Wiretap and Gelfand-Pinsker Channels Analogy and its Applications
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A framework of analogy between wiretap channels (WTCs) and state-dependent point-to-point channels with non-causal encoder channel state information (referred to as Gelfand-Pinker channels (GPCs)) is proposed. A good (reliable and secure) sequence of wiretap codes is shown to induce a good (reliable) sequence of codes for a corresponding GPC. Consequently, the framework enables exploiting existing results for GPCs to produce converse proofs for their wiretap analogs. The fundamental limits of communication of analogous wiretap and GP models are characterized by the same rate bounds; the optimization domains may differ. The analogy readily extends to multiuser broadcasting scenarios, encompassing broadcast channels (BCs) with deterministic components, degradation ordering between users, and BCs with cooperative receivers. Given a wiretap BC (WTBC) with two receivers and one eavesdropper, an analogous Gelfand-Pinsker BC (GPBC) is constructed by converting the eavesdropper's observations to a state sequence with an appropriate product distribution, and non-causally revealing the states to the encoder. The transition matrix of the (state-dependent) GPBC is the appropriate conditional marginal of the WTBC's transition law, with the eavesdropper's output playing the role of the channel state. Past capacity results for the semi-deterministic (SD) GPBC and the physically-degraded (PD) GPBC with an informed receiver are leveraged to furnish analogy-based converse proofs for the analogous WTBC setups. This characterizes the secrecy-capacity regions of the SD-WTBC, which was an open problem until this work, as well as the PD-WTBC where the stronger receiver also observes the eavesdropper's channel output. These new derivations exemplify the strength of the wiretap-GP analogy as a research tool through which results on one problem directly translate into advances in the study of the other.
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