ANALYSIS OF EUCLIDEAN ALGORITHMS FOR POLYNOMIALS OVER FINITE-FIELDS

This paper analyzes the Euclidean algorithm and some variants of it for computingthe greatest common divisor of two univariate polynomials over a finite field. The minimum, maximum and average number of arithmetic operations both on polynomials and in the ground field are derived. We consider five different algorithms to compute gcd(A1, A2) where A1, A2∈Z2[x] have degrees m≥n≥0. Compared with the classical Euclidean algorithm that needs on average 1/2n+1 polynomial divisions, two algorithms involving divisions need on average 1/3n+O(1) and 1/4n+O(1) polynomial divisions; two other algorithms use an average of 1/2m+1/3n+O(1) and 1/4m+2/9n+O(1) polynomial substractions and no divisions. © 1990, Academic Press Limited. All rights reserved.