Good and bad uses of elliptic curves in cryptography

In the first part of this article I describe the construction of cryptosystems using elliptic curves, discuss the Elliptic Curve Discrete Logarithm Problem (upon which the security of all elliptic curve cryptosystems rests), and survey the different types of elliptic curves that can be chosen for cryptographic applications. In the second part I describe three unsuccessful approaches to breaking various cryptosystems by means of liftings to global elliptic curves. I explain how the failure of these attacks is caused by fundamental properties of the global curves.