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.

翻译：本研究探讨了源自容错学习问题的最短向量问题的求解方法。容错学习问题是在模环上定义的线性方程组系统，其右侧添加了一个扰动向量。此类问题具有重要研究价值，因为破解基于格密码系统（如美国国家标准与技术研究院于2024年发布的基于模块格的密钥封装机制）必须解决容错学习问题。基于此背景，学界已研究了多种经典算法与量子算法用于求解最短向量问题。在简化给定最短向量问题方面，两种著名算法分别是Lenstra-Lenstra-Lovász算法与分块Korkine-Zolotarev算法。这两种算法通过构造基向量组来精确计算或近似求解最短向量问题。本文系统研究了两种算法在不同规模与模环参数下的性能表现。若算法生成的基向量组包含最短向量问题的解向量，则判定该算法对给定问题的求解是成功的。