On arithmetical algorithms over finite fields

Abstract

Standard methods for calculating over G F( p n ), the finite field of p n elements, require an irreducible polynomial of degree n with coefficients in G F( p). Such a polynomial is usually obtained by choosing it randomly and then verifying that it is irreducible, using a probabilistic algorithm. If it is not, the procedure is repeated. Here we given an explicit basis, with multiplication table, for the fields G F( p p k ), for k = 0, 1, 2,…, and their union. This leads to efficient computational methods, not requiring the preliminary calculation of irreducible polynomials over finite fields and, at the same time, yields a simple recursive formula for irreducible polynomials which generate the fields. The fast Fourier transform (FFT) is a method for efficiently evaluating (or interpolating) a polynomial of degree < n at all of the nth roots of unity, i.e., on the finite multiplicate subgroups of F, in O( nlog n) operations in the underlying field. We give an analogue of the fast Fourier transform which efficiently evaluates a polynomial on some of the additive subgroups of F. This yields new “fast” algorithms for polynomial computation.