Gowers U2 norm as a measure of nonlinearity for Boolean functions and their generalizations
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In this paper, we investigate the Gowers U2 norm for generalized Boolean func- tions, and Z-bent functions. The Gowers U2 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 pro- vide a framework for employing the Gowers U2 norm in the context of generalized Boolean functions with cryptographic significance, in particular, we give a recurrence rule for the Gowers U2 norms, and an evaluation of the Gowers U2 norm of functions that are affine over spreads. We also give an introduction to Z-bent functions, as proposed by Dobbertin and Leander , to provide a recursive framework to study bent functions. In the second part of the paper, we concentrate on Z-bent functions and their U2 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 Z-bent functions to be bent, in terms of the Gowers U2 norms of its components.
The article of record as published may be found at https://doi.org/10.3934/amc.2019038
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