Compressed Oracles

We formalize the compressed quantum random oracle methodology by Zhandry (Crypto 2019). This is a formalism for modeling quantum random oracles to make quantum cryptographic proofs feasible. Our definition of the compressed oracles is loosely based on the presentation from Unruh (arXiv 2021), but with a considerable amount of new definitions and results. In particular, we make extensive use of the quantum references formalism (Unruh, arXiv 2024, AFP 2025) to enable reasoning about queries on arbitrary subsystems, something which is left very informal in pen-and-paper formalizations of the compressed oracles. We use the developed formalism to prove that finding $x$ with $H(x)=0$, and finding collisions in $H$, is hard for quantum adversaries with oracle access to a random function $H$.