Reducibility and Completeness in Private Computations

We de ne the notions of reducibility and completeness in (two party and multi-party) private computations. Let g be an n-argument function. We say that a function f is reducible to a function g if n honest-but-curious players can compute the function f n-privately, given a black-box for g (for which they secretly give inputs and get the result of operating g on these inputs). We say that g is complete (for private computations) if every function f is reducible to g. In this paper, we characterize the complete boolean functions: we show that a boolean function g is complete if and only if g itself cannot be computed n-privately (when there is no black-box available). Namely, for boolean functions, the notions of completeness and n-privacy are complementary . This characterization gives a huge collection of complete functions (any non-private boolean function!) compared to very few examples given (implicitly) in previous work. On the other hand, for non-boolean functions, we show that these two notions are not complementary.