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Bent Functions
Results and Applications to Cryptography
1st Edition - August 6, 2015
Author: Natalia Tokareva
Language: English
Paperback ISBN:9780128023181
9 7 8 - 0 - 1 2 - 8 0 2 3 1 8 - 1
eBook ISBN:9780128025550
9 7 8 - 0 - 1 2 - 8 0 2 5 5 5 - 0
Bent Functions: Results and Applications to Cryptography offers a unique survey of the objects of discrete mathematics known as Boolean bent functions. As these maximal, nonlinear…Read more
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Bent Functions: Results and Applications to Cryptography
offers a unique survey of the objects of discrete mathematics known as Boolean bent functions. As these maximal, nonlinear Boolean functions and their generalizations have many theoretical and practical applications in combinatorics, coding theory, and cryptography, the text provides a detailed survey of their main results, presenting a systematic overview of their generalizations and applications, and considering open problems in classification and systematization of bent functions.
The text is appropriate for novices and advanced researchers, discussing proofs of several results, including the automorphism group of bent functions, the lower bound for the number of bent functions, and more.
Provides a detailed survey of bent functions and their main results, presenting a systematic overview of their generalizations and applications
Presents a systematic and detailed survey of hundreds of results in the area of highly nonlinear Boolean functions in cryptography
Appropriate coverage for students from advanced specialists in cryptography, mathematics, and creators of ciphers
Researchers in discrete math and cryptography; students and professors of math and IT departments
Foreword
Preface
Notation
Notation
Chapter 1: Boolean Functions
Abstract
Introduction
1.1 Definitions
1.2 Algebraic Normal Form
1.3 Boolean Cube and Hamming Distance
1.4 Extended Affinely Equivalent Functions
1.5 Walsh-Hadamard Transform
1.6 Finite Field and Boolean Functions
1.7 Trace Function
1.8 Polynomial Representation of a Boolean Function
1.9 Trace Representation of a Boolean Function
1.10 Monomial Boolean Functions
Chapter 2: Bent Functions: An Introduction
Abstract
Introduction
2.1 Definition of a Nonlinearity
2.2 Nonlinearity of a Random Boolean Function
2.3 Definition of a Bent Function
2.4 If n Is Odd?
2.5 Open Problems
2.6 Surveys
Chapter 3: History of Bent Functions
Abstract
Introduction
3.1 Oscar Rothaus
3.2 V.A. Eliseev and O.P. Stepchenkov
3.3 From the 1970s to the Present
Chapter 4: Applications of Bent Functions
Abstract
Introduction
4.1 Cryptography: Linear Cryptanalysis and Boolean Functions
4.2 Cryptography: One Historical Example
4.3 Cryptography: Bent Functions in CAST
4.4 Cryptography: Bent Functions in Grain
4.5 Cryptography: Bent Functions in HAVAL
4.6 Hadamard Matrices and Graphs
4.7 Links to Coding Theory
4.8 Bent Sequences
4.9 Mobile Networks, CDMA
4.10 Remarks
Chapter 5: Properties of Bent Functions
Abstract
Introduction
5.1 Degree of a Bent Function
5.2 Affine Transformations of Bent Functions
5.3 Rank of a Bent Function
5.4 Dual Bent Functions
5.5 Other Properties
Chapter 6: Equivalent Representations of Bent Functions
Abstract
Introduction
6.1 Hadamard Matrices
6.2 Difference Sets
6.3 Designs
6.4 Linear Spreads
6.5 Sets of Subspaces
6.6 Strongly Regular Graphs
6.7 Bent Rectangles
Chapter 7: Bent Functions with a Small Number of Variables
Abstract
Introduction
7.1 Two and Four Variables
7.2 Six Variables
7.3 Eight Variables
7.4 Ten and More Variables
7.5 Algorithms for Generation of Bent Functions
7.6 Concluding Remarks
Chapter 8: Combinatorial Constructions of Bent Functions
Abstract
Introduction
8.1 Rothaus’s Iterative Construction
8.2 Maiorana-McFarland Class
8.3 Partial Spreads: PS+, PS−
8.4 Dillon’s Bent Functions: PSap
8.5 Dobbertin’s Construction
8.6 More Iterative Constructions
8.7 Minterm Iterative Constructions
8.8 Bent Iterative Functions: BI
8.9 Other Constructions
Chapter 9: Algebraic Constructions of Bent Functions
Abstract
Introduction
9.1 An Algebraic Approach
9.2 Bent Exponents: General Properties
9.3 Gold Bent Functions
9.4 Dillon Exponent
9.5 Kasami Bent Functions
9.6 Canteaut-Leander Bent Functions (MF-1)
9.7 Canteaut-Charpin-Kuyreghyan Bent Functions (MF-2)
9.8 Niho Exponents
9.9 General Algebraic Approach
9.10 Other Constructions
Chapter 10: Bent Functions and Other Cryptographic Properties
Abstract
Introduction
10.1 Cryptographic Criteria
10.2 High Degree and Balancedness
10.3 Correlation Immunity and Resiliency
10.4 Algebraic Immunity
10.5 Vectorial Bent Functions, Almost Bent Functions, and Almost Perfect Nonlinear Functions
Chapter 11: Distances Between Bent Functions
Abstract
Introduction
11.1 The Minimal Possible Distance Between Bent Functions
11.2 Classification of Bent Functions at the Minimal Distance from the Quadratic Bent Function
11.3 Upper Bound for the Number of Bent Functions at the Minimal Distance from an Arbitrary Bent Function
11.4 Bent Functions at the Minimal Distance from a McFarland Bent Function
11.5 Locally Metrically Equivalent Bent Functions
11.6 The Graph of Minimal Distances of Bent Functions
Chapter 12: Automorphisms of the Set of Bent Functions
Abstract
Introduction
12.1 Preliminaries
12.2 Shifts of the Class of Bent Functions
12.3 Duality Between Definitions of Bent and Affine Functions
12.4 Automorphisms of the Set of Bent Functions
12.5 Metrically Regular Sets
Chapter 13: Bounds on the Number of Bent Functions
Abstract
Introduction
13.1 Preliminaries
13.2 The Number of Bent Functions for Small n
13.3 Upper Bounds
13.4 Direct Lower Bounds
13.5 Iterative Lower Bounds
13.6 Lower Bound from the Bent Iterative Functions
13.7 Testing of the Lower Bound for Small n
13.8 Asymptotic Problem and Hypotheses
Chapter 14: Bent Decomposition Problem
Abstract
Introduction
14.1 Preliminaries
14.2 Partial Results
14.3 Boolean Function as the Sum of a Constant Number of Bent Functions
14.4 Any Cubic Boolean Function in Eight Variables is the Sum of at Most Four Bent Functions
14.5 Decomposition of Dual Bent Functions
Chapter 15: Algebraic Generalizations of Bent Functions
Abstract
Introduction
15.1 Preliminaries
15.2 The q-Valued Bent Functions
15.3 The p-ary Bent Functions
15.4 Bent Functions Over a Finite Field
15.5 Bent Functions Over Quasi-Frobenius Local Rings
15.6 Generalized Boolean Bent Functions (of Schmidt)
15.7 Bent Functions from a Finite Abelian Group into the Set of Complex Numbers on the Unit Circle
15.8 Bent Functions from a Finite Abelian Group into a Finite Abelian Group
15.9 Non-Abelian Bent Functions
15.10 Vectorial G-Bent Functions
15.11 Multidimensional Bent Functions on a Finite Abelian Group
Chapter 16: Combinatorial Generalizations of Bent Functions
Abstract
Introduction
16.1 Symmetric Bent Functions
16.2 Homogeneous Bent Functions
16.3 Rotation-Symmetric Bent Functions
16.4 Normal Bent Functions
16.5 Self-Dual and Anti-Self-Dual Bent Functions
16.6 Partially Defined Bent Functions
16.7 Plateaued Functions
16.8 Z-Bent Functions
16.9 Negabent Functions, Bent4-Functions, and I-Bent Functions
Chapter 17: Cryptographic Generalizations of Bent Functions
Abstract
Introduction
17.1 Semibent Functions (Near-Bent Functions)
17.2 Balanced (Semi-) Bent Functions
17.3 Partially Bent Functions
17.4 Hyperbent Functions
17.5 Bent Functions of Higher Order
17.6 k-Bent Functions
References
Index
No. of pages: 220
Language: English
Edition: 1
Published: August 6, 2015
Imprint: Academic Press
Paperback ISBN: 9780128023181
eBook ISBN: 9780128025550
NT
Natalia Tokareva
Dr. Natalia Tokareva is a senior researcher at the Laboratory of Discrete Analysis in the Sobolev Institute of Mathematics and she teaches courses in cryptology in the Department of Mathematics and Mechanics at Novosibirsk State University. She has studied bent functions and their applications for several years, publishing one monograph (in Russian) and more than 12 articles. She has been a participant of many international conferences and seminars and presentations in the area of bent functions, particularly with applications in cryptography. Her research interests include Boolean functions in cryptography, bent functions, block and stream ciphers, cryptanalysis, coding theory, combinatorics, and algebra. She is chief of the seminar "Cryptography and Cryptanalysis" at the Sobolev Institute of Mathematics and she supervises BS, MS, and PhD students in discrete mathematics and cryptology.
Affiliations and expertise
Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russian Federation
Department of Mathematics and Mechanics, Novosibirsk State University, Novosibirsk, Russian Federation