The review on elliptic curves as cryptographic pairing groups

Elliptic curve is a set of two variable points on polynomials of degree 3 over a field acted by an addition operation that forms a group structure. The motivation of this study is that the mathematics behind that elliptic curve to the applicability within a cryptosystem. Nowadays, pairings bilinear maps on elliptic curves are popular to construct cryptographic protocol pairings to help to transform a discrete logarithm problem on an elliptic curve to the discrete logarithm problem in finite fields. The purpose of this paper is to introduce elliptic curve, bilinear pairings on elliptic curves as based on pairing cryptography. Also this investigation serves as a basis in guiding anyone interested to understand one of the applications group theory in cryptosystem.