Quantum Proofs of Deletion for Learning with Errors
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Quantum information has the property that measurement is an inherently destructive process. This feature is most apparent in the principle of complementarity, which states that mutually incompatible observables cannot be measured at the same time. Recent work by Broadbent and Islam (TCC 2020) builds on this aspect of quantum mechanics to realize a cryptographic notion called certified deletion. While this remarkable notion enables a classical verifier to be convinced that a (private-key) quantum ciphertext has been deleted by an untrusted party, it offers no additional layer of functionality. In this work, we augment the proof-of-deletion paradigm with fully homomorphic encryption (FHE). This results in a new and powerful cryptographic notion called fully homomorphic encryption with certified deletion – an interactive protocol which enables an untrusted quantum server to compute on encrypted data and, if requested, to simultaneously prove data deletion to a client. Our main technical ingredient is an interactive protocol by which a quantum prover can convince a classical verifier that a sample from the Learning with Errors (LWE) distribution in the form of a quantum state was deleted. We introduce an encoding based on Gaussian coset states which is highly generic and suggests that essentially any LWE-based cryptographic primitive admits a classically-verifiable quantum proof of deletion. As an application of our protocol, we construct a Dual-Regev public-key encryption scheme with certified deletion, which we then extend towards a (leveled) FHE scheme of the same type. Our construction achieves indistinguishable ciphertexts in the semi-honest adversarial model, even if the secret key is later revealed after deletion has taken place.
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