An LLL algorithm with symmetries

TL;DR

This paper generalizes the LLL lattice reduction algorithm to incorporate symmetries from arbitrary reductive groups, providing explicit algorithms for classical and exceptional groups, enhancing lattice reduction techniques with symmetry considerations.

Contribution

It introduces a novel generalization of the LLL algorithm applicable to reductive groups, with explicit implementations for classical and exceptional groups.

Findings

Generalized LLL algorithm for reductive groups

Explicit algorithms for classical groups Sp and SO

Implementation for the exceptional group G2

Abstract

We give a generalisation of the Lenstra-Lenstra-Lov\'asz (LLL) lattice-reduction algorithm that is valid for an arbitrary (split, semisimple) reductive group $G$. This can be regarded as `lattice reduction with symmetries'. We make this algorithm explicit for the classical groups $G = \mathrm{Sp}_{2g}$, $\mathrm{SO}_{2g}$, and for the exceptional group $G = G_2$.