Multi-party private comparison protocol for a malicious third party without pre-shared keys

Most existing quantum private comparison protocols are designed for the scenario of a semi-honest third party, but in actual situations, the case of a malicious third party also needs to be considered. In this paper, we propose a multi-party quantum private comparison protocol based on d-dimensional GHZ state in the presence of a malicious third party TP. With the help of a malicious third party TP, multiple participants use a portion of GHZ state particles to compare the equality of their secrets and another portion of GHZ state particles to verify the honesty of the malicious TP. Multiple participants encode their secret information into the corresponding particles in turn using unitary operations, thereby achieving the goal of comparing secret equality and verifying the honesty of the malicious TP in a single protocol execution. The malicious TP in the proposed protocol does not know the comparison results and cannot perform malicious actions without being detected. In addition, the proposed protocol in this paper does not require pre-shared keys. Security analysis shows that the proposed protocol can resist the entangle-measure attack, intercept-resend attack, and internal attacks such as third-party TP attacks and participant attacks.