Super-Isolated Elliptic Curves and Abelian Surfaces in Cryptography

We call a simple abelian variety over Fp super-isolated if its (Fp-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 Fp with prime order and p ≤ N , is roughly Θ̃( √ N). In contrast, we prove that there are only 2 super-isolated surfaces of cryptographic size and nearprime order.