Perfect Omniscience, Perfect Secrecy and Steiner Tree Packing

TL;DR

This paper explores the capacity for perfect secret key generation in pairwise independent networks, linking it to Steiner tree packing and perfect omniscience, and provides algorithms and conditions for optimality.

Contribution

It introduces a novel connection between secret key capacity and Steiner tree packing, along with efficient algorithms and optimality conditions for helper-assisted scenarios.

Findings

The secret key capacity is characterized by perfect omniscience.

An efficient Steiner tree packing algorithm attains capacity in certain cases.

Necessary and sufficient conditions for helper-assisted optimality are established.

Abstract

We consider perfect secret key generation for a ``pairwise independent network'' model in which every pair of terminals share a random binary string, with the strings shared by distinct terminal pairs being mutually independent. The terminals are then allowed to communicate interactively over a public noiseless channel of unlimited capacity. All the terminals as well as an eavesdropper observe this communication. The objective is to generate a perfect secret key shared by a given set of terminals at the largest rate possible, and concealed from the eavesdropper. First, we show how the notion of perfect omniscience plays a central role in characterizing perfect secret key capacity. Second, a multigraph representation of the underlying secrecy model leads us to an efficient algorithm for perfect secret key generation based on maximal Steiner tree packing. This algorithm attains capacity when all the terminals seek to share a key, and, in general, attains at least half the capacity. Third, when a single ``helper'' terminal assists the remaining ``user'' terminals in generating a perfect secret key, we give necessary and sufficient conditions for the optimality of the algorithm; also, a ``weak'' helper is shown to be sufficient for optimality.