The beauty and the beasts—The hard cases in LLL reduction

Saed Alsayigh, Jintai Ding, Tsuyoshi Takagi, Yuntao Wang

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Citation (Scopus)


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.

Original languageEnglish
Title of host publicationAdvances in Information and Computer Security - 12th International Workshop on Security, IWSEC 2017, Proceedings
EditorsSatoshi Obana, Koji Chida
PublisherSpringer Verlag
Number of pages17
ISBN (Print)9783319641997
Publication statusPublished - 2017
Event12th International Workshop on Security, IWSEC 2017 - Hiroshima, Japan
Duration: Aug 30 2017Sept 1 2017

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume10418 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Other12th International Workshop on Security, IWSEC 2017

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)


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