Comparative analysis of image encryption based on 1D maps and their integrated chaotic maps

The Maximum Lyapunov Exponent(MLE) measures the sensitivity of a chaotic system to its initial conditions. In chaos theory, systems with positive MLE values are considered chaotic, and larger MLE values generally indicate more robust chaos. A higher MLE suggests a more chaotic and complex behavior, which could be beneficial for encryption as it might provide enhanced security due to increased sensitivity to initial conditions and resistance to attacks. This paper searches for whether a 1D seed chaotic map or a 1D integrated chaotic map is preferred for image encryption. Does the MLE have any role? Firstly, three 1D integrated chaotic maps were designed: Sine-Cubic Integrated Map(SCIM), Tent-Logistic Integrated Map(TLIM), and Sine-Logistic Integrated Map(SLIM). These integrated chaotic maps are designed using the four available seed maps: sine, logistics, cubic, and tent. Thus, we have considered seven 1D chaotic maps to analyse and answer the question. Secondly, image encryption and decryption are performed using the considered seven 1D chaotic maps, one after the other, and the security measures of the encrypted image are analysed using various available tools. The image encryption is performed using block shuffling as diffusion and bit-Xor operation as the confusion process. A comparative analysis is performed using the six quantitative security analysis tools. According to the encryption correlation coefficient value of 0.0017, the Pick Signal-To-Noise Ratio(PSNR) value of 9.204, the Mean Square Error(MSE) value of 7809.1, the Number of Pixel Change Rate(NPCR) value of 95.3903, the Unified Average Change Intensity(UACI) value of 33.3676, and the information entropy value of 7.9635, the sine map is ranked first in security. The comparative analysis result reveals that seed maps give better encrypted image security than integrated chaotic maps. Therefore, integrating 1D chaotic maps is not guaranteed to get a better-secured encrypted image. Further analysis is made to understand if the MLE directly impacts the security of the encrypted process. It is found that the integrated chaotic maps provide a higher MLE. However, in this analysis, we couldn't observe the direct relationship between the MLE and the security of the encrypted image. This suggests that other factors beyond just MLE contribute to the security of the encryption process.