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New Kloosterman sums identities over \(\mathbb F_{2^m}\) for all \(m\). (English) Zbl 1081.11077
Summary: Recently, [Discrete Math. 268, No. 1-3, 337–341 (2003; Zbl 1049.11134)] D. J. Shin and W. Sung found new identities for Kloosterman sums over \(\mathbb F_{2^m}\) with odd \(m\). They posed the question whether similar results could be obtained for even \(m\). In this paper, we give a positive answer to this question. We present new results that hold for any \(m\) and include as special cases the results of Shin and Sung in the case where \(m\) is odd.

11T24 Other character sums and Gauss sums
11L05 Gauss and Kloosterman sums; generalizations
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[1] Berlekamp, E.R.; Rumsey, H.; Solomon, G., On the solution of algebraic equations over finite fields, Inform. and control, 10, 553-564, (1967) · Zbl 0166.04803
[2] Carlitz, L., Kloosterman sums and finite field extensions, Acta arith., XVI, 179-193, (1969) · Zbl 0194.07902
[3] Fisher, B., Distinctness of Kloosterman sums, (), 81-102 · Zbl 0797.11095
[4] Helleseth, T.; Zinoviev, V., On Z4-linear goethals codes and Kloosterman sums, Des. codes cryptogr., 17, 269-288, (1999) · Zbl 0951.11026
[5] T. Helleseth, V. Zinoviev, On a new identity for Kloosterman sums over finite field of characteristic 2, Discrete Math. (2000), submitted for publication. · Zbl 1048.11096
[6] Helleseth, T.; Zinoviev, V., On coset weight distributions of the Z4-linear goethals codes, IEEE trans. inform. theory, 47, 1758-1772, (2001) · Zbl 0998.94023
[7] D. Shin, P.V. Kumar, T. Helleseth, 3-Designs from the Z4 Goethals code via a new Kloosterman identity, Des. Codes Cryptogr., to appear.
[8] D. Shin, W. Sung, A new Kloosterman identity over F2m with odd m, Discrete Math. (1999), submitted for publication.
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