A new characterization of dual bases in finite fields and its applications

Abstract: We propose a new characterization of dual bases in finite fields. Let be a basis of over and its dual basis with the transition matrix such that . We show that holds for all , where satisfies . Conversely, suppose and for some and , then B is equivalent to . As applications, we can construct the dual basis of a given basis or determine whether the dual basis of satisfies the desired conditions from . This generalizes the results obtained by Liao and Sun for normal bases. Furthermore, we give a simple proof of the theorem of Gollmann, Wang and Blake for polynomial bases. [Copyright &y& Elsevier]