IMAGE ENCRYPTION BASED ON CHAOTIC MAP AND REVERSIBLE INTEGER WAVELET TRANSFORM

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1 Journal of ELECTRICAL ENGINEERING, VOL. 65, NO., 014, IMAGE ENCRYPTION BASED ON CHAOTIC MAP AND REVERSIBLE INTEGER WAVELET TRANSFORM Xiaopeng Wei Bin Wang Qiang Zang Cao Ce In recent years, tere as been growing interesting in image encryption based on caotic maps and wavelet transform. In tis paper, a novel sceme for image encryption based on caotic maps and reversible integer wavelet transform is proposed. Firstly, te ciper key wic is related to plain-image is used to generate different parameters and initial values of caotic maps. Ten te plain-image is permuted by te order from caotic maps, and processed by integer wavelet transform. A part of transform coefficient is diffused by te orbits of caotic maps. Finally, te ciper image is obtained by inverse integer wavelet transform based on te diffused coefficient. Numerical experimental results and comparing wit previous works sow tat te proposed sceme possesses iger security tan previous works, wic is suitable for protecting te image information. Keywords: integer wavelet transform; image encryption; caotic map; caos 1 INTRODUCTION Along wit significant improvements in computer and internet tecnology, it as created an environment in wic it is very easy to disclose important information by illegal users, suc as personal image. For tis reason, te study of image encryption as become an important aspect of protecting security of image information. Since digital image possesses some inerent features suc as bulk data capacity and ig correlation among adjacent pixels, te algoritms of image encryption are different from te traditional text cryptograpic metod. At te same time, due to teir features of ergodicity, sensitivity to initial conditions and control parameters, etc, caotic maps ave good potential for information encryption, especially image encryption. Inspired by te subtle similarity between caotic map and cryptograpy, a large number of caos-based image encryption algoritms ad been proposed [1 8]. In [1], te autors generalized two-dimensional caotic cat map to 3D for designing a real-time secure symmetric encryption sceme, used 3D cat map to permute te position of image pixels in te permutation stage and employed logistic caotic system to diffuse te permuted image in te diffusion stage. In [], te autors firstly analyzed te parameter sensitivity of standard map, and compared te secret key space of standard map wit tat of cat map and baker map. Ten an improved standard map was used to realize position permutation, wile te diffusion function consisted of logistic map tat was used to realize te diffusion of image. In [3], te autors introduced a certain diffusion effect in te permutation stage by simple sequential add-and-sift operations. Altoug tat led to a longer processing time in a single round, te overall encryption time was reduced as fewer rounds were required. In[4], te autors proposed a caos-based image encryption algoritm wit variable control parameters. Te control parameters used in te permutation stage and te keystream employed in te diffusion stage were generated from two caotic maps related to te plainimage. A fast image encryption algoritm combined wit permutation and diffusion was proposed in [5], te image was partitioned into blocks of pixels. Ten, spatiotemporal caos was employed to suffle te blocks, and at te same time, to cange te pixel values. Meanwile, an efficient metod for generating pseudorandom numbers from spatiotemporal caos was suggested, wic furter increases te encryption speed. A novel image encryption algoritm based on a tree dimensional (3D) caotic map tat could defeat te aforementioned attack among oter existing attacks was proposed in [6]. Te design of te proposed algoritm was simple and efficient, and based on tree pases wic provided te necessary properties for a secure image encryption algoritm including te confusion and diffusion properties. In [7], tis paper proposed a novel caos-based image encryption algoritm to encrypt color images by using a Coupled Two-dimensional Piecewise Nonlinear Caotic Map, called CTPNCM, and a masking process. Distinct caracteristics of te algoritm were ig security, ig sensitivity, and ig speed tat could be applied in encryption of color images. In [9], te autors designed a new suffling scemes tat could efficiently destroy redundancy in te visual data ensuring its secured transmission and distribution over public networks. In [10], te issues pertaining wit efficient, fast, cost effective and secured image transmission were addressed in totality. Te proposed model employs Compressed Hybrid Cryptosystem constituted compression, encryption and secured session key excange along wit te transmission of image. In te proposed work, te algoritm ad been designed to generate diffusion template using 3D Standard map. Key Laboratory of Advanced Design and Intelligent Computing (Dalian University), Ministry of Education, Dalian, 1166, Cina, zangq@dlu.edu.cn DOI: /jee , Print ISSN , On-line ISSN X, c 014 FEI STU

2 Journal of ELECTRICAL ENGINEERING 65, NO., Fig. 1. Flowcart of permutation-diffusion type of caos-based image cryptosystems As an important metod, wavelet transforms are widely used in image processing, suc as image compression and image watermarking. A number of works on image compression and image watermarking are based on reversible integer wavelet transform [11 14]. In [11], te autors presented te factors affecting te compression performance of reversible integer-to-integer wavelet transforms, and supported by bot experimental data and teoretical arguments. In [1], te autor used te reversible integer wavelet transform to implement te tecnique of difference expansion, wic was widely employed in image watermarking. Te Ref. [13] proposed a ig capacity reversible image watermarking sceme based on integer-to-integer wavelet transforms. It divided an input image into non-overlapping blocks and embedded a watermark into te ig-frequency wavelet coefficients of eac block. An intelligent reversible watermarking approac GA-RevWM for medical images was proposed in [14]. GA-RevWM was based on te concept of blockbased embedding using genetic algoritm (GA) and integer wavelet transform (IWT). However, in recent years, many proposed caotic cryptosystems are broken by some kinds of cryptanalysis [1, 15 17]. Te main broken reasons can be briefly stated as follows: (1) in te different rounds, te same control parameters for permutation are used; () te key stream obtained from te caotic map only depends on te key, namely te parameters or initial value of caotic map. Using te first loopole, te attacker can easily divide te permutation-diffusion process into two unrelated stages by coosing te plain-image wit identical pixels [4]. Due to te second loopole, te attacker can obtain te key stream, wic is extracted from te caotic map in te diffusion, by te cryptoanalytic solution, suc as knownplaintext and cosen-plaintext attacks [18 0]. In tis paper, a novel sceme for image encryption based on caotic map and reversible integer wavelet transform is proposed. Te initial key is randomly generated, and relates to plain-image to generate ciper key, wic is used as te different parameters and initial values of caotic maps. It can overcome te flaws above. Te plainimage is firstly permuted by te order from caotic maps. Ten te permuted image is processed by integer wavelet transform. In order to diffuse te image, a part of coefficient is diffused by te obits of caotic maps. Finally, te inverse integer wavelet transform is used to obtain te ciper image based on te diffused coefficient. Numerical experimental results and comparing wit previous works sow tat te proposed sceme possesses iger security tan previous works, wic is suitable for protecting te image information. Te paper is organized as follows. In te next section, te related works are described in detail. In Section 3, te process of encryption and decryption is described in detail. Performance analyses and simulation results are reported in Section 4. Finally, conclusions are drawn in Section 5. RELATED WORKS In [1], te autors proposed a general cryptograpic caos-based arcitecture for image encryption, namely permutation-diffusion arcitecture, wic s sown as Fig. 1. Tis arcitecture includes two iterative stages, namely permutation stage and diffusion stage. Te former permutes te plain-image but not cange te value of pixel. Te latter canges te value of pixel but not cange te position of pixel. In order to improve encryption effect of algoritms, te wole permutation-diffusion round will be repeated R times..1 Caotic maps used in image encryption Tere are tree types of two-dimensional caotic maps wic are widely used in te permutation stage, namely Standard caotic map, Cat caotic map and generalized Baker caotic map [1,, 10]. Altoug tese tree caotic maps can effectively confuse te position of image pixel in te permutation stage, some parameters used in te in te permutation stage may cause a secure loopole [4,19,0]. Tere are many works on tese caotic maps, suc as parameter space, key space, and iteration time. More details of tis step can be obtained from [,1 3]. In te diffusion stage, logistic map and tent map are frequently employed to generate te key stream or subkey for te stream ciper or block cipers, respectively, and cange te value of pixels wic ave been permuted [,3,17]. It can be denoted as follows x i+1 = µx i (1 x i ). (1)

3 9 Bin Wang Xiaopeng Wie Qiang Zang Cao Ce: IMAGE ENCRYPTION BASED ON CHAOTIC MAP AND... Te reversible integer transforms set up a one-to-one correspondence between (x, y) and (l, ) [1]. For example, we ave a pair pixels x = 156,y = 141. Ten, by (), l = = 148, = = 15. Recover- = = 156, ing te original pixels x = l + y = l = = 141 by (3)..3 Preventing te overflow and underflow problems For te grayimage, to preventte overflowand underflowproblems,itmustrestrict x,y interangeof [0.55]. From (3), it is equivalent to ave 0 l , and 0 l 55.Ten, +1 l 55 +1, and l 55+. For te different values of, tere is l 55 l 55+, 0,, < 0. (4) Fig.. Te flowcart of encryption Here, µ is control parameter for caotic map, x i and x i+1 are te it and te (i+1)t state of caotic map, respectively. Some oter caotic map can be employed in te diffusion stage, suc as Cen s caotic system, Lorentz caotic system and so on [16, 3, 4]. However, tese caotic systems are more complex te logistic map and tent map, wic makes te rise of runtime of caosbased image encryption. Designing fast image encryption arcitecture, te complex caotic map is not advised to use. A number of works related on te logistic map and tent map ave been publised, including parameter sensitivity, initial value sensitivity, statistical properties and degradation penomenon [5,6]. So we use two logistic maps wit different parameters and initial value to implement te permutation-diffusion arcitecture in tis paper.. Reversible integer wavelet transform Recently, reversible integer wavelet transforms are widely used in image compression and image watermarking. A reversible integer wavelet transform is used in tis paper. It can be denoted as follows Inverse transform of () is x+y l =, = x y. () +1 x = l+, y = l (3) Te reversible integer transforms () and (3) are also called integer Haar wavelet transform, or te S transform. One can derive tat te above inequalities are equivalent to: 0 l 55 0 l , 0, 3 THE PROCESS OF ENCRYPTION AND DECRYPTION 3.1 Encryption algoritm, < 0. (5) According to te caracteristic of logistic map, logistic map is cosen as te caotic map in te proposed arcitecture. Te different parameters and initial values for Eq (1) are denoted as µ 1, µ, x 1 (0) and x (0), respectively, were µ 1,µ [3.9,4] and x 1 (0),x (0) (0,1). Ikey is denoted as te initial key. Te detailed of encryption is described as follows: Step 1. Randomly generating te initial key and obtaining ciper key by relating to te plain-image; Step. Evenly dividing ciper key into four parts as te parameters and initial values of logistic maps, iterating logistic maps for Q times to get rid of te transient effect, were Q = 100; Step 3. Sorting te caotic orbit obtained from previous step from small to large, and permuting te plainimage by tis order only once Mim(i) = permute ( P(i),Order(i) ), i = 1,,...M N ; (6) Step 4. Processing te permuted image by integer wavelet transform, namely by (), and obtaining te transform coefficient l, ;

4 Journal of ELECTRICAL ENGINEERING 65, NO., Fig. 3. (a) Plain-image of Lena, (b) Encrypted image by key , (c) Encrypted image by key , (d) Difference image Step 5. Obtaining new coefficient l based on (7); l = l, 0, l = l+, < 0; (7) Step 3. Obtaining Orbit(i) and Order(i); Step 4. Implementing integer wavelet transform and obtaining te coefficient Cl, ; Step 5. Recovering te l(i) by (10) l(i) = Cl(i) Orbit(i), i = 1,,...,M N/; (10) Step 6. Diffusing te coefficient l once by te logistic caotic orbits Cl(i) = l (i) Orbit(i), i = 1,,...,M N/; (8) Step 7. Recovering image by inverse integer wavelet transform wit te diffused coefficient l, and obtaining te ciper-image; Step 8. Output te ciper-image. Were P(i) is te original image pixel value, Mim(i) is te pixel value permuted by te order, Order(i) is te ordered position of P(i), Orbit(i) are te logistic caotic orbit from step, and Cl(i) is te diffused coefficient l. M and N are te widt and eigt of te plain-image. denotes te xor operator. Note tat implementing te Step 5 is convenient to deal wit te overflow and underflow problems wile calculating (8). If Cl(i) out off te constraintfrom (5), Cl(i) is processed by ( mod Cl(i),55 ( mod Cl(i), ), 0, ), < 0. (9) And te sceme saves te location of i for decrypting te ciper image in one piece. Te detailed of encryption is illustrated in Fig.. 3. Decryption algoritm Te decryption process is similar to tat of encryption procedure in te reversed order. It can be briefly stated as follows Step 1. Iterating te logistic maps for Q times to get rid of te transient effect; Step. Concurrently generating te caotic orbits and order as encryption process; Step 6. Implementing inverse wavelet transform by te coefficient l and, and obtaining te Mim(i); Step 7. Recovering te P(i) by (11) P ( Order(i) ) = Mim(i), i = 1,,...,M N ; (11) Step 8. Outputting te plain-image. 4 PERFORMANCE ANALYSES AND SIMULATION 4.1 Te space of key In a good image cryptosystem, te space of key sould be enoug large to make brute-force attack infeasible. In tis proposed arcitecture, te Ikey consists of 16 elements, namely Ikey = {x i }, i = 1,,...,16, x i [0,55]. So te key space of te proposed arcitecture is equal to , wic is sufficiently large to meet te need for practical application. 4. Key sensitivity In tis part, te key sensitivity will be performed as follows: Step 1. Calculate te Imkey of te standard test image Lena; Step. Encrypt te test image by Ikey ; Step 3. Sligtly cange te generated Ikey , and encrypt te same plain-image; Step 4. Compare te ciper-image wic is encrypted by different key. Te results are: te image encrypted by te key as 99.59% of different from te image encrypted by te key in terms of

5 94 Bin Wang Xiaopeng Wie Qiang Zang Cao Ce: IMAGE ENCRYPTION BASED ON CHAOTIC MAP AND... Fig. 4. (a) plain-image of Lena, (b) istogram of te plain-image, (c) ciper-image, (d) istogram of te ciper-image Fig. 5. (a) te correlation of vertical adjacent two pixels for original image, (b) te correlation of vertical adjacent two pixels for ciper-image pixel values, altoug tere is only one bit difference in te two keys. Figure 3 sows te test result. Moreover, wen a key is used to encrypt an image wile anoter trivially canged key is used to decrypt te cipered image, te decryption also completely fails. 4.3 Statistical analysis As Sannon said: It is possible to solve many kinds of cipers by statistical analysis [7]. Terefore, e suggested two metods of diffusion and confusion sould be used in any cryptosystems. In tis proposed arcitecture, te standard Lena test image of size is selected to test te property of resisting statistical analysis. Te istograms of encrypted image are sown in Fig. 4. From te figure, one can see tat te istogram of te cipered image is fairly uniform and is significantly differentfromtatofteoriginalimage.totesttecorrelation between two vertically adjacent pixels, two orizontally adjacent pixels, and two diagonally adjacent pixels, respectively, in a cipered image, te following procedure is carried out. First, randomly select 1000 pairs of two adjacent pixels from an image. Ten, calculate te correlation coefficient of eac pair by using te following two

6 Journal of ELECTRICAL ENGINEERING 65, NO., formulas [1] Table 1. Comparing results NPCR UACI Proposed sceme 99.59% 8.1% Wang s work 44.7% % Gupta s work 99.6% 17.30% cov(x,y) = E {( x E(x) )( y E(y) )}, (1) R xy = cov(x, y) D(x) D(y) (13) Were x and y are grey-scale values of two adjacent pixels in te image. In numerical computation, te following discrete formulas are employed E(x) = 1 N D(x) = 1 N cov(x,y) = 1 N N x i, (14) i=1 N ( xi E(x) ), (15) i=1 N {( xi E(x) )( y i E(y) )}. (16) i=1 Figure 5 sows te correlation distribution of two vertically adjacent pixels in te plain-image and tat in te ciper-image: te correlation coefficients are and , respectively, wic are far apart. 4.4 Differential attack To test te property of resisting differential attack of te proposed arcitecture, two common quantitative criteria are employed: number of pixels cange rate (NPCR) and unified average canging intensity (UACI). Te NPCR and UACI are defined as follows [10,8] i,j NPCR = D(i,j) 100%, (17) UACI = 1 [ W H i,j W H C 1 ( i,j) C (i,j) ] % (18) were C 1 and C are te two ciper-image wose corresponding Ikeys ave only difference, te grey-scale values of te pixels at poison (I,j) of C 1 and C are denoted as C 1 (i,j) and C (i,j), respectively; W and H are te widt and eigt of te ciper-image; D(i, j) is determined by C 1 (i,j) and C (i,j), namely, if C 1 (i,j) = C (i,j) ten D(i,j) = 0 oterwise, D(i,j) = 1. In te Table 1, we compare our results of NPCR wit te Wang [5] and Gupta [10] works. Te wole permutation-diffusion round is repeated only once. Our results are te average of ten trials. Comparing wit previous works, it suggests tat our sceme as iger security. 4.5 Resistance to known-plaintext and cosenplaintext attacks In te proposed sceme, tis sceme uses different parameters and values of caotic maps wic are used in te permutation and diffusion stage to overcome te first flow. At te same time, it makes te ciper key related to te permuted image to overcome te second flow. So te different control conditions, key streams and non-identical ciper-images will be generated by distinct plain-images. Te attacker cannot obtain useful information by encrypting some special images since te resultant information is only related to tose cosen-images. Terefore, te proposed algoritm can well resist te knownplaintext and te cosen-plaintext attacks. 5 CONCLUSIONS A novel sceme for image encryption based on caotic maps and reversible integer wavelet transform as been proposed in tis paper. Te different parameters and initial values of caotic maps are obtained by ciper key wic is related to te plain-image. Ten te order is from caotic maps tat are used to permute plain-image. Subsequently, te permuted image is processed by integer wavelet transform. Te coefficient is diffused by te orbits of caotic maps. Finally, te ciper image is obtained by inverse integer wavelet transform based on te diffused coefficient. Numerical experimental results and comparing wit previous works sow tat te proposed sceme possesses iger security tan previous works, wic is suitable for protecting te image information. Acknowledgment We would like to tank te anonymous reviewers for elpful comments. Tis work is supported by te National Hig Tecnology Researc and Development Program ( 863 Program) of Cina (No. 009AA01Z416) te National Natural Science Foundation of Cina( , ), Program for Cangjiang Scolars and Innovative Researc Team in University(IRT1109), te Program for Liaoning Innovative Researc Team in University(LT011018), te Program for Liaoning Excellent Talents in University (LR01003), te Program for Liaoning Science and Tecnology Researc in University (LS010179) and te open fund of Key Laboratory of Advanced Design and Intelligent Computing (Dalian University), Ministry of Education, Dalian University (ADIC01001).

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CAI, Y.: Statistical Properties of Digital Piecewise Linear Caotic Maps and teir Roles in Cryptograpy and Pseudo-Random Coding, Cryptograpy and Coding (001), [6] LI, S. CHEN, G. MOU, X.: On te Dynamical Degradation of Digital Piecewise Linear Caotic Maps, International JournAL of Bifurcation and Caos in Applied Sciences and Engineering 15 (005), [7] SHANNON, C. E.: Communication Teory of Secrecy Systems, MD Computing 15 (1998), [8] KWOK, H. TANG, W. K. S.: A Fast Image Encryption System based on Caotic Maps wit Finite Precision Representation, Caos, Solitons & Fractals 3 (007), Received 4 January 013 Xiaopeng Wei is a professor at Dalian University of Tecnology and Dalian University, Dalian, Cina. His researc areas include computer animation, intelligent CAD. So far, e as (co-) autored about 160 papers publised. Bin Wang is a doctoral student of Scool of Mecanical and Engineering at Dalian University of Tecnology Dalian, Cina. His researc areas include DNA Computing, DNA Sequence designing, DNA Cryptograpy and Biological Network. So far, e as (co-) autored about 16 papers publised. Qiang Zang is a professor at Dalian University, Dalian, Cina. His researc interests are computer animation, intelligent computing. Now e as served as editorial boards of seven international journals and as edited special issues in journals suc as Neurocomputing and International Journal of Computer Applications in Tecnology. Cao Ce is a lectuer at Dalian University, Dalian, Cina. He as received is PD degree in 010 from Dalian University of Tecnology.