Ternary Kloosterman Sums Modulo 18 Using Stickelberger’s Theorem

  • Faruk Göloğlu
  • Gary McGuire
  • Richard Moloney
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6338)

Abstract

A result due to Helleseth and Zinoviev characterises binary Kloosterman sums modulo 8. We give a similar result for ternary Kloosterman sums modulo 9. This leads to a complete characterisation of values that ternary Kloosterman sums assume modulo 18. The proof uses Stickelberger’s theorem and Fourier analysis.

Keywords

Kloosterman sums Stickelberger’s theorem 

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References

  1. 1.
    Charpin, P., Helleseth, T., Zinoviev, V.: The divisibility modulo 24 of Kloosterman sums on GF(2m), m odd. Journal of Combinatorial Theory 114, 332–338 (2007)MathSciNetGoogle Scholar
  2. 2.
    Dillon, J.F.: Elementary Hadamard Difference Sets. PhD thesis, University of Maryland (1974)Google Scholar
  3. 3.
    Garaschuk, K., Lisoněk, P.: On ternary Kloosterman sums modulo 12. Finite Fields Appl. 14(4), 1083–1090 (2008)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Garaschuk, K., Lisoněk, P.: On binary Kloosterman sums divisible by 3. Designs, Codes and Cryptography 49, 347–357 (2008)MATHCrossRefGoogle Scholar
  5. 5.
    Gross, B.H., Koblitz, N.: Gauss sums and the p-adic Γ-function. Ann. of Math. (2) 109(3), 569–581 (1979)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Helleseth, T., Kholosha, A.: Monomial and quadratic bent functions over the finite fields of odd characteristic. IEEE Trans. Inform. Theory 52(5), 2018–2032 (2006)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Helleseth, T., Zinoviev, V.: On ℤ4-linear Goethals codes and Kloosterman sums. Designs, Codes and Cryptography 17, 269–288 (1999)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Katz, N., Livné, R.: Sommes de Kloosterman et courbes elliptiques universelles caractéristiques 2 et 3. C. R. Acad. Sci. Paris Sér. I. Math. 309(11), 723–726 (1989)MATHGoogle Scholar
  9. 9.
    Katz, N.M.: Gauss sums, Kloosterman sums, and monodromy groups. Annals of Mathematics Studies, vol. 116. Princeton University Press, Princeton (1988)MATHGoogle Scholar
  10. 10.
    Lachaud, G., Wolfmann, J.: The weights of the orthogonals of the extended quadratic binary Goppa codes. IEEE Trans. Inform. Theory 36(3), 686–692 (1990)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Langevin, P., Leander, G.: Monomial bent functions and Stickelberger’s theorem. Finite Fields and Their Applications 14, 727–742 (2008)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Lidl, R., Niederreiter, H.: Introduction to Finite Fields and Their Applications. Cambridge University Press, Cambridge (1986)MATHGoogle Scholar
  13. 13.
    Lisoněk, P.: On the connection between Kloosterman sums and elliptic curves. In: Golomb, S.W., Parker, M.G., Pott, A., Winterhof, A. (eds.) SETA 2008. LNCS, vol. 5203, pp. 182–187. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  14. 14.
    Lisoněk, P., Moisio, M.: On zeros of Kloosterman sums (to appear 2009)Google Scholar
  15. 15.
    Moisio, M.: The divisibility modulo 24 of Kloosterman sums on GF(2m), m even. Finite Fields and Their Applications 15, 174–184 (2009)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Robert, A.: The Gross-Koblitz formula revisited. Rendiconti del Seminario Matematico della Università di Padova 105, 157–170 (2001)Google Scholar
  17. 17.
    van der Geer, G., van der Vlugt, M.: Kloosterman sums and the p-torsion of certain Jacobians. Math. Ann. 290(3), 549–563 (1991)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Wan, D.Q.: Minimal polynomials and distinctness of Kloosterman sums. Finite Fields Appl. 1(2), 189–203 (1995); Special issue dedicated to Leonard Carlitz MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Washington, L.C.: Introduction to Cyclotomic Fields. Springer, Heidelberg (1982)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Faruk Göloğlu
    • 1
  • Gary McGuire
    • 1
  • Richard Moloney
    • 1
  1. 1.School of Mathematical SciencesUniversity College DublinIreland

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