Multi-party quantum privacy comparison of size based on d-level GHZ states

Quantum privacy comparison(QPC) plays an important role in secret ballot elections, private auctions and so on. To date, many multi-party QPC(MQPC) protocols have been proposed to compare the equality of $k(k\geq 3)$ participants. However, there are few examples of MQPC used to compare the sizes or values of their privacies. In this paper, we propose a MQPC protocol by which any $k(k\geq 3)$ participants can compare the sizes of their privacies with executing the protocol just once. The proposed MQPC protocol takes the $d-level$ GHZ states as quantum resources, and a semi-honest $TP$ is introduced to help the participants to determine the relationship of their privacies. Further more, only single-particle unitary transformations and measurements are involved, and the participants need not to share common secrets with each other beforehand which makes the proposed protocol much more efficient. Analysis shows that our protocol is secure against internal and external attack in theory.