Skip to main content

Random polynomials and polynomial factorization

  • Session 5: Algebraic Complexity
  • Conference paper
  • First Online:
Automata, Languages and Programming (ICALP 1996)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1099))

Included in the following conference series:

Abstract

We give a precise average-case analysis of a complete polynomial factorization chain over finite fields by methods based on generating functions and singularity analysis.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Berlekamp, E. R. Algebraic Coding Theory. Mc Graw-Hill, 1968. Revised edition, 1984.

    Google Scholar 

  2. Buchmann, J. Complexity of algorithms in number theory. In Number Theory: Proceedings of the First Conference of the Canadian Number Theory Association (1990), Walter de Gruyter, pp. 37–53.

    Google Scholar 

  3. Cantor, D. G., and Zassenhauss, H. A new algorithm for factoring polynomials over finite fields. Mathematics of Computation 36 (1981), 587–592.

    Google Scholar 

  4. Car, M. Factorisation dans Fq[x]. Comptes-Rendus de l'Académie des Sciences 294 (Ser. I) (1982), 147–150.

    Google Scholar 

  5. Carlitz, L. The arithmetic of polynomials in a Galois field. American Journal of Mathematics 54 (1932), 39–50.

    Google Scholar 

  6. Chor, B., and Rivest, R. A knapsack type public key cryptosystem based on on arithmetics over finite fields. IEEE Transactions on Information Theory 34 (1988), 901–909.

    Google Scholar 

  7. Comtet, L.Advanced Combinatorics. Reidel, Dordrecht, 1974.

    Google Scholar 

  8. Dedekind, R. Abriss einer Theorie der höhern Congruenzen in Bezug auf einen reellen Primzahlmodulus. Journal für die reine und angewandte Mathematik 54 (1857), 1–26.

    Google Scholar 

  9. Flajolet, P., and Odlyzko, A. M. Singularity analysis of generating functions. SIAM Journal on Discrete Mathematics 3, 2 (1990), 216–240.

    Article  Google Scholar 

  10. Flajolet, P., and Soria, M. Gaussian limiting distributions for the number of components in combinatorial structures. Journal of Combinatorial Theory, Series A 53 (1990), 165–182.

    Google Scholar 

  11. Gauss, C. F.Untersuchungen úber höhere Mathematik. Chelsea, New York, 1889.

    Google Scholar 

  12. Geddes, K. O., Czapor, S. R., and Labahn, G.Algorithms for Computer Algebra. Kluwer Academic Publishers, Boston, 1992.

    Google Scholar 

  13. Greene, D. H., and Knuth, D. E.Mathematics for the analysis of algorithms, second ed. Birkhauser, Boston, 1982.

    Google Scholar 

  14. Knopfmacher, J., and Knopfmacher, A. Counting irreducible factors of polynomials over a finite field. Discrete Mathematics 112 (1993), 103–118.

    Google Scholar 

  15. Knuth, D. E. The Art of Computer Programming, vol. 3: Sorting and Searching. Addison-Wesley, 1973.

    Google Scholar 

  16. Knuth, D. E. The Art of Computer Programming, 2nd ed., vol. 2: Seminumerical Algorithms. Addison-Wesley, 1981.

    Google Scholar 

  17. Knuth, D. E., and Pardo, L. T. Analysis of a simple factorization algorithm. Theoretical Computer Science 3 (1976), 321–348.

    Google Scholar 

  18. Lenstra, H. W. On the Chor Rivest cryptosystem. Journal of Cryptology 3 (1991), 149–155.

    Google Scholar 

  19. Lidl, R., and Niederreiter, H. Finite Fields, vol. 20 of Encyclopedia of Mathematics and its Applications. Addison-Wesley, 1983.

    Google Scholar 

  20. Odlyzko, A. M. Discrete logarithms and their cryptographic significance. In Advances in Cryptology (1985), Lecture Notes in Computer Science, Springer Verlag, pp. 224–314.

    Google Scholar 

  21. Odlyzko, A. M. Asymptotic enumeration methods. In Handbook of Combinatorics, M. G. R. Graham and L. Lovász, Eds., vol. II. Elsevier, Amsterdam, 1995, pp. 1063–1229.

    Google Scholar 

  22. Postnikov, A. G. Tauberian theory and its applications, vol. 144 of Proceedings of the Steklov Institute of Mathematics. American Mathematical Society, 1980.

    Google Scholar 

  23. Shepp, L. A., and Lloyd, S. P. Ordered cycle lengths in a random permutation. Transactions of the American Mathematical Society 121 (1966), 340–357.

    Google Scholar 

  24. Shoup, V. A new polynomial factorization algorithm and its implementation. Preprint, 1994.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Friedhelm Meyer Burkhard Monien

Rights and permissions

Reprints and permissions

Copyright information

© 1996 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Flajolet, P., Gourdon, X., Panario, D. (1996). Random polynomials and polynomial factorization. In: Meyer, F., Monien, B. (eds) Automata, Languages and Programming. ICALP 1996. Lecture Notes in Computer Science, vol 1099. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61440-0_131

Download citation

  • DOI: https://doi.org/10.1007/3-540-61440-0_131

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-61440-1

  • Online ISBN: 978-3-540-68580-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics