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Image encryption based on new Beta chaotic maps
Optics and Lasers in Engineering 96 (2017) 39–49
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Image encryption based on new Beta chaotic maps ⁎
Rim Zahmoul , Ridha Ejbali, Mourad Zaied
MARK
Research Team in Intelligent Machines, National School of Engineers of Gabes, B.P. W 6072 Gabes, Tunisia
A R T I C L E I N F O
A B S T R A C T
Keywords: Beta function Beta chaotic maps Image encryption
In this paper, we created new chaotic maps based on Beta function. The use of these maps is to generate chaotic sequences. Those sequences were used in the encryption scheme. The proposed process is divided into three stages: Permutation, Diffusion and Substitution. The generation of different pseudo random sequences was carried out to shuffle the position of the image pixels and to confuse the relationship between the encrypted the original image, so that significantly increasing the resistance to attacks. The acquired results of the different types of analysis indicate that the proposed method has high sensitivity and security compared to previous schemes.
1. Introduction 1.1. Research background As a result of the advanced developments in communications and computer technologies, the Internet has become more and more used for the purpose of supporting client and server services. However, one of the major problems with data transmission over the network is the ‘security’. Data security refers to the protection of the information from unauthorized users or attackers. Encryption forms an efficient methodology to make this data secure. A cryptographic algorithm mechanism leads with the combination of a key a word, number, or expression to encrypt the original data. The identical plaintext encrypts to dissimilar cipher text with unlike keys. Due to its fundamental role in diverse applications, the security of images becomes an important topic for most image and data processing researchers. Cryptographic algorithm is the mathematical function used for encrypting and decrypting process, this mechanism leads to encrypt the original data using different combination of a key a word, number, or expression. The encrypted data security is completely reliant on two important aspects; the key confidentiality and the cryptographic algorithm strength. A cryptosystem is designate due to the presence of cryptographic algorithm, along with all the working protocols and all potential keys. Therefore, many encryption techniques have been proposed all over the years [1–10]. They chiefly incorporate optical encryption [1,2], blocks-based [3], permutation and shuffling encryption [4,5], public and secret key encryption [6,7], DNA and genetic encryption [8–10].
⁎
One of the most known encryption methods is the chaotic encryption [11]. Cryptography and chaos have some regular peculiarities, which is debated in consequent segment. Chaotic encryption illustrates the use of chaos hypothesis to accomplish diverse cryptographic tasks. Many researches have used the logistic map [11] in the encryption process because its easier and more effective, but they realize that it has small key space and weak security. Due to its drawbacks [12], they try to create new chaotic maps [5,13] which have more security and better performances. 1.2. Previous research The similarity between chaotic systems and cryptosystems has led to several chaos-based cryptosystem schemes in which researchers used different methods and features to obtain a secure encryption algorithm. In what follows, some of those works were reviewed. Belazi et al. [5]. proposed a novel image encryption method based on permutationsubstitution (SP) network. Wang et al. [14] proposed a color image encryption approach based on chaotic system. Fouda et al. [3] presented a fast chaotic block cipher for image encryption in which they generated a chaotic sequence formed of integer numbers. Wang et al. [15] proposed a novel image encryption algorithm for chaotic block images, using the technique of dynamic random growth. Using chaotic map approach, Rathore et al. [16] presented a proficient image encryption method; they used the logistic map to generate pseudo random sequence which serves to the encryption cryptosystem in which they used Arnold cat map to scramble the positions of image pixels. Then, chaotic map is used to generate pseudorandom key generation. Hua et al. Alkher et al. introduced Securing Images Using Chaotic-based
Corresponding author. E-mail addresses: [email protected] (R. Zahmoul), [email protected] (R. Ejbali), [email protected] (M. Zaied).
http://dx.doi.org/10.1016/j.optlaseng.2017.04.009 Received 21 January 2017; Received in revised form 6 April 2017; Accepted 19 April 2017 0143-8166/ © 2017 Elsevier Ltd. All rights reserved.
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image for substitution [17]. Murillo-Escobar et al. presented an image encryption scheme, based chaos and plain image characteristics, for color images [18]. Chen et al. presented an encryption algorithm using a dynamic diffusion key stream generated from the permutation matrix [19]. Li et al. [20] proposed a new image encryption scheme based on hybrid cellular automata (CA) and depth-conversion integral imaging, which have better performances, trying to satisfy the needs of secure image transmission. Guesmi et al. proposed a color image encryption scheme based on chaos, crossover operator and the Secure Hash Algorithm (SHA-2) and used one-time keys [21]. 1.3. Our contribution To further increase the protection of the image encryption schemes based on chaos, we create new chaotic maps based on Beta function. Those created maps have plenty of advantages; like the large range of bifurcation parameter, the strong chaotic behavior, better pseudo random chaotic sequences, and the high number of parameters. We applied a several combinations of these maps in the proposed encryption process in order to generate chaotic pseudo random sequences. Those sequences were used in the key generation; they were used to shuffle the coordinates of the image pixels and to make a confusing relationship between the encrypted and the original image. Therefore, a high resistance against the attacks was noticed. The obtained results show that our method has the advantages of high security analysis. The rest of the paper is structured as follow: a preliminary of three gives a description of our new created Beta maps. The fourth section presents in a summary manner the proposed encryption algorithm. In section five, we organized the obtained results. In the end, we conclude the proposed work.
Fig. 2. Bifurcation diagram for the tent map.
and
After r exceeds 3.57,The chaotic behavior appears. 2.2. Tent map A Tent map is defined by an iterated function associated to a dynamical system. It has a chaotic behavior and it is represented by the following equation.
⎧ μx , x ≤ ⎪ Tμ (x ) = ⎨ ⎪μ(1 − x ), ⎩
2. Review of the most known chaotic maps 2.1. The logistic map
1 2
(4)
2.3. Piecewise linear chaotic maps (PWLCMs)
(1) The PWLCMs are simple dynamical non-linear systems. Those maps have perfect behavior and high dynamical properties like the invariant distribution, ergodicity, auto-correlation function, mixing property and large positive Lyapunov exponent [24]. The generation of an orbit, which is a real numbers sequence between 0 and 1, was made by the iteration of the PWLCM with control parameters and initial value. It is also called skew tent map due to the similarity of its Eq. (5) with the Tent map's Eq. (4).
The logistic map depends essentially on two parameters (x0 and r), it has a random behavior, seems like an irregular jumble of dots, obtained by changing the value of one or both of these parameters. The main idea to create the logistic map was based on an iterations function, where the previous output xn − 1 value influence the current one xn. Fig. 1 shows the Bifurcation diagram of logistic map. The logistic map parameters x0 (2) and r (3) represent the initial conditions. (2)
x 0 ∈ [0, 1]
1 2
Its diagram is comparable to the logistic map's diagram, but with a corner [23]. It is shown in Fig. 2.
Proposed by Pierre Verhulst in 1845, the logistic map was one of the simplest and the most famous maps. It was defined by the following equation [22].
xn +1 = rxn (1 − xn )
(3)
r ∈ [0, 4]
3. Our new created chaotic maps 3.1. Beta function Inspired from the Beta function, which is often found in mathematical statistics and probability theory, we created our new chaotic maps. According to [25–27], the Beta function is defined as follow
⎧ ⎛ (x − x ) ⎞ p ⎛ (x − x ) ⎞ q 1 2 ⎪⎜ ⎟ ⎜ ⎟ if x ∈ ] x1, x2 [ Beta (x; p , q , x1, x2 ) = ⎨ ⎝ (xc − x1) ⎠ ⎝ (x2 − xc ) ⎠ ⎪ ⎩0 else
(5)
With p, q, x1 and x2 ∈ R , x1 < x2 and xc:
xc =
(px 2 + qx1) (p + q )
(6)
The divers shapes of Beta function were given in Fig. 3. We noticed that this function can take similar shapes to that of a trapezoidal,
Fig. 1. Bifurcation diagram for the logistic map.
40
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Fig. 3. Different shapes of the Beta function.
high number of parameters. Thus, the encryption procedures become more and more efficient and could resist most of the attacks. Figs. 4–7 shows the different Beta maps. The map in Fig. 4 looks like the logistic map, while the Fig. 5 shows an inverted one. The map in Fig. 6 has a large parameter range and in Fig. 7 we obtained a specified chaotic map's shape.
Gaussian and triangular, functions. Beta function is applicable to several engineering applications, such as image compression [25], bio-medical signal compression [26,28], image object detection, tracking and recognition [29], and for many other applications [30,31,37,38]. 3.2. Beta chaotic map The new chaotic maps were inspired from the Beta function. They are a polynomial mapping and they reflect an example of how complex, chaotic behavior can appear from a simple non-linear dynamical equations. Mathematically, a Beta map is written as indicate:
xn +1 = k × Beta (xn ; x1, x2 , p , q )
(7)
where
p = b1 + c1 × a
(8)
and
q = b 2 + c2 × a
(9)
with b1,c1,b2 and c2 are the adequately chosen constants, The k parameter multiplied chaotic map, whose role is to control the amplitude of the Beta map and a denote the bifurcation parameter. Although the Beta map's equation is simple, a variation in some of its parameters produces a new chaotic maps with different shapes. Besides, those maps have a large range of bifurcation parameter, a strong chaotic behavior, better pseudo random chaotic sequences, and a
Fig. 4. a=[−0.8:0.7], x 0 = −1, x1 = 1, k=0.85, b1 = 5, c1 = 1, b 2 = 3 and c2 = −1.
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The key used in this approach was composed of the two chosen maps parameters (x01, x11, x21, a1, b11, c11, b21, c21, k1; x02; x12; x22, a2, b12, c12, b22, c22, k2) and the parameters of the congruential generator (A, T) [21]. The two maps parameters are denoted as fractional parts for simple precision float number with 23 bit length. Thus, the key length is about 512 bits. Therefore, those initial values were used to generate a long chaotic sequence which had the same length as the original image P, so the key was constructed to control the pseudo random sequence from the Beta chaotic maps. 4.1. Algorithm of image encryption The encryption process of the proposed scheme consisted of the following steps: Step 1: Consider an image of size M*N. Resize the chosen image into square dimension. Step 2: Generate two different pseudo random sequences after making several combinations of Beta chaotic maps, Due to the sensitivity of chaotic function to the slight variation of the initial condition, many random sequences could be generated. Step 3: Permutation stage: At this point, the generated sequences of the Beta chaotic maps are used to shuffle the plaintext image's rows and columns. Sorting elements of Qseq and Q1seq whose number is M *N in matrix form and obtain Q and Q1 matrices with M*N size. The permutation process affect the original image pixels by permuting them within columns, using Q1 matrice element's positions. The elements of the resulting matrix are permuted,using Q2 within rows. Step 4: Substitution stage: Divide the resulting matrix into four blocks of equal size. Translate each block to a pseudo random matrix W where each matrix further changed by the functions (10), (11), (12) and (13) given below:
Fig. 5. a=[0:1], x 0 = −1, x1 = 1, k=0.88, b1 = 5, c1 = −1.5, b 2 = 2 and c2=0.5.
fN (d ) = T (d )mod G
(10)
fR (d ) = T⌊( d )⌋ mod G
(11)
fS (d ) = T (d 2 )mod G
(12)
fD (d ) = T (2d )mod G
(13)
Fig. 6. a=[500:600], x 0 = −1, x1 = 1, k=0.9, b1 = 5, c1=0.01, b 2 = 3 and c2 = −0.01.
⎡ fN (B1.1) ⎢ f (B2.1) W = ⎢⎢ R f (B3.1) ⎢ S ⎣ fD (B4.1)
fR (B1.2 ) fS (B1.3) fS (B2.2 ) fD (B2.3) fD (B3.2 ) fN (B3.3) fN (B4.2 ) fR (B4.3)
fR (B1.4 ) ⎤ ⎥ fN (B2.4 )⎥ fR (B3.4 ) ⎥ ⎥ fS (B4.4 ) ⎦
(14)
The function T denotes a truncation of a decimal to form an integer for each number of the resulting matrix W. G denotes the image type,for (G = 256) it is an 8-bit gray image, and for (G = 2) it is a binary image. So we got a new random integer matrix I. Then, we obtain a ciphertext image C with the following equation (for encryption): C = (P + I )mod G And the plaintext P in decryption is given by: P = (C − I )mod G Step 5: Diffusion stage: Diffusion reflects the property that the redundancy in the statistics and data of the plaintext is dissipated in the those of the ciphertext. It is achieved by changing each pixel in the main image over the finite field GF (28). Depending on the reality that the diffusion process was applied, the size of the image block differs according to the image type. Any pixel change in the plaintext image leads to a change in the pixel block in each round. To end with,The encryption algorithm ameliorates the security of images greatly. The decryption system of the algorithm can be
Fig. 7. a=[−0.4:0.7], x 0 = −0.7 , x1 = 1, k=0.93, b1 = 8 , c1 = 1, b 2 = 3 and c2 = −1.
4. The proposed encryption algorithm The proposed image cryptosystem scheme is composed of three parts. Since our new created maps have a wide intervals of parameters values for which the chaos is fulfilled, we randomly select two Beta maps with different initial values and parameters to constitute the key, which is used to implement the encryption architecture. Firstly, we introduced our Key Schedule: 42
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Fig. 8. Respectively (from left to right): original images of ‘Lena’, ‘Cameraman’, ‘Boat’, ‘Clock’, and their histograms plain images of ‘Lena’, ‘Cameraman’, ‘Boat’, ‘Clock’, and their histograms.
Fig. 9. Respectively (from left to right): ciphered images of ‘Lena’, ‘Cameraman’, ‘Boat’, ‘Clock’, and their histograms.
5.1. Histogram analysis
described as the inverse of the encryption system using the same key.
In this part we have made analyze of the histograms of different original and encrypted images. The histogram of the original image contains large spikes and it is much tilted. Histograms of ciphered images are very flat, uniform and incomparably different from those of the plaintext images with no statistical resemblance. Fig. 9 shows encrypted images histograms and Fig. 8 shows those of the plain images. Thus, by comparing the histograms of both original and ciphered images, we conclude that the ciphered images are randomlike.
5. Experimental results Most encryption algorithms are broken by using statistical analysis. Those analyses are used to find a relationship between the encrypted and the original image. As a result, the secret key could be determined. To avoid the statistical attacks, we have made several tests including the number of pixel change rate (NPCR), unified average changing intensity (UACI), correlation analysis, PSNR (Peak Signal to Noise Ratio) and MSE (mean square error) tests and information entropy evaluation. Some of the created Beta maps are not suitable for all the analysis tests. In our work, we use the Beta maps that have a high resistance to all kind of attacks. The tests are preformed using several gray images from USC-SIPI Image Database which were used as plain images. A comparison is done with the recent algorithms mentioned in [5,14,15,33,34]. Results were discussed in the following paragraphs.
5.2. Key sensitivity analysis A major impact on the security of cryptosystem depends on the key sensitivity. Therefore, a slight variation of only one bit within the key should produce a different ciphered images. The proposed algorithm proves the high sensitivity of the encryption key as shown in Fig. 10. It shows also the key sensitivity of the proposed algorithm with respect to encryption and decryption where each K1 and K2 and K2 and K3 are 43
Optics and Lasers in Engineering 96 (2017) 39–49
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Fig. 10. Key sensitivity results. (a) plaintext image P; (b) ciphertext image C1=Enc(P,K1); (c) ciphertext image C2=Enc (P,K2); (d) ciphertext image difference C1 − C 2 ; (e) deciphertext image D1=Dec(C1,K1); (f) deciphertext image D2=Dec(C1,K2); (g) deciphertext image D3=Dec(C1,K3); (h) deciphertext image difference D3 − D2 (K1 and K2 are different only in one bit; K2 and K3 are also different only in one bit; and K1 ≠ K 3).
Fig. 11. Number of pixel change rate (NPCR) in the plain and encrypted images of ‘Elaine’: (a) plaintext image P1; (b) plaintext image P2 different from P1 with only for one bit; (c) plaintext image difference P1 − P 2 (d) ciphertext image C1 of P1; (e) ciphertext image C2 of P2; (f) ciphertext image difference C1 − C 2 .
5.3. Key's space analysis
different only for one bit. These results show the sensitivity of the ciphered images for both encryption and decryption process using our new Beta maps, to the encryption key (Fig. 11). Otherwise stated,the designed cipher, refereeing to [39], has good confusion. properties.
A good encryption algorithm is characterized by its large key space in order to enhance its resistance to different attacks. Therefore, our key which is about 512 bit length is sufficient enough to overcome the brute force attack. It is composed of several parts, and the parameters of the 44
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7.1. NPCR and UACI tests
congruential generator (A, T ). Those parameters are denoted as a fractional parts for simple precision float number with 23 bit length and (A, T ) [32] each have a 49 bit. Therefore, an encryption key used in the proposed method is of 18*23 + (49*2) = 512 bit length. The encryption key space is comparable and even better than the encryption standards mentioned in [40,41]. In consequence, it has high immunity against brute force attacks [42].
The Number of Pixel Change Rate (NPCR) and Unified Average Changing Intensity (UACI) are two metrics used to test the impact of changing one pixel in both the plain and the encrypted images [35]. The NPCR is used to measure the number of distinct pixels between the two images, while the UACI is used to measure the average of intensity. Mathematically, the NPCR and the UACI scores between two encrypted images C1 and C2, whose original images are a little different, can be defined as indicate Eqs. (18)–(20), respectively. Consider two images C1 denotes the original image and C2 is the encrypted image. D denotes the bipolar array with equivalent size as images C1 and C2. Then, D(i,j) is determined by C1 (i , j ) and C2 (i , j ), to be specific, if C1 (i , j )=C2 (i , j ) then D(i,j)=0; otherwise, D(i,j)=1. NPCR is represented as follow:
6. Information entropy analysis Information entropy is one of the quantitative measurement kinds that evinces how random a signal source. Particularly, the information entropy can be used to indicate the randomness of the image as shown in Eq. (15), where X represent the test image, xi symbolize the ith possible value in X, and Pr (xi ) indicates the probability of X = xi , i.e. the probability of picking a random pixel in X and it has xi as value. The max of H(X)value is reached when X is uniformly distributed as shown in Eq. (16), i.e. X has a complete at histogram. Also, G is the number of allowed intensity scales related to the image format.
M
NPCR =
i=1 j=1
∑ Pr(xi ) log2 Pr(xi ) i =1
Pr(X = xi ) = 1/ G
(16)
⎡M N ⎤ C (i, j) − C2 (i, j) ⎥ 100% UACI = ⎢∑ ∑ 1 × ⎢⎣ ⎥⎦ M × N 255 i=1 j=1
rxy =
M ⎛ ∑i =1 ⎜xi − ⎝
⎞⎛ M ∑ j =1 xj ⎟ ⎜yi ⎠⎝
−
1 M
7.2. Mean square error analysis Mean Square Error (MSE) is a parameter used to define the difference between the plain and the ciphered image in which pixels are expressed between 0 and 255. MSE is denoted by:
⎞ M ∑ j =1 yj ⎟
⎞2 M M ⎛ 1 ∑ x ⎟ ∑i =1 ⎜yi − j =1 j M ⎝ ⎠
⎠
1 M
⎞2 M ∑ j =1 yj ⎟ ⎠
(20)
The NPCR value of an 8 bit grayscale image is 99.6094, while its UACI is about 33.4635. Using our approach and other proposed ones, we obtained the NPCR and UACI of a large number of images. The results are mentioned in Tables 4 and 5. First, We made a comparison between the NPCR of our scheme and other previous works in [5,14,15,33,34], we observe an NPCR average of 99.61%, we conclude that our encryption scheme is very sensitive to small pixel changes in the plain-image. Then, the UACI average of our method, which is about 33.50%, is higher than 33%. consequently, the rate of influence related to singlepixel change in plain image is very high. Results shows that the proposed encryption approach and those studied in [5,14] and [15] have better NPCR performance than other algorithms. In addition, the UACI performance of the proposed scheme is in harmony with those of schemes in [5,14,15,33,34]. Results indicate that our algorithm is not vulnerable to the differential attacks.
One of the special characteristics of an original image is the fact that its pixels have a high neighborhood correlations, which means that their adjacent pixels are strongly interconnected. Whereas in an encrypted image, to consider that the used method is secure and efficient, the correlation between pixels and its adjacent ones should not exist. We have made a correlation analysis of neighborhood pixels in the cipher and plain images. Therefore, we randomly selected 2000 pairs of adjacent pixels in each direction from the original image and the encrypted image. Eq. (17) indicates the correlation coefficient between each pair. 1 M
(19)
UACI measures the average intensity of differences between the two images. It is defined by the following equation:
7. Correlation analysis
⎛ ⎜x i − ⎝
(18)
(15)
We applied the information entropy test for many ciphered images. The results were recapped in Table 1 prove that the entropies of the encrypted images are very close to 8. Hence, we can deducted that our proposed method using the new created Beta maps has an entropy average of 7.9977 which is comparable to that studied in [5] and much better than those of the schemes in [14,15,33,34]. Therefore, the proposed encryption scheme is strong against entropy attacks.
M ∑i =1
100% M×N
⎧ 0 if C1 (i , j ) = C2 (i , j ) ⎨ ⎩1 if C1 (i , j ) ≠ C2 (i , j )
n
H (x ) =
N
∑ ∑ D(i, j) ×
MSE =
(17)
1 MN
N
M
∑ ∑ [C (i, j ) − C′(i, j )] i =1 j =1
(21)
where C (i , j ) original image pixel, C′(i , j ) is encrypted image pixel and M and N are the size of the original or encrypted image. In case of image encryption, MSE should be as high as possible. Higher value of MSE between the original and the encrypted image represents more immunity to attacks. Table 6 shows the MSE of encrypted image using 2D logistic map Ref. [36] and encrypted image using Beta map. From the following results. We conclude that the proposed encryption algorithm is better than the one using 2D logistic map.
where xi and yi from ith pair of horizontal, vertical and diagonal adjacent pixels, M denotes the total number of pairs of the adjacent pixels. The correlation coefficients in the three directions of adjacent pixels for two pairs of the original images mentioned above and their related encrypted images are shown in Tables 2 and 3. Tables 2 and 3 indicate that the obtained coefficients of correlation are close to 1, whereas those of the ciphered images are approximately null. The average correlations in the three directions obtained from the proposed algorithm for all tested images are smaller than those measured with algorithms studied in the references [5,14,15,33,34]. Consequently, we obtain an image with de-correlated adjacent pixels generated from the proposed encryption scheme. So the proposed method is secure against statistical attacks.
7.3. Peak signal to noise ratio analysis The Peak Signal to Noise Ratio (PSNR) is the ratio between 45
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Table 1 Entropies of encrypted images for algorithms in [5,14,15,33,34]. Image name
Our approach
Ref. [5]
Ref. [14]
Ref. [15]
Ref. [33]
Ref. [34]
Airport Airplane Barbara Boat Camera man Chemical plant Clock Couple Elaine House Lena Man Moon surface Peppers Tank Test pattern Average
7.9998 7.9972 7.9993 7.9993 7.9972 7.9971 7.9975 7.9993 7.9992 7.9992 7.9991 7.9998 7.9975 7.9992 7.9994 7.9998 7.9977
7.9960 7.9974 7.9978 7.9980 7.9985 7.9964 7.9992 7.9976 7.9985 7.9981 7.9963 7.9975 7.9987 7.9985 7.9969 7.9986 7.9977
7.9966 7.9965 7.9964 7.9979 7.9966 7.9986 7.9974 7.9987 7.9972 7.9989 7.9991 7.9974 7.9979 7.9973 7.9990 7.9988 7.9978
7.9966 7.9926 7.9937 7.9960 7.9955 7.9947 7.9984 7.9951 7.9939 7.9978 7.9951 7.9990 7.9951 7.9965 7.9976 7.9967 7.9959
7.9969 7.9954 7.9957 7.9959 7.9964 7.9990 7.9956 7.9980 7.9971 7.9952 7.9965 7.9965 7.9954 7.9958 7.9965 7.9984 7.9965
7.9957 7.9966 7.9964 7.9985 7.9990 7.9940 7.9977 7.9956 7.9951 7.9958 7.9964 7.9949 7.9969 7.9971 7.9997 7.9977 7.9967
Table 2 Comparison of NPCR and UACI with other algorithms. Image
Direction
Plain image
Encrypted image Our scheme
Ref. [5]
Ref. [14]
Ref. [15]
Ref. [33]
Ref. [34]
Airplane
Horizontal Vertical Diagonal
0.9570 0.9365 0.8926
0.0033 0.0002 −0.0019
−0.0002 −0.0090 −0.0066
0.0011 −0.0035 −0.0029
−0.0164 −0.0181 0.0004
0.0180 0.0061 −0.0079
0.0045 −0.0215 −0.0015
Airport
Horizontal Vertical Diagonal
0.9099 0.9033 0.8590
−0.0021 0.0002 0.0003
−0.0275 −0.0154 −0.0187
0.0040 −0.0174 −0.0135
−0.0061 −0.0079 −0.0001
0.0094 −0.0107 −0.0007
−0.0142 −0.0141 0.0002
Barbara
Horizontal Vertical Diagonal
0.8953 0.9588 0.8830
0.0033 0.0032 0.0025
−0.0033 −0.0269 −0.0121
−0.0052 −0.0067 0.0068
−0.0212 −0.0161 −0.0110
−0.0187 −0.0016 0.0001
0.0037 −0.0202 0.0046
Boat
Horizontal Vertical Diagonal
0.9381 0.9713 0.9221
0.0025 0.0018 0.0005
−0.0100 −0.0124 −0.0185
−0.0054 −0.0009 0.0026
−0.0189 0.0003 −0.0204
−0.0295 −0.0150 −0.0224
−0.0138 −0.0199 −0.0057
Camera man
Horizontal Vertical Diagonal
0.9334 0.9592 0.9086
0.0047 −0.0054 0.0016
−0.0095 −0.0170 −0.0119
−0.0211 −0.0103 0.0054
0.0063 −0.0142 0.0168
−0.0047 −0.0195 0.0279
−0.0009 −0.0223 0.0025
Chemical plant
Horizontal Vertical Diagonal
0.9466 0.8984 0.8529
−0.0030 −0.0019 −0.0063
−0.0134 −0.0005 −0.0033
−0.0073 −0.0073 −0.0115
−0.0069 −0.0100 −0.0078
−0.0091 −0.0029 −0.0092
0.0072 −0.0015 −0.0040
Clock
Horizontal Vertical Diagonal
0.9564 0.9740 0.9389
0.0008 0.0013 −0.009
0.0024 −0.0246 −0.0081
−0.0140 −0.0139 −0.0175
−0.0248 −0.0172 −0.0025
−0.0143 0.0097 0.0120
−0.0123 −0.0041 −0.0027
Couple
Horizontal Vertical Diagonal
0.9370 0.8926 0.8558
0.0012 0.0011 0.0013
−0.0251 −0.0213 −0.0078
−0.0178 −0.0025 0.0001
−0.0122 0.0262 −0.0257
−0.0236 −0.0045 0.0016
−0.0106 −0.0047 0.0262
Elaine
Horizontal Vertical Diagonal
0.9756 0.9730 0.9692
0.0018 0.0005 −0.0053
−0.0232 −0.0420 −0.0030
−0.0065 −0.0096 −0.0148
−0.0191 −0.0130 −0.0096
−0.0066 −0.0019 0.0007
−0.0062 −0.0197 −0.0169
House
Horizontal Vertical Diagonal
0.9484 0.9577 0.9135
0.0004 −0.0012 −0.0006
−0.0095 −0.0259 −0.0094
−0.0126 −0.0097 −0.0123
0.0023 −0.0187 −0.0225
−0.0339 0.0186 −0.0001
0.0109 −0.0173 −0.0002
Lena
Horizontal Vertical Diagonal
0.9719 0.9850 0.9593
−0.0047 0.0015 0.0030
−0.0048 −0.0112 −0.0045
−0.0086 −0.0102 −0.0125
−0.0066 −0.0089 0.0424
0.0011 0.0098 −0.0227
−0.0063 −0.0109 −0.0154
46
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Table 3 Comparison of NPCR and UACI with other algorithms. Image
Direction
Plain image
Encrypted image Our scheme
Ref. [5]
Ref. [14]
Ref. [15]
Ref. [33]
Ref. [34]
Man
Horizontal Vertical Diagonal
0.9774 0.9812 0.9671
−0.0010 0.0029 −0.0005
−0.0155 −0.0276 −0.0157
−0.0190 −0.0095 −0.0141
−0.0083 −0.0180 0.0250
0.0022 −0.0226 0.0060
−0.0100 −0.0027 −0.0195
Moon surface
Horizontal Vertical Diagonal
0.9020 0.9389 0.9037
−0.0061 0.0001 0.0021
−0.0062 −0.0075 −0.0044
−0.0288 −0.0194 −0.0137
−0.0065 0.0063 0.0135
−0.0212 −0.0166 0.0138
−0.0063 0.0142 −0.0091
Peppers
Horizontal Vertical Diagonal
0.9767 0.9792 0.9639
−0.0023 −0.0021 0.0003
−0.0056 −0.0162 −0.0113
−0.0089 −0.0113 0.0045
0.0194 −0.0091 0.0123
0.0071 −0.0065 −0.0165
0.0038 −0.0082 0.0078
Tank
Horizontal Vertical Diagonal
0.9456 0.9320 0.9017
0.0033 0.0002 −0.0019
−0.0202 −0.0200 −0.0013
0.0049 −0.0128 −0.0011
−0.0185 −0.0119 −0.0249
−0.0383 −0.0362 0.0220
−0.0159 −0.0114 0.0126
Test-pattern
Horizontal Vertical Diagonal
0.7592 0.7991 0.6978
−0.0008 −0.0008 −0.0007
−0.0087 0.0002 −0.0220
0.0079 0.0015 −0.0134
−0.0075 0.0284 −0.0250
0.0039 0.0022 −0.0064
0.0060 −0.0010 0.0312
Average
Horizontal Vertical Diagonal
– – –
0.0014 0.0001 −0.0146
−0.0113 −0.0173 −0.0099
−0.0086 −0.0090 −0.0067
−0.0091 −0.0064 −0.0024
−0.0099 −0.0057 −0.0001
−0.0038 −0.0103 0.0006
Table 4 The NPCR of encrypted images for our approach and algorithms in [5,14,15,33,34]. Image name
Our method
Ref. [5]
Ref. [14]
Ref. [15]
Ref. [33]
Ref. [34]
Airport Airplane Barbara Boat Camera man Chemical plant Clock Couple Elaine House Lena Man Moon surface Pepper Tank Test pattern Average
99.5849 99.6459 99.6215 99.6227 99.6124 99.6200 99.5727 99.6185 99.6292 99.6265 99.6253 99.6189 99.6276 99.6040 99.6307 99.6105 99.6105
99.6150 99.6083 99.6092 99.6102 99.6205 99.6121 99.6102 99.6399 99.6143 99.6200 99.6228 99.6070 99.6139 99.6319 99.6290 99.6195 99.6177
99.6107 99.6077 99.6162 99.6281 99.6292 99.6131 99.6218 99.6292 99.6107 99.6257 99.6146 99.6084 99.6168 99.6092 99.6131 99.6177 99.6170
99.5662 99.5565 99.5227 99.5609 99.5749 99.5325 99.5910 99.5606 99.5431 99.5332 99.5511 99.5417 99.5684 99.5808 99.5794 99.5679 99.5582
99.6180 99.6231 99.6402 99.6209 99.6105 99.6258 99.6126 99.6312 99.6299 99.6182 99.6092 99.6211 99.6172 99.6236 99.6687 99.6541 99.6265
99.6201 99.5499 99.5178 99.5743 99.6216 99.6185 99.5728 99.5468 99.5529 99.6078 99.6231 99.5514 99.6033 99.5728 99.6078 99.6185 99.5850
Table 5 The UACI of encrypted images for our approach and algorithms in [5,14,15,33,34]. Image name
Our method
Ref. [5]
Ref. [14]
Ref. [15]
Ref. [33]
Ref. [34]
Airport Airplane Barbara Boat Camera man Chemical plant Clock Couple Elaine House Lena Man Moon surface Pepper Tank Test pattern Average
33.6829 33.4188 33.4691 33.5137 33.6551 33.4291 33.4444 33.3615 33.4799 33.6284 33.4807 33.4815 33.4408 33.4628 33.5212 33.6803 33.5094
33.5625 33.5359 33.7431 33.5367 33.7786 33.6068 33.5820 33.8085 33.5160 33.4914 33.7041 33.7905 33.6898 33.6923 33.9213 33.7515 33.6694
33.6632 33.4143 33.5776 33.6145 33.7050 33.8543 33.4836 33.6446 33.6163 33.7022 33.5561 33.6744 33.5333 33.6284 33.7155 33.5046 33.6180
33.3716 33.4063 33.3890 33.4176 33.3691 33.3872 33.4147 33.3723 33.3853 33.3912 33.3461 33.3684 33.3758 33.3540 33.4045 33.3975 33.3844
33.6183 33.5817 33.6714 33.6018 33.6862 33.3825 33.5793 33.7252 33.6232 33.7240 33.6322 33.4720 33.6993 33.7386 33.5224 33.6340 33.6183
33.5961 33.6640 33.5052 33.3975 33.7326 33.5831 33.9272 33.8911 33.7063 33.7656 33.8144 33.6949 33.8063 33.4723 33.7415 33.4487 33.6717
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References
Table 6 MSE value of encrypted image using our algorithm and Ref. [36]. Image
MSE (our new approach)
MSE (Ref. [36])
Airport Airplane Barbara Boat Cameraman Chemical plant Clock Couple Elaine House Lena Man Moon surface Pepper Tank Test pattern Average
8571.54 10,988.48 9275.24 7532.69 9488.81 7792.67 12,220.64 7083.14 7673.11 8757.26 7694.30 10,288.25 6252.33 8397.10 6235.69 9678.48 9145.94
8465.15 10,990.67 8285.23 7530.09 9376.41 7778.31 12,211.31 6955.55 7588.56 8794.40 7779.56 10,137.09 6229.09 8345.43 6203.26 9604.36 8517.15
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In: Proceedings of the second international conference on artificial intelligence, modelling and simulation; 2014. p. 3–14. http://dx.doi.org/10.1109/AIMS.2014.65〉. [8] Gupta R, Jain A. A new image encryption algorithm based on DNA approach. Int J Comput Appl 2014;85(18):27–31. [ISSN: 0975 - 8887]. [9] Zhang Q, Guo L, Wei X. Image encryption using DNA addition combining with chaotic maps. In: Proceedings of the international conference on bio-inspired computing: theory and applications. Vol. 52, issue no. 11–12; Dec 2010. p. 2028–35. [10] Li X, Wang L, Yan Y, Liu P. An improvement color image encryption algorithm based on DNA operations and real and complex chaotic systems. Opt-Int J Light Electron Opt 2016;127(5):2558–65. [11] Maqableh M, Samsudin AB, Alia MA. New hash function based on chaos theory (CHA-1). IJCSNS Int J Comput Sci Netw Secur 2008;8(2):20–6. [12] Ausloos M. The logistic map and the route to chaos: from the beginnings to modern applications; 2006. XVI.411p. [13] Moysis L, Azar AT. New discrete time 2D chaotic maps. Int J Syst Dyn Appl (IJSDA) 2017;6(1):77–104. [14] Wang X, Teng L, Qin X. A novel colour image encryption algorithm based on chaos. Signal Process 2012;92(4):1101–8. [15] Wang X, Liu L, Zhang Y. A novel chaotic block image encryption algorithm based on dynamic random growth technique. Opt Lasers Eng 2015;66:10–8. [16] Rathore D, Suryavansh A. A proficient image encryption using chaotic map approach. Int J Comput Appl 2016;134(10):20–4. [17] Alkher AA, El-Bakry HM, Fathalla SM. Securing images using chaotic-based image encryption cryptosystem; 2016. [18] Murillo-Escobar MA, Cruz-Hernndez C, Abundiz-Prez F, Lpez-Gutirrez RM, Acosta Del Campo OR. A RGB image encryption algorithm based on total plain image characteristics and chaos. Signal Process 2015;109:119–31. [19] Chen JX, Zhu ZL, Fu C, Yu H, Zhang Y. Reusing the permutation matrix dynamically for efficient image cryptographic algorithm. Signal Process 2015;111:294–307. [20] Li X, Li C, Lee IK. Chaotic image encryption using pseudo-random masks and pixel mapping. Signal Process 2016;125:48–63. [21] Guesmi R, Farah MAB, Kachouri A, Samet M. Hash key-based image encryption using crossover operator and chaos. Multimed Tools Appl 2015:1–17. [22] Alligood KT, Sauer TD, Yorke JA. Chaos an introduction to dynamical systems. first ed. New York: Springer-Verlag; 1996. [23] Zeng X, Pielke RA, Eykholt R. Chaos theory and its application to the atmosphere. Bull Am Meteorol Soc 1993;74(4):631–9. [24] Peterson Gabriel. Arnold's cat map survey. Math 45. Linear algebra, Fall; 1997. p. 1–7. [25] Amar CB, Zaied M, Alimi A. Beta wavelets. Synthesis and application to lossy image compression. Adv Eng Softw 2005;36(7):459–74. [26] Kumar Ranjeet, Kumar A, Pandey Rajesh K. Electrocardiogram signal compression using beta wavelets. J Math Model Algorithms 2012;11:235–48. [27] Zaied Mourad, Ben Amar, C, Alimi MA. Award a new wavelet based beta function. In: Proceedings of the international conference on signal, system and design, SSD03; vol. 1; 2003. [28] Kumar Ranjeet, Kumar A, Pandey Rajesh K. Beta wavelet based ECG signal compression using loss-less encoding with modified thresholding. Comput Electr Eng 2013;39(1):130–40. [29] Zaied M, Jemai O, Ben Amar C. Training of the Beta wavelet networks by the frames theory: application to face recognition. Image Process Theory Tools Appl 2008. [30] Alimi MA. Beta neuro-fuzzy systems. In: Duch W, Rutkowska D, editors. TASK Q J, Spec Issue Neural Netw; 7(1); 2003. p. 23–41. [31] Ejbali R, Zaied M, Ben Amar C. Face recognition based on Beta 2D elastic bunch graph matching. In: Proceedings of the 13th international conference on hybrid intelligent systems, p. 89–93. [32] Menezes A, Oorschot Pv, Vanstone S. Handbook of applied cryptography. FL: CRC Press, Boca Raton; 1997. [33] Hua Z, Zhou Y, Pun CM, Philip Chen CL. 2D sine logistic modulation map for image encryption. Inf Sci 2015;297:80–94. [34] Wang XY, Yang L, Liu R, Kadir A. A chaotic image encryption algorithm based on perceptron model. Nonlinear Dyn 2010;62(3):615–21. [35] Wu Y, Noonan JP, Agaian S. NPCR and UACI randomness tests for image encryption. Cyber J: Multidiscip J Sci Technol J Sel Areas Telecommun (JSAT) 2011:31–8. [36] Wu Y, Yang G, Jin H, Noonan JP. Image encryption using the two dimensional logistic chaotic map. J Electron Imaging 2012;21(1). [013014–1].
Table 7 PSNR value of encrypted image using our algorithm and Ref. [36]. Image
PSNR (our new approach)
PSNR (Ref. [36])
Airport Airplane Barbara Boat Cameraman Chemical plant Clock Couple Elaine House Lena Man Moon surface Pepper Tank Test pattern Average
8.83 7.76 8.49 9.40 8.39 9.25 7.29 9.66 9.32 8.74 9.30 8.04 10.20 8.92 10.22 8.31 8.88
8.89 7.75 8.98 9.40 8.44 9.26 7.30 9.74 9.36 8.72 9.26 8.11 10.22 8.95 10.24 8.34 8.93
corrupting noise and maximum possible power that influence representation of image. PSNR is usually defined as decibel scale. It is mainly used as measure of quality reconstruction of image. It is calculated by the Eq. (22).
⎡ R2 ⎤ PSNR = 10 × log10 ⎢ ⎥ ⎣ MSE ⎦
(22)
where R is the maximum possible pixel value of image. The signal in this case is original data and the noise is the error introduced. High value of PSNR proves a high image quality. Therefore, the difference between two images psnr should be as small as possible to ensure the efficiency of the proposed method. The psnr value between the original image and the encrypted one was listed in Table 7. Our approach indicates a smaller psnr then than the one computed in Ref. [36]. We conclude that the proposed algorithm have a better resistance to statistical attacks. 8. Conclusion In order to improve encryption quality and performances, we have created new chaotic maps “Beta maps” based on Beta function which is a simple mathematic tool. The proposed image encryption approach adopts a permutation-substitution network structure with good confusion and diffusion properties. In such a way, the proposed scheme is able to prevent many existing cryptography attacks and cryptanalysis techniques. 48
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and simulation (HPCS); July 2011. p. 369–75. [39] Shannon CE. Communication theory of secrecy systems. Bell Labs Tech J 1949;20(4):656–715. [40] Stinson DR. Cryptography: theory and practice. Chapman and Hall CRC Press; 2006. [41] FIPS PUB 46. Data encryption standard; 1977. [42] FIPS PUB 197. Advanced encryption standard; 2001.
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49
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Image encryption based on new Beta chaotic maps ⁎
Rim Zahmoul , Ridha Ejbali, Mourad Zaied
MARK
Research Team in Intelligent Machines, National School of Engineers of Gabes, B.P. W 6072 Gabes, Tunisia
A R T I C L E I N F O
A B S T R A C T
Keywords: Beta function Beta chaotic maps Image encryption
In this paper, we created new chaotic maps based on Beta function. The use of these maps is to generate chaotic sequences. Those sequences were used in the encryption scheme. The proposed process is divided into three stages: Permutation, Diffusion and Substitution. The generation of different pseudo random sequences was carried out to shuffle the position of the image pixels and to confuse the relationship between the encrypted the original image, so that significantly increasing the resistance to attacks. The acquired results of the different types of analysis indicate that the proposed method has high sensitivity and security compared to previous schemes.
1. Introduction 1.1. Research background As a result of the advanced developments in communications and computer technologies, the Internet has become more and more used for the purpose of supporting client and server services. However, one of the major problems with data transmission over the network is the ‘security’. Data security refers to the protection of the information from unauthorized users or attackers. Encryption forms an efficient methodology to make this data secure. A cryptographic algorithm mechanism leads with the combination of a key a word, number, or expression to encrypt the original data. The identical plaintext encrypts to dissimilar cipher text with unlike keys. Due to its fundamental role in diverse applications, the security of images becomes an important topic for most image and data processing researchers. Cryptographic algorithm is the mathematical function used for encrypting and decrypting process, this mechanism leads to encrypt the original data using different combination of a key a word, number, or expression. The encrypted data security is completely reliant on two important aspects; the key confidentiality and the cryptographic algorithm strength. A cryptosystem is designate due to the presence of cryptographic algorithm, along with all the working protocols and all potential keys. Therefore, many encryption techniques have been proposed all over the years [1–10]. They chiefly incorporate optical encryption [1,2], blocks-based [3], permutation and shuffling encryption [4,5], public and secret key encryption [6,7], DNA and genetic encryption [8–10].
⁎
One of the most known encryption methods is the chaotic encryption [11]. Cryptography and chaos have some regular peculiarities, which is debated in consequent segment. Chaotic encryption illustrates the use of chaos hypothesis to accomplish diverse cryptographic tasks. Many researches have used the logistic map [11] in the encryption process because its easier and more effective, but they realize that it has small key space and weak security. Due to its drawbacks [12], they try to create new chaotic maps [5,13] which have more security and better performances. 1.2. Previous research The similarity between chaotic systems and cryptosystems has led to several chaos-based cryptosystem schemes in which researchers used different methods and features to obtain a secure encryption algorithm. In what follows, some of those works were reviewed. Belazi et al. [5]. proposed a novel image encryption method based on permutationsubstitution (SP) network. Wang et al. [14] proposed a color image encryption approach based on chaotic system. Fouda et al. [3] presented a fast chaotic block cipher for image encryption in which they generated a chaotic sequence formed of integer numbers. Wang et al. [15] proposed a novel image encryption algorithm for chaotic block images, using the technique of dynamic random growth. Using chaotic map approach, Rathore et al. [16] presented a proficient image encryption method; they used the logistic map to generate pseudo random sequence which serves to the encryption cryptosystem in which they used Arnold cat map to scramble the positions of image pixels. Then, chaotic map is used to generate pseudorandom key generation. Hua et al. Alkher et al. introduced Securing Images Using Chaotic-based
Corresponding author. E-mail addresses: [email protected] (R. Zahmoul), [email protected] (R. Ejbali), [email protected] (M. Zaied).
http://dx.doi.org/10.1016/j.optlaseng.2017.04.009 Received 21 January 2017; Received in revised form 6 April 2017; Accepted 19 April 2017 0143-8166/ © 2017 Elsevier Ltd. All rights reserved.
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image for substitution [17]. Murillo-Escobar et al. presented an image encryption scheme, based chaos and plain image characteristics, for color images [18]. Chen et al. presented an encryption algorithm using a dynamic diffusion key stream generated from the permutation matrix [19]. Li et al. [20] proposed a new image encryption scheme based on hybrid cellular automata (CA) and depth-conversion integral imaging, which have better performances, trying to satisfy the needs of secure image transmission. Guesmi et al. proposed a color image encryption scheme based on chaos, crossover operator and the Secure Hash Algorithm (SHA-2) and used one-time keys [21]. 1.3. Our contribution To further increase the protection of the image encryption schemes based on chaos, we create new chaotic maps based on Beta function. Those created maps have plenty of advantages; like the large range of bifurcation parameter, the strong chaotic behavior, better pseudo random chaotic sequences, and the high number of parameters. We applied a several combinations of these maps in the proposed encryption process in order to generate chaotic pseudo random sequences. Those sequences were used in the key generation; they were used to shuffle the coordinates of the image pixels and to make a confusing relationship between the encrypted and the original image. Therefore, a high resistance against the attacks was noticed. The obtained results show that our method has the advantages of high security analysis. The rest of the paper is structured as follow: a preliminary of three gives a description of our new created Beta maps. The fourth section presents in a summary manner the proposed encryption algorithm. In section five, we organized the obtained results. In the end, we conclude the proposed work.
Fig. 2. Bifurcation diagram for the tent map.
and
After r exceeds 3.57,The chaotic behavior appears. 2.2. Tent map A Tent map is defined by an iterated function associated to a dynamical system. It has a chaotic behavior and it is represented by the following equation.
⎧ μx , x ≤ ⎪ Tμ (x ) = ⎨ ⎪μ(1 − x ), ⎩
2. Review of the most known chaotic maps 2.1. The logistic map
1 2
(4)
2.3. Piecewise linear chaotic maps (PWLCMs)
(1) The PWLCMs are simple dynamical non-linear systems. Those maps have perfect behavior and high dynamical properties like the invariant distribution, ergodicity, auto-correlation function, mixing property and large positive Lyapunov exponent [24]. The generation of an orbit, which is a real numbers sequence between 0 and 1, was made by the iteration of the PWLCM with control parameters and initial value. It is also called skew tent map due to the similarity of its Eq. (5) with the Tent map's Eq. (4).
The logistic map depends essentially on two parameters (x0 and r), it has a random behavior, seems like an irregular jumble of dots, obtained by changing the value of one or both of these parameters. The main idea to create the logistic map was based on an iterations function, where the previous output xn − 1 value influence the current one xn. Fig. 1 shows the Bifurcation diagram of logistic map. The logistic map parameters x0 (2) and r (3) represent the initial conditions. (2)
x 0 ∈ [0, 1]
1 2
Its diagram is comparable to the logistic map's diagram, but with a corner [23]. It is shown in Fig. 2.
Proposed by Pierre Verhulst in 1845, the logistic map was one of the simplest and the most famous maps. It was defined by the following equation [22].
xn +1 = rxn (1 − xn )
(3)
r ∈ [0, 4]
3. Our new created chaotic maps 3.1. Beta function Inspired from the Beta function, which is often found in mathematical statistics and probability theory, we created our new chaotic maps. According to [25–27], the Beta function is defined as follow
⎧ ⎛ (x − x ) ⎞ p ⎛ (x − x ) ⎞ q 1 2 ⎪⎜ ⎟ ⎜ ⎟ if x ∈ ] x1, x2 [ Beta (x; p , q , x1, x2 ) = ⎨ ⎝ (xc − x1) ⎠ ⎝ (x2 − xc ) ⎠ ⎪ ⎩0 else
(5)
With p, q, x1 and x2 ∈ R , x1 < x2 and xc:
xc =
(px 2 + qx1) (p + q )
(6)
The divers shapes of Beta function were given in Fig. 3. We noticed that this function can take similar shapes to that of a trapezoidal,
Fig. 1. Bifurcation diagram for the logistic map.
40
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Fig. 3. Different shapes of the Beta function.
high number of parameters. Thus, the encryption procedures become more and more efficient and could resist most of the attacks. Figs. 4–7 shows the different Beta maps. The map in Fig. 4 looks like the logistic map, while the Fig. 5 shows an inverted one. The map in Fig. 6 has a large parameter range and in Fig. 7 we obtained a specified chaotic map's shape.
Gaussian and triangular, functions. Beta function is applicable to several engineering applications, such as image compression [25], bio-medical signal compression [26,28], image object detection, tracking and recognition [29], and for many other applications [30,31,37,38]. 3.2. Beta chaotic map The new chaotic maps were inspired from the Beta function. They are a polynomial mapping and they reflect an example of how complex, chaotic behavior can appear from a simple non-linear dynamical equations. Mathematically, a Beta map is written as indicate:
xn +1 = k × Beta (xn ; x1, x2 , p , q )
(7)
where
p = b1 + c1 × a
(8)
and
q = b 2 + c2 × a
(9)
with b1,c1,b2 and c2 are the adequately chosen constants, The k parameter multiplied chaotic map, whose role is to control the amplitude of the Beta map and a denote the bifurcation parameter. Although the Beta map's equation is simple, a variation in some of its parameters produces a new chaotic maps with different shapes. Besides, those maps have a large range of bifurcation parameter, a strong chaotic behavior, better pseudo random chaotic sequences, and a
Fig. 4. a=[−0.8:0.7], x 0 = −1, x1 = 1, k=0.85, b1 = 5, c1 = 1, b 2 = 3 and c2 = −1.
41
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The key used in this approach was composed of the two chosen maps parameters (x01, x11, x21, a1, b11, c11, b21, c21, k1; x02; x12; x22, a2, b12, c12, b22, c22, k2) and the parameters of the congruential generator (A, T) [21]. The two maps parameters are denoted as fractional parts for simple precision float number with 23 bit length. Thus, the key length is about 512 bits. Therefore, those initial values were used to generate a long chaotic sequence which had the same length as the original image P, so the key was constructed to control the pseudo random sequence from the Beta chaotic maps. 4.1. Algorithm of image encryption The encryption process of the proposed scheme consisted of the following steps: Step 1: Consider an image of size M*N. Resize the chosen image into square dimension. Step 2: Generate two different pseudo random sequences after making several combinations of Beta chaotic maps, Due to the sensitivity of chaotic function to the slight variation of the initial condition, many random sequences could be generated. Step 3: Permutation stage: At this point, the generated sequences of the Beta chaotic maps are used to shuffle the plaintext image's rows and columns. Sorting elements of Qseq and Q1seq whose number is M *N in matrix form and obtain Q and Q1 matrices with M*N size. The permutation process affect the original image pixels by permuting them within columns, using Q1 matrice element's positions. The elements of the resulting matrix are permuted,using Q2 within rows. Step 4: Substitution stage: Divide the resulting matrix into four blocks of equal size. Translate each block to a pseudo random matrix W where each matrix further changed by the functions (10), (11), (12) and (13) given below:
Fig. 5. a=[0:1], x 0 = −1, x1 = 1, k=0.88, b1 = 5, c1 = −1.5, b 2 = 2 and c2=0.5.
fN (d ) = T (d )mod G
(10)
fR (d ) = T⌊( d )⌋ mod G
(11)
fS (d ) = T (d 2 )mod G
(12)
fD (d ) = T (2d )mod G
(13)
Fig. 6. a=[500:600], x 0 = −1, x1 = 1, k=0.9, b1 = 5, c1=0.01, b 2 = 3 and c2 = −0.01.
⎡ fN (B1.1) ⎢ f (B2.1) W = ⎢⎢ R f (B3.1) ⎢ S ⎣ fD (B4.1)
fR (B1.2 ) fS (B1.3) fS (B2.2 ) fD (B2.3) fD (B3.2 ) fN (B3.3) fN (B4.2 ) fR (B4.3)
fR (B1.4 ) ⎤ ⎥ fN (B2.4 )⎥ fR (B3.4 ) ⎥ ⎥ fS (B4.4 ) ⎦
(14)
The function T denotes a truncation of a decimal to form an integer for each number of the resulting matrix W. G denotes the image type,for (G = 256) it is an 8-bit gray image, and for (G = 2) it is a binary image. So we got a new random integer matrix I. Then, we obtain a ciphertext image C with the following equation (for encryption): C = (P + I )mod G And the plaintext P in decryption is given by: P = (C − I )mod G Step 5: Diffusion stage: Diffusion reflects the property that the redundancy in the statistics and data of the plaintext is dissipated in the those of the ciphertext. It is achieved by changing each pixel in the main image over the finite field GF (28). Depending on the reality that the diffusion process was applied, the size of the image block differs according to the image type. Any pixel change in the plaintext image leads to a change in the pixel block in each round. To end with,The encryption algorithm ameliorates the security of images greatly. The decryption system of the algorithm can be
Fig. 7. a=[−0.4:0.7], x 0 = −0.7 , x1 = 1, k=0.93, b1 = 8 , c1 = 1, b 2 = 3 and c2 = −1.
4. The proposed encryption algorithm The proposed image cryptosystem scheme is composed of three parts. Since our new created maps have a wide intervals of parameters values for which the chaos is fulfilled, we randomly select two Beta maps with different initial values and parameters to constitute the key, which is used to implement the encryption architecture. Firstly, we introduced our Key Schedule: 42
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Fig. 8. Respectively (from left to right): original images of ‘Lena’, ‘Cameraman’, ‘Boat’, ‘Clock’, and their histograms plain images of ‘Lena’, ‘Cameraman’, ‘Boat’, ‘Clock’, and their histograms.
Fig. 9. Respectively (from left to right): ciphered images of ‘Lena’, ‘Cameraman’, ‘Boat’, ‘Clock’, and their histograms.
5.1. Histogram analysis
described as the inverse of the encryption system using the same key.
In this part we have made analyze of the histograms of different original and encrypted images. The histogram of the original image contains large spikes and it is much tilted. Histograms of ciphered images are very flat, uniform and incomparably different from those of the plaintext images with no statistical resemblance. Fig. 9 shows encrypted images histograms and Fig. 8 shows those of the plain images. Thus, by comparing the histograms of both original and ciphered images, we conclude that the ciphered images are randomlike.
5. Experimental results Most encryption algorithms are broken by using statistical analysis. Those analyses are used to find a relationship between the encrypted and the original image. As a result, the secret key could be determined. To avoid the statistical attacks, we have made several tests including the number of pixel change rate (NPCR), unified average changing intensity (UACI), correlation analysis, PSNR (Peak Signal to Noise Ratio) and MSE (mean square error) tests and information entropy evaluation. Some of the created Beta maps are not suitable for all the analysis tests. In our work, we use the Beta maps that have a high resistance to all kind of attacks. The tests are preformed using several gray images from USC-SIPI Image Database which were used as plain images. A comparison is done with the recent algorithms mentioned in [5,14,15,33,34]. Results were discussed in the following paragraphs.
5.2. Key sensitivity analysis A major impact on the security of cryptosystem depends on the key sensitivity. Therefore, a slight variation of only one bit within the key should produce a different ciphered images. The proposed algorithm proves the high sensitivity of the encryption key as shown in Fig. 10. It shows also the key sensitivity of the proposed algorithm with respect to encryption and decryption where each K1 and K2 and K2 and K3 are 43
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Fig. 10. Key sensitivity results. (a) plaintext image P; (b) ciphertext image C1=Enc(P,K1); (c) ciphertext image C2=Enc (P,K2); (d) ciphertext image difference C1 − C 2 ; (e) deciphertext image D1=Dec(C1,K1); (f) deciphertext image D2=Dec(C1,K2); (g) deciphertext image D3=Dec(C1,K3); (h) deciphertext image difference D3 − D2 (K1 and K2 are different only in one bit; K2 and K3 are also different only in one bit; and K1 ≠ K 3).
Fig. 11. Number of pixel change rate (NPCR) in the plain and encrypted images of ‘Elaine’: (a) plaintext image P1; (b) plaintext image P2 different from P1 with only for one bit; (c) plaintext image difference P1 − P 2 (d) ciphertext image C1 of P1; (e) ciphertext image C2 of P2; (f) ciphertext image difference C1 − C 2 .
5.3. Key's space analysis
different only for one bit. These results show the sensitivity of the ciphered images for both encryption and decryption process using our new Beta maps, to the encryption key (Fig. 11). Otherwise stated,the designed cipher, refereeing to [39], has good confusion. properties.
A good encryption algorithm is characterized by its large key space in order to enhance its resistance to different attacks. Therefore, our key which is about 512 bit length is sufficient enough to overcome the brute force attack. It is composed of several parts, and the parameters of the 44
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7.1. NPCR and UACI tests
congruential generator (A, T ). Those parameters are denoted as a fractional parts for simple precision float number with 23 bit length and (A, T ) [32] each have a 49 bit. Therefore, an encryption key used in the proposed method is of 18*23 + (49*2) = 512 bit length. The encryption key space is comparable and even better than the encryption standards mentioned in [40,41]. In consequence, it has high immunity against brute force attacks [42].
The Number of Pixel Change Rate (NPCR) and Unified Average Changing Intensity (UACI) are two metrics used to test the impact of changing one pixel in both the plain and the encrypted images [35]. The NPCR is used to measure the number of distinct pixels between the two images, while the UACI is used to measure the average of intensity. Mathematically, the NPCR and the UACI scores between two encrypted images C1 and C2, whose original images are a little different, can be defined as indicate Eqs. (18)–(20), respectively. Consider two images C1 denotes the original image and C2 is the encrypted image. D denotes the bipolar array with equivalent size as images C1 and C2. Then, D(i,j) is determined by C1 (i , j ) and C2 (i , j ), to be specific, if C1 (i , j )=C2 (i , j ) then D(i,j)=0; otherwise, D(i,j)=1. NPCR is represented as follow:
6. Information entropy analysis Information entropy is one of the quantitative measurement kinds that evinces how random a signal source. Particularly, the information entropy can be used to indicate the randomness of the image as shown in Eq. (15), where X represent the test image, xi symbolize the ith possible value in X, and Pr (xi ) indicates the probability of X = xi , i.e. the probability of picking a random pixel in X and it has xi as value. The max of H(X)value is reached when X is uniformly distributed as shown in Eq. (16), i.e. X has a complete at histogram. Also, G is the number of allowed intensity scales related to the image format.
M
NPCR =
i=1 j=1
∑ Pr(xi ) log2 Pr(xi ) i =1
Pr(X = xi ) = 1/ G
(16)
⎡M N ⎤ C (i, j) − C2 (i, j) ⎥ 100% UACI = ⎢∑ ∑ 1 × ⎢⎣ ⎥⎦ M × N 255 i=1 j=1
rxy =
M ⎛ ∑i =1 ⎜xi − ⎝
⎞⎛ M ∑ j =1 xj ⎟ ⎜yi ⎠⎝
−
1 M
7.2. Mean square error analysis Mean Square Error (MSE) is a parameter used to define the difference between the plain and the ciphered image in which pixels are expressed between 0 and 255. MSE is denoted by:
⎞ M ∑ j =1 yj ⎟
⎞2 M M ⎛ 1 ∑ x ⎟ ∑i =1 ⎜yi − j =1 j M ⎝ ⎠
⎠
1 M
⎞2 M ∑ j =1 yj ⎟ ⎠
(20)
The NPCR value of an 8 bit grayscale image is 99.6094, while its UACI is about 33.4635. Using our approach and other proposed ones, we obtained the NPCR and UACI of a large number of images. The results are mentioned in Tables 4 and 5. First, We made a comparison between the NPCR of our scheme and other previous works in [5,14,15,33,34], we observe an NPCR average of 99.61%, we conclude that our encryption scheme is very sensitive to small pixel changes in the plain-image. Then, the UACI average of our method, which is about 33.50%, is higher than 33%. consequently, the rate of influence related to singlepixel change in plain image is very high. Results shows that the proposed encryption approach and those studied in [5,14] and [15] have better NPCR performance than other algorithms. In addition, the UACI performance of the proposed scheme is in harmony with those of schemes in [5,14,15,33,34]. Results indicate that our algorithm is not vulnerable to the differential attacks.
One of the special characteristics of an original image is the fact that its pixels have a high neighborhood correlations, which means that their adjacent pixels are strongly interconnected. Whereas in an encrypted image, to consider that the used method is secure and efficient, the correlation between pixels and its adjacent ones should not exist. We have made a correlation analysis of neighborhood pixels in the cipher and plain images. Therefore, we randomly selected 2000 pairs of adjacent pixels in each direction from the original image and the encrypted image. Eq. (17) indicates the correlation coefficient between each pair. 1 M
(19)
UACI measures the average intensity of differences between the two images. It is defined by the following equation:
7. Correlation analysis
⎛ ⎜x i − ⎝
(18)
(15)
We applied the information entropy test for many ciphered images. The results were recapped in Table 1 prove that the entropies of the encrypted images are very close to 8. Hence, we can deducted that our proposed method using the new created Beta maps has an entropy average of 7.9977 which is comparable to that studied in [5] and much better than those of the schemes in [14,15,33,34]. Therefore, the proposed encryption scheme is strong against entropy attacks.
M ∑i =1
100% M×N
⎧ 0 if C1 (i , j ) = C2 (i , j ) ⎨ ⎩1 if C1 (i , j ) ≠ C2 (i , j )
n
H (x ) =
N
∑ ∑ D(i, j) ×
MSE =
(17)
1 MN
N
M
∑ ∑ [C (i, j ) − C′(i, j )] i =1 j =1
(21)
where C (i , j ) original image pixel, C′(i , j ) is encrypted image pixel and M and N are the size of the original or encrypted image. In case of image encryption, MSE should be as high as possible. Higher value of MSE between the original and the encrypted image represents more immunity to attacks. Table 6 shows the MSE of encrypted image using 2D logistic map Ref. [36] and encrypted image using Beta map. From the following results. We conclude that the proposed encryption algorithm is better than the one using 2D logistic map.
where xi and yi from ith pair of horizontal, vertical and diagonal adjacent pixels, M denotes the total number of pairs of the adjacent pixels. The correlation coefficients in the three directions of adjacent pixels for two pairs of the original images mentioned above and their related encrypted images are shown in Tables 2 and 3. Tables 2 and 3 indicate that the obtained coefficients of correlation are close to 1, whereas those of the ciphered images are approximately null. The average correlations in the three directions obtained from the proposed algorithm for all tested images are smaller than those measured with algorithms studied in the references [5,14,15,33,34]. Consequently, we obtain an image with de-correlated adjacent pixels generated from the proposed encryption scheme. So the proposed method is secure against statistical attacks.
7.3. Peak signal to noise ratio analysis The Peak Signal to Noise Ratio (PSNR) is the ratio between 45
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Table 1 Entropies of encrypted images for algorithms in [5,14,15,33,34]. Image name
Our approach
Ref. [5]
Ref. [14]
Ref. [15]
Ref. [33]
Ref. [34]
Airport Airplane Barbara Boat Camera man Chemical plant Clock Couple Elaine House Lena Man Moon surface Peppers Tank Test pattern Average
7.9998 7.9972 7.9993 7.9993 7.9972 7.9971 7.9975 7.9993 7.9992 7.9992 7.9991 7.9998 7.9975 7.9992 7.9994 7.9998 7.9977
7.9960 7.9974 7.9978 7.9980 7.9985 7.9964 7.9992 7.9976 7.9985 7.9981 7.9963 7.9975 7.9987 7.9985 7.9969 7.9986 7.9977
7.9966 7.9965 7.9964 7.9979 7.9966 7.9986 7.9974 7.9987 7.9972 7.9989 7.9991 7.9974 7.9979 7.9973 7.9990 7.9988 7.9978
7.9966 7.9926 7.9937 7.9960 7.9955 7.9947 7.9984 7.9951 7.9939 7.9978 7.9951 7.9990 7.9951 7.9965 7.9976 7.9967 7.9959
7.9969 7.9954 7.9957 7.9959 7.9964 7.9990 7.9956 7.9980 7.9971 7.9952 7.9965 7.9965 7.9954 7.9958 7.9965 7.9984 7.9965
7.9957 7.9966 7.9964 7.9985 7.9990 7.9940 7.9977 7.9956 7.9951 7.9958 7.9964 7.9949 7.9969 7.9971 7.9997 7.9977 7.9967
Table 2 Comparison of NPCR and UACI with other algorithms. Image
Direction
Plain image
Encrypted image Our scheme
Ref. [5]
Ref. [14]
Ref. [15]
Ref. [33]
Ref. [34]
Airplane
Horizontal Vertical Diagonal
0.9570 0.9365 0.8926
0.0033 0.0002 −0.0019
−0.0002 −0.0090 −0.0066
0.0011 −0.0035 −0.0029
−0.0164 −0.0181 0.0004
0.0180 0.0061 −0.0079
0.0045 −0.0215 −0.0015
Airport
Horizontal Vertical Diagonal
0.9099 0.9033 0.8590
−0.0021 0.0002 0.0003
−0.0275 −0.0154 −0.0187
0.0040 −0.0174 −0.0135
−0.0061 −0.0079 −0.0001
0.0094 −0.0107 −0.0007
−0.0142 −0.0141 0.0002
Barbara
Horizontal Vertical Diagonal
0.8953 0.9588 0.8830
0.0033 0.0032 0.0025
−0.0033 −0.0269 −0.0121
−0.0052 −0.0067 0.0068
−0.0212 −0.0161 −0.0110
−0.0187 −0.0016 0.0001
0.0037 −0.0202 0.0046
Boat
Horizontal Vertical Diagonal
0.9381 0.9713 0.9221
0.0025 0.0018 0.0005
−0.0100 −0.0124 −0.0185
−0.0054 −0.0009 0.0026
−0.0189 0.0003 −0.0204
−0.0295 −0.0150 −0.0224
−0.0138 −0.0199 −0.0057
Camera man
Horizontal Vertical Diagonal
0.9334 0.9592 0.9086
0.0047 −0.0054 0.0016
−0.0095 −0.0170 −0.0119
−0.0211 −0.0103 0.0054
0.0063 −0.0142 0.0168
−0.0047 −0.0195 0.0279
−0.0009 −0.0223 0.0025
Chemical plant
Horizontal Vertical Diagonal
0.9466 0.8984 0.8529
−0.0030 −0.0019 −0.0063
−0.0134 −0.0005 −0.0033
−0.0073 −0.0073 −0.0115
−0.0069 −0.0100 −0.0078
−0.0091 −0.0029 −0.0092
0.0072 −0.0015 −0.0040
Clock
Horizontal Vertical Diagonal
0.9564 0.9740 0.9389
0.0008 0.0013 −0.009
0.0024 −0.0246 −0.0081
−0.0140 −0.0139 −0.0175
−0.0248 −0.0172 −0.0025
−0.0143 0.0097 0.0120
−0.0123 −0.0041 −0.0027
Couple
Horizontal Vertical Diagonal
0.9370 0.8926 0.8558
0.0012 0.0011 0.0013
−0.0251 −0.0213 −0.0078
−0.0178 −0.0025 0.0001
−0.0122 0.0262 −0.0257
−0.0236 −0.0045 0.0016
−0.0106 −0.0047 0.0262
Elaine
Horizontal Vertical Diagonal
0.9756 0.9730 0.9692
0.0018 0.0005 −0.0053
−0.0232 −0.0420 −0.0030
−0.0065 −0.0096 −0.0148
−0.0191 −0.0130 −0.0096
−0.0066 −0.0019 0.0007
−0.0062 −0.0197 −0.0169
House
Horizontal Vertical Diagonal
0.9484 0.9577 0.9135
0.0004 −0.0012 −0.0006
−0.0095 −0.0259 −0.0094
−0.0126 −0.0097 −0.0123
0.0023 −0.0187 −0.0225
−0.0339 0.0186 −0.0001
0.0109 −0.0173 −0.0002
Lena
Horizontal Vertical Diagonal
0.9719 0.9850 0.9593
−0.0047 0.0015 0.0030
−0.0048 −0.0112 −0.0045
−0.0086 −0.0102 −0.0125
−0.0066 −0.0089 0.0424
0.0011 0.0098 −0.0227
−0.0063 −0.0109 −0.0154
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Table 3 Comparison of NPCR and UACI with other algorithms. Image
Direction
Plain image
Encrypted image Our scheme
Ref. [5]
Ref. [14]
Ref. [15]
Ref. [33]
Ref. [34]
Man
Horizontal Vertical Diagonal
0.9774 0.9812 0.9671
−0.0010 0.0029 −0.0005
−0.0155 −0.0276 −0.0157
−0.0190 −0.0095 −0.0141
−0.0083 −0.0180 0.0250
0.0022 −0.0226 0.0060
−0.0100 −0.0027 −0.0195
Moon surface
Horizontal Vertical Diagonal
0.9020 0.9389 0.9037
−0.0061 0.0001 0.0021
−0.0062 −0.0075 −0.0044
−0.0288 −0.0194 −0.0137
−0.0065 0.0063 0.0135
−0.0212 −0.0166 0.0138
−0.0063 0.0142 −0.0091
Peppers
Horizontal Vertical Diagonal
0.9767 0.9792 0.9639
−0.0023 −0.0021 0.0003
−0.0056 −0.0162 −0.0113
−0.0089 −0.0113 0.0045
0.0194 −0.0091 0.0123
0.0071 −0.0065 −0.0165
0.0038 −0.0082 0.0078
Tank
Horizontal Vertical Diagonal
0.9456 0.9320 0.9017
0.0033 0.0002 −0.0019
−0.0202 −0.0200 −0.0013
0.0049 −0.0128 −0.0011
−0.0185 −0.0119 −0.0249
−0.0383 −0.0362 0.0220
−0.0159 −0.0114 0.0126
Test-pattern
Horizontal Vertical Diagonal
0.7592 0.7991 0.6978
−0.0008 −0.0008 −0.0007
−0.0087 0.0002 −0.0220
0.0079 0.0015 −0.0134
−0.0075 0.0284 −0.0250
0.0039 0.0022 −0.0064
0.0060 −0.0010 0.0312
Average
Horizontal Vertical Diagonal
– – –
0.0014 0.0001 −0.0146
−0.0113 −0.0173 −0.0099
−0.0086 −0.0090 −0.0067
−0.0091 −0.0064 −0.0024
−0.0099 −0.0057 −0.0001
−0.0038 −0.0103 0.0006
Table 4 The NPCR of encrypted images for our approach and algorithms in [5,14,15,33,34]. Image name
Our method
Ref. [5]
Ref. [14]
Ref. [15]
Ref. [33]
Ref. [34]
Airport Airplane Barbara Boat Camera man Chemical plant Clock Couple Elaine House Lena Man Moon surface Pepper Tank Test pattern Average
99.5849 99.6459 99.6215 99.6227 99.6124 99.6200 99.5727 99.6185 99.6292 99.6265 99.6253 99.6189 99.6276 99.6040 99.6307 99.6105 99.6105
99.6150 99.6083 99.6092 99.6102 99.6205 99.6121 99.6102 99.6399 99.6143 99.6200 99.6228 99.6070 99.6139 99.6319 99.6290 99.6195 99.6177
99.6107 99.6077 99.6162 99.6281 99.6292 99.6131 99.6218 99.6292 99.6107 99.6257 99.6146 99.6084 99.6168 99.6092 99.6131 99.6177 99.6170
99.5662 99.5565 99.5227 99.5609 99.5749 99.5325 99.5910 99.5606 99.5431 99.5332 99.5511 99.5417 99.5684 99.5808 99.5794 99.5679 99.5582
99.6180 99.6231 99.6402 99.6209 99.6105 99.6258 99.6126 99.6312 99.6299 99.6182 99.6092 99.6211 99.6172 99.6236 99.6687 99.6541 99.6265
99.6201 99.5499 99.5178 99.5743 99.6216 99.6185 99.5728 99.5468 99.5529 99.6078 99.6231 99.5514 99.6033 99.5728 99.6078 99.6185 99.5850
Table 5 The UACI of encrypted images for our approach and algorithms in [5,14,15,33,34]. Image name
Our method
Ref. [5]
Ref. [14]
Ref. [15]
Ref. [33]
Ref. [34]
Airport Airplane Barbara Boat Camera man Chemical plant Clock Couple Elaine House Lena Man Moon surface Pepper Tank Test pattern Average
33.6829 33.4188 33.4691 33.5137 33.6551 33.4291 33.4444 33.3615 33.4799 33.6284 33.4807 33.4815 33.4408 33.4628 33.5212 33.6803 33.5094
33.5625 33.5359 33.7431 33.5367 33.7786 33.6068 33.5820 33.8085 33.5160 33.4914 33.7041 33.7905 33.6898 33.6923 33.9213 33.7515 33.6694
33.6632 33.4143 33.5776 33.6145 33.7050 33.8543 33.4836 33.6446 33.6163 33.7022 33.5561 33.6744 33.5333 33.6284 33.7155 33.5046 33.6180
33.3716 33.4063 33.3890 33.4176 33.3691 33.3872 33.4147 33.3723 33.3853 33.3912 33.3461 33.3684 33.3758 33.3540 33.4045 33.3975 33.3844
33.6183 33.5817 33.6714 33.6018 33.6862 33.3825 33.5793 33.7252 33.6232 33.7240 33.6322 33.4720 33.6993 33.7386 33.5224 33.6340 33.6183
33.5961 33.6640 33.5052 33.3975 33.7326 33.5831 33.9272 33.8911 33.7063 33.7656 33.8144 33.6949 33.8063 33.4723 33.7415 33.4487 33.6717
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References
Table 6 MSE value of encrypted image using our algorithm and Ref. [36]. Image
MSE (our new approach)
MSE (Ref. [36])
Airport Airplane Barbara Boat Cameraman Chemical plant Clock Couple Elaine House Lena Man Moon surface Pepper Tank Test pattern Average
8571.54 10,988.48 9275.24 7532.69 9488.81 7792.67 12,220.64 7083.14 7673.11 8757.26 7694.30 10,288.25 6252.33 8397.10 6235.69 9678.48 9145.94
8465.15 10,990.67 8285.23 7530.09 9376.41 7778.31 12,211.31 6955.55 7588.56 8794.40 7779.56 10,137.09 6229.09 8345.43 6203.26 9604.36 8517.15
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Table 7 PSNR value of encrypted image using our algorithm and Ref. [36]. Image
PSNR (our new approach)
PSNR (Ref. [36])
Airport Airplane Barbara Boat Cameraman Chemical plant Clock Couple Elaine House Lena Man Moon surface Pepper Tank Test pattern Average
8.83 7.76 8.49 9.40 8.39 9.25 7.29 9.66 9.32 8.74 9.30 8.04 10.20 8.92 10.22 8.31 8.88
8.89 7.75 8.98 9.40 8.44 9.26 7.30 9.74 9.36 8.72 9.26 8.11 10.22 8.95 10.24 8.34 8.93
corrupting noise and maximum possible power that influence representation of image. PSNR is usually defined as decibel scale. It is mainly used as measure of quality reconstruction of image. It is calculated by the Eq. (22).
⎡ R2 ⎤ PSNR = 10 × log10 ⎢ ⎥ ⎣ MSE ⎦
(22)
where R is the maximum possible pixel value of image. The signal in this case is original data and the noise is the error introduced. High value of PSNR proves a high image quality. Therefore, the difference between two images psnr should be as small as possible to ensure the efficiency of the proposed method. The psnr value between the original image and the encrypted one was listed in Table 7. Our approach indicates a smaller psnr then than the one computed in Ref. [36]. We conclude that the proposed algorithm have a better resistance to statistical attacks. 8. Conclusion In order to improve encryption quality and performances, we have created new chaotic maps “Beta maps” based on Beta function which is a simple mathematic tool. The proposed image encryption approach adopts a permutation-substitution network structure with good confusion and diffusion properties. In such a way, the proposed scheme is able to prevent many existing cryptography attacks and cryptanalysis techniques. 48
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