A Combined Circuit for Multiplication and Inversion in ${\rm GF}(2^{m})$

A combined circuit for multiplication and inversion in GF(2^m) is proposed. In order to develop a combined circuit, we start with combining the most significant bit first multiplication algorithm and the modified extended Euclid´s algorithm by focusing on the similarities between them. Since almost all hardware components of the circuits are shared by multiplication and inversion, the combined circuit can be implemented with significantly smaller hardware than that necessary to implement both multiplication and inversion separately. By logic synthesis, the area of the proposed circuit is estimated to be approximately over 15% smaller than that of previously proposed combined multiplication/division circuits.