On Tightness of Mutual Dependence Upperbound for Secret-key Capacity of Multiple Terminals

TL;DR

This paper investigates the tightness of an upper bound on secret-key capacity for multiple terminals, proving tightness in the all-active case and providing a counter-example when some users are inactive.

Contribution

It establishes the tightness of the mutual dependence upper bound for secret-key capacity in the all-active scenario using polymatroidal structure, and shows the bound may not be tight otherwise.

Findings

Bound is tight when all users are active

Counter-example with partial activity shows bound may not be tight

Polymatroidal structure underpins the proof

Abstract

Csiszar and Narayan[3] defined the notion of secret key capacity for multiple terminals, characterized it as a linear program with Slepian-Wolf constraints of the related source coding problem of communication for omniscience, and upper bounded it by some information divergence expression from the joint to the product distribution of the private observations. This paper proves that the bound is tight for the important case when all users are active, using the polymatroidal structure[6] underlying the source coding problem. When some users are not active, the bound may not be tight. This paper gives a counter-example in which 3 out of the 6 terminals are active.