Abelian varieties in coding and cryptography

Abstract

Algebraic curves over a finite field have played a central role in both coding theory and cryptography over the past three decades. In coding theory the use of algebraic curves led to the discovery of asymptotically good codes whose parameters lie above the Varshamov-Gilbert bound in certain cases while in cryptography the use of elliptic curves led to public key cryptosystems that are more efficient, in some sense, for a given level of security than integer factorization based ones. It would seem natural that the use of higher dimensional varieties might lead to even better results for both applications. Such has not so far been the case in any dramatic way. The purpose of this talk is to review the situation on the use of Abelian varieties in these two areas.