The Beauty and the Beasts—The Hard Cases in LLL Reduction View Full Text


Ontology type: schema:Chapter     


Chapter Info

DATE

2017-07-27

AUTHORS

Saed Alsayigh , Jintai Ding , Tsuyoshi Takagi , Yuntao Wang

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. More... »

PAGES

19-35

References to SciGraph publications

  • 2001. A 3-Dimensional Lattice Reduction Algorithm in CRYPTOGRAPHY AND LATTICES
  • 2014. Rounding and Chaining LLL: Finding Faster Small Roots of Univariate Polynomial Congruences in PUBLIC-KEY CRYPTOGRAPHY – PKC 2014
  • 2000. Worst-Case Complexity of the Optimal LLL Algorithm in LATIN 2000: THEORETICAL INFORMATICS
  • 2008. Predicting Lattice Reduction in ADVANCES IN CRYPTOLOGY – EUROCRYPT 2008
  • 1994-08. Lattice basis reduction: Improved practical algorithms and solving subset sum problems in MATHEMATICAL PROGRAMMING
  • 2010. On Ideal Lattices and Learning with Errors over Rings in ADVANCES IN CRYPTOLOGY – EUROCRYPT 2010
  • 1982-12. Factoring polynomials with rational coefficients in MATHEMATISCHE ANNALEN
  • 2010. Lattice Enumeration Using Extreme Pruning in ADVANCES IN CRYPTOLOGY – EUROCRYPT 2010
  • Book

    TITLE

    Advances in Information and Computer Security

    ISBN

    978-3-319-64199-7
    978-3-319-64200-0

    Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/978-3-319-64200-0_2

    DOI

    http://dx.doi.org/10.1007/978-3-319-64200-0_2

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1090965459


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