On the structure of the privacy hierarchy

An N argument function f(x1,...,xN) is called t-private if a protocol for computing f exists so that no coalition of at most t parties can infer any additional information from the execution, other than the value of the function. The motivation of this work is to understand what levels of privacy are attainable. So far, only two levels of privacy are known for N argument functions which are defined over finite domains: functions that are N-private and functions that are ⌊(N − 1)/2⌋-private but not ⌈N/2⌉-private.In this work we show that the privacy hierarchy for N-argument functions which are defined over finite domains, has exactly ⌈(N + 1)/2⌉ levels. We prove this by constructing, for any ⌈N/2⌉ ≤ t ≤ N − 2, an N-argument function which is t-private but not (t + 1)-private.