Fast transforms over finite fields of characteristic two

TL;DR

This paper introduces efficient algorithms for basis conversions in finite fields of characteristic two, improving polynomial evaluation techniques and enabling faster computations for arbitrary and specially constructed subspaces.

Contribution

It generalizes basis conversion algorithms to arbitrary subspaces and constructs subspaces with smooth degrees for enhanced performance.

Findings

New algorithms outperform existing methods on certain subspaces.

Efficient basis conversions facilitate faster polynomial evaluations.

Applicable to arbitrary subspaces with improved computational complexity.

Abstract

An additive fast Fourier transform over a finite field of characteristic two efficiently evaluates polynomials at every element of an $\mathbb{F}_2$-linear subspace of the field. We view these transforms as performing a change of basis from the monomial basis to the associated Lagrange basis, and consider the problem of performing the various conversions between these two bases, the associated Newton basis, and the '' novel '' basis of Lin, Chung and Han (FOCS 2014). Existing algorithms are divided between two families, those designed for arbitrary subspaces and more efficient algorithms designed for specially constructed subspaces of fields with degree equal to a power of two. We generalise techniques from both families to provide new conversion algorithms that may be applied to arbitrary subspaces, but which benefit equally from the specially constructed subspaces. We then construct subspaces of fields with smooth degree for which our algorithms provide better performance than existing algorithms.