The selection of secure elliptic curves and the scalar multiplications of elliptic curves are two important problems in the practice of efficiently implementing an elliptic curve cryptosystems.In this paper,we study those two problems jointly,give a class of secure elliptic curves mainly based on the computer words,describe a detailed process of how to selecting those curves,and present a new method,which is based on the idea of "baby step-giant step",of computing the scalar multiplication concerning those curves.With the new method,the amount of scalar multiplications based on those curves can be reduced greatly. Besides,when those curves are used,special representation method for the elements in the base field is no longer needed,and all the arithmetic in the field can be quickly accomplished.