Title: Super-Isolated Elliptic Curves and Abelian Surfaces in Cryptography,
Abstract: We call a simple abelian variety over $\mathbb{F}_p$ super-isolated if its
($\mathbb{F}_p$-rational) isogeny class contains no other varieties. The
motivation for considering these varieties comes from concerns about isogeny
based attacks on the discrete log problem. We heuristically estimate that the
number of super-isolated elliptic curves over $\mathbb{F}_p$ with prime order
and $p \leq N$, is roughly $\tilde{\Theta}(\sqrt{N})$. In contrast, we prove
that there are only 2 super-isolated surfaces of cryptographic size and
near-prime order.