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.
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.
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.
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 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.
In this paper, the Gaussian wiretap feedback channel is revisited, and some new results on its secrecy capacity are obtained. To be specific, first, we show that the Schalkwijk–Kailath (SK) feedback scheme, which achieves the secrecy capacity of the degraded Gaussian wiretap feedback channel, also achieves the secrecy capacity of the non-degraded Gaussian wiretap feedback channel. Second, applying the existing secret key-based feedback schemes to Gaussian wiretap feedback channels, we derive some new lower bounds on the secrecy capacities of these models. Finally, we compare the performances of the above feedback schemes in the degraded and non-degraded Gaussian wiretap feedback channels and show which feedback scheme performs better for these channel models.
In the literature, it has been shown that the secrecy capacity of the additive white Gaussian noise (AWGN) wiretap channel with noise-free feedback equals the capacity of the same model without secrecy constraint, and the classical Schalkwijk-Kailath (SK) scheme achieves the secrecy capacity. In this paper, we show that in finite blocklength regime, the SK scheme is not optimal, and propose a modified SK scheme which may perform better than the classical one. Besides this, this paper establishes a finite blocklength converse for the AWGN wiretap channel with feedback, which can also be viewed as a converse for the same model without secrecy constraint. To the best of the authors' knowledge, this is the first paper to address such a problem, and the results of this paper are further explained via numerical examples.
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.