crypto/bn/kronecker.c

/* ====================================================================
 * Copyright (c) 1998-2000 The OpenSSL Project.  All rights reserved.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions
 * are met:
 *
 * 1. Redistributions of source code must retain the above copyright
 *    notice, this list of conditions and the following disclaimer.
 *
 * 2. Redistributions in binary form must reproduce the above copyright
 *    notice, this list of conditions and the following disclaimer in
 *    the documentation and/or other materials provided with the
 *    distribution.
 *
 * 3. All advertising materials mentioning features or use of this
 *    software must display the following acknowledgment:
 *    "This product includes software developed by the OpenSSL Project
 *    for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
 *
 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
 *    endorse or promote products derived from this software without
 *    prior written permission. For written permission, please contact
 *    openssl-core@openssl.org.
 *
 * 5. Products derived from this software may not be called "OpenSSL"
 *    nor may "OpenSSL" appear in their names without prior written
 *    permission of the OpenSSL Project.
 *
 * 6. Redistributions of any form whatsoever must retain the following
 *    acknowledgment:
 *    "This product includes software developed by the OpenSSL Project
 *    for use in the OpenSSL Toolkit (http://www.openssl.org/)"
 *
 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
 * PURPOSE ARE DISCLAIMED.  IN NO EVENT SHALL THE OpenSSL PROJECT OR
 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
 * OF THE POSSIBILITY OF SUCH DAMAGE.
 * ====================================================================
 *
 * This product includes cryptographic software written by Eric Young
 * (eay@cryptsoft.com).  This product includes software written by Tim
 * Hudson (tjh@cryptsoft.com). */

#include <openssl/bn.h>

#include "internal.h"


/* least significant word */
#define BN_lsw(n) (((n)->top == 0) ? (BN_ULONG) 0 : (n)->d[0])

/* Returns -2 for errors because both -1 and 0 are valid results. */
int BN_kronecker(const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx) {
  int i;
  int ret = -2;
  BIGNUM *A, *B, *tmp;
  /* In 'tab', only odd-indexed entries are relevant:
   * For any odd BIGNUM n,
   *     tab[BN_lsw(n) & 7]
   * is $(-1)^{(n^2-1)/8}$ (using TeX notation).
   * Note that the sign of n does not matter. */
  static const int tab[8] = {0, 1, 0, -1, 0, -1, 0, 1};

  BN_CTX_start(ctx);
  A = BN_CTX_get(ctx);
  B = BN_CTX_get(ctx);
  if (B == NULL) {
    goto end;
  }

  if (!BN_copy(A, a) ||
      !BN_copy(B, b)) {
    goto end;
  }

  /* Kronecker symbol, imlemented according to Henri Cohen,
   * "A Course in Computational Algebraic Number Theory"
   * (algorithm 1.4.10). */

  /* Cohen's step 1: */

  if (BN_is_zero(B)) {
    ret = BN_abs_is_word(A, 1);
    goto end;
  }

  /* Cohen's step 2: */

  if (!BN_is_odd(A) && !BN_is_odd(B)) {
    ret = 0;
    goto end;
  }

  /* now B is non-zero */
  i = 0;
  while (!BN_is_bit_set(B, i)) {
    i++;
  }
  if (!BN_rshift(B, B, i)) {
    goto end;
  }
  if (i & 1) {
    /* i is odd */
    /* (thus B was even, thus A must be odd!)  */

    /* set 'ret' to $(-1)^{(A^2-1)/8}$ */
    ret = tab[BN_lsw(A) & 7];
  } else {
    /* i is even */
    ret = 1;
  }

  if (B->neg) {
    B->neg = 0;
    if (A->neg) {
      ret = -ret;
    }
  }

  /* now B is positive and odd, so what remains to be done is to compute the
   * Jacobi symbol (A/B) and multiply it by 'ret' */

  while (1) {
    /* Cohen's step 3: */

    /* B is positive and odd */
    if (BN_is_zero(A)) {
      ret = BN_is_one(B) ? ret : 0;
      goto end;
    }

    /* now A is non-zero */
    i = 0;
    while (!BN_is_bit_set(A, i)) {
      i++;
    }
    if (!BN_rshift(A, A, i)) {
      goto end;
    }
    if (i & 1) {
      /* i is odd */
      /* multiply 'ret' by  $(-1)^{(B^2-1)/8}$ */
      ret = ret * tab[BN_lsw(B) & 7];
    }

    /* Cohen's step 4: */
    /* multiply 'ret' by  $(-1)^{(A-1)(B-1)/4}$ */
    if ((A->neg ? ~BN_lsw(A) : BN_lsw(A)) & BN_lsw(B) & 2) {
      ret = -ret;
    }

    /* (A, B) := (B mod |A|, |A|) */
    if (!BN_nnmod(B, B, A, ctx)) {
      ret = -2;
      goto end;
    }
    tmp = A;
    A = B;
    B = tmp;
    tmp->neg = 0;
  }

end:
  BN_CTX_end(ctx);
  return ret;
}