/*

* Number Theory Functions

* (C) 1999-2011,2016,2018,2019 Jack Lloyd

* (C) 2007,2008 Falko Strenzke, FlexSecure GmbH

*

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

*/

#include <botan/numthry.h>

#include <botan/reducer.h>

#include <botan/internal/monty.h>

#include <botan/internal/divide.h>

#include <botan/rng.h>

#include <botan/internal/ct_utils.h>

#include <botan/internal/mp_core.h>

#include <botan/internal/monty_exp.h>

#include <botan/internal/primality.h>

#include <algorithm>

namespace Botan {

namespace {

void sub_abs(BigInt& z, const BigInt& x, const BigInt& y)

   {

   const size_t x_sw = x.sig_words();

   const size_t y_sw = y.sig_words();

   z.resize(std::max(x_sw, y_sw));

   bigint_sub_abs(z.mutable_data(),

                  x.data(), x_sw,

                  y.data(), y_sw);

   }

}

/*

* Tonelli-Shanks algorithm

*/

BigInt ressol(const BigInt& a, const BigInt& p)

   {

   if(p <= 1 || p.is_even())

      throw Invalid_Argument("ressol: invalid prime");

   if(a == 0)

      return 0;

   else if(a < 0)

      throw Invalid_Argument("ressol: value to solve for must be positive");

   else if(a >= p)

      throw Invalid_Argument("ressol: value to solve for must be less than p");

   if(p == 2)

      return a;

   if(jacobi(a, p) != 1) // not a quadratic residue

      return -BigInt(1);

   if(p % 4 == 3) // The easy case

      {

      return power_mod(a, ((p+1) >> 2), p);

      }

   size_t s = low_zero_bits(p - 1);

   BigInt q = p >> s;

   q -= 1;

   q >>= 1;

   Modular_Reducer mod_p(p);

   BigInt r = power_mod(a, q, p);

   BigInt n = mod_p.multiply(a, mod_p.square(r));

   r = mod_p.multiply(r, a);

   if(n == 1)

      return r;

   // find random quadratic nonresidue z

   word z = 2;

   for(;;)

      {

      if(jacobi(z, p) == -1) // found one

         break;

      z += 1; // try next z

      /*

      * The expected number of tests to find a non-residue modulo a

      * prime is 2. If we have not found one after 256 then almost

      * certainly we have been given a non-prime p.

      */

      if(z >= 256)

         return -BigInt(1);

      }

   BigInt c = power_mod(z, (q << 1) + 1, p);

   while(n > 1)

      {

      q = n;

      size_t i = 0;

      while(q != 1)

         {

         q = mod_p.square(q);

         ++i;

         if(i >= s)

            {

            return -BigInt(1);

            }

         }

      c = power_mod(c, BigInt::power_of_2(s-i-1), p);

      r = mod_p.multiply(r, c);

      c = mod_p.square(c);

      n = mod_p.multiply(n, c);

      s = i;

      }

   return r;

   }

/*

* Calculate the Jacobi symbol

*/

int32_t jacobi(const BigInt& a, const BigInt& n)

   {

   if(n.is_even() || n < 2)

      throw Invalid_Argument("jacobi: second argument must be odd and > 1");

   BigInt x = a % n;

   BigInt y = n;

   int32_t J = 1;

   while(y > 1)

      {

      x %= y;

      if(x > y / 2)

         {

         x = y - x;

         if(y % 4 == 3)

            J = -J;

         }

      if(x.is_zero())

         return 0;

      size_t shifts = low_zero_bits(x);

      x >>= shifts;

      if(shifts % 2)

         {

         word y_mod_8 = y % 8;

         if(y_mod_8 == 3 || y_mod_8 == 5)

            J = -J;

         }

      if(x % 4 == 3 && y % 4 == 3)

         J = -J;

      std::swap(x, y);

      }

   return J;

   }

/*

* Square a BigInt

*/

BigInt square(const BigInt& x)

   {

   BigInt z = x;

   secure_vector<word> ws;

   z.square(ws);

   return z;

   }

/*

* Return the number of 0 bits at the end of n

*/

size_t low_zero_bits(const BigInt& n)

   {

   size_t low_zero = 0;

   auto seen_nonempty_word = CT::Mask<word>::cleared();

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

      {

      const word x = n.word_at(i);

      // ctz(0) will return sizeof(word)

      const size_t tz_x = ctz(x);

      // if x > 0 we want to count tz_x in total but not any

      // further words, so set the mask after the addition

      low_zero += seen_nonempty_word.if_not_set_return(tz_x);

      seen_nonempty_word |= CT::Mask<word>::expand(x);

      }

   // if we saw no words with x > 0 then n == 0 and the value we have

   // computed is meaningless. Instead return 0 in that case.

   return seen_nonempty_word.if_set_return(low_zero);

   }

/*

* Calculate the GCD

*/

BigInt gcd(const BigInt& a, const BigInt& b)

   {

   if(a.is_zero() || b.is_zero())

      return 0;

   if(a == 1 || b == 1)

      return 1;

   // See https://gcd.cr.yp.to/safegcd-20190413.pdf fig 1.2

   BigInt f = a;

   BigInt g = b;

   f.const_time_poison();

   g.const_time_poison();

   f.set_sign(BigInt::Positive);

   g.set_sign(BigInt::Positive);

   const size_t common2s = std::min(low_zero_bits(f), low_zero_bits(g));

   CT::unpoison(common2s);

   f >>= common2s;

   g >>= common2s;

   f.ct_cond_swap(f.is_even(), g);

   int32_t delta = 1;

   const size_t loop_cnt = 4 + 3*std::max(f.bits(), g.bits());

   BigInt newg, t;

   for(size_t i = 0; i != loop_cnt; ++i)

      {

      sub_abs(newg, f, g);

      const bool need_swap = (g.is_odd() && delta > 0);

      // if(need_swap) delta *= -1

      delta *= CT::Mask<uint8_t>::expand(need_swap).select(0, 2) - 1;

      f.ct_cond_swap(need_swap, g);

      g.ct_cond_swap(need_swap, newg);

      delta += 1;

      g.ct_cond_add(g.is_odd(), f);

      g >>= 1;

      }

   f <<= common2s;

   f.const_time_unpoison();

   g.const_time_unpoison();

   return f;

   }

/*

* Calculate the LCM

*/

BigInt lcm(const BigInt& a, const BigInt& b)

   {

   return ct_divide(a * b, gcd(a, b));

   }

/*

* Modular Exponentiation

*/

BigInt power_mod(const BigInt& base, const BigInt& exp, const BigInt& mod)

   {

   if(mod.is_negative() || mod == 1)

      {

      return 0;

      }

   if(base.is_zero() || mod.is_zero())

      {

      if(exp.is_zero())

         return 1;

      return 0;

      }

   Modular_Reducer reduce_mod(mod);

   const size_t exp_bits = exp.bits();

   if(mod.is_odd())

      {

      const size_t powm_window = 4;

      auto monty_mod = std::make_shared<Montgomery_Params>(mod, reduce_mod);

      auto powm_base_mod = monty_precompute(monty_mod, reduce_mod.reduce(base), powm_window);

      return monty_execute(*powm_base_mod, exp, exp_bits);

      }

   /*

   Support for even modulus is just a convenience and not considered

   cryptographically important, so this implementation is slow ...

   */

   BigInt accum = 1;

   BigInt g = reduce_mod.reduce(base);

   BigInt t;

   for(size_t i = 0; i != exp_bits; ++i)

      {

      t = reduce_mod.multiply(g, accum);

      g = reduce_mod.square(g);

      accum.ct_cond_assign(exp.get_bit(i), t);

      }

   return accum;

   }

BigInt is_perfect_square(const BigInt& C)

   {

   if(C < 1)

      throw Invalid_Argument("is_perfect_square requires C >= 1");

   if(C == 1)

      return 1;

   const size_t n = C.bits();

   const size_t m = (n + 1) / 2;

   const BigInt B = C + BigInt::power_of_2(m);

   BigInt X = BigInt::power_of_2(m) - 1;

   BigInt X2 = (X*X);

   for(;;)

      {

      X = (X2 + C) / (2*X);

      X2 = (X*X);

      if(X2 < B)

         break;

      }

   if(X2 == C)

      return X;

   else

      return 0;

   }

/*

* Test for primality using Miller-Rabin

*/

bool is_prime(const BigInt& n,

              RandomNumberGenerator& rng,

              size_t prob,

              bool is_random)

   {

   if(n == 2)

      return true;

   if(n <= 1 || n.is_even())

      return false;

   const size_t n_bits = n.bits();

   // Fast path testing for small numbers (<= 65521)

   if(n_bits <= 16)

      {

      const uint16_t num = static_cast<uint16_t>(n.word_at(0));

      return std::binary_search(PRIMES, PRIMES + PRIME_TABLE_SIZE, num);

      }

   Modular_Reducer mod_n(n);

   if(rng.is_seeded())

      {

      const size_t t = miller_rabin_test_iterations(n_bits, prob, is_random);

      if(is_miller_rabin_probable_prime(n, mod_n, rng, t) == false)

         return false;

      if(is_random)

         return true;

      else

         return is_lucas_probable_prime(n, mod_n);

      }

   else

      {

      return is_bailie_psw_probable_prime(n, mod_n);

      }

   }

}