crypto/fipsmodule/bn/internal.h

// Copyright 1995-2016 The OpenSSL Project Authors. All Rights Reserved.
// Copyright (c) 2002, Oracle and/or its affiliates. All rights reserved.
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
//     https://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.

#ifndef OPENSSL_HEADER_CRYPTO_FIPSMODULE_BN_INTERNAL_H
#define OPENSSL_HEADER_CRYPTO_FIPSMODULE_BN_INTERNAL_H

#include <openssl/bn.h>

#if defined(OPENSSL_X86_64) && defined(_MSC_VER)
#include <intrin.h>
#pragma intrinsic(__umulh, _umul128)
#endif

#include "../../internal.h"


BSSL_NAMESPACE_BEGIN

#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_BITS2_LG 6
#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_BITS2_LG 5
#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

// MOD_EXP_CTIME_ALIGN is the alignment needed for `BN_mod_exp_mont_consttime`'s
// tables.
//
// TODO(davidben): Historically, this alignment came from cache line
// assumptions, which we've since removed. Is 64-byte alignment still necessary
// or ideal? The true alignment requirement seems to now be 32 bytes, coming
// from RSAZ's use of VMOVDQA to a YMM register. Non-x86_64 has even fewer
// requirements.
#define MOD_EXP_CTIME_ALIGN 64

// MOD_EXP_CTIME_STORAGE_LEN is 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 \
  }

// bn_minimal_width returns the minimal number of words needed to represent
// `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->width` 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_assert_fits_in_bytes asserts that `bn` fits in `num` bytes. This is a
// no-op in release builds, but triggers an assert in debug builds, and
// declassifies all bytes which are therefore known to be zero in constant-time
// validation.
void bn_assert_fits_in_bytes(const BIGNUM *bn, size_t num);

// bn_secret marks `bn`'s contents, but not its width or sign, as secret. See
// `CONSTTIME_SECRET` for details.
inline void bn_secret(BIGNUM *bn) {
  CONSTTIME_SECRET(bn->d, bn->width * sizeof(BN_ULONG));
}

// bn_declassify marks `bn`'s value as public. See `CONSTTIME_DECLASSIFY` for
// details.
inline void bn_declassify(BIGNUM *bn) {
  CONSTTIME_DECLASSIFY(bn->d, bn->width * sizeof(BN_ULONG));
}

#if !defined(OPENSSL_NO_ASM) && defined(OPENSSL_X86)
// See asm/bn-586.pl.
#define BN_ADD_ASM
#define BN_MUL_ASM
#endif

#if !defined(OPENSSL_NO_ASM) && defined(OPENSSL_X86_64) && \
    (defined(__GNUC__) || defined(__clang__))
// See asm/x86_64-gcc.c
#define BN_ADD_ASM
#define BN_MUL_ASM
#endif

#if !defined(OPENSSL_NO_ASM) && defined(OPENSSL_AARCH64)
// See asm/bn-armv8.pl.
#define BN_ADD_ASM
#endif

// 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.
#if defined(BN_MUL_ASM)
extern "C"
#endif
    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.
#if defined(BN_MUL_ASM)
extern "C"
#endif
    BN_ULONG bn_mul_words(BN_ULONG *rp, const BN_ULONG *ap, size_t num,
                          BN_ULONG w);

// bn_sqr_add_words computes `tmp` where `tmp[2*i]` and `tmp[2*i+1]` are
// `ap[i]`'s square, for all `i` up to `num`, and adds the result to `rp`. If
// the result does not fit in `2*num` words, the final carry bit is truncated.
// `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`.
#if defined(BN_MUL_ASM)
extern "C"
#endif
    void bn_sqr_add_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.

#if defined(BN_ADD_ASM)
extern "C"
#endif
    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.
#if defined(BN_ADD_ASM)
extern "C"
#endif
    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`.
#if defined(BN_MUL_ASM)
extern "C"
#endif
    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`.
#if defined(BN_MUL_ASM)
extern "C"
#endif
    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.
#if defined(BN_MUL_ASM)
extern "C"
#endif
    void bn_sqr_comba8(BN_ULONG r[16], const BN_ULONG a[8]);

// bn_sqr_comba4 sets `r` to `a`^2.
#if defined(BN_MUL_ASM)
extern "C"
#endif
    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);

// BN_MONTGOMERY_MAX_WORDS is the maximum number of words allowed in a `BIGNUM`
// used with Montgomery reduction. Ideally this limit would be applied to all
// `BIGNUM`s, in `bn_wexpand`, but the exactfloat library needs to create 8 MiB
// values for other operations.
//
// This limit is set so that one number fits within 2 KiB, giving room to
// allocate a few of them on the stack. It is also set to limit the DoS impact
// of large RSA, DH, and DSA keys, which scale cubically.
#define BN_MONTGOMERY_MAX_WORDS (16384 / BN_BITS2)

BSSL_NAMESPACE_END

struct bn_mont_ctx_st {
  // RR is R^2, reduced modulo `N`. It is used to convert to Montgomery form. It
  // is guaranteed to have the same width as `N`.
  BIGNUM RR;
  // N is the modulus. It is always stored in minimal form, so `N.width`
  // determines R.
  BIGNUM N;
  BN_ULONG n0[BN_MONT_CTX_N0_LIMBS];  // least significant words of (R*Ri-1)/N
};

BSSL_NAMESPACE_BEGIN

#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_words 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`.
//
// If at least one of `ap` or `bp` is fully reduced, `rp` will be fully reduced.
// If neither is fully-reduced, the output may not be either.
//
// This function allocates up to 2 * `num` words (plus a constant allocation) on
// the stack, so `num` should be at most `BN_MONTGOMERY_MAX_WORDS`.
// Additionally, `num` must be at least 128 / `BN_BITS2`.
//
// 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
// the `BN_MONTGOMERY_MAX_WORDS` bound makes this moot.
//
// See also discussion in `ToWord` in abi_test.h for notes on smaller-than-word
// inputs.
extern "C" void bn_mul_mont_words(BN_ULONG *rp, const BN_ULONG *ap,
                                  const BN_ULONG *bp, const BN_ULONG *np,
                                  const BN_ULONG n0[BN_MONT_CTX_N0_LIMBS],
                                  size_t num);

#if defined(OPENSSL_X86_64)
inline int bn_mulx_adx_capable() {
  // MULX is in BMI2.
  return CRYPTO_is_BMI2_capable() && CRYPTO_is_ADX_capable();
}
extern "C" void bn_mul_mont_nohw(BN_ULONG *rp, const BN_ULONG *ap,
                                 const BN_ULONG *bp, const BN_ULONG *np,
                                 const BN_ULONG n0[BN_MONT_CTX_N0_LIMBS],
                                 size_t num);
inline int bn_mul4x_mont_capable(size_t num) {
  return num >= 8 && (num & 3) == 0;
}
extern "C" void bn_mul4x_mont(BN_ULONG *rp, const BN_ULONG *ap,
                              const BN_ULONG *bp, const BN_ULONG *np,
                              const BN_ULONG n0[BN_MONT_CTX_N0_LIMBS],
                              size_t num);
inline int bn_mulx4x_mont_capable(size_t num) {
  return bn_mul4x_mont_capable(num) && bn_mulx_adx_capable();
}
extern "C" void bn_mulx4x_mont(BN_ULONG *rp, const BN_ULONG *ap,
                               const BN_ULONG *bp, const BN_ULONG *np,
                               const BN_ULONG n0[BN_MONT_CTX_N0_LIMBS],
                               size_t num);
inline int bn_sqr8x_mont_capable(size_t num) {
  return num >= 8 && (num & 7) == 0;
}
extern "C" void bn_sqr8x_mont(BN_ULONG *rp, const BN_ULONG *ap,
                              BN_ULONG mulx_adx_capable, const BN_ULONG *np,
                              const BN_ULONG n0[BN_MONT_CTX_N0_LIMBS],
                              size_t num);
#elif defined(OPENSSL_ARM)
inline int bn_mul8x_mont_neon_capable(size_t num) {
  return (num & 7) == 0 && CRYPTO_is_NEON_capable();
}
extern "C" void bn_mul8x_mont_neon(BN_ULONG *rp, const BN_ULONG *ap,
                                   const BN_ULONG *bp, const BN_ULONG *np,
                                   const BN_ULONG n0[BN_MONT_CTX_N0_LIMBS],
                                   size_t num);
extern "C" void bn_mul_mont_nohw(BN_ULONG *rp, const BN_ULONG *ap,
                                 const BN_ULONG *bp, const BN_ULONG *np,
                                 const BN_ULONG n0[BN_MONT_CTX_N0_LIMBS],
                                 size_t num);
#endif

#endif  // OPENSSL_BN_ASM_MONT

#if !defined(OPENSSL_NO_ASM) && defined(OPENSSL_X86_64)
#define OPENSSL_BN_ASM_MONT5

// The following functions implement `bn_mul_mont_gather5`. See
// `bn_mul_mont_gather5` for details.
inline int bn_mul4x_mont_gather5_capable(int num) { return (num & 7) == 0; }
extern "C" void bn_mul4x_mont_gather5(BN_ULONG *rp, const BN_ULONG *ap,
                                      const BN_ULONG *table, const BN_ULONG *np,
                                      const BN_ULONG n0[BN_MONT_CTX_N0_LIMBS],
                                      int num, int power);

inline int bn_mulx4x_mont_gather5_capable(int num) {
  return bn_mul4x_mont_gather5_capable(num) && CRYPTO_is_ADX_capable() &&
         CRYPTO_is_BMI1_capable() && CRYPTO_is_BMI2_capable();
}
extern "C" void bn_mulx4x_mont_gather5(BN_ULONG *rp, const BN_ULONG *ap,
                                       const BN_ULONG *table,
                                       const BN_ULONG *np,
                                       const BN_ULONG n0[BN_MONT_CTX_N0_LIMBS],
                                       int num, int power);

extern "C" void bn_mul_mont_gather5_nohw(
    BN_ULONG *rp, const BN_ULONG *ap, const BN_ULONG *table, const BN_ULONG *np,
    const BN_ULONG n0[BN_MONT_CTX_N0_LIMBS], 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 and is treated as
// public. `table` must be 32*`num` words long. `table` must be aligned to at
// least 16 bytes.
extern "C" 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 and
// is treated as secret. `table` must be aligned to at least 16 bytes.
extern "C" void bn_gather5(BN_ULONG *out, size_t num, const BN_ULONG *table,
                           size_t power);

// The following functions implement `bn_power5`. See `bn_power5` for details.
extern "C" void bn_power5_nohw(BN_ULONG *rp, const BN_ULONG *ap,
                               const BN_ULONG *table, const BN_ULONG *np,
                               const BN_ULONG n0[BN_MONT_CTX_N0_LIMBS], int num,
                               int power);

inline int bn_power5_capable(int num) { return (num & 7) == 0; }

inline int bn_powerx5_capable(int num) {
  return bn_power5_capable(num) && CRYPTO_is_ADX_capable() &&
         CRYPTO_is_BMI1_capable() && CRYPTO_is_BMI2_capable();
}
extern "C" void bn_powerx5(BN_ULONG *rp, const BN_ULONG *ap,
                           const BN_ULONG *table, const BN_ULONG *np,
                           const BN_ULONG n0[BN_MONT_CTX_N0_LIMBS], int num,
                           int power);

#endif  // !OPENSSL_NO_ASM && OPENSSL_X86_64

uint64_t bn_mont_n0(const BIGNUM *n);

// bn_mont_ctx_set_RR_consttime initializes `mont->RR`. It returns one on
// success and zero on error. `mont->N` and `mont->n0` must have been
// initialized already. The bit width of `mont->N` is assumed public, but
// `mont->N` is otherwise treated as secret.
int bn_mont_ctx_set_RR_consttime(BN_MONT_CTX *mont, 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, size_t 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
// 186-5, section B.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. `divisor_min_bits` is a
// public lower bound for `BN_num_bits(divisor)`. When `divisor`'s bit width is
// public, this can speed up the operation.
//
// 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,
                                    unsigned divisor_min_bits, 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);

// bn_mont_ctx_init zero-initialies `mont`.
void bn_mont_ctx_init(BN_MONT_CTX *mont);

// bn_mont_ctx_cleanup releases memory associated with `mont`, without freeing
// `mont` itself.
void bn_mont_ctx_cleanup(BN_MONT_CTX *mont);


// 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);

// BN_MONT_CTX_set_locked takes `lock` and checks whether `*pmont` is NULL. If
// so, it creates a new `BN_MONT_CTX` and sets the modulus for it to `mod`. It
// then stores it as `*pmont`. It returns one on success and zero on error. Note
// this function assumes `mod` is public.
//
// If `*pmont` is already non-NULL then it does nothing and returns one.
int BN_MONT_CTX_set_locked(UniquePtr<BN_MONT_CTX> *pmont, Mutex *lock,
                           const BIGNUM *mod, BN_CTX *bn_ctx);


// 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_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`. `num_p`, measured in bits,
// must fit in `size_t`. `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);


// Word-based byte conversion functions.

// bn_big_endian_to_words interprets `in_len` bytes from `in` as a big-endian,
// unsigned integer and writes the result to `out_len` words in `out`. `out_len`
// must be large enough to represent any `in_len`-byte value. That is, `in_len`
// must be at most `BN_BYTES * out_len`.
void bn_big_endian_to_words(BN_ULONG *out, size_t out_len, const uint8_t *in,
                            size_t in_len);

// bn_words_to_big_endian represents `in_len` words from `in` as a big-endian,
// unsigned integer in `out_len` bytes. It writes the result to `out`. `out_len`
// must be large enough to represent `in` without truncation.
//
// Note `out_len` may be less than `BN_BYTES * in_len` if `in` is known to have
// leading zeros.
void bn_words_to_big_endian(uint8_t *out, size_t out_len, const BN_ULONG *in,
                            size_t in_len);

BSSL_NAMESPACE_END


#endif  // OPENSSL_HEADER_CRYPTO_FIPSMODULE_BN_INTERNAL_H