// Copyright 1995-2016 The OpenSSL Project Authors. 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.

#include <openssl/bn.h>

#include <iterator>

#include <openssl/err.h>

#include <openssl/mem.h>

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

#include "../../mem_internal.h"

#include "internal.h"

using namespace bssl;

// kPrimes contains the first 1024 primes.

static const uint16_t kPrimes[] = {

    2,    3,    5,    7,    11,   13,   17,   19,   23,   29,   31,   37,

    41,   43,   47,   53,   59,   61,   67,   71,   73,   79,   83,   89,

    97,   101,  103,  107,  109,  113,  127,  131,  137,  139,  149,  151,

    157,  163,  167,  173,  179,  181,  191,  193,  197,  199,  211,  223,

    227,  229,  233,  239,  241,  251,  257,  263,  269,  271,  277,  281,

    283,  293,  307,  311,  313,  317,  331,  337,  347,  349,  353,  359,

    367,  373,  379,  383,  389,  397,  401,  409,  419,  421,  431,  433,

    439,  443,  449,  457,  461,  463,  467,  479,  487,  491,  499,  503,

    509,  521,  523,  541,  547,  557,  563,  569,  571,  577,  587,  593,

    599,  601,  607,  613,  617,  619,  631,  641,  643,  647,  653,  659,

    661,  673,  677,  683,  691,  701,  709,  719,  727,  733,  739,  743,

    751,  757,  761,  769,  773,  787,  797,  809,  811,  821,  823,  827,

    829,  839,  853,  857,  859,  863,  877,  881,  883,  887,  907,  911,

    919,  929,  937,  941,  947,  953,  967,  971,  977,  983,  991,  997,

    1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069,

    1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163,

    1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249,

    1259, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321,

    1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439,

    1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511,

    1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601,

    1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693,

    1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783,

    1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877,

    1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987,

    1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069,

    2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143,

    2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267,

    2269, 2273, 2281, 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347,

    2351, 2357, 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423,

    2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539, 2543,

    2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, 2621, 2633, 2647, 2657,

    2659, 2663, 2671, 2677, 2683, 2687, 2689, 2693, 2699, 2707, 2711, 2713,

    2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801,

    2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903,

    2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011,

    3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119,

    3121, 3137, 3163, 3167, 3169, 3181, 3187, 3191, 3203, 3209, 3217, 3221,

    3229, 3251, 3253, 3257, 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323,

    3329, 3331, 3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413,

    3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511, 3517, 3527,

    3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593, 3607,

    3613, 3617, 3623, 3631, 3637, 3643, 3659, 3671, 3673, 3677, 3691, 3697,

    3701, 3709, 3719, 3727, 3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797,

    3803, 3821, 3823, 3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907,

    3911, 3917, 3919, 3923, 3929, 3931, 3943, 3947, 3967, 3989, 4001, 4003,

    4007, 4013, 4019, 4021, 4027, 4049, 4051, 4057, 4073, 4079, 4091, 4093,

    4099, 4111, 4127, 4129, 4133, 4139, 4153, 4157, 4159, 4177, 4201, 4211,

    4217, 4219, 4229, 4231, 4241, 4243, 4253, 4259, 4261, 4271, 4273, 4283,

    4289, 4297, 4327, 4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4409,

    4421, 4423, 4441, 4447, 4451, 4457, 4463, 4481, 4483, 4493, 4507, 4513,

    4517, 4519, 4523, 4547, 4549, 4561, 4567, 4583, 4591, 4597, 4603, 4621,

    4637, 4639, 4643, 4649, 4651, 4657, 4663, 4673, 4679, 4691, 4703, 4721,

    4723, 4729, 4733, 4751, 4759, 4783, 4787, 4789, 4793, 4799, 4801, 4813,

    4817, 4831, 4861, 4871, 4877, 4889, 4903, 4909, 4919, 4931, 4933, 4937,

    4943, 4951, 4957, 4967, 4969, 4973, 4987, 4993, 4999, 5003, 5009, 5011,

    5021, 5023, 5039, 5051, 5059, 5077, 5081, 5087, 5099, 5101, 5107, 5113,

    5119, 5147, 5153, 5167, 5171, 5179, 5189, 5197, 5209, 5227, 5231, 5233,

    5237, 5261, 5273, 5279, 5281, 5297, 5303, 5309, 5323, 5333, 5347, 5351,

    5381, 5387, 5393, 5399, 5407, 5413, 5417, 5419, 5431, 5437, 5441, 5443,

    5449, 5471, 5477, 5479, 5483, 5501, 5503, 5507, 5519, 5521, 5527, 5531,

    5557, 5563, 5569, 5573, 5581, 5591, 5623, 5639, 5641, 5647, 5651, 5653,

    5657, 5659, 5669, 5683, 5689, 5693, 5701, 5711, 5717, 5737, 5741, 5743,

    5749, 5779, 5783, 5791, 5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849,

    5851, 5857, 5861, 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939,

    5953, 5981, 5987, 6007, 6011, 6029, 6037, 6043, 6047, 6053, 6067, 6073,

    6079, 6089, 6091, 6101, 6113, 6121, 6131, 6133, 6143, 6151, 6163, 6173,

    6197, 6199, 6203, 6211, 6217, 6221, 6229, 6247, 6257, 6263, 6269, 6271,

    6277, 6287, 6299, 6301, 6311, 6317, 6323, 6329, 6337, 6343, 6353, 6359,

    6361, 6367, 6373, 6379, 6389, 6397, 6421, 6427, 6449, 6451, 6469, 6473,

    6481, 6491, 6521, 6529, 6547, 6551, 6553, 6563, 6569, 6571, 6577, 6581,

    6599, 6607, 6619, 6637, 6653, 6659, 6661, 6673, 6679, 6689, 6691, 6701,

    6703, 6709, 6719, 6733, 6737, 6761, 6763, 6779, 6781, 6791, 6793, 6803,

    6823, 6827, 6829, 6833, 6841, 6857, 6863, 6869, 6871, 6883, 6899, 6907,

    6911, 6917, 6947, 6949, 6959, 6961, 6967, 6971, 6977, 6983, 6991, 6997,

    7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079, 7103, 7109, 7121,

    7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207, 7211, 7213, 7219, 7229,

    7237, 7243, 7247, 7253, 7283, 7297, 7307, 7309, 7321, 7331, 7333, 7349,

    7351, 7369, 7393, 7411, 7417, 7433, 7451, 7457, 7459, 7477, 7481, 7487,

    7489, 7499, 7507, 7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561,

    7573, 7577, 7583, 7589, 7591, 7603, 7607, 7621, 7639, 7643, 7649, 7669,

    7673, 7681, 7687, 7691, 7699, 7703, 7717, 7723, 7727, 7741, 7753, 7757,

    7759, 7789, 7793, 7817, 7823, 7829, 7841, 7853, 7867, 7873, 7877, 7879,

    7883, 7901, 7907, 7919, 7927, 7933, 7937, 7949, 7951, 7963, 7993, 8009,

    8011, 8017, 8039, 8053, 8059, 8069, 8081, 8087, 8089, 8093, 8101, 8111,

    8117, 8123, 8147, 8161,

};

// BN_prime_checks_for_size returns the number of Miller-Rabin iterations

// necessary for generating a 'bits'-bit candidate prime.

//

//

// This table is generated using the algorithm of FIPS PUB 186-5

// Digital Signature Standard (DSS), section C.1, page 72.

// (https://doi.org/10.6028/NIST.FIPS.186-5).

// The following magma script was used to generate the output:

// securitybits:=125;

// k:=1024;

// for t:=1 to 65 do

//   for M:=3 to Floor(2*Sqrt(k-1)-1) do

//     S:=0;

//     // Sum over m

//     for m:=3 to M do

//       s:=0;

//       // Sum over j

//       for j:=2 to m do

//         s+:=(RealField(32)!2)^-(j+(k-1)/j);

//       end for;

//       S+:=2^(m-(m-1)*t)*s;

//     end for;

//     A:=2^(k-2-M*t);

//     B:=8*(Pi(RealField(32))^2-6)/3*2^(k-2)*S;

//     pkt:=2.00743*Log(2)*k*2^-k*(A+B);

//     seclevel:=Floor(-Log(2,pkt));

//     if seclevel ge securitybits then

//       printf "k: %5o, security: %o bits  (t: %o, M: %o)\n",k,seclevel,t,M;

//       break;

//     end if;

//   end for;

//   if seclevel ge securitybits then break; end if;

// end for;

//

// It can be run online at: http://magma.maths.usyd.edu.au/calc

// And will output:

// k:  1024, security: 129 bits  (t: 6, M: 23)

// k is the number of bits of the prime, securitybits is the level we want to

// reach.

// prime length | RSA key size | # MR tests | security level

// -------------+--------------|------------+---------------

//  (b) >= 6394 |     >= 12788 |          3 |        256 bit

//  (b) >= 3747 |     >=  7494 |          3 |        192 bit

//  (b) >= 1345 |     >=  2690 |          4 |        128 bit

//  (b) >= 1080 |     >=  2160 |          5 |        128 bit

//  (b) >=  852 |     >=  1704 |          5 |        112 bit

//  (b) >=  476 |     >=   952 |          5 |         80 bit

//  (b) >=  400 |     >=   800 |          6 |         80 bit

//  (b) >=  347 |     >=   694 |          7 |         80 bit

//  (b) >=  308 |     >=   616 |          8 |         80 bit

//  (b) >=   55 |     >=   110 |         27 |         64 bit

//  (b) >=    6 |     >=    12 |         34 |         64 bit

static int BN_prime_checks_for_size(int bits) {

  if (bits >= 3747) {

    return 3;

  }

  if (bits >= 1345) {

    return 4;

  }

  if (bits >= 476) {

    return 5;

  }

  if (bits >= 400) {

    return 6;

  }

  if (bits >= 347) {

    return 7;

  }

  if (bits >= 308) {

    return 8;

  }

  if (bits >= 55) {

    return 27;

  }

  return 34;

}

// num_trial_division_primes returns the number of primes to try with trial

// division before using more expensive checks. For larger numbers, the value

// of excluding a candidate with trial division is larger.

static size_t num_trial_division_primes(const BIGNUM *n) {

  if (n->width * BN_BITS2 > 1024) {

    return std::size(kPrimes);

  }

  return std::size(kPrimes) / 2;

}

// BN_PRIME_CHECKS_BLINDED is the iteration count for blinding the constant-time

// primality test. See |BN_primality_test| for details. This number is selected

// so that, for a candidate N-bit RSA prime, picking |BN_PRIME_CHECKS_BLINDED|

// random N-bit numbers will have at least |BN_prime_checks_for_size(N)| values

// in range with high probability.

//

// The following Python script computes the blinding factor needed for the

// corresponding iteration count.

/*

import math

# We choose candidate RSA primes between sqrt(2)/2 * 2^N and 2^N and select

# witnesses by generating random N-bit numbers. Thus the probability of

# selecting one in range is at least sqrt(2)/2.

p = math.sqrt(2) / 2

# Target around 2^-8 probability of the blinding being insufficient given that

# key generation is a one-time, noisy operation.

epsilon = 2**-8

def choose(a, b):

  r = 1

  for i in xrange(b):

    r *= a - i

    r /= (i + 1)

  return r

def failure_rate(min_uniform, iterations):

  """ Returns the probability that, for |iterations| candidate witnesses, fewer

      than |min_uniform| of them will be uniform. """

  prob = 0.0

  for i in xrange(min_uniform):

    prob += (choose(iterations, i) *

             p**i * (1-p)**(iterations - i))

  return prob

for min_uniform in (3, 4, 5, 6, 8, 13, 19, 28):

  # Find the smallest number of iterations under the target failure rate.

  iterations = min_uniform

  while True:

    prob = failure_rate(min_uniform, iterations)

    if prob < epsilon:

      print min_uniform, iterations, prob

      break

    iterations += 1

Output:

  3 9 0.00368894873911

  4 11 0.00363319494662

  5 13 0.00336215573898

  6 15 0.00300145783158

  8 19 0.00225214119331

  13 27 0.00385610026955

  19 38 0.0021410539126

  28 52 0.00325405801769

16 iterations suffices for 400-bit primes and larger (6 uniform samples needed),

which is already well below the minimum acceptable key size for RSA.

*/

#define BN_PRIME_CHECKS_BLINDED 16

static int probable_prime(BIGNUM *rnd, int bits);

static int probable_prime_dh(BIGNUM *rnd, int bits, const BIGNUM *add,

                             const BIGNUM *rem, BN_CTX *ctx);

static int probable_prime_dh_safe(BIGNUM *rnd, int bits, const BIGNUM *add,

                                  const BIGNUM *rem, BN_CTX *ctx);

BN_GENCB *BN_GENCB_new() { return NewZeroed<BN_GENCB>(); }

void BN_GENCB_free(BN_GENCB *callback) { Delete(callback); }

void BN_GENCB_set(BN_GENCB *callback,

                  int (*f)(int event, int n, struct bn_gencb_st *), void *arg) {

  callback->callback = f;

  callback->arg = arg;

}

int BN_GENCB_call(BN_GENCB *callback, int event, int n) {

  if (!callback) {

    return 1;

  }

  return callback->callback(event, n, callback);

}

void *BN_GENCB_get_arg(const BN_GENCB *callback) { return callback->arg; }

int BN_generate_prime_ex(BIGNUM *ret, int bits, int safe, const BIGNUM *add,

                         const BIGNUM *rem, BN_GENCB *cb) {

  BIGNUM *t;

  int i, j, c1 = 0;

  int checks = BN_prime_checks_for_size(bits);

  if (bits < 2) {

    // There are no prime numbers this small.

    OPENSSL_PUT_ERROR(BN, BN_R_BITS_TOO_SMALL);

    return 0;

  } else if (bits == 2 && safe) {

    // The smallest safe prime (7) is three bits.

    OPENSSL_PUT_ERROR(BN, BN_R_BITS_TOO_SMALL);

    return 0;

  }

  UniquePtr<BN_CTX> ctx(BN_CTX_new());

  if (ctx == nullptr) {

    return 0;

  }

  BN_CTXScope scope(ctx.get());

  t = BN_CTX_get(ctx.get());

  if (!t) {

    return 0;

  }

loop:

  // make a random number and set the top and bottom bits

  if (add == nullptr) {

    if (!probable_prime(ret, bits)) {

      return 0;

    }

  } else {

    if (safe) {

      if (!probable_prime_dh_safe(ret, bits, add, rem, ctx.get())) {

        return 0;

      }

    } else {

      if (!probable_prime_dh(ret, bits, add, rem, ctx.get())) {

        return 0;

      }

    }

  }

  if (!BN_GENCB_call(cb, BN_GENCB_GENERATED, c1++)) {

    // aborted

    return 0;

  }

  if (!safe) {

    i = BN_is_prime_fasttest_ex(ret, checks, ctx.get(), 0, cb);

    if (i == -1) {

      return 0;

    } else if (i == 0) {

      goto loop;

    }

  } else {

    // for "safe prime" generation, check that (p-1)/2 is prime. Since a prime

    // is odd, We just need to divide by 2

    if (!BN_rshift1(t, ret)) {

      return 0;

    }

    // Interleave |ret| and |t|'s primality tests to avoid paying the full

    // iteration count on |ret| only to quickly discover |t| is composite.

    //

    // TODO(davidben): This doesn't quite work because an iteration count of 1

    // still runs the blinding mechanism.

    for (i = 0; i < checks; i++) {

      j = BN_is_prime_fasttest_ex(ret, 1, ctx.get(), 0, nullptr);

      if (j == -1) {

        return 0;

      } else if (j == 0) {

        goto loop;

      }

      j = BN_is_prime_fasttest_ex(t, 1, ctx.get(), 0, nullptr);

      if (j == -1) {

        return 0;

      } else if (j == 0) {

        goto loop;

      }

      if (!BN_GENCB_call(cb, BN_GENCB_PRIME_TEST, i)) {

        return 0;

      }

      // We have a safe prime test pass

    }

  }

  // we have a prime :-)

  return 1;

}

static int bn_trial_division(uint16_t *out, const BIGNUM *bn) {

  const size_t num_primes = num_trial_division_primes(bn);

  for (size_t i = 1; i < num_primes; i++) {

    // During RSA key generation, |bn| may be secret, but only if |bn| was

    // prime, so it is safe to leak failed trial divisions.

    if (constant_time_declassify_int(bn_mod_u16_consttime(bn, kPrimes[i]) ==

                                     0)) {

      *out = kPrimes[i];

      return 1;

    }

  }

  return 0;

}

int bssl::bn_odd_number_is_obviously_composite(const BIGNUM *bn) {

  uint16_t prime;

  return bn_trial_division(&prime, bn) && !BN_is_word(bn, prime);

}

int bssl::bn_miller_rabin_init(BN_MILLER_RABIN *miller_rabin,

                               const BN_MONT_CTX *mont, BN_CTX *ctx) {

  // This function corresponds to steps 1 through 3 of FIPS 186-5, B.3.1.

  const BIGNUM *w = &mont->N;

  // Note we do not call |BN_CTX_start| in this function. We intentionally

  // allocate values in the containing scope so they outlive this function.

  miller_rabin->w1 = BN_CTX_get(ctx);

  miller_rabin->m = BN_CTX_get(ctx);

  miller_rabin->one_mont = BN_CTX_get(ctx);

  miller_rabin->w1_mont = BN_CTX_get(ctx);

  if (miller_rabin->w1 == nullptr ||        //

      miller_rabin->m == nullptr ||         //

      miller_rabin->one_mont == nullptr ||  //

      miller_rabin->w1_mont == nullptr) {

    return 0;

  }

  // See FIPS 186-5, B.3.1, steps 1 through 3.

  if (!bn_usub_consttime(miller_rabin->w1, w, BN_value_one())) {

    return 0;

  }

  miller_rabin->a = BN_count_low_zero_bits(miller_rabin->w1);

  if (!bn_rshift_secret_shift(miller_rabin->m, miller_rabin->w1,

                              miller_rabin->a, ctx)) {

    return 0;

  }

  miller_rabin->w_bits = BN_num_bits(w);

  // Precompute some values in Montgomery form.

  if (!bn_one_to_montgomery(miller_rabin->one_mont, mont, ctx) ||

      // w - 1 is -1 mod w, so we can compute it in the Montgomery domain, -R,

      // with a subtraction. (|one_mont| cannot be zero.)

      !bn_usub_consttime(miller_rabin->w1_mont, w, miller_rabin->one_mont)) {

    return 0;

  }

  return 1;

}

int bssl::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) {

  // This function corresponds to steps 4.3 through 4.5 of FIPS 186-5, B.3.1.

  BN_CTXScope scope(ctx);

  // Step 4.3. We use Montgomery-encoding for better performance and to avoid

  // timing leaks.

  const BIGNUM *w = &mont->N;

  BIGNUM *z = BN_CTX_get(ctx);

  crypto_word_t is_possibly_prime;

  if (z == nullptr ||

      !BN_mod_exp_mont_consttime(z, b, miller_rabin->m, w, ctx, mont) ||

      !BN_to_montgomery(z, z, mont, ctx)) {

    return 0;

  }

  // is_possibly_prime is all ones if we have determined |b| is not a composite

  // witness for |w|. This is equivalent to going to step 4.7 in the original

  // algorithm. To avoid timing leaks, we run the algorithm to the end for prime

  // inputs.

  is_possibly_prime = 0;

  // Step 4.4. If z = 1 or z = w-1, b is not a composite witness and w is still

  // possibly prime.

  is_possibly_prime = BN_equal_consttime(z, miller_rabin->one_mont) |

                      BN_equal_consttime(z, miller_rabin->w1_mont);

  is_possibly_prime = 0 - is_possibly_prime;  // Make it all zeros or all ones.

  // Step 4.5.

  //

  // To avoid leaking |a|, we run the loop to |w_bits| and mask off all

  // iterations once |j| = |a|.

  for (int j = 1; j < miller_rabin->w_bits; j++) {

    if (constant_time_declassify_w(constant_time_eq_int(j, miller_rabin->a) &

                                   ~is_possibly_prime)) {

      // If the loop is done and we haven't seen z = 1 or z = w-1 yet, the

      // value is composite and we can break in variable time.

      break;

    }

    // Step 4.5.1.

    if (!BN_mod_mul_montgomery(z, z, z, mont, ctx)) {

      return 0;

    }

    // Step 4.5.2. If z = w-1 and the loop is not done, this is not a composite

    // witness.

    crypto_word_t z_is_w1_mont = BN_equal_consttime(z, miller_rabin->w1_mont);

    z_is_w1_mont = 0 - z_is_w1_mont;    // Make it all zeros or all ones.

    is_possibly_prime |= z_is_w1_mont;  // Go to step 4.7 if |z_is_w1_mont|.

    // Step 4.5.3. If z = 1 and the loop is not done, the previous value of z

    // was not -1. There are no non-trivial square roots of 1 modulo a prime, so

    // w is composite and we may exit in variable time.

    if (constant_time_declassify_w(

            BN_equal_consttime(z, miller_rabin->one_mont) &

            ~is_possibly_prime)) {

      break;

    }

  }

  *out_is_possibly_prime = constant_time_declassify_w(is_possibly_prime) & 1;

  return 1;

}

int BN_primality_test(int *out_is_probably_prime, const BIGNUM *w, int checks,

                      BN_CTX *ctx, int do_trial_division, BN_GENCB *cb) {

  // This function's secrecy and performance requirements come from RSA key

  // generation. We generate RSA keys by selecting two large, secret primes with

  // rejection sampling.

  //

  // We thus treat |w| as secret if turns out to be a large prime. However, if

  // |w| is composite, we treat this and |w| itself as public. (Conversely, if

  // |w| is prime, that it is prime is public. Only the value is secret.) This

  // is fine for RSA key generation, but note it is important that we use

  // rejection sampling, with each candidate prime chosen independently. This

  // would not work for, e.g., an algorithm which looked for primes in

  // consecutive integers. These assumptions allow us to discard composites

  // quickly. We additionally treat |w| as public when it is a small prime to

  // simplify trial decryption and some edge cases.

  //

  // One RSA key generation will call this function on exactly two primes and

  // many more composites. The overall cost is a combination of several factors:

  //

  // 1. Checking if |w| is divisible by a small prime is much faster than

  //    learning it is composite by Miller-Rabin (see below for details on that

  //    cost). Trial division by p saves 1/p of Miller-Rabin calls, so this is

  //    worthwhile until p exceeds the ratio of the two costs.

  //

  // 2. For a random (i.e. non-adversarial) candidate large prime and candidate

  //    witness, the probability of false witness is very low. (This is why FIPS

  //    186-5 only requires a few iterations.) Thus composites not discarded by

  //    trial decryption, in practice, cost one Miller-Rabin iteration. Only the

  //    two actual primes cost the full iteration count.

  //

  // 3. A Miller-Rabin iteration is a modular exponentiation plus |a| additional

  //    modular squares, where |a| is the number of factors of two in |w-1|. |a|

  //    is likely small (the distribution falls exponentially), but it is also

  //    potentially secret, so we loop up to its log(w) upper bound when |w| is

  //    prime. When |w| is composite, we break early, so only two calls pay this

  //    cost. (Note that all calls pay the modular exponentiation which is,

  //    itself, log(w) modular multiplications and squares.)

  //

  // 4. While there are only two prime calls, they multiplicatively pay the full

  //    costs of (2) and (3).

  //

  // 5. After the primes are chosen, RSA keys derive some values from the

  //    primes, but this cost is negligible in comparison.

  *out_is_probably_prime = 0;

  if (BN_cmp(w, BN_value_one()) <= 0) {

    return 1;

  }

  if (!BN_is_odd(w)) {

    // The only even prime is two.

    *out_is_probably_prime = BN_is_word(w, 2);

    return 1;

  }

  // Miller-Rabin does not work for three.

  if (BN_is_word(w, 3)) {

    *out_is_probably_prime = 1;

    return 1;

  }

  if (do_trial_division) {

    // Perform additional trial division checks to discard small primes.

    uint16_t prime;

    if (bn_trial_division(&prime, w)) {

      *out_is_probably_prime = BN_is_word(w, prime);

      return 1;

    }

    if (!BN_GENCB_call(cb, BN_GENCB_PRIME_TEST, -1)) {

      return 0;

    }

  }

  if (checks == BN_prime_checks_for_generation) {

    checks = BN_prime_checks_for_size(BN_num_bits(w));

  }

  UniquePtr<BN_CTX> new_ctx;

  if (ctx == nullptr) {

    new_ctx.reset(BN_CTX_new());

    if (new_ctx == nullptr) {

      return 0;

    }

    ctx = new_ctx.get();

  }

  // See B.3.1 from FIPS 186-5.

  BN_CTXScope scope(ctx);

  BIGNUM *b = BN_CTX_get(ctx);

  UniquePtr<BN_MONT_CTX> mont(BN_MONT_CTX_new_consttime(w, ctx));

  BN_MILLER_RABIN miller_rabin;

  crypto_word_t uniform_iterations = 0;

  if (b == nullptr || mont == nullptr ||

      // Steps 1-3.

      !bn_miller_rabin_init(&miller_rabin, mont.get(), ctx)) {

    return 0;

  }

  // The following loop performs in inner iteration of the Miller-Rabin

  // Primality test (Step 4).

  //

  // The algorithm as specified in FIPS 186-5 leaks information on |w|, the RSA

  // private key. Instead, we run through each iteration unconditionally,

  // performing modular multiplications, masking off any effects to behave

  // equivalently to the specified algorithm.

  //

  // We also blind the number of values of |b| we try. Steps 4.1–4.2 say to

  // discard out-of-range values. To avoid leaking information on |w|, we use

  // |bn_rand_secret_range| which, rather than discarding bad values, adjusts

  // them to be in range. Though not uniformly selected, these adjusted values

  // are still usable as Miller-Rabin checks.

  //

  // Miller-Rabin is already probabilistic, so we could reach the desired

  // confidence levels by just suitably increasing the iteration count. However,

  // to align with FIPS 186-5, we use a more pessimal analysis: we do not count

  // the non-uniform values towards the iteration count. As a result, this

  // function is more complex and has more timing risk than necessary.

  //

  // We count both total iterations and uniform ones and iterate until we've

  // reached at least |BN_PRIME_CHECKS_BLINDED| and |iterations|, respectively.

  // If the latter is large enough, it will be the limiting factor with high

  // probability and we won't leak information.

  //

  // Note this blinding does not impact most calls when picking primes because

  // composites are rejected early. Only the two secret primes see extra work.

  // Using |constant_time_lt_w| seems to prevent the compiler from optimizing

  // this into two jumps.

  for (int i = 1; constant_time_declassify_w(

           (i <= BN_PRIME_CHECKS_BLINDED) |

           constant_time_lt_w(uniform_iterations, checks));

       i++) {

    // Step 4.1-4.2

    int is_uniform;

    if (!bn_rand_secret_range(b, &is_uniform, 2, miller_rabin.w1)) {

      return 0;

    }

    uniform_iterations += is_uniform;

    // Steps 4.3-4.5

    int is_possibly_prime = 0;

    if (!bn_miller_rabin_iteration(&miller_rabin, &is_possibly_prime, b,

                                   mont.get(), ctx)) {

      return 0;

    }

    if (!is_possibly_prime) {

      // Step 4.6. We did not see z = w-1 before z = 1, so w must be composite.

      *out_is_probably_prime = 0;

      return 1;

    }

    // Step 4.7

    if (!BN_GENCB_call(cb, BN_GENCB_PRIME_TEST, i - 1)) {

      return 0;

    }

  }

  declassify_assert(uniform_iterations >= (crypto_word_t)checks);

  *out_is_probably_prime = 1;

  return 1;

}

int BN_is_prime_ex(const BIGNUM *candidate, int checks, BN_CTX *ctx,

                   BN_GENCB *cb) {

  return BN_is_prime_fasttest_ex(candidate, checks, ctx, 0, cb);

}

int BN_is_prime_fasttest_ex(const BIGNUM *a, int checks, BN_CTX *ctx,

                            int do_trial_division, BN_GENCB *cb) {

  int is_probably_prime;

  if (!BN_primality_test(&is_probably_prime, a, checks, ctx, do_trial_division,

                         cb)) {

    return -1;

  }

  return is_probably_prime;

}

int BN_enhanced_miller_rabin_primality_test(

    enum bn_primality_result_t *out_result, const BIGNUM *w, int checks,

    BN_CTX *ctx, BN_GENCB *cb) {

  // Enhanced Miller-Rabin is only valid on odd integers greater than 3.

  if (!BN_is_odd(w) || BN_cmp_word(w, 3) <= 0) {

    OPENSSL_PUT_ERROR(BN, BN_R_INVALID_INPUT);

    return 0;

  }

  if (checks == BN_prime_checks_for_generation) {

    checks = BN_prime_checks_for_size(BN_num_bits(w));

  }

  BN_CTXScope scope(ctx);

  BIGNUM *w1 = BN_CTX_get(ctx);

  if (w1 == nullptr || !BN_copy(w1, w) || !BN_sub_word(w1, 1)) {

    return 0;

  }

  // Write w1 as m*2^a (Steps 1 and 2).

  int a = 0;

  while (!BN_is_bit_set(w1, a)) {

    a++;

  }

  BIGNUM *m = BN_CTX_get(ctx);

  if (m == nullptr || !BN_rshift(m, w1, a)) {

    return 0;

  }

  BIGNUM *b = BN_CTX_get(ctx);

  BIGNUM *g = BN_CTX_get(ctx);

  BIGNUM *z = BN_CTX_get(ctx);

  BIGNUM *x = BN_CTX_get(ctx);

  BIGNUM *x1 = BN_CTX_get(ctx);

  if (b == nullptr || g == nullptr || z == nullptr || x == nullptr ||

      x1 == nullptr) {

    return 0;

  }

  // Montgomery setup for computations mod w

  UniquePtr<BN_MONT_CTX> mont(BN_MONT_CTX_new_for_modulus(w, ctx));

  if (mont == nullptr) {

    return 0;

  }

  // The following loop performs in inner iteration of the Enhanced Miller-Rabin

  // Primality test (Step 4).

  for (int i = 1; i <= checks; i++) {

    // Step 4.1-4.2

    if (!BN_rand_range_ex(b, 2, w1)) {

      return 0;

    }

    // Step 4.3-4.4

    if (!BN_gcd(g, b, w, ctx)) {

      return 0;

    }

    if (BN_cmp_word(g, 1) > 0) {

      *out_result = bn_composite;

      return 1;

    }

    // Step 4.5

    if (!BN_mod_exp_mont(z, b, m, w, ctx, mont.get())) {

      return 0;

    }

    // Step 4.6

    if (BN_is_one(z) || BN_cmp(z, w1) == 0) {

      goto loop;

    }

    // Step 4.7

    for (int j = 1; j < a; j++) {

      if (!BN_copy(x, z) || !BN_mod_mul(z, x, x, w, ctx)) {

        return 0;

      }

      if (BN_cmp(z, w1) == 0) {

        goto loop;

      }

      if (BN_is_one(z)) {

        goto composite;

      }

    }

    // Step 4.8-4.9

    if (!BN_copy(x, z) || !BN_mod_mul(z, x, x, w, ctx)) {

      return 0;

    }

    // Step 4.10-4.11

    if (!BN_is_one(z) && !BN_copy(x, z)) {

      return 0;

    }

  composite:

    // Step 4.12-4.14

    if (!BN_copy(x1, x) || !BN_sub_word(x1, 1) || !BN_gcd(g, x1, w, ctx)) {

      return 0;

    }

    if (BN_cmp_word(g, 1) > 0) {

      *out_result = bn_composite;

    } else {

      *out_result = bn_non_prime_power_composite;

    }

    return 1;

  loop:

    // Step 4.15

    if (!BN_GENCB_call(cb, BN_GENCB_PRIME_TEST, i - 1)) {

      return 0;

    }

  }

  *out_result = bn_probably_prime;

  return 1;

}

static int probable_prime(BIGNUM *rnd, int bits) {

  do {

    if (!BN_rand(rnd, bits, BN_RAND_TOP_TWO, BN_RAND_BOTTOM_ODD)) {

      return 0;

    }

  } while (bn_odd_number_is_obviously_composite(rnd));

  return 1;

}

static int probable_prime_dh(BIGNUM *rnd, int bits, const BIGNUM *add,

                             const BIGNUM *rem, BN_CTX *ctx) {

  BN_CTXScope scope(ctx);

  BIGNUM *t1;

  if ((t1 = BN_CTX_get(ctx)) == nullptr) {

    return 0;

  }

  if (!BN_rand(rnd, bits, BN_RAND_TOP_ONE, BN_RAND_BOTTOM_ODD)) {

    return 0;

  }

  // we need ((rnd-rem) % add) == 0

  if (!BN_mod(t1, rnd, add, ctx)) {

    return 0;

  }

  if (!BN_sub(rnd, rnd, t1)) {

    return 0;

  }

  if (rem == nullptr) {

    if (!BN_add_word(rnd, 1)) {

      return 0;

    }

  } else {

    if (!BN_add(rnd, rnd, rem)) {

      return 0;

    }

  }

  // we now have a random number 'rand' to test.

  size_t num_primes = num_trial_division_primes(rnd);

loop:

  for (size_t i = 1; i < num_primes; i++) {

    // check that rnd is a prime

    if (bn_mod_u16_consttime(rnd, kPrimes[i]) <= 1) {

      if (!BN_add(rnd, rnd, add)) {

        return 0;

      }

      goto loop;

    }

  }

  return 1;

}

static int probable_prime_dh_safe(BIGNUM *p, int bits, const BIGNUM *padd,

                                  const BIGNUM *rem, BN_CTX *ctx) {

  bits--;

  BN_CTXScope scope(ctx);

  BIGNUM *t1 = BN_CTX_get(ctx);

  BIGNUM *q = BN_CTX_get(ctx);

  BIGNUM *qadd = BN_CTX_get(ctx);

  if (qadd == nullptr) {

    return 0;

  }

  if (!BN_rshift1(qadd, padd)) {

    return 0;

  }

  if (!BN_rand(q, bits, BN_RAND_TOP_ONE, BN_RAND_BOTTOM_ODD)) {

    return 0;

  }

  // we need ((rnd-rem) % add) == 0

  if (!BN_mod(t1, q, qadd, ctx)) {

    return 0;

  }

  if (!BN_sub(q, q, t1)) {

    return 0;

  }

  if (rem == nullptr) {

    if (!BN_add_word(q, 1)) {

      return 0;

    }

  } else {

    if (!BN_rshift1(t1, rem)) {

      return 0;

    }

    if (!BN_add(q, q, t1)) {

      return 0;

    }

  }

  // we now have a random number 'rand' to test.

  if (!BN_lshift1(p, q)) {

    return 0;

  }

  if (!BN_add_word(p, 1)) {

    return 0;

  }

  size_t num_primes = num_trial_division_primes(p);

loop:

  for (size_t i = 1; i < num_primes; i++) {

    // check that p and q are prime

    // check that for p and q

    // gcd(p-1,primes) == 1 (except for 2)

    if (bn_mod_u16_consttime(p, kPrimes[i]) == 0 ||

        bn_mod_u16_consttime(q, kPrimes[i]) == 0) {

      if (!BN_add(p, p, padd)) {

        return 0;

      }

      if (!BN_add(q, q, qadd)) {

        return 0;

      }

      goto loop;

    }

  }

  return 1;

}