On the Capacity of Private Nonlinear Computation for Replicated Databases

TL;DR

This paper analyzes the maximum rate at which a user can privately compute nonlinear functions of replicated data across multiple databases without revealing the function, providing theoretical capacity bounds and numerical comparisons.

Contribution

It derives the first information-theoretic achievable private computation rate for nonlinear functions in distributed storage systems, including capacity characterization and bounds.

Findings

Achieves the capacity for large message sizes when computing nonlinear functions.

Provides an outer bound on the private computation rate as the number of messages grows.

Numerically compares private monomial computation rates to the outer bound for finite messages.

Abstract

We consider the problem of private computation (PC) in a distributed storage system. In such a setting a user wishes to compute a function of $f$ messages replicated across $n$ noncolluding databases, while revealing no information about the desired function to the databases. We provide an information-theoretically accurate achievable PC rate, which is the ratio of the smallest desired amount of information and the total amount of downloaded information, for the scenario of nonlinear computation. For a large message size the rate equals the PC capacity, i.e., the maximum achievable PC rate, when the candidate functions are the $f$ independent messages and one arbitrary nonlinear function of these. When the number of messages grows, the PC rate approaches an outer bound on the PC capacity. As a special case, we consider private monomial computation (PMC) and numerically compare the achievable PMC rate to the outer bound for a finite number of messages.