The c-differential properties of a class of power functions

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 \(p\neq3\) is an odd prime and \(n\) is a positive integer. In this paper, for all primes \(p\neq3\), 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 \(p^n \equiv 3 \pmod 4\), 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 \(p^n \equiv 1 \pmod 4\), 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.