# Boosted KZ and LLL Algorithms

## TL;DR

This paper introduces boosted KZ and LLL algorithms that improve basis reduction performance and efficiency, addressing issues of vector length increase and suboptimality in classical methods, with applications in MIMO communication systems.

## Contribution

The paper presents novel boosted KZ and LLL algorithms with enhanced performance bounds and efficiency for lattice reduction, applicable to MIMO systems.

## Key findings

- Boosted algorithms outperform classical KZ and LLL in basis quality.
- Enhanced performance bounds for KZ reduction.
- Simulations show improved rate and complexity in MIMO applications.

## Abstract

There exist two issues among popular lattice reduction (LR) algorithms that should cause our concern. The first one is Korkine-Zolotarev (KZ) and Lenstra-Lenstra-Lovasz (LLL) algorithms may increase the lengths of basis vectors. The other is KZ reduction suffers much worse performance than Minkowski reduction in terms of providing short basis vectors, despite its superior theoretical upper bounds. To address these limitations, we improve the size reduction steps in KZ and LLL to set up two new efficient algorithms, referred to as boosted KZ and LLL, for solving the shortest basis problem (SBP) with exponential and polynomial complexity, respectively. Both of them offer better actual performance than their classic counterparts, and the performance bounds for KZ are also improved. We apply them to designing integer-forcing (IF) linear receivers for multi-input multi-output (MIMO) communications. Our simulations confirm their rate and complexity advantages.