Polynomial-time factorization of multivariate polynomials over finite fields

  • J. von zur Gathen
  • E. Kaltofen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 154)


We present a probabilistic algorithm that finds the irreducible factors of a bivariate polynomial with coefficients from a finite field in time polynomial in the input size, i.e. in the degree of the polynomial and log (cardinality of field). The algorithm generalizes to multivariate polynomials and has polynomial running time for densely encoded inputs. Also a deterministic version of the algorithm is discussed whose running time is polynomial in the degree of the input polynomial and the size of the field.


Finite Field Total Degree Input Size Irreducible Factor Univariate Polynomial 
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  1. A.V. Aho, J.E. Hopcroft and J.D. Ullman, The design and analysis of computer algorithms. Addison-Wesley, Reading MA, 1974.Google Scholar
  2. M. Ben-Or, Probabilistic algorithms in finite fields. Proc. 22nd Symp. Foundations Comp. Sci. IEEE, 1981, 394–398.Google Scholar
  3. E.R. Berlekamp, Factoring polynomials over finite fields. Bell System Tech. J. 46 (1967), 1853–1859.Google Scholar
  4. E.R. Berlekamp, Factoring polynomials over large finite fields. Math. Comp. 24 (1970), 713–735.Google Scholar
  5. W.S. Brown, On Euclid's algorithm and the computation of polynomial Greatest Common Divisors. J. ACM 18 (1971), 478–504.Google Scholar
  6. D.G. Cantor and H. Zassenhaus, On algorithms for factoring polynomials over finite fields. Math. Comp. 36 (1981), 587–592.Google Scholar
  7. A.L. Chistov and D.Yu. Grigoryev, Polynomial-time factoring of the multivariable polynomials over a global field. LOMI preprint E-5-82, Leningrad, 1982.Google Scholar
  8. J.H. Davenport and B.M. Trager, Factorization over finitely generated fields. Proc. 1981 ACM Symp. Symbolic and Algebraic Computation, ed. by P. Wang, 1981, 200–205.Google Scholar
  9. J. von zur Gathen, Hensel and Newton methods in valuation rings. Tech. Report 155(1981), Dept. of Computer Science, University of Toronto. To appear in Math. Comp.Google Scholar
  10. J. von zur Gathen, Parallel algorithms for algebraic problems. Proc. 15th ACM Symp. Theory of Computing, Boston, 1983.Google Scholar
  11. J. von zur Gathen [83a], Factoring sparse multivariate polynomials. Manuscript, 1983.Google Scholar
  12. G.H. Hardy and E.M. Wright, An introduction to the theory of numbers. Clarendon Press, Oxford, 1962.Google Scholar
  13. E. Kaltofen, A Polynomial Time Reduction from Bivariate to Univariate Integral Polynomial Factorization. Proc. 23rd Symp. Foundations of Comp. Sci., IEEE, 1982, 57–64.Google Scholar
  14. E. Kaltofen, Polynomial-time Reduction from Multivariate to Bivariate and Univariate Integer Polynomial Factorization. Manuscript, 1983, submitted to SIAM J. Comput.Google Scholar
  15. D.E. Knuth, The Art of Computer Programming, Vol.2, 2nd Ed. Addison-Wesley, Reading MA, 1981.Google Scholar
  16. A. Lempel, G. Seroussi and S. Winograd, On the complexity of multiplication in finite fields. Theor. Comp. Science 22 (1983), 285–296.Google Scholar
  17. A.K. Lenstra, Factoring multivariate polynomials over finite fields. Proc. 15th ACM Symp. Theory of Computing, Boston, 1983.Google Scholar
  18. A.K. Lenstra, H.W. Lenstra, and L. Lovász, Factoring polynomials with rational coefficients. Math. Ann. 261 (1982), 515–534.Google Scholar
  19. D.R. Musser, Algorithms for Polynomial Factorization. Ph.D. thesis and TR 134, Univ. of Wisconsin, 1971.Google Scholar
  20. M.O. Rabin, Probabilistic algorithms in finite fields. SIAM J. Comp. 9 (1980), 273–280.Google Scholar
  21. T. Schönemann, Grundzüge einer allgemeinen Theorie der höheren Congruenzen, deren Modul eine reelle Primzahl ist. J. f. d. reine u. angew. Math. 31 (1846), 269–325.Google Scholar
  22. B.L. van der Waerden, Modern Algebra, vol. 1. Ungar, New York, 1953.Google Scholar
  23. H. Weyl, Algebraic theory of numbers. Princeton University Press, 1940.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1983

Authors and Affiliations

  • J. von zur Gathen
    • 1
  • E. Kaltofen
    • 1
  1. 1.Department of Computer ScienceUniversity of TorontoTorontoCanada

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