Improvement of multi-party quantum private comparison protocol using d-dimensional basis states without entanglement swapping

Recently, Gu et al. (2019, arXiv:1907.02656) and Zhang et al. (Quantum Inf Process (2019)18:336) independently pointed out that in Yang et al.’s quantum summation protocol (Quantum Inf Process (2018) 17:129), the party in charge of preparing the initial quantum states is able to steal other parties’ private integer strings without being discovered by launching two special kinds of attack. In this paper, it is pointed out that in Liu et al.’s MQPC protocol, if the third party (TP) named Calvin launches the similar attacks, she can also steal the private secrets from n parties without being discovered. Then, two methods are put forward to avoid this drawback, i.e., the first one is to add the process of checking whether the quantum states shared among n parties are genuinely in the state of |W〉12⋯n while the second one is to encrypt the private secrets from n parties with a private key sequence shared among them beforehand.