Elliptic butterflies

TL;DR

This paper explores recursive algorithms for elliptic functions, enabling efficient evaluation and interpolation with applications in finite fields, coding, and cryptography.

Contribution

It introduces a recursive approach based on elliptic butterflies, extending classical methods to elliptic functions with complexity scaling as d log d.

Findings

Existence of straight-line programs with complexity d log d

Applications to finite field arithmetic, coding theory, cryptography

Extension of classical butterfly algorithms to elliptic functions

Abstract

We study natural evaluation and interpolation problems for elliptic functions and prove that they allow a recursive treatment using a variant of classical butterflies first introduced by Gauss. We deduce the existence of straight-line programs with complexity scaling with $d\log(d)$ for these problems and present applications to finite field arithmetic, coding theory and cryptography.