# On the Communication Cost of Determining an Approximate Nearest Lattice Point

## TL;DR

This paper analyzes the communication costs and error probabilities of the approximate nearest lattice point problem in distributed networks, focusing on the Babai nearest-plane algorithm and basis selection.

## Contribution

It provides new bounds on communication costs and error probabilities, and introduces an algorithm to reduce communication in arbitrary dimensions.

## Key findings

- Communication cost bounds for the Babai algorithm in distributed settings
- Error probability analysis for reduced lattice bases in 2D
- A new algorithm to reduce communication in high-dimensional lattices

## Abstract

We consider the closest lattice point problem in a distributed network setting and study the communication cost and the error probability for computing an approximate nearest lattice point, using the nearest-plane algorithm, due to Babai. Two distinct communication models, centralized and interactive, are considered. The importance of proper basis selection is addressed. Assuming a reduced basis for a two-dimensional lattice, we determine the approximation error of the nearest plane algorithm. The communication cost for determining the Babai point, or equivalently, for constructing the rectangular nearest-plane partition, is calculated in the interactive setting. For the centralized model, an algorithm is presented for reducing the communication cost of the nearest plane algorithm in an arbitrary number of dimensions.