Elliptic curve is a set of two variable points on polynomials of degree 3 over a field actedby an addition operation that forms a group structure. The motivation of this study is that themathematics behind that elliptic curve to the applicability within a cryptosystem. Nowadays, pair-ings bilinear maps on elliptic curve are popular to construct cryptographic protocol pairings help totransform a discrete logarithm problem on an elliptic curve to the discrete logarithm problem in nite elds. The purpose of this paper is to introduce elliptic curve, bilinear pairings on elliptic curves asbased on pairing cryptography. Also this investigation serves as a basis in guiding anyone interestedto understand one of the applications of group theory in cryptosystem.