F 0 1 2 3 2 N ? 2 2 N ? 1 0 0 0 0 0 0 0 1 1 2 2 2 2 2

13 Proof: R private (f) C private (f) since in any protocol the parties send at least one bit per round. On the other hand, for every privately computable function f the bound R private (f) is achieved by the protocol A f. In any round of this protocol the parties send a message with size at most n. (Note that in steps (2a) and (2b) of A f we may save one bit per round by omitting the most signiicant bit which is always '1'). The claim follows. Corollary 8: Randomization does not help to reduce the number of rounds needed in private computations. This is true since the protocol A f is a deterministic protocol. We remark that A f besides being optimal in its rounds complexity, is also optimal in its communication complexity among the deterministic protocols. To prove that we observe that if A is a deterministic protocol then the number of diierent communication strings is equal to the number of leaves in the decomposition tree that corresponds to A. We conclude not only that randomization does not help to compute more functions privately, but also that it does not help to reduce the number of rounds needed in private computations. 5 Discussion As it is pointed out in 8], the characterization of two-party private functions can be used to give necessary conditions for privacy in a multi-party scenario. t-private, for t j N ?1 2 k + 1, if for any partition of the N parties into two sets of participants T and T, where T is of size t, the two-argument function ^ f deened as ^ f(fx i g i2T ; fx i g i2 T) def = f(x 1 ; x 2 ; : : :; x N) is private. Using the above observation and Theorem 1, we get from any such partition a necessary condition for f to be t-private (t j N ?1 2 k + 1). In addition, from any such partition and using Corollary 5, we get lower bounds on the rounds complexity and the communication complexity of computing f t-privately. The problem of giving an exact characterization of the t-private (general) functions, for j N ?1 2 k < t N is still open. 6 Acknowledgments I would like to thank Benny Chor and Oded Goldreich for helpful discussions on the topics of this paper, and Avi …