Sampling Methods for Shortest Vectors, Closest Vectors and Successive Minima
Johannes Blömer and Stefanie Naewe
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 l_p 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 l_2 norm to arbitrary l_p norms.