Cryptographic Boolean Functions and Applications

Chapter 1: A Bit of History

• Abstract
• 1.1. George Boole (1815–1864)
• 1.2. Claude Elwood Shannon (1916–2001)
• References

Chapter 2: Fourier Analysis of Boolean Functions

• Abstract
• 2.1. Basic Definitions for Boolean Functions
• 2.2. Walsh Transform
• 2.3. Autocorrelation Function
• 2.4. Walsh Transform on Subspaces
• 2.5. Linear Transformations and the Sign Function
• 2.6. Parseval Equation
• 2.7. Asymptotic Results on Walsh Coefficients
• 2.8. Probability Distributions
• 2.9. Hadamard Matrices and Nonlinearity Bounds
• 2.10. Fast Walsh Transform
• 2.11. LFSRs and Linear Complexity
• 2.12. The Berlekamp–Massey Algorithm
• 2.13. De Bruijn Sequences
• References

Chapter 3: Avalanche and Propagation Criteria

• Abstract
• 3.1. Introduction
• 3.2. Counting SAC Functions
• 3.3. Counting Balanced SAC Functions
• 3.4. Higher Order SAC
• 3.5. Propagation Criteria
• 3.6. Higher Order PC(k)
• 3.7. Construction of SAC(k) and PC(k) Functions
• References

Chapter 4: Correlation Immune and Resilient Boolean Functions

• Abstract
• 4.1. Introduction
• 4.2. Basic Properties of Correlation Immunity
• 4.3. LFSRs and Correlation Immunity
• 4.4. Counting Correlation Immune Functions
• 4.5. Resilient Functions
• 4.6. Tradeoff Between Correlation Immunity and Degree
• 4.7. Connections with Orthogonal Arrays
• 4.8. Constructing Correlation Immune Functions
• 4.9. Tradeoff Between Correlation Immunity and Nonlinearity
• 4.10. Some Computational Data
• References

Chapter 5: Bent Boolean Functions

• Abstract
• 5.1. Introduction
• 5.2. Definitions and Background
• 5.3. Characterizations of the Bent Property
• 5.4. Meier and Staffelbach's Approach
• 5.5. Degree of a Bent Function
• 5.6. New From Old Bent Functions
• 5.7. Rothaus's Construction
• 5.8. Maiorana and McFarland's Construction
• 5.9. Dillon's Construction
• 5.10. Dobbertin's Construction
• 5.11. Carlet's Construction
• 5.12. Normal and Nonnormal Bent Functions
• 5.13. Counting Bent Functions
• 5.14. Partially Bent Functions
• 5.15. Semi-bent Functions
• References

Chapter 6: Special Types of Boolean Functions

• Abstract
• 6.1. Symmetric Functions
• 6.2. Rotation Symmetric Functions
• 6.3. k-Rotation Symmetric Functions
• 6.4. Balanced Functions
• 6.5. Cryptographic Boolean Functions with Biased Inputs
• References

Chapter 7: Stream Cipher Design

• Abstract
• 7.1. Introduction
• 7.2. Boolean Functions in Pseudorandom Bit Generators
• 7.3. Nonlinear Combination Generators
• 7.4. Nonlinear Filter Generators
• 7.5. Multiplexer Generator
• 7.6. Irregularly Clocked LFSRs in Generators
• 7.7. Algebraic and Linearization Attacks
• 7.8. The eStream Project
• 7.9. AIDA and Cube Attacks on Tweakable Symmetric Ciphers
• References

Chapter 8: Block Ciphers

• Abstract
• 8.1. Some History
• 8.2. Introduction
• 8.3. Block Ciphers' Modes of Operation
• 8.4. Design Approaches
• 8.5. Notable Symmetric Ciphers
• 8.6. Periods of Rijndael Transformations
• 8.7. Algebraic Representations of Rijndael/AES
• 8.8. Embedding AES in BES
• 8.9. Further Embeddings of AES
• References

Chapter 9: Boolean Cayley Graphs

• Abstract
• 9.1. Introduction
• 9.2. Spectra of Boolean Cayley Graphs
• 9.3. Few Spectral Coefficients of Boolean Functions
• 9.4. Bent Boolean Cayley Graphs
• 9.5. Coloring the Boolean Cayley Graph
• 9.6. Avalanche Features of the Cayley Graphs
• 9.7. Sensitivity of Hamming Weight of f to Spec(Γf)
• 9.8. Boolean Cayley Graphs Under Affine Transformations
• References