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 wellknown 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.
 Publication:

International Journal of Mathematical Education in Science and Technology
 Pub Date:
 January 2011
 DOI:
 10.1080/0020739X.2010.510215
 Bibcode:
 2011IJMES..42...91T
 Keywords:

 Galois Field;
 logarithm tables;
 multiplication;
 algorithm