A New Composite Fractal Function and Its Application in Image Encryption

Abstract

Fractal's spatially nonuniform phenomena and chaotic nature highlight the function utilization in fractal cryptographic applications. This paper proposes a new composite fractal function (CFF) that combines two different Mandelbrot set (MS) functions with one control parameter. The CFF simulation results demonstrate that the given map has high initial value sensitivity, complex structure, wider chaotic region, and more complicated dynamical behavior. By considering the chaotic properties of a fractal, an image encryption algorithm using a fractal-based pixel permutation and substitution is proposed. The process starts by scrambling the plain image pixel positions using the Henon map so that an intruder fails to obtain the original image even after deducing the standard confusion-diffusion process. The permutation phase uses a Z-scanned random fractal matrix to shuffle the scrambled image pixel. Further, two different fractal sequences of complex numbers are generated using the same function i.e. CFF. The complex sequences are thus modified to a double datatype matrix and used to diffuse the scrambled pixels in a row-wise and column-wise manner, separately. Security and performance analysis results confirm the reliability, high-security level, and robustness of the proposed algorithm against various attacks, including brute-force attack, known/chosen-plaintext attack, differential attack, and occlusion attack.

Keywords: composite fractal function; diffusion; henon map; permutation; random fractal matrix; z-scan.

Figure 1

Fractal images generated from MS: (a) Equation (1); (b) Equation (2).

Figure 2

Fractal images generated from CFF: (a) $\beta$ = 0.2; (b) $\beta$ = 0.3; (c) $\beta$ = 0.5.

Figure 3

Zoomed version of fractal images generated from CFF: (a) $\beta$ = 0.3; (b) $\beta$ = 0.5; (c) $\beta$ = 0.5.

Figure 4

Fractal image trajectory map generated from: (a) MS function; (b) CFF with $\beta$ = 0.3; (c) CFF with $\beta$ = 0.5.

Figure 5

Lyapunov map of fractal images generated from: (a) MS function; (b) CFF with $\beta$ = 0.3; (c) CFF with $\beta$ = 0.5.

Figure 6

Proposed Image encryption method block diagram.

Figure 7

(a) Sample CFF matrix; (b) Z-Scanned one-dimensional array.

Figure 8

Scrambling process example using 4 × 4 matrix. (a) Actual pixel position and its corresponding shuffled pixel position; (b) pixels mapping between P1 and SP; (c) Matrix view of the scrambling process using S, P1 and SP respectively.

Figure 9

(a) Plain image; (b) Encrypted image; (c) Decrypted image.

Figure 10

(a) Plain image; (b) Histogram of (a); (c) Cipher image; (d) Histogram of (c).

Figure 11

Elaine image pixel distribution in a different direction. (a–c) Original image pixel distribution in horizontal, vertical, and diagonal direction; (d–f) Cipher image pixel distribution in horizontal, vertical, and diagonal direction.

Figure 12

(a) Actual cipher image and decrypted image; (b) Noisy cipher image with 2% salt and pepper noise and its decrypted image; (c–e) Cipher images and their corresponding decrypted images with different percentages of data loss.

Figure 13

(a–d) Encryption sensitivity using the image and its histogram, (a) Elaine image; (b) Encrypted image using correct secret key; (c) Encrypted images using modified secret key; (d) Image difference between encrypted images using correct and modified secret key (e–h); Decryption sensitivity using an image and its histogram; (e) Decrypted images using the correct secret key; (f–g) Decrypted images using two different modified secret keys; (h) Image difference between decrypted images using two different modified secret keys.

Figure 14

Image autocorrelation plots: (a) Original image; (b) Autocorrelation of original image; (c) Cipher image; (d) Autocorrelation of cipher image.