# C++ Program to Implement the Rabin-Miller Primality Test to Check if a Given Number is Prime

Rabin-Miller Primality Test is used to check if a given Number is Prime or not. It is similar to the format primality and the Solovay-Stressen test. this test first was discovered by Russian Mathematician M. M. Artjuhov.

## Algorithm

Begin
ll mulmod(ll a, ll b, ll m)
ll x = 0,y = a mod m
while (b > 0)
if (b mod 2 == 1)
compute x = (x + y) mod m
y = (y * 2) mod m
b = b/ 2
return x mod m.
End

Begin
ll modulo(ll base, ll e, ll m)
Initialize:
ll x = 1
ll y = base
while (e > 0)
if (e mod 2 == 1)
x = (x * y) mod m
y = (y * y) mod m
e = e / 2;
return x mod m
End

Begin
bool Miller(ll p, int iteration)
if (p < 2)
return false
if (p != 2 and p mod 2==0)
return false;
Compute: ll s = p - 1
while (s mod 2 == 0)
s = s/ 2;
for i = 0 to iteration - 1
Do
ll a = rand() mod (p - 1) + 1, temp = s
ll mod = modulo(a, temp, p)
while (temp != p - 1 and mod != 1 and mod != p - 1)
mod = mulmod(mod, mod, p);
temp *= 2;
if (mod != p - 1 && temp % 2 == 0)
return false
else
return true
End

## Example Code

#include <iostream>
#include<stdlib.h>
#define ll long long
using namespace std;
ll mulmod(ll a, ll b, ll m)//It returns true if number is prime otherwise false {
ll x = 0,y = a % m;
while (b > 0) {
if (b % 2 == 1) {
x = (x + y) % m;
}
y = (y * 2) % m;
b /= 2;
}
return x % m;
}

ll modulo(ll base, ll e, ll m) {
ll x = 1;
ll y = base;
while (e > 0) {
if (e % 2 == 1)
x = (x * y) % m;
y = (y * y) % m;
e = e / 2;
}
return x % m;
}

bool Miller(ll p, int iteration) {
if (p < 2) {
return false;
}
if (p != 2 && p % 2==0) {
return false;
}
ll s = p - 1;
while (s % 2 == 0) {
s /= 2;
}
for (int i = 0; i < iteration; i++) {
ll a = rand() % (p - 1) + 1, temp = s;
ll mod = modulo(a, temp, p);
while (temp != p - 1 && mod != 1 && mod != p - 1) {
mod = mulmod(mod, mod, p);
temp *= 2;
}
if (mod != p - 1 && temp % 2 == 0) {
return false;
}
}
return true;
}

int main() {
int iteration = 10;
ll num;
cout<<"Enter integer to test primality: ";
cin>>num;
if (Miller(num, iteration))
cout<<num<<" is prime"<<endl;
else
cout<<num<<" is not prime"<<endl;
return 0;
}

## Output

Enter integer to test primality: 26
26 is not prime