Near bent functions on a hyperplane
The authors show that the Kasami-Welch near bent function \(f(x) = \text{Tr}(x^{4^k-2^k+1})\) from \(\mathbb F_{2^n}\) to \(\mathbb F_2\) (\(n\) odd) restricted to the hyperplane \(H = \{x \in \mathbb F_{2^n}\mid\text{Tr}(x) = 0\}\) of \(\mathbb F_{2^n}\) is bent for \(n = 3k \pm 1\). Moreover \(f(x) = \text{Tr}(x^{4^k-2^k+1})\) is not bent when restricted to any other hyperplane. In order to prove this result a criterion for near bent functions to be bent on hyperplanes is given. With this criterion the authors also reprove a similar result on bent restrictions to a hyperplane of Gold functions first shown in \textit{J. Lahtonen, G. McGuire} and \textit{H. N. Ward} [Adv. Math. Commun. 1, 243--250 (2007)].
scientific article; zbMATH DE number 5316923