Bounds on the Communication Rate Needed to Achieve SK Capacity in the Hypergraphical Source Model

TL;DR

This paper introduces a new polynomial-time computable upper bound on the communication rate needed for secret key capacity in hypergraphical sources, improving upon the omniscience-based bound, with conjectures on tightness.

Contribution

It derives a novel upper bound for the SK communication rate in hypergraphical sources using fractional hyperedge removal, and provides a lower bound for graphical sources.

Findings

New upper bound on R_SK for hypergraphical sources

Bound is computable in polynomial time

Counterexample shows lower bound is not tight

Abstract

In the multiterminal source model of Csisz$\text{\'a}$r and Narayan, the communication complexity, $R_{\text{SK}}$, for secret key (SK) generation is the minimum rate of communication required to achieve SK capacity. An obvious upper bound to $R_{\text{SK}}$ is given by $R_{\text{CO}}$, which is the minimum rate of communication required for \emph{omniscience}. In this paper we derive a better upper bound to $R_{\text{SK}}$ for the hypergraphical source model, which is a special instance of the multiterminal source model. The upper bound is based on the idea of fractional removal of hyperedges. It is further shown that this upper bound can be computed in polynomial time. We conjecture that our upper bound is tight. For the special case of a graphical source model, we also give an explicit lower bound on $R_{\text{SK}}$. This bound, however, is not tight, as demonstrated by a counterexample.