Generation of Ternary Bent Functions by Spectral Invariant Operations in the Generalized Reed-Muller Domain

Boolean functions expressing some particular properties often appear in engineering practice. Therefore, a lot of research efforts are put into exploring different approaches towards classification of Boolean functions with respect to various criteria that are typically selected to serve some specific needs of the intended applications. A classification is considered to be strong if there is a reasonably small number of different classes for a given number of variables n and it it desir able that classification rules are simple. A classification with respect to Walsh spectral coefficients, introduced formerly for digital system design purposes, appears to be useful in the context of Boolean functions used in cryptography, since it is in a way compatible with characterization of cryptographically interesting functions through Walsh spectral coefficients. This classification is performed in terms of certain spectral invariant operations. We show by introducing a new spectral invariant operation in the Walsh domain, that by starting from n?5, some classes of Boolean functions can be merged which makes the classification stronger, and from the theoretical point of view resolves a problem raised already in seventies of the last century. Further, this new spectral invariant operation can be used in constructing bent functions from bent functions represented by quadratic forms.

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The Lower Bounds on the Second Order Nonlinearity of Bent Functions and Semi-bent Functions

JOURNAL OF ELECTRONICS INFORMATION TECHNOLOGY ◽

10.3724/sp.j.1146.2010.00191 ◽

2010 ◽

Vol 32 (10) ◽

pp. 2521-2525

Author(s):

Xue-lian Li ◽

Yu-pu Hu ◽

Jun-tao Gao

Keyword(s):

Lower Bounds ◽

Second Order ◽

Bent Functions ◽

Second Order Nonlinearity

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Further study on constructing bent functions outside the completed Maiorana–McFarland class

IET Information Security ◽

10.1049/iet-ifs.2018.5425 ◽

2020 ◽

Vol 14 (6) ◽

pp. 654-660

Author(s):

Shishi Liu ◽

Fengrong Zhang ◽

Enes Pasalic ◽

Shixiong Xia ◽

Zepeng Zhuo

Keyword(s):

Bent Functions

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Systematic Constructions of Rotation Symmetric Bent Functions, 2-Rotation Symmetric Bent Functions, and Bent Idempotent Functions

IEEE Transactions on Information Theory ◽

10.1109/tit.2016.2621751 ◽

2017 ◽

Vol 63 (7) ◽

pp. 4658-4667 ◽

Cited By ~ 8

Author(s):

Sihong Su ◽

Xiaohu Tang

Keyword(s):

Bent Functions ◽

Rotation Symmetric

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Translation uniqueness of phase retrieval and magnitude retrieval of band-limited signals

Journal of Applied Analysis ◽

10.1515/jaa-2021-2052 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Lung-Hui Chen

Keyword(s):

Fourier Transform ◽

Entire Function ◽

Convex Hull ◽

Scattering Theory ◽

Phase Retrieval ◽

Indicator Function ◽

Band Limited ◽

The Fourier Transform ◽

Complex Valued ◽

Spectral Invariant

Abstract In this paper, we discuss how to partially determine the Fourier transform F ⁢ ( z ) = ∫ - 1 1 f ⁢ ( t ) ⁢ e i ⁢ z ⁢ t ⁢ 𝑑 t , z ∈ ℂ , F(z)=\int_{-1}^{1}f(t)e^{izt}\,dt,\quad z\in\mathbb{C}, given the data | F ⁢ ( z ) | {\lvert F(z)\rvert} or arg ⁡ F ⁢ ( z ) {\arg F(z)} for z ∈ ℝ {z\in\mathbb{R}} . Initially, we assume [ - 1 , 1 ] {[-1,1]} to be the convex hull of the support of the signal f. We start with reviewing the computation of the indicator function and indicator diagram of a finite-typed complex-valued entire function, and then connect to the spectral invariant of F ⁢ ( z ) {F(z)} . Then we focus to derive the unimodular part of the entire function up to certain non-uniqueness. We elaborate on the translation of the signal including the non-uniqueness associates of the Fourier transform. We show that the phase retrieval and magnitude retrieval are conjugate problems in the scattering theory of waves.

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