Trace self-orthogonal relations of normal bases

Normal bases with specific trace self-orthogonal relations over finite fields have been found to be very useful for many fast arithmetic computations, especially when the extensions of finite fields have no self-dual normal basis. Recent work in [1] has given the necessary and sufficient conditions for the existence of normal bases of GF(2n)GF(2n) with a prescribed trace vector when n is odd or n is a power of two. However, the methods in [1] cannot work in general cases. In this paper, using methods different from [1], we give a complete characterization of the trace self-orthogonal relations of arbitrary normal bases. Furthermore, we provide a combination method to construct normal elements with the prescribed trace vectors. These generalize the results in [1] to general cases. The main result of this paper is shown as follows.Let a̲=(a0,a1,⋯,an−1) be a prescribed n  -vector over GF(q)GF(q), with corresponding polynomial fa(x)=∑i=0n−1aixi. We present that there exists a normal element α   of GF(qn)GF(qn) over GF(q)GF(q), with trace vector a̲, such that ai=Trqn|q(α1+qi)ai=Trqn|q(α1+qi) for all 0≤i≤n−10≤i≤n−1, if and only if ai=an−iai=an−i for all 1≤i≤n−11≤i≤n−1 and1)when q   is odd, fa(x)fa(x) is prime to xn−1xn−1; fa(1)fa(1) is a quadratic residue in GF(q)GF(q); for even n  , if fa(−1)≠0fa(−1)≠0, then fa(−1)fa(−1) is not a quadratic residue in GF(q)GF(q);2)when q   is even, fa(x)fa(x) is prime to xn−1xn−1; for even n  , an/2=0an/2=0 and if 4|n4|n, then Trq|2(∑i=0n/4−1a0−1a2i+1)=1.