In this letter, a generalized syndrome polynomial is proposed from which several decoding key-equations for Reed-Solomon codes can be derived systematically. These equations are always solved by the extended Euclidean algorithm.

In this paper, we will propose a modified Euclidean algorithm which can be obtained by examining thoroughly the equivalence between conventional Euclidean algorithm and Berlekamp-Massey algorithm. Furthermore, we will show the design method for BCH decoders and Inverters based on the modified Euclidean algorithm in which the minimum number register realization is given.

The arithmetic operations over the finite field GF(pn) has many important applications. There is a need for a fast inverter circuit over the finite field GF(pn), because inversion' is must complicated and lengthy operation. In this paper, we will propose a design method of fast inverter circuit that is based on Euclid's algorithm. The inverter circuit is composed of a number of identical basic circuits. Therefore the designed circuit has a regular and expandable feature and has a modular structure.