The number of hyperelliptic curves over a finite field: recent results

Abstract

In this paper we survey the state of art of the use of hyperelliptic curves in Cryptology. We recall the definition of public-key cryptosystems based on the discrete logarithm problem and, specially, the particular case for which the base group of the cryptosystem is the Jacobian variety of a hyperelliptic curve. Moreover, we present some of the open problems related to this kind of cryptosystems and some of the recent results about them. In particular, those related to the number of hyperelliptic curves over a finite field.