/* ==================================================================== * Copyright (c) 1998-2000 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 "internal/cryptlib.h" #include "bn_lcl.h" /* least significant word */ #define BN_lsw(n) (((n)->top == 0) ? (BN_ULONG) 0 : (n)->d[0]) /* Returns -2 for errors because both -1 and 0 are valid results. */ int BN_kronecker(const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx) { int i; int ret = -2; /* avoid 'uninitialized' warning */ int err = 0; BIGNUM *A, *B, *tmp; /*- * In 'tab', only odd-indexed entries are relevant: * For any odd BIGNUM n, * tab[BN_lsw(n) & 7] * is $(-1)^{(n^2-1)/8}$ (using TeX notation). * Note that the sign of n does not matter. */ static const int tab[8] = { 0, 1, 0, -1, 0, -1, 0, 1 }; bn_check_top(a); bn_check_top(b); BN_CTX_start(ctx); A = BN_CTX_get(ctx); B = BN_CTX_get(ctx); if (B == NULL) goto end; err = !BN_copy(A, a); if (err) goto end; err = !BN_copy(B, b); if (err) goto end; /* * Kronecker symbol, imlemented according to Henri Cohen, * "A Course in Computational Algebraic Number Theory" * (algorithm 1.4.10). */ /* Cohen's step 1: */ if (BN_is_zero(B)) { ret = BN_abs_is_word(A, 1); goto end; } /* Cohen's step 2: */ if (!BN_is_odd(A) && !BN_is_odd(B)) { ret = 0; goto end; } /* now B is non-zero */ i = 0; while (!BN_is_bit_set(B, i)) i++; err = !BN_rshift(B, B, i); if (err) goto end; if (i & 1) { /* i is odd */ /* (thus B was even, thus A must be odd!) */ /* set 'ret' to $(-1)^{(A^2-1)/8}$ */ ret = tab[BN_lsw(A) & 7]; } else { /* i is even */ ret = 1; } if (B->neg) { B->neg = 0; if (A->neg) ret = -ret; } /* * now B is positive and odd, so what remains to be done is to compute * the Jacobi symbol (A/B) and multiply it by 'ret' */ while (1) { /* Cohen's step 3: */ /* B is positive and odd */ if (BN_is_zero(A)) { ret = BN_is_one(B) ? ret : 0; goto end; } /* now A is non-zero */ i = 0; while (!BN_is_bit_set(A, i)) i++; err = !BN_rshift(A, A, i); if (err) goto end; if (i & 1) { /* i is odd */ /* multiply 'ret' by $(-1)^{(B^2-1)/8}$ */ ret = ret * tab[BN_lsw(B) & 7]; } /* Cohen's step 4: */ /* multiply 'ret' by $(-1)^{(A-1)(B-1)/4}$ */ if ((A->neg ? ~BN_lsw(A) : BN_lsw(A)) & BN_lsw(B) & 2) ret = -ret; /* (A, B) := (B mod |A|, |A|) */ err = !BN_nnmod(B, B, A, ctx); if (err) goto end; tmp = A; A = B; B = tmp; tmp->neg = 0; } end: BN_CTX_end(ctx); if (err) return -2; else return ret; }