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Symmetric quantum fully homomorphic encryption with perfect security

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Abstract

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 \(n\)-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.

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Authors and Affiliations

  1. Data Communication Science and Technology Research Institute, Beijing, 100191, China

    Min Liang

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  1. Min Liang
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Correspondence to Min Liang.

Appendix

Appendix

This section lists some commutation rules about the operators \(X,Y,Z,H, CNOT, R_z(\theta ),R_y(\theta )\). Here, \(j,k,l,m\) are single-bit numbers.

$$\begin{aligned}&Z^kX^j=(-1)^{j\cdot k}X^jZ^k, \\&R_z(\theta )X^j=X^jR_z((-1)^j\theta ),\forall \theta \in [0,2\pi ),\\&R_y(\theta )X^j=X^jR_y((-1)^j\theta ),\forall \theta \in [0,2\pi ),\\&R_y(\theta )Z^k=Z^kR_y((-1)^k\theta ),\forall \theta \in [0,2\pi ),\\&R_y(\theta )H^j=H^jR_y((-1)^j\theta ),\forall \theta \in [0,2\pi ),\\&CNOT(X^j \otimes I)=(X^j \otimes X^j)CNOT, \\&CNOT(Z^k \otimes I)=(Z^k \otimes I)CNOT, \\&CNOT(I \otimes X^l)=(I \otimes X^l)CNOT, \\&CNOT(I \otimes Z^m)=(Z^m \otimes Z^m)CNOT. \end{aligned}$$

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Liang, M. Symmetric quantum fully homomorphic encryption with perfect security. Quantum Inf Process 12, 3675–3687 (2013). https://doi.org/10.1007/s11128-013-0626-5

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