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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
##### Keywords:
Kloosterman sum; finite fields; trace; Kloosterman polynomials
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##### References:
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