The below cryptographic scheme has many applications, both in security-related systems and in building other cryptographic schemes. Please help suggesting more applications!
Ring signature is a "signer-ambiguous" signature, the verifier can only ensure the message is signed by 1 out of n purported users, but not exactly whom. For signing, (i.e. to produce a ring signature), the actual signer declares an arbitrary set of possible signers that
must include himself, and computes the signature entirely by himself using only
his secret key and the others' public keys. In particular, the other possible "members of this group"may be completely unaware that their public keys are used by a
stranger to produce such a ring signature on a message they have never seen and
would not wish to sign.
You may assume additional properties that you find useful, e.g. 1. perfect anonymity, 2. there exists a guy who can revoke the anonymity of the signature; or 3. we can tell if two signatures are signed by the same signer, but with the identity remains hidden. You name it, I (try to) build it!