Teleportation-based quantum homomorphic encryption scheme with quasi-compactness and perfect security

Abstract

Quantum homomorphic encryption (QHE) is an important cryptographic technology for delegated quantum computation. It enables remote Server performing quantum computation on encrypted quantum data, and the specific algorithm performed by Server is unnecessarily known by Client. Quantum fully homomorphic encryption (QFHE) is a QHE that satisfies both compactness and F-homomorphism, which is homomorphic for any quantum circuits. However, Yu et al.[Phys. Rev. A 90, 050303(2014)] proved a negative result: assume interaction is not allowed, it is impossible to construct perfectly secure QFHE scheme. So this article focuses on non-interactive and perfectly secure QHE scheme with loosen requirement, specifically quasi-compactness. This article defines encrypted gate, which is denoted by EG[U ] : |α → ((a, b), Enc a,b (U |α )). We present a gate-teleportation-based two-party computation scheme for EG[U ], where one party gives arbitrary quantum state |α as input and obtains the encrypted U -computing result Enc a,b (U |α ), and the other party obtains the random bits a, b. Based on EG[P x ](x ∈ {0, 1}), we propose a method to remove the P -error generated in the homomorphic evaluation of T /T † -gate. Using this method, we design two non-interactive and perfectly secure QHE schemes named GT and VGT. Both of them are F-homomorphic and quasi-compact (the decryption complexity depends on the T /T † -gate complexity). Assume F-homomorphism, non-interaction and perfect security are necessary property, the quasi-compactness is proved to be bounded by O(M ), where M is the total number of T /T † -gates in the evaluated circuit. VGT is proved to be optimal and has M -quasi-compactness. According to our QHE schemes, the decryption would be inefficient if the evaluated circuit contains exponential number of T /T † -gates. Thus our schemes are suitable for homomorphic evaluation of any quantum circuit with low T /T † -gate complexity, such as any polynomial-size quantum circuit or any quantum circuit with polynomial number of T /T † -gates.

Quantum Ciphertext Dimension Reduction Scheme for Homomorphic Encrypted Data

“…It can be seen that it is simply derived from the unit matrix 𝐼 and the pauli Y matrix. Therefore, it also meets the correctness requirements of the GT scheme [25]. For arbitrary n-qubit quantum circuits 𝐶 and n-qubit qubit data 𝜌.…”

“…1.This scheme has been proved to be correct [24]. The scheme points out that if the logic gates in the quantum circuit in the evaluation function in the encryption process are all composed of 𝑆 = 𝑋, 𝑍, 𝐻, 𝑆, 𝑃, 𝐶𝑁𝑂𝑇 ,𝑇 ,𝑇 †…”

“…According to this definition, 'U-rotate Bell basis'. Liang [24] proposed a quantum homomorphic encryption scheme based on teleportation.…”

“…This scheme can be used to evaluate the homomorphism of general circuits. the 𝑆 𝑎 rotation measurement can be postponed until other quantum circuits are completed, and the measurement base Φ (𝑆 𝑎 ) only depends on the secret key 𝑎.The specific update steps of the T gate of the update algorithm can be found in [3,24].…”

“…• In this paper, this paper proposes a quantum principal component extraction algorithm (QPCE) for quantum cloud computing, The construction of the quantum circuit is completed, and a simple example is calculated on the IBMQ computing platform to verify the correctness of the proposed algorithm. • Based on the scheme [25], a dimensionality reduction scheme based on quantum homomorphic encryption (QHEDR) is proposed for the first time by combining the GT-QHE scheme with our quantum dimensionality reduction algorithm in quantum cloud computing. This scheme does not require interaction to ensure security.…”