Elliptic curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields . In comparison with other cryptographic systems based on finite fields, ECC provides the same level of security with keys of smaller size. Elliptic curves are applicable for encryption , digital signatures , pseudo-random generators and other tasks . They are also used in several integer factorization algorithms that have applications in cryptography , such as Lenstra elliptic curve factorization. Discrete logarithm problem can be applied over the group of points of an elliptic curve defined over a finited field . Elliptic curve cryptosystems are more approperiate for using in systems with limited memory bandwidth and limited computational ability rather than other public-key cryptosystems . The fundamental operation in computation of the elliptic curves cryptography systems is the problem of computing scalar multiplication of a point of the elliptic curve . Therefore, researchers have been trying to find new methods for increasing the speed of this computation . The basic methods are the double-and-add method , window method and comb method . In this theses , we will introduce the GLV method of Gallant , Lambert , and Vanstone that is used for computing scalar multiplication of a point of prime order of the elliptic curve with an efficient endomorphism. This is done by considering a two dimensional decomposition using the concepts of lattices and efficient endomorphisms . The GLS method of Galbraith , Lin , and Scott is the extension of the GLV method over a larger