void fe448_cmov(sword32* a, const sword32* b, int c)
{
sword32 m = -(sword32)c;
sword32 t0 = m & (a[0] ^ b[0]);
sword32 t1 = m & (a[1] ^ b[1]);
sword32 t2 = m & (a[2] ^ b[2]);
sword32 t3 = m & (a[3] ^ b[3]);
sword32 t4 = m & (a[4] ^ b[4]);
sword32 t5 = m & (a[5] ^ b[5]);
sword32 t6 = m & (a[6] ^ b[6]);
sword32 t7 = m & (a[7] ^ b[7]);
sword32 t8 = m & (a[8] ^ b[8]);
sword32 t9 = m & (a[9] ^ b[9]);
sword32 t10 = m & (a[10] ^ b[10]);
sword32 t11 = m & (a[11] ^ b[11]);
sword32 t12 = m & (a[12] ^ b[12]);
sword32 t13 = m & (a[13] ^ b[13]);
sword32 t14 = m & (a[14] ^ b[14]);
sword32 t15 = m & (a[15] ^ b[15]);
a[0] ^= t0;
a[1] ^= t1;
a[2] ^= t2;
a[3] ^= t3;
a[4] ^= t4;
a[5] ^= t5;
a[6] ^= t6;
a[7] ^= t7;
a[8] ^= t8;
a[9] ^= t9;
a[10] ^= t10;
a[11] ^= t11;
a[12] ^= t12;
a[13] ^= t13;
a[14] ^= t14;
a[15] ^= t15;
}
#endif /* HAVE_ED448 */
#endif
#endif /* HAVE_CURVE448 || HAVE_ED448 */
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