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arXiv:2606.25351

Inner and Outer Bounds on the Secrecy Capacity of Degraded Broadcast Channels with RMSI and Transmitter CSI

Saeid Pakravan, Mohsen Ahmadzadeh, Ghosheh Abed Hodtani

Published Jun 24, 2026• 0 citations
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Abstract

This paper studies the secrecy capacity of a class of degraded broadcast channels in the presence of an external eavesdropper, where a transmitter aims to deliver two independent confidential messages to two legitimate receivers. The transmitter is assumed to have non-causal access to the channel state information (CSI), and each legitimate receiver possesses prior knowledge of the other receiver's message, referred to as receiver message side information (RMSI). We consider two distinct scenarios: complementary RMSI, where each receiver knows only the other's message, and non-complementary RMSI, where the side information does not perfectly align. For both scenarios, we derive novel inner bounds on the achievable secrecy rate region and present tight outer bounds, establishing the secrecy capacity region for the considered degraded channel settings. Unlike prior works, which primarily address general broadcast settings without secrecy constraints or omit key interactions between RMSI and CSI, our results provide a complete characterization of the secure communication limits under these conditions. Moreover, we extend our analysis to the Gaussian degraded broadcast channel, highlighting the pivotal role of CSI in enhancing secure transmission performance. Our findings demonstrate that the combination of RMSI and CSI can be strategically leveraged to expand the secrecy capacity region, thus offering new insights into secure multiuser communication system design.

DOI
10.48550/arxiv.2606.25351

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This paper studies the secrecy capacity of a class of degraded broadcast channels in the presence of an external eavesdropper, where a transmitter aims to deliver two independent confidential messages to two legitimate receivers. The transmitter is assumed to have non-causal access to the channel state information (CSI), and each legitimate receiver possesses prior knowledge of the other receiver's message, referred to as receiver message side information (RMSI). We consider two distinct scenarios: complementary RMSI, where each receiver knows only the other's message, and non-complementary RMSI, where the side information does not perfectly align. For both scenarios, we derive novel inner bounds on the achievable secrecy rate region and present tight outer bounds, establishing the secrecy capacity region for the considered degraded channel settings. Unlike prior works, which primarily address general broadcast settings without secrecy constraints or omit key interactions between RMSI and CSI, our results provide a complete characterization of the secure communication limits under these conditions. Moreover, we extend our analysis to the Gaussian degraded broadcast channel, highlighting the pivotal role of CSI in enhancing secure transmission performance. Our findings demonstrate that the combination of RMSI and CSI can be strategically leveraged to expand the secrecy capacity region, thus offering new insights into secure multiuser communication system design.
In this article, a broadcast channel in the degraded state with two legitimate receivers in the attendance of an eavesdropper has been studied, where, the sender demands to forward two independent confidential messages that ought to be kept as secret as feasible of the eavesdropper to the legitimate receivers. In the studied system, it has been supposed that the non-causal channel state information has been considered at the transmitter side and every one of receivers has the message forwarded via source demanded via the further one. The model studied in this paper is further generalized into the formerly considered broadcast channel with confidential messages and considering side information in the receiver. It has been supplied an outer bound by considering confidentially conditions for proposed system.
We study the two-transmitter two-receiver channel with confidential messages channel under a secrecy constraint, exploring the effect of a single eavesdropper on the secure communication rate. A general achievable secrecy rate region and an outer bound are derived for the degraded version of this channel. An example is provided in which the inner and outer bound meet. For the general non-degraded channel, another outer bound is derived and a special case is highlighted in which the inner and outer bound meet. Our inner bound recovers the known inner bounds for the multiple-access wiretap channel, broadcast channel with confidential messages, and the compound MAC channel.
We study secure communication in which two confidential messages are transmitted over a broadcast channel to two legitimate receivers, while keeping an eavesdropper ignorant. Each legitimate receiver is interested in decoding one confidential message, while having the other one as side information. In order to measure the secrecy of the communication, we investigate two different secrecy criteria: joint secrecy and individual secrecy. For both criteria, we provide an achievable rate region and a matching multi-letter outer bound presenting a multi-letter description for the capacity region. We further investigate the class of more capable channels and provide a single-letter converse establishing the secrecy capacity region, not only for more capable channels but less noisy and degraded channels as well. Our results indicate that the secrecy capacity for individual secrecy is higher than the one for joint secrecy, as one message can be used as a secret key for the other one.
In this paper, we consider a scenario where a source node wishes to broadcast two confidential messages for two respective receivers, while a wire-tapper also receives the transmitted signal. This model is motivated by wireless communications, where individual secure messages are broadcast over open media and can be received by any illegitimate receiver. The secrecy level is measured by equivocation rate at the eavesdropper. We first study the general (non-degraded) broadcast channel with confidential messages. We present an inner bound on the secrecy capacity region for this model. The inner bound coding scheme is based on a combination of random binning and the Gelfand-Pinsker bining. This scheme matches the Marton's inner bound on the broadcast channel without confidentiality constraint. We further study the situation where the channels are degraded. For the degraded broadcast channel with confidential messages, we present the secrecy capacity region. Our achievable coding scheme is based on Cover's superposition scheme and random binning. We refer to this scheme as Secret Superposition Scheme. In this scheme, we show that randomization in the first layer increases the secrecy rate of the second layer. This capacity region matches the capacity region of the degraded broadcast channel without security constraint. It also matches the secrecy capacity for the conventional wire-tap channel. Our converse proof is based on a combination of the converse proof of the conventional degraded broadcast channel and Csiszar lemma. Finally, we assume that the channels are Additive White Gaussian Noise (AWGN) and show that secret superposition scheme with Gaussian codebook is optimal. The converse proof is based on the generalized entropy power inequality.
We study the degraded multi-receiver wiretap channel with public and confidential messages. In this channel, there is a transmitter that wishes to communicate with two legitimate users in the presence of an external eavesdropper. The legitimate users and the eavesdropper satisfy a certain degradation order. In this paper, we study this channel model for the scenario where the transmitter sends a pair of public and confidential messages to each legitimate user. While there are no secrecy constraints on the public messages, confidential messages need to be transmitted in perfect secrecy. First, we obtain an inner bound for the capacity region of the discrete memoryless channel by using an achievable scheme that uses superposition coding and binning. Next, we obtain an outer bound for the capacity region of the degraded discrete memoryless channel. We show that this outer bound partially matches the inner bound. Consequently, we provide a partial characterization of the capacity region of the degraded discrete memoryless channel. Finally, we consider the degraded Gaussian multi-input multi-output (MIMO) wiretap channel with public and confidential messages. We show that, to evaluate both the inner and outer bounds for the Gaussian MIMO case, considering only jointly Gaussian auxiliary random variables and channel input is sufficient. Since the inner and outer bounds provided earlier for the degraded discrete memoryless channel partially match, these sufficiency results provide a partial characterization of the capacity region of the degraded Gaussian MIMO channel.
This paper studies the problem of secure communication over the broadcast channel with receiver-side information under the lens of individual secrecy constraints, that is, the transmitter wants to send two independent messages to two receivers, which have, respectively, the desired message of the other receiver as side information, while keeping the eavesdropper ignorant of each message (i.e., the information leakage rate from each message to the eavesdropper is made vanishing). Building upon one-time pad, secrecy coding, and broadcasting schemes, achievable rate regions are investigated, and the capacity region for special cases of either a weak or strong eavesdropper (compared to both legitimate receivers) is characterized. Interestingly, the capacity region for the former corresponds to a line and the latter corresponds to a rectangle with missing corners; a phenomenon occurring due to the coupling between user's rates. Moreover, the individual secrecy capacity region is also fully characterized for the case where the eavesdropper's channel is deterministic. In addition to discrete memoryless setup, Gaussian scenarios are studied. For the Gaussian model, in addition to the strong and weak eavesdropper cases, the capacity region is characterized for the low and high SNR regimes when the eavesdropper's channel is stronger than one receiver but weaker than the other. Remarkably, positive secure transmission rates are always guaranteed under the individual secrecy constraint, unlike the case of the joint secrecy constraint (i.e., the information leakage rate from both messages to the eavesdropper is made vanishing). Thus, this notion of secrecy serves as an appropriate candidate for trading off secrecy level and transmission rate, making secrecy more affordable but still acceptable to the end user.