On the Concurrent Composition of Quantum Zero-Knowledge

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 number of verifiers that can simultaneously interact with the prover. Under the same assumption, we also show that there exists a quantum zero-knowledge proof system for QMA in the bounded concurrency setting. -Quantum Proofs of Knowledge: Assuming statistical receiver-private post-quantum oblivious transfer, there exists a bounded concurrent zero-knowledge proof system for NP satisfying quantum proof of knowledge property. We then instantiate the oblivious transfer protocol based on a new assumption, called learning with errors with cloning security. Our extraction mechanism simultaneously allows for extraction probability to be negligibly close to acceptance probability (extractability) and also ensures that the prover's state after extraction is statistically close to the prover's state after interacting with the verifier (simulatability). The seminal work of [Unruh EUROCRYPT'12], and all its followups, satisfied a weaker version of extractability property and moreover, did not achieve simulatability. Our result yields a proof of quantum knowledge system for QMA with better parameters than prior works.