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 $m\ge 4$ 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.