# Realizing Efficient Security and Privacy in IoT Networks
## 4 System Model
### Optimization of Swift Privacy Rate in a MIMOME
\begin{array}{l}h(\mathbf{z}_{r},P_{m},P_{B})=-\log\left|\mathbf{M}_{B}+\mathbf{ G}\mathbf{V}_{1}\mathbf{Z}_{r}\mathbf{V}_{1}^{\mathrm{G}}\mathbf{G}^{\mathrm{G}} \right|+\log\left|\mathbf{M}_{B}\right|\\ +A_{E}\Omega\left(\bar{\beta}_{3},\bar{\Theta}_{3},\bar{\sigma}_{3}\right)-A_{E }\Omega\left(\bar{\beta}_{4},\bar{\Theta}_{4},\bar{\sigma}_{4}\right),\end{array}  \tag{31}\]  
where \(\mathbf{M}_{B}\) implies that the receiver has full knowledge of her covariance matrix of noise and interference, \(\bar{\beta}_{3}=\frac{A_{T_{N}}+A_{Rx}}{A_{E}}\), \(\bar{\beta}_{4}=\frac{A_{T_{N}}-r+A_{Rx}}{A_{E}}\), while \(\bar{\sigma}_{3}\) and \(\bar{\sigma}_{4}\) are solutions to the formulated problem. In order to optimize the swift privacy rate, we proposed an optimization problem as stated below. \[ \begin{array}{l}\min\limits_{\mathbf{r}}\mathbf{z}_{r},\bar{P}_{m}^{\min},P _{B}\,h\left(\mathbf{z}_{r},P_{m},P_{B}\right)\\ \text{s.t.}\sum\limits_{i=1}^{r}\mathbf{z}_{r}(i)+P_{m}\leq P_{A}^{\max}\\ \mathbf{z}_{r}(i)\geq 0,\forall i=1,...,r\\ P_{m}\geq 0\\ 0\leq P_{B}\leq P_{B}^{\max}\end{array}  \tag{32}\]  
In the optimization problem above, the constraints are in their convex form; nevertheless, \(h(. )\) is yet to assume a convex form. We can achieve this by rewriting the function  
\[ \begin{array}{l}h(\mathbf{z}_{r},P_{m},P)=-\log\left|\mathbf{M}_{B}+\mathbf{ G}\mathbf{V}_{1}\mathbf{Z}_{r}\mathbf{V}_{1}^{\mathrm{G}}\mathbf{G}^{\mathrm{G}} \right|+\log\left|\mathbf{M}_{B}\right|\\ +A_{E}\bar{\beta}_{3}v_{\Theta_{3}}(\bar{\sigma}_{3})-A_{E}\bar{\beta}_{4}v_{ \Theta_{4}}(\bar{\sigma}_{4})\\ +A_{E}\log\left(\frac{\bar{\sigma}_{4}}{\bar{\sigma}_{3}}\right)+A_{E}\left( \bar{\sigma}_{3}-\bar{\sigma}_{4}\right)\log(c)\\ =-\log\left|\mathbf{M}_{B}+\mathbf{G}\mathbf{V}_{1}\mathbf{Z}_{r}\mathbf{V}_{ 1}^{\mathrm{G}}\mathbf{G}^{\mathrm{G}}\right|+\log\left|\mathbf{M}_{B}\right| \\ +\sum\limits_{k=1}^{N_{\Theta_{3}}}\log\left(1+\bar{\sigma}_{3}(\bar{\Theta}_ {3})k,k\right)-\sum\limits_{k=1}^{N_{\Theta_{4}}}\log\left(1+\bar{\sigma}_{4} (\bar{\Theta}_{4})k,k\right)\\ +A_{E}\log\left(\frac{\bar{\sigma}_{4}}{\bar{\sigma}_{3}}\right)+A_{E}\left( \bar{\sigma}_{3}-\bar{\sigma}_{4}\right)\log(c)\end{array}  \tag{33}\]  
From Equation (33), we can linearize \(+\sum\limits_{k=1}^{N_{\Theta_{3}}}\log\left(1+\bar{\sigma}_{3}(\bar{\Theta} _{3})k,k\right)\), which is the convex form of \(h(. )\) at every iteration point of the optimization algorithm using the expansion method of first-order Taylor series. Similarly, we can resolve the reliance state of \(h(. )\) on the \(\bar{\sigma}_{3},\bar{\sigma}_{4}\) parameters by making them constant with respect to their values and the upgrading them at the end to achieve the following,  
\[1-\bar{\sigma}_{i}^{j+1}=\frac{\bar{\beta}_{i}\bar{\sigma}_{i}^{j+1}}{N_{\bar{ \Theta}_{i}}}\sum\limits_{k=1}^{N_{\Theta_{i}}}\frac{(\bar{\Theta}_{i}^{j+1}) k,k}{1+\bar{\sigma}_{i}^{j+1}(\bar{\Theta}_{i}^{j+1})k,k},i=3,4. \tag{34}\]  
note that \(j+1\) represents the parameter value at \(j+1\) iteration. Thus, we optimized the preceding convex function at \(j\) iteration,  
\[ \begin{array}{l}g^{j}(\mathbf{y}_{r})=-\log\left|\mathbf{M}_{B}+\mathbf{G} \mathbf{V}_{1}\mathbf{Z}_{r}\mathbf{V}_{1}^{\mathrm{G}}\mathbf{G}^{\mathrm{G}} \right|\\ -\sum\limits_{k=1}^{N_{\Theta_{4}}}\log\left(1+\bar{\sigma}_{3}^{j}(\bar{ \Theta}_{4})k,k\right)+\left(\mathbf{y}_{r}-\mathbf{y}_{r}^{j}\right)^{ \dagger}\
abla_{\mathbf{y}_{r}}f^{j}\left|\mathbf{{}_{y_{r}=y_{r}^{j}}}\right.,\end{array}  \tag{35}\]denoting \(\mathbf{y}_{r}=\left[\mathbf{z}_{r}^{I},P_{m},P_{B}\right]^{I}\) and \(f^{j}(\mathbf{y}_{r})=\log|\mathbf{M}_{B}|+\sum\limits_{k=1}^{N_{\Theta_{3}}} \log\left(1+\vartheta_{3}^{j}(\bar{\Theta}_{3})k,k\right)\). Recall that terms that are constant were not used in (25) because they do not have any effect in the optimization. However, at this point, \(h(. )\) has assumed a convex function and we can then optimize it using the preceding optimization  
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