Recent active studies have demonstrated that cryptography without one-way functions (OWFs) could be possible in the quantum world. Many fundamental primitives that are natural quantum analogs of OWFs or pseudorandom generators (PRGs) have been introduced, and their mutual relations and applications have been studied. Among them, pseudorandom function-like state generators (PRFSGs) [Ananth, Qian, and Yuen, Crypto 2022] are one of the most important primitives. PRFSGs are a natural quantum analogue of pseudorandom functions (PRFs), and imply many applications such as IND-CPA secret-key encryption (SKE) and EUF-CMA message authentication code (MAC). However, only known constructions of (many-query-secure) PRFSGs are ones from OWFs or pseudorandom unitaries (PRUs).
In this paper, we construct classically-accessible adaptive secure PRFSGs in the invertible quantum Haar random oracle (QHRO) model which is introduced in [Chen and Movassagh, Quantum]. The invertible QHRO model is an idealized model where any party can access a public single Haar random unitary and its inverse, which can be considered as a quantum analog of the random oracle model. Our PRFSG constructions resemble the classical Even-Mansour encryption based on a single permutation, and are secure against any unbounded polynomial number of queries to the oracle and construction. To our knowledge, this is the first application in the invertible QHRO model without any assumption or conjecture. The previous best constructions in the idealized model are PRFSGs secure up to \(o(\lambda /\log \lambda )\) queries in the common Haar state model [Ananth, Gulati, and Lin, TCC 2024] and (inverseless) PRUs in a relaxed QRHO model without inverse access [Ananth, Bostanci, Gulati, and Lin, Eurocrypt 2025].
We develop new techniques on Haar random unitaries to prove the selective and adaptive security of our PRFSGs. For selective security, we introduce a new formula, which we call the Haar twirl approximation formula. For adaptive security, we show the unitary reprogramming lemma and the unitary resampling lemma along with the several technical tools for unitary oracle security proof with pure state queries. These have their own interest, and may have many further applications. In particular, by using the approximation formula, we give an alternative proof of the non-adaptive security of the PFC ensemble [Metger, Poremba, Sinha, and Yuen, FOCS 2024] as an additional result.
Finally, we prove that our construction is not PRUs or quantum-accessible non-adaptive PRFSGs by presenting quantum polynomial time attacks. Our attack is based on generalizing the hidden subgroup problem where the relevant function outputs quantum states.
