## 3 Identification Scheme with Privacy-Preserving
### Identification of Single Secondary User  
#### 3.2.1 Likelihood Ratio Test in Ideal Scenarios  
By comparing the likelihood of data under two hypotheses, the Likelihood Ratio Test (LRT) is a classical method for solving binary hypothesis testing problems. Specifically, the LRT calculates the ratio of likelihood functions under the two hypotheses and compares it with a predefined threshold. If the ratio exceeds the threshold, hypothesis \(H_{1}\) is accepted; otherwise, hypothesis \(H_{0}\) is accepted. According to the Neyman-Pearson lemma [27], the LRT provides an optimal decision rule under ideal conditions, where model parameters are known, and it is effective in addressing tasks such as source camera identification. In this paper, according to hypothesis testing theory and the privacy-preserving strategy, source camera identification can be regarded as a binary hypothesis testing problem in an encrypted environment. In an ideal scenario where all parameters are known, we apply  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Number of Secondary Users** & **Computational Time** & **Memory Usage** \\ \hline
3 & 1.54 h & 10.91 GB \\
7 & 3.61 h & 25.63 GB \\
12 & 6.17 h & 44.11 GB \\ \hline \hline \end{tabular}
\end{table}
Table 1: Performance metrics for identifying whether the same image set is from a Nikon D90 [26] using a distributed scheme with varying numbers of secondary users.  
the likelihood ratio test (LRT) to solve different situations. Specifically, the LRT can be defined as  
\[\delta_{LRT}(\dot{M})=\begin{cases}H_{0},\text{if}\Lambda_{LR}(\dot{M})<\Theta, \\ H_{1},\text{if}\Lambda_{LR}(\dot{M})\geq\Theta,\end{cases} \tag{10}\]  
where \(\Lambda_{LR}(\dot{M})=\sum_{h=1}^{H}\sum_{i=1}^{S_{h}}\Lambda_{LR}(M_{h,i})\) denotes the likelihood ratio (LR) of the image \(\dot{M}\). The threshold \(\Theta\) is set empirically. The LR of \(\dot{M}_{h,i}\) can be formulized as  
\[\Lambda(\dot{M}_{h,i})=\frac{1}{2}\log(\frac{\sigma_{h,0}^{2}}{\sigma_{h,1}^{ 2}})+\frac{1}{2}(\frac{1}{\sigma_{h,0}^{2}}-\frac{1}{\sigma_{h,1}^{2}})(\dot{ M}_{h,i}-\mu_{h})^{2}, \tag{11}\]  
where \(\log(\cdot)\) represents the natural logarithm function, \(\sigma_{h,0}^{2}\) and \(\sigma_{h,1}^{2}\) indicate the variances of \(f(\mu_{h};c_{0},d_{0})\) and \(f(\mu_{h};c_{1},d_{1})\) corresponding to hypotheses \(H_{0}\) and \(H_{1}\), respectively.  
In order to analytically evaluate the statistical performance of \(\delta_{LRT}(\dot{M})\), the statistical properties of the likelihood ratio \(\Lambda_{LR}(\dot{M})\) are examined, which can be represented as a statistical model.  
\[\Lambda_{LR}(\dot{M})\overset{d}{\longrightarrow}N(m_{j},v_{j}) \tag{12}\]  
where \(m_{j}\) and \(v_{j}\) represent the expectation and variance of \(\Lambda_{LR}(\dot{M})\) under hypothesis \(H_{j}\), respectively, as detailed in [12].  
Considering the inherent variability of natural images, the likelihood ratio \(\Lambda_{LR}(\dot{M})\) is normalized under the hypothesis, and can be expressed as follows:  
\[\bar{\Lambda}_{LR}(\dot{M})=\frac{\Lambda_{LR}(\dot{M})-m_{0}}{\sqrt{v_{0}^{2 }}}, \tag{13}\]  
The expression for \(\delta_{LRT}(\dot{M})\) given in Equation (10) can be redefined as follows:  
\[\bar{\delta}_{LRT}(\dot{M})=\begin{cases}H_{0},\text{if}\ \bar{\Lambda}_{LR}( \dot{M})<\bar{\Theta},\\ H_{1},\text{if}\ \bar{\Lambda}_{LR}(\dot{M})\geq\bar{\Theta},\end{cases} \tag{14}\]  
where the threshold \(\bar{\Theta}\) is set empirically.