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...

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Main Authors: Wei Zhao, Guoyou Qian
Format: Article
Language:English
Published: Wiley 2024-01-01
Series:Journal of Mathematics
Online Access:http://dx.doi.org/10.1155/2024/7777881
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author Wei Zhao
Guoyou Qian
author_facet Wei Zhao
Guoyou Qian
author_sort Wei Zhao
collection DOAJ
description 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.
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issn 2314-4785
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publishDate 2024-01-01
publisher Wiley
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series Journal of Mathematics
spelling doaj-art-4883a0e3a9d1436c94ecda45018a781c2025-02-03T05:56:54ZengWileyJournal of Mathematics2314-47852024-01-01202410.1155/2024/7777881The Second and Fourth Moments of Discrete Gaussian Distributions over LatticesWei Zhao0Guoyou Qian1Science and Technology on Communication Security LaboratoryMathematical CollegeLet 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.http://dx.doi.org/10.1155/2024/7777881
spellingShingle Wei Zhao
Guoyou Qian
The Second and Fourth Moments of Discrete Gaussian Distributions over Lattices
Journal of Mathematics
title The Second and Fourth Moments of Discrete Gaussian Distributions over Lattices
title_full The Second and Fourth Moments of Discrete Gaussian Distributions over Lattices
title_fullStr The Second and Fourth Moments of Discrete Gaussian Distributions over Lattices
title_full_unstemmed The Second and Fourth Moments of Discrete Gaussian Distributions over Lattices
title_short The Second and Fourth Moments of Discrete Gaussian Distributions over Lattices
title_sort second and fourth moments of discrete gaussian distributions over lattices
url http://dx.doi.org/10.1155/2024/7777881
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