Uma introdução à corpos finitos e suas aplicações

Abstract

This monograph presents an introductory study on finite fields, connecting their algebraic foundations to essential applications in information security and data transmission. The first part of the work establishes the theoretical basis, reviewing the structures of groups and rings to construct and characterize finite fields as extensions of prime fields. The existence and uniqueness of finite fields, the structure of the multiplicative group, and the theory of irreducible and cyclotomic polynomials are detailed. The second part is dedicated to the practical applications of these structures. In the realm of Cryptography, public-key systems are analyzed, with an emphasis on the RSA algorithm and the complexity of the Discrete Logarithm problem. Finally, a brief introduction to Coding Theory for error correction is presented, addressing the concepts of linear codes and the algebraic characterization of cyclic codes, culminating in the study of BCH codes.