# Super-Isolated Elliptic Curves and Abelian Surfaces in Cryptography

## TL;DR

This paper introduces the concept of super-isolated abelian varieties, especially elliptic curves and surfaces, analyzing their rarity and potential cryptographic significance due to their unique isogeny class properties.

## Contribution

It defines super-isolated varieties, heuristically estimates their abundance among elliptic curves, and proves their extreme scarcity among surfaces of cryptographic size.

## Key findings

- Number of super-isolated elliptic curves with prime order is roughly proportional to ()
- Only 2 super-isolated surfaces of cryptographic size exist
- Super-isolated varieties are rare and potentially resistant to isogeny-based attacks

## 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.