Fermat Method of Primality Test

Last Updated : 01 Jun, 2023

Given a number n, check if it is prime or not. We have introduced and discussed the School method for primality testing in Set 1.
Introduction to Primality Test and School Method
In this post, Fermat’s method is discussed. This method is a probabilistic method and is based on Fermat’s Little Theorem.

Fermat's Little Theorem:
If n is a prime number, then for every a, 1 < a < n-1,

a^{n-1 ? 1 (mod n)}
^OR
a^n-1% n = 1 
 

Example: Since 5 is prime, 2⁴ ? 1 (mod 5) [or 2⁴%5 = 1],
         3⁴ ? 1 (mod 5) and 4⁴ ? 1 (mod 5) 

         Since 7 is prime, 2⁶ ? 1 (mod 7),
         3⁶ ? 1 (mod 7), 4⁶ ? 1 (mod 7) 
         5⁶ ? 1 (mod 7) and 6⁶ ? 1 (mod 7) 

Refer this for different proofs.

If a given number is prime, then this method always returns true. If the given number is composite (or non-prime), then it may return true or false, but the probability of producing incorrect results for composite is low and can be reduced by doing more iterations.

Below is algorithm:

// Higher value of k indicates probability of correct
// results for composite inputs become higher. For prime
// inputs, result is always correct
1)  Repeat following k times:
      a) Pick a randomly in the range [2, n - 2]
      b) If gcd(a, n) ? 1, then return false
      c) If a^n-1 &nequiv; 1 (mod n), then return false
2) Return true [probably prime].

Below is the implementation of the above algorithm. The code uses power function from Modular Exponentiation

C++

// C++ program to find the smallest twin in given range
#include <bits/stdc++.h>
using namespace std;
 
/* Iterative Function to calculate (a^n)%p in O(logy) */
int power(int a, unsigned int n, int p)
{
    int res = 1;      // Initialize result
    a = a % p;  // Update 'a' if 'a' >= p
 
    while (n > 0)
    {
        // If n is odd, multiply 'a' with result
        if (n & 1)
            res = (res*a) % p;
 
        // n must be even now
        n = n>>1; // n = n/2
        a = (a*a) % p;
    }
    return res;
}
 
/*Recursive function to calculate gcd of 2 numbers*/
int gcd(int a, int b)
{
    if(a < b)
        return gcd(b, a);
    else if(a%b == 0)
        return b;
    else return gcd(b, a%b);  
}
 
// If n is prime, then always returns true, If n is
// composite than returns false with high probability
// Higher value of k increases probability of correct
// result.
bool isPrime(unsigned int n, int k)
{
   // Corner cases
   if (n <= 1 || n == 4)  return false;
   if (n <= 3) return true;
 
   // Try k times
   while (k>0)
   {
       // Pick a random number in [2..n-2]        
       // Above corner cases make sure that n > 4
       int a = 2 + rand()%(n-4);  
 
       // Checking if a and n are co-prime
       if (gcd(n, a) != 1)
          return false;
  
       // Fermat's little theorem
       if (power(a, n-1, n) != 1)
          return false;
 
       k--;
    }
 
    return true;
}
 
// Driver Program to test above function
int main()
{
    int k = 3;
    isPrime(11, k)?  cout << " true\n": cout << " false\n";
    isPrime(15, k)?  cout << " true\n": cout << " false\n";
    return 0;
}

Java

// Java program to find the 
// smallest twin in given range
 
import java.io.*;
import java.math.*;
 
class GFG {
     
    /* Iterative Function to calculate
    // (a^n)%p in O(logy) */
    static int power(int a,int n, int p)
    {
        // Initialize result
        int res = 1;
         
        // Update 'a' if 'a' >= p
        a = a % p; 
     
        while (n > 0)
        {
            // If n is odd, multiply 'a' with result
            if ((n & 1) == 1)
                res = (res * a) % p;
     
            // n must be even now
            n = n >> 1; // n = n/2
            a = (a * a) % p;
        }
        return res;
    }
     
    // If n is prime, then always returns true, 
    // If n is composite than returns false with 
    // high probability Higher value of k increases
    //  probability of correct result.
    static boolean isPrime(int n, int k)
    {
    // Corner cases
    if (n <= 1 || n == 4) return false;
    if (n <= 3) return true;
     
    // Try k times
    while (k > 0)
    {
        // Pick a random number in [2..n-2]     
        // Above corner cases make sure that n > 4
        int a = 2 + (int)(Math.random() % (n - 4)); 
     
        // Fermat's little theorem
        if (power(a, n - 1, n) != 1)
            return false;
     
        k--;
        }
     
        return true;
    }
     
    // Driver Program 
    public static void main(String args[])
    {
        int k = 3;
        if(isPrime(11, k))
            System.out.println(" true");
        else
            System.out.println(" false");
        if(isPrime(15, k))
            System.out.println(" true");
        else
            System.out.println(" false");
             
    }
}
 
// This code is contributed by Nikita Tiwari.

Python3

# Python3 program to find the smallest
# twin in given range 
import random
 
# Iterative Function to calculate 
# (a^n)%p in O(logy) 
def power(a, n, p):
     
    # Initialize result 
    res = 1
     
    # Update 'a' if 'a' >= p 
    a = a % p  
     
    while n > 0:
         
        # If n is odd, multiply 
        # 'a' with result 
        if n % 2:
            res = (res * a) % p
            n = n - 1
        else:
            a = (a ** 2) % p
             
            # n must be even now 
            n = n // 2
             
    return res % p
     
# If n is prime, then always returns true,
# If n is composite than returns false with
# high probability Higher value of k increases
# probability of correct result
def isPrime(n, k):
     
    # Corner cases
    if n == 1 or n == 4:
        return False
    elif n == 2 or n == 3:
        return True
     
    # Try k times 
    else:
        for i in range(k):
             
            # Pick a random number 
            # in [2..n-2]      
            # Above corner cases make 
            # sure that n > 4 
            a = random.randint(2, n - 2)
             
            # Fermat's little theorem 
            if power(a, n - 1, n) != 1:
                return False
                 
    return True
             
# Driver code
k = 3
if isPrime(11, k):
  print("true")
else:
  print("false")
   
if isPrime(15, k):
  print("true")
else:
  print("false")
 
# This code is contributed by Aanchal Tiwari

C#

// C# program to find the 
// smallest twin in given range
using System;
class GFG {
     
    /* Iterative Function to calculate
    // (a^n)%p in O(logy) */
    static int power(int a,int n, int p)
    {
        // Initialize result
        int res = 1;
          
        // Update 'a' if 'a' >= p
        a = a % p; 
      
        while (n > 0)
        {
            // If n is odd, multiply 'a' with result
            if ((n & 1) == 1)
                res = (res * a) % p;
      
            // n must be even now
            n = n >> 1; // n = n/2
            a = (a * a) % p;
        }
        return res;
    }
      
    // If n is prime, then always returns true, 
    // If n is composite than returns false with 
    // high probability Higher value of k increases
    //  probability of correct result.
    static bool isPrime(int n, int k)
    {
        // Corner cases
        if (n <= 1 || n == 4) return false;
        if (n <= 3) return true;
          
        // Try k times
        while (k > 0)
        {
            // Pick a random number in [2..n-2]     
            // Above corner cases make sure that n > 4
            Random rand = new Random(); 
            int a = 2 + (int)(rand.Next() % (n - 4)); 
          
            // Fermat's little theorem
            if (power(a, n - 1, n) != 1)
                return false;
          
            k--;
        }
      
        return true;
    }
     
  static void Main() {
        int k = 3;
        if(isPrime(11, k))
            Console.WriteLine(" true");
        else
            Console.WriteLine(" false");
        if(isPrime(15, k))
            Console.WriteLine(" true");
        else
            Console.WriteLine(" false");
  }
}
 
// This code is contributed by divyesh072019

PHP

<?php
// PHP program to find the 
// smallest twin in given range
 
// Iterative Function to calculate
// (a^n)%p in O(logy) 
function power($a, $n, $p)
{
     
    // Initialize result
    $res = 1; 
     
    // Update 'a' if 'a' >= p
    $a = $a % $p; 
 
    while ($n > 0)
    {
         
        // If n is odd, multiply 
        // 'a' with result
        if ($n & 1)
            $res = ($res * $a) % $p;
 
        // n must be even now
        $n = $n >> 1; // n = n/2
        $a = ($a * $a) % $p;
    }
    return $res;
}
 
// If n is prime, then always 
// returns true, If n is
// composite than returns 
// false with high probability
// Higher value of k increases
// probability of correct
// result.
function isPrime($n, $k)
{
     
    // Corner cases
    if ($n <= 1 || $n == 4) 
    return false;
    if ($n <= 3) 
    return true;
     
    // Try k times
    while ($k > 0)
    {
         
        // Pick a random number 
        // in [2..n-2] 
        // Above corner cases 
        // make sure that n > 4
        $a = 2 + rand() % ($n - 4); 
     
        // Fermat's little theorem
        if (power($a, $n-1, $n) != 1)
            return false;
     
        $k--;
    }
 
    return true;
}
 
// Driver Code
$k = 3;
$res = isPrime(11, $k) ? " true\n": " false\n";
echo($res);
 
$res = isPrime(15, $k) ? " true\n": " false\n";
echo($res);
 
// This code is contributed by Ajit.
?>

Javascript

<script>
 
// Javascript program to find the
// smallest twin in given range
 
     
/* Iterative Function to calculate
// (a^n)%p in O(logy) */
function power( a, n, p)
{
    // Initialize result
    let res = 1;
         
    // Update 'a' if 'a' >= p
    a = a % p;
     
    while (n > 0)
    {
        // If n is odd, multiply 'a' with result
        if ((n & 1) == 1)
            res = (res * a) % p;
     
        // n must be even now
        n = n >> 1; // n = n/2
        a = (a * a) % p;
    }
    return res;
}
     
// If n is prime, then always returns true,
// If n is composite than returns false with
// high probability Higher value of k increases
// probability of correct result.
function isPrime( n, k)
{
// Corner cases
if (n <= 1 || n == 4) return false;
if (n <= 3) return true;
     
// Try k times
while (k > 0)
{
    // Pick a random number in [2..n-2]    
    // Above corner cases make sure that n > 4
    let a = Math.floor(Math.random()* (n-1 - 2) + 2);
     
    // Fermat's little theorem
    if (power(a, n - 1, n) != 1)
        return false;
     
    k--;
    }
     
    return true;
}
 
 
// Driver Code
 
let k = 3;
if(isPrime(11, k))
    document.write(" true" + "</br>");
else
    document.write(" false"+ "</br>");
if(isPrime(15, k))
    document.write(" true"+ "</br>");
else
    document.write(" false"+ "</br>");
 
</script>

Output:

true
false

Time complexity: O(k Log n). Note that the power function takes O(Log n) time.

Auxiliary Space: O(min(log a, log b))
Note that the above method may fail even if we increase the number of iterations (higher k). There exist some composite numbers with the property that for every a < n and gcd(a, n) = 1 we have a^n-1 ? 1 (mod n). Such numbers are called Carmichael numbers. Fermat’s primality test is often used if a rapid method is needed for filtering, for example in the key generation phase of the RSA public key cryptographic algorithm.

We will soon be discussing more methods for Primality Testing.

References:
https://en.wikipedia.org/wiki/Fermat_primality_test
https://en.wikipedia.org/wiki/Prime_number
http://www.cse.iitk.ac.in/users/manindra/presentations/FLTBasedTests.pdf
https://en.wikipedia.org/wiki/Primality_test

Primality Test | Set 3 (Miller–Rabin)

Ajay

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Fermat Method of Primality Test

C++

Java

Python3

C#

PHP

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