Storing data on an external server with information-theoretic security, while using a key shorter than the data itself, is impossible. As an alternative, we propose a scheme that achieves information-theoretically secure tamper evidence: The server is able to obtain information about the stored data, but not while staying undetected. Moreover, the client only needs to remember a key whose length is much shorter than the data. We provide a security proof for our scheme, based on an entropic uncertainty relation, similar to QKD proofs. Our scheme works if Alice is able to (reversibly) randomise the message to almost-uniformity with only a short key. By constructing an explicit attack we show that short-key unconditional tamper evidence cannot be achieved without this randomisability.