The AKS Primality Test

Abstract

In 2002 Manindra Agrawal, Neeraj Kayal, and Nitin Saxena discovered an algorithm to test a number for primality that is both deterministic and runs in polynomial time. The AKS algorithm hinges on a calculated value they call r, which is defined for a given integer n >1 as the least value for which the order of n modulo r is greater than log 2 over 2 n. This r has a proven upper bound of log 5 over 2 n. In this paper, we prove that 2 + log 2 over 2 n is a lower bound of the value r, and if n is a square, there is a lower bound of 1 + 2 log 2 over 2 n . We also present data suggesting that 3 log 2 over 2 n is a smaller upper bound of r. If this is indeed an upper bound, the AKS Primality Test is shown to have a time complexity of O(log 6 + ε over 2 n) in bit operations for any small ε greater than 0. Data also suggests a number n is a square if and only if its corresponding r is greater than 3 log 2 over 2 n.