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 "cryptlib.h" 57 #include "bn_lcl.h" 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_check_top(a); 78 bn_check_top(b); 79 80 BN_CTX_start(ctx); 81 A = BN_CTX_get(ctx); 82 B = BN_CTX_get(ctx); 83 if (B == NULL) goto end; 84 85 err = !BN_copy(A, a); 86 if (err) goto end; 87 err = !BN_copy(B, b); 88 if (err) goto end; 89 90 /* 91 * Kronecker symbol, imlemented according to Henri Cohen, 92 * "A Course in Computational Algebraic Number Theory" 93 * (algorithm 1.4.10). 94 */ 95 96 /* Cohen's step 1: */ 97 98 if (BN_is_zero(B)) 99 { 100 ret = BN_abs_is_word(A, 1); 101 goto end; 102 } 103 104 /* Cohen's step 2: */ 105 106 if (!BN_is_odd(A) && !BN_is_odd(B)) 107 { 108 ret = 0; 109 goto end; 110 } 111 112 /* now B is non-zero */ 113 i = 0; 114 while (!BN_is_bit_set(B, i)) 115 i++; 116 err = !BN_rshift(B, B, i); 117 if (err) goto end; 118 if (i & 1) 119 { 120 /* i is odd */ 121 /* (thus B was even, thus A must be odd!) */ 122 123 /* set 'ret' to $(-1)^{(A^2-1)/8}$ */ 124 ret = tab[BN_lsw(A) & 7]; 125 } 126 else 127 { 128 /* i is even */ 129 ret = 1; 130 } 131 132 if (B->neg) 133 { 134 B->neg = 0; 135 if (A->neg) 136 ret = -ret; 137 } 138 139 /* now B is positive and odd, so what remains to be done is 140 * to compute the Jacobi symbol (A/B) and multiply it by 'ret' */ 141 142 while (1) 143 { 144 /* Cohen's step 3: */ 145 146 /* B is positive and odd */ 147 148 if (BN_is_zero(A)) 149 { 150 ret = BN_is_one(B) ? ret : 0; 151 goto end; 152 } 153 154 /* now A is non-zero */ 155 i = 0; 156 while (!BN_is_bit_set(A, i)) 157 i++; 158 err = !BN_rshift(A, A, i); 159 if (err) goto end; 160 if (i & 1) 161 { 162 /* i is odd */ 163 /* multiply 'ret' by $(-1)^{(B^2-1)/8}$ */ 164 ret = ret * tab[BN_lsw(B) & 7]; 165 } 166 167 /* Cohen's step 4: */ 168 /* multiply 'ret' by $(-1)^{(A-1)(B-1)/4}$ */ 169 if ((A->neg ? ~BN_lsw(A) : BN_lsw(A)) & BN_lsw(B) & 2) 170 ret = -ret; 171 172 /* (A, B) := (B mod |A|, |A|) */ 173 err = !BN_nnmod(B, B, A, ctx); 174 if (err) goto end; 175 tmp = A; A = B; B = tmp; 176 tmp->neg = 0; 177 } 178 end: 179 BN_CTX_end(ctx); 180 if (err) 181 return -2; 182 else 183 return ret; 184 } 185