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IEEE Information Theory Workshop (ITW)

Linear perfect secret key agreement

Chung Chan

Published Oct 1, 2011• 28 citations
Publisher

Abstract

A linear scheme is proposed for multiterminal secret key agreement under a private finite linear source model with public discussion. With a wiretapper observing the public discussion and a subset of the source components, it attains the secrecy capacity perfectly and non-asymptotically with a finite block length. Upper bounds on the block length and public discussion rate are given.

DOI
10.1109/itw.2011.6089530

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Allerton Conference on Communication, Control, and Computing•Sep 1, 2011•10 citations
A linear scheme is proposed for multiterminal secret key agreement under a private finite linear source model with public discussion. With a wiretapper observing the public discussion and a subset of the source components, it attains the secrecy capacity perfectly and non-asymptotically with a finite block length. Upper bounds on the block length and public discussion rate are given.
We consider a non-asymptotic (one-shot) version of the multiterminal secret key agreement problem on a finite linear source model. In this model, the observation of each terminal is a linear function of an underlying random vector composed of finitely many i.i.d. uniform random variables. By restricting the public discussion to be a linear function of the terminals' observations, we obtain a characterization of the communication complexity (minimum number of symbols of public discussion) of generating a secret key of maximum length. More precisely, we show that the minimum discussion can be achieved by a non-interactive protocol in which each terminal first does a linear processing of its own private observations, following which the terminals all execute a discussion-optimal communication-for-omniscience protocol. The secret key can be chosen to be a linear function of the vector of all observations.
We consider the secret key agreement problem under the multiterminal source model proposed by Csiszár and Narayan. A single-letter characterization of the secrecy capacity is desired but remains unknown except in the extreme case with unlimited public discussion and without wiretapper's side information. Taking the problem to the opposite extreme by requiring the public discussion rate to be zero asymptotically, we obtain the desired characterization under surprisingly general setting with wiretapper's side information, silent users, trusted and untrusted helpers. An immediate consequence of the result is that the capacity with nearly no discussion is the same as the capacity with no discussion, resolving a previous conjecture in the affirmative. The idea of the proof is to characterize the capacity in the special case with neither wiretapper's side information nor untrusted helpers using a multivariate extension of Gács-Körner common information, and then extend the result to the general setting by a change of scenario that turns untrusted helpers into trusted helpers. We further show how to evaluate the capacity explicitly for finite linear sources and discuss how the current result can be extended to improve and unify existing bounds on the capacity for strictly positive discussion rates.
This work considers the problem of multiterminal secret key agreement by limited total public discussion under the hypergraphical source model. The secrecy capacity as a function of the total discussion rate is completely characterized by a polynomial-time computable linear program. Compared to the existing solution for a particular hypergraphical source model called the pairwise independent network (PIN) model, the current result is a non-trivial extension as it applies to a strictly larger class of sources and a more general scenario involving helpers and wiretapper's side information. In particular, while the existing solution by tree-packing can be strictly suboptimal for the PIN model with helpers and the hypergraphical source model in general, we can show that decremental secret key agreement and linear network coding is optimal, resolving a previous conjecture in the affirmative. The converse is established by a single-letter upper bound on the secrecy capacity for discrete memoryless multiple sources and individual discussion rate constraints. The minimax optimization involved in the bound can be relaxed to give the best existing upper bounds on secrecy capacities such as the lamination bounds for hypergraphical sources, helper-set bound for general sources, the bound at asymptotically zero discussion rate via the multivariate Gác--Körner common information, and the lower bound on communication complexity via a multivariate extension of the Wyner common information. These reductions unify existing bounding techniques and reveal surprising connections between seemingly different information-theoretic notions. Further challenges are posed in this work along with a simple example of finite linear source where the current converse techniques fail even though the proposed achieving scheme remains optimal.
For the multiterminal secret key agreement problem, new single-letter lower bounds are obtained on the minimum public discussion rate required to achieve any given secret key rate below the secrecy capacity. The results apply to the general source model without helpers or wiretapper's side information, but can be strengthened for hypergraphical sources. In particular, for the pairwise independent network, our results yield a complete characterization of the maximum secret key rate achievable under a constraint on the total discussion rate.
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, RLR_{\mathop{\mathrm{L}}}RL​, in terms of the wiretap secret key capacity, CWC_{\mathop{\mathrm{W}}}CW​. 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 CWC_{\mathop{\mathrm{W}}}CW​ 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.