Construction of self-dual normal bases and their complexity

TL;DR

This paper presents explicit constructions of self-dual normal bases in finite field extensions, analyzes their complexity through computational searches, and finds they often have minimal complexity compared to general normal bases.

Contribution

It provides a constructive method for self-dual normal bases and identifies their low complexity in small extensions via computer search.

Findings

Self-dual normal bases often have the lowest complexity among normal bases.

Explicit construction methods for self-dual normal bases are developed.

Computational results for small degree extensions are presented.

Abstract

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.