1 Introduction

1.1 Cryptography based on number theory problems

1.2 Post-quantum cryptography

1.3 Lattice-based cryptosystems

2. Lattices

2.1 Definition

2.2 Discrete Gaussian

2.3 Small Integer Solutions

2.4 Learning with Errors

2.5 Trapdoor Generation

3. Public-key encryption

3.1 Definition

3.2 PKE against CPA attacks

3.2 Achieving CCA-Security in the random oracle model

3.3 CCA-secure PKE in the standard model

4. Identity-based encryption

4.1 Definition

4.2 The GPV IBE scheme

4.3 Short IBE in the standard model

5. Attribute-based encryption

5.1 Definition

5.2 ABE supporting and-gate policy

5.3 ABE supporting flexible threshold policy

5.4 Functional encryption

6. Key Exchanges

6.1 Definition

6.2 Key exchange against passive attacks

6.3 Implicit authenticated key exchange

6.4 Password-based authenticate key exchange

7. Digital signatures

7.1 Definition

7.2 Signatures in the random oracle model

7.3 Short signatures with tight security

7.4 Group signatures

This book focuses on lattice-based cryptosystems, widely considered to be one of the most promising post-quantum cryptosystems and provides fundamental insights into how to construct provably secure cryptosystems from hard lattice problems. The concept of provable security is used to inform the choice of lattice tool for designing cryptosystems, including public-key encryption, identity-based encryption, attribute-based encryption, key change and digital signatures. Given its depth of coverage, the book especially appeals to graduate students and young researchers who plan to enter this research area.