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.
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.
For the multiterminal secret key agreement problem under a private source model, it is known that the communication complexity required to achieve the capacity can be strictly smaller than the minimum rate of communication for omniscience, but a single-letter characterization is not known. We obtain a single-letter lower bound on the communication complexity as well as some conditions for the communication complexity to be maximal (equal to the smallest rate of communication for omniscience). The results are are stated and derived using a meaningful multivariate mutual information measure. They are stronger than existing ones because 1) they apply to a general discrete memoryless multiple source rather than a special source model, 2) the problem formulation allows private randomization by individual users, 3) the bound is single-letter and the condition can be checked easily, and so 4) more scenarios in which the communication complexity is maximal are discovered. We conjecture that the lower bound can be further improved by giving a concrete example.
This dissertation is primarily concerned with the communication required to achieve secret key (SK) capacity in a multiterminal source model. The multiterminal source model introduced by Csiszar and Narayan consists of a group of remotely located terminals with access to correlated sources and a noiseless public channel. The terminals wish to secure their communication by agreeing upon a group secret key. The key agreement protocol involves communicating over the public channel, and agreeing upon an SK secured from eavesdroppers listening to the public communication. The SK capacity, i.e., the maximum rate of an SK that can be agreed upon by the terminals, has been characterized by Csiszar and Narayan. Their capacity-achieving key generation protocol involved terminals communicating to attain omniscience, i.e., every terminal gets to recover the sources of the other terminals. While this is a very general protocol, it often requires larger rates of public communication than is necessary to achieve SK capacity.
The primary focus of this dissertation is to characterize the communication complexity, i.e., the minimum rate of public discussion needed to achieve SK capacity. A lower bound to communication complexity is derived for a general multiterminal source, although it turns out to be loose in general. While the minimum rate of communication for omniscience is always an upper bound to the communication complexity, we derive tighter upper bounds to communication complexity for a special class of multiterminal sources, namely, the hypergraphical sources. This upper bound yield a complete characterization of hypergraphical sources where communication for omniscience is a rate-optimal protocol for SK generation, i.e., the communication complexity equals the minimum rate of communication for omniscience.
Another aspect of the public communication touched upon by this dissertation is the necessity of omnivocality, i.e., all terminals communicating, to achieve the SK capacity. It is well known that in two-terminal sources, only one terminal communicating success to generate a maximum rate secret key. However, we are able to show that for three or more terminals, omnivocality is indeed required to achieve SK capacity if a certain condition is met. For the specific case of three terminals, we show that this condition is also necessary to ensure omnivocality is essential in generating a SK of maximal rate. However, this condition is no longer necessary when there are four or more terminals.
A certain notion of common information, namely, the Wyner common information, plays a central role in the communication complexity problem. This dissertation thus includes a study of multiparty versions of the two widely used notions of common information, namely, Wyner common information and Gacs-Korner (GK) common information. While evaluating these quantities is difficult in general, we are able to derive explicit expressions for both types of common information in the case of hypergraphical sources.
We also study fault-tolerant SK…
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 an eavesdropper having 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 eavesdropper. The starting point of our work is a lower bound on the minimum leakage rate for omniscience,RL, in terms of the wiretap secret key capacity,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 eavesdropper. Duality also holds for any FLS model in whichCWis 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.
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.