Remove the permutation requirement in the one-query impossibility result

Establish an information-theoretic impossibility result for one-query block-cipher constructions whose output is only computationally indistinguishable from a permutation, rather than required to be an actual permutation for every public permutation and key.

Background

The paper proves that any deterministic one-query cipher constructed from a public permutation is distinguishable from a random cipher with a small number of quantum queries, under the explicit requirement that the constructed cipher be a permutation for every public permutation and every key. The proof uses Choi states and is information-theoretic, although the resulting distinguisher need not be computationally efficient.

The authors note that this permutation requirement is a technical limitation: computationally secure quantum pseudorandom permutations can be constructed in the equal-size case, but such constructions need not satisfy the stronger condition of being an exact permutation for every underlying public permutation and key. Removing this restriction would extend the scope of the one-query impossibility theorem.