Normal Bases, Optimal Normal Bases And Their Dual-Bases Over Finite Fields

This paper consists of three chapters. In chapter 1, we have given two very usefulalgorithms to compute multiplication tables of optimal normal bases over finite fields.Through comparing these algorithms with other well–known two algorithms, we foundthat these agorithms are very beautiful.In chapter 2, we have discussed the relationship of normal bases and their dualnormal bases over finite fields. We not only got some sufficient and necessary con-ditions for which a normal basis or an optimal normal basis is equivalent to its dualnormal basis over finite fields, and obtained all self-dual optimal normal bases over fi-nite fields, but also discussed other two extensions of self-dual normal bases which areweakly normal bases and the normal basis whose dual basis is generated by its gen-erator's linearly combinations, and obtained some properties for their multiplicationtables and complexities.In chapter 3, we proved that the distribution of primitive elements of F = Fqnin NF-/1Fq(β) is uniformly , where NF?/1Fq(β) = {α∈F : NF/Fq(α) = β} is theinverse function of the norm function NF/Fq(α) of αover Fq. We also obtained severalsufficient and necessary conditions for primitive elements and studied some particularcases for optimal normal bases whose generator is a primitive element of F.