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arXiv:2007.12923

Multi-theorem (Malicious) Designated-Verifier NIZK for QMA

Omri Shmueli

Published Jul 25, 2020• 6 citations
arXivHTMLPDF

Abstract

We present the first non-interactive zero-knowledge argument system for QMA with multi-theorem security. Our protocol setup constitutes an additional improvement and is constructed in the malicious designated-verifier (MDV-NIZK) model (Quach, Rothblum, and Wichs, EUROCRYPT 2019), where the setup consists of a trusted part that includes only a common uniformly random string and an untrusted part of classical public and secret verification keys, which even if sampled maliciously by the verifier, the zero knowledge property still holds. The security of our protocol is established under the Learning with Errors Assumption. Our main technical contribution is showing a general transformation that compiles any sigma protocol into a reusable MDV-NIZK protocol, using NIZK for NP. Our technique is classical but works for quantum protocols and allows the construction of a reusable MDV-NIZK for QMA.

DOI
10.48550/arxiv.2007.12923

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Non-interactive quantum statistical zero-knowledge (NIQSZK), introduced by Kobayashi in 2002 via shared EPR pairs replacing a classical common reference string, is often invoked as a general tool for quantum-native authentication, but the term spans results of very different scope. We give an account of what Kobayashi’s model actually establishes – non-interactive proofs for one promise problem, Quantum State Closeness to Identity, and, by reduction, Graph Non-Automorphism – and contrast it with the independent, more recent line achieving non-interactive zero-knowledge arguments for all of QMA from quantum-hardness of Learning With Errors (LWE). Using a Software-Defined Quantum Network (SDQN) control-plane authentication case study, we compare the realistically available proof systems along a resource-cost model grounded in published entanglement-distribution benchmarks and lattice-parameter estimation, and argue classical post-quantum NIZK dominates unless the credential itself is an inherently quantum object.

Simpósio Brasileiro de Cibersegurança (SBSeg)•Sep 1, 2026•0 citations
We present the first non-interactive zero-knowledge argument system for QMA with multi-theorem security. Our protocol setup constitutes an additional improvement and is constructed in the malicious designated-verifier (MDV-NIZK) model (Quach, Rothblum, and Wichs, EUROCRYPT 2019), where the setup consists of a trusted part that includes only a common uniformly random string and an untrusted part of classical public and secret verification keys, which even if sampled maliciously by the verifier, the zero knowledge property still holds. The security of our protocol is established under the Learning with Errors Assumption. Our main technical contribution is showing a general transformation that compiles any sigma protocol into a reusable MDV-NIZK protocol, using NIZK for NP. Our technique is classical but works for quantum protocols and allows the construction of a reusable MDV-NIZK for QMA.
We initiate the study of non-interactive zero-knowledge (NIZK) arguments for languages in QMA. Our first main result is the following: if Learning With Errors (LWE) is hard for quantum computers, then any language in QMA has an NIZK argument with preprocessing. The preprocessing in our argument system consists of (i) the generation of a CRS and (ii) a single (instance-independent) quantum message from verifier to prover. The instance-dependent phase of our argument system involves only a single classical message from prover to verifier. Importantly, verification in our protocol is entirely classical, and the verifier needs not have quantum memory; its only quantum actions are in the preprocessing phase. Our second contribution is to extend the notion of a classical proof of knowledge to the quantum setting. We introduce the notions of arguments and proofs of quantum knowledge (AoQK/PoQK), and we show that our non-interactive argument system satisfies the definition of an AoQK. In particular, we explicitly construct an extractor which can recover a quantum witness from any prover which is successful in our protocol. Finally, we show that any language in QMA has an (interactive) proof of quantum knowledge.
An important open question in quantum cryptography is the construction of publicly-verifiable NIZKs for QMA. Classically, one can construct NIZKs for NP in the random oracle model (and sometimes in the standard model) by compiling an honest-verifier ZK (HVZK) ΣΣΣ-protocol for NP using the Fiat-Shamir transformation. Broadbent and Grilo introduced a quantum analog of a ΣΣΣ-protocol (which they call a ΞΞΞ-protocol) in which the prover's first message is quantum, and show that HVZK ΞΞΞ-protocols exist for QMA. However, it is not clear how to compile such protocols into NIZKs in the (Q)ROM, because the Fiat-Shamir transformation seems to be incompatible with quantum messages. In this work we give formal evidence that this is indeed the case: we show that if generic "Fiat-Shamir-like" compilers for quantum protocols exist in the QROM (with small completeness and soundness error) then QMA = BQP.
Succinct argument systems are of central importance to modern crytpography, enabling the efficient verification of computational claims. In the classical setting, Kilian (STOC 92) established that any probabilistically checkable proof for NP can be transformed into a succinct argument system for NP using only collision-resistant hash functions. In the quantum setting, recent works have established the feasibility of (classically-verifiable) succinct arguments for QMA, capturing statements that require *quantum* proofs. However, known constructions all rely on the highly structured assumption of learning with errors (LWE), which stands in stark contrast with the unstructured assumptions that suffice for NP. In this work, we develop a new framework that broadens the cryptographic foundations of succinct arguments for QMA. We assume the existence of (i) an oblivious state preparation (OSP) protocol, which in turn can be constructed from *plain* trapdoor claw-free functions, and (ii) collapsing hash functions, the quantum analogue of collision-resistance. In particular, we obtain the first succinct, classically-verifiable argument system for QMA which does not rely on the hardness of LWE. Our construction proceeds in two steps. First, we design a *round-efficient* classically-verifiable argument system for QMA based only on the assumption of OSP. Second, we introduce a *generalized communication compression compiler*, which, assuming collapsing hash functions, transforms any TTT-round interactive protocol into one in which the communication size is bounded by $T \cdot \poly(\secp)$ for some fixed $\poly$ independent of the original size of each message. Our compiler extends a quantum rigidity-based communication compression technique of Zhang (QCrypt 25), and may be of independent interest.
We propose three constructions of classically verifiable non-interactive proofs (CV-NIP) and non-interactive zero-knowledge proofs and arguments (CV-NIZK) for QMA in various preprocessing models. - We construct an information theoretically sound CV-NIP for QMA in the secret parameter model where a trusted party generates a quantum proving key and classical verification key and gives them to the corresponding parties while keeping it secret from the other party. Alternatively, we can think of the protocol as one in a model where the verifier sends an instance-independent quantum message to the prover as preprocessing. - We construct a CV-NIZK for QMA in the secret parameter model. It is information theoretically sound and zero-knowledge. - Assuming the quantum hardness of the leaning with errors problem, we construct a CV-NIZK for QMA in a model where a trusted party generates a CRS and the verifier sends an instance-independent quantum message to the prover as preprocessing. This model is the same as one considered in the recent work by Coladangelo, Vidick, and Zhang (CRYPTO '20). Our construction has the so-called dual-mode property, which means that there are two computationally indistinguishable modes of generating CRS, and we have information theoretical soundness in one mode and information theoretical zero-knowledge property in the other. This answers an open problem left by Coladangelo et al, which is to achieve either of soundness or zero-knowledge information theoretically. To the best of our knowledge, ours is the first dual-mode NIZK for QMA in any kind of model.
We study non-interactive zero-knowledge proofs (NIZKs) for NP satisfying: 1) statistical soundness, 2) computational zero-knowledge (ZK) and 3) certified-everlasting zero-knowledge (CE-ZK). The CE-ZK property allows a verifier of a quantum proof to revoke the proof in a way that can be checked (certified) by the prover. Conditioned on successful certification, the verifier's state can be efficiently simulated with only the statement, in a statistically indistinguishable way. Our contributions regarding these certified-everlasting NIZKs (CE-NIZKs) are as follows: - We identify a barrier to obtaining CE-NIZKs in the CRS model via generalizations of known interactive ZK proofs that satisfy CE-ZK. - We circumvent this by constructing CE-NIZK using non-black-box use of NIZK for NP satisfying certain properties, along with OWFs. As a result, we obtain CE-NIZKs for NP in the CRS model, based on polynomial hardness of the learning with errors (LWE) assumption. - In addition, we observe that the aforementioned barrier does not apply to the shared EPR model. We leverage this to construct CE-NIZK for NP in this model based on any statistical binding hidden bits generator, which is known from LWE. The only quantum computation here involves single-qubit measurements of the EPR pairs.
Non-interactive quantum statistical zero-knowledge (NIQSZK), introduced by Kobayashi in 2002 via shared EPR pairs replacing a classical common reference string, is often invoked as a general tool for quantum-native authentication, but the term spans results of very different scope. We give an account of what Kobayashi’s model actually establishes – non-interactive proofs for one promise problem, Quantum State Closeness to Identity, and, by reduction, Graph Non-Automorphism – and contrast it with the independent, more recent line achieving non-interactive zero-knowledge arguments for all of QMA from quantum-hardness of Learning With Errors (LWE). Using a Software-Defined Quantum Network (SDQN) control-plane authentication case study, we compare the realistically available proof systems along a resource-cost model grounded in published entanglement-distribution benchmarks and lattice-parameter estimation, and argue classical post-quantum NIZK dominates unless the credential itself is an inherently quantum object.