The subject of this work is the possibility of private distributed computations of n-argument functions defined over the integers. A function f is t-private if there exists a protocol for computing f, so that no coalition of at most t participants can infer any additional information from the execution of the protocol. It is known that over finite domains every function can be computed $\lfloor (n - 1)/2 \rfloor $-privately. Some functions, like addition, are even n-private. We prove that this result cannot be extended to infinite domains. The possibility of privately computing f is shown to be closely related to the communication complexity of f. By using this relation, we show, for example, that n-argument addition is $\lfloor (n - 1)/2 \rfloor $-private over the nonnegative integers, but not even 1-private over all the integers. Finally, a complete characterization of t-private Boolean functions over countable domains is given. A Boolean function is 1-private if and only if its communication complexity is bounded. This characterization enables us to prove that every Boolean function falls into one of the following three categories: It is either n-private, $\lfloor (n - 1)/2 \rfloor $-private but not $\lceil n/2 \rceil$-private, or not 1-private.

We study the problem of interprocessor communication in a parallel computer model suggested by recent Their algorithm is complicated as opposed to ours that we believe to be practical.Optimal interprocessor communication algorithms in realistic models of parallel computers are not only interesting in their own sake, they are one of the main components in optimally simulating PRAMs as well aa implementing Valiant's bulk-synchronous parallel model [13, 19, 18].

An optimal lower bound on the average time required by any algorithm that merges two sorted lists on Valiant’s parallel computation tree model is proven.

The possibility of private distributed computations of n-argument functions defined over the integers is considered. A function f is t-private if there exists a protocol for computing f so that no coalition of>

This paper proves that in a one-round fair coin flipping scheme with n participants, either the average influence of all participants is at least $3/n - o( 1/n )$, or there is at least one participant whose influence is $\Omega ( n^{ - 5 /6} )$.

We prove a worst case lower bound of Ω(log log n) for randomized algorithms merging two sorted lists of length n in parallel using n processors on Valiant's parallel computation tree model. We show how to strengthen this result to a lower bound for the expected time taken by any algorithm on the uniform distribution. Finally, bounds are given for the average time required for the problem when the number of processors is less than and greater than n.