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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.

MSC:
11L05 Gauss and Kloosterman sums; generalizations
11T24 Other character sums and Gauss sums
11T06 Polynomials over finite fields
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