Construction of logarithm tables for Galois Fields

Abstract

A branch of mathematics commonly used in cryptography is Galois Fields GF(p n ). Two basic operations performed in GF(p n ) are the addition and the multiplication. While the addition is generally easy to compute, the multiplication requires a special treatment. A well-known method to compute the multiplication is based on logarithm and antilogarithm tables. A primitive element of a GF(p n ) is a key part in the construction of such tables, but it is generally hard to find a primitive element for arbitrary values of p and n. This article presents a naive algorithm that can simultaneously find a primitive element of GF(p n ) and construct its corresponding logarithm and antilogarithm tables. The proposed algorithm was tested in GF(p n ) for several values of p and n; the results show a good performance, having an average time of 0.46 seconds to find the first primitive element of a given GF(p n ) for values of n = {2, 3, 4, 5, 8, 12} and prime values p between 2 and 97.