Communication-Efficient Search for an Approximate Closest Lattice Point

TL;DR

This paper develops communication-efficient protocols for approximating the closest lattice point in a distributed setting, analyzing error probabilities and communication costs for low-dimensional lattices, with implications for high-dimensional cases.

Contribution

It introduces protocols for distributed approximate lattice point search with minimized communication, and provides analytical and computational error estimates for 2D and 3D lattices.

Findings

Error probability increases with lattice packing density in 2D and 3D.

Communication cost is characterized for lattices of dimension n>1.

Protocols are efficient for low-dimensional lattices, with potential extensions to higher dimensions.

Abstract

We consider the problem of finding the closest lattice point to a vector in n-dimensional Euclidean space when each component of the vector is available at a distinct node in a network. Our objectives are (i) minimize the communication cost and (ii) obtain the error probability. The approximate closest lattice point considered here is the one obtained using the nearest-plane (Babai) algorithm. Assuming a triangular special basis for the lattice, we develop communication-efficient protocols for computing the approximate lattice point and determine the communication cost for lattices of dimension n>1. Based on available parameterizations of reduced bases, we determine the error probability of the nearest plane algorithm for two dimensional lattices analytically, and present a computational error estimation algorithm in three dimensions. For dimensions 2 and 3, our results show that the error probability increases with the packing density of the lattice.