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arXiv:1701.05008 · IEEE International Symposium on Information Theory (ISIT)

Secret key agreement under discussion rate constraints

Chung Chan, Manuj Mukherjee, Navin Kashyap, Qiaoqiao Zhou

Published Jun 1, 2017• 17 citations
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Abstract

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.

DOI
10.1109/isit.2017.8006783

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IEEE International Symposium on Information Theory (ISIT)•Jun 1, 2018•7 citations
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.
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 under a private source model, it is known that the maximum key rate, i.e., the secrecy capacity, can be achieved through communication for omniscience, but the omniscience strategy can be strictly suboptimal in terms of minimizing the public discussion rate. While a single-letter characterization is not known for the minimum discussion rate needed for achieving the secrecy capacity, we derive single-letter lower bounds that yield some simple conditions for omniscience to be discussion-rate optimal. These conditions turn out to be enough to deduce the optimality of omniscience for a large class of sources, including the hypergraphical sources. We also extend our results to more general class of multiterminal sources with helpers and silent users.
In the multiterminal secret key agreement problem, a set of users want to discuss with each other until they share a common secret key independent of their discussion. We want to characterize the maximum secret key rate, called the secrecy capacity, asymptotically when the total discussion rate goes to zero. In the case of only two users, the capacity is equal to the Gács-Körner common information. However, when there are more than two users, the capacity is unknown. It is plausible that a multivariate extension of the Gács-Kömer common information is the capacity, however, proving the converse is challenging. We resolved this for the hypergraphical sources and finite linear sources, and provide efficiently computable characterizations. We also give some ideas of extending the techniques to more general source models.
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 paper investigates the secret key generation in the multiterminal source model, where users observing correlated sources discuss interactively under limited rates to agree on a secret key. We focus on a class of sources representable by minimally connected hypergraphs. For such sources, we give a single-letter explicit characterization of the region of achievable secret key rate and public discussion rate tuple. This is the first result that completely characterizes the achievable rate region for a multiterminal source model, which is beyond the PIN model on a tree. We also obtain an explicit formula for the maximum achievable secret key rate, called the constrained secrecy capacity, as a function of the total discussion rate.
We investigate the secret key generation in the multiterminal source model, where the users discuss under limited rate. For the minimally connected hypergraphical sources, we give an explicit formula of the maximum achievable secret key rate, called the secrecy capacity, under any given total discussion rate. Besides, we also partially characterize the region of achievable secret key rate and discussion rate tuple. When specializes to the hypertree sources, our results give rise to a complete characterization of the region.