  In his seminal work on recording quantum queries [Crypto 2019], Zhandry
studied interactions between quantum query algorithms and the quantum oracle
corresponding to random functions. Zhandry presented a framework for
interpreting various states in the quantum space of the oracle that can be used
to provide security proofs in quantum cryptography.
  In this paper, we introduce a similar interpretation for the case when the
oracle corresponds to random permutations instead of random functions. Because
both random functions and random permutations are highly significant in
security proofs, we hope that the present framework will find applications in
quantum cryptography. Additionally, we show how this framework can be used to
prove that the success probability for a k-query quantum algorithm that
attempts to invert a random N-element permutation is at most O(k^2/N).
