# Feedback Approach for the Relay Channel with Noisy Feedback and Its Security Analysis
## Appendix C Proof of Theorem 3
\begin{split}&\text{(II)}\quad\text{To analyze secrecy level }\Delta_{II}=\frac{H(M|\mathbf{Z}_{1}^{N}\mathcal{Z}_{f,1}^{N-1},\mathbf{V}_{1}^{N-1}, \mathbf{h},\mathbf{g})}{H(M)},\,H(M|\mathbf{Z}_{1}^{N},\mathbf{Z}_{f,1}^{N-1}, \mathbf{V}_{1}^{N-1},\mathbf{h},\mathbf{g})\text{ can be bounded by}\\ &\quad H(M|\mathbf{Z}_{1}^{N},\mathbf{Z}_{f,1}^{N-1},\mathbf{V}_{1 }^{N-1},\mathbf{h},\mathbf{g})\\ \overset{(\text{a})}{\geq}& H(\Theta|\mathbf{Z}_{1} ^{N},\mathbf{Z}_{f,1}^{N-1},\mathbf{V}_{1}^{N-1},\mathbf{h},\mathbf{g}, \boldsymbol{\eta}_{h})\\ \overset{(\text{b})}{=}& H(\Theta|\underbrace{\frac{ \mathbf{g}_{\text{se}}\mathbf{h}_{\text{sd}}^{\mathcal{H}}}{\|\mathbf{h}_{ \text{sd}}\|}X_{1}^{\prime}+\boldsymbol{\eta}_{e,1}}_{Z_{1}},\underbrace{ \frac{\mathbf{g}_{\text{se}}\mathbf{h}_{\text{sd}}^{\mathcal{H}}}{\|\mathbf{ h}_{\text{sd}}\|}X_{2}^{\prime}+\boldsymbol{\eta}_{e,2}}_{Z_{2}},\underbrace{\frac{ \mathbf{g}_{\text{se}}\mathbf{h}_{\text{sd}}^{\mathcal{H}}}{\|\mathbf{h}_{ \text{sd}}\|}X_{3}^{\prime}+\frac{\mathbf{g}_{\text{re}}\mathbf{h}_{rd}^{ \mathcal{H}}}{|\mathbf{h}_{rd}|}X_{r,n}^{{}^{\prime}}+\boldsymbol{\eta}_{e,3}},\cdots,\\ &\underbrace{\frac{\mathbf{g}_{\text{se}}\mathbf{h}_{\text{sd}}^{ \mathcal{H}}}{\|\mathbf{h}_{\text{sd}}\|}X_{N}^{\prime}+\frac{\mathbf{g}_{ \text{re}}\mathbf{h}_{rd}^{\mathcal{H}}}{|\mathbf{h}_{rd}|}X_{r,N}^{{}^{\prime }}+\boldsymbol{\eta}_{e,N},\mathbf{Z}_{f,1}^{N-1},\mathbf{V}_{1}^{N-1}, \mathbf{h},\mathbf{g},\boldsymbol{\eta}_{h})\\ \overset{(\text{c})}{=}& H(\Theta)\underbrace{\frac{ \mathbf{g}_{\text{se}}\mathbf{h}_{\text{sd}}^{\mathcal{H}}}{\|\mathbf{h}_{ \text{sd}}\|}X_{1}^{\prime}+\boldsymbol{\eta}_{e,1},\boldsymbol{\eta}_{e,2}^{N },\mathbf{Z}_{f,1}^{N-1},\mathbf{V}_{1}^{N-1},\mathbf{h},\mathbf{g}, \boldsymbol{\eta}_{h})\\ \overset{(\text{d})}{=}& H(\Theta)+h(\boldsymbol{\eta}_{e,1} )-h\left(\frac{\mathbf{g}_{\text{se}}\mathbf{h}_{\text{sd}}^{\mathcal{H}}}{\| \mathbf{h}_{\text{sd}}\|}X_{1}^{\prime}+\boldsymbol{\eta}_{e,1}\right)+h \Big{(}\boldsymbol{\eta}_{fe,1}^{N-1}\Big{)}-h\Big{(}\mathbf{Z}_{f,1}^{N-1} \Big{)}\\ \geq& H(\Theta)+h(\boldsymbol{\eta}_{e,1})-h\left( \frac{\mathbf{g}_{\text{se}}\mathbf{h}_{\text{sd}}^{\mathcal{H}}}{\|\mathbf{h}_{ \text{sd}}\|}X_{1}^{\prime}+\boldsymbol{\eta}_{e,1}\right)+\sum_{n=1}^{N-1} \!\left(h\Big{(}\boldsymbol{\eta}_{fe,1}\Big{)}-h\Big{(}\mathbf{Z}_{f,1}\Big{)} \right)\\ \overset{(\text{e})}{\geq}& H(\[PERSON])-\underbrace{ \log\!\left(1+\frac{\|\mathbf{h}_{\text{sd}}\cdot\mathbf{g}_{\text{se}}^{ \mathcal{H}}}{\sigma_{e}^{2}}\right)}_{Z}\quad-\underbrace{(N-1)\log\!\left(1+ \frac{\|\mathbf{g}_{\text{de}}\|^{2}P_{f}}{\sigma_{fe}^{2}}\right)}_{Z_{2}} \end{split} ,\] (124) where \((N-1)\log\!\left(1+\frac{\|\mathbf{g}_{\text{de}}\|^{2}P_{f}}{\sigma_{fe}^{2}}\right)\)follows from the fact that \(\Theta\) is a deterministic function of \(M\), \(\mathbf{\eta}_{h}=(\eta_{1}^{N},\eta_{r,1}^{N},\mathbf{\eta}_{f,1}^{N-1})\), and conditioning reduces entropy; 2. follows from (A47); 3. follows from \(X_{r,1}^{\prime}=X_{r,2}^{\prime}=0\), \(X_{n}^{\prime}(2\leq n\leq N)\) and \(X_{r,n}{}^{\prime}(3\leq n\leq N)\) are deterministic functions of \((\eta_{n},\eta_{r,n},\mathbf{\eta}_{f,n},\mathbf{\eta})\); 4. follows from the fact that \(X_{1}^{\prime}\), \(\mathbf{\eta}_{e,1}\) is independent of \(\mathbf{Z}_{f,1}^{N-1}\), for \(n(1\leq n\leq N-1)\), \(Z_{f,n}=\mathbf{g}_{\text{de}}X_{f,n}+\mathbf{\eta}_{fe,n}\) and \(X_{f,n}=\mathbb{M}_{\Lambda}[\gamma_{n}\widehat{\Theta}_{n}+V_{n}]\) are both deterministic functions of \((\Theta,V_{1}^{N-1},\mathbf{h},\mathbf{g},\mathbf{\eta}_{h})\); 5. follows from \[