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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Wiley
2024-01-01
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| 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. |
| format | Article |
| id | doaj-art-4883a0e3a9d1436c94ecda45018a781c |
| institution | Kabale University |
| issn | 2314-4785 |
| language | English |
| publishDate | 2024-01-01 |
| publisher | Wiley |
| record_format | Article |
| 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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