The discrete logarithm problem in the class group of a curve over a finite field can be used for constructing cryptographic primitives. The case of elliptic curves is particularly well known. However, one needs to choose the curve with some care, as there are various techniques for computing discrete logarithms that work well on special classes of curves.

In general, one needs to use curves of low genus, as high genus curves are subject to index calculus attacks. In certain cases, it is possible to cover a low genus curve with a higher genus one in such a way that the index calculus attack also affects the security of the low genus curve. This kind of attack is called a cover attack.

In the first talk, I'll give a brief introduction to curve cryptography, and I'll discuss some aspects of index calculus, including some recent developments. In the second talk, I'll discuss various constructions of covering curves that weaken the discrete logarithm problem on low genus curves.