Parallelization of matrix algorithms for Gröbner basis computation

  • D. E. Alexandrov
  • V. V. Galkin
  • A. I. Zobnin
  • M. V. Levin


Sequential and parallel implementations of the F4 algorithm for computing Gröbner bases of polynomial ideals are discussed.


Nonzero Element Message Passing Interface Polynomial Ideal Critical Pair Matrix Algorithm 


  1. 1.
    G. Attardi and C. Traverso, “Strategy-accurate parallel Buchberger algorithm,” J. Symb. Comput., 21, No. 4–6, 411–425 (1996).MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    T. Becker and V.Weispfenning, Gröbner Bases. A Computational Approach to Commutative Algebra, Grad. Texts Math., Vol. 141, Springer, New York (1993).MATHGoogle Scholar
  3. 3.
    B. Buchberger, “Gröbner bases: An algorithmic method in polynomial ideal theory,” in: Multidimensional Systems Theory, Reidel, Dordrecht (1985), pp. 184–232.Google Scholar
  4. 4.
    D. Cox, J. Little, and D. O’Shea, Ideals, Varieties and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, Springer, New York (1998).Google Scholar
  5. 5.
    J.-C. Faugère, “Parallelization of Gröbner bases,” in: Parallel and Symbolic Computation, Lect. Notes Comput., Vol. 5, World Scientific (1994), pp. 124–132.Google Scholar
  6. 6.
    J.-C. Faugère, “A new efficient algorithm for computing Gröbner bases (F4),” J. Pure Appl. Algebra, 139, No. 1-3, 61–88 (1999).MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    J.-C. Faugère, “A new efficient algorithm for computing Gröbner bases without reduction to zero (F5),” in: Proc. of the 2002 Int. Symp. on Symbolic and Algebraic Computation (ISSAC), ACM Press (2002), pp. 75–83.Google Scholar
  8. 8.
    R. Gebauer and H. M. Möller, “On an installation of Buchberger’s algorithm,” J. Symbolic Comput., 6, 275–286 (1988).MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    V. P. Gerdt, “Gröbner bases and involutive methods for algebraic and differential equations,” Math. Comput. Modelling, 25, No. 8/9, 75–90 (1997).MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    V. P. Gerdt and D. A. Yanovich, “Parallel computation of involutive and Gröbner bases,” in: V. G. Ganza, E. W. Mayr, and E. V. Vorozhtsov, eds., Proc. of CASC 2003, Institute of Informatics, Technical University of Munich, Garching (2004), pp. 185–194.Google Scholar
  11. 11.
    V. P. Gerdt, D. A. Yanovich, and Yu. A. Blinkov, “Fast search for the Janet divisor,” Program. Comput. Software, 27, No. 1, 22–24 (2001).MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    V. P. Gerdt and M. V. Zinin, “Involutive method for computing Gröbner bases over F 2,” Program. Comput. Software, 34, No. 4, 191–203 (2008).MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    A. Kipnis and A. Shamir, “Cryptanalysis of the HFE public key cryptosystem by relinearization,” in: Advances in Cryptology—Crypto’99, Lect. Notes Comput. Sci., Vol. 1666, Springer (1999), pp. 19–30.Google Scholar
  14. 14.
    V. A. Mityunin and E. V. Pankratiev, “Parallel algorithms for Gröbner bases construction,” J. Math. Sci., 142, No. 4, 2248–2266 (2007).MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    H. M. Möller, T. Mora, C. Traverso, “Gröbner bases computation using syzygies,” P. S. Wang, ed., Int. Symp. on Symbolic and Algebraic Computation 92. ISSAC 92. Berkeley, CA, USA, July 27–29, 1992, ACM Press, Baltimore (1992), pp. 320–328.CrossRefGoogle Scholar
  16. 16.
    E. V. Pankratiev, Elements of Computer Algebra [in Russian], Internet-Univ. Inform. Tekhnol., Moscow (2007).Google Scholar
  17. 17.
    A. A. Reeves, “A parallel implementation of Buchberger’s algorithm over ℤp for p = 31991,” J. Symbolic Comput., 26, No. 2, 229–244 (1998).MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    B. H. Roune, The F4 Algorithm: Speeding up Gröbner Basis Computation Using Linear Algebra, (2005).
  19. 19.
    A. J. M. Segers, Algebraic Attacks from a Gröbner Bases Perspective, MSc Thesis, Technische Universiteit Eindhoven (2004).Google Scholar
  20. 20.
    T. Stegers, Faugère’s F5 Algorithm Revisited, Diploma Thesis, Technische Universität Darmstadt (2005),
  21. 21.
    Q.-N. Tran, “Parallel computation and Gröbner bases: An application for converting bases with the Gröbner walk,” in: B. Buchberger and F. Winkler, eds., Gröbner Bases and Applications, Cambridge Univ. Press (1998), pp. 519–534.Google Scholar
  22. 22.
    D. A. Yanovich, “Parallelization of an algorithm for computation of involutive Janet bases,” Program. Comput. Software, 28, No. 2, 66–69 (2002).MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    D. A. Yanovich, “Efficiency estimate for distributed computation of Gröbner bases and involutive bases,” Program. Comput. Software, 34, No. 4, 210–215 (2008).MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  • D. E. Alexandrov
    • 1
  • V. V. Galkin
    • 1
  • A. I. Zobnin
    • 1
  • M. V. Levin
    • 1
  1. 1.Moscow State UniversityMoscowRussia

Personalised recommendations