Inherent connections are known to exist between common randomness generated by multiterminal source coding and channel coding on the one hand, and information theoretic secrecy generation by means of public communication on the other hand. Based on these connections, points of contact have been found recently also between secrecy generation for a special model for the latter and the combinatorial problem of tree packing in a multigraph. This paper provides a brief (and partial) survey of these links, especially those between secrecy generation and combinatorial tree packing. Our objective is to highlight specific connections between common randomness, secrecy and tree packing.
Inherent connections are known to exist between common randomness generated by multiterminal source coding and channel coding on the one hand, and information theoretic secrecy generation by means of public communication on the other hand. Based on these connections, points of contact have been found recently also between secrecy generation for a special model for the latter and the combinatorial problem of tree packing in a multigraph. This paper provides a brief (and partial) survey of these links, especially those between secrecy generation and combinatorial tree packing. Our objective is to highlight specific connections between common randomness, secrecy and tree packing.
This monograph describes principles of information theoretic secrecy generation by legitimate parties with public discussion in the presence of an eavesdropper. The parties are guaranteed secrecy in the form of independence from the eavesdropper’s observation of the communication. Part I develops basic technical tools for secrecy generation, many of which are potentially of independent interest beyond secrecy settings. Various information theoretic and cryptographic notions of secrecy are compared. Emphasis is placed on central themes of interactive communication and common randomness as well as on core methods of balanced coloring and leftover hash for extracting secret uniform randomness. Achievability and converse results are shown to emerge from “single shot” incarnations that serve to explain essential structure. Part II applies the methods of Part I to secrecy generation in two settings: a multiterminal source model and a multiterminal channel model, in both of which the legitimate parties are afforded privileged access to correlated observations of which the eavesdropper has only partial knowledge. Characterizations of secret key capacity bring out inherent connections to the data compression concept of omniscience and, for a specialized source model, to a combinatorial problem of maximal spanning tree packing in a multigraph. Interactive common information is seen to govern the minimum rate of communication needed to achieve secret key capacity in the two-terminal source model. Furthermore, necessary and sufficient conditions are analyzed for the secure computation of a given function in the multiterminal source model. Based largely on known recent results, this self-contained monograph also includes new formulations with associated new proofs. Supplementing each chapter in Part II are descriptions of several open problems.
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.
This dissertation concerns the secure processing of distributed data by multi-ple terminals, using interactive public communication among themselves, in order to accomplish a given computational task. In the setting of a probabilistic multiterminal source model in which several terminals observe correlated random signals, we analyze secure distributed data processing protocols that harness the correlation in the data. The specific tasks considered are: computing functions of the data under secrecy requirements; generating secretly shared bits with minimal rate of public communication; and securely sharing bits in presence of a querying eavesdropper. In studying these various secure distributed processing tasks, we adopt a unified approach that entails examining the form of underlying common randomness (CR) that is generated at the terminals during distributed processing. We make the case that the exact form of established CR is linked inherently to the data processing task at hand, and its characterization can lead to a structural understanding of the associated algorithms. An identification of the underlying CR and its decomposition into independent components, each with a different operational significance, is
This monograph describes the principles of information theoretic secrecy generation by legitimate parties with public discussion in the presence of an eavesdropper. The parties are guaranteed secrecy in the form of independence from the eavesdropper’s observation of the communication. The focus is on secrecy generation in two settings: a multiterminal source model and a multiterminal channel model, in both of which the legitimate parties are given privileged access to correlated data of which the eavesdropper has only partial knowledge. Part I is concerned with basic technical tools for secrecy generation, many of which are potentially of independent interest beyond secrecy settings. Part II applies the methods of Part I to secrecy generation for the multiterminal source and channel models. Based largely on known recent results, this self-contained tutorial also includes new formulations with associated new proofs.
We investigate perfect secret key generation for a ¿pairwise independent network¿ model in which every pair of terminals observes correlated sources that are independent of sources observed by all other pairs of terminals. The terminals are then allowed to communicate interactively in multiple rounds over a public noiseless channel of unlimited capacity. This communication is observed by all the terminals as well as by an eavesdropper. The objective is to generate a perfect secret key shared by a given set of terminals at the largest rate possible. All the terminals cooperate in generating the secret key, with perfect secrecy being required from the eavesdropper. For this model, we introduce the concept of communication for perfect omniscience using which we first obtain a single-letter characterization of the perfect secret key capacity. Moreover, this perfect secret key capacity is shown to be achieved by linear noninteractive communication, and coincides with the (standard) secret key capacity. Our second contribution, exploiting the notion of communication for perfect omniscience, is a new nonasymptotic and computable upper bound for the combinatorial problem of maximal Steiner tree packing in a multigraph. Thus, our work establishes certain connections among perfect secrecy generation and communication for perfect omniscience for the pairwise independent network model, 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. Our results yield new bounds for the maximum size and rate of Steiner tree packing, and are of independent interest from a graph theoretic viewpoint. 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.