jacobi.c

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/* ====================================================================
 * 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:
 *
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 *    notice, this list of conditions and the following disclaimer.
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 * 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/)"
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 *    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
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 *
 * 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 <openssl/err.h>
 
#include "internal.h"
 
 
// least significant word
#define BN_lsw(n) (((n)->width == 0) ? (BN_ULONG) 0 : (n)->d[0])
 
int bn_jacobi(const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx) {
  // 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};
 
  // The Jacobi symbol is only defined for odd modulus.
  if (!BN_is_odd(b)) {
    OPENSSL_PUT_ERROR(BN, BN_R_CALLED_WITH_EVEN_MODULUS);
    return -2;
  }
 
  // Require b be positive.
  if (BN_is_negative(b)) {
    OPENSSL_PUT_ERROR(BN, BN_R_NEGATIVE_NUMBER);
    return -2;
  }
 
  int ret = -2;
  BN_CTX_start(ctx);
  BIGNUM *A = BN_CTX_get(ctx);
  BIGNUM *B = BN_CTX_get(ctx);
  if (B == NULL) {
    goto end;
  }
 
  if (!BN_copy(A, a) ||
      !BN_copy(B, b)) {
    goto end;
  }
 
  // Adapted from logic to compute the Kronecker symbol, originally implemented
  // according to Henri Cohen, "A Course in Computational Algebraic Number
  // Theory" (algorithm 1.4.10).
 
  ret = 1;
 
  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
    int i = 0;
    while (!BN_is_bit_set(A, i)) {
      i++;
    }
    if (!BN_rshift(A, A, i)) {
      ret = -2;
      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;
    }
    BIGNUM *tmp = A;
    A = B;
    B = tmp;
    tmp->neg = 0;
  }
 
end:
  BN_CTX_end(ctx);
  return ret;
}