Low complexity normal bases in

We investigate low complexity normal bases in finite fields of the form . First, we prove that if two normal elements have the same multiplication table, then they are conjugates. Then, we provide a partial converse to the known fact that if α generates a Type I optimal normal basis in , then its dual basis has complexity 3n−3. Finally, we determine the multiplication tables of low complexity normal elements that arise from products of normal elements in subfields of .