In recent years, China has elevated the low-altitude economy as a vital emerging strategic industry, with policies accelerating the development of China UAV technologies. However, the broadcast nature of wireless channels and the near-line-of-sight propagation in low-altitude scenarios make information susceptible to eavesdropping and interference. Ensuring the security of China UAV communications is therefore crucial for the healthy growth of this economy. Physical layer key generation (PLKG) has emerged as a lightweight and efficient solution, especially for resource-constrained platforms like small China UAVs. In this work, we propose an OTFS-based key generation framework designed specifically for high-mobility China UAV scenarios, addressing challenges such as channel time variability, sparsity, and slow variation.
Our approach integrates Sparse Bayesian Learning (SBL) for accurate channel estimation in the delay-Doppler (DD) domain, Karhunen-Loève Transform (KLT) for decorrelation of channel components, and a joint multi-parameter quantization scheme that leverages path coefficients, Doppler shifts, and delays. This enables high key generation rates with low bit disagreement rates and enhanced entropy, providing a robust security solution for next-generation China UAV networks.
System Model
We consider a typical low-altitude network scenario where a China UAV (airborne node) communicates with a ground node using OTFS modulation. The DD domain channel impulse response with \(P\) paths is modeled as:
$$
h(\nu, \tau) = \sum_{i=1}^{P} h_i \, \delta(\nu – \nu_i) \, \delta(\tau – \tau_i),
$$
where \(h_i\), \(\nu_i\), and \(\tau_i\) denote the complex gain, Doppler shift, and delay of the \(i\)-th path, respectively. The OTFS frame structure employs a pilot symbol at position \((k_p, l_p)\) with guard intervals, and the received signal in the DD domain is:
$$
y(k,l) = \sum_{k’} \sum_{l’} x(k’,l’) \, h_w(k-k’, l-l’) + z(k,l),
$$
where \(h_w\) is the effective impulse response incorporating the sampling functions \(w_\nu\) and \(w_\tau\). The channel is assumed quasi-static within a single frame, but varies across frames following an autoregressive (AR) model to capture slow variation:
$$
x_t = \rho_x x_{t-1} + \sqrt{1-\rho_x^2} \, \Delta x,
$$
with \(\rho_x\) being the autocorrelation coefficient. For complex gains \(h\) we use a complex Gaussian distribution, while for real-valued delays \(\tau\) and Doppler \(\nu\) we use real Gaussian distributions.
Channel Estimation via Sparse Bayesian Learning
Accurate channel estimation in the DD domain is critical for reliable key generation. The input-output relation can be written in matrix form as:
$$
\mathbf{y} = \mathbf{\Phi}(\mathbf{k}_\nu, \mathbf{l}_\tau) \mathbf{h} + \mathbf{z},
$$
where \(\mathbf{y} \in \mathbb{C}^{MN \times 1}\) is the received signal, \(\mathbf{h} = [h_1,\ldots,h_P]^T\) are the path gains, and \(\mathbf{\Phi}\) is the sensing matrix depending on the Doppler indices \(\mathbf{k}_\nu\) and delay indices \(\mathbf{l}_\tau\). The unknown off-grid parameters cause model mismatch; we adopt a first-order linear approximation to handle this.
We employ a variational Bayesian inference framework. The prior for \(\mathbf{h}\) is \(\mathcal{CN}(\mathbf{0}, \mathbf{\Lambda}^{-1})\) with \(\mathbf{\Lambda}=\text{Diag}(\boldsymbol{\alpha})\), and gamma hyperpriors for precision parameters. The noise is white Gaussian with precision \(\alpha_0\). The posterior distributions are updated iteratively using the following key steps:
| Step | Operation |
|---|---|
| 1 | Compute posterior mean \(\boldsymbol{\mu}_h = \alpha_0 \boldsymbol{\Sigma}_h \mathbf{\Phi}^H \mathbf{y}\) and covariance \(\boldsymbol{\Sigma}_h = (\alpha_0 \mathbf{H}_h + \mathbf{\Lambda})^{-1}\). |
| 2 | Update \(\boldsymbol{\Lambda}\) via \(\alpha_i = \tilde{a}_i / \tilde{b}_i\) with \(\tilde{a}_i = a+1\), \(\tilde{b}_i = b + \|h_i\|^2\). |
| 3 | Update noise precision \(\alpha_0 = \tilde{c} / \tilde{d}\). |
| 4 | Update means and variances for \(\mathbf{k}_\nu\) and \(\mathbf{l}_\tau\) using first-order gradient. |
The computational complexity per iteration is \(\mathcal{O}(P (MN)^2 + P^3)\). With a single-path channel and proper initialization, the algorithm converges quickly, typically within 20 iterations with a tolerance of \(10^{-6}\). The estimated parameters \(\hat{\mathbf{h}}\), \(\hat{\mathbf{k}}_\nu\), and \(\hat{\mathbf{l}}_\tau\) are used for subsequent key generation.
Key Generation Protocol
Karhunen-Loève Transform for Decorrelation
The DD domain channel often exhibits slow variation and temporal correlation, leading to insufficient randomness in raw measurements. To address this, we apply the discrete Karhunen-Loève Transform (KLT) on a set of channel measurements. Given \(L\) groups of \(S\)-length measurement vectors \(\mathbf{x}\), we estimate the covariance matrix \(\mathbf{R}_x\) and perform eigen-decomposition:
$$
\mathbf{R}_x = \mathbf{U} \mathbf{\Lambda} \mathbf{U}^H.
$$
The KLT applies \(\mathbf{y} = \mathbf{U}^H (\mathbf{x} – \hat{\boldsymbol{\mu}}_x)\). The legitimate nodes Alice and Bob use the same transformation matrix \(\mathbf{U}\) (publicly transmitted by Alice) to produce decorrelated components \(\mathbf{y}_a\) and \(\mathbf{y}_b\). This significantly reduces correlation and increases entropy of subsequent quantization.
Multi-bit Adaptive Quantization
We adopt a multi-bit adaptive quantization (MAQ) scheme to convert channel components into bits with high consistency. Alice acts as the leading node. For each component with cumulative distribution function \(F(y)\), we define \(K = 4 \times 2^{m_l}\) equiprobable intervals using quantiles:
$$
\eta_k = F^{-1}\left( \frac{k}{4 \times 2^{m_l}} \right), \quad k=1,\ldots,K-1.
$$
The interval index \(k(l)\) is determined by the measurement value. A binary flag \(e(k)\) is defined as:
$$
e(k) = \begin{cases} 1 & \text{if } k \bmod 4 \ge 2, \\ 0 & \text{otherwise.} \end{cases}
$$
Alice transmits the vector \(\mathbf{e} = [e(k(1)),\ldots,e(k(L))]^T\) publicly. Based on \(e(k)\), both nodes encode using either codeword set \(d_1\) or \(d_0\) from a Gray-coded list. The final key bits are concatenated. This process ensures that even if Eve knows \(\mathbf{e}\), she gains no information about the actual key bits because the mapping depends on Alice’s secret measurement.
Bit Disagreement Rate Control
The bit disagreement rate (BDR) is a critical metric. Assuming the joint distribution of Alice and Bob’s components is bivariate normal, we derive an analytical bound:
$$
P_{BD} \approx \frac{P_{CD}}{m_l}, \quad P_{CD} = 1 – P_{CA},
$$
where \(P_{CA}\) is the probability of codeword agreement. Given a target BDR threshold \(BDR_T\), the required correlation coefficient can be estimated. We compute the correlation coefficient \(\hat{\rho}\) between Alice and Bob’s components using the noise variance estimated during SBL. Figure below shows the BDR as a function of correlation \(\rho\) for different quantization bits \(m_l\).
| \(\rho\) | \(m_l=1\) | \(m_l=2\) | \(m_l=3\) |
|---|---|---|---|
| 0.95 | 0.010 | 0.005 | 0.002 |
| 0.97 | 0.004 | 0.002 | 0.001 |
| 0.99 | 0.001 | 0.0005 | 0.0002 |
In practice, we set \(BDR_T = 10^{-3}\) for most China UAV applications, ensuring that information reconciliation can correct remaining errors with low overhead.
Information Reconciliation and Privacy Amplification
We employ the Cascade protocol for reconciliation, which corrects bit mismatches through multiple rounds of parity checks. The leaked information during this process is accounted for. After reconciliation, we apply privacy amplification using a universal hash function to eliminate any partial information that Eve might have obtained. This step ensures that the final key is uniformly random and independent of any disclosed data.
Security Analysis
The public transmission of the KLT matrix \(\mathbf{U}\) introduces a controlled amount of information leakage. Suppose Alice and Bob each collect \(L\) measurement blocks of length \(S\). The entropy of the transformed data for one block is \(H(\mathbf{Y}_a) = \sum_{s=1}^S L \log_2 Q_s\). After Eve learns the covariance matrix (or equivalently \(\mathbf{Y}_a \mathbf{Y}_a^H\)), the conditional entropy reduces to:
$$
H(\mathbf{Y}_a | \mathbf{U}) \approx \sum_{s=1}^S (L-1) \log_2 Q_s.
$$
Thus, the mutual information leakage is:
$$
I(\mathbf{Y}_a; \mathbf{U}) \approx \sum_{s=1}^S \log_2 Q_s,
$$
and the leakage ratio \(\theta = 1/L\). By choosing a sufficiently large \(L\) (e.g., \(L=500\)), the relative leakage becomes negligible (0.2%).
Furthermore, consider an eavesdropper Eve whose channel measurements are correlated with Bob’s with coefficient \(\rho_e\). For a given component, the mutual information between Bob’s measurement \(y_{b,s}\) and Eve’s \(y_{e,s}\) is:
$$
I(y_{b,s}; y_{e,s}) = \log_2\left(1 + \frac{\rho_e^2 \lambda_s}{1-\rho_e^2}\right).
$$
The total leakage over all components and time blocks is:
$$
I_e = \sum_{i=1}^S L \log_2\left(1 + \frac{\rho_e^2 \lambda_i}{1-\rho_e^2}\right).
$$
In typical China UAV scenarios, decorrelation due to spatial separation ensures \(\rho_e\) is very small (e.g., 0.2 or less). Combined with the large \(L\), the total leakage remains under 5% as shown in our simulations.
Performance Evaluation
We simulate a single-path double-scatterer channel with 105 OTFS frames. System parameters: \(M=32, N=32, l_{\max}=2k_{\max}=32\). Channel statistics: \(\mu_h=0, \sigma_h=1, \mu_\tau = 4/(M\Delta f), \sigma_\tau=1/(M\Delta f), \mu_\nu=0, \sigma_\nu=1/(NT)\). We compare our proposed scheme with two baselines: Spike Location (SL) and Entropy Coding (EC). Two additional baselines omit SBL or KLT to isolate their contributions.
| Scheme | BDRT=10-3 | BDRT=10-4 |
|---|---|---|
| Proposed (full) | 8.2e-4 | 9.1e-5 |
| Proposed w/o SBL | 2.3e-3 | 5.4e-4 |
| Proposed w/o KLT | 1.1e-3 | 3.2e-4 |
| SL [10] | 6.5e-3 | 1.8e-3 |
| EC [12] | 8.9e-3 | 2.7e-3 |
The proposed scheme achieves BDR consistently below the threshold, while baselines exhibit higher error rates. The KLT component reduces correlation-induced mismatches, and SBL improves estimation accuracy, leading to lower BDR.
| \(\rho_x\) | Proposed | Proposed w/o SBL | SL [10] |
|---|---|---|---|
| 0.5 | 48.2 | 32.5 | 18.4 |
| 0.7 | 42.1 | 28.3 | 15.2 |
| 0.9 | 31.5 | 21.0 | 11.7 |
The joint quantization of path gain, Doppler, and delay yields significantly longer keys compared to single-parameter methods. The KLT further boosts useful bits by decorrelation. At higher autocorrelation \(\rho_x\), the key length decreases, but our scheme still outperforms alternatives.
| Scheme | SNR=10 dB | SNR=15 dB | SNR=20 dB |
|---|---|---|---|
| Proposed | 0.986 | 0.991 | 0.995 |
| Proposed w/o KLT | 0.742 | 0.768 | 0.781 |
| SL [10] | 0.653 | 0.672 | 0.689 |
| EC [12] | 0.588 | 0.603 | 0.614 |
Our proposed scheme achieves near-optimal entropy close to 1, indicating excellent randomness. The KLT decorrelation is key to this achievement. NIST statistical tests confirm the cryptographic quality: all applicable tests passed with \(p\)-values > 0.01, with pass rates meeting or exceeding minimum requirements.
| Test | Pass/Total | p-value |
|---|---|---|
| Frequency | 20/20 | 0.012650 |
| Block Frequency | 20/20 | 0.739918 |
| Cusum | 20/20 | 0.066882 |
| Runs | 20/20 | 0.534146 |
| Longest Run | 20/20 | 0.637119 |
| Rank | 19/20 | 0.637119 |
| FFT | 18/20 | 0.122325 |
| Overlapping Template | 20/20 | 0.350485 |
| Universal | 20/20 | 0.834308 |
| Approximate Entropy | 18/20 | 0.162606 |
| Linear Complexity | 20/20 | 0.534146 |
Conclusion
In this work, we have presented an OTFS-based physical layer key generation framework tailored for high-mobility China UAV communications. By combining SBL-based accurate DD domain channel estimation, KLT decorrelation, and joint multi-parameter quantization, our scheme achieves low BDR, high key generation rate, and strong randomness. The controlled information leakage analysis demonstrates that the public transmission of the KLT matrix incurs minimal loss when using a sufficiently large measurement block count. Our approach significantly outperforms existing methods such as Spike Location and Entropy Coding, making it a promising candidate for securing the next generation of China UAV networks. Future work will extend this framework to millimeter-wave massive MIMO systems, where similar sparsity and off-grid challenges arise.