Normal bases and self-dual normal bases over finite fields have been found to be very useful in many fast arithmetic computations. It is well-known that there exists a self-dual normal basis of F2n over F2 if and only if 4 ∤ n. In this paper, we prove there exists a normal element α of F2n over F2 corresponding to a prescribed vector a = (a0, a1, · · · , an−1) ∈ F n 2 such that ai = Tr 2 n |2 (α 1+2 i ) for 0 ≤ i ≤ n − 1, where n is a 2-power or odd, if and only if the given vector a is symmetric (ai = an−i for all i, 1 ≤ i ≤ n − 1), and one of the following is true. 1) n = 2 s ≥ 4, a0 = 1, a n/2 = 0,Furthermore we give an algorithm to obtain normal elements corresponding to prescribed vectors in the above two cases. For a general positive integer n with 4|n, some necessary conditions for a vector to be the corresponding vector of a normal element of F2n over F2 are given. And for all n with 4|n, we prove that there exists a normal element of F2n over F2 such that the Hamming weight of its corresponding vector is 3, which is the lowest possible Hamming weight.
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