bn_gf2m.c revision 9dc0df1bac950d6e491f9a7c7e4888f2b301cb15
/* ====================================================================
* 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,
* 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
* 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"
/* Maximum number of iterations before BN_GF2m_mod_solve_quad_arr should fail. */
#define MAX_ITERATIONS 50
{ 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) \
#define SQR0(w) \
#endif
#ifdef THIRTY_TWO_BIT
#define SQR1(w) \
#define SQR0(w) \
#endif
#ifdef SIXTEEN_BIT
#define SQR1(w) \
#define SQR0(w) \
#endif
#ifdef EIGHT_BIT
#define SQR1(w) \
#define SQR0(w) \
SQR_tb[(w) & 15]
#endif
/* 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 EIGHT_BIT
{
register BN_ULONG h, l, s;
s = tab[b & 0x3]; l = s;
/* compensate for the top bit of a */
}
#endif
#ifdef SIXTEEN_BIT
{
register BN_ULONG h, l, s;
s = tab[b & 0x3]; l = s;
/* compensate for the top bit of a */
}
#endif
#ifdef THIRTY_TWO_BIT
{
register BN_ULONG h, l, s;
s = tab[b & 0x7]; l = s;
/* compensate for the top two bits of a */
}
#endif
#if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG)
{
register BN_ULONG h, l, s;
s = tab[b & 0xF]; l = s;
/* compensate for the top three bits of a */
}
#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)
{
/* r[3] = h1, r[2] = h0; r[1] = l1; r[0] = l0 */
/* Correction on m1 ^= l1 ^ h1; m0 ^= l0 ^ h0; */
}
/* 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 i;
bn_check_top(a);
bn_check_top(b);
{
}
{
r->d[i] = at->d[i];
}
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 j, k;
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];
}
}
z = r->d;
/* start reduction */
{
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];
n /= BN_BITS2;
}
/* reducing component t^0 */
n = dN;
}
/* final round of reduction */
while (j == dN)
{
if (zz == 0) break;
z[0] ^= zz; /* reduction t^0 component */
for (k = 1; p[k] != 0; k++)
{
/* reducing component t^p[k]*/
n = p[k] / BN_BITS2;
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 ret = 0;
const int max = BN_num_bits(p);
bn_check_top(a);
bn_check_top(p);
{
goto err;
}
bn_check_top(r);
err:
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 unsigned int p[], BN_CTX *ctx)
{
BIGNUM *s;
bn_check_top(a);
bn_check_top(b);
if (a == b)
{
return BN_GF2m_mod_sqr_arr(r, a, p, ctx);
}
for (i = 0; i < zlen; i++) s->d[i] = 0;
for (j = 0; j < b->top; j += 2)
{
y0 = b->d[j];
for (i = 0; i < a->top; i += 2)
{
x0 = a->d[i];
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:
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 ret = 0;
const int max = BN_num_bits(p);
bn_check_top(a);
bn_check_top(b);
bn_check_top(p);
{
goto err;
}
bn_check_top(r);
err:
return ret;
}
/* Square a, reduce the result mod p, and store it in a. r could be a. */
{
int i, ret = 0;
BIGNUM *s;
bn_check_top(a);
for (i = a->top - 1; i >= 0; i--)
{
s->d[2*i ] = SQR0(a->d[i]);
}
bn_correct_top(s);
if (!BN_GF2m_mod_arr(r, s, p)) goto err;
bn_check_top(r);
ret = 1;
err:
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 ret = 0;
const int max = BN_num_bits(p);
bn_check_top(a);
bn_check_top(p);
{
goto err;
}
bn_check_top(r);
err:
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 ret = 0;
bn_check_top(a);
bn_check_top(p);
b = BN_CTX_get(ctx);
c = BN_CTX_get(ctx);
u = BN_CTX_get(ctx);
v = BN_CTX_get(ctx);
if (!BN_GF2m_mod(u, a, p)) goto err;
if (BN_is_zero(u)) goto err;
while (1)
{
while (!BN_is_odd(u))
{
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))
{
}
if (!BN_GF2m_add(u, u, v)) goto err;
if (!BN_GF2m_add(b, b, c)) goto err;
}
bn_check_top(r);
ret = 1;
err:
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 ret = 0;
bn_check_top(r);
err:
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 ret = 0;
bn_check_top(y);
bn_check_top(x);
bn_check_top(p);
bn_check_top(r);
ret = 1;
err:
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".
*/
{
BIGNUM *a, *b, *u, *v;
int ret = 0;
bn_check_top(y);
bn_check_top(x);
bn_check_top(p);
a = BN_CTX_get(ctx);
b = BN_CTX_get(ctx);
u = BN_CTX_get(ctx);
v = BN_CTX_get(ctx);
/* reduce x and y mod p */
if (!BN_GF2m_mod(u, y, p)) goto err;
if (!BN_GF2m_mod(a, x, p)) goto err;
while (!BN_is_odd(a))
{
if (!BN_rshift1(a, a)) 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_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_rshift1(u, u)) goto err;
} while (!BN_is_odd(a));
}
} while (1);
bn_check_top(r);
ret = 1;
err:
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 unsigned int p[], BN_CTX *ctx)
{
int ret = 0;
bn_check_top(r);
err:
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 unsigned 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))
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_is_bit_set(b, i))
{
}
}
bn_check_top(r);
ret = 1;
err:
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 ret = 0;
const int max = BN_num_bits(p);
bn_check_top(a);
bn_check_top(b);
bn_check_top(p);
{
goto err;
}
bn_check_top(r);
err:
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 ret = 0;
BIGNUM *u;
bn_check_top(a);
if (!p[0])
{
/* reduction mod 1 => return 0 */
BN_zero(r);
return 1;
}
bn_check_top(r);
err:
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 ret = 0;
const int max = BN_num_bits(p);
bn_check_top(a);
bn_check_top(p);
{
goto err;
}
bn_check_top(r);
err:
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.
*/
{
unsigned int j;
if (!p[0])
{
/* reduction mod 1 => return 0 */
BN_zero(r);
return 1;
}
a = BN_CTX_get(ctx);
z = BN_CTX_get(ctx);
w = BN_CTX_get(ctx);
if (BN_is_zero(a))
{
BN_zero(r);
ret = 1;
goto err;
}
if (p[0] & 0x1) /* m is odd */
{
/* compute half-trace of a */
for (j = 1; j <= (p[0] - 1) / 2; j++)
{
if (!BN_GF2m_add(z, z, a)) goto err;
}
}
else /* m is even */
{
do
{
BN_zero(z);
for (j = 1; j <= p[0] - 1; j++)
{
}
count++;
if (BN_is_zero(w))
{
goto err;
}
}
if (!BN_GF2m_add(w, z, w)) goto err;
if (BN_GF2m_cmp(w, a))
{
goto err;
}
bn_check_top(r);
ret = 1;
err:
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 ret = 0;
const int max = BN_num_bits(p);
bn_check_top(a);
bn_check_top(p);
if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) *
{
goto err;
}
bn_check_top(r);
err:
return ret;
}
/* Convert the bit-string representation of a polynomial
* ( \sum_{i=0}^n a_i * x^i , where a_0 is *not* zero) into an array
* of integers corresponding to the bits with non-zero coefficient.
* Up to max elements of the array will be filled. Return value is total
* number of coefficients that would be extracted if array was large enough.
*/
{
int i, j, k = 0;
if (BN_is_zero(a) || !BN_is_bit_set(a, 0))
/* a_0 == 0 => return error (the unsigned int array
* must be terminated by 0)
*/
return 0;
for (i = a->top - 1; i >= 0; i--)
{
if (!a->d[i])
/* skip word if a->d[i] == 0 */
continue;
for (j = BN_BITS2 - 1; j >= 0; j--)
{
if (a->d[i] & mask)
{
k++;
}
mask >>= 1;
}
}
return k;
}
/* Convert the coefficient array representation of a polynomial to a
* bit-string. The array must be terminated by 0.
*/
int BN_GF2m_arr2poly(const unsigned int p[], BIGNUM *a)
{
int i;
bn_check_top(a);
BN_zero(a);
for (i = 0; p[i] != 0; i++)
{
BN_set_bit(a, p[i]);
}
BN_set_bit(a, 0);
bn_check_top(a);
return 1;
}