Multi-mediated semi-quantum key distribution protocol with cyclic topology

Mediated semi-quantum key distribution (M-SQKD) is a class of quantum cryptographic protocols that allow two or more legitimate classical users to share a string of secure keys with the help of a third-party quantum server. Research has shown that these protocols remain valid even if the server is an adversary. Recently, two circular M-SQKD (CM-SQKD) protocols have emerged, where the server and all legitimate users form a topological loop when transmitting qubits. In this paper, we extend two existing CM-SQKD protocols, one based on a single state and the other on an entangled state, into multi-mediated versions. In the asymptotic case, we derive new bounds for the key rate of the two protocols and give noise tolerances, thus proving their unconditional security. In particular, we improve on previous results when only one mediator works.