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.
- 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