crypto/fipsmodule/bn/gcd.c - boringssl - Git at Google

 /* 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;
 }
	/* 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 = Aa - Bn
	// v = Dn - Ca
	// 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,
	// -signXa == B (mod \|n\|),
	// signYa == 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) -signXa == B (mod \|n\|),
	// (2) signYa == 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) -signXa == B (mod \|n\|),
	// (2) signYa == 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
	// signYa == 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;
	}