text: /* Copyright (C) 1995-1997 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) 1998-2006 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).
   *
   */
  /* ====================================================================
   * Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED.
   *
   * Portions of the attached software ("Contribution") are developed by
   * SUN MICROSYSTEMS, INC., and are contributed to the OpenSSL project.
   *
   * The Contribution is licensed pursuant to the Eric Young open source
   * license provided above.
   *
   * The binary polynomial arithmetic software is originally written by
   * Sheueling Chang Shantz and Douglas Stebila of Sun Microsystems
   * Laboratories. */
  
  #ifndef OPENSSL_HEADER_BN_INTERNAL_H
  #define OPENSSL_HEADER_BN_INTERNAL_H
  
  #include <CCryptoBoringSSL_bn.h>
  
  #if defined(OPENSSL_X86_64) && defined(_MSC_VER)
  OPENSSL_MSVC_PRAGMA(warning(push, 3))
  #include <intrin.h>
  OPENSSL_MSVC_PRAGMA(warning(pop))
  #pragma intrinsic(__umulh, _umul128)
  #endif
  
  #include "../../internal.h"
  
  #if defined(__cplusplus)
  extern "C" {
  #endif
  
  #if defined(OPENSSL_64_BIT)
  
  #if defined(BORINGSSL_HAS_UINT128)
  // MSVC doesn't support two-word integers on 64-bit.
  #define BN_ULLONG uint128_t
  #if defined(BORINGSSL_CAN_DIVIDE_UINT128)
  #define BN_CAN_DIVIDE_ULLONG
  #endif
  #endif
  
  #define BN_BITS2 64
  #define BN_BYTES 8
  #define BN_BITS4 32
  #define BN_MASK2 (0xffffffffffffffffUL)
  #define BN_MASK2l (0xffffffffUL)
  #define BN_MASK2h (0xffffffff00000000UL)
  #define BN_MASK2h1 (0xffffffff80000000UL)
  #define BN_MONT_CTX_N0_LIMBS 1
  #define BN_DEC_CONV (10000000000000000000UL)
  #define BN_DEC_NUM 19
  #define TOBN(hi, lo) ((BN_ULONG)(hi) << 32 | (lo))
  
  #elif defined(OPENSSL_32_BIT)
  
  #define BN_ULLONG uint64_t
  #define BN_CAN_DIVIDE_ULLONG
  #define BN_BITS2 32
  #define BN_BYTES 4
  #define BN_BITS4 16
  #define BN_MASK2 (0xffffffffUL)
  #define BN_MASK2l (0xffffUL)
  #define BN_MASK2h1 (0xffff8000UL)
  #define BN_MASK2h (0xffff0000UL)
  // On some 32-bit platforms, Montgomery multiplication is done using 64-bit
  // arithmetic with SIMD instructions. On such platforms, |BN_MONT_CTX::n0|
  // needs to be two words long. Only certain 32-bit platforms actually make use
  // of n0[1] and shorter R value would suffice for the others. However,
  // currently only the assembly files know which is which.
  #define BN_MONT_CTX_N0_LIMBS 2
  #define BN_DEC_CONV (1000000000UL)
  #define BN_DEC_NUM 9
  #define TOBN(hi, lo) (lo), (hi)
  
  #else
  #error "Must define either OPENSSL_32_BIT or OPENSSL_64_BIT"
  #endif
  
  #if !defined(OPENSSL_NO_ASM) && (defined(__GNUC__) || defined(__clang__))
  #define BN_CAN_USE_INLINE_ASM
  #endif
  
  // |BN_mod_exp_mont_consttime| is based on the assumption that the L1 data
  // cache line width of the target processor is at least the following value.
  #define MOD_EXP_CTIME_MIN_CACHE_LINE_WIDTH 64
  
  // The number of |BN_ULONG|s needed for the |BN_mod_exp_mont_consttime| stack-
  // allocated storage buffer. The buffer is just the right size for the RSAZ
  // and is about ~1KB larger than what's necessary (4480 bytes) for 1024-bit
  // inputs.
  #define MOD_EXP_CTIME_STORAGE_LEN \
    (((320u * 3u) + (32u * 9u * 16u)) / sizeof(BN_ULONG))
  
  #define STATIC_BIGNUM(x)                                    \
    {                                                         \
      (BN_ULONG *)(x), sizeof(x) / sizeof(BN_ULONG),          \
          sizeof(x) / sizeof(BN_ULONG), 0, BN_FLG_STATIC_DATA \
    }
  
  #if defined(BN_ULLONG)
  #define Lw(t) ((BN_ULONG)(t))
  #define Hw(t) ((BN_ULONG)((t) >> BN_BITS2))
  #endif
  
  // bn_minimal_width returns the minimal value of |bn->top| which fits the
  // value of |bn|.
  int bn_minimal_width(const BIGNUM *bn);
  
  // bn_set_minimal_width sets |bn->width| to |bn_minimal_width(bn)|. If |bn| is
  // zero, |bn->neg| is set to zero.
  void bn_set_minimal_width(BIGNUM *bn);
  
  // bn_wexpand ensures that |bn| has at least |words| works of space without
  // altering its value. It returns one on success or zero on allocation
  // failure.
  int bn_wexpand(BIGNUM *bn, size_t words);
  
  // bn_expand acts the same as |bn_wexpand|, but takes a number of bits rather
  // than a number of words.
  int bn_expand(BIGNUM *bn, size_t bits);
  
  // bn_resize_words adjusts |bn->top| to be |words|. It returns one on success
  // and zero on allocation error or if |bn|'s value is too large.
  OPENSSL_EXPORT int bn_resize_words(BIGNUM *bn, size_t words);
  
  // bn_select_words sets |r| to |a| if |mask| is all ones or |b| if |mask| is
  // all zeros.
  void bn_select_words(BN_ULONG *r, BN_ULONG mask, const BN_ULONG *a,
                       const BN_ULONG *b, size_t num);
  
  // bn_set_words sets |bn| to the value encoded in the |num| words in |words|,
  // least significant word first.
  int bn_set_words(BIGNUM *bn, const BN_ULONG *words, size_t num);
  
  // bn_set_static_words acts like |bn_set_words|, but doesn't copy the data. A
  // flag is set on |bn| so that |BN_free| won't attempt to free the data.
  //
  // The |STATIC_BIGNUM| macro is probably a better solution for this outside of
  // the FIPS module. Inside of the FIPS module that macro generates rel.ro data,
  // which doesn't work with FIPS requirements.
  void bn_set_static_words(BIGNUM *bn, const BN_ULONG *words, size_t num);
  
  // bn_fits_in_words returns one if |bn| may be represented in |num| words, plus
  // a sign bit, and zero otherwise.
  int bn_fits_in_words(const BIGNUM *bn, size_t num);
  
  // bn_copy_words copies the value of |bn| to |out| and returns one if the value
  // is representable in |num| words. Otherwise, it returns zero.
  int bn_copy_words(BN_ULONG *out, size_t num, const BIGNUM *bn);
  
  // bn_mul_add_words multiples |ap| by |w|, adds the result to |rp|, and places
  // the result in |rp|. |ap| and |rp| must both be |num| words long. It returns
  // the carry word of the operation. |ap| and |rp| may be equal but otherwise may
  // not alias.
  BN_ULONG bn_mul_add_words(BN_ULONG *rp, const BN_ULONG *ap, size_t num,
                            BN_ULONG w);
  
  // bn_mul_words multiples |ap| by |w| and places the result in |rp|. |ap| and
  // |rp| must both be |num| words long. It returns the carry word of the
  // operation. |ap| and |rp| may be equal but otherwise may not alias.
  BN_ULONG bn_mul_words(BN_ULONG *rp, const BN_ULONG *ap, size_t num, BN_ULONG w);
  
  // bn_sqr_words sets |rp[2*i]| and |rp[2*i+1]| to |ap[i]|'s square, for all |i|
  // up to |num|. |ap| is an array of |num| words and |rp| an array of |2*num|
  // words. |ap| and |rp| may not alias.
  //
  // This gives the contribution of the |ap[i]*ap[i]| terms when squaring |ap|.
  void bn_sqr_words(BN_ULONG *rp, const BN_ULONG *ap, size_t num);
  
  // bn_add_words adds |ap| to |bp| and places the result in |rp|, each of which
  // are |num| words long. It returns the carry bit, which is one if the operation
  // overflowed and zero otherwise. Any pair of |ap|, |bp|, and |rp| may be equal
  // to each other but otherwise may not alias.
  BN_ULONG bn_add_words(BN_ULONG *rp, const BN_ULONG *ap, const BN_ULONG *bp,
                        size_t num);
  
  // bn_sub_words subtracts |bp| from |ap| and places the result in |rp|. It
  // returns the borrow bit, which is one if the computation underflowed and zero
  // otherwise. Any pair of |ap|, |bp|, and |rp| may be equal to each other but
  // otherwise may not alias.
  BN_ULONG bn_sub_words(BN_ULONG *rp, const BN_ULONG *ap, const BN_ULONG *bp,
                        size_t num);
  
  // bn_mul_comba4 sets |r| to the product of |a| and |b|.
  void bn_mul_comba4(BN_ULONG r[8], const BN_ULONG a[4], const BN_ULONG b[4]);
  
  // bn_mul_comba8 sets |r| to the product of |a| and |b|.
  void bn_mul_comba8(BN_ULONG r[16], const BN_ULONG a[8], const BN_ULONG b[8]);
  
  // bn_sqr_comba8 sets |r| to |a|^2.
  void bn_sqr_comba8(BN_ULONG r[16], const BN_ULONG a[8]);
  
  // bn_sqr_comba4 sets |r| to |a|^2.
  void bn_sqr_comba4(BN_ULONG r[8], const BN_ULONG a[4]);
  
  // bn_less_than_words returns one if |a| < |b| and zero otherwise, where |a|
  // and |b| both are |len| words long. It runs in constant time.
  int bn_less_than_words(const BN_ULONG *a, const BN_ULONG *b, size_t len);
  
  // bn_in_range_words returns one if |min_inclusive| <= |a| < |max_exclusive|,
  // where |a| and |max_exclusive| both are |len| words long. |a| and
  // |max_exclusive| are treated as secret.
  int bn_in_range_words(const BN_ULONG *a, BN_ULONG min_inclusive,
                        const BN_ULONG *max_exclusive, size_t len);
  
  // bn_rand_range_words sets |out| to a uniformly distributed random number from
  // |min_inclusive| to |max_exclusive|. Both |out| and |max_exclusive| are |len|
  // words long.
  //
  // This function runs in time independent of the result, but |min_inclusive| and
  // |max_exclusive| are public data. (Information about the range is unavoidably
  // leaked by how many iterations it took to select a number.)
  int bn_rand_range_words(BN_ULONG *out, BN_ULONG min_inclusive,
                          const BN_ULONG *max_exclusive, size_t len,
                          const uint8_t additional_data[32]);
  
  // bn_range_secret_range behaves like |BN_rand_range_ex|, but treats
  // |max_exclusive| as secret. Because of this constraint, the distribution of
  // values returned is more complex.
  //
  // Rather than repeatedly generating values until one is in range, which would
  // leak information, it generates one value. If the value is in range, it sets
  // |*out_is_uniform| to one. Otherwise, it sets |*out_is_uniform| to zero,
  // fixing up the value to force it in range.
  //
  // The subset of calls to |bn_rand_secret_range| which set |*out_is_uniform| to
  // one are uniformly distributed in the target range. Calls overall are not.
  // This function is intended for use in situations where the extra values are
  // still usable and where the number of iterations needed to reach the target
  // number of uniform outputs may be blinded for negligible probabilities of
  // timing leaks.
  //
  // Although this function treats |max_exclusive| as secret, it treats the number
  // of bits in |max_exclusive| as public.
  int bn_rand_secret_range(BIGNUM *r, int *out_is_uniform, BN_ULONG min_inclusive,
                           const BIGNUM *max_exclusive);
  
  #if !defined(OPENSSL_NO_ASM) &&                         \
      (defined(OPENSSL_X86) || defined(OPENSSL_X86_64) || \
       defined(OPENSSL_ARM) || defined(OPENSSL_AARCH64))
  #define OPENSSL_BN_ASM_MONT
  // bn_mul_mont writes |ap| * |bp| mod |np| to |rp|, each |num| words
  // long. Inputs and outputs are in Montgomery form. |n0| is a pointer to the
  // corresponding field in |BN_MONT_CTX|. It returns one if |bn_mul_mont| handles
  // inputs of this size and zero otherwise.
  //
  // TODO(davidben): The x86_64 implementation expects a 32-bit input and masks
  // off upper bits. The aarch64 implementation expects a 64-bit input and does
  // not. |size_t| is the safer option but not strictly correct for x86_64. But
  // this function implicitly already has a bound on the size of |num| because it
  // internally creates |num|-sized stack allocation.
  //
  // See also discussion in |ToWord| in abi_test.h for notes on smaller-than-word
  // inputs.
  int bn_mul_mont(BN_ULONG *rp, const BN_ULONG *ap, const BN_ULONG *bp,
                  const BN_ULONG *np, const BN_ULONG *n0, size_t num);
  #endif
  
  #if !defined(OPENSSL_NO_ASM) && defined(OPENSSL_X86_64)
  #define OPENSSL_BN_ASM_MONT5
  
  // bn_mul_mont_gather5 multiples loads index |power| of |table|, multiplies it
  // by |ap| modulo |np|, and stores the result in |rp|. The values are |num|
  // words long and represented in Montgomery form. |n0| is a pointer to the
  // corresponding field in |BN_MONT_CTX|.
  void bn_mul_mont_gather5(BN_ULONG *rp, const BN_ULONG *ap,
                           const BN_ULONG *table, const BN_ULONG *np,
                           const BN_ULONG *n0, int num, int power);
  
  // bn_scatter5 stores |inp| to index |power| of |table|. |inp| and each entry of
  // |table| are |num| words long. |power| must be less than 32. |table| must be
  // 32*|num| words long.
  void bn_scatter5(const BN_ULONG *inp, size_t num, BN_ULONG *table,
                   size_t power);
  
  // bn_gather5 loads index |power| of |table| and stores it in |out|. |out| and
  // each entry of |table| are |num| words long. |power| must be less than 32.
  void bn_gather5(BN_ULONG *out, size_t num, BN_ULONG *table, size_t power);
  
  // bn_power5 squares |ap| five times and multiplies it by the value stored at
  // index |power| of |table|, modulo |np|. It stores the result in |rp|. The
  // values are |num| words long and represented in Montgomery form. |n0| is a
  // pointer to the corresponding field in |BN_MONT_CTX|. |num| must be divisible
  // by 8.
  void bn_power5(BN_ULONG *rp, const BN_ULONG *ap, const BN_ULONG *table,
                 const BN_ULONG *np, const BN_ULONG *n0, int num, int power);
  
  // bn_from_montgomery converts |ap| from Montgomery form modulo |np| and writes
  // the result in |rp|, each of which is |num| words long. It returns one on
  // success and zero if it cannot handle inputs of length |num|. |n0| is a
  // pointer to the corresponding field in |BN_MONT_CTX|.
  int bn_from_montgomery(BN_ULONG *rp, const BN_ULONG *ap,
                         const BN_ULONG *not_used, const BN_ULONG *np,
                         const BN_ULONG *n0, int num);
  #endif  // !OPENSSL_NO_ASM && OPENSSL_X86_64
  
  uint64_t bn_mont_n0(const BIGNUM *n);
  
  // bn_mod_exp_base_2_consttime calculates r = 2**p (mod n). |p| must be larger
  // than log_2(n); i.e. 2**p must be larger than |n|. |n| must be positive and
  // odd. |p| and the bit width of |n| are assumed public, but |n| is otherwise
  // treated as secret.
  int bn_mod_exp_base_2_consttime(BIGNUM *r, unsigned p, const BIGNUM *n,
                                  BN_CTX *ctx);
  
  #if defined(_MSC_VER)
  #if defined(OPENSSL_X86_64)
  #define BN_UMULT_LOHI(low, high, a, b) ((low) = _umul128((a), (b), &(high)))
  #elif defined(OPENSSL_AARCH64)
  #define BN_UMULT_LOHI(low, high, a, b) \
    do {                                 \
      const BN_ULONG _a = (a);           \
      const BN_ULONG _b = (b);           \
      (low) = _a * _b;                   \
      (high) = __umulh(_a, _b);          \
    } while (0)
  #endif
  #endif  // _MSC_VER
  
  #if !defined(BN_ULLONG) && !defined(BN_UMULT_LOHI)
  #error "Either BN_ULLONG or BN_UMULT_LOHI must be defined on every platform."
  #endif
  
  // bn_jacobi returns the Jacobi symbol of |a| and |b| (which is -1, 0 or 1), or
  // -2 on error.
  int bn_jacobi(const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx);
  
  // bn_is_bit_set_words returns one if bit |bit| is set in |a| and zero
  // otherwise.
  int bn_is_bit_set_words(const BN_ULONG *a, size_t num, unsigned bit);
  
  // bn_one_to_montgomery sets |r| to one in Montgomery form. It returns one on
  // success and zero on error. This function treats the bit width of the modulus
  // as public.
  int bn_one_to_montgomery(BIGNUM *r, const BN_MONT_CTX *mont, BN_CTX *ctx);
  
  // bn_less_than_montgomery_R returns one if |bn| is less than the Montgomery R
  // value for |mont| and zero otherwise.
  int bn_less_than_montgomery_R(const BIGNUM *bn, const BN_MONT_CTX *mont);
  
  // bn_mod_u16_consttime returns |bn| mod |d|, ignoring |bn|'s sign bit. It runs
  // in time independent of the value of |bn|, but it treats |d| as public.
  OPENSSL_EXPORT uint16_t bn_mod_u16_consttime(const BIGNUM *bn, uint16_t d);
  
  // bn_odd_number_is_obviously_composite returns one if |bn| is divisible by one
  // of the first several odd primes and zero otherwise.
  int bn_odd_number_is_obviously_composite(const BIGNUM *bn);
  
  // A BN_MILLER_RABIN stores state common to each Miller-Rabin iteration. It is
  // initialized within an existing |BN_CTX| scope and may not be used after
  // that scope is released with |BN_CTX_end|. Field names match those in FIPS
  // 186-4, section C.3.1.
  typedef struct {
    // w1 is w-1.
    BIGNUM *w1;
    // m is (w-1)/2^a.
    BIGNUM *m;
    // one_mont is 1 (mod w) in Montgomery form.
    BIGNUM *one_mont;
    // w1_mont is w-1 (mod w) in Montgomery form.
    BIGNUM *w1_mont;
    // w_bits is BN_num_bits(w).
    int w_bits;
    // a is the largest integer such that 2^a divides w-1.
    int a;
  } BN_MILLER_RABIN;
  
  // bn_miller_rabin_init initializes |miller_rabin| for testing if |mont->N| is
  // prime. It returns one on success and zero on error.
  OPENSSL_EXPORT int bn_miller_rabin_init(BN_MILLER_RABIN *miller_rabin,
                                          const BN_MONT_CTX *mont, BN_CTX *ctx);
  
  // bn_miller_rabin_iteration performs one Miller-Rabin iteration, checking if
  // |b| is a composite witness for |mont->N|. |miller_rabin| must have been
  // initialized with |bn_miller_rabin_setup|. On success, it returns one and sets
  // |*out_is_possibly_prime| to one if |mont->N| may still be prime or zero if
  // |b| shows it is composite. On allocation or internal failure, it returns
  // zero.
  OPENSSL_EXPORT int bn_miller_rabin_iteration(
      const BN_MILLER_RABIN *miller_rabin, int *out_is_possibly_prime,
      const BIGNUM *b, const BN_MONT_CTX *mont, BN_CTX *ctx);
  
  // bn_rshift1_words sets |r| to |a| >> 1, where both arrays are |num| bits wide.
  void bn_rshift1_words(BN_ULONG *r, const BN_ULONG *a, size_t num);
  
  // bn_rshift_words sets |r| to |a| >> |shift|, where both arrays are |num| bits
  // wide.
  void bn_rshift_words(BN_ULONG *r, const BN_ULONG *a, unsigned shift,
                       size_t num);
  
  // bn_rshift_secret_shift behaves like |BN_rshift| but runs in time independent
  // of both |a| and |n|.
  OPENSSL_EXPORT int bn_rshift_secret_shift(BIGNUM *r, const BIGNUM *a,
                                            unsigned n, BN_CTX *ctx);
  
  // bn_reduce_once sets |r| to |a| mod |m| where 0 <= |a| < 2*|m|. It returns
  // zero if |a| < |m| and a mask of all ones if |a| >= |m|. Each array is |num|
  // words long, but |a| has an additional word specified by |carry|. |carry| must
  // be zero or one, as implied by the bounds on |a|.
  //
  // |r|, |a|, and |m| may not alias. Use |bn_reduce_once_in_place| if |r| and |a|
  // must alias.
  BN_ULONG bn_reduce_once(BN_ULONG *r, const BN_ULONG *a, BN_ULONG carry,
                          const BN_ULONG *m, size_t num);
  
  // bn_reduce_once_in_place behaves like |bn_reduce_once| but acts in-place on
  // |r|, using |tmp| as scratch space. |r|, |tmp|, and |m| may not alias.
  BN_ULONG bn_reduce_once_in_place(BN_ULONG *r, BN_ULONG carry, const BN_ULONG *m,
                                   BN_ULONG *tmp, size_t num);
  
  
  // Constant-time non-modular arithmetic.
  //
  // The following functions implement non-modular arithmetic in constant-time
  // and pessimally set |r->width| to the largest possible word size.
  //
  // Note this means that, e.g., repeatedly multiplying by one will cause widths
  // to increase without bound. The corresponding public API functions minimize
  // their outputs to avoid regressing calculator consumers.
  
  // bn_uadd_consttime behaves like |BN_uadd|, but it pessimally sets
  // |r->width| = |a->width| + |b->width| + 1.
  int bn_uadd_consttime(BIGNUM *r, const BIGNUM *a, const BIGNUM *b);
  
  // bn_usub_consttime behaves like |BN_usub|, but it pessimally sets
  // |r->width| = |a->width|.
  int bn_usub_consttime(BIGNUM *r, const BIGNUM *a, const BIGNUM *b);
  
  // bn_abs_sub_consttime sets |r| to the absolute value of |a| - |b|, treating
  // both inputs as secret. It returns one on success and zero on error.
  OPENSSL_EXPORT int bn_abs_sub_consttime(BIGNUM *r, const BIGNUM *a,
                                          const BIGNUM *b, BN_CTX *ctx);
  
  // bn_mul_consttime behaves like |BN_mul|, but it rejects negative inputs and
  // pessimally sets |r->width| to |a->width| + |b->width|, to avoid leaking
  // information about |a| and |b|.
  int bn_mul_consttime(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx);
  
  // bn_sqrt_consttime behaves like |BN_sqrt|, but it pessimally sets |r->width|
  // to 2*|a->width|, to avoid leaking information about |a| and |b|.
  int bn_sqr_consttime(BIGNUM *r, const BIGNUM *a, BN_CTX *ctx);
  
  // bn_div_consttime behaves like |BN_div|, but it rejects negative inputs and
  // treats both inputs, including their magnitudes, as secret. It is, as a
  // result, much slower than |BN_div| and should only be used for rare operations
  // where Montgomery reduction is not available.
  //
  // Note that |quotient->width| will be set pessimally to |numerator->width|.
  OPENSSL_EXPORT int bn_div_consttime(BIGNUM *quotient, BIGNUM *remainder,
                                      const BIGNUM *numerator,
                                      const BIGNUM *divisor, BN_CTX *ctx);
  
  // bn_is_relatively_prime checks whether GCD(|x|, |y|) is one. On success, it
  // returns one and sets |*out_relatively_prime| to one if the GCD was one and
  // zero otherwise. On error, it returns zero.
  OPENSSL_EXPORT int bn_is_relatively_prime(int *out_relatively_prime,
                                            const BIGNUM *x, const BIGNUM *y,
                                            BN_CTX *ctx);
  
  // bn_lcm_consttime sets |r| to LCM(|a|, |b|). It returns one and success and
  // zero on error. |a| and |b| are both treated as secret.
  OPENSSL_EXPORT int bn_lcm_consttime(BIGNUM *r, const BIGNUM *a, const BIGNUM *b,
                                      BN_CTX *ctx);
  
  
  // Constant-time modular arithmetic.
  //
  // The following functions implement basic constant-time modular arithmetic.
  
  // bn_mod_add_words sets |r| to |a| + |b| (mod |m|), using |tmp| as scratch
  // space. Each array is |num| words long. |a| and |b| must be < |m|. Any pair of
  // |r|, |a|, and |b| may alias.
  void bn_mod_add_words(BN_ULONG *r, const BN_ULONG *a, const BN_ULONG *b,
                        const BN_ULONG *m, BN_ULONG *tmp, size_t num);
  
  // bn_mod_add_consttime acts like |BN_mod_add_quick| but takes a |BN_CTX|.
  int bn_mod_add_consttime(BIGNUM *r, const BIGNUM *a, const BIGNUM *b,
                           const BIGNUM *m, BN_CTX *ctx);
  
  // bn_mod_sub_words sets |r| to |a| - |b| (mod |m|), using |tmp| as scratch
  // space. Each array is |num| words long. |a| and |b| must be < |m|. Any pair of
  // |r|, |a|, and |b| may alias.
  void bn_mod_sub_words(BN_ULONG *r, const BN_ULONG *a, const BN_ULONG *b,
                        const BN_ULONG *m, BN_ULONG *tmp, size_t num);
  
  // bn_mod_sub_consttime acts like |BN_mod_sub_quick| but takes a |BN_CTX|.
  int bn_mod_sub_consttime(BIGNUM *r, const BIGNUM *a, const BIGNUM *b,
                           const BIGNUM *m, BN_CTX *ctx);
  
  // bn_mod_lshift1_consttime acts like |BN_mod_lshift1_quick| but takes a
  // |BN_CTX|.
  int bn_mod_lshift1_consttime(BIGNUM *r, const BIGNUM *a, const BIGNUM *m,
                               BN_CTX *ctx);
  
  // bn_mod_lshift_consttime acts like |BN_mod_lshift_quick| but takes a |BN_CTX|.
  int bn_mod_lshift_consttime(BIGNUM *r, const BIGNUM *a, int n, const BIGNUM *m,
                              BN_CTX *ctx);
  
  // bn_mod_inverse_consttime sets |r| to |a|^-1, mod |n|. |a| must be non-
  // negative and less than |n|. It returns one on success and zero on error. On
  // failure, if the failure was caused by |a| having no inverse mod |n| then
  // |*out_no_inverse| will be set to one; otherwise it will be set to zero.
  //
  // This function treats both |a| and |n| as secret, provided they are both non-
  // zero and the inverse exists. It should only be used for even moduli where
  // none of the less general implementations are applicable.
  OPENSSL_EXPORT int bn_mod_inverse_consttime(BIGNUM *r, int *out_no_inverse,
                                              const BIGNUM *a, const BIGNUM *n,
                                              BN_CTX *ctx);
  
  // bn_mod_inverse_prime sets |out| to the modular inverse of |a| modulo |p|,
  // computed with Fermat's Little Theorem. It returns one on success and zero on
  // error. If |mont_p| is NULL, one will be computed temporarily.
  int bn_mod_inverse_prime(BIGNUM *out, const BIGNUM *a, const BIGNUM *p,
                           BN_CTX *ctx, const BN_MONT_CTX *mont_p);
  
  // bn_mod_inverse_secret_prime behaves like |bn_mod_inverse_prime| but uses
  // |BN_mod_exp_mont_consttime| instead of |BN_mod_exp_mont| in hopes of
  // protecting the exponent.
  int bn_mod_inverse_secret_prime(BIGNUM *out, const BIGNUM *a, const BIGNUM *p,
                                  BN_CTX *ctx, const BN_MONT_CTX *mont_p);
  
  
  // Low-level operations for small numbers.
  //
  // The following functions implement algorithms suitable for use with scalars
  // and field elements in elliptic curves. They rely on the number being small
  // both to stack-allocate various temporaries and because they do not implement
  // optimizations useful for the larger values used in RSA.
  
  // BN_SMALL_MAX_WORDS is the largest size input these functions handle. This
  // limit allows temporaries to be more easily stack-allocated. This limit is set
  // to accommodate P-521.
  #if defined(OPENSSL_32_BIT)
  #define BN_SMALL_MAX_WORDS 17
  #else
  #define BN_SMALL_MAX_WORDS 9
  #endif
  
  // bn_mul_small sets |r| to |a|*|b|. |num_r| must be |num_a| + |num_b|. |r| may
  // not alias with |a| or |b|.
  void bn_mul_small(BN_ULONG *r, size_t num_r, const BN_ULONG *a, size_t num_a,
                   const BN_ULONG *b, size_t num_b);
  
  // bn_sqr_small sets |r| to |a|^2. |num_a| must be at most |BN_SMALL_MAX_WORDS|.
  // |num_r| must be |num_a|*2. |r| and |a| may not alias.
  void bn_sqr_small(BN_ULONG *r, size_t num_r, const BN_ULONG *a, size_t num_a);
  
  // In the following functions, the modulus must be at most |BN_SMALL_MAX_WORDS|
  // words long.
  
  // bn_to_montgomery_small sets |r| to |a| translated to the Montgomery domain.
  // |r| and |a| are |num| words long, which must be |mont->N.width|. |a| must be
  // fully reduced and may alias |r|.
  void bn_to_montgomery_small(BN_ULONG *r, const BN_ULONG *a, size_t num,
                              const BN_MONT_CTX *mont);
  
  // bn_from_montgomery_small sets |r| to |a| translated out of the Montgomery
  // domain. |r| and |a| are |num_r| and |num_a| words long, respectively. |num_r|
  // must be |mont->N.width|. |a| must be at most |mont->N|^2 and may alias |r|.
  //
  // Unlike most of these functions, only |num_r| is bounded by
  // |BN_SMALL_MAX_WORDS|. |num_a| may exceed it, but must be at most 2 * |num_r|.
  void bn_from_montgomery_small(BN_ULONG *r, size_t num_r, const BN_ULONG *a,
                                size_t num_a, const BN_MONT_CTX *mont);
  
  // bn_mod_mul_montgomery_small sets |r| to |a| * |b| mod |mont->N|. Both inputs
  // and outputs are in the Montgomery domain. Each array is |num| words long,
  // which must be |mont->N.width|. Any two of |r|, |a|, and |b| may alias. |a|
  // and |b| must be reduced on input.
  void bn_mod_mul_montgomery_small(BN_ULONG *r, const BN_ULONG *a,
                                   const BN_ULONG *b, size_t num,
                                   const BN_MONT_CTX *mont);
  
  // bn_mod_exp_mont_small sets |r| to |a|^|p| mod |mont->N|. It returns one on
  // success and zero on programmer or internal error. Both inputs and outputs are
  // in the Montgomery domain. |r| and |a| are |num| words long, which must be
  // |mont->N.width| and at most |BN_SMALL_MAX_WORDS|. |a| must be fully-reduced.
  // This function runs in time independent of |a|, but |p| and |mont->N| are
  // public values. |a| must be fully-reduced and may alias with |r|.
  //
  // Note this function differs from |BN_mod_exp_mont| which uses Montgomery
  // reduction but takes input and output outside the Montgomery domain. Combine
  // this function with |bn_from_montgomery_small| and |bn_to_montgomery_small|
  // if necessary.
  void bn_mod_exp_mont_small(BN_ULONG *r, const BN_ULONG *a, size_t num,
                             const BN_ULONG *p, size_t num_p,
                             const BN_MONT_CTX *mont);
  
  // bn_mod_inverse0_prime_mont_small sets |r| to |a|^-1 mod |mont->N|. If |a| is
  // zero, |r| is set to zero. |mont->N| must be a prime. |r| and |a| are |num|
  // words long, which must be |mont->N.width| and at most |BN_SMALL_MAX_WORDS|.
  // |a| must be fully-reduced and may alias |r|. This function runs in time
  // independent of |a|, but |mont->N| is a public value.
  void bn_mod_inverse0_prime_mont_small(BN_ULONG *r, const BN_ULONG *a,
                                        size_t num, const BN_MONT_CTX *mont);
  
  
  #if defined(__cplusplus)
  }  // extern C
  #endif
  
  #endif  // OPENSSL_HEADER_BN_INTERNAL_H
  