A characterization of a class of maximum nonlinear functions

TL;DR

This paper characterizes a class of maximum nonlinear functions over finite fields, focusing on their distance to subspace characteristic functions, with implications for cryptography.

Contribution

It provides a new characterization of Gold power functions based on their coordinate functions' distance to subspace characteristic functions.

Findings

Gold power functions are characterized by their distance to subspace characteristic functions.

Maximum nonlinear functions have large Hamming distance to hyperplane characteristic functions.

The characterization aids in understanding cryptographic properties of these functions.

Abstract

Maximum nonlinear functions on finite fields are widely used in cryptography because the coordinate functions have large distance to linear functions. More precisely, the Hamming distance to the characteristic functions of hyperplanes is large. One class of maximum nonlinear functions are the Gold power functions We characterize these functions in terms of the distance of their coordinate functions to characteristic functions of subspaces of codimension 2.