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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.

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.

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.

An invented form of Mandelbrot set came into existence in 1992 when Michelitsch and Rossler have applied seemingly small changes in complex analytic Mandelbrot set function and got an image resembled to a ship going into flame. He named it burning ship. Our goal in this paper is to apply Mann Iteration method to burning ship function and produce a collection of stunning images. We have also calculated the fixed points of such images to measure the convergence rate and those fixed points can be further useful in various fractal applications such as fractal cryptography. Hence we are in position to examining numerically the stability of the fractals.

Antipolynomial of a complex polynomial is generated by applying iteration on a function zd+c for d>=2. This complex function has been intense area for researcher. If we use transcendental function like sine, cosine etc with antipolynomial, i.e. sin zd+c, it becomes a more elite area to design beautiful images of fractal. This paper emphasizes on the creation and analysis of fractals and their convergent to fixed point.

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.