Isolated Curves for Hyperelliptic Curve Cryptography

TL;DR

This paper introduces isolated genus two curves in hyperelliptic cryptography, showing they may offer enhanced security due to the difficulty of constructing isogenies, supported by theoretical and computational evidence.

Contribution

It defines isolated genus two curves, analyzes their endomorphism rings, and provides heuristic and computational results on their distribution and security implications.

Findings

Isolated genus two curves have prime or almost prime index in endomorphism rings.

Computational results match heuristic predictions within acceptable errors.

Heuristic asymptotic results describe the density of isolated curves for given CM fields.

Abstract

We introduce the notion of isolated genus two curves. As there is no known efficient algorithm to explicitly construct isogenies between two genus two curves with large conductor gap, the discrete log problem (DLP) cannot be efficiently carried over from an isolated curve to a large set of isogenous curves. Thus isolated genus two curves might be more secure for DLP based hyperelliptic curve cryptography. We establish results on explicit expressions for the index of an endomorphism ring in the maximal CM order, and give conditions under which the index is a prime number or an almost prime number for three different categories of quartic CM fields. We also derived heuristic asymptotic results on the densities and distributions of isolated genus two curves with CM by any fixed quartic CM field. Computational results, which are also shown for three explicit examples, agree with heuristic prediction with errors within a tolerable range.