Efficient multivariate factorization over finite fields

  • Laurent Bernardin
  • Michael B. Monagan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1255)


We describe the Maple [23] implementation of multivariate factorization over general finite fields. Our first implementation is available in Maple V Release 3. We give selected details of the algorithms and show several ideas that were used to improve its efficiency. Most of the improvements presented here are incorporated in Maple V Release 4.

In particular, we show that we needed a general tool for implementing computations in GF(p k )[x1, x2,..., x v ]. We also needed an efficient implementation of our algorithms in ℤ p [y][x] because any multivariate factorization may depend on several bivariate factorizations.

The efficiency of our implementation is illustrated by the ability to factor bivariate polynomials with over a million monomials over a small prime field.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bernardin, L. Fast dense Hensel lifting, manuscript, ETH Zürich, 1995. Available via Scholar
  2. 2.
    Bernardin, L. On square-free factorization of multivariate polynomials over a finite field. Theoretical Computer Science 187 (1997). to appear.Google Scholar
  3. 3.
    Cantor, D. G., and Zassenhaus, H. A new algorithm for factoring polynomials over finite fields. Mathematics of Computation 36, 154 (1981), 587–592.Google Scholar
  4. 4.
    Czapor, S. R. Solving algebraic equations:combining Buchberger's algorithm with multivariate factorization. Journal of Symbolic Computation 7, 1 (January 1989), 49–54.Google Scholar
  5. 5.
    da Rosa, R. M. Private communication, February 1996.Google Scholar
  6. 6.
    Geddes, K. O., Czapor, S. R., and Labahn, G.Algorithms for Computer Algebra. Kluwer Academic Publishers, Boston, 1992.Google Scholar
  7. 7.
    Jenks, R., and Sutor, R. AXIOM: The Scientific Computation System. Springer Verlag, 1992.Google Scholar
  8. 8.
    Kaltofen, E. Sparse Hensel lifting. In Proceedings of Eurocal '85, Vol. II (1985), B. F. Caviness, Ed., vol. 204 of Lecture Notes in Computer Science, Springer-Verlag, pp. 4–17.Google Scholar
  9. 9.
    Kaltofen, E., Musser, D. R., and Saunders, B. D. A generalized class of polynomials that are hard to factor. SIAM Journal on Computing 12, 3 (1983), 473–485.Google Scholar
  10. 10.
    Kaltofen, E., and Trager, B. M. Computing with polynomials given by black boxes for their evaluations: Greatest common divisors, factorization, separation of numerators and denominators. Journal of Symbolic Computation 9, 3 (March 1990), 300–320.Google Scholar
  11. 11.
    Knuth, D. E. Seminumerical Algorithms, vol. 2 of The Art of Computer Programming. Addison Wesley, 1981.Google Scholar
  12. 12.
    Lucks, M. A fast implementation of polynomial factorization. In SYMSAC '86: Proceedings of the 1986 ACM Symposium on Symbolic and Algebraic Computation (1986), pp. 228–232.Google Scholar
  13. 13.
    Monagan, M. B. Gauss: A parameterized domain of computation system with support for signature functions. In Proceedings of DISCO '93 (1993), vol. 722 of Lecture Notes in Computer Science, Springer-Verlag, pp. 81–94.Google Scholar
  14. 14.
    Monagan, M. B. In-place arithmetic for polynomials over Zn. In Proceedings of DISCO '92 (1993), vol. 721 of Lecture Notes in Computer Science, Springer-Verlag, pp. 22–34.Google Scholar
  15. 15.
    Popp, H. Moduli Theory and Classification Theory of Algebraic Varieties, vol. 620 of Lecture Notes in Mathematics. Springer-Verlag, 1977.Google Scholar
  16. 16.
    Shoup, V. A new polynomial factorization algorithm and its implementation. Journal of Symbolic Computation 20, 4 (1995), 363–397.Google Scholar
  17. 17.
    Stoutmyer, D. R. Which polynomial representation is best? In Proceedings of the 1984 MACSYMA User's Conference (1984), V. E. Golden, Ed., pp. 221–243.Google Scholar
  18. 18.
    Swanson, S. L. On the Factorization of Multivariate Polynomials over Finite Fields. PhD thesis, Purdue University, 1993.Google Scholar
  19. 19.
    von Zur Gathen, J. Factoring sparse multivariate polynomials. In Proceedings of the 24th IEEE Symposium on Foundations of Computer Science (1983), pp. 172–179.Google Scholar
  20. 20.
    von Zur Gathen, J. Irreducibility of multivariate polynomials. Journal of Computer and System Sciences 31 (1985), 225–264.Google Scholar
  21. 21.
    von Zur Gathen, J., and Kaltofen, E. Polynomial-time factorization of multivariate polynomials over finite fields. In Proceedings of ICALP '83 (1983), vol. 154 of Lecture Notes in Computer Science, Springer-Verlag, pp. 250–262.Google Scholar
  22. 22.
    von Zur Gathen, J., and Kaltofen, E. Factoring sparse multivariate polynomials. Journal of Computer and System Sciences 31 (1985), 265–287.Google Scholar
  23. 23.
    Waterloo Maple Inc. Maple V learning guide. Springer-Verlag, 1996.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Laurent Bernardin
    • 1
  • Michael B. Monagan
    • 2
  1. 1.Institut für Wissenschaftliches RechnenETHZürichSwitzerland
  2. 2.Center for Experimental and Constructive Mathematics Department of Mathematics and StatisticsSimon Fraser UniversityCanada

Personalised recommendations