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.

翻译：具有低$c$-差分一致性的幂函数不仅因其对乘法差分攻击的强抵抗性而被广泛研究，还因其在硬件实现中的低成本优势。此外，函数的$c$-差分谱能更精确地表征其$c$-差分性质。设$f(x)=x^{\frac{p^n+3}{2}}$为有限域$\mathbb{F}_{p^{n}}$上的幂函数，其中$p\neq3$为奇素数，$n$为正整数。本文针对所有素数$p\neq3$，通过研究椭圆曲线上的特征和并计算$\mathbb{F}_{p^{n}}$上方程组解的数量，采用统一方法明确确定了$f$的$(-1)$-差分谱。我们证明：若$p^n \equiv 3 \pmod 4$，则$f$是差分$(-1,3)$-一致函数，除非$p^n\in\{7,19,23\}$时$f$为APcN函数；若$p^n \equiv 1 \pmod 4$，则$f$的$(-1)$-差分一致性等于$4$。此外，本文还给出了$f$的$c$-差分一致性的一个上界。