Paper 2024/1847

Notions of Quantum Reductions and Impossibility of Statistical NIZK

Abstract

Non-Interactive Zero-Knowledge Arguments (NIZKs) are cryptographic protocols that enable a prover to demonstrate the validity of an $\mathsf{NP}$ statement to a verifier with a single message, without revealing any additional information. The soundness and zero-knowledge properties of a NIZK correspond to security against a malicious prover and a malicious verifier respectively. Statistical NIZKs (S-NIZKs) are a variant of NIZKs for which the zero-knowledge property is guaranteed to hold information-theoretically. Previous works have shown that S-NIZKs satisfying a weak version of soundness known as static soundness exist based on standard assumptions. However, the work of Pass (TCC 2013) showed that S-NIZKs with the stronger \emph{adaptive} soundness property are inherently challenging to obtain. The work proved that standard (black-box) proof techniques are insufficient to prove the security of an S-NIZK based on any standard (falsifiable) assumption. We extend this result to the setting where parties can perform quantum computations and communicate using quantum information, while the quantum security reduction is restricted to query the adversary classically. To this end, we adapt the well-known meta-reduction paradigm for showing impossibility results to the quantum setting. Additionally, we reinterpret our result using a new framework for studying quantum reductions, which we believe to be of independent interest.