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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},

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