Privacy of Dense Symmetric Functions

Abstract

An n argument function, f, is called t — private if there exists a distributed protocol for computing f, so that no coalition of ≤ 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 $\left[ {\frac{{n - 1}}{2}} \right]$-private. The general question of t — privacy (for t ≥ $\left[ {\frac{n}{2}} \right]$) is still unresolved.

In this work we relate the question of $\left[ {\frac{n}{2}} \right]$-privacy for the class of symmetric functions of Boolean arguments f: {0,1}ⁿ → {0,1,..., n} to the structure of weights in f ^-1(b) (b ∈ {0,1,..., n}). We show that if f is $\left[ {\frac{n}{2}} \right]$-private, then every set of 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 $\left[ {\frac{n}{2}} \right]$ — privacy of f: Every collection of such arithmetic progressions must yield distinct remainders, when computed modulo the greatest common divisor of their differences. This condition is used to show that for dense symmetric functions, $\left[ {\frac{n}{2}} \right]$-privacy implies n-privacy.

Research supported by US-Israel Binational Science Foundation grant 88–00282.

Supported in part by the Julius and Dorothea Harbrand and by the Edwards fellowships.