Security analysis of the semi-quantum secret-sharing protocol of specific bits and its improvement

Recently, Tian et al. (Quantum Inf. Process., 20(6), 217(2021)) proposed an efficient semi-quantum secret-sharing protocol of specific bits. In their protocol, the dealer can split a specific secret such that two classical agents can efficiently and corporately reconstruct the secret by applying the simple operations such as Z-base measurement and reflecting operation. The qubit efficiency of their protocol can be up to 50%. Unfortunately, according to our security analysis, their protocol is vulnerable to the eavesdropping attacks. We prove that the eavesdropper can steal the dealer's secret without being detected by applying the double-controlled NOT attack (DCNA) to the quantum channels. Then, some improvements are proposed, which can overcome the security flaw of the old protocol. We prove the improved protocol has enhanced security against DCNA, intercept-resend attack, entangle-measure attack and Trojan horse attack. In particular, the security proof shows that inserting decoy photons into the quantum message and reordering the quantum sequence is very helpful in detecting the adversary's entangle-measure attack, even the adversary uses two auxiliary probes to eavesdrop the quantum channel. Furthermore, the improved eavesdropping check strategy can effectively increase the length of the shared secret.