International Journal of Research in Computer Science eISSN 2249-8265 Volume 3 Issue 1 (2013) pp. 27- 34 www.ijorcs.org, A...
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Design a New Image Encryption using Fuzzy Integral Permutation with Coupled Chaotic Maps                                  ...
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Design a New Image Encryption using Fuzzy Integral Permutation with Coupled Chaotic Maps                                  ...
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Design a New Image Encryption using Fuzzy Integral Permutation with Coupled Chaotic Maps                               33 ...
34                                                                                                            Yasaman Hash...
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Design a New Image Encryption using Fuzzy Integral Permutation with Coupled Chaotic Maps

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This article introduces a novel image encryption algorithm based on DNA addition combining and coupled two-dimensional piecewise nonlinear chaotic map. This algorithm consists of two parts. In the first part of the algorithm, a DNA sequence matrix is obtained by encoding each color component, and is divided into some equal blocks and then the generated sequence of Sugeno integral fuzzy and the DNA sequence addition operation is used to add these blocks. Next, the DNA sequence matrix from the previous step is decoded and the complement operation to the result of the added matrix is performed by using Sugeno fuzzy integral. In the second part of the algorithm, the three modified color components are encrypted in a coupling fashion in such a way to strengthen the cryptosystem security. It is observed that the histogram, the correlation and avalanche criterion, can satisfy security and performance requirements (Avalanche criterion > 0.49916283). The experimental results obtained for the CVG-UGR image databases reveal the fact that the proposed algorithm is suitable for practical use to protect the security of digital image information over the Internet.

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Design a New Image Encryption using Fuzzy Integral Permutation with Coupled Chaotic Maps

  1. 1. International Journal of Research in Computer Science eISSN 2249-8265 Volume 3 Issue 1 (2013) pp. 27- 34 www.ijorcs.org, A Unit of White Globe Publications doi: 10.7815/ijorcs. 31.2013.058 DESIGN A NEW IMAGE ENCRYPTION USING FUZZY INTEGRAL PERMUTATION WITH COUPLED CHAOTIC MAPS Yasaman Hashemi Department of Computer Engineering, Islamic Azad University Varamin-Pishva Branch, IRAN Email: yasaman.hashemi@iauvaramin.ac.irAbstract: This article introduces a novel image The fundamental ideas for image encryption can beencryption algorithm based on DNA addition classified into three major types for spatial domain: thecombining and coupled two-dimensional piecewise value transformation, the position substitution, andnonlinear chaotic map. This algorithm consists of two their combining form. There already exist severalparts. In the first part of the algorithm, a DNA image encryption methods in spatial domain, amongsequence matrix is obtained by encoding each color which 2D Cellular Automata(CA)-based methods [5,component, and is divided into some equal blocks and 6], chaos-based methods[7, 8], the tree structure-basedthen the generated sequence of Sugeno integral fuzzy methods[9], are most popular.and the DNA sequence addition operation is used to Cellular automata [5, 6], is applied into encryptingadd these blocks. Next, the DNA sequence matrix from images due to having several strengths. First, largethe previous step is decoded and the complement number of CA evolution rules has made manyoperation to the result of the added matrix is techniques available for producing a sequence of CAperformed by using Sugeno fuzzy integral. In the data encryption and decrypting images. Second, localsecond part of the algorithm, the three modified color interactions have made CA suitable for many physicalcomponents are encrypted in a coupling fashion in streams such as those in image processing and datasuch a way to strengthen the cryptosystem security. It encryption. Third, recursive substitution in CA isis observed that the histogram, the correlation and computationally simple by the sole use of integeravalanche criterion, can satisfy security and arithmetic and/or logic operations. An imageperformance requirements (Avalanche criterion > encryption algorithm is recently proposed in [6] using0.49916283). The experimental results obtained for the recursive cellular automata substitution. The CACVG-UGR image databases reveal the fact that the structure in the proposed algorithms allows an infiniteproposed algorithm is suitable for practical use to number of blocks to be linked together in a chain.protect the security of digital image information over There are two potential disadvantages associated withthe Internet. using this chain. First, the entire chain could beKeywords: Fuzzy integral, Chaotic Maps, Image garbled by an error anywhere during the transmission,Encryption. and second, the system security could be weakened by excessive link dilution. I. INTRODUCTION Chaos-based cryptographic scheme [10] is an With the rapid development of the networked efficient encryption first presented in 1989. It hasmultimedia, communication and propagation many brilliant characteristics different from othertechniques, transmission of a wide range of digital algorithms such as the sensitive dependence on initialdata, from digital images to audio and video files, over conditions, non-periodicity, non-convergence andthe internet or through wireless networks has been control parameters [11, 12]. The one-dimensionalincreased. One of the major issues considering the chaos system has the advantages of simplicity and highdigital transmission in this virtual environment is the security [13, 14], and many studies [7, 15-17] weresecurity of digital images, audio, and video files. proposed to adopt and improve it. Some of them useHence, a growing number of researches have been high-dimensional dynamical systems [14], othercarried out to prevent illegal access to legal data. studies use couple maps [18-20]; however, all researchEncryption is usually one good way to ensure high is mainly to increase parameters to increase thesecurity. Due to some intrinsic features of image such security. As we know, an ideal image encryptionas bulk data capacity and high correlation among scheme should be sensitive to the secret key and keypixels, encryption of images is different from that of space should be large enough to make the brute-forcetexts. Various digital methods have been proposed for attack infeasible [21].image encryption [1-4]. www.ijorcs.org
  2. 2. 28 Yasaman Hashemi Furthermore, it must have some ability to resist This condition reads as g(X) = 1 , hence, the value of λouter attack from statistical analysis. However, too can be found by solving the following:many parameters will influence the implementation n ∏ (1 + λg ) (2)and will increase computation; for example, the λ +1 = iproposed scheme in [13] uses eight parameters and i =1combines two chaotic maps to shuffle the image pixels. The speed and efficiency will also be low using more Where λ ∈ ( −1, +∞), λ ≠ 0 , and g i = g({x i })than one encryption scheme to an image. is the value of the fuzzy density function. Eq. (2) can In this paper, we have proposed a novel color image be calculated by solving the (n-1)th degree polynomialencryption algorithm based on DNA sequence addition and finding the unique root greater than -1.combining and coupled two-dimensional piecewisenonlinear chaotic map. The architecture of the Let A i = {x1 , x 2 ,....., x i } . The values of g(A i ) byalgorithm consists of two stages: The permutation the fuzzy measure over the corresponding subsets ofstage uses the generated keystream of Sugeno Fuzzy elements can be determined recursively as follows:Integral (SFI) to change the position and values ofDNA strings. In the diffusion stage, the three = g({x1}) g1 (3) g(A1 ) =components (Red, Green and Blue) of modified imageare encrypted in a coupling fashion in such a way to g(A i ) g i + g(A i −1 ) + λg i g(A i −1 ) 1 < i ≤ n (4) =strengthen the cryptosystem security. Therefore, in order to obtain λ -fuzzy measure, The rest of this paper is organized as follows. there are n parameters g1 , g 2 , g 3 , g 4 needed to beSection II describes the fuzzy measure and fuzzy determined in advance.integral. In Section III, theory of the DNA sequenceoperation is explained. The proposed coupled two B. Sugeno Fuzzy Integraldimensional piecewise nonlinear chaotic map is Sugeno fuzzy integral of function h computed overdiscussed in Section IV. In Section V, the proposed X with respect to a fuzzy measure g is defined in theencryption and decryption algorithm is introduced.Simulation results and security analysis are provided in form:Section VI. Finally, the conclusions are drawn in nSection VII. E g (h) = max[min(h(yi ), g(A i ))] (5) i =1 II. FUZZY MEASURE AND FUZZY INTEGRAL Where h(x1 ) ≤ h(x 2 ) ≤ ... ≤ h(x n ) , and h(x 0 ) = 0 . From Eqs. (4) and (5), it is obvious that the calculationA. Fuzzy Measure of the Sugeno fuzzy integral with respect to λ-fuzzy = Let X {x1 , x 2 ,….., x n } be a finite set associated measure requires the knowledge of the fuzzy density g and the input value h.with n attributes on information source space, anddonate P(X) as the power set X consisting of all III. DNA SEQUENCE DESCRIPTIONsubsets of X . A set function [22] g : P ( x ) → [ 0,1] iscalled a fuzzy measure if the following conditions are In DNA computing, a DNA string is represented bysatisfied: a sequence of four basic nucleotides and is usually described by letters A(Adenine), T(Thymine),1. Boundary conditions: = 0, g(X) 1; g(φ) = G(Guanine), C(Cytosine), where the pairs (A,T) and (G,C) are complementary. In the binary, 0 and 1 are2. Monotonicity: g(A) ≤ g(B) , if A ⊂ B and complementary, so 00 and 11 are complementary, 01 A, B ∈ P(X); and 10 are also complementary. In this paper, we use3. Continuity: limi →∞ g(A i ) = g(limi →∞ A i ) , if C, A, T, G to denote 00, 01, 10, 11, respectively. For 8 ∞ bit grey images, each pixel can be expressed a DNA {A i }i =1 is an increasing sequence of sequence whose length is 4. For instance: If the first measurable sets. pixel value of the original image is 220, convert it into a binary stream as [11011100], by using the below Segeno presented the so-called λ fuzzy measure DNA encoding rule to encode the stream, we can get a[22] satisfying the following additional property that DNA sequence GAGC. Whereas using 00, 01, 10, 11for all A, B ⊂ X with A ∩ B = : φ to denote C, A, T, G, respectively, to decode the aboveg(A ∪ B) g(A) + g(B) + λg(A)g(B) λ > −1 (1) = DNA sequence, we can get a binary sequence [11011100]. In general, the value of λ can be determined owingto the boundary condition of the g λ -fuzzy measure. www.ijorcs.org
  3. 3. Design a New Image Encryption using Fuzzy Integral Permutation with Coupled Chaotic Maps 29 In recent years, algebraic operations such as the coupled two dimensional piecewise nonlinear chaotic addition operation, based on the DNA sequence have maps. In order to have good chaotic properties, T and ε been proposed by researchers [4, 23].For example, 11 are chosen as 3 and 0.02 respectively [29, 30]. + 10 = 01 or G + T=A. V. THE PROPOSED CRYPTOSYSTEM IV. COUPLED 2D PIECEWISE NONLINEAR CHAOTIC MAP A. Generation of the initial conditions and parameters The proposed image encryption process utilizes a A. Nearest-Neighboring Coupled-Map Lattices 256 bit-long external secret key (K) in permutation and A general nearest-neighboring spatiotemporal chaos diffusion stages. This key is divided into 8-bit blocks system, also called nearest-neighboring coupled-map (ki) referred to as session keys. The 256-bit external lattices (NCML) [24], can be described by: secret-key (K) is given by: z n +1 ( j)= (1 − ε)f (z n ( j)) + εf (z n ( j + 1)) (6) K = k1 , k 2 , k 3 ,..., k 32 (12) Where n = 1,2,.... , is the time index; j = 1,2,...,T is the Based on the analysis presented in section II, there lattice state index; f is a chaotic map, and ε ∈ (0,1) is are four initial inputs of fuzzy integrals and four membership grades for generating pseudo-random a coupling constant. The periodic boundary condition number by the SFI. z n (j + T) = is imposed into this system. z n (j) For the image of size - W × H , the initial inputs B. Two Dimensional Piecewise Nonlinear Chaotic h1 , h 2 , h 3 , h 4 are derived as follows, respectively: Map i =32 = (((k ((m −1)×8) +1 ⊕ .... ⊕ k ((m −1)×8) +8 ) + ∑ (k i )) mod W) / W hm Two dimensional piecewise nonlinear chaotic map i =1 with invariant measure are defined as (see [25, 26], for (13) the detail): Where the values h1 , h 2 , h 3 , h 4 must be rearranged in 4α x n (1 − x n ) 2 increasing order [31] and then using these values, this Φ1 (x n , α) x n = = +1 (7) 1 + 4(α 2 − 1)x n (1 − x n ) scheme generates four membership grades. These membership grades can be determined in many  4b12 y n (1 − y n ) 1 different ways. Here, we follow the method introduced =  y n +1 x n ∈ [0, ] (a)  1 + 4(b12 − 1)y n (1 − y n ) 2 (8) in [32], namely: Φ 2 (y n , b1 , b 2 ) =  y 4b 2 y n (1 − y n ) 2 1 1 = x ∈ [ ,1] (b)  n +1 1 + 4(b 2 2 − 1)y n (1 − y n ) n 2  gm = m=1, 2, 3,4 (14) 1+ hm The corresponding invariant measure is (a similar In diffusion stage, the initial conditions and the calculation has been presented in [27]): parameters (α,b1 ,b 2 ) of the coupled chaotic map are 1 β1 1 β2 derived as follows:=µ(x, y) × π x(1 − x) (β1 + (1 − β1x)) π y(1 − y) (β2 + (1 − β2 y)) i =32 (9) = (((k (i-1)×4+1 ⊕ .... ⊕ k (i-1)×4+4 ) + ∑ (k i )) / W)mod1 x 0 ( j) i =1 According to [28], these maps are ergodic in [0, 1]. (15) i =32 C. Applying the Coupling Structure to the Two = (((k (i+2)×4+1 ⊕ .... ⊕ k (i+2)×4+4 ) + ∑ (k i )) / W)mod1 y 0 (j) Dimensional Piecewise Nonlinear Chaotic Maps i =1 (16) In the proposed algorithm, we apply the coupling i =32 structure to the two dimensional piecewise nonlinear = (((k (m-1)×4+25 ⊕ ... ⊕ k (m-1)×4+28 ) + ∑ (k i )) / W)mod1 bm= 1,2 chaotic maps as follows: i =1 (17) x n+1 (j) = (1− ε)Φ1 (x n (j), α j ) + εΦ1 (x n (j + 1), α j ) α = (y 0 (1) + x 0 (2) + y 0 (2) + x 0 (3) + y 0 (3) + b1 + b 2 )mod1 (10) (18) y n+1 (j) = (1− ε)Φ2 (y n (j), b1 , b2 ) + εΦ2 (y n (j + 1), b1 , b2 ) j j j j (11) B. Design of the Encryption Algorithm n = 1, 2, 3,....., L; j = 1, 2,..., T According to Fig. 1, the proposed encryption algorithm can be divided into the following steps: Where ε, α, b1 and b 2 are the system parameters; x 0 (j) and y0 (j) are the initial conditions of the www.ijorcs.org
  4. 4. 30 Yasaman Hashemi Step 1. First of all, convert the image of size W × H DNA B (i, j) , i = 1, 2,....., W j = 1, 2,...., H , where into its RGB components, next, transform each component into binary matrix, and then carry the size of blocks is 1× 4 . out DNA encoding for the binary matrix Step 3. Apply external secret key and produce the according to Section III, to obtain encoding initial inputs h1 , h 2 , h 3 , h 4 and the membership matrices DNAR, DNAG and DNAB. The size grades g1 , g 2 , g 3 , g 4 according to section V, then, of the DNAR, DNAG and DNAB is generate sequences X n (i), Yn ( j), M(i,1), N(1, j), (W × (H × 4)) . n = 1, 2,3, 4 . Step 2. Divide each of matrices DNAR, DNAG and DNAB into blocks DNA R (i, j) , DNA G (i, j) , Step 4. Use the new initial data to iterate SFI once using Eqs. (4) and (5): Secret Key Key Generator Fuzzy Integral Permutation Diffusion Recombine Blocks & Original Image Encoding DNA Matrix Blocks Add Blocks Complement Coupled Chaotic Map Decoding DNA Cipher Image Fig. 1. Architecture of permutation–diffusion type of proposed image cryptosystem M(i,1) = E i mod1 (19) Step 10. Sorting X n and Yn in descending order, we= (ARS(Int((E i mod1) × 1014 ), (m − 1) × 4)) mod WX m =1,...,4 (i) get two new sequences X n and Y n . (20) Step 11. Let the location value of sequences X n , Y n Remark: ARS(y, x) performs the x-bit right- be row coordinates and column coordinates of arithmetic-shift operation on the binary sequence y. DNA R (i, j) , DNA G (i, j) , DNA B (i, j) , in other words, it can be expressed as DNA R (xp , yq ) , Step 5. Update h1 , h 2 , h 3 , h 4 as follow: DNA G (xp , yq ) , DNA B (x p , y q ) . hm = 1, 2,3, 4 (21) X m (i) / W, m Step 12. Add DNA R (i, j) and DNA R (xp , yq ) , Remark: The values of h n must be rearranged in DNA G (i, j) and DNA G (xp , yq ) , increasing order[31]. DNA B (i, j) and DNA B (x , yq ) , according to p Step 6. Set i = i + 1 , If i ≤ W , go to step 4, otherwise, the rules in Section III, obtaining the result as Set j=1, and continue the process from step blocks BR, BG and BB, respectively. Step 7. Step 13. Carry out the inverse process of the step 1 for Step 7. Use the new initial data to iterate SFI once the decoding matrices BR, BG and BB, we using Eqs. (4) and (5): will obtain a real value matrices PR, PG and PB. M(1,j) = E j mod1 (22) Step 14. Two sequences M (W×1) N (1×H) are and= (ARS(Int((E j mod1) × 1014 ), (m − 1) × 4)) mod WYm =1,2,3,4 ( j) produced by SFI, Performing the multiply (23) operation for M (W×1) and N (1×H) , we obtain the Step 8. Update h1 , h 2 , h 3 , h 4 as follow: matrix Z whose size is W × H .Use the following threshold function f (x) to get a hm = 1, 2,3, 4 (24) Ym ( j) / W, m binary matrix: Step 9. Set j=j+1 , If j<=H , go to step 7, otherwise, continue the process from step 10. www.ijorcs.org
  5. 5. Design a New Image Encryption using Fuzzy Integral Permutation with Coupled Chaotic Maps 31 0 , 0 < Z(i, j) ≤ 0.5 Step 16. Set i=1 , F= false, Bitxor1= 0, Bitxor2= 0,f (x) =  (25) Bitxor3= 0, C0(1)= 0, C0(2)= 0 and C0(3)= 0.  1 , 0.5 < Z(i, j) ≤ 1 Apply external secret key and generate the If Z(i, j) = 1, PR , PG and PB is complemented, otherwise initial conditions x 0 (1), y0 (1), x 0 (2), y0 (2), x 0 (3), y0 (3)it is unchanged. After the complementing operation, and the parameters α,b1 and b2 according towe get matrices PR, PG, and PB. section IV.Step 15. Matrices PR, PG, and PB are transformed into Step 17. Step 17: Use the new initial conditions to matrices R (W×H)×1 , G (W×H)×1 and B(W×H)×1 , iterate coupled two dimensional piecewise respectively. nonlinear chaotic maps once. (a) 3500 (b) 3000 (c) 3500 3000 2500 3000 2500 2000 2500 2000 2000 1500 1500 1500 1000 1000 1000 500 500 500 0 0 0 0 50 100 150 200 250 0 50 100 150 200 250 0 50 100 150 200 250 (d) 2500 (e) 2500 (f) 2500 2000 2000 2000 1500 1500 1500 1000 1000 1000 500 500 500 0 0 0 0 50 100 150 200 250 0 50 100 150 200 250 0 50 100 150 200 250Figure 2: Histogram of the original image of Lena in the (a) red (b) green (c) blue, components, Histogram of the encrypted image of Lena in the (d) red (e) green (f) blue, components.Step 18. In this step, R i , R i +1 , G i , G i +1 , Bi and Bi+1 are b1 = (x i (2) + yi (2) + b1 )mod 1 new old (33) encrypted and the results are stored in the b new = (x i (3) + yi (3) + b ) mod 1 old (34) 2 2 matrices Ci (1), Ci +1 (1), Ci (2), Ci +1 (2), Ci (3) and Bitxorm =1,2,3 = Cm (r) ⊕ Cm+1 (r) (35) Ci +1 (3) using equations (11) and (12).Step 19. Step 21. Set i = i + 2 . Go to step (17) until the elements inCi (1) = (R i + int(x i (1) × L) + Ci-1 (1) + SBox(Bitxor1 ))mod 256 the R (W×H)×1 , G (W×H)×1 and B(W×H)×1 (26) exhaust.Ci +1 (1) (R i +1 + int(yi (1) × L) + Ci (1) + SBox(Bitxor1 ))mod 256 = Step 22. If value of the variable F=false, it is set equal (27) to true and the operation will start from stepCi (2) = (G i + int(x i (2) × L) + Ci-1 (2) + SBox(Bitxor2 ))mod 256 23, otherwise, the process will be continued (28) from step 24.Ci +1 (2) (G i +1 + int(yi (2) × L) + Ci (2) + SBox(Bitxor2 ))mod 256 = Step 23. Set R ( W × H )×1 , G ( W × H )×1 , B( W × H )×1 equal to (29) reverse of matrices C(1), C(2) and C(3)Ci (3) = (Bi + int(x i (3) × L) + Ci-1 (3) + SBox(Bitxor3 ))mod 256 respectively ,next set C0(1)=0, C0(2)=0 and (30) C0(3)=0, Bitxor1= 0, Bitxor2= 0, Bitxor3= 0, Ci +1 (3) (Bi +1 + int(yi (3) × L) + Ci (3) + SBox(Bitxor3 ))mod 256 = and jump to step 17. (31) Step 24. Transform the elements of the matricesRemark: SBox performs nonlinear transform S-box C(W×H)×1 (1), C(W×H)×1 (2), C(W×H)×1 (3)[33] on variable a. into C W×H (1), C W×H (2), C W×H (3) andStep 20. Update α,b1 , b 2 , Bitxor1 , Bitxor2 , Bitxor3 and b 2 as output them as color cipher image. follow:α new = (x i (1) + yi (1) + α old ) mod 1 (32) www.ijorcs.org
  6. 6. 32 Yasaman Hashemi It is obvious that the generation of the keystreams Remark1: Since the decryption process requires thedepends on the keys of cryptosystem, the length of the same keystreams as for decryption of the color image,plaintext (L), the width of color image (W) and the the same secret key K = k1 , k 2 , k 3 ,..., k 32 should be usedplaintext through all the color components (R, G, and for decryption. Hence, According to section V, it isB). Besides, since coupled two-dimensional piecewise possible to set the same initial conditions for SFI andnonlinear chaotic map has coupling and nonlinear coupled two dimensional piecewise nonlinear chaoticstructure, the stream cipher of color components maps.depend on each other. These features will in turn,strengthen the security of cryptosystem. By employing Remark2: We can rewrite encryption equations to givethis method, we can strongly claim that our proposed the pixels’ values as follows:algorithm does not suffer from the cryptosystem R i (Ci (1) − int(x i (1) × L) − Ci-1 (1) - SBox(Bitxor1))mod 256 =weaknesses. (36)C. Design of the Decryption Algorithm G i (Ci (2) − int(x i (2) × L) − Ci-1 (2) -SBox(Bitxor2))mod 256 = The decryption procedure is similar to that of the (37)encryption process except that some steps are followed Bi (Ci (3) − int(x i (3) × L) − Ci-1 (3) - SBox(Bitxor3))mod 256 =in a reversed order. Therefore, some remarks should be (38)considered in the decryption process as follows: 250 250 (a) 300 (b) (c) 250 200 200 200 150 150 150 100 100 100 50 50 50 0 0 0 0 50 100 150 200 250 300 0 50 100 150 200 250 0 50 100 150 200 250 300 300 300 (d) (e) (f) 250 250 250 200 200 200 150 150 150 100 100 100 50 50 50 0 0 0 0 50 100 150 200 250 300 0 50 100 150 200 250 300 0 50 100 150 200 250 300Figure 3: Correlation analysis of two horizontally adjacent pixels: Frames (a), (b) and (c), respectively, show thedistribution of two horizontally adjacent pixels in the plain image of Lena in the (a) red (b) green (c) blue, components.Frames (d), (e) and (f) , respectively, show the distribution of two horizontally adjacent pixels in the encrypted image ofLena in the (a) red (b) green (c) blue , components ;obtained using the proposed scheme.VI. SECURITY ANALYSIS FOR THE PROPOSED Fig. 3 is the horizontal relevance of adjacent ALGORITHM elements in image before and after encryption. Fig. 3 shows significant reduction in relevance of adjacentA. Histogram elements. Image histogram is a very important feature in C. Avalanche Criterionimage analysis. From Fig. 2, it is obvious that thehistograms of the encrypted image are nearly uniform As we know the change of one bit in the plaintextand significantly different from the histograms of the should result in theoretically 50% difference in theoriginal image. Hence, it does not provide any clue to cipher’s bits. Hence, for proving the so calledemploy any statistical analysis attack on the encrypted sensitivity to plaintext, two plain images are generatedimage. with just one-pixel difference. The bits change rate of the cipher obtained by proposed algorithm isB. Correlation Analysis of Two Adjacent Pixels 49.916283%. Hence, the avalanche criterion of is very We have analyzed the correlation between two close to the ideal value of 50%. The results of the testsvertically adjacent pixels, two horizontally adjacent comparing the avalanche criterion of the proposedpixels, and two diagonally adjacent pixels in an image. algorithm and other encryption algorithms are shown5000 pairs of two adjacent (in vertical, horizontal, and in Table I.diagonal direction) pixels from plain-image andciphered image were randomly selected and thecorrelation coefficients were calculated. www.ijorcs.org
  7. 7. Design a New Image Encryption using Fuzzy Integral Permutation with Coupled Chaotic Maps 33 Table 1: Avalanche Test Conference on Bio-Inspired Computing (BIC-TA 09), pp. 1-5, 2009. doi: 10.1109/BICTA.2009.5338151 AvalancheAlgorithm [5] O. Lafe, "Data compression and encryption using criterion cellular automata transform," Engineering ApplicationsProposed 0.49916283 of Artificial Intelligence, vol 10, issue 6, pp. 581–591,Ref.[19] 0.49820101 1997. doi: 10.1016/S0952-1976(97)00040-7Ref. [20] 0.00000095 [6] Rong-Jian Chen, Jui-Lin Lai, "Image security system using recursive cellular automata substitution," Pattern VII. CONCLUSION Recognition, vol 40, issue 5, pp. 1621-1631, 2007. doi: 10.1016/j.patcog.2006.11.011 A novel algorithm based on DNA additioncombining and coupled two-dimensional piecewise [7] Shiguo Lian, Jinsheng Sun, Zhiquan Wang, "A blocknonlinear chaotic map has been proposed for the cipher based on a suitable use of the chaotic standard map," Chaos, Solitons & Fractals, vol 26, issue 1, pp.permutation–diffusion architecture. The proposed 117-129, 2005. doi: 10.1016/j.chaos.2004.11.096algorithm combines good permutation and diffusionproperties which can be applied to the encryption of [8] H.S. Kwok, Wallace K.S. Tang, "A fast image encryption system based on chaotic maps with finitecolor images. Combination of image pixels in precision representation," Chaos, Solitons & Fractals,permutation stage based on the generated keystreams vol 32, issue 4, pp. 1518-1529, 2007. doi:by Sugeno fuzzy integral is performed. Meanwhile, the 10.1016/j.chaos.2005.11.090.keystreams used in the diffusion stage is extracted [9] H. Cheng, Xiaobo Li, "Partial encryption of compressedfrom a two-dimensional piecewise nonlinear chaotic images and videos," IEEE Transactions on Signalmap. Processive, vol 48, no. 8, pp. 2439-2451, 2000. doi: The generation of the keystreams depends on the 10.1109/78.852023keys of cryptosystem, the length of the plaintext (L), [10] Robert Matthews, "One the derivation of a chaoticthe width of color image and the plaintext through all encryption algorithm," Cryptologia, vol 13, issue 1, pp.the color components (R, G, and B). Besides, since 29-42, 1989. doi: 10.1080/0161-118991863745coupled two-dimensional piecewise nonlinear chaotic [11] M. S. Baptista, "Cryptography with chaos," Physicsmap has coupling and nonlinear structure, the stream Letters A, vol 240, issues 1-2, pp. 50-54, 1998. doi:cipher of color components depend on each other. 10.1016/S0375-9601(98)00086-3These features will in turn, strengthen the security of [12] Kuo-Liang Chung, Lung-Chun Chang, "Largecryptosystem. By employing this method, we can encrypting binary images with higher security," Patternstrongly claim that our proposed algorithm does not Recognition Letters, vol 19, issue 5-6, pp. 461-468,suffer from the cryptosystem weaknesses. 1998. doi: 10.1016/S0167-8655(98)00017-8 [13] Tiegang Gao, Zengqiang Chen, "Image encryption Simulation results demonstrate that satisfactory based on a new total shuffling algorithm," Chaos,performance is achievable in our proposed algorithm. Solitons & Fractals, vol 38, issue 1, pp. 213-220, 2008.Hence, based on the achieved results, we have come to doi: 10.1016/j.chaos.2006.11.009a conclusion that the new image encryption algorithm [14] Fuyan Sun, Shutang Liu, Zhongqin Li, Zongwang Lü,with speed and high security can be useful for the real- "A novel image encryption scheme based on spatialtime secure image and video communication chaos map," Chaos, Solitons & Fractals, vol 38, issue 3,applications. pp. 631-640, 2008. doi: 10.1016/j.chaos.2008.01.028 [15] Linhua Zhang, Xiaofeng Liao, Xuebing Wang, "An VIII. 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