Sampling Methods for Shortest Vectors, Closest Vectors and Successive Minima

  • Johannes Blömer
  • Stefanie Naewe
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4596)

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 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 ℓp norms.

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References

  1. 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
  2. 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
  3. Ajtai, M., Kumar, R., Sivakumar, D.: Sampling short lattice vectors and the closest lattice vector problem. In: Proceedings of the 17th IEEE Annual Conference on Computational Complexity - CCC, pp. 53–57. IEEE Computer Society Press, Los Alamitos (2002)CrossRefGoogle Scholar
  4. Babai, L.: On Lovász’ lattice reduction and the nearest lattice point problem. Combinatorica 6(1), 1–13 (1986)MATHCrossRefMathSciNetGoogle Scholar
  5. Blömer, J.: Closest vectors, successive minima, and dual HKZ-bases of lattices. In: Welzl, E., Montanari, U., Rolim, J.D.P. (eds.) ICALP 2000. LNCS, vol. 1853, pp. 248–259. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  6. 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
  7. Dyer, M., Frieze, A., Kannan, R.: A random polynomial time algorithm for approximating the volume of convex bodies. Journal of the ACM 38(1), 1–17 (1991)MATHCrossRefMathSciNetGoogle Scholar
  8. Dinur, I., Kindler, G., Raz, R., Safra, S.: Approximating CVP to within almost-polynomial factors in NP-hard. Combinatorica 23(2), 205–243 (2003)MATHCrossRefMathSciNetGoogle Scholar
  9. Kannan, R.: Algorithmic geometry of numbers. Annual Reviews in Computer Science 2, 231–267 (1987)CrossRefMathSciNetGoogle Scholar
  10. Kannan, R.: Minkowski’s convex body theorem and integer programming. Mathematics of Operations Research 12(3), 415–440 (1987)MATHMathSciNetCrossRefGoogle Scholar
  11. Khot, S.: Hardness of approximating the shortest vector problem in lattices. Journal of the ACM (JACM) 52(5), 789–808 (2005)CrossRefMathSciNetGoogle Scholar
  12. Lenstra, A.K., Lenstra, H.W., Lovász, L.: Factoring polynomials with rational coefficients. Mathematische Annalen 261, 515–534 (1982)MATHCrossRefMathSciNetGoogle Scholar
  13. Micciancio, D., Goldwasser, S.: Complexity of Lattice Problems - A Cryptographic Perspective. Kluwer Academic Publishers, Dordrecht (2002)MATHGoogle Scholar
  14. Micciancio, D.: The shortest vector in a lattice is hard to approximate to within some constant. SIAM Journal on Computing 30(6), 2008–2035 (2000)CrossRefMathSciNetGoogle Scholar
  15. Regev, O.: Lecture note on lattices in computer science, lecture 8: 2O(n)-time algorithm for SVP (2004)Google Scholar
  16. Schnorr, C.-P.: A hierarchy of polynomial time lattice basis reduction algorithms. Theoretical Computer Science 53, 201–224 (1987)MATHCrossRefMathSciNetGoogle Scholar
  17. Schnorr, C.-P.: Block reduced lattice bases and successive minima. Combinatorics, Probability & Computing 3, 507–522 (1994)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Johannes Blömer
    • 1
  • Stefanie Naewe
    • 1
  1. 1.Department of Computer Science, University of Paderborn 

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