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 ≥ 3 ) participants. However, few examples of MQPC can be used to compare the sizes or values of their privacies. In this paper, we propose a MQPC protocol by which any k ( k ≥ 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. Furthermore, only single-particle unitary transformations and measurements are involved, and the participants need not to share a private key beforehand which makes the proposed protocol much more efficient. Analysis shows that our protocol is secure against internal and external attack in theory.