Primality Test | Set 2 (Fermat Method)

3.6

Given a number n, check if it is prime or not. We have introduced and discussed School method for primality testing in Set 1.

Primality Test | Set 1 (Introduction and School Method)

In this post, Fermat’s method is discussed. This method is a probabilistic method and is based on below Fermat’s Little Theorem.

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

an-1 ≡ 1 (mod n)
 OR 
an-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 given number is composite (or non-prime), then it may return true or false, but the probability of producing incorrect result 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 an-1 ≢ 1 (mod n), then return false
2) Return true [probably prime]. 

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

// 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;
}

// 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);  

       // 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;
}

Output:

true
false

Time complexity of this solution is O(k Log n). Note that power function takes O(Log n) time.

Note that the above method may fail even if we increase number of iterations (higher k). There exist sum composite numbers with the property that for every a < n, an-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 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

This article is contributed by Ajay. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above

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