> If someone was trying to be dishonest then they could fake any attribute of f() that you could likely verify.
Not necessarily. For example: you can't fake bitcoin transactions, as each one requires significant amount of comuptation. Basically faking would require exactly the same work (modulo problems with SHA1 itself, but that's another matter).
I'm not familiar with the details of bitcoin, but in that case aren't (some of) the inputs to the function hidden not the function itself.
Proof of work techniques are relying on forcing the other party to run a known function in order to prevent various attacks. If the other party can't verify the result they would be no use.
Not necessarily. For example: you can't fake bitcoin transactions, as each one requires significant amount of comuptation. Basically faking would require exactly the same work (modulo problems with SHA1 itself, but that's another matter).
This is used for other purposes as well: http://en.wikipedia.org/wiki/Proof_of_work