Elliptic periods for finite fields

We construct two new families of basis for finite field extensions. Basis in the first family, the so-called elliptic basis, are not quite normal basis, but they allow very fast Frobenius exponentiation while preserving sparse multiplication formulas. Basis in the second family, the so-called normal elliptic basis are normal basis and allow fast (quasi linear) arithmetic. We prove that all extensions admit models of this kind

Abelian Groups, Gauss Periods, and Normal Bases

AbstractA result on finite abelian groups is first proved and then used to solve problems in finite fields. Particularly, all finite fields that have normal bases generated by general Gauss periods are characterized and it is shown how to find normal bases of low complexity

Fast Encoding and Decoding of Gabidulin Codes

Gabidulin codes are the rank-metric analogs of Reed-Solomon codes and have a major role in practical error control for network coding. This paper presents new encoding and decoding algorithms for Gabidulin codes based on low-complexity normal bases. In addition, a new decoding algorithm is proposed based on a transform-domain approach. Together, these represent the fastest known algorithms for encoding and decoding Gabidulin codes.Comment: 5 pages, 1 figure, to be published at ISIT 200

Construction of self-dual normal bases and their complexity

Recent work of Pickett has given a construction of self-dual normal bases for extensions of finite fields, whenever they exist. In this article we present these results in an explicit and constructive manner and apply them, through computer search, to identify the lowest complexity of self-dual normal bases for extensions of low degree. Comparisons to similar searches amongst normal bases show that the lowest complexity is often achieved from a self-dual normal basis