Elliptic periods for finite fields

TL;DR

This paper introduces two novel basis constructions for finite field extensions, enabling faster arithmetic operations and Frobenius exponentiation, with broad applicability across all extension types.

Contribution

It presents the elliptic basis and normal elliptic basis, new structures that improve computational efficiency in finite fields.

Findings

Elliptic basis allows fast Frobenius exponentiation.

Normal elliptic basis enables quasi-linear arithmetic.

All finite field extensions admit these basis models.

Abstract

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.