Assisted Common Information: Further Results

TL;DR

This paper extends the concept of assisted common information, linking it to Gray-Wyner systems, and demonstrates its effectiveness in deriving tight bounds for secure two-party sampling protocols.

Contribution

It establishes a connection between assisted residual information and Gray-Wyner regions, providing improved bounds for secure sampling and new interpretations for source coding quantities.

Findings

The assisted residual information region is the increasing hull of the Gray-Wyner region under an affine map.

The new bounds for secure sampling are significantly tighter and sometimes optimal.

The work offers new insights into the cryptographic primitives related to oblivious transfer.

Abstract

We presented assisted common information as a generalization of G\'acs-K\"orner (GK) common information at ISIT 2010. The motivation for our formulation was to improve upperbounds on the efficiency of protocols for secure two-party sampling (which is a form of secure multi-party computation). Our upperbound was based on a monotonicity property of a rate-region (called the assisted residual information region) associated with the assisted common information formulation. In this note we present further results. We explore the connection of assisted common information with the Gray-Wyner system. We show that the assisted residual information region and the Gray-Wyner region are connected by a simple relationship: the assisted residual information region is the increasing hull of the Gray-Wyner region under an affine map. Several known relationships between GK common information and Gray-Wyner system fall out as consequences of this. Quantities which arise in other source coding contexts acquire new interpretations. In previous work we showed that assisted common information can be used to derive upperbounds on the rate at which a pair of parties can {\em securely sample} correlated random variables, given correlated random variables from another distribution. Here we present an example where the bound derived using assisted common information is much better than previously known bounds, and in fact is tight. This example considers correlated random variables defined in terms of standard variants of oblivious transfer, and is interesting on its own as it answers a natural question about these cryptographic primitives.