Primality Test | Set 3 (Miller–Rabin)
Last Updated :
23 Jul, 2025
Given a number n, check if it is prime or not. We have introduced and discussed School and Fermat methods for primality testing.
Primality Test | Set 1 (Introduction and School Method)
Primality Test | Set 2 (Fermat Method)
In this post, the Miller-Rabin method is discussed. This method is a probabilistic method ( like Fermat), but it is generally preferred over Fermat's method.
Algorithm:
// It returns false if n is composite and returns true if n
// is probably prime. k is an input parameter that determines
// accuracy level. Higher value of k indicates more accuracy.
bool isPrime(int n, int k)
1) Handle base cases for n < 3
2) If n is even, return false.
3) Find an odd number d such that n-1 can be written as d*2r.
Note that since n is odd, (n-1) must be even and r must be
greater than 0.
4) Do following k times
if (millerTest(n, d) == false)
return false
5) Return true.
// This function is called for all k trials. It returns
// false if n is composite and returns true if n is probably
// prime.
// d is an odd number such that d*2r = n-1 for some r>=1
bool millerTest(int n, int d)
1) Pick a random number 'a' in range [2, n-2]
2) Compute: x = pow(a, d) % n
3) If x == 1 or x == n-1, return true.
// Below loop mainly runs 'r-1' times.
4) Do following while d doesn't become n-1.
a) x = (x*x) % n.
b) If (x == 1) return false.
c) If (x == n-1) return true.
Example:
Input: n = 13, k = 2.
1) Compute d and r such that d*2r = n-1,
d = 3, r = 2.
2) Call millerTest k times.
1st Iteration:
1) Pick a random number 'a' in range [2, n-2]
Suppose a = 4
2) Compute: x = pow(a, d) % n
x = 43 % 13 = 12
3) Since x = (n-1), return true.
IInd Iteration:
1) Pick a random number 'a' in range [2, n-2]
Suppose a = 5
2) Compute: x = pow(a, d) % n
x = 53 % 13 = 8
3) x neither 1 nor 12.
4) Do following (r-1) = 1 times
a) x = (x * x) % 13 = (8 * 8) % 13 = 12
b) Since x = (n-1), return true.
Since both iterations return true, we return true.
Implementation:
Below is the implementation of the above algorithm.
C++
// C++ program Miller-Rabin primality test
#include <bits/stdc++.h>
using namespace std;
// Utility function to do modular exponentiation.
// It returns (x^y) % p
int power(int x, unsigned int y, int p)
{
int res = 1; // Initialize result
x = x % p; // Update x if it is more than or
// equal to p
while (y > 0)
{
// If y is odd, multiply x with result
if (y & 1)
res = (res*x) % p;
// y must be even now
y = y>>1; // y = y/2
x = (x*x) % p;
}
return res;
}
// This function is called for all k trials. It returns
// false if n is composite and returns true if n is
// probably prime.
// d is an odd number such that d*2 = n-1
// for some r >= 1
bool miillerTest(int d, int n)
{
// Pick a random number in [2..n-2]
// Corner cases make sure that n > 4
int a = 2 + rand() % (n - 4);
// Compute a^d % n
int x = power(a, d, n);
if (x == 1 || x == n-1)
return true;
// Keep squaring x while one of the following doesn't
// happen
// (i) d does not reach n-1
// (ii) (x^2) % n is not 1
// (iii) (x^2) % n is not n-1
while (d != n-1)
{
x = (x * x) % n;
d *= 2;
if (x == 1) return false;
if (x == n-1) return true;
}
// Return composite
return false;
}
// It returns false if n is composite and returns true if n
// is probably prime. k is an input parameter that determines
// accuracy level. Higher value of k indicates more accuracy.
bool isPrime(int n, int k)
{
// Corner cases
if (n <= 1 || n == 4) return false;
if (n <= 3) return true;
// Find r such that n = 2^d * r + 1 for some r >= 1
int d = n - 1;
while (d % 2 == 0)
d /= 2;
// Iterate given number of 'k' times
for (int i = 0; i < k; i++)
if (!miillerTest(d, n))
return false;
return true;
}
// Driver program
int main()
{
int k = 4; // Number of iterations
cout << "All primes smaller than 100: \n";
for (int n = 1; n < 100; n++)
if (isPrime(n, k))
cout << n << " ";
return 0;
}
// Java program Miller-Rabin primality test
import java.io.*;
import java.math.*;
class GFG {
// Utility function to do modular
// exponentiation. It returns (x^y) % p
static int power(int x, int y, int p) {
int res = 1; // Initialize result
//Update x if it is more than or
// equal to p
x = x % p;
while (y > 0) {
// If y is odd, multiply x with result
if ((y & 1) == 1)
res = (res * x) % p;
// y must be even now
y = y >> 1; // y = y/2
x = (x * x) % p;
}
return res;
}
// This function is called for all k trials.
// It returns false if n is composite and
// returns false if n is probably prime.
// d is an odd number such that d*2<sup>r</sup>
// = n-1 for some r >= 1
static boolean miillerTest(int d, int n) {
// Pick a random number in [2..n-2]
// Corner cases make sure that n > 4
int a = 2 + (int)(Math.random() % (n - 4));
// Compute a^d % n
int x = power(a, d, n);
if (x == 1 || x == n - 1)
return true;
// Keep squaring x while one of the
// following doesn't happen
// (i) d does not reach n-1
// (ii) (x^2) % n is not 1
// (iii) (x^2) % n is not n-1
while (d != n - 1) {
x = (x * x) % n;
d *= 2;
if (x == 1)
return false;
if (x == n - 1)
return true;
}
// Return composite
return false;
}
// It returns false if n is composite
// and returns true if n is probably
// prime. k is an input parameter that
// determines accuracy level. Higher
// value of k indicates more accuracy.
static boolean isPrime(int n, int k) {
// Corner cases
if (n <= 1 || n == 4)
return false;
if (n <= 3)
return true;
// Find r such that n = 2^d * r + 1
// for some r >= 1
int d = n - 1;
while (d % 2 == 0)
d /= 2;
// Iterate given number of 'k' times
for (int i = 0; i < k; i++)
if (!miillerTest(d, n))
return false;
return true;
}
// Driver program
public static void main(String args[]) {
int k = 4; // Number of iterations
System.out.println("All primes smaller "
+ "than 100: ");
for (int n = 1; n < 100; n++)
if (isPrime(n, k))
System.out.print(n + " ");
}
}
/* This code is contributed by Nikita Tiwari.*/
# Python3 program Miller-Rabin primality test
import random
# Utility function to do
# modular exponentiation.
# It returns (x^y) % p
def power(x, y, p):
# Initialize result
res = 1;
# Update x if it is more than or
# equal to p
x = x % p;
while (y > 0):
# If y is odd, multiply
# x with result
if (y & 1):
res = (res * x) % p;
# y must be even now
y = y>>1; # y = y/2
x = (x * x) % p;
return res;
# This function is called
# for all k trials. It returns
# false if n is composite and
# returns false if n is
# probably prime. d is an odd
# number such that d*2<sup>r</sup> = n-1
# for some r >= 1
def miillerTest(d, n):
# Pick a random number in [2..n-2]
# Corner cases make sure that n > 4
a = 2 + random.randint(1, n - 4);
# Compute a^d % n
x = power(a, d, n);
if (x == 1 or x == n - 1):
return True;
# Keep squaring x while one
# of the following doesn't
# happen
# (i) d does not reach n-1
# (ii) (x^2) % n is not 1
# (iii) (x^2) % n is not n-1
while (d != n - 1):
x = (x * x) % n;
d *= 2;
if (x == 1):
return False;
if (x == n - 1):
return True;
# Return composite
return False;
# It returns false if n is
# composite and returns true if n
# is probably prime. k is an
# input parameter that determines
# accuracy level. Higher value of
# k indicates more accuracy.
def isPrime( n, k):
# Corner cases
if (n <= 1 or n == 4):
return False;
if (n <= 3):
return True;
# Find r such that n =
# 2^d * r + 1 for some r >= 1
d = n - 1;
while (d % 2 == 0):
d //= 2;
# Iterate given number of 'k' times
for i in range(k):
if (miillerTest(d, n) == False):
return False;
return True;
# Driver Code
# Number of iterations
k = 4;
print("All primes smaller than 100: ");
for n in range(1,100):
if (isPrime(n, k)):
print(n , end=" ");
# This code is contributed by mits
// C# program Miller-Rabin primality test
using System;
class GFG
{
// Utility function to do modular
// exponentiation. It returns (x^y) % p
static int power(int x, int y, int p)
{
int res = 1; // Initialize result
// Update x if it is more than
// or equal to p
x = x % p;
while (y > 0)
{
// If y is odd, multiply x with result
if ((y & 1) == 1)
res = (res * x) % p;
// y must be even now
y = y >> 1; // y = y/2
x = (x * x) % p;
}
return res;
}
// This function is called for all k trials.
// It returns false if n is composite and
// returns false if n is probably prime.
// d is an odd number such that d*2<sup>r</sup>
// = n-1 for some r >= 1
static bool miillerTest(int d, int n)
{
// Pick a random number in [2..n-2]
// Corner cases make sure that n > 4
Random r = new Random();
int a = 2 + (int)(r.Next() % (n - 4));
// Compute a^d % n
int x = power(a, d, n);
if (x == 1 || x == n - 1)
return true;
// Keep squaring x while one of the
// following doesn't happen
// (i) d does not reach n-1
// (ii) (x^2) % n is not 1
// (iii) (x^2) % n is not n-1
while (d != n - 1)
{
x = (x * x) % n;
d *= 2;
if (x == 1)
return false;
if (x == n - 1)
return true;
}
// Return composite
return false;
}
// It returns false if n is composite
// and returns true if n is probably
// prime. k is an input parameter that
// determines accuracy level. Higher
// value of k indicates more accuracy.
static bool isPrime(int n, int k)
{
// Corner cases
if (n <= 1 || n == 4)
return false;
if (n <= 3)
return true;
// Find r such that n = 2^d * r + 1
// for some r >= 1
int d = n - 1;
while (d % 2 == 0)
d /= 2;
// Iterate given number of 'k' times
for (int i = 0; i < k; i++)
if (miillerTest(d, n) == false)
return false;
return true;
}
// Driver Code
static void Main()
{
int k = 4; // Number of iterations
Console.WriteLine("All primes smaller " +
"than 100: ");
for (int n = 1; n < 100; n++)
if (isPrime(n, k))
Console.Write(n + " ");
}
}
// This code is contributed by mits
<?php
// PHP program Miller-Rabin primality test
// Utility function to do
// modular exponentiation.
// It returns (x^y) % p
function power($x, $y, $p)
{
// Initialize result
$res = 1;
// Update x if it is more than or
// equal to p
$x = $x % $p;
while ($y > 0)
{
// If y is odd, multiply
// x with result
if ($y & 1)
$res = ($res*$x) % $p;
// y must be even now
$y = $y>>1; // $y = $y/2
$x = ($x*$x) % $p;
}
return $res;
}
// This function is called
// for all k trials. It returns
// false if n is composite and
// returns false if n is
// probably prime. d is an odd
// number such that d*2<sup>r</sup> = n-1
// for some r >= 1
function miillerTest($d, $n)
{
// Pick a random number in [2..n-2]
// Corner cases make sure that n > 4
$a = 2 + rand() % ($n - 4);
// Compute a^d % n
$x = power($a, $d, $n);
if ($x == 1 || $x == $n-1)
return true;
// Keep squaring x while one
// of the following doesn't
// happen
// (i) d does not reach n-1
// (ii) (x^2) % n is not 1
// (iii) (x^2) % n is not n-1
while ($d != $n-1)
{
$x = ($x * $x) % $n;
$d *= 2;
if ($x == 1) return false;
if ($x == $n-1) return true;
}
// Return composite
return false;
}
// It returns false if n is
// composite and returns true if n
// is probably prime. k is an
// input parameter that determines
// accuracy level. Higher value of
// k indicates more accuracy.
function isPrime( $n, $k)
{
// Corner cases
if ($n <= 1 || $n == 4) return false;
if ($n <= 3) return true;
// Find r such that n =
// 2^d * r + 1 for some r >= 1
$d = $n - 1;
while ($d % 2 == 0)
$d /= 2;
// Iterate given number of 'k' times
for ($i = 0; $i < $k; $i++)
if (!miillerTest($d, $n))
return false;
return true;
}
// Driver Code
// Number of iterations
$k = 4;
echo "All primes smaller than 100: \n";
for ($n = 1; $n < 100; $n++)
if (isPrime($n, $k))
echo $n , " ";
// This code is contributed by ajit
?>
<script>
// Javascript program Miller-Rabin primality test
// Utility function to do
// modular exponentiation.
// It returns (x^y) % p
function power(x, y, p)
{
// Initialize result
let res = 1;
// Update x if it is more than or
// equal to p
x = x % p;
while (y > 0)
{
// If y is odd, multiply
// x with result
if (y & 1)
res = (res*x) % p;
// y must be even now
y = y>>1; // y = y/2
x = (x*x) % p;
}
return res;
}
// This function is called
// for all k trials. It returns
// false if n is composite and
// returns false if n is
// probably prime. d is an odd
// number such that d*2<sup>r</sup> = n-1
// for some r >= 1
function miillerTest(d, n)
{
// Pick a random number in [2..n-2]
// Corner cases make sure that n > 4
let a = 2 + Math.floor(Math.random() * (n-2)) % (n - 4);
// Compute a^d % n
let x = power(a, d, n);
if (x == 1 || x == n-1)
return true;
// Keep squaring x while one
// of the following doesn't
// happen
// (i) d does not reach n-1
// (ii) (x^2) % n is not 1
// (iii) (x^2) % n is not n-1
while (d != n-1)
{
x = (x * x) % n;
d *= 2;
if (x == 1)
return false;
if (x == n-1)
return true;
}
// Return composite
return false;
}
// It returns false if n is
// composite and returns true if n
// is probably prime. k is an
// input parameter that determines
// accuracy level. Higher value of
// k indicates more accuracy.
function isPrime( n, k)
{
// Corner cases
if (n <= 1 || n == 4) return false;
if (n <= 3) return true;
// Find r such that n =
// 2^d * r + 1 for some r >= 1
let d = n - 1;
while (d % 2 == 0)
d /= 2;
// Iterate given number of 'k' times
for (let i = 0; i < k; i++)
if (!miillerTest(d, n))
return false;
return true;
}
// Driver Code
// Number of iterations
let k = 4;
document.write("All primes smaller than 100: <br>");
for (let n = 1; n < 100; n++)
if (isPrime(n, k))
document.write(n , " ");
// This code is contributed by gfgking
</script>
Output:
All primes smaller than 100:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
61 67 71 73 79 83 89 97
Time Complexity: O(k*logn)
Auxiliary Space: O(1)
How does this work?
Below are some important facts behind the algorithm:
- Fermat's theorem states that, If n is a prime number, then for every a, 1 <= a < n, an-1 % n = 1
- Base cases make sure that n must be odd. Since n is odd, n-1 must be even. And an even number can be written as d * 2s where d is an odd number and s > 0.
- From the above two points, for every randomly picked number in the range [2, n-2], the value of ad*2r % n must be 1.
- As per Euclid's Lemma, if x2 % n = 1 or (x2 - 1) % n = 0 or (x-1)(x+1)% n = 0. Then, for n to be prime, either n divides (x-1) or n divides (x+1). Which means either x % n = 1 or x % n = -1.
- From points 2 and 3, we can conclude
For n to be prime, either
ad % n = 1
OR
ad*2i % n = -1
for some i, where 0 <= i <= r-1.
Next Article :
Primality Test | Set 4 (Solovay-Strassen)
This article is contributed Ruchir Garg.
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