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)


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.


Kloosterman sums Stickelberger’s theorem 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Garaschuk, K., Lisoněk, P.: On binary Kloosterman sums divisible by 3. Designs, Codes and Cryptography 49, 347–357 (2008)zbMATHCrossRefGoogle 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)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Helleseth, T., Zinoviev, V.: On ℤ4-linear Goethals codes and Kloosterman sums. Designs, Codes and Cryptography 17, 269–288 (1999)zbMATHCrossRefMathSciNetGoogle 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)zbMATHGoogle Scholar
  9. 9.
    Katz, N.M.: Gauss sums, Kloosterman sums, and monodromy groups. Annals of Mathematics Studies, vol. 116. Princeton University Press, Princeton (1988)zbMATHGoogle 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)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Langevin, P., Leander, G.: Monomial bent functions and Stickelberger’s theorem. Finite Fields and Their Applications 14, 727–742 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Lidl, R., Niederreiter, H.: Introduction to Finite Fields and Their Applications. Cambridge University Press, Cambridge (1986)zbMATHGoogle 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)zbMATHCrossRefMathSciNetGoogle 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)zbMATHCrossRefMathSciNetGoogle 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 zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Washington, L.C.: Introduction to Cyclotomic Fields. Springer, Heidelberg (1982)zbMATHGoogle 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

Personalised recommendations