On differential properties of a class of Niho-type power function

Abstract

This paper deals with Niho functions which are one of the most important classes of functions thanks to their close connections with a wide variety of objects from mathematics, such as spreads and oval polynomials or from applied areas, such as symmetric cryptography, coding theory and sequences. In this paper, we investigate specifically the $c$-differential uniformity of the power function $F(x)=x^{s(2^m-1)+1}$ over the finite field ${\mathbb{F}}_{2^n}$, where $n=2m$, $m$ is odd and $s=(2^k+1{)}^{-1}$ is the multiplicative inverse of $2^k+1$ modulo $2^m+1$, and show that the $c$-differential uniformity of $F(x)$ is $2^{\gcd (k,m)}+1$ by carrying out some subtle manipulation of certain equations over ${\mathbb{F}}_{2^n}$. Notably, $F(x)$ has a very low $c$-differential uniformity equals $3$ when $k$ and $m$ are coprime.