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.