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，这类函数因其与数学中多种对象（如扩域和椭圆多项式）以及应用领域（如对称密码学、编码理论和序列）的紧密联系而成为最重要的一类函数。本文具体研究了有限域$\mathbb{F}_{2^n}$上幂函数$F(x)=x^{s(2^m-1)+1}$的$c$-微分均匀性，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，并通过巧妙处理$\mathbb{F}_{2^n}$上的某些方程，证明$F(x)$的$c$-微分均匀性为$2^{\gcd(k,m)}+1$。值得注意地，当$k$与$m$互素时，$F(x)$具有非常低的$c$-微分均匀性，等于$3$。