An Equivalent Representation of Generalized Differentials

TL;DR

This paper introduces an equivalent formula for higher-order derivatives in the context of Generalized Almost Perfect Nonlinear functions over finite fields, using combinatorial counting methods.

Contribution

It provides a novel combinatorial approach to represent higher-order derivatives, enhancing understanding of their structure in finite field functions.

Findings

Derived an equivalent formula for higher-order derivatives

Connected derivative properties to subset sum counts in prime fields

Explored diversity questions of higher-order derivatives

Abstract

We propose an equivalent formula for the higher-order derivatives used in the study of Generalized Almost Perfect Nonlinear functions over an arbitrary finite field of characteristic $p$. The result is obtained by counting the number of subsets of the prime field with a fixed cardinality for which the sum of their elements is constant. We then ask related questions regarding the diversity of higher-order derivatives.