Manuj Mukherjee

Achieving SK capacity in the source model: When must all terminals talk?

In this paper, we address the problem of characterizing the instances of the multiterminal source model of Csiszár and Narayan in which communication from all terminals is needed for establishing a secret key of maximum rate. We give an information-theoretic sufficient condition for identifying such instances. We believe that our sufficient condition is in fact an exact characterization, but we are only able to prove this in the case of the three-terminal source model.

On the Public Communication Needed to Achieve SK Capacity in the Multiterminal Source Model

The focus of this paper is on the public communication required for generating a maximal-rate secret key (SK) within the multiterminal source model of Csiszár and Narayan. Building on the prior work of Tyagi for the two-terminal scenario, we derive a lower bound on the communication complexity, RSK, defined to be the minimum rate of public communication needed to generate a maximal-rate SK. It is well known that the minimum rate of communication for omniscience, denoted by RCO, is an upper bound on RSK. For the class of pairwise independent network (PIN) models defined on uniform hypergraphs, we show that a certain Type S condition, which is verifiable in polynomial time, guarantees that our lower bound on RSK meets the RCO upper bound. Thus, the PIN models satisfying our condition are RSK-maximal, indicating that the upper bound RSK ≤ RCO holds with equality. This allows us to explicitly evaluate RSK for such PIN models. We also give several examples of PIN models that satisfy our Type S condition. Finally, we prove that for an arbitrary multiterminal source model, a stricter version of our Type S condition implies that communication from all terminals (omnivocality) is needed for establishing an SK of maximum rate. For three-terminal source models, the converse is also true: omnivocality is needed for generating a maximal-rate SK only if the strict Type S condition is satisfied. However, for the source models with four or more terminals, counterexamples exist showing that the converse does not hold in general.

When is omniscience a rate-optimal strategy for achieving secret key capacity?

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.

On the Communication Complexity of Secret Key Generation in the Multiterminal Source Model

Communication complexity refers to the minimum rate of public communication required for generating a maximal-rate secret key (SK) in the multiterminal source model of Csiszár and Narayan. Tyagi recently characterized this communication complexity for a two-terminal system. We extend the ideas in Tyagis work to derive a lower bound on communication complexity in the general multiterminal setting. In the important special case of the complete graph pairwise independent network (PIN) model, our bound allows us to determine the exact linear communication complexity, i.e., the communication complexity when the communication and SK are restricted to be linear functions of the randomness available at the terminals.

The communication complexity of achieving SK capacity in a class of PIN models

The communication complexity of achieving secret key (SK) capacity in the multiterminal source model of Csiszár and Narayan is the minimum rate of public communication required to generate a maximal-rate SK. It is well known that the minimum rate of communication for omniscience, denoted by RCO, is an upper bound on the communication complexity, denoted by RSK. A source model for which this upper bound is tight is called RSK-maximal. In this paper, we establish a sufficient condition for RSK-maximality within the class of pairwise independent network (PIN) models defined on hypergraphs. This allows us to compute RSK exactly within the class of PIN models satisfying this condition. On the other hand, we also provide a counterexample that shows that our condition does not in general guarantee RSK-maximality for sources beyond PIN models.

Bounds on the communication rate needed to achieve SK capacity in the hypergraphical source model

In the multiterminal source model of Csiszár and Narayan, the communication complexity, RSK, for secret key (SK) generation is the minimum rate of communication required to achieve SK capacity. An obvious upper bound to RSK is given by RCO, which is the minimum rate of communication required for omniscience. In this paper we derive a better upper bound to RSK for the hypergraphical source model, which is a special instance of the multiterminal source model. The upper bound is based on the idea of fractional removal of hyperedges. It is further shown that this upper bound can be computed in polynomial time. We conjecture that our upper bound is tight. For the special case of a graphical source model, we also give an explicit lower bound on RSK. This bound, however, is not tight, as demonstrated by a counterexample.

Secret key agreement under discussion rate constraints

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 wiretappers 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.

On the secret key capacity of the Harary graph PIN model

A pairwise independent network (PIN) model consists of pairwise secret keys (SKs) distributed among m terminals. The goal is to generate, through public communication among the terminals, a group SK that is information-theoretically secure from an eavesdropper. In this paper, we study the Harary graph PIN model, which has useful fault-tolerant properties. We derive the exact SK capacity for a regular Harary graph PIN model. Lower and upper bounds on the fault-tolerant SK capacity of the Harary graph PIN model are also derived.

On the Optimality of Secret Key Agreement via Omniscience

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.