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URN: urn:nbn:de:0030-drops-17380
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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 =		{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: 2008

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