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This 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.
Random Lattice Theory
Reduction Principle of Ajtai
Learning with Error
LWE Public Key Cryptosystem
Cyclic Lattices and Ideal Lattices
Fully Homomorphic Encryption
A Generalization of NTRUencrypt

For integer factorization and discrete logarithm calculation, P.W.Shor published an effective quantum calculation in SIAM Journal on Computing in 1997, which is called the Shor algorithm in academic circles. Classical public key cryptosystems such as RSA, ECC and so on could not resist the attack of the Shor algorithm, so the major security risks of public key cryptosystems are completely exposed to the Shor algorithm and quantum computer.
In the past 20 years, the rise and development of post-quantum cryptography have close relation with the lattice cryptosystems. The academic community believes that the hard problems on lattice, such as the shortest vector problem (SVP), the continuous shortest vector problem (SIVP) and the determination of the shortest vector problem (GapSVP) can resist quantum computing effectively, so the public key cryptosystems based on the hard problems on lattice become the core theory and technology of the post-quantum cryptography. At present, there are six kinds of published post-quantum cryptosystems:
Ajtai-Dwork cryptosystem
GGH/HNF cryptosystem
NTRU cryptosystem
MacElience/Niderreiter cryptosystem
LWE cryptosystem
Fully homomorphic encryption (FHE)
In the book Modern Cryptography, we give a detailed introduction to the basic theory of lattice and the first four kinds of lattice-based cryptosystems