In this study, first of all, we categorized encryption algorithms and exposed the structure and specifications of the symmetric and asymmetric algorithms. We explained that asymmetric encryption algorithms are based on hard problems (NP) and explained what these problems are. In addition, we showed how addition and doubling are realized on the elliptic curves and which theorems are used for that. We also showed how we can use Elliptic curves, which are very important development for asymmetric cryptosystems, in encryption process and explained how ECC algorithm using elliptic curves realize encryption and decryption process. As a result, we developed ECC application showing numerical results to compare with other asymmetric encryption algorithms in view of security