Algorithms and cryptography in algebraic function fields

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The last few decades have seen an unprecedented increase in the use of computing and communication technology. Throughout the world, the amount of data that is being transmitted over unprotected channels, such as the internet and mobile telephones, is increasing at a staggering rate. In addition, the number of large scale databases containing possibly sensitive records is continuing to grow. Unfortunately, this development is coupled with an ever rising threat of unauthorized access to (and possibly modification of) information that was assumed to be safe from such intrusions. Sadly, the public has gotten quite accustomed to news flashes about virus-infected computers, internet worms, defaced web sites, and cases of identity theft. One of the most effective ways of protecting data against unauthorized access is to scrample (encrypt) it before sending or storing it, thereby rendering it unintelligible to prying eyes. Only authorized parties are able to unscramble (decrypt) the information with the aid of a secret key known only to them. A cryptographic system, or cryptosystem for short, is a method for effecting this process. The security of many modern cryptosystems relies on certain mathematical problems that experts believe to be extremely difficult to solve. The idea underlying the design of such a system is that an adversary would have to solve an instance of one of these very hard problems in order to crack the scheme. It is therefore of great interest, both theoretical and practical, to thoroughly study these types of problems and devise new cryptosystems that make use of them. To this end, I propose to investigate a type of mathematical structure called an algebraic function field. Due to their fast arithmetic and their wealth of built-in hard mathematical problems, these structures provide an excellent framework for efficient and very secure cryptosystems. Some function field based schemes -- namely the so-called elliptic curve based cryptosystems -- are already used commercially on a variety of electronic devices, such as computers, personal data organizers, and cell phones.