Efficient quantum homomorphic encryption scheme with flexible evaluators and its simulation

Quantum homomorphic encryption (QHE) allows computation on encrypted data by employing the principles of quantum mechanics. Usually, only one evaluator is chosen to complete such computation and it is easy to get overburdened in network. In addition, users sometimes do not trust only one evalutor. Recently, Chen et al. proposed a very flexible QHE scheme based on the idea of (k, n)-threshold quantum state sharing where d evaluators can finish the required operations by cooperating together as long as k≤d≤n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ k \le d \le n$$\end{document}. But it can only calculate some of single-qubit unitary operations when k≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\ge 2$$\end{document} and the quantum capability of each evaluator is a bit demanding. In this paper, we propose an improved flexible QHE scheme which extends the operations that can be computed in the QHE scheme proposed by Chen et al. to involve all single-qubit unitary operations even if k≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k \ge 2$$\end{document} and reduces the quantum capability of at least d-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d-1$$\end{document} evaluators. We also give an example to show the feasibility of the improved scheme and simulate it on the IBM’s cloud quantum computing platform.