Paper 2023/037 Original: eprint.iacr.org/2023/037 ↗

Efficient Isogeny Proofs Using Generic Techniques

Abstract

Generating supersingular elliptic curves of unknown endomorphism ring has been a problem vexing isogeny-based cryptographers for several years. A recent development has proposed a trusted setup protocol to generate such a curve, where each participant generates and proves knowledge of an isogeny. Thus, the construction of efficient proofs of knowledge of isogeny has developed new interest. Historically, the isogeny community has assumed that obtaining isogeny proofs of knowledge from generic proof systems, such as zkSNARKs, was not a practical approach. We contribute the first concrete result in this area by applying Aurora (EUROCRYPT'19), Ligero (CCS'17) and Limbo (CCS'21) to an isogeny path relation, and comparing their performance to a state-of-the-art, tailor-made protocol for the same relation. In doing so, we show that modern generic proof systems are competitive when applied to isogeny assumptions, and provide an order of magnitude ($3\textrm{-}10\times$) improvement to proof and verification times, with similar proof sizes. In addition, these proofs provide a stronger notion of soundness, and statistical zero-knowledge; a property that has only recently been achieved in isogeny PoKs. Independently, this technique shows promise as a component in the design of future isogeny-based or other post-quantum protocols.