## 2 Problem Formulation and System Model
### System Model  
#### 2.2.1 Unencrypted Hybrid Fingerprint Model  
A noise model can characterize an image because it is related to the image acquisition and post-acquisition processes. The classical noise model usually uses two parameters to characterize camera fingerprints. One of the most widely used statistical noise model is  
\[\sigma_{z_{i}}^{2}\overset{\triangle}{=}f(\mu_{z_{i}};a,b)=a\mu_{z_{i}}^{2}+b\mu _{z_{i}}+\frac{\Delta^{2}}{12}, \tag{1}\]  
where \(z_{i}\) represents the \(i\)-th pixel of an image, \(i=1,\ldots,I\), and \(I\) denotes the number of pixels. \(\sigma_{z_{i}}^{2}\) is the variance of the \(i\)-th pixel. \(\mu_{z_{i}}\) is the expectation of the \(i\)-th pixel. \((a,b)\) represent the camera fingerprints. \(\frac{\Delta^{2}}{12}\) is the quantized noise with step \(\Delta\), and we set \(\Delta=1\) in this paper [12]. We note that the fingerprints \((a,b)\) of different camera models have little difference, which leads to insufficient differentiation between different camera models. Moreover, the statistical noise model \(\sigma_{z_{i}}^{2}\) cannot identify different devices of a specific camera model. Figure 2: The process of identifying the image source camera by a single secondary user. To solve the above two problems, we introduce a tag into the statistical noise model to propose a new hybrid fingerprint model, which is  
\[\sigma_{m_{k}}^{2}=\sigma_{z_{i}}^{2}+\sigma_{g}^{2}\overset{\triangle}{=}f(\mu_{ m_{k}};c,d)=c\mu_{m_{k}}^{2}+d\mu_{m_{k}}+\frac{\Delta^{2}}{12}, \tag{2}\]  
where \(m_{k}\) denotes the \(k\)-th pixel of an image, \(k=1,\ldots,K\), and \(K\) is the number of pixels. \(\sigma_{m_{k}}^{2}\) is the variance of the \(k\)-th pixel. \(\mu_{m_{k}}\) represents the expectation of the \(k\)-th pixel. \((c,d)\) denote the camera fingerprints and \((c,d)\
eq(a,b)\). The designed tag is a zero-mean Gaussian noise that is embedded into the image during the fingerprint extraction process. Denoting the tag as \(g\), it follows the distribution:  
\[g\sim\mathcal{N}(0,\sigma_{g}^{2}) \tag{3}\]  
where \(\sigma_{g}^{2}\) represents the variance of the tag. We chose zero-mean Gaussian noise as the artificial tag primarily due to its well-understood statistical properties, compatibility with existing statistical noise models, and ease of integration into the pixel-wise variance framework. Gaussian noise preserves the assumption of normality commonly used in likelihood-based detectors such as GLRT, thereby maintaining theoretical consistency. While we considered alternative synthetic patterns such as uniform noise, Laplacian noise, and deterministic pseudo-random sequences, these either deviated from the Gaussianity assumption crucial for likelihood computation, or exhibited inferior performance in empirical identification tasks. As a result, Gaussian noise was selected as a suitable and effective design choice for the hybrid fingerprint model. Note that our designed tag helps to improve the identification rate between different camera models when conducting source camera identification. Moreover, our proposed scheme also enables the identification of different devices of a specific camera model. To prove the above two statements, we randomly selected several camera models from the Dresden dataset [24]. First, we conducted an ablation study to explicitly evaluate the impact of introducing the tag into the fingerprint extraction process. Specifically, we plot in Figure 3a,b the results of camera fingerprints extracted from different camera models under the original statistical noise model and our proposed hybrid fingerprint model, respectively. In Figure 3a, the camera fingerprints \((a,b)\) obtained without the tag show significant overlap across different camera models, indicating poor separability. In contrast, Figure 3b shows the camera fingerprints \((c,d)\) extracted using the hybrid fingerprint model, where the overlap areas are substantially reduced.