## Monday, April 27, 2015

### Miller-Rabin Primality Test

Miller-Rabin Algorithm:

Modular Arithmetic:

Source (C++):

```#include <iostream>
#include <cstring>
#include <cstdlib>
#include <stdio.h>
#define ull unsigned long long
using namespace std;

/*
* calculates (a * b) % c taking into account that a * b might overflow
*/
ull mulmod(ull a, ull b, ull mod)
{
ull x = 0,y = a % mod;
while (b > 0)
{
if (b % 2 == 1)
{
x = (x + y) % mod;
}
y = (y * 2) % mod;
b /= 2;
}
return x % mod;
}

/*
* modular exponentiation
*/
ull modulo(ull base, ull exponent, ull mod)
{
ull x = 1;
ull y = base;
while (exponent > 0)
{
if (exponent % 2 == 1)x = mulmod(x, y, mod);
y = mulmod(y, y, mod);
exponent = exponent / 2;
}
return x % mod;
}

/*
* Miller-Rabin primality test, iteration signifies the accuracy
*/
bool MillerRabin(ull p,int iteration)
{
if (p < 2)
{
return false;
}
if (p != 2 && p % 2==0)
{
return false;
}
ull s = p - 1;
while (s % 2 == 0)
{
s /= 2;
}
for (int i = 0; i < iteration; i++)
{
ull a = rand() % (p - 1) + 1, temp = s;
ull mod = modulo(a, temp, p);
if(mod==1 || mod == p-1)continue;
while (temp != p - 1 && mod != 1 && mod != p - 1)
{
mod = mulmod(mod, mod, p);
temp *= 2;
}
if (mod != p - 1)return false;
}

return true;
}

int main()
{
int tests = 5;
ull num;
scanf("%llu", &num);
if (MillerRabin(num, tests))printf("YES\n");
else printf("NO\n");
}```