Sampling methods for shortest vectors, closest vectors and successive minima

Abstract

We study four problems from the geometry of numbers, the shortest vector problem ( Svp ) , the closest vector problem ( Cvp ) , the successive minima problem ( Smp ) , and the shortest independent vectors problem ( Sivp ). Extending and generalizing results of Ajtai, Kumar, and Sivakumar we present probabilistic single exponential time algorithms for all four problems for all ℓ p norms. The results on Smp and Sivp are new for all norms. The results on Svp and Cvp generalize previous results of Ajtai et al. for the Euclidean ℓ 2 norm to arbitrary ℓ p norms. We achieve our results by introducing a new lattice problem, the generalized shortest vector problem ( GSvp ). 1 1 In the original conference version of this paper, we called this problem the subspace avoiding problem ( Sap ). We describe a single exponential time algorithm for GSvp . We also describe polynomial time reductions from Svp , Cvp , Smp , and Sivp to GSvp , establishing single exponential time algorithms for the four classical lattice problems. This approach leads to a unified algorithmic treatment of the lattice problems Svp , Cvp , Smp , and Sivp .