On differential properties of a class of Niho-type power functions
Zhexin Wang, Nian Li, Xiangyong Zeng · Advances in Mathematics of Communications · 2025
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 coding theory, symmetric cryptography, and sequences. In this paper, we investigate specifically the $ c $-differential uniformity of the Niho 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} $. It is noteworthy that $ F(x) $ has a very low $ c $-differential uniformity that equals $ 3 $ when $ k $ and $ m $ are coprime.