Given hash(x) and y, can I verify z = hash(x + y)? Here's the problem I'm trying to solve:


y is public, it's a timestamp
there is A who knows x - it's a file
A can send B some information about the file (a hash?), but not the full file (it's too big)
A then sends the file x to C


The role of B now, is to periodically verify that at given time y C still has the file x. C could send something like hash(x, y) to B, and B (who doesn't have the full x) needs a way to verify that C indeed has the full file x at time y.



Here's the original question I've asked (it includes the solution approach I've been trying to take):

Say I only know hash(x) and y.

I receive z that is supposed to be hash(x + y)? Is there any way I can verify that without knowing x?

I'm interested in any mechanism that could make such check possible rather than particular hash and + operations.
The technical term for what you're looking for is "publicly verifiable proof of retrievability". Here is a question on crypto.SE that discusses existing implementations of PoRs.