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The Beauty and the Beasts—The Hard Cases in LLL Reduction

Saed Alsayigh Department of Mathematical Sciences, University of Cincinnati, Cincinnati, USA Jintai Ding Department of Mathematical Sciences, University of Cincinnati, Cincinnati, USA Tsuyoshi Takagi Institute of Mathematics for Industry, Kyushu University, Fukuoka, Japan; CREST, Japan Science and Technology Agency, Kawaguchi, Japan Yuntao Wang Graduate School of Mathematics, Kyushu University, Fukuoka, Japan

TBD mathscidoc:2207.43075

IWSEC 2017, 19–35, 2017.8
In this paper, we will systematically study who indeed are the hard lattice cases in LLL reduction. The “hard” cases here mean for their special geometric structures, with a comparatively high “failure probability” that LLL can not solve SVP even by using a powerful relaxation factor. We define the perfect lattice as the “Beauty”, which is given by basis of vectors of the same length with the mutual angles of any two vectors to be exactly 60°. Simultaneously the “Beasts” lattice is defined as the lattice close to the Beauty lattice. There is a relatively high probability (e.g. 15.0% in 3 dimensions) that our “Beasts” bases can withstand the exact-arithmetic LLL reduction (relaxation factors δ close to 1), comparing to the probability (corresponding <0.01%) when apply same LLL on random bases from TU Darmstadt SVP Challenge. Our theoretical proof gives us a direct explanation of this phenomenon. Moreover, we give rational Beauty bases of 3 and 8 dimensions, an irrational Beauty bases of general high dimensions. We also give a general way to construct Beasts lattice bases from the Beauty ones. Experimental results show the Beasts bases derived from Beauty can withstand LLL reduction by a stable probability even for high dimensions. Our work in a way gives a simple and direct way to explain how to build a hard lattice in LLL reduction.
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@inproceedings{saed2017the,
  title={The Beauty and the Beasts—The Hard Cases in LLL Reduction},
  author={Saed Alsayigh, Jintai Ding, Tsuyoshi Takagi, and Yuntao Wang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220715102546641470654},
  booktitle={IWSEC 2017},
  pages={19–35},
  year={2017},
}
Saed Alsayigh, Jintai Ding, Tsuyoshi Takagi, and Yuntao Wang. The Beauty and the Beasts—The Hard Cases in LLL Reduction. 2017. In IWSEC 2017. pp.19–35. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220715102546641470654.
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