ABSTRACT
Elliptic curves are an essential tool in today’s cryptography. They are often used to construct cryptosystems but there are also some cryptanalytic applications where the use of elliptic curves is necessary. In this chapter, we give a self-contained introduction to elliptic curves, together with a description of the Weil pairing. This pairing was initially introduced in cryptography as a cryptanalytic tool to attack the discrete logarithm problem on some special elliptic curves. It is interesting to note that the Weil pairing1 has now become an essential tool for constructing new cryptosystems.
