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arXiv:1406.2456 · Physical Review A

Limitations on information-theoretically-secure quantum homomorphic encryption

Li Zhi Yu, Carlos A. Pérez-Delgado, Joseph F. Fitzsimons

Published Nov 10, 2014• 75 citations
arXivHTMLPDF

Abstract

Homomorphic encryption is a form of encryption which allows computation to be carried out on the encrypted data without the need for decryption. The success of quantum approaches to related tasks in a delegated computation setting has raised the question of whether quantum mechanics may be used to achieve information-theoretically-secure fully homomorphic encryption. Here we show, via an information localization argument, that deterministic fully homomorphic encryption necessarily incurs exponential overhead if perfect security is required.

DOI
10.1103/physreva.90.050303

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Homomorphic encryption is a form of encryption which allows computation to be carried out on the encrypted data without the need for decryption. The success of quantum approaches to related tasks in a delegated computation setting has raised the question of whether quantum mechanics may be used to achieve information-theoretically-secure fully homomorphic encryption. Here we show, via an information localization argument, that deterministic fully homomorphic encryption necessarily incurs exponential overhead if perfect security is required.
Homomorphic encryption is an encryption scheme that allows computations to be evaluated on encrypted inputs without knowledge of their raw messages. Recently Ouyang et al. constructed a quantum homomorphic encryption (QHE) scheme for Clifford circuits with statistical security (or information-theoretic security (IT-security)). It is desired to see whether an information-theoretically-secure (ITS) quantum FHE exists. If not, what other nontrivial class of quantum circuits can be homomorphically evaluated with IT-security? We provide a limitation for the first question that an ITS quantum FHE necessarily incurs exponential overhead. As for the second one, we propose a QHE scheme for the instantaneous quantum polynomial-time (IQP) circuits. Our QHE scheme for IQP circuits follows from the one-time pad.
Suppose some data have been encrypted, can you compute with the data without decrypting them? This problem has been studied as homomorphic encryption and blind computing. We consider this problem in the context of quantum information processing, and present the definitions of quantum homomorphic encryption (QHE) and quantum fully homomorphic encryption (QFHE). Then, based on quantum one-time pad (QOTP), we construct a symmetric QFHE scheme, where the evaluate algorithm depends on the secret key. This scheme permits any unitary transformation on any nnn-qubit state that has been encrypted. Compared with classical homomorphic encryption, the QFHE scheme has perfect security. Finally, we also construct a QOTP-based symmetric QHE scheme, where the evaluate algorithm is independent of the secret key.
Homomorphic encryption enables computation on encrypted data while maintaining secrecy. This leads to an important open question whether quantum computation can be delegated and verified in a non-interactive manner or not. In this paper, we affirmatively answer this question by constructing the quantum homomorphic encryption scheme with quantum obfuscation. It takes advantage of the interchangeability of the unitary operator, and exchanges the evaluation operator and the encryption operator by means of equivalent multiplication to complete homomorphic encryption. The correctness of the proposed scheme is proved theoretically. The evaluator does not know the decryption key and does not require a regular interaction with a user. Because of key transmission after quantum obfuscation, the encrypting party and the decrypting party can be different users. The output state has the property of complete mixture, which guarantees the scheme security. Moreover, the security level of the quantum homomorphic encryption scheme depends on quantum obfuscation and encryption operators.
Homomorphic encryption is a cryptographic primitive that enables information processing on encrypted data. Such primitives are useful in a delegated computing setting in which a client delegates processing to a server, but does not trust the server fully with her information. Here, I discuss a way of performing quantum homomorphic encryption. A homomorphic encryption protocol comprises of four algorithms: random key generation, encryption, evaluation, and decryption. Our technique requires that a pair of commuting operators be used to implement the encryption and evaluation. The client randomly selects the former, while the server implements the latter. Then post-evaluation, if the client decrypts using the inverse of the encryption, he retrieves the input state with the desired computation implemented. The code space has to be chosen carefully so that non-trivial evaluations can be performed on the logical qubits. Since the encryption key is not known to the server, the randomness introduced by key generation hides some information about the encoded data despite it being put into the hands of the server. This framework is powerful because it gives a general description for implementing quantum homomorphic encryption. For example, families of commuting groups of operators are known to exist via the Schur-Weyl duality. However, it is limited by a no-go theorem that states that an exponential overhead is needed if arbitrary computation and perfect security are desired. Nonetheless, our protocols are still interesting as an application for near-term quantum computers, especially when the computational tasks require only low-depth circuits.
In this brief note, we review and extend existing limitations on information-theoretically (IT) secure quantum fully homomorphic encryption (QFHE). The essential ingredient remains Nayak's bound, which provides a tradeoff between the number of homomorphically implementable functions of an IT-secure QHE scheme and its efficiency. Importantly, the bound is robust to imperfect IT-security guarantees. We summarize these bounds in the context of existing QHE schemes, and discuss subtleties of the imposed restrictions.
Abstract Quantum computers promise not only to outperform classical machines for certain important tasks, but also to preserve privacy of computation. For example, the blind quantum computing protocol enables secure delegated quantum computation, where a client can protect the privacy of their data and algorithms from a quantum server assigned to run the computation. However, this security comes with the practical limitation that the client and server must communicate after each step of computation. A practical alternative is homomorphic encryption, which does not require any interactions, while providing quantum-enhanced data security for a variety of computations. In this scenario, the server specifies the computation to be performed, and the client provides only the input data, thus enabling secure noninteractive computation. Here, we demonstrate homomorphic-encrypted quantum computing with unitary transformations of individual qubits, as well as multi-qubit quantum walk computations using single-photon states and non-birefringent integrated optics. The client encrypts their input in the photons’ polarization state, while the server performs the computation using the path degree of freedom. Our demonstration using integrated quantum photonics underlines the applicability of homomorphic-encrypted quantum computations, and shows the potential for delegated quantum computing using photons.