Quantum Merkle Trees (2112.14317v4)

Abstract: Committing to information is a central task in cryptography, where a party (typically called a prover) stores a piece of information (e.g., a bit string) with the promise of not changing it. This information can be accessed by another party (typically called the verifier), who can later learn the information and verify that it was not meddled with. Merkle trees are a well-known construction for doing so in a succinct manner, in which the verifier can learn any part of the information by receiving a short proof from the honest prover. Despite its significance in classical cryptography, there was no quantum analog of the Merkle tree. A direct generalization using the Quantum Random Oracle Model (QROM) does not seem to be secure. In this work, we propose the quantum Merkle tree. It is based on what we call the Quantum Haar Random Oracle Model (QHROM). In QHROM, both the prover and the verifier have access to a Haar random quantum oracle $G$ and its inverse. Using the quantum Merkle tree, we propose a succinct quantum argument for the Gap-$k$-Local-Hamiltonian problem. Assuming the Quantum PCP conjecture is true, this succinct argument extends to all of QMA. This work raises a number of interesting open research problems.

• The paper introduces the Quantum Haar Random Oracle Model (QHROM) as a novel foundation for developing secure quantum commitment schemes.
• The paper presents a construction of Quantum Merkle Trees that efficiently verifies individual qubits using Haar random oracles.
• The paper applies the QMT framework to design a succinct quantum argument protocol for the Gap-k-Local-Hamiltonian problem, extending classical methods into quantum settings.

Quantum Merkle Trees: An Exploration of Cryptographic Innovation in Quantum Settings

This paper presents an innovative extension of the classical Merkle tree—of central importance in cryptography—into the quantum domain via the introduction of the Quantum Merkle Tree (QMT). The authors tackle the absence of a quantum analog for Merkle trees by proposing a model based on the Quantum Haar Random Oracle Model (QHROM), seeking to establish commitment schemes suitable for quantum states.

Key Contributions

1. Quantum Haar Random Oracle Model (QHROM): The authors introduce QHROM as a foundation for their construction. Unlike the classical Random Oracle Model (ROM) or the Quantum Random Oracle Model (QROM), QHROM utilizes a Haar random quantum oracle, offering a potentially higher level of security against quantum adversaries by leveraging the inherent randomness of the Haar measure on unitary matrices.
2. Quantum Merkle Tree Construction: The paper details a strategy to construct QMTs by instantiating quantum states and applying Haar random oracles. This construction allows for commitment schemes on quantum data, a significant divergence from classical bit string commitments, and encompasses a novel approach where both the prover and verifier can verify individual qubits efficiently.
3. Application to Quantum Arguments: Utilizing the QMT, the authors propose a succinct quantum argument protocol for the Gap-$k$-Local-Hamiltonian problem, with conjectural extensions to all of QMA, assuming the Quantum PCP Conjecture holds. This extends classical Kilian's succinct argument methods to the quantum setting, potentially impacting quantum verification processes.

Theoretical and Practical Implications

• Security Analysis: A theoretical security analysis posits that the construction should be resilient against semi-honest adversaries within QHROM. However, security against fully malicious adversaries remains conjectural, highlighting the need for further heuristics and advances in quantum-cryptographic tools akin to the compressed oracle technique for QROM.
• Efficiency and Succinctness: The protocol's communication complexity is intricately tied to the parameters of block size and the depth of the commitment tree, achieving efficient verification through minimal communication overhead. This efficiency makes it viable for practical quantum verification of computational tasks, with applications potentially extending to protocols requiring limited interaction.

Future Directions and Open Problems

The paper highlights several open problems:

• Developing Compressed Oracle Techniques for QHROM: Similar to recent advances in analyzing classical protocols against quantum adversaries, a comprehensive framework for understanding the interactions within QHROM is necessary to establish rigorous security proofs.
• Exploration of Non-interactive Succinct Arguments: Nevertheless, the landscape for Zero-Knowledge or Non-Interactive arguments in quantum settings remains underexplored, posing significant challenges yet remarkable opportunities for research exploration.
• Species and Security Models: Given the profound implications for cryptographic systems and quantum computation models, understanding the utility and limits of QMTs under various quantum adversarial models requires deeper exploration.

In summary, "Quantum Merkle Trees" represents a substantial step forward in bringing classical cryptographic tools into quantum applications. While conjectural in its security assurances in certain contexts, the proposed protocols underscore an exciting frontier where classical cryptographic principles intersect with quantum computation, promising innovations in secure quantum communication, cryptographic commitments, and quantum computational problem verification.