Complexities of Erez self-dual normal bases

TL;DR

This paper investigates the complexity of Erez self-dual normal bases by analyzing the number of non-simple points in minimal Besicovitch arrangements, comparing theoretical and experimental values across many primes.

Contribution

It introduces a novel method to determine the complexity of self-dual normal bases using geometric arrangements and compares these with random arrangements for numerous primes.

Findings

Expected and actual counts of non-simple points are compared.

Certain minimal arrangements share geometrical properties with those used for complexity analysis.

Results provide insights into the structure and complexity of self-dual normal bases.

Abstract

The complexities of self-dual normal bases, which are candidates for the lowest complexity basis of some defined extensions, are determined with the help of the number of all but the simple points in well chosen minimal Besicovitch arrangements. In this article, these values are first compared with the expected value of the number of all but the simple points in a minimal randomly selected Besicovitch arrangement in F d 2 for the first 370 prime numbers d. Then, particular minimal Besicovitch arrangements which share several geometrical properties with the arrangements considered to determine the complexity will be considered in two distinct cases.