ABSTRACT
Finite fields have received a lot of attention because of their important and practical applications in cryptography, coding theory, switching theory and digital signal processing. For example, one of the major application of finite fields is algebraic coding theory – the theory of error-correcting and error-detecting codes. Typically, the message transmitted over a discrete communication channel consists of a finite sequence of symbols that are elements of some finite alphabet. In general, the alphabet is assumed to be a finite field. For instance, if the alphabet consists of 0 and 1, it is considered as the finite field GF(2). Since noise is unavoidable in a communication channel, error-control codes have been extensively used in digital communication and computer systems to achieve efficient and reliable digital transmission and storage, and have become an essential part of digital communication and recording system. Due to the inherent properties of digital transmission over a communication channel, most error-control encoding and decoding algorithms are based on finite field arithmetic operations, such as Reed-Solomon codes, Golay codes, etc. [1] [4]. In practice, the finite fields of characteristic 2, GF(2 m ), are generally used and the multiplications over GF(2 m ) are the major building blocks in many error-control coders.
