Normal Bases And Their Dual-bases Over Galois Extension Fields.

With the development of Computer Science and Cryptology, the studyof finite fields, particularly the normal bases over finite fields, become more and moreimportant. Because the hardware and software may be implemented quickly if wechoose proper normal bases to represent finite fields. In this paper, we first introducesome basic definitions and facts on normal bases over finite fields, then extend theseresults to any Galois extension fields. We obtain a necessary and su?cient conditionon equivalent normal bases over Galois extension fields. Consequently we generalizethe equivalent condition on normal bases over finite fields to Galois extensions of anyfield. Comparatively our approach is more elementary and straightforward.