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IEEE Convention of Electrical and Electronics Engineers in Israel

The Gaussian wiretap channel with correlated sources at the terminals: Secret communication and key generation

Alexander Bunin, Pablo Piantanida, Shlomo Shitz Shamai

Published Nov 1, 2016• 4 citations
Publisher

Abstract

This paper considers the setup of Alice, Bob and Eve sharing a Gaussian Broadcast channel, and correlated Gaussian sources, assuming these are independent of the channel components. Both secret key generation and secret message transmission are considered. A closed form expression is presented characterizing the optimal trade-off between the secret key and the secret message rates. A few known results are shown to be accepted as special cases.

DOI
10.1109/icsee.2016.7806128

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Alice and Bob want to share a secret key and to communicate an independent message, both of which they desire to be kept secret from an eavesdropper Eve. This problem of secret communication and secret-key generation when two resources are available-correlated sources at Alice, Bob, and Eve, and a noisy broadcast channel from Alice to Bob and Eve which is independent of the sources is studied. The goal is to characterize the fundamental tradeoff between the rates of the secret message and secret key. An achievable solution and proof of its optimality for the parallel channels and sources case when each subchannel and source component satisfies a degradation order (either in favor of the legitimate receiver or the eavesdropper) is presented. This includes the case of jointly Gaussian sources and an additive Gaussian channel, for which the secrecy region is evaluated.
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Alice wants to send Bob a secret message in the presence of an eavesdropper Eve. Alice has two resources available: a noisy broadcast channel and source observations correlated with observations at Bob and Eve. We consider two cases, the first in which the source observations and channel are independent and the second in which they can depend on each other. For the independent case, we consider a strategy that exploits the sources and channels separately and show this separation strategy is optimal for a class of sources and channels that includes a Gaussian example. For the case in which the sources and channels depend on each other, this separation strategy does not work, and an optimal strategy has yet to be established. For this setting, we improve on the best known upper bound to the secret message capacity.
A joint-source-channel setup for secret-key generation between remote terminals is considered. The sender communicates to the receiver over a discrete memoryless wiretap channel and the sender and receiver observe a pair of correlated discrete memoryless sources. Lower and upper bounds for the secret-key rate are presented and shown to coincide for the case when the underlying channel is a reversely degraded parallel channel. Our setup also provides an operational significance to the rate-equivocation tradeoff of the wiretap channel, and this is illustrated in detail for the Gaussian case.
In this paper, we consider the problem of secret key generation with one-way communication through both a rate-limited public channel and a rate-limited secure channels where the public channel is from Alice to Bob and Eve and the secure channel is from Alice to Bob. In this model, we do not pose any constraints on the sources, i.e. Bob is not degraded to or less noisy than Eve. We obtain the optimal secret key rate in this problem, both for the discrete memoryless sources and vector Gaussian sources. The vector Gaussian characterization is derived by suitably applying the enhancement argument, and proving a new extremal inequality. The extremal inequality can be seen as coupling of two extremal inequalities, which are related to the degraded compound MIMO Gaussian broadcast channel, and the vector generalization of Costa's entropy power inequality, accordingly.