The Second and Fourth Moments of Discrete Gaussian Distributions over Lattices

Let Λ be an n-dimensional lattice. For any n-dimensional vector c and positive real number s, let Ds,c and DΛ,s,c denote the continuous Gaussian distribution and the discrete Gaussian distribution over Λ, respectively. In this paper, we establish the exact relationship between the second and fourth moments centered around c of the discrete Gaussian distribution DΛ,s,c and those of the continuous Gaussian distribution Ds,c, respectively. This provides a quantization form of the result obtained by Micciancio and Regev on the second and fourth moments of discrete Gaussian distribution. Using the relationship, we also derive an uncertainty principle for Gaussian functions, which extend the result of Zheng, Zhao, and Xu. Our proof is based on combination of the idea of Micciancio and Regev and the idea of Zheng, Zhao, and Xu, where the main tool is high-dimensional Fourier transform.