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

Csiszàr and Narayan[3] show that the secret-key capacity CSK(A) with unlimited public discussion and the smallest achievable rate RCO(A) of communication for omniscience of a group A of at least two active users sum up to the entropy rate H(XM) of the discrete multiple memoryless sources (Xi, i ∈ M) for all terminals M ⊃ A. They then derive a heuristically appealing upperbound[3, (26)] on the secret-key capacity in the form of the information divergence from joint to product probability measure commonly interpreted as the mutual dependence of a set of random variables. Tightness of this bound would confirm its heuristic interpretation with the operational meaning of the secret-key capacity, i.e. the maximum mutual consensus among the active users that need not be explicitly described in public. While one can easily check that the bound is tight for any system with three or less users, testing the case with more users quickly becomes unmanageable. Yet, there is no apparent reason, other than its heuristic interpretation, that the bound is tight, nor is there a counter-example that suggests otherwise. This paper proves that the bound is indeed tight when all users are active, i.e. A = M, as a consequence of the polymatroidal structure[6] underlying the source coding problem. This already confirms the heuristic interpretation of the bound as a measure of mutual dependence of random variables. For the other case when some users are helpers, i.e. A ( M, there is a counterexample with three active users and three helpers for which the bound is loose.