## 3 Identification Scheme with Privacy-Preserving
### Identification of Single Secondary User
#### 3.2.2 Generalized Likelihood Ratio Test in Real Scenarios  
In practical applications, the parameters of all camera models are typically unknown, so the LRT proposed for ideal scenarios is not applicable in real-world situations. The Generalized Likelihood Ratio Test (GLRT) is an extension of the LRT, designed for cases where parameters are not fully known. In the GLRT, because certain parameters (such as the noise characteristics or statistical distribution parameters of the camera model) may be unknown, it estimates these unknown parameters by maximizing the likelihood function, and then computes the likelihood ratio. Unlike the LRT, the GLRT provides effective inference when parameters are partially unknown, making it especially suitable for the complex scenarios encountered in real-world applications. In this paper, we assume that the camera model \(C_{0}\) is known, and that the camera fingerprint \((c_{0},\dot{d}_{0})\) can be accurately estimated from images of model \(C_{0}\). Specifically, we randomly select 10 images from the known camera model \(C_{0}\) and estimate the camera fingerprint \((c_{0},d_{0})\) in advance. Therefore, in this paper, the task is to determine whether an image comes from the known camera model \(C_{0}\), where \(\dot{M}\) denotes the encrypted image \(M\). Based on existing research (e.g., [28; 29]), the GLRT is considered the most effective method to address source camera identification in real-world scenarios. The GLRT can be expressed as  
\[\delta_{GLRT}(\dot{M})= \begin{cases}H_{0},\text{if}\Lambda_{GLR}(\dot{M})<\Theta^ {*},\\ H_{1},\text{if}\Lambda_{GLR}(\dot{M})\geq\Theta^{*},\end{cases}  \tag{15}\]  
where \(\Lambda_{GLR}(\dot{M})=\sum_{h=1}^{H}\sum_{i=1}^{s_{h}}\Lambda_{GLR}(\dot{M}_{ h,i})\) denotes the generalized likelihood ratio (GLR) of the image \(\dot{M}\). \(\Theta^{*}\) is a preset threshold set empirically. According to (15), the GLR of \(\dot{M}_{h,i}\) can be written as  
\[\Lambda_{GLR}(\dot{M}_{h,j})=\frac{1}{2}\log(\frac{\widetilde{\sigma}_{h,0}^{2 }}{\widetilde{\sigma}_{h,1}^{2}})+\frac{1}{2}(\frac{1}{\widetilde{\sigma}_{h, 0}^{2}}-\frac{1}{\widetilde{\sigma}_{h,1}^{2}})(\dot{M}_{h,j}-\widetilde{ \mu}_{h})^{2}, \tag{16}\]  
where \(\widetilde{\sigma}_{h,0}^{2}\) and \(\widetilde{\sigma}_{h,1}^{2}\) denote the estimated variances under hypothesis \(H_{0}\) and \(H_{1}\), respectively. The local expectation \(\widetilde{\mu}_{h}=\frac{1}{s_{h}}\sum_{i=1}^{s_{h}}m_{h,i}^{app}\) and \(m_{h,i}^{app}\) denotes the denoised pixel in the \(h\)-th part with index \(i\) of image \(\dot{M}\). Based on our hybrid fingerprint model, \(\widetilde{\sigma}_{h,0}^{2}\) corresponds to \(f(\widetilde{\mu}_{h};c_{0},d_{0})\) and \(\widetilde{\sigma}_{h,1}^{2}\) can be acquired through maximum likelihood (ML) estimation, formulated as  
\[\widetilde{\sigma}_{h,1}^{2}=\frac{1}{s_{h}-1}\sum_{i=1}^{s_{h}}\left(m_{h,i}^ {res}-\bar{m}_{h}^{res}\right)^{2} \tag{17}\]  
where \(m_{h,i}^{res}\) means the noise in the \(h\)-th part with index \(i\) of image \(\dot{M}\) and \(\bar{m}_{h}^{res}=\frac{1}{s_{h}}\sum_{i=1}^{s_{h}}m_{h,i}^{res}\). In addition, the variation in the estimated unknown fingerprint parameters under \(H1\) conditions was considered to fully evaluate the statistical performance of GLRT for source camera identification. The GLR \(\Lambda_{GLR}(\dot{M})\) under hypothesis \(H_{j}\) can be expressed as  
\[\Lambda_{GLR}(\dot{M})\overset{d}{\longrightarrow}N(m_{j}^{*},v_{j}^{*}) \tag{18}\]  
where \(m_{j}^{*}\) and \(v_{j}^{*}\) represent the expectation and variance of the GLR \(\Lambda_{GLR}(\dot{M})\), as detailed in [12]. In order to avoid the image content affecting the selection of thresholds, we use a normalization process for \(\Lambda_{GLR}(\dot{M})\).