## 3 Preliminaries
### Review of the PHY-Layer Secret Key Extraction Scheme in [5]
Thus, [PERSON]'s received signal can be expressed as  
\[ \begin{split}& r_{b}\big{(}t^{\prime}_{0}\big{)}=\sum_{i=1}^{N} \sqrt{\frac{2|h_{i}|^{2}E_{S}}{T}}e^{j\left(\mathcal{T}^{\prime}_{2E_{i}}( \theta_{i})+\xi_{b,i}\right)}+N_{i}\\ & r_{b}\big{(}t^{\prime}_{0}+\Delta t\big{)}=\sum_{i=1}^{N}\sqrt{ \frac{2|h^{\prime}_{i}|^{2}E_{S}}{T}}e^{j\left(\mathcal{T}^{\prime}_{t_{i}}( \theta_{i})+\mathcal{T}^{\prime}_{b,i}\right)+}N^{\prime}_{i},\end{split}  \tag{3}\]  
where \(\{|h_{i}|,|h^{\prime}_{i}|\}\) and \(\{\xi_{i},\xi^{\prime}_{i}\}\) are the channel-fading coefficients and phase responses of the \(i\)th subcarrier at times \(\{t^{\prime}_{0},t^{\prime}_{0}+\Delta t\}\), respectively and \(\{N_{i},N^{\prime}_{i}\}\) are complex additive Gaussian noises \(\mathcal{CN}(0,\sigma^{2}_{n})\) with zero means and \(\sigma^{2}_{n}\) variances. It is noteworthy to mention that the channel responses \(\{|h_{i}|,\xi_{i}\}\) are highly correlated with \(\{|h^{\prime}_{i}|,\xi^{\prime}_{i}\}\) for \(\Delta t\leq T_{c}\). Similarly, [PERSON] replies by sending an authenticated probing packet as in (2) with phases \(\{T^{\prime}_{2m_{i}}(\theta_{i}),T^{\prime}_{m_{i}}(\theta_{i})\}\) at times \(\{t^{\prime}_{1},t^{\prime}_{1}+\Delta t\}\). Then, both terminals, [PERSON] and [PERSON], equalize their received signals by computing \(e_{a}(t)=r_{a}\big{(}t^{\prime}_{1}\big{)}r_{a}\big{(}t^{\prime}_{1}+\Delta t \big{)}^{*}\) and \(e_{b}(t)=r_{b}\big{(}t^{\prime}_{0}\big{)}r_{b}\big{(}t^{\prime}_{0}+\Delta t \big{)}^{*}\), respectively. Hence, the phases of \(e_{a}(t)\) and \(e_{b}(t)\) of the \(i\)th subcarrier can be formulated as  
\[ \begin{split}&\angle e_{a,i}(t)=m_{i}\theta_{i}+\big{(}\xi_{a,i}- \xi^{\prime}_{a,i}\big{)}+\big{(}\omega_{a,i}-\omega^{\prime}_{a,i}\big{)}\\ &\angle e_{b,i}(t)=n_{i}\theta_{i}+\big{(}\xi_{b,i}-\xi^{\prime}_ {b,i}\big{)}+\Big{(}\omega_{b,i}-\omega^{\prime}_{b,i}\big{)},\end{split}  \tag{4}\]  
where \(\{\omega_{a,i},\omega^{\prime}_{a,i}\}\) and \(\{\omega_{b,i},\omega^{\prime}_{b,i}\}\) are the noisy added estimates result from \(\{N_{i},N^{\prime}_{i}\}\) in (3) at the sides of Alice and Bob, respectively with Gaussian distributions \(\mathcal{N}(0,\sigma^{2})\). Accordingly, both terminals use the \(Round\) function to obtain \(\hat{T}^{\prime}_{m_{i}}(\theta_{i})\) and \(\hat{T}^{\prime}_{n_{i}}(\theta_{i})\) as  
\[ \begin{split}&\hat{T}^{\prime}_{m_{i}}(\theta_{i})\quad=Round \big{(}\angle e_{a,i}(t)\big{)}=Round\big{(}m_{i}\theta_{i}+\big{(}\xi_{a,i}- \xi^{\prime}_{a,i}\big{)}+\big{(}\omega_{a,i}-\omega^{\prime}_{a,i}\big{)} \big{)}\\ &\hat{T}^{\prime}_{n_{i}}(\theta_{i})\quad=Round\big{(}\angle e_{b,i}(t)\big{)}=Round\Big{(}n_{i}\theta_{i}+\Big{(}\xi_{b,i}-\xi^{\prime}_{b,i} \Big{)}+\Big{(}\omega_{b,i}-\omega^{\prime}_{b,i}\Big{)}\Big{)},\end{split}  \tag{5}\]  
where the function \(Round(x)\) is used to round \(x\) to the nearest multiple of \(2\pi/2^{r}\) for \(r\in\{1,2,3\}\). Then, [PERSON] and [PERSON] compute \(T^{\prime}_{n_{i}m_{i}}(\theta_{i})|_{[PERSON]}=T^{\prime}_{n_{i}}\Big{(}\hat{T}^{ \prime}_{m_{i}}(\theta_{i})\Big{)}\) and \(T^{\prime}_{n_{i}m_{i}}(\theta_{i})|_{[PERSON]}=T^{\prime}_{m_{i}}\Big{(}\hat{T}^{ \prime}_{n_{i}}(\theta_{i})\Big{)}\), respectively. The use of the \(Round\) function in the context is important to avoid the significant error results from multiplying the negligible value of \(((\xi-\xi^{\prime})+(\omega-\omega^{\prime}))\) by the large integer number \(n_{i}\) or \(m_{i}\). Finally, both terminals quantize their estimates to convert them into bit streams using a mapping operation \(\mathcal{M}^{-1}(. )\) of order \(r\). For clarity, a Gray code mapping operation of order 2 can be expressed as  
\[\mathcal{M}^{-1}\Big{(}T^{\prime}_{n,m_{i}}(\theta_{i})\Big{)}= \begin{cases}00&T^{ \prime}_{n,m_{i}}(\theta_{i})\in[-\frac{\pi}{4},\frac{\pi}{4})\\ 01&T^{\prime}_{n,m_{i}}(\theta_{i})\in[\frac{\pi}{4},\frac{3\pi}{4})\\ 11&T^{\prime}_{n,m_{i}}(\theta_{i})\in[\frac{3\pi}{4},-\frac{3\pi}{4})\\ 10&T_{n,m_{i}}(\theta_{i})\in[-\frac{3\pi}{4},-\frac{\pi}{4})\end{cases}