A Zero-One Law for Boolean Privacy

Abstract

A Boolean function $f:A_1 \times A_2 \times \cdots \times A_n \to \{ 0,1 \}$ is t-private if there exists a protocol for computing f so that no coalition of size $\leqq t$ can infer any additional information from the execution, other than the value of the function. It is shown that f is $\lceil n/2 \rceil $-private if and only if it can be represented as \[ f ( x_1 ,x_2 , \cdots ,x_n ) = f_1 ( x_1 ) \oplus f_2 ( x_2 ) \oplus \cdots \oplus f_n ( x_n ), \] where the $f_i $ are arbitrary Boolean functions. It follows that if f is $\lceil n/2 \rceil $-private, then it is also n-private. Combining this with a result of Ben-Or, Goldwasser, and Wigderson, and of Chaum, Crepeau, and Damgard, [Proc. 20th Symposium on Theory of Computing, 1988, pp. 1–10 and pp. 11–19] an interesting “zero-one” law for private distributed computation of Boolean functions is derived: every Boolean function defined over a finite domain is either n-private, or it is $\lfloor ( n - 1 )/2 \rfloor $-private but not $\lceil n/2 \rceil $-private. A weaker notion of privacy is also investigated, where (a) coalitions are allowed to infer a limited amount of additional information, and (b) there is a probability of error in the final output of the protocol. It is shown that the same characterization of $\lceil n/2 \rceil $-private Boolean functions holds, even under these weaker requirements.In particular, this implies that for Boolean functions, the strong and the weak notions of privacy are equivalent.