Analysis of Rabin's irreducibility test for polynomials over finite fields

Abstract

We give a precise average-case analysis of Rabin's algorithm for testing the irreducibility of polynomials over finite fields. The main technical contribution of the paper is the study of the probability that a random polynomial of degree n contains an irreducible factor of degree dividing several maximal divisors of the degree n. We study the expected value and the variance of the number of operations performed by the algorithm. We present a exact analysis when n=p1 and n=p1p2 for p1,p2 prime numbers, and as asymptotic analysis for the general case. Our method generalizes to other algorithms that deal with similar divisor conditions. In particular, we analyze the average-case number of operations for two variants of Rabin's algorithm, and determine the ordering of prime divisors of n that minimizes the leading factor.