有限域小波及其在密码学和译码中的应用

1.3 Linear Algebra Background
1.3.1 Involution
1.3.2 Sesquilinear Form
1.3.3 Unitary Matrix
1.3.4 Paraunitary Matrix
I Finite-Field Wavelets
2 Background Review and Motivation
2.1 Wavelets for Discrete-Time Signals
2.2 Cyclic Wavelet Transforms
2.3 Review of Transforms over Finite Fields
2.3.1 Discrete Fourier Transform over Finite Fields
2.3.2 Base-Field Transforms over Finite Fields
2.3.3 Related Work on Finite-Field Wavelets
3 Finite-Field Wavelet Basis Functions
3.1 Finite-Field Discrete-Time Basis
3.1.1 Non-Degenerate Bilinear Form
3.1.2 Orthonormal Wavelet Basis over Finite Fields
3.1.3 Completeness of the Orthonormal Set
3.2 Construction of Mother Wavelet and Scaling Function
3.3 Summary
4 Theory of Paraunitary Filter Banks over Fields of Characteristic 2
4.1 Background Review
4.1.1 Degree-1 Paraunitary Building Block over GF(2)
4.1.2 Degree-2 Paraunitary Building Blocks over GF(2)
4.1.3 Lapped Orthogonal Transforms over G F(2)
4.2 Unitary Matrices over GF(2r)
4.3 Paraunitary Matrices over Fields of Characteristic 2
4.3.1 Properties of 2 x 2 Paraunitary Matrices over GF (2r)
4.4 Factorization of Paraunitary Matrices over GF (2r)
4.4.1 Degree-1 Paraunitary Building Block over GF (2r)
4.4.2 Degree-2 Paraunitary Building Block over GF (2r)
4.4.3 Degree-2r Paraunitary Building Block over GF (2r)
4.4.4 Factorization of 2 x 2 Paraunitary Matrices over GF (2r)
4.4.5 Degree-Mr Paraunitary Building Block over GF (2r)
4.4.6 Factorization ofM x M Paraunitary Matrices over GF (2r)
4.5 Summary
II Multivariate Cryptography
5 Introduction
5.1 Historical Background and Motivation
5.2 RSA
5.3 Elliptic Curve Cryptography
5.4 Multivariate Cryptography
6 Wavelet Self-Synchronizing Stream Cipher
6.1 Background Review
6.1.1 Classification of Stream Ciphers
6.2 Wavelet Self-Synchronizing Stream Cipher (WSSC)
6.2.1 Modified Wavelet Transform
6.2.2 Basic Round of the WSSC
6.2.3 Multiple Rounds of the WSSC
6.2.4 Key Setup
6.3 Cryptanalysis of the WSSC
6.3.1 Interpolation Attack
6.3.2 Algebraic Attacks
6.3.3 Delta Attack
6.3.4 Time-Memory Tradeoff Attack
6.3.5 Divide-and-Conquer Attack
6.3.6 Correlation and Distinguishing Attacks
6.4 Performance Evaluation
6.5 Summary
7 Wavelet Block Cipher
7.1 Background Review
7.1.1 Feistel Cipher and Data Encryption Standard (DES)
7.1.2 Advanced Encryption Standard (AES)
7.2 Wavelet Block Cipher (WBC)
7.2.1 Linear Components of the WBC
7.2.2 Nonlinear Components of the WBC
7.3 Two-Round Wavelet Block Cipher
7.3.1 Key Setup
7.4 Cryptanalysis of the WBC
7.4.1 Differential and Linear Attacks
7.4.2 Divide-and-Conquer Attack
7.4.3 Interpolation Attack
7.4.4 Delta Attack
7.5 Performance Evaluation
7.6 Summary
8 Paraunitary Public-Key Cryptography
8.1 Background Review
8.1.1 Signature Based on Birational Permutations
8.1.2 Tame Transformation Methods
8.1.3 Tractable Rational Map Cryptosystem
8.1.4 C* Algorithm and its Variants
8.2 Paraunitary Asymmetric Cryptosystem (PAC)
8.2.1 Bijective Mappings
8.2.2 Polynomial Vector
8.2.3 Setup Algorithms
8.3 Probabilistic PAC
8.4 On the Computational Security of the PAC
8.5 A Practical Instance of the PAC
8.5.1 Constructing the Polynomial Vector
8.5.2 Complexity of the PAC
8.6 Cryptanalysis of the Instance of the PAC
8.6.1 Grobner Basis
8.6.2 Univariate Polynomial Representation of the Public Polynomials
8.6.3 XL and FXL Algorithms
8.6.4 An Attack for Small r
8.7 Paraunitary Digital Signature Scheme (PDSS)
8.7.1 Polynomial Vector
8.7.2 Setup Algorithm
8.7.3 A Practical Instance of the PDSS
8.8 Summary
III Error-Control Coding
9 Some Basic Concepts of Error-Control Coding
10 Double-Circulant Wavelet Block Codes
10.1 Structure of Double-Circulant Wavelet Coding
10.1.1 Wavelet Structures for Encoding and Decoding
10.2 Maximum-Distance Separable Codes
10.3 Double-Circulant Self-Dual Codes
10.3.1 Fundamental Structure of Self-Dual Wavelet Codes
10.3.2 Maximum-Distance Separable Self-Dual Codes
10.4 Decoding Wavelet Codes
10.4.1 Bounded-Distance Decoding of (20, 10, 6) Double-Circulant Wavelet Code
10.4.2 Bounded-Distance Decoding of the Wavelet-Golay Code
10.5 Summary
11 Arbitrary-Rate Wavelet Block Codes
11.1 Structure of Wavelet Coding
11.1.1 Wavelet Structure for Encoding and Decoding
11.2 Rate-1/L Maximum-Distance Separable Codes
11.3 Arbitrary-Rate Wavelet Block Codes
11.4 Arbitrary-Rate Maximum-Distance Separable Codes
11.5 Decoding Arbitrary-Rate Wavelet Block Codes
11.5.1 Bounded-Distance Decoding of the (12, 4, 6) Wavelet Code
11.5.2 Symbol Error Correction in the (7, 3, 5) MDS Code
11.5.3 Tail-Biting Trellises for Wavelet Block Codes
11.6 Summary
12 Wavelet Convolutional Codes
12.1 Structure of Wavelet Convolutional Codes
12.2 Algebraic Properties of Wavelet Convolutional Encoders
12.3 Syndrome Generators and Dual Encoders
12.4 Self-Dual and Self-Orthogonal Convolutional Codes
12.5 Time-Varying Wavelet Convolutional Codes and Bipartite Trellises
12.6 Summary
Appendices
A Proofs of Chap. 4 in Part I
B Efficient Generation of PU Matrices
C Toy Examples of the PAC and the PDSS
D Proofs of Chap. 10 in Part III
E Brief Review of Trellis Structures
Bibliography
Index