International Association
for Cryptologic Research

Cryptographic protocols such as zkSNARKs use 2-cycles of elliptic curves for efficiency, often relying on pairing computations. However, 2-cycles of pairing-friendly curves are hard to find, and the only known cases consist of an MNT4 and an MNT6 curve. In this work, we prove that a 2-cycle containing an MNT3 curve cannot be pairing-friendly. For other curve families, we have a similar result for cryptographically attractive field sizes. Thus we cannot hope to find new pairing-friendly 2-cycles using the current methods.

Furthermore, we show that there are no SNARK-friendly 2-chains of elliptic curves from combinations of MNT, Freeman and BN curves of reasonable size, except for the (MNT4, MNT6) chains.