# Feedback Approach for the Relay Channel with Noisy Feedback and Its Security Analysis
## 4 An SK-Type Scheme for the MISO Fading Relay Channel with Noisy Feedback and Its Security Analysis
### Security Analysis of the Proposed Extended Scheme  
The communication scenario is similar to Figure 1b, except that the existence of Eavesdropper equipped with \(B\) (\(B>1\)) antennas. The information-theoretic schematic diagram is depicted in Figure 3. We assume that Eavesdropper can simultaneously eavesdrop on the signals of both the forward and feedback channels. Here, the channel gains of the Source-Eavesdropper channel, the Relay-Eavesdropper channel and the Destination-Eavesdropper channel are, respectively, denoted by \(\mathbf{g}_{\text{se}}\in\mathbb{C}^{D\times B},\mathbf{g}_{\text{re}}\in \mathbb{C}^{D\times 1},\mathbf{g}_{\text{de}}\in\mathbb{C}^{D\times 1}\). We assume that these three links are independent of each other and experience quasi-static fading. So, the signals received by Eavesdropper are given by  
\[ \begin{split}&\mathbf{Z}_{n}=\mathbf{g}_{\text{se}}\mathbf{X}_{n} +\mathbf{g}_{\text{re}}X_{r,n}+\boldsymbol{\eta}_{e,n},1\leq n\leq N,\\ &\mathbf{Z}_{f,n}=\mathbf{g}_{\text{de}}X_{f,n}+\boldsymbol{\eta }_{fe,n},1\leq n\leq N-1,\end{split}  \tag{62}\]where \(\mathbf{Z}_{n},\mathbf{Z}_{f,n}\in\mathbb{C}^{D\times 1}\), \(\mathbf{X}_{n}\in\mathbb{C}^{B\times 1},X_{r,n}\in\mathbb{C}^{1\times 1},X_{f,n}\in \mathbb{C}^{1\times 1}\) represent the input of Source, Relay and Destination, respectively. \(\eta_{e,n}\sim\mathcal{CN}(0,\sigma_{e}^{2}),\mathbf{\eta}_{fe,n}\sim\mathcal{CN }(0,\sigma_{fe}^{2})\) are the channel noise, which are i.i.d. across the time index \(n\). Definition 7: Following the secrecy criteria defined in [33], a normalized eavesdropper's equivocation (also called the secrecy level) is adopted in this paper and is given by  
\[\Delta=\frac{H\Big{(}M|\mathbf{Z}_{1}^{N},\mathbf{Z}_{f,1}^{N-1},\mathbf{h},\mathbf{g }\Big{)}}{H(M)},0\leq\Delta\leq 1, \tag{63}\]  
where \(\mathbf{Z}_{1}^{N}=(\mathbf{Z}_{1},\mathbf{Z}_{2},\cdots,\mathbf{Z}_{N})\), \(\mathbf{Z}_{f,1}^{N-1}=(\mathbf{Z}_{f,1},\mathbf{Z}_{f,2},\cdots,\mathbf{Z}_{f,N-1})\), \(\mathbf{h}=(\mathbf{h}_{\text{sr}},h_{rd},\mathbf{h}_{\text{sd}},\mathbf{h}_{ \text{ds}})\), \(\mathbf{g}=(\mathbf{g}_{\text{se}},\mathbf{g}_{\text{re}},\mathbf{g}_{\text{de }})\). Here, note that \(\Delta=1\) corresponds to perfect secrecy, indicating that Eavesdropper cannot obtain any useful information, which was first introduced in [34], and \(\Delta=0\) corresponds to no secrecy constraint. Furthermore, in terms of whether Eavesdropper is powerful enough to acquire the complex dither signal sequence \(V_{1}^{N-1}\) shared between legal parties in our scheme, we define the following two cases. * _Case I: Eavesdropper does not know the complex dither signal sequence \(V_{1}^{N-1}\), and the secrecy level is defined by_ \[\Delta_{I}=\frac{H\Big{(}M|\mathbf{Z}_{1}^{N},\mathbf{Z}_{f,1}^{N-1},\mathbf{h}, \mathbf{g}\Big{)}}{H(M)},0\leq\Delta_{I}\leq 1.\] (64)
* _Case II: Eavesdropper knows the complex dither signal sequence \(V_{1}^{N-1}\), and the secrecy level is defined by_ \[\Delta_{II}=\frac{H\Big{(}M|\mathbf{Z}_{1}^{N},\mathbf{Z}_{f,1}^{N-1},V_{1}^{N-1}, \mathbf{h},\mathbf{g}\Big{)}}{H(M)},0\leq\Delta_{II}\leq 1.\] (65)  
Theorem 3.1: _For given coding blocklength \(N\) and decoding error probability \(\epsilon\), the secrecy levels \(\Delta_{I}\) and \(\Delta_{II}\) of the proposed scheme are, respectively, lower-bounded by_  
\[\Delta_{I}\geq 1-\frac{\kappa_{1}}{NR_{\text{FRC}}^{\text{NRC}}(N,\epsilon)}, \Delta_{II}\geq 1-\frac{\kappa_{1}+\kappa_{2}}{NR_{\text{FRC}}^{\text{NRC}}(N, \epsilon)}, \tag{66}\]  
Figure 3: The MISO FRC with noisy feedback and a multi-antenna external eavesdropper.