A new efficient factorization algorithm for polynomials over small finite fields

  • Harald Niederreiter
Article

Abstract

We present a new deterministic factorization algorithm for polynomials over a finite prime fieldF p . As in other factorization algorithms for polynomials over finite fields such as the Berlekamp algorithm, the key step is the “linearization” of the factorization problem, i.e., the reduction of the problem to a system of linear equations. The theoretical justification for our algorithm is based on a study of the differential equationy(p−1)+y p =0 of orderp−1 in the rational function fieldFp(x). In the casep=2 the new algorithm is more efficient than the Berlekamp algorithm since there is no set-up cost for the coefficient matrix of the system of linear equations.

Keywords

Factorization of polynomials over finite fields Differential equations over rational function fields 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Berlekamp, E. R.: Factoring polynomials over finite fields. Bell System Tech. J.46, 1853–1859 (1967)Google Scholar
  2. 2.
    Camion, P.: A deterministic algorithm for factorizing polynomials ofF q[X]. Ann. Discrete Math.17, 149–157 (1983)Google Scholar
  3. 3.
    Knuth, D. E.: The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 2nd ed. Reading, MA: Addison-Wesley 1981Google Scholar
  4. 4.
    Lidl, R., Niederreiter, H.: Finite Fields. Reading, MA: Addison-Wesley 1983Google Scholar
  5. 5.
    Mignotte, M.: Mathématiques pour le calcul formel. Paris: Presses Universitaires de France 1989Google Scholar
  6. 6.
    Willett, M.: Factoring polynomials over a finite field. SIAM J. Appl. Math.35, 333–337 (1978)Google Scholar
  7. 7.
    Göttfert, R.: The Niederreiter factorization algorithm is polynomial time in characteristic 2. Preprint, 1992Google Scholar
  8. 8.
    Niederreiter, H.: Factorization of polynomials and some linear algebra problems over finite fields. Preprint, 1992Google Scholar
  9. 9.
    Niederreiter, H.: Factoring polynomials over finite fields using differential equations and normal bases. Preprint, 1992Google Scholar
  10. 10.
    Niederreiter, H., Göttfert, R.: Factorization of polynomials over finite fields and characteristic sequences. Preprint, 1992Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Harald Niederreiter
    • 1
  1. 1.Institute for Information ProcessingAustrian Academy of SciencesViennaAustria

Personalised recommendations