Assisted Common Information: Further Results
Vinod M. Prabhakaran, Manoj M. Prabhakaran

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…
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