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Some Sieving Algorithms for Lattice Problems Arvind, V. Joglekar, Pushkar S. Lattice problems sieving algorithm closest vector problem 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik V. Arvind and Pushkar S. Joglekar 2008 Is Part Of LIPIcs, Volume 2, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2008) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.FSTTCS.2008.1738 urn:nbn:de:0030-drops-17380 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2008.1738 eng https://creativecommons.org/licenses/by-nc-nd/3.0/legalcode