Coded Cooperative Data Exchange for a Secret Key

We consider a coded cooperative data exchange problem with the goal of generating a secret key. Specifically, we investigate the number of public transmissions required for a set of clients to agree on a secret key with probability one, subject to the constraint that it remains private from an eavesdropper. Although the problems are closely related, we prove that secret key generation with fewest number of linear transmissions is NP-hard, while it is known that the analogous problem in traditional cooperative data exchange can be solved in polynomial time. In doing this, we completely characterize the best possible performance of linear coding schemes, and also prove that linear codes can be strictly suboptimal. Finally, we extend the single-key results to characterize the minimum number of public transmissions required to generate a desired integer number of statistically independent secret keys.Comment: Full version of a paper that appeared at ISIT 2014. 19 pages, 2 figure

Successive Local and Successive Global Omniscience

This paper considers two generalizations of the cooperative data exchange problem, referred to as the successive local omniscience (SLO) and the successive global omniscience (SGO). The users are divided into $\ell$ nested sub-groups. Each user initially knows a subset of packets in a ground set $X$ of size $k$, and all users wish to learn all packets in $X$. The users exchange their packets by broadcasting coded or uncoded packets. In SLO or SGO, in the $l$th ($1\leq l\leq \ell$) round of transmissions, the $l$th smallest sub-group of users need to learn all packets they collectively hold or all packets in $X$, respectively. The problem is to find the minimum sum-rate (i.e., the total transmission rate by all users) for each round, subject to minimizing the sum-rate for the previous round. To solve this problem, we use a linear-programming approach. For the cases in which the packets are randomly distributed among users, we construct a system of linear equations whose solution characterizes the minimum sum-rate for each round with high probability as $k$ tends to infinity. Moreover, for the special case of two nested groups, we derive closed-form expressions, which hold with high probability as $k$ tends to infinity, for the minimum sum-rate for each round.Comment: Accepted for publication in Proc. ISIT 201

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 Csisz{\'a}r and Narayan. Building on the prior work of Tyagi for the two-terminal scenario, we derive a lower bound on the communication complexity, $R_{\text{SK}}$, 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 $R_{\text{CO}}$, is an upper bound on $R_{\text{SK}}$. For the class of pairwise independent network (PIN) models defined on uniform hypergraphs, we show that a certain "Type $\mathcal{S}$" condition, which is verifiable in polynomial time, guarantees that our lower bound on $R_{\text{SK}}$ meets the $R_{\text{CO}}$ upper bound. Thus, PIN models satisfying our condition are $R_{\text{SK}}$-maximal, meaning that the upper bound $R_{\text{SK}} \le R_{\text{CO}}$ holds with equality. This allows us to explicitly evaluate $R_{\text{SK}}$ for such PIN models. We also give several examples of PIN models that satisfy our Type $\mathcal S$ condition. Finally, we prove that for an arbitrary multiterminal source model, a stricter version of our Type $\mathcal S$ condition implies that communication from \emph{all} terminals ("omnivocality") is needed for establishing a 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 $\mathcal S$ condition is satisfied. Counterexamples exist that show that the converse is not true in general for source models with four or more terminals.Comment: Submitted to the IEEE Transactions on Information Theory. arXiv admin note: text overlap with arXiv:1504.0062

Cooperative Data Exchange with Unreliable Clients

Consider a set of clients in a broadcast network, each of which holds a subset of packets in the ground set X. In the (coded) cooperative data exchange problem, the clients need to recover all packets in X by exchanging coded packets over a lossless broadcast channel. Several previous works analyzed this problem under the assumption that each client initially holds a random subset of packets in X. In this paper we consider a generalization of this problem for settings in which an unknown (but of a certain size) subset of clients are unreliable and their packet transmissions are subject to arbitrary erasures. For the special case of one unreliable client, we derive a closed-form expression for the minimum number of transmissions required for each reliable client to obtain all packets held by other reliable clients (with probability approaching 1 as the number of packets tends to infinity). Furthermore, for the cases with more than one unreliable client, we provide an approximation solution in which the number of transmissions per packet is within an arbitrarily small additive factor from the value of the optimal solution.Comment: 8 pages; in Proc. 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton 2015

Compressed Secret Key Agreement: Maximizing Multivariate Mutual Information Per Bit

The multiterminal secret key agreement problem by public discussion is formulated with an additional source compression step where, prior to the public discussion phase, users independently compress their private sources to filter out strongly correlated components for generating a common secret key. The objective is to maximize the achievable key rate as a function of the joint entropy of the compressed sources. Since the maximum achievable key rate captures the total amount of information mutual to the compressed sources, an optimal compression scheme essentially maximizes the multivariate mutual information per bit of randomness of the private sources, and can therefore be viewed more generally as a dimension reduction technique. Single-letter lower and upper bounds on the maximum achievable key rate are derived for the general source model, and an explicit polynomial-time computable formula is obtained for the pairwise independent network model. In particular, the converse results and the upper bounds are obtained from those of the related secret key agreement problem with rate-limited discussion. A precise duality is shown for the two-user case with one-way discussion, and such duality is extended to obtain the desired converse results in the multi-user case. In addition to posing new challenges in information processing and dimension reduction, the compressed secret key agreement problem helps shed new light on resolving the difficult problem of secret key agreement with rate-limited discussion, by offering a more structured achieving scheme and some simpler conjectures to prove

A Monetary Mechanism for Stabilizing Cooperative Data Exchange with Selfish Users

This paper considers the problem of stabilizing cooperative data exchange with selfish users. In this setting, each user has a subset of packets in the ground set $X$, and wants all other packets in $X$. The users can exchange their packets by broadcasting coded or uncoded packets over a lossless broadcast channel, and monetary transactions are allowed between any pair of users. We define the utility of each user as the sum of two sub-utility functions: (i) the difference between the total payment received by the user and the total transmission rate of the user, and (ii) the difference between the total number of required packets by the user and the total payment made by the user. A rate-vector and payment-matrix pair $(r,p)$ is said to stabilize the grand coalition (i.e., the set of all users) if $(r,p)$ is Pareto optimal over all minor coalitions (i.e., all proper subsets of users who collectively know all packets in $X$). Our goal is to design a stabilizing rate-payment pair with minimum total sum-rate and minimum total sum-payment for any given instance of the problem. In this work, we propose two algorithms that find such a solution. Moreover, we show that both algorithms maximize the sum of utility of all users (over all solutions), and one of the algorithms also maximizes the minimum utility among all users (over all solutions).Comment: 7 page