A fast algorithm to compute irreducible and primitive polynomials in finite fields

Abstract

In this paper we present a method to computeall the irreducible and primitive polynomials of degreem over the finite fieldGF(q). Our method finds each new irreducible or primitive polynomial with a complexity ofO(m) arithmetic operations inGF(q). The best previously known methods [3], [10] use the Berlekamp-Massey algorithm [7] and they have a complexityO(m2). We reach mis improvement taking into account a systolic implementation [2] of the extended Euclidean algorithm instead of using the Berlekamp-Massey algorithm.