Deterministic chaos game: A new fractal based pseudo-random number generator and its cryptographic application

Abstract

In this paper, a digital image encryption algorithm is proposed based on the generalized model of the chaos game. The chaos game is a well-known fractal, which acts as a pseudo-random number generator (PRNG) in the proposed encryption algorithm. The foundation of the chaos game is based on basic points and its distance ratio that determine the basis of how they distribute random values in 2D or 3D space. These basic points are entered by the user interface and are the result of an encrypted image with a fractal structure. The use of the bifurcation diagram and Lyapunov exponent analysis showed that the proposed chaos game has the dynamical behavior, and fully chaotic characteristic, and can be used as a secure PRNG in cryptography systems. In the proposed method, the region of interest is determined by a number of Bases, and the fractal mechanism of chaos game for the encryption process is performed on the image. This process is very sensitive to any changes in keys and refers to confusion. The evaluation results of security and performance analysis on standard images confirm the efficiency of the proposed method and demonstrate that the proposed method is robust against attacks.

Introduction

With the development of the Internet networks and free access to private information by unauthorized users, the use of an encryption system to protect multimedia systems such as image, video, and audio seems to be necessary. The image is widely used as one of the most important multimedia systems on personal computers and mobile phones. Several algorithms and methods have been used to encrypt the digital image in the last three decades, and the challenges are still ongoing [1].

Image encryption algorithms are divided into two categories: the spatial domain and the transform domain [2]. Spatial domain techniques directly encrypt the original image pixels, that have high speed and more security and easy to use. But the main problem in pixel-based methods is the high capacity of the encrypted image. Transform-based methods are used to encrypt compressed images, such as JPEG and JPEG2000, which has many uses and benefits in saving memory consumption. These algorithms are widely used in DCT and DWT transform domains [3], [4].

On the other hand, cryptosystem algorithms are divided into two class of block and stream cipher [5]. The block cipher has high security and less speed than the stream cipher. Particular examples of block cipher techniques are algorithms such as DES [6] and AES [7], which are also used in image encryption [8]. In most block and cipher stream algorithms, the pseudo-random number generator is used to ensure data security. The PRNGs are deterministic sequences that are generated with an initial condition that these initial conditions play as the secret key in the cryptosystem [9]. Providing a secure PRNG requires the passing of standard statistical tests such as NIST [10], Diehard [11], [12], ENT [13], TESTU01 [14] and etc.

The use of chaos theory as a secure cryptosystem in the last two decades has been at the forefront of dynamical systems. The sensitivity to the initial condition and the deterministic chaotic sequences in discrete chaotic maps are very important properties in cryptographic systems [15]. For this reason, chaotic iterated maps have been used as pseudo-random number generators along with encryption operators for image and video encryption [16], [17], [18]. A variety of applications such as image and video encryption [19], [20], [21], [22], [23], digital stenography [24], [25], [26], image and video watermarking [27], [28], [29], [30], [31], [32] and information hiding [33], [34], [35] are applications that use chaotic discrete maps in the security element of applied methods.

Classic maps such as logistic map [36], [37], [38], [39], Tent map [40], Arnold cat map [41], [42], Henon Map [43], [44], and Sine map [45] are used in many image encryption algorithms. Classical logistics or tent maps, due to their smaller keyspace, have low security which has been investigated in cryptanalysis of chaotic maps [46], [47].

In the research literature, in order to increase the key space, there are various chaotic maps such as DNA sequences[48], [49], hyper chaos [50], mixed map [51], [52], [53], [54], [55], chaotic neural network [56], Josephus Scrambling[57], and Quantum map [58]. Also, one of the most important uses of chaotic mapping in the last two decades has been the substitution boxes in block cipher methods [59], [60]. Using the science of cryptanalysis, a great deal of study has been done on the security of chaotic maps and the efficiency of these PRNGs [61], [62], [63], [64], [65].

A fractal is a geometric structure consisting of components that are obtained by increasing each component to a certain scale, the same basic structure. In other words, the fractal is a structure that each component of a structure is similar to the whole structure [66]. Fractals are found in many natural structures such as snow, mountains, clouds, roots, trunks and leaves of trees, the growth of crystals in igneous rocks, the network of waterways and rivers, electrochemical deposition, bacterial mass growth and blood vessel systems, DNA and..., which can be described, interpreted and predicted by many natural phenomena [67], [68]. Many classic fractals have been raised in real and complex domains, such as the Cantor set, the Koch curve, the Sierpinski gasket, the Julia set and the Mandelbrot set [69]. Fractal networks can be used to produce chaotic attractors that have sensitivity to the initial conditions and control parameters [70].

The chaos game as a self-similar fractal was presented in 1988 by Barnsley [71]. The Barnsley’s chaos game is a deterministic fractal shape which uses random numbers to produce a geometric structure. In this paper, a modified model of the chaos game is presented which is sensitive to initial conditions and has a fully chaotic behavior. To illustrate the chaotic behaviors of the proposed dynamic system, the Lyapunov exponent and bifurcation diagram are used. The standard randomness tests are used to display the randomness of the chaos game as a safe PRNG. Finally, a database of standard images is used to evaluate the performance of the proposed encryption algorithm and the obtained results of the proposed algorithm are compared with other algorithms.

The rest of the paper is organized as follows. In Section 2, the proposed chaos game will be introduced as a new dynamical system. The encryption and decryption process will be described in Section 3. The experimental results of the proposed method and the evaluation of efficiency parameters in the proposed algorithm are discussed in Section 4. Section 5 will be the conclusion section of this paper.