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.

翻译：本文研究Niho函数，这是一类最重要的函数，因为它们与数学中的多种对象（如铺展和卵形多项式）或应用领域（如对称密码学、编码理论和序列）密切相关。本文具体研究了有限域$\mathbb{F}_{2^n}$上的幂函数$F(x)=x^{s(2^m-1)+1}$的$c$-差分均匀性，其中$n=2m$，$m$为奇数，$s=(2^k+1)^{-1}$是$2^k+1$模$2^m+1$的乘法逆元。通过对$\mathbb{F}_{2^n}$上某些方程进行精细操作，我们证明了$F(x)$的$c$-差分均匀性为$2^{\gcd(k,m)}+1$。值得注意的是，当$k$与$m$互素时，$F(x)$的$c$-差分均匀性非常低，等于$3$。