Year: 2025
Type: dissertation
Abstract: Lattice-based cryptography has garnered increasing attention in recent years, largely due to the NIST standardization process, which selected lattice-based algorithms for key exchange and digital signatures.The algorithms are known as Kyber and Dilithium respectively.To understand where the security comes from for the algorithms, one must know about the complexitytheoretic problems of Module Learning With Errors (Module-LWE) and Module Short Integer Solution (Module-SIS).Alternatively, one may study a variant of the Shortest Vector Problem, namely GapSVP, for which there exists a known quantum probabilistic polynomial-time reduction to the LWE problem.This connection allows the thesis to focus on the SVP while maintaining relevance to the cryptographic applications of lattices.The first objective for this thesis is to provide an up-to-date overview of the complexity-theoretic results related to the Shortest Vector Problem (SVP) and the Closest Vector Problem (CVP).For this reason, the different variants of these problems are discussed.Also, the progression of results over time is reviewed.To ensure rigour and accessibility for readers less familiar with the subject, the necessary background in linear algebra and properties of lattices is provided at the outset.The second objective of this thesis is to reinforce the theoretical foundation through practical implementation.This means solving the SVP or the CVP in practice, with a focus placed on the former.From the two primary approaches in the literature, sieving and enumeration, the enumeration-based strategy is used.Accordingly, relevant lattice enumeration algorithms are examined theoretically.To enhance the efficiency of enumeration, the theoretical discussion is extended to include basis reduction techniques and the concept of pruning.The experimental work was conducted in Python with the addition of the fpyLLL library.The Schnorr-Euchner enumeration algorithm was implemented with later additions of LLL reduction and the extreme pruning provided by fpyLLL.The heuristic time complexity of the enumeration was evaluated against both increasing dimension and bit size.Additionally, the accuracy of the Gaussian heuristic was assessed with a result of low accuracy in low dimensions.The final measurements involved fpyLLL's pruning class to determine whether rough parameter tuning could yield performance improvements.In the end, parameter adjustments did not yield significant speed-ups in low-dimensional and low-bit-size instances.
Author: Mauri, Jouni
Language: English
Subfield: Computer Networks and Communications
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7133067045