Privacy and communication complexity

Each of two parties P₁ and P₂ holds an n-bit input, x and y, respectively. They wish to compute privately the value of f(x,y). Two questions are considered: (1) Which functions can be privately computed? (2) What is the communication complexity of protocols that privately compute a function f (in the case in which such protocols exist)? A complete combinatorial characterization of privately computable functions is given. This characterization is used to derive tight bounds on the rounds complexity of any privately computable function and to design optimal private protocols that compute these functions. It is shown that for every 1⩽g(n)⩽2×2ⁿ there are functions that can be privately computed with g(n) rounds of communication, but not with g(n)-1 rounds of communication