Arithmetic Operators over Finite Field GF($2^m$) in BCH and Reed-Solomon Codes

Galois field arithmetic circuits are utilized in various areas such as error correction codes, communications, signal processing, and security engineering. In this chapter, we will explain the significance of error detection and correction technique, while also examining the fundamental principles and wide range of techniques available. Moreover, explaining the mathematical details of BCH and Reed-Solomon codes necessitates a comprehensive utilization of GF($2^m$) arithmetic. Therefore, the primary contribution of this chapter entails an investigation of the arithmetic operations over finite field that are indispensable for the implementation of BCH and Reed-Solomon codes. These operations involve addition, subtraction, multiplication, squaring, square roots, multiplicative inverses, division, and exponentiation.