  Sampling from the lattice Gaussian distribution is emerging as an important
problem in coding and cryptography. In this paper, the classic
Metropolis-Hastings (MH) algorithm from Markov chain Monte Carlo (MCMC) methods
is adapted for lattice Gaussian sampling. Two MH-based algorithms are proposed,
which overcome the restriction suffered by the default Klein's algorithm. The
first one, referred to as the independent Metropolis-Hastings-Klein (MHK)
algorithm, tries to establish a Markov chain through an independent proposal
distribution. We show that the Markov chain arising from the independent MHK
algorithm is uniformly ergodic, namely, it converges to the stationary
distribution exponentially fast regardless of the initial state. Moreover, the
rate of convergence is explicitly calculated in terms of the theta series,
leading to a predictable mixing time. In order to further exploit the
convergence potential, a symmetric Metropolis-Klein (SMK) algorithm is
proposed. It is proven that the Markov chain induced by the SMK algorithm is
geometrically ergodic, where a reasonable selection of the initial state is
capable to enhance the convergence performance.
