Put only the implementation of BigInt in src/bigint, mathematical functions

on top of BigInt go to src/math. Some prototypes were moved in order to
complete the split, in particular random_integer() is now a static member
of BigInt instead of being a global function, and divide() is in divide.h
instead of numthry.h

18 files changed, 2369 insertions, 0 deletions

diff --git a/src/math/blinding.cpp b/src/math/blinding.cpp
new file mode 100644
index 000000000..740904d10
--- /dev/null
+++ b/src/math/blinding.cpp
@@ -0,0 +1,47 @@
+/*************************************************
+* Blinder Source File                            *
+* (C) 1999-2007 Jack Lloyd                       *
+*************************************************/
+
+#include <botan/blinding.h>
+#include <botan/numthry.h>
+
+namespace Botan {
+
+/*************************************************
+* Blinder Constructor                            *
+*************************************************/
+Blinder::Blinder(const BigInt& e, const BigInt& d, const BigInt& n)
+   {
+   if(e < 1 || d < 1 || n < 1)
+      throw Invalid_Argument("Blinder: Arguments too small");
+
+   reducer = Modular_Reducer(n);
+   this->e = e;
+   this->d = d;
+   }
+
+/*************************************************
+* Blind a number                                 *
+*************************************************/
+BigInt Blinder::blind(const BigInt& i) const
+   {
+   if(!reducer.initialized())
+      return i;
+
+   e = reducer.square(e);
+   d = reducer.square(d);
+   return reducer.multiply(i, e);
+   }
+
+/*************************************************
+* Unblind a number                               *
+*************************************************/
+BigInt Blinder::unblind(const BigInt& i) const
+   {
+   if(!reducer.initialized())
+      return i;
+   return reducer.multiply(i, d);
+   }
+
+}
diff --git a/src/math/blinding.h b/src/math/blinding.h
new file mode 100644
index 000000000..958686fb1
--- /dev/null
+++ b/src/math/blinding.h
@@ -0,0 +1,32 @@
+/*************************************************
+* Blinder Header File                            *
+* (C) 1999-2007 Jack Lloyd                       *
+*************************************************/
+
+#ifndef BOTAN_BLINDER_H__
+#define BOTAN_BLINDER_H__
+
+#include <botan/bigint.h>
+#include <botan/reducer.h>
+
+namespace Botan {
+
+/*************************************************
+* Blinding Function Object                       *
+*************************************************/
+class BOTAN_DLL Blinder
+   {
+   public:
+      BigInt blind(const BigInt&) const;
+      BigInt unblind(const BigInt&) const;
+
+      Blinder() {}
+      Blinder(const BigInt&, const BigInt&, const BigInt&);
+   private:
+      Modular_Reducer reducer;
+      mutable BigInt e, d;
+   };
+
+}
+
+#endif
diff --git a/src/math/def_powm.h b/src/math/def_powm.h
new file mode 100644
index 000000000..c91ff002c
--- /dev/null
+++ b/src/math/def_powm.h
@@ -0,0 +1,62 @@
+/*************************************************
+* Modular Exponentiation Header File             *
+* (C) 1999-2007 Jack Lloyd                       *
+*************************************************/
+
+#ifndef BOTAN_DEFAULT_MODEXP_H__
+#define BOTAN_DEFAULT_MODEXP_H__
+
+#include <botan/pow_mod.h>
+#include <botan/reducer.h>
+#include <vector>
+
+namespace Botan {
+
+/*************************************************
+* Fixed Window Exponentiator                     *
+*************************************************/
+class BOTAN_DLL Fixed_Window_Exponentiator : public Modular_Exponentiator
+   {
+   public:
+      void set_exponent(const BigInt&);
+      void set_base(const BigInt&);
+      BigInt execute() const;
+
+      Modular_Exponentiator* copy() const
+         { return new Fixed_Window_Exponentiator(*this); }
+
+      Fixed_Window_Exponentiator(const BigInt&, Power_Mod::Usage_Hints);
+   private:
+      Modular_Reducer reducer;
+      BigInt exp;
+      u32bit window_bits;
+      std::vector<BigInt> g;
+      Power_Mod::Usage_Hints hints;
+   };
+
+/*************************************************
+* Montgomery Exponentiator                       *
+*************************************************/
+class BOTAN_DLL Montgomery_Exponentiator : public Modular_Exponentiator
+   {
+   public:
+      void set_exponent(const BigInt&);
+      void set_base(const BigInt&);
+      BigInt execute() const;
+
+      Modular_Exponentiator* copy() const
+         { return new Montgomery_Exponentiator(*this); }
+
+      Montgomery_Exponentiator(const BigInt&, Power_Mod::Usage_Hints);
+   private:
+      BigInt exp, modulus;
+      BigInt R2, R_mod;
+      std::vector<BigInt> g;
+      word mod_prime;
+      u32bit mod_words, exp_bits, window_bits;
+      Power_Mod::Usage_Hints hints;
+   };
+
+}
+
+#endif
diff --git a/src/math/dsa_gen.cpp b/src/math/dsa_gen.cpp
new file mode 100644
index 000000000..baaba66ee
--- /dev/null
+++ b/src/math/dsa_gen.cpp
@@ -0,0 +1,134 @@
+/*************************************************
+* DSA Parameter Generation Source File           *
+* (C) 1999-2007 Jack Lloyd                       *
+*************************************************/
+
+#include <botan/dl_group.h>
+#include <botan/numthry.h>
+#include <botan/lookup.h>
+#include <botan/parsing.h>
+#include <algorithm>
+#include <memory>
+
+namespace Botan {
+
+namespace {
+
+/*************************************************
+* Check if this size is allowed by FIPS 186-3    *
+*************************************************/
+bool fips186_3_valid_size(u32bit pbits, u32bit qbits)
+   {
+   if(qbits == 160)
+      return (pbits == 512 || pbits == 768 || pbits == 1024);
+
+   if(qbits == 224)
+      return (pbits == 2048);
+
+   if(qbits == 256)
+      return (pbits == 2048 || pbits == 3072);
+
+   return false;
+   }
+
+}
+
+/*************************************************
+* Attempt DSA prime generation with given seed   *
+*************************************************/
+bool DL_Group::generate_dsa_primes(RandomNumberGenerator& rng,
+                                   BigInt& p, BigInt& q,
+                                   u32bit pbits, u32bit qbits,
+                                   const MemoryRegion<byte>& seed_c)
+   {
+   if(!fips186_3_valid_size(pbits, qbits))
+      throw Invalid_Argument(
+         "FIPS 186-3 does not allow DSA domain parameters of " +
+         to_string(pbits) + "/" + to_string(qbits) + " bits long");
+
+   if(qbits == 224)
+      throw Invalid_Argument(
+         "DSA parameter generation with a q of 224 bits not supported");
+
+   if(seed_c.size() * 8 < qbits)
+      throw Invalid_Argument(
+         "Generating a DSA parameter set with a " + to_string(qbits) +
+         "long q requires a seed at least as many bits long");
+
+   std::auto_ptr<HashFunction> hash(get_hash("SHA-" + to_string(qbits)));
+
+   const u32bit HASH_SIZE = hash->OUTPUT_LENGTH;
+
+   class Seed
+      {
+      public:
+         Seed(const MemoryRegion<byte>& s) : seed(s) {}
+
+         operator MemoryRegion<byte>& () { return seed; }
+
+         Seed& operator++()
+            {
+            for(u32bit j = seed.size(); j > 0; --j)
+               if(++seed[j-1])
+                  break;
+            return (*this);
+            }
+      private:
+         SecureVector<byte> seed;
+      };
+
+   Seed seed(seed_c);
+
+   q.binary_decode(hash->process(seed));
+   q.set_bit(qbits-1);
+   q.set_bit(0);
+
+   if(!is_prime(q, rng))
+      return false;
+
+   const u32bit n = (pbits-1) / (HASH_SIZE * 8),
+                b = (pbits-1) % (HASH_SIZE * 8);
+
+   BigInt X;
+   SecureVector<byte> V(HASH_SIZE * (n+1));
+
+   for(u32bit j = 0; j != 4096; ++j)
+      {
+      for(u32bit k = 0; k <= n; ++k)
+         {
+         ++seed;
+         hash->update(seed);
+         hash->final(V + HASH_SIZE * (n-k));
+         }
+
+      X.binary_decode(V + (HASH_SIZE - 1 - b/8),
+                      V.size() - (HASH_SIZE - 1 - b/8));
+      X.set_bit(pbits-1);
+
+      p = X - (X % (2*q) - 1);
+
+      if(p.bits() == pbits && is_prime(p, rng))
+         return true;
+      }
+   return false;
+   }
+
+/*************************************************
+* Generate DSA Primes                            *
+*************************************************/
+SecureVector<byte> DL_Group::generate_dsa_primes(RandomNumberGenerator& rng,
+                                                 BigInt& p, BigInt& q,
+                                                 u32bit pbits, u32bit qbits)
+   {
+   SecureVector<byte> seed(qbits/8);
+
+   while(true)
+      {
+      rng.randomize(seed, seed.size());
+
+      if(generate_dsa_primes(rng, p, q, pbits, qbits, seed))
+         return seed;
+      }
+   }
+
+}
diff --git a/src/math/info.txt b/src/math/info.txt
new file mode 100644
index 000000000..6ce3583c8
--- /dev/null
+++ b/src/math/info.txt
@@ -0,0 +1,29 @@
+realname "Math Functions"
+
+load_on auto
+
+define BIGINT_MATH
+
+<requires>
+bigint
+</requires>
+
+<add>
+blinding.cpp
+blinding.h
+def_powm.h
+dsa_gen.cpp
+jacobi.cpp
+make_prm.cpp
+mp_numth.cpp
+numthry.cpp
+numthry.h
+pow_mod.cpp
+pow_mod.h
+powm_fw.cpp
+powm_mnt.cpp
+primes.cpp
+reducer.cpp
+reducer.h
+ressol.cpp
+</add>
diff --git a/src/math/jacobi.cpp b/src/math/jacobi.cpp
new file mode 100644
index 000000000..57c78508a
--- /dev/null
+++ b/src/math/jacobi.cpp
@@ -0,0 +1,51 @@
+/*************************************************
+* Jacobi Function Source File                    *
+* (C) 1999-2007 Jack Lloyd                       *
+*************************************************/
+
+#include <botan/numthry.h>
+
+namespace Botan {
+
+/*************************************************
+* Calculate the Jacobi symbol                    *
+*************************************************/
+s32bit jacobi(const BigInt& a, const BigInt& n)
+   {
+   if(a.is_negative())
+      throw Invalid_Argument("jacobi: first argument must be non-negative");
+   if(n.is_even() || n < 2)
+      throw Invalid_Argument("jacobi: second argument must be odd and > 1");
+
+   BigInt x = a, y = n;
+   s32bit 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;
+
+      u32bit 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;
+   }
+
+}
diff --git a/src/math/make_prm.cpp b/src/math/make_prm.cpp
new file mode 100644
index 000000000..30ac9a623
--- /dev/null
+++ b/src/math/make_prm.cpp
@@ -0,0 +1,95 @@
+/*************************************************
+* Prime Generation Source File                   *
+* (C) 1999-2007 Jack Lloyd                       *
+*************************************************/
+
+#include <botan/numthry.h>
+#include <botan/parsing.h>
+#include <algorithm>
+
+namespace Botan {
+
+/*************************************************
+* Generate a random prime                        *
+*************************************************/
+BigInt random_prime(RandomNumberGenerator& rng,
+                    u32bit bits, const BigInt& coprime,
+                    u32bit equiv, u32bit modulo)
+   {
+   if(bits <= 1)
+      throw Invalid_Argument("random_prime: Can't make a prime of " +
+                             to_string(bits) + " bits");
+   else if(bits == 2)
+      return ((rng.next_byte() % 1) ? 2 : 3);
+   else if(bits == 3)
+      return ((rng.next_byte() % 1) ? 5 : 7);
+   else if(bits == 4)
+      return ((rng.next_byte() % 1) ? 11 : 13);
+
+   if(coprime <= 0)
+      throw Invalid_Argument("random_prime: coprime must be > 0");
+   if(modulo % 2 == 1 || modulo == 0)
+      throw Invalid_Argument("random_prime: Invalid modulo value");
+   if(equiv >= modulo || equiv % 2 == 0)
+      throw Invalid_Argument("random_prime: equiv must be < modulo, and odd");
+
+   while(true)
+      {
+      BigInt p(rng, bits);
+      p.set_bit(bits - 2);
+      p.set_bit(0);
+
+      if(p % modulo != equiv)
+         p += (modulo - p % modulo) + equiv;
+
+      const u32bit sieve_size = std::min(bits / 2, PRIME_TABLE_SIZE);
+      SecureVector<u32bit> sieve(sieve_size);
+
+      for(u32bit j = 0; j != sieve.size(); ++j)
+         sieve[j] = p % PRIMES[j];
+
+      u32bit counter = 0;
+      while(true)
+         {
+         if(counter == 4096 || p.bits() > bits)
+            break;
+
+         bool passes_sieve = true;
+         ++counter;
+         p += modulo;
+
+         if(p.bits() > bits)
+            break;
+
+         for(u32bit j = 0; j != sieve.size(); ++j)
+            {
+            sieve[j] = (sieve[j] + modulo) % PRIMES[j];
+            if(sieve[j] == 0)
+               passes_sieve = false;
+            }
+
+         if(!passes_sieve || gcd(p - 1, coprime) != 1)
+            continue;
+         if(passes_mr_tests(rng, p))
+            return p;
+         }
+      }
+   }
+
+/*************************************************
+* Generate a random safe prime                   *
+*************************************************/
+BigInt random_safe_prime(RandomNumberGenerator& rng, u32bit bits)
+   {
+   if(bits <= 64)
+      throw Invalid_Argument("random_safe_prime: Can't make a prime of " +
+                             to_string(bits) + " bits");
+
+   BigInt p;
+   do
+      p = (random_prime(rng, bits - 1) << 1) + 1;
+   while(!is_prime(p, rng));
+   return p;
+   }
+
+}
diff --git a/src/math/mp_numth.cpp b/src/math/mp_numth.cpp
new file mode 100644
index 000000000..b024d2e2d
--- /dev/null
+++ b/src/math/mp_numth.cpp
@@ -0,0 +1,69 @@
+/*************************************************
+* Fused and Important MP Algorithms Source File  *
+* (C) 1999-2007 Jack Lloyd                       *
+*************************************************/
+
+#include <botan/numthry.h>
+#include <botan/mp_core.h>
+#include <botan/util.h>
+#include <algorithm>
+
+namespace Botan {
+
+/*************************************************
+* Square a BigInt                                *
+*************************************************/
+BigInt square(const BigInt& x)
+   {
+   const u32bit x_sw = x.sig_words();
+
+   BigInt z(BigInt::Positive, round_up(2*x_sw, 16));
+   SecureVector<word> workspace(z.size());
+
+   bigint_sqr(z.get_reg(), z.size(), workspace,
+              x.data(), x.size(), x_sw);
+   return z;
+   }
+
+/*************************************************
+* Multiply-Add Operation                         *
+*************************************************/
+BigInt mul_add(const BigInt& a, const BigInt& b, const BigInt& c)
+   {
+   if(c.is_negative() || c.is_zero())
+      throw Invalid_Argument("mul_add: Third argument must be > 0");
+
+   BigInt::Sign sign = BigInt::Positive;
+   if(a.sign() != b.sign())
+      sign = BigInt::Negative;
+
+   const u32bit a_sw = a.sig_words();
+   const u32bit b_sw = b.sig_words();
+   const u32bit c_sw = c.sig_words();
+
+   BigInt r(sign, std::max(a.size() + b.size(), c_sw) + 1);
+   SecureVector<word> workspace(r.size());
+
+   bigint_mul(r.get_reg(), r.size(), workspace,
+              a.data(), a.size(), a_sw,
+              b.data(), b.size(), b_sw);
+   const u32bit r_size = std::max(r.sig_words(), c_sw);
+   bigint_add2(r.get_reg(), r_size, c.data(), c_sw);
+   return r;
+   }
+
+/*************************************************
+* Subtract-Multiply Operation                    *
+*************************************************/
+BigInt sub_mul(const BigInt& a, const BigInt& b, const BigInt& c)
+   {
+   if(a.is_negative() || b.is_negative())
+      throw Invalid_Argument("sub_mul: First two arguments must be >= 0");
+
+   BigInt r = a;
+   r -= b;
+   r *= c;
+   return r;
+   }
+
+}
diff --git a/src/math/numthry.cpp b/src/math/numthry.cpp
new file mode 100644
index 000000000..ffd523e82
--- /dev/null
+++ b/src/math/numthry.cpp
@@ -0,0 +1,339 @@
+/*************************************************
+* Number Theory Source File                      *
+* (C) 1999-2008 Jack Lloyd                       *
+*************************************************/
+
+#include <botan/numthry.h>
+#include <botan/bit_ops.h>
+#include <algorithm>
+
+namespace Botan {
+
+namespace {
+
+/*************************************************
+* Miller-Rabin Iterations                        *
+*************************************************/
+u32bit miller_rabin_test_iterations(u32bit bits, bool verify)
+   {
+   struct mapping { u32bit bits; u32bit verify_iter; u32bit check_iter; };
+
+   static const mapping tests[] = {
+      {   50, 55, 25 },
+      {  100, 38, 22 },
+      {  160, 32, 18 },
+      {  163, 31, 17 },
+      {  168, 30, 16 },
+      {  177, 29, 16 },
+      {  181, 28, 15 },
+      {  185, 27, 15 },
+      {  190, 26, 15 },
+      {  195, 25, 14 },
+      {  201, 24, 14 },
+      {  208, 23, 14 },
+      {  215, 22, 13 },
+      {  222, 21, 13 },
+      {  231, 20, 13 },
+      {  241, 19, 12 },
+      {  252, 18, 12 },
+      {  264, 17, 12 },
+      {  278, 16, 11 },
+      {  294, 15, 10 },
+      {  313, 14,  9 },
+      {  334, 13,  8 },
+      {  360, 12,  8 },
+      {  392, 11,  7 },
+      {  430, 10,  7 },
+      {  479,  9,  6 },
+      {  542,  8,  6 },
+      {  626,  7,  5 },
+      {  746,  6,  4 },
+      {  926,  5,  3 },
+      { 1232,  4,  2 },
+      { 1853,  3,  2 },
+      {    0,  0,  0 }
+   };
+
+   for(u32bit j = 0; tests[j].bits; ++j)
+      {
+      if(bits <= tests[j].bits)
+         {
+         if(verify)
+            return tests[j].verify_iter;
+         else
+            return tests[j].check_iter;
+         }
+      }
+   return 2;
+   }
+
+}
+
+/*************************************************
+* Return the number of 0 bits at the end of n    *
+*************************************************/
+u32bit low_zero_bits(const BigInt& n)
+   {
+   if(n.is_negative() || n.is_zero()) return 0;
+
+   u32bit low_zero = 0;
+
+   if(n.is_positive() && n.is_nonzero())
+      {
+      for(u32bit i = 0; i != n.size(); ++i)
+         {
+         word x = n[i];
+
+         if(x)
+            {
+            low_zero += ctz(x);
+            break;
+            }
+         else
+            low_zero += BOTAN_MP_WORD_BITS;
+         }
+      }
+
+   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;
+
+   BigInt x = a, y = b;
+   x.set_sign(BigInt::Positive);
+   y.set_sign(BigInt::Positive);
+   u32bit shift = std::min(low_zero_bits(x), low_zero_bits(y));
+
+   x >>= shift;
+   y >>= shift;
+
+   while(x.is_nonzero())
+      {
+      x >>= low_zero_bits(x);
+      y >>= low_zero_bits(y);
+      if(x >= y) { x -= y; x >>= 1; }
+      else       { y -= x; y >>= 1; }
+      }
+
+   return (y << shift);
+   }
+
+/*************************************************
+* Calculate the LCM                              *
+*************************************************/
+BigInt lcm(const BigInt& a, const BigInt& b)
+   {
+   return ((a * b) / gcd(a, b));
+   }
+
+/*************************************************
+* Find the Modular Inverse                       *
+*************************************************/
+BigInt inverse_mod(const BigInt& n, const BigInt& mod)
+   {
+   if(mod.is_zero())
+      throw BigInt::DivideByZero();
+   if(mod.is_negative() || n.is_negative())
+      throw Invalid_Argument("inverse_mod: arguments must be non-negative");
+
+   if(n.is_zero() || (n.is_even() && mod.is_even()))
+      return 0;
+
+   BigInt x = mod, y = n, u = mod, v = n;
+   BigInt A = 1, B = 0, C = 0, D = 1;
+
+   while(u.is_nonzero())
+      {
+      u32bit zero_bits = low_zero_bits(u);
+      u >>= zero_bits;
+      for(u32bit j = 0; j != zero_bits; ++j)
+         {
+         if(A.is_odd() || B.is_odd())
+            { A += y; B -= x; }
+         A >>= 1; B >>= 1;
+         }
+
+      zero_bits = low_zero_bits(v);
+      v >>= zero_bits;
+      for(u32bit j = 0; j != zero_bits; ++j)
+         {
+         if(C.is_odd() || D.is_odd())
+            { C += y; D -= x; }
+         C >>= 1; D >>= 1;
+         }
+
+      if(u >= v) { u -= v; A -= C; B -= D; }
+      else       { v -= u; C -= A; D -= B; }
+      }
+
+   if(v != 1)
+      return 0;
+
+   while(D.is_negative()) D += mod;
+   while(D >= mod) D -= mod;
+
+   return D;
+   }
+
+/*************************************************
+* Modular Exponentiation                         *
+*************************************************/
+BigInt power_mod(const BigInt& base, const BigInt& exp, const BigInt& mod)
+   {
+   Power_Mod pow_mod(mod);
+   pow_mod.set_base(base);
+   pow_mod.set_exponent(exp);
+   return pow_mod.execute();
+   }
+
+/*************************************************
+* Do simple tests of primality                   *
+*************************************************/
+s32bit simple_primality_tests(const BigInt& n)
+   {
+   const s32bit NOT_PRIME = -1, UNKNOWN = 0, PRIME = 1;
+
+   if(n == 2)
+      return PRIME;
+   if(n <= 1 || n.is_even())
+      return NOT_PRIME;
+
+   if(n <= PRIMES[PRIME_TABLE_SIZE-1])
+      {
+      const word num = n.word_at(0);
+      for(u32bit j = 0; PRIMES[j]; ++j)
+         {
+         if(num == PRIMES[j]) return PRIME;
+         if(num <  PRIMES[j]) return NOT_PRIME;
+         }
+      return NOT_PRIME;
+      }
+
+   u32bit check_first = std::min(n.bits() / 32, PRIME_PRODUCTS_TABLE_SIZE);
+   for(u32bit j = 0; j != check_first; ++j)
+      if(gcd(n, PRIME_PRODUCTS[j]) != 1)
+         return NOT_PRIME;
+
+   return UNKNOWN;
+   }
+
+/*************************************************
+* Fast check of primality                        *
+*************************************************/
+bool check_prime(const BigInt& n, RandomNumberGenerator& rng)
+   {
+   return run_primality_tests(rng, n, 0);
+   }
+
+/*************************************************
+* Test for primality                             *
+*************************************************/
+bool is_prime(const BigInt& n, RandomNumberGenerator& rng)
+   {
+   return run_primality_tests(rng, n, 1);
+   }
+
+/*************************************************
+* Verify primality                               *
+*************************************************/
+bool verify_prime(const BigInt& n, RandomNumberGenerator& rng)
+   {
+   return run_primality_tests(rng, n, 2);
+   }
+
+/*************************************************
+* Verify primality                               *
+*************************************************/
+bool run_primality_tests(RandomNumberGenerator& rng,
+                         const BigInt& n, u32bit level)
+   {
+   s32bit simple_tests = simple_primality_tests(n);
+   if(simple_tests) return (simple_tests == 1) ? true : false;
+   return passes_mr_tests(rng, n, level);
+   }
+
+/*************************************************
+* Test for primaility using Miller-Rabin         *
+*************************************************/
+bool passes_mr_tests(RandomNumberGenerator& rng,
+                     const BigInt& n, u32bit level)
+   {
+   const u32bit PREF_NONCE_BITS = 40;
+
+   if(level > 2)
+      level = 2;
+
+   MillerRabin_Test mr(n);
+
+   if(!mr.passes_test(2))
+      return false;
+
+   if(level == 0)
+      return true;
+
+   const u32bit NONCE_BITS = std::min(n.bits() - 1, PREF_NONCE_BITS);
+
+   const bool verify = (level == 2);
+
+   u32bit tests = miller_rabin_test_iterations(n.bits(), verify);
+
+   BigInt nonce;
+   for(u32bit j = 0; j != tests; ++j)
+      {
+      if(verify) nonce.randomize(rng, NONCE_BITS);
+      else       nonce = PRIMES[j];
+
+      if(!mr.passes_test(nonce))
+         return false;
+      }
+   return true;
+   }
+
+/*************************************************
+* Miller-Rabin Test                              *
+*************************************************/
+bool MillerRabin_Test::passes_test(const BigInt& a)
+   {
+   if(a < 2 || a >= n_minus_1)
+      throw Invalid_Argument("Bad size for nonce in Miller-Rabin test");
+
+   BigInt y = pow_mod(a);
+   if(y == 1 || y == n_minus_1)
+      return true;
+
+   for(u32bit j = 1; j != s; ++j)
+      {
+      y = reducer.square(y);
+
+      if(y == 1)
+         return false;
+      if(y == n_minus_1)
+         return true;
+      }
+   return false;
+   }
+
+/*************************************************
+* Miller-Rabin Constructor                       *
+*************************************************/
+MillerRabin_Test::MillerRabin_Test(const BigInt& num)
+   {
+   if(num.is_even() || num < 3)
+      throw Invalid_Argument("MillerRabin_Test: Invalid number for testing");
+
+   n = num;
+   n_minus_1 = n - 1;
+   s = low_zero_bits(n_minus_1);
+   r = n_minus_1 >> s;
+
+   pow_mod = Fixed_Exponent_Power_Mod(r, n);
+   reducer = Modular_Reducer(n);
+   }
+
+}
diff --git a/src/math/numthry.h b/src/math/numthry.h
new file mode 100644
index 000000000..a271a428a
--- /dev/null
+++ b/src/math/numthry.h
@@ -0,0 +1,100 @@
+/*************************************************
+* Number Theory Functions Header File            *
+* (C) 1999-2007 Jack Lloyd                       *
+*************************************************/
+
+#ifndef BOTAN_NUMBER_THEORY_H__
+#define BOTAN_NUMBER_THEORY_H__
+
+#include <botan/bigint.h>
+#include <botan/reducer.h>
+#include <botan/pow_mod.h>
+#include <botan/rng.h>
+
+namespace Botan {
+
+/*************************************************
+* Fused Arithmetic Operations                    *
+*************************************************/
+BigInt BOTAN_DLL mul_add(const BigInt&, const BigInt&, const BigInt&);
+BigInt BOTAN_DLL sub_mul(const BigInt&, const BigInt&, const BigInt&);
+
+/*************************************************
+* Number Theory Functions                        *
+*************************************************/
+inline BigInt abs(const BigInt& n) { return n.abs(); }
+
+void BOTAN_DLL divide(const BigInt&, const BigInt&, BigInt&, BigInt&);
+
+BigInt BOTAN_DLL gcd(const BigInt&, const BigInt&);
+BigInt BOTAN_DLL lcm(const BigInt&, const BigInt&);
+
+BigInt BOTAN_DLL square(const BigInt&);
+BigInt BOTAN_DLL inverse_mod(const BigInt&, const BigInt&);
+s32bit BOTAN_DLL jacobi(const BigInt&, const BigInt&);
+
+BigInt BOTAN_DLL power_mod(const BigInt&, const BigInt&, const BigInt&);
+
+/*************************************************
+* Compute the square root of x modulo a prime    *
+* using the Shanks-Tonnelli algorithm            *
+*************************************************/
+BigInt ressol(const BigInt& x, const BigInt& p);
+
+/*************************************************
+* Utility Functions                              *
+*************************************************/
+u32bit BOTAN_DLL low_zero_bits(const BigInt&);
+
+/*************************************************
+* Primality Testing                              *
+*************************************************/
+bool BOTAN_DLL check_prime(const BigInt&, RandomNumberGenerator&);
+bool BOTAN_DLL is_prime(const BigInt&, RandomNumberGenerator&);
+bool BOTAN_DLL verify_prime(const BigInt&, RandomNumberGenerator&);
+
+s32bit BOTAN_DLL simple_primality_tests(const BigInt&);
+
+bool BOTAN_DLL passes_mr_tests(RandomNumberGenerator&,
+                               const BigInt&, u32bit = 1);
+
+bool BOTAN_DLL run_primality_tests(RandomNumberGenerator&,
+                                   const BigInt&, u32bit = 1);
+
+/*************************************************
+* Random Number Generation                       *
+*************************************************/
+BigInt BOTAN_DLL random_prime(RandomNumberGenerator&,
+                              u32bit bits, const BigInt& coprime = 1,
+                              u32bit equiv = 1, u32bit equiv_mod = 2);
+
+BigInt BOTAN_DLL random_safe_prime(RandomNumberGenerator&,
+                                   u32bit);
+
+/*************************************************
+* Prime Numbers                                  *
+*************************************************/
+const u32bit PRIME_TABLE_SIZE = 6541;
+const u32bit PRIME_PRODUCTS_TABLE_SIZE = 256;
+
+extern const u16bit BOTAN_DLL PRIMES[];
+extern const u64bit PRIME_PRODUCTS[];
+
+/*************************************************
+* Miller-Rabin Primality Tester                  *
+*************************************************/
+class BOTAN_DLL MillerRabin_Test
+   {
+   public:
+      bool passes_test(const BigInt&);
+      MillerRabin_Test(const BigInt&);
+   private:
+      BigInt n, r, n_minus_1;
+      u32bit s;
+      Fixed_Exponent_Power_Mod pow_mod;
+      Modular_Reducer reducer;
+   };
+
+}
+
+#endif
diff --git a/src/math/pow_mod.cpp b/src/math/pow_mod.cpp
new file mode 100644
index 000000000..17ca7b796
--- /dev/null
+++ b/src/math/pow_mod.cpp
@@ -0,0 +1,155 @@
+/*************************************************
+* Modular Exponentiation Proxy Source File       *
+* (C) 1999-2007 Jack Lloyd                       *
+*************************************************/
+
+#include <botan/pow_mod.h>
+#include <botan/engine.h>
+
+namespace Botan {
+
+/*************************************************
+* Power_Mod Constructor                          *
+*************************************************/
+Power_Mod::Power_Mod(const BigInt& n, Usage_Hints hints)
+   {
+   core = 0;
+   set_modulus(n, hints);
+   }
+
+/*************************************************
+* Power_Mod Copy Constructor                     *
+*************************************************/
+Power_Mod::Power_Mod(const Power_Mod& other)
+   {
+   core = 0;
+   if(other.core)
+      core = other.core->copy();
+   }
+
+/*************************************************
+* Power_Mod Assignment Operator                  *
+*************************************************/
+Power_Mod& Power_Mod::operator=(const Power_Mod& other)
+   {
+   delete core;
+   core = 0;
+   if(other.core)
+      core = other.core->copy();
+   return (*this);
+   }
+
+/*************************************************
+* Power_Mod Destructor                           *
+*************************************************/
+Power_Mod::~Power_Mod()
+   {
+   delete core;
+   }
+
+/*************************************************
+* Set the modulus                                *
+*************************************************/
+void Power_Mod::set_modulus(const BigInt& n, Usage_Hints hints) const
+   {
+   delete core;
+   core = ((n == 0) ? 0 : Engine_Core::mod_exp(n, hints));
+   }
+
+/*************************************************
+* Set the base                                   *
+*************************************************/
+void Power_Mod::set_base(const BigInt& b) const
+   {
+   if(b.is_zero() || b.is_negative())
+      throw Invalid_Argument("Power_Mod::set_base: arg must be > 0");
+
+   if(!core)
+      throw Internal_Error("Power_Mod::set_base: core was NULL");
+   core->set_base(b);
+   }
+
+/*************************************************
+* Set the exponent                               *
+*************************************************/
+void Power_Mod::set_exponent(const BigInt& e) const
+   {
+   if(e.is_negative())
+      throw Invalid_Argument("Power_Mod::set_exponent: arg must be > 0");
+
+   if(!core)
+      throw Internal_Error("Power_Mod::set_exponent: core was NULL");
+   core->set_exponent(e);
+   }
+
+/*************************************************
+* Compute the result                             *
+*************************************************/
+BigInt Power_Mod::execute() const
+   {
+   if(!core)
+      throw Internal_Error("Power_Mod::execute: core was NULL");
+   return core->execute();
+   }
+
+namespace {
+
+/*************************************************
+* Choose potentially useful hints                *
+*************************************************/
+Power_Mod::Usage_Hints choose_base_hints(const BigInt& b, const BigInt& n)
+   {
+   if(b == 2)
+      return Power_Mod::Usage_Hints(Power_Mod::BASE_IS_2 |
+                                    Power_Mod::BASE_IS_SMALL);
+
+   const u32bit b_bits = b.bits();
+   const u32bit n_bits = n.bits();
+
+   if(b_bits < n_bits / 32)
+      return Power_Mod::BASE_IS_SMALL;
+   if(b_bits > n_bits / 4)
+      return Power_Mod::BASE_IS_LARGE;
+
+   return Power_Mod::NO_HINTS;
+   }
+
+/*************************************************
+* Choose potentially useful hints                *
+*************************************************/
+Power_Mod::Usage_Hints choose_exp_hints(const BigInt& e, const BigInt& n)
+   {
+   const u32bit e_bits = e.bits();
+   const u32bit n_bits = n.bits();
+
+   if(e_bits < n_bits / 32)
+      return Power_Mod::BASE_IS_SMALL;
+   if(e_bits > n_bits / 4)
+      return Power_Mod::BASE_IS_LARGE;
+   return Power_Mod::NO_HINTS;
+   }
+
+}
+
+/*************************************************
+* Fixed_Exponent_Power_Mod Constructor           *
+*************************************************/
+Fixed_Exponent_Power_Mod::Fixed_Exponent_Power_Mod(const BigInt& e,
+                                                   const BigInt& n,
+                                                   Usage_Hints hints) :
+   Power_Mod(n, Usage_Hints(hints | EXP_IS_FIXED | choose_exp_hints(e, n)))
+   {
+   set_exponent(e);
+   }
+
+/*************************************************
+* Fixed_Base_Power_Mod Constructor               *
+*************************************************/
+Fixed_Base_Power_Mod::Fixed_Base_Power_Mod(const BigInt& b, const BigInt& n,
+                                           Usage_Hints hints) :
+   Power_Mod(n, Usage_Hints(hints | BASE_IS_FIXED | choose_base_hints(b, n)))
+   {
+   set_base(b);
+   }
+
+}
diff --git a/src/math/pow_mod.h b/src/math/pow_mod.h
new file mode 100644
index 000000000..37e0871da
--- /dev/null
+++ b/src/math/pow_mod.h
@@ -0,0 +1,91 @@
+/*************************************************
+* Modular Exponentiator Header File              *
+* (C) 1999-2007 Jack Lloyd                       *
+*************************************************/
+
+#ifndef BOTAN_POWER_MOD_H__
+#define BOTAN_POWER_MOD_H__
+
+#include <botan/bigint.h>
+
+namespace Botan {
+
+/*************************************************
+* Modular Exponentiator Interface                *
+*************************************************/
+class BOTAN_DLL Modular_Exponentiator
+   {
+   public:
+      virtual void set_base(const BigInt&) = 0;
+      virtual void set_exponent(const BigInt&) = 0;
+      virtual BigInt execute() const = 0;
+      virtual Modular_Exponentiator* copy() const = 0;
+      virtual ~Modular_Exponentiator() {}
+   };
+
+/*************************************************
+* Modular Exponentiator Proxy                    *
+*************************************************/
+class BOTAN_DLL Power_Mod
+   {
+   public:
+      enum Usage_Hints {
+         NO_HINTS        = 0x0000,
+
+         BASE_IS_FIXED   = 0x0001,
+         BASE_IS_SMALL   = 0x0002,
+         BASE_IS_LARGE   = 0x0004,
+         BASE_IS_2       = 0x0008,
+
+         EXP_IS_FIXED    = 0x0100,
+         EXP_IS_SMALL    = 0x0200,
+         EXP_IS_LARGE    = 0x0400
+      };
+
+      void set_modulus(const BigInt&, Usage_Hints = NO_HINTS) const;
+      void set_base(const BigInt&) const;
+      void set_exponent(const BigInt&) const;
+
+      BigInt execute() const;
+
+      Power_Mod& operator=(const Power_Mod&);
+
+      Power_Mod(const BigInt& = 0, Usage_Hints = NO_HINTS);
+      Power_Mod(const Power_Mod&);
+      ~Power_Mod();
+   private:
+      mutable Modular_Exponentiator* core;
+      Usage_Hints hints;
+   };
+
+/*************************************************
+* Fixed Exponent Modular Exponentiator Proxy     *
+*************************************************/
+class BOTAN_DLL Fixed_Exponent_Power_Mod : public Power_Mod
+   {
+   public:
+      BigInt operator()(const BigInt& b) const
+         { set_base(b); return execute(); }
+
+      Fixed_Exponent_Power_Mod() {}
+      Fixed_Exponent_Power_Mod(const BigInt&, const BigInt&,
+                               Usage_Hints = NO_HINTS);
+   };
+
+/*************************************************
+* Fixed Base Modular Exponentiator Proxy         *
+*************************************************/
+class BOTAN_DLL Fixed_Base_Power_Mod : public Power_Mod
+   {
+   public:
+      BigInt operator()(const BigInt& e) const
+         { set_exponent(e); return execute(); }
+
+      Fixed_Base_Power_Mod() {}
+      Fixed_Base_Power_Mod(const BigInt&, const BigInt&,
+                           Usage_Hints = NO_HINTS);
+   };
+
+}
+
+#endif
diff --git a/src/math/powm_fw.cpp b/src/math/powm_fw.cpp
new file mode 100644
index 000000000..c29b9f311
--- /dev/null
+++ b/src/math/powm_fw.cpp
@@ -0,0 +1,102 @@
+/*************************************************
+* Fixed Window Exponentiation Source File        *
+* (C) 1999-2007 Jack Lloyd                       *
+*************************************************/
+
+#include <botan/def_powm.h>
+#include <botan/numthry.h>
+#include <vector>
+
+namespace Botan {
+
+namespace {
+
+/*************************************************
+* Try to choose a good window size               *
+*************************************************/
+u32bit choose_window_bits(u32bit exp_bits, u32bit,
+                          Power_Mod::Usage_Hints hints)
+   {
+   static const u32bit wsize[][2] = {
+      { 2048, 7 }, { 1024, 6 }, { 256, 5 }, { 128, 4 }, { 64, 3 }, { 0, 0 }
+   };
+
+   u32bit window_bits = 3;
+
+   if(exp_bits)
+      {
+      for(u32bit j = 0; wsize[j][0]; ++j)
+         {
+         if(exp_bits >= wsize[j][0])
+            {
+            window_bits += wsize[j][1];
+            break;
+            }
+         }
+      }
+
+   if(hints & Power_Mod::EXP_IS_FIXED)
+      window_bits += 2;
+   if(hints & Power_Mod::EXP_IS_LARGE)
+      window_bits += 2;
+   if(hints & Power_Mod::BASE_IS_FIXED)
+      ++window_bits;
+
+   return window_bits;
+   }
+
+}
+
+/*************************************************
+* Set the exponent                               *
+*************************************************/
+void Fixed_Window_Exponentiator::set_exponent(const BigInt& e)
+   {
+   exp = e;
+   }
+
+/*************************************************
+* Set the base                                   *
+*************************************************/
+void Fixed_Window_Exponentiator::set_base(const BigInt& base)
+   {
+   window_bits = choose_window_bits(exp.bits(), base.bits(), hints);
+
+   g.resize((1 << window_bits) - 1);
+   g[0] = base;
+   for(u32bit j = 1; j != g.size(); ++j)
+      g[j] = reducer.multiply(g[j-1], g[0]);
+   }
+
+/*************************************************
+* Compute the result                             *
+*************************************************/
+BigInt Fixed_Window_Exponentiator::execute() const
+   {
+   const u32bit exp_nibbles = (exp.bits() + window_bits - 1) / window_bits;
+
+   BigInt x = 1;
+   for(u32bit j = exp_nibbles; j > 0; --j)
+      {
+      for(u32bit k = 0; k != window_bits; ++k)
+         x = reducer.square(x);
+
+      u32bit nibble = exp.get_substring(window_bits*(j-1), window_bits);
+      if(nibble)
+         x = reducer.multiply(x, g[nibble-1]);
+      }
+   return x;
+   }
+
+/*************************************************
+* Fixed_Window_Exponentiator Constructor         *
+*************************************************/
+Fixed_Window_Exponentiator::Fixed_Window_Exponentiator(const BigInt& n,
+   Power_Mod::Usage_Hints hints)
+   {
+   reducer = Modular_Reducer(n);
+   this->hints = hints;
+   window_bits = 0;
+   }
+
+}
diff --git a/src/math/powm_mnt.cpp b/src/math/powm_mnt.cpp
new file mode 100644
index 000000000..6091d467a
--- /dev/null
+++ b/src/math/powm_mnt.cpp
@@ -0,0 +1,178 @@
+/*************************************************
+* Montgomery Exponentiation Source File          *
+* (C) 1999-2007 Jack Lloyd                       *
+*************************************************/
+
+#include <botan/def_powm.h>
+#include <botan/numthry.h>
+#include <botan/mp_core.h>
+
+namespace Botan {
+
+namespace {
+
+/*************************************************
+* Try to choose a good window size               *
+*************************************************/
+u32bit choose_window_bits(u32bit exp_bits, u32bit,
+                          Power_Mod::Usage_Hints hints)
+   {
+   static const u32bit wsize[][2] = {
+      { 2048, 4 }, { 1024, 3 }, { 256, 2 }, { 128, 1 }, { 0, 0 }
+   };
+
+   u32bit window_bits = 1;
+
+   if(exp_bits)
+      {
+      for(u32bit j = 0; wsize[j][0]; ++j)
+         {
+         if(exp_bits >= wsize[j][0])
+            {
+            window_bits += wsize[j][1];
+            break;
+            }
+         }
+      }
+
+   if(hints & Power_Mod::BASE_IS_FIXED)
+      window_bits += 2;
+   if(hints & Power_Mod::EXP_IS_LARGE)
+      ++window_bits;
+
+   return window_bits;
+   }
+
+/*************************************************
+* Montgomery Reduction                           *
+*************************************************/
+inline void montgomery_reduce(BigInt& out, MemoryRegion<word>& z_buf,
+                              const BigInt& x_bn, u32bit x_size, word u)
+   {
+   const word* x = x_bn.data();
+   word* z = z_buf.begin();
+   u32bit z_size = z_buf.size();
+
+   bigint_monty_redc(z, z_size, x, x_size, u);
+
+   out.get_reg().set(z + x_size, x_size + 1);
+   }
+
+}
+
+/*************************************************
+* Set the exponent                               *
+*************************************************/
+void Montgomery_Exponentiator::set_exponent(const BigInt& exp)
+   {
+   this->exp = exp;
+   exp_bits = exp.bits();
+   }
+
+/*************************************************
+* Set the base                                   *
+*************************************************/
+void Montgomery_Exponentiator::set_base(const BigInt& base)
+   {
+   window_bits = choose_window_bits(exp.bits(), base.bits(), hints);
+
+   g.resize((1 << window_bits) - 1);
+
+   SecureVector<word> z(2 * (mod_words + 1));
+   SecureVector<word> workspace(z.size());
+
+   g[0] = (base >= modulus) ? (base % modulus) : base;
+   bigint_mul(z.begin(), z.size(), workspace,
+              g[0].data(), g[0].size(), g[0].sig_words(),
+              R2.data(), R2.size(), R2.sig_words());
+
+   montgomery_reduce(g[0], z, modulus, mod_words, mod_prime);
+
+   const BigInt& x = g[0];
+   const u32bit x_sig = x.sig_words();
+
+   for(u32bit j = 1; j != g.size(); ++j)
+      {
+      const BigInt& y = g[j-1];
+      const u32bit y_sig = y.sig_words();
+
+      z.clear();
+      bigint_mul(z.begin(), z.size(), workspace,
+                 x.data(), x.size(), x_sig,
+                 y.data(), y.size(), y_sig);
+
+      montgomery_reduce(g[j], z, modulus, mod_words, mod_prime);
+      }
+   }
+
+/*************************************************
+* Compute the result                             *
+*************************************************/
+BigInt Montgomery_Exponentiator::execute() const
+   {
+   const u32bit exp_nibbles = (exp_bits + window_bits - 1) / window_bits;
+
+   BigInt x = R_mod;
+   SecureVector<word> z(2 * (mod_words + 1));
+   SecureVector<word> workspace(2 * (mod_words + 1));
+
+   for(u32bit j = exp_nibbles; j > 0; --j)
+      {
+      for(u32bit k = 0; k != window_bits; ++k)
+         {
+         z.clear();
+         bigint_sqr(z.begin(), z.size(), workspace,
+                    x.data(), x.size(), x.sig_words());
+
+         montgomery_reduce(x, z, modulus, mod_words, mod_prime);
+         }
+
+      u32bit nibble = exp.get_substring(window_bits*(j-1), window_bits);
+      if(nibble)
+         {
+         const BigInt& y = g[nibble-1];
+
+         z.clear();
+         bigint_mul(z.begin(), z.size(), workspace,
+                    x.data(), x.size(), x.sig_words(),
+                    y.data(), y.size(), y.sig_words());
+
+         montgomery_reduce(x, z, modulus, mod_words, mod_prime);
+         }
+      }
+
+   z.clear();
+   z.copy(x.data(), x.size());
+
+   montgomery_reduce(x, z, modulus, mod_words, mod_prime);
+   return x;
+   }
+
+/*************************************************
+* Montgomery_Exponentiator Constructor           *
+*************************************************/
+Montgomery_Exponentiator::Montgomery_Exponentiator(const BigInt& mod,
+   Power_Mod::Usage_Hints hints)
+   {
+   if(!mod.is_positive())
+      throw Exception("Montgomery_Exponentiator: modulus must be positive");
+   if(mod.is_even())
+      throw Exception("Montgomery_Exponentiator: modulus must be odd");
+
+   window_bits = 0;
+   this->hints = hints;
+   modulus = mod;
+
+   mod_words = modulus.sig_words();
+
+   BigInt mod_prime_bn(BigInt::Power2, MP_WORD_BITS);
+   mod_prime = (mod_prime_bn - inverse_mod(modulus, mod_prime_bn)).word_at(0);
+
+   R_mod = BigInt(BigInt::Power2, MP_WORD_BITS * mod_words);
+   R_mod %= modulus;
+
+   R2 = BigInt(BigInt::Power2, 2 * MP_WORD_BITS * mod_words);
+   R2 %= modulus;
+   }
+
+}
diff --git a/src/math/primes.cpp b/src/math/primes.cpp
new file mode 100644
index 000000000..d005167e5
--- /dev/null
+++ b/src/math/primes.cpp
@@ -0,0 +1,674 @@
+/*************************************************
+* Small Primes Table                             *
+* (C) 1999-2007 Jack Lloyd                       *
+*************************************************/
+
+#include <botan/numthry.h>
+
+namespace Botan {
+
+const u16bit PRIMES[PRIME_TABLE_SIZE+1] = {
+    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,
+ 8167,  8171,  8179,  8191,  8209,  8219,  8221,  8231,  8233,  8237,  8243,
+ 8263,  8269,  8273,  8287,  8291,  8293,  8297,  8311,  8317,  8329,  8353,
+ 8363,  8369,  8377,  8387,  8389,  8419,  8423,  8429,  8431,  8443,  8447,
+ 8461,  8467,  8501,  8513,  8521,  8527,  8537,  8539,  8543,  8563,  8573,
+ 8581,  8597,  8599,  8609,  8623,  8627,  8629,  8641,  8647,  8663,  8669,
+ 8677,  8681,  8689,  8693,  8699,  8707,  8713,  8719,  8731,  8737,  8741,
+ 8747,  8753,  8761,  8779,  8783,  8803,  8807,  8819,  8821,  8831,  8837,
+ 8839,  8849,  8861,  8863,  8867,  8887,  8893,  8923,  8929,  8933,  8941,
+ 8951,  8963,  8969,  8971,  8999,  9001,  9007,  9011,  9013,  9029,  9041,
+ 9043,  9049,  9059,  9067,  9091,  9103,  9109,  9127,  9133,  9137,  9151,
+ 9157,  9161,  9173,  9181,  9187,  9199,  9203,  9209,  9221,  9227,  9239,
+ 9241,  9257,  9277,  9281,  9283,  9293,  9311,  9319,  9323,  9337,  9341,
+ 9343,  9349,  9371,  9377,  9391,  9397,  9403,  9413,  9419,  9421,  9431,
+ 9433,  9437,  9439,  9461,  9463,  9467,  9473,  9479,  9491,  9497,  9511,
+ 9521,  9533,  9539,  9547,  9551,  9587,  9601,  9613,  9619,  9623,  9629,
+ 9631,  9643,  9649,  9661,  9677,  9679,  9689,  9697,  9719,  9721,  9733,
+ 9739,  9743,  9749,  9767,  9769,  9781,  9787,  9791,  9803,  9811,  9817,
+ 9829,  9833,  9839,  9851,  9857,  9859,  9871,  9883,  9887,  9901,  9907,
+ 9923,  9929,  9931,  9941,  9949,  9967,  9973, 10007, 10009, 10037, 10039,
+10061, 10067, 10069, 10079, 10091, 10093, 10099, 10103, 10111, 10133, 10139,
+10141, 10151, 10159, 10163, 10169, 10177, 10181, 10193, 10211, 10223, 10243,
+10247, 10253, 10259, 10267, 10271, 10273, 10289, 10301, 10303, 10313, 10321,
+10331, 10333, 10337, 10343, 10357, 10369, 10391, 10399, 10427, 10429, 10433,
+10453, 10457, 10459, 10463, 10477, 10487, 10499, 10501, 10513, 10529, 10531,
+10559, 10567, 10589, 10597, 10601, 10607, 10613, 10627, 10631, 10639, 10651,
+10657, 10663, 10667, 10687, 10691, 10709, 10711, 10723, 10729, 10733, 10739,
+10753, 10771, 10781, 10789, 10799, 10831, 10837, 10847, 10853, 10859, 10861,
+10867, 10883, 10889, 10891, 10903, 10909, 10937, 10939, 10949, 10957, 10973,
+10979, 10987, 10993, 11003, 11027, 11047, 11057, 11059, 11069, 11071, 11083,
+11087, 11093, 11113, 11117, 11119, 11131, 11149, 11159, 11161, 11171, 11173,
+11177, 11197, 11213, 11239, 11243, 11251, 11257, 11261, 11273, 11279, 11287,
+11299, 11311, 11317, 11321, 11329, 11351, 11353, 11369, 11383, 11393, 11399,
+11411, 11423, 11437, 11443, 11447, 11467, 11471, 11483, 11489, 11491, 11497,
+11503, 11519, 11527, 11549, 11551, 11579, 11587, 11593, 11597, 11617, 11621,
+11633, 11657, 11677, 11681, 11689, 11699, 11701, 11717, 11719, 11731, 11743,
+11777, 11779, 11783, 11789, 11801, 11807, 11813, 11821, 11827, 11831, 11833,
+11839, 11863, 11867, 11887, 11897, 11903, 11909, 11923, 11927, 11933, 11939,
+11941, 11953, 11959, 11969, 11971, 11981, 11987, 12007, 12011, 12037, 12041,
+12043, 12049, 12071, 12073, 12097, 12101, 12107, 12109, 12113, 12119, 12143,
+12149, 12157, 12161, 12163, 12197, 12203, 12211, 12227, 12239, 12241, 12251,
+12253, 12263, 12269, 12277, 12281, 12289, 12301, 12323, 12329, 12343, 12347,
+12373, 12377, 12379, 12391, 12401, 12409, 12413, 12421, 12433, 12437, 12451,
+12457, 12473, 12479, 12487, 12491, 12497, 12503, 12511, 12517, 12527, 12539,
+12541, 12547, 12553, 12569, 12577, 12583, 12589, 12601, 12611, 12613, 12619,
+12637, 12641, 12647, 12653, 12659, 12671, 12689, 12697, 12703, 12713, 12721,
+12739, 12743, 12757, 12763, 12781, 12791, 12799, 12809, 12821, 12823, 12829,
+12841, 12853, 12889, 12893, 12899, 12907, 12911, 12917, 12919, 12923, 12941,
+12953, 12959, 12967, 12973, 12979, 12983, 13001, 13003, 13007, 13009, 13033,
+13037, 13043, 13049, 13063, 13093, 13099, 13103, 13109, 13121, 13127, 13147,
+13151, 13159, 13163, 13171, 13177, 13183, 13187, 13217, 13219, 13229, 13241,
+13249, 13259, 13267, 13291, 13297, 13309, 13313, 13327, 13331, 13337, 13339,
+13367, 13381, 13397, 13399, 13411, 13417, 13421, 13441, 13451, 13457, 13463,
+13469, 13477, 13487, 13499, 13513, 13523, 13537, 13553, 13567, 13577, 13591,
+13597, 13613, 13619, 13627, 13633, 13649, 13669, 13679, 13681, 13687, 13691,
+13693, 13697, 13709, 13711, 13721, 13723, 13729, 13751, 13757, 13759, 13763,
+13781, 13789, 13799, 13807, 13829, 13831, 13841, 13859, 13873, 13877, 13879,
+13883, 13901, 13903, 13907, 13913, 13921, 13931, 13933, 13963, 13967, 13997,
+13999, 14009, 14011, 14029, 14033, 14051, 14057, 14071, 14081, 14083, 14087,
+14107, 14143, 14149, 14153, 14159, 14173, 14177, 14197, 14207, 14221, 14243,
+14249, 14251, 14281, 14293, 14303, 14321, 14323, 14327, 14341, 14347, 14369,
+14387, 14389, 14401, 14407, 14411, 14419, 14423, 14431, 14437, 14447, 14449,
+14461, 14479, 14489, 14503, 14519, 14533, 14537, 14543, 14549, 14551, 14557,
+14561, 14563, 14591, 14593, 14621, 14627, 14629, 14633, 14639, 14653, 14657,
+14669, 14683, 14699, 14713, 14717, 14723, 14731, 14737, 14741, 14747, 14753,
+14759, 14767, 14771, 14779, 14783, 14797, 14813, 14821, 14827, 14831, 14843,
+14851, 14867, 14869, 14879, 14887, 14891, 14897, 14923, 14929, 14939, 14947,
+14951, 14957, 14969, 14983, 15013, 15017, 15031, 15053, 15061, 15073, 15077,
+15083, 15091, 15101, 15107, 15121, 15131, 15137, 15139, 15149, 15161, 15173,
+15187, 15193, 15199, 15217, 15227, 15233, 15241, 15259, 15263, 15269, 15271,
+15277, 15287, 15289, 15299, 15307, 15313, 15319, 15329, 15331, 15349, 15359,
+15361, 15373, 15377, 15383, 15391, 15401, 15413, 15427, 15439, 15443, 15451,
+15461, 15467, 15473, 15493, 15497, 15511, 15527, 15541, 15551, 15559, 15569,
+15581, 15583, 15601, 15607, 15619, 15629, 15641, 15643, 15647, 15649, 15661,
+15667, 15671, 15679, 15683, 15727, 15731, 15733, 15737, 15739, 15749, 15761,
+15767, 15773, 15787, 15791, 15797, 15803, 15809, 15817, 15823, 15859, 15877,
+15881, 15887, 15889, 15901, 15907, 15913, 15919, 15923, 15937, 15959, 15971,
+15973, 15991, 16001, 16007, 16033, 16057, 16061, 16063, 16067, 16069, 16073,
+16087, 16091, 16097, 16103, 16111, 16127, 16139, 16141, 16183, 16187, 16189,
+16193, 16217, 16223, 16229, 16231, 16249, 16253, 16267, 16273, 16301, 16319,
+16333, 16339, 16349, 16361, 16363, 16369, 16381, 16411, 16417, 16421, 16427,
+16433, 16447, 16451, 16453, 16477, 16481, 16487, 16493, 16519, 16529, 16547,
+16553, 16561, 16567, 16573, 16603, 16607, 16619, 16631, 16633, 16649, 16651,
+16657, 16661, 16673, 16691, 16693, 16699, 16703, 16729, 16741, 16747, 16759,
+16763, 16787, 16811, 16823, 16829, 16831, 16843, 16871, 16879, 16883, 16889,
+16901, 16903, 16921, 16927, 16931, 16937, 16943, 16963, 16979, 16981, 16987,
+16993, 17011, 17021, 17027, 17029, 17033, 17041, 17047, 17053, 17077, 17093,
+17099, 17107, 17117, 17123, 17137, 17159, 17167, 17183, 17189, 17191, 17203,
+17207, 17209, 17231, 17239, 17257, 17291, 17293, 17299, 17317, 17321, 17327,
+17333, 17341, 17351, 17359, 17377, 17383, 17387, 17389, 17393, 17401, 17417,
+17419, 17431, 17443, 17449, 17467, 17471, 17477, 17483, 17489, 17491, 17497,
+17509, 17519, 17539, 17551, 17569, 17573, 17579, 17581, 17597, 17599, 17609,
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+43391, 43397, 43399, 43403, 43411, 43427, 43441, 43451, 43457, 43481, 43487,
+43499, 43517, 43541, 43543, 43573, 43577, 43579, 43591, 43597, 43607, 43609,
+43613, 43627, 43633, 43649, 43651, 43661, 43669, 43691, 43711, 43717, 43721,
+43753, 43759, 43777, 43781, 43783, 43787, 43789, 43793, 43801, 43853, 43867,
+43889, 43891, 43913, 43933, 43943, 43951, 43961, 43963, 43969, 43973, 43987,
+43991, 43997, 44017, 44021, 44027, 44029, 44041, 44053, 44059, 44071, 44087,
+44089, 44101, 44111, 44119, 44123, 44129, 44131, 44159, 44171, 44179, 44189,
+44201, 44203, 44207, 44221, 44249, 44257, 44263, 44267, 44269, 44273, 44279,
+44281, 44293, 44351, 44357, 44371, 44381, 44383, 44389, 44417, 44449, 44453,
+44483, 44491, 44497, 44501, 44507, 44519, 44531, 44533, 44537, 44543, 44549,
+44563, 44579, 44587, 44617, 44621, 44623, 44633, 44641, 44647, 44651, 44657,
+44683, 44687, 44699, 44701, 44711, 44729, 44741, 44753, 44771, 44773, 44777,
+44789, 44797, 44809, 44819, 44839, 44843, 44851, 44867, 44879, 44887, 44893,
+44909, 44917, 44927, 44939, 44953, 44959, 44963, 44971, 44983, 44987, 45007,
+45013, 45053, 45061, 45077, 45083, 45119, 45121, 45127, 45131, 45137, 45139,
+45161, 45179, 45181, 45191, 45197, 45233, 45247, 45259, 45263, 45281, 45289,
+45293, 45307, 45317, 45319, 45329, 45337, 45341, 45343, 45361, 45377, 45389,
+45403, 45413, 45427, 45433, 45439, 45481, 45491, 45497, 45503, 45523, 45533,
+45541, 45553, 45557, 45569, 45587, 45589, 45599, 45613, 45631, 45641, 45659,
+45667, 45673, 45677, 45691, 45697, 45707, 45737, 45751, 45757, 45763, 45767,
+45779, 45817, 45821, 45823, 45827, 45833, 45841, 45853, 45863, 45869, 45887,
+45893, 45943, 45949, 45953, 45959, 45971, 45979, 45989, 46021, 46027, 46049,
+46051, 46061, 46073, 46091, 46093, 46099, 46103, 46133, 46141, 46147, 46153,
+46171, 46181, 46183, 46187, 46199, 46219, 46229, 46237, 46261, 46271, 46273,
+46279, 46301, 46307, 46309, 46327, 46337, 46349, 46351, 46381, 46399, 46411,
+46439, 46441, 46447, 46451, 46457, 46471, 46477, 46489, 46499, 46507, 46511,
+46523, 46549, 46559, 46567, 46573, 46589, 46591, 46601, 46619, 46633, 46639,
+46643, 46649, 46663, 46679, 46681, 46687, 46691, 46703, 46723, 46727, 46747,
+46751, 46757, 46769, 46771, 46807, 46811, 46817, 46819, 46829, 46831, 46853,
+46861, 46867, 46877, 46889, 46901, 46919, 46933, 46957, 46993, 46997, 47017,
+47041, 47051, 47057, 47059, 47087, 47093, 47111, 47119, 47123, 47129, 47137,
+47143, 47147, 47149, 47161, 47189, 47207, 47221, 47237, 47251, 47269, 47279,
+47287, 47293, 47297, 47303, 47309, 47317, 47339, 47351, 47353, 47363, 47381,
+47387, 47389, 47407, 47417, 47419, 47431, 47441, 47459, 47491, 47497, 47501,
+47507, 47513, 47521, 47527, 47533, 47543, 47563, 47569, 47581, 47591, 47599,
+47609, 47623, 47629, 47639, 47653, 47657, 47659, 47681, 47699, 47701, 47711,
+47713, 47717, 47737, 47741, 47743, 47777, 47779, 47791, 47797, 47807, 47809,
+47819, 47837, 47843, 47857, 47869, 47881, 47903, 47911, 47917, 47933, 47939,
+47947, 47951, 47963, 47969, 47977, 47981, 48017, 48023, 48029, 48049, 48073,
+48079, 48091, 48109, 48119, 48121, 48131, 48157, 48163, 48179, 48187, 48193,
+48197, 48221, 48239, 48247, 48259, 48271, 48281, 48299, 48311, 48313, 48337,
+48341, 48353, 48371, 48383, 48397, 48407, 48409, 48413, 48437, 48449, 48463,
+48473, 48479, 48481, 48487, 48491, 48497, 48523, 48527, 48533, 48539, 48541,
+48563, 48571, 48589, 48593, 48611, 48619, 48623, 48647, 48649, 48661, 48673,
+48677, 48679, 48731, 48733, 48751, 48757, 48761, 48767, 48779, 48781, 48787,
+48799, 48809, 48817, 48821, 48823, 48847, 48857, 48859, 48869, 48871, 48883,
+48889, 48907, 48947, 48953, 48973, 48989, 48991, 49003, 49009, 49019, 49031,
+49033, 49037, 49043, 49057, 49069, 49081, 49103, 49109, 49117, 49121, 49123,
+49139, 49157, 49169, 49171, 49177, 49193, 49199, 49201, 49207, 49211, 49223,
+49253, 49261, 49277, 49279, 49297, 49307, 49331, 49333, 49339, 49363, 49367,
+49369, 49391, 49393, 49409, 49411, 49417, 49429, 49433, 49451, 49459, 49463,
+49477, 49481, 49499, 49523, 49529, 49531, 49537, 49547, 49549, 49559, 49597,
+49603, 49613, 49627, 49633, 49639, 49663, 49667, 49669, 49681, 49697, 49711,
+49727, 49739, 49741, 49747, 49757, 49783, 49787, 49789, 49801, 49807, 49811,
+49823, 49831, 49843, 49853, 49871, 49877, 49891, 49919, 49921, 49927, 49937,
+49939, 49943, 49957, 49991, 49993, 49999, 50021, 50023, 50033, 50047, 50051,
+50053, 50069, 50077, 50087, 50093, 50101, 50111, 50119, 50123, 50129, 50131,
+50147, 50153, 50159, 50177, 50207, 50221, 50227, 50231, 50261, 50263, 50273,
+50287, 50291, 50311, 50321, 50329, 50333, 50341, 50359, 50363, 50377, 50383,
+50387, 50411, 50417, 50423, 50441, 50459, 50461, 50497, 50503, 50513, 50527,
+50539, 50543, 50549, 50551, 50581, 50587, 50591, 50593, 50599, 50627, 50647,
+50651, 50671, 50683, 50707, 50723, 50741, 50753, 50767, 50773, 50777, 50789,
+50821, 50833, 50839, 50849, 50857, 50867, 50873, 50891, 50893, 50909, 50923,
+50929, 50951, 50957, 50969, 50971, 50989, 50993, 51001, 51031, 51043, 51047,
+51059, 51061, 51071, 51109, 51131, 51133, 51137, 51151, 51157, 51169, 51193,
+51197, 51199, 51203, 51217, 51229, 51239, 51241, 51257, 51263, 51283, 51287,
+51307, 51329, 51341, 51343, 51347, 51349, 51361, 51383, 51407, 51413, 51419,
+51421, 51427, 51431, 51437, 51439, 51449, 51461, 51473, 51479, 51481, 51487,
+51503, 51511, 51517, 51521, 51539, 51551, 51563, 51577, 51581, 51593, 51599,
+51607, 51613, 51631, 51637, 51647, 51659, 51673, 51679, 51683, 51691, 51713,
+51719, 51721, 51749, 51767, 51769, 51787, 51797, 51803, 51817, 51827, 51829,
+51839, 51853, 51859, 51869, 51871, 51893, 51899, 51907, 51913, 51929, 51941,
+51949, 51971, 51973, 51977, 51991, 52009, 52021, 52027, 52051, 52057, 52067,
+52069, 52081, 52103, 52121, 52127, 52147, 52153, 52163, 52177, 52181, 52183,
+52189, 52201, 52223, 52237, 52249, 52253, 52259, 52267, 52289, 52291, 52301,
+52313, 52321, 52361, 52363, 52369, 52379, 52387, 52391, 52433, 52453, 52457,
+52489, 52501, 52511, 52517, 52529, 52541, 52543, 52553, 52561, 52567, 52571,
+52579, 52583, 52609, 52627, 52631, 52639, 52667, 52673, 52691, 52697, 52709,
+52711, 52721, 52727, 52733, 52747, 52757, 52769, 52783, 52807, 52813, 52817,
+52837, 52859, 52861, 52879, 52883, 52889, 52901, 52903, 52919, 52937, 52951,
+52957, 52963, 52967, 52973, 52981, 52999, 53003, 53017, 53047, 53051, 53069,
+53077, 53087, 53089, 53093, 53101, 53113, 53117, 53129, 53147, 53149, 53161,
+53171, 53173, 53189, 53197, 53201, 53231, 53233, 53239, 53267, 53269, 53279,
+53281, 53299, 53309, 53323, 53327, 53353, 53359, 53377, 53381, 53401, 53407,
+53411, 53419, 53437, 53441, 53453, 53479, 53503, 53507, 53527, 53549, 53551,
+53569, 53591, 53593, 53597, 53609, 53611, 53617, 53623, 53629, 53633, 53639,
+53653, 53657, 53681, 53693, 53699, 53717, 53719, 53731, 53759, 53773, 53777,
+53783, 53791, 53813, 53819, 53831, 53849, 53857, 53861, 53881, 53887, 53891,
+53897, 53899, 53917, 53923, 53927, 53939, 53951, 53959, 53987, 53993, 54001,
+54011, 54013, 54037, 54049, 54059, 54083, 54091, 54101, 54121, 54133, 54139,
+54151, 54163, 54167, 54181, 54193, 54217, 54251, 54269, 54277, 54287, 54293,
+54311, 54319, 54323, 54331, 54347, 54361, 54367, 54371, 54377, 54401, 54403,
+54409, 54413, 54419, 54421, 54437, 54443, 54449, 54469, 54493, 54497, 54499,
+54503, 54517, 54521, 54539, 54541, 54547, 54559, 54563, 54577, 54581, 54583,
+54601, 54617, 54623, 54629, 54631, 54647, 54667, 54673, 54679, 54709, 54713,
+54721, 54727, 54751, 54767, 54773, 54779, 54787, 54799, 54829, 54833, 54851,
+54869, 54877, 54881, 54907, 54917, 54919, 54941, 54949, 54959, 54973, 54979,
+54983, 55001, 55009, 55021, 55049, 55051, 55057, 55061, 55073, 55079, 55103,
+55109, 55117, 55127, 55147, 55163, 55171, 55201, 55207, 55213, 55217, 55219,
+55229, 55243, 55249, 55259, 55291, 55313, 55331, 55333, 55337, 55339, 55343,
+55351, 55373, 55381, 55399, 55411, 55439, 55441, 55457, 55469, 55487, 55501,
+55511, 55529, 55541, 55547, 55579, 55589, 55603, 55609, 55619, 55621, 55631,
+55633, 55639, 55661, 55663, 55667, 55673, 55681, 55691, 55697, 55711, 55717,
+55721, 55733, 55763, 55787, 55793, 55799, 55807, 55813, 55817, 55819, 55823,
+55829, 55837, 55843, 55849, 55871, 55889, 55897, 55901, 55903, 55921, 55927,
+55931, 55933, 55949, 55967, 55987, 55997, 56003, 56009, 56039, 56041, 56053,
+56081, 56087, 56093, 56099, 56101, 56113, 56123, 56131, 56149, 56167, 56171,
+56179, 56197, 56207, 56209, 56237, 56239, 56249, 56263, 56267, 56269, 56299,
+56311, 56333, 56359, 56369, 56377, 56383, 56393, 56401, 56417, 56431, 56437,
+56443, 56453, 56467, 56473, 56477, 56479, 56489, 56501, 56503, 56509, 56519,
+56527, 56531, 56533, 56543, 56569, 56591, 56597, 56599, 56611, 56629, 56633,
+56659, 56663, 56671, 56681, 56687, 56701, 56711, 56713, 56731, 56737, 56747,
+56767, 56773, 56779, 56783, 56807, 56809, 56813, 56821, 56827, 56843, 56857,
+56873, 56891, 56893, 56897, 56909, 56911, 56921, 56923, 56929, 56941, 56951,
+56957, 56963, 56983, 56989, 56993, 56999, 57037, 57041, 57047, 57059, 57073,
+57077, 57089, 57097, 57107, 57119, 57131, 57139, 57143, 57149, 57163, 57173,
+57179, 57191, 57193, 57203, 57221, 57223, 57241, 57251, 57259, 57269, 57271,
+57283, 57287, 57301, 57329, 57331, 57347, 57349, 57367, 57373, 57383, 57389,
+57397, 57413, 57427, 57457, 57467, 57487, 57493, 57503, 57527, 57529, 57557,
+57559, 57571, 57587, 57593, 57601, 57637, 57641, 57649, 57653, 57667, 57679,
+57689, 57697, 57709, 57713, 57719, 57727, 57731, 57737, 57751, 57773, 57781,
+57787, 57791, 57793, 57803, 57809, 57829, 57839, 57847, 57853, 57859, 57881,
+57899, 57901, 57917, 57923, 57943, 57947, 57973, 57977, 57991, 58013, 58027,
+58031, 58043, 58049, 58057, 58061, 58067, 58073, 58099, 58109, 58111, 58129,
+58147, 58151, 58153, 58169, 58171, 58189, 58193, 58199, 58207, 58211, 58217,
+58229, 58231, 58237, 58243, 58271, 58309, 58313, 58321, 58337, 58363, 58367,
+58369, 58379, 58391, 58393, 58403, 58411, 58417, 58427, 58439, 58441, 58451,
+58453, 58477, 58481, 58511, 58537, 58543, 58549, 58567, 58573, 58579, 58601,
+58603, 58613, 58631, 58657, 58661, 58679, 58687, 58693, 58699, 58711, 58727,
+58733, 58741, 58757, 58763, 58771, 58787, 58789, 58831, 58889, 58897, 58901,
+58907, 58909, 58913, 58921, 58937, 58943, 58963, 58967, 58979, 58991, 58997,
+59009, 59011, 59021, 59023, 59029, 59051, 59053, 59063, 59069, 59077, 59083,
+59093, 59107, 59113, 59119, 59123, 59141, 59149, 59159, 59167, 59183, 59197,
+59207, 59209, 59219, 59221, 59233, 59239, 59243, 59263, 59273, 59281, 59333,
+59341, 59351, 59357, 59359, 59369, 59377, 59387, 59393, 59399, 59407, 59417,
+59419, 59441, 59443, 59447, 59453, 59467, 59471, 59473, 59497, 59509, 59513,
+59539, 59557, 59561, 59567, 59581, 59611, 59617, 59621, 59627, 59629, 59651,
+59659, 59663, 59669, 59671, 59693, 59699, 59707, 59723, 59729, 59743, 59747,
+59753, 59771, 59779, 59791, 59797, 59809, 59833, 59863, 59879, 59887, 59921,
+59929, 59951, 59957, 59971, 59981, 59999, 60013, 60017, 60029, 60037, 60041,
+60077, 60083, 60089, 60091, 60101, 60103, 60107, 60127, 60133, 60139, 60149,
+60161, 60167, 60169, 60209, 60217, 60223, 60251, 60257, 60259, 60271, 60289,
+60293, 60317, 60331, 60337, 60343, 60353, 60373, 60383, 60397, 60413, 60427,
+60443, 60449, 60457, 60493, 60497, 60509, 60521, 60527, 60539, 60589, 60601,
+60607, 60611, 60617, 60623, 60631, 60637, 60647, 60649, 60659, 60661, 60679,
+60689, 60703, 60719, 60727, 60733, 60737, 60757, 60761, 60763, 60773, 60779,
+60793, 60811, 60821, 60859, 60869, 60887, 60889, 60899, 60901, 60913, 60917,
+60919, 60923, 60937, 60943, 60953, 60961, 61001, 61007, 61027, 61031, 61043,
+61051, 61057, 61091, 61099, 61121, 61129, 61141, 61151, 61153, 61169, 61211,
+61223, 61231, 61253, 61261, 61283, 61291, 61297, 61331, 61333, 61339, 61343,
+61357, 61363, 61379, 61381, 61403, 61409, 61417, 61441, 61463, 61469, 61471,
+61483, 61487, 61493, 61507, 61511, 61519, 61543, 61547, 61553, 61559, 61561,
+61583, 61603, 61609, 61613, 61627, 61631, 61637, 61643, 61651, 61657, 61667,
+61673, 61681, 61687, 61703, 61717, 61723, 61729, 61751, 61757, 61781, 61813,
+61819, 61837, 61843, 61861, 61871, 61879, 61909, 61927, 61933, 61949, 61961,
+61967, 61979, 61981, 61987, 61991, 62003, 62011, 62017, 62039, 62047, 62053,
+62057, 62071, 62081, 62099, 62119, 62129, 62131, 62137, 62141, 62143, 62171,
+62189, 62191, 62201, 62207, 62213, 62219, 62233, 62273, 62297, 62299, 62303,
+62311, 62323, 62327, 62347, 62351, 62383, 62401, 62417, 62423, 62459, 62467,
+62473, 62477, 62483, 62497, 62501, 62507, 62533, 62539, 62549, 62563, 62581,
+62591, 62597, 62603, 62617, 62627, 62633, 62639, 62653, 62659, 62683, 62687,
+62701, 62723, 62731, 62743, 62753, 62761, 62773, 62791, 62801, 62819, 62827,
+62851, 62861, 62869, 62873, 62897, 62903, 62921, 62927, 62929, 62939, 62969,
+62971, 62981, 62983, 62987, 62989, 63029, 63031, 63059, 63067, 63073, 63079,
+63097, 63103, 63113, 63127, 63131, 63149, 63179, 63197, 63199, 63211, 63241,
+63247, 63277, 63281, 63299, 63311, 63313, 63317, 63331, 63337, 63347, 63353,
+63361, 63367, 63377, 63389, 63391, 63397, 63409, 63419, 63421, 63439, 63443,
+63463, 63467, 63473, 63487, 63493, 63499, 63521, 63527, 63533, 63541, 63559,
+63577, 63587, 63589, 63599, 63601, 63607, 63611, 63617, 63629, 63647, 63649,
+63659, 63667, 63671, 63689, 63691, 63697, 63703, 63709, 63719, 63727, 63737,
+63743, 63761, 63773, 63781, 63793, 63799, 63803, 63809, 63823, 63839, 63841,
+63853, 63857, 63863, 63901, 63907, 63913, 63929, 63949, 63977, 63997, 64007,
+64013, 64019, 64033, 64037, 64063, 64067, 64081, 64091, 64109, 64123, 64151,
+64153, 64157, 64171, 64187, 64189, 64217, 64223, 64231, 64237, 64271, 64279,
+64283, 64301, 64303, 64319, 64327, 64333, 64373, 64381, 64399, 64403, 64433,
+64439, 64451, 64453, 64483, 64489, 64499, 64513, 64553, 64567, 64577, 64579,
+64591, 64601, 64609, 64613, 64621, 64627, 64633, 64661, 64663, 64667, 64679,
+64693, 64709, 64717, 64747, 64763, 64781, 64783, 64793, 64811, 64817, 64849,
+64853, 64871, 64877, 64879, 64891, 64901, 64919, 64921, 64927, 64937, 64951,
+64969, 64997, 65003, 65011, 65027, 65029, 65033, 65053, 65063, 65071, 65089,
+65099, 65101, 65111, 65119, 65123, 65129, 65141, 65147, 65167, 65171, 65173,
+65179, 65183, 65203, 65213, 65239, 65257, 65267, 65269, 65287, 65293, 65309,
+65323, 65327, 65353, 65357, 65371, 65381, 65393, 65407, 65413, 65419, 65423,
+65437, 65447, 65449, 65479, 65497, 65519, 65521, 0 };
+
+const u64bit PRIME_PRODUCTS[PRIME_PRODUCTS_TABLE_SIZE] = {
+0xFF658BDE2F2A43DF, 0xFEEB94CD535119ED, 0xFA921839EC24DDD5, 0xFDDA766C77E1E605,
+0xFF3024B0EB4EE333, 0xFEEE350BBC92F4DF, 0xFFC724B7D011D01B, 0xFEED34B826C33B05,
+0xFE69D8DE3F85C6E3, 0xFE3B48909250918F, 0xFF8EC0CE9C632429, 0xFFD92A5C78226D6B,
+0xFFB4BFB0C65133CF, 0xFE77113704902C57, 0xFF8A21D222EA81FD, 0xFEDA1299661CF5AB,
+0xFF4CE86187737D0D, 0xFFD26443A07F519D, 0xFFA817B7191D7967, 0xFF00EDC142868873,
+0xFFB9C6D7F7A239B7, 0xFFE76D3481E98E39, 0xFF76D5432584120D, 0xFFAA499F071EC705,
+0xFEB5198F05722E59, 0xFF7E0431CA41107F, 0xFFCFD52FEDDC928F, 0xFE0EA42537BC6ABF,
+0xFF64937896876925, 0xFC6FC87E811607D3, 0xFFBF600E6CDD0F4F, 0xFF022700FE658243,
+0xFF2E21166779D6B9, 0xFFC224624C665C33, 0xFF1372F41FF177AD, 0xFF31E57E972D0C13,
+0xFFA891F866404D23, 0xFF7BF13EF716E9A3, 0xFE51CAFD9466E733, 0xFDA1CF55F6D6336F,
+0xFFAF6C040ED0950F, 0xFFAA1725F40BA269, 0xFEC593BC3570BEEB, 0xFEE05B35B426F413,
+0xFFCA5209A08890F9, 0xFFED8AF70EB0CC89, 0xFF3F98E3E27860A5, 0xFF92FECD017FF9F7,
+0xFEFA655B2609018F, 0xFFFC51D15AAC7B77, 0xFEF5007E71420DB1, 0xFFEC4784141332D1,
+0xFE8384ED4E1D21CD, 0xFFD3FF614D3ECC47, 0xFFDE5166FD540313, 0xFF5320ECED04B26F,
+0xFF223980F122FF75, 0xFF19C1F27CB1B4A5, 0xFF0F1DFC9DA9523B, 0xFF82DE7B387F5427,
+0xFF9A026BA87314E3, 0xFFAC7FF3ACE64E77, 0xFF808EB2FD5873C3, 0xFE983ED5BB363301,
+0xFF714856DB2CFE95, 0xFF84E1510CF3EB9F, 0xFF29D04C1DA0B115, 0xFFBCF3BF9433552F,
+0xFF32203D58A4C473, 0xFFF00910A15021C3, 0xFDE93041F28240AD, 0xFFC518BCD81C03C5,
+0xFEF504CD8BB9CBDD, 0xFEB8FFBFFF116A6B, 0xFF7642E0785ADA23, 0xFFECF068800FD50D,
+0xFFD703577CA247A7, 0xFF54C0ECAD2C9691, 0xFFC031706B8C72F5, 0xFFE59E5CA58BBDF5,
+0xFFF31FAFFD3B331D, 0xFF64DDF32349FF6D, 0xFFE38309D0BD4A51, 0xFF8C934F76B3C737,
+0xFFDC80B4BAEAFC1F, 0xFFCC1FE4C856FBD9, 0xFFDB5976DDF601FD, 0xFFD3DD25F424433D,
+0xFFC00FA367E746C7, 0xFFE08BF011CC854F, 0xFFC3F21982468F6D, 0xFFDA6C52478A76DF,
+0xFFC67D95AADED363, 0xFFD605D18C3AFC65, 0xFFE828C9D698F1DF, 0xFFBE5098D83B7737,
+0xFF79EB34474ABFB9, 0xFFD27AEED0786363, 0xFFD0FE27B77C271F, 0xFFFBB6563BD065EF,
+0xFFF3638F8635E1EB, 0xFFBE862C22C9F065, 0xFF44712D8488A01D, 0xFF7EEC97F9913111,
+0xFFC23CC78CB12AB1, 0xFFF390FE85F81D3D, 0xFFE8EA21A0FB9931, 0xFFB9D42D17A93385,
+0xFFCDB63AB21E904D, 0xFF5EB7F2210D33DF, 0xFFE6F6C7BB60C9DF, 0xFFAD4CA8DC26D699,
+0xFF7BE75BD21DCA51, 0xFEF89CE23CB61789, 0xFF40ECA3CCA22CE5, 0xF52BDF080F7ABA6F,
+0xEC8F38C8B28E0493, 0xE68E732A2ABED62F, 0xE21A13779E0CCDC7, 0xD823C075C325191B,
+0xD1B284C91EED248B, 0xCBA5A08068E8C1F7, 0xC483EE5A2228985D, 0xBCAEE9F787AC75EB,
+0xB782DAB1B77D3E09, 0xB0D77226F15E387B, 0xAA2A8727D47941CD, 0xA4A45682E9CE533D,
+0x9CAF15AF4CE7FCF7, 0x94C051DD15537305, 0x9006D2FBD933A297, 0x8C4DED05F19B7399,
+0x884FD7A270AD1B1B, 0x83C687D33F238D4B, 0xFF62E2BAE50C6C16, 0x7A59E1FD9D203DBB,
+0x764F1DC07B0E442D, 0x72732FE1F2023153, 0x6E373B550764872F, 0x680FFFD267C5F3FF,
+0x6206BFEC14F1CFC5, 0x5FA6F70CFD587265, 0x5CC7A1B4F6DF9823, 0x599291B29311407F,
+0xFF3CEBD359B67EF9, 0x51C573C14F289F6D, 0x4FA265B31B73C6DF, 0x4B3154ACBD077DED,
+0x4785C96B29A1E437, 0x451F887F646CF763, 0x429DC254C5490571, 0x408410840EAE2883,
+0x3E12CC83606624F3, 0x3A70D774B821DA71, 0x37A21449A196A825, 0x34C5D056E2278B81,
+0xFA0C6CAB29D8E297, 0x2FA5AEC982A5972B, 0x2D6831749426068F, 0x2B7F876418155CA7,
+0x2A1B897ED2AB433D, 0x28C9430D0F92132F, 0x26DF879EBF12E103, 0xFD2FAB4CA364D43B,
+0x22B5B4FC40D4C35F, 0x209298AA84D7E6A1, 0x1ED4B9F11445F1E7, 0x1DC6D2DD416CC91D,
+0x1C1517A52E37C3EF, 0x1A808916125AEF2F, 0x197A2FB2938FF13D, 0x1814AA6C087B561D,
+0xFB3B173E72947609, 0x1571187A8E3D4D6B, 0x13D306D29263C139, 0xF8AEC6ADA137E865,
+0x123EA204BAB48731, 0x11012099D202F297, 0x10290E15797C21BD, 0x0F3AB38E679D6317,
+0xF50B5505D593FCF9, 0xFF23754F7F2052B5, 0x0CC52D96BC2E5A2D, 0x0BF80EAD87B228E5,
+0x0B59A623082C9171, 0xF44E28B9221A433B, 0x09DB5CDD2505EABD, 0x09638C123BCAB351,
+0xFDB9AE6935254CD3, 0xFFE30D7E4F02F163, 0x07CF1FC053B9C61F, 0x0789244FF1705821,
+0x06FBD05649B0B9C7, 0xEF9713EC6A0C250B, 0xF47691AD6AA9F0DB, 0xF2A8EB02CB08CA51,
+0xF9559D40380A20E1, 0x04E15138A5B9BF43, 0xEECD739EA48F3ABB, 0xF76E7E7530574E79,
+0xF8393D2E42D7D277, 0xF666F9AD3A16D173, 0xF403C629749F3ED5, 0xFBD7EC45F220A473,
+0xFA8AFF7491B234FD, 0xFF471CE534D1F537, 0xF4BEBFDD9C54CEC9, 0xDD04722310A6CE9D,
+0xFD8071236214FA05, 0xFBCA07B399A482DD, 0xFD9642C104864C17, 0xFA525105AADEFA39,
+0xF71122156406E645, 0xFF415FDFD1247539, 0xFB709936F52446AF, 0xFF7734CCB806CDA7,
+0xF801E9A88CD3D70D, 0xFC0C00AC9BCC5491, 0xFF462CD8E52ED221, 0xFC97426300FCE331,
+0xFEB3049C5E37A059, 0xFFFC8AB1E05051CD, 0xFE5F4621F2D9FE63, 0xFE931DB54FC5D521,
+0xFFDE43D960FE42A5, 0xFFDBFAD1B802BDB5, 0xFF23C485F6B7BF53, 0xFFC98F169C8DF21B,
+0xF1609D0E2E564D01, 0xCB10B976C333834B, 0x9B52037A38DAB8F9, 0x800E88FF5E929095,
+0x55A9AD1C21F5E173, 0x3D1A64E4E555D699, 0x2A5D1D73694F7B93, 0x198F4260D8807623,
+0x140D45BB525C35EB, 0x102F4743FF914EEB, 0x0CB114936A734FBF, 0x096D97150B7B0A71,
+0x06F06B90B850C2E5, 0x053B17A0D7F7386B, 0xE3AD1CE3C82FE6A5, 0xDAE968B4B710E857,
+0xFA2DC15B2C96BE77, 0xF1FF5F22AF135BD9, 0xFC65C5CAAA878A13, 0xFB9427EB08CF9C11,
+0xCCB12B6FEBFE285D, 0x5BAADA462B48F999, 0x2E53167EC64B703B, 0x1264ED670CD61961,
+0x071F216A9AB74E2D, 0xEE26503C1266CE55, 0x4C6004C7E404E4B5, 0xCB649E41ECE95F85
+};
+
+}
diff --git a/src/math/reducer.cpp b/src/math/reducer.cpp
new file mode 100644
index 000000000..47c5c20fc
--- /dev/null
+++ b/src/math/reducer.cpp
@@ -0,0 +1,95 @@
+/*************************************************
+* Modular Reducer Source File                    *
+* (C) 1999-2007 Jack Lloyd                       *
+*************************************************/
+
+#include <botan/reducer.h>
+#include <botan/numthry.h>
+#include <botan/mp_core.h>
+
+namespace Botan {
+
+/*************************************************
+* Modular_Reducer Constructor                    *
+*************************************************/
+Modular_Reducer::Modular_Reducer(const BigInt& mod)
+   {
+   if(mod <= 0)
+      throw Invalid_Argument("Modular_Reducer: modulus must be positive");
+
+   modulus = mod;
+   mod_words = modulus.sig_words();
+
+   modulus_2 = Botan::square(modulus);
+   mod2_words = modulus_2.sig_words();
+
+   mu = BigInt(BigInt::Power2, 2 * MP_WORD_BITS * mod_words) / modulus;
+   mu_words = mu.sig_words();
+   }
+
+/*************************************************
+* Barrett Reduction                              *
+*************************************************/
+BigInt Modular_Reducer::reduce(const BigInt& x) const
+   {
+   if(mod_words == 0)
+      throw Invalid_State("Modular_Reducer: Never initalized");
+
+   BigInt t1 = x;
+   t1.set_sign(BigInt::Positive);
+
+   if(t1 < modulus)
+      {
+      if(x.is_negative() && t1.is_nonzero())
+         return modulus - t1;
+      return x;
+      }
+
+   if(t1 >= modulus_2)
+      return (x % modulus);
+
+   t1 >>= (MP_WORD_BITS * (mod_words - 1));
+   t1 *= mu;
+   t1 >>= (MP_WORD_BITS * (mod_words + 1));
+
+   t1 *= modulus;
+   t1.mask_bits(MP_WORD_BITS * (mod_words+1));
+
+   BigInt t2 = x;
+   t2.set_sign(BigInt::Positive);
+   t2.mask_bits(MP_WORD_BITS * (mod_words+1));
+
+   t1 = t2 - t1;
+
+   if(t1.is_negative())
+      {
+      BigInt b_to_k1(BigInt::Power2, MP_WORD_BITS * (mod_words+1));
+      t1 += b_to_k1;
+      }
+
+   while(t1 >= modulus)
+      t1 -= modulus;
+
+   if(x.is_negative() && t1.is_nonzero())
+      t1 = modulus - t1;
+
+   return t1;
+   }
+
+/*************************************************
+* Multiply, followed by a reduction              *
+*************************************************/
+BigInt Modular_Reducer::multiply(const BigInt& x, const BigInt& y) const
+   {
+   return reduce(x * y);
+   }
+
+/*************************************************
+* Square, followed by a reduction                *
+*************************************************/
+BigInt Modular_Reducer::square(const BigInt& x) const
+   {
+   return reduce(Botan::square(x));
+   }
+
+}
diff --git a/src/math/reducer.h b/src/math/reducer.h
new file mode 100644
index 000000000..48008e73b
--- /dev/null
+++ b/src/math/reducer.h
@@ -0,0 +1,34 @@
+/*************************************************
+* Modular Reducer Header File                    *
+* (C) 1999-2007 Jack Lloyd                       *
+*************************************************/
+
+#ifndef BOTAN_MODARITH_H__
+#define BOTAN_MODARITH_H__
+
+#include <botan/bigint.h>
+
+namespace Botan {
+
+/*************************************************
+* Modular Reducer                                *
+*************************************************/
+class BOTAN_DLL Modular_Reducer
+   {
+   public:
+      BigInt multiply(const BigInt&, const BigInt&) const;
+      BigInt square(const BigInt&) const;
+      BigInt reduce(const BigInt&) const;
+
+      bool initialized() const { return (mod_words != 0); }
+
+      Modular_Reducer() { mod_words = 0; }
+      Modular_Reducer(const BigInt&);
+   private:
+      BigInt modulus, modulus_2, mu;
+      u32bit mod_words, mod2_words, mu_words;
+   };
+
+}
+
+#endif
diff --git a/src/math/ressol.cpp b/src/math/ressol.cpp
new file mode 100644
index 000000000..0cd2b988a
--- /dev/null
+++ b/src/math/ressol.cpp
@@ -0,0 +1,82 @@
+/*************************************************
+* Shanks-Tonnelli (RESSOL) Source File           *
+* (C) 2007-2008 Falko Strenzke, FlexSecure GmbH  *
+* (C) 2008 Jack Lloyd                            *
+*************************************************/
+
+#include <botan/numthry.h>
+#include <botan/reducer.h>
+
+#include <iostream>
+
+namespace Botan {
+
+/*************************************************
+* Shanks-Tonnelli algorithm                      *
+*************************************************/
+BigInt ressol(const BigInt& a, const BigInt& p)
+   {
+   if(a < 0)
+      throw Invalid_Argument("ressol(): a to solve for must be positive");
+   if(p <= 1)
+      throw Invalid_Argument("ressol(): prime must be > 1");
+
+   if(a == 0)
+      return 0;
+   if(p == 2)
+      return a;
+
+   if(jacobi(a, p) != 1) // not a quadratic residue
+      return -BigInt(1);
+
+   if(p % 4 == 3)
+      return power_mod(a, ((p+1) >> 2), p);
+
+   u32bit 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 non quadratic residue z
+   BigInt z = 2;
+   while(jacobi(z, p) == 1) // while z quadratic residue
+      ++z;
+
+   BigInt c = power_mod(z, (q << 1) + 1, p);
+
+   while(n > 1)
+      {
+      q = n;
+
+      u32bit i = 0;
+      while(q != 1)
+         {
+         q = mod_p.square(q);
+         ++i;
+         }
+      u32bit t = s;
+
+      if(t <= i)
+         return -BigInt(1);
+
+      c = power_mod(c, BigInt(BigInt::Power2, t-i-1), p);
+      r = mod_p.multiply(r, c);
+      c = mod_p.square(c);
+      n = mod_p.multiply(n, c);
+      s = i;
+      }
+
+   return r;
+   }
+
+}