A class of locally differentially $4$-uniform power functions with Niho exponents

TL;DR

This paper analyzes a specific Niho exponent power function over finite fields, determining its differential spectrum and establishing it as locally 4-uniform, which advances understanding in cryptography and sequence design.

Contribution

It explicitly computes the differential spectrum of the power function with Niho exponent $3q - 2$, showing it is locally 4-uniform, a novel result in the study of such functions.

Findings

The power function $F(x) = x^{3q - 2}$ is locally differentially 4-uniform.

The differential spectrum of $F$ is explicitly determined.

This work extends the understanding of differential properties of Niho exponent functions.

Abstract

Niho exponents have found important applications in sequence design, coding theory, and cryptography. Determining the differential spectrum of a power function with Niho exponent is a topic of considerable interest. In this paper, we investigate the power function $F(x) = x^{3q - 2}$ over $\mathbb{F}_{q^2}$, where $q = 2^m$ and $m\geq 4$ is an even integer. Notably, the exponent $3q - 2$ is a Niho exponent. By analyzing the properties of certain polynomials over $\mathbb{F}_{q^2}$, we determine the differential spectrum of $F$. Our results show that $F$ is locally differentially $4$-uniform, which complements existing results on the differential spectra of power functions with Niho exponents.