Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-26T19:26:24.593Z Has data issue: false hasContentIssue false
  • English
  • Français

Dickson Polynomials Over Finite Fields and Complete Mappings

Published online by Cambridge University Press:  20 November 2018

Gary L. Mullen  and
Harald Niederreiter
Gary L. Mullen
Affiliation:
Department of Mathematics, The Pennsylvania State UniversityUniversity Park, PA 16802, U.S.A.
Harald Niederreiter
Affiliation:
Mathematical Institute Austrian Academy of SciencesD R. Ignaz-Seipel-Platz 2 A-1010ViennaAustria
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Dickson polynomials over finite fields are familiar examples of permutation polynomials, i.e. of polynomials for which the corresponding polynomial mapping is a permutation of the finite field. We prove that a Dickson polynomial can be a complete mapping polynomial only in some special cases. Complete mapping polynomials are of interest in combinatorics and are defined as polynomials f(x) over a finite field for which both f(x) and f(x) + x are permutation polynomials. Our result also verifies a special case of a conjecture of Chowla and Zassenhaus on permutation polynomials.

Keywords

12C05
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 30 , Issue 1 , 01 March 1987 , pp. 19 - 27
Copyright
Copyright © Canadian Mathematical Society 01

References

1. Atkin, A. O. L., Hay, L., and Larson, R.G., Enumeration and construction of pandiagonal latin squares of prime order, Computers and Math, with Appl., 9 (1983), 267–292.Google Scholar
2. Chowla, S. and Zassenhaus, H., Some conjectures concerning finite fields, Norske Vid. Selsk. Forh. (Trondheim), 41 (1968), 34– 35.Google Scholar
3. Dénes, J. and Keedwell, A.D., Latin Squares and Their Applications, Academic Press, New York, 1974.Google Scholar
4. Dickson, L.E., Linear Groups with an Exposition of the Galois Field Theory, Dover, New York, 1958.Google Scholar
5. Hsu, D. F. and Keedwell, A.D., Generalized complete mappings, neofields, sequenceable groups and block designs. I, Pacific J. Math., 111 (1984), 317–332.Google Scholar
6. Keedwell, A.D., Sequenceable groups, generalized complete mappings, neofields and block designs, Combinatorial Mathematics X (Adelaide, 1982), pp. 49– 71 , Lecture Notes in Math., vol. 1036, Springer-Verlag, Berlin-Heidelberg-New York, 1983.Google Scholar
7. Lang, S. and Weil, A., Number of points of varieties in finite fields, Amer. J. Math., 76 (1954), 819-827.Google Scholar
8. Lausch, H. and Nöbauer, W., Algebra of Polynomials, North-Holland, Amsterdam, 1973.Google Scholar
9. Lidl, R. and Niederreiter, H., Finite Fields, Encyclopedia of Math, and Its Appl., vol. 20, Addison-Wesley, Reading, Mass., 1983.Google Scholar
10. Mann, H.B., The construction of orthogonal latin squares, Ann. Math. Statist., 13 (1942), 418-423.Google Scholar
11. Niederreiter, H. and Robinson, K.H., Bol loops of orderpq, Math. Proc. Cambridge Philos. Soc, 89 (1981), 241-256.Google Scholar
12. Niederreiter, H. and Robinson, K.H., Complete mappings of finite fields, J. Austral. Math. Soc. Ser. A, 33 (1982), 197–212.Google Scholar
13. Schmidt, W.M., Equations over Finite Fields, Lecture Notes in Math., vol. 536, Springer-Verlag, Berlin-Heidelberg-New York, 1976.Google Scholar
14. Williams, K.S., Note on Dickson s permutation polynomials, Duke Math. J., 38 (1971), 659–665.Google Scholar