book

Bent Functions

Name: Bent Functions
Author: Natalia Tokareva
ISBN: 9780128025550

by Natalia Tokareva

August 2015

Intermediate to advanced

220 pages

7h 50m

English

Academic Press

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Cover image
Title page
Table of Contents
Copyright
Foreword
Preface
Notation
Notation
Chapter 1: Boolean Functions
AbstractIntroduction1.1 Definitions1.2 Algebraic Normal Form1.3 Boolean Cube and Hamming Distance1.4 Extended Affinely Equivalent Functions1.5 Walsh-Hadamard Transform1.6 Finite Field and Boolean Functions1.7 Trace Function1.8 Polynomial Representation of a Boolean Function1.9 Trace Representation of a Boolean Function1.10 Monomial Boolean Functions
Chapter 2: Bent Functions: An Introduction
AbstractIntroduction2.1 Definition of a Nonlinearity2.2 Nonlinearity of a Random Boolean Function2.3 Definition of a Bent Function2.4 If n Is Odd?2.5 Open Problems2.6 Surveys
Chapter 3: History of Bent Functions
AbstractIntroduction3.1 Oscar Rothaus3.2 V.A. Eliseev and O.P. Stepchenkov3.3 From the 1970s to the Present

Chapter 4: Applications of Bent Functions
AbstractIntroduction4.1 Cryptography: Linear Cryptanalysis and Boolean Functions4.2 Cryptography: One Historical Example4.3 Cryptography: Bent Functions in CAST4.4 Cryptography: Bent Functions in Grain4.5 Cryptography: Bent Functions in HAVAL4.6 Hadamard Matrices and Graphs4.7 Links to Coding Theory4.8 Bent Sequences4.9 Mobile Networks, CDMA4.10 Remarks
Chapter 5: Properties of Bent Functions
AbstractIntroduction5.1 Degree of a Bent Function5.2 Affine Transformations of Bent Functions5.3 Rank of a Bent Function5.4 Dual Bent Functions5.5 Other Properties
Chapter 6: Equivalent Representations of Bent Functions
AbstractIntroduction6.1 Hadamard Matrices6.2 Difference Sets6.3 Designs6.4 Linear Spreads6.5 Sets of Subspaces6.6 Strongly Regular Graphs6.7 Bent Rectangles
Chapter 7: Bent Functions with a Small Number of Variables
AbstractIntroduction7.1 Two and Four Variables7.2 Six Variables7.3 Eight Variables7.4 Ten and More Variables7.5 Algorithms for Generation of Bent Functions7.6 Concluding Remarks
Chapter 8: Combinatorial Constructions of Bent Functions
AbstractIntroduction8.1 Rothaus’s Iterative Construction8.2 Maiorana-McFarland Class8.3 Partial Spreads: PS+, PS−8.4 Dillon’s Bent Functions: PSap8.5 Dobbertin’s Construction8.6 More Iterative Constructions8.7 Minterm Iterative Constructions8.8 Bent Iterative Functions: BI8.9 Other Constructions
Chapter 9: Algebraic Constructions of Bent Functions
AbstractIntroduction9.1 An Algebraic Approach9.2 Bent Exponents: General Properties9.3 Gold Bent Functions9.4 Dillon Exponent9.5 Kasami Bent Functions9.6 Canteaut-Leander Bent Functions (MF-1)9.7 Canteaut-Charpin-Kuyreghyan Bent Functions (MF-2)9.8 Niho Exponents9.9 General Algebraic Approach9.10 Other Constructions
Chapter 10: Bent Functions and Other Cryptographic Properties
AbstractIntroduction10.1 Cryptographic Criteria10.2 High Degree and Balancedness10.3 Correlation Immunity and Resiliency10.4 Algebraic Immunity10.5 Vectorial Bent Functions, Almost Bent Functions, and Almost Perfect Nonlinear Functions
Chapter 11: Distances Between Bent Functions
AbstractIntroduction11.1 The Minimal Possible Distance Between Bent Functions11.2 Classification of Bent Functions at the Minimal Distance from the Quadratic Bent Function11.3 Upper Bound for the Number of Bent Functions at the Minimal Distance from an Arbitrary Bent Function11.4 Bent Functions at the Minimal Distance from a McFarland Bent Function11.5 Locally Metrically Equivalent Bent Functions11.6 The Graph of Minimal Distances of Bent Functions
Chapter 12: Automorphisms of the Set of Bent Functions
AbstractIntroduction12.1 Preliminaries12.2 Shifts of the Class of Bent Functions12.3 Duality Between Definitions of Bent and Affine Functions12.4 Automorphisms of the Set of Bent Functions12.5 Metrically Regular Sets
Chapter 13: Bounds on the Number of Bent Functions
AbstractIntroduction13.1 Preliminaries13.2 The Number of Bent Functions for Small n13.3 Upper Bounds13.4 Direct Lower Bounds13.5 Iterative Lower Bounds13.6 Lower Bound from the Bent Iterative Functions13.7 Testing of the Lower Bound for Small n13.8 Asymptotic Problem and Hypotheses
Chapter 14: Bent Decomposition Problem
AbstractIntroduction14.1 Preliminaries14.2 Partial Results14.3 Boolean Function as the Sum of a Constant Number of Bent Functions14.4 Any Cubic Boolean Function in Eight Variables is the Sum of at Most Four Bent Functions14.5 Decomposition of Dual Bent Functions
Chapter 15: Algebraic Generalizations of Bent Functions
AbstractIntroduction15.1 Preliminaries15.2 The q-Valued Bent Functions15.3 The p-ary Bent Functions15.4 Bent Functions Over a Finite Field15.5 Bent Functions Over Quasi-Frobenius Local Rings15.6 Generalized Boolean Bent Functions (of Schmidt)15.7 Bent Functions from a Finite Abelian Group into the Set of Complex Numbers on the Unit Circle15.8 Bent Functions from a Finite Abelian Group into a Finite Abelian Group15.9 Non-Abelian Bent Functions15.10 Vectorial G-Bent Functions15.11 Multidimensional Bent Functions on a Finite Abelian Group
Chapter 16: Combinatorial Generalizations of Bent Functions
AbstractIntroduction16.1 Symmetric Bent Functions16.2 Homogeneous Bent Functions16.3 Rotation-Symmetric Bent Functions16.4 Normal Bent Functions16.5 Self-Dual and Anti-Self-Dual Bent Functions16.6 Partially Defined Bent Functions16.7 Plateaued Functions16.8 Z-Bent Functions16.9 Negabent Functions, Bent4-Functions, and I-Bent Functions
Chapter 17: Cryptographic Generalizations of Bent Functions
AbstractIntroduction17.1 Semibent Functions (Near-Bent Functions)17.2 Balanced (Semi-) Bent Functions17.3 Partially Bent Functions17.4 Hyperbent Functions17.5 Bent Functions of Higher Order17.6 k-Bent Functions
References
Index

Content preview from Bent Functions

Chapter 13

Bounds on the Number of Bent Functions

Abstract

One of the most important questions in the theory of bent functions is how many there are. If n is equal to 2, 4, 6, or 8, the number of bent functions in n variables is 8, 896, about 2^32.3, and about 2^106.29, respectively. If n > 10, the exact number of bent functions in n variables is still unknown. Moreover, there is a large gap between the lower ( $2^{2^{(n / 2) + {log}_{2} (n - 2) - 1}}$ ) and upper ( $2^{2^{n - 1} + \frac{1}{2} (\binom{n}{n / 2})}$ ) bounds for this number. There have been several improvements of these bounds, but they are ...

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Bent Functions

by Natalia Tokareva

Bounds on the Number of Bent Functions

Abstract

Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
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