1 /* crypto/bn/bn_kron.c */ 2 /* ==================================================================== 3 * Copyright (c) 1998-2000 The OpenSSL Project. All rights reserved. 4 * 5 * Redistribution and use in source and binary forms, with or without 6 * modification, are permitted provided that the following conditions 7 * are met: 8 * 9 * 1. Redistributions of source code must retain the above copyright 10 * notice, this list of conditions and the following disclaimer. 11 * 12 * 2. Redistributions in binary form must reproduce the above copyright 13 * notice, this list of conditions and the following disclaimer in 14 * the documentation and/or other materials provided with the 15 * distribution. 16 * 17 * 3. All advertising materials mentioning features or use of this 18 * software must display the following acknowledgment: 19 * "This product includes software developed by the OpenSSL Project 20 * for use in the OpenSSL Toolkit. (http://www.openssl.org/)" 21 * 22 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to 23 * endorse or promote products derived from this software without 24 * prior written permission. For written permission, please contact 25 * openssl-core@openssl.org. 26 * 27 * 5. Products derived from this software may not be called "OpenSSL" 28 * nor may "OpenSSL" appear in their names without prior written 29 * permission of the OpenSSL Project. 30 * 31 * 6. Redistributions of any form whatsoever must retain the following 32 * acknowledgment: 33 * "This product includes software developed by the OpenSSL Project 34 * for use in the OpenSSL Toolkit (http://www.openssl.org/)" 35 * 36 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY 37 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 38 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR 39 * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR 40 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, 41 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT 42 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; 43 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 44 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, 45 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) 46 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED 47 * OF THE POSSIBILITY OF SUCH DAMAGE. 48 * ==================================================================== 49 * 50 * This product includes cryptographic software written by Eric Young 51 * (eay@cryptsoft.com). This product includes software written by Tim 52 * Hudson (tjh@cryptsoft.com). 53 * 54 */ 55 56 #include "bn_lcl.h" 57 58 59 /* least significant word */ 60 #define BN_lsw(n) (((n)->top == 0) ? (BN_ULONG) 0 : (n)->d[0]) 61 62 /* Returns -2 for errors because both -1 and 0 are valid results. */ 63 int BN_kronecker(const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx) 64 { 65 int i; 66 int ret = -2; /* avoid 'uninitialized' warning */ 67 int err = 0; 68 BIGNUM *A, *B, *tmp; 69 /* In 'tab', only odd-indexed entries are relevant: 70 * For any odd BIGNUM n, 71 * tab[BN_lsw(n) & 7] 72 * is $(-1)^{(n^2-1)/8}$ (using TeX notation). 73 * Note that the sign of n does not matter. 74 */ 75 static const int tab[8] = {0, 1, 0, -1, 0, -1, 0, 1}; 76 77 BN_CTX_start(ctx); 78 A = BN_CTX_get(ctx); 79 B = BN_CTX_get(ctx); 80 if (B == NULL) goto end; 81 82 err = !BN_copy(A, a); 83 if (err) goto end; 84 err = !BN_copy(B, b); 85 if (err) goto end; 86 87 /* 88 * Kronecker symbol, imlemented according to Henri Cohen, 89 * "A Course in Computational Algebraic Number Theory" 90 * (algorithm 1.4.10). 91 */ 92 93 /* Cohen's step 1: */ 94 95 if (BN_is_zero(B)) 96 { 97 ret = BN_abs_is_word(A, 1); 98 goto end; 99 } 100 101 /* Cohen's step 2: */ 102 103 if (!BN_is_odd(A) && !BN_is_odd(B)) 104 { 105 ret = 0; 106 goto end; 107 } 108 109 /* now B is non-zero */ 110 i = 0; 111 while (!BN_is_bit_set(B, i)) 112 i++; 113 err = !BN_rshift(B, B, i); 114 if (err) goto end; 115 if (i & 1) 116 { 117 /* i is odd */ 118 /* (thus B was even, thus A must be odd!) */ 119 120 /* set 'ret' to $(-1)^{(A^2-1)/8}$ */ 121 ret = tab[BN_lsw(A) & 7]; 122 } 123 else 124 { 125 /* i is even */ 126 ret = 1; 127 } 128 129 if (B->neg) 130 { 131 B->neg = 0; 132 if (A->neg) 133 ret = -ret; 134 } 135 136 /* now B is positive and odd, so what remains to be done is 137 * to compute the Jacobi symbol (A/B) and multiply it by 'ret' */ 138 139 while (1) 140 { 141 /* Cohen's step 3: */ 142 143 /* B is positive and odd */ 144 145 if (BN_is_zero(A)) 146 { 147 ret = BN_is_one(B) ? ret : 0; 148 goto end; 149 } 150 151 /* now A is non-zero */ 152 i = 0; 153 while (!BN_is_bit_set(A, i)) 154 i++; 155 err = !BN_rshift(A, A, i); 156 if (err) goto end; 157 if (i & 1) 158 { 159 /* i is odd */ 160 /* multiply 'ret' by $(-1)^{(B^2-1)/8}$ */ 161 ret = ret * tab[BN_lsw(B) & 7]; 162 } 163 164 /* Cohen's step 4: */ 165 /* multiply 'ret' by $(-1)^{(A-1)(B-1)/4}$ */ 166 if ((A->neg ? ~BN_lsw(A) : BN_lsw(A)) & BN_lsw(B) & 2) 167 ret = -ret; 168 169 /* (A, B) := (B mod |A|, |A|) */ 170 err = !BN_nnmod(B, B, A, ctx); 171 if (err) goto end; 172 tmp = A; A = B; B = tmp; 173 tmp->neg = 0; 174 } 175 176 end: 177 BN_CTX_end(ctx); 178 if (err) 179 return -2; 180 else 181 return ret; 182 } 183