Privacy, additional information, and communication

Two parties, each holding one input of a two-variable function, communicate in order to determine the value of the function. Each party wants to expose as little of its input as possible to the other party. We prove tight bounds on the minimum amount of information about the individual inputs that must be revealed in the computation of most functions and of some specific ones, and show that a computation that reveals little information about the individual inputs may require many more message exchanges than a more revealing computation.