  In this paper, we put forward a multi-party quantum private comparison (MQPC)
protocol with two semi-honest third parties (TPs) by adopting d-dimensional
Bell states, which can judge the size relationship of private integers from
more than two users within one execution of protocol. Each TP is permitted to
misbehave on her own but cannot collude with others. In the proposed MQPC
protocol, TPs are only required to apply d-dimensional single-particle
measurements rather than d-dimensional Bell state measurements. There are no
quantum entanglement swapping and unitary operations required in the proposed
MQPC protocol. The security analysis validates that the proposed MQPC protocol
can resist both the outside attacks and the participant attacks. The proposed
MQPC protocol is adaptive for the case that users want to compare the size
relationship of their private integers under the control of two supervisors.
Furthermore, the proposed MQPC protocol can be used in the strange user
environment, because there are not any communication and pre-shared key between
each pair of users.
