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On divisibility of exponential sums of polynomials of special type over fields of characteristic 2

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Abstract

We study the divisibility by eight of exponential sums of several classes of functions over finite fields of characteristic two. For the binary classical Kloosterman sums K(a) over such fields we give a simple recurrent algorithm for finding the largest k, such that 2k divides the Kloosterman sum K(a). This gives a simple description of zeros of such Kloosterman sums.

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References

  1. Bassalygo L.A., Zinoviev V.A.: Polynomials of special form over finite fields with given values of trigonometric sum. Math. Notes. 82(1–2), 3–9 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Carlitz L.: Explicit evaluation of certain exponential sums. Math. Scand. 44, 5–16 (1979)

    MathSciNet  MATH  Google Scholar 

  3. Carlitz L.: Kloosterman sums and finite field extensions. Acta Arith. XVI: 179(−193), 179–193 (1969)

    Google Scholar 

  4. Charpin P., Helleseth T., Zinoviev V.A.: Propagation characteristics of x −1. Finite Fields Appl. 13, 366–381 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Charpin P., Helleseth T., Zinoviev V.A.: The divisibility modulo 24 of Kloosterman sums on GF(2m), m odd. J. Combinatorial Theory Ser. A. 114(2), 322–338 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Charpin P., Helleseth T., Zinoviev V.A.: On divisibility properties of classical binary Kloosterman sums. Discret. Math. 309(12), 3975–3984 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Charpin P., Gong G.: Hyperbent functions, Kloosterman sums, and Dickson polynomials. IEEE Transac. Inf. Theory 54(9), 4230–4238 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dillon J.F.: Elementary Hadamard difference sets. In: Proceedings of Sixth Southeastern Conference on Combinatorics, Graph theory, and Computing. Congressus Numerantium, No. XIV, Utilitas Math., Winnipeg, Man. pp. 237–249 (1975).

  9. Enge A.: Elliptic curves and their applications to cryptography: an introduction. Klumer Academic Publishers, Boston (1999)

    Book  Google Scholar 

  10. Göloğlu F., Lisonek P., McGuire G., Moloney R.: Binary Kloosterman sums modulo 128 and coefficients of the characteristic polynomial. IEEE Transac. Inf. Theory (submitted) (2010).

  11. Göloğlu F., McGuire G., Moloney R.: Binary Kloosterman sums using Stickelberger’s theorem and the Gross-Koblitz formula. Acta Arith. (to appear).

  12. Göloğlu F., McGuire G., Moloney R.: Some results on Kloosterman sums and their minimal polynomials. In: Augot D., Canteaut A. (eds.) Proceedings of Seventh International Workshop on Coding and Cryptography, WCC 2011, April 11–15, 2011, pp. 403–412 Paris, France (2011).

  13. van der Geer G., van der Vlugt M.: Kloosterman sums and the p-torsion of certain Jacobians. Math. Ann. 290(3), 549–563 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  14. Helleseth T., Zinoviev V.A.: On Z 4-linear Goethals codes and Kloosterman sums. Des. Codes Cryptogr. 17(1–3), 246–262 (1999)

    MathSciNet  Google Scholar 

  15. Lachaud G., Wolfmann J.: The weights of the orthogonals of the extended quadratic binary Goppa codes. IEEE Trans. Inf. Theory 36(3), 686–692 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  16. Leander N.G.: Monomial bent functions. IEEE Trans. Inf. Theory. 52(2), 738–743 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Leonard P.A., Williams K.S.: Quartics over GF(2m). Proc. Am. Math. Soc. 36, 347–350 (1972)

    MathSciNet  Google Scholar 

  18. Lidl R., Niederreiter H.: Finite Fields, Encyclopedia of Mathematics and Its Applications vol. 20. Addison Wesley, Reading (1983)

    Google Scholar 

  19. Lisonek P.: On the connection between Kloosterman sums and elliptic curves. In: Golomb et al. (eds.) Proceedings of the 5th International Conference on Sequences and Their Applications (SETA 2008), Lecture Notes in Computer Science, Springer, vol. 5203, pp. 182–187 (2008).

  20. Lisonek P., Moisio M.: On zeros of Kloosterman sums. Des. Codes Cryptogr. 59(1–3), 223–230 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Menezes A.: Elliptic curve public key cryptosystems. Kluwer Academic Publishers, Boston (1993)

    Book  MATH  Google Scholar 

  22. Moisio A.: The divisibility modulo 24 of Kloosterman sums on GF(2m), m even. Finite Fields Appl. 15, 174–184 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ahmadi O., Granger R.: An efficient deterministic test for Kloosterman sum zeros. Math(NT). arXiv:1104.3882v1 (2011).

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Authors and Affiliations

  1. A. A. Kharkevich Institute for Problems of Information Transmission, Russian Academy of Sciences, Bol’shoi Karetnyi per.19, GSP-4, Moscow, 127994, Russia

    L. A. Bassalygo & V. A. Zinoviev

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  1. L. A. Bassalygo
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  2. V. A. Zinoviev
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Correspondence to V. A. Zinoviev.

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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”.

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Bassalygo, L.A., Zinoviev, V.A. On divisibility of exponential sums of polynomials of special type over fields of characteristic 2. Des. Codes Cryptogr. 66, 129–143 (2013). https://doi.org/10.1007/s10623-012-9669-3

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  • DOI: https://doi.org/10.1007/s10623-012-9669-3

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