Gowers $ U_2 $ norm as a measure of nonlinearity for Boolean functions and their generalizations
Sugata Gangopadhyay, Constanza Riera, Pantelimon Stănică · Advances in Mathematics of Communications · 2020
In this paper, we investigate the Gowers \begin{document}$ U_2 $\end{document} norm for generalized Boolean functions, and \begin{document}$ \mathbb{Z} $\end{document} -bent functions. The Gowers \begin{document}$ U_2 $\end{document} norm of a function is a measure of its resistance to affine approximation. Although nonlinearity serves the same purpose for the classical Boolean functions, it does not extend easily to generalized Boolean functions. We first provide a framework for employing the Gowers \begin{document}$ U_2 $\end{document} norm in the context of generalized Boolean functions with cryptographic significance, in particular, we give a recurrence rule for the Gowers \begin{document}$ U_2 $\end{document} norms, and an evaluation of the Gowers \begin{document}$ U_2 $\end{document} norm of functions that are affine over spreads. We also give an introduction to \begin{document}$ \mathbb{Z} $\end{document} -bent functions, as proposed by Dobbertin and Leander [ 8 ], to provide a recursive framework to study bent functions. In the second part of the paper, we concentrate on \begin{document}$ \mathbb{Z} $\end{document} -bent functions and their \begin{document}$ U_2 $\end{document} norms. As a consequence of one of our results, we give an alternate proof to a known theorem of Dobbertin and Leander, and also find necessary and sufficient conditions for a function obtained by gluing \begin{document}$ \mathbb{Z} $\end{document} -bent functions to be bent, in terms of the Gowers \begin{document}$ U_2 $\end{document} norms of its components.