Resumen de Construction of logarithm tables for Galois Fields

José Torres-Jiménez, Nelson Rangel, Ana Loreto González-Hernández, Himer Avila-George

• A branch of mathematics commonly used in cryptography is Galois Fields GF(pn). Two basic operations performed in GF(pn) 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(pn) 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(pn) and construct its corresponding logarithm and antilogarithm tables. The proposed algorithm was tested in GF(pn) 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(pn) for values of n = {2, 3, 4, 5, 8, 12} and prime values p between 2 and 97.