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Secrecy Capacity Region of the AWGN MAC with External Eavesdropper and Feedback

Haoheng Yuan, Han Deng, Guangfen Xie, Chuanchuan Yang, Bin Dai

Published Dec 4, 2023• 2 citations
Publisher

Abstract

For the point-to-point additive white Gaussian noise (AWGN) channel with eavesdropper and feedback, it has already been shown that the secrecy capacity can be achieved by a secret key based feedback scheme, where the channel feedback is used for secret sharing, and then encrypting the transmitted message by the shared key, any capacity-achieving coding scheme for the AWGN channel without feedback is secure by itself, which indicates that the capacity of the same model without secrecy constraint is also an achievable secrecy rate of the AWGN channel with eavesdropper and feedback. Then it is natural to ask: is the secret key based feedback scheme still an optimal scheme for the AWGN multiple-access channel (MAC) with external eavesdropper and channel feedback (AWGN-MAC-E-CF), namely, achieving the secrecy capacity region of the AWGN-MAC-E-CF? In this paper, we show that the answer to the aforementioned question is no, and propose an optimal feedback coding scheme for the AWGN-MAC-E-CF, which combines an existing linear feedback scheme for the AWGN MAC with feedback and the secret key scheme in the literature. The study of this paper provides a way to find optimal coding schemes for AWGN multi-user channels in the presence of external eavesdronner and channel feedback.

DOI
10.1109/globecom54140.2023.10436998

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For the point-to-point additive white Gaussian noise (AWGN) channel with eavesdropper and feedback, it has already been shown that the secrecy capacity can be achieved by a secret key based feedback scheme, where the channel feedback is used for secret sharing, and then encrypting the transmitted message by the shared key, any capacity-achieving coding scheme for the AWGN channel without feedback is secure by itself, which indicates that the capacity of the same model without secrecy constraint is also an achievable secrecy rate of the AWGN channel with eavesdropper and feedback. Then it is natural to ask: is the secret key based feedback scheme still an optimal scheme for the AWGN multiple-access channel (MAC) with external eavesdropper and channel feedback (AWGN-MAC-E-CF), namely, achieving the secrecy capacity region of the AWGN-MAC-E-CF? In this paper, we show that the answer to the aforementioned question is no, and propose an optimal feedback coding scheme for the AWGN-MAC-E-CF, which combines an existing linear feedback scheme for the AWGN MAC with feedback and the secret key scheme in the literature. The study of this paper provides a way to find optimal coding schemes for AWGN multi-user channels in the presence of external eavesdronner and channel feedback.
For the point-to-point additive white Gaussian noise (AWGN) channel with an eavesdropper and feedback, it has already been shown that the secrecy capacity can be achieved by a secret key-based feedback scheme, where the channel feedback is used for secret sharing, and then encrypting the transmitted message by the shared key. By secret sharing, any capacity-achieving coding scheme for the AWGN channel without feedback can be secure by itself, which indicates that the capacity of the same model without the secrecy constraint also affords an achievable secrecy rate to the AWGN channel with an eavesdropper and feedback. Then it is natural to ask: is the secret key-based feedback scheme still the optimal scheme for the AWGN multiple-access channel (MAC) with an external eavesdropper and channel feedback (AWGN-MAC-E-CF), namely, achieving the secrecy capacity region of the AWGN-MAC-E-CF? In this paper, we show that the answer to the aforementioned question is no, and propose the optimal feedback coding scheme for the AWGN-MAC-E-CF, which combines an existing linear feedback scheme for the AWGN MAC with feedback and the secret key scheme in the literature. This paper provides a way to find optimal coding schemes for AWGN multi-user channels in the presence of an external eavesdropper and channel feedback.
In this paper, the finite-order autoregressive moving average (ARMA) Gaussian wiretap channel with noiseless causal feedback is considered, in which an eavesdropper receives noisy observations of signals in both forward and feedback channels. It is shown that the generalized Schalkwijk-Kailath scheme, a capacity-achieving coding scheme for the Gaussian feedback channel, achieves the same maximum rate for the same channel even with the presence of an eavesdropper. Therefore, the secrecy capacity is equal to the feedback capacity without the presence of an eavesdropper for the Gaussian feedback channel. Furthermore, the results are extended to the additive white Gaussian noise (AWGN) channel with quantized feedback. It is shown that the proposed coding scheme achieves a positive secrecy rate. Our result implies that as the amplitude of the quantization noise decreases to zero, the secrecy rate converges to the capacity of the AWGN channel.
In this paper, the k-th order autoregressive moving average (ARMA(k)) Gaussian wiretap channel with noiseless causal feedback is considered, in which an eavesdropper receives noisy observations of the signals in both forward and feedback channels. It is shown that a variant of the generalized Schalkwijk-Kailath scheme, a capacity-achieving coding scheme for the feedback Gaussian channel, achieves the same maximum rate for the same channel with the presence of an eavesdropper. Therefore, the secrecy capacity is equal to the feedback capacity without the presence of an eavesdropper for the feedback channel. Furthermore, the results are extended to the additive white Gaussian noise (AWGN) channel with quantized feedback. It is shown that the proposed coding scheme achieves a positive secrecy rate. As the amplitude of the quantization noise decreases to zero, the secrecy rate converges to the capacity of the AWGN channel.
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.
In this paper, the finite-order autoregressive moving average (ARMA) Gaussian wiretap channel with noiseless causal feedback is considered, in which an eavesdropper receives noisy observations of the signals in both forward and feedback channels. It is shown that the generalized Schalkwijk-Kailath scheme, a capacity-achieving coding scheme for the feedback Gaussian channel, achieves the same maximum rate for the same channel with the presence of an eavesdropper. Therefore, the secrecy capacity is equal to the feedback capacity without the presence of an eavesdropper for the feedback channel. Furthermore, the results are extended to the additive white Gaussian noise (AWGN) channel with quantized feedback. It is shown that the proposed coding scheme achieves a positive secrecy rate. As the amplitude of the quantization noise decreases to zero, the secrecy rate converges to the capacity of the AWGN channel.
It is commonly known that physical layer security is achieved with a trade-off in terms of the achievable rate. Hence, security constraints generate rate losses in wiretap channels. To mitigate such rate losses in multi-user channels, we propose a coding/decoding scheme for multi-user multiple access wiretap channel (MAC-WT), where previously transmitted messages are used as a secret key to enhance the secrecy rates of the transmitting users, until the usual Shannon capacity region of a multiple access channel (MAC) is achieved without the secrecy constraint. With this coding scheme, all messages transmitted in the recent past are secure with respect to all the information of the eavesdropper till now. To achieve this goal, we introduce secret key buffers at both the users and the legitimate receiver. Finally, we consider a fading MAC-WT and show that with this coding/decoding scheme, we can achieve the capacity region of a fading MAC channel (in the ergodic sense).