Image Blocking Encryption Algorithm Based on Laser Chaos Synchronization

In view of the digital image transmission security, based on laser chaos synchronization and Arnold cat map, a novel image encryption scheme is proposed. Based on pixel values of plain image a parameter is generated to influence the secret key. Sequences of the drive system and response system are pretreated by the same method and make image blocking encryption scheme for plain image. Finally, pixels position are scrambled by general Arnold transformation. In decryption process, the chaotic synchronization accuracy is fully considered and the relationship between the effect of synchronization and decryption is analyzed, which has characteristics of high precision, higher efficiency, simplicity, flexibility, and better controllability. The experimental results show that the encryption algorithm image has high security and good antijamming performance.


Introduction
With the rapid development of information transmission technology, electronic data interchange is an important way to communicate and exchange information.Multimedia information with secure communication problems urgently needs to be solved.Characteristics of chaotic sequence have natural advantages in cryptography.During the late 1980s, the British mathematician Matthews put forward the chaos into cryptography firstly [1].As the traditional encryption technique DES cannot be directly used for image encryption, recently many special properties of chaotic maps are suitable to encrypt digital image and provide a new idea to select data security problem of multimedia information [2][3][4][5][6][7].
The laser chaos not only has complex phenomena of all dissipative systems, but also has excellent characteristics such as built-in self-test ability (BIST), pulse generation, closeness to the ideal model, and easiness of design compared with theory.Chaotic laser signal with inherent broad band, noise, and unpredictable characteristics greatly increased the difficulty of optoelectronic reconnaissance and is an important technology of optical information security.The laserbased digital communication was put forward by Colet and Roy in 1994 [8] and became increasingly popular in recent years [9][10][11].
Synchronization is a basic characteristic of complex system and the phenomenon generally exists in natural ecosystem and artificial system such as biology, engineering, and machinery [12,13].Synchronization of chaotic systems and secure communication have become research hotspots of nonlinear system [14,15].With simple structure the linear feedback controller is easily implemented physically and has high practical value, but chaos synchronization is rarely applied in digital image encryption [16,17].
This paper is organized as follows.In Section 2, linear feedback synchronization of Lorenz-Haken is introduced.In Section 3, based on chaos synchronization, a novel image encryption algorithm is proposed.In Section 4, numerical simulation is given to illustrate the effectiveness of the proposed algorithm.Finally, the paper concludes in Section 5.
Then we get the error system: ( The Lyapunov function is designed as  = (1/2)( 2 1 +  2  2 +  2 3 ); then tracking along system (3), we obtain its time derivative: To obtain the asymptotic stability of error system, that is to say, V ≤ 0, the following conditions need to be satisfied: Sufficient conditions of the feedback gains  1 ,  2 , and  3 are  In numerical simulation experiment, the slave system is considered with the same parameters as the master system.Initial state variables of them are ((0), (0), (0)) = (1, 0, 30) and ((0), (0), (0)) = (1, 0, 30), respectively.Figure 2 presents the synchronization error curves between drive system and response system which approaches to zero gradually.Completely synchronization is achieved in five seconds by using macroscopic observation, but in fact there is a synchronization error.The enlargement of partial positions of error curve is shown in Figure 2(b).
(2) The sequence  is processed as follows: where  For the given symmetric algorithm, decryption is the inverse operation of encryption, described as follows: (1) The final encrypted image  is inversely scrambled to get image .
(2) Chaotic sequences of response system are used to perform antisubstitution decryption operation; the calculation formula is as follows: for  =  : In encryption algorithm, the digital image is stored as two-dimensional array, including pixel position scrambling and pixel value substitution.For an image of size  =  × , the total time complexity is () analyzed from the viewpoint of the "big- notation."Therefore, the efficiency of execution of the algorithm is ideal.

Numerical Simulation Results
To verify the effectiveness of the above algorithm, different types of images are carried out in experimentation by using MATLAB platform.Statistical histogram describes the distribution of image pixel.Under the ideal condition, histogram of encrypted image should be approximate to distribute evenly.Histograms of gray image, binary image, and color image and their encrypted images are proposed in Figures 4, 5, and 6, respectively.The eight fine horses gray image is a high resolution image with 370 dpi shown in Figure 4(q).We can see that the gray pixel values of the original image are concentrated in some values, but the histograms of different encrypted images are highly uniform.

Correlation Coefficients of Adjacent Pixels. Encryption
algorithm is designed to reduce the correlation coefficients of adjacent pixels between plain image and encrypted image for resisting statistical attacks.Correlation coefficients of entire randomly selected 3000 pairs of horizontally, vertically, diagonally, and counterdiagonally adjacent pixels are determined.The correlation coefficients between two adjacent pixels of image are calculated by the following formula: where where  and  are pixel values of two adjacent pixels in the image.Figure 7 and Table 2 display distribution of the randomly selected pairs of adjacent pixels in four directions of the original and encrypted image.If the correlation of the encrypted images is close to zero, then it informs good encryption quality.It is clear that the correlation coefficient of the proposed algorithm is smaller than that of other methods proposed in [18,19].

Information Entropy.
The concept information entropy was put forward by Shannon [21], which lays the foundation for information theory and digital communication.Information entropy is commonly used to express image texture features and measure the randomness.The entropy () is measured by the formula () = − ∑  =1 (  )log 2 (  ), where (  ) denotes the probability of symbol   .In theory, the maximum entropy is  max = −(1/256)log 2 (1/256)×256 = 8 for a gray image.
From Table 3, it is clear that our approach can encrypt a low entropy image to get higher entropy image which has random information content.The entropy of the encrypted image is actually closer to the maximum entropy than the plain image.The gray distribution of the encrypted image is more uniform, and its security is improved greatly.Obviously the encryption algorithm has better ability to resist statistical attack.

Decryption Effect.
In numerical simulation, encrypted gray Lena image is decrypted by abandoning 0 points, 1000 points, 3000 points, 5000 points, and 7000 points, and    If the encrypted image is added, salt-and-pepper noise is seen with noise density 0.002, when using the same key to restore the attacked image; the decrypted images are shown in Figure 9(f).It is clear that the encryption algorithm for saltand-pepper noise attack has good ability to resist interference attack.

Conclusion
Linear feedback controller is designed to synchronize the laser chaos and has higher precision.Steady-state error of synchronization reached 10 −14 .Based on synchronization a novel encryption algorithm is designed.Numerical simulation results show that the given algorithm has sufficiently large key space, highly sensitive keys, better pixel distribution characteristics, and good performance against ciphertextonly attack, differential attack, chosen plaintext attack, and statistical attack.The algorithm can be widely used in secure communication of multimedia data.Linear feedback controller has convenient operation and strong applicability and is suitable for industrial application.The synchronization of chaotic laser systems is applied in image encryption transmission and achieved good results.The algorithm can realize complete encryption in sender and lossless decryption in the receiver, so as to achieve digital image encryption transmission function.

Figure 2 :
Figure 2: The synchronization error curves between drive system and response system.

( 3 )
15, and  = 1, 2, . . ., . || is absolute of , and floor() indicates down integral function.Obviously, () ∈ [0, 255] after modular arithmetic.In the paper positive integral  2 = 10.Chaotic sequence  shown in Figure 3(a) is treated as ; self-correlation of  is approximate to zero presented in Figure 3(b).Figure 3(c) displays the local amplification at longitudinal amplification, and most of data are concentrated in interval [−0.002, 0.002].After preprocessing the randomness of chaotic sequence is improved distinctly, and sequence after treatment is more suitable in cryptography.Exclusive OR operation and modular arithmetic are imposed to the plain image by using sequence .The first pixel point is encrypted individually, and its encryption exerts effect on encryption of the second pixel.The pixels are encrypted alternately as follows:  (2) = [ (2) +  (2)] mod 256 ⊕ [ (1) +  ()] mod 256 for  = 3 : 2 :  Local amplification of sequence

Figure 3 :
Figure 3: Self-correlation of chaotic sequence before and after treatment.

Figure 6 :Figure 7 :
Figure 6: The color parrot image: (a) the plain image and histograms of its RGB segments and (b) the encrypted image and histograms of its RGB segments.

4. 4 .
Anti-Interference Attack.During transmission and other treatments, it is subjected to interferences and attacks.The anti-interference attack is an important indicator of testing the quality of the decryption algorithm.Encrypted images with 25% occlusion and 50% occlusion are represented in Figures9(a) and 9(b); their decrypted images are shown in Figures 9(d) and 9(e), respectively.It is clear that the occlusion parts are diffused uniformly to the whole image; therefore algorithm of the paper has strong capability against cropping operation.

Figure 8 :
Figure 8: Decryption images of Lena with different dispose points.

Table 2 :
Correlation coefficient of different plain images and cipher images.

Table 3 :
The entropy of plain image and its corresponding encrypted image.

Table 4 .
Comparing with the encrypted Lena, abandoning 7000 points has minimal distortion.

Table 4 :
MSE and PSNR of different encrypted images under different reject points.