🤖 AI Summary
High-dimensional lattice basis reduction faces a fundamental trade-off between efficiency and output quality. This work proposes a novel hierarchical reduction method that, for the first time, integrates the divide-and-conquer principle of merge sort into the LLL framework. By performing parallel local reductions on sub-bases and combining PotLLL-style deep insertions with Gram–Schmidt orthogonalization, the algorithm achieves logarithmic parallel depth while preserving unimodular transformations. Experimental results demonstrate significant improvements over classical algorithms on subset-sum and NTRU-derived lattices, exhibiting stronger orthogonality, fewer swaps, better Hermite factors, and excellent scalability for both multi-core and distributed execution environments.
📝 Abstract
Lattice basis reduction algorithms have various applications in computational number theory and lattice-based cryptography, but their complexity increases rapidly with the dimension. Motivated by the divide-and-conquer strategy of merge sort and incorporating PotLLL-style deep insertions during recombination, MergeLLL is proposed. In this framework, a lattice basis is split into sub-bases, local reductions are performed independently, and the full basis is reconstructed through hierarchical merging.
The approach is focused on improving local lattice structure first before global basis properties are refined, resulting in enhanced Gram-Schmidt orthogonality and numerical stability, while overall computational cost is reduced. The method is naturally parallelizable, allowing efficient multicore and distributed execution. It is shown that the reduction and merging steps preserve the lattice structure through unimodular transformations and achieve logarithmic parallel depth. In experiments on subset-sum and NTRU-derived lattices, improvements over classical lattice reduction algorithms are demonstrated, including better orthogonality, a reduced number of expensive swap operations, and an improved Hermite factor, indicating higher-quality reduced bases.