Non-Local Box Complexity and Secure Function Evaluation

TL;DR

This paper explores the complexity of non-local boxes in computing Boolean functions, their application to secure function evaluation, and establishes new bounds and simulation results related to quantum and classical computation.

Contribution

It provides tight bounds on non-local box complexity for Boolean functions and links this to secure function evaluation and quantum measurement simulation.

Findings

Tight bounds on non-local box complexity for Boolean functions.

Relation between non-local box complexity and secure function evaluation.

Simulation of certain quantum measurements with a finite number of non-local boxes.

Abstract

A non-local box is an abstract device into which Alice and Bob input bits x and y respectively and receive outputs a and b respectively, where a, b are uniformly distributed and the parity of a+b equals xy. Such boxes have been central to the study of quantum or generalized non-locality as well as the simulation of non-signaling distributions. In this paper, we start by studying how many non-local boxes Alice and Bob need in order to compute a Boolean function f. We provide tight upper and lower bounds in terms of the communication complexity of the function both in the deterministic and randomized case. We then proceed to show that the study of non-local box complexity has interesting applications for classical computation as well. In particular, we look at secure function evaluation, and study the question posed by Beimel and Malkin of how many Oblivious Transfer calls Alice and Bob need in order to securely compute a function f. We show that this question is related to the non-local box complexity of the function and conclude by greatly improving their bounds. Finally, another consequence of our results is that traceless two-outcome measurements on maximally entangled states can be simulated with 3 non-local boxes, while no finite bound was previously known.