An LP Characterization of the Secret-message Capacity of Three Erasure Networks With Feedback

László Czap, Suhas Diggavi, Christina Fragouli, Vinod Malathidevi Prabhakaran
Ieee-Inst Electrical Electronics Engineers Inc, 2016
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Résumé

This paper presents exact capacity characterizations for the case, when a principal, Alice, wants to securely send a message to another principal, Bob, over three network configurations: the parallel edges network, the V-network, and the triangle network. We assume that: 1) a passive eavesdropper, Eve, overhears any one edge in the network; 2) each edge corresponds to an independent broadcast packet erasure channel with arbitrary erasure probabilities; and 3) all legitimate nodes can publicly but causally acknowledge whether they received each packet or not. We develop optimal achievability schemes that are expressed as linear programs (LPs) and share a two-phase structure, where at the first phase, we create secret keys, and at the second phase, we use them to encrypt the transmitted message. Our outer bounds are also expressed through LP formulations. We prove that our schemes are optimal by showing that the optimal solution of the outer bound LP and the optimal solution of the achievability scheme LP coincide.

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https://infoscience.epfl.ch/record/219760?ln=fr

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The security and efficiency of communication are two of the main concerns for networks of today and the future. Our understanding of how to efficiently send information over various channels and networks has significantly increased in the past decade (see e.g., [1–3]), whereas our understanding of how to securely send information has not yet reached the same level. In this thesis, we advance the theory of secure communication by deriving capacity results and by developing coding schemes that provide information-theoretic security for erasure networks. We characterize the highest achievable secret-message rate in the presence of an eavesdropping adversary in various settings, where communication takes place over erasure channels with state-feedback. Our results provide such a characterization for a point-to-point erasure channel, for a broadcast erasure channel with multiple receivers, for a network with multiple parallel channels, a V-network and for a triangle network. We introduce several two-phase secure coding schemes that consist of a key generation phase and an encrypted message sending phase. Our schemes leverage several resources for security: channel erasures, feedback, common randomness and the topology of the network. We present coding schemes for all the above mentioned settings as well as for erasure networks with arbitrary topology. In all the cases where we provide exact characterization, a two-phase scheme achieves the secret-message capacity. All our proposed coding schemes use only linear operations and thus can serve as a basis for practical code designs. For networks, we develop a linear programming framework for describing secure coding schemes and for deriving new outer bounds. We use linear programs to describe our schemes and to prove their optimality. We derive new information theoretic outer bounds. In our intuitive interpretation, our proofs find the connection between the rate of the message and the rate of a secret key that is required to secure the message. Our results reveal nontrivial characteristics of secure communication in erasure networks. We find that – in contrast to non-secure communication – the secret message capacity of a cut does not simplify to the sum of the capacities of the channels that form the cut, moreover, the secret message capacity of a network does not simplify to the minimum secret message capacity of its cuts.

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