Bridging Early and Modern Lattice Cryptosystems: A Theoretical Journey through SVP, LWE and Kyber

Abstract

This thesis offers an in-depth theoretical study of lattice-based cryptography, tracing its evolution from early foundational systems to modern standardized constructions. Initially, we examine and emphasize the significance of the Ajtai-Dwork cryptosystem and its foundational worst-case to average-case reductions based on lattice problems. The thesis then details Regev's Learning with Errors (LWE) problem and shows its impact on the development of practical public-key schemes while maintaining the security guarantees from the well-studied worst-case lattice problems.

Further, we discuss the structured variants such as Ring-LWE and Module-LWE, showing how they improve the efficiency and scalability of the lattice-based schemes while maintaining the security foundations of the worst-case to average-case reductions. This analysis then culminates in the study of ML-KEM, a lattice-based scheme recently standardized by NIST, examining its specific design choices, optimizations, and security proofs. Finally, we investigate algorithmic methods for solving the fundamental lattice problems, analysing exact Shortest Vector Problem (SVP) solvers including Kannan's enumeration algorithm and the AKS sieve algorithm, in order to understand their implications on the hardness assumptions underlying the lattice-based cryptographic security.

In summary, this work shows that lattice-based cryptography offers a secure and efficient foundation for (post-quantum) cryptographic schemes, with strong theoretical bases and practical implementations.