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ON PAIRWISE GAUSSIAN BASES AND LLL ALGORITHM FOR THREE DIMENSIONAL LATTICES
- Title
- ON PAIRWISE GAUSSIAN BASES AND LLL ALGORITHM FOR THREE DIMENSIONAL LATTICES
- Authors
- Kim, Kitae; Lee, Hyang-Sook; Lim, Seongan; Park, Jeongeun; Yie, Ikkwon
- Ewha Authors
- 이향숙
- SCOPUS Author ID
- 이향숙
- Issue Date
- 2022
- Journal Title
- JOURNAL OF THE KOREAN MATHEMATICAL SOCIETY
- ISSN
- 0304-9914
2234-3008
- Citation
- JOURNAL OF THE KOREAN MATHEMATICAL SOCIETY vol. 59, no. 6, pp. 1047 - 1065
- Keywords
- Lattice; pairwise-Gaussian basis; lattice basis reduction; LLL
- Publisher
- KOREAN MATHEMATICAL SOC
- Indexed
- SCIE; SCOPUS; KCI
- Document Type
- Article
- Abstract
- For two dimensional lattices, a Gaussian basis achieves all two successive minima. For dimension larger than two, constructing a pairwise Gaussian basis is useful to compute short vectors of the lattice. For three dimensional lattices, Semaev showed that one can convert a pairwise Gaussian basis to a basis achieving all three successive minima by one simple reduction. A pairwise Gaussian basis can be obtained from a given basis by executing Gauss algorithm for each pair of basis vectors repeatedly until it returns a pairwise Gaussian basis. In this article, we prove a necessary and sufficient condition for a pairwise Gaussian basis to achieve the first k successive minima for three dimensional lattices for each k is an element of {1, 2, 3} by modifying Semaev's condition. Our condition directly checks whether a pairwise Gaussian basis contains the first k shortest independent vectors for three dimensional lattices. LLL is the most basic lattice basis reduction algorithm and we study how to use LLL to compute a pairwise Gaussian basis. For delta >= 0.9, we prove that LLL(delta) with an additional simple reduction turns any basis for a three dimensional lattice into a pairwise SV-reduced basis. By using this, we convert an LLL reduced basis to a pairwise Gaussian basis in a few simple reductions. Our result suggests that the LLL algorithm is quite effective to compute a basis with all three successive minima for three dimensional lattices.
- DOI
- 10.4134/JKMS.j210496
- Appears in Collections:
- 자연과학대학 > 수학전공 > Journal papers
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