Character sums of quadratic forms over finite fields and the number of rational points for some classes of artin-schreier type curves

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2017
Coşgun, Ayhan
Exponential sums of quadratic forms over finite fields have many applications to various areas such as coding theory and cryptography. As an example to these applications, there is an organic connection between exponential sums of quadratic forms and the number of rational points of algebraic curves defined over finite fields. This connection is central in the application of algebraic geometry to coding theory and cryptography. In this thesis, different facts and techniques of theory of finite fields are combined properly in order to improve and generalize some of the results in the existing literature on evaluation of exponential sums of certain quadratic forms. These evaluations also correspond to the Walsh-Hadamard transforms of Boolean functions in characteristic two. As a result of these evaluations, the number of rational points are computed for some classes of Artin-Schreier type curves over finite fields.