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arXiv:2411.06339 · IEEE Journal on Selected Areas in Communications

Probabilistic Shaped Multilevel Polar Coding for Wiretap Channel

Li Shen, Yongpeng Wu, Peihong Yuan, Chengshan Xiao, Xiang‐Gen Xia, Wenjun Zhang

Published Jan 30, 2025• 0 citations
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Abstract

A wiretap channel is served as the fundamental model of physical layer security techniques, where the secrecy capacity of the Gaussian wiretap channel is proven to be achieved by Gaussian input. However, there remains a gap between the Gaussian secrecy capacity and the secrecy rate with conventional uniformly distributed discrete constellation input, e.g. amplitude shift keying (ASK) and quadrature amplitude modulation (QAM). In this paper, we propose a probabilistic shaped multilevel polar coding scheme to bridge the gap. Specifically, the input distribution optimization problem for maximizing the secrecy rate with ASK/QAM input is solved. Numerical results show that the resulting sub-optimal solution can still approach the Gaussian secrecy capacity. Then, we investigate the polarization of multilevel polar codes for the asymmetric discrete memoryless wiretap channel, and thus propose a multilevel polar coding scheme integration with probabilistic shaping. It is proved that the scheme can achieve the secrecy capacity of the Gaussian wiretap channel with discrete constellation input, and satisfies the reliability condition and weak security condition. A security-oriented polar code construction method to natively satisfies the leakage-based security condition is also investigated. Simulation results show that the proposed scheme achieves more efficient and secure transmission than the uniform constellation input case over both the Gaussian wiretap channel and the Rayleigh fading wiretap channel.

DOI
10.1109/jsac.2025.3531539

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A wiretap channel is served as the fundamental model of physical layer security techniques, where the secrecy capacity of the Gaussian wiretap channel is proven to be achieved by Gaussian input. However, there remains a gap between the Gaussian secrecy capacity and the secrecy rate with conventional uniformly distributed discrete constellation input, e.g. amplitude shift keying (ASK) and quadrature amplitude modulation (QAM). In this paper, we propose a probabilistic shaped multilevel polar coding scheme to bridge the gap. Specifically, the input distribution optimization problem for maximizing the secrecy rate with ASK/QAM input is solved. Numerical results show that the resulting sub-optimal solution can still approach the Gaussian secrecy capacity. Then, we investigate the polarization of multilevel polar codes for the asymmetric discrete memoryless wiretap channel, and thus propose a multilevel polar coding scheme integration with probabilistic shaping. It is proved that the scheme can achieve the secrecy capacity of the Gaussian wiretap channel with discrete constellation input, and satisfies the reliability condition and weak security condition. A security-oriented polar code construction method to natively satisfies the leakage-based security condition is also investigated. Simulation results show that the proposed scheme achieves more efficient and secure transmission than the uniform constellation input case over both the Gaussian wiretap channel and the Rayleigh fading wiretap channel.
Suppose Alice wishes to send messages to Bob through a communication channel C_1, but her transmissions also reach an eavesdropper Eve through another channel C_2. The goal is to design a coding scheme that makes it possible for Alice to communicate both reliably and securely. Reliability is measured in terms of Bob's probability of error in recovering the message, while security is measured in terms of Eve's equivocation ratio. Wyner showed that the situation is characterized by a single constant C_s, called the secrecy capacity, which has the following meaning: for all ε>0ε> 0ε>0, there exist coding schemes of rate R≥Cs−εR \ge C_s - εR≥Cs​−ε that asymptotically achieve both the reliability and the security objectives. However, his proof of this result is based upon a nonconstructive random-coding argument. To date, despite a considerable research effort, the only case where we know how to construct coding schemes that achieve secrecy capacity is when Eve's channel C_2 is an erasure channel, or a combinatorial variation thereof. Polar codes were recently invented by Arikan; they approach the capacity of symmetric binary-input discrete memoryless channels with low encoding and decoding complexity. Herein, we use polar codes to construct a coding scheme that achieves the secrecy capacity for a wide range of wiretap channels. Our construction works for any instantiation of the wiretap channel model, as long as both C_1 and C_2 are symmetric and binary-input, and C_2 is degraded with respect to C_1. Moreover, we show how to modify our construction in order to provide strong security, in the sense defined by Maurer, while still operating at a rate that approaches the secrecy capacity. In this case, we cannot guarantee that the reliability condition will be satisfied unless the main channel C_1 is noiseless, although we believe it can be always satisfied in practice.
This paper studies a practical design of probabilistic constellation shaping (PCS) for enhancing the performance of physical layer security in visible light communications (VLC). In particular, we consider a wiretap VLC channel employing a probabilistically shapedM-ary pulse amplitude modulation (PAM) constellation. Considering the requirements for reliability of the legitimate user’s channel, flickering-free transmission, and symmetric constellation distribution, the optimal constellation distributions to maximize modulation-constrained secrecy capacity or the bit error rate (BER) of eavesdropper’s channel are investigated for both scenarios of known and unknown eavesdropper’s channel state information (CSI). To formulate the constraint on the channel reliability, tractable closed-form expressions for the upper bound and approximate BER ofM-ary PAM under an arbitrary symbol probability are derived. The design problem is shown to be non-convex due to the non-convex BER constraint. By proving that the upper bound BER is a concave function of the constellation distribution, a suboptimal solution based on the convex-concave procedure (CCCP) is presented. Our findings reveal that while the uniform signaling can only satisfy the BER constraint when the optical power is beyond a certain value, the proposed PCS design works in the entire region of the optical power. Some insights into the optimal constellation distribution with respect to the emitted optical power are also discussed.
Orthogonal frequency division multiplexing (OFDM) systems have enjoyed widespread adoption as the physical layer standard for high data rate wired and wireless networks, due to their ability to efficiently cope with slowly varying dispersive channels. Therefore in searching for feasible implementations of physical layer security techniques, it is appropriate to analyze how existing information theoretic results can be applied to the OFDM structure This paper considers the information theoretic secrecy rates that are achievable in a wiretap OFDM channel when transmitting quadrature amplitude modulation (QAM) constellation symbols. The loss with respect to the secrecy rates obtained with Gaussian distributed inputs is evaluated for both nite constellation cardinalities and in the asymptotic approximation of arbitrarily high cardinality. Moving from the insight gained with this analysis, we propose bit-loading strategies to efficiently allocate the appropriate number of bits in each subchannel, by considering the twofold objective of minimizing the loss with respect to the Gaussian input secrecy capacity and minimizing the total bit load.
Polar codes have been proven to be capacity achieving for any binary-input discrete memoryless channel, while at the same time they can reassure secure and reliable transmission over the single-input single-output wireless channel. However, the use of polar codes to secure multiple-antenna transmission and reception has not yet been reported in the open literature. In this paper, we assume a multiple-input multiple-output wiretap channel, where the legitimate receiver and the eavesdropper are equipped with the same number of antennas. We introduce a protocol that exploits the properties of both physical and media access control layer security by employing polar coding and encryption techniques in a hybrid manner in order to guarantee secure transmission. A novel security technique is also proposed, where a cryptographic key is generated based on the information transmitted and renewed every transmission block without the need for a separate key exchange method. Finally, to illustrate the effectiveness of the proposed protocol, we prove the weak and strong security conditions, and we provide a practical method to achieve computational security for the cases where these conditions cannot be established.
This paper proposes a method to provide secrecy for digital communications with arbitrarily large quadrature amplitude modulation (QAM) constellations for transmission over a Gaussian fading wiretap channel. This is accomplished by breaking the constellation down into nested quadrature phase-shift keying (QPSK) symbols and randomizing the assignment between message bits and modulated symbols using channel state information (CSI). If enough random bits can be generated from CSI it becomes possible to uniquely map an arbitrary message to any symbol in the large QAM constellation. The proposed method can thereby provide perfect secrecy while maintaining high reliability by exclusively assigning minimum-distance-mapped constellations through the randomization for use by the legitimate decoder.
Channel state information (CSI) is differently available at each terminal in state-dependent wiretap channels (SD-WTCs). Considering a random channel state non-causally available only at the encoder, this paper investigates an explicit polar coding scheme for the discrete and memoryless SD-WTC, where Alice aims to transmit a secret message (SM) and a secret key (SK) to Bob while concealing them from Eve. Based on a two-layer superposition coding, which includes an inner layer and an outer layer, the proposed polar coding scheme can achieve the inner bound of the current optimal SM-SK capacity region for the discrete and memoryless SD-WTC with CSI non-causally available only at the encoder. A cross-layer construction is proposed to address the issue where the regular chaining construction used in wiretap polar codes cannot be applied in the inner layer. Results show that the decoding error probability (DEP) of the legitimate decoder vanishes to zero as the block length increases. In addition, the proposed scheme is proven to satisfy strong secrecy.