, Volume 77, Issue 2, pp 179–203 | Cite as

A New Sparse Gaussian Elimination Algorithm and the Niederreiter Linear System for Trinomials over F2



An important factorization algorithm for polynomials over finite fields was developed by Niederreiter. The factorization problem is reduced to solving a linear system over the finite field in question, and the solutions are used to produce the complete factorization of the polynomial into irreducibles. One charactersistic feature of the linear system arising in the Niederreiter algorithm is the fact that, if the polynomial to be factorized is sparse, then so is the Niederreiter matrix associated with it. In this paper, we investigate the special case of factoring trinomials over the binary field. We develop a new algorithm for solving the linear system using sparse Gaussian elmination with the Markowitz ordering strategy. Implementing the new algorithm to solve the Niederreiter linear system for trinomials over F2 suggests that, the system is not only initially sparse, but also preserves its sparsity throughout the Gaussian elimination phase. When used with other methods for extracting the irreducible factors using a basis for the solution set, the resulting algorithm provides a more memory efficient and sometimes faster sequential alternative for achieving high degree trinomial factorizations over F2.

AMS Subject Classifications

11T06 15-04 68-04 68W05 68W30 68W40 


Finite fields polynomial factorization Niederreiter algorithm sparse Gaussian elimination 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Roelse, P. 1999Factoring high-degree polynomials over F2 with Niederreiter's algorithm on the IBM SP2Math. Comp.68869880CrossRefMATHMathSciNetGoogle Scholar
  2. Fleischmann, P., Holder, M., Roelse, P. 2003The black-box Niederreiter algorithm and its implementation over the binary fieldMath. Comp.7218871899CrossRefMathSciNetGoogle Scholar
  3. Gao, S., von zur Gathen, J. 1994Berlekamp's and Niederreiter's polynomial factorization algorithmsContemp. Math.168101116MathSciNetGoogle Scholar
  4. Berlekamp, E. R. 1967Factoring polynomials over finite fieldsBell Sys. Tech. J.4618531859MATHMathSciNetGoogle Scholar
  5. Niederreiter, H. 1993A new efficient factorization algorithm for polynomials over small finite fieldsAAECC48187CrossRefMATHMathSciNetGoogle Scholar
  6. Niederreiter, H. 1993Factorization of polynomials and some linear algebra problems over finite fieldsLin. Alg. Appl.192301328CrossRefMATHMathSciNetGoogle Scholar
  7. Niederreiter, H. 1994Factoring polynomials over finite fields using differential equations and normal basesMath. Comp.62819830MATHMathSciNetGoogle Scholar
  8. Niederreiter, H., Göttfert, R. 1993Factorization of polynomials over finite fields and characteristic sequencesJ. Symb. Comp.16401412Google Scholar
  9. Gustavson, F. G. 1972

    Some basic techniques for solving sparse systems of linear equations

    Rose, D. J.Willoughby, R. A. eds. Sparse matrices and their applicationsPlenum PressNew York4152
    Google Scholar
  10. Göttfert, R. 1994An acceleration of the Niederreiter factorization algorithm in characteristic 2Math. Comp.62831839MATHMathSciNetGoogle Scholar
  11. von zur Gathen, J., Gerhard, J. 1999Modern Computer algebraCambridge University PressCambridgeGoogle Scholar
  12. Fleischmann, P. 1993Connections between the algorithms of Berlekamp and Niederreiter for factoring polynomials over F q Lin. Alg. Appl.192101108CrossRefMATHMathSciNetGoogle Scholar
  13. Lee, T. C. Y., Vanstone, S. A. 1995Subspaces and polynomial factorizations over finite fieldsAAECC6147157CrossRefMathSciNetGoogle Scholar
  14. Duff, I. S., Erisman, A. M., Reid, J. K. 1986Direct methods for sparse matricesOxford University PressNew YorkGoogle Scholar
  15. Stappen, A. F., Bisseling, R. H., Vorst, J. G. G. 1993Parallel sparse LU decomposition on a mesh network of transputersSIAM J. Matrix Anal. Appl.14853879MathSciNetGoogle Scholar
  16. Markowitz, H. M. 1957The elimination form of the inverse and its application to linear programmingManagement Sci.3255269MATHMathSciNetGoogle Scholar
  17. Curtis, A. R., Reid, J. K. 1971The solution of large sparse unsymmetric systems of linear equationsJ. Inst. Math. Appl.8344353Google Scholar
  18. Abu Salem, F.: A BSP parallel model of the Göttfert algorithm for polynomial factorization over F2. PPAM 2003, Lecture Notes in Computer Science, vol. 3019, 217–224 (2004).Google Scholar
  19. von zur Gathen, J., Gerhard, J.: Arithmetic and factorization of polynomials over F2. Proc. ISSAC 96. Zurich: ACM Press, pp. 1–9.Google Scholar
  20. Bonorden, O., von zur Gathen, J., Gerhard, J., Muller, O., Nocker, M. 2001Factoring a binary polynomial of degree over one millionACM SIGSAM Bull.351618Google Scholar

Copyright information

© Springer-Verlag Wien 2006

Authors and Affiliations

  1. 1.Computer Science DepartmentAmerican University of BeirutBeirutLebanon

Personalised recommendations