Noise Assisted Image Encryption and Decryption using 2-D Chaotic System

A BSTRACT : In this paper, a new two-dimensional (2-D) chaos-based colour image encryption and decryption scheme is proposed in which the noise signal is selected randomly to set the initial values for a chaotic system which also enhances the security of the system. The 256-bit hash value of noise is transformed into one-time initial values for the state variables of this proposed chaotic system. XOR operation is further carried out to diffuse the pixels. Finally, statistical and security analyses are performed for understanding the effectiveness of the proposed system. Experimental results confirm that the proposed chaos-based cryptosystem is efficient and suitable for information (image) transmission in a highly secured way.


I. INTRODUCTION
In recent years, many researchers have shown interest in Chaos theory and its applications in wide spread areas, especially in the field of engineering and communication due to the unique features of chaotic sequence such as highly sensitive to the initial conditions, broad-band power spectrum, ergodicity in randomness. Even though there is consistency in the sustained development in the field of communication, increased security aspects have been the main concern for the users. To fulfil this requirement, chaos theory is capable of providing solutions with its inherent cryptographic properties. Cryptography is the technique of converting the plaintext in a non-readable form and vice-versa. Therefore, it is difficult for the attacker to decode the original information. The data can be of any form such as text, audio, video or image. Conventional encryption algorithm such as AES (Advanced Encryption Standard), DES (Data Encryption Standard), RSA (Rivest, Shamir, Adleman) etc. cannot be used for the multimedia data transmission due to bulk data volume and high correlation among the pixels of an original plain image. Also, it is seen that it is more complicated and difficult to implement.
Many algorithms and schemes [1][2][3] have been proposed time to time for image encryption/decryption [4][5]. The discrete multi-dimension Chaotic System has increased application in image encryption [6][7] because of their complex structures and multiple parameters. According to Shannon's theory of secrecy systems [8][9][10], cryptography based on one-time key is theoretically unbreakable, but in practical implementation there exist some problems. For example, one time key requires the key stream generated using true random number generator (TRNG) never be reused, the key stream and the plain image must be of same length. Also, it is impractical to send the whole key stream generated by TRNG to the receiver. To solve this problem in this work, it is preferred to take several true random numbers from environmental noise as the pseudo-random generator (PRNG) to produce key stream dynamically which makes it is extremely difficult for the attacker to produce two identical noise signals in finite time.
Recently Chaos-based image encryption algorithm has been proposed with some chaotic maps, such as logistic map [6], tent map and Chebyshev map [10], Lorenz system [11]. But security aspects of the transmission are still need to be addressed.
In this paper, a new version of chaotic system is proposed and a chaos-based color image encryption and decryption technique is discussed which uses the one-time key from the environmental noise that enhances the security but also reduces the encryption rounds making the known-plaintext attack, ciphertext attack and differential attack ineffective. The hash value of the noise is applied to generate the initial conditions of the chaotic system. This chaotic system will generate sequence with good randomness to encrypt the Red (R), Green (G), Blue (B) components of the color image to fit for XOR operation.
Section II introduces the dynamical and numerical simulation of the new chaotic system. In Section III, the encryption and decryption algorithm are discussed. Section IV gives the MATLAB simulation results of the proposed cryptosystem. In Section V the security analysis of the system is analyzed. Finally, the results are discussed and concluded in the last section.

II. THE NEW CHAOTIC SYSTEM
A new chaotic system with additional parameters helps to set more random chaotic behaviors and larger key space which increases the security in transmission line. The state equations of the new version of (Henon Map) are given below: x n+1 = ax n -x n y n y n+1 = bx n -cy n (1) The parameter values are set as a=2.3, b=0.4 and c=0.8. Then Chaotic system is modelled using MATLAB Simulink in Fig. 1 to understand the dynamics of the system.
Theoretical calculation of the system is derived and results are given below: The system has two equilibrium points E1 and E2: To find the Eigen value of the given system: Where, Try Two roots: Eigen Vector:  (17) The real parts of the eigen value are positive, which states that the presence of chaos and the equilibrium points are unstable. Thus, the system orbits around the two equilibrium points.    From the above graph we can say that at point 2.24 the bifurcation diagram is shifted with the addition of noise which will enhance the security and the maximum noise amplitude in time domain is given in Fig. 4.
Lyapunov exponent is a quantity that characterizes the points of divergence or convergence in phase space. When the control parameter a=2.3 and b=0.4 of the system are set, the largest Lyapunov exponent of c is [-0.5,0.5] is shown in Fig.5. One of the properties of chaos is sensitive to the initial values and control parameter, so in this cryptosystem, the control parameter c is varied dynamically in the interval c [-0.5,0.5] for randomness.   To enhance the randomness in the system three disturbance part is added in eq (1), where h is the step length.

Where
In this cryptosystem, the pseudo random sequence is generated by (18) produced by the hash value of noise will encrypt the colour components, give initial values to the system and control parameter c.

III. IMAGE ENCRYPTION SCHEME a. Generating the initial values and control
The proposed cryptosystem utilizes 256-bit external key H, which is the common hash value calculated by SHA-256 of the noise which is randomly sampled for each encryption. The 256bit hash value can be expressed as a hexadecimal number array H which is expressed as: The initial value for x, y, control parameter c, and pre-iteration 0 can be obtained by: Where N is the noise signal array.

b. Design of encryption algorithm
Input: Colour image P of size with randomly sampled noise signal (N).
Output: Ciphered colour image C of the same size.
Step1: the hash value of N is obtained to calculate the initial value for x, y, control parameter c and 0 by solving equation (21).
Step2: solve eq(19) using Runge-Katta method with a step length of h = 0.008911. Then pre-iterate 0 times for L = to get a sequence of X and Y with the same length of L as shown in eq (22) And then generate the three sequence of , and by eq (23).
Step3: Apply these sequences to encrypt red, green and blue component of P by eq (24) to get Ciphered image C obtained by C R ,C G and C B .
Where  is the XOR operation.
The flowchart for the encryption algorithm is shown in Fig. 8. Output: Deciphered image P.
Step1: Same as Step 2 of the encryption algorithm above, to get three floating-point number sequences of XR, XG and XB.
Step 2: Apply XR, XG and XB to decrypt the red, green and blue components of image C by (25), to get PR, PG and PB, and then combine them into the deciphered image P Where the symbol  denotes bitwise XOR operation.
The flowchart of the decryption algorithm is shown in Fig. 9.

IV. SIMULATION RESULTS
The colour plain image of Lena with size 512 X 512 is used for encryption using the discrete chaotic system. The initial condition and parameters used are x=12.0277, y=120.277, N=0.250577, 0 =1309.02 and c=2.1. The algorithm implemented in MATLAB. The Colour plain image is sown in Fig.10. The R, G, B component of the colour image is given in Fig. 10a, 10b and 10c and the encryption results of the R, G, B component of the plain colour image is shown in Fig. 11a, Fig.  11b and Fig. 11c below:

V. SECURITY ANALYSIS
An efficient encryption process must be sensitive to the secret keys and should resist against all kinds of attacks, such as a plaintext attack, cipher-text attack, differential attack, brute force attack etc. The security analysis of the proposed encryption scheme is discussed based on Key Space analysis, Key Sensitivity analysis, Histogram analysis, Differential attack analysis, Correlation coefficient analysis, Information Entropy Analysis and Speed performance analysis.

a. Key Space Analysis
Key space size is the total number of different keys that can be used for encryption. The key must be large and very sensitive so that it can resist against Brute force attack. In this proposed cryptosystem the initial conditions and parameter for x, y, c and 0 can be used as keys as well as the initial iteration. Generally, the initial condition is set to 10 −14 [11] by which the key space size can reach up to 10 56 . The 0 key space can reach up to 10 3 . Therefore, the total key space is 10 56 x 10 3 which is bigger than 2 128 [12,13] which is large enough to resist against brute force attack.

b. Key Sensitivity Analysis
For an ideal encryption scheme the secret key must be very sensitive so that if there is any change in the bit, the secret key should produce a completely different encrypted image.  Fig.13.b. the differences between the Figs.12 and 13.b. is shown in Fig.13.c. From the figures, we can say that their ciphered image are clearly different which states that small change in the key will generate a completely different decryption result and it will be difficult to obtain the correct plain image.

c. Histogram Analysis
An image histogram is the graphical depiction of the number of pixels distributed at each colour intensity level. It is one of the security analyses that explain about the statistical properties of a ciphered image. The Histogram of the original image and its encrypted form in Fig.14   The results shows that the pixel of the encrypted image is relatively uniform which makes the statistical attack difficult.

d. Differential Attack Analysis
The differential attack in which the attacker attempts to recover some information by comparing the cipher image with the corresponding plain image. To check the effect of change in one bit or one-pixel NPCR (net pixel change rate) and UACI (unified average changing intensity) measurements are used. They are defined as follows [14]: (26) Where, W and H are the width and height of the image.
C i,j and C i,j ' are the cipher image before and after one pixel of the plain image is changed. From the above Table 1. we can see that the results are satisfactory and better than the existing techniques in ref [12,16], so this scheme based on one-time key is secure against differential attack.

e. Correlation Coefficient Analysis
In the image, the pixels are highly correlated among each other. In order to inspect this, we compare the correlation of the adjacent NPCR% UACI% Original encrypted image of Lena (Fig.10 pixels in a horizontal, vertical and diagonal direction for the plain and ciphered image. The correlation between the adjacent pixels can be calculated by [16]: Where N is the total number of pixels, ̅ , ̅ are the mean values of and . To analyse the correlation coefficient, we selected 1000 pairs of adjacent pixels from the horizontal, vertical and diagonal direction of the plain image and cipher image the of Lena (512 X 512) in Fig. 10 and Fig. 12. The correlation graph and the values of the plain and ciphered image are given in Fig. 16 and Fig. 17 below:  From the above figures we can say that the two adjacent pixels are highly correlated for the original image and is concentrated in a certain region which can be seen in Fig.16 whereas there is a very negligible correlation between the two adjacent pixels in the cipher image as the pixels are scattered in Fig.17. Table 2 gives the calculations of the correlation coefficients of two adjacent pixels in Fig. 10 and Fig. 12. The results indicate that the correlation of the adjacent pixels in plain image is very significant, while that of ciphered image is very small, so the encryption is satisfactory.

f. Information Entropy Analysis
Information entropy is the measure of randomness that defines the texture in an image. It is calculated by [17]: Where, p ( ) is the probability of message . The ideal value of the entropy of an encrypted image is 8. The Table 3 below shows the information entropy values of three colour components; the results are all close to the ideal value of 8 and better than in ref [15,16] which makes the algorithm resist against entropy attack. g. Speed Performance Analysis Speed of an algorithm is an important factor for an ideal encryption scheme. In this we have measured encryption speed of colour image of different sizes. The speed analysis is done in MATLAB 2018a on a desktop having specification as Windows 10 AMD A9 7 th generation, 4GB RAM with 64-bit operating system. The encryption/decryption time taken by the proposed cryptosystem for different sized image is shown in Table 4. below:  From the above graph we can say that the relationship between size and time is nearly linear, which means size of the image is proportional to time taken for encryption/decryption.

h. Noise Attack Analysis
A good encryption scheme should be resistant against noise jamming. Here we add salt & Pepper noise and white Gaussian noise, respectively, to the ciphered image Lena of Fig. 12. The deciphered image below in Fig. 19 a,b shows that as we increase salt & Pepper noise density from 0.01 to 0.1, more noise points appear in the deciphered Lena, but the deciphered Lena can still be distinguishable. Similarly, we add the white Gaussian noise, when the mean value is set to zero, more noise points appear in the deciphered Lena with the increase of variance, from 0.001 to 0.01 and the deciphered Lena is still distinguishable. The overall effect of resisting against noise is better than that in Ref. [18].

VI. CONCLUSION
In this paper, a new chaos-based cryptosystem is proposed which generates one-time-key using the hash value of the noise followed by XOR operation. The advantage of using a one-time password is that even if the attacker retrieves the plaintext and ciphertext pairs, it is very difficult to decrypt the next cipher as the same key cannot be re-used for the next encryption process. The cryptosystem is ineffective plaintext, ciphertext, differential and brute force attack. A detailed analysis of the proposed cryptosystem is carried out using MATLAB and the experimental results such as correlation coefficient, information entropy, noise attack analysis suggest that the proposed cryptosystem is efficient and can be used for real-time applications.
Declaration *Funding -No funding was received for conducting this study.

*Conflicts of interest/Competing interests-The authors
declare that there is no conflict of interest.
*Availability of data material -The authors confirm that the data supporting the findings of this study are available within the article and in its supplementary materials. The data that support the findings of this study are available on request from the corresponding author I. Raja Mohamed.
*Code availability-MATLAB software used for the analysis and is available in supplementary file.
*Author's contributions -All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Namrata Biswas under the supervision of I. Raja Mohamed. The first draft of the manuscript was written by Namrata Biswas and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.