Private Computations over the Integers

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 $\left\lfloor{(n-1)/2}\right\rfloor$-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 $f$. Using this relation, we show, for example, that $n$-argument addition is $\left\lfloor{(n-1)/2}\right\rfloor$-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, $\left\lfloor{(n-1)/2}\right\rfloor$-private but not $\left\lceil{n/2}\right\rceil $-private, or not $1$-private.