Perfect Omniscience, Perfect Secrecy and Steiner Tree Packing

We consider perfect secret key generation for a ``pairwise independent network'' model in which every pair of terminals share a random binary string, with the strings shared by distinct terminal pairs being mutually independent. The terminals are then allowed to communicate interactively over a public noiseless channel of unlimited capacity. All the terminals as well as an eavesdropper observe this communication. The objective is to generate a perfect secret key shared by a given set of terminals at the largest rate possible, and concealed from the eavesdropper. First, we show how the notion of perfect omniscience plays a central role in characterizing perfect secret key capacity. Second, a multigraph representation of the underlying secrecy model leads us to an efficient algorithm for perfect secret key generation based on maximal Steiner tree packing. This algorithm attains capacity when all the terminals seek to share a key, and, in general, attains at least half the capacity. Third, when a single ``helper'' terminal assists the remaining ``user'' terminals in generating a perfect secret key, we give necessary and sufficient conditions for the optimality of the algorithm; also, a ``weak'' helper is shown to be sufficient for optimality.Comment: accepted to the IEEE Transactions on Information Theor

INFORMATION THEORETIC SECRET KEY GENERATION: STRUCTURED CODES AND TREE PACKING

This dissertation deals with a multiterminal source model for secret key generation by multiple network terminals with prior and privileged access to a set of correlated signals complemented by public discussion among themselves. Emphasis is placed on a characterization of secret key capacity, i.e., the largest rate of an achievable secret key, and on algorithms for key construction. Various information theoretic security requirements of increasing stringency: weak, strong and perfect secrecy, as well as different types of sources: finite-valued and continuous, are studied. Specifically, three different models are investigated. First, we consider strong secrecy generation for a discrete multiterminal source model. We discover a connection between secret key capacity and a new source coding concept of ``minimum information rate for signal dissemination,'' that is of independent interest in multiterminal data compression. Our main contribution is to show for this discrete model that structured linear codes suffice to generate a strong secret key of the best rate. Second, strong secrecy generation is considered for models with continuous observations, in particular jointly Gaussian signals. In the absence of suitable analogs of source coding notions for the previous discrete model, new techniques are required for a characterization of secret key capacity as well as for the design of algorithms for secret key generation. Our proof of the secret key capacity result, in particular the converse proof, as well as our capacity-achieving algorithms for secret key construction based on structured codes and quantization for a model with two terminals, constitute the two main contributions for this second model. Last, we turn our attention to perfect secrecy generation for fixed signal observation lengths as well as for their asymptotic limits. In contrast with the analysis of the previous two models that relies on probabilistic techniques, perfect secret key generation bears the essence of ``zero-error information theory,'' and accordingly, we rely on mathematical techniques of a combinatorial nature. The model under consideration is the ``Pairwise Independent Network'' (PIN) model in which every pair of terminals share a random binary string, with the strings shared by distinct pairs of terminals being mutually independent. This model, which is motivated by practical aspects of a wireless communication network in which terminals communicate on the same frequency, results in three main contributions. First, the concept of perfect omniscience in data compression leads to a single-letter formula for the perfect secret key capacity of the PIN model; moreover, this capacity is shown to be achieved by linear noninteractive public communication, and coincides with strong secret key capacity. Second, taking advantage of a multigraph representation of the PIN model, we put forth an efficient algorithm for perfect secret key generation based on a combinatorial concept of maximal packing of Steiner trees of the multigraph. When all the terminals seek to share perfect secrecy, the algorithm is shown to achieve capacity. When only a subset of terminals wish to share perfect secrecy, the algorithm is shown to achieve at least half of it. Additionally, we obtain nonasymptotic and asymptotic bounds on the size and rate of the best perfect secret key generated by the algorithm. These bounds are of independent interest from a purely graph theoretic viewpoint as they constitute new estimates for the maximum size and rate of Steiner tree packing of a given multigraph. Third, a particular configuration of the PIN model arises when a lone ``helper'' terminal aids all the other ``user'' terminals generate perfect secrecy. This model has special features that enable us to obtain necessary and sufficient conditions for Steiner tree packing to achieve perfect secret key capacity

Secret Key Agreement under Discussion Rate Constraints

For the multiterminal secret key agreement problem, new single-letter lower bounds are obtained on the public discussion rate required to achieve any given secret key rate below the secrecy capacity. The results apply to general source model without helpers or wiretapper's side information but can be strengthened for hypergraphical sources. In particular, for the pairwise independent network, the results give rise to a complete characterization of the maximum secret key rate achievable under a constraint on the total discussion rate

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

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 Csiszar and Narayan. Tyagi recently characterized this communication complexity for a two-terminal system. We extend the ideas in Tyagi's 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.Comment: A 5-page version of this manuscript will be submitted to the 2014 IEEE International Symposium on Information Theory (ISIT 2014

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

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