Given a number n, check if it is prime or not. We have introduced and discussed School and Fermat methods for primality testing.
In this page, 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.