  Semi-quantum key distribution (SQKD) protocols allow for the establishment of
a secret key between two users Alice and Bob, when one of the two users
(typically Bob) is limited or "classical" in nature. Recently it was shown that
protocols exists when both parties are limited/classical in nature if they
utilize the services of a quantum server. These protocols are called mediated
SQKD protocols. This server, however, is untrusted and, in fact, adversarial.
In this paper, we reconsider a mediated SQKD protocol and derive a new proof of
unconditional security for it. In particular, we derive a new lower bound on
its key rate in the asymptotic scenario. Furthermore, we show this new lower
bound is an improvement over prior work, thus showing that the protocol in
question can tolerate higher rates of error than previously thought.
