Secure Quantum Extraction Protocols

Knowledge extraction, typically studied in the classical setting, is at the heart of several cryptographic protocols. We introduce the notion of secure quantum extraction protocols. A secure quantum extraction protocol for an NP relation $\mathcal{R}$ is a classical interactive protocol between a sender and a receiver, where the sender gets the instance $z$ and a witness $w$, while the receiver only gets the instance $z$. For any efficient quantum adversarial sender (who follows the protocol but can choose its own randomness), there exists a quantum extractor that can extract a witness $w'$ such that $(z,w') \in \mathcal{R}$ while a malicious receiver should not be able to output any valid witness. We study and construct two types of secure quantum extraction protocols. (1) Quantum extraction protocols secure against quantum malicious receivers based on quantum fully homomorphic encryption satisfying some mild properties and quantum hardness of learning with errors. In this construction, we introduce a non black box technique in the quantum setting. All previous extraction techniques in the quantum setting were solely based on quantum rewinding. (2) Quantum extraction protocols secure against classical malicious receivers based on quantum hardness of learning with errors. As an application, based on the quantum hardness of learning with errors, we present a construction of constant round quantum zero-knowledge argument systems for NP that guarantee security even against quantum malicious verifiers; however, our soundness only holds against classical probabilistic polynomial time adversaries. Prior to our work, such protocols were known based, additionally, on the assumptions of decisional Diffie-Hellman (or other cryptographic assumptions that do not hold against polynomial time quantum algorithms).