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arXiv:2403.04922 · Physical Review Research

Implementing the Grover algorithm in homomorphic encryption schemes

Pablo Fernández, Miguel Angel Martin-Delgado

Published Nov 6, 2024• 13 citations
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Abstract

We apply quantum homomorphic encryption (QHE) schemes suitable for circuits with a polynomial number of T + T † gates to Grover's algorithm, performing a simulation in Qiskit of a Grover circuit that contains three qubits. The T + T † -gate complexity of Grover's algorithm is also analyzed in order to show that any Grover circuit can be evaluated homomorphically in an efficient manner. We discuss how to apply these QHE schemes to allow for the efficient homomorphic evaluation of any Grover circuit composed of n qubits using n − 2 extra ancilla qubits. We also show how the homomorphic evaluation of the special case where there is only one marked item can be implemented using an algorithm that makes the decryption process more efficient compared with the standard Grover algorithm. Published by the American Physical Society 2024

DOI
10.1103/physrevresearch.6.043109

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We apply quantum homomorphic encryption (QHE) schemes suitable for circuits with a polynomial number of T + T † gates to Grover's algorithm, performing a simulation in Qiskit of a Grover circuit that contains three qubits. The T + T † -gate complexity of Grover's algorithm is also analyzed in order to show that any Grover circuit can be evaluated homomorphically in an efficient manner. We discuss how to apply these QHE schemes to allow for the efficient homomorphic evaluation of any Grover circuit composed of n qubits using n − 2 extra ancilla qubits. We also show how the homomorphic evaluation of the special case where there is only one marked item can be implemented using an algorithm that makes the decryption process more efficient compared with the standard Grover algorithm. Published by the American Physical Society 2024
Growing interest in the field of quantum computing is fueled by quantum computers projected ”quantum supremacy” in speed and security. The potential for ultra-high speeds may produce a dramatic change in data science, machine learning, analytics, and information processing. This research study will focus on encryption algorithms where quantum computing may affect protocols and deciphering codes. Specifically, homomorphic encryption (HE) enables mathematical operations to be performed on encrypted data without having to decrypt the data in the process. Quantum homomorphic encryption (QHE) enables quantum circuits to be performed on encrypted qubits. In this research experience for undergraduates (REU) study, we design quantum circuits to implement QHE on a quantum teleportation circuit. The teleportation algorithm is profiled in terms of performance and complexity and comparative results are provided for encoded versus unencoded circuits. This work serves as a building block for encrypting more complex quantum algorithms such as Quantum Neural Networks (QNN).
Abstract We introduce a class of circuits that solve a particular case of the Bernstein–Vazirani recursive problem for second-level recursion. This class of circuits allows for the implementation of the oracle using a number of T-gates that grows linearly with the number of qubits in the problem. We find an application of this scheme to quantum homomorphic encryption (QHE), which is an important cryptographic technology useful for delegated quantum computing, allowing a remote server to perform quantum computations on encrypted quantum data, so that the server cannot know anything about the client’s data. Liang’s QHE schemes are suitable for circuits with a polynomial number of gates T / T † . Thus, the simplified circuits we have constructed can be evaluated homomorphically in an efficient manner.
We present a new scheme for quantum homomorphic encryption which is compact and allows for efficient evaluation of arbitrary quantum circuits, as long as those circuits are of size, polynomial in the security parameter. Building on the framework of Broadbent and Jeffery (CRYPTO'15) and recent results in the area of instantaneous non-local quantum computation by Speelman (TQC'16), we show how to construct quantum gadgets that allow perfect correction of the errors which occur during the homomorphic evaluation of T\mathsf{T}T gates on encrypted quantum data. Our scheme can be based on any classical (leveled) fully homomorphic encryption (FHE) scheme and requires no computational assumptions besides those already used by the classical scheme. The size of our quantum gadget depends on the space complexity of the classical decryption function -- which aligns well with the current efforts to minimize the complexity of the decryption function. Our scheme (or slight variants of it) offers a number of additional advantages such as ideal compactness, the ability to supply gadgets “on demand,” circuit privacy for the evaluator against passive adversaries, and a three-round scheme for blind delegated quantum computation which puts only very limited demands on the quantum abilities of the client. A conference version of this paper appeared in Advances in Cryptology: Proceedings of the 36th International Cryptology Conference (CRYPTO 2016).
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.
Abstract As cloud services continue to expand, the security of private data stored and processed in these environments has become paramount. This work delves into quantum homomorphic encryption (QHE), an emerging technology that facilitates secure computation on encrypted quantum data without revealing the underlying information. We reinterpret QHE schemes through classical-quantum circuits (CQC), enhancing efficiency and addressing previous limitations related to key computations. Our approach eliminates the need for exponential key preparation by calculating keys in real-time during simulation, leading to a linear complexity in classically controlled gates. We also investigate the T / T † -gate complexity associated with various quantum walks, particularly Szegedy quantum and semiclassical algorithms, demonstrating efficient homomorphic implementations across different graph structures. Our simulations, conducted in Qiskit, validate the effectiveness of QHE for both standard and semiclassical walks. The rules for the homomorphic evaluation of the reset and intermediate measurement operations have also been included to perform the QHE of semiclassical walks. Additionally, we introduce the CQC-QHE library, a comprehensive tool that simplifies the construction and simulation of CQC tailored for QHE. Future work will focus on optimizing classical functions within this framework and exploring broader graph types to enhance QHE applications in practical scenarios.
Quantum homomorphic encryption (QHE) can allow clients to directly perform quantum computation on encrypted data with the assistance of a remote quantum server. However, existing QHE schemes often overlook the crucial property of verifiability which enables clients to validate the correctness of computation results provided by the server. In this paper, we propose a verifiable QHE scheme based on the universal quantum gate set {H,P,Toffoli}. At first, a specialized gadget is designed to eliminate the errors that may arise during the homomorphic evaluation of non-Clifford Toffoli gates in a non-interactive manner. Furthermore, the designed gadget is versatile and can be seamlessly integrated into an existing QHE scheme that implements quantum gates in another universal quantum gate set {H,T,CNOT} to homomorphically evaluate more quantum gates. Subsequently, a verifiable method is introduced to the QHE scheme based on {H,P,Toffoli} for the client to detect whether the server is honest during the homomorphic computation by employing three types of indistinguishable quantum circuits.