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In this paper, we address the problem for enumerating the number of finitefield elements with prescribed trace and co-trace in case of arbitrarycharacteristic $p$. We show that this problem can be reduced to solving asystem of $p-1$ linear equations with matrix of coefficients a slightmodification of circulant matrix formed by the Kloosterman sums over the field$\MF_{p}$. The presented approach is illustrated in the cases of characteristic$p = 2,3$ and $5$.

Computing the Number of Finite Field Elements with Prescribed Trace and Co-trace