Statistical Analysis of the Negative–Positive Transformation in Image Encryption

The negative–positive transformation (NPT) is a widely employed technique for encrypting images on pixel blocks, commonly integrated into cryptosystems compatible with compression algorithms. The existing literature on NPT analysis can be categorized into two types: theoretical analyses with results that apply to any image, primarily focused on compression compatibility, and numerical analyses that report empirical results from specific images, some without explaining the causes of the security results, while others are only related to the compression performance. Consequently, there is a significant gap in understanding the implications of applying the NPT for data protection. For that reason, this paper conducts a theoretical statistical analysis, presenting, demonstrating, and verifying six theorems to understand the security contributions of NPT. Two theorems examine the shape of the image histogram and the scatter plot of adjacent pixels after the NPT application. The subsequent four theorems explore the influence of NPT on the mean, variance, covariance, and correlation within each pixel block. The findings indicate that the NPT generates images with symmetrical histograms, the correlation of pixel blocks remains invariant, and distinct vertical and horizontal reflections manifest on the scatter plot. These theorems are verified by encrypting the Lena image with four pixel-block sizes. The histogram symmetry passed the goodness-of-fit test at a significance level of 5%, revealing consistent results. The correlation of pixel blocks remained unchanged, and the scatter plot exhibited an x-shaped pattern. Therefore, as the NPT alone does not achieve desirable encryption results, such as uniform histograms, scatter plots, and decreasing correlation, cryptosystems should complement it with additional techniques.