The cooperative data exchange problem is studied for the fully connected network. In this problem, each node initially only possesses a subset of the @math packets making up the file. Nodes make broadcast transmissions that are received by all other nodes. The goal is for each node to recover the full file. In this paper, we present a polynomial-time deterministic algorithm to compute the optimal (i.e., minimal) number of required broadcast transmissions and to determine the precise transmissions to be made by the nodes. A particular feature of our approach is that each of the @math transmissions is a linear combination of exactly @math packets, and we show how to optimally choose the value of @math We also show how the coefficients of these linear combinations can be chosen by leveraging a connection to Maximum Distance Separable (MDS) codes. Moreover, we show that our method can be used to solve cooperative data exchange problems with weighted cost as well as the so-called successive local omniscience problem.
The CDE problem for the fully connected network is also related to the secret key generation problem, which was introduced in @cite_17 and was formulated as a maximization problem over all partitions of the node set. Tyagi @cite_4 leveraged this to derive an algorithm which achieves local omniscience in each step and outputs a sub-optimal solution. The weakly secure data exchange problem was introduced in @cite_7 . The goal is to achieve universal recovery while revealing as little information as possible. In contrast to the coding scheme in @cite_7 in which each transmission is a linear combination of as many packet as possible, our scheme considers a fixed number of packets for every transmission. In general cases, the communication rate in the secret key generation problem is asymptotic which makes the problem NP-hard, while the minimal number of required transmissions in the CDE problem is integer so that it can be solved by polynomial-time algorithms (as previously mentioned).
@cite_4 @cite_7 @cite_17
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Multiple parties observing correlated data seek to recover each other’s data and attain omniscience. To that end, they communicate interactively over a noiseless broadcast channel - each bit transmitted over this channel is received by all the parties. We give a universal interactive communication protocol, termed the recursive data exchange protocol (RDE), which attains omniscience for any sequence of data observed by the parties and provide an individual sequence guarantee of performance. As a by-product, for observations of length @math , we show the universal rate optimality of RDE up to an @math term in a generative setting where the data sequence is independent and identically distributed (in time). Furthermore, drawing on the duality between omniscience and secret key agreement due to Csiszar and Narayan, we obtain a universal protocol for generating a multiparty secret key of rate at most @math less than the maximum rate possible. A key feature of RDE is its recursive structure whereby when a subset @math of parties recover each-other’s data, the rates appear as if the parties have been executing the protocol in an alternative model where the parties in @math are collocated. We study the generation of a secret key of maximum rate by a pair of terminals observing correlated sources and with the means to communicate over a noiseless public communication channel. Our main result establishes a structural equivalence between the generation of a maximum rate secret key and the generation of a common randomness that renders the observations of the two terminals conditionally independent. The minimum rate of such common randomness, termed interactive common information, is related to Wyner's notion of common information, and serves to characterize the minimum rate of interactive public communication required to generate an optimum rate secret key. This characterization yields a single-letter expression for the aforementioned communication rate when the number of rounds of interaction are bounded. An application of our results shows that interaction does not reduce this rate for binary symmetric sources. Further, we provide an example for which interaction does reduce the minimum rate of communication. Also, certain invariance properties of common information quantities are established that may be of independent interest. It has been shown that the radio channel impulse response for a pair of legitimate Ultra-wide band (UWB) transceivers can be used to generate secret keys for secure communications. Past proposed secret key generation algorithms under-exploited the available number of secret key bits from the radio channel. This paper proposes a new efficient method for generation of the shared key where the transceivers use LDPC decoders to resolve the differences in their channel impulse response measurements caused by measurement noise. To ensure secret key agreement, a method of public discussion between the two users is performed using the syndrome from Hamming (7,3) binary codes. An algorithm is proposed to check the equality of generated keys for both legitimate users, and ensure error-free secure communication. The security of this algorithm has been verified by AVISPA. Comparisons are performed with previous work on secret key generation and it has been shown that this algorithm reliably generates longer secret keys in standard UWB radio channels. We design and analyze a method to extract secret keys from the randomness inherent to wireless channels. We study a channel model for a multipath wireless channel and exploit the channel diversity in generating secret key bits. We compare the key extraction methods based both on entire channel state information (CSI) and on single channel parameter such as the received signal strength indicators (RSSI). Due to the reduction in the degree-of-freedom when going from CSI to RSSI, the rate of key extraction based on CSI is far higher than that based on RSSI. This suggests that exploiting channel diversity and making CSI information available to higher layers would greatly benefit the secret key generation. We propose a key generation system based on low-density parity-check (LDPC) codes and describe the design and performance of two systems: one based on binary LDPC codes and the other (useful at higher signal-to-noise ratios) based on four-ary LDPC codes.