In this thesis, we introduce some of the fundamental concepts of elliptic curves over the field of rational numbers \(\mathbb{Q}\) and finite fields \(\mathbb{F}_p\). We define the rank of an elliptic curve, a notion of significant importance in the study of elliptic curves and their application in cryptography. Various methods for determining the rank of an elliptic curve are described, and an elliptic curve of rank \(\geq 8\) is constructed using the Mestre-Nagao sums. Through the study of elliptic curves, we utilize concepts of algebraic structures, which are outlined in the first chapter. The applications of elliptic curves in cryptography are a highly researched topic. We discuss the discrete logarithm problem and the difficulty of solving this problem in the context of elliptic curves. We describe some of the cryptosystems utilizing elliptic curves, and the connection between the challenge of finding elliptic curves with high rank and their practical application.