from the wikipedia article:
Public-key cryptography is based on the intractability of certain mathematical problems. Early public-key systems are secure assuming that it is difficult to factor a large integer composed of two or more large prime factors. For elliptic-curve-based protocols, it is assumed that finding the discrete logarithm of a random elliptic curve element with respect to a publicly known base point is infeasible. The size of the elliptic curve determines the difficulty of the problem.
We will be discussing the base concepts required for ECC - namely discrete log and finite fields (or modular number systems), followed by the application to cryptography. At the end, we will discuss the underpinnings that make ECC difficult to reverse and what could change that.