Gowers U2 norm as a measure of nonlinearity for Boolean functions and their generalizations

Sugata Gangopadhyay, Constanza Riera, Pantelimon Stanica

Abstract: In this paper, we investigate the Gowers $ U_2 $ norm for generalized Boolean functions, and $ \mathbb{Z} $-bent functions. The Gowers $ U_2 $ 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 $ U_2 $ norm in the context of generalized Boolean functions with cryptographic significance, in particular, we give a recurrence rule for the Gowers $ U_2 $ norms, and an evaluation of the Gowers $ U_2 $ norm of functions that are affine over spreads. We also give an introduction to $ \mathbb{Z} $-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 $ \mathbb{Z} $-bent functions and their $ U_2 $ 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 $ \mathbb{Z} $-bent functions to be bent, in terms of the Gowers $ U_2 $ norms of its components.