Curvas elípticas sobre corpos finitos

Abstract

Much has been said about elliptic curves and their applications in cryptography. This text regards their algebraic aspects. We shall approach elliptic curves as plane projective curves possessing an operation that turns the set of its points into an abelian group. The first chapter deals with properly defining elliptic curves, their operations and concrete formulas for calculating in them. It is shown how to determine the Weierstrass form and results about the structure, like Nagell-Lutz’s and Mordell’s theorems are presented. The second chapter begins the work on elliptic curves over finite fields, with the purpose of briefly exposing how they are used in cryptography. The final chapter uses more sophisticated algebraic methods to display results about the existence and structure of the groups in elliptic curves over finite fields.