Lattice reduction by cubification

TL;DR

This paper introduces cubification, a novel lattice reduction method that avoids Gram-Schmidt calculations, outperforming LLL in speed and effectiveness, with potential applications beyond crystallography.

Contribution

The paper presents cubification, a new lattice reduction approach that is faster and more effective than LLL, without relying on Gram-Schmidt orthogonalization.

Findings

Cubification outperforms LLL in reducing columnar matrices.

Python implementation of cubification is ten times faster than reference LLL code.

Cubification may have broader applications beyond crystallography.

Abstract

Lattice reduction is a NP-hard problem well known in computer science and cryptography. The Lenstra-Lenstra-Lovasz (LLL) algorithm based on the calculation of orthogonal Gram-Schmidt (GS) bases is efficient and gives a good solution in polynomial time. Here, we present a new approach called cubification that does not require the calculation of the GS bases. It relies on complementary directional and hyperplanar reductions. The deviation from cubicity at each step of the reduction process is evaluated by a parameter called lattice rhombicity, which is simply the sum of the absolute values of the metric tensor. Cubification seems to equal LLL; it even outperforms it in the reduction of columnar matrices. We wrote a Python program that is ten times faster than a reference Python LLL code. This work may open new perspectives for lattice reduction and may have implications and applications beyond crystallography.