An efficient controlled semi-quantum secret sharing protocol with entangled state

In this paper, we present an entangled state controlled semi-quantum secret sharing CSQSS protocol for the first time. In this scheme, with the permission of a trusted classical user, Bob1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Bob_{1}$$\end{document}, Alice, as a quantum user, can share a one-bit specific message with n classical users, and the secret can only be recovered by the cooperation of all classical users. Then, the protocol is extended where m-bit specific messages, K(k1,k2,...,km)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(k_{1}, k_{2},..., k_{m})$$\end{document}, can be shared with n classical users. The security of the proposed protocol against common attacks is analysed in detail, which shows that the proposed protocol is theoretically secure. Compared with previous SQSS protocols, the proposed protocol can achieve a lower cost because it does not use returning qubits for producing the secret message, uses fewer returning qubits for eavesdropping check, and does not perform entangled state measurement. Moreover, the proposed protocol has the highest qubit efficiency among the previous SQSS schemes.