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Cyclic Lattices, Ideal Lattices and Bounds for the Smoothing Parameter

¹ Engineering Research Center of Ministry of Education for Financial Computing and Digital Engineering, Renmin University of China, Beijing, China
² Artificial Intelligence Research Institute, Beihang University of China, Beijing, China
³ Engineering Research Center of Ministry of Education for Financial Computing and Digital Engineering, Renmin University of China, Beijing, China
⁴ Engineering Research Center of Ministry of Education for Financial Computing and Digital Engineering, Renmin University of China, Beijing, China

Journal of Information Security·Volume 13 (2022)·Pages 272–293·Published 23 August 2022·DOI10.4236/jis.2022.134015

Abstract

In this article, we introduce the discrete subgroup in ℝ n as preliminaries first. Then we provide some theories of cyclic lattice s and ideal lattices. By regarding the cyclic lattices and ideal lattices as the correspondences of finitely generated R -modules, we prove our main theorem, i.e. the correspondence between cyclic lattices in ℝ n and finitely generated R -modules is one - to - one. Finally, we give an explicit and countable upper bound for the smoothing parameter of cyclic lattice s .

KeywordsCyclic LatticeIdeal LatticeFinitely Generated &lti&gtR&lt/i&gt-ModuleSmoothing Parameter

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Cyclic Lattices, Ideal Lattices and Bounds for the Smoothing Parameter

Abstract

References (21)