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New characterizations of generalized Boolean functions

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Applicable Algebra in Engineering, Communication and Computing Aims and scope

Abstract

This paper focuses on providing the characteristics of generalized Boolean functions from a new perspective. We first generalize the classical Fourier transform and correlation spectrum into what we will call the \(\rho\)-Walsh–Hadamard transform (\(\rho\)-WHT) and the \(\rho\)-correlation spectrum, respectively. Then a direct relationship between the \(\rho\)-correlation spectrum and the \(\rho\)-WHT is presented. We investigate the characteristics and properties of generalized Boolean functions based on the \(\rho\)-WHT and the \(\rho\)-correlation spectrum, as well as the sufficient (or also necessary) conditions and subspace decomposition of \(\rho\)-bent functions. We also derive the \(\rho\)-autocorrelation for a class of generalized Boolean functions on \((n+2)\)-variables. Secondly, we present a characterization of a class of generalized Boolean functions with \(\rho\)-WHT in terms of the classical Boolean functions. Finally, we demonstrate that \(\rho\)-bent functions can be obtained from a class of composite construction if and only if \(\rho =1\). Some examples of non-affine \(\rho\)-bent functions are also provided.

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Notes

  1. The set of all Boolean functions in n variables is denoted by \(\mathcal {B}_{n}\), \(f\in \mathcal {B}_{n}\).

  2. The set of all generalized Boolean functions in n variables is denoted by \(\mathcal{G}\mathcal{B}_{n}^{q}\), \(\mathfrak {f}\in \mathcal{G}\mathcal{B}_{n}^{q}\).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Huaibei Normal University, Huaibei, 235000, Anhui, China

    Zhiyao Yang

  2. School of Mathematics and Statistics, Fujian Normal University, Fuzhou, 350117, Fujian, China

    Zhiyao Yang & Pinhui Ke

  3. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, Henan, China

    Zuling Chang

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  1. Zhiyao Yang
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  2. Pinhui Ke
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  3. Zuling Chang
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Correspondence to Pinhui Ke.

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This work is supported by the National Natural Science Foundation of China (Nos. 62272420, 61902140); Provincial Natural Science Foundation of Fujian (No. 2023J01535).

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Yang, Z., Ke, P. & Chang, Z. New characterizations of generalized Boolean functions. AAECC (2024). https://doi.org/10.1007/s00200-024-00650-w

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