Image Encryption Based on Hyperchaotic Liu system Algorithm

: In this paper a proposed method to encrypt images based Liu Hyper Chaotic System (LHCS) is presented. In the proposed algorithm is used to generate a key for diffusion process, to confuse relationships between the input image and encrypted image. Besides of the four initial state (initial keys) in LHCS, six control parameters were used to increase the key space. Simulation results show that the proposed method has large key space (key space is 10 70 ), and it is responsive to a trivial change of the key. Therefore it with stands different types of attacks such as brute force, differential, and chosen-cipher text attacks.

2 algorithms have been submitted [1]. Recently, there are many encryption schemes for the digital image information. Chaos based image encryption is the one which more interesting to researcher because the cryptographic properties in chaotic system such as control parameters, sensitivity to initial conditions, and unknown behavior [2]. Selective Encryption (SE) is a technique in multimedia encryption used to reduce computation process [3]. In this paper, proposed image encryption algorithm with diffusion operation using key generate based on LHCS. LHCS has a high level of disorder than existing chaotic systems due to the high value of its two positive Lyapunov Exponents [4] .

Liu Hyperchaotic System(LHCS)
The LHCS is one of the important models of four dimensional hyperchaotic systems which described mathematically is by system equation-1 , [4,5]:

Discrete Cosine Transform(DCT)
domain. It uses the existing DCT on the block, because this algorithm reduces the amount of data required for re-digital image. The optical properties of the image at low frequencies, while the details are placed in the higher frequencies. The human visual system (HVS) is less than the most sensitive to higher frequencies. In the upper left corner of the matrix DCT contains a value that is always of a very large size and low frequency coefficient is called direct current (DC). All others represent an (1)

Figure (1):LHCS behavior
3 increase in the vertical and horizontal frequencies higher, called Alternating Current (AC) coefficient and become less size as they move from left to right or from top to bottom. This means that during the performance of DCT on the image in the spatial domain, the representation of focus in the upper left corner of the matrix DCT, with the lower right transactions contain less important information [6].
Assuming an 8x8 (image block) the forward (2-D) discrete cosine transforms equation (2) [3]: where C f is C i , C j and P xy are the values of image component i,j= 0,1,...,7, x,y = 0,1,…,7. the Inverse DCT (IDCT), using equation (4), where Ci, C j as indicated by equation-3 [6] Quantization Quantization is the process of minimize the number of bits required to store an integer value by reducing the precision of an integer. Each position in the matrix element DCT, which is the corresponding value in the quantization matrix gives the quantum value. Actual formula is represented by equation (4 )and (5) represents the inverse quantization.

The Proposed Image Encryption Method
The proposed image encryption method consiste of two main steps, including: Key Generation Based on LHCS, selective image encryption. The details of each step are described in the following subsections.

4
To explain the key generation based on LHCS solving via fourth order Runge-Kutta(Rk-4) in detail, several steps will be illustrated . a.
Numerical Method For LHCS The 4 th order Runge-Kutta (RK-4) is used for resolving the continuous LHCS because it produces a more accurate estimate of the solution.The numerical solution is described by the following steps: Step 1: Initialization This is covered by system equation (1), where n number of iteration time, where n is a constant. h= time step=0.0005, X 0 =X, Y0=Y, Z0=Z, W0=W as initial values, initial state of the Liu chaotic system, a, b, c , d, e as initial control parameters.
Step 2: In this step, Pre-iterate LHCS is for n. In order to compute the solutions of the Liu hyperchaotic system the following sub steps is used: Step 2.1: Compute the initial slopes kj, mj, lj, and pj, where j=1, and the step h is chosen as 0.0005, defined by system equations (7): At the next clock cycle, the system passes to the next step 2.2.

Step2.3:
Compute the initial slopes kj, mj, lj, and pj, where j=4, and the step h is chosen as 0.0005 defined by system equations (9):

5
Step2. 4 In the final sub step, compute the next hyper chaotic solution values, defined by system equations (10): All steps are repeated so that chaotic digital are obtained at the output. b.
Key Generation In this stage, the LHCS is iterated continuously to obtains four keys stream elements from the current state according to the following system equations (11): Where N number of iterations, abs(x n ) returns the absolute value of X. Floor(x) finds the nearest integer less than or equal to X, mod(x, y) get remainder after division [8,9]. All the variables are declared as 64-bit double-precision type, which has a 15-digit precision according to the IEEE floating-point standard, and therefore the decimal fractions of the variable is multiplied by 10 14 (11) , and The function assumed that the 256 gray scale image for this reason mod on 256.The output from this stage four keys for each iteration time to decrypted first (DC, 3AC) for each block. The output from this stage is four keys for each iteration time (n) used in partial encryption as the next sub section in the diffusion process.

Selective Image Encryption
A proposed selective encryption approach(shown in figure2) is applied to first 2×2 sub block (first DC and the first three AC coefficients) for each block using the key generated via LHCS Diffusion process steps are: 1.
Prepare image (Loading and Reading a bitmap image, then Transforming RGB format into YCrCb color space format).

3.
Transforming each block from spatial to frequency domain by adopting DCT .

4.
Quantizing DCT coefficients to the nearest integer value.
Encryption the first (2×2) sub block for each block (DC and the first three AC coefficients (diffuse coefficients using Xor operation ) 7.
Reconstructing encrypted Y' component form 8×8 blocks, Finally, Transforming Y'CrCb color space format into an encrypted RGB image format. The decryption algorithm is similar to the encryption algorithm. Figure 3 show decryption algorithm (diffusion process) based on LHCS.

Experimental Results
The proposed algorithm was implemented and the experimental analysis of the proposed algorithm presented in this paper has been done with several images. Figure  Figure 4(b) is its encrypted image with the encryption initial key (x 0 , y0, z0, w0)= (30,30,30,30),and control parameter(a,b,c,d,e,and f)= (10,40,-2.5,-10,1,4) with as can be see, the encrypted image is rough and unknowable. Figure 4(c) is the decrypted image by use of the decryption algorithm with the same key (correct key). It can be seen that the decrypted image is clear and correct without any distortion. But if we use the wrong key, we will get an unexpected image. For example, Figure4(d) shows the decrypted image using the wrong initial key (x 0 , y0, z0, w0)= (30.00000000001,30,30,30) ,and control parameter(a,b,c,d,e,and f)= (10,40,-2.5,-10,1,4). So it can be concluded that LHCS is responsive to the key, a little change of the key will generate a completely different decryption result and cannot get the correct original image.
The encryption algorithm proposed in the paper has the ability to resist brute-force attack since the key space for the proposed algorithm is (10 14×5 =10 70 ), where 5 for the hyperchaotic initial value (x 0 , y 0 , z 0 , w 0 , N) as the original key, and each digit has 14 digital numbers, therefore, it can be seen that the new chaotic image encryption algorithm is good at resisting brute-force attack. to prove that the proposed algorithm has the ability to hide the pure image information and in the same time when reconstruct the original image from deciphered one without loss any information a set of test measurements are used and the result shown in table (1).
Measurements which used for test are MSE (Mean Square Error), SNR (Signal to Noise Ratio) and PSNR (Peak Signal to Noise Ratio). Large values of MSE stand for the key is able to hide pure image information while the small values of SNR and PSNR stand for the key caused large error result in high image suppression of original image information [12].

Conclusions
To overcome the drawbacks of small key space and weak security in the widely used image encryption, this paper propose an image encryption algorithm based on LHCS with SE technique. Experimental analysis demonstrates that the proposed algorithm shows advantages of large key space and high-level security (MSE, SNR and PSNR). The algorithm presented in this paper can be widely applied in any information security fields.