This paper proposes a multiplication algorithm for $F_{p^m}$, which can be efficiently applied to many pairs of characteristic p and extension degree m except for the case that 8p divides m(p-1). It uses a special class of type-<k, m> Gauss period normal bases. This algorithm has several advantages: it is easily parallelized; Frobenius mapping is easily carried out since its basis is a normal basis; its calculation cost is clearly given; and it is sufficiently practical and useful when parameters k and m are small.