New Lower Bounds For Privacy In Communication Protocols

Communication complexity is a central model of computation introduced by Yao [22], where two players, Alice and Bob, receive inputs x and y respectively and want to compute f(x, y) for some fixed function f with the least amount of communication. Recently people have revisited the question of the privacy of such protocols: is it possible for Alice and Bob to compute f(x, y) without revealing too much information about their inputs? There are two types of privacy for communication protocols that have been proposed: first, an information theoretic definition ([9, 15]), which for Boolean functions is equivalent to the notion of information cost introduced by [12] and that has since found many important applications; second, a combinatorial definition introduced by [13] and further developed by [1].We provide new results for both notions of privacy, as well as the relation between them. Our new lower bound techniques both for the combinatorial and the information-theoretic definitions enable us to give tight bounds for the privacy of several functions, including Equality, Disjointness, Inner Product, Greater Than. In the process we also prove tight bounds (up to 1 or 2 additive bits) for the external information complexity of these functions.We also extend the definitions of privacy to bounded-error randomized protocols and provide a relation between the two notions and the communication complexity. Again, we are able to prove tight bounds for the above-mentioned functions as well as the Vector in Subspace and Gap Hamming Distance problems.