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Polar-Domain Multi-User Key Generation in Near-Field Communications

Tianyu Lu, Liquan Chen, Junqing Zhang, Weicheng Zhang, Michail Matthaiou

Published Dec 8, 2025• 0 citations
Publisher

Abstract

With the substantial increase in the number of antennas, polar-domain channel modeling for extremely large-scale antenna array (ELAA) systems has been introduced to capture both angular and distance information in near-field environments. The fine-grained polar-domain channel provides additional sources of randomness, making it well-suited for physical layer key generation (PLKG). To minimize the pilot overhead in multi-user key generation and leverage the randomness from the polar-domain channel paths, we herein design a zero-forcing (ZF)-based precoding scheme to mitigate inter-path and inter-user interference. Using ZF precoding, we derive an analytical expression for the sum secret key rate (SKR) as a function of power allocation variables, and then optimize these variables in the presence of eavesdroppers. Our simulations validate the proposed precoding design and power allocation methods in terms of sum SKR versus the transmit power, antenna configurations, and spatial correlation between legitimate and eavesdropping channels.

DOI
10.1109/globecom59602.2025.11432070

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With the substantial increase in the number of antennas, polar-domain channel modeling for extremely large-scale antenna array (ELAA) systems has been introduced to capture both angular and distance information in near-field environments. The fine-grained polar-domain channel provides additional sources of randomness, making it well-suited for physical layer key generation (PLKG). To minimize the pilot overhead in multi-user key generation and leverage the randomness from the polar-domain channel paths, we herein design a zero-forcing (ZF)-based precoding scheme to mitigate inter-path and inter-user interference. Using ZF precoding, we derive an analytical expression for the sum secret key rate (SKR) as a function of power allocation variables, and then optimize these variables in the presence of eavesdroppers. Our simulations validate the proposed precoding design and power allocation methods in terms of sum SKR versus the transmit power, antenna configurations, and spatial correlation between legitimate and eavesdropping channels.
Extremely large-scale antenna arrays (ELAA) require near-field spherical wave modeling due to the substantial increase in the number of antennas, which introduces new spatial dimensions to physical layer key generation (PLKG). We investigate multi-user PLKG in near-field environments, where a base station with an ELAA simultaneously generates secret keys with multiple users. We derive an analytical expression for the key rate (KR). By utilizing spatial dimensions of distance and angle in near-field environments, we apply eigenvalue decomposition and singular value decomposition to design precoding matrices to reduce interference among user equipments (UEs) and extract uncorrelated subchannels. Given that the KR is non-convex, we approximate it and optimize the precoding matrix to increase the KR. After precoding design, the KR depends on the transmit power allocated to the subchannels. Two optimization problems are formulated to further optimize transmit power allocation. The first problem focuses on maximizing the sum KR. We apply the Lagrange multiplier method to determine the optimal power allocation variables by searching the Lagrange multiplier. To reduce computational complexity, a supervised feedforward neural network (FNN) is designed to capture the relationship between the power allocation variables and the Lagrange multiplier. The second optimization problem focuses on KR fairness. By introducing a slack variable that is smaller than the KRs of all users, we use the CVX toolbox to find optimal power allocation variables that maximize this slack variable. To further reduce complexity, the Lagrange multiplier method offers an analytical solution for power allocation variables in terms of Lagrange multipliers determined by the slack variable in the high-power case. We employ a bisection algorithm to find the slack variable. Furthermore, we propose an FNN to map transmit power to the slack variable. Simulations demonstrate that the proposed methods efficiently leverage near-field effects for multi-user PLKG, reducing pilot overhead.
Physical-layer key generation (PKG) based on channel reciprocity has recently emerged as a new technique to establish secret keys between devices. Most works focus on pairwise communication scenarios with single or small-scale antennas. However, the fifth generation (5G) wireless communications employ massive multiple-input multiple-output (MIMO) to support multiple users simultaneously, bringing serious overhead of reciprocal channel acquisition. This paper presents a multi-user secret key generation in massive MIMO wireless networks. We provide a beam domain channel model, in which different elements represent the channel gains from different transmit directions to different receive directions. Based on this channel model, we analyze the secret key rate and derive a closed-form expression under independent channel conditions. To maximize the sum secret key rate, we provide the optimal conditions for the Kronecker product of the precoding and receiving matrices and propose an algorithm to generate these matrices with pilot reuse. The proposed optimization design can significantly reduce the pilot overhead of the reciprocal channel state information acquisition. Furthermore, we analyze the security under the channel correlation between user terminals (UTs), and propose a low overhead multi-user secret key generation with non-overlapping beams between UTs. Simulation results demonstrate the near-optimal performance of the proposed precoding and receiving matrices design and the advantages of the non-overlapping beam allocation.
The next generation of communication systems are expected to operate at high frequency bands such as millimetre wave (mmWave) and terahertz (THz) bands, and use extremely large-scale multiple-input-multiple-output (XL-MIMO). This brings a paradigm shift from far-field to near-field communications. In this paper, we investigate physical-layer key generation in near-field XL-MIMO communications and focus on the most challenging line-of-sight (LoS) propagation scenario. To be specific, we introduce artificial randomness to enhance secret key generation and enable theoretical analysis of secret key rate (SKR). We provide the zero-forcing (ZF) precoding solution that can null the received signal at the eavesdropper. We show that the ZF precoding leads to a low SKR in challenging scenarios of low transmit powers and small eavesdropping distances. To improve the SKR in these challenging scenarios, we propose a novel low-complexity machine learning-based beam focusing (MLBF) scheme. Simulation results show that the proposed MLBF scheme achieves a higher SKR than the benchmark methods.
This paper investigates an artificial randomness-based key generation scheme in near-field extremely large-scale multiple-input multiple-output (XL-MIMO) communications, aiming to overcome the limitations of static wireless chan-nels in generating secret keys. The proposed approach introduces artificial randomness via the pre coding vector, enabling legitimate users to generate secret keys while preventing eavesdropping. To maximize the secret key rate (SKR), the singular value decomposition (SVD) method is utilized to identify the legitimate subspace for generating secret keys and its orthogonal subspace to thwart eavesdropping. Additionally, a Dinkelbach method-based power allocation algorithm is designed to inject noise into both the legitimate subspace and its orthogonal subspace. Our simulation results validate the efficiency of the proposed key generation scheme, demonstrating its superiority over existing approaches.
Secret key generation in multi-user massive multiple-input multiple-output (MIMO) communication suffers from high computation complexity, pilot overhead, and inter-user interference caused by the large dimension of the massive MIMO channel. To address these problems, this paper proposes a novel key generation approach that exploits the beam domain channel for channel feature extraction. The proposed approach can effectively reduce the channel feature dimension and mitigate the inter-user interference, via the well-designed power allocation and beam scheduling algorithms. Specifically, the multi-user key generation model is built and the design problem of channel feature extraction matrices is formulated as a sum key rate maximization problem. To tackle this problem, we derive an upper bound of the secret key rate and prove that the upper bound maximization is a convex optimization. After solving the upper bound maximization, the base station (BS) obtains the sum beams for all the users. To allocate the selected beams to each user, an efficient beam scheduling algorithm is proposed, which converges to a locally optimal solution with low computational complexity. Numerical results illustrate that in the proposed scheme, the bit disagreement ratio (BDR) of legitimate users achieves 10−2, while that of the eavesdropper tends to 0.5, confirming the feasibility and security of the proposed scheme.
Physical-layer key generation (PKG) based on wireless channels is a lightweight technique to establish secure keys between legitimate communication nodes. Recently, intelligent reflecting surfaces (IRSs) have been leveraged to enhance the performance of PKG in terms of secret key rate (SKR), as it can reconfigure the wireless propagation environment and introduce more channel randomness. In this paper, we investigate an IRS-assisted PKG system, taking into account the channel spatial correlation at both the base station (BS) and the IRS. Based on the considered system model, the closed-form expression of SKR is derived analytically considering correlated eavesdropping channels. Aiming to maximize the SKR, a joint design problem of the BS’s precoding matrix and the IRS’s phase shift vector is formulated. To address this high-dimensional non-convex optimization problem, we propose a novel unsupervised deep neural network (DNN)-based algorithm with a simple structure. Different from most previous works that adopt iterative optimization to solve the problem, the proposed DNN-based algorithm directly obtains the BS precoding and IRS phase shifts as the output of the DNN. Simulation results reveal that the proposed DNN-based algorithm outperforms the benchmark methods with regard to SKR.