Analysis of decreasing squared-sum of Gram-Schmidt lengths for short lattice vectors

Masaya Yasuda, Kazuhiro Yokoyama, Takeshi Shimoyama, Jun Kogure, Takeshi Koshiba

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

In 2015, Fukase and Kashiwabara proposed an efficient method to find a very short lattice vector. Their method has been applied to solve Darmstadt shortest vector problems of dimensions 134 to 150. Their method is based on Schnorr's random sampling, but their preprocessing is different from others. It aims to decrease the sum of the squared lengths of the Gram-Schmidt vectors of a lattice basis, before executing random sampling of short lattice vectors. The effect is substantiated from their statistical analysis, and it implies that the smaller the sum becomes, the shorter sampled vectors can be. However, no guarantee is known to strictly decrease the sum. In this paper, we study Fukase-Kashiwabara's method in both theory and practice, and give a heuristic but practical condition that the sum is strictly decreased. We believe that our condition would enable one to monotonically decrease the sum and to find a very short lattice vector in fewer steps.

Original languageEnglish
Pages (from-to)1-24
Number of pages24
JournalJournal of Mathematical Cryptology
Volume11
Issue number1
DOIs
Publication statusPublished - 2017 Mar 1
Externally publishedYes

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Vector Lattice
Random Sampling
Decrease
Strictly
Sampling
Statistical Analysis
Preprocessing
Heuristics
Statistical methods
Imply

Keywords

  • LLL algorithm
  • random sampling
  • SVP

ASJC Scopus subject areas

  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics

Cite this

Analysis of decreasing squared-sum of Gram-Schmidt lengths for short lattice vectors. / Yasuda, Masaya; Yokoyama, Kazuhiro; Shimoyama, Takeshi; Kogure, Jun; Koshiba, Takeshi.

In: Journal of Mathematical Cryptology, Vol. 11, No. 1, 01.03.2017, p. 1-24.

Research output: Contribution to journalArticle

Yasuda, Masaya ; Yokoyama, Kazuhiro ; Shimoyama, Takeshi ; Kogure, Jun ; Koshiba, Takeshi. / Analysis of decreasing squared-sum of Gram-Schmidt lengths for short lattice vectors. In: Journal of Mathematical Cryptology. 2017 ; Vol. 11, No. 1. pp. 1-24.
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