zbMATH — the first resource for mathematics

Kloosterman sum identities over \(\mathbb F_{2^{m}}\). (English) Zbl 1099.11040
Let \(E\) denote the finite field of \(2^m\) elements and let \(\text{Tr}\) denote the trace from \(E\) to \(\mathbb F_2\). A Kloosterman sum over \(E\) is \(K(a,b)=\sum_{x\in E^*} (-1)^{\text{Tr}(ax+b/x)}\), where \(a,b\in E\). The sum \(K(1,b)\) is denoted by \(K(b)\). The authors prove the identities, for all \(b\in E\), \[ \begin{aligned} K(b^n(b+1)) &=K(b(b+1)^n),\quad n=3,5\\ K(b^8(b^4+b)) &=K((b+1)^8(b^4+b)). \end{aligned} \] The first identity (for both \(n=3\) and \(n=5\)) was previously proven by T. Helleseth and V. Zinoviev [Finite Fields Appl. 9, No. 2, 187–193 (2003; Zbl 1081.11077)]. The proof here depends on what the authors call Kloosterman polynomials. These are \(L(x)=f(x)+g(1/x)\), where \(f,g\) are linearized polynomials, \(g\) has even weight, and \(L\) is injective on the set of elements of \(E\) of trace 1. The paper ends with two conjectures which, if true, essentially classify all Kloosterman polynomials.

11L05 Gauss and Kloosterman sums; generalizations
11T24 Other character sums and Gauss sums
11T06 Polynomials over finite fields
Full Text: DOI
[1] Fisher, B., Distinctness of Kloosterman sums, (), 81-102 · Zbl 0797.11095
[2] Fisher, B., Kloosterman sums as algebraic integers, Math. ann., 301, 485-505, (1995) · Zbl 0816.11065
[3] Helleseth, T.; Zinoviev, V., New Kloosterman sums identities over \( F2\^{}\{m\}\) for all m, Finite fields their appl., 9, 187-193, (2003) · Zbl 1081.11077
[4] M.R.S. Kojo, Modular curves and identities of classical Kloosterman sums, preprint.
[5] Lidl, R.; Niederreiter, H., Finite fields, (1997), Cambridge University Press Cambridge
[6] L. Ojala, Graduate Thesis, University of Turku, 2001.
[7] Shin, Dong-Joon; Kumar, P.V.; Helleseth, T., 3-designs from the \( Z4\)-goethals codes via a new Kloosterman sum identity, Des. codes cryptography, 28, 247-263, (2003) · Zbl 1028.94032
[8] Shin, Dong-Joon; Sung, Wonjin, A new Kloosterman sum identity over \( F2\^{}\{m\}\) for odd m, Discrete math., 268, 337-341, (2003) · Zbl 1049.11134
[9] Wan, Daqing, Minimal polynomials and distinctness of Kloosterman sums, Finite fields their appl., 1, 189-203, (1995) · Zbl 0841.11062
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.