Recently, Gallant, Lambert arid Vanstone introduced a method for speeding up the scalar multiplication on a family of elliptic curves over prime fields that have efficiently-computable endomorphisms. It really depends on decomposing an integral scalar in terms of an integer eigenvalue of the characteristic polynomial of such an endomorphism. In this paper, by using an element in the endomorphism ring of such an elliptic curve, we present an alternate method for decomposing a scalar. The proposed algorithm is more efficient than that of Gallant`s and an upper bound on the lengths of the components is explicitly given.