The Coupling of a Multiplicative Group and the Theory of Chaos in the Encryptions of Images

Abstract

This study is centered on a significant scientific contribution within the realm of image cryptography. The chosen approach involves employing the bidimensional Arnold Cat Map transformation to reposition and modify pixel locations, guided by parameters derived from the original image. The construction of the multiplicative group Z/nZ, comprising equivalence classes modulo n, relies on a hyper-chaotic sequence derived from the 2D sinusoidal logistic modulation map. The correlation between this sequence and the preceding step yields an unpredictable blurring pattern, effectively altering the statistical properties of resulting matrices and distributing the influence of individual bits across the entire encrypted image. For each pixel, the encryption process entails an XOR operation with the Z/nZ group, followed by a right shift based on the three Least Significant Bits (LSB) of the preceding pixel. This meticulous procedure is iterated for every pixel, leaving no trace of similarity or association with the original plaintext image, effectively rendering it blurred and indecipherable. To gauge the efficacy of our algorithm, we subjected it to thorough evaluation utilizing diverse criteria, including histogram analysis, which unveiled a nearly uniform pattern in the encrypted images. Entropy values were found to be close to 8, while the correlation analysis exhibited a pronounced proximity to 0. Moreover, we subjected our approach to differential attacks, and the calculated values of the Number of Changing Pixel Rate (NPCR > 99.6) and the Unified Averaged Changed Intensity (UACI > 33.2) corroborated the strength and resilience of our methodology. In addition, to establish its comparative standing, we undertook a comprehensive assessment, meticulously comparing our method to various existing approaches from the literature, including those proposed by Hua, Es-sabry, and Faragallah. This systematic process accentuated the high level of responsiveness and sensitivity inherent in our approach, thus underscoring its innovative and promising nature