# Private Sequential Function Computation

## TL;DR

This paper investigates how to privately compute a specific ordered composition of linear basic functions across multiple servers, optimizing the number of queries while ensuring the servers learn nothing about the function order.

## Contribution

It introduces the problem of private sequential function computation, derives capacity bounds, and proposes a method to achieve the desired privacy with optimal query efficiency.

## Key findings

- Capacity bounds: (1 - 1/N) / (1 - 1/ max(K,N)) ≤ C ≤ 1
- Achievability: proper inquiry order retrieves desired composition while maintaining privacy
- Upper bound: information-theoretic converse on capacity

## Abstract

Consider a system, including a user, $N$ servers, and $K$ basic functions which are known at all of the servers. Using the combination of those basic functions, it is possible to construct a wide class of functions. The user wishes to compute a particular combination of the basic functions, by offloading the computation to $N$ servers, while the servers should not obtain any information about which combination of the basic functions is to be computed. The objective is to minimize the total number of queries asked by the user from the servers to achieve the desired result. As a first step toward this problem, in this paper, we consider the case where the user is interested in a class of functions which are composition of the basic functions, while each basic function appears in the composition exactly once. This means that in this case, to ensure privacy, we only require to hide to the order of the basic functions in the desired composition of the user. We further assume that the basic functions are linear and can be represented by (possibly large-scale) matrices. We call this problem as private sequential function computation. We study the capacity $C$, defined as the supremum of the number of desired computations, normalized by the number of computations done at the servers, subject to the privacy constraint. We prove that $(1-\frac{1}{N})/ (1-\frac{1}{\max(K,N)}) \le C \le 1$. For the achievability, we show that the user can retrieve the desired order of composition, by choosing a proper order of inquiries among different servers, while keeping the order of computations for each server fixed, irrespective of the desired order of composition. In the end, we develop an information-theoretic converse which results in an upper bound on the capacity.