src/crypto/bn/gcd.c

/* Copyright (C) 1995-1998 Eric Young (eay@cryptsoft.com)
 * All rights reserved.
 *
 * This package is an SSL implementation written
 * by Eric Young (eay@cryptsoft.com).
 * The implementation was written so as to conform with Netscapes SSL.
 *
 * This library is free for commercial and non-commercial use as long as
 * the following conditions are aheared to.  The following conditions
 * apply to all code found in this distribution, be it the RC4, RSA,
 * lhash, DES, etc., code; not just the SSL code.  The SSL documentation
 * included with this distribution is covered by the same copyright terms
 * except that the holder is Tim Hudson (tjh@cryptsoft.com).
 *
 * Copyright remains Eric Young's, and as such any Copyright notices in
 * the code are not to be removed.
 * If this package is used in a product, Eric Young should be given attribution
 * as the author of the parts of the library used.
 * This can be in the form of a textual message at program startup or
 * in documentation (online or textual) provided with the package.
 *
 * 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 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 acknowledgement:
 *    "This product includes cryptographic software written by
 *     Eric Young (eay@cryptsoft.com)"
 *    The word 'cryptographic' can be left out if the rouines from the library
 *    being used are not cryptographic related :-).
 * 4. If you include any Windows specific code (or a derivative thereof) from
 *    the apps directory (application code) you must include an acknowledgement:
 *    "This product includes software written by Tim Hudson (tjh@cryptsoft.com)"
 *
 * THIS SOFTWARE IS PROVIDED BY ERIC YOUNG ``AS IS'' AND
 * ANY EXPRESS 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 AUTHOR OR 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.
 *
 * The licence and distribution terms for any publically available version or
 * derivative of this code cannot be changed.  i.e. this code cannot simply be
 * copied and put under another distribution licence
 * [including the GNU Public Licence.]
 */
/* ====================================================================
 * Copyright (c) 1998-2001 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 <openssl/err.h>

#include "internal.h"

static BIGNUM *euclid(BIGNUM *a, BIGNUM *b) {
  BIGNUM *t;
  int shifts = 0;

  /* 0 <= b <= a */
  while (!BN_is_zero(b)) {
    /* 0 < b <= a */

    if (BN_is_odd(a)) {
      if (BN_is_odd(b)) {
        if (!BN_sub(a, a, b)) {
          goto err;
        }
        if (!BN_rshift1(a, a)) {
          goto err;
        }
        if (BN_cmp(a, b) < 0) {
          t = a;
          a = b;
          b = t;
        }
      } else {
        /* a odd - b even */
        if (!BN_rshift1(b, b)) {
          goto err;
        }
        if (BN_cmp(a, b) < 0) {
          t = a;
          a = b;
          b = t;
        }
      }
    } else {
      /* a is even */
      if (BN_is_odd(b)) {
        if (!BN_rshift1(a, a)) {
          goto err;
        }
        if (BN_cmp(a, b) < 0) {
          t = a;
          a = b;
          b = t;
        }
      } else {
        /* a even - b even */
        if (!BN_rshift1(a, a)) {
          goto err;
        }
        if (!BN_rshift1(b, b)) {
          goto err;
        }
        shifts++;
      }
    }
    /* 0 <= b <= a */
  }

  if (shifts) {
    if (!BN_lshift(a, a, shifts)) {
      goto err;
    }
  }

  return a;

err:
  return NULL;
}

int BN_gcd(BIGNUM *r, const BIGNUM *in_a, const BIGNUM *in_b, BN_CTX *ctx) {
  BIGNUM *a, *b, *t;
  int ret = 0;

  BN_CTX_start(ctx);
  a = BN_CTX_get(ctx);
  b = BN_CTX_get(ctx);

  if (a == NULL || b == NULL) {
    goto err;
  }
  if (BN_copy(a, in_a) == NULL) {
    goto err;
  }
  if (BN_copy(b, in_b) == NULL) {
    goto err;
  }

  a->neg = 0;
  b->neg = 0;

  if (BN_cmp(a, b) < 0) {
    t = a;
    a = b;
    b = t;
  }
  t = euclid(a, b);
  if (t == NULL) {
    goto err;
  }

  if (BN_copy(r, t) == NULL) {
    goto err;
  }
  ret = 1;

err:
  BN_CTX_end(ctx);
  return ret;
}

/* solves ax == 1 (mod n) */
static BIGNUM *BN_mod_inverse_no_branch(BIGNUM *out, const BIGNUM *a,
                                        const BIGNUM *n, BN_CTX *ctx);

BIGNUM *BN_mod_inverse(BIGNUM *out, const BIGNUM *a, const BIGNUM *n,
                       BN_CTX *ctx) {
  BIGNUM *A, *B, *X, *Y, *M, *D, *T, *R = NULL;
  BIGNUM *ret = NULL;
  int sign;

  if ((a->flags & BN_FLG_CONSTTIME) != 0 ||
      (n->flags & BN_FLG_CONSTTIME) != 0) {
    return BN_mod_inverse_no_branch(out, a, n, ctx);
  }

  BN_CTX_start(ctx);
  A = BN_CTX_get(ctx);
  B = BN_CTX_get(ctx);
  X = BN_CTX_get(ctx);
  D = BN_CTX_get(ctx);
  M = BN_CTX_get(ctx);
  Y = BN_CTX_get(ctx);
  T = BN_CTX_get(ctx);
  if (T == NULL) {
    goto err;
  }

  if (out == NULL) {
    R = BN_new();
  } else {
    R = out;
  }
  if (R == NULL) {
    goto err;
  }

  BN_zero(Y);
  if (!BN_one(X) || BN_copy(B, a) == NULL || BN_copy(A, n) == NULL) {
    goto err;
  }
  A->neg = 0;
  if (B->neg || (BN_ucmp(B, A) >= 0)) {
    if (!BN_nnmod(B, B, A, ctx)) {
      goto err;
    }
  }
  sign = -1;
  /* From  B = a mod |n|,  A = |n|  it follows that
   *
   *      0 <= B < A,
   *     -sign*X*a  ==  B   (mod |n|),
   *      sign*Y*a  ==  A   (mod |n|).
   */

  if (BN_is_odd(n) && (BN_num_bits(n) <= (BN_BITS <= 32 ? 450 : 2048))) {
    /* Binary inversion algorithm; requires odd modulus.
     * This is faster than the general algorithm if the modulus
     * is sufficiently small (about 400 .. 500 bits on 32-bit
     * sytems, but much more on 64-bit systems) */
    int shift;

    while (!BN_is_zero(B)) {
      /*      0 < B < |n|,
       *      0 < A <= |n|,
       * (1) -sign*X*a  ==  B   (mod |n|),
       * (2)  sign*Y*a  ==  A   (mod |n|) */

      /* Now divide  B  by the maximum possible power of two in the integers,
       * and divide  X  by the same value mod |n|.
       * When we're done, (1) still holds. */
      shift = 0;
      while (!BN_is_bit_set(B, shift)) {
        /* note that 0 < B */
        shift++;

        if (BN_is_odd(X)) {
          if (!BN_uadd(X, X, n)) {
            goto err;
          }
        }
        /* now X is even, so we can easily divide it by two */
        if (!BN_rshift1(X, X)) {
          goto err;
        }
      }
      if (shift > 0) {
        if (!BN_rshift(B, B, shift)) {
          goto err;
        }
      }

      /* Same for A and Y. Afterwards, (2) still holds. */
      shift = 0;
      while (!BN_is_bit_set(A, shift)) {
        /* note that 0 < A */
        shift++;

        if (BN_is_odd(Y)) {
          if (!BN_uadd(Y, Y, n)) {
            goto err;
          }
        }
        /* now Y is even */
        if (!BN_rshift1(Y, Y)) {
          goto err;
        }
      }
      if (shift > 0) {
        if (!BN_rshift(A, A, shift)) {
          goto err;
        }
      }

      /* We still have (1) and (2).
       * Both  A  and  B  are odd.
       * The following computations ensure that
       *
       *     0 <= B < |n|,
       *      0 < A < |n|,
       * (1) -sign*X*a  ==  B   (mod |n|),
       * (2)  sign*Y*a  ==  A   (mod |n|),
       *
       * and that either  A  or  B  is even in the next iteration. */
      if (BN_ucmp(B, A) >= 0) {
        /* -sign*(X + Y)*a == B - A  (mod |n|) */
        if (!BN_uadd(X, X, Y)) {
          goto err;
        }
        /* NB: we could use BN_mod_add_quick(X, X, Y, n), but that
         * actually makes the algorithm slower */
        if (!BN_usub(B, B, A)) {
          goto err;
        }
      } else {
        /*  sign*(X + Y)*a == A - B  (mod |n|) */
        if (!BN_uadd(Y, Y, X)) {
          goto err;
        }
        /* as above, BN_mod_add_quick(Y, Y, X, n) would slow things down */
        if (!BN_usub(A, A, B)) {
          goto err;
        }
      }
    }
  } else {
    /* general inversion algorithm */

    while (!BN_is_zero(B)) {
      BIGNUM *tmp;

      /*
       *      0 < B < A,
       * (*) -sign*X*a  ==  B   (mod |n|),
       *      sign*Y*a  ==  A   (mod |n|) */

      /* (D, M) := (A/B, A%B) ... */
      if (BN_num_bits(A) == BN_num_bits(B)) {
        if (!BN_one(D)) {
          goto err;
        }
        if (!BN_sub(M, A, B)) {
          goto err;
        }
      } else if (BN_num_bits(A) == BN_num_bits(B) + 1) {
        /* A/B is 1, 2, or 3 */
        if (!BN_lshift1(T, B)) {
          goto err;
        }
        if (BN_ucmp(A, T) < 0) {
          /* A < 2*B, so D=1 */
          if (!BN_one(D)) {
            goto err;
          }
          if (!BN_sub(M, A, B)) {
            goto err;
          }
        } else {
          /* A >= 2*B, so D=2 or D=3 */
          if (!BN_sub(M, A, T)) {
            goto err;
          }
          if (!BN_add(D, T, B)) {
            goto err; /* use D (:= 3*B) as temp */
          }
          if (BN_ucmp(A, D) < 0) {
            /* A < 3*B, so D=2 */
            if (!BN_set_word(D, 2)) {
              goto err;
            }
            /* M (= A - 2*B) already has the correct value */
          } else {
            /* only D=3 remains */
            if (!BN_set_word(D, 3)) {
              goto err;
            }
            /* currently  M = A - 2*B,  but we need  M = A - 3*B */
            if (!BN_sub(M, M, B)) {
              goto err;
            }
          }
        }
      } else {
        if (!BN_div(D, M, A, B, ctx)) {
          goto err;
        }
      }

      /* Now
       *      A = D*B + M;
       * thus we have
       * (**)  sign*Y*a  ==  D*B + M   (mod |n|). */

      tmp = A; /* keep the BIGNUM object, the value does not matter */

      /* (A, B) := (B, A mod B) ... */
      A = B;
      B = M;
      /* ... so we have  0 <= B < A  again */

      /* Since the former  M  is now  B  and the former  B  is now  A,
       * (**) translates into
       *       sign*Y*a  ==  D*A + B    (mod |n|),
       * i.e.
       *       sign*Y*a - D*A  ==  B    (mod |n|).
       * Similarly, (*) translates into
       *      -sign*X*a  ==  A          (mod |n|).
       *
       * Thus,
       *   sign*Y*a + D*sign*X*a  ==  B  (mod |n|),
       * i.e.
       *        sign*(Y + D*X)*a  ==  B  (mod |n|).
       *
       * So if we set  (X, Y, sign) := (Y + D*X, X, -sign),  we arrive back at
       *      -sign*X*a  ==  B   (mod |n|),
       *       sign*Y*a  ==  A   (mod |n|).
       * Note that  X  and  Y  stay non-negative all the time. */

      /* most of the time D is very small, so we can optimize tmp := D*X+Y */
      if (BN_is_one(D)) {
        if (!BN_add(tmp, X, Y)) {
          goto err;
        }
      } else {
        if (BN_is_word(D, 2)) {
          if (!BN_lshift1(tmp, X)) {
            goto err;
          }
        } else if (BN_is_word(D, 4)) {
          if (!BN_lshift(tmp, X, 2)) {
            goto err;
          }
        } else if (D->top == 1) {
          if (!BN_copy(tmp, X)) {
            goto err;
          }
          if (!BN_mul_word(tmp, D->d[0])) {
            goto err;
          }
        } else {
          if (!BN_mul(tmp, D, X, ctx)) {
            goto err;
          }
        }
        if (!BN_add(tmp, tmp, Y)) {
          goto err;
        }
      }

      M = Y; /* keep the BIGNUM object, the value does not matter */
      Y = X;
      X = tmp;
      sign = -sign;
    }
  }

  /* The while loop (Euclid's algorithm) ends when
   *      A == gcd(a,n);
   * we have
   *       sign*Y*a  ==  A  (mod |n|),
   * where  Y  is non-negative. */

  if (sign < 0) {
    if (!BN_sub(Y, n, Y)) {
      goto err;
    }
  }
  /* Now  Y*a  ==  A  (mod |n|).  */

  if (BN_is_one(A)) {
    /* Y*a == 1  (mod |n|) */
    if (!Y->neg && BN_ucmp(Y, n) < 0) {
      if (!BN_copy(R, Y)) {
        goto err;
      }
    } else {
      if (!BN_nnmod(R, Y, n, ctx)) {
        goto err;
      }
    }
  } else {
    OPENSSL_PUT_ERROR(BN, BN_mod_inverse, BN_R_NO_INVERSE);
    goto err;
  }
  ret = R;

err:
  if (ret == NULL && out == NULL) {
    BN_free(R);
  }
  BN_CTX_end(ctx);
  return ret;
}

/* BN_mod_inverse_no_branch is a special version of BN_mod_inverse.
 * It does not contain branches that may leak sensitive information. */
static BIGNUM *BN_mod_inverse_no_branch(BIGNUM *out, const BIGNUM *a,
                                        const BIGNUM *n, BN_CTX *ctx) {
  BIGNUM *A, *B, *X, *Y, *M, *D, *T, *R = NULL;
  BIGNUM local_A, local_B;
  BIGNUM *pA, *pB;
  BIGNUM *ret = NULL;
  int sign;

  BN_CTX_start(ctx);
  A = BN_CTX_get(ctx);
  B = BN_CTX_get(ctx);
  X = BN_CTX_get(ctx);
  D = BN_CTX_get(ctx);
  M = BN_CTX_get(ctx);
  Y = BN_CTX_get(ctx);
  T = BN_CTX_get(ctx);
  if (T == NULL) {
    goto err;
  }

  if (out == NULL) {
    R = BN_new();
  } else {
    R = out;
  }
  if (R == NULL) {
    goto err;
  }

  BN_zero(Y);
  if (!BN_one(X) || BN_copy(B, a) == NULL || BN_copy(A, n) == NULL) {
    goto err;
  }
  A->neg = 0;

  if (B->neg || (BN_ucmp(B, A) >= 0)) {
    /* Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
     * BN_div_no_branch will be called eventually.
     */
    pB = &local_B;
    BN_with_flags(pB, B, BN_FLG_CONSTTIME);
    if (!BN_nnmod(B, pB, A, ctx)) {
      goto err;
    }
  }
  sign = -1;
  /* From  B = a mod |n|,  A = |n|  it follows that
   *
   *      0 <= B < A,
   *     -sign*X*a  ==  B   (mod |n|),
   *      sign*Y*a  ==  A   (mod |n|).
   */

  while (!BN_is_zero(B)) {
    BIGNUM *tmp;

    /*
     *      0 < B < A,
     * (*) -sign*X*a  ==  B   (mod |n|),
     *      sign*Y*a  ==  A   (mod |n|)
     */

    /* Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
     * BN_div_no_branch will be called eventually.
     */
    pA = &local_A;
    BN_with_flags(pA, A, BN_FLG_CONSTTIME);

    /* (D, M) := (A/B, A%B) ... */
    if (!BN_div(D, M, pA, B, ctx)) {
      goto err;
    }

    /* Now
     *      A = D*B + M;
     * thus we have
     * (**)  sign*Y*a  ==  D*B + M   (mod |n|).
     */

    tmp = A; /* keep the BIGNUM object, the value does not matter */

    /* (A, B) := (B, A mod B) ... */
    A = B;
    B = M;
    /* ... so we have  0 <= B < A  again */

    /* Since the former  M  is now  B  and the former  B  is now  A,
     * (**) translates into
     *       sign*Y*a  ==  D*A + B    (mod |n|),
     * i.e.
     *       sign*Y*a - D*A  ==  B    (mod |n|).
     * Similarly, (*) translates into
     *      -sign*X*a  ==  A          (mod |n|).
     *
     * Thus,
     *   sign*Y*a + D*sign*X*a  ==  B  (mod |n|),
     * i.e.
     *        sign*(Y + D*X)*a  ==  B  (mod |n|).
     *
     * So if we set  (X, Y, sign) := (Y + D*X, X, -sign),  we arrive back at
     *      -sign*X*a  ==  B   (mod |n|),
     *       sign*Y*a  ==  A   (mod |n|).
     * Note that  X  and  Y  stay non-negative all the time.
     */

    if (!BN_mul(tmp, D, X, ctx)) {
      goto err;
    }
    if (!BN_add(tmp, tmp, Y)) {
      goto err;
    }

    M = Y; /* keep the BIGNUM object, the value does not matter */
    Y = X;
    X = tmp;
    sign = -sign;
  }

  /*
   * The while loop (Euclid's algorithm) ends when
   *      A == gcd(a,n);
   * we have
   *       sign*Y*a  ==  A  (mod |n|),
   * where  Y  is non-negative.
   */

  if (sign < 0) {
    if (!BN_sub(Y, n, Y)) {
      goto err;
    }
  }
  /* Now  Y*a  ==  A  (mod |n|).  */

  if (BN_is_one(A)) {
    /* Y*a == 1  (mod |n|) */
    if (!Y->neg && BN_ucmp(Y, n) < 0) {
      if (!BN_copy(R, Y)) {
        goto err;
      }
    } else {
      if (!BN_nnmod(R, Y, n, ctx)) {
        goto err;
      }
    }
  } else {
    OPENSSL_PUT_ERROR(BN, BN_mod_inverse_no_branch, BN_R_NO_INVERSE);
    goto err;
  }
  ret = R;

err:
  if (ret == NULL && out == NULL) {
    BN_free(R);
  }

  BN_CTX_end(ctx);
  return ret;
}