Introducing an Analysis in Finite Fields

TL;DR

This paper develops a new analytical framework for finite fields, introducing derivatives, series expansions, and applications to functions relevant in coding theory and digital signal processing.

Contribution

It introduces the negacyclic Hasse derivative and finite field Taylor series, expanding analytical tools available for finite field functions.

Findings

Defined a new negacyclic Hasse derivative.

Derived finite field Taylor series and alpha-adic expansions.

Applied methods to exponential and trigonometric functions.

Abstract

Looking forward to introducing an analysis in Galois Fields, discrete functions are considered (such as transcendental ones) and MacLaurin series are derived by Lagrange's Interpolation. A new derivative over finite fields is defined which is based on the Hasse Derivative and is referred to as negacyclic Hasse derivative. Finite field Taylor series and alpha-adic expansions over GF(p), p prime, are then considered. Applications to exponential and trigonometric functions are presented. Theses tools can be useful in areas such as coding theory and digital signal processing.