Computing an LLL-reduced basis of the orthogonal lattice

TL;DR

This paper introduces a new technique for bounding the number of iterations in the LLL lattice basis reduction algorithm when applied to orthogonal lattices, using a novel potential function.

Contribution

It proposes a new method to estimate iteration bounds for LLL on orthogonal lattices, enhancing understanding of its convergence behavior.

Findings

New upper bounds on LLL iterations for orthogonal lattices

A variant of LLL potential applicable to other lattice families

Insights into LLL algorithm efficiency and convergence

Abstract

As a typical application, the Lenstra-Lenstra-Lovasz lattice basis reduction algorithm (LLL) is used to compute a reduced basis of the orthogonal lattice for a given integer matrix, via reducing a special kind of lattice bases. With such bases in input, we propose a new technique for bounding from above the number of iterations required by the LLL algorithm. The main technical ingredient is a variant of the classical LLL potential, which could prove useful to understand the behavior of LLL for other families of input bases.