Elliptic Gauss sums and applications to point counting

Elliptic Gauss sums and applications to point counting

We shall first briefly review some general facts on elliptic curves over finite fields, and the algorithms of Schoof and Schoof Elkies Atkin (SEA) for counting points. We refer to text books as Cox, Silverman or Washington [Cox, Si, Wa] for the general topics and to the original papers of Schoof [Sch, Sch1] for a presentation of the algorithms. Let p be an odd prime. Consider the elliptic curve:

Point Counting on Elliptic and Hyperelliptic Curves

Rings of Integers, Gauss-jacobi Sums, and Their Applications

In this paper we shall explore the structure of the ring of algebraic integers in any quadratic extension of the field of rational numbers Q, develop the concepts of Gauss and Jacobi sums, and apply the theory of algebraic integers and that of Gauss-Jacobi sums to solving problems involving power congruences and power sums as well as to proving the quadratic and cubic reciprocity laws. In parti...

Exponential Sums, Gauss Sums and Cyclic Codes

The dissertation consists of three articles in which the evaluation of certain exponential sums and Gauss sums and bounds for the absolute values of exponential sums are considered. The summary part of the thesis provides interpretations in terms of coding theory for the results obtained in the articles.

A hybrid mean value involving a new Gauss sums and Dedekind sums

‎In this paper‎, ‎we introduce a new sum‎ ‎analogous to Gauss sum‎, ‎then we use the properties of the‎ ‎classical Gauss sums and analytic method to study the hybrid mean‎ ‎value problem involving this new sums and Dedekind sums‎, ‎and‎ ‎give an interesting identity for it.