Question

Devise a recursive algorithm for finding xn (mod m) whenever n, x and m are positive integers. Also prove that your algorithm is correct. You should use the fact that xn (mod m) = (xn−1 (mod m) · x (mod m)) (mod m).

Answer 1

The student's question involves creating a recursive algorithm for finding x^n mod m and proving its correctness. The algorithm is based on the fact that x^n mod m can be represented as a product of x^(n-1) mod m and x mod m modulo m. The correctness can be proven using mathematical induction.

Step-by-step explanation:

The student is asking for a recursive algorithm to compute x^n mod m, where n, x, and m are positive integers. The recursive relation is based on the fact that x^n mod m is equivalent to the product of x^(n-1) mod m and x mod m, all taken modulo m.

A recursive algorithm would look like this:

To prove the algorithm is correct, we can use induction. For the base case when n = 0, the result is 1, which is correct. Assuming the calculation is correct for n-1, for n we have x^n mod m = (x^(n-1) mod m * x mod m) mod m, which is exactly the recurrence used in the algorithm.