Efficient Proofs of Possession for Legacy Signatures

Digital signatures underpin identity, authenticity, and trust in modern computer systems. Cryptography research has shown that it is possible to prove possession of a valid message and signature for some public key, without revealing the message or signature. These proofs of possession work only for specially-designed signature schemes. Though these proofs of possession have many useful applications to improving security, privacy, and anonymity, they are not currently usable for widely deployed, legacy signature schemes—like RSA, ECDSA, and Ed25519. Unlocking practical proofs of possession for these legacy signature schemes requires closing a huge efficiency gap. This work brings proofs of possession for legacy signature schemes very close to practicality. Our design strategy is to encode the signature's verification algorithm as a rank-one constraint system (R1CS), then use a zkSNARK to prove knowledge of a solution. To do this efficiently we (1) design and analyze a new zkSNARK called Dorian that supports randomized computations, (2) introduce several new techniques for encoding hashes, elliptic curve operations, and modular arithmetic, (3) give a new approach that allows performing the most expensive parts of ECDSA and Ed25519 verifications outside R1CS, and (4) generate a novel elliptic curve that allows expressing Ed25519 curve operations very efficiently. Our techniques reduce R1CS sizes by up to 200× and prover times by more than 20×. We can generate a 240-byte proof of possession of an RSA signature over a message the size of a typical TLS certificate—two kilobytes—in only three seconds.