Sampling Methods for Shortest Vectors, Closest Vectors and Successive Minima
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 Open image in new window , the closest vector problem Open image in new window , the successive minima problem Open image in new window , and the shortest independent vectors problem ( Open image in new window ) 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 Open image in new window and Open image in new window are new for all norms. The results on Open image in new window and Open image in new window generalize previous results of Ajtai et al. for the ℓ2 norm to arbitrary ℓ p norms.
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- Ajtai, M.: The shortest vector problem in l 2 is NP-hard for randomized reductions. In: Proceedings of the 30th ACM Symposium on Theory of Computing, pp. 10–19. ACM Press, New York (1998)Google Scholar
- Ajtai, M., Kumar, R., Sivakumar, D.: A sieve algorithm for the shortest lattice vector problem. In: Proceedings of the 33th ACM Symposium on Theory of Computing, pp. 601–610. ACM Press, New York (2001)Google Scholar
- Blömer, J., Seifert, J.-P.: The complexity of computing short linearly independent vectors and sort bases in a lattice. In: Proceedings of the 21th Symposium on Theory of Computing, pp. 711–720 (1999)Google Scholar
- Regev, O.: Lecture note on lattices in computer science, lecture 8: 2O(n)-time algorithm for SVP (2004)Google Scholar