A Note on the Higher–order Nonlinearity of Niho Function

Abstract

Sun and Wu investigated the lower bound of higher-order nonlinearity of the niho Boolean function $f(x)=T{r}_1^n(\lambda x^d)$ over ᵓ_2ⁿ; where $\lambda \in {\mathbb{F}}_{2^n}^{*},d=2^{\frac{n-1}{2}}+2^{\frac{3n-1}{4}}-1$ when n ≡ 3 mod 4 in second case of Theorem 3.6 of the paper titled higher-order nonlinearity of niho functions published in Fundamenta Informaticae 137 (2015) 403–412. Unfortunately, the proof of finding the lower bound of higher-order nonlinearities of the niho Boolean function f(x) is not correct in the above mentioned paper. In this paper the author gives the correct proof of lower bound of higher-order nonlinearities of the niho Boolean function f(x).