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/*
* (C) 2016,2018 Jack Lloyd
*
* Botan is released under the Simplified BSD License (see license.txt)
*/
#include <botan/internal/primality.h>
#include <botan/internal/monty_exp.h>
#include <botan/bigint.h>
#include <botan/monty.h>
#include <botan/reducer.h>
#include <botan/rng.h>
#include <algorithm>
namespace Botan {
bool is_lucas_probable_prime(const BigInt& C, const Modular_Reducer& mod_C)
{
if(C <= 1)
return false;
else if(C == 2)
return true;
else if(C.is_even())
return false;
else if(C == 3 || C == 5 || C == 7 || C == 11 || C == 13)
return true;
BigInt D = 5;
for(;;)
{
int32_t j = jacobi(D, C);
if(j == 0)
return false;
if(j == -1)
break;
// Check 5, -7, 9, -11, 13, -15, 17, ...
if(D.is_negative())
{
D.flip_sign();
D += 2;
}
else
{
D += 2;
D.flip_sign();
}
if(D == 17 && is_perfect_square(C).is_nonzero())
return false;
}
const BigInt K = C + 1;
const size_t K_bits = K.bits() - 1;
BigInt U = 1;
BigInt V = 1;
BigInt Ut, Vt, U2, V2;
for(size_t i = 0; i != K_bits; ++i)
{
const uint8_t k_bit = K.get_bit(K_bits - 1 - i);
Ut = mod_C.multiply(U, V);
Vt = mod_C.reduce(mod_C.square(V) + mod_C.multiply(D, mod_C.square(U)));
if(Vt.is_odd())
Vt += C;
Vt >>= 1;
Vt = mod_C.reduce(Vt);
U = Ut;
V = Vt;
U2 = mod_C.reduce(Ut + Vt);
if(U2.is_odd())
U2 += C;
U2 >>= 1;
V2 = mod_C.reduce(Vt + Ut*D);
if(V2.is_odd())
V2 += C;
V2 >>= 1;
U.ct_cond_assign(k_bit, U2);
V.ct_cond_assign(k_bit, V2);
}
return (U == 0);
}
bool is_bailie_psw_probable_prime(const BigInt& n, const Modular_Reducer& mod_n)
{
auto monty_n = std::make_shared<Montgomery_Params>(n, mod_n);
return passes_miller_rabin_test(n, mod_n, monty_n, 2) && is_lucas_probable_prime(n, mod_n);
}
bool is_bailie_psw_probable_prime(const BigInt& n)
{
Modular_Reducer mod_n(n);
return is_bailie_psw_probable_prime(n, mod_n);
}
bool passes_miller_rabin_test(const BigInt& n,
const Modular_Reducer& mod_n,
const std::shared_ptr<Montgomery_Params>& monty_n,
const BigInt& a)
{
BOTAN_ASSERT_NOMSG(n > 1);
const BigInt n_minus_1 = n - 1;
const size_t s = low_zero_bits(n_minus_1);
const BigInt nm1_s = n_minus_1 >> s;
const size_t n_bits = n.bits();
const size_t powm_window = 4;
auto powm_a_n = monty_precompute(monty_n, a, powm_window);
BigInt y = monty_execute(*powm_a_n, nm1_s, n_bits);
if(y == 1 || y == n_minus_1)
return true;
for(size_t i = 1; i != s; ++i)
{
y = mod_n.square(y);
if(y == 1) // found a non-trivial square root
return false;
/*
-1 is the trivial square root of unity, so ``a`` is not a
witness for this number - give up
*/
if(y == n_minus_1)
return true;
}
return false;
}
bool is_miller_rabin_probable_prime(const BigInt& n,
const Modular_Reducer& mod_n,
RandomNumberGenerator& rng,
size_t test_iterations)
{
BOTAN_ASSERT_NOMSG(n > 1);
auto monty_n = std::make_shared<Montgomery_Params>(n, mod_n);
for(size_t i = 0; i != test_iterations; ++i)
{
const BigInt a = BigInt::random_integer(rng, 2, n);
if(!passes_miller_rabin_test(n, mod_n, monty_n, a))
return false;
}
// Failed to find a counterexample
return true;
}
size_t miller_rabin_test_iterations(size_t n_bits, size_t prob, bool random)
{
const size_t base = (prob + 2) / 2; // worst case 4^-t error rate
/*
* If the candidate prime was maliciously constructed, we can't rely
* on arguments based on p being random.
*/
if(random == false)
return base;
/*
* For randomly chosen numbers we can use the estimates from
* http://www.math.dartmouth.edu/~carlp/PDF/paper88.pdf
*
* These values are derived from the inequality for p(k,t) given on
* the second page.
*/
if(prob <= 128)
{
if(n_bits >= 1536)
return 4; // < 2^-133
if(n_bits >= 1024)
return 6; // < 2^-133
if(n_bits >= 512)
return 12; // < 2^-129
if(n_bits >= 256)
return 29; // < 2^-128
}
/*
If the user desires a smaller error probability than we have
precomputed error estimates for, just fall back to using the worst
case error rate.
*/
return base;
}
}
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