Abelian Varieties with Small Isogeny Class and Applications to Cryptography

Abstract

An elliptic curve $E$ over a finite field $\FF_q$ is called isolated if it admits few efficiently computable $\FF_q$-isogenies from $E$ to a non-isomorphic curve. We present a variation on the CM method that constructs isolated curves. Assuming the Bateman-Horn conjecture, we show that there is negligible probability that a curve of cryptographic size constructed via this method is vulnerable to any known attack on the ECDLP. A special case of isolated curves is when the $\FF_q$-isogeny class contains only one $\FF_q$-isomorphism class. We call an elliptic curve, or abelian variety, super-isolated if it has this property. We give a simple characterization of super-isolated elliptic curves, and several examples of cryptographic size. We prove that there are only $2$ super-isolated surfaces suitable for cryptographic use. Finally, we show that for any $g \geq 3$, there are only finitely many super-isolated ordinary simple abelian varities of genus $g$. Essentially, we have an existence result in the practical range for genus $g \leq 2$, and a non-existence result for the impractical genera $g \geq 3$.