Quantum Cryptography and Post-Quantum Cryptography

9.

Lattice-Based Cryptography

9.1.

Mathematical Foundations

9.1.1.

Lattice Theory Basics

9.1.1.1.

Lattice Definition

9.1.1.2.

Basis and Fundamental Region

9.1.1.3.

Lattice Problems

9.1.2.

Shortest Vector Problem

9.1.2.1.

Problem Definition

9.1.2.2.

Computational Hardness

9.1.2.3.

Approximation Variants

9.1.3.

Closest Vector Problem

9.1.3.1.

Problem Definition

9.1.3.2.

Relationship to SVP

9.1.3.3.

Hardness Results

9.1.4.

Learning with Errors Problem

9.1.4.1.

Problem Definition

9.1.4.2.

Hardness Assumptions

9.1.4.3.

Variants and Generalizations

9.1.5.

Ring Learning with Errors

9.1.5.1.

Polynomial Ring Structure

9.1.5.2.

Efficiency Advantages

9.1.5.3.

Security Considerations

9.1.6.

Module Learning with Errors

9.1.6.1.

Generalization of LWE and Ring-LWE

9.1.6.2.

Security and Efficiency Trade-offs

9.2.

Cryptographic Constructions

9.2.1.

Public-Key Encryption

9.2.1.1.

LWE-Based Encryption

9.2.1.2.

Ring-LWE-Based Encryption

9.2.1.3.

Security Analysis

9.2.2.

Key Exchange Mechanisms

9.2.2.1.

Lattice-Based Key Agreement

9.2.2.2.

Error Reconciliation

9.2.2.3.

Security Proofs

9.2.3.

Digital Signatures

9.2.3.1.

Hash-and-Sign Approaches

9.2.3.2.

Fiat-Shamir Signatures

9.2.3.3.

Rejection Sampling Techniques

9.3.

Specific Algorithms

9.3.1.

9.3.1.1.

Algorithm Description

9.3.1.2.

Parameter Selection

9.3.1.3.

Security Analysis

9.3.1.4.

Implementation Considerations

9.3.2.

9.3.2.1.

Key Encapsulation Mechanism

9.3.2.2.

Module-LWE Foundation

9.3.2.3.

NIST Standardization

9.3.2.4.

Performance Characteristics

9.3.3.

9.3.3.1.

Signature Scheme Design

9.3.3.2.

Module-LWE Security

9.3.3.3.

Implementation Features

9.3.3.4.

Standardization Status

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10. Code-Based Cryptography