The privacy of dense symmetric functions

An n argument function, f, is called t-private if there exists a distributed protocol for computing f so that no coalition of at most t processors can infer any additional information from the execution of the protocol. It is known that every function defined over a finite domain is [(n-1)/2]-private. The general question of t-privacy (for t≥[n/2]) is still unresolved. In this work, we relate the question of [n/2]-privacy for the class of symmetric functions of Boolean arguments f: {0, 1}ⁿ→{0, 1,..., n} to the structure of Hamming weights in f^-1(b) (b∈{0, 1, ..., n}). We show that if f is [n/2]-private, then every set of Hamming weights f^-1(b) must be an arithmetic progression. For the class of dense symmetric functions (defined in the sequel), we refine this to the following necessary and sufficient condition for [n/2]-privacy of f: Every collection of such arithmetic progressions must yield non-identical remainders, when computed modulo the greatest common divisor of their differences. This condition is used to show that for dense symmetric functions, [n/2]-privacy implies n-privacy.