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-rw-r--r--src/math/numbertheory/blinding.cpp47
-rw-r--r--src/math/numbertheory/blinding.h32
-rw-r--r--src/math/numbertheory/def_powm.h62
-rw-r--r--src/math/numbertheory/dsa_gen.cpp134
-rw-r--r--src/math/numbertheory/info.txt29
-rw-r--r--src/math/numbertheory/jacobi.cpp51
-rw-r--r--src/math/numbertheory/make_prm.cpp95
-rw-r--r--src/math/numbertheory/mp_numth.cpp69
-rw-r--r--src/math/numbertheory/numthry.cpp339
-rw-r--r--src/math/numbertheory/numthry.h100
-rw-r--r--src/math/numbertheory/pow_mod.cpp155
-rw-r--r--src/math/numbertheory/pow_mod.h91
-rw-r--r--src/math/numbertheory/powm_fw.cpp102
-rw-r--r--src/math/numbertheory/powm_mnt.cpp178
-rw-r--r--src/math/numbertheory/primes.cpp674
-rw-r--r--src/math/numbertheory/reducer.cpp95
-rw-r--r--src/math/numbertheory/reducer.h34
-rw-r--r--src/math/numbertheory/ressol.cpp82
18 files changed, 2369 insertions, 0 deletions
diff --git a/src/math/numbertheory/blinding.cpp b/src/math/numbertheory/blinding.cpp
new file mode 100644
index 000000000..740904d10
--- /dev/null
+++ b/src/math/numbertheory/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/numbertheory/blinding.h b/src/math/numbertheory/blinding.h
new file mode 100644
index 000000000..958686fb1
--- /dev/null
+++ b/src/math/numbertheory/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/numbertheory/def_powm.h b/src/math/numbertheory/def_powm.h
new file mode 100644
index 000000000..c91ff002c
--- /dev/null
+++ b/src/math/numbertheory/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/numbertheory/dsa_gen.cpp b/src/math/numbertheory/dsa_gen.cpp
new file mode 100644
index 000000000..baaba66ee
--- /dev/null
+++ b/src/math/numbertheory/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/numbertheory/info.txt b/src/math/numbertheory/info.txt
new file mode 100644
index 000000000..6ce3583c8
--- /dev/null
+++ b/src/math/numbertheory/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/numbertheory/jacobi.cpp b/src/math/numbertheory/jacobi.cpp
new file mode 100644
index 000000000..57c78508a
--- /dev/null
+++ b/src/math/numbertheory/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/numbertheory/make_prm.cpp b/src/math/numbertheory/make_prm.cpp
new file mode 100644
index 000000000..30ac9a623
--- /dev/null
+++ b/src/math/numbertheory/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/numbertheory/mp_numth.cpp b/src/math/numbertheory/mp_numth.cpp
new file mode 100644
index 000000000..b024d2e2d
--- /dev/null
+++ b/src/math/numbertheory/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/numbertheory/numthry.cpp b/src/math/numbertheory/numthry.cpp
new file mode 100644
index 000000000..ffd523e82
--- /dev/null
+++ b/src/math/numbertheory/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/numbertheory/numthry.h b/src/math/numbertheory/numthry.h
new file mode 100644
index 000000000..a271a428a
--- /dev/null
+++ b/src/math/numbertheory/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/numbertheory/pow_mod.cpp b/src/math/numbertheory/pow_mod.cpp
new file mode 100644
index 000000000..17ca7b796
--- /dev/null
+++ b/src/math/numbertheory/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/numbertheory/pow_mod.h b/src/math/numbertheory/pow_mod.h
new file mode 100644
index 000000000..37e0871da
--- /dev/null
+++ b/src/math/numbertheory/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/numbertheory/powm_fw.cpp b/src/math/numbertheory/powm_fw.cpp
new file mode 100644
index 000000000..c29b9f311
--- /dev/null
+++ b/src/math/numbertheory/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/numbertheory/powm_mnt.cpp b/src/math/numbertheory/powm_mnt.cpp
new file mode 100644
index 000000000..6091d467a
--- /dev/null
+++ b/src/math/numbertheory/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/numbertheory/primes.cpp b/src/math/numbertheory/primes.cpp
new file mode 100644
index 000000000..d005167e5
--- /dev/null
+++ b/src/math/numbertheory/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,
+17623, 17627, 17657, 17659, 17669, 17681, 17683, 17707, 17713, 17729, 17737,
+17747, 17749, 17761, 17783, 17789, 17791, 17807, 17827, 17837, 17839, 17851,
+17863, 17881, 17891, 17903, 17909, 17911, 17921, 17923, 17929, 17939, 17957,
+17959, 17971, 17977, 17981, 17987, 17989, 18013, 18041, 18043, 18047, 18049,
+18059, 18061, 18077, 18089, 18097, 18119, 18121, 18127, 18131, 18133, 18143,
+18149, 18169, 18181, 18191, 18199, 18211, 18217, 18223, 18229, 18233, 18251,
+18253, 18257, 18269, 18287, 18289, 18301, 18307, 18311, 18313, 18329, 18341,
+18353, 18367, 18371, 18379, 18397, 18401, 18413, 18427, 18433, 18439, 18443,
+18451, 18457, 18461, 18481, 18493, 18503, 18517, 18521, 18523, 18539, 18541,
+18553, 18583, 18587, 18593, 18617, 18637, 18661, 18671, 18679, 18691, 18701,
+18713, 18719, 18731, 18743, 18749, 18757, 18773, 18787, 18793, 18797, 18803,
+18839, 18859, 18869, 18899, 18911, 18913, 18917, 18919, 18947, 18959, 18973,
+18979, 19001, 19009, 19013, 19031, 19037, 19051, 19069, 19073, 19079, 19081,
+19087, 19121, 19139, 19141, 19157, 19163, 19181, 19183, 19207, 19211, 19213,
+19219, 19231, 19237, 19249, 19259, 19267, 19273, 19289, 19301, 19309, 19319,
+19333, 19373, 19379, 19381, 19387, 19391, 19403, 19417, 19421, 19423, 19427,
+19429, 19433, 19441, 19447, 19457, 19463, 19469, 19471, 19477, 19483, 19489,
+19501, 19507, 19531, 19541, 19543, 19553, 19559, 19571, 19577, 19583, 19597,
+19603, 19609, 19661, 19681, 19687, 19697, 19699, 19709, 19717, 19727, 19739,
+19751, 19753, 19759, 19763, 19777, 19793, 19801, 19813, 19819, 19841, 19843,
+19853, 19861, 19867, 19889, 19891, 19913, 19919, 19927, 19937, 19949, 19961,
+19963, 19973, 19979, 19991, 19993, 19997, 20011, 20021, 20023, 20029, 20047,
+20051, 20063, 20071, 20089, 20101, 20107, 20113, 20117, 20123, 20129, 20143,
+20147, 20149, 20161, 20173, 20177, 20183, 20201, 20219, 20231, 20233, 20249,
+20261, 20269, 20287, 20297, 20323, 20327, 20333, 20341, 20347, 20353, 20357,
+20359, 20369, 20389, 20393, 20399, 20407, 20411, 20431, 20441, 20443, 20477,
+20479, 20483, 20507, 20509, 20521, 20533, 20543, 20549, 20551, 20563, 20593,
+20599, 20611, 20627, 20639, 20641, 20663, 20681, 20693, 20707, 20717, 20719,
+20731, 20743, 20747, 20749, 20753, 20759, 20771, 20773, 20789, 20807, 20809,
+20849, 20857, 20873, 20879, 20887, 20897, 20899, 20903, 20921, 20929, 20939,
+20947, 20959, 20963, 20981, 20983, 21001, 21011, 21013, 21017, 21019, 21023,
+21031, 21059, 21061, 21067, 21089, 21101, 21107, 21121, 21139, 21143, 21149,
+21157, 21163, 21169, 21179, 21187, 21191, 21193, 21211, 21221, 21227, 21247,
+21269, 21277, 21283, 21313, 21317, 21319, 21323, 21341, 21347, 21377, 21379,
+21383, 21391, 21397, 21401, 21407, 21419, 21433, 21467, 21481, 21487, 21491,
+21493, 21499, 21503, 21517, 21521, 21523, 21529, 21557, 21559, 21563, 21569,
+21577, 21587, 21589, 21599, 21601, 21611, 21613, 21617, 21647, 21649, 21661,
+21673, 21683, 21701, 21713, 21727, 21737, 21739, 21751, 21757, 21767, 21773,
+21787, 21799, 21803, 21817, 21821, 21839, 21841, 21851, 21859, 21863, 21871,
+21881, 21893, 21911, 21929, 21937, 21943, 21961, 21977, 21991, 21997, 22003,
+22013, 22027, 22031, 22037, 22039, 22051, 22063, 22067, 22073, 22079, 22091,
+22093, 22109, 22111, 22123, 22129, 22133, 22147, 22153, 22157, 22159, 22171,
+22189, 22193, 22229, 22247, 22259, 22271, 22273, 22277, 22279, 22283, 22291,
+22303, 22307, 22343, 22349, 22367, 22369, 22381, 22391, 22397, 22409, 22433,
+22441, 22447, 22453, 22469, 22481, 22483, 22501, 22511, 22531, 22541, 22543,
+22549, 22567, 22571, 22573, 22613, 22619, 22621, 22637, 22639, 22643, 22651,
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+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/numbertheory/reducer.cpp b/src/math/numbertheory/reducer.cpp
new file mode 100644
index 000000000..47c5c20fc
--- /dev/null
+++ b/src/math/numbertheory/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/numbertheory/reducer.h b/src/math/numbertheory/reducer.h
new file mode 100644
index 000000000..48008e73b
--- /dev/null
+++ b/src/math/numbertheory/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/numbertheory/ressol.cpp b/src/math/numbertheory/ressol.cpp
new file mode 100644
index 000000000..0cd2b988a
--- /dev/null
+++ b/src/math/numbertheory/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;
+ }
+
+}
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