A new proof to complexity of dual basis of a type I optimal normal basis

TL;DR

This paper presents a new proof for the complexity of the dual basis of a type I optimal normal basis in finite fields, simplifying previous results by leveraging polynomial basis duality.

Contribution

It provides a novel proof for the known complexity of dual bases in type I optimal normal bases using polynomial basis duality techniques.

Findings

Confirmed the complexity as 3n-3 for even q

Confirmed the complexity as 3n-2 for odd q

Simplified the proof of existing results

Abstract

The complexity of dual basis of a type I optimal normal basis of $\mathbb{F}_{q^n}$ over $\mathbb{F}_{q}$ was determined to be $3n-3$ or $3n-2$ according as $q$ is even or odd, respectively, by Z.-X. Wan and K. Zhou in 2007. We give a new proof to this result by clearly deriving the dual of a type I optimal normal basis with the aid of a lemma on the dual of a polynomial basis.