Wiretap Secret Key Agreement Via Secure Omniscience

TL;DR

This paper investigates the relationship between secret key agreement and secure omniscience in multiterminal source models with wiretappers, establishing duality results for certain linear models and exploring conditions where this duality holds or fails.

Contribution

It identifies classes of sources where secure omniscience and secret key agreement duality holds, especially in linear models, and discusses conditions where the duality breaks down.

Findings

Duality holds for certain finite linear source models.

Duality is achieved when the wiretap secret key capacity is attained by a perfect linear scheme.

Counterexamples show duality may not hold in non-linear or limited communication scenarios.

Abstract

In this paper, we explore the connection between secret key agreement and secure omniscience within the setting of the multiterminal source model with a wiretapper who has side information. While the secret key agreement problem considers the generation of a maximum-rate secret key through public discussion, the secure omniscience problem is concerned with communication protocols for omniscience that minimize the rate of information leakage to the wiretapper. The starting point of our work is a lower bound on the minimum leakage rate for omniscience, $R_{\mathop{\mathrm{L}}}$, in terms of the wiretap secret key capacity, $C_{\mathop{\mathrm{W}}}$. Our interest is in identifying broad classes of sources for which this lower bound is met with equality, in which case we say that there is a duality between secure omniscience and secret key agreement. We show that this duality holds in the case of certain finite linear source (FLS) models, such as two-terminal FLS models and pairwise independent network models on trees with a linear wiretapper. Duality also holds for any FLS model in which $C_{\mathop{\mathrm{W}}}$ is achieved by a perfect linear secret key agreement scheme. We conjecture that the duality in fact holds unconditionally for any FLS model. On the negative side, we give an example of a (non-FLS) source model for which duality does not hold if we limit ourselves to communication-for-omniscience protocols with at most two (interactive) communications. We also address the secure function computation problem and explore the connection between the minimum leakage rate for computing a function and the wiretap secret key capacity.