On Communication for Distributed Babai Point Computation

TL;DR

This paper introduces a communication-efficient distributed protocol for approximating the nearest lattice point, analyzes error probabilities across dimensions and lattice types, and discusses conditions under which the protocol reliably finds the true nearest point.

Contribution

It proposes an optimal distributed protocol for Babai point computation and analyzes its error probability behavior across different lattice dimensions and distributions.

Findings

Protocol minimizes sum rate for independent components

Error probability increases with dimension and packing density

Gaussian noise can reduce error probability in high dimensions

Abstract

We present a communication-efficient distributed protocol for computing the Babai point, an approximate nearest point for a random vector ${\bf X}\in\mathbb{R}^n$ in a given lattice. We show that the protocol is optimal in the sense that it minimizes the sum rate when the components of ${\bf X}$ are mutually independent. We then investigate the error probability, i.e. the probability that the Babai point does not coincide with the nearest lattice point. In dimensions two and three, this probability is seen to grow with the packing density. For higher dimensions, we use a bound from probability theory to estimate the error probability for some well-known lattices. Our investigations suggest that for uniform distributions, the error probability becomes large with the dimension of the lattice, for lattices with good packing densities. We also consider the case where $\mathbf{X}$ is obtained by adding Gaussian noise to a randomly chosen lattice point. In this case, the error probability goes to zero with the lattice dimension when the noise variance is sufficiently small. In such cases, a distributed algorithm for finding the approximate nearest lattice point is sufficient for finding the nearest lattice point.