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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