Lattices and their application in Cryptography

Inhaltsverzeichnis (Table of Contents)

• Introduction
  • Outline
• Mathematical Background
  • Lattices and Lattice Reduction Problems
    • Definitions and Properties
    • Lattice Problems
    • Lattice-Reduction.
  • Asymmetric Cryptosystems and Digital Signatures
    • Asymmetric Cryptography
    • Digital Signatures
• Lattice-based Cryptosystems
  • GGH-Cryptosystem.
    • Construction
    • Attacks
  • Comparison to other cryptosystems.
• Conclusion and Future Work

Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)

This thesis aims to explore the characteristics of lattice-based cryptosystems, providing a comprehensive analysis of their construction, security, and potential applications. The primary focus is on the use of lattices to address vulnerabilities and inefficiencies in traditional encryption and signature schemes, particularly those vulnerable to quantum computing attacks. The thesis examines how lattice-based cryptography leverages worst-case lattice problems to achieve strong security guarantees.

• The security and hardness of lattice problems in cryptography.
• The construction and analysis of lattice-based cryptosystems.
• The comparison of lattice-based cryptosystems with other existing methods.
• The potential of lattice-based cryptography in addressing contemporary security challenges.
• The relationship between linear algebra, complexity theory, and public-key cryptography in the context of lattice-based cryptosystems.

Zusammenfassung der Kapitel (Chapter Summaries)

• Chapter 1: Introduction - This chapter introduces the importance of secure communication in today's digital world, highlighting the need for robust and reliable cryptography. It discusses the emergence of public-key cryptography and its limitations, particularly the vulnerabilities of traditional methods such as RSA to attacks by quantum computers. The chapter then presents lattices as an alternative cryptographic foundation, suggesting their potential to address these vulnerabilities. It concludes by outlining the structure of the thesis.
• Chapter 2: Mathematical Background - This chapter lays out the fundamental mathematical concepts necessary to understand lattice-based cryptosystems. It delves into the properties and definitions of lattices, exploring various lattice problems and their complexity. The chapter also provides an overview of lattice-reduction techniques and their relevance in cryptography. Further, it introduces asymmetric cryptosystems and digital signatures, providing context for the application of lattices in cryptography.
• Chapter 3: Lattice-based Cryptosystems - This chapter dives into the specific construction and analysis of lattice-based cryptosystems. It focuses on the GGH-Cryptosystem, examining its construction and known attacks. The chapter also includes a comparative analysis of lattice-based cryptosystems with other types of cryptosystems, highlighting their advantages and limitations.

Schlüsselwörter (Keywords)

The key terms and focus topics of this thesis include: lattice-based cryptography, worst-case lattice problems, asymmetric cryptography, digital signatures, GGH-Cryptosystem, quantum computing, public-key cryptography, linear algebra, complexity theory, and security analysis.