# A High-Utility Differentially Private Mechanism for Space Information Networks
## 3 Proposed Mechanism
### Privacy Analyses
\begin{split}&\text{Pr}\{M(D_{1}\cup\ldots D_{i+x}\ldots\cup D_{n})=a\} \\ =&\text{Pr}\{S(D_{1})=\bar{D}_{1}\}\ldots\text{Pr}\{S(D_ {i+x})=\bar{D}_{i+x}\}\ldots\text{Pr}\{S(D_{n})=\bar{D}_{n}\}\text{Pr}\{Lap( \bar{D}_{1}\cup\ldots\bar{D}_{i+x}\ldots\cup\bar{D}_{n})=a\}\\ +&\text{Pr}\{S(D_{1})=\bar{D}_{1}\}\ldots\text{Pr}\{S(D_ {i+x})=\bar{D}_{i-x}\}\ldots\text{Pr}\{S(D_{n})=\bar{D}_{n}\}\text{Pr}\{Lap( \bar{D}_{1}\cup\ldots\bar{D}_{i-x}\ldots\cup\bar{D}_{n})=a\}\\ =&\text{Pr}\{S(D_{1}\cup\ldots D_{i+x}\ldots\cup D_{n})= \bar{D}_{1}\cup\ldots\bar{D}_{i+x}\ldots\cup\bar{D}_{n}\}\text{Pr}\{Lap(\bar{D }_{-x})=a\}\\ =&\text{Pr}\{S(D_{1}\cup\ldots D_{i+x}\ldots\cup D_{ n})=\bar{D}_{1}\cup\ldots\bar{D}_{i-x}\ldots\cup\bar{D}_{n}\}\text{Pr}\{Lap(\bar{D }_{-x})=a\}\\ =&\text{Pr}\{S(D_{+x})=\bar{D}_{+x}\}\text{Pr}\{ Lap(\bar{D}_{+x})=a\}+\text{Pr}\{S(D_{+x})=\bar{D}_{-x}\}\text{Pr}\{Lap(\bar{D }_{-x})=a\}\\ =&\text{$\pi_{x}$}\text{Pr}\{S(D_{-x})=\bar{D}_{- x}\}\text{Pr}\{Lap(\bar{D}_{+x})=a\}+(1-\text{$\pi_{x}$})\text{Pr}\{S(D_{-x})= \bar{D}_{-x}\}\text{Pr}\{Lap(\bar{D}_{-x})=a\}\\ \leq&\text{$\pi_{x}$}\text{Pr}\{S(D_{-x})=\bar{D}_ {-x}\}\text{$\epsilon^{\frac{LRS_{x}}{c}}$}\text{Pr}\{Lap(\bar{D}_{-x})=a\}+(1 -\text{$\pi_{x}$})\text{Pr}\{S(D_{-x})=\bar{D}_{-x}\}\text{Pr}\{Lap(\bar{D}_{ -x})=a\}\\ =&(1-\text{$\pi_{x}$}+\text{$\pi_{x}$}\text{$ \epsilon^{\frac{LRS_{x}}{c}}$})\text{Pr}\{S(D_{-x})=\bar{D}_{-x}\}\text{Pr}\{ Lap(\bar{D}_{-x})=a\}\\ =&(1-\text{$\pi_{x}$}+\text{$\pi_{x}$}\text{$ \epsilon^{\frac{LRS_{x}}{c}}$})\text{Pr}\{M(D_{-x})=a\}\\ \end{split}  \tag{19}\]  
For \(LRS_{x}\geq c_{i}\), the \(\text{$\pi_{x}$}=\frac{1-\text{$\epsilon^{\frac{LRS_{x}}{c}}$}}{1-\text{$ \epsilon^{\frac{LRS_{x}}{c}}$}}\). We have  
\[ \begin{split}& 1-\text{$\pi_{x}$}+\text{$\pi_{x}$}\text{$ \epsilon^{\frac{LRS_{x}}{c}}$}\\ =& 1-\frac{1-\text{$\epsilon^{\epsilon}$}}{1-\text{$ \epsilon^{\frac{LRS_{x}}{c}}$}}+\frac{1-\text{$\epsilon^{\epsilon}$}}{1-\text{$ \epsilon^{\frac{LRS_{x}}{c}}$}}\text{$\epsilon^{\frac{LRS_{x}}{c}}$}\text{$ \epsilon^{\frac{LRS_{x}}{c}}$}\\ \leq& 1-\frac{1-\text{$\epsilon^{\epsilon}$}}{1-\text{$ \epsilon^{\frac{LRS_{x}}{c}}$}}+\frac{1-\text{$\epsilon^{\epsilon}$}}{1-\text{$ \epsilon^{\frac{LRS_{x}}{c}}$}}\text{$\epsilon^{\frac{LRS_{x}}{c_{i}}}$}\\ =&\text{$\epsilon^{\epsilon}$}\\ \end{split}  \tag{20}\]  
For \(LRS_{x}<c_{i}\), the \(\text{$\pi_{x}$}=1\) we have  
\[ \begin{split}& 1-\text{$\pi_{x}$}+\text{$\pi_{x}$}\text{$ \epsilon^{\frac{LRS_{x}}{c}}$}\\ =&\text{$\epsilon^{\frac{LRS_{x}}{c}}$}\\ \leq&\text{$\epsilon^{\frac{LRS_{x}}{c_{i}}}$}\\ <&\text{$\epsilon^{\epsilon}$}\\ \end{split}  \tag{21}\]  
Therefore, we have  
\[\text{$Pr}\{M(D_{+x})=a\}\leq\text{$\epsilon^{\epsilon}$}\text{Pr}\{M(D_{-x} )=a\} \tag{22}\]  
Because of the symmetry, we have  
\[\text{$Pr}\{M(D_{-x})=a\}\leq\text{$\epsilon^{\epsilon}$}\text{Pr}\{M(D_{+x})=a\} \tag{23}\]  
Consequently, the suggested mechanism \(M\) is a distinct private mechanism that complies with \(\epsilon\). Theorem 2: _The output of each satellite satisfies \(\text{$\epsilon^{\frac{LRS_{x}}{c}}$}\text{$\epsilon^{\frac{LRS_{x}}{c}}$}\) differential privacy._  
Proof: For output \(a_{i}\) of \(i\)th satellite, we have  
\[a_{i}=q(\bar{D}_{i})+\sqrt{B_{n-1}}l_{i} \tag{24}\]  
The record \(x\) is indicated by the index \(+x\) of datasets as being in those datasets, while the record \(x\) is indicated by the index \(-x\) as not being in those datasets. In the event that the dataset acquired from \(D_{i}\) by sampling procedure \(S\) is \(\bar{D}_{i}\), let \(Lap_{i}(\bar{D}_{i})\) represent the output \(a_{i}\). We have Equation (25) when record \(x\) is present in dataset \(\bar{D}_{i}\). \[Pr\{Lap_{i}(\bar{D}_{i+x})=a_{i}=q(\bar{D}_{i+x})+\sqrt{B_{n-1}}l_{i}\}=Pr\{l_{i}= \frac{Lap_{i}(\bar{D}_{i+x})-q(\bar{D}_{i+x})}{\sqrt{B_{n-1}}}\} \tag{24}\]  
Due to \(l_{i}\) complies with probability density function \(f(x)=\frac{e}{2c}e^{-\frac{e(x)}{c}}\) and the Laplace distribution, where \(c=max(c_{1},\ldots,c_{i},\ldots,c_{n})\).