# Privacy-Preserving Data Analytics in Internet of Medical Things
## 4 Differential Privacy
### Mechanisms of Differential Privacy  
#### 4.1.1 Laplace Mechanism  
This mechanism used in differential privacy to add a controlled amount of noise to the output of computations [33]. The Laplace mechanism can be applied to achieve differential privacy to make sure that the presence or absence of an individual will not significantly alter the result of a calculation or analysis. The amount of noise added in computation's output is evaluated according to the Laplace distribution using the Laplace mechanism [34].  
**Definition 2**.: _Given a function (\(f:O\rightarrow\mathbb{R}\)) that operates on a dataset (\(O\)), the Laplace mechanism perturbs the output of \(f(O)\) to ensure \(\epsilon\)-differential privacy. The perturbed output (\(P(O)\)) is defined as follows:_  
\[P(O)=f(O)+\mathrm{Lap}\bigg{(}\frac{\Delta f}{\epsilon}\bigg{)} \tag{3}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Ref.** & **System Model** & **Goals** & **Limitations** & **Privacy-Preserving Technique(s)** & **Trust Model** \\ \hline \multirow{6}{*}{[PERSON], et al., 2020 [22]} & Publishing of & Privacy & Increased & Enhanced l-diverse & \multirow{2}{*}{
\begin{tabular}{} \end{tabular} } \\  & healthcare data & preservation when & computational & slicing for the & \\  & using l-diverse & releasing EHR & complexity with & grouping of & \\  & slicing and a & data, providing & large datasets and & attributes and & \\  & fixed-interval & maximum data & vulnerability to & fixed-interval & \\  & technique for & utility while & certain privacy & anonymization for & \\  & attribute-focused & ensuring privacy & attacks with fixed & numerical & \\  & anonymization & & methods & attributes & \\ \hline \hline \end{tabular}
\end{table}
Table 2: _Cont._The likelihood density function of the Laplace distribution, which is employed in this mechanism for differential privacy, is represented by this expression [35].  
\[f(u)=\frac{\epsilon}{2b}\exp\left(-\frac{\epsilon|u|}{b}\right) \tag{4}\]  
The privacy budget \(\epsilon\) is a privacy parameter that is responsible obtaining differential privacy [36]. A lesser value of \(\epsilon\) provides higher privacy protection. In Figure 1, \(b\) is a scale parameter used to determines the spread of the distribution (\(b>0\)). The larger the value of \(b\), the greater the increase in the amount of added noise, leading to more fluctuations in the final results. \(|u|\) defines the absolute value of \(u\) to ensure that the Laplace distribution is symmetric around its mean. The value of \(|u|\) is often calculated as  
\[|u|=\frac{b\Delta f}{\epsilon} \tag{5}\]  
Here, privacy and utility are both trade-offs with values larger than \(|u|\), providing stronger privacy but reducing the utility of the output and vice versa. The change in the output of a function applied to two adjacent datasets, i.e., neighbouring datasets that vary in terms of the presence or absence of a single individual's record, is known as the sensitivity of the \(\Delta f\) function.  
\[\Delta f=\max_{d_{1},d_{2}:|d_{1}\Delta d_{2}|=1}||f(d_{1})-f(d_{2})||_{1} \tag{6}\]