Lattice-Based Cryptosystems : A Design Perspective

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