Constructing Permutation Rational Functions From Isogenies

TL;DR

This paper presents an efficient method to generate permutation rational functions over large finite fields using elliptic curve isogenies, with potential cryptographic applications.

Contribution

It introduces a novel algorithm leveraging elliptic curve isogenies and Fried's modular interpretation to construct permutation rational functions.

Findings

Algorithm efficiently generates permutation rational functions

Applicable to large finite fields

Potential cryptographic uses discussed

Abstract

A permutation rational function $f\in \mathbb{F}_q(x)$ is a rational function that induces a bijection on $\mathbb{F}_q$, that is, for all $y\in\mathbb{F}_q$ there exists exactly one $x\in\mathbb{F}_q$ such that $f(x)=y$. Permutation rational functions are intimately related to exceptional rational functions, and more generally exceptional covers of the projective line, of which they form the first important example. In this paper, we show how to efficiently generate many permutation rational functions over large finite fields using isogenies of elliptic curves, and discuss some cryptographic applications. Our algorithm is based on Fried's modular interpretation of certain dihedral exceptional covers of the projective line (Cont. Math., 1994).