Division and bit-serial multiplication over GF(qm)

Division and bit-serial multiplication in finite fields are considered. Using co-ordinates of the supporting elements it is shown that, when field elements are represented by polynomials, division over GF(q^m) can be performed by solving a system of m linear equations over GF(q). For a canonical basis representation, a relationship between the division and the discrete-time Wiener-Hopf equation of degree m over GF(q) is derived. This relationship leads to a bit-serial multiplication scheme that can be easily realised for all irreducible polynomials.