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DOI: 10.4230/LIPIcs.FSTTCS.2008.1738
URN: urn:nbn:de:0030-drops-17380
URL: http://drops.dagstuhl.de/opus/volltexte/2008/1738/
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Arvind, V. ; Joglekar, Pushkar S.

Some Sieving Algorithms for Lattice Problems

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Abstract

We study the algorithmic complexity of lattice problems based on the sieving technique due to Ajtai, Kumar, and Sivakumar~\cite{aks}. Given a $k$-dimensional subspace $M\subseteq \R^n$ and a full rank integer lattice $\L\subseteq \Q^n$, the \emph{subspace avoiding problem} SAP, defined by Bl\"omer and Naewe \cite{blomer}, is to find a shortest vector in $\L\setminus M$. We first give a $2^{O(n+k \log k)}$ time algorithm to solve \emph{the subspace avoiding problem}. Applying this algorithm we obtain the following results. \begin{enumerate} \item We give a $2^{O(n)}$ time algorithm to compute $i^{th}$ successive minima of a full rank lattice $\L\subset \Q^n$ if $i$ is $O(\frac{n}{\log n})$. \item We give a $2^{O(n)}$ time algorithm to solve a restricted \emph{closest vector problem CVP} where the inputs fulfil a promise about the distance of the input vector from the lattice. \item We also show that unrestricted CVP has a $2^{O(n)}$ exact algorithm if there is a $2^{O(n)}$ time exact algorithm for solving CVP with additional input $v_i\in \L, 1\leq i\leq n$, where $\|v_i\|_p$ is the $i^{th}$ successive minima of $\L$ for each $i$. \end{enumerate} We also give a new approximation algorithm for SAP and the \emph{Convex Body Avoiding problem} which is a generalization of SAP. Several of our algorithms work for \emph{gauge} functions as metric, where the gauge function has a natural restriction and is accessed by an oracle.

BibTeX - Entry

@InProceedings{arvind_et_al:LIPIcs:2008:1738,
  author =	{V. Arvind and Pushkar S. Joglekar},
  title =	{{Some Sieving Algorithms for Lattice Problems}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science},
  pages =	{25--36},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-08-8},
  ISSN =	{1868-8969},
  year =	{2008},
  volume =	{2},
  editor =	{Ramesh Hariharan and Madhavan Mukund and V Vinay},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2008/1738},
  URN =		{urn:nbn:de:0030-drops-17380},
  doi =		{http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2008.1738},
  annote =	{Keywords: Lattice problems, sieving algorithm, closest vector problem}
}

Keywords: Lattice problems, sieving algorithm, closest vector problem
Seminar: IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science
Issue Date: 2008
Date of publication: 05.12.2008


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