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URN: urn:nbn:de:0030-drops-21256
URL: http://drops.dagstuhl.de/opus/volltexte/2009/2125/
Buchmann, Johannes A. ; Lindner, Richard

Density of Ideal Lattices

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Abstract

The security of many emph{efficient} cryptographic constructions, e.g.~collision-resistant hash functions, digital signatures, and identification schemes, has been proven assuming the hardness of emph{worst-case} computational problems in ideal lattices. These lattices correspond to ideals in the ring of integers of some fixed number field $K$. In this paper we show that the density of $n$-dimensional ideal lattices with determinant $le b$ among all lattices under the same bound is in $O(b^{1-n})$. So for lattices of dimension $> 1$ with bounded determinant, the subclass of ideal lattices is always vanishingly small.

BibTeX - Entry

@InProceedings{buchmann_et_al:DSP:2009:2125,
  author =	{Johannes A. Buchmann and Richard Lindner},
  title =	{Density of Ideal Lattices},
  booktitle =	{Algorithms and Number Theory },
  year =	{2009},
  editor =	{Johannes A. Buchmann and John Cremona and Michael E. Pohst},
  number =	{09221},
  series =	{Dagstuhl Seminar Proceedings},
  ISSN =	{1862-4405},
  publisher =	{Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2009/2125},
  annote =	{Keywords: Post-quantum cryptography, provable security, ideal lattices}
}

Keywords: Post-quantum cryptography, provable security, ideal lattices
Seminar: 09221 - Algorithms and Number Theory
Issue date: 2009
Date of publication: 20.08.2009

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