Results on the Gowers U2 Norm of Generalized Boolean Functions

Abstract

Recently, a framework for employing the Gowers [Formula: see text] norm in the context of (generalized) Boolean functions with cryptographic significance was introduced. In this paper, we first give tight bounds on the Gowers [Formula: see text] norm of generalized Boolean functions via the (generalized) sum-of-squares indicator. Secondly, we provide a framework for the generalized signal-to-noise ratio ([Formula: see text]) of generalized [Formula: see text]-functions. We characterize the [Formula: see text] in terms of the Gowers [Formula: see text] norm. In particular, we present a direct link between the [Formula: see text] of a class of generalized Boolean functions and the [Formula: see text] of its component Boolean functions. Finally, the expressions of the Gowers [Formula: see text] norm of generalized Boolean functions from some well-known secondary constructions (the concatenation and Carlet’s construction) are obtained.