In this paper we introduce a new lattice problem, the sub-space avoiding problem (Sap). We describe a probabilistic single exponential time algorithm for Sap for arbitrary d_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 e_2 norm to arbitrary e_p norms.
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