Explicit formula for gram-schmidt vectors in LLL with deep insertions and its applications

Junpei Yamaguchi, Masaya Yasuda

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

3 Citations (Scopus)

Abstract

Lattice basis reduction algorithms have been used as a strong tool for cryptanalysis. The most famous one is LLL, and its typical improvements are BKZ and LLL with deep insertions (DeepLLL). In LLL and DeepLLL, at every time to replace a lattice basis, we need to recompute the Gram-Schmidt orthogonalization (GSO) for the new basis. Compared with LLL, the form of the new GSO vectors is complicated in DeepLLL, and no formula has been known. In this paper, we give an explicit formula for GSO in DeepLLL, and also propose an efficient method to update GSO in DeepLLL. As another work, we embed DeepLLL into BKZ as a subroutine instead of LLL, which we call “DeepBKZ”, in order to find a more reduced basis. By using our DeepBKZ with blocksizes up to β = 50, we have found a number of new solutions for the Darmstadt SVP challenge in dimensions from 102 to 123.

Original languageEnglish
Title of host publicationNumber-Theoretic Methods in Cryptology - 1st International Conference, NuTMiC 2017, Revised Selected Papers
EditorsJosef Pieprzyk, Josef Pieprzyk, Jerzy Kaczorowski, Jacek Pomykała
PublisherSpringer Verlag
Pages142-160
Number of pages19
ISBN (Print)9783319766195
DOIs
Publication statusPublished - Jan 1 2018
Event1st International Conference on Number-Theoretic Methods in Cryptology, NuTMiC 2017 - Warsaw, Poland
Duration: Sep 11 2017Sep 13 2017

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume10737 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other1st International Conference on Number-Theoretic Methods in Cryptology, NuTMiC 2017
CountryPoland
CityWarsaw
Period9/11/179/13/17

Fingerprint

Orthogonalization
Subroutines
Insertion
Explicit Formula
Lattice Basis Reduction
Cryptanalysis
Update

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Yamaguchi, J., & Yasuda, M. (2018). Explicit formula for gram-schmidt vectors in LLL with deep insertions and its applications. In J. Pieprzyk, J. Pieprzyk, J. Kaczorowski, & J. Pomykała (Eds.), Number-Theoretic Methods in Cryptology - 1st International Conference, NuTMiC 2017, Revised Selected Papers (pp. 142-160). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 10737 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-319-76620-1_9

Explicit formula for gram-schmidt vectors in LLL with deep insertions and its applications. / Yamaguchi, Junpei; Yasuda, Masaya.

Number-Theoretic Methods in Cryptology - 1st International Conference, NuTMiC 2017, Revised Selected Papers. ed. / Josef Pieprzyk; Josef Pieprzyk; Jerzy Kaczorowski; Jacek Pomykała. Springer Verlag, 2018. p. 142-160 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 10737 LNCS).

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

Yamaguchi, J & Yasuda, M 2018, Explicit formula for gram-schmidt vectors in LLL with deep insertions and its applications. in J Pieprzyk, J Pieprzyk, J Kaczorowski & J Pomykała (eds), Number-Theoretic Methods in Cryptology - 1st International Conference, NuTMiC 2017, Revised Selected Papers. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 10737 LNCS, Springer Verlag, pp. 142-160, 1st International Conference on Number-Theoretic Methods in Cryptology, NuTMiC 2017, Warsaw, Poland, 9/11/17. https://doi.org/10.1007/978-3-319-76620-1_9
Yamaguchi J, Yasuda M. Explicit formula for gram-schmidt vectors in LLL with deep insertions and its applications. In Pieprzyk J, Pieprzyk J, Kaczorowski J, Pomykała J, editors, Number-Theoretic Methods in Cryptology - 1st International Conference, NuTMiC 2017, Revised Selected Papers. Springer Verlag. 2018. p. 142-160. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-319-76620-1_9
Yamaguchi, Junpei ; Yasuda, Masaya. / Explicit formula for gram-schmidt vectors in LLL with deep insertions and its applications. Number-Theoretic Methods in Cryptology - 1st International Conference, NuTMiC 2017, Revised Selected Papers. editor / Josef Pieprzyk ; Josef Pieprzyk ; Jerzy Kaczorowski ; Jacek Pomykała. Springer Verlag, 2018. pp. 142-160 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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