**Super-Isolated Elliptic Curves and Abelian Surfaces in Cryptography**

*Travis Scholl*

**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.

**Category / Keywords: **public-key cryptography / elliptic curve cryptosystem, number theory

**Date: **received 1 May 2017

**Contact author: **tscholl2 at uw edu

**Available format(s): **PDF | BibTeX Citation

**Version: **20170504:115115 (All versions of this report)

**Short URL: **ia.cr/2017/383

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