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A New Simple Digital Image Cryptography. Technique Based on Multi-Scroll Chaotic Delay. Differential Equation. S. Maksuanpan and W. San-Um. Intelligent ...

2013 5th International Conference on Knowledge and Smart Technology (KST)

A New Simple Digital Image Cryptography Technique Based on Multi-Scroll Chaotic Delay Differential Equation S. Maksuanpan and W. San-Um Intelligent Electronic Systems Research Laboratory (IES) Faculty of Engineering, Thai-Nichi Institute of Technology (TNI) Patthanakarn, Suanlaung, Bangkok, Thailand, 10250. Fax :(+662)-763-2700, Tel :(+662)-763-2600 E-mail addresses: [email protected], [email protected]

Abstract—A new simple digital image cryptography technique based on multi-scroll chaotic Delay Differential Equation (DDE) is presented. The proposed cryptography technique realizes a XOR operation between separated planes of binary gray-scale image and a shuffled multi-scroll DDE chaotic attractor image. The security keys are parameters in DDE, including initial conditions, time constants, and simulation time that sets final states of an attractor. Experimental results are performed in MATLAB. Nonlinear dynamics of DDE are described in terms of equilibrium points and an infinite-dimensional system of Ordinary Differential Equation (ODE) with demonstrations of 3scroll attractors in both time and phase-space domains. Encryption and decryption security performances of a gray-scale Lena image with 512x512 pixels are evaluated through histograms, 2-dimensional power spectrums, image correlation plots and coefficients. Demonstrations of wrong-key decrypted image are also included. The proposed technique offers a potential alternative to simple-but-highly-secured image transmissions in information privacy protection applications.

image encryption due to large computational time and high computing power, especially for the images with large data capacity and high correlation among pixels [2]. Recently, the utilization of chaotic systems has extensively been suggested as one of a potential alternative cryptography in secured image transmissions. As compared with those of conventional encryption algorithms, chaos-based encryptions are sensitive to initial conditions and parameters whilst conventional algorithms are sensitive to designated keys. Furthermore, chaos-based encryptions spread the initial region over the entire phase space, but cryptographic algorithms shuffle and diffuse data by rounds of encryption [3]. Therefore, the security of chaos-based encryptions is defined on real numbers through mathematical models of nonlinear dynamics while conventional encryption operations are defined on finite sets. Such chaos-based encryption aspects consequently offer high flexibility in encryption design processes and acceptable privacy due to vast numbers of chaotic system variants and numerous possible encryption keys.

Keywords-component; Cryptography, Attractor Image, Security Keys, Delay Differential Equation, 2-dimensional power spectrums.


Chaos-based encryption algorithms are performed in two stages, i.e. the confusion stage that permutes the image pixels and the diffusion stage that spreads out pixels over the entire space. Most existing chaos-based encryptions based on such two-stage operations employ both initial conditions and control parameters of 1-D, 2-D, and 3-D chaotic maps such as Baker map [4,5], Arnold cat map [6,7], and Standard map [8, 9] for secret key generations. Furthermore, the combinations of two or three different maps have been suggested [10,11] in order to achieve higher security levels. Despite the fact that such maps offer satisfactory security levels, iterations of maps require specific conditions of chaotic behaviors through a narrow region of parameters and initial conditions. Consequently, the use of iteration maps has become typical for most of proposed ciphers and complicated techniques in pixel confusion and diffusion are ultimately required.


Recent advances in communication technologies have led to great demand for secured image transmissions through wired and wireless networks in a variety of particular applications such as in medical, industrial and military imaging systems. The secured image transmissions greatly require reliable, fast and robust security systems, and can be achieved through cryptography, which is a technique of information privacy protection under hostile conditions [1]. Image cryptography may be classified into two categories, i.e. (1) pixel value substitution which focuses on the change in pixel values so that original pixel information cannot be read, and (2) pixel location scrambling which focuses on the change in pixel position. Conventional encryption algorithms for such cryptography, for example, Data Encryption Standard (DES), International Data Encryption Algorithm (IDEA), Advanced Encryption Standard (AES), and RSA algorithm may not applicable in real-time

The DDE has emerged in mathematical models of natural systems whose time evolution depends explicitly on a past state, and can be described by an infinite-dimensional system that can exhibit complex chaotic behaviors with a relatively

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(a) Image to be encrypted



(b) DDE Attractor

8-Bit Binary Number per Pixel









1-Bit Binary Number per Pixel

XOR Operations

9 10 11 12 13 14 15 16


(c) Encrypted Image






8-Bit Binary Number per Pixel

 1  G ( x  1)  G ( x  1)   

1-Bit Binary Number per Pixel

where δ(·) is a Dirac delta function. The eigenvalues evaluated at each fixed point are all equal at -1, which are negative real values, indicating that the three equilibrium points are all stable nodes when τ=0. In the case where τ>0, the characteristic equation of DDE generally has infinitely many roots while the number of characteristic roots of ODEs coincides with the dimension of the system. Therefore, the DDE in (3) can be approximated by an infinite-dimensional system of ODEs as

Figure 1. Proposed encryption and detection algorithms using XOR operation between separated planes of binary gray-scale image and a shuffled multi-scroll DDE chaotic attractor image.

simple first-order differential equation. Existing DDEs include the prominent Mackey-Glass DDE [12], which models the production of white blood cells, and the Ikeda DDE [13], which models a passive optical resonator system. Recently, In recent years, further chaotic DDEs [14-16] based on the Mackey-Glass DDE have been reported through the use of piecewise-linear nonlinearities corresponding to a complex two-scroll and multi-scroll attractors. In addition, the simplest DDE with a sinusoidal nonlinearity [17] based on the Ikeda DDE has also been presented.



where 1

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