Security analysis for single-state circular mediated semi-quantum key distribution

In this paper, we prove the unconditional security of a single-state circular mediated semi-quantum key distribution protocol. It starts with a third-party quantum server, TP, sending a string of qubits sequentially through two classical users Alice and Bob, and finally back to TP. We derive an expression for the key rate in the asymptotic scenario. When the communication parties Alice and Bob observe that the error rate is less than a determined threshold, the key rate is always greater than 0, so they can distill a string of secure secret keys. At the end of this paper, we extend this protocol to the multiple users case, where more than two classical users establish a string of security keys.