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An efficient parallel block-reduction algorithm

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Algorithmic Number Theory (ANTS 1998)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1423))

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Abstract

In this paper, we present a new parallel block-reduction algorithm for reducing lattice bases which allows the use of an arbitrarily chosen block-size between two and n where n denotes the dimension of the lattice. Thus, we are building a hierarchy of parallel lattice basis reduction algorithms between the known parallel all-swap algorithm which is a parallelization for block-size two and the reduction algorithm for block-size n which corresponds to the known sequential lattice basis reduction algorithm. We show that even though the parallel all-swap algorithm as well as the parallel block-reduction algorithm have the same asymptotic complexity in respect to arithmetic operations in theory, in practice neither block-size two nor block-size n are a priori the best choices. The optimal block-size in respect to minimizing the reduction time rather depends strongly on the used parallel system and the corresponding communication costs.

The research was done while the author was a member of the Graduiertenkolleg Informatik at the UniversitÄt des Saarlandes (Saarbrücken), a fellowship program of the DFG (Deutsche Forschungsgemeinschaft).

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References

  1. Biehl, I., Buchmann, J., and Papanikolaou, T.: LiDIA: A Library for Computational Number Theory. Technical Report 03/95, SFB 124, UniversitÄt des Saarlandes (1995).

    Google Scholar 

  2. Buchmann, J., and Kessler, V.: Computing a Reduced Lattice Basis from a Generating System. Preprint, UniversitÄt des Saarlandes, Saarbrücken (1992).

    Google Scholar 

  3. Cohen, H.: A Course in Computational Algebraic Number Theory. Second Edition, Springer Verlag Heidelberg (1993).

    MATH  Google Scholar 

  4. Coster, M.J., LaMacchia, B.A., Odlyzko, A.M., and Schnorr, C.P.: An Improved Low-density Subset Sum Algorithm. Proceedings EUROCRYPT '91, Springer Lecture Notes in Computer Science LNCS 547, pp. 54–67 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  5. Golub, G.H., and van Loan, C.F.: Matrix Computations. John Hopkins University Press Baltimore (1996).

    MATH  Google Scholar 

  6. Grötschel, M., Lovász, L., and Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Second Edition, Springer Verlag Heidelberg (1993).

    MATH  Google Scholar 

  7. Heckler, C.: Automatische Parallelisierung und parallele Gitterbasisreduktion. PhD Thesis, UniversitÄt des Saarlandes, Saarbrücken, Germany (1995).

    Google Scholar 

  8. Heckler, C., and Thiele, L.: On the Time Complexity of Parallel Algorithms for Lattice Basis Reduction. Technical Report 05/93, SFB 124, UniversitÄt des Saarlandes (1995).

    Google Scholar 

  9. Heckler, C., and Thiele, L.: A Parallel Lattice Basis Reduction for Mesh-Connected Processor Arrays and Parallel Complexity. Proceedings SPDP '93, pp. 400–407 (1993).

    Google Scholar 

  10. Heckler, C., and Thiele, L.: Parallel Complexity of Lattice Basis Reduction and a Floating-Point Parallel Algorithm. Proceedings PARLE'93, Springer Lecture Notes in Computer Science LNCS 694, pp. 744–747 (1993).

    Google Scholar 

  11. Heckler, C., and Thiele, L.: Complexity Analysis of a Parallel Lattice Basis Reduction Algorithm. To appear in SIAM J. Comput. (1998).

    Google Scholar 

  12. Joux, A.: A Fast Parallel Lattice Reduction Algorithm. Proceedings Second Gauss Symposium, pp. 1–15 (1993).

    Google Scholar 

  13. Joux, A.: La Réduction des Réseaux en Cryptographie. PhD Thesis Laboratoire d'Informatique de L'Ecole Normale Superieure LIENS, Paris, France (1993).

    Google Scholar 

  14. Joux, A., and Stern, J.: Lattice Reduction: A Toolbox for the Cryptanalyst. Preprint (1994).

    Google Scholar 

  15. Kaltofen, E.: On the Complexity of Finding Short Vectors in Integer Lattices. Computer Algebra, Springer Lecture Notes in Computer Science LNCS 162, pp. 236–244 (1983).

    MATH  MathSciNet  Google Scholar 

  16. Lagarias, J.C., and Odlyzko, A.M.: Solving Low-Density Subset Sum Problems. JACM 32, pp. 229–246 (1985).

    Article  MATH  MathSciNet  Google Scholar 

  17. LaMacchia, B.A.: Basis Reduction Algorithms and Subset Sum Problems. Master's Thesis MIT, (1991).

    Google Scholar 

  18. Lenstra, A.K., Lenstra, H.W., and Lovász, L.: Factoring Polynomials with Rational Coefficients. Math. Ann. 261, pp. 515–534 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  19. Lenstra, H.W.: Integer Programming With a Fixed Number of Variables. Mathematics Operations Research, pp. 538–548 (1983).

    Google Scholar 

  20. LiDIA Group: LiDIA Manual. UniversitÄt des Saarlandes/TU Darmstadt, see LiDIA homepage: http://www.informatik.tu-darmstadt.de/TI/LiDIA (1997).

    Google Scholar 

  21. Papanikolaou, T.: Software-Entwicklung in der Computer-Algebra am Beispiel einer objektorientierten Bibliothek für algorithmische Zahlentheorie. PhD Thesis, UniversitÄt des Saarlandes, Saarbrücken, Germany (1997).

    Google Scholar 

  22. Pohst, M.E.: A Modification of the LLL Reduction Algorithm. Journal of Symbolic Computation 4, pp. 123–127 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  23. Pohst, M.E., and Zassenhaus, H.J.: Algorithmic Algebraic Number Theory. Cambridge University Press (1989).

    Google Scholar 

  24. Roch, J.L., and Villard, G.: Parallel Gcd and Lattice Basis Reduction. Proceedings CONPAR '92, Springer Lecture Notes in Computer Science LNCS 634, pp. 557–564 (1992).

    Google Scholar 

  25. Schnorr, C.P.: A More Efficient Algorithm for Lattice Basis Reduction. Journal of Algorithms 9, pp. 47–62 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  26. Schnorr, C.P., and Euchner, M.: Lattice Basis Reduction: Improved Practical Algorithms and Solving Subset Sum Problems. Proceedings of Fundamentals of Computation Theory '91, Springer Lecture Notes in Computer Science LNCS 529, pp. 68–85 (1991).

    MATH  MathSciNet  Google Scholar 

  27. Schönhage, A.: Factorization of Univariate Integer Polynomials by Diophantine Approximation and an Improved Basis Reduction Algorithm. Proceedings ICALP '84, Springer Lecture Notes in Computer Science LNCS 172, pp. 436–447 (1984).

    MATH  Google Scholar 

  28. Schrijver, A.: Theory of Linear and Integer Programming. J. Wiley & Sons, New York (1986).

    MATH  Google Scholar 

  29. Villard, G.: Parallel Lattice Basis Reduction. Proceedings ISSAC '92, ACM Press, pp. 269–277 (1992).

    Google Scholar 

  30. Wetzel, S.: Lattice Basis Reduction Algorithms and their Applications. PhD Thesis, UniversitÄt des Saarlandes, Saarbrücken, Germany, submitted (1998).

    Google Scholar 

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Authors and Affiliations

  1. FTK/A, Daimler Benz AG, HPC 0507, D-70564, Stuttgart, Germany

    Susanne Wetzel

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  1. Susanne Wetzel
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Joe P. Buhler

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© 1998 Springer-Verlag Berlin Heidelberg

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Wetzel, S. (1998). An efficient parallel block-reduction algorithm. In: Buhler, J.P. (eds) Algorithmic Number Theory. ANTS 1998. Lecture Notes in Computer Science, vol 1423. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0054872

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  • DOI: https://doi.org/10.1007/BFb0054872

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64657-0

  • Online ISBN: 978-3-540-69113-6

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