FUNCTION FIELDS OVER FINITE FIELDS AND THEIR APPLICATIONS TO CRYPTOGRAPHY

Abstract

It is well known that algebraic function fields over finite fields have many applications in coding theory, and the latter is closely related to cryptography. This has led researchers in a natural way to consider methods based on some specified function fields in order to construct cryptographic schemes, such as schemes for unconditionally secure authentication, traitor tracing, secret sharing, broadcast encryption and secure multicast, just to mention a few. There is no doubt that the sophisticated techniques of function fields and their cryptographic applications have become a new, promising research direction. Over the past few years, a lot of research in this new direction has been carried out and the results are fruitful. This paper surveys some initial efforts in this new emerging direction. We describe several interesting links among function fields, cryptography and combinatorics. As examples we show that constructions based on function fields for authentication codes, frameproof codes, perfect hash families, cover-free families and sequences with high linear complexity outperform the previously existing results, thus yielding in turn, directly or indirectly, cryptographic schemes with better performance. It should be noted that this paper is far from exhaustive, as it presents only a portion of results that appeared in the literature. We hope that the paper will nevertheless stimulate further research in this new and promising direction of applying function fields to cryptography.