MergeLLL: A Hierarchical Divide-and-Conquer Framework for LLL-Based Lattice Reduction

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 (blocks), where local reductions are performed within each block using KZ reduction, approximated in practice by BKZ with block size equal to the sublattice dimension. The full basis is then reconstructed through a hierarchical merging process. The approach focuses on improving local lattice structure before refining global basis properties. The method is naturally parallelizable, enabling efficient multicore and distributed execution. The reduction and merging steps preserve the lattice structure via unimodular transformations and admit logarithmic parallel depth. In experiments on subset-sum and NTRU-style lattices, the method is evaluated against PotLLL and BKZ 2.0, showing comparable computation time with PotLLL while achieving a lower root Hermite factor, indicating higher-quality reduced bases.