Some Sieving Algorithms for Lattice Problems

We study the algorithmic complexity of lattice problems based on the sieving technique due to Ajtai, Kumar, and Sivakumar [AKS01]. Given a k-dimensional subspace M ⊆ Rn and a full rank integer lattice L ⊆ Qn, the subspace avoiding problem SAP, defined by Blömer and Naewe [BN07], is to find a shortest vector in L \ M. We first give a 2O(n+k log k) time algorithm to solve the subspace avoiding problem. Applying this algorithm we obtain the following results. 1. We give a 2O(n) time algorithm to compute ith successive minima of a full rank lattice L ⊂ Qn if i is O( n log n ). 2. We give a 2O(n) time algorithm to solve a restricted closest vector problem CVPwhere the inputs fulfil a promise about the distance of the input vector from the lattice. 3. We also show that unrestricted CVP has a 2O(n) exact algorithm if there is a 2O(n) time exact algorithm for solving CVP with additional input vi ∈ L, 1 ≤ i ≤ n, where ‖vi‖p is the ith successive minima of L for each i. We also give a new approximation algorithm for SAP and the Convex Body Avoiding problem which is a generalization of SAP. Several of our algorithms work for gauge functions as metric, where the gauge function has a natural restriction and is accessed by an oracle.