A strong construction of S-box using Mandelbrot set an image encryption scheme

Abstract

The substitution box (S-box) plays a vital role in creating confusion during the encryption process of digital data. The quality of encryption schemes depends upon the S-box. There have been several attempts to enhance the quality of the S-box by using fractal chaotic mechanisms. However, there is still weakness in the robustness against cryptanalysis of fractal-based S-boxes. Due to their chaotic behavior, fractals are frequently employed to achieve randomness by confusion and diffusion process. A complex number-based S-box and a chaotic map diffusion are proposed to achieve high nonlinearity and low correlation. This study proposed a Mandelbrot set S-box construction based on the complex number and Chen chaotic map for resisting cryptanalytic attacks by creating diffusion in our proposed algorithm. The cryptosystem was built on the idea of substitution permutation networks (SPN). The complex nature of the proposed S-box makes it more random than other chaotic maps. The robustness of the proposed system was analyzed by different analysis properties of the S-box, such as nonlinearity, strict avalanche criterion, Bit independent criterion, and differential and linear probability. Moreover, to check the strength of the proposed S-box against differential and brute force attacks, we performed image encryption with the proposed S-box. The security analysis was performed, including statistical attack analysis and NIST analysis. The analysis results show that the proposed system achieves high-security standards than existing schemes.

Keywords: Chen chaotic system; Cryptosystem; Image encryption; Mandelbrot set; Substitution-box.

Figure 1. Proposed encryption model.

Figure 2. Proposed S-box construction model flowchart.

Figure 3. Formal modeling of Mandelbrot set.

Figure 4. (A–B) Chen’s chaotic system (Chen & Ueta, 1999).

Figure 5. Nonlinearity comparison chart.

Figure 6. Differential probability comparison with literature.

Figure 7. Original image (A, D, G), and histogram analysis of the original image (B, E, H), and cipher images histogram (C, F, I).

Figure 8. RGB channel separation. (B, F, J) red channel, (C, G, K) green channel, (D, H, L) blue channel of airplane, owl and fruits test images.

Figure 9. Histogram of RGB channel of owl (B, C, D); encrypted image channel histogram (F, G, H), histogram of RGB channel of fruits (J, K, L), encrypted image channel histogram (N, O, P).

Figure 10. (A–X) Correlation coefficient of original and cipher images airplane, owl and fruits.

Figure 11. Layer wise correlation of airplane original and cipher image.

(B, F, J, N, R, V) red, green and blue Horizontal plane correlation; (C, G, K, O, S, W) red, green and blue diagonal plane correlation; (D, H, L, P, T, X) shows red, green and blue vertical plane correlation.