Hybrid protocols for multi-party semiquantum private comparison, multiplication and summation without a pre-shared key based on d-dimensional single-particle states

In this paper, by utilizing d-dimensional single-particle states, three semiquantum cryptography protocols, i.e., the multi-party semiquantum private comparison (MSQPC) protocol, the multi-party semiquantum multiplication (MSQM) protocol and the multi-party semiquantum summation (MSQS) protocol, can be achieved simultaneously under the assistance of two semi-honest quantum third parties (TPs). Here, the proposed MSQPC scheme is the only protocol which is devoted to judging the size relationship of secret integers from more than two semiquantum participants without a pre-shared key. And the proposed MSQM protocol absorbs the innovative concept of semiquantumness into quantum multiplication for the first time, which can calculate the modulo d multiplication of private inputs from more than two semiquantum users. As for the proposed MSQS protocol, it is the only semiquantum summation protocol which aims to accomplish the modulo d addition of more than three semiquantum users’ private integers. Neither quantum entanglement swapping nor unitary operations are necessary in the three proposed protocols. The security analysis verifies in detail that both the external attacks and the internal attacks can be resisted in the three proposed protocols.