A new approximation algorithm for maximum weighted matching in general edge-weighted graphs is presented. It calculates a matching with an edge weight of at least of the edge weight of a maximum weighted matching. Its time complexity is O(|E|), with |E| being the number of edges in the graph. This improves over the previously known -approximation algorithms for maximum weighted matching which require O(|E| log(|V|)) steps, where |V| is the number of vertices.

Linear Time 1/2-Approximation Algorithm for Maximum Weighted Matching in General Graphs

We address secure network coding over networks with unequal link capacities in the presence of a wiretapper who has only access to a restricted number of k links in the network. Previous results show that for the case of equal link capacities and unrestricted wiretapping sets, the secrecy capacity is given by the cut-set bound, whether or not the location of the wiretapped links is known. The cut-set bound can be achieved by injecting k random keys at the source which are decoded at the sink along with the message. In contrast, for the case where the wiretapping set is restricted, or where link capacities are not equal, we show that the cut-set bound is not achievable in general. Finally, it is shown that determining the secrecy capacity is a NP-hard problem.

/pdf/on-secure-network-coding-with-unequal-link-capacities-and-n2o82sod2o.pdf

On secure network coding with unequal link capacities and restricted wiretapping sets

Secure network coding assumes that the underlying network channels are error-free; thus, if our channels introduce errors, we need to first apply a channel code to correct them, and then build security on top of the resulting error-free network. In this paper, we develop achievability protocols and outer bounds for the secure network coding setting, where the edges are subject to packet erasures, and public feedback of the channel state is available to both Eve and the legitimate network nodes. We show that by leveraging erasures and feedback, we can achieve secrecy rates that are in some cases multiple times higher than the alternative of separate channel-error-correction followed by secure network coding; moreover, we develop outer bounds and prove optimality of our proposed schemes in some special cases.

/pdf/secure-network-coding-with-erasures-and-feedback-265hlkdur0.pdf

Secure Network Coding With Erasures and Feedback

Secure network coding assumes that the underlying network links are lossless, thus it can be applied over lossy networks after channel error correction. Yet it is well known that channel losses, such as packet erasures, can be constructively used for secrecy over a link. We address here the challenge of extending these results for arbitrary networks. We provide achievability schemes over erasure networks with feedback, that outperform the alternative approach of channel error correction followed by secure message transmission separation. We derive outer bounds on the securely achievable rate and as a consequence we show optimality of our proposed scheme in some special cases.

/pdf/secure-network-coding-with-erasures-and-feedback-3uxywiqlqw.pdf

Secure network coding with erasures and feedback

We characterize the secret message capacity of the triangle network, that consists of a source, a relay and a destination connected through orthogonal erasure channels. A passive eavesdropper, Eve, wiretaps any one of the three channels. The source and the relay can each generate unlimited private randomness; the relay and the destination can publicly provide strictly causal channel state information. Our achievable scheme is expressed through a linear program (LP) with 11 inequalities that captures a minimal set of secret key generation methods and the use of them for message encryption. Our outer bound is expressed also through a linear program, in this case with 41 constraints, constructed from general information inequalities. We prove that the optimal value of the outer bound LP is no larger than that of the scheme LP, which implies that the solution of the achievable scheme LP is the capacity. We find that equipping the relay with private randomness increases the secrecy rate by more than 40% in some cases and that cut-set bounds, directly applied in the network, are not always tight. Because the derivation of the inner and outer bound are both lengthy, we describe in this paper the achievability scheme, outline the outer bound, and provide the full derivations online [1]. We also make available Matlab functions that take as input the erasure probabilities and evaluate the inner and outer bounds.

/pdf/triangle-network-secrecy-2r9ohdzhbn.pdf

Triangle Network Secrecy

In the first part of this thesis, we exploit wireless broadcast across different layers in wireless networks. The wireless channel is distinguished by its broadcast nature. Wireless broadcast provides a fertile ground to improve the efficiency of existing wireless networks and design new ones. Specifically, we first consider relaying strategies for memoryless two-way relay channels at the physical layer. We generalize networking layer network coding operating on a finite field to physical layer network coding, which is a mapping from the relay's received signal to its transmitted signal. We analyze the symbol-error performance of several relay strategies, and optimize the relay function via functional analysis. Our results indicate that the interference caused by wireless broadcast can be exploited to improve the spectrum efficiency. We then develop a cross-layer framework with wireless broadcast, which integrates rate control, network coding and scheduling in transport, network and link layers. Under the primary interference model, we show that the link scheduling problem is the maximum weighted hypergraph matching problem, which is NP-complete. We propose several distributed approximation algorithms and bound their worst case performance. Next, we describe a new class of medium access control (MAC) protocol, which uses successive interference cancelation to resolve packet collision due to wireless broadcast. Each user is allowed to transmit at different data rates chosen randomly from an appropriately determined set of rates. We characterize the throughput of the proposed protocol compared to that with a centralized controller. A game-theoretic framework along with the dynamic algorithms is proposed to achieve the desired throughput optimal equilibrium, which provides a valuable perspective to understand existing MAC protocols and a general framework to design new ones to improve the system performance. In the second part of this thesis, we consider the problem of secure transmission in the presence of a wiretapper. Due to wireless broadcast, wireless signals are particularly easy to jam and intercept. We derive the secrecy capacity region for the case when the location of the wiretapped links is known and propose several achievable strategies for the case when such information is unknown. We give an example to show that the secrecy capacities of the two cases are generally unequal and show that in both cases computing the secrecy capacity is NP-complete.

Coding for Wireless Broadcast and Network Secrecy

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Coding for Wireless Broadcast and Network Secrecy