Research On Some Properties Of Generators In Finite Fields

Finite fields are among the most important tools in computer cryptography and figure communication. With the swift and violent development of the technology of the computer, Operation in finite fields is getting easier and easier. Research on properties and construction of finite fields is promoted enormously. Finite fields can be regard as linear space in its base fields. So researching on properties of finite fields can be conversed to study all kinds of bases in finite fields. The normal bases with the rapidly calculating character arouse many people's interests. Research on constructing the finite fields is of necessity to look for the multiple generators, i.e., primitive elements of finite fields. Therefore, it is natural of theoretical and application important to study the primitive element and its associated primitive polynomial, the normal base and its associated normal elements. In this paper, we settle the properties of some generators of finite fields with character sums. The main results are as follows:We shall first give a brief introduction about the research history and primary application of some generators over finite fields and present some main results which have been obtained by former scholars.Next, this paper estimates the number of certain generators in the following three cases by making use of Gauss sums and Kloosterman sums: (i)ξis a power residual normal element in GF ( q~n); (ii) bothξand 1ξ?are power residual normal elements; (iii) bothξand 1ξ?are power residual normal elements which satisfies the trace ofξand 1ξ? is given. We provide some sufficient conditions for the existence of the elements in the above three cases.Finally, we discuss the properties of primitive elements with the formα+ x. Weil sum is extended in this paper. We resolve the question of whether there exists a primitive elementαof GF ( q~n) such thatα+ x is also a primitive element for a certain nonzero element x in GF ( q~n). We estimate the number of such elements and give a positive answer to this question under some conditions. These results not only enrich the known research efforts on normal elements and primitive elements, but also make a big progress in the study of the properties and the constructions of finite fields.