 Solution

For encrypt-and-authenticate, the transmitted pair is $(E_k(x),s)$ with $s=f(sk,x)$. The recovery map is public, so any observer computesNo symmetric key, exhaustive key search or oracle is needed. Publishing this message-recovery digital signature reveals the entire plaintext and breaks cryptographic confidentiality outright. A digital signature of a suitable digest would have different leakage properties, but the printed construction signs the raw block.