text: /* 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 <assert.h>
  
  #include <openssl/err.h>
  
  #include "internal.h"
  
  
  static BN_ULONG word_is_odd_mask(BN_ULONG a) { return (BN_ULONG)0 - (a & 1); }
  
  static void maybe_rshift1_words(BN_ULONG *a, BN_ULONG mask, BN_ULONG *tmp,
                                  size_t num) {
    bn_rshift1_words(tmp, a, num);
    bn_select_words(a, mask, tmp, a, num);
  }
  
  static void maybe_rshift1_words_carry(BN_ULONG *a, BN_ULONG carry,
                                        BN_ULONG mask, BN_ULONG *tmp,
                                        size_t num) {
    maybe_rshift1_words(a, mask, tmp, num);
    if (num != 0) {
      carry &= mask;
      a[num - 1] |= carry << (BN_BITS2-1);
    }
  }
  
  static BN_ULONG maybe_add_words(BN_ULONG *a, BN_ULONG mask, const BN_ULONG *b,
                                  BN_ULONG *tmp, size_t num) {
    BN_ULONG carry = bn_add_words(tmp, a, b, num);
    bn_select_words(a, mask, tmp, a, num);
    return carry & mask;
  }
  
  static int bn_gcd_consttime(BIGNUM *r, unsigned *out_shift, const BIGNUM *x,
                              const BIGNUM *y, BN_CTX *ctx) {
    size_t width = x->width > y->width ? x->width : y->width;
    if (width == 0) {
      *out_shift = 0;
      BN_zero(r);
      return 1;
    }
  
    // This is a constant-time implementation of Stein's algorithm (binary GCD).
    int ret = 0;
    BN_CTX_start(ctx);
    BIGNUM *u = BN_CTX_get(ctx);
    BIGNUM *v = BN_CTX_get(ctx);
    BIGNUM *tmp = BN_CTX_get(ctx);
    if (u == NULL || v == NULL || tmp == NULL ||
        !BN_copy(u, x) ||
        !BN_copy(v, y) ||
        !bn_resize_words(u, width) ||
        !bn_resize_words(v, width) ||
        !bn_resize_words(tmp, width)) {
      goto err;
    }
  
    // Each loop iteration halves at least one of |u| and |v|. Thus we need at
    // most the combined bit width of inputs for at least one value to be zero.
    unsigned x_bits = x->width * BN_BITS2, y_bits = y->width * BN_BITS2;
    unsigned num_iters = x_bits + y_bits;
    if (num_iters < x_bits) {
      OPENSSL_PUT_ERROR(BN, BN_R_BIGNUM_TOO_LONG);
      goto err;
    }
  
    unsigned shift = 0;
    for (unsigned i = 0; i < num_iters; i++) {
      BN_ULONG both_odd = word_is_odd_mask(u->d[0]) & word_is_odd_mask(v->d[0]);
  
      // If both |u| and |v| are odd, subtract the smaller from the larger.
      BN_ULONG u_less_than_v =
          (BN_ULONG)0 - bn_sub_words(tmp->d, u->d, v->d, width);
      bn_select_words(u->d, both_odd & ~u_less_than_v, tmp->d, u->d, width);
      bn_sub_words(tmp->d, v->d, u->d, width);
      bn_select_words(v->d, both_odd & u_less_than_v, tmp->d, v->d, width);
  
      // At least one of |u| and |v| is now even.
      BN_ULONG u_is_odd = word_is_odd_mask(u->d[0]);
      BN_ULONG v_is_odd = word_is_odd_mask(v->d[0]);
      assert(!(u_is_odd & v_is_odd));
  
      // If both are even, the final GCD gains a factor of two.
      shift += 1 & (~u_is_odd & ~v_is_odd);
  
      // Halve any which are even.
      maybe_rshift1_words(u->d, ~u_is_odd, tmp->d, width);
      maybe_rshift1_words(v->d, ~v_is_odd, tmp->d, width);
    }
  
    // One of |u| or |v| is zero at this point. The algorithm usually makes |u|
    // zero, unless |y| was already zero on input. Fix this by combining the
    // values.
    assert(BN_is_zero(u) || BN_is_zero(v));
    for (size_t i = 0; i < width; i++) {
      v->d[i] |= u->d[i];
    }
  
    *out_shift = shift;
    ret = bn_set_words(r, v->d, width);
  
  err:
    BN_CTX_end(ctx);
    return ret;
  }
  
  int BN_gcd(BIGNUM *r, const BIGNUM *x, const BIGNUM *y, BN_CTX *ctx) {
    unsigned shift;
    return bn_gcd_consttime(r, &shift, x, y, ctx) &&
           BN_lshift(r, r, shift);
  }
  
  int bn_is_relatively_prime(int *out_relatively_prime, const BIGNUM *x,
                             const BIGNUM *y, BN_CTX *ctx) {
    int ret = 0;
    BN_CTX_start(ctx);
    unsigned shift;
    BIGNUM *gcd = BN_CTX_get(ctx);
    if (gcd == NULL ||
        !bn_gcd_consttime(gcd, &shift, x, y, ctx)) {
      goto err;
    }
  
    // Check that 2^|shift| * |gcd| is one.
    if (gcd->width == 0) {
      *out_relatively_prime = 0;
    } else {
      BN_ULONG mask = shift | (gcd->d[0] ^ 1);
      for (int i = 1; i < gcd->width; i++) {
        mask |= gcd->d[i];
      }
      *out_relatively_prime = mask == 0;
    }
    ret = 1;
  
  err:
    BN_CTX_end(ctx);
    return ret;
  }
  
  int bn_lcm_consttime(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx) {
    BN_CTX_start(ctx);
    unsigned shift;
    BIGNUM *gcd = BN_CTX_get(ctx);
    int ret = gcd != NULL &&
              bn_mul_consttime(r, a, b, ctx) &&
              bn_gcd_consttime(gcd, &shift, a, b, ctx) &&
              bn_div_consttime(r, NULL, r, gcd, ctx) &&
              bn_rshift_secret_shift(r, r, shift, ctx);
    BN_CTX_end(ctx);
    return ret;
  }
  
  int bn_mod_inverse_consttime(BIGNUM *r, int *out_no_inverse, const BIGNUM *a,
                               const BIGNUM *n, BN_CTX *ctx) {
    *out_no_inverse = 0;
    if (BN_is_negative(a) || BN_ucmp(a, n) >= 0) {
      OPENSSL_PUT_ERROR(BN, BN_R_INPUT_NOT_REDUCED);
      return 0;
    }
    if (BN_is_zero(a)) {
      if (BN_is_one(n)) {
        BN_zero(r);
        return 1;
      }
      *out_no_inverse = 1;
      OPENSSL_PUT_ERROR(BN, BN_R_NO_INVERSE);
      return 0;
    }
  
    // This is a constant-time implementation of the extended binary GCD
    // algorithm. It is adapted from the Handbook of Applied Cryptography, section
    // 14.4.3, algorithm 14.51, and modified to bound coefficients and avoid
    // negative numbers.
    //
    // For more details and proof of correctness, see
    // https://github.com/mit-plv/fiat-crypto/pull/333. In particular, see |step|
    // and |mod_inverse_consttime| for the algorithm in Gallina and see
    // |mod_inverse_consttime_spec| for the correctness result.
  
    if (!BN_is_odd(a) && !BN_is_odd(n)) {
      *out_no_inverse = 1;
      OPENSSL_PUT_ERROR(BN, BN_R_NO_INVERSE);
      return 0;
    }
  
    // This function exists to compute the RSA private exponent, where |a| is one
    // word. We'll thus use |a_width| when available.
    size_t n_width = n->width, a_width = a->width;
    if (a_width > n_width) {
      a_width = n_width;
    }
  
    int ret = 0;
    BN_CTX_start(ctx);
    BIGNUM *u = BN_CTX_get(ctx);
    BIGNUM *v = BN_CTX_get(ctx);
    BIGNUM *A = BN_CTX_get(ctx);
    BIGNUM *B = BN_CTX_get(ctx);
    BIGNUM *C = BN_CTX_get(ctx);
    BIGNUM *D = BN_CTX_get(ctx);
    BIGNUM *tmp = BN_CTX_get(ctx);
    BIGNUM *tmp2 = BN_CTX_get(ctx);
    if (u == NULL || v == NULL || A == NULL || B == NULL || C == NULL ||
        D == NULL || tmp == NULL || tmp2 == NULL ||
        !BN_copy(u, a) ||
        !BN_copy(v, n) ||
        !BN_one(A) ||
        !BN_one(D) ||
        // For convenience, size |u| and |v| equivalently.
        !bn_resize_words(u, n_width) ||
        !bn_resize_words(v, n_width) ||
        // |A| and |C| are bounded by |m|.
        !bn_resize_words(A, n_width) ||
        !bn_resize_words(C, n_width) ||
        // |B| and |D| are bounded by |a|.
        !bn_resize_words(B, a_width) ||
        !bn_resize_words(D, a_width) ||
        // |tmp| and |tmp2| may be used at either size.
        !bn_resize_words(tmp, n_width) ||
        !bn_resize_words(tmp2, n_width)) {
      goto err;
    }
  
    // Each loop iteration halves at least one of |u| and |v|. Thus we need at
    // most the combined bit width of inputs for at least one value to be zero.
    unsigned a_bits = a_width * BN_BITS2, n_bits = n_width * BN_BITS2;
    unsigned num_iters = a_bits + n_bits;
    if (num_iters < a_bits) {
      OPENSSL_PUT_ERROR(BN, BN_R_BIGNUM_TOO_LONG);
      goto err;
    }
  
    // Before and after each loop iteration, the following hold:
    //
    //   u = A*a - B*n
    //   v = D*n - C*a
    //   0 < u <= a
    //   0 <= v <= n
    //   0 <= A < n
    //   0 <= B <= a
    //   0 <= C < n
    //   0 <= D <= a
    //
    // After each loop iteration, u and v only get smaller, and at least one of
    // them shrinks by at least a factor of two.
    for (unsigned i = 0; i < num_iters; i++) {
      BN_ULONG both_odd = word_is_odd_mask(u->d[0]) & word_is_odd_mask(v->d[0]);
  
      // If both |u| and |v| are odd, subtract the smaller from the larger.
      BN_ULONG v_less_than_u =
          (BN_ULONG)0 - bn_sub_words(tmp->d, v->d, u->d, n_width);
      bn_select_words(v->d, both_odd & ~v_less_than_u, tmp->d, v->d, n_width);
      bn_sub_words(tmp->d, u->d, v->d, n_width);
      bn_select_words(u->d, both_odd & v_less_than_u, tmp->d, u->d, n_width);
  
      // If we updated one of the values, update the corresponding coefficient.
      BN_ULONG carry = bn_add_words(tmp->d, A->d, C->d, n_width);
      carry -= bn_sub_words(tmp2->d, tmp->d, n->d, n_width);
      bn_select_words(tmp->d, carry, tmp->d, tmp2->d, n_width);
      bn_select_words(A->d, both_odd & v_less_than_u, tmp->d, A->d, n_width);
      bn_select_words(C->d, both_odd & ~v_less_than_u, tmp->d, C->d, n_width);
  
      bn_add_words(tmp->d, B->d, D->d, a_width);
      bn_sub_words(tmp2->d, tmp->d, a->d, a_width);
      bn_select_words(tmp->d, carry, tmp->d, tmp2->d, a_width);
      bn_select_words(B->d, both_odd & v_less_than_u, tmp->d, B->d, a_width);
      bn_select_words(D->d, both_odd & ~v_less_than_u, tmp->d, D->d, a_width);
  
      // Our loop invariants hold at this point. Additionally, exactly one of |u|
      // and |v| is now even.
      BN_ULONG u_is_even = ~word_is_odd_mask(u->d[0]);
      BN_ULONG v_is_even = ~word_is_odd_mask(v->d[0]);
      assert(u_is_even != v_is_even);
  
      // Halve the even one and adjust the corresponding coefficient.
      maybe_rshift1_words(u->d, u_is_even, tmp->d, n_width);
      BN_ULONG A_or_B_is_odd =
          word_is_odd_mask(A->d[0]) | word_is_odd_mask(B->d[0]);
      BN_ULONG A_carry =
          maybe_add_words(A->d, A_or_B_is_odd & u_is_even, n->d, tmp->d, n_width);
      BN_ULONG B_carry =
          maybe_add_words(B->d, A_or_B_is_odd & u_is_even, a->d, tmp->d, a_width);
      maybe_rshift1_words_carry(A->d, A_carry, u_is_even, tmp->d, n_width);
      maybe_rshift1_words_carry(B->d, B_carry, u_is_even, tmp->d, a_width);
  
      maybe_rshift1_words(v->d, v_is_even, tmp->d, n_width);
      BN_ULONG C_or_D_is_odd =
          word_is_odd_mask(C->d[0]) | word_is_odd_mask(D->d[0]);
      BN_ULONG C_carry =
          maybe_add_words(C->d, C_or_D_is_odd & v_is_even, n->d, tmp->d, n_width);
      BN_ULONG D_carry =
          maybe_add_words(D->d, C_or_D_is_odd & v_is_even, a->d, tmp->d, a_width);
      maybe_rshift1_words_carry(C->d, C_carry, v_is_even, tmp->d, n_width);
      maybe_rshift1_words_carry(D->d, D_carry, v_is_even, tmp->d, a_width);
    }
  
    assert(BN_is_zero(v));
    if (!BN_is_one(u)) {
      *out_no_inverse = 1;
      OPENSSL_PUT_ERROR(BN, BN_R_NO_INVERSE);
      goto err;
    }
  
    ret = BN_copy(r, A) != NULL;
  
  err:
    BN_CTX_end(ctx);
    return ret;
  }
  
  int BN_mod_inverse_odd(BIGNUM *out, int *out_no_inverse, const BIGNUM *a,
                         const BIGNUM *n, BN_CTX *ctx) {
    *out_no_inverse = 0;
  
    if (!BN_is_odd(n)) {
      OPENSSL_PUT_ERROR(BN, BN_R_CALLED_WITH_EVEN_MODULUS);
      return 0;
    }
  
    if (BN_is_negative(a) || BN_cmp(a, n) >= 0) {
      OPENSSL_PUT_ERROR(BN, BN_R_INPUT_NOT_REDUCED);
      return 0;
    }
  
    BIGNUM *A, *B, *X, *Y;
    int ret = 0;
    int sign;
  
    BN_CTX_start(ctx);
    A = BN_CTX_get(ctx);
    B = BN_CTX_get(ctx);
    X = BN_CTX_get(ctx);
    Y = BN_CTX_get(ctx);
    if (Y == NULL) {
      goto err;
    }
  
    BIGNUM *R = out;
  
    BN_zero(Y);
    if (!BN_one(X) || BN_copy(B, a) == NULL || BN_copy(A, n) == NULL) {
      goto err;
    }
    A->neg = 0;
    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|).
  
    // 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 systems, 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;
        }
      }
    }
  
    if (!BN_is_one(A)) {
      *out_no_inverse = 1;
      OPENSSL_PUT_ERROR(BN, BN_R_NO_INVERSE);
      goto err;
    }
  
    // 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|).
  
    // 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;
      }
    }
  
    ret = 1;
  
  err:
    BN_CTX_end(ctx);
    return ret;
  }
  
  BIGNUM *BN_mod_inverse(BIGNUM *out, const BIGNUM *a, const BIGNUM *n,
                         BN_CTX *ctx) {
    BIGNUM *new_out = NULL;
    if (out == NULL) {
      new_out = BN_new();
      if (new_out == NULL) {
        OPENSSL_PUT_ERROR(BN, ERR_R_MALLOC_FAILURE);
        return NULL;
      }
      out = new_out;
    }
  
    int ok = 0;
    BIGNUM *a_reduced = NULL;
    if (a->neg || BN_ucmp(a, n) >= 0) {
      a_reduced = BN_dup(a);
      if (a_reduced == NULL) {
        goto err;
      }
      if (!BN_nnmod(a_reduced, a_reduced, n, ctx)) {
        goto err;
      }
      a = a_reduced;
    }
  
    int no_inverse;
    if (!BN_is_odd(n)) {
      if (!bn_mod_inverse_consttime(out, &no_inverse, a, n, ctx)) {
        goto err;
      }
    } else if (!BN_mod_inverse_odd(out, &no_inverse, a, n, ctx)) {
      goto err;
    }
  
    ok = 1;
  
  err:
    if (!ok) {
      BN_free(new_out);
      out = NULL;
    }
    BN_free(a_reduced);
    return out;
  }
  
  int BN_mod_inverse_blinded(BIGNUM *out, int *out_no_inverse, const BIGNUM *a,
                             const BN_MONT_CTX *mont, BN_CTX *ctx) {
    *out_no_inverse = 0;
  
    if (BN_is_negative(a) || BN_cmp(a, &mont->N) >= 0) {
      OPENSSL_PUT_ERROR(BN, BN_R_INPUT_NOT_REDUCED);
      return 0;
    }
  
    int ret = 0;
    BIGNUM blinding_factor;
    BN_init(&blinding_factor);
  
    if (!BN_rand_range_ex(&blinding_factor, 1, &mont->N) ||
        !BN_mod_mul_montgomery(out, &blinding_factor, a, mont, ctx) ||
        !BN_mod_inverse_odd(out, out_no_inverse, out, &mont->N, ctx) ||
        !BN_mod_mul_montgomery(out, &blinding_factor, out, mont, ctx)) {
      OPENSSL_PUT_ERROR(BN, ERR_R_BN_LIB);
      goto err;
    }
  
    ret = 1;
  
  err:
    BN_free(&blinding_factor);
    return ret;
  }
  
  int bn_mod_inverse_prime(BIGNUM *out, const BIGNUM *a, const BIGNUM *p,
                           BN_CTX *ctx, const BN_MONT_CTX *mont_p) {
    BN_CTX_start(ctx);
    BIGNUM *p_minus_2 = BN_CTX_get(ctx);
    int ok = p_minus_2 != NULL &&
             BN_copy(p_minus_2, p) &&
             BN_sub_word(p_minus_2, 2) &&
             BN_mod_exp_mont(out, a, p_minus_2, p, ctx, mont_p);
    BN_CTX_end(ctx);
    return ok;
  }
  
  int bn_mod_inverse_secret_prime(BIGNUM *out, const BIGNUM *a, const BIGNUM *p,
                                  BN_CTX *ctx, const BN_MONT_CTX *mont_p) {
    BN_CTX_start(ctx);
    BIGNUM *p_minus_2 = BN_CTX_get(ctx);
    int ok = p_minus_2 != NULL &&
             BN_copy(p_minus_2, p) &&
             BN_sub_word(p_minus_2, 2) &&
             BN_mod_exp_mont_consttime(out, a, p_minus_2, p, ctx, mont_p);
    BN_CTX_end(ctx);
    return ok;
  }
  