Kayhan CELİK. Erol KURT. x ' ay ax. y ' xz rx y. z ' xy bz

Transcription

1 ECAI Internatonal Conference 8th Edton Electroncs, Computers and Artfcal Intellgence 3 June -2 July, 216, Bucharest, ROMÂIA A ew Image Encrypton Algorthm Based on Lorenz System Kayhan CELİK Gaz Unversty, Faculty of Technology, Electrcal and Electroncs Engneerng, Ankara, Turkey kayhancelk@gaz.edu.tr Abstract In ths paper, a new mage encrypton algorthm has been ntroduced. The algorthm ncludes the Lorenz system n order to produce a chaotc data and uses t to mask the mages for the secure encrypton. Both the hgh senstvty to the ntal condtons and unpredctablty have ncreased the capablty of encryptng the mages. The algorthm has been tested by dfferent methods n order to evaluate the performance by varous analyses such as the secret key sze and senstvty, hstogram, correlaton, dfferental (plan mage senstvty), nformaton entropy analyss. The expermental results prove that the proposed encrypton approach provdes an effcent and secure approach for the mage encrypton. Keywords- mage encrypton; lorenz chaotc system; chaos; securty analyss. I. ITRODUCTIO The ncrease of communcaton networks and of unsecure meda n the nternet envronment cause to rase the demand on the tools for the secure communcaton and encrypton [1-3]. Ths s a requrement for the socety, whch uses the dgtzed nformaton n daly lfe. In fact, people get and send much nformaton va the nternet for the busness and prvate purposes, frequently. Thus the wde usage of the nternet enhances the securty problems on transmttng and/or recevng of nformaton, safely [4]. ot only the personal nformaton but also the mltary or governmental nformaton should be transmtted and receved. The nformaton may not be only n text format, but t can be n mage, vdeo or sound formats. In the lterature, there exst several well-approved effcent algorthms for the text message encrypton, such as: Data Encrypton Standard (DES), Advanced Encrypton Standard (AES), Internatonal Data Encrypton Algorthm (IDEA) and the Rvest-Shamr-Adleman cryptosystem (RSA). However those algorthms are not well-apprecated for mage or vdeo fle encrypton applcatons [5]. Snce the mages dffer from the text data n many aspects such as hgh redundancy and correlaton, the local structure and the characterstcs of ampltude frequency, the conventonal encrypton methods cannot be applcable to the mages [2]. In that pont, the chaotc systems can serve as a tool to generate hghly-randomzed data for the encrypton, due to ts characterstc features such as the senstve dependence on the ntal condtons, hgh number of system parameters, pseudo-random property, unpredctablty, aperodcty and topologcal Erol KURT Gaz Unversty, Faculty of Technology, Electrcal and Electroncs Engneerng, Ankara, Turkey ekurt@gaz.edu.tr transtvty [6-1]. All these features are ndeed desred for an encrypton applcaton and become necessary for an excellent encrypton system [1]. Therefore, the chaos-based mage encrypton algorthms have wdely flourshed n the last decade for the usage n mage encrypton [1,3,4]. In the present study, we ntroduce a new mage encrypton algorthm based on the chaotc Lorenz system. Accordng to the present method, the chaotc system runs for the generaton of pseudo-random sequences, then these pseudo-random sequences are used both n order to shuffle the postons of mage pxels and to change the gray values of the mage pxels. The paper s organzed as follows: The chaotc Lorenz system s explaned n the next secton. Secton III presents the new mage encrypton algorthm based on the chaotc map. Later on, the man expermental results and related securty analyss are reported n Secton IV. Fnally, the paper ends wth a concluson. II. CHAOTIC LOREZ SYSTEM The Lorenz equaton s nothng else than a model of thermally nduced flud convecton n the atmosphere. The model was frst reported and publshed by E. Lorenz n 1963 [11-13]. It s among the classcal chaotc systems and mples as the cause of the butterfly effect n the scentfc studes due to the fact that the attractor has two wngs as the butterfles [12, 14]. Therefore, t has been wdely studed n chaos theory, dynamc system modelng, chaotc control and synchronzaton phenomenon. Lorenz chaotc equaton s a 3D dynamcal system, whch s defned by x, y and z. The equaton system gves a chaotc behavor wth regard to the ntal system parameters. Apart from any 1D or 2D chaotc systems, the Lorenz system has a much complcated chaotc behavor. The equaton system contans three dfferental equatons: x ' ay ax (1) y ' xz rx y (2) z ' xy bz (3) Here x, y and z are the functons of tme wth the dervatve forms (.e. x, y and z ) and a, b, r are the system parameters for the determnstc system.

2 2 Kayhan CELIK and Erol KURT between and 255 are converted to a 8-bt bnary data for the pxel value transformaton. Secondly, the double values of chaotc sequences are preprocessed as follows: k(, j) mod (( abs( k(, j)) 15 floor( abs( k(, j)))) x 1, 256) (4) a) b) c) Fgure 1. The projectons of attractor on the planes (a) x-y, (b) x-z and (c) y-z. In order to obtan chaos, a, b and r are usually defned as 1, 8/3 and 28, respectvely. However, dfferent dynamcal behavors can be observed, when these parameters are changed. Fg. 1 (a, b, c) presents a typcal chaotc attractor for the ntal condtons (x, y, z) = (1.2, 1.3, 1.6). In the smulatons, dt s adjusted as.5. III. IMAGE ECRYPTIO ALGORITHM The proposed mage encrypton algorthm conssts of varous steps. Step 1: Frstly, the ntal condtons x, y z and, system parameters a, b, r are determned as a secret key. Then, tme teratons of Eqs. (1, 2, 3) are realzed. In order to obtan a more pseudo-random sequence and to remove the transent effect, the results of frst n teratons (.e. n=34) have been gnored. Then 6 dfferent pseudo-random sequences have been produced by usng dfferent teratons as a double value wth 15 1 precson for cryptography. Step 2: In ths step, x, y, z chaotc sequences are used for rotatng pxel values of red, green and blue channels of the plan mage respectvely. The szes of chaotc sequences are M x, where M and are row and column numbers of the plan mage, respectvely. Intally, the pxel values of the plan mage changng Here, k shows x, y or z sequence and k (, j) s the (,j) th element of the sequence whch sze s Mx and abs(k(,j)) returns the absolute value. floor(k(,j)) gves the value of k(, j) to the nearest ntegers less than or equal to k(, j). mod(x, y) gves the remander after dvson as usual. Thrdly, l(,j) value s calculated for each k(,j), whch shows the sum of the dgts of k(,j). Then one can defne m, n and p(,j) as follows: m mod ( l(, j), 2) (5) n mod ( l(, j), 9) (6) p (, j ) ( p (, j ), m, n ) (7) n o All new color values from the orgnal mage s calculated accordng to Eq. 7. Here, p (, j ) shows the o orgnal and p (, j ) s the transformed color value. If n m=, the color s rotated to left and f m=1, the color s rotated to rght. Here n denotes the rotaton amount n terms of bts. Step 3: Intal column vectors have been generated from the chaotc system for red, green and blue color maps. Therefore, ther szes are Mx1. Then, all elements of the frst columns of red, green and blue components of mage have been made xor wth the elements of ntal column vector mutually. Xor results were adjusted to frst column of the mage. Later on, the frst column was made xor wth the second one and adjusted to the second column and all columns were made xor n smlar manner as n Fg 2. Step 4: For the pxel shufflng process, two chaotc sequences (szes M and ) have been used. Ther values are, of course, unrepeatable n the range -M and -, respectvely. These two sequences have been used for the creaton of Mx sze matrx, whch leads to new pxel postons. Thus, all the color values of red, green and blue channels are transformed randomly to new postons. Fgure 2. Xor operatons appled to the columns

3 A ew Image Encrypton Algorthm Based on Lorenz System 3 Step 5: In ths step, the pxel value transformaton s made one more tme as n step 2. All of the pxel values of the plan mage have been rotated randomly by usng new chaotc sequences. Step 6: Row xor operaton s the same as n step 3. Frstly, the ntal row vectors have been generated from the chaotc system for red, green and blue color maps wth the szes 1x. Then, xor has been appled to all elements of the frst row of mage wth the ntal row vector s elements mutually. After that, xor has been appled to the frst row wth the second row and adjusted to second row and all rows have been made xor respectvely as n Fg. 3. Fgure 4. Plan Lena Image Fgure 5. Cphered Lena Image In fact, the secret key sze, secret key senstvty, hstogram, correlaton, dfferental (plan mage senstvty), nformaton entropy analyss are appled. The results show that the proposed encrypton algorthm can be mplemented n real-tme applcatons wth good securty results. Fgure 6. Plan Baboon Image Fgure 7. Cphered Baboon Image Fgure 3. Xor operatons appled to the rows Step 7: In ths step, pxel shufflng has been appled one more tme for the dffuson and confuson processes as n step 4. As a result, the last form of the encrypted mage s obtaned. The decrypton algorthm s just the nverse process of encrypton. Therefore for decryptng the mage, same parameters and the ntal values have been used. The encrypton and the decrypton processes have same algorthmc complexty. At the end of the decrypton process, the output of the decrypton system has gven the orgnal mage as we expect. IV. EXPERIMETAL RESULTS For ths study, the ntal values and the parameters are adjusted as follows wth a good accuracy: a 1., b , r 28., dt.5, x 1.1, y 1.3, z 1.5. In ths study, we used 512x512 colorful Lena and Baboon mages for the demonstraton of the effcency of the proposed mage encrypton algorthm. Whle Fgs. 4 and 6 present these orgnal mages, the cphered mages are gven n Fgs. 5 and 7, respectvely. Below, the robustness of the proposed algorthm s verfed wth the results of certan analyss by applyng the cphered mages (Fgs. 5 and 7). A. The Secret Key And Secret Key Senstvty Analyss In order to test the proposed mage encrypton scheme, the secret key analyss s mportant. In ths analyss, the key space should be large enough to make the brute-force attack nfeasble. In the algorthm, the ntal condtons of the chaotc system have been used as the secret keys wth the precson The key space of the proposed system s The key space should be more than 2 to provde a suffcent securty n order to avod brute-force attacks [1]. Accordng to ths, the key space s large enough for the secret keys. Ths s enough to resst all knds of brute-force attacks. An deal mage encrypton algorthm should be senstve to the secret key. It means that the small change n the secret key must produce a completely dfferent encrypted and decrypted mages n order to hde the mage [1]. The plan mage n Fg. 6 has been encrypted and decrypted wth the same ntal condtons except x. A very small change has been appled to x (.e. x =1.1, x =1.11 ) for the key and encrypted and decrypted Baboon mages of these keys are shown n Fg. 8 and Fg. 9. The dfference between these mages has been measured wth Mean Square Error (MSE) metrc as follows: M MSE f ( x, y) h( x, y) (8) M x y Here f and h are the mages and (x,y) shows the pxel coordnates. 2

4 4 Kayhan CELIK and Erol KURT Fgure 8. The encrypted Baboon wth the closed ntal condtons of x =1.1 and x =1.11 respectvely. TABLE I. MSE RESULTS OF RED, GREE AD BLUE COMPOETS OF THE ECRYPTED BABOO AD LEA IMAGES WITH THE CLOSED IITIAL VALUES. Lena Image Baboon Image Red Green Blue Fgure 1. The hstograms of Lena mage: red, green and blue components from top to bottom respectvely Fg. 9. The encrypted and decrypted Baboon wth the closed ntal condtons of x 1.1 and The hstogram of the orgnal mage contans several peaks. However, the hstogram of the cphered mage s unform. Hence, t s proven that the hstograms of the encrypted mage do not gve any clue to operate any statstcal attack to recover the orgnal one. x =1.11 respectvely. TABLE II. MSE RESULTS OF RED, GREE AD BLUE COMPOETS OF THE DECRYPTED BABOO AD LEA IMAGES WITH THE WROG IITIAL VALUES. Lena Image Baboon Image Red Green Blue The encrypted mages wth the closed ntal condtons are shown n Fg. 8 and decrypted mage wth wrong ntal condton s shown n Fg. 9. It s obvous from Tables 1 and 2 that the mages are completely dstnctve from each other. As a result, the proposed algorthm s hghly senstve to secret key. B. Hstogram Analyss The mage hstogram determnes the dstrbuton of each color ntensty level of the ndvdual pxels of an mage. An deal hstogram for an encrypted mage should be flat, mplyng the unform dstrbuton on the mage range and the complete randomness. In fact, the flatness of a hstogram could better prevent the leakage of the nformaton by the statstcal attacks. Fgs. 1 and 11 show the hstograms of the orgnal Lena mage and ts encrypted one. Fgure 11. The hstograms of cphered Lena mage: red, green and blue components from top to bottom respectvely C. Correlaton analyss In ths secton, the horzontal, vertcal and dagonal correlaton coeffcents of the cphered pxels have been calculated. In ths manner, 3 pars of the adjacent pxels (horzontal, vertcal and dagonal) have been selected from the orgnal and the encrypted

5 A ew Image Encrypton Algorthm Based on Lorenz System 5 mages. To test the correlaton between two adjacent pxels n the orgnal and the encrypted mage, the followng equatons have been used [15]: 1 E( x) x (9) 1 D 1 ( x ) x E ( x ) 2 (1) 1 Cov 1 ( x, y ) x E ( x ) y E ( y ) (11) 1 ( xy, ) Cov( x, y) D( x). D( y) (12) Here x and y are color value of two adjacent pxels. Cov(x,y), D(x) and E(x) denote the covarance, varance and the mean. For nstance, Fgs. 12 and 13 show the correlatons of two horzontal pxels of the orgnal mage and the encrypted mage, respectvely. It s obvous that the correlaton between the adjacent pxels s totally reduced. The detals of the analyss are presented n Table 3. Accordng to Table 3, the correlaton coeffcents of the adjacent pxels n the encrypted mage are very small. It s clear that the proposed mage encrypton algorthm has strong ablty to resst the statstcal attacks. TABLE III. Drecton Horzontal Vertcal Dagonal CORRELATIO COEFFICIETS OF THE ORIGIAL AD CIPHERED IMAGES. RGB Plan mage Cphered mage R G B R G B R G B Fgure 13. Correlaton analyss of the cphered Lena mage: Red, green and blue components from top to bottom respectvely. mages as follows: Fgure 12. Correlaton analyss of the Lena mage: Red, green and blue components from top to bottom respectvely. D. Dfferental analyss A good cryptography system should be senstve to small changes (for nstance, one bt) n the plan mage and that must create bg dfference n the encrypted mage. If the encrypton has ths feature, the cryptosystem can resst toward the dfferental attack, whch s bascally a chosen plan mage attack amed to fnd the secret key. Two metrcs are used to measure how much the cryptosystem at the plan mage s senstve: et Pxel Change Rate (PCR) and Unfed Average Changng Intensty (UACI). Whle PCR represents the change rate of the cphered mage va only one pxel of plan mage changes, UACI measures the average ntensty of the dfferences between the orgnal and cphered PCR 1 UACI Mx D (, j ) j 1% Mx, (13) x, j C (, j C (, j) 1 2 x1% 255 (14) where M and denote the row and column numbers of the mages, respectvely. C and C 1 2are two cphered mages of the correspondng orgnal mages havng only one pxel dfference. D(,j) s determned by C 1 (, j) and C 2 (, j ). amely, f C (, j) C (, j) 1 2 exst, then D(,j)=1, otherwse, D(,j) = stands. PCR and UACI are tested on two Lena and Baboon mages by a pxel dfference, respectvely. Ther testng results are shown n Table 4. From these results, t can be concluded that our algorthm s not

6 6 Kayhan CELIK and Erol KURT very robust aganst the dfferental attacks, however, t can be mproved further to gve hgher dfferental analyss results. TABLE IV. LEA BABOO DIFFERETIAL AALYSIS OF THE IMAGES I FIGS. 4 AD 6. PCR UACI R G B R G B E. Informaton Entropy Analyss The nformaton entropy measures the strength of the cryptosystem and gves the equally probable gray levels for a good mage encrypton algorthm by applyng t to the encrypted mage. In ths sense, entropy s gven by, 2 1 H( m) p( m )log 2 1 pm ( ) (15) Here s the number of bts of the pxel value (m) and p(m) represents the probablty of m. If there are 256 state of m and they have the same probablty, H(m) wll be theoretcally equal to 8. The smaller entropy value means greater degree of predctablty, whch threatens the encrypton system securty. TABLE V. I FIGS. 4 R G B Orgnal lena THE RESULTS OF ETROPY AALYSIS FOR IMAGES Cphered lena Orgnal baboon Cphered baboon Table 5 gves the entropy values of two orgnal mages and ther encrypted ones. Accordng to the results, the nformaton entropy s ncreased by the encrypton system and the nformaton entropes of the encrypted mages are nearly 8. Ths proves that the proposed encrypton algorthm s hghly resstve to the nformaton leakage. V. COCLUSIOS A new mage encrypton algorthm based on the Lorenz chaotc system has been ntroduced. The man features of the chaotc systems, namely the unpredctablty and the senstvty to the ntal condtons, have contrbuted at the results of the cphered mages. The man tests have been performed n order to prove the strength of the new algorthm. Whle the secret key analyses gve dfference between the mages, thereby prevent the predcton of orgnal mage, the color-dependent correlaton tests yeld to uncorrelated results. The nformaton entropy values and the hstogram results have proven the unpredctablty of the cphered mages. Regardng the dfferental analyss tests, the results are not as good as we expect. The proposed algorthm therefore should be mproved further to get hgher scores from dfferental analyss tests. To conclude, ths new chaotc based algorthm can be used n encrypton studes n ts current form due to the good results obtaned from the other analyses. REFERECES [1] Alvarez, G., & L, S. (26). Some basc cryptographc requrements for chaos-based cryptosystems. Internatonal Journal of Bfurcaton and Chaos, 16(8), [2] Lu, H., Kadr, A., & u, Y. (214). Chaos-based color mage block encrypton scheme usng S-box. AEU- Internatonal Journal of Electroncs and Communcatons, 68(7), [3] L, J., & Lu, H. (213). Colour mage encrypton based on advanced encrypton standard algorthm wth twodmensonal chaotc map. Informaton Securty, IET, 7(4), [4] Akhshan, A., Behna, S., Akhavan, A., Lm, S. C., & Hassan, Z. (214), An Image Encrypton Approach Usng Quantum Chaotc Map. Int. J. Adv. Computer Sc. Appl. ( ).4. [5] Sngh, K., & Kaur, K. (211). Image Encrypton usng Chaotc Maps and DA Addton Operaton and ose Effects on t, Int. J. Computer Applcatons ( ) Vol, 23. [6] Moon, F. C. (28). Chaotc and Fractal Dynamcs: Introducton for Appled Scentsts and Engneers. John Wley & Sons. [7] Kurt, E. (26). onlneartes from a non-autonomous chaotc crcut wth a non-autonomous model of Chua's dode. Physca Scrpta, 74(1), 22. [8] Kurt, E., & Canturk, M. (29). Chaotc dynamcs of resstvely coupled DC-drven dstnct Josephson junctons and the effects of crcut parameters. Physca D: onlnear Phenomena, 238(22), [9] Kurt, E., & Canturk, M. (21). Bfurcatons and Hyperchaos from a DC Drven ondentcal Josephson Juncton System. Int. J. Bfurcaton and Chaos, 2(11), [1] Kurt, E., Cylan, B., Taskan, O. O., & Kurt, H. H. (214). Bfurcaton analyss of a resstor-double nductor and double dode crcut and a comparson wth a resstor-nductor-dode crcut n phase space and parametrcal responses. Scenta Iranca Trans. D: Computer Scence & Engneerng, Electrcal, 21(3), 935. [11] Song, W., & Lang, J. (213). Dfference equaton of Lorenz system. Internatonal Journal of Pure and Appled Mathematcs, 83(1), [12] Kurt, E., Kasap, R. (211), Karmasanın Blm Kaos, obel Publ., Ankara, Turkey, [13] Ott, E. (22). Chaos n dynamcal systems. Cambrdge unversty press. [14] Psarchk, A.., Flores-Carmona,. J., & Carpo-Valadez, M. (26). Encrypton and decrypton of mages wth chaotc map lattces. Chaos: An Interdscplnary Journal of onlnear Scence, 16(3), [15] El-Same, F. E. A., Ahmed, H. E. H., Elashry, I. F., Shaheen, M. H., Faragallah, O. S., El-Rabae, E. S. M., & Alshebel, S. A. (213). Image Encrypton: A Communcaton Perspectve, CRC Press.