We present a fair and optimistic quantum contract signing protocol between
two clients that requires no communication with the third trusted party during
the exchange phase. We discuss its fairness and show that it is possible to
design such a protocol for which the probability of a dishonest client to cheat
becomes negligible, and scales as N^{-1/2}, where N is the number of messages
exchanged between the clients. Our protocol is not based on the exchange of
signed messages: its fairness is based on the laws of quantum mechanics. Thus,
it is abuse-free, and the clients do not have to generate new keys for each
message during the Exchange phase. We discuss a real-life scenario when the
measurement errors and qubit state corruption due to noisy channels occur and
argue that for real, good enough measurement apparatus and transmission
channels, our protocol would still be fair. Our protocol could be implemented
by today's technology, as it requires in essence the same type of apparatus as
the one needed for BB84 cryptographic protocol. Finally, we briefly discuss two
alternative versions of the protocol, one that uses only two states (based on
B92 protocol) and the other that uses entangled pairs, and show that it is
possible to generalize our protocol to an arbitrary number of clients.Comment: 11 pages, 2 figure