On Cycles of Pairing-Friendly Abelian Varieties

One of the most promising avenues for realising scalable proof systems relies on the existence of 2-cycles of pairing-friendly elliptic curves. Such a cycle consists of two elliptic curves \(\ensuremath {\mathcal {E}}/\ensuremath {\mathbb {F}}_p\) and \(\ensuremath {\mathcal {E}}'/\ensuremath {\mathbb {F}}_q\) that both have a low embedding degree and also satisfy \(q = \#\ensuremath {\mathcal {E}}(\ensuremath {\mathbb {F}}_p)\) and \(p = \#\ensuremath {\mathcal {E}}'(\ensuremath {\mathbb {F}}_q)\) . These constraints turn out to be rather restrictive; in the decade that has passed since 2-cycles were first proposed for use in proof systems, no new constructions of 2-cycles have been found. In this paper, we generalise the notion of cycles of pairing-friendly elliptic curves to study cycles of pairing-friendly abelian varieties, with a view towards realising more efficient pairing-based SNARKs. We show that considering abelian varieties of dimension larger than 1 unlocks a number of interesting possibilities for finding pairing-friendly cycles, and we give several new constructions that can be instantiated at any security level.