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@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@cryptsoft.com). This product includes software written by Tim
51 * Hudson (tjh@cryptsoft.com). */
52
53 #include <openssl/bn.h>
54
55 #include <openssl/err.h>
56
57 #include "internal.h"
58
59
60 // least significant word
61 #define BN_lsw(n) (((n)->width == 0) ? (BN_ULONG) 0 : (n)->d[0])
62
bn_jacobi(const BIGNUM * a,const BIGNUM * b,BN_CTX * ctx)63 int bn_jacobi(const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx) {
64 // In 'tab', only odd-indexed entries are relevant:
65 // For any odd BIGNUM n,
66 // tab[BN_lsw(n) & 7]
67 // is $(-1)^{(n^2-1)/8}$ (using TeX notation).
68 // Note that the sign of n does not matter.
69 static const int tab[8] = {0, 1, 0, -1, 0, -1, 0, 1};
70
71 // The Jacobi symbol is only defined for odd modulus.
72 if (!BN_is_odd(b)) {
73 OPENSSL_PUT_ERROR(BN, BN_R_CALLED_WITH_EVEN_MODULUS);
74 return -2;
75 }
76
77 // Require b be positive.
78 if (BN_is_negative(b)) {
79 OPENSSL_PUT_ERROR(BN, BN_R_NEGATIVE_NUMBER);
80 return -2;
81 }
82
83 int ret = -2;
84 BN_CTX_start(ctx);
85 BIGNUM *A = BN_CTX_get(ctx);
86 BIGNUM *B = BN_CTX_get(ctx);
87 if (B == NULL) {
88 goto end;
89 }
90
91 if (!BN_copy(A, a) ||
92 !BN_copy(B, b)) {
93 goto end;
94 }
95
96 // Adapted from logic to compute the Kronecker symbol, originally implemented
97 // according to Henri Cohen, "A Course in Computational Algebraic Number
98 // Theory" (algorithm 1.4.10).
99
100 ret = 1;
101
102 while (1) {
103 // Cohen's step 3:
104
105 // B is positive and odd
106 if (BN_is_zero(A)) {
107 ret = BN_is_one(B) ? ret : 0;
108 goto end;
109 }
110
111 // now A is non-zero
112 int i = 0;
113 while (!BN_is_bit_set(A, i)) {
114 i++;
115 }
116 if (!BN_rshift(A, A, i)) {
117 ret = -2;
118 goto end;
119 }
120 if (i & 1) {
121 // i is odd
122 // multiply 'ret' by $(-1)^{(B^2-1)/8}$
123 ret = ret * tab[BN_lsw(B) & 7];
124 }
125
126 // Cohen's step 4:
127 // multiply 'ret' by $(-1)^{(A-1)(B-1)/4}$
128 if ((A->neg ? ~BN_lsw(A) : BN_lsw(A)) & BN_lsw(B) & 2) {
129 ret = -ret;
130 }
131
132 // (A, B) := (B mod |A|, |A|)
133 if (!BN_nnmod(B, B, A, ctx)) {
134 ret = -2;
135 goto end;
136 }
137 BIGNUM *tmp = A;
138 A = B;
139 B = tmp;
140 tmp->neg = 0;
141 }
142
143 end:
144 BN_CTX_end(ctx);
145 return ret;
146 }
147