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 E / F p and E ′ / F q that both have a low embedding degree and also satisfy q = # E ( F p ) and p = # E ′ ( 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 un-locks a number of interesting possibilities for finding pairing-friendly cycles, and we give several new constructions that can be instantiated at any security level.