Gowers U_2 norm as a measure of nonlinearity for Boolean functions and their generalizations

Abstract

<p style='text-indent:20px;'>In this paper, we investigate the Gowers <inline-formula><tex-math id="M2">\begin{document}$ U_2 $\end{document}</tex-math></inline-formula> norm for generalized Boolean functions, and <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula>-bent functions. The Gowers <inline-formula><tex-math id="M4">\begin{document}$ U_2 $\end{document}</tex-math></inline-formula> norm of a function is a measure of its resistance to affine approximation. Although nonlinearity serves the same purpose for the classical Boolean functions, it does not extend easily to generalized Boolean functions. We first provide a framework for employing the Gowers <inline-formula><tex-math id="M5">\begin{document}$ U_2 $\end{document}</tex-math></inline-formula> norm in the context of generalized Boolean functions with cryptographic significance, in particular, we give a recurrence rule for the Gowers <inline-formula><tex-math id="M6">\begin{document}$ U_2 $\end{document}</tex-math></inline-formula> norms, and an evaluation of the Gowers <inline-formula><tex-math id="M7">\begin{document}$ U_2 $\end{document}</tex-math></inline-formula> norm of functions that are affine over spreads. We also give an introduction to <inline-formula><tex-math id="M8">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula>-bent functions, as proposed by Dobbertin and Leander [<xref ref-type="bibr" rid="b8">8</xref>], to provide a recursive framework to study bent functions. In the second part of the paper, we concentrate on <inline-formula><tex-math id="M9">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula>-bent functions and their <inline-formula><tex-math id="M10">\begin{document}$ U_2 $\end{document}</tex-math></inline-formula> norms. As a consequence of one of our results, we give an alternate proof to a known theorem of Dobbertin and Leander, and also find necessary and sufficient conditions for a function obtained by <i>gluing</i> <inline-formula><tex-math id="M11">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula>-bent functions to be bent, in terms of the Gowers <inline-formula><tex-math id="M12">\begin{document}$ U_2 $\end{document}</tex-math></inline-formula> norms of its components.