Super-isolated elliptic curves and abelian surfaces in cryptography

An abelian variety is super-isolated if it is alone in its isogeny class. The motivation for these varieties comes from concerns about isogeny based attacks on the discrete log problem. In this talk, I'll outline methods for finding super-isolated elliptic curves and abelian surfaces. We will see that there are many more super-isolated curves than there are surfaces. In fact, there are only two super-isolated surfaces of cryptographic size and near-prime order.