Minimal condition for shortest vectors in lattices of low dimension

Q2 Mathematics
Juhee Lee , Seunghwan Chang , Taewan Kim , Hyang-Sook Lee , Seongan Lim
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引用次数: 0

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.

低维格中最短向量的最小条件
对于一个晶格,寻找一个非零最短向量通常在计算上是困难的。即使晶格的维数是5,这个问题也变得相当复杂。约简基有两个相关的概念,即minkowski -约简基和greedy-约简基。当维数变为d = 5时,存在未达到第一个最小值的贪婪约简基,而任何minkowski -约简基都包含最短的四个线性无关向量。这表明minkowski -约简基的概念是比较强的,而贪婪约简基的概念对于一个基来说是太弱了,不能达到晶格的第一个最小值。在这项工作中,我们研究了一个更合适的条件,使基达到d = 5的第一个最小值。我们给出了五维晶格基达到第一极小值的最小充分条件APG+,即所需不等式的任何适当子集都不足以达到第一极小值。
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来源期刊
Electronic Notes in Discrete Mathematics
Electronic Notes in Discrete Mathematics Mathematics-Discrete Mathematics and Combinatorics
CiteScore
1.30
自引率
0.00%
发文量
0
期刊介绍: Electronic Notes in Discrete Mathematics is a venue for the rapid electronic publication of the proceedings of conferences, of lecture notes, monographs and other similar material for which quick publication is appropriate. Organizers of conferences whose proceedings appear in Electronic Notes in Discrete Mathematics, and authors of other material appearing as a volume in the series are allowed to make hard copies of the relevant volume for limited distribution. For example, conference proceedings may be distributed to participants at the meeting, and lecture notes can be distributed to those taking a course based on the material in the volume.
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