Feedback Capacities of Gaussian Multiple-Access Wiretap Channels
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The feedback capacities of the Gaussian multiple-access channel (GMAC) and the GMAC with noncausal channel state information at the transmitters (NCSIT) are achieved by Ozarow's Schalkwijk-Kailath (SK) type feedback scheme and its extension (also called Rosenzweig's feedback scheme), respectively. In this paper, first, we show that the above Ozarow's and Rosenzweig's feedback schemes, which are not designed with the consideration of secrecy, already achieve the secrecy capacity regions (capacity regions with perfect weak secrecy constraint) of the Gaussian multiple-access wiretap channel (GMAC-WT) with feedback and the GMAC-WT-NCSIT with feedback, respectively. This indicates that for Gaussian multiple-access wiretap channels, the traditional secret key based feedback schemes are not necessary. Next, we show that the feedback capacities of the GMAC with degraded message sets (GMAC-DMS) and the GMAC-NCSIT with degraded message sets (GMAC-NCSIT-DMS) can be achieved by two-step SK type feedback schemes. Furthermore, we show that the above capacity-achieving feedback schemes also achieve the secrecy capacity regions of the GMAC-WT-DMS with feedback and the GMAC-WT-NCSIT-DMS with feedback, i.e., the secrecy capacity regions of the GMAC-WT-DMS with feedback and the GMAC-WT-NCSIT-DMS with feedback equal the capacity regions of the same models without secrecy constraint. Finally, we derive outer bounds on the secrecy capacity regions of GMAC-WT, GMAC-WT-DMS and GMAC-WT-NCSIT-DMS, and show that feedback helps to increase the secrecy capacity regions of GMAC-WT and GMAC-WT-DMS, and in some cases helps to increase the secrecy capacity regions of GMAC-WT-NCSIT and GMAC-WT-NCSIT-DMS. Overall, the main contribution of this paper is to find the inherent secrecy nature of SK-type feedback schemes for the GMACs.
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