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)

Abstract

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
Pages19-35
Number of pages17
ISBN (Print)9783319641997
DOIs
Publication statusPublished - Jan 1 2017
Event12th International Workshop on Security, IWSEC 2017 - Hiroshima, Japan
Duration: Aug 30 2017Sep 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

Other

Other12th International Workshop on Security, IWSEC 2017
CountryJapan
CityHiroshima
Period8/30/179/1/17

Fingerprint

Beast
Higher Dimensions
Failure Probability
Geometric Structure
Angle
Experimental Results

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Alsayigh, S., Ding, J., Takagi, T., & Wang, Y. (2017). The beauty and the beasts—The hard cases in LLL reduction. In S. Obana, & K. Chida (Eds.), Advances in Information and Computer Security - 12th International Workshop on Security, IWSEC 2017, Proceedings (pp. 19-35). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 10418 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-319-64200-0_2

The beauty and the beasts—The hard cases in LLL reduction. / Alsayigh, Saed; Ding, Jintai; Takagi, Tsuyoshi; Wang, Yuntao.

Advances in Information and Computer Security - 12th International Workshop on Security, IWSEC 2017, Proceedings. ed. / Satoshi Obana; Koji Chida. Springer Verlag, 2017. p. 19-35 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 10418 LNCS).

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

Alsayigh, S, Ding, J, Takagi, T & Wang, Y 2017, The beauty and the beasts—The hard cases in LLL reduction. in S Obana & K Chida (eds), Advances in Information and Computer Security - 12th International Workshop on Security, IWSEC 2017, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 10418 LNCS, Springer Verlag, pp. 19-35, 12th International Workshop on Security, IWSEC 2017, Hiroshima, Japan, 8/30/17. https://doi.org/10.1007/978-3-319-64200-0_2
Alsayigh S, Ding J, Takagi T, Wang Y. The beauty and the beasts—The hard cases in LLL reduction. In Obana S, Chida K, editors, Advances in Information and Computer Security - 12th International Workshop on Security, IWSEC 2017, Proceedings. Springer Verlag. 2017. p. 19-35. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-319-64200-0_2
Alsayigh, Saed ; Ding, Jintai ; Takagi, Tsuyoshi ; Wang, Yuntao. / The beauty and the beasts—The hard cases in LLL reduction. Advances in Information and Computer Security - 12th International Workshop on Security, IWSEC 2017, Proceedings. editor / Satoshi Obana ; Koji Chida. Springer Verlag, 2017. pp. 19-35 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
@inproceedings{7a5073e8627b4d65b052e19e6cfc2c64,
title = "The beauty and the beasts—The hard cases in LLL reduction",
abstract = "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.",
author = "Saed Alsayigh and Jintai Ding and Tsuyoshi Takagi and Yuntao Wang",
year = "2017",
month = "1",
day = "1",
doi = "10.1007/978-3-319-64200-0_2",
language = "English",
isbn = "9783319641997",
series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",
publisher = "Springer Verlag",
pages = "19--35",
editor = "Satoshi Obana and Koji Chida",
booktitle = "Advances in Information and Computer Security - 12th International Workshop on Security, IWSEC 2017, Proceedings",
address = "Germany",

}

TY - GEN

T1 - The beauty and the beasts—The hard cases in LLL reduction

AU - Alsayigh, Saed

AU - Ding, Jintai

AU - Takagi, Tsuyoshi

AU - Wang, Yuntao

PY - 2017/1/1

Y1 - 2017/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85028471590&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85028471590&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-64200-0_2

DO - 10.1007/978-3-319-64200-0_2

M3 - Conference contribution

AN - SCOPUS:85028471590

SN - 9783319641997

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 19

EP - 35

BT - Advances in Information and Computer Security - 12th International Workshop on Security, IWSEC 2017, Proceedings

A2 - Obana, Satoshi

A2 - Chida, Koji

PB - Springer Verlag

ER -