Sampling methods for shortest vectors, closest vectors and successive minima

Abstract

In this paper we introduce a new lattice problem, the subspace avoiding problem (SAP). We describe a probabilistic single exponential time algorithm for SAP for arbitrary ℓp norms. We also describe polynomial time reductions for four classical 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) to SAP, establishing probabilistic single exponential time algorithms for them. The result generalize and extend previous results of Ajtai, Kumar and Sivakumar. 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 ℓ2 norm to arbitrary ℓp norms. © Springer-Verlag Berlin Heidelberg 2007.