Robust Encryption System Based on Novel Chaotic Sequence 1

The aim of this study is proposing a novel chaotic system in which the variables of the system are treated as encryption keys to achieve secure transmission of digital images. The proposed system has high sensitivity to the variation of its initial values and parameters producing unpredictable trajectories. The novel system successfully passes the 0-1 test and FIPS PUB 140-1 statistical tests that check the distribution of the distinguished elements by generated chaotic sequences. The results obtained proved the robustness of the image encryption/decryption system based on sequences generated from the proposed chaotic generator.


INTRODUCTION
Many scientists and scholars for the last decades try to investigate the issue of data encryption and decryption with different methods and systems to promote communication security.Because of the properties of chaotic systems such as highly sensitivity to initial condition and parameters and the unpredictable trajectory of chaos, the application of chaotic systems in encryption and communication security become more and more used (Cuomo et al., 1993).Chaotic systems can be used in two ways in cryptography: -generate pseudo-random sequences, which are then used as key streams to mask the plaintext in manifold ways, the plaintext is used as initial state and the cipher text follows from the orbit being generated (Merah et al., 2013).
Many chaotic systems had been proposed to increase the security of encryption system such as Lorenz (Moghtadaei and Hashemi Golpayegani, 2012), Chua (Aziz and Faraj, 2012), R ssler (Sambas et al., 2012), Nien (Nien et al., 2007), Lu (Vaidyanathan, 2011) and chen (Sprott, 2015).To improve security, the researchers suggested using hybrid chaotic systems.However, the previous works about proposing hybrid chaotic systems for encryption are few.The reason behind that is having hybrid chaotic system is not a straightforward operation by combining two different types of chaotic systems because the chaotic behavior will be lost unless a careful design is considered.Examples of successful hybrid chaotic systems are those presented by combining LIU and Lu systems (Nien et al., 2007).
In this study a novel chaotic system is proposed with its attracter characteristics.The system had been tested by using 0-1 test (Gottwald and Melbourne, 2009) and the FIPS PUB 140-1 (FIPS, 2002) standard test.Then, the novel system has been used later to create encryption keys and applied to encryption and decryption of digital color images.Because of the greater scatter and unpredictability characteristics of encryption keys, images lose original outline, colors and characteristics after encryption.After the encryption process, the encryption image could be sent in unsecure lines.Wider key space and the unpredictability of the system's trajectory make the predicting of the encryption key by an intruder unreachable.

MATERIALS AND METHODS
Before presenting the materials and methods of this study, we would like to point out that this study is a part of preparing for a Ph.D. thesis preparing at electrical and electronic engineering department, university of technology since 2015.
According to the prescription of FIPSPUB 140-1, the data (any random number (key) generator) must pass the following four tests for randomness.The numbers must be converted to a single bit stream of 20,000 bits then pass though the tests.If any of the following test fail, then the module shall enter an error state (in other word, the system is predictable).

The monobit (frequency) test:
This test counts the number of ones in the 20,000 bit stream.The system The runs test: Runs test is the 3rd test which is used to find a maximal sequence of consecutive bits of either all ones or all zeros, which is part of the 20,000 bit sample stream.The incidences of runs of all lengths in the sample stream should be counted and stored.The system pass the test if it is applied according to the values in Table 1 (FIPS, 2002).

The long run test:
A long run is defined to be a run of length 34 or more (of either zeros or ones).On the sample of 20,000 bits, the test is passed if there are NO long runs (FIPS, 2002) The 0-1test: A 0-1 test is used to define the system as chaotic or not, which is proposed by Gottwald and Melbourne (2009).The advantages of this test over Lyapunov exponent (Dawes and Freeland, 2008) is that this test applied directly on the key generated from the system and there is no need to reconstruct the phase space of the system.The test generates a number between zero and one.The system is called a chaotic system if the result very near to one.Let f(n) is a set of data sampled at time n = 1, 2, 3 … N which represents a one dimensional data set: ∑ cos (2) where, a is a real positive number The mean square displacement is calculated from the following equation: The asymptotic growth rate is obtained using: Now if k ≈ 0, the system is predictable (regular) while k ≈ 1 the system is chaotic (Gottwald and Melbourne, 2009) The correlation coefficient: The correlation coefficient is a statistical analysis which is used to obtain the relationship between two random variables or data sets.In image processing, it is used to obtain the similarity between two images.So, for image encryption, it is one of statistical analysis that is used to calculate the encryption algorithm strength.When the result of it equals to zero, that's mean the images are totally different (original and the encrypted images).If it is equal to one, that's mean the encryption process failed to hide the details of the original image.The correlation coefficient could be obtained as following: where, ∑ , x and y are the pixel's value of the original and the encrypted image respectively (Song and Qiao, 2015) Maximum deviation analysis: Maximum Deviation analysis is a statistical analysis which is used to calculate the deviation between the original and the encrypted images.This analysis depends could be calculated according to the following equation: where, H and H' are the histogram distribution of the original and the encrypted image.Higher MD means higher encryption degree and the encrypted image faraway (deviated) from the original one (Ahmad and Ahmed, 2012) Irregular deviation analysis: One of the effective factors to calculate the encryption quality is Irregular Deviation analysis.The irregular deviation used to measure how the encryption process changed the values of the original image irregularly and randomize it in uniform manner to make the statistical distribution of changing the pixels' values uniformly.So, for good encryption process, the Irregular deviation is near to uniform distribution.
The irregular deviation analysis can be obtained from the following steps: • Calculate the absolute difference between the original and the encrypted images:

D=|O-E|
where O and E represent the original and encrypted images respectively.
• Obtain the histogram of the absolute difference: • Calculate the average value of the histogram deviation: • Evaluate the absolute deference between the histogram deviation and the average deviation value: | | • Compute the irregular deviation factor value: ∑ Smaller I D means higher encryption strength, where it is a pointer for the uniform distribution between the original and encrypted pixels' values (Ahmad and Ahmed, 2012) Entropy: One of the well-known analyses for randomness and encryption quality measurements is Information Entropy Analysis.The encryption quality could be measured by calculating the entropy of the plain image and the entropy of the cipher image, then comparing between them.The entropy if the image could be evaluated by the following equation:

∑ log
where, P(i) means the symbol i probability which Expressed in bits.So, for images with gay level of 256 (0 to 255), the maximum entropy equal 8 and it is referred to an ideal case of randomness.In general, the entropy of practical image is less than the maximum entropy.
In encryption process, the entropy of the encrypted image should be ideally equal to 8. When the entropy of it is less than 8, it confirmed degree predictability.
To resist the entropy attacking, the entropy of the encrypted image should be close to the maximum entropy value (Song and Qiao, 2015) Peak Signal-to-Noise Ratio (PSNR): Peak Signal-to-Noise Ratio (PSNR) can be used to evaluate the encryption scheme strength by indicating the of pixel's value between the original image and the encrypted image.It can calculated by using the following equation: where, O and E are the original and the encrypted image respectively, (i, j) is the coordinate of the pixel and M, N are the image size.
Lower PSNR means higher encryption effectively (Gorji et al., 2015) Measurement based on the value changing: Plainimage pixels values change after image encryption as compared to their original values before encryption.Such change may be irregular.This means that the higher the change in pixels values, the more effective will be the image encryption and hence the encryption quality.So the encryption quality may be expressed in terms of the total changes in pixels values between the plain-image and the cipher-image.Ahmed and Ahmad (2012) proposed a measure for encryption quality that is expressed as the deviation between the original and encrypted image.This method is determined as follows: Let P, C denote the original image (plain-image) and the encrypted image (cipher-image) respectively, each of size W×H pixels with L grey levels.P(x, y), C (x, y) {0,..., L −1} are the grey levels of the images P, C at position (x, y), 0 <x< W-1, 0 <y < H-1.We will define HL(P) as the number of occurrence for each grey level L in the original image (plain-image) and HL(C) as the number of occurrence for each grey level L in the encrypted image (cipher-image).The encryption quality represents the average number of changes to each grey level L and it can be expressed mathematically as:

∑ | 256
Another measurement is proposed by Luo (Jolfaei and Mirghadri, 2010) by computing the relative error, which for an image of H×W is defined as: Which gives the average relative error of a pixel (Jolfaei and Mirghadri, 2010) The NovelChaotic system and encryption algorithm: Many chaotic systems are proposed to be used in encryption algorithms like Rӧssler, Chua, Lorenz and Nien.Among these systems we will focus on the systems that are used in the proposed system which are Chua and Lorenz systems.The differential equations of these systems are: Chua system: x a y x h x y x y z Lorenz system: x a y x y x c z y (7) z xy bz where, a = 10, c = 24, b = 8/3, The proposed chaotic system is hybrid system generated by mixing bands of Chua and Lorenz chaotic systems, so we will call it CL system.The new CL chaotic system has a new chaotic trajectory and improved randomness and it will be proved by 0-1 test and the FIPS PUB 140-1 statistical tests.The differential equations of the CL system are defined as: In Eq. ( 8) above, x is taken from Chau system while y and z are taken from Lorenz system.
Figure 1 shows the trajectory of the proposed system.It can be seen that this trajectory has a strange shape which refer to a chaotic behavior.
The encryption algorithm based on novel CL chaotic system is as follows: • Set the initial conditions X 0 , Y 0 , Z 0 and the parameters for CL chaotic system.• Decompose the RGB image A into three level matrices which are A1, A2 and A3 with the same dimension.
• Determine the dimension of matrix A1 and retrieve x, y, z of the same dimension and create chaotic encryption keys using: 10 256, 10 256, 10 256 • Implement the encryption key on the digital image by: The decryption process has the same steps as in the encryption process.

RESULTS AND DISCUSSION
The results are divided into two parts, first testing the novel CL system presented in section III to verify its chaotic and randomness behavior.Then testing the encryption system based on CL system.After implementing the 0-1 test to the CL system, we have obtained the following results x = 0.9987, y = 0.9983; z = 0.9983.According to the test, all system variables produce numbers very closed to one, then it is a chaotic system and the chaotic behavior can be obtained from anyone of its outputs.Table 2 shows the result of implementing of the FIPS PUB 140-1 statistical tests to check the system randomness.The results obtained in the table strongly prove that CL chaotic system fulfills the randomness requirements needed for reliable encryption algorithm.To evaluate the performance of the encryption algorithm it is tested with Lina image.Figure 2 shows the results of implementing the algorithm using CL system.The figure shows the encrypted and decrypted image when the same initial conditions are used (in both Tx and Rx sides) and the decrypted image when atiny difference in any of initial conditions or parameters (when intruder tries to attack the transmission) with the corresponding histograms.The parameters of CL system are a = 10, b = 8/3, c = 24, m0 = -1.27;m1 = -0.68.In Fig. 2, it is very clear that the novel chaotic system has high sensitivity to the initial conditions and to the system parameters as well to any chaotic system.So any attacker use different systems, initial conditions or parameters even with a very little difference, it is impossible to achieve correct decryption process and obtain the original data before encryption.
The tests images that are used in the system performance evaluation are in size of 256*256 and shown in Fig. 3.
As explained before, the CL system is used for encryption process by changing the values of the image's pixels.Then, the tests' results are evaluated after the CL implementation on these images and   compared with the traditional systems implementation's results as shown in Table 3.
According to the tests results in Table 3, it could be notice that: • In all cases, The CL performance is either very near from the best test result of the traditional system or it has the better result.• The CL performance is the best system for image pixel's value changing when the image is more regular, which is mean that the system is almost has the highest randomness behavior between all chaotic systems.

CONCLUSION
A novel chaotic system designed by mixing two chaotic systems is proposed in this study.The system has been tested using 0-1 test and FIPS PUB 140-1 statistical tests to test its chaotic characteristics and to check its randomness characteristics.The encryption key generated from the proposed system is used for digital RGB image encryption by using the XOR logic.If an intruder stole the encrypted image, it is impossible to reconstruct the original image because the CL system is unpredictable and has higher sensitivity to the initial conditions and the variable (parameter) values.

Table 1 :
Passing values for runs test

Table 2 :
The

Table 3 :
CL a