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arXiv:1805.03110 · IEEE Transactions on Information Theory

Secret Key Generation for Minimally Connected Hypergraphical Sources

Qiaoqiao Zhou, Chung Chan

Published Feb 12, 2020• 9 citations
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Abstract

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.

DOI
10.1109/tit.2020.2971215

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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.
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.
Hypergraphical sources are a natural class of sources for secret key generation, within which different subsets of terminals sharing secrets are allowed to discuss publicly in order to agree upon a global secret key. While their secrecy capacity, i.e., the maximum rate of a secret key that can be agreed upon by the entire set of terminals, is well-understood, what remains open is the maximum rate of a secret key that can be generated when there is a restriction on the overall rate of public discussion allowed. In this paper, we obtain a family of explicitly computable upper bounds on the number of bits of secret key that can be generated per bit of public discussion. These upper bounds are derived using a lamination technique based on the submodularity of the entropy function. In particular, a specific instance of these upper bounds, called the edge-partition bound, is shown to be tight for the pairwise independent network model, a special case of the hypergraphical source when the hypergraph is a graph. The secret key generation scheme achieving this upper bound is the tree-packing protocol of Nitinawarat et al., thereby resolving in the affirmative the discussion rate optimality of the tree-packing protocol.