Abstract
We explore the known connection of Kloosterman sums on fields of characteristic 2 and 3 with the number of points on certain elliptic curves over these fields. We use this connection to prove results on the divisibility of Kloosterman sums, and to compute numerical examples of zeros of Kloosterman sums on binary and ternary fields of large orders. We also show that this connection easily yields some formulas due to Carlitz that were recently used to prove certain non-existence results on Kloosterman zeros in subfields.
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References
Bosma, W., Cannon, J., Playoust, C.: The Magma Algebra System. I. The User Language. J. Symbolic Comput. 24, 235–265 (1997)
Carlitz, C.: Kloosterman Sums and Finite Field Extensions. Acta Arithmetica XVI, 179–193 (1969)
Charpin, P., Gong, G.: Hyperbent Functions, Kloosterman Sums and Dickson Polynomials. IEEE Trans. Inform. Theory (to appear)
Charpin, P., Helleseth, T., Zinoviev, V.: Propagation Characteristics of \(x\mapsto x\sp {-1}\) and Kloosterman Sums. Finite Fields Appl. 13, 366–381 (2007)
Enge, A.: Elliptic Curves and Their Applications to Cryptography: An Introduction. Kluwer Academic Publishers, Boston (1999)
Garaschuk, K., Lisoněk, P.: On Ternary Kloosterman Sums modulo 12. Finite Fields Appl. (to appear)
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)
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)
Lercier, R., Lubicz, D., Vercauteren, F.: Point Counting on Elliptic and Hyperelliptic Curves. In: Cohen, H., Frey, G. (eds.) Handbook of Elliptic and Hyperelliptic Curve Cryptography. Chapman & Hall/CRC, Boca Raton (2006)
Menezes, A.: Elliptic Curve Public Key Cryptosystems. Kluwer Academic Publishers, Boston (1993)
Moisio, M.: Kloosterman Sums, Elliptic Curves, and Irreducible Polynomials with Prescribed Trace and Norm. Acta Arithmetica (to appear)
Moisio, M., Ranto, K.: Kloosterman Sum Identities and Low-weight Codewords in a Cyclic Code with Two Zeros. Finite Fields Appl. 13, 922–935 (2007)
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Lisoněk, P. (2008). On the Connection between Kloosterman Sums and Elliptic Curves. In: Golomb, S.W., Parker, M.G., Pott, A., Winterhof, A. (eds) Sequences and Their Applications - SETA 2008. SETA 2008. Lecture Notes in Computer Science, vol 5203. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85912-3_17
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DOI: https://doi.org/10.1007/978-3-540-85912-3_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-85911-6
Online ISBN: 978-3-540-85912-3
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