The rapid evolution of imaging and communication technologies has transformed images into a widespread data type. Different types of data, such as personal medical information, official correspondence, or governmental and military documents, are saved and transmitted in the form of images over public networks. Hence, a fast and secure cryptosystem is needed for highresolution images. In this paper, a novel encryption scheme is presented for securing images based on Arnold cat and Henon chaotic maps. The scheme uses Arnold cat map for bit and pixellevel permutations on plain and secret images, while Henon map creates secret images and specific parameters for the permutations. Both the encryption and decryption processes are explained, formulated, and graphically presented. The results of security analysis of five different images demonstrate the strength of the proposed cryptosystem against statistical, brute force and differential attacks. The evaluated running time for both encryption and decryption processes guarantee that the cryptosystem can work effectively in realtime applications.
Some researchers utilized conventional cryptosystems to directly encrypting images. But this is not advisable due to large data size and realtime constraints of image data. Conventional cryptosystems require a lot of time to directly encrypt thousands of image pixels value. On the other hand, unlike textual data, a decrypted image is usually acceptable even if it contains small levels of distortion. For all the above mentioned reasons, the algorithms that function well for textual data may not be suitable for multimedia data [
Chaotic maps are simple functions and are iterated quickly. Chaosbased image encryption systems are therefore fast enough for realtime applications. Chaos is a natural phenomenon discovered by Edward Lorenz in 1963 while studying the butterfly effect in dynamical systems. The butterfly effect describes the sensitivity of a system to initial conditions as mentioned in Lorenz’s paper titled “Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas?” [
In the 1990s, numerous researchers found that there are some relationships between properties that have counterparts in chaos and cryptography. A high sensitivity to initial conditions, with deterministic pseudorandom behavior, is an interesting similarity between chaotic maps and cryptographic algorithms. Furthermore, confusion and diffusion are two general principles in the design of cryptography algorithms that lead to the concealing of the statistical structure of pixels in a plain image and to a decrease in the statistical dependence of a plain image and the corresponding encrypted one. Applying a mixing property on chaosbased encryption algorithms will increase the complexity of the cipher image.
Chaotic maps are assigned to discrete and continuous timedomains. Discrete maps are usually in the form of iterated functions, which corresponded to rounds in cryptosystems. This similarity between cryptography and discrete chaotic dynamic systems is utilized to propose chaotic cryptosystems. Each map has some parameters that are equivalent to the encryption keys in cryptography. In stream cipher, a chaotic system is applied to generate a pseudorandom key stream but in block ciphers, the plaintext or the secret key(s) are used as the initial and control parameters. Finally, some iteration is applied on the chaotic systems to obtain the ciphertext. Security and complexity are significant concerns in cryptosystems. These should be considered when selecting a map and its parameters for use in cryptography [
The first chaosbased cryptosystem was proposed by Matthews in 1989 [
The algorithm proposed by Wang et al. [
Many similar works have failed in security analysis. Hence, when designing and implementing a chaosbased cryptographic system, some important requirements should be kept in mind. A common framework was proposed by Alvarez and Li [
Chaosbased encryption algorithms are based on diverse types of chaotic maps and also on discrete maps. Most of these are a combination of two or more chaotic maps to achieve a greater level of complexity, security, and expanded key space. A combination of the Arnold cat map and the Chen map was the work of Guan et al. [
To overcome the disadvantages of permutationonly cryptosystems, Fu et al. [
Xu et al. [
To overcome the drawback of timeconsuming real number arithmetic calculations in chaosbased image encryption techniques, a block cipher cryptosystem was proposed by Fouda et al. [
The scheme of Chen et al. [
Table
Applied chaos maps in some proposed image encryption techniques.
ACM  Logistic  Henon  Lorenz  Baker  Chen  Tent  CML  Standard map  

Zhu et al. [ 
×  ×  
Xu et al. [ 
×  
Zhang and Cao [ 
×  ×  
Fu et al. [ 
×  
Zhang et al. [ 
×  ×  ×  
Ghebleh et al. [ 
×  ×  
Elshamy et al. [ 
×  
Ye and Zhou [ 
×  ×  
Ye and Zhou [ 
×  ×  
Wang et al. [ 
×  
AlMaadeed et al. [ 
×  
Patidar et al. [ 
×  ×  
Wong et al. [ 
×  
Guanghuia et al. [ 
×  
Zhang et al. [ 
×  
Liu et al. [ 
×  × 
Henon is a twodimensional dynamic system proposed [
Henon map attractor after (a) 500, (b) 5000, and (c) 50000 iterations with initial parameters
ACM is a mixing discrete ergodic system that performs an area preserving stretch and fold mapping discovered by V. Arnold in 1968 using the image of a cat. This 2D transformation is based on a matrix with a determinant of 1 that makes this transformation reversible and described as
In addition to
Proposed encryption scheme architecture.
Creating the secret images and a set of parameters
The permutation steps by the Arnold cat map are based on the parameters
Figure
For a graylevel image with a size of
Splitting an image to 8 subimages.
After creating the
Concurrent with bit permutation of the plain image, the secret image is permuted for
The concluding phase is a sequence XOR to modify the pixel values. This step will cause extreme changes in the pixels of cipher image with even one bit change in a pixel in plain image. This step is based on the shuffled secret image and the bitlevel permuted plain image. Equations (
Figure
Decryption model architecture.
Experiments are carried out to analyze the proposed algorithm and evaluate their security and robustness. A vigorous encryption algorithm is resistant to attacks by an opponent or to unauthorized access. Due to the various types of attacks, a comprehensive security analysis is inevitable. This section analyzes the results of simulated attacks such as a statistical attack, a differential attack, a bruteforce attack, and a known/chosen plaintext attack to demonstrate the strength of the proposed technique. Key space, encryption time, and decryption time are additional parameters that will affect the decisionmaking regarding the choice of applied cryptosystem. The selected images for the experiments are “Peppers,” “Baboon,” and “Fingerprint,” which are 512
The Henon map is the main function in this cryptosystem and the generated chaotic sequence is employed to produce both the secret image and the parameters for the Arnold cat map. Its initial value is one of the secret keys in this scheme. The chosen point (1.210000001, 0.360000001) is the starting point for generating the Henon chaotic sequence.
Pixel permutation, bit permutation, and pixel modification are three key functions in this cryptosystem. The encryption time depends on the run time of each function and the number of rounds. Table
Evaluated running time for main functions in proposed cryptosystem.
Variable  Function  Running time for one round (ms) 


Bit permutation running time  62 

Pixel permutation running time  13 

Pixel modification running time  4 

Inverse bit permutation running time  75 

Inverse pixel permutation running time  23 

Inverse pixel modification running time  4 
Comparison of encryption time for proposed algorithm with recent similar works.
Proposed scheme  [ 
[ 
[ 
[ 
[  

Encryption time (ms)  19.75  23  22  20.79  32  52 
The image histogram is the graphical illustration of the pixel distribution at different gray levels. A great deal of statistical information regarding the image is extractable from its histogram [
(a) Plain image and (b) its histogram.
(a) Plain image after four rounds of bit permutation based on Zhu’s algorithm and (b) its histogram.
(a) Plain subimages, (b) plain image after four rounds of proposed bit permutation and (c) its histogram.
(a) Encrypted image and (b) its histogram.
(a) Decrypted image and (b) its histogram.
The total number of possible keys that an attacker must try to break a cryptosystem is called key space and it should be large enough to prevent bruteforce attack. In the proposed cryptosystem, the initial point
Secret parameters length in bit.
Parameter  Length (bit) 


10 

10 

10 

48 

48 

24 

24 

64 

64 
Comparison of encryption time for proposed algorithm with recent similar works.
Proposed scheme  [ 
[ 
[ 
[ 
[ 
[  

Key space 







In addition to a sufficiently large key space to protect an encrypted image from bruteforce attacks, a strength algorithm should also be absolutely sensitive to both encryption and decryption keys. Changing even one bit in a secret key will cause a completely different result in either the encrypted image or the decrypted image. Key sensitivity is analyzed in both the encryption and the decryption phase. In the encryption phase, the cipher image that results from changing even one bit in any one of the initial values is compared with the encrypted image that resulted before changing the key. The results are given in Table
Difference rates of two encrypted images with slight change in a parameter.
Parameter  Initial value  Changed value  Encrypted images difference rate 


6  7  99.59% 

2  1  99.60% 

2  1  99.62% 

12345678  12345679  99.58% 

87654321  87654320  99.62% 

12345  12346  99.63% 

67890  67891  99.60% 

1.21000001  1.21  99.59% 

0.36000001  0.36  99.60% 
In the decryption phase, key sensitivity means that the encrypted image cannot be decrypted by slight variations in the secret key. Based on the results in Table
Difference rate of two decrypted images with slight change in a parameter.
Parameter  Encryption parameters  Decryption parameters  Decrypted images difference rate 


6  5  99.59% 

2  1  99.62% 

2  1  99.59% 

12345678  12345677  99.62% 

87654321  87654322  99.58% 

12345  12344  99.59% 

67890  67889  99.60% 

1.21000001  1.21000002  99.60% 

0.36000001  0.36000002  99.62% 
Statistical analysis can extract the relationships between the original and the encrypted image. Shannon in his theory of information and communication [
Two adjoining pixels in a regular image are strongly correlated in horizontal, vertical, and diagonal positions. Scatter plots in Figures
Correlation of plain image’s pixels in (a) horizontal, (b) vertical, and (c) diagonal position.
Correlation of cipher image’s pixels in (a) horizontal, (b) vertical, and (c) diagonal position.
Entropy is a statistical parameter that is defined to measure the uncertainly and randomness of a bundle of data. According to Shannon theory, image entropy is the number of bits that is necessary to encode every pixel of the image. The optimal value for entropy of an encrypted image is ~8. This quantity describes the random pattern and texture of pixels in an encrypted image and is calculated by
For the purpose of differential attack, an attacker changes a specific pixel in the plain image and traces the differences in the analogous encrypted image to find a meaningful relation. This is also known as a chosenplaintext attack. A robust encrypted image must be sensitive to minor changes and even changing one bit in the plain image should result in a wide range of changes in the cipher image.
The NPCR measures the number of pixels change rate in an encrypted image when 1 bit is changed in the plain image. This parameter is calculated by (
The UACI in differential analysis is the unified average changing intensity between two encrypted images with a difference in only one bit in corresponding plain images. The UACI can be calculated by (
Calculated UACI and NPCR for different combinations of

1  2  3  4  5  6  7  8  9  10  

1  UACI  5.1163  2.7810  2.0331  1.1860  1.5810  6.1305  1.4349  3.7728  3.3215  0.0564 
NPCR  81.5414  44.3218  32.4032  18.9026  25.1965  97.7051  22.8695  60.1284  52.9366  0.8984  
2  UACI  5.1163  2.7810  2.0331  1.1860  1.5810  6.1305  1.4349  3.7728  3.3215  0.0564 
NPCR  81.5414  44.3218  32.4032  18.9026  25.1965  97.7051  22.8695  60.1284  52.9366  0.8984  
3  UACI  5.1163  2.7810  2.0331  1.1860  1.5810  6.1305  1.4349  3.7728  3.3215  0.0564 
NPCR  81.5414  44.3218  32.4032  18.9026  25.1965  97.7051  22.8695  60.1284  52.9366  0.8984  
4  UACI  5.1163  2.7810  2.0331  1.1860  1.5810  6.1305  1.4349  3.7728  3.3215  0.0564 
NPCR  81.5414  44.3218  32.4032  18.9026  25.1965  97.7051  22.8695  60.1284  52.9366  0.8984  
5  UACI  5.1163  2.7810  2.0331  1.1860  1.5810  6.1305  1.4349  3.7728  3.3215  0.0564 
NPCR  81.5414  44.3218  32.4032  18.9026  25.1965  97.7051  22.8695  60.1284  52.9366  0.8984  
6  UACI  5.1163  2.7810  2.0331  1.1860  1.5810  6.1305  1.4349  3.7728  3.3215  0.0564 
NPCR  81.5414  44.3218  32.4032  18.9026  25.1965  97.7051  22.8695  60.1284  52.9366  0.8984  
7  UACI  5.1163  2.7810  2.0331  1.1860  1.5810  6.1305  1.4349  3.7728  3.3215  0.0564 
NPCR  81.5414  44.3218  32.4032  18.9026  25.1965  97.7051  22.8695  60.1284  52.9366  0.8984  
8  UACI  5.1163  2.7810  2.0331  1.1860  1.5810  6.1305  1.4349  3.7728  3.3215  0.0564 
NPCR  81.5414  44.3218  32.4032  18.9026  25.1965  97.7051  22.8695  60.1284  52.9366  0.8984  
9  UACI  5.1163  2.7810  2.0331  1.1860  1.5810  6.1305  1.4349  3.7728  3.3215  0.0564 
NPCR  81.5414  44.3218  32.4032  18.9026  25.1965  97.7051  22.8695  60.1284  52.9366  0.8984  
10  UACI  5.1163  2.7810  2.0331  1.1860  1.5810  6.1305  1.4349  3.7728  3.3215  0.0564 
NPCR  81.5414  44.3218  32.4032  18.9026  25.1965  97.7051  22.8695  60.1284  52.9366  0.8984 
Calculated UACI and NPCR for different combinations of

1  2  3  4  5  6  7  8  9  10  

1  UACI  5.1163  16.2147 








NPCR  81.5414  89.1617 









2  UACI  5.1163  16.2173 








NPCR  81.5414  89.1617 









3  UACI  5.1163  16.2191 








NPCR  81.5414  89.1617 









4  UACI  5.1163  16.2003 








NPCR  81.5414  89.1617 









5  UACI  5.1163  16.1772 








NPCR  81.5414  89.1617 









6  UACI  5.1163  16.1845 








NPCR  81.5414  89.1617 









7  UACI  5.1163  16.1885 








NPCR  81.5414  89.1617 









8  UACI  5.1163  16.1882 








NPCR  81.5414  89.1617 









9  UACI  5.1163  16.1954 








NPCR  81.5414  89.1617 









10  UACI  5.1163  16.2147 








NPCR  81.5414  89.1617 








Calculated UACI and NPCR for different combinations of

1  2  3  4  5  6  7  8  9  10  

1  UACI  5.1163  2.7810  2.0331  1.1860  1.5810  6.1305  1.4349  3.7728  3.3215  0.0564 
NPCR  81.5414  44.3218  32.4032  18.9026  25.1965  97.7051  22.8695  60.1284  52.9366  0.8984  
2  UACI  16.2119  2.0258  0.9736  0.4904  0.8663  32.6779  0.4922  3.8314  4.1271  0.1998 
NPCR  89.1617  85.5148  81.7390  75.0000  76.3233  94.8944  75.3403  86.6261  89.8193  50.9499  
3  UACI 










NPCR 











4  UACI 










NPCR 











5  UACI 










NPCR 











6  UACI 










NPCR 











7  UACI 










NPCR 











8  UACI 










NPCR 











9  UACI 










NPCR 











10  UACI 










NPCR 










In addition to Peppers, the same experiments were performed on Baboon, Figure
Calculated UACI and NPCR for different combinations of

1  2  3  4  5  6  7  8  9  10  

1  UACI  34.5764  18.5622  1.9700  39.7193  43.0302  12.3125  4.4068  40.1488  48.6447  20.7776 
NPCR  68.8828  36.9793  3.9246  79.1283  85.7243  24.5289  8.7791  79.9839  96.9093  41.3929  
2  UACI  7.9196  0.9566  0.1960  15.6506  16.3404  0.5035  0.1959  14.2109  32.8500  1.7190 
NPCR  87.5687  81.3938  49.9817  93.7302  93.2835  76.1761  49.9420  86.6169  93.3601  67.2352  
3  UACI 










NPCR 











4  UACI 










NPCR 











5  UACI 










NPCR 











6  UACI 










NPCR 











7  UACI 










NPCR 











8  UACI 










NPCR 











9  UACI 










NPCR 











10  UACI 










NPCR 










Calculated UACI and NPCR for different combinations of

1  2  3  4  5  6  7  8  9  10  

1  UACI  0.1102  0.2833  0.2429  0.3697  0.3291  0.4672  0.6168  0.1744  0.0714  0.1102 
NPCR  14.0533  36.1149  30.9715  47.1390  41.9651  59.5711  78.6449  22.2366  9.0992  14.0533  
2  UACI  0.4539  1.0997  1.0502  2.0179  2.0101  4.0957  16.3335  0.4905  0.1923  0.4539 
NPCR  71.9505  85.4733  86.7638  85.9703  84.3555  88.2622  89.8018  75.0031  49.0311  71.9505  
3  UACI 










NPCR 











4  UACI 










NPCR 











5  UACI 










NPCR 











6  UACI 










NPCR 











7  UACI 










NPCR 











8  UACI 










NPCR 











9  UACI 










NPCR 











10  UACI 










NPCR 










Calculated UACI and NPCR for different combinations of

1  2  3  4  5  6  7  8  9  10  

1  UACI  7.4904  2.6620  5.1333  11.4982  3.7695  5.6794  5.3902  11.9543  8.4184  7.4904 
NPCR  29.8447  10.6064  20.4529  45.8130  15.0192  22.6288  21.4767  47.6303  33.5419  29.8447  
2  UACI  1.0214  0.1943  0.4910  1.9258  0.3592  0.5052  0.4911  1.8747  1.0055  1.0214 
NPCR  83.5632  49.5514  75.0793  86.7004  63.8062  76.2024  75.0595  84.4025  81.1432  83.5632  
3  UACI 










NPCR 











4  UACI 










NPCR 











5  UACI 










NPCR 











6  UACI 










NPCR 











7  UACI 










NPCR 











8  UACI 










NPCR 











9  UACI 










NPCR 











10  UACI 










NPCR 










Calculated UACI and NPCR for different combinations of

1  2  3  4  5  6  7  8  9  10  

1  UACI  49.7511  38.9683  0.6828  0.3194  17.8328  9.1234  1.8446  0.6682  24.8476  45.3508 
NPCR  99.1135  77.6321  2.7206  81.4575  71.0526  72.7020  14.6988  85.1974  99.0021  90.3473  
2  UACI  30.8111  26.3993  34.1747  33.5065  33.3622  33.4134  33.0488  33.5821  33.5865  33.5562 
NPCR  72.5723  90.6128  99.3164  99.3210  99.5193  99.5987  99.4949  99.6674  99.5697  99.6399  
3  UACI 










NPCR 











4  UACI 










NPCR 











5  UACI 










NPCR 











6  UACI 










NPCR 











7  UACI 










NPCR 











8  UACI 










NPCR 











9  UACI 










NPCR 











10  UACI 










NPCR 










Results of security analysis.
Image name 



Plain entropy  Cipher entropy  Plain image correlations  Cipher image correlations  UACI  NPCR  

HC  VC  DC  HC  VC  DC  
Cameraman 
1  1  1  7.0097  7.9969  0.8390  0.7189  0.6973 





3  1  1  7.9976 




 
1  3  1  7.9971 




 
1  1  3  7.9972 




 


Chessplate 
1  1  1  1  7.9970  0.9775  0.9800  0.9637 





3  1  1  7.9974 




 
1  3  1  7.9972 




 
1  1  3  7.9973 




 


Baboon 
1  1  1  7.3579  7.9993  0.8644  0.7587  0.7261 





3  1  1  7.9993 




 
1  3  1  7.9993 




 
1  1  3  7.9993 




 


Peppers 
1  1  1  7.5714  7.9993  0.8642  0.7587  0.7261 





3  1  1  7.9993 




 
1  3  1  7.9993 




 
1  1  3  7.9993 




 


Fingerprint 
1  1  1  6.7279  7.9993  0.8644  0.7587  0.7261 





3  1  1  7.9992 




 
1  3  1  7.9994 




 
1  1  3  7.9993 





(a) Baboon image and (b) its histograms and (c) encrypted image and (d) its histogram.
(a) Fingerprint image and (b) its histograms and (c) encrypted image and (d) its histogram.
(a) Cameraman image and (b) its histograms and (c) encrypted image and (d) its histogram.
(a) Chessplate image and (b) its histograms and (c) encrypted image and (d) its histogram.
In this paper, a new chaosbased cryptosystem has been proposed for encrypting images. The Arnold cat map and the Henon map are two discrete chaotic maps that are used in this scheme. Bit shuffling and pixel shuffling are reversible transformations that are performed using the Arnold cat map with various secret parameters. Improving the randomness of transformation and the efficiency of bit permutation are two advantages of this cryptosystem that increases the strength of the ciphered image in comparison with previous works. Iterating the Arnold cat map with different parameters at each round prevents undesirable reconstruction of the input image. These parameters are generated by the Henon map with secret initial values. The points generated by the Henon map are also applied to create secret images for more confusion and diffusion and to increase the key space. Sequential XOR of the bitpermuted plain image and the pixelpermuted secret image is another phase of modifying the pixels values. This creates a slight distortion in the plain image to prevent successful differential attacks. The results of security analysis of five images demonstrate the resistance of the encrypted image to statistical attacks and to the chosenplaintext attack. In addition, a sufficiently large key space makes a brute force attack impractical. As the future work, the proposed cryptosystem in this paper will combine with a public key technique such as ECC or RSA to propose a hybrid encryption method. This technique is a chaotic asymmetric cryptosystem.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors wish to thank Universiti Kebangsaan Malaysia (UKM) and Ministry of Higher Education Malaysia for supporting this work by research Grants FRGS/1/2012/SG05/UKM/02/1 and ERGS/1/2012/STG07/UKM/02/9.