1 /* ==================================================================== 2 * Copyright (c) 1998-2000 The OpenSSL Project. All rights reserved. 3 * 4 * Redistribution and use in source and binary forms, with or without 5 * modification, are permitted provided that the following conditions 6 * are met: 7 * 8 * 1. Redistributions of source code must retain the above copyright 9 * notice, this list of conditions and the following disclaimer. 10 * 11 * 2. Redistributions in binary form must reproduce the above copyright 12 * notice, this list of conditions and the following disclaimer in 13 * the documentation and/or other materials provided with the 14 * distribution. 15 * 16 * 3. All advertising materials mentioning features or use of this 17 * software must display the following acknowledgment: 18 * "This product includes software developed by the OpenSSL Project 19 * for use in the OpenSSL Toolkit. (http://www.openssl.org/)" 20 * 21 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to 22 * endorse or promote products derived from this software without 23 * prior written permission. For written permission, please contact 24 * openssl-core (at) openssl.org. 25 * 26 * 5. Products derived from this software may not be called "OpenSSL" 27 * nor may "OpenSSL" appear in their names without prior written 28 * permission of the OpenSSL Project. 29 * 30 * 6. Redistributions of any form whatsoever must retain the following 31 * acknowledgment: 32 * "This product includes software developed by the OpenSSL Project 33 * for use in the OpenSSL Toolkit (http://www.openssl.org/)" 34 * 35 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY 36 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 37 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR 38 * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR 39 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, 40 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT 41 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; 42 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 43 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, 44 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) 45 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED 46 * OF THE POSSIBILITY OF SUCH DAMAGE. 47 * ==================================================================== 48 * 49 * This product includes cryptographic software written by Eric Young 50 * (eay (at) cryptsoft.com). This product includes software written by Tim 51 * Hudson (tjh (at) cryptsoft.com). */ 52 53 #include <openssl/bn.h> 54 55 #include "internal.h" 56 57 58 /* least significant word */ 59 #define BN_lsw(n) (((n)->top == 0) ? (BN_ULONG) 0 : (n)->d[0]) 60 61 /* Returns -2 for errors because both -1 and 0 are valid results. */ 62 int BN_kronecker(const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx) { 63 int i; 64 int ret = -2; 65 BIGNUM *A, *B, *tmp; 66 /* In 'tab', only odd-indexed entries are relevant: 67 * For any odd BIGNUM n, 68 * tab[BN_lsw(n) & 7] 69 * is $(-1)^{(n^2-1)/8}$ (using TeX notation). 70 * Note that the sign of n does not matter. */ 71 static const int tab[8] = {0, 1, 0, -1, 0, -1, 0, 1}; 72 73 BN_CTX_start(ctx); 74 A = BN_CTX_get(ctx); 75 B = BN_CTX_get(ctx); 76 if (B == NULL) { 77 goto end; 78 } 79 80 if (!BN_copy(A, a) || 81 !BN_copy(B, b)) { 82 goto end; 83 } 84 85 /* Kronecker symbol, imlemented according to Henri Cohen, 86 * "A Course in Computational Algebraic Number Theory" 87 * (algorithm 1.4.10). */ 88 89 /* Cohen's step 1: */ 90 91 if (BN_is_zero(B)) { 92 ret = BN_abs_is_word(A, 1); 93 goto end; 94 } 95 96 /* Cohen's step 2: */ 97 98 if (!BN_is_odd(A) && !BN_is_odd(B)) { 99 ret = 0; 100 goto end; 101 } 102 103 /* now B is non-zero */ 104 i = 0; 105 while (!BN_is_bit_set(B, i)) { 106 i++; 107 } 108 if (!BN_rshift(B, B, i)) { 109 goto end; 110 } 111 if (i & 1) { 112 /* i is odd */ 113 /* (thus B was even, thus A must be odd!) */ 114 115 /* set 'ret' to $(-1)^{(A^2-1)/8}$ */ 116 ret = tab[BN_lsw(A) & 7]; 117 } else { 118 /* i is even */ 119 ret = 1; 120 } 121 122 if (B->neg) { 123 B->neg = 0; 124 if (A->neg) { 125 ret = -ret; 126 } 127 } 128 129 /* now B is positive and odd, so what remains to be done is to compute the 130 * Jacobi symbol (A/B) and multiply it by 'ret' */ 131 132 while (1) { 133 /* Cohen's step 3: */ 134 135 /* B is positive and odd */ 136 if (BN_is_zero(A)) { 137 ret = BN_is_one(B) ? ret : 0; 138 goto end; 139 } 140 141 /* now A is non-zero */ 142 i = 0; 143 while (!BN_is_bit_set(A, i)) { 144 i++; 145 } 146 if (!BN_rshift(A, A, i)) { 147 goto end; 148 } 149 if (i & 1) { 150 /* i is odd */ 151 /* multiply 'ret' by $(-1)^{(B^2-1)/8}$ */ 152 ret = ret * tab[BN_lsw(B) & 7]; 153 } 154 155 /* Cohen's step 4: */ 156 /* multiply 'ret' by $(-1)^{(A-1)(B-1)/4}$ */ 157 if ((A->neg ? ~BN_lsw(A) : BN_lsw(A)) & BN_lsw(B) & 2) { 158 ret = -ret; 159 } 160 161 /* (A, B) := (B mod |A|, |A|) */ 162 if (!BN_nnmod(B, B, A, ctx)) { 163 ret = -2; 164 goto end; 165 } 166 tmp = A; 167 A = B; 168 B = tmp; 169 tmp->neg = 0; 170 } 171 172 end: 173 BN_CTX_end(ctx); 174 return ret; 175 } 176
