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

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.