  Mediated semi-quantum key distribution (M-SQKD) permits two limited
"semi-quantum" or "classical" users to establish a secret key with the help of
a third party (TP), in which TP has fully quantum power and may be untrusted.
Several protocols have been studied recently for two-party scenarios, but no
one has considered M-SQKD for multi-party scenarios. In this paper, we design a
circular M-SQKD protocol based on Bell states, which offers an approach to
realizing multiple "classical" users' key distribution. Then, we prove the
protocol is unconditional security in the asymptotic scenario. The protocol's
key rate and noise tolerance can be derived by utilizing the parameters
observed in the channel. The results show that our protocol may hold similar
security to a fully quantum one. We also compare the proposed protocol with
similar protocols in terms of noise tolerance, qubit efficiency, communication
cost, and scalability. Finally, the security proof method of this paper may
contribute to studying the security of other circular semi-quantum cryptography
protocols.
