/*

* Prime Generation

* (C) 1999-2007,2018,2019 Jack Lloyd

*

* Botan is released under the Simplified BSD License (see license.txt)

*/

#include <botan/numthry.h>

#include <botan/rng.h>

#include <botan/internal/bit_ops.h>

#include <botan/loadstor.h>

#include <botan/reducer.h>

#include <botan/internal/primality.h>

#include <algorithm>

namespace Botan {

namespace {

class Prime_Sieve final

   {

   public:

      Prime_Sieve(const BigInt& init_value, size_t sieve_size) :

         m_sieve(std::min(sieve_size, PRIME_TABLE_SIZE))

         {

         for(size_t i = 0; i != m_sieve.size(); ++i)

            m_sieve[i] = static_cast<uint16_t>(init_value % PRIMES[i]);

         }

      void step(word increment)

         {

         for(size_t i = 0; i != m_sieve.size(); ++i)

            {

            m_sieve[i] = (m_sieve[i] + increment) % PRIMES[i];

            }

         }

      bool passes(bool check_2p1 = false) const

         {

         for(size_t i = 0; i != m_sieve.size(); ++i)

            {

            /*

            In this case, p is a multiple of PRIMES[i]

            */

            if(m_sieve[i] == 0)

               return false;

            if(check_2p1)

               {

               /*

               In this case, 2*p+1 will be a multiple of PRIMES[i]

               So if potentially generating a safe prime, we want to

               avoid this value because 2*p+1 will certainly not be prime.

               See "Safe Prime Generation with a Combined Sieve" M. Wiener

               https://eprint.iacr.org/2003/186.pdf

               */

               if(m_sieve[i] == (PRIMES[i] - 1) / 2)

                  return false;

               }

            }

         return true;

         }

   private:

      std::vector<uint16_t> m_sieve;

   };

}

/*

* Generate a random prime

*/

BigInt random_prime(RandomNumberGenerator& rng,

                    size_t bits, const BigInt& coprime,

                    size_t equiv, size_t modulo,

                    size_t prob)

   {

   if(bits <= 1)

      {

      throw Invalid_Argument("random_prime: Can't make a prime of " +

                             std::to_string(bits) + " bits");

      }

   if(coprime.is_negative() || (!coprime.is_zero() && coprime.is_even()) || coprime.bits() >= bits)

      {

      throw Invalid_Argument("random_prime: invalid coprime");

      }

   if(modulo == 0)

      {

      throw Invalid_Argument("random_prime: Invalid modulo value");

      }

   equiv %= modulo;

   if(equiv == 0)

      throw Invalid_Argument("random_prime Invalid value for equiv/modulo");

   // Handle small values:

   if(bits <= 16)

      {

      if(equiv != 1 || modulo != 2 || coprime != 0)

         throw Not_Implemented("random_prime equiv/modulo/coprime options not usable for small primes");

      if(bits == 2)

         {

         return ((rng.next_byte() % 2) ? 2 : 3);

         }

      else if(bits == 3)

         {

         return ((rng.next_byte() % 2) ? 5 : 7);

         }

      else if(bits == 4)

         {

         return ((rng.next_byte() % 2) ? 11 : 13);

         }

      else

         {

         for(;;)

            {

            // This is slightly biased, but for small primes it does not seem to matter

            const uint8_t b0 = rng.next_byte();

            const uint8_t b1 = rng.next_byte();

            const size_t idx = make_uint16(b0, b1) % PRIME_TABLE_SIZE;

            const uint16_t small_prime = PRIMES[idx];

            if(high_bit(small_prime) == bits)

               return small_prime;

            }

         }

      }

   const size_t MAX_ATTEMPTS = 32*1024;

   while(true)

      {

      BigInt p(rng, bits);

      // Force lowest and two top bits on

      p.set_bit(bits - 1);

      p.set_bit(bits - 2);

      p.set_bit(0);

      // Force p to be equal to equiv mod modulo

      p += (modulo - (p % modulo)) + equiv;

      Prime_Sieve sieve(p, bits);

      for(size_t attempt = 0; attempt <= MAX_ATTEMPTS; ++attempt)

         {

         p += modulo;

         sieve.step(modulo);

         // p can be even if modulo is odd, continue on in that case

         if(p.is_even() || sieve.passes(true) == false)

            continue;

         Modular_Reducer mod_p(p);

         /*

         First do a single M-R iteration to quickly elimate most non-primes,

         before doing the coprimality check which is expensive

         */

         if(is_miller_rabin_probable_prime(p, mod_p, rng, 1) == false)

            continue;

         if(coprime > 1)

            {

            /*

            * Check if gcd(p - 1, coprime) != 1 by attempting to compute a

            * modular inverse. We are assured coprime is odd (since if it was

            * even, it would always have a common factor with p - 1), so

            * take off the factors of 2 from (p-1) then compute the inverse

            * of coprime modulo that integer.

            */

            BigInt p1 = p - 1;

            p1 >>= low_zero_bits(p1);

            if(ct_inverse_mod_odd_modulus(coprime, p1).is_zero())

               {

               BOTAN_DEBUG_ASSERT(gcd(p - 1, coprime) > 1);

               continue;

               }

            BOTAN_DEBUG_ASSERT(gcd(p - 1, coprime) == 1);

            }

         if(p.bits() > bits)

            break;

         const size_t t = miller_rabin_test_iterations(bits, prob, true);

         if(is_miller_rabin_probable_prime(p, mod_p, rng, t) == false)

            continue;

         if(prob > 32 && !is_lucas_probable_prime(p, mod_p))

            continue;

         return p;

         }

      }

   }

BigInt generate_rsa_prime(RandomNumberGenerator& keygen_rng,

                          RandomNumberGenerator& prime_test_rng,

                          size_t bits,

                          const BigInt& coprime,

                          size_t prob)

   {

   if(bits < 512)

      throw Invalid_Argument("generate_rsa_prime bits too small");

   /*

   * The restriction on coprime <= 64 bits is arbitrary but generally speaking

   * very large RSA public exponents are a bad idea both for performance and due

   * to attacks on small d.

   */

   if(coprime <= 1 || coprime.is_even() || coprime.bits() > 64)

      throw Invalid_Argument("generate_rsa_prime coprime must be small odd positive integer");

   const size_t MAX_ATTEMPTS = 32*1024;

   while(true)

      {

      BigInt p(keygen_rng, bits);

      // Force lowest and two top bits on

      p.set_bit(bits - 1);

      p.set_bit(bits - 2);

      p.set_bit(0);

      Prime_Sieve sieve(p, bits);

      const word step = 2;

      for(size_t attempt = 0; attempt <= MAX_ATTEMPTS; ++attempt)

         {

         p += step;

         sieve.step(step);

         if(sieve.passes() == false)

            continue;

         Modular_Reducer mod_p(p);

         /*

         * Do a single primality test first before checking coprimality, since

         * currently a single Miller-Rabin test is faster than computing modular

         * inverse, and this eliminates almost all wasted modular inverses.

         */

         if(is_miller_rabin_probable_prime(p, mod_p, prime_test_rng, 1) == false)

            continue;

         /*

         * Check if p - 1 and coprime are relatively prime by computing the inverse.

         *

         * We avoid gcd here because that algorithm is not constant time.

         * Modular inverse is (for odd modulus anyway).

         *

         * We earlier verified that coprime argument is odd. Thus the factors of 2

         * in (p - 1) cannot possibly be factors in coprime, so remove them from p - 1.

         * Using an odd modulus allows the const time algorithm to be used.

         *

         * This assumes coprime < p - 1 which is always true for RSA.

         */

         BigInt p1 = p - 1;

         p1 >>= low_zero_bits(p1);

         if(ct_inverse_mod_odd_modulus(coprime, p1).is_zero())

            {

            BOTAN_DEBUG_ASSERT(gcd(p - 1, coprime) > 1);

            continue;

            }

         BOTAN_DEBUG_ASSERT(gcd(p - 1, coprime) == 1);

         if(p.bits() > bits)

            break;

         const size_t t = miller_rabin_test_iterations(bits, prob, true);

         if(is_miller_rabin_probable_prime(p, mod_p, prime_test_rng, t) == true)

            return p;

         }

      }

   }

/*

* Generate a random safe prime

*/

BigInt random_safe_prime(RandomNumberGenerator& rng, size_t bits)

   {

   if(bits <= 64)

      throw Invalid_Argument("random_safe_prime: Can't make a prime of " +

                             std::to_string(bits) + " bits");

   BigInt q, p;

   for(;;)

      {

      /*

      Generate q == 2 (mod 3)

      Otherwise [q == 1 (mod 3) case], 2*q+1 == 3 (mod 3) and not prime.

      Initially allow a very high error prob (1/2**8) to allow for fast checks,

      then if 2*q+1 turns out to be a prime go back and strongly check q.

      */

      q = random_prime(rng, bits - 1, 0, 2, 3, 8);

      p = (q << 1) + 1;

      if(is_prime(p, rng, 128, true))

         {

         // We did only a weak check before, go back and verify q before returning

         if(is_prime(q, rng, 128, true))

            return p;

         }

      }

   }

}