On the geometric ergodicity of Gibbs algorithm for lattice Gaussian sampling

Sampling from the lattice Gaussian distribution is emerging as an important problem in coding and cryptography. In this paper, the conventional Gibbs sampling algorithm is demonstrated to be geometrically ergodic in tackling with lattice Gaussian sampling, which means its induced Markov chain converges exponentially fast to the stationary distribution. Moreover, as the exponential convergence rate is dominated by the spectral radius of the forward operator of the Markov chain, a comprehensive analysis is given and we show that the convergence performance can be further enhanced by usages of blocked sampling strategy and choices of selection probabilities.

I. Introduction

Nowadays, lattice Gaussian distribution has drawn a lot of attentions in various research fields. In mathematics, Ba-naszczyk firstly applied it to prove the transference theorems for lattices [1]. In coding, lattice Gaussian distribution was employed to obtain the full shaping gain for lattice coding [2], and to achieve the capacity of the Gaussian channel and the secrecy capacity of the Gaussian wiretap channel, respectively [3]. Meanwhile, lattice Gaussian distribution is also applied to relay network under the compute-and-forward strategy for the physical layer security [4]. In cryptography, the lattice Gaussian distribution has already become a central tool in the construction of many primitives. Specifically, Micciancio and Regev used it to propose lattice-based cryptosystems based on the worst-case hardness assumptions [5]. Meanwhile, it also has underpinned the fully-homomorphic encryption for cloud computing [6]. Algorithmically, lattice Gaussian sampling with a suitable variance allows to solve the shortest vector problem (SVP) and the closest vector problem (CVP) [7]; for example, it has led to efficient lattice decoding for multi-input multi-output (MIMO) systems [8].