### 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 language | English |
---|---|

Pages (from-to) | 1-24 |

Number of pages | 24 |

Journal | Journal of Mathematical Cryptology |

Volume | 11 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2017 Mar 1 |

Externally published | Yes |

### Fingerprint

### Keywords

- LLL algorithm
- random sampling
- SVP

### ASJC Scopus subject areas

- Computer Science Applications
- Computational Mathematics
- Applied Mathematics

### Cite this

*Journal of Mathematical Cryptology*,

*11*(1), 1-24. https://doi.org/10.1515/jmc-2016-0008

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

Research output: Contribution to journal › Article

*Journal of Mathematical Cryptology*, vol. 11, no. 1, pp. 1-24. https://doi.org/10.1515/jmc-2016-0008

}

TY - JOUR

T1 - Analysis of decreasing squared-sum of Gram-Schmidt lengths for short lattice vectors

AU - Yasuda, Masaya

AU - Yokoyama, Kazuhiro

AU - Shimoyama, Takeshi

AU - Kogure, Jun

AU - Koshiba, Takeshi

PY - 2017/3/1

Y1 - 2017/3/1

N2 - 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.

AB - 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.

KW - LLL algorithm

KW - random sampling

KW - SVP

UR - http://www.scopus.com/inward/record.url?scp=85014658891&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85014658891&partnerID=8YFLogxK

U2 - 10.1515/jmc-2016-0008

DO - 10.1515/jmc-2016-0008

M3 - Article

VL - 11

SP - 1

EP - 24

JO - Journal of Mathematical Cryptology

JF - Journal of Mathematical Cryptology

SN - 1862-2976

IS - 1

ER -