Secret Key Generation in the Multiterminal Source Model : Communication and Other Aspects
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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 Csiszár 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 Csiszár 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 Gács-Körner (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 capacity in this dissertation. The maximum rate of SK that can be generated even if an arbitrary subset of terminals drops out is called a fault-tolerant SK capacity. Now, suppose we have a fixed number of pairwise SKs. How should one distribute them amongpairs of terminals, to ensure good fault tolerance behavior in generating a groupSK? We show that the distribution of the pairwise keys according to a Harary graph provides a certain degree of fault tolerance, and bounds are obtained on its fault-tolerant SK capacity.
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