Monomial generalized almost perfect nonlinear functions

TL;DR

This paper investigates monomial GAPN functions over finite fields of odd characteristic, identifying those with maximum or minimum algebraic degrees, thus extending the understanding of nonlinear functions in cryptography.

Contribution

It characterizes all monomial GAPN functions with extremal algebraic degrees in odd characteristic fields, a novel extension of previous work on APN functions.

Findings

Identified all monomial GAPN functions with maximum algebraic degree.

Identified all monomial GAPN functions with minimum algebraic degree.

Extended the classification of nonlinear functions to odd characteristic fields.

Abstract

Generalized almost perfect nonlinear (GAPN) functions were defined to satisfy some generalizations of basic properties of almost perfect nonlinear (APN) functions for even characteristic. In this paper, we study monomial GAPN functions for odd characteristic. In particular, we give all monomial GAPN functions whose algebraic degree are maximum or minimum on a finite field of odd characteristic.