Rate-1 Quantum Fully Homomorphic Encryption

Secure function evaluation (SFE) allows Alice to publish an encrypted version of her input m such that Bob (holding a circuit C) can send a single message that reveals C(m) to Alice, and nothing more. Security is required to hold against malicious parties, that may behave arbitrarily. In this work we study the notion of SFE in the quantum setting, where Alice outputs an encrypted quantum state ∣∣????⟩ | ψ ⟩ and learns ????(∣∣????⟩) C ( | ψ ⟩ ) after receiving Bob’s message. We show that, assuming the quantum hardness of the learning with errors problem (LWE), there exists an SFE protocol for quantum computation with communication complexity (|∣∣????⟩|+|????(∣∣????⟩)|)⋅(1+????(1)) which is nearly optimal. This result is obtained by two main technical steps, which might be of independent interest. Specifically, we show (i) a construction of a rate-1 quantum fully-homomorphic encryption and (ii) a generic transformation to achieve malicious circuit privacy in the quantum setting.