Markov Chain Monte Carlo Algorithms for Lattice Gaussian Sampling

TL;DR

This paper introduces MCMC-based algorithms, including Gibbs and Gibbs-Klein sampling, for lattice Gaussian distribution sampling, ensuring convergence regardless of standard deviation and improving efficiency over Klein's algorithm.

Contribution

It proposes novel MCMC algorithms for lattice Gaussian sampling that work for any standard deviation and enhance convergence speed compared to existing methods.

Findings

Gibbs sampling converges to the lattice Gaussian distribution for all standard deviations.

Gibbs-Klein sampling improves convergence rate by block sampling.

The proposed algorithms require less restrictive conditions than Klein's algorithm.

Abstract

Sampling from a lattice Gaussian distribution is emerging as an important problem in various areas such as coding and cryptography. The default sampling algorithm --- Klein's algorithm yields a distribution close to the lattice Gaussian only if the standard deviation is sufficiently large. In this paper, we propose the Markov chain Monte Carlo (MCMC) method for lattice Gaussian sampling when this condition is not satisfied. In particular, we present a sampling algorithm based on Gibbs sampling, which converges to the target lattice Gaussian distribution for any value of the standard deviation. To improve the convergence rate, a more efficient algorithm referred to as Gibbs-Klein sampling is proposed, which samples block by block using Klein's algorithm. We show that Gibbs-Klein sampling yields a distribution close to the target lattice Gaussian, under a less stringent condition than that of the original Klein algorithm.