Efficient symmetric image encryption by using a novel 2D chaotic system

IET Image ProcessingVolume 14, Issue 6 p. 1157-1163 Research ArticleFree Access Efficient symmetric image encryption by using a novel 2D chaotic system Huiqing Huang, Corresponding Author Huiqing Huang hq-huang2@126.com School of Mathematics, Jiaying University, Meizhou, 514015 Guangdong, People's Republic of ChinaSearch for more papers by this authorShouzhi Yang, Shouzhi Yang Department of Mathematics, Shantou University, Shantou, 515063 Guangdong, People's Republic of ChinaSearch for more papers by this authorRuisong Ye, Ruisong Ye Department of Mathematics, Shantou University, Shantou, 515063 Guangdong, People's Republic of ChinaSearch for more papers by this author Huiqing Huang, Corresponding Author Huiqing Huang hq-huang2@126.com School of Mathematics, Jiaying University, Meizhou, 514015 Guangdong, People's Republic of ChinaSearch for more papers by this authorShouzhi Yang, Shouzhi Yang Department of Mathematics, Shantou University, Shantou, 515063 Guangdong, People's Republic of ChinaSearch for more papers by this authorRuisong Ye, Ruisong Ye Department of Mathematics, Shantou University, Shantou, 515063 Guangdong, People's Republic of ChinaSearch for more papers by this author First published: 08 April 2020 https://doi.org/10.1049/iet-ipr.2019.0551Citations: 27AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract The authors know that a common and effective way to protect digital images security are to encrypt these images into white noise image. In this study, the authors have designed a new two-dimensional (2D) chaotic system which is derived from two existing one-dimensional (1D) chaotic maps. The simulation results show that the new 2D chaotic system is able to produce many 2D chaotic maps by selecting different 1D chaotic maps, and which have the wider chaotic ranges and more complex chaotic behaviours compared with the existing 1D chaotic maps. In order to investigate its applications, using the proposed 2D chaotic maps, the authors design a novel image encryption algorithm. First of all, the original image is scrambled by using the chaotic sequences which are generated by new 2D chaotic maps, Arnold transform and Hilbert curve. Then the scrambled image is confused and diffused by chaotic sequences. Finally, the performance of the proposed encryption algorithm is simulated and the experimental results show that the validity and reliability of the proposed algorithm is validated. 1 Introduction With the rapid development of network and communication technology, hundreds of millions of multimedia information are generated and spread on the Internet every day. As digital image include such traits as intuitive, visualise, understandability and vivid, therefore, has become one of the major carriers that people capture information. And due to Internet's opening to the outside world, the security of digital images is becoming increasingly important. To protect private images information from the illegal acquisition, use and dissemination of others, researchers have proposed many image encryption algorithms [1-8]. The chaotic system has a series of characteristics such as sensitivity to initial conditions and control parameters, pseudo-randomness and unpredictability, and these characteristics are exactly consistent with the requirements of cryptography. So, the image encryption algorithm based on chaotic theory has become a hot researching field recently [9-12]. As the 1D chaotic systems have the advantages of simple structure and easy implement, 1D chaotic maps are usually employed in chaos-based image encryption algorithms, firstly. Such as, in [13] Behnia et al. proposed an implementation of digital image encryption scheme based on the mixture of chaotic systems. In 2010, Liu and Wang proposed a novel colour image encryption scheme based on one-time keys and robust chaotic maps [14]. Elhoseny et al. proposed an efficient approach for image encryption which is based on feedback stream cipher using a developed chaotic maps [15]. In 2014, Zhou et al. design a simple and effective chaotic system which is generated by using a combination of two existing 1D chaotic maps (seed maps), and then proposed a novel image encryption algorithm based on the new 1D chaotic maps [16]. In [17], Dhall et al. demonstrate cryptanalysis of an image encryption scheme based on a new 1D chaotic system proposed by Zhou et al. [16]. However, because the key space of one-dimensional chaotic system is small and its sensitivity is weak, the image encryption algorithm based on 1D chaotic system cannot effectively protect the security of information. To improve the security of algorithm, lots of researchers naturally consider increasing the dimensionality of chaotic systems [18-23]. In [18], Liu and Wang designed an effective colour image encryption based on spatial bit-level permutation and high-dimension chaotic system. In 2014, Hua et al. inspired by Zhou et al. [16] introduce a new 2D Logistic-Sine map (2D-LSM), which has excellent chaotic performance and its outputs are difficult to predict, and using 2D-LSM they propose a new image encryption algorithm [19]. In [21], Akgu et al. proposed a 3D chaotic system without equilibrium points and thus realised different engineering applications especially for encryption studies. Subsequently, Sahari et al. proposed a novel 3D chaotic map obtained by coupling the piecewise and logistic maps. Considering the low complexity and security of one-dimensional chaotic system, and inspired by the authors of [16, 19, 20, 22], we present a new 2D chaotic system which is generated by two existing 1D chaotic maps. The performance estimations demonstrate that they have excellent chaos characteristics, such as complex chaotic behaviour, sensitivity of initial value and a wider chaotic interval. Further, using the proposed 2D chaotic maps and Arnold transform, an efficient symmetric image encryption algorithm is proposed. Input the initial condition of the proposed 2D chaotic map, and then the chaotic sequence is generated by iterating the chaotic map. Utilising the chaotic sequences, Arnold transform [24] and Hilbert curve [25, 26], we scrambled the original image. Next, the chaotic sequences are used to do confusion and diffuse operations to the scrambled image. Finally, we simulated and analysed the various performance of the encryption algorithm, the results indicate that the proposed algorithm can effectively reduce the correlation of adjacent pixels in the original image and resist various typical attacks. 2 Preliminary In this section, we review three 1D classic chaotic systems: the Logistic, Sine and Tent maps, which are the basic components of generating new 2D chaotic system. 2.1 Logistic map The Logistic map is a classical model to study the behaviour of complex systems such as chaotic fractal of dynamical systems. The mathematical definition can be expressed as follows [16]: (1)where r is control parameter and Logistic map is in chaotic state when [27], is the output chaotic sequence. The bifurcation and Lyapunov exponent diagrams of the Logistic map are shown in Figs. 1a and d, respectively. Fig. 1Open in figure viewerPowerPoint Bifurcation and Lyapunov exponent of the chaotic systems (a) Rhe bifurcation of the Logistic map, (b) The bifurcation of the Sine map, (c) The bifurcation of the Tent map, (d) The Lyapunov exponent of the Logistic map, (e) The Lyapunov exponent of the Sine map, (f) The Lyapunov exponent of the Tent map 2.2 Sine map A class of 1D Sine discrete systems are a commonly used 1D chaotic maps. The mathematical definition is described by the following equation [16]: (2)where parameter and Sine map is chaotic when . Figs. 1b and e show the bifurcation and Lyapunov exponent diagrams of Sine map, respectively. 2.3 Tent map In mathematics, the Tent map is a piecewise linear mapping, so named because its function image resembles a tent. Its mathematical definition is shown as follows [16]: (3)where parameter . In order to observe its chaotic behaviours, the bifurcation and Lyapunov exponent diagrams of Tent map are shown in Figs. 1c and f, respectively. The Logistic, Sine and Tent maps have a simple iterative structure, which is conducive to the rapid generation of chaotic sequences. In addition, the iteration structure is simple, and there are many disadvantages such as simple behaviour and frail chaotic intervals, and these disadvantages narrow down the applications of these chaotic maps. 3 New 2D chaotic system To solve these disadvantages of Logistic, Sine and Tent maps, we propose a new 2D chaotic system in this section. It couples two different 1D chaotic maps which are considered as seed maps, and the new 2D chaotic system can be defined by the following equation: (4)where and are two 1D chaotic maps (seed maps) whose parameters are a and b; and the integer n is the iteration number. As can be observed from its definition, the new chaotic system is obtained by two seed maps, firstly adding the two seed maps, and then performing a sine or cosine transform to the result, lastly, extending the dimension from 1D to 2D. In this way, the new 2D chaotic system shows a mixed chaotic property, which can obtain complex chaotic behaviour. Here, the sine and cosine operation is to ensure its output data within . It can be seen from the above 2D chaotic system definition that by selecting different seed maps, we can get different 2D chaotic systems. Compared to its seed maps, the new 2D chaotic system has more complex chaotic properties and wider chaotic region. 3.1 Examples of the proposed 2D chaotic system In order to illustrate the perfect performance of the proposed new 2D chaotic system, we introduce and analyse three examples in this section. 3.1.1 2D logistic-sine map The first example is derived from Logistic and Sine maps, the new chaotic system is called the 2D-LSM, which can be defined as (5)where a and b are the control parameters. To show its good chaotic behaviour, we analyse its bifurcation, Lyapunov exponent and attractor. The bifurcation and Lyapunov exponent diagrams of the 2D-LSM are shown in Figs. 2a and d, respectively. From its bifurcation and Lyapunov exponent we can see that 2D-LSM is chaotic when , which is much larger than the control parameter interval of the Logistic or Sine maps. Comparing Fig. 1 with Fig. 2, we can find that the Lyapunov exponent of 2D-LSM is larger than the Lyapunov exponent of Logistic or Sine maps, thus, the proposed 2D-LSM can generate more random output sequences than the Logistic and Sine maps. The chaos attractor of 2D-LSM is shown in Fig. 3a with parameters and . Fig. 2Open in figure viewerPowerPoint Bifurcation and Lyapunov exponent of the 2D chaotic systems (a) The bifurcation of the 2D-LSM, (b) The bifurcation of the 2D-LTM, (c) The bifurcation of the 2D-STM, (d) The Lyapunov exponent of the 2D-LSM, (e) The Lyapunov exponent of the 2D-LTM, (f) The Lyapunov exponent of the 2D-STM Fig. 3Open in figure viewerPowerPoint Attractor of six 2D chaotic maps (a) 2D-LSM with parameters and , (b) 2D-LTM with parameters and , (c) 2D-STM with parameters and , (d) 2D Logistic map with parameter , (e) 2D-SLMM with parameters and , (f) 2D-LASM with parameter 3.1.2 2D logistic-tent map When and , the proposed 2D chaotic system in (4) becomes a new 2D chaotic system, which is called the 2D Logistic-Tent map (2D-LTM). The 2D-LTM can be defined as the following equation: (6)where a and b are the control parameters. Figs. 2b and e show the bifurcation and Lyapunov exponent diagram of the 2D-LTM, respectively. And when the control parameters and , the attractor of the 2D-LTM is shown in Fig. 3b. 3.1.3 2D sine-tent map The 2D Sine-Tent map (2D-STM) is the third example, which uses the Sine and Tent maps as seed maps, and its mathematical definition can be defined as the following equation: (7)where a and b are the control parameters. As shown in Figs. 2c, f and 3c, the 2D-STM has complex chaotic behaviour similar with the 2D-LSM and 2D-LTM. Figs. 3d–f show the attractors of 2D Logistic map [28], 2D Sine Logistic modulation map (2D-SLMM) [20] and 2D Logistic-adjusted-Sine map (2D-LASM) [29]. We can see from these attractors that the output pairs of the proposed 2D chaotic maps distribute in the whole region of size , and that the output pairs of the 2D Logistic map, 2D-SLMM and 2D-LASM only distribute in the region of size . From Figs. 3d and e we can find that the attractors of the 2D Logistic map and 2D-SLMM only occupy a small space in the region of size . So, the chaotic regions of the proposed 2D chaotic maps are larger than other 2D chaotic maps. That is to say, the proposed 2D chaotic maps can generate chaotic sequences with more complex chaotic behaviour. 4 Novel image encryption scheme In order to study the application of the proposed 2D chaotic systems mentioned above in information security, in this section, the 2D-LSM is taken as an example; we propose a new image encryption algorithm with 2D-LSM. For convenience, assume that the size of original image I is , and the detailed steps of the proposed image encryption algorithm can be described as follows: Step 1. Input the initial values and parameters of 2D-LSM (, , a and b). Step 2. Two chaotic sequences and are generated by the 2D-LSM with the initial values and parameters. Step 3. From sequence X or Y, we can intercept a sequence of length L, a sequence (here ) is obtained, where . Step 4. The three sequences X, Y and H are preprocessed, and can get three integer sequences U, V and W, as follows: (8)where (9)where returns the nearest integer less than or equal to x, and mod returns the remainder after division. Step 5. Scrambled original image I by using sequence W, Hilbert curves and two-dimensional Arnold transform. In the traversal sequence W, if , two-dimensional Arnold transform is performed on the digital image; otherwise, the Hilbert curve is used to scramble the image, therefore, after the two-dimensional Arnold transformation and Hilbert scrambling, the scrambled image T was obtained. Step 6. Arrange the pixels of the scrambled image T from row to column, we can get a sequence of A, as follows: (10) Step 7. According to (12), the pixel values of the scrambled image are diffused and confused, then we can get B(11)where (12)Here, and initial values and are keys, the symbol represents the bit-level XOR operation. Step 8. Reshape the sequence B, we can obtain the encrypted image E with the same size as the original image. The chaotic image encryption algorithm proposed in this paper is symmetric, that is to say, both encryption and decryption use the same keys, so the decryption is the reverse process of the encryption. 5 Numerical simulation and analysis In order to verify the good encryption effect of the proposed algorithm, in this section we have carried out a series of simulation experiments on the proposed image encryption algorithm. The grey images ‘Lena’, ‘Cameraman’ and colour image ‘Lena’ will be encrypted by the proposed algorithm. In the experiment, the encryption keys is fixed at . The encryption and decryption results are shown in Figs. 4 and 5. Comparing the encrypted images in Figs. 4 and 5 with the original images, we can see that the encrypted images are completely different from the original images, no valid information about the original images can be obtained from their corresponding encrypted images. Fig. 4Open in figure viewerPowerPoint Encryption and decryption results of the grey images ‘Lena’ and ‘Cameraman’ (a) Original image ‘Lena’, (b) Encrypted image of ‘Lena’, (c) Decryption of ‘Lena’, (d) Original image ‘Cameraman’, (e) Encrypted image of ‘Cameraman’, (f) Decryption of ‘Cameraman’ Fig. 5Open in figure viewerPowerPoint Encryption and decryption results of the colour image ‘Lena’ (a) Original image ‘Lena’, (b) Encrypted image of ‘Lena’, (c) Decryption of ‘Lena’ 5.1 Statistical analysis It is well known that statistical analysis attack is a method by which the attacker deciphers the cipher according to the statistical rules of plaintext and ciphertext. Therefore, in order to demonstrate that the proposed algorithm can effectively resist statistical attacks, we test the histograms and the correlations of adjacent pixels in the encrypted image. 5.1.1 Histogram analysis The histogram is used to graphically summarise and display the pixel distribution of the image. To make statistical attack more difficult, the histogram of ciphertext should be uniformly distributed than that of plaintext. Fig. 6 shows the histograms of grey images ‘Lena’, ‘Cameraman’ and their encrypted images obtained by the proposed algorithm, and the histograms of R, G, B components of colour image ‘Lena’ and its encrypted image are shown in Fig. 7. It is observed that although the histograms of the original images are completely different, but the histograms of the encrypted images are similar uniform-distributed histograms, so it becomes impossible to obtain the original image information through the histogram. Fig. 6Open in figure viewerPowerPoint Histograms (a) Original image ‘Lena’, (b) Encrypted image of ‘Lena’, (c) Original image ‘Cameraman’, (d) Encrypted image of ‘Cameraman’ Fig. 7Open in figure viewerPowerPoint Histogram of R, G, B components of original and encrypted image ‘Lena’ (a) Histogram of R component of original image ‘Lena’, (b) Histogram of G component of original image ‘Lena’, (c) Histogram of B component of original image ‘Lena’, (d) Histogram of R component of encrypted image ‘Lena’, (e) Histogram of G component of encrypted image ‘Lena’, (f) Histogram of B component of encrypted image ‘Lena’ However, we can only estimate the uniformity of the image roughly by observing the histogram of an image. So, in order to measure how nearly uniform the image is, we introduced the Chi-square test. The mathematical expression of the Chi-square test formula is as follows: (13)where represents the frequency of pixel value i (0 to 255) occur in the image, is the ideal occurrence frequency of the uniform distribution and is calculated for (here M and N represent the height and width of the image, respectively). So, the smaller the is, the closer the pixel value occurrence frequency of the image is to the uniform distribution. The of the plain images and encrypted images are shown in Table 1. It is seen from Table 1, compared with the values of the original images, the values of the encrypted images is much smaller, and the values of the proposed algorithm are around 250, which indicate that the encrypted images are close to the uniform distribution. Table 1. values for plain and encrypted images Lena Cameraman Colour Lena R G B plain image 47,518 110,970 65,012 30,363 92,313 encrypted image 232.7266 221.6016 249.8672 268.875 249.1641 5.1.2 Correlation analysis We know that visually meaningful images have high correlations between pixels and their adjacent pixels. This feature undoubtedly provides a good breakthrough for attackers, so one aim of image encryption is to break these pixel correlations in the original image. The correlation coefficient of any two adjacent pixels can be calculated by the following equation [30]: (14)where and . To test the performance of the proposed image encryption algorithm, the correlation between adjacent pixels (in horizontal, vertical and diagonal direction) of plain-images and cipher-images were tested. First, we randomly select 8000 pairs of two adjacent pixels from grey images ‘Lena’, ‘Cameraman’ and their decrypted images obtained by the proposed algorithm, then by calculating, we can obtain the correlation coefficients of plain-images and cipher-images. The correlation coefficients of the proposed algorithm and the algorithm in [7] are compiled in Table 2. From Table 2 we can know that the correlation of two adjacent pixels of the original images are close to 1 in all directions, while after encryption, the correlation between adjacent pixel pairs in different directions was seriously damaged, and the correlation of the encrypted images are close to 0 in each direction. This shows the excellent encryption effect of the proposed image encryption algorithm. The correlation distribution of two vertical adjacent pixels in the original image and the encrypted image are shownn in Fig. 8. As can be seen from Fig. 8, the correlation distributions between adjacent pixels in the original image are similar to a linear-like area in the vertical direction, and the correlation distribution between adjacent pixels in ciphertext is uniformly distributed throughout the region. This further proves that attackers cannot obtain useful information through statistical analysis. Table 2. Correlation coefficients of two adjacent pixels in original image andencrypted image Image Horizontal Vertical Diagonal Lena 0.9692 0.9462 0.9243 Cameraman 0.9567 0.9396 0.9025 proposed algorithm encrypted Lena 0.0004 0.0008 encrypted Cameraman 0.0015 Ref. [7] encrypted Lena 0.0005 encrypted Cameraman 0.0017 0.0021 Fig. 8Open in figure viewerPowerPoint Correlation distribution of two vertical adjacent pixels in (a) Original ‘Lena’, (b) Encrypted ‘Lena’ 5.1.3 Information entropy analysis Information entropy is used to measure the expected value of a random variable, and a message source m shooting L symbols , the mathematical representation of the entropy can be expressed as follows [31]: (15)here, stands for the probability of and represents the base 2 logarithm so that the entropy is expressed in bits. For a greyscale image with a data range of , its maximum information entropy is 8 bits. After applying the proposed algorithm, the information entropy of the encrypted images are listed in Table 3, which are compared with the results in [5, 6]. It indicates that the encrypted images are close to 8, this means that the encrypted images of this proposed algorithm have a good random distributions. Table 3. Comparison of information entropy Image Information entropy proposed algorithm encrypted image ‘Lena’ 7.9971 encrypted image ‘Cameraman’ 7.9976 Ref. [5] encrypted image ‘Lena’ 7.9874 encrypted image ‘Cameraman’ 7.9780 Ref. [6] encrypted image ‘Lena’ 7.9970 encrypted image ‘Cameraman’ 7.9972 5.2 Key space and sensitivity analysis To against brute force attacks of password crackers, large enough key space and high sensitivity to keys are necessary for an effective encryption algorithm. The aim of this subsection is to check the key space and sensitiveness of the proposed encryption. 5.2.1 Key space analysis For an image encryption algorithm to resist brute force attack effectively, the key space should be at least as large as [32, 33]. As mentioned earlier, the keys of the proposed image encryption algorithm in the encryption and decryption processes are composed of four parameters . So, the total key space of the proposed image encryption algorithm S can be expressed as (16)where is the key subspace of the ith key, . By calculating the mean absolute error (MAE) of the two sequences, we can obtain the key space of , the two sequences are generated with initial values and ( is the deviation of the ), respectively. Mathematically, the MAE between the two sequences can be defined as follows [34]: (17)When is the right value subject to , then the key space of is equal to . The numerical simulation results show that the space of is , and with the same way we can get the space of , a and b is about . So the total key space of the proposed algorithm is , which is sufficiently large to withstand the brute-force attack. 5.2.2 Sensitivity analysis As we know, a highly key sensitive encryption algorithm can effectively protect the encrypted data. To measure the sensitivity of the proposed encryption algorithm to the keys, the mean square error (MSE) is introduced in this section. The mathematical representation of the MSE between the original image and its decrypted image is defined as follows [35]: (18)here I and H denote the original and decrypted image, respectively. Fig. 9 shows the MSE curves for the parameters a and b. As can be seen from the figure, the MSE is very large with a little deviation to a or b and the MSE is close to 0 only when the main keys are correct. This means that the encrypted image can be properly decrypted only if the keys are correct, i.e. the designed algorithm is very sensitive to the keys. Fig. 9Open in figure viewerPowerPoint MSE curves (a) Parameter a, (b) Parameter b In the decryption process, it is amazing how such a tiny change in the one parameter of , the decrypted images are completely unrecognisable, and the simulation results are shown in Fig. 10. It is illustrated that even if a tiny change of , the decrypted images are absolutely different from the plain image. Fig. 10Open in figure viewerPowerPoint Decrypted ‘Lena’ with (a) Incorrect independent parameter , (b) Incorrect decryption key , (c) Incorrect decryption key a, (d) Incorrect decryption key b 5.3 Differential analysis The ciphertext only, known plaintext, chosen plaintext and chosen ciphertext are four efficient and widely used security attack models in cryptanalysis. In these attacks, the attack power of the chosen plaintext attack is the most powerful. So, if the algorithm can resist the chosen plaintext attack, it can resist the other three types of attack [36]. For an image encryption algorithm, the more sensitive the algorithm is to plaintext, the more resistant it is to chosen plaintext attack. The sensitivity of encryption algorithm to plaintext can be measured by the number of pixel changing rate (NPCR) or the unified averaged changed intensity (UACI). For two encrypted images, and , they are obtained by encrypting two original images with one-pixel difference with the same key, respectively. Mathematically, the NPCR and UACI are defined as the following equation [37]: (19)and (20)respectively, where N and M are the width and height of or , H represents the largest allowed pixel value in the image, and (21) In the test, even with a one-bit difference in the plain image ‘Lena’ (here, we randomly choose a pixel at position (225,80)), the NPCR and UACI are equal to 99.63 and 33.4%, respectively. That is, for the proposed scheme, a slight change in the plaintext image can have a huge impact on the cipher image. Fig. 11a shows the cipher image corresponding to only one-pixel difference in the original ‘Lena’, and the difference image between Figs. 4b and 11a is shown in Fig. 11b. Experimental results show that the proposed algorithm can effectively withstand chosen plaintext attack and differential attack. Fig. 11Open in figure viewerPowerPoint Cipher image corresponding only one-pixel difference in the original ‘Lena’ and difference image (a) Cipher image, (b) The difference image between Fig. 4b and Fig. 11a 5.4 Robustness to data loss It is well known that digital images are easy to lose data during transmission and storage. Therefore, a good image encryption algorithm should have good robustness to resist data loss. So in this section, we test the tolerance against the loss of encrypted data. Fig. 12 shows the simulation results of data loss. Fig. 4b is applied with a data cut with size of (Fig. 12a), (Fig. 10b) and (Fig. 12c). And the corresponding decryption images are shown in Figs. 10d–f, respectively. We note that the decrypted image of the encrypt image can still be recognised under the condition of loss, although the noise of the decrypted image increases with the proportion of loss, but the contour of the image can still be recognised. The experiment simulation demonstrates the excellent performance of the proposed algorithm against the data loss attack. Fig. 12Open in figure viewerPowerPoint Occluded encrypted images of Fig. 4b and the corresponding recovered images with correct keys (a) Encrypted image with a data loss, (b) Encrypted image with a data loss, (c) Encrypted image with a data loss, (d) Decrypted output for Fig. 12a, (e) Decrypted output for Fig. 12b, (f) Decrypted output for Fig. 12c 5.5 Speed analysis In the real-time transmission process, we know that the speed requirement of the algorithm is very high. So, in this section, the speed performance of the proposed algorithm is tested and compared with other algorithms. In the test, we measured the running speed of the proposed encryption algorithm and compared with other encryption algorithms. In [38], the algorithm is deployed on the platform of Visual C++6.0 on Windows XP Professional operating system with Pentium 4 CPU of 3.0 GHz, 512 MB RAM. In [39], the algorithm is deployed on the platform of MATLAB on 64-bit Ultimate Windows 7 system with 8.00 GB RAM, Intel(R) Core(TM) and 2 Duo CPU T9400 @ 2.53 GHz. But in my experiment, we run our proposed algorithm deployed on the platform of MATLAB on 64-bit Windows 10 system with 8.00GB RAM, Intel(R) Core(TM) and 4 Duo CPU i7-8550U @ 1.8 GHz. The time cost to encrypt an grey image ‘Lena’ of size is shown in Table 4. It shows that although the test image is larger but our running time is less than 0.66 s, therefore, the running speed of our algorithm can meet the requirements of practical application. Table 4. Time complexity comparison Algorithms Proposed algorithm Ref. [38] Ref. [39] times, s 0.6576 0.1750 4.307 6 Conclusion In this paper, we presented a new 2D chaotic system, which is derived from the existing 1D chaotic maps. Many new 2D chaotic maps can be obtained by the new 2D chaotic system. These new 2D chaotic maps have better chaos performance than 1D chaotic maps, therefore they are more suitable for the design of image encryption algorithm. To show the proposed 2D chaotic system in image application on being safe, we further proposed a novel image encryption algorithm with the new 2D chaotic system. In this new algorithm, the new 2D chaotic system, Arnold transform and Hilbert curve are employed to scramble the original image first and then the scrambled image is diffused by using the chaotic sequence generated by the 2D chaotic system. The performance is analysed with numerical simulations, and it is shown that the proposed algorithm can encrypt the grey or colour image effectively and resist various attacks such as the statistical attack, differential attack and so on. 7 Acknowledgments This work was supported by the National Natural Science Foundation of China (grant nos. 11071152, 11601188), the Natural Science Foundation of Guangdong Province (grant no. 2018A030307024). 8 References 1Yeo J., and Guo J.: ‘Efficient hierarchical chaotic image encryption algorithm and its vlsi realisation’, IEE Proc.-Vis. Image Signal Process., 2000, 147, (2), pp. 167– 175 2Gao T., and Chen Z.: ‘A new image encryption algorithm based on hyper-chaos’, Phys. Lett. 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