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Quantum Fully Homomorphic Encryption with neither Clifford Gate Decomposition nor Real Representation

Guangsheng Ma, Hongbo Li

Published Dec 8, 2020• 0 citations
OpenAlex

Abstract

We present a novel quantum fully homomorphic encryption (QFHE) scheme, which allows to perform conditional rotations with the control bit in encrypted form. In our scheme, any quantum circuit can be directly evaluated with no need to decompose into Clifford/non-Clifford gates, nor to be transformed into real representation. Our scheme is able to evaluate quantum Fourier transform algorithm faster than previous QFHE schemes in the worst case. The security of our scheme relies on the hardness of the underlying quantum capable FHE scheme, and the latter sets its security on the learning with errors problem and the circular security assumption.

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We present a novel quantum fully homomorphic encryption (QFHE) scheme, which allows to perform conditional rotations with the control bit in encrypted form. In our scheme, any quantum circuit can be directly evaluated with no need to decompose into Clifford/non-Clifford gates, nor to be transformed into real representation. Our scheme is able to evaluate quantum Fourier transform algorithm faster than previous QFHE schemes in the worst case.
The security of our scheme relies on the hardness of the underlying quantum capable FHE scheme, and the latter sets its security on the learning with errors problem and the circular security assumption.
Quantum fully homomorphic encryption (QFHE) allows to evaluate quantum circuits on encrypted data. We present a novel QFHE scheme, which extends Pauli one-time pad encryption by relying on the quaternion representation of SU(2). With the scheme, evaluating 1-qubit gates is more efficient, and evaluating general quantum circuits is polynomially improved in asymptotic complexity. Technically, a new encrypted multi-bit control technique is proposed, which allows to perform any 1-qubit gate whose parameters are given in the encrypted form. With this technique, we establish a conversion between the new encryption and previous Pauli one-time pad encryption, bridging our QFHE scheme with previous ones. Also, this technique is useful for private quantum circuit evaluation. The security of the scheme relies on the hardness of the underlying quantum capable FHE scheme, and the latter sets its security on the learning with errors problem and the circular security assumption.
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 is a recent and fundamental breakthrough in modern cryptography, which allows the performance of operations on encrypted data without unveiling the data. Leveraging quantum mechanics principles, quantum computers can potentially solve certain computational problems exponentially faster than classical computers. This immense computational power offers new possibilities for various fields, including cryptography. The rapid evolution of both these fields has led to the development of quantum fully homomorphic encryption (QFHE), which makes the capabilities of classical HE extend into the quantum domain. However, many existing QFHE schemes require significant memory due to complex calculations and fault-tolerance needs. This paper contributes in two ways. First, we provide a comprehensive survey of two specific QFHE schemes, discussing their underlying principles, mathematical frameworks, security aspects, and practical applications. We also explore the challenges posed by quantum computing and how QFHE addresses these to achieve both security and computational efficiency. Second, we propose a new hierarchical memory management system for QFHE, which includes a “quantum cache” (a specialized memory storage for quantum data) and a “reinforcement learning agent” (an intelligent system that learns from experience to optimize decisions). This system dynamically manages data movement between the cache and classical memory, improving memory efficiency and potentially boosting computational performance.
The recent discovery of fully homomorphic classical encryption schemes has had a dramatic effect on the direction of modern cryptography. Such schemes, however, implicitly rely on the assumption that solving certain computation problems is intractable. Here we present a quantum encryption scheme which is homomorphic for arbitrary classical and quantum circuits which have at most some constant number of non-Clifford gates. Unlike classical schemes, the security of the scheme we present is information theoretic and hence independent of the computational power of an adversary.
Homomorphic encryption has giant advantages in the protection of privacy information. In this paper, we present a new kind of probabilistic quantum homomorphic encryption scheme for the universal quantum circuit evaluation. Firstly, the pre-shared non-maximally entangled states are utilized as auxiliary resources, which lower the requirements of the quantum channel, to correct the errors in non-Clifford gate evaluation. By using the set synthesized by Clifford gates and T gates, it is feasible to perform the arbitrary quantum computation on the encrypted data. Secondly, our scheme is different from the previous scheme described by the quantum homomorphic encryption algorithm. From the perspective of application, a two-party probabilistic quantum homomorphic encryption scheme is proposed. It is clear what the computation and operation that the client and the server need to perform respectively, as well as the permission to access the data. Finally, the security of probabilistic quantum homomorphic encryption scheme is analyzed in detail. It demonstrates that the scheme has favorable security in three aspects, including privacy data, evaluated data and encryption and decryption keys.
Quantum fully homomorphic encryption (QFHE) is a kind of secure method which allows random evaluation computations performed by each server on the cipher states and do not decrypt them. In order to extend the QFHE scheme for three-participant, an extend QFHE scheme based on brown state is proposed in this paper. Firstly, the conception of extend means we use a state which are familiar in quantum secret sharing procedure to extend the former QFHE scheme with three-part. Secondly, universal quantum circuit (UQC) enables random quantum transformations on the cipher states which promise the security, and the key-updating algorithms are also secure because of the former UQC. Finally, due to the property of QFHE, we do not need any security check to make it efficient for the secret sharing procedure.