# Feedback Approach for the Relay Channel with Noisy Feedback and Its Security Analysis
## Appendix A Coefficient's Derivation and Error Probability Analysis of Theorem 1  
Let \(\widehat{M}\) be the estimate of the transmitted message \(M\), the decoder declares an error if \(\widehat{M}\
eq M\). To determine the parameters, we need to analyze the error event in \(P_{e}\). And the error events of the SK-Type FBL approach for the GRC with noisy feedback are described in Section 3.2.3, consisting of  
1. A modulo-aliasing error occurs in Source at time instant \(n(1\leq n\leq N-1)\). The event is defined as \[C_{n}\triangleq\bigg{\{}\gamma_{n}\varepsilon_{n}+\frac{\eta_{f,n}}{h_{\rm ds }}\
otin\bigg{[}-\frac{d}{2},\frac{d}{2}\bigg{)}\bigg{\}}.\] (A1)
2. A decoding error occurs in Destination at time instant \(N\). The event is defined as \[C_{N}\triangleq\bigg{\{}\varepsilon_{N}\
otin\bigg{[}-\frac{0.5}{2^{NR(N, \varepsilon)}},\frac{0.5}{2^{NR(N,\varepsilon)}}\bigg{)}\bigg{\}}.\] (A2)Hence, by applying the union bound, we can derive  
\[ \begin{split} P_{e}&\leq\Pr\left(\bigcup_{n=1}^{N}C_{n} \right)\\ &=\Pr\left(\bigcup_{n=1}^{N-1}C_{n}\right)+\Pr\left(\bigcap_{n=1}^ {N-1}\overline{C}_{n}\cap C_{N}\right)\\ &=\sum_{n=1}^{N-1}\Pr\left(\bigcap_{i=1}^{n-1}\overline{C}_{i} \cap C_{n}\right)+\Pr\left(\bigcap_{n=1}^{N-1}\overline{C}_{n}\cap C_{N} \right)\\ &\triangleq\sum_{n=1}^{N-1}\Pr\left(\widetilde{C}_{n}\right)+ \Pr\left(\widetilde{C}_{N}\right),\end{split}  \tag{10}\]  
where \(\overline{C}\) denotes the complement of the set \(C\), \(\widetilde{C}_{n}\triangleq\bigcap_{i=1}^{n-1}\overline{C}_{i}\cap C_{n}\), \(\widetilde{C}_{N}=\bigcap_{n=1}^{N-1}\overline{C}_{n}\cap C_{N}\). \(\Pr(\widetilde{C}_{n})\)\((1\leq n\leq N-1)\) represents the error probability that a modulo-aliasing occurs in Source at time instant \(n\) when there is no error occurring in all previous times, and \(\Pr(\widetilde{C}_{N})\) represents the error probability of the final decoding without any modulo-aliasing error occurring in Source before time instance \(N\). For a given decoding error probability \(\epsilon\), let  
\[\Pr\left(\widetilde{C}_{N}\right)=\frac{\epsilon}{2},\Pr\left(\widetilde{C}_ {n}\right)=\frac{\epsilon}{2(N-1)}. \tag{11}\]  
For the event \(\widetilde{C}_{n}\)\((1\leq n\leq N-1)\), since no modulo-aliasing error occurs before time instant \(n\), from (10) and the fact that \(\eta_{f,n}\) is Gaussian-distributed, we conclude that \(\gamma_{n}\epsilon_{n}+\eta_{f,n}/h_{\text{ds}}\sim\mathcal{N}(0,\gamma_{n}^{2 }\alpha_{n}+\sigma_{f}^{2}/h_{\text{ds}}^{2})\). From (10), (11) and \(d=\sqrt{12P_{f}}\), we have  
\[ \begin{split}\Pr\left(\widetilde{C}_{n}\right)&=\Pr \left\{\gamma_{n}\epsilon_{n}+\frac{\eta_{f,n}}{h_{\text{ds}}}\
otin\left[- \frac{d}{2},\frac{d}{2}\right)\right\}\\ &=2Q\left(\sqrt{\frac{3P_{f}}{\gamma_{n}^{2}\alpha_{n}+\frac{ \sigma_{f}^{2}}{h_{\text{ds}}^{2}}}}\right).\end{split}  \tag{12}\]  
To simplify the calculation, we introduce a looseness parameter \(L_{\uparrow}\), which is denoted by  
\[L_{\uparrow}\triangleq\frac{P_{f}}{\gamma_{n}^{2}\alpha_{n}+\frac{\sigma_{f}^ {2}}{h_{\text{ds}}^{2}}}=\frac{1}{3}\bigg{(}Q^{-1}\bigg{(}\frac{\epsilon}{4(N -1)}\bigg{)}\bigg{)}^{2}. \tag{13}\]  
Substituting (11) and (13) into (12), we determine the appropriate selection of coefficient \(\gamma_{n}\)\((1\leq n\leq N-1)\),  
\[\gamma_{n}=\sqrt{\left(\frac{P_{f}}{L_{\uparrow}}-\frac{\sigma_{f}^{2}}{h_{ \text{ds}}^{2}}\right)\frac{1}{\alpha_{n}}}. \tag{14}\]  
According to (38), we obtain  
\[\alpha_{f,n}\triangleq\text{Var}\Big{(}\epsilon_{f,n}\Big{)}\overset{\text{(a )}}{=}\alpha_{n}+\frac{\sigma_{f}^{2}}{\gamma_{n}^{2}h_{\text{ds}}^{2}} \overset{\text{(b)}}{=}\alpha_{n}\frac{P_{f}h_{\text{ds}}^{2}}{P_{f}h_{\text{ ds}}^{2}-L_{\uparrow}\sigma_{f}^{2}}\overset{\text{(c)}}{=}\alpha_{n}\phi, \tag{15}\]  
where \((a)\) follows from the fact that \(\eta_{f,n}\) is dependent on \(\epsilon_{n}\), \((b)\) follows from (14), and \((c)\) follows from \(\phi\triangleq\frac{P_{f}h_{\text{ds}}^{2}}{P_{f}h_{\text{ds}}^{2}-L_{\uparrow} \sigma_{f}^{2}}\).