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].