On a new identity for Kloosterman sums and nonlinear system of equations over finite fields of characteristic 2. (Q1421515)

Let \(F\) be a finite field of even order \(2^m\). For \(a\in F^*\) the Kloosterman sum \(K(a)\) is defined by \[ K(a)=\sum_{x\in F^*} (-1)^{\text{ Tr}(ax+1/x)}, \] where \(\text{ Tr}\) is the absolute trace function of \(F\), \[ \text{ Tr}(x)=x+x^2+\ldots+x^{2^{m-1}}. \] The main result of the paper is the equality \[ K(a^3(a+1))=K(a(a+1)^3),\quad a\in F^*. \] For odd \(m\) this result follows from \textit{D. J. Shin} and \textit{W. Sung} [Discrete Math. 268, 337--341 (2003; Zbl 1049.11134); see also \textit{D. J. Shin}, \textit{P. V. Kumar} and \textit{T. Helleseth}, Des. Codes Cryptography 28, 247--263 (2003; Zbl 1028.94032) and \textit{K. Ranto}, SIAM J. Discrete Math. 15, 289--304 (2002; Zbl 1007.05020)], but for even \(m\) a proof has not been known yet. The proof is based on the expression of the number of solutions of a nonlinear system of equations over \(F\) in terms of Kloosterman sums.