Gauss sums play an important role in number theory and arithmetic geometry. The main objects of study in this paper are Gauss sums over the finite field with q elements. Recently, the problem of explicit evaluation of Gauss sums in the small index case has been studied in several papers. In the process of the evaluation, it is realized that a sign （or a root of unity） ambiguity unavoidably occurs. These papers determined the ambiguities by the congruences modulo L, where L is certain divisor of the order of Gauss sum. However, such method is unavailable in some situations. This paper presents a new method to determine the sign （root of unity） ambiguities of Gauss sums in the index 2 case and index 4 case, which is not only suitable for all the situations with q being odd, but also comparatively more efficient and uniform than the previous method.
In this paper, explicit determination of the cyclotomic numbers of order l and 2l, for odd prime l ≡ 3 (mod 4), over finite field Fq in the index 2 case are obtained, utilizing the explicit formulas on the corresponding Gauss sums. The main results in this paper are related with the number of rational points of certain elliptic curve, called "Legendre curve", and the properties and value distribution of such number are also presented.
Let Fq be a finite field of odd characteristic, m, v the integers with 1 ≤ m ≤ v and K a 2v × 2v nonsingular alternate matrix over Fq. In this paper, the generalized symplectic graph GSp2v（q, m） relative to K over Fq is introduced. It is the graph with m-dimensional totally isotropic subspaces of the 2v-dimensional symplectic space Fq（2v） as its vertices and two vertices P and Q are adjacent if and only if the rank of PKQw is 1 and the dimension of P ∩ Q is m - 1. It is proved that the full automorphism group of the graph GSp2v（q, m） is the projective semilinear symplectic group P∑p（2v, q）.
In the present paper we propose a new hierarchical management scheme within suitable extensions of 2-adic numbers, in which the subfields are treated as classes, and the communications are realized by computation on elliptic curves. The security of our system is much stronger than that of the traditional public key cryptographic systems.
