Finite Fields and their Applications
Nadiya M. Gubareni · 2021
Finite fields, which are also called Galois fields, are very important algebraic structures. First, they were considered by Evariste Galois, who was one of the most outstanding mathematicians in history. Galois fields play a significant role in various fields of mathematics: Number theory, group theory, projective geometry and others. They also have many interesting practical applications, in particular, in coding theory and cryptography. Galois fields are the basis for different cryptographic systems and correcting codes which are used in various applications of data transmission. This chapter considers some main properties of finite fields and their applications in coding theory and cryptographic algorithms, such as Diffie Hellman&s;s: algorithm and the El Gamal algorithm. It examines that there are no other finite fields up to isomorphism. The chapter presents one of the most difficult computational problems: The problem of discrete logarithm. It then introduces the notions of primitive roots and indexes and prove some their basic properties.