numthry.cpp

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/*
* 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/rng.h>
#include <botan/internal/barrett.h>
#include <botan/internal/ct_utils.h>
#include <botan/internal/divide.h>
#include <botan/internal/monty.h>
#include <botan/internal/monty_exp.h>
#include <botan/internal/mp_core.h>
#include <botan/internal/primality.h>
#include <algorithm>
 
namespace Botan {
 
/*
* Tonelli-Shanks algorithm
*/
BigInt sqrt_modulo_prime(const BigInt& a, const BigInt& p) {
   BOTAN_ARG_CHECK(p > 1, "invalid prime");
   BOTAN_ARG_CHECK(a < p, "value to solve for must be less than p");
   BOTAN_ARG_CHECK(a >= 0, "value to solve for must not be negative");
 
   // some very easy cases
   if(p == 2 || a <= 1) {
      return a;
   }
 
   BOTAN_ARG_CHECK(p.is_odd(), "invalid prime");
 
   if(jacobi(a, p) != 1) {  // not a quadratic residue
      return BigInt::from_s32(-1);
   }
 
   auto mod_p = Barrett_Reduction::for_public_modulus(p);
   const Montgomery_Params monty_p(p, mod_p);
 
   // If p == 3 (mod 4) there is a simple solution
   if(p % 4 == 3) {
      return monty_exp_vartime(monty_p, a, ((p + 1) >> 2)).value();
   }
 
   // Otherwise we have to use Shanks-Tonelli
   size_t s = low_zero_bits(p - 1);
   BigInt q = p >> s;
 
   q -= 1;
   q >>= 1;
 
   BigInt r = monty_exp_vartime(monty_p, a, q).value();
   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(BigInt::from_word(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::from_s32(-1);
      }
   }
 
   BigInt c = monty_exp_vartime(monty_p, BigInt::from_word(z), (q << 1) + 1).value();
 
   while(n > 1) {
      q = n;
 
      size_t i = 0;
      while(q != 1) {
         q = mod_p.square(q);
         ++i;
 
         if(i >= s) {
            return BigInt::from_s32(-1);
         }
      }
 
      BOTAN_ASSERT_NOMSG(s >= (i + 1));  // No underflow!
      c = monty_exp_vartime(monty_p, c, BigInt::power_of_2(s - i - 1)).value();
      r = mod_p.multiply(r, c);
      c = mod_p.square(c);
      n = mod_p.multiply(n, c);
 
      // s decreases as the algorithm proceeds
      BOTAN_ASSERT_NOMSG(s >= i);
      s = i;
   }
 
   return r;
}
 
/*
* Calculate the Jacobi symbol
*
* See Algorithm 2.149 in Handbook of Applied Cryptography
*/
int32_t jacobi(BigInt a, BigInt n) {
   BOTAN_ARG_CHECK(n.is_odd() && n >= 3, "Argument n must be an odd integer >= 3");
 
   if(a < 0 || a >= n) {
      a %= n;
   }
 
   if(a == 0) {
      return 0;
   }
   if(a == 1) {
      return 1;
   }
 
   int32_t s = 1;
 
   for(;;) {
      const size_t e = low_zero_bits(a);
      a >>= e;
      const word n_mod_8 = n.word_at(0) % 8;
      const word n_mod_4 = n_mod_8 % 4;
 
      if(e % 2 == 1 && (n_mod_8 == 3 || n_mod_8 == 5)) {
         s = -s;
      }
 
      if(n_mod_4 == 3 && a % 4 == 3) {
         s = -s;
      }
 
      /*
      * The HAC presentation of the algorithm uses recursion, which is not
      * desirable or necessary.
      *
      * Instead we loop accumulating the product of the various jacobi()
      * subcomputations into s, until we reach algorithm termination, which
      * occurs in one of two ways.
      *
      * If a == 1 then the recursion has completed; we can return the value of s.
      *
      * Otherwise, after swapping and reducing, check for a == 0 [this value is
      * called `n1` in HAC's presentation]. This would imply that jacobi(n1,a1)
      * would have the value 0, due to Line 1 in HAC 2.149, in which case the
      * entire product is zero, and we can immediately return that result.
      */
 
      if(a == 1) {
         return s;
      }
 
      std::swap(a, n);
 
      BOTAN_ASSERT_NOMSG(n.is_odd());
 
      a %= n;
 
      if(a == 0) {
         return 0;
      }
   }
}
 
/*
* 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 BigInt::zero() in that case.
   return static_cast<size_t>(seen_nonempty_word.if_set_return(low_zero));
}
 
/*
* Calculate the GCD in constant time
*/
BigInt gcd(const BigInt& a, const BigInt& b) {
   if(a.is_zero()) {
      return abs(b);
   }
   if(b.is_zero()) {
      return abs(a);
   }
 
   const size_t sz = std::max(a.sig_words(), b.sig_words());
   auto u = BigInt::with_capacity(sz);
   auto v = BigInt::with_capacity(sz);
   u += a;
   v += b;
 
   CT::poison_all(u, v);
 
   u.set_sign(BigInt::Positive);
   v.set_sign(BigInt::Positive);
 
   // In the worst case we have two fully populated big ints. After right
   // shifting so many times, we'll have reached the result for sure.
   const size_t loop_cnt = u.bits() + v.bits();
 
   // This temporary is big enough to hold all intermediate results of the
   // algorithm. No reallocation will happen during the loop.
   // Note however, that `ct_cond_assign()` will invalidate the 'sig_words'
   // cache, which _does not_ shrink the capacity of the underlying buffer.
   auto tmp = BigInt::with_capacity(sz);
   secure_vector<word> ws(sz * 2);
   size_t factors_of_two = 0;
   for(size_t i = 0; i != loop_cnt; ++i) {
      auto both_odd = CT::Mask<word>::expand_bool(u.is_odd()) & CT::Mask<word>::expand_bool(v.is_odd());
 
      // Subtract the smaller from the larger if both are odd
      auto u_gt_v = CT::Mask<word>::expand_bool(bigint_cmp(u._data(), u.size(), v._data(), v.size()) > 0);
      bigint_sub_abs(tmp.mutable_data(), u._data(), v._data(), sz, ws.data());
      u.ct_cond_assign((u_gt_v & both_odd).as_bool(), tmp);
      v.ct_cond_assign((~u_gt_v & both_odd).as_bool(), tmp);
 
      const auto u_is_even = CT::Mask<word>::expand_bool(u.is_even());
      const auto v_is_even = CT::Mask<word>::expand_bool(v.is_even());
      BOTAN_DEBUG_ASSERT((u_is_even | v_is_even).as_bool());
 
      // When both are even, we're going to eliminate a factor of 2.
      // We have to reapply this factor to the final result.
      factors_of_two += (u_is_even & v_is_even).if_set_return(1);
 
      // remove one factor of 2, if u is even
      bigint_shr2(tmp.mutable_data(), u._data(), sz, 1);
      u.ct_cond_assign(u_is_even.as_bool(), tmp);
 
      // remove one factor of 2, if v is even
      bigint_shr2(tmp.mutable_data(), v._data(), sz, 1);
      v.ct_cond_assign(v_is_even.as_bool(), tmp);
   }
 
   // The GCD (without factors of two) is either in u or v, the other one is
   // zero. The non-zero variable _must_ be odd, because all factors of two were
   // removed in the loop iterations above.
   BOTAN_DEBUG_ASSERT(u.is_zero() || v.is_zero());
   BOTAN_DEBUG_ASSERT(u.is_odd() || v.is_odd());
 
   // make sure that the GCD (without factors of two) is in u
   u.ct_cond_assign(u.is_even() /* .is_zero() would not be constant time */, v);
 
   // re-apply the factors of two
   u.ct_shift_left(factors_of_two);
 
   CT::unpoison_all(u, v);
 
   return u;
}
 
/*
* Calculate the LCM
*/
BigInt lcm(const BigInt& a, const BigInt& b) {
   if(a == b) {
      return a;
   }
 
   auto ab = a * b;
   ab.set_sign(BigInt::Positive);  // ignore the signs of a & b
   const auto g = gcd(a, b);
   return ct_divide(ab, g);
}
 
/*
* Modular Exponentiation
*/
BigInt power_mod(const BigInt& base, const BigInt& exp, const BigInt& mod) {
   if(mod.is_negative() || mod == 1) {
      return BigInt::zero();
   }
 
   if(base.is_zero() || mod.is_zero()) {
      if(exp.is_zero()) {
         return BigInt::one();
      }
      return BigInt::zero();
   }
 
   auto reduce_mod = Barrett_Reduction::for_secret_modulus(mod);
 
   const size_t exp_bits = exp.bits();
 
   if(mod.is_odd()) {
      const Montgomery_Params monty_params(mod, reduce_mod);
      return monty_exp(monty_params, ct_modulo(base, mod), exp, exp_bits).value();
   }
 
   /*
   Support for even modulus is just a convenience and not considered
   cryptographically important, so this implementation is slow ...
   */
   BigInt accum = BigInt::one();
   BigInt g = ct_modulo(base, mod);
   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 BigInt::one();
   }
 
   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 BigInt::zero();
   }
}
 
/*
* 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);
   }
 
   auto mod_n = Barrett_Reduction::for_secret_modulus(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)) {
         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);
   }
}
 
}  // namespace Botan