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An Overview of the Sieve Algorithm for the Shortest Lattice Vector Problem

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Cryptography and Lattices (CaLC 2001)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2146))

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Abstract

We present an overview of a randomized 2g(n) time algorithm to compute a shortest non-zero vector in an n-dimensional rational lattice. The complete details of this algorithm can be found in [2].

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References

  1. M. Ajtai. The shortest vector problem in L 2 is NP-hard for randomized reductions. Proc. 30th ACM Symposium on Theory of Computing, pp. 10–19, 1998.

    Google Scholar 

  2. M. Ajtai, R. Kumar, and D. Sivakumar. A sieve algorithm for the shortest lattice vector problem. Proc. 33rd ACM Symposium on Theory of Computing, 2001. To appear.

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  3. P. van Emde Boas. Another NP-complete partition problem and the complexity of computing short vectors in lattices. Mathematics Department, University of Amsterdam, TR 81-04, 1981.

    Google Scholar 

  4. C. F. Gauss. Disquisitiones Arithmeticae. English edition, (Translated by A. A. Clarke) Springer-Verlag, 1966.

    Google Scholar 

  5. O. Goldreich and S. Goldwasser. On the limits of nonapproximability of lattice problems. Journal of Computer and System Sciences, 60(3):540–563, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  6. B. Helfrich. Algorithms to construct Minkowski reduced and Hermite reduced bases. Theoretical Computer Science, 41:125–139, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  7. C. Hermite. Second letter to Jacobi, Oeuvres, I, Journal für Mathematik, 40:122–135, 1905.

    Google Scholar 

  8. R. Kannan. Minkowski’s convex body theorem and integer programming. Mathematics of Operations Research, 12:415–440, 1987. Preliminary version in ACM Symposium on Theory of Computing 1983.

    Article  MATH  MathSciNet  Google Scholar 

  9. R. Kumar and D. Sivakumar. On polynomial approximations to the shortest lattice vector length. Proc. 12th Symposium on Discrete Algorithms, 2001.

    Google Scholar 

  10. A. K. Lenstra, H. W. Lenstra, and L. Lovász. Factoring polynomials with rational coefficients. Mathematische Annalen, 261:515–534, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  11. J. C. Lagarias, H. W. Lenstra, and C. P. Schnorr. Korkine-Zolotarev bases and successive minima of a lattice and its reciprocal lattice. Combinatorica, 10:333–348, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  12. D. Micciancio. The shortest vector in a lattice is hard to approximate to within some constant. Proc. 39th IEEE Symposium on Foundations of Computer Science, pp. 92–98, 1998.

    Google Scholar 

  13. H. Minkowski. Geometrie der Zahlen. Leipzig, Teubner, 1990.

    Google Scholar 

  14. C. P. Schnorr. A hierarchy of polynomial time basis reduction algorithms. Theoretical Computer Science, 53:201–224, 1987.

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. IBM Almaden Research Center, 650 Harry Road, San Jose, CA, 95120

    Miklós Ajtai, Ravi Kumar & Dandapani Sivakumar

Authors
  1. Miklós Ajtai
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  2. Ravi Kumar
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  3. Dandapani Sivakumar
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Editor information

Editors and Affiliations

  1. Mathematics Department, Brown University, Box 1917, Providence, RI, 02912, USA

    Joseph H. Silverman

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© 2001 Springer-Verlag Berlin Heidelberg

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Ajtai, M., Kumar, R., Sivakumar, D. (2001). An Overview of the Sieve Algorithm for the Shortest Lattice Vector Problem. In: Silverman, J.H. (eds) Cryptography and Lattices. CaLC 2001. Lecture Notes in Computer Science, vol 2146. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44670-2_1

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  • DOI: https://doi.org/10.1007/3-540-44670-2_1

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42488-8

  • Online ISBN: 978-3-540-44670-5

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