Upper Bounds via Lamination on the Constrained Secrecy Capacity of Hypergraphical Sources

TL;DR

This paper introduces explicit upper bounds on the secret key rate for hypergraphical sources with discussion rate constraints, using lamination techniques, and confirms the optimality of the tree-packing protocol in certain cases.

Contribution

It develops a family of computable upper bounds on secret key rates under discussion constraints using lamination, and proves the optimality of the tree-packing protocol for specific models.

Findings

Derived explicit upper bounds on secret key rates with discussion constraints.

Proved the tightness of the edge-partition bound for the pairwise independent network model.

Confirmed the optimality of the tree-packing protocol for certain hypergraphical sources.

Abstract

Hypergraphical sources are a natural class of sources for secret key generation, within which different subsets of terminals sharing secrets are allowed to discuss publicly in order to agree upon a global secret key. While their secrecy capacity, i.e., the maximum rate of a secret key that can be agreed upon by the entire set of terminals, is well-understood, what remains open is the maximum rate of a secret key that can be generated when there is a restriction on the overall rate of public discussion allowed. In this work, we obtain a family of explicitly computable upper bounds on the number of bits of secret key that can be generated per bit of public discussion. These upper bounds are derived using a lamination technique based on the submodularity of the entropy function. In particular, a specific instance of these upper bounds, called the edge-partition bound, is shown to be tight for the pairwise independent network model, a special case of the hypergraphical source. The secret key generation scheme achieving this upper bound is the tree-packing protocol of Nitinawarat et al., thereby resolving in the affirmative the discussion rate optimality of the tree packing protocol.