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International Symposium on Information Theory and its Applications

On the Capacity of Gaussian Multiple-Access Wiretap Channels with Feedback

Bin Dai, Chong Li, Yingbin Liang, Zheng Ma, Shlomo Shitz Shamai

Published Oct 24, 2020• 2 citations
Publisher

Abstract

ISITA2020, Kapolei, Hawai'i, USA, October 24-27, 2020
DOI
10.5281/zenodo.4699137

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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.

•Jul 29, 2020•1 citation
It has been shown that the SK scheme, which was proposed by Schalkwijk and Kailath, is a self-secure capacity-achieving (SSCA) feedback scheme for the Gaussian wiretap channel, i.e., the SK scheme not only achieves the feedback capacity of the Gaussian channel, but also is secure by itself and achieves the feedback secrecy capacity of the Gaussian wiretap channel. For the multi-user wiretap channels, very recently, it has been shown that Ozarow’s capacity-achieving feedback scheme for the two-user Gaussian multiple-access channel (GMAC) is the SSCA feedback scheme for the two-user Gaussian multiple-access wiretap channel (GMAC-WT). In this paper, first, we propose a SSCA feedback scheme for the two-user GMAC-WT with degraded message sets (GMAC-WT-DMS). Next, we extend the above scheme to the two-user GMAC-WT-DMS with noncausal channel state information at the transmitters (NCSIT), and show that the extended scheme is also a SSCA feedback scheme. Finally, we derive outer bounds on the secrecy capacity regions of the two-user GMAC-WT-DMS with or without NCSIT, and numerical results show the rate gains by the feedback.
The Schalkwijk-Kailath (SK) scheme, which achieves the capacity of the point-to-point white Gaussian channel with feedback, is secure by itself and also achieves the secrecy capacity of the Gaussian wiretap channel with feedback, i.e., the SK scheme is a self-secure capacity-achieving (SSCA) feedback scheme for the Gaussian wiretap channel. For the multi-user wiretap channels, recently, it has been shown that Ozarow's capacity-achieving feedback scheme for the two-user Gaussian multiple-access channel (GMAC) is the SSCA feedback scheme for the two-user Gaussian multiple-access wiretap channel (GMAC-WT). In this paper, first, we propose a capacity-achieving feedback scheme for the two-user GMAC with degraded message sets (GMAC-DMS), and show that this scheme is the SSCA feedback scheme for the two-user GMAC-WT with degraded message sets (GMAC-WT-DMS). Next, we extend the above scheme to the two-user GMAC-DMS with noncausal channel state information at the transmitters (NCSIT), and show that the extended scheme is capacity-achieving and also a SSCA feedback scheme for the two-user GMAC-WT-DMS with NCSIT. Finally, we derive outer bounds on the secrecy capacity regions of the two-user GMAC-WT-DMS with or without NCSIT, and numerical results show the rate gains by the feedback.
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.