The Capacity of Private Computation

TL;DR

This paper introduces the problem of private computation with multiple servers and datasets, characterizes its capacity for linear functions, and shows it matches PIR capacity, extending to non-linear functions as datasets grow large.

Contribution

It generalizes private information retrieval to include arbitrary functions, providing capacity results for linear and non-linear cases, revealing no rate loss compared to PIR.

Findings

Capacity of private computation matches PIR capacity for linear functions.

Allowing arbitrary linear computations does not reduce the communication rate.

Non-linear computations have the same capacity as PIR when datasets tend to infinity.

Abstract

We introduce the problem of private computation, comprised of $N$ distributed and non-colluding servers, $K$ independent datasets, and a user who wants to compute a function of the datasets privately, i.e., without revealing which function he wants to compute, to any individual server. This private computation problem is a strict generalization of the private information retrieval (PIR) problem, obtained by expanding the PIR message set (which consists of only independent messages) to also include functions of those messages. The capacity of private computation, $C$, is defined as the maximum number of bits of the desired function that can be retrieved per bit of total download from all servers. We characterize the capacity of private computation, for $N$ servers and $K$ independent datasets that are replicated at each server, when the functions to be computed are arbitrary linear combinations of the datasets. Surprisingly, the capacity, $C=\left(1+1/N+\cdots+1/N^{K-1}\right)^{-1}$, matches the capacity of PIR with $N$ servers and $K$ messages. Thus, allowing arbitrary linear computations does not reduce the communication rate compared to pure dataset retrieval. The same insight is shown to hold even for arbitrary non-linear computations when the number of datasets $K\rightarrow\infty$.