The c-differential properties of a class of power functions

Abstract

Power functions with low $c$-differential uniformity have been widely studied not only because of their strong resistance to multiplicative differential attacks, but also low implementation cost in hardware. Furthermore, the $c$-differential spectrum of a function gives a more precise characterization of its $c$-differential properties. Let $f(x)=x^{\frac{p^n+3}{2}}$ be a power function over the finite field ${\mathbb{F}}_{p^n}$, where $p\ne 3$ is an odd prime and $n$ is a positive integer. In this paper, for all primes $p\ne 3$, by investigating certain character sums with regard to elliptic curves and computing the number of solutions of a system of equations over ${\mathbb{F}}_{p^n}$, we determine explicitly the $(-1)$-differential spectrum of $f$ with a unified approach. We show that if $p^n\equiv 3(\mathrm{mod}4)$, then $f$ is a differentially $(-1,3)$-uniform function except for $p^n\in \{7,19,23\}$ where $f$ is an APcN function, and if $p^n\equiv 1(\mathrm{mod}4)$, the $(-1)$-differential uniformity of $f$ is equal to $4$. In addition, an upper bound of the $c$-differential uniformity of $f$ is also given.