Complete characterization of the differential spectrum of a Niho type power function

Abstract

Power functions with Niho exponents have attracted considerable attention due to their important applications in sequence design, coding theory, and cryptography. This paper investigates the differential properties of Niho type power functions of the form $F(x)=x^{s(2^m-1)+1}$ over ${\mathbb{F}}_{2^{2m}}$ with $2\le s\le 2^m$. We first establish a general characterization of the differential spectrum of $F(x)$ having at most three nonzero values via its Walsh spectrum. Focusing subsequently on the case $s=(2^k+1{)}^{-1}(\mathrm{mod}{2}^m+1)$ where $\gcd (k,m)=e$, we employ a refined analysis of the number of solutions to certain equations over finite fields. Specifically, it is proved that $F(x)$ is locally differentially $2^e$-uniform when $\gcd (2^k-1,2^m+1)=2^e+1$ and locally differentially $(2^{2e}-2^e)$-uniform when $\gcd (2^k-1,2^m+1)=1$, and their differential spectra are completely determined. These results completely characterize the differential properties of this family and yield new infinite families of locally differentially $4$-uniform power functions.