Breaking and repairing optimistic fair exchange from PODC 2003

In PODC 2003, Park, Chong, Siegel and Ray proposed an optimistic protocol for fair exchange, based on RSA signatures. We show that their protocol is totally breakable already in the registration phase: the honest-but-curious arbitrator can easily determine the signer's secret key. On a positive note, the authors of informally introduced a connection between fair exchange and "sequential two-party multisignature schemes" (which we call two-signatures), but used an insecure two-signature scheme in their actual construction. Nonetheless, we show that this connection can be properly formalized to imply provably secure fair exchange protocols. By utilizing the state-of-the-art non-interactive two-signature of Boldyreva, we obtain an efficient and provably secure (in the random oracle model) fair exchange protocol, which is based on GDH signatures. Of independent interest, we introduce a unified model for non-interactive fair exchange protocols, which results in a new primitive we call verifiably committed signatures. Verifiably committed signatures generalize (non-interactive) verifiably encrypted signatures and two-signatures, both of which are sufficient for fair exchange.