Analysis of DeepBKZ reduction for finding short lattice vectors

Abstract

Lattice basis reduction is a mandatory tool for solving lattice problems such as the shortest vector problem. The Lenstra–Lenstra–Lovasz 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 $$\beta $$ , 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 $$\beta $$ . 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.