# Lattice Gaussian Sampling by Markov Chain Monte Carlo: Bounded Distance Decoding and Trapdoor Sampling

## TL;DR

This paper advances MCMC-based lattice Gaussian sampling, deriving spectral gaps, analyzing decoding performance, and proposing new algorithms that improve convergence and enable parallel implementation for practical cryptographic applications.

## Contribution

It introduces new spectral gap bounds, analyzes bounded distance decoding trade-offs, and proposes the MTMK algorithm for faster convergence in lattice Gaussian sampling.

## Key findings

- Independent MHK converges faster due to spectral gap bounds.
- Decoding performance shows a trade-off between radius and complexity.
- MTMK enhances convergence rate and supports parallel implementation.

## Abstract

Sampling from the lattice Gaussian distribution plays an important role in various research fields. In this paper, the Markov chain Monte Carlo (MCMC)-based sampling technique is advanced in several fronts. Firstly, the spectral gap for the independent Metropolis-Hastings-Klein (MHK) algorithm is derived, which is then extended to Peikert's algorithm and rejection sampling; we show that independent MHK exhibits faster convergence. Then, the performance of bounded distance decoding using MCMC is analyzed, revealing a flexible trade-off between the decoding radius and complexity. MCMC is further applied to trapdoor sampling, again offering a trade-off between security and complexity. Finally, the independent multiple-try Metropolis-Klein (MTMK) algorithm is proposed to enhance the convergence rate. The proposed algorithms allow parallel implementation, which is beneficial for practical applications.