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. */
63int 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