A new polynomial-time variant of LLL with deep insertions for decreasing the squared-sum of Gram–Schmidt lengths

Masaya Yasuda, Junpei Yamaguchi

Research output: Contribution to journalArticle

Abstract

Lattice basis reduction algorithms have been used in cryptanalysis. The most famous algorithm is LLL, proposed by Lenstra, Lenstra, Lovász, and one of its typical improvements is LLL with deep insertions (DeepLLL). A DeepLLL-reduced basis is LLL-reduced, and hence its quality is at least as good as LLL. In practice, DeepLLL often outputs a more reduced basis than LLL, but no theoretical result is known. First, we show provable output quality of DeepLLL, strictly better than that of LLL. Second, as a main work of this paper, we propose a new variant of DeepLLL. The squared-sum of Gram–Schmidt lengths of a basis is related with the computational hardness of lattice problems such as the shortest vector problem (SVP). Given an input basis, our variant monotonically decreases the squared-sum by a given factor at every deep insertion. This guarantees that our variant runs in polynomial-time.

Original languageEnglish
JournalDesigns, Codes, and Cryptography
DOIs
Publication statusPublished - Jan 1 2019

Fingerprint

Insertion
Polynomial time
Lattice Basis Reduction
Polynomials
Output
Cryptanalysis
Hardness
Strictly
Decrease

All Science Journal Classification (ASJC) codes

  • Computer Science Applications
  • Applied Mathematics

Cite this

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abstract = "Lattice basis reduction algorithms have been used in cryptanalysis. The most famous algorithm is LLL, proposed by Lenstra, Lenstra, Lov{\'a}sz, and one of its typical improvements is LLL with deep insertions (DeepLLL). A DeepLLL-reduced basis is LLL-reduced, and hence its quality is at least as good as LLL. In practice, DeepLLL often outputs a more reduced basis than LLL, but no theoretical result is known. First, we show provable output quality of DeepLLL, strictly better than that of LLL. Second, as a main work of this paper, we propose a new variant of DeepLLL. The squared-sum of Gram–Schmidt lengths of a basis is related with the computational hardness of lattice problems such as the shortest vector problem (SVP). Given an input basis, our variant monotonically decreases the squared-sum by a given factor at every deep insertion. This guarantees that our variant runs in polynomial-time.",
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