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. */
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_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		}
178end:
179	BN_CTX_end(ctx);
180	if (err)
181		return -2;
182	else
183		return ret;
184	}
185