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An increasing demand of secure data transmission over internet leads to the challenge of implementing a consistent cryptosystem. In 2004, USA navy published the patent which highlights the importance of fractal as an encryption/decryption key in a cryptosystem [1]. Fractal possess butterfly effect i.e. sensitivity to initial condition, due to which small change in input produces a major change in output. This paper summarizes the various recent image encryption techniques in which fractal key is used to encrypt/decrypt followed by substitution, scrambling and diffusion techniques to provide strong cryptosystem. The algorithms covered both private key encryption as well as public key encryption technique in the paper. The analysed algorithms include a set of fractal function such as Mandelbrot set, Julia set, Hilbert curve, 3D fractal, multi-fractal, IFS and chaotic function to generate a complex key used in the encryption process. Corresponding performance of each algorithm is analysed by PSNR test, key space, sensitivity analysis and correlation coefficient value between the adjacent pixels of both images (Original image and encrypted image) which shows significant improvement in performance over the traditional encryption methods.

Fractal&rsquo, 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.

This paper proposes a fractal-based image encryption algorithm which follows permutation-substitution structure to maintain confusion and diffusion properties. The scheme consists of three phases: key generation process; pixel permutation using the Knuth shuffle method; and the dynamic diffusion of scrambled image. A burning ship fractal function is employed to generate a secret key sequence which is further scanned using the Hilbert transformation method to increase the randomness. The chaotic behavior of the fractal strengthens the key sensitivity towards its initial condition. In the permutation phase, the Knuth shuffle method is applied to a noisy plain image to change the index value of each pixel. To substitute the pixel values, a dynamic diffusion is suggested in which each scrambled pixel change its value by using the current key pixel and the previously ciphered image pixel. To enhance the security of the cryptosystem, the secret key is also modified at each encryption step by performing algebraic transformations. The visual and numerical analysis demonstrates that the proposed scheme is reliable to secure transmission of gray as well as color images.

The study introduces a key generation scheme using a burning ship fractal function, Hilbert transformation and an external key. The burning ship function is a modified version of a well-known Mandelbrot set function in which absolute value of a complex variable is considered. The process starts with the scrambling of the fractal image pixels by applying a Hilbert curve scanning. To enhance the randomness and complexity, an external key is obtained using a pseudo random number generator (PRNG), whose length depends on the size of the used fractal image. Further, a covering module is applied in which eight different types of operations are performed recursively to cover the scrambled fractal image pixels using eight different keys. At each iteration, a modified external key was used to perform the respective operation to the remaining image pixels. Moreover, to ensure the robustness of the proposed scheme, each block of tempkey (created in previous step) permuted using the sorting indexes of the modified tempkey blocks. The performance analysis of the given method is carried out in terms of the key space, key sensitivity, key generation time, histogram, and correlation coefficient. The results indicate that the proposed method is reliable and secure with great potential to be further use in the image encryption applications.

A chaos-based cryptosystem has been suggested and investigated since last decade because of its sensitivity to the initial condition, unpredictability and ergodicity properties. The paper introduces a new chaotic map which helps to enhance the security of image transmission by blending the superior fractal function with a new 2D-Sine Tent composite map (2D-STCM) to generate a key stream. A trajectory map of a proposed 2D-STCM show a wider chaotic range implies better unpredictability and ergodicity feature, suitable to design a cryptosystem. A fractal based image encryption increases the key space of the security key up-to hundreds of bits, thus secure the proposed cryptosystem from brute-force attack. The requirement of confusion and diffusion are fulfilled by applying chaotic circular pixel shuffling (CCPS) to change the pixel position repeatedly and the execution of an improved XOR operation i.e., complex XOR, designed to increase the encryption quality. The proposed cryptosystem has been analyzed using statistical analysis, key sensitivity, differential analysis and key space analysis. The experimental result proves that the new scheme has a high security level to protect the image transmission over the network.