# A High-Utility Differentially Private Mechanism for Space Information Networks
## 3 Proposed Mechanism
### Proposed Differentially Private Mechanism  
In order to safeguard the privacy of the Space Information Networks, this study integrates customized sampling technology into local differentially private processes and makes use of distributed noise. Based on personalized sampling and distributed noise, this paper proposes a new differentially private mechanism with high data utility. Figure 4 shows how the proposed mechanism handles queries from the user.  
In particular, the user sends a query \(q\) to the master satellite. Then, the master satellite draws a number \(B_{n-1}\) obeying the Beta distribution. The master satellite sends the query \(q\) together with the square root of drawn number \(B_{n-1}\) to other satellites.  
The number \(B_{n-1}\) is used to generate a number obeying a particular Laplace distribution in a distributed way. Specifically, generating a number obeying a particular Laplace distribution can be simulated by generating other numbers obeying the same Laplace distribution, as defined by the following formula [26]  
\[l=\sqrt{B_{n-1}}\sum_{i}^{n}l_{i} \tag{4}\]  
Notably, the target Laplace distribution's mean needs to be zero. The number \(B_{n-1}\) follows the Beta distribution and has a probability density function  
Figure 4: The process of proposed mechanism handles queries from the user.  
Figure 3: System structure diagram of this work.  
The specific formula of calculating sampling probability of record \(x\in D_{i}\) is  
\[\pi_{x}=\left\{\begin{array}{cc}\frac{1-e^{c}}{1-e^{\frac{LBS_{x}}{c_{i}}}},&LRS _{x}\geq c_{i}\\ 1&,LRS_{x}<c_{i}\end{array}\right. \tag{6}\]  
Here, \(c_{i}\) is named balance parameter, which is used to maximize the utility of the proposed mechanism, and it may be determined using Equation (7) if the original dataset is \(D_{i}\) and the dataset generated by sampling is \(\bar{D_{i}}\)[21, 22].  
\[c_{i}=\arg\max_{c}\frac{1}{2}\sum_{D_{i}\in\{S(D_{i})\}}\prod_{x\in\bar{D_{i} }\setminus D_{c}}\frac{1-e^{c}}{1-e^{\frac{LBS_{x}}{c}}}\prod_{x\in D_{i} \setminus D_{i}}(\frac{e^{c}-e^{\frac{LBS_{x}}{c}}}{1-e^{\frac{LBS_{x}}{c}}} )\cdot(e^{-\frac{c(a+q(D_{i})-q(D_{i})))}{c}}+e^{-\frac{c(a-q(D_{i})+q(D_{i}) ))}{c}}) \tag{7}\]  
Here, \(D_{c}\) represents the set of data records with \(LRS>c_{i}\). \(\alpha\) is the error boundary.  
The \(i\)th satellite samples the dataset \(D_{i}\) into \(\bar{D_{i}}\) through the sampling process \(S\) with calculated sampling probabilities \(\{\pi_{x}|x\in D_{i}\}\) and calculates \(q(\bar{D_{i}})\). The \(i\)th satellite then generates a number with probability density function \(f(x)=\frac{e}{2c}e^{-\frac{c(x)}{c}}\) and obeys the Laplace distribution, where \(c=max(c_{1},\ldots,c_{i},\ldots,c_{n})\). Finally, the \(i\)th satellite calculates \(a_{i}=q(\bar{D_{i}})+\sqrt{B_{n-1}}l_{i}\) and then sends \(a_{i}\) to the master satellite.  
After receiving \(a_{1},\ldots,a_{i},\ldots,a_{n}\) from all \(n\) satellites, the master satellite calculates \(a=\sum_{i=0}^{n}a_{i}\) and returns the calculated sum to the user as the answer to the query submitted by the user.  
Notably, the query \(q\) is the linear query. For the linear query, the following Equation (8)  
\[q(D_{1}\cup D_{2}\cdots\cup D_{i}\cdots\cup D_{n})=q(D_{1})+q(D_{2})+\cdots+q( D_{i})+\cdots+q(D_{n}) \tag{8}\]  
holds provided that \(\forall i\
eq j,0<i\leq n,0<j\leq n\), \(D_{i}\cap D_{j}=\emptyset\). This equation makes it possible to calculate the correct answer to query \(q(D)\) by the master satellite through returned values \(q(D_{1})\), \(q(D_{2})\), \(\ldots\), \(q(D_{i})\), \(\ldots\), \(q(D_{n})\).  
For the non-linear query, the proposed differentially private mechanism also works if existing an aggregation function \(f\) such that  
\[q(D_{1}\cup D_{2}\cdots\cup D_{i}\cdots\cup D_{n})\approx f(q(D_{1}),q(D_{2}), \ldots,q(D_{i}),\ldots,q(D_{n})) \tag{9}\]  
The characteristic of the non-linear query determines which aggregation function should be used. The average function, mod function, median function, and other aggregation functions are often employed.