A parameter study for LLL and BKZ with application to shortest vector problems

TL;DR

This paper investigates the effectiveness of LLL and BKZ algorithms in solving shortest vector problems related to learning with error problems, which are crucial for cryptanalysis of lattice-based cryptosystems.

Contribution

It provides a comparative analysis of LLL and BKZ algorithms' performance on SVPs across various sizes and modular rings, relevant for cryptography.

Findings

LLL and BKZ can produce solutions for SVPs in different settings

Performance varies with problem size and ring parameters

Results inform cryptographic security assessments

Abstract

In this work, we study the solution of shortest vector problems (SVPs) arising in terms of learning with error problems (LWEs). LWEs are linear systems of equations over a modular ring, where a perturbation vector is added to the right-hand side. This type of problem is of great interest, since LWEs have to be solved in order to be able to break lattice-based cryptosystems as the Module-Lattice-Based Key-Encapsulation Mechanism published by NIST in 2024. Due to this fact, several classical and quantum-based algorithms have been studied to solve SVPs. Two well-known algorithms that can be used to simplify a given SVP are the Lenstra-Lenstra-Lov\'asz (LLL) algorithm and the Block Korkine-Zolotarev (BKZ) algorithm. LLL and BKZ construct bases that can be used to compute or approximate solutions of the SVP. We study the performance of both algorithms for SVPs with different sizes and modular rings. Thereby, application of LLL or BKZ to a given SVP is considered to be successful if they produce bases containing a solution vector of the SVP.