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

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 article 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 then study the expected value and the variance of the number of operations performed by the algorithm. We present an exact analysis when n = p(1) and n = p(1)p(2) for p(1), p(2) prime numbers, and an 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. (C) 2001 John Wiley & Sons, Inc.