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PotLLL: a polynomial time version of LLL with deep insertions

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Abstract

Lattice reduction algorithms have numerous applications in number theory, algebra, as well as in cryptanalysis. The most famous algorithm for lattice reduction is the LLL algorithm. In polynomial time it computes a reduced basis with provable output quality. One early improvement of the LLL algorithm was LLL with deep insertions (DeepLLL). The output of this version of LLL has higher quality in practice but the running time seems to explode. Weaker variants of DeepLLL, where the insertions are restricted to blocks, behave nicely in practice concerning the running time. However no proof of polynomial running time is known. In this paper PotLLL, a new variant of DeepLLL with provably polynomial running time, is presented. We compare the practical behavior of the new algorithm to classical LLL, BKZ as well as blockwise variants of DeepLLL regarding both the output quality and running time.

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Notes

  1. http://www.latticechallenge.org/svp-challenge

  2. To be able to compute confidence intervals, we assume that the data is distributed normally. We did some more experiments in dimensions 40, 50 and 60 to verify this conjecture: in case of the logarithm of the running time, this conjecture is quite accurate for most experiments; in case of the \(n\)-th root Hermite factor, it seems to be fine for most values, but there is some deviation at the tails.

    Fig. 1
    figure 1

    Average \(n\)-th root Hermite factor (\(y\) axis) for dimension \(n\) (\(x\) axis) from 40 to 400

    Full size image

  3. http://perso.ens-lyon.fr/damien.stehle/fplll/

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Acknowledgments

This work was supported by CASED (http://www.cased.de). Michael Schneider was supported by project BU 630/23-1 of the German Research Foundation (DFG). Urs Wagner and Felix Fontein are supported by SNF Grant no. 132256. The authors would like to thank the anonymous referees for their helpful comments. F. F. would also like to thank Kornelius Walter for the helpful discussions about statistics.

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Authors and Affiliations

  1. Universität Zürich, Zurich, Switzerland

    Felix Fontein & Urs Wagner

  2. Technische Universität Darmstadt, Darmstadt, Germany

    Michael Schneider

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  1. Felix Fontein
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  2. Michael Schneider
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Correspondence to Felix Fontein.

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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”.

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Fontein, F., Schneider, M. & Wagner, U. PotLLL: a polynomial time version of LLL with deep insertions. Des. Codes Cryptogr. 73, 355–368 (2014). https://doi.org/10.1007/s10623-014-9918-8

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  • DOI: https://doi.org/10.1007/s10623-014-9918-8

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