text: /*
   * Copyright 1995-2020 The OpenSSL Project Authors. All Rights Reserved.
   *
   * Licensed under the Apache License 2.0 (the "License").  You may not use
   * this file except in compliance with the License.  You can obtain a copy
   * in the file LICENSE in the source distribution or at
   * https://www.openssl.org/source/license.html
   */
  
  #include "internal/cryptlib.h"
  #include "bn_local.h"
  
  /*
   * bn_mod_inverse_no_branch is a special version of BN_mod_inverse. It does
   * not contain branches that may leak sensitive information.
   *
   * This is a static function, we ensure all callers in this file pass valid
   * arguments: all passed pointers here are non-NULL.
   */
  static ossl_inline
  BIGNUM *bn_mod_inverse_no_branch(BIGNUM *in,
                                   const BIGNUM *a, const BIGNUM *n,
                                   BN_CTX *ctx, int *pnoinv)
  {
      BIGNUM *A, *B, *X, *Y, *M, *D, *T, *R = NULL;
      BIGNUM *ret = NULL;
      int sign;
  
      bn_check_top(a);
      bn_check_top(n);
  
      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 (in == NULL)
          R = BN_new();
      else
          R = in;
      if (R == NULL)
          goto err;
  
      BN_one(X);
      BN_zero(Y);
      if (BN_copy(B, a) == NULL)
          goto err;
      if (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.
           */
           {
              BIGNUM local_B;
              bn_init(&local_B);
              BN_with_flags(&local_B, B, BN_FLG_CONSTTIME);
              if (!BN_nnmod(B, &local_B, A, ctx))
                  goto err;
              /* Ensure local_B goes out of scope before any further use of B */
          }
      }
      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.
           */
          {
              BIGNUM local_A;
              bn_init(&local_A);
              BN_with_flags(&local_A, A, BN_FLG_CONSTTIME);
  
              /* (D, M) := (A/B, A%B) ... */
              if (!BN_div(D, M, &local_A, B, ctx))
                  goto err;
              /* Ensure local_A goes out of scope before any further use of A */
          }
  
          /*-
           * 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 {
          *pnoinv = 1;
          /* caller sets the BN_R_NO_INVERSE error */
          goto err;
      }
  
      ret = R;
      *pnoinv = 0;
  
   err:
      if ((ret == NULL) && (in == NULL))
          BN_free(R);
      BN_CTX_end(ctx);
      bn_check_top(ret);
      return ret;
  }
  
  /*
   * This is an internal function, we assume all callers pass valid arguments:
   * all pointers passed here are assumed non-NULL.
   */
  BIGNUM *int_bn_mod_inverse(BIGNUM *in,
                             const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx,
                             int *pnoinv)
  {
      BIGNUM *A, *B, *X, *Y, *M, *D, *T, *R = NULL;
      BIGNUM *ret = NULL;
      int sign;
  
      /* This is invalid input so we don't worry about constant time here */
      if (BN_abs_is_word(n, 1) || BN_is_zero(n)) {
          *pnoinv = 1;
          return NULL;
      }
  
      *pnoinv = 0;
  
      if ((BN_get_flags(a, BN_FLG_CONSTTIME) != 0)
          || (BN_get_flags(n, BN_FLG_CONSTTIME) != 0)) {
          return bn_mod_inverse_no_branch(in, a, n, ctx, pnoinv);
      }
  
      bn_check_top(a);
      bn_check_top(n);
  
      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 (in == NULL)
          R = BN_new();
      else
          R = in;
      if (R == NULL)
          goto err;
  
      BN_one(X);
      BN_zero(Y);
      if (BN_copy(B, a) == NULL)
          goto err;
      if (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) <= 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 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;
              }
          }
      } 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 {
          *pnoinv = 1;
          goto err;
      }
      ret = R;
   err:
      if ((ret == NULL) && (in == NULL))
          BN_free(R);
      BN_CTX_end(ctx);
      bn_check_top(ret);
      return ret;
  }
  
  /* solves ax == 1 (mod n) */
  BIGNUM *BN_mod_inverse(BIGNUM *in,
                         const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx)
  {
      BN_CTX *new_ctx = NULL;
      BIGNUM *rv;
      int noinv = 0;
  
      if (ctx == NULL) {
          ctx = new_ctx = BN_CTX_new_ex(NULL);
          if (ctx == NULL) {
              BNerr(BN_F_BN_MOD_INVERSE, ERR_R_MALLOC_FAILURE);
              return NULL;
          }
      }
  
      rv = int_bn_mod_inverse(in, a, n, ctx, &noinv);
      if (noinv)
          BNerr(BN_F_BN_MOD_INVERSE, BN_R_NO_INVERSE);
      BN_CTX_free(new_ctx);
      return rv;
  }
  
  /*-
   * This function is based on the constant-time GCD work by Bernstein and Yang:
   * https://eprint.iacr.org/2019/266
   * Generalized fast GCD function to allow even inputs.
   * The algorithm first finds the shared powers of 2 between
   * the inputs, and removes them, reducing at least one of the
   * inputs to an odd value. Then it proceeds to calculate the GCD.
   * Before returning the resulting GCD, we take care of adding
   * back the powers of two removed at the beginning.
   * Note 1: we assume the bit length of both inputs is public information,
   * since access to top potentially leaks this information.
   */
  int BN_gcd(BIGNUM *r, const BIGNUM *in_a, const BIGNUM *in_b, BN_CTX *ctx)
  {
      BIGNUM *g, *temp = NULL;
      BN_ULONG mask = 0;
      int i, j, top, rlen, glen, m, bit = 1, delta = 1, cond = 0, shifts = 0, ret = 0;
  
      /* Note 2: zero input corner cases are not constant-time since they are
       * handled immediately. An attacker can run an attack under this
       * assumption without the need of side-channel information. */
      if (BN_is_zero(in_b)) {
          ret = BN_copy(r, in_a) != NULL;
          r->neg = 0;
          return ret;
      }
      if (BN_is_zero(in_a)) {
          ret = BN_copy(r, in_b) != NULL;
          r->neg = 0;
          return ret;
      }
  
      bn_check_top(in_a);
      bn_check_top(in_b);
  
      BN_CTX_start(ctx);
      temp = BN_CTX_get(ctx);
      g = BN_CTX_get(ctx);
  
      /* make r != 0, g != 0 even, so BN_rshift is not a potential nop */
      if (g == NULL
          || !BN_lshift1(g, in_b)
          || !BN_lshift1(r, in_a))
          goto err;
  
      /* find shared powers of two, i.e. "shifts" >= 1 */
      for (i = 0; i < r->dmax && i < g->dmax; i++) {
          mask = ~(r->d[i] | g->d[i]);
          for (j = 0; j < BN_BITS2; j++) {
              bit &= mask;
              shifts += bit;
              mask >>= 1;
          }
      }
  
      /* subtract shared powers of two; shifts >= 1 */
      if (!BN_rshift(r, r, shifts)
          || !BN_rshift(g, g, shifts))
          goto err;
  
      /* expand to biggest nword, with room for a possible extra word */
      top = 1 + ((r->top >= g->top) ? r->top : g->top);
      if (bn_wexpand(r, top) == NULL
          || bn_wexpand(g, top) == NULL
          || bn_wexpand(temp, top) == NULL)
          goto err;
  
      /* re arrange inputs s.t. r is odd */
      BN_consttime_swap((~r->d[0]) & 1, r, g, top);
  
      /* compute the number of iterations */
      rlen = BN_num_bits(r);
      glen = BN_num_bits(g);
      m = 4 + 3 * ((rlen >= glen) ? rlen : glen);
  
      for (i = 0; i < m; i++) {
          /* conditionally flip signs if delta is positive and g is odd */
          cond = (-delta >> (8 * sizeof(delta) - 1)) & g->d[0] & 1
              /* make sure g->top > 0 (i.e. if top == 0 then g == 0 always) */
              & (~((g->top - 1) >> (sizeof(g->top) * 8 - 1)));
          delta = (-cond & -delta) | ((cond - 1) & delta);
          r->neg ^= cond;
          /* swap */
          BN_consttime_swap(cond, r, g, top);
  
          /* elimination step */
          delta++;
          if (!BN_add(temp, g, r))
              goto err;
          BN_consttime_swap(g->d[0] & 1 /* g is odd */
                  /* make sure g->top > 0 (i.e. if top == 0 then g == 0 always) */
                  & (~((g->top - 1) >> (sizeof(g->top) * 8 - 1))),
                  g, temp, top);
          if (!BN_rshift1(g, g))
              goto err;
      }
  
      /* remove possible negative sign */
      r->neg = 0;
      /* add powers of 2 removed, then correct the artificial shift */
      if (!BN_lshift(r, r, shifts)
          || !BN_rshift1(r, r))
          goto err;
  
      ret = 1;
  
   err:
      BN_CTX_end(ctx);
      bn_check_top(r);
      return ret;
  }
  