We consider the following problem that arises in outsourced storage: a user stores her data $x$ on a remote server but wants to audit the server at some later point to make sure it actually did store $x$. The goal is to design a (randomized) verification protocol that has the property that if the server passes the verification with some reasonably high probability then the user can rest assured that the server is storing $x$.
While doing this, we need to minimize the user's storage and the total amount of communication between the server and the user. This is the data possession problem and is closely related to the problem of obtaining a 'proof of retrievability.' Existing schemes with provable guarantees mostly use cryptographic primitives and can only guarantee that the server is storing a constant fraction of the amount of information that it is supposed to store.

In this work we present an optimal solution (in terms of the user's storage and communication) while at the same time ensuring that a server that passes the verification protocol with any reasonable probability will store, to within a small \textit{additive} factor, $C(x)$ bits of information, where $C(x)$ is the plain Kolmogorov complexity of $x$. (Since we cannot prevent the server from compressing $x$, $C(x)$ is a natural upper bound.) The proof of security of our protocol combines Kolmogorov complexity with list decoding and unlike previous work that relies upon cryptographic assumptions, we allow the server to have unlimited computational power. To the best of our knowledge, this is the first work that combines Kolmogorov complexity and list decoding.

Our framework is general enough to capture extensions where the user splits up $x$ and stores the fragment across multiple servers and our verification protocol can handle non-responsive servers and colluding servers. As a by-product, we also get a proof of retrievability. Finally, our results also have an application in `storage enforcement' schemes, which in turn have an application in trying to update a remote server that is potentially infected with a virus.