Miden's private multi-signature scheme uses an interesting approach—moving the signature collection off-chain, and only submitting a zero-knowledge proof on-chain at the end.

The core design is as follows: all signers maintain the latest version of the same private state, signatures are performed entirely off-chain, and when it comes to on-chain submission, only a verifiable proof is needed. This method ensures the security of multi-party collaboration while avoiding exposing private data on the public chain. Compared to traditional multi-signature methods that submit signatures one by one on-chain, this scheme significantly improves both efficiency and privacy.