On communication protocols that compute almost privately

We further investigate and generalize the approximate privacy model recently introduced by Feigenbaum et al. (2010) [7]. We explore the privacy properties of a natural class of communication protocols that we refer to as "dissection protocols". Informally, in a dissection protocol the communicating parties are restricted to answering questions of the form "Is your input between the values α and β (under a pre-defined order over the possible inputs)?". We prove that for a large class of functions, called tiling functions, there always exists a dissection protocol that provides a constant average-case privacy approximation ratio for uniform or "almost uniform" probability distributions over inputs. To establish this result we present an interesting connection between the approximate privacy framework and basic concepts in computational geometry. We show that such a good privacy approximation ratio for tiling functions does not, in general, exist in the worst case. We also discuss extensions of the basic setup to more than two parties and to non-tiling functions, and provide calculations of privacy approximation ratios for two functions of interest.