Characterizing Linear Size Circuits in Terms of Pricacy

In this paper we prove a perhaps unexpected relationship between the complexity class of the boolean functions that have linear size circuits andn-party private protocols. Specifically, letfbe a boolean function. We show thatfhas a linear size circuit if and only iffhas a 1-privaten-party protocol in which the total number of random bits used byallplayers is constant. From the point of view of complexity theory, our result gives a characterization of the class of linear size circuits in terms of another class of a very different nature. From the point of view of privacy, this result provides 1-private protocols that use a constant number of random bits, for many important functions for which no such protocol was previously known. On the other hand, our result suggests that proving, for anyNPfunction, that it has no 1-private constant-random protocol, might be difficult.