Quantumly Accessible Pseudorandom Oracle Model

• The paper introduces a quantumly accessible pseudorandom oracle model that generalizes the classical random oracle paradigm to support quantum superposition queries.
• It details formal security definitions, novel constructions, and simulation techniques ensuring cryptographic primitives resist quantum adversaries.
• The work explores applications in quantum cryptography, including protocol design, obfuscation, and separation results, highlighting both practical implementations and theoretical limits.

The quantumly accessible pseudorandom oracle model is a foundational framework in quantum cryptography, generalizing the classical random oracle paradigm to support adversarial and honest parties making quantum superposition queries. This article presents technical details, key constructions, security definitions, simulation techniques, separation results, and cryptographic consequences as established in recent research.

1. Formal Definition and Motivation

The classical random oracle model (ROM) is defined by universal access to a random function $H:\{0,1\}^n \to \{0,1\}^m$, with only classical queries permitted. The quantumly accessible random oracle model (QROM) extends this by allowing adversaries and honest parties to query $H$ with quantum superpositions. Formally, the black-box operation exposed is the unitary: $$U_H: |x\rangle\,|y\rangle \mapsto |x\rangle\,|y \oplus H(x)\rangle.$$ This extension is necessary to capture quantum attacks such as Grover's search, and more generally, adversaries capable of quantum information processing (Katz et al., 2024). The model is generalized further to quantum-accessible pseudorandom oracles (QAPO/QAPOM), where the random oracle is instantiated by a keyed function (typically a pseudorandom function, PRF) that remains indistinguishable from random even under quantum superposition queries (Huang et al., 13 Jan 2026, Don et al., 2022).

2. Security Definitions: PRGs, PRFs, PRSs, PRUs in Quantum Oracle Models

Let $G^H$ be a construction that (possibly itself) makes (quantum or classical) queries to an oracle $H$. Security in the quantumly accessible model is typically defined by requiring that no quantum polynomial-time (QPT) adversary making polynomially many superposition queries to the oracle can distinguish the output of the construction from corresponding random objects.

Primitive	Oracle Model	Security Notion (quantum)
PRG ($G^H$)	QROM/QAPOM	$$\mathrm{Adv}^{\mathrm{PRG}}_{A,G} := |\Pr[A^H(G^H(s))=1] - \Pr[A^H(g)=1]|$$ negligible (Katz et al., 2024)
PRF ($F^H$)	QAPOM/QHROM	$F_k$ indistinguishable from random via QPT superposition queries (Don et al., 2022)
PRS ($V_f$)	QAPOM/QHROM	For all QPT $A, t$: $$|\Pr_k[A(\ket{\phi_k}^{\otimes t})=1]-\Pr_\psi[A(\psi^{\otimes t})=1]|$$ negligible (Batra et al., 30 Jul 2025)
PRU ($G_k$)	QHROM/iQHROM	For all QPT $A$: $$|\Pr_k[A^{G_k,G_k^\dag}=1]-\mathbb{E}_U[A^{U,U^\dag}=1]|$$ negligible (Ananth et al., 29 Sep 2025, Ananth et al., 2024)

These definitions admit variations such as classical-access-only queries, adaptive queries, or restrictions on the number/type of oracle accesses.

3. Quantum Lifting Theorem and Simulation Techniques

The quantum lifting theorem provides a central result: unconditional classical-ROM security for PRGs lifts automatically to quantum security in the QROM, even for adversaries making quantum queries. More formally, Katz and Sela prove that for every quantum adversary $A$, there exists a classical adversary $B$ making only polynomially more queries with at least half the distinguishing advantage of $A$:

$$\mathrm{Adv}^{\mathrm{PRG}}_{B,G} \geq \frac{1}{2}\,\mathrm{Adv}^{\mathrm{PRG}}_{A,G} - \mathrm{negl}(n)$$

This applies to deterministic classical-query constructions and pseudo-deterministic quantum-query constructions (Katz et al., 2024). The proof involves identifying "useful" points (high query magnitude), hybrid reprogramming via the Swapping and Reprogramming Lemmas, and explicit simulation algorithms for pseudo-deterministic quantum-oracle routines, reducing query access to poly-bounded classical interaction.

4. Explicit Constructions and Quantum Oracle Instantiations

Several models instantiate the quantumly accessible pseudorandom oracle:

• QAPO (Quantumly Accessible Pseudorandom Oracle): Keyed families $\{O_k\}$ generated from QAP-secure PRFs and ideal-model obfuscation, allowing for superposition and inverse queries. Obfuscation of arbitrary quantum circuits is possible under this instantiation (Huang et al., 13 Jan 2026).
• QHROM (Quantum Haar Random Oracle Model): Oracle is a single Haar random unitary; parties obtain forward/inverse access. Strong PRUs (requiring two sequential calls) and unbounded-query secure multi-copy PRSGs are constructed; path-recording isometry techniques provide simulation and security proofs (Ananth et al., 29 Sep 2025, Ananth et al., 2024).
• CHRS (Common Haar Random State): Parties receive copies of a single Haar random state via an oracle, useful for exploring separations between 1-copy PRS and multi-copy PRS (Chen et al., 2024).

Quantum security reductions routinely depend on such instantiations: for example, scalable PRS and adaptive PRFS generators are built using isometric procedures evaluated on (superposition) quantum oracle access to QAP-secure PRFs (Batra et al., 30 Jul 2025).

5. Separation Results, Limitations, and Oracle Lower Bounds

Recent works establish sharp separation results between primitives in quantumly accessible models:

• PRG vs. PRS/PRFS: It is impossible to construct a PRG from PRS (both linear- and logarithmic-sized) via black-box quantum reductions, given access to a unitary quantum oracle with inverse access. Thus, logarithmic-PRS, though cryptographically powerful, do not suffice for building PRGs, digital signatures, or quantum public-key encryption via black-box arguments (Barhoush, 23 Oct 2025).
• Query Complexity Limits: In the iQHROM, single-query constructions cannot yield unbounded-query secure PRUs; two queries are minimally required. This is established via path-recording isometry arguments and swap-test adversaries (Ananth et al., 2024).
• Statistical vs. Computational Oracle Security: In models such as CHRS and QHROM, unconditional statistical security is achievable for certain "weaker" primitives (e.g., 1PRS), while full PRS requires stronger assumptions or additional oracle structure (Chen et al., 2024).

These separation and security bounds are crucial for characterizing the cryptographic landscape in quantum settings.

6. Applications and Extensions in Quantum Cryptography

The quantumly accessible pseudorandom oracle model is now a canonical tool for:

• Designing quantum-secure protocols: Including commitments, signatures, encryption, and obfuscation (Huang et al., 13 Jan 2026, Batra et al., 30 Jul 2025).
• Proving the security of classical constructions: Katz and Sela's lifting theorem demonstrates that classical random-oracle-based PRGs are secure against quantum adversaries, reducing the burden for post-quantum security proofs (Katz et al., 2024).
• Providing candidate quantum obfuscators: By combining QAPO-secure primitives, unitary obfuscators, and subspace-preserving PRUs, it is possible to achieve ideal-model obfuscation for arbitrary quantum circuits (Huang et al., 13 Jan 2026).
• Exploring pseudorandom quantum states and unitaries: The frameworks allow scalable PRS/PRFS constructions, efficient simulations, and separation of cryptographic tasks at a quantum level (Batra et al., 30 Jul 2025, Ananth et al., 29 Sep 2025).

A schematic summary:

Model	Key Constructions	Security Regime	Applications
QROM/QAPOM	PRG, split-key PRF, iO, PRS	info-theoretic/comp.	KEM combiner, proof lifting
QHROM/iQHROM	PRU (2-query), PRSG/PRFS (1-query)	unbounded-query/statistical	quantum money, zero-knowledge, separation proofs
CHRS	1PRS, bit-commitment	statistical	separations in quantum pseudorandomness

7. Open Problems and Future Directions

Major research questions include extension of the lifting theorem to full-fledged pseudorandom functions (PRFs) under quantum access, developing efficient unitary lazy-sampling for arbitrary output sizes, and identifying further separation phenomena among quantum primitives. Notably, composability of quantumly accessible oracles, practical instantiations balancing key size and output length, and extending quantum security proofs to broader classes of cryptographic schemes remain active areas of inquiry (Katz et al., 2024, Barhoush, 23 Oct 2025, Hhan et al., 2024).

In summary, the quantumly accessible pseudorandom oracle model provides a rigorous, versatile framework for analyzing quantum security, constructing quantum-native cryptographic primitives, and resolving core complexity-theoretic and cryptographic separation questions. Its continued development is central to the evolution of quantum cryptography.