 Solution

For authenticate-then-encrypt, write the intercepted pair as $(c,t)=(E_k(x),E_k(s))$, where $s=f(sk,x)$. Each trial key $k^{\prime }$ givesThe adversary can check the public relationThe true key always passes. An incorrect key will generally fail, and further intercepted pairs can eliminate accidental matches. This gives a ciphertext-only attack with a useful key-testing predicate even when the plaintext has no recognisable format. It makes exhaustive key search applicable with less prior information; it does not reduce the generic key-space size below $2^m$.

The usual approximate false-pass chance $2^{-n}$ is an ideal-behaviour estimate, not a consequence of the bare permutation identities alone. The decisive point is that the encrypted digital signature supplies publicly checkable redundancy once a candidate decryption key is tried.