The privacy of dense symmetric functions

Ann argument function,f, is calledt-private if there exists a distributed protocol for computingf so that no coalition of at mostt 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 oft-privacy (fort[n/2]) is still unresolved.In this work, we relate the question of [n/2]-privacy for the class of symmetric functions of Boolean argumentsf: {0, 1}ⁿ{0, 1,...,n} to the structure of Hamming weights inf^–1(b) (b{0, 1, ...,n}). We show that iff is [n/2]-private, then every set of Hamming weightsf^–1(b) must be an arithmetic progression. For the class ofdense symmetric functions (defined in the sequel), we refine this to the following necessary and sufficient condition for [n/2]-privacy off: 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 impliesn-privacy.