We consider a public hash function $H$, assumed collision-resistant and preimage-resistant (for both first and second preimage), similar in construction to SHA-1 or SHA-256.
Alice discloses a value $h$, claiming that she (or/and parties she can communicate with or/and devices they have access to) knows a message $m$ such that $H(m)=h$. Can some protocol convince Bob of this claim without help of a third party/device that Bob trusts, nor allowing Bob to find $m$?
At Crypto 98 rump session, Hal Finney made a 7-minutes presentation A zero-knowledge proof of possession of a pre-image of a SHA-1 hash which seems to be intended for that. This remarkable result is occasionally stated as fact, including recently here and next door. But I do not get how it is supposed to work.
Update: This talk mentions using the protocol in the Crypto'98 paper of Ronald Cramer and Ivan B. Damgård: Zero-Knowledge Proofs for Finite Field Arithmetic or: Can Zero-Knowledge be for Free? (this freely downloadable version is very similar, or there is this earlier, longer version).