Abstract
A Kloosterman zero is a non-zero element of \({{\mathbb F}_q}\) for which the Kloosterman sum on \({{\mathbb F}_q}\) attains the value 0. Kloosterman zeros can be used to construct monomial hyperbent (bent) functions in even (odd) characteristic, respectively. We give an elementary proof of the fact that for characteristic 2 and 3, no Kloosterman zero in \({{\mathbb F}_q}\) belongs to a proper subfield of \({{\mathbb F}_q}\) with one exception that occurs at q = 16. It was recently proved that no Kloosterman zero exists in a field of characteristic greater than 3. We also characterize those binary Kloosterman sums that are divisible by 16 as well as those ternary Kloosterman sums that are divisible by 9. Hence we provide necessary conditions that Kloosterman zeros must satisfy.
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Bosma W., Cannon J., Playoust C.: The Magma algebra system. I. The user language. J. Symbolic Comput. 24, 235–265 (1997)
Charpin P., Gong G.: Hyperbent functions, Kloosterman sums, and Dickson polynomials. IEEE Trans. Inform. Theory 54, 4230–4238 (2008)
Charpin P., Helleseth T., Zinoviev V.: Propagation characteristics of \({x\mapsto x\sp {-1}}\) and Kloosterman sums. Finite Fields Appl. 13, 366–381 (2007)
Dillon J.F.: Elementary Hadamard difference sets. Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic University, Boca Raton, FL, 1975), pp. 237–249. Congressus Numerantium, No. XIV, Utilitas Math., Winnipeg, MB (1975).
Enge A.: Elliptic Curves and Their Applications to Cryptography: An Introduction. Kluwer, Boston (1999)
Helleseth T., Kholosha A.: Monomial and quadratic bent functions over the finite fields of odd characteristic. IEEE Trans. Inform. Theory 52, 2018–2032 (2006)
Helleseth T., Zinoviev V.: On Z 4-linear Goethals codes and Kloosterman sums. Des. Codes Cryptogr. 17, 269–288 (1999)
Hirschfeld J.W.P.: Projective Geometries over Finite Fields, 2nd edn. The Clarendon Press, Oxford University Press, New York (1998)
Kononen K., Rinta-aho M., Väänänen K.: On integer values of Kloosterman sums. IEEE Trans. Inform. Theory 56(8), 4011–4013 (2010)
Lachaud G., Wolfmann J.: The weights of the orthogonals of the extended quadratic binary Goppa codes. IEEE Trans. Inform. Theory 36, 686–692 (1990)
Leonard P.A., Williams K.S.: Quartics over GF(2n). Proc. Amer. Math. Soc. 36, 347–350 (1972)
Lidl R., Niederreiter H.: Finite Fields. Cambridge University Press, Cambridge (1997)
Lisoněk P.: On the connection between Kloosterman sums and elliptic curves. In: Golomb S. et al. (Eds) Proceedings of the 5th International Conference on Sequences and Their Applications (SETA 2008) Lecture Notes in Computer Science, vol. 5203, pp. 182–187, Springer (2008).
Moisio M.: Kloosterman sums, elliptic curves, and irreducible polynomials with prescribed trace and norm. Acta Arith. 132, 329–350 (2008)
Moisio M.: On certain values of Kloosterman sums. IEEE Trans. Inform. Theory 55, 3563–3564 (2009)
Silverman J.H.: The Arithmetic of Elliptic Curves, 2nd edn. Springer, Dordrecht (2009)
Stichtenoth H.: Algebraic Function Fields and Codes. Springer, Berlin (1993)
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Research partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).
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Lisoněk, P., Moisio, M. On zeros of Kloosterman sums. Des. Codes Cryptogr. 59, 223–230 (2011). https://doi.org/10.1007/s10623-010-9457-x
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DOI: https://doi.org/10.1007/s10623-010-9457-x