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Closest Vectors, Successive Minima, and Dual HKZ-Bases of Lattices

  • Johannes Blömer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1853)

Abstract

In this paper we introduce a new technique to solve lattice problems. The technique is based on dual HKZ-bases. Using this technique we show how to solve the closest vector problem in lattices with rank n in time n! · s O(1), where s is the input size of the problem. This is an exponential improvement over an algorithm due to Kannan and Helfrich [16,15]. Based on the new technique we also show how to compute the successive minima of a lattice in time n! · 3n · s O(1), where n is the rank of the lattice and s is the input size of the lattice. The problem of computing the successive minima plays an important role in Ajtai’s worst-case to average-case reduction for lattice problems. Our results reveal a close connection between the closest vector problem and the problem of computing the successive minima of a lattice.

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References

  1. 1.
    M. Ajtai, “Generating Hard Instances of Lattice Problems”, Proc. 28th Symposium on Theory of Computing 1996, pp. 99–108.Google Scholar
  2. 2.
    M. Ajtai, “Worst-Case Complexity, Average-Case Complexity and Lattice Problems”, Proc. International Congress of Mathematicians 1998, Vol. III, pp. 421–428.MathSciNetGoogle Scholar
  3. 3.
    S. Arora, C. Lund, “Hardness of Approximations”, in D. S. Hochbaum (ed.), Approximation Algorithms for NP-Hard Problems, PWS Publishing, 1997.Google Scholar
  4. 4.
    S. Arora, L. Babai, J. Stern, Z. Sweedyk, “The Hardness of Approximate Optima in Lattices, Codes, and Systems of Linear Equations”, Journal of Computer and System Sciences Vol. 54, pp. 317–331, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    M. Ajtai, C. Dwork“A Public-Key Cryptosystem with Worst-Case/Average-Case Equivalence”, Proc. 29th Symposium on Theory of Computing 1997, pp. 284–293.Google Scholar
  6. 6.
    L. Babai, “On Lovasz Lattice Reduction Reduction and the Nearest Lattice Point Problem”, Combinatorica, Vol. 6, pp. 1–6, 1986.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    W. Banaszczyk, “New Bounds in Some Transference Theorems in the Geometry of Numbers”, Mathematische Annalen Vol. 296, pp. 625–635, 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    J. Blömer, J.-P. Seifert, “The Complexity of Computing Short Linearly Independent Vectors and Short Bases in a Lattice”, Proc. 21stth Symposium on Theory of Computing 1999, pp. 711–720.Google Scholar
  9. 9.
    J. W. S. Cassels, An Introduction to the Geometry of Numbers, Springer-Verlag, 1971.Google Scholar
  10. 10.
    J. Y. Cai, “A New Transference Theorem and Applications to Ajtai’s Connection Factor”, Proc. 14th Conference on Computational Complexity 1999, pp. 205–214.Google Scholar
  11. 11.
    J. Y. Cai, “Some Recent Progress on the Complexity of Lattice Problems”, Proc. 14th Conference on Computational Complexity 1999, pp. 158–177.Google Scholar
  12. 12.
    J. Y. Cai, A. P. Nerurkar, “An Improved Worst-Case to Average-Case Reduction for Lattice Problems”, Proc. 38th Symposium on Foundations of Computer Science 1997, pp. 468–477.Google Scholar
  13. 13.
    I. Dinur, G. Kindler, S. Safra, “Approximating CVP to Within Almost-Polynomial Factors is NP-Hard”, Proc. 39th Symposium on Foundations of Computer Science 1998, pp 99–109.Google Scholar
  14. 14.
    O. Goldreich, S. Goldwasser, “On the Limits of Non-Approximability of Lattice Problems”, Proc. 30th Symposium on Theory of Computing 1998, pp. 1–9.Google Scholar
  15. 15.
    B. Helfrich, “Algorithms to Construct Minkowski Reduced and Hermite Reduced Lattice Basis”, Theoretical Computer Science, Vol. 41, pp. 125–139, 1985.zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    R. Kannan, “Minkowski’s Convex Body Theorem and Integer Programming”, Mathematics of Operations Research, Vol. 12, No. 3, pp. 415–440, 1987.zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    L. Lovasz, An Algorithmic Theory of Graphs, Numbers and Convexity, SIAM, 1986.Google Scholar
  18. 18.
    J. Lagarias, H. W. Lenstra, C.P. Schnorr, “Korkin-Zolotarev Bases and Successive Minima of a Lattice and its Reciprocal Lattice”, Combinatorica Vol. 10, No. 4, pp. 333–348, 1990.zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    M. Pohst, “On the Computation of Lattice Vectors of Minimal Length, Successive Minima and Reduced Bases with Applications”, SIGSAM Bulletin Vol. 15, No. 1, pp. 37–44, 1981.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Johannes Blömer
    • 1
  1. 1.Department of Mathematics and Computer ScienceUniversity of PaderbornPaderbornGermany

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