This paper studies the complexity classes QZK and HVQZK of problems having a quantum computational zero-knowledge proof system and an honest-verifier quantum computational zero-knowledge proof system, respectively. The results proved in this paper include: (a) HVQZK = QZK, (b) any problem in QZK has a public-coin quantum computational zero-knowledge proof system, (c) any problem in QZK has a quantum computational zero-knowledge proof system of perfect completeness, and (d) any problem in QZK has a three-message public-coin quantum computational zero-knowledge proof system of perfect completeness with arbitrarily small constant error in soundness. All the results above are unconditional and do not rely any computational assumptions. For the classes QPZK, HVQPZK, and QSZK of problems having a quantum perfect zero-knowledge proof system, an honest-verifier quantum perfect zero-knowledge proof system, and a quantum statistical zero-knowledge proof system, respectively, the following new properties are proved: (e) HVQPZK = QPZK, (f) any problem in QPZK has a public-coin quantum perfect zero-knowledge proof system, (g) any problem in QSZK has a quantum statistical zero-knowledge proof system of perfect completeness, and (h) any problem in QSZK has a three-message public-coin quantum statistical zero-knowledge proof system of perfect completeness with arbitrarily small constant error in soundness. It is stressed that our proofs are direct and do not use complete promise problems or those equivalents. This gives a unified framework that works well for all of quantum perfect, statistical, and computational zero-knowledge proofs, and enables us to prove properties even on the computational and perfect zero-knowledge proofs for which no complete promise problems are known.
Telecom Paris, Institut Polytechnique de Paris（巴黎理工学院电信学院）；Beijing University of Posts and Telecommunications（北京邮电大学）；Beijing Institute of Technology（北京理工大学）
We define the notion of a proof of knowledge in the setting where the verifier is classical, but the prover is quantum, and where the witness that the prover holds is in general a quantum state. We establish simple properties of our definition, including that, if a nondestructive classical proof of quantum knowledge exists for some state, then that state can be cloned by an unbounded adversary, and that, under certain conditions on the parameters in our definition, a proof of knowledge protocol for a hard-to-clone state can be used as a (destructive) quantum money verification protocol. In add…
We study the notion of zero-knowledge secure against quantum polynomial-time verifiers (referred to as quantum zero-knowledge) in the concurrent composition setting. Despite being extensively studied in the classical setting, concurrent composition in the quantum setting has hardly been studied. We initiate a formal study of concurrent quantum zero-knowledge. Our results are as follows: -Bounded Concurrent QZK for NP and QMA: Assuming post-quantum one-way functions, there exists a quantum zero-knowledge proof system for NP in the bounded concurrent setting. In this setting, we fix a priori the…
Recently Chen and Movassagh proposed the quantum Merkle tree, which is a quantum analogue of the well-known classical Merkle tree. It gives a succinct verification protocol for quantum state commitment. Although they only proved security against semi-honest provers, they conjectured its general security. Using the proposed quantum Merkle tree, they gave a quantum analogue of Kilian's succinct argument for NP, which is based on probabilistically checkable proofs (PCPs). A nice feature of Kilian's argument is that it can be extended to a zero-knowledge succinct argument for NP, if the underlying…
We investigate the existence of constant-round post-quantum black-box zero-knowledge protocols for $\mathbf{NP}$. As a main result, we show that there is no constant-round post-quantum black-box zero-knowledge argument for $\mathbf{NP}$ unless $\mathbf{NP}\subseteq \mathbf{BQP}$. As constant-round black-box zero-knowledge arguments for $\mathbf{NP}$ exist in the classical setting, our main result points out a fundamental difference between post-quantum and classical zero-knowledge protocols. Combining previous results, we conclude that unless $\mathbf{NP}\subseteq \mathbf{BQP}$, constant-round…
In known constructions of classical zero-knowledge protocols for NP, either of zero-knowledge or soundness holds only against computationally bounded adversaries. Indeed, achieving both statistical zero-knowledge and statistical soundness at the same time with classical verifier is impossible for NP unless the polynomial-time hierarchy collapses, and it is also believed to be impossible even with a quantum verifier. In this work, we introduce a novel compromise, which we call the certified everlasting zero-knowledge proof for QMA. It is a computational zero-knowledge proof for QMA, but the ver…
We introduce Prove-It, a Python-based general-purpose interactive theorem-proving assistant designed with the goal of making formal theorem proving as easy and natural as informal theorem proving (with moderate training). Prove-It uses a highly-flexible Jupyter notebook-based user interface that documents interactions and proof steps using LaTeX. We review Prove-It's highly expressive representation of expressions, judgments, theorems, and proofs; demonstrate the system by constructing a traditional proof-by-contradiction that $\sqrt{2}\notin\mathbb{Q}$; and discuss how the system avoids incon…
Protecting secrets is a key challenge in our contemporary information-based era. In common situations, however, revealing secrets appears unavoidable, for instance, when identifying oneself in a bank to retrieve money. In turn, this may have highly undesirable consequences in the unlikely, yet not unrealistic, case where the bank's security gets compromised. This naturally raises the question of whether disclosing secrets is fundamentally necessary for identifying oneself, or more generally for proving a statement to be correct. Developments in computer science provide an elegant solution via …
We provide several advances to the understanding of the class of Quantum Merlin-Arthur proof systems (QMA), the quantum analogue of NP. Our central contribution is proving a longstanding conjecture that the Consistency of Local Density Matrices (CLDM) problem is QMA-hard under Karp reductions. The input of CLDM consists of local reduced density matrices on sets of at most k qubits, and the problem asks if there is an n-qubit global quantum state that is consistent with all of the k-qubit local density matrices. The containment of this problem in QMA and the QMA-hardness under Turing reductions…
A major difficulty in quantum rewinding is the fact that measurement is destructive: extracting information from a quantum state irreversibly changes it. This is especially problematic in the context of zero-knowledge simulation, where preserving the adversary's state is essential. In this work, we develop new techniques for quantum rewinding in the context of extraction and zero-knowledge simulation: (1) We show how to extract information from a quantum adversary by rewinding it without disturbing its internal state. We use this technique to prove that important interactive protocols, such as…
