  Since the negative result of Lo (Physical Review A, 1997), it has been left
open whether there exist some functions that can be securely computed in
two-party setting in quantum domain when one of the parties is malicious. In
this paper, we for the first time, show that there are some functions for which
secure two-party quantum computation is indeed possible for non-simultaneous
channel model. This is in sharp contrast with the impossibility result of Ben
-Or et al. (FOCS, 2006) in broadcast channel model. The functions we study are
of two types - one is any function without an embedded XOR, and the other one
is a particular function containing an embedded XOR. Contrary to classical
solutions, security against adversaries with unbounded power of computation is
achieved by the quantum protocols due to entanglement. Further, in the context
of secure multi-party quantum computation, for the first time we introduce
rational parties, each of whom tries to maximize its utility by obtaining the
function output alone. We adapt our quantum protocols for both the above types
of functions in rational setting to achieve fairness and strict Nash
equilibrium.
