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

Sugata Gangopadhyay; Constanza Riera; Pantelimon Stănică

doi:10.3934/amc.2020056

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2021, Volume 15, Issue 2: 241-256. Doi: 10.3934/amc.2020056

This issue Previous Article Construction of minimal linear codes from multi-variable functions Next Article Verifying solutions to LWE with implications for concrete security

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

1.
Department of Computer Science and Engineering, Indian Institute of Technology Roorkee, Roorkee 247667, India
2.
Department of Computer Science, Electrical Engineering and Mathematical Sciences, Western Norway University of Applied Sciences, 5020 Bergen, Norway
3.
Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA 93943, USA

^* Corresponding author: Pantelimon Stăniă
^* Corresponding author: Pantelimon Stăniă

Received: July 2019

Revised: October 2019

Early access: January 2020

Published: May 2021

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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.

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Mathematics Subject Classification: Primary: 94C10; Secondary: 06E30.

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