Super-Isolated Elliptic Curves and Abelian Surfaces in Cryptography

Travis Scholl

Paper 2017/383

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.

Metadata

Available format(s): PDF
Category: Public-key cryptography
Publication info: Preprint. MINOR revision.
Keywords: elliptic curve cryptosystem number theory
Contact author(s): tscholl2 @ uw edu
History: 2017-05-04: received
Short URL: https://ia.cr/2017/383
License: CC BY

BibTeX

@misc{cryptoeprint:2017/383,
      author = {Travis Scholl},
      title = {Super-Isolated Elliptic Curves and Abelian Surfaces in Cryptography},
      howpublished = {Cryptology {ePrint} Archive, Paper 2017/383},
      year = {2017},
      url = {https://eprint.iacr.org/2017/383}
}