Bent Functions and Permutation Methods: Binary and Multiple-Valued Bent Functions (Synthesis Lectures on Engineering, Science, and Technology)

Preface
Contents
List of Figures
List of Tables
1 Basic Concepts and Notations
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	1.1 Discrete Functions
	1.2 Functional Expressions
	1.3 Reed–Muller–Fourier Expressions
		1.3.1 RMF-Expressions for Ternary Functions
		1.3.2 RMF-Expressions for Quaternary Functions
	1.4 Fourier Transforms on Finite Abelian Groups
	1.5 Fast Fourier Transform
	1.6 Binary Bent Functions
	1.7 Ternary Bent Functions
	1.8 Quaternary Bent Functions
	1.9 Distribution of Function Values and Bentness
	1.10 Spectral Invariant Operations
2 Gibbs Derivatives on Finite Abelian Groups
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		2.1 Gibbs Derivative for Binary Functions
		2.2 Computing the Dyadic Gibbs Derivative
		2.3 Gibbs Derivative for Ternary Functions
		2.4 Galois Field Gibbs Derivative for Ternary Functions
		2.5 Gibbs Derivatives for Quaternary Functions
		2.6 Gibbs RMF-Derivative for Quaternary Functions
3 Gibbs Characterization of Binary Bent Functions
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	3.1 Properties of the Gibbs Dyadic Derivative of Bent Functions
	3.2 Checking if a Binary Function is Bent by Using the Gibbs Dyadic Derivative
	3.3 Gibbs Permutation Matrices
	3.4 Structure of Gibbs Permutation Matrices
	3.5 Binary Bent Functions in Four Variables
4 Gibbs Characterization of Ternary Bent Functions
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	4.1 Ternary Bent Functions for n equals 1n=1
	4.2 Spectral Invariant Operations and Ternary Bent Functions
	4.3 Gibbs Characterization of Ternary Bent Functions
		4.3.1 Ternary Bent Functions for n equals 1n=1 and Gibbs Derivatives
		4.3.2 Ternary Bent Functions for n equals 2n=2 Characterized by the VC-Gibbs Derivative
	4.4 Gibbs Permutation Matrices for Ternary Functions
	4.5 Extensions to Any Number of Variables
	4.6 Construction Algorithm
	4.7 The Galois Field Gibbs Derivatives and Ternary Bent Functions
	4.8 Gibbs Characterization of Ternary Functions and Distribution  of Function Values
5 Gibbs Characterization of a Class of Quaternary Bent Functions
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	5.1 Quaternary Bent Functions for n equals 1n=1
		5.1.1 Classes of Quaternary Bent Functions for n equals 1n=1 in Terms of the RMF-Gibbs Derivative
		5.1.2 Classes of Quaternary Bent Functions for n equals 1n=1 in Terms of the VC-Gibbs Derivative
	5.2 Quaternary Bent Functions for n equals 2n=2
	5.3 Experiments for Quaternary Bent Functions in n equals 2n=2 Variables
	5.4 Binary and Quaternary Bent Functions
	5.5 Generalized Four-Valued Bent Functions
	5.6 Construction of Generalized Boolean Bent Functions
		5.6.1 Straightforward Algorithm
		5.6.2 Construction by Permutation Matrices
		5.6.3 Construction of Bent Functions by Combination of Permutation Matrices
6 Matrix-Valued Binary Bent Functions
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	6.1 Matrix-Valued Functions
		6.1.1 Classification Method
	6.2 Classes of Binary Bent Functions for n equals 4n=4
	6.3 Classes of Binary Bent Functions for n equals 6n=6
		6.3.1 left parenthesis 2 times 2 right parenthesis(2times2)-spectra for Binary Bent Functions for n equals 6n=6
		6.3.2 left parenthesis 4 times 4 right parenthesis(4times4)-spectra for Binary Bent Functions for n equals 6n=6
	6.4 Classes for Binary Bent Functions for n equals 8n=8
		6.4.1 left parenthesis 2 times 2 right parenthesis(2times2)-spectra for Functions in n equals 8n=8 Variables
		6.4.2 left parenthesis 4 times 4 right parenthesis(4times4)-spectra for Functions in n equals 8n=8 Variables
7 Matrix-Valued Ternary Bent Functions
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	7.1 Matrix-Valued Equivalents of Bent Functions
		7.1.1 Similarity of Ternary Bent Functions
		7.1.2 Matrix-Valued Vilenkin–Chrestenson Coefficients of Linear Ternary Functions
	7.2 Classification Approach for Ternary Bent Functions
	7.3 Classes of Ternary Bent Functions for n equals 3n=3 and n equals 4n=4
		7.3.1 left parenthesis 3 times 3 right parenthesis(3times3)-spectra for Ternary Bent Functions of the Degree 33  for n equals 3n=3
		7.3.2 left parenthesis 3 times 3 right parenthesis(3times3)-spectra for Ternary Bent Functions of the Degree 44 for n equals 3n=3
		7.3.3 left parenthesis 3 times 3 right parenthesis(3times3)-spectra for Ternary Functions for n equals 4n=4
		7.3.4 Ternary Linear Functions and Ternary Bent Functions
	7.4 Construction of Ternary Bent Functions from Ternary Linear Functions
8 Construction of Bent Functions by FFT-like Permutation Matrices
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	8.1 FFT-like Permutation Matrices for Binary Bent Functions
	8.2 Permutation Matrices for Disjoint Spectral Translation
	8.3 Construction of Binary Bent Functions by FFT-like Permutation Matrices
	8.4 Gibbs Matrices and FFT-like Permutation Matrices for Binary Functions
	8.5 FFT-like Permutation Matrices for Ternary Bent Functions
		8.5.1 Kronecker Product Representable Matrices
	8.6 Computation with Permutation Matrices
	8.7 Extensions to Functions in Arbitrary Number of Variables
		8.7.1 Permutation Matrices for Disjoint Spectral Translation
		8.7.2 Permutation Matrices for Permutation of Variables
		8.7.3 Block Diagonal Permutation Matrices
		8.7.4 Block Diagonal Permutation Matrices for n equals 4n=4
		8.7.5 Shift-Based Permutation Matrices
	8.8 Construction of Ternary Bent Functions by FFT-like Permutation Matrices
	8.9 Generalizations
		8.9.1 Permutation of Subvectors
	8.10 Gibbs and FFT-like Permutation Matrices for Ternary Functions
9 Construction of Ternary Bent Functions From Matrix Representations
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	9.1 Matrix Representations of Ternary Bent Functions
	9.2 Construction of Ternary Bent Functions from Matrix …