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IEEE Transactions on Information Forensics and Security

Two-User Gaussian Broadcast Wiretap Channel With Common Message and Feedback: Revisit

Haoheng Yuan, Yanghe Feng, Chuanchuan Yang, Zhuojun Zhuang, Bin Dai

Published Sep 25, 2023• 5 citations
Publisher

Abstract

The two-user Gaussian broadcast wiretap channel with common message and feedback (GBC-WTC-CM-F) is revisited. Traditionally, achievable secrecy rate of this model is achieved by combining Marton’s coding scheme for the two-user broadcast channel (BC) and the secret-key based feedback scheme, where both of the two feedback links are used to transmit secret keys shared between the transceivers. Recently, it has been shown that for the Gaussian wiretap channel with feedback, the Schalkwijk-Kailath (SK) feedback scheme achieves its secrecy capacity. Then it is natural to ask: can we do better when applying the SK-type scheme to the GBC-WTC-CM-F? In this paper, we answer this question by proposing two kinds of SK-type schemes. Specifically, first, we propose a hybrid scheme where one feedback link is used to transmit a secret key, and the other one is used for SK-type coding. We show that this hybrid scheme may perform better than the existing one in some cases. Next, we show that Ozarow’s extended SK scheme for the two-user Gaussian BC with feedback, where both feedback links are used for SK-type coding, is self-secure (satisfying perfect weak secrecy constraint by itself) and may perform the best. We further show that Ozarow’s scheme is in fact a secure finite blocklength coding scheme, and extend it to the static fading SISO and SIMO cases. Finally, the results of this paper are further explained by numerical examples.

DOI
10.1109/tifs.2023.3318948

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The two-user Gaussian broadcast wiretap channel with common message and feedback (GBC-WTC-CM-F) is revisited. Traditionally, achievable secrecy rate of this model is achieved by combining Marton’s coding scheme for the two-user broadcast channel (BC) and the secret-key based feedback scheme, where both of the two feedback links are used to transmit secret keys shared between the transceivers. Recently, it has been shown that for the Gaussian wiretap channel with feedback, the Schalkwijk-Kailath (SK) feedback scheme achieves its secrecy capacity. Then it is natural to ask: can we do better when applying the SK-type scheme to the GBC-WTC-CM-F? In this paper, we answer this question by proposing two kinds of SK-type schemes. Specifically, first, we propose a hybrid scheme where one feedback link is used to transmit a secret key, and the other one is used for SK-type coding. We show that this hybrid scheme may perform better than the existing one in some cases. Next, we show that Ozarow’s extended SK scheme for the two-user Gaussian BC with feedback, where both feedback links are used for SK-type coding, is self-secure (satisfying perfect weak secrecy constraint by itself) and may perform the best. We further show that Ozarow’s scheme is in fact a secure finite blocklength coding scheme, and extend it to the static fading SISO and SIMO cases. Finally, the results of this paper are further explained by numerical examples.
In the literature, it has already been shown that a feedback control based coding scheme enhances the secrecy capacity of the Gaussian broadcast wiretap channel (GBC-WTC) with common message, and it performs better than the conventional random key based feedback scheme. However, note that the previously proposed scheme is based on the assumption that both the two receivers can send their received signals back to the transmitter via two feedback channels, which may not be available when a certain feedback channel fails to work. In this paper, we study the GBC-WTC with common message and partial feedback, where only one legitimate receiver can send his received signal back to the sender via a perfect feedback channel. A hybrid scheme, which combines the classical Schalkwijk-Kailath (SK) feedback control scheme, a dirty paper pre-coding scheme and the conventional random binning scheme is proposed for this model. Numerical results show that even in such a partial feedback case, our proposed scheme still works well and outperforms the existing secret-key based scheme in the literature.
The Schalkwijk-Kailath (SK) feedback scheme is a capacity-achieving coding scheme for the point-to-point white Gaussian channel with feedback. Recently, it has been shown that the SK scheme, which is not designed with consideration of secrecy, already achieves perfect weak secrecy by itself, i.e., the secrecy capacity of the Gaussian wiretap channel with feedback equals the capacity of the same model without secrecy constraint. In this paper, we propose a capacity-achieving SK type feedback scheme for the two-user Gaussian multiple-access channel with degraded message sets (GMAC-DMS). Similarly to the inherent secrecy nature of the classical SK scheme, we show that the proposed scheme is also secure by itself, which indicates that the feedback secrecy capacity of the two-user Gaussian multiple-access wiretap channel with degraded message sets (GMAC-WT-DMS) equals the capacity of the same model without secrecy constraint.
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 current feedback coding strategy for the physical layer security (PLS) mainly focuses on generating secret key used for protecting the transmitted message, and this strategy has been proved only optimal for several degraded cases. Our previous work has proposed a new hybrid feedback strategy for the classical PLS model-the wiretap channel (WTC), where the feedback is not only used to generate key, but also used to produce cooperative information helping the legitimate parties to improve the communication between them. It has been shown that for the general WTC, this hybrid feedback strategy performs better than the already existing secret key based feedback scheme. In this paper, we extend the previous proposed feedback strategy for the WTC to a multi-terminal case, and apply this generalized feedback strategy to the broadcast channel with mutual secrecy requirement (BC-MSR), which constitutes the essence of physical layer security (PLS) in the down-link of the wireless communication systems. From a non-degraded example (Dueck-type broadcast channel), we show that for the BC-MSR, this generalized feedback strategy may perform better than the already existing one.
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.
Broadcast channel (BC) faces critical physical layer security (PLS) challenges due to their inherent openness. Traditional encryption methods are costly and unsuitable for low-latency applications. Recent studies have shown that Schalk-wijk-Kailath (SK) feedback coding scheme naturally satisfies PLS and achieves double-exponential error decay. This paper investigates two-user Single-Input Multiple-Output (SIMO) broad-cast wiretap channel with partial feedback. A SK-type feedback control scheme based on scalarization and channel splitting is proposed for the two-user SIMO broadcast wiretap channel. Finially, numerical results demonstrate that the proposed scheme outperforms existing secret-key-based methods in the literature.