This open access book covers the most cutting-edge and hot research topics and fields of post-quantum cryptography. The main purpose of this book is to focus on the computational complexity theory of lattice ciphers, especially the reduction principle of Ajtai, in order to fill the gap that post-quantum ciphers focus on the implementation of encryption and decryption algorithms, but the theoretical proof is insufficient. In Chapter 3, Chapter 4 and Chapter 6, author introduces the theory and technology of LWE distribution, LWE cipher and homomorphic encryption in detail. When using random analysis tools, there is a problem of "ambiguity" in both definition and algorithm. The greatest feature of this book is to use probability distribution to carry out rigorous mathematical definition and mathematical demonstration for various unclear or imprecise expressions, so as to make it a rigorous theoretical system for classroom teaching and dissemination. Chapters 5 and 7 further expand and improve the theory of cyclic lattice, ideal lattice and generalized NTRU cryptography.
This book is used as a professional book for graduate students majoring in mathematics and cryptography, as well as a reference book for scientific and technological personnel engaged in cryptography research.
Author(s): Zhiyong Zheng, Kun Tian, Fengxia Liu
Series: Financial Mathematics and Fintech
Publisher: Springer
Year: 2022
Language: English
Pages: 202
City: Singapore
Preface
Contents
Notations
1 Random Lattice Theory
1.1 Fourier Transform
1.2 Discrete Gauss Measure
1.3 Smoothing Parameter
1.4 Some Properties of Discrete Gauss Distribution
2 Reduction Principle of Ajtai
2.1 Random Linear System
2.2 SIS Problem
2.3 INCGDD Problem
2.4 Reduction Principle
3 Learning with Error
3.1 Circulant Matrix
3.2 SIS and Knapsack Problem on Ring
3.3 LWE Problem
3.4 Proof of the Main Theorem
3.4.1 From LWE to DGS
3.4.2 From DGS to Hard Problems on Lattice
3.4.3 From D-LWE to LWE
4 LWE Public Key Cryptosystem
4.1 LWE Cryptosystem of Regev
4.2 The Proof of Security
4.3 Properties of Rounding Function
4.4 General LWE-Based Cryptosystem
4.5 Probability of Decryption Error for General Disturbance
5 Cyclic Lattices and Ideal Lattices
5.1 Some Basic Properties of Lattice
5.2 Ideal Matrices
5.3 φ-Cyclic Lattice
5.4 Improved Upper Bound for Smoothing Parameter
6 Fully Homomorphic Encryption
6.1 Definitions and Examples
6.2 Gadget Matrix and Gadget Technique
6.3 Bounded Fully Homomorphic Encryption
6.4 Construction of Gentry
6.5 Attribute-Based Encryption
7 A Generalization of NTRUencrypt
7.1 φ-Cyclic Code
7.2 A Generalization of NTRUencrypt
Appendix References
