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.
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)
Σ-protocol for NP using the Fiat-Shamir transformation. Broadbent and Grilo introduced a quantum analog of a
Σ-protocol (which they call a
Ξ-protocol) in which the prover's first message is quantum, and show that HVZK
Ξ-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.
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.
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.