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

Abstract. In this paper, we address the problem for enumerating the number of finite field elements with prescribed trace and co-trace in case of arbitrary characteristic p. We show that this problem can be reduced to solving a system of p − 1 linear equations with matrix of coefficients a slight modification of circulant matrix formed by the Kloosterman sums over the field Fp. The presented approach is illustrated in the cases of characteristic p = 2, 3 and 5.