## 3 Preliminaries
### Review of the PHY-Layer Secret Key Extraction Scheme in [5]  
The work introduced in [5] proposes a novel Diffie-Hellman channel-probing mechanism that uses the extended Chebyshev chaotic mapping operation to exchange probing signals in an interleaved fashion. Specifically, the extended Chebyshev mapping operation for the OFDM system of \(N\) subcarriers is formulated as  
\[T^{\prime}_{n_{i}}(\theta_{i})= \begin{cases}n_{i}\cdot\theta_{i}\bmod p,&\theta _{i}\in[0,2\pi)\\ n_{i}\cdot\cos^{-1}(x_{i})\bmod p,&x_{i}=\cos(\theta_{i})\end{cases} \quad \text{for }i=1,\ldots,N, \tag{1}\]  
where \(p=2\pi\), \(n_{i}\) is a large integer number, and \(\theta_{i}=\frac{2\Pi}{2^{2}}\) for \(r\in\{1,2,3\}\) is the primitive root of the \(i\)th subcarrier. The primitive root \(\theta_{i}\) is a generator of the group \(\mathbb{G}\) such that its multiples generate the entire group. For example, let \(r=2\), then \(\theta_{i}=\frac{\Pi}{2}\). Thus, the cyclic group elements are \(\mathbb{G}_{2}=\{0,\frac{\Pi}{2},\Pi,\frac{3\Pi}{2}\}\). For \(r=3\), \(\theta_{i}=\frac{\Pi}{4}\). Thus, the cyclic group elements are \(\mathbb{G}_{3}=\{0,\frac{\Pi}{4},\frac{1}{4},\frac{3\Pi}{4},\frac{1}{4},\frac{ 3\Pi}{4},\frac{7\Pi}{4}\}\). We have considered a scenario where two \begin{table}
\begin{tabular}{l l} \hline \hline
**Symbol** & **Definition** \\ \hline \(\theta_{i}\) & The generator of the cyclic group \(\mathbb{G}\) for the \(i\)th subcarrier of the OFDM symbol \\ \hline \(n_{i},m_{i}\) & The private integer numbers at the sides of Alice and Bob, respectively \\ \hline \(\Delta t\) & The transmission time interval between two subsequent OFDM symbols \\ \hline \(|h_{i}|,\xi_{i}\) & The wireless-channel amplitude and phase responses, respectively \\ \hline \(\hat{T}^{\prime}_{m_{i}}(\theta_{i}),\hat{T}^{\prime}_{n_{i}}(\theta_{i})\) & The equalized phase estimates at the sides of [PERSON] and [PERSON], respectively \\ \hline \(\mathcal{M}^{-1}\) & The Gray code mapping operation that converts final estimates into bit streams \\ \hline \(r\) & The order of the generator \(\theta_{i}\) and the mapping operation \(\mathcal{M}^{-1}\) \\ \hline \(\phi(x),\text{erf}(z)\) & The cumulative distribution function and the error function \\ \hline \(P_{e}\) & The probability of error in the extracted key between two parties \\ \hline \(I\) & The secret key capacity \\ \hline \(HD\) & The Hadamard codebook used for optimizing the RIS configuration \\ \hline \(\overline{\text{SNR}_{i}^{[PERSON]}},\overline{\text{SNR}_{i}^{[PERSON]}}\) & The average SNR of the signals transmitted from [PERSON] and [PERSON], respectively \\ \hline \(H_{opt}\) & The optimal configuration for the RIS’s reflecting units \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of notations. parties ([PERSON] and [PERSON]) are in the same communication range and want to establish a secure communication link. In this context, [PERSON] and [PERSON] exchange authenticated probing packets at times \(t_{0}\) and \(t_{1}\), respectively. Based on the received probing packets, both terminals can extract a high-entropy secret key, which is used to secure subsequent transmissions using the upper layer's crypto-based approaches. Figure 1 reviews the steps involved in the secret key extraction process in [5]. Generally, the extraction process comprises channel probing and quantization, information reconciliation, and privacy amplification. In the former, [PERSON] sends the probing packet in the form of two OFDM symbols of \(N\) subcarriers, which can be represented in a simplified form as:  
\[ \begin{split}& s_{a}(t_{0})=\sum_{i=1}^{N}\sqrt{\frac{2E_{S}}{T}}e^{ \left(\mathcal{T}^{\prime}_{t_{i}}(\theta_{i})\right)}=\sum_{i=1}^{N}\sqrt{ \frac{2E_{S}}{T}}e^{j(2n_{i}\theta_{i})}\\ & s_{a}(t_{0}+\Delta t)=\sum_{i=1}^{N}\sqrt{\frac{2E_{S}}{T}}e^{j \left(\mathcal{T}^{\prime}_{t_{i}}(\theta_{i})\right)}=\sum_{i=1}^{N}\sqrt{ \frac{2E_{S}}{T}}e^{j(n_{i}\theta_{i})},\end{split}  \tag{2}\]  
where the transmission time interval between both OFDM symbols is \(\Delta t\leq T_{c}\).