  Mediated semi-quantum key distribution involves the use of two end-users who
have very restricted, almost classical, capabilities, who wish to establish a
shared secret key using the help of a fully-quantum server who may be
adversarial. In this paper, we introduce a new mediated semi-quantum key
distribution protocol, extending prior work, which has asymptotically perfect
efficiency. Though this comes at the cost of decreased noise tolerance, our
protocol is backwards compatible with prior work, so users may easily switch to
the old (normally less efficient) protocol if the noise level is high enough to
justify it. To prove security, we show an interesting reduction from the
mediated semi-quantum scenario to a fully-quantum entanglement based protocol
which may be useful when proving the security of other multi-user QKD
protocols.
