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Minimal condition for shortest vectors in lattices of low dimension

Title
Minimal condition for shortest vectors in lattices of low dimension
Authors
Lee J.Chang S.Kim T.Lee H.-S.Lim S.
Ewha Authors
이향숙임선간
SCOPUS Author ID
이향숙scopus; 임선간scopusscopus
Issue Date
2019
Journal Title
Electronic Notes in Discrete Mathematics
ISSN
1571-0653JCR Link
Citation
Electronic Notes in Discrete Mathematics vol. 71, pp. 9 - 14
Keywords
greedy-reduced basislatticeMinkowski-reduced basisshortest vector problem
Publisher
Elsevier B.V.
Indexed
SCOPUS scopus
Document Type
Article
Abstract
For a lattice, finding a nonzero shortest vector is computationally difficult in general. The problem becomes quite complicated even when the dimension of the lattice is five. There are two related notions of reduced bases, say, Minkowski-reduced basis and greedy-reduced basis. When the dimension becomes d = 5, there are greedy-reduced bases without achieving the first minimum while any Minkowski-reduced basis contains the shortest four linearly independent vectors. This suggests that the notion of Minkowski-reduced basis is somewhat strong and the notion of greedy-reduced basis is too weak for a basis to achieve the first minimum of the lattice. In this work, we investigate a more appropriate condition for a basis to achieve the first minimum for d = 5. We present a minimal sufficient condition, APG + , for a five dimensional lattice basis to achieve the first minimum in the sense that any proper subset of the required inequalities is not sufficient to achieve the first minimum. © 2019 Elsevier B.V.
DOI
10.1016/j.endm.2019.02.002
Appears in Collections:
자연과학대학 > 수학전공 > Journal papers
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