Chaos-based model for encryption and decryption of ...

Chaos-based model for encryption and decryption of digital images

Fatma Elgendy, Amany M. Sarhan, Tarek E. Eltobely, S. F. El-Zoghdy, Hala S. El-sayed & Osama S. Faragallah Multimedia Tools and Applications An International Journal ISSN 1380-7501 Multimed Tools Appl DOI 10.1007/s11042-015-2883-z

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Author's personal copy Multimed Tools Appl DOI 10.1007/s11042-015-2883-z

Chaos-based model for encryption and decryption of digital images Fatma Elgendy 1 & Amany M. Sarhan 1 & Tarek E. Eltobely 1 & S. F. El-Zoghdy 2,5 & Hala S. El-sayed 3 & Osama S. Faragallah 4,5

Received: 22 August 2014 / Revised: 20 July 2015 / Accepted: 12 August 2015 # Springer Science+Business Media New York 2015

Abstract This paper introduces a secure chaos-based model for ciphering and deciphering of digital images. The proposed approach is composed of successive confusion and diffusion stages. The confusion stage is repeated n rounds using a different key in each round. The output of the confusion stage is subjected to diffusion stage which is repeated m rounds with a different key for each round. The nested iterations in the confusion and diffusion stages with a different key for each round enlarges the key space which enhances the proposed image cryptosystem security level. A security investigation is done on a family of 2D chaotic confusion maps to select the one with highest security level to be used with the proposed image cryptosystem. The results demonstrated that the Standard map has the highest security level among the examined 2D chaotic confusion maps because it is more complicated and it has a large key space. The proposed image cryptosystem is compared to other three recent image cryptosystems using different security analysis factors including statistical tests, key space analysis, information entropy test, maximum deviation analysis, irregular deviation analysis, and avalanche effect differential analysis. The results demonstrated that, the proposed image cryptosystem with Standard map outperforms all of the other examined image encryption techniques from security point of view.

* Osama S. Faragallah [email protected]; [email protected] 1

Department of Computers and Automatic Control Engineering, Faculty of Engineering, Tanta, Egypt

2

Department of Mathematics and Computer Science, Faculty of Science, Menoufia University, Shebin El-Kom, Egypt

3

Department of Electrical Engineering, Faculty of Engineering, Menoufia University, Shebin El-kom 32511, Egypt

4

Department of Computer Science and Engineering, Faculty of Electronic Engineering, Menoufia University, Menouf 32952, Egypt

5

Department of Information Technology, College of Computers and Information Technology, Taif University, Al-Hawiya 21974, Kingdom of Saudi Arabia

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Keywords Chaotic map . Confusion . Diffusion . Security evaluation

1 Introduction The advances in the networking technology coupled with popularity of the internet leads to dramatical increase in the amount of exchanged data/information in all of our daily life activities. Such activities may include but not limited to banking, internet communication, education, medication, training, and military applications, and so on. The exchanged data/information could be in the form of text, images, videos, and other forms. Exchanging sensitive and confidential information on such public, open and shared networks facilitate the task of exploiting such information on the attackers [7, 12, 18, 25, 40]. Therefore, the secure transmission of sensitive and confidential information on networks has become a major research problem. Cryptographic schemes can be employed in solving this serious problem. Cryptography is a tool for secure communication in presence of adversaries. Conventional cryptographic schemes such as IDEA, DES, RSA and ELGamal etc. are classified into key cryptosystems and double key cryptosystems. These cryptosystems are designed on the basis of complex mathematic problems. These conventional cryptosystems are implemented to be used only for text data encryption [41]. Compared to text, images have many fundamental characteristics like large bulky data capacity redundancy. These special characteristics cannot be handled using conventional encryption algorithms. One of the major obstacles in designing a powerful image cryptosystem based on conventional encryption techniques is that it is very hard to diffuse and shuffle the image data using such techniques [27]. Chaotic maps have unique, distinct and special characteristics such as quasi-randomness, ergodicity, elevated sensitivity to system parameters and initial conditions. These special characteristics make chaos a very good promising alternative for conventional cryptographic techniques in securing information especially image encryption [7, 9, 12–14, 18, 19, 21, 25, 27, 31, 37, 40, 41]. Also such distinct properties enable chaos’s systems to satisfy the confusion and diffusion classic Shannon requirement’s [37]. In contrast to traditional cryptographic techniques that are based on discrete mathematics, chaotic-based cryptographic techniques are mainly based on nonlinear complex dynamics maps that are simple and deterministic. As a result, chaotic-based ciphering techniques can give fast and secure data protection means that is very important for exchanging secure data among users over open, public, and shared communication networks such as broadband internet communication. Based on the two main principals used in encryption [37]; namely confusion and diffusion, the full motivations of special characteristics of chaotic nonlinear dynamics should be taken in order to implement the cryptosystem’s confusion and diffusion stages to obtain a high security level. This kind of cryptosystems is based on either discrete or continuous chaotic systems. The advances in the research of pseudo random sequence generator and its 1D chaotic based system’s applications performed a lot of achievements especially in the field of cryptographic techniques. But because of its structure simplicity, these encryption techniques can easily be cryptanalyzed by many methods such as phase regression image, space reconstruction, or nonlinear prediction. As a result of the emergence of the multi-dimensional image processing technologies, and multivariable networks, the cryptosystem’s designers developed multi-dimensional chaos-based image cryptosystems [8], hyper-chaos-based image cryptosystems [36]. However, these image cryptosystems have been cryptanalyzed and demonstrated insecurity and they also exposed some inner flaws [2, 8, 21, 23, 34, 35, 39].

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This paper proposes a secure chaotic-based model for image encryption. The proposed image cryptosystem consists of alternative confusion and diffusion stages. The confusion stage is repeated n rounds using a different key in each round. The output of the confusion stage is subjected to diffusion stage which is repeated m rounds with a different key for each round. The nested iterations in the confusion and diffusion stages with a different key for each iteration enlarges the key space using the proposed image cryptosystem with Standard map from [(N2)!]L to [(N2)!]nmLnm for an N×N-sized image with gray-level L. During the confusion stage, a chaotic map is utilized to determine control parameters in every confusion cycle, and another chaotic map is utilized in the diffusion cycle to extract the keystream. In the proposed method, control parameters and keystream are correlated with many factors including the plainimage, the key, number of rounds in the confusion stage (n), and number of rounds in the confusion-diffusion stage (m). This kind of multi-parameter correlation efficiently handles the flaws mentioned above and strongly improves the performance of the confusion and diffusion stages. The relation between the utilized chaotic map (confusion stage), the diffusion function (diffusion stage), and the proposed system’s security is studied deeply. Also, the security of proposed image cryptosystem is evaluated against several kinds of attacks including cryptanalytic, differential, statistical, and brute-force attacks. Experimental results ensured the effectiveness of proposed image cryptosystem using different mixtures of chaotic maps in confusion and diffusion stages. This mixed application of confusion and diffusion chaotic functions in the proposed image cryptosystem enlarged the key space and therefore, the system’s security is improved. The encrypted image using the proposed image cryptosystem has the same size as the plainimage. This property makes the proposed image cryptosystem applicable in securing confidential information transmitted over open, public, and shared networks. The rest of paper is arranged as follows: the related works with respect to image cryptosystems is presented in section 2. Section 3 gives the architecture of proposed image cryptosystem. Section 4 illustrates the analysis of the chaotic confusion functions, chaotic diffusion functions, and their relation to the system’s security. Section 5 discusses the obtained results. In section 6, the security evaluation of the proposed method for image cryptosystem is discussed. Finally, section 7 summarizes the paper.

2 Related works The explosive growth in networking technologies combined with the strong and fast achievements in the digital image processing field leads to dramatic increasing in the amount of exchanged data between users. This shard data may contain sensitive and confidential information. Exchanging sensitive and confidential information over open, public and shared networks like the internet facilitates the hacker’s task for exploiting the information. Therefore sensitive and confidential information should to be protected while transmitting it on public networks. The problem of securing digital image during transmission is studied by many scientists. Using chaotic-based systems for encrypting digital image is not new idea. Several chaoticbased image cryptosystems have been proposed. One direct method for designing chaoticbased image cryptosystems is conducted by adopting the stream cipher architecture. In this

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method, continuous chaotic circuits or chaotic maps are utilized to produce a pseudo-random sequence which in turn is utilized to mask the pixel values. Majid Khan et al. [21] presented a general survey of image encryption techniques in frequency, spatial, and hybrid domains. Based on chaos, Chen et al. [8] introduced a 3D symmetric image encryption algorithm. To have a real-time security in his algorithm, he generalizes the 2D chaotic cat map into a 3D one. Then, the algorithm uses that 3D cat map in shuffling the image pixel’s placement and the grey values (if desired) as well. To enhance security level of the cipher with respect to differential and statistical attacks, the relation between the plainimage and its resulted cipherimage one is confused using another chaotic map. In [29], M. Mishra et al. presented an image cryptosystem which combines pixel shuffling and three chaotic maps. Their algorithm starts by decomposing the plainimage into 8x8 size blocks and then based on these blocks, it shuffles image. The permuted image is ciphered after that using a chaotic sequence produced using another chaotic map. S. S. Askar et al. [4] introduced an image ciphering algorithm depending on a chaotic economic map, and implemented it on a plainimage. Results demonstrated that, with the same security keys, their algorithm can encrypt and decrypt images successfully. In Y. Mao et al. [28], the authors extended the 2D Baker map to a 3D one, and then they used it in designing their image encryption algorithm. The purpose of extending Baker map is to speed up the encryption process while keeping the same security level. Also, they analyzed the security of their algorithm. In Gao et al. [16] another image encryption technique is proposed. In this technique, a power and tangent function is utilized in generating a chaotic pseudo-random sequence which in turn is employed in masking the pixels of the plainimage. In [22], the authors introduced a chaotic digital image cryptosystem. They generated the key-stream for masking pixels of plainimage by cascading a high-dimensional cat map and the skewed tent map. Nien et al. [30] introduced a digital color image ciphering algorithm. The idea is to separately encrypt the three RGB components by using a third-order RLC chaotic function. In Behnia’s et al. [6] image encryption scheme, the masking bits are generated by utilizing a onedimensional chaotic map coupled with a two-dimensional map. In addition to this masking technique, there are many other chaotic based image encryption techniques having their own masking structure. For example, in Pisarchik et al. [33], the image pixels are converted to chaotic maps coupled unidirectionally forming a chaotic map lattice (CML). The CML is then iterated a number of rounds with secret system parameters to obtain the encrypted image. Pareek et al. [32] uses the logistic map output and an external key in the process of selecting one of the eight possible types of operations for encrypting an image pixel. In Zhang’s et al. [42] technique, an exponentially discrete chaotic map is utilized to permute pixels and a logic XOR function is utilized for diffusion. The image encryption technique presented by Lian et al. [26] employed two chaotic maps namely; a Standard map and a quantized logistic map in confusion and diffusion stages respectively. In each round, a key-stream is generated which in turn is employed in determining the parameters of the two utilized chaotic maps. Using chaos theory, M. Amin el al [3] presented a one-way hash function algorithm, and theoretically analyzed it. The simulation results showed that their algorithm satisfies all of the hash function performance requirements in an efficient and flexible manner. Asmita et al. [5] introduced a chaotic-based image cryptosystem which employs an external secret key and two different chaotic maps in the encryption process. The secret key is utilized in deriving chaotic map’s initial conditions. Using these initial conditions, the two chaotic maps are employed in confusing the connection between the plainimage, and cipherimage. Yuling Tiegang G. et al. [15] introduced a digital image cryptosystem. In this cryptosystem, the positions of image pixels are shuffled using an image shuffling matrix. After that, the cipher uses a hyper-chaotic

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system with confusion process. Using spatiotemporal chaos, Shiguo et al. [24] presented a digital image/video cryptosystem. The algorithm utilizes the chaotic system in generating pseudorandom sequences, and then it sequentially encrypts the image blocks. The chaotic maps are iterated for certain cycles to generate the pseudorandom sequences aiming to increase their randomness and sensitivity to initial-value. The algorithm uses the pseudorandom-bits to encrypt the Alternating Current coefficients Signs as well as the Direct Current coefficient. Nanrun et al. [43] presented a nonlinear optical image cryptosystem based on fractional Mellin transform FrMT. Compared to linear encryption techniques, their optical image cryptosystem is less vulnerable to attacks. Their main objective from utilizing the FrMT is to strengthen the cryptosystem’s capability against most commonly known security attacks. Hone-Ene et al., [20] proposed a new effective color image cryptosystem. Their proposed cryptosystem can securely encrypt color image with multiple choice of parameters for Fresnel diffraction as keys. Lihua et al. [17] presented a multiple-image cryptosystem which combines, in the fractional Fourier domain, log-polar transform with DRPE. Their system starts by utilizing the inverse log-polar transform to transform plain images to annular domains. Second, the annular domains are combined into a single image. Finally, this combined image is ciphered by using the traditional DRPE technique. According to Arnold and discrete Hartley transform in gyrator transform domain, a color image cryptosystem is proposed by Muhammad et al. [1]. Nanrun et al. [44] presented a hybrid image compression–encryption technique which achieves compression and encryption simultaneously. The plainimage is divided into 4 blocks to be compressed and encrypted, after that, the pixels of every two adjacent blocks are exchanged randomly by random matrices. By using the circulant matrices, the measurement matrices in compressive sensing are constructed and the logistic map is used for controlling the original row vectors of the circulant matrices.

3 Structure of the proposed image cryptosystem The proposed image cryptosystem structure is depicted in Fig. 1. It is composed of confusion and diffusion stages. Confusion stage employs a 2D chaotic confusion map to permute the plainimage pixels without performing any change in their values while diffusion stage changes pixel values such that a little change in any pixel is diffused to all-of its surrounding pixels. During confusion stage, the chaotic map parameters have been utilized as the confusion key. Similarly, in diffusion stage, parameters such as, diffusion function’s control parameter or the initial value have been employed as the diffusion key. The relationship between the adjacent pixels is de-correlated using n>1 permutation rounds in the confusion process. The proposed image cryptosystem repeats the whole confusion-diffusion round m times to realize a satisfactory security level. Also, to grantee a higher security level, the chaotic map parameters (controlling the permutation) in confusion and diffusion processes should better be changed in every cycle. In the proposed image cryptosystem, chaotic map parameter updating is done through employing round key generator that uses a seed secret key as an input. Various 2D chaotic maps including Baker map, Cat map and Standard map can be utilized in confusion process. To satisfy the confusion of all pixels, the chaotic maps must be discretized over the image lattice. The discretized Standard map, Cat map and

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m rounds

n rounds

Diffusion (Sequential Pixel Value Modification)

Confusion (Pixel Permutation)

Cipherimage

Plainimage

Key Generator Secret Key Fig. 1 Typical architecture of the proposed image cryptosystem

Baker map can be defined for an N ×N image lattice by Eqs. (1), (2), and (3) respectively.


xiþ1 yiþ1





xiþ1 yiþ1





3 ðxi þ yi Þmod N   5 xiþ1 N ¼4 yi þ K sin mod N 2π

xiþ1 yiþ1

2




¼

1 p q pq þ 1




 xi mod N yi

8 3 N N K 1 þ K 2 þ ……: þ K¼N ; > > ð x −N Þ þ y mod < i 6 Ki i i 7 N K i ¼ K 1 þ K 2 þ …… þ K i−1 ; i   7 where ¼6 4 Ki 5 N N i ≤xi < N i þ K i ; > > y −y mod þ Ni : 0≤yi < N : N i i Ki

ð1Þ

ð2Þ

2

ð3Þ

Here, p and q are parameters, (xi ,yi), and (xi+1,yi+1) are the ith and (i+1)th states respectively. The key in confusion process consists of parameter k, parameters p and q, and K=[k1, k2, …, kt] in the Standard, Cat map, and Baker map respectively. In Baker map, the key k satisfies condition given in Eq. (3). Beginning with the upper left corner, the diffusion process sequentially scans every pixel of the permuted 2D image. In this process, any of Chen’s chaotic map or Hénon map, defined by Eqs. (4) and (5), can be employed to satisfy the diffusion effect in the diffusion stage.


xiþ1 yiþ1




¼

1−ax2i þ yi bxi

where;i=1,2,…….., and the values of a=0.3 and b=1.4.

 ð4Þ

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xiþ1 ¼ cðyi −xi Þ yiþ1 ¼ ðe−cÞxi −xi zi þ eyi ziþ1 ¼ xi yi −dzi

ð5Þ

where; c, d and e are parameters. When c=35, d=3, and e ∈ [20, 28], the system is chaotic according to the experimental results conducted in [8]. Also, the results of the simulation experiments conducted in [8] demonstrated that, the parameter c strongly affects the system orbit i.e., the system orbit is strongly sensitive with respect to any change in the parameter c. Therefore, the proposed image cryptosystem utilizes the c parameter in controlling the cipher key generation process.

4 Security description The proposed chaos-based model for digital image encryption is realized through confusion and diffusion processes. It utilizes a 2D chaotic map for confusion stage and a chaotic diffusion function for diffusion stage. The proposed image cryptosystem can be stated as: m

Y ¼ ½DðC ðX ; K 1 Þn ; K 2 Þ ;

ð6Þ

where X, Y, K1, K2, C(), D(), n and m represent plainimage, cipherimage, confusion key, diffusion key, confusion process, diffusion process, confusion iteration times and whole confusiondiffusion iteration times respectively. Form Eq. (4), one can see that the proposed image cryptosystem security is affected according to the utilized chaotic map for confusion stage C(), chaotic diffusion map D(), n confusion iteration times, and m confusion-diffusion iteration times. The security of proposed image cryptosystem depends on the characteristics of the utilized chaotic maps in the confusion and diffusion stages. There is a close relationship between parameters sensitivity and key sensitivity. So, the greater the sensitivity of parameter is, the greater the sensitivity of key will be, and therefore the superior the proposed image cryptosystem is. Therefore, it is preferred to use a chaotic confusion map that has higher parameter sensitivity since confusion key is generated from such confusion map parameters. The strength of the confusion process is characterized by the sensitivity of selected state ergodicity and initial-value of utilized confusion map. In confusion stage, initial value gives the initial first pixel location. So, sensitivity to initial-value results in minimizing correlation coefficients of adjacent pixels and achieving greater randomization for the permuted cipherimage. The state ergodicity implies changing a pixel location to another location with equal probability. Hence, greater state ergodicity implies best randomness for confusion process. It is preferably to use a chaotic map with greater sensitivity with respect to initial-value and state ergodicity. With respect to the diffusion stage, The chaotic diffusion function make sure that the proposed image cryptosystem is hypersensitive to tiny alternation in plainimage since change of one pixel will be diffused to other pixels. With respect to diffusion stage, the more changed pixels in one round, the greater the effect of diffusion function. In addition to confusion and diffusion stages, the iteration times, n and m have a great effect on cryptosystem’s security. The increase in iteration times implies increasing the key space of image cryptosystem’s when different keys are utilized with different iterations. So, larger iteration times result in best image cryptosystem’s encryption power. In the proposed cryptosystem, high security level of is achieved using large values for n and m.

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5 Encryption results and discussion Encryption performance of the proposed image cryptosystem is examined through several experiments to test and to prove its applicability to images. Figures 2a, 3a, 4a, 5a, 6a, 7a, 8a and 9a showed the test images that are used as plainimages during the experiments. The results demonstrated that, the encrypted images areas are completely invisible and give no information about the original images as shown in Figs. 2b, 3b, 4b, 5b, 6b, 7b, 8b and 9b. The deciphered images are illustrated in Figs. 2c, 3c, 4c, 5c, 6c, 7c, 8c and 9c. Test images in Figs. 2c, 3c, 4c, 5c, 6c, 7c, 8c and 9c and 2a, 3a, 4a, 5a, 6a, 7a, 8a and 9a show that the obtained images after decryption are identical to original images that ensured the success of decryption process. The visual inspection of the figures indicates the success of implementing the proposed chaos-based image cryptosystem for images. This ensures its superiority for securing information included in them.

6 Security evaluation An efficient cipher must be immune to all types of attacks like brute-force attacks, statistical attacks, differential attacks, and cryptanalytic attacks [3, 5, 15, 26, 42]. Security evaluation have applied to the proposed chaos-based image cryptosystem, with several tests such as key space, statistical, information entropy, maximum deviation, irregular deviation, and avalanche effect differential analysis. The results of all of these kinds of security analysis ensured superior security for the presented chaos-based image cryptosystem. Details of the performed security analysis are given below.

6.1 Key space evaluation Confusion-diffusion processes are performed with the main core of the proposed image cryptosystem. So, key space of presented chaos-based image cryptosystem is estimated by multiplying key spaces of confusion and diffusion processes. Assuming that, S1 is key space for iteration in confusion process. Since the confusion process in the proposed cryptosystem is repeated n times, then the key space of the whole confusion process will be Sn1. Here in the confusion process, identical same key space is utilized with different iterations. Practically,

a) Original image

b) Encrypted image

c) Decrypted image

Fig. 2 Application of the proposed image cryptosystem to ancient-egypt-pyramids image

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a) Original image

b) Encrypted image

c) Decrypted image

Fig. 3 Application of the proposed image cryptosystem to baboon image

dissimilar keys can be employed with different iterations. Also, suppose that the key space for diffusion process is S2. Since confusion-diffusion processes are iterated m times in the proposed image cryptosystem, then whole cryptosystem key space is given by:  m S ¼ S n1 :S 2

ð7Þ

From previous calculation, it is clear that confusion space is estimated by parameter space of utilized confusion function. Also, diffusion space is estimated by the initial-value space of utilized diffusion function. From Eq. (7), the key space S of cryptosystem depends on parameter space S1, number of iterations of the confusion process n, initial-value space S2, and number of iteration of confusion-diffusion process m. The key space S of the proposed cryptosystem is increased by increasing any or all of the parameters S1, n, S2, and m. Through them, the initial-value space is estimated using image pixels gray level, parameter space is set through choosing efficient chaotic map. Also, the number of iterations (n and m) can be selected according to security requirements. For example, choosing N×N image, and assuming that L is the gray-level, then cryptosystem key spaces using Standard, Baker and Cat chaotic maps are estimated Table 1. For all chaotic maps, the key space increases as different key is utilized with different iteration. Among the investigated chaotic maps, the Standard map is the best as it obtains largest parameter space. Cat map is the worst as it obtains smallest parameter

a) Original image

b) Encrypted image

Fig. 4 Application of the proposed image cryptosystem to eiffel tower image

c) Decrypted image

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a) Original image

b) Encrypted image

c) Decrypted image

Fig. 5 Application of the proposed image cryptosystem to full black image

space, and the Baker map is between them. Based on calculated results, Standard and Baker maps are more superior with respect to Cat map for application in the proposed chaos-based image cryptosystem. Also based on this result, for a certain chaotic map, increasing number of iterations n and m increases the key space which means higher security. Another performance measure that must be taken when evaluating the chaotic maps, which specifies the complexity of any encryption algorithm, is the number of computation (processes) required by the algorithm. For example, choosing N×N image, and assuming that L is the gray-level, a is an addition or subtraction process and b is a multiplication or division process, then the complexity of the cryptosystems using Standard, Cat, and Baker maps is shown in Table 2.

6.2 Statistical analysis The statistical analysis using several statistical attacks is performed to successfully test and analyze the proposed cryptosystem. A good cryptosystem must be immune against any type of statistical attacks. To ensure the immunity of the proposed cryptosystem, statistical tests are employed through estimating plainimage/ cipherimage histograms and correlation coefficients of two adjacent pixels in the plainimage/cipherimage.

a) Original image

b) Encrypted image

Fig. 6 Application of the proposed image cryptosystem to man image

c) Decrypted image

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a) Original image

b) Encrypted image

c) Decrypted image

Fig. 7 Application of the proposed image cryptosystem to plane image

6.2.1 Histograms analysis To avoid information loss to the attacker, the cipherimage must have small or no statistical similarity with respect to plainimage. The image histogram shows how pixels in an image are spread by drawing the number of pixels at each color intensity level. The histograms of the several cipherimages and their plainimages have constructed and calculated. The results illustrated that they are completely different. Examples are given in Figs. 10 and 11. Figure 10b shows the histogram of 1024× 1024 black plainimage, and its corresponding cipherimage histogram is given in Fig. 10d. From two subfigures, the cipherimage histogram is just uniform and different from that of plainimage. So, it is very difficult to employ statistical attacks on the proposed image cryptosystem. Figure 11 illustrates four plainimages histograms (Lena, Baboon, Plane and Man) against the histograms of their cipherimages using the presented chaos model for image cryptosystem. It is clear that cipherimages histograms are uniformly distributed among all color intensity levels and completely different from the histograms of their corresponding plainimages. This again confirmed that no statistical resemblance to the plainimages. So, it is very difficult to perform any statistical attacks on the proposed image cryptosystem.

a) Original image

b) Encrypted image

Fig. 8 Application of the proposed image cryptosystem to Lena image

c) Decrypted image

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a) Original image

b) Encrypted image

c) Decrypted image

Fig. 9 Application of the proposed image cryptosystem to jelly beanns image

6.2.2 Correlation coefficients of adjacent pixels The correlation coefficients of adjacent pixels for both plainimage/cipherimage are also analyzed. This is done through estimating the correlation amoong two vertically adjacent pixels, two horizontally adjacent pixels and two diagonally adjacent pixels in original and corresponding encrypted image. This is done by selecting randomly 1000 pairs of two adjacent pixels from the image. Then, estimate correlation coefficients by the following formulas: covðx; yÞ ¼ E ðx−E ðxÞÞðy−E ðyÞÞ;

ð8Þ

covðx; yÞ rxy ¼ pffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffi ; DðxÞ DðyÞ

ð9Þ

where x and y are grey-scale amounts of two adjacent pixels in the image. With calculations, the following formulas were used: N 1X xi N i¼1

ð10Þ

N 1X ðxi −E ðxÞÞ2 ; N i¼1

ð11Þ

E ðxÞ ¼

DðxÞ ¼

Table 1 Relationship between the control parameter’s properties and the cryptosystem’s security with different chaotic maps Chaotic map

Parameter space

Key space (the same key in different iteration)

Key space (different key in different iteration)

Cat map Baker map

N2 2(N−1) 2N−1

N2L 2(N−1)L

N2nm ⋅Lnm 2nm(N−1) ⋅Lnm

Standard map

(N2)!

[(N2)!]L

[(N2)!]nm ⋅Lnm

Author's personal copy Multimed Tools Appl Table 2 Computing complexity comparison for standard, baker and cat maps Operation

Cat map

Baker map

Standard map

Addition/subtraction

2N2a

2N2a

2N2a

2

3N b

Multiplication/division

covðx; yÞ ¼

2

2N b

4N2b

N 1X ðxi −E ðxÞÞðyi −EðyÞÞ; N i¼1

ð12Þ

Figure 12 illustrates the correlation distribution for 1000 pairs of horizontally adjacent pixels in one plainimage (Lena), and its corresponding cipherimage with the presented chaos model for image cryptosystem. A comparison between correlation coefficients for horizontal, diagonal and vertical directions of both plainimage and cipherimage for four digital images (Lena, Baboon, Plane and Man) using the proposed image cryptosystem with Standard, Baker and Cat Maps versus Guan et al. [18], Nanrun et al. [43], and Elashry et al. [10] image encryption schemes is presented in Table 3. From that Table, one can notice that, the correlation coefficients

a) Black image

b) Black histogram

c) Encrypted image

d) Histogram of the encrypted image

Fig. 10 Histogram of the original and encrypted images using the proposed image cryptosystem

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Fig. 11 Histogram of: a) original image, b) encrypted image using the proposed image cryptosystem

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Fig. 12 Correlation of 1000 pairs of horizontally adjacent pixels in plainimage/cipherimage, respectively

of both plainimage and cipherimage for the four tested images are far apart. The results presented in Table 3, and Fig. 12 illustrate that the two adjacent pixels in the plainimage are strongly correlated while in the cipherimage; there is negligible correlation between the two adjacent pixels for all images tested using all investigated image encryption techniques. Also, it can easily be seen from Table 3 that, among all the examined image encryption schemes, the proposed image cryptosystem with standard map gives minimum absolute value of correlation coefficient in all cases and for all tested images which means higher security level.

Table 3 Correlation coefficients comparison of the proposed image cryptosystem with standard, baker and cat maps for lena, baboon, plane, and man images and different encryption schemes Image name

Lena

Direction of Plainimage Cipher adjacent pixels Proposed Proposed Proposed Guan Elashry Nanrun standard map cat map baker map et al. [18] et al. [10] et al. [43] Horizontal

0.9962

0.0276

0.0303

0.0341

0.0517

0.0406

0.0355

Vertical Diagonal

0.9928 0.9891

0.0259 0.0222

0.0347 0.0315

0.0372 0.0354

0.0566 0.0537

0.0434 0.0419

0.0337 0.0353

0.9090

0.0126

0.0156

0.0159

0.0361

−0.0272

−0.0211

Vertical

0.8250

0.0130

0.0139

0.0177

−0.0382

0.0239

0.0235

Diagonal

0.7331

0.0111

0.0128

0.0180

−0.0379

−0.0240

0.0212

Horizontal

0.9938

0.0170

0.0219

0.0234

0.0236

0.0219

0.0211

Vertical

0.9901

0.0108

0.0251

0.0254

0.0250

0.0248

0.0234

Diagonal

0.9848

0.0157

0.0190

0.0226

0.0226

0.0221

0.0217

Horizontal Vertical

0.9878 0.9783

0.0133 0.0135

0.0211 0.0226

0.0277 0.0229

0.0372 0.0358

0.0256 0.0230

0.0156 0.0177

Diagonal

0.9687

0.0130

0.0271

0.0292

0.0303

0.0285

0.0161

Baboon Horizontal

Plane

Man

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6.3 Information entropy evaluation Information theory is founded in 1949 by C.E. Shannon [37]. It represents the mathematical theory for data communication and storage. Now, information theory is interested with correction of errors, compression of data, cryptography, and communications systemss. The entropy H (m) of a source m is estimated as: 2 −1 X N

H ðmÞ ¼

Pðmi Þlog2

i¼0

1 bits Pðmi Þ

ð13Þ

where p(mi) is the probability of symbol mi and the entropy is measured in bits. Assuming that the source emits 28 symbols with equal probability, i.e., m ¼ fm1 ; m2 ; :::::; m28 g after calculating Eq. (13), we have its entropy H (m)=8, with respect to a truly random source. Given an information source that produces random messages, its entropy value is lower than the ideal one. When the messages are ciphered, their entropy should ideally equal to 8. If the output of such a cipher emits symbols with entropy less than 8, there exists certain degree of predictability, which cancelled its security. A comparison of entropy analysis test is conducted on the four examined encrypted images (Lena, Baboon, Plane and Man) with the proposed image cryptosystem using Standard, Baker and Cat Maps versus Guan et al. [18], Nanrun et al.[43], and Elashry et al. [10] image encryption schemes. The number of repetition for each cipherimage block is reported and the probability of occurrence is estimated. The obtained results are listed in Table 4. From that table, one can notice that, for all examined images, the entropy values obtained from the proposed image cryptosystem using Standard, Baker and Cat Maps, and that of Nanrun et al. [43] image cryptosystem are nearly reaching the theoretically proved entropy value which equals 8. Also, the result demonstrated that the proposed image cryptosystem with standard map gives the best entropy values compared with the other examined image cryptosystems. This ensures that information loss caused by ciphering process is negligible and hence, the proposed image cryptosystem is securely immune with respect to the entropy attack.

6.4 The avalanche effect analysis A strong cipher must be very sensitive to slight small changes in the plainimage such as change of single bit in plainimage [4, 28]. Generally, the attacker may perform small change

Table 4 Entropy analysis comparison of the proposed image cryptosystem with standard, baker and cat maps for lena, baboon, plane, and man images and different encryption schemes Cipher

Proposed standard map

Proposed cat map

Proposed baker map

Guan et al. [18]

Elashry et al. [10]

Nanrun et al. [43]

Lena

7.9993

7.9884

7.9831

7.6470

7.7480

7.8980

Baboon Plane

7.9715 7.8331

7.8871 7.8199

7.8657 7.8151

7.6126 7.6357

7.7117 7.7331

7.8932 7.8199

Man

7.8112

7.7953

7.7932

7.6010

7.7111

7.8605

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such as changing just one pixel of the plainimage, and then keep an eye on the modification of the result. So, he may obtain a significant relationship between the original image and its corresponding encrypted image. Since only one small change in the original image would result in a meaningful alternation for the cipherimage. So, this attack would become very insufficient and practically worthless. As in [10, 38], the number of pixels change rate (NPCR) measure the percentage of changed pixels in original image, and the Unified Average Changing Intensity (UACI) estimate the average difference intensity between the original image and encrypted image. Both of measures are utilized as factors to evaluate the effect of just only single pixel changing in the plainimage on the whole image encrypted using the presented image cryptosystem. To explain how NPCR and UACI are calculated, let C1 and C2 be two ciphered images with the same size whose corresponding plainimages have just single pixel difference. Pixels gray-scale values of the two cipherimages C1 and C2 at grid (i,j) are known as C1(i,j) and C2(i,j), respectively. Define a bipolar array D that have the same size as cipherimages C1 and C2. Then, D(i,j) is calculated from C1(i,j) and C2(i,j), namely, if C1(i,j)=C2(i,j) then D(i,j)=1; otherwise, D(i,j)=0. NPCR and UACI are estimated using the following formulas: X NPCR ¼

i; j

Dði; jÞ

W H

 100%;


X  1 C 1 ði; jÞ−C 2 ði; jÞ UACI ¼  100%; i; j W H 255

ð14Þ

ð15Þ

where W and H are the width and height of C1 or C2. The larger the values of NPCR and UACI are, the better the cipher. The NPCR and UACI estimation is performed to compute the single-pixel change influence with the proposed image cryptosystem using Standard, Baker and Cat Maps, and Guan et al. [18], Nanrun et al. [43], and Elashry et al. [10] image encryption schemes. The test is performed on the four examined digital images; Lena, Baboon, Plane and Man with size 512×512. The results, presented in Table 5, indicate that a small change in the plainimage will cause a meaningful change in the cipherimage using all examined image cryptosystems, and the highest change is achieved using the proposed image cryptosystem with standard map. So, the proposed image cryptosystem is securely resistant against differential attack which guarantees a higher security level.

6.5 The maximum deviation analysis The maximum deviation estimates the encryption strength by how much it expands the difference between the plainimage and the cipherimage images [45]. It is performed with the following steps: 1. For each gray scale value in both plainimage and cipherimage ranging from 0 to 255; calculate the number of pixels, and present the results graphically. 2. Estimate the absolute difference through the two curves and represent it graphically.

99.6 %

90.5 %

91.1 %

90.9 %

Baboon

Plane

Man

28.6 %

25.2 %

27.8 %

26.9 %

92.4 %

89.5 %

89.2 %

89.4 % 25.0 %

27.5 %

26.5 %

27.2 %

UACI

NPCR

NPCR

UACI

Proposed cat map

Proposed standard map

Cipher

Lena

Image name

89.2 %

79.1 %

80.1 %

87.3 %

NPCR

24.9 %

26.9 %

26.4 %

25.6 %

UACI

Proposed baker map

84.4 %

75.1 %

76.3 %

83.8 %

NPCR

22.4 %

24.1 %

22.8 %

19.7 %

UACI

Guan et al. [18]

86.5 %

76.4 %

77.5 %

84.7 %

NPCR

22.8 %

24.9 %

24.8 %

19.8 %

UACI

Elashry et al. [10]

87.9 %

77.2 %

79.4 %

86.9 %

NPCR

23.1 %

25.5 %

26.3 %

20.8 %

UACI

Nanrun et al. [43]

Table 5 NPCR and UACI comparison of the proposed image cryptosystem with standard, baker and cat maps for lena, baboon, plane, and man images and different encryption schemes

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3. Calculate the area under the absolute difference curve, which is the sum of differences (D) and this represents the encryption quality. D is calculated using the following equation: D¼

h0 þ h255 X 254 þ h i¼1 i 2

ð16Þ

where hi is the amplitude of the absolute difference curve at value i. The greater the value of D, the best the cipherimage is deviated from its corresponding plainimage [45]. So, higher value of the measure D indicates good encryption efficiency for the cipher. The efficiency for the proposed image cryptosystem with Standard, Baker and Cat maps is estimated and compared with that of Guan et al. [18], Nanrun et al. [43], and Elashry et al. [10] image encryption schemes using maximum deviation test. The test is conducted on the four studied digital images; Lena, Baboon, Plane and Man. Results obtained in Table 6, illustrate that the maximum deviation value is relatively high for all studied image cryptosystems. It is also noticed that, the proposed image cryptosystem with Standard map gives the highest maximum deviation value which means the best encryption efficiency compared to all of the other examined image cryptosystems.

6.6 The irregular deviation analysis The Irregular Deviation factor depends on how much the divergence caused by cryptosystem on the encrypted image is irregular [11]. It yields an attention to each individual pixel value and the divergence caused at every location of the original image before obtaining the histogram as given in [11]. It does not preserve any information about the location of the pixels. The main steps of this measure can be summarized as: 1. Estimate the ‘D’ matrix which gives the absolute values of the difference between each pixel values before and after ciphering. So, D can be given as: D ¼ jI− J j

ð17Þ

where I is the input plainimage, and J is the cipherimage. 2. Estimate the histogram distribution ‘H’ of the absolute deviation between the plainimage and the cipherimage. So, H=histogram (D).

Table 6 Maximum deviation comparison of the proposed image cryptosystem with standard, baker and cat maps for lena, baboon, plane, and man images and different encryption schemes Image name Cipher Proposed Proposed Proposed Guan et al. [18] Elashry et al. [10] Nanrun et al. [43] standard map cat map baker map Lena

15694

15523

15428

15194

15228

15323

Baboon Plane

14817 17324

14729 17263

14565 17110

14217 16759

14366 16910

14429 17051

Man

16348

16058

15867

15258

15385

15443

Author's personal copy Multimed Tools Appl Table 7 Irregular deviation comparison of the proposed image cryptosystem with standard, baker and cat maps for Lena, baboon, plane, and man images and different encryption schemes Image name

Cipher name Proposed standard Proposed cat map map

Proposed baker map

Guan et al.

Elashry et al.

Nanrun et al.

[18]

[10]

[43]

Lena

17855

17996

18024

19255

19024

18996

Baboon

27135

27564

27986

28426

28280

28164

Plane

31180

31523

32145

32380

32145

31351

Man

32166

32658

32984

32546

32495

33213

3. Take the average value of how many pixels are deviated at every deviation value. This average (DC) value can be estimated as: DC ¼

1 X 255 h i¼0 i 256

ð18Þ

where hi is the amplitude of the absolute difference histogram at the value i. 4. Obtain the difference between this average and the deviation histogram, and estimate the absolute value of the result. ACðiÞ ¼ jH ðiÞ−DC j

ð19Þ

5. Estimate the area under the absolute AC curve, which represents the sum of variations of the deviation histogram from the uniformly distributed histogram. ID ¼

X 255 i¼0

ACðiÞ

ð20Þ

The minimum the ID value indicate the superiority of the cryptosystem [11]. The encryption efficiency for the proposed image cryptosystem with Standard, Baker and Cat maps is estimated and compared with that of Guan et al. [18], Nanrun et al. [43], and Elashry et al. [10] image encryption schemes using irregular deviation test. The test is conducted on the four digital images; Lena, Baboon, Plane and Man images. The obtained results presented in Table 7, demonstrate that, among all the examined image cryptosystems, Table 8 Running time comparison of the proposed image cryptosystem with standard, baker and cat maps for Lena image and different encryption schemes Image size (in pixels)

Image size on disk

Encryption time (sec.) Proposed Standard Map

Proposed cat map

Proposed baker map

Guan et al. [18]

Elashry et al. [10]

Nanrun et al. [43]

512×512

257 KB

0.007

0.046

0.031

0.062

0.058

0.051

1024×1024

1.00 MB

0.015

0.054

0.047

0.069

0.063

0.056

2048×2048 4096×4096

4.00 MB 16.00 MB

0.093 0.500

0.281 1.054

0.171 0.703

0.316 1.251

0.302 1.159

0.289 1.079

8192×8192

64.00 MB

1.906

4.193

2.786

4.792

4.589

4.386

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the proposed image cryptosystem using standard map has minimum irregular deviation and so better encryption efficiency.

6.7 Performance of the proposed image cryptosystem Beside security evaluations for image cryptosystem schemes, the running time consideration is very important factor especially for real time multimedia applications. The running time tests are conducted for estimating the performance of the proposed image cryptosystem. An indexed image for BLena^ is utilized as a plainimage. The specifications for utilized PC in all software programs and tests were 3 GHz Pentium IV with 160 G hard-disk and 512 MB of memory. The computed average time consumed by encryption/decryption on 512×512 grey-scale images is about 0.007 s. Table 8 compares the performance of the proposed image cryptosystem with Standard Map, Cat Map, and Baker Map against that of Guan et al. [18], Nanrun et al. [43], and Elashry et al. [10] image encryption schemes on different sizes grey-scale Lena image. The results ensure that the proposed image cryptosystem using Standard map has minimum running time compared among all of the examined image cryptosystems.

7 Conclusion In this research, a secure chaos image cryptosystem method is reported. The proposed method for image cryptosystem is designed using two confusion and diffusion stages. A different key is used for every round in confusion and diffusion stages. Confusion stage works by permuting plainimage pixels without changing their values using a 2D chaotic map while, diffusion stage works by changing pixel values serially so small modification in any pixel is diffused to its surrounding pixels. Control parameters and keystream are correlated with many factors including the plainimage, the key, number of rounds (n) in the confusion stage, and number of rounds (m) in the confusion-diffusion stage which effectively enhances the confusion and diffusion performance, and enlarge the key space. The relation between the utilized chaotic map (confusion stage), the diffusion function (diffusion stage), and the proposed image cryptosystem’s security is studied in details. The results show that Standard map has the highest security level among the examined 2D chaotic confusion maps because it is more complicated and it has a large key space. In the diffusion stage, the Henon map is used as a diffusion function. A course of experiments is conducted on the proposed image cryptosystem with Standard, Baker and Cat maps, and other three recent image cryptosystems. Results illustrated that the proposed image cryptosystem with Standard map has good cryptographic and perceptual security, fast encryption speed compared to all of the other examined image cryptosystems.

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Author's personal copy Multimed Tools Appl 34. Rhouma R, Belghith S (2008) Cryptanalysis of a new image encryption algorithm based on hyper-chaos. Phys Lett A 372:5973–5978 35. Rhouma R, Belghith S (2008) Cryptanalysis of a spatiotemporal chaotic image/video cryptosystem. Phys Lett A 372:5790–5794 36. Sathishkumar GA, Bhoopathybagan K, Sriraam N, Venkatachalam SP, Vignesh R (2011) A novel image encryption algorithm using two chaotic maps for medical application. Adv Comput Commun Comput Inf Sci 133:290–299, Berlin-Heidelberg: Springer-Verlag 37. Shannon CE (1949) Communication theory of secrecy system. Bell Syst Technol J 28:656–715 38. Sun F, Liu S, Li Z, Lü Z (2008) A novel image encryption algorithm based on spatial chaos map. Chaos, Solitons Fractals 38:631–640 39. Wang K, Pei L, Zou A, Song ZH (2005) On the security of 3D Cat map based symmetric image encryption scheme. Phys Lett A 343(6):432–439 40. Wong KW, Kwok BS, Lawk WS (2007) A fast image encryption scheme based on chaotic standard map. Phys Lett A. doi:10.1016/ j.physleta.2007.12.026 41. Xiang T, Wong KW, Liao X (2007) Selective image encryption using a spatiotemporal chaotic system. Chaos 17:023115-1–12 42. Zhang LH, Liao XF, Wang XB (2005) An image encryption approach based on chaotic maps. Chaos, Solitons Fractals 24:759–65 43. Zhou N, Wang Y, Gong L (2011) Novel optical image encryption scheme based on fractional Mellin transform. Opt Commun 284(13):3234–3242 44. Zhou N, Zhang A, Zheng F, Gong L (2014) Novel Image compression-encryption hybrid algorithm based on key-controlled measurement matrix in compressive sensing. Opt Laser Technol 62: 152–160 45. Ziedan IE, Fouad MM, Salem DH (2003) Application of data encryption standard to bitmap and JPEG images. In Proc. 12th National Radio Science Conference (NRSC2003), pp. C16/1–C16 Fatma Elgendy received the B. Sc degree in Electronics engineering, and M.Sc. in Computer Science and Automatic Control from the Faculty of Engineering, Tanta University, in 2005, and 2014, respectively. Her interests are in the areas of: cryptography, image encryption, and chaos theory.

Amany M. Sarhan received the B. Sc degree in Electronics engineering, and M.Sc. in Computer Science and Automatic Control from the Faculty of Engineering, Mansoura University, in 1990, and 1997, respectively. She awarded the PhD degree as a joint research between Tanta Univ., Egypt and Univ. of Connecticut, USA. She is working now as an Associated Prof. at Computers and Automatic Control Dept., Tanta Univ., Egypt. Her interests are in the areas of: Software restructuring, Object-oriented Database, Fragmentation and allocation of databases, Parallel and distributed systems, Garbage collection, wireless security and Computations. She can be reached at [email protected]

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Tarek E. Eltobely received the B. Sc degree in Computer Science and Engineering from Menoufia University, Menouf, Egypt, in 1991, and M.Sc. in Computer Science and Automatic Control from the Faculty of Engineering, Mansoura University, in 1995. He awarded the PhD degree from Department of Intelligent Systems, Graduate School of Information Science and Electrical Engineering, Kyushu University, Japan in 2003. He is working now as an Associated Prof. at Computers and Automatic Control Dept., Tanta Univ., Egypt. His interests are in the areas of: Higher order neural networks, Neural networks modeling and learning, Blind source separation, Brain imaging, Images classification and segmentation, Human-Computer-Interaction applications, and Object tracking and motion capture.

S. F. Elzoghdy received the B.Sc. (Hons.), M.Sc., degrees in Computer Science from Faculty of Science, Menoufia University, Shebi El-kom, Egypt, in 1993, 1997, respectively. He awarded the PhD degree from Operating System & Distributed/Parallel Processing (OSDP) laboratory, Institute of Information Sciences and Electronics (IISE), University of Tsukuba, Tsukuba Science City, Japan in 2004. He is currently Associate Professor with the Department of Computer Science, Faculty of Science, Menoufia University, where he was a Demonstrator from 1994 to 1997 and has been Assistant Lecturer from 1997 to 2004 and since 2004 he has been a Teaching Staff Member with the Department of Computer Science, Faculty of Science, Menoufia University. His current research interests include load balancing in distributed and parallel computer systems, modeling and simulation, grid computing load balancing, security and cryptography, design and analysis of parallel algorithms. Email: [email protected].

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Hala S. El-sayed received the B.Sc.(Hons.), M.Sc., and Ph.D. degrees in Electrical Engineering from Menoufia University, Shebin El-kom, Egypt, in 2000, 2004, and 2010, respectively. She is currently Assistant Professor with the Department of Electrical Engineering, Faculty of Engineering, Menoufia University, where she was a Demonstrator from 2002 to 2004 and has been Assistant Lecturer from 2004 to 2010 and since 2010 she has been a Teaching Staff Member with the Department of Electrical Engineering, Faculty of Engineering, Menoufia University. Her research interests cover database Security, network security, data hiding, image encryption, signal processing, wireless sensor network, robotics, secure building automation systems, and biometrics. Email: [email protected], Mobile: 20201003184654.

Osama S. Faragallah received the B.Sc. (Hons.), M.Sc., and Ph.D. degrees in Computer Science and Engineering from Menoufia University, Menouf, Egypt, in 1997, 2002, and 2007, respectively. He is currently Associate Professor with the Department of Computer Science and Engineering, Faculty of Electronic Engineering, Menoufia University, where he was a Demonstrator from 1997 to 2002 and has been Assistant Lecturer from 2002 to 2007 and since 2007 he has been a Teaching Staff Member with the Department of Computer Science and Engineering, Faculty of Electronic Engineering, Menoufia University. He is a coauthor of about 100 papers in international journals and conference proceedings, and two textbooks. His current research interests include network security, cryptography, internet security, multimedia security, image encryption, watermarking, steganography, data hiding, medical image processing, and chaos theory. Email: [email protected], Mobile: 20201000456572.