This paper describes an algorithm for computing elliptic scalar multiplications on non-supersingular elliptic curves defined over GF (2 m). The algorithm is an optimized version of a method described in [1], which is based on Montgomery's method [8]. Our algorithm is easy to implement in both hardware and software, works for any elliptic curve over GF (2 m), requires no precomputed multiples of a point, and is faster on average than the addition-subtraction method described in draft standard IEEE P1363. In addition, the method requires