/* crypto/bn/bn_gf2m.c */
/* ====================================================================
* Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED.
*
* The Elliptic Curve Public-Key Crypto Library (ECC Code) included
* herein is developed by SUN MICROSYSTEMS, INC., and is contributed
* to the OpenSSL project.
*
* The ECC Code is licensed pursuant to the OpenSSL open source
* license provided below.
*
* In addition, Sun covenants to all licensees who provide a reciprocal
* covenant with respect to their own patents if any, not to sue under
* current and future patent claims necessarily infringed by the making,
* using, practicing, selling, offering for sale and/or otherwise
* disposing of the ECC Code as delivered hereunder (or portions thereof),
* provided that such covenant shall not apply:
* 1) for code that a licensee deletes from the ECC Code;
* 2) separates from the ECC Code; or
* 3) for infringements caused by:
* i) the modification of the ECC Code or
* ii) the combination of the ECC Code with other software or
* devices where such combination causes the infringement.
*
* The software is originally written by Sheueling Chang Shantz and
* Douglas Stebila of Sun Microsystems Laboratories.
*
*/
/*
* NOTE: This file is licensed pursuant to the OpenSSL license below and may
* be modified; but after modifications, the above covenant may no longer
* apply! In such cases, the corresponding paragraph ["In addition, Sun
* covenants ... causes the infringement."] and this note can be edited out;
* but please keep the Sun copyright notice and attribution.
*/
/* ====================================================================
* Copyright (c) 1998-2002 The OpenSSL Project. All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
*
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
*
* 2. 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.
*
* 3. All advertising materials mentioning features or use of this
* software must display the following acknowledgment:
* "This product includes software developed by the OpenSSL Project
* for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
*
* 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
* endorse or promote products derived from this software without
* prior written permission. For written permission, please contact
* openssl-core@openssl.org.
*
* 5. Products derived from this software may not be called "OpenSSL"
* nor may "OpenSSL" appear in their names without prior written
* permission of the OpenSSL Project.
*
* 6. Redistributions of any form whatsoever must retain the following
* acknowledgment:
* "This product includes software developed by the OpenSSL Project
* for use in the OpenSSL Toolkit (http://www.openssl.org/)"
*
* THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
* EXPRESSED 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 OpenSSL PROJECT OR
* ITS 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.
* ====================================================================
*
* This product includes cryptographic software written by Eric Young
* (eay@cryptsoft.com). This product includes software written by Tim
* Hudson (tjh@cryptsoft.com).
*
*/
#include <assert.h>
#include <limits.h>
#include <stdio.h>
#include "cryptlib.h"
#include "bn_lcl.h"
#ifndef OPENSSL_NO_EC2M
/*
* Maximum number of iterations before BN_GF2m_mod_solve_quad_arr should
* fail.
*/
# define MAX_ITERATIONS 50
static const BN_ULONG SQR_tb[16] = { 0, 1, 4, 5, 16, 17, 20, 21,
64, 65, 68, 69, 80, 81, 84, 85
};
/* Platform-specific macros to accelerate squaring. */
# if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG)
# define SQR1(w) \
SQR_tb[(w) >> 60 & 0xF] << 56 | SQR_tb[(w) >> 56 & 0xF] << 48 | \
SQR_tb[(w) >> 52 & 0xF] << 40 | SQR_tb[(w) >> 48 & 0xF] << 32 | \
SQR_tb[(w) >> 44 & 0xF] << 24 | SQR_tb[(w) >> 40 & 0xF] << 16 | \
SQR_tb[(w) >> 36 & 0xF] << 8 | SQR_tb[(w) >> 32 & 0xF]
# define SQR0(w) \
SQR_tb[(w) >> 28 & 0xF] << 56 | SQR_tb[(w) >> 24 & 0xF] << 48 | \
SQR_tb[(w) >> 20 & 0xF] << 40 | SQR_tb[(w) >> 16 & 0xF] << 32 | \
SQR_tb[(w) >> 12 & 0xF] << 24 | SQR_tb[(w) >> 8 & 0xF] << 16 | \
SQR_tb[(w) >> 4 & 0xF] << 8 | SQR_tb[(w) & 0xF]
# endif
# ifdef THIRTY_TWO_BIT
# define SQR1(w) \
SQR_tb[(w) >> 28 & 0xF] << 24 | SQR_tb[(w) >> 24 & 0xF] << 16 | \
SQR_tb[(w) >> 20 & 0xF] << 8 | SQR_tb[(w) >> 16 & 0xF]
# define SQR0(w) \
SQR_tb[(w) >> 12 & 0xF] << 24 | SQR_tb[(w) >> 8 & 0xF] << 16 | \
SQR_tb[(w) >> 4 & 0xF] << 8 | SQR_tb[(w) & 0xF]
# endif
# if !defined(OPENSSL_BN_ASM_GF2m)
/*
* Product of two polynomials a, b each with degree < BN_BITS2 - 1, result is
* a polynomial r with degree < 2 * BN_BITS - 1 The caller MUST ensure that
* the variables have the right amount of space allocated.
*/
# ifdef THIRTY_TWO_BIT
static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a,
const BN_ULONG b)
{
register BN_ULONG h, l, s;
BN_ULONG tab[8], top2b = a >> 30;
register BN_ULONG a1, a2, a4;
a1 = a & (0x3FFFFFFF);
a2 = a1 << 1;
a4 = a2 << 1;
tab[0] = 0;
tab[1] = a1;
tab[2] = a2;
tab[3] = a1 ^ a2;
tab[4] = a4;
tab[5] = a1 ^ a4;
tab[6] = a2 ^ a4;
tab[7] = a1 ^ a2 ^ a4;
s = tab[b & 0x7];
l = s;
s = tab[b >> 3 & 0x7];
l ^= s << 3;
h = s >> 29;
s = tab[b >> 6 & 0x7];
l ^= s << 6;
h ^= s >> 26;
s = tab[b >> 9 & 0x7];
l ^= s << 9;
h ^= s >> 23;
s = tab[b >> 12 & 0x7];
l ^= s << 12;
h ^= s >> 20;
s = tab[b >> 15 & 0x7];
l ^= s << 15;
h ^= s >> 17;
s = tab[b >> 18 & 0x7];
l ^= s << 18;
h ^= s >> 14;
s = tab[b >> 21 & 0x7];
l ^= s << 21;
h ^= s >> 11;
s = tab[b >> 24 & 0x7];
l ^= s << 24;
h ^= s >> 8;
s = tab[b >> 27 & 0x7];
l ^= s << 27;
h ^= s >> 5;
s = tab[b >> 30];
l ^= s << 30;
h ^= s >> 2;
/* compensate for the top two bits of a */
if (top2b & 01) {
l ^= b << 30;
h ^= b >> 2;
}
if (top2b & 02) {
l ^= b << 31;
h ^= b >> 1;
}
*r1 = h;
*r0 = l;
}
# endif
# if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG)
static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a,
const BN_ULONG b)
{
register BN_ULONG h, l, s;
BN_ULONG tab[16], top3b = a >> 61;
register BN_ULONG a1, a2, a4, a8;
a1 = a & (0x1FFFFFFFFFFFFFFFULL);
a2 = a1 << 1;
a4 = a2 << 1;
a8 = a4 << 1;
tab[0] = 0;
tab[1] = a1;
tab[2] = a2;
tab[3] = a1 ^ a2;
tab[4] = a4;
tab[5] = a1 ^ a4;
tab[6] = a2 ^ a4;
tab[7] = a1 ^ a2 ^ a4;
tab[8] = a8;
tab[9] = a1 ^ a8;
tab[10] = a2 ^ a8;
tab[11] = a1 ^ a2 ^ a8;
tab[12] = a4 ^ a8;
tab[13] = a1 ^ a4 ^ a8;
tab[14] = a2 ^ a4 ^ a8;
tab[15] = a1 ^ a2 ^ a4 ^ a8;
s = tab[b & 0xF];
l = s;
s = tab[b >> 4 & 0xF];
l ^= s << 4;
h = s >> 60;
s = tab[b >> 8 & 0xF];
l ^= s << 8;
h ^= s >> 56;
s = tab[b >> 12 & 0xF];
l ^= s << 12;
h ^= s >> 52;
s = tab[b >> 16 & 0xF];
l ^= s << 16;
h ^= s >> 48;
s = tab[b >> 20 & 0xF];
l ^= s << 20;
h ^= s >> 44;
s = tab[b >> 24 & 0xF];
l ^= s << 24;
h ^= s >> 40;
s = tab[b >> 28 & 0xF];
l ^= s << 28;
h ^= s >> 36;
s = tab[b >> 32 & 0xF];
l ^= s << 32;
h ^= s >> 32;
s = tab[b >> 36 & 0xF];
l ^= s << 36;
h ^= s >> 28;
s = tab[b >> 40 & 0xF];
l ^= s << 40;
h ^= s >> 24;
s = tab[b >> 44 & 0xF];
l ^= s << 44;
h ^= s >> 20;
s = tab[b >> 48 & 0xF];
l ^= s << 48;
h ^= s >> 16;
s = tab[b >> 52 & 0xF];
l ^= s << 52;
h ^= s >> 12;
s = tab[b >> 56 & 0xF];
l ^= s << 56;
h ^= s >> 8;
s = tab[b >> 60];
l ^= s << 60;
h ^= s >> 4;
/* compensate for the top three bits of a */
if (top3b & 01) {
l ^= b << 61;
h ^= b >> 3;
}
if (top3b & 02) {
l ^= b << 62;
h ^= b >> 2;
}
if (top3b & 04) {
l ^= b << 63;
h ^= b >> 1;
}
*r1 = h;
*r0 = l;
}
# endif
/*
* Product of two polynomials a, b each with degree < 2 * BN_BITS2 - 1,
* result is a polynomial r with degree < 4 * BN_BITS2 - 1 The caller MUST
* ensure that the variables have the right amount of space allocated.
*/
static void bn_GF2m_mul_2x2(BN_ULONG *r, const BN_ULONG a1, const BN_ULONG a0,
const BN_ULONG b1, const BN_ULONG b0)
{
BN_ULONG m1, m0;
/* r[3] = h1, r[2] = h0; r[1] = l1; r[0] = l0 */
bn_GF2m_mul_1x1(r + 3, r + 2, a1, b1);
bn_GF2m_mul_1x1(r + 1, r, a0, b0);
bn_GF2m_mul_1x1(&m1, &m0, a0 ^ a1, b0 ^ b1);
/* Correction on m1 ^= l1 ^ h1; m0 ^= l0 ^ h0; */
r[2] ^= m1 ^ r[1] ^ r[3]; /* h0 ^= m1 ^ l1 ^ h1; */
r[1] = r[3] ^ r[2] ^ r[0] ^ m1 ^ m0; /* l1 ^= l0 ^ h0 ^ m0; */
}
# else
void bn_GF2m_mul_2x2(BN_ULONG *r, BN_ULONG a1, BN_ULONG a0, BN_ULONG b1,
BN_ULONG b0);
# endif
/*
* Add polynomials a and b and store result in r; r could be a or b, a and b
* could be equal; r is the bitwise XOR of a and b.
*/
int BN_GF2m_add(BIGNUM *r, const BIGNUM *a, const BIGNUM *b)
{
int i;
const BIGNUM *at, *bt;
bn_check_top(a);
bn_check_top(b);
if (a->top < b->top) {
at = b;
bt = a;
} else {
at = a;
bt = b;
}
if (bn_wexpand(r, at->top) == NULL)
return 0;
for (i = 0; i < bt->top; i++) {
r->d[i] = at->d[i] ^ bt->d[i];
}
for (; i < at->top; i++) {
r->d[i] = at->d[i];
}
r->top = at->top;
bn_correct_top(r);
return 1;
}
/*-
* Some functions allow for representation of the irreducible polynomials
* as an int[], say p. The irreducible f(t) is then of the form:
* t^p[0] + t^p[1] + ... + t^p[k]
* where m = p[0] > p[1] > ... > p[k] = 0.
*/
/* Performs modular reduction of a and store result in r. r could be a. */
int BN_GF2m_mod_arr(BIGNUM *r, const BIGNUM *a, const int p[])
{
int j, k;
int n, dN, d0, d1;
BN_ULONG zz, *z;
bn_check_top(a);
if (!p[0]) {
/* reduction mod 1 => return 0 */
BN_zero(r);
return 1;
}
/*
* Since the algorithm does reduction in the r value, if a != r, copy the
* contents of a into r so we can do reduction in r.
*/
if (a != r) {
if (!bn_wexpand(r, a->top))
return 0;
for (j = 0; j < a->top; j++) {
r->d[j] = a->d[j];
}
r->top = a->top;
}
z = r->d;
/* start reduction */
dN = p[0] / BN_BITS2;
for (j = r->top - 1; j > dN;) {
zz = z[j];
if (z[j] == 0) {
j--;
continue;
}
z[j] = 0;
for (k = 1; p[k] != 0; k++) {
/* reducing component t^p[k] */
n = p[0] - p[k];
d0 = n % BN_BITS2;
d1 = BN_BITS2 - d0;
n /= BN_BITS2;
z[j - n] ^= (zz >> d0);
if (d0)
z[j - n - 1] ^= (zz << d1);
}
/* reducing component t^0 */
n = dN;
d0 = p[0] % BN_BITS2;
d1 = BN_BITS2 - d0;
z[j - n] ^= (zz >> d0);
if (d0)
z[j - n - 1] ^= (zz << d1);
}
/* final round of reduction */
while (j == dN) {
d0 = p[0] % BN_BITS2;
zz = z[dN] >> d0;
if (zz == 0)
break;
d1 = BN_BITS2 - d0;
/* clear up the top d1 bits */
if (d0)
z[dN] = (z[dN] << d1) >> d1;
else
z[dN] = 0;
z[0] ^= zz; /* reduction t^0 component */
for (k = 1; p[k] != 0; k++) {
BN_ULONG tmp_ulong;
/* reducing component t^p[k] */
n = p[k] / BN_BITS2;
d0 = p[k] % BN_BITS2;
d1 = BN_BITS2 - d0;
z[n] ^= (zz << d0);
if (d0 && (tmp_ulong = zz >> d1))
z[n + 1] ^= tmp_ulong;
}
}
bn_correct_top(r);
return 1;
}
/*
* Performs modular reduction of a by p and store result in r. r could be a.
* This function calls down to the BN_GF2m_mod_arr implementation; this wrapper
* function is only provided for convenience; for best performance, use the
* BN_GF2m_mod_arr function.
*/
int BN_GF2m_mod(BIGNUM *r, const BIGNUM *a, const BIGNUM *p)
{
int ret = 0;
int arr[6];
bn_check_top(a);
bn_check_top(p);
ret = BN_GF2m_poly2arr(p, arr, sizeof(arr) / sizeof(arr[0]));
if (!ret || ret > (int)(sizeof(arr) / sizeof(arr[0]))) {
BNerr(BN_F_BN_GF2M_MOD, BN_R_INVALID_LENGTH);
return 0;
}
ret = BN_GF2m_mod_arr(r, a, arr);
bn_check_top(r);
return ret;
}
/*
* Compute the product of two polynomials a and b, reduce modulo p, and store
* the result in r. r could be a or b; a could be b.
*/
int BN_GF2m_mod_mul_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b,
const int p[], BN_CTX *ctx)
{
int zlen, i, j, k, ret = 0;
BIGNUM *s;
BN_ULONG x1, x0, y1, y0, zz[4];
bn_check_top(a);
bn_check_top(b);
if (a == b) {
return BN_GF2m_mod_sqr_arr(r, a, p, ctx);
}
BN_CTX_start(ctx);
if ((s = BN_CTX_get(ctx)) == NULL)
goto err;
zlen = a->top + b->top + 4;
if (!bn_wexpand(s, zlen))
goto err;
s->top = zlen;
for (i = 0; i < zlen; i++)
s->d[i] = 0;
for (j = 0; j < b->top; j += 2) {
y0 = b->d[j];
y1 = ((j + 1) == b->top) ? 0 : b->d[j + 1];
for (i = 0; i < a->top; i += 2) {
x0 = a->d[i];
x1 = ((i + 1) == a->top) ? 0 : a->d[i + 1];
bn_GF2m_mul_2x2(zz, x1, x0, y1, y0);
for (k = 0; k < 4; k++)
s->d[i + j + k] ^= zz[k];
}
}
bn_correct_top(s);
if (BN_GF2m_mod_arr(r, s, p))
ret = 1;
bn_check_top(r);
err:
BN_CTX_end(ctx);
return ret;
}
/*
* Compute the product of two polynomials a and b, reduce modulo p, and store
* the result in r. r could be a or b; a could equal b. This function calls
* down to the BN_GF2m_mod_mul_arr implementation; this wrapper function is
* only provided for convenience; for best performance, use the
* BN_GF2m_mod_mul_arr function.
*/
int BN_GF2m_mod_mul(BIGNUM *r, const BIGNUM *a, const BIGNUM *b,
const BIGNUM *p, BN_CTX *ctx)
{
int ret = 0;
const int max = BN_num_bits(p) + 1;
int *arr = NULL;
bn_check_top(a);
bn_check_top(b);
bn_check_top(p);
if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL)
goto err;
ret = BN_GF2m_poly2arr(p, arr, max);
if (!ret || ret > max) {
BNerr(BN_F_BN_GF2M_MOD_MUL, BN_R_INVALID_LENGTH);
goto err;
}
ret = BN_GF2m_mod_mul_arr(r, a, b, arr, ctx);
bn_check_top(r);
err:
if (arr)
OPENSSL_free(arr);
return ret;
}
/* Square a, reduce the result mod p, and store it in a. r could be a. */
int BN_GF2m_mod_sqr_arr(BIGNUM *r, const BIGNUM *a, const int p[],
BN_CTX *ctx)
{
int i, ret = 0;
BIGNUM *s;
bn_check_top(a);
BN_CTX_start(ctx);
if ((s = BN_CTX_get(ctx)) == NULL)
goto err;
if (!bn_wexpand(s, 2 * a->top))
goto err;
for (i = a->top - 1; i >= 0; i--) {
s->d[2 * i + 1] = SQR1(a->d[i]);
s->d[2 * i] = SQR0(a->d[i]);
}
s->top = 2 * a->top;
bn_correct_top(s);
if (!BN_GF2m_mod_arr(r, s, p))
goto err;
bn_check_top(r);
ret = 1;
err:
BN_CTX_end(ctx);
return ret;
}
/*
* Square a, reduce the result mod p, and store it in a. r could be a. This
* function calls down to the BN_GF2m_mod_sqr_arr implementation; this
* wrapper function is only provided for convenience; for best performance,
* use the BN_GF2m_mod_sqr_arr function.
*/
int BN_GF2m_mod_sqr(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
{
int ret = 0;
const int max = BN_num_bits(p) + 1;
int *arr = NULL;
bn_check_top(a);
bn_check_top(p);
if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL)
goto err;
ret = BN_GF2m_poly2arr(p, arr, max);
if (!ret || ret > max) {
BNerr(BN_F_BN_GF2M_MOD_SQR, BN_R_INVALID_LENGTH);
goto err;
}
ret = BN_GF2m_mod_sqr_arr(r, a, arr, ctx);
bn_check_top(r);
err:
if (arr)
OPENSSL_free(arr);
return ret;
}
/*
* Invert a, reduce modulo p, and store the result in r. r could be a. Uses
* Modified Almost Inverse Algorithm (Algorithm 10) from Hankerson, D.,
* Hernandez, J.L., and Menezes, A. "Software Implementation of Elliptic
* Curve Cryptography Over Binary Fields".
*/
int BN_GF2m_mod_inv(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
{
BIGNUM *b, *c = NULL, *u = NULL, *v = NULL, *tmp;
int ret = 0;
bn_check_top(a);
bn_check_top(p);
BN_CTX_start(ctx);
if ((b = BN_CTX_get(ctx)) == NULL)
goto err;
if ((c = BN_CTX_get(ctx)) == NULL)
goto err;
if ((u = BN_CTX_get(ctx)) == NULL)
goto err;
if ((v = BN_CTX_get(ctx)) == NULL)
goto err;
if (!BN_GF2m_mod(u, a, p))
goto err;
if (BN_is_zero(u))
goto err;
if (!BN_copy(v, p))
goto err;
# if 0
if (!BN_one(b))
goto err;
while (1) {
while (!BN_is_odd(u)) {
if (BN_is_zero(u))
goto err;
if (!BN_rshift1(u, u))
goto err;
if (BN_is_odd(b)) {
if (!BN_GF2m_add(b, b, p))
goto err;
}
if (!BN_rshift1(b, b))
goto err;
}
if (BN_abs_is_word(u, 1))
break;
if (BN_num_bits(u) < BN_num_bits(v)) {
tmp = u;
u = v;
v = tmp;
tmp = b;
b = c;
c = tmp;
}
if (!BN_GF2m_add(u, u, v))
goto err;
if (!BN_GF2m_add(b, b, c))
goto err;
}
# else
{
int i;
int ubits = BN_num_bits(u);
int vbits = BN_num_bits(v); /* v is copy of p */
int top = p->top;
BN_ULONG *udp, *bdp, *vdp, *cdp;
if (!bn_wexpand(u, top))
goto err;
udp = u->d;
for (i = u->top; i < top; i++)
udp[i] = 0;
u->top = top;
if (!bn_wexpand(b, top))
goto err;
bdp = b->d;
bdp[0] = 1;
for (i = 1; i < top; i++)
bdp[i] = 0;
b->top = top;
if (!bn_wexpand(c, top))
goto err;
cdp = c->d;
for (i = 0; i < top; i++)
cdp[i] = 0;
c->top = top;
vdp = v->d; /* It pays off to "cache" *->d pointers,
* because it allows optimizer to be more
* aggressive. But we don't have to "cache"
* p->d, because *p is declared 'const'... */
while (1) {
while (ubits && !(udp[0] & 1)) {
BN_ULONG u0, u1, b0, b1, mask;
u0 = udp[0];
b0 = bdp[0];
mask = (BN_ULONG)0 - (b0 & 1);
b0 ^= p->d[0] & mask;
for (i = 0; i < top - 1; i++) {
u1 = udp[i + 1];
udp[i] = ((u0 >> 1) | (u1 << (BN_BITS2 - 1))) & BN_MASK2;
u0 = u1;
b1 = bdp[i + 1] ^ (p->d[i + 1] & mask);
bdp[i] = ((b0 >> 1) | (b1 << (BN_BITS2 - 1))) & BN_MASK2;
b0 = b1;
}
udp[i] = u0 >> 1;
bdp[i] = b0 >> 1;
ubits--;
}
if (ubits <= BN_BITS2) {
if (udp[0] == 0) /* poly was reducible */
goto err;
if (udp[0] == 1)
break;
}
if (ubits < vbits) {
i = ubits;
ubits = vbits;
vbits = i;
tmp = u;
u = v;
v = tmp;
tmp = b;
b = c;
c = tmp;
udp = vdp;
vdp = v->d;
bdp = cdp;
cdp = c->d;
}
for (i = 0; i < top; i++) {
udp[i] ^= vdp[i];
bdp[i] ^= cdp[i];
}
if (ubits == vbits) {
BN_ULONG ul;
int utop = (ubits - 1) / BN_BITS2;
while ((ul = udp[utop]) == 0 && utop)
utop--;
ubits = utop * BN_BITS2 + BN_num_bits_word(ul);
}
}
bn_correct_top(b);
}
# endif
if (!BN_copy(r, b))
goto err;
bn_check_top(r);
ret = 1;
err:
# ifdef BN_DEBUG /* BN_CTX_end would complain about the
* expanded form */
bn_correct_top(c);
bn_correct_top(u);
bn_correct_top(v);
# endif
BN_CTX_end(ctx);
return ret;
}
/*
* Invert xx, reduce modulo p, and store the result in r. r could be xx.
* This function calls down to the BN_GF2m_mod_inv implementation; this
* wrapper function is only provided for convenience; for best performance,
* use the BN_GF2m_mod_inv function.
*/
int BN_GF2m_mod_inv_arr(BIGNUM *r, const BIGNUM *xx, const int p[],
BN_CTX *ctx)
{
BIGNUM *field;
int ret = 0;
bn_check_top(xx);
BN_CTX_start(ctx);
if ((field = BN_CTX_get(ctx)) == NULL)
goto err;
if (!BN_GF2m_arr2poly(p, field))
goto err;
ret = BN_GF2m_mod_inv(r, xx, field, ctx);
bn_check_top(r);
err:
BN_CTX_end(ctx);
return ret;
}
# ifndef OPENSSL_SUN_GF2M_DIV
/*
* Divide y by x, reduce modulo p, and store the result in r. r could be x
* or y, x could equal y.
*/
int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x,
const BIGNUM *p, BN_CTX *ctx)
{
BIGNUM *xinv = NULL;
int ret = 0;
bn_check_top(y);
bn_check_top(x);
bn_check_top(p);
BN_CTX_start(ctx);
xinv = BN_CTX_get(ctx);
if (xinv == NULL)
goto err;
if (!BN_GF2m_mod_inv(xinv, x, p, ctx))
goto err;
if (!BN_GF2m_mod_mul(r, y, xinv, p, ctx))
goto err;
bn_check_top(r);
ret = 1;
err:
BN_CTX_end(ctx);
return ret;
}
# else
/*
* Divide y by x, reduce modulo p, and store the result in r. r could be x
* or y, x could equal y. Uses algorithm Modular_Division_GF(2^m) from
* Chang-Shantz, S. "From Euclid's GCD to Montgomery Multiplication to the
* Great Divide".
*/
int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x,
const BIGNUM *p, BN_CTX *ctx)
{
BIGNUM *a, *b, *u, *v;
int ret = 0;
bn_check_top(y);
bn_check_top(x);
bn_check_top(p);
BN_CTX_start(ctx);
a = BN_CTX_get(ctx);
b = BN_CTX_get(ctx);
u = BN_CTX_get(ctx);
v = BN_CTX_get(ctx);
if (v == NULL)
goto err;
/* reduce x and y mod p */
if (!BN_GF2m_mod(u, y, p))
goto err;
if (!BN_GF2m_mod(a, x, p))
goto err;
if (!BN_copy(b, p))
goto err;
while (!BN_is_odd(a)) {
if (!BN_rshift1(a, a))
goto err;
if (BN_is_odd(u))
if (!BN_GF2m_add(u, u, p))
goto err;
if (!BN_rshift1(u, u))
goto err;
}
do {
if (BN_GF2m_cmp(b, a) > 0) {
if (!BN_GF2m_add(b, b, a))
goto err;
if (!BN_GF2m_add(v, v, u))
goto err;
do {
if (!BN_rshift1(b, b))
goto err;
if (BN_is_odd(v))
if (!BN_GF2m_add(v, v, p))
goto err;
if (!BN_rshift1(v, v))
goto err;
} while (!BN_is_odd(b));
} else if (BN_abs_is_word(a, 1))
break;
else {
if (!BN_GF2m_add(a, a, b))
goto err;
if (!BN_GF2m_add(u, u, v))
goto err;
do {
if (!BN_rshift1(a, a))
goto err;
if (BN_is_odd(u))
if (!BN_GF2m_add(u, u, p))
goto err;
if (!BN_rshift1(u, u))
goto err;
} while (!BN_is_odd(a));
}
} while (1);
if (!BN_copy(r, u))
goto err;
bn_check_top(r);
ret = 1;
err:
BN_CTX_end(ctx);
return ret;
}
# endif
/*
* Divide yy by xx, reduce modulo p, and store the result in r. r could be xx
* * or yy, xx could equal yy. This function calls down to the
* BN_GF2m_mod_div implementation; this wrapper function is only provided for
* convenience; for best performance, use the BN_GF2m_mod_div function.
*/
int BN_GF2m_mod_div_arr(BIGNUM *r, const BIGNUM *yy, const BIGNUM *xx,
const int p[], BN_CTX *ctx)
{
BIGNUM *field;
int ret = 0;
bn_check_top(yy);
bn_check_top(xx);
BN_CTX_start(ctx);
if ((field = BN_CTX_get(ctx)) == NULL)
goto err;
if (!BN_GF2m_arr2poly(p, field))
goto err;
ret = BN_GF2m_mod_div(r, yy, xx, field, ctx);
bn_check_top(r);
err:
BN_CTX_end(ctx);
return ret;
}
/*
* Compute the bth power of a, reduce modulo p, and store the result in r. r
* could be a. Uses simple square-and-multiply algorithm A.5.1 from IEEE
* P1363.
*/
int BN_GF2m_mod_exp_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b,
const int p[], BN_CTX *ctx)
{
int ret = 0, i, n;
BIGNUM *u;
bn_check_top(a);
bn_check_top(b);
if (BN_is_zero(b))
return (BN_one(r));
if (BN_abs_is_word(b, 1))
return (BN_copy(r, a) != NULL);
BN_CTX_start(ctx);
if ((u = BN_CTX_get(ctx)) == NULL)
goto err;
if (!BN_GF2m_mod_arr(u, a, p))
goto err;
n = BN_num_bits(b) - 1;
for (i = n - 1; i >= 0; i--) {
if (!BN_GF2m_mod_sqr_arr(u, u, p, ctx))
goto err;
if (BN_is_bit_set(b, i)) {
if (!BN_GF2m_mod_mul_arr(u, u, a, p, ctx))
goto err;
}
}
if (!BN_copy(r, u))
goto err;
bn_check_top(r);
ret = 1;
err:
BN_CTX_end(ctx);
return ret;
}
/*
* Compute the bth power of a, reduce modulo p, and store the result in r. r
* could be a. This function calls down to the BN_GF2m_mod_exp_arr
* implementation; this wrapper function is only provided for convenience;
* for best performance, use the BN_GF2m_mod_exp_arr function.
*/
int BN_GF2m_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *b,
const BIGNUM *p, BN_CTX *ctx)
{
int ret = 0;
const int max = BN_num_bits(p) + 1;
int *arr = NULL;
bn_check_top(a);
bn_check_top(b);
bn_check_top(p);
if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL)
goto err;
ret = BN_GF2m_poly2arr(p, arr, max);
if (!ret || ret > max) {
BNerr(BN_F_BN_GF2M_MOD_EXP, BN_R_INVALID_LENGTH);
goto err;
}
ret = BN_GF2m_mod_exp_arr(r, a, b, arr, ctx);
bn_check_top(r);
err:
if (arr)
OPENSSL_free(arr);
return ret;
}
/*
* Compute the square root of a, reduce modulo p, and store the result in r.
* r could be a. Uses exponentiation as in algorithm A.4.1 from IEEE P1363.
*/
int BN_GF2m_mod_sqrt_arr(BIGNUM *r, const BIGNUM *a, const int p[],
BN_CTX *ctx)
{
int ret = 0;
BIGNUM *u;
bn_check_top(a);
if (!p[0]) {
/* reduction mod 1 => return 0 */
BN_zero(r);
return 1;
}
BN_CTX_start(ctx);
if ((u = BN_CTX_get(ctx)) == NULL)
goto err;
if (!BN_set_bit(u, p[0] - 1))
goto err;
ret = BN_GF2m_mod_exp_arr(r, a, u, p, ctx);
bn_check_top(r);
err:
BN_CTX_end(ctx);
return ret;
}
/*
* Compute the square root of a, reduce modulo p, and store the result in r.
* r could be a. This function calls down to the BN_GF2m_mod_sqrt_arr
* implementation; this wrapper function is only provided for convenience;
* for best performance, use the BN_GF2m_mod_sqrt_arr function.
*/
int BN_GF2m_mod_sqrt(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
{
int ret = 0;
const int max = BN_num_bits(p) + 1;
int *arr = NULL;
bn_check_top(a);
bn_check_top(p);
if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL)
goto err;
ret = BN_GF2m_poly2arr(p, arr, max);
if (!ret || ret > max) {
BNerr(BN_F_BN_GF2M_MOD_SQRT, BN_R_INVALID_LENGTH);
goto err;
}
ret = BN_GF2m_mod_sqrt_arr(r, a, arr, ctx);
bn_check_top(r);
err:
if (arr)
OPENSSL_free(arr);
return ret;
}
/*
* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns
* 0. Uses algorithms A.4.7 and A.4.6 from IEEE P1363.
*/
int BN_GF2m_mod_solve_quad_arr(BIGNUM *r, const BIGNUM *a_, const int p[],
BN_CTX *ctx)
{
int ret = 0, count = 0, j;
BIGNUM *a, *z, *rho, *w, *w2, *tmp;
bn_check_top(a_);
if (!p[0]) {
/* reduction mod 1 => return 0 */
BN_zero(r);
return 1;
}
BN_CTX_start(ctx);
a = BN_CTX_get(ctx);
z = BN_CTX_get(ctx);
w = BN_CTX_get(ctx);
if (w == NULL)
goto err;
if (!BN_GF2m_mod_arr(a, a_, p))
goto err;
if (BN_is_zero(a)) {
BN_zero(r);
ret = 1;
goto err;
}
if (p[0] & 0x1) { /* m is odd */
/* compute half-trace of a */
if (!BN_copy(z, a))
goto err;
for (j = 1; j <= (p[0] - 1) / 2; j++) {
if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx))
goto err;
if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx))
goto err;
if (!BN_GF2m_add(z, z, a))
goto err;
}
} else { /* m is even */
rho = BN_CTX_get(ctx);
w2 = BN_CTX_get(ctx);
tmp = BN_CTX_get(ctx);
if (tmp == NULL)
goto err;
do {
if (!BN_rand(rho, p[0], 0, 0))
goto err;
if (!BN_GF2m_mod_arr(rho, rho, p))
goto err;
BN_zero(z);
if (!BN_copy(w, rho))
goto err;
for (j = 1; j <= p[0] - 1; j++) {
if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx))
goto err;
if (!BN_GF2m_mod_sqr_arr(w2, w, p, ctx))
goto err;
if (!BN_GF2m_mod_mul_arr(tmp, w2, a, p, ctx))
goto err;
if (!BN_GF2m_add(z, z, tmp))
goto err;
if (!BN_GF2m_add(w, w2, rho))
goto err;
}
count++;
} while (BN_is_zero(w) && (count < MAX_ITERATIONS));
if (BN_is_zero(w)) {
BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR, BN_R_TOO_MANY_ITERATIONS);
goto err;
}
}
if (!BN_GF2m_mod_sqr_arr(w, z, p, ctx))
goto err;
if (!BN_GF2m_add(w, z, w))
goto err;
if (BN_GF2m_cmp(w, a)) {
BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR, BN_R_NO_SOLUTION);
goto err;
}
if (!BN_copy(r, z))
goto err;
bn_check_top(r);
ret = 1;
err:
BN_CTX_end(ctx);
return ret;
}
/*
* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns
* 0. This function calls down to the BN_GF2m_mod_solve_quad_arr
* implementation; this wrapper function is only provided for convenience;
* for best performance, use the BN_GF2m_mod_solve_quad_arr function.
*/
int BN_GF2m_mod_solve_quad(BIGNUM *r, const BIGNUM *a, const BIGNUM *p,
BN_CTX *ctx)
{
int ret = 0;
const int max = BN_num_bits(p) + 1;
int *arr = NULL;
bn_check_top(a);
bn_check_top(p);
if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL)
goto err;
ret = BN_GF2m_poly2arr(p, arr, max);
if (!ret || ret > max) {
BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD, BN_R_INVALID_LENGTH);
goto err;
}
ret = BN_GF2m_mod_solve_quad_arr(r, a, arr, ctx);
bn_check_top(r);
err:
if (arr)
OPENSSL_free(arr);
return ret;
}
/*
* Convert the bit-string representation of a polynomial ( \sum_{i=0}^n a_i *
* x^i) into an array of integers corresponding to the bits with non-zero
* coefficient. Array is terminated with -1. Up to max elements of the array
* will be filled. Return value is total number of array elements that would
* be filled if array was large enough.
*/
int BN_GF2m_poly2arr(const BIGNUM *a, int p[], int max)
{
int i, j, k = 0;
BN_ULONG mask;
if (BN_is_zero(a))
return 0;
for (i = a->top - 1; i >= 0; i--) {
if (!a->d[i])
/* skip word if a->d[i] == 0 */
continue;
mask = BN_TBIT;
for (j = BN_BITS2 - 1; j >= 0; j--) {
if (a->d[i] & mask) {
if (k < max)
p[k] = BN_BITS2 * i + j;
k++;
}
mask >>= 1;
}
}
if (k < max) {
p[k] = -1;
k++;
}
return k;
}
/*
* Convert the coefficient array representation of a polynomial to a
* bit-string. The array must be terminated by -1.
*/
int BN_GF2m_arr2poly(const int p[], BIGNUM *a)
{
int i;
bn_check_top(a);
BN_zero(a);
for (i = 0; p[i] != -1; i++) {
if (BN_set_bit(a, p[i]) == 0)
return 0;
}
bn_check_top(a);
return 1;
}
#endif
