Pollard’s rho algorithm is an algorithm for integer factorization. It is particularly effective at splitting composite numbers with small factors. The Rho algorithm’s most remarkable success was the factorization of eighth Fermat number: 1238926361552897 * 93461639715357977769163558199606896584051237541638188580280321. This algorithm was a good choice for F8 because the prime factor p = 1238926361552897 is much smaller than the other factor.
Example:
Input: n = 315 Output: 3 [OR 3 OR 5 OR 7] Input: n = 10 Output: 2 [OR 5 ]
Approach:
- The algorithm takes as its inputs n.
- The integer N to be factored, and g(x).
- A polynomial in x computed modulo n.
g(x) = (x^2 + 1) % n
The output is either a non-trivial factor of n or failure.
Example: Let us suppose n = 187, y = x = 2 and c = 1, Hence, our g(x) = x^2 + 1.
11 is a non-trivial factor of 187.
Below is a Java program to Implement Pollard Rho Algorithm:
Java
// Java Program to implement Pollard’s Rho Algorithmimport java.io.*; class GFG { int n = 315; // function to return gcd of a and b public int gcd(int a, int b) { // initialise gcd = 0 int gcd = 0; for (int i = 1; i <= a || i <= b; i++) { if (a % i == 0 && b % i == 0) { gcd = i; } } return gcd; } /* Function to calculate (base^exponent)%modulus */ int g(int x, int n) { return ((x * x) - 1) % n; } public static void main(String args[]) { GFG gfg = new GFG(); int n = 315; int x = 2, y = 2, d = 1; while (d == 1) { // Tortoise Move x = gfg.g(x, n); // Hare Move: y = gfg.g(gfg.g(y, n), n); /* check gcd of |x-y| and n */ d = gfg.gcd((x - y), gfg.n); } // if the algorithm fails to find prime factor if (d == gfg.n) { System.out.println( "GCD cannot be found for this element"); } else { System.out.println("One of the prime factor of " + n + " is " + d); } }} |
Output
One of the prime factor of 315 is 5
Time Complexity: O(sqrt(n))
