Tangent-Chebyshev rational maps and Redei functions

TL;DR

This paper demonstrates that Lima and Campello de Souza's new rational functions over finite fields are conjugate to Redei functions, establishes their properties, and introduces analogous functions and a new finite field trigonometry system.

Contribution

It reveals the conjugacy between new rational functions and Redei functions, proves their properties, and extends the framework to characteristic 2 fields with a novel trigonometry system.

Findings

New properties of Redei and related rational functions

Conjugacy between new functions and Redei functions

Introduction of finite field trigonometry for characteristic 2

Abstract

Recently Lima and Campello de Souza introduced a new class of rational functions over odd-order finite fields, and explained their potential usefulness in cryptography. We show that these new functions are conjugate to the classical family of Redei rational functions, so that the properties of the new functions follow from properties of Redei functions. We also prove new properties of these functions, and introduce analogous functions in characteristic 2, while also introducing a new version of trigonometry over finite fields of even order, which is of independent interest.