Elliptic curve cryptography, in essence, entails using the group of points on an elliptic curve as the underlying number system for public key cryptography. There are two main reasons for using elliptic curves as a basis for public key cryptosystems. The first reason is that elliptic curve based cryptosystems appear to provide better security than traditional cryptosystems for a given key size. One can take advantage of this fact to increase security, or (more often) to increase performance by reducing the key size while keeping the same security. The second reason is that the additional structure on an elliptic curve can be exploited to construct cryptosystems with interesting features which are difficult or impossible to achieve in any other way. A notable example of this phenomenon is the development of identity-based encryption and the accompanying emergence of pairing-based cryptographic protocols.