Finite Fields, Modular Arithmetic, Linear Algebra, and Number Theory

This chapter presents a rigorous treatment of modular arithmetic. This leads into linear algebra, which is important for its applicability to coding, especially the class of linear codes. The chapter exhibits a construction procedure for finite fields. It discusses polynomials in terms of this construction procedure, and then provides a few results regarding polynomials that are of use in other contexts. The chapter presents a discussion on complexity issues and describes a factoring algorithm. The principle of Rivest, Shamir, Adleman algorithm is applied to obtain two famous results in number theory. Various “probabilistic algorithms” have been used with some success such as the index calculus method for the Diffie–Hellman problem. The Euclidean algorithm is used to express the greatest common divisor of two numbers as a linear combination of two numbers. However, using the method described can be tiresome as we are forced to work backward after having found “the bottom.”