| Recently, the study of elliptic curves has branched into many different disciplines. An application of the properties of these curves has been found in cryptography. This paper aims to gives an overview of the mathematics behind elliptic curves, including properties of finite fields. It will discuss some basic cryptography in general and then describe a particular cryptographic use of elliptic curves in identity-based encryption. This type of encryption came about to allow arbitrary length strings (such as e-mail addresses) to be used as the encryption-key in a Public-Key cryptosystem. This allows the identity of the receiving party of a message to be used within the encryption itself. Other applications of identity-based encryption are briefly discussed as well. Finally, there will be a discussion of a specific algorithm by Dan Boneh and Matthew Franklin for a concrete identity-based encryption scheme based on the Weil Pairing for elliptic curves. |
