Secret Key Generation for a Pairwise Independent Network Model

TL;DR

This paper introduces a new model for secret key generation in pairwise independent networks, providing a formula for capacity, a Steiner tree packing algorithm, and conditions under which the capacity is achieved.

Contribution

It establishes a single-letter formula for secret key capacity in pairwise independent networks and proposes an explicit Steiner tree packing algorithm for key generation.

Findings

The formula links secret key capacity to Steiner tree packing in a multigraph.

The Steiner tree packing algorithm achieves capacity in special cases.

The proposed method provides a lower bound for secret key capacity.

Abstract

We consider secret key generation for a "pairwise independent network" model in which every pair of terminals observes correlated sources that are independent of sources observed by all other pairs of terminals. The terminals are then allowed to communicate publicly with all such communication being observed by all the terminals. The objective is to generate a secret key shared by a given subset of terminals at the largest rate possible, with the cooperation of any remaining terminals. Secrecy is required from an eavesdropper that has access to the public interterminal communication. A (single-letter) formula for secret key capacity brings out a natural connection between the problem of secret key generation and a combinatorial problem of maximal packing of Steiner trees in an associated multigraph. An explicit algorithm is proposed for secret key generation based on a maximal packing of Steiner trees in a multigraph; the corresponding maximum rate of Steiner tree packing is thus a lower bound for the secret key capacity. When only two of the terminals or when all the terminals seek to share a secret key, the mentioned algorithm achieves secret key capacity in which case the bound is tight.