Lattice basis reduction is a mandatory tool for solving lattice problems such as the shortest vector problem. The Lenstra–Lenstra–Lovász reduction algorithm (LLL) is the most famous, and its typical improvements are the block Korkine–Zolotarev algorithm and LLL with deep insertions (DeepLLL), both proposed by Schnorr and Euchner. In BKZ with blocksize $$\beta $$ β , LLL is called many times to reduce a lattice basis before enumeration to find a shortest non-zero vector in every block lattice of dimension $$\beta $$ β . Recently, “DeepBKZ” was proposed as a mathematical improvement of BKZ, in which DeepLLL is called as a subroutine alternative to LLL. In this paper, we analyze the output quality of DeepBKZ in both theory and practice. Specifically, we give provable upper bounds specific to DeepBKZ. We also develop “DeepBKZ 2.0”, an improvement of DeepBKZ like BKZ 2.0, and show experimental results that it finds shorter lattice vectors than BKZ 2.0 in practice.

中文翻译：

---

分析 DeepBKZ 约简以寻找短点阵向量

晶格基约简是解决晶格问题（例如最短向量问题）的必备工具。Lenstra-Lenstra-Lovász 约简算法（LLL）最为著名，其典型的改进是块 Korkine-Zolotarev 算法和深度插入的 LLL（DeepLLL），均由 Schnorr 和 Euchner 提出。在块大小为 $$\beta $$ β 的 BKZ 中，多次调用 LLL 以在枚举之前减少格基，以在维度 $$\beta $$ β 的每个块格中找到最短的非零向量。最近，“DeepBKZ”被提出作为BKZ的数学改进，其中DeepLLL被称为LLL的子程序替代。在本文中，我们从理论和实践两方面分析了 DeepBKZ 的输出质量。具体来说，我们给出了特定于 DeepBKZ 的可证明上限。我们还开发了“DeepBKZ 2.0”，