1 /* $OpenBSD: bn_kron.c,v 1.6 2015/02/09 15:49:22 jsing Exp $ */ 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 /* 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 63 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 70 /* In 'tab', only odd-indexed entries are relevant: 71 * For any odd BIGNUM n, 72 * tab[BN_lsw(n) & 7] 73 * is $(-1)^{(n^2-1)/8}$ (using TeX notation). 74 * Note that the sign of n does not matter. 75 */ 76 static const int tab[8] = {0, 1, 0, -1, 0, -1, 0, 1}; 77 78 bn_check_top(a); 79 bn_check_top(b); 80 81 BN_CTX_start(ctx); 82 if ((A = BN_CTX_get(ctx)) == NULL) 83 goto end; 84 if ((B = BN_CTX_get(ctx)) == NULL) 85 goto end; 86 87 err = !BN_copy(A, a); 88 if (err) 89 goto end; 90 err = !BN_copy(B, b); 91 if (err) 92 goto end; 93 94 /* 95 * Kronecker symbol, imlemented according to Henri Cohen, 96 * "A Course in Computational Algebraic Number Theory" 97 * (algorithm 1.4.10). 98 */ 99 100 /* Cohen's step 1: */ 101 102 if (BN_is_zero(B)) { 103 ret = BN_abs_is_word(A, 1); 104 goto end; 105 } 106 107 /* Cohen's step 2: */ 108 109 if (!BN_is_odd(A) && !BN_is_odd(B)) { 110 ret = 0; 111 goto end; 112 } 113 114 /* now B is non-zero */ 115 i = 0; 116 while (!BN_is_bit_set(B, i)) 117 i++; 118 err = !BN_rshift(B, B, i); 119 if (err) 120 goto end; 121 if (i & 1) { 122 /* i is odd */ 123 /* (thus B was even, thus A must be odd!) */ 124 125 /* set 'ret' to $(-1)^{(A^2-1)/8}$ */ 126 ret = tab[BN_lsw(A) & 7]; 127 } else { 128 /* i is even */ 129 ret = 1; 130 } 131 132 if (B->neg) { 133 B->neg = 0; 134 if (A->neg) 135 ret = -ret; 136 } 137 138 /* now B is positive and odd, so what remains to be done is 139 * to compute the Jacobi symbol (A/B) and multiply it by 'ret' */ 140 141 while (1) { 142 /* Cohen's step 3: */ 143 144 /* B is positive and odd */ 145 146 if (BN_is_zero(A)) { 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) 157 goto end; 158 if (i & 1) { 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) 172 goto end; 173 tmp = A; 174 A = B; 175 B = tmp; 176 tmp->neg = 0; 177 } 178 179 end: 180 BN_CTX_end(ctx); 181 if (err) 182 return -2; 183 else 184 return ret; 185 } 186