## 4 Results and Discussion
### Histogram Analysis
Mean Squared Error, Peak Signal-to-Noise Ratio, and Structural Similarity Index Measure (SSIM) Analysis  
In our analysis, the following parameters implemented for qualitative evaluation, are the mean squared error (MSE) and peak signal-to-noise ratio (PSNR), which are defined by the equations [58]:  
\[\text{MSE}=\frac{1}{\text{M}\times\text{N}}\sum_{i=1}^{\text{M}}\sum_{j=1}^{ \text{N}}[\text{M}(\text{i},\text{j})-\text{F}(\text{i},\text{j})]^{2} \tag{3}\]  
\[\text{PSNR}=10^{\text{x}}\text{ln}(f_{\text{max}}/\text{MSE})^{2} \tag{4}\]  
where M represents the matrix features of the original biometric image, F represents the matrix features related to the encrypted image, m represents the number of rows of pixels of the image, i represents the index of each row, n represents the number of columns of  
Figure 3: Histogram graphical representation of users 1, 2, and 3. (**a**) Original biometric image. (**b**) Encrypted image using the BFV algorithm. (**c**) Encrypted image using the BGV algorithm. pixels of the image, j represents the index of each column, and f\({}_{\text{max}}\) is the maximum value of the signal that exists in our original image. The PSNR is a full reference metric that comprises the noise ratio between the original biometric image and the encrypted image using the homomorphic encryption algorithm. In our current analysis, we are interested in obtaining lower values for the PSNR parameter which indicates a higher level of noise. Consequently, a significant difference identified between the original biometric image and its encrypted version indicates better encryption algorithms, while higher values of PSNR indicate better image quality and accuracy, but not a good encryption algorithm. In contrast to the PSNR, the MSE is a regression factor that calculates the average squared difference between the pixels of the biometric image and its encrypted version, and lower MSE values approaching 0 indicate better accuracy and a lower level of errors between images, which is a mandatory condition for an efficient algorithm. Another important parameter used to assess the performance of our proposed algorithms is the structural similarity index measure (SSIM), which measures the structural similarity between the original image and its modified version, capitalizing on three main components derived from the structure of these images: luminance (l), contrast (c), and structure (s). This parameter and its components are mathematically represented by the following equations [59]:  
\[\text{SSIM}(\text{x},\text{y})=\left[\text{l}(\text{x},\text{y})\right]^{ \alpha}\cdot\left[\text{c}(\text{x},\text{y})\right]^{\beta}\cdot\left[\text{ s}(\text{x},\text{y})\right]^{\gamma} \tag{5}\]  
\[\text{l}(\text{x},\text{y})=\frac{2\mu_{\text{x}}\mu_{\text{y}}+\text{C}_{1}} {\mu_{\text{x}}{}^{2}+\mu_{\text{y}}{}^{2}+\text{C}_{1}} \tag{6}\]  
\[\text{c}(\text{x},\text{y})=\frac{2\sigma_{\text{x}}\sigma_{\text{y}}+\text{C} _{2}}{\sigma_{\text{x}}{}^{2}+\sigma_{\text{y}}{}^{2}+\text{C}_{2}} \tag{7}\]  
\[\text{s}(\text{x},\text{y})=\frac{\sigma_{\text{xy}}+\text{C}_{3}}{\sigma_{ \text{x}}\sigma_{\text{y}}+\text{C}_{3}} \tag{8}\]  
where x is the reference image, y is a modified version of the original image, \(\mu\) calculates the mean intensity of x and y, \(\sigma\) is the standard deviation, \(\alpha\), \(\beta\), and \(\gamma\) are parameters used to adjust the relative importance of the three components luminance (l), contrast (c), and structure (s), and C\({}_{1}\), C\({}_{2}\), C\({}_{3}\) are numerical stabilizing constants. As shown in Table 2, no remarkable differences between the resulting values were identified, and the MSE and the PSNR values calculated using the homomorphic encryption algorithms are very tight. The lower PSNR values indicate a higher level of noise and there is a weak correlation between the two types of images, highlighting the encryption performance of the cryptographic algorithms.