License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
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DOI: 10.4230/DagSemProc.09221.2
URN: urn:nbn:de:0030-drops-21256
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Buchmann, Johannes A. ; Lindner, Richard

Density of Ideal Lattices

09221.LindnerRichard.Paper.2125.pdf (0.2 MB)


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

  author =	{Buchmann, Johannes A. and Lindner, Richard},
  title =	{{Density of Ideal Lattices}},
  booktitle =	{Algorithms and Number Theory},
  pages =	{1--6},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2009},
  volume =	{9221},
  editor =	{Johannes A. Buchmann and John Cremona and Michael E. Pohst},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-21256},
  doi =		{10.4230/DagSemProc.09221.2},
  annote =	{Keywords: Post-quantum cryptography, provable security, ideal lattices}

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

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