Normal Bases using 1-dimensional Algebraic Groups

TL;DR

This paper explores geometric methods for constructing normal bases in finite fields using algebraic groups, providing algorithms with quasi-linear complexity for efficient arithmetic operations.

Contribution

It introduces novel constructions of normal bases via additive groups, multiplicative groups, and Lucas tori, along with efficient multiplication algorithms.

Findings

Normal bases can be constructed using algebraic groups.

Algorithms achieve quasi-linear complexity for multiplication.

Methods improve efficiency of finite field arithmetic.

Abstract

This paper surveys and illustrates geometric methods for constructing normal bases allowing efficient finite field arithmetic. These bases are constructed using the additive group, the multiplicative group and the Lucas torus. We describe algorithms with quasi-linear complexity to multiply two elements given in each one of the bases.