A Key Generation Method Based on Atomic Norm Minimization For Reconfigurable Intelligent Surface-Assisted Millimeter Wave MIMO Communication Systems
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摘要: 针对毫米波大规模多输入多输出(MIMO)系统中智能超表面(RIS)辅助的密钥生成方法面临的导频开销高、信道信息估计误差大和信道稀疏度依赖性强等问题，该文提出一种基于原子范数最小化(ANM)的RIS辅助密钥生成方案。现有基于压缩感知(CS)的密钥生成方案需预设信道稀疏度作为先验知识，且受限于网格化离散建模，易导致密钥失配问题。该方案通过引入ANM技术，将基站(Base Station, BS)与用户(UE)间的级联信道估计问题转化为无限分辨率的稀疏信号恢复问题，结合多信号分类(MUSIC)算法联合估计虚拟离开角(AoDs)与到达角(AoAs)，突破传统网格化约束并消除对稀疏度的显式假设，从而提取高精度信道参数作为密钥源。仿真结果表明，与基于传统CS的方案相比，该方案在5 dB的信噪比条件下密钥不一致率降低了47.7%，同时显著减少了导频开销；随着天线规模扩大，其性能优势进一步凸显。该方案为RIS辅助毫米波MIMO系统的密钥生成提供了一种无需先验稀疏度、抗网格误差的可靠解决思路。Abstract:
Objective The reciprocity, time variability, and unpredictability of wireless channels enable physical-layer key generation, a promising technology for B5G/6G systems due to its independence from third-party involvement and inherent quantum-resistant properties. In millimeter-wave Multiple Input Multiple Output (MIMO) systems, channel sparsity imposes stringent constraints on key capacity, particularly in quasi-static propagation environments. While Reconfigurable Intelligent Surface (RIS) technology enhances channel time variability and increases key capacity, it also leads to an exponential increase in pilot overhead with the number of transceiver antennas and RIS elements. To mitigate pilot overhead, Compressive Sensing (CS) techniques have been employed by leveraging channel sparsity and reformulating channel estimation as a sparse signal recovery problem. However, existing CS-based key generation schemes require prior knowledge of channel sparsity, which may not reflect actual dynamic channel conditions. Additionally, these approaches typically rely on grid-based discrete modeling, where Angles of Departure (AoDs) and Angles of Arrival (AoAs) are quantized into predefined grids, leading to key mismatches. To address these challenges, this study proposes an RIS-assisted key generation scheme based on Atomic Norm Minimization (ANM) for RIS-assisted millimeter-wave MIMO systems. Methods The proposed method presents a novel key extraction approach based on virtual AoDs and virtual AoAs for RIS-assisted millimeter-wave MIMO systems. First, the problem of virtual channel parameter estimation in RIS-assisted millimeter-wave MIMO cascaded channels is formulated as a continuous sparse signal recovery problem. An optimization problem is then constructed using ANM, where ANM serves as the objective function and pilot observation error as the constraint. The Multiple Signal Classification (MUSIC) algorithm is integrated to enhance channel sparsity and achieve super-resolution angle estimation, thereby extracting high-precision virtual AoDs and AoAs as key parameters. Based on these parameters, a comprehensive key generation scheme is proposed, incorporating quantization, information reconciliation, and privacy amplification. Additionally, the key capacity of the proposed scheme is theoretically derived, with a closed-form expression provided based on the distribution of virtual AoDs/AoAs. Finally, Monte Carlo simulations are conducted to validate the effectiveness of the proposed scheme. Comparative analysis with existing schemes demonstrates its advantages in terms of key inconsistency, mutual information per bit, key generation rate, and pilot overhead. Results and Discussions Analysis of the simulation results indicates that the proposed scheme improves pilot overhead, Bit Disagreement Rate (BDR), mutual information per bit, and Secret Key Rate (SKR). These metrics primarily assess channel information extraction and key generation performance. For channel estimation accuracy, the Normalized Mean Square Error (NMSE) of the estimated virtual angles is used as an evaluation metric, where a lower NMSE indicates higher accuracy. Compared to other schemes, the proposed approach consistently achieves a lower NMSE, particularly for short pilot lengths. Even with $ {N_{\text{p}}} = 4 $, the NMSE remains below 0.1 ( Fig. 3 ), demonstrating superior handling of sparse signals. This contributes to reduced pilot overhead and improved estimation accuracy. Key generation performance is evaluated using BDR, mutual information per bit, and SKR. Compared to schemes using the channel response matrix, employing virtual AoAs and AoDs as random keys results in a lower BDR (Fig. 4 ) and higher bit-wise mutual information (Fig. 6 ) across various Signal-to-Noise Ratio (SNR) conditions, demonstrating robustness in both high and low SNR environments. This advantage arises from the inherent sparsity of millimeter-wave channels, where primary propagation paths are clearly distinguishable. Unlike the channel response matrix, angle information is less susceptible to environmental factors and minor physical variations, providing a more stable key source. Compared with traditional CS-based schemes, the proposed approach overcomes grid constraints, reducing the key inconsistency rate by 47.7% under low SNR conditions (5 dB). Additionally, when the number of propagation paths remains constant, BDR decreases as the number of antennas increases. Conversely, when the number of antennas is fixed, BDR increases as the number of paths (L) grows (Fig. 5 ). This occurs because a higher number of paths increases the complexity of distinguishing AoAs and AoDs, leading to greater estimation error. Furthermore, a larger number of paths generates more key bits, causing BDR accumulation across paths, which raises the overall BDR. However, as the number of antennas increases, the sparsity of millimeter-wave MIMO channels becomes more pronounced (L is smaller), further amplifying the advantages of the proposed scheme. Additionally, by utilizing virtual angles as the key source, the proposed scheme maintains a high SKR even under low SNR conditions, further demonstrating its potential for practical applications (Fig. 7 ).Conclusions The proposed method employs ANM to formulate the cascaded channel estimation problem between the Base Station (BS) and User Equipment (UE) as a gridless sparse signal recovery problem. By integrating the MUSIC algorithm, the method jointly estimates virtual AoDs and AoAs, overcoming traditional grid-based constraints and eliminating the explicit assumption of channel sparsity. Therefore, high-precision channel parameters are extracted as key sources. Simulation results demonstrate that, compared to conventional CS-based methods, the proposed scheme reduces the BDR by 47.7% at an SNR of 5 dB while significantly lowering pilot overhead. Additionally, its performance advantage becomes more pronounced as the antenna array size increases. The proposed scheme offers a robust solution for key generation in RIS-assisted millimeter-wave MIMO systems, eliminating the need for prior sparsity knowledge and mitigating grid quantization errors. -
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