A new efficient factorization algorithm for polynomials over small finite fields

  • Harald Niederreiter


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.


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


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  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

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