AN EXTENDED IMAGE ENCRYPTION WITH MARKOV PROCESSES IN SOLUTIONS IMAGES DYNAMICAL SYSTEM OF NON-LINEAR DIFFERENTIAL EQUATIONS

1Department of Mathematics, Air University, Islamabad 44000, Pakistan 2Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan 3Department of Mathematics, Preston University, Islamabad 44000, Pakistan 4Division of Computational Science, Prince of Songkla University, Hat Yai, Songkhla 90110, Thailand 5Centre of Excellence in Mathematics, Si Ayuthaya Road, Bangkok 10400, Thailand


INTRODUCTION
The idea of embedding images into images has been a very versatile and hot topic over a decade. The chaotic behavior of many mathematical functions has attracted for such embedding. This phenomena is inherited non-repeating and non-periodicity. An encryption tool is an effective approach to protect such information when sending and receiving data through multiple ways of communications [5,13,16]. Techniques based on chaos theory were developed encryption with confusion and diffusion into multiple rounds ( [6,8,9,11,15,19,20,22,24]).
Non-linear dynamical systems have been based on Chaos [10]. A non-linear chaotic map together with genetic algorithm was employed for text and images encryption [20,21]. The analysis of such embedding was presented in [4]. Various techniques were developed in order to secure these digital images. Researchers are working to develop new and innovative techniques using chaotic behavior of solutions of non-linear differential equations. In the present analysis, an idea is presented to use solutions of dynamical system for image encryption. Image encryption algorithm based on chaotic economic model were considered in [3]. Image encryption based on Lorenz and Rossler was studied in [1] and [2]. Solutions of Lorenz and Rossler are highly non-linear parametric and sensitive to initial conditions. These solutions behaves like series solutions of multiple equations, in which fixed and bifurcation points can be plotted accordingly. Solutions have become chaotic for certain parameter values. When it is chaotic, called Lorenz or Rossler attractors and have certain chaotic properties. The attractors can effectively be used for text and image encryption through mathematical data manipulations. The purpose of this paper is to provide analytical expressions for the variables involved in Lorenz and Rossler's equations. An investigation is carried out for the influence of parameters over the state variables in these equations. Further to our previous attempt, this extended work is embedded pictures into phase solutions of Lorenz and Rossler together with Markov processes on image matrices. Markov processes have applied to image matrices of Lena, Vegetable and Fruits and obtained encrypted images which are absolutely unrecognizable. These encrypted images are then embedded into solution matrices of Lorenz and Rossler. Decryption process is obtained by decrypt the encrypted pictures from solution matrices followed by the inverse Markov processes. Markov processes have confused/diffused target images and made difficult for intruders to attack for retrieval. Statistical analysis has shown reliability of the method with no extra computational time.

A CLASSICAL SOLUTION SET OF NON-LINEAR ORDINARY DIFFERENTIAL EQUA-TIONS
Non-linear ordinary differential equations have great importance while modeling physical problems. The solutions of these equations have in close agreement with experimental but sometimes they behaved strangely. A series of solutions are obtained with different initial conditions. Lorenz [14] has found strange chaotic behavior in solutions of differential equations. He then successfully has implemented in weather forecasting as the weather continuously changes with various initial conditions. Rossler has also obtained the similar behavior in his solutions.
He has implemented these solutions to explain problems in engineering, geography and even in stock market.

Lorenz Equations.
Lorenz has introduced ordinary differential equations whose solutions can be modeled the simplest example of deterministic non periodic flow and finite amplitude convection. He has explained the work of meteorologist and physicists while incorporating several physical phenomena ( [21]). He has found variables to construct a simple model based on the 2-dimensional representation of the earth's atmosphere. Consider the following differential equations, (1) dx dt = −ax + ay, where a = 15, b = 28, c = 8/3 are parameters. Initial conditions for above equations at t = 0, are x(t) = y(t) = z(t) = 1.

Rossler's Equations.
Similar to Lorenz equations, Rossler has developed the nonlinear ordinary differential equations in 3-dimensions with a different combination of variables and initial conditions. He has also found chaotic behavior in solutions which can be used in various chaotic problems. The Rossler's ordinary differential equations are as follows: where d = 0.15, e = 0.2, f = 10 are parameters. Initial conditions for above equations at t = 0,

Numerical Solutions of Lorenz and Rossler's Equations. For numerical solutions, we
reconsider the Lorenz equations after linearization about the origin as which decoupled the z-motion. In a special case with σ = 15, b = 8/3, γ = 28 and the initial conditions as (x 0 , y 0 , z 0 ) = (0, 1, 0), the resulting solutions y(t) is given in Figure 1.  In Figure 2, it is seen that the trajectory to cross itself repeatedly, but that is just an artifact of projecting the three dimensional trajectory onto a two dimensional plane. In 3-D no crossing has occurred. The number of circuits appeared on each side varies unpredictably from one cycle to the next. The sequence of cycles is random and useful from encryption purposes. The 3-D plot of these trajectories are appeared to be a pair of butterfly wings known as Attractor ( Figure 2).
The unique theorem means trajectories can not be crossed or merged, surfaces of the Attractor can only be appeared to merge [14]. The infinite complex surfaces is called Fractal.    • Obtained Lorenz or Rossler solutions, X(t).
• Consider one of the phase vector say, x(t), add unrecognizable image of single array reduced function dd(L, 1) to get a new phase vector x (t). It is mandatory that the size of dd(L, 1) must be less than or equal to the size of original phase vector x(t), x (t) = x(t) + dd(L, 1).
• The new phase surfaces have encapsulated the unrecognizable image p(i, j) under consideration (original image processed with Markov image).

Decryption.
• Read RGB encrypted image which was produced by (x (t), z(t)).
• Extract the phase vector x (t) from the RGB encrypted image and obtained dd(L, 1) by subtracting x(t) from x (t).
• Apply the inverse reduction function onto dd(L, 1) to get the original single array d(L, 1).
• Finally the single array d(L, 1) extended into unrecognizable matrix.   and encrypted all three pictures. we have also calculated PSNR for all sub steps given in Table   2.

PERFORMANCE ANALYSIS OF INDIVIDUAL ALGORITHMIC STEPS
In order to analyze the proposed algorithm, the conventional standards have been analyzed in detail in the following sub section.

Randomness Test for Cipher
System. The essential part of this algorithm is a random test. In every crypto system, the security of the system must have some random tests, for instance, periodicity, uniform distribution, high intricacy and productivity. In order to fulfill the prerequisites, NIST SP 800-22 for testing the haphazardness of digital images is used. A number of tests are included in the testing tool but some of the tests are performed over the chosen digital images (Lena, Vegetable and fruit). The effects of these tests are reported in Table  2. It is shown in the table that our anticipated digital image encryption mechanism effectively pass the NIST tests. In the light of accomplished outcomes, the proposed random cipher in our encryption algorithm can be taken as irregular. Mathematically it is given as

Histograms
where n is total number of observations and k is the total number of bins, m i is the histogram.
The histogram of butterfly, embedded Lena into butterfly is given in Figure 8

PSNR Analysis.
To evaluate the image quality based on pixels, we have to calculate PSNR and MSE value as follows