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A cascade function is designed by combining two seed maps that resultantly has more parameters, high complexity, randomness, and more unpredictable behavior. In the paper, a cascade fractal function, i.e. cascade-PLMS is proposed by considering the phoenix and lambda fractal functions. The constructed cascade-PLMS exhibits the required fractal features such as fractional dimension, self-similar structure, and covering entire phase space by the data sequence in addition to the chaotic properties. Due to the chaotic behavior, the proposed function is utilized to generate a pseudo-random number sequence in both integer and binary format. This is the result of an extreme scalability feature of a fractal function that can be implemented on a large scale. A sequence generator is designed by performing the linear function operation to the real and imaginary part of a cascade-PLMS, cascade-PLJS separately, and the iteration number at which the cascade-PLJS converges to the fixed point. The performance analysis results show that the given method has a large keyspace, fast key generation speed, high key sensitivity, and strong randomness. Therefore, the scheme can be efficiently used further to design a secure cryptosystem with the ability to withstand various attacks.

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.

The chapter intends to propose a hybrid cryptosystem based on a chaotic map and a fractal function. The sequential order of process execution provides a computationally less expensive and simple approach that still designed a secure cryptosystem. A one-dimensional Ricker map and its modified form are employed to initially shuffle the image pixels twice, and also a pseudo-random sequence is generated using both maps. The algorithm implemented a sequence of pixel confusion-diffusion steps using the image rotation and a transcendental anti-Mandelbrot fractal function (TAMFF) and its Mann-iterated fractal function (Sup-TAMFF). Finally, the pixel value of an image obtained in the last step and the recent two pixels of the encrypted image is XORed with the corresponding pseudo-random matrix value to get the cipher image. Subsequently, various performance tests are conducted to verify the suitability of the given method to be used in real-world information transmission.

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.

Mandelbrot set with transcendental function presents an out of the ordinary field of research because of its magnificence and geometrical complexities. Earlier many researchers used feedback system to study it and reveal new concepts unexplored the geometry of transcendental Mandelbrot set with new iterative approach. The objective of the paper is to analyze the work done by several researchers to visualize beautiful fractal graphics of Mandelbar set with sine, cosine functions using Mann iteration method. Also the rate of convergence of function in the presence of transcendental function and conjugacy with Mandelbrot set and related complex structure of superior Julia set.

A rapid growth of data in recent time, Industries and academia required an intelligent data analysis tool that would be helpful to satisfy the need to analysis a huge amount of data. MapReduce framework is basically designed to compute data intensive applications to support effective decision making. Since its introduction, remarkable research efforts have been put to make it more familiar to the users subsequently utilized to support the execution of massive data intensive applications. Our survey paper emphasizes the state of the art in improving the performance of various applications using recent MapReduce models and how it is useful to process large scale dataset. A comparative study of given models corresponds to Apache Hadoop and Phoenix will be discussed primarily based on execution time and fault tolerance. At the end, a high-level discussion will be done about the enhancement of the MapReduce computation in specific problem area such as Iterative computation, continuous query processing, hybrid database etc.

These days Mandelbrot set with transcendental function is an interesting area for mathematicians. New equations have been created for Mandelbrot set using trigonometric, logarithmic and exponential functions. Earlier, Ishikawa iteration has been applied to these equations and generate new fractals named as Relative Superior Mandelbrot Set with transcendental function. In this paper, the Mann iteration is being applied on Mandelbrot set with sine function i.e. sin(zn)+c and new fractals with the concept of Superior Transcendental Mandelbrot Set will be shown. Our goal is to focus on the less number of iterations which are required to obtain fixed point of function sin(zn)+c.