Chaos-Based Image Encryption Using an Improved Quadratic Chaotic Map

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1 America Joural of Sigal Processig 26, 6(): -3 DOI:.5923/j.ajsp.266. Chaos-Based Image Ecryptio Usig a Improved Quadratic Chaotic Map Noha Ramada,*, Hossam Eldi H. Ahmed, Said E. Elkhamy 2, Fathi E. Abd El-Samie Commuicatio, Faculty of Electroic Egieerig, Meofia Uiversity, Egypt 2 Electrical Egieerig, Faculty of Egieerig, Alexadria Uiversity, Egypt Abstract I recet years, chaos-based image ecryptio has become a efficiet way to ecrypt images due to its high security. I this paper, we improve the classical Quadratic chaotic map to ehace its chaotic properties ad use it for image ecryptio. Compared with the classical Quadratic map, the proposed Quadratic map demostrates better chaotic properties for ecryptio such as a much larger maximal Lyapuov expoet. The proposed image ecryptio scheme is based o two chaotic maps. The first map is the Chepyshev chaotic map, which is used for the permutatio of the pixels of the image. The permuted image is subjected to the diffusio process usig the improved Quadratic map i a efficiet ecryptio algorithm which its key is related to the origial image. The mai advatages of the proposed scheme are the large key space ad its resistace to various attacks. Simulatio results show that the proposed scheme has a high security level with low computatioal complexity, which makes it suitable for real-time applicatios. Keywords Chaos ecryptio, Quadratic map, Chepyshev, Lyapuov expoet. Itroductio I recet years, the trasmissio of a large amout of data such as video cofereces, medical ad military images over commuicatio media was highly developed. The security of iformatio trasmitted is a vital issue. Most covetioal secure ciphers such as Data Ecryptio Stadard (DES) [] ad Advaced Ecryptio Stadard (AES) [2] are ot suitable for fast ecryptio of a large data volume i real time. The implemetatio of traditioal algorithms for image ecryptio is eve more complicated because of the high correlatio betwee image pixels. Therefore, there is still a lot of work to be doe for the developmet of sophisticated ecryptio methods. May researchers have poited out the existece of a strog relatioship betwee chaos ad cryptography. The idea of usig chaos i cryptography ca be traced back to Shao o secrecy systems [3]. Although the word "chaos" was ot mited till the 97s [4], the use of chaos i cryptography seems quite atural. The two basic properties of a good cipher; cofusio ad diffusio are strogly related to the fudametal characteristics of chaos such as a broadbad spectrum, ergodicity ad high sesitivity to iitial coditios. The implemetatio of Shao's idea had to wait till the developmet of chaos theory i the 98s. * Correspodig author: eg_oharamada@yahoo.com (Noha Ramada) Published olie at Copyright 26 Scietific & Academic Publishig. All Rights Reserved I the first scietific paper o chaotic cryptography that appeared i 989, Matthews [5] came up with the idea of a stream cipher based o oe-dimesioal chaotic map. Afterwards, chaotic cryptography has spread ad more papers about digital chaotic ciphers have bee published [6-8]. Traditioally, ecryptio is based o discrete umber theory, so that data has to be digitized before ay ecryptio process ca take place. I order to ecrypt a cotiuous voice or a video i the old fashio, digitalizatio ad ecryptio ca pose a heavy computatioal process. The use of chaotic commuicatio eables to ecrypt the message waveform without a eed to digitalize it [9]. Based o stregths ad weakesses of already existig algorithms, Kelber ad Schwarz formulated several geeral rules to desig a good chaos-based cryptosystem [9-]: Either use a suitable chaotic map, which preserves importat properties durig discretizatio for block cipher or use a balaced combiig fuctio ad a suitable key-stream geerator for a stream cipher. Use a large key space. Avoid simple permutatios of idetical system parameters. Use the same precisio for sub-key values ad their correspodig system parameters. Use a complex iput key trasformatio. Use a dyamical system. Use complex oliearities. Modify oliearities i terms of key ad sigal values.

2 2 Noha Ramada et al.: Chaos-Based Image Ecryptio Usig a Improved Quadratic Chaotic Map Use several rouds of operatio for block ciphers. The basic properties of chaotic systems are the determiisticity, the sesitivity to iitial coditios ad parameters ad the ergodicity. Determiisticity meas that chaotic systems have some determiig mathematical equatios rulig their behavior. The sesitivity to iitial coditios meas that whe a chaotic map is iteratively applied to two iitially close poits, the iteratios quickly diverge, ad become ucorrelated i the log term. Sesitivity to parameters causes the properties of the map to chage quickly, whe slightly disturbig the parameters, o which the map depeds. Hece, a chaotic system ca be used as a pseudo-radom umber geerator. The ergodicity property of a chaotic map meas that if the state space is partitioed ito a fiite umber of regios, o matter how may, ay orbit of the map will pass through all these regios. A umber of traditioal chaotic maps such as Quadratic map [] ad Logistic map [2] have limited properties ad may o loger satisfy our eeds. Without improvemet of chaotic maps, our applicatios will remai uchaged ad might be subject to differet attacks i the future. Hece, there is a bad eed for more improvemets i the chaotic maps. I this paper, a improved Quadratic chaotic map is first itroduced. We use the maximal Lyapuov expoet [3] ad the bifurcatio diagram to determie the performace of the map. A ew image ecryptio scheme based o this improved Quadratic map is preseted cotaiig two mai processes; permutatio ad diffusio. The permutatio process breaks the strog relatioship betwee adjacet pixels. The permutatio operatio oly shuffles the pixel positios without chagig values. The shuffled ad origial images have the same etropy, ad therefore the shuffled image is weak agaist statistical attack ad kow plai-text attack. I the diffusio process, the pixel values are altered. Most researchers focus o security improvemets [4-5], while oly a few are dealig with efficiecy issues [6-7]. For most of the security improvemets, researchers eed at least three rouds of the diffusio process to obtai a satisfactory performace. Researchers focusig o efficiecy improvemet oly eed oe roud of the diffusio process to achieve the high security level ad speed up the performace. However, some of the proposed algorithms lead to a loger processig time i a sigle roud. Therefore, the key problem of desigig a efficiet image cryptosystem is how to reduce the computatioal complexity with efficietly to avoid the large umber of rouds i the geeratio of diffusio ad permutatio keys ad the achieve high speed performace. At the permutatio step, we sort the chaotic sequeces of the Chepyshev map i order to shuffle the etire image. At the diffusio step, the shuffled image is ecrypted with a key related to plai image. 2. Aalysis of Quadratic Map Quadratic map is a basic example of a chaotic system. The equatio of the classical Quadratic map is [8]: x + = r ( x ) 2 () where r is the chaotic parameter ad is the umber of iteratios. The system of the Quadratic map is chaotic, because it is oliear. It is determiistic sice it has a equatio that determies the behavior of the system. Also, a very slight chage of the iitial value x ca lead to a sigificatly differet behavior of the map. I the followig subsectios, may plots for the aalysis of the Quadratic chaotic map will be studied such as the iteratio property, the bifurcatio diagram, ad the Lyapuov expoet. 2.. Iteratio Property The iteratio diagram plots the relatio betwee the umber of iteratios ad the Quadratic chaotic map at differet values of the chaotic parameter r ad at a specific iitial value x. The parameter r ca be divided ito three regios, which ca be examied by simulatio usig MATLAB as followig: Whe r [,.74], as show i Fig. (a), the calculated value come to the same result after several iteratios without ay chaotic behavior. Whe r [.74,.5], as show i Fig. (b), the system appears as havig a periodic behavior. Whe r [.5, 2], it becomes a chaotic system as show i Fig. (c) Bifurcatio Diagram Bifurcatio is usually referred to as the qualitative trasitio from regular to chaotic behavior by chagig the cotrol parameter. The bifurcatio diagram is used to study the chaotic system as a fuctio of the values of the cotrol parameters. This diagram allows kowig the regios of the system displayig covergece, bifurcatio, ad chaos depedig o the values of the cotrol parameters [9]. Fig. (d) shows the bifurcatio diagram of the classical Quadratic map. This diagram has three regios. The covergece regio is at r [,.74]. The bifurcatio regio is at r [.74,.5]. The chaos regio is at r [.5, 2], where the chaotic behavior occurs Lyapuov Expoet Lyapuov expoet represets the features of a chaotic system ad ca largely express the overall performace of chaotic maps. It is used as a quatitative measure for the sesitive depedece o iitial coditios. For a discrete system x + =f(x ) ad for a orbit startig with x, the Lyapuov expoet ca be defied as follows [2-2]: λ (x ) = lim i= l f (xi) (2) where f is the derivative of the fuctio f. If λ is egative,

3 Lyapuov expoet X+ X+ X+ America Joural of Sigal Processig 26, 6(): -3 3 the system is ot chaotic. If λ is zero, this meas that the system is eutrally stable ad is i steady state mode. If λ is positive, the evolutio is sesitive to iitial coditios ad therefore chaotic. Also, it is commo to refer to the largest defied by Eq.(2) as the Maximal Lyapuov Expoet (MLE), because it determies a otio of predictability for a chaotic system. The larger MLE is, the more chaotic the map is ad the smaller the umber of iteratios ecessary to achieve the required degree of diffusio or cofusio of iformatio is, ad this meas a better chaotic map. Fig. (e) shows the Lyapuov expoet of the classical Quadratic map. It is obviously clear that whe r [,.5], all Lyapuov expoets are less tha or equal to zero. Whe r [.5,2], the Lyapuov expoets are positive, ad hece chaotic. The maximal Lyapuov expoet of the Quadratic map is The Proposed Quadratic Map The equatio of the proposed quadratic map is: X + = (r+ (-ax ) 2 ) mod (3) We will replace -(x ) 2 i Eq.() with the term (-ax ) 2 ad module. Now, we will examie the proposed Quadratic maps ad plot the iteratio property, bifurcatio diagram, ad Lyapuov expoet at three differet values of r =.25 a = 2, 4, Aalysis of the Proposed Quadratic Map The equatio of the proposed Quadratic map at a= 2 is: X + = (r+ (-2x ) 2 ) mod (4) Fig. 2 shows the aalysis of the proposed Quadratic map. From Fig. 2 (a), (b), (c), ad (d), it is clear that there are several covergece, bifurcatio, ad chaos regios. These regios are exteded to ifiity. The bifurcatio regios are at r [.5,.3], r [.5,.3], etc. to ifiity. The covergece regios are at r [.3,.56], r [.2,.56], etc. to ifiity. The chaos regios are at r [,.4], r [.56, 2.4], r [2.56,3.4], etc. to ifiity, where the chaotic behavior occurs. Now, it is obvious that the chaotic rage of the proposed Quadratic map is larger tha the chaotic rage for the classical oe, ad hece this will icrease the available chaotic value of parameter r that ca be used i ecryptio. I Fig. 2 (e), the Lyapuov expoet has a positive value at r [,.4], r [.56,2.4], r [2.56,3.4], etc, ad hece the proposed Quadratic map exhibits a chaotic behavior at these periods. The MLE of the proposed Quadratic map is.6732, which is greater tha the classical Quadratic map but still small. r =.8 r = (a) Iteratio property at r=.25. (b) Iteratio property at r=.8. (c) Iteratio property at r= r (d) Bifurcatio diagram. (e) Lyapuov expoet. Figure. Aalysis of the classical Quadratic map at x =.2

4 Lyapuov expoet X+ X+ X+ 4 Noha Ramada et al.: Chaos-Based Image Ecryptio Usig a Improved Quadratic Chaotic Map r = r =.25 r = (a) Iteratio property at r=. (b) Iteratio property at r=.25. (c) Iteratio property at r= r (d) Bifurcatio diagram. (e) Lyapuov expoet. Figure 2. Aalysis of the proposed Quadratic map at x = Aalysis of the Proposed Quadratic Map 2 The equatio of the proposed Quadratic map 2 at a= 4 is: X + = (r+ (-4x ) 2 ) mod (5) Fig. 3 shows the aalysis of the proposed Quadratic map 2. From Fig. 3 (a), (b), (c), ad (d), it is clear that the covergece ad bifurcatio regios have become very small ad chaos regios are icreased. These regios are exteded to ifiity. The bifurcatio regios are at r [.38,.4], r [.7,.9], r [.38,.4], r [.7,.9] etc. to ifiity. The covergece regios are at r [.2,.266], r [.2,.266], etc. to ifiity. The chaos regios are at r [,.37], r [.4,2.4], r [.4, 3.4], etc. to ifiity, except the small regios of covergece ad bifurcatio, where the chaotic behavior occurs. I Fig. 3 (e), the Lyapuov expoet has a positive value at all values of r except small rages of covergece ad bifurcatio. Hece, the proposed Quadratic map 2 exhibits a chaotic behavior i the rest of the rage. The MLE of the proposed Quadratic map 2 is 2.257, which is much greater tha the classical Quadratic map ad proposed Quadratic map Aalysis of the Proposed Quadratic Map 3 The equatio of the proposed Quadratic map 3 at a= 8 is: X + = (r+(-8x ) 2 )mod (6) Fig. 4 shows the aalysis of the proposed Quadratic map 3. From Fig. 4 (a), (b), (c), ad (d), it is clear that the proposed Quadratic map 3 exhibits a chaotic behavior at all values of r except the values.,., etc. to ifiity. I Fig. 4 (e), the Lyapuov expoet has a positive value at all values of r except the values of.,., etc. So, the proposed Quadratic map 3 exhibits a chaotic behavior at the rest of the rage. The MLE of the proposed Quadratic map 3 is 3.479, which is much greater tha the classical Quadratic map ad the proposed Quadratic map ad 2. Table () summarized the aalysis of the classical ad proposed quadratic maps. It shows the improvemet i both the chaotic parameter rage r ad MLE.

5 Lyapuov expoet X+ X+ X+ Lyapuov expoet X+ X+ X+ America Joural of Sigal Processig 26, 6(): -3 5 r = r =.8 r = (a) Iteratio property at r=. (b) Iteratio property at r=.8. (c) Iteratio property at r= (d) Bifurcatio diagram r (e) Lyapuov expoet. Figure 3. Aalysis of the proposed Quadratic map 2 at x =.2 r = 2 r =. r = (a) Iteratio property at r=. (b) Iteratio property at r=.. (c) Iteratio property at r= (d) Bifurcatio diagram r (e) Lyapuov expoet. Figure 4. Aalysis of the proposed Quadratic map 3 at x =.2

6 Lyapuov expoet 6 Noha Ramada et al.: Chaos-Based Image Ecryptio Usig a Improved Quadratic Chaotic Map Table (). Compariso betwee the classical ad proposed Quadratic maps Chaotic map Equatio Chaotic parameter rage MLE Classical Quadratic Map X + = r (x ) 2 r [.4, 2].672 Proposed Quadratic Map X + = (r+(-2x ) 2 )mod r [,.4], r [.56, 2.4], r [2.56,3.4] to.6732 Proposed Quadratic Map 2 X + = (r+(-4x ) 2 )mod r [,.37], r [.4, 2.4], r [.4,3.4] to Proposed Quadratic Map 3 X + = (r+(-8x ) 2 )mod All values to except (r=.,.,etc ) Applicatio o Image Ecryptio Now, we will itroduce the proposed image ecryptio scheme based o two chaotic maps; the Chepyshev map ad the proposed Quadratic maps we have just costructed. 4.. Chepyshev Chaotic Map The Chepyshev chaotic map is a oe dimesioal chaotic system with oe iitial coditio x ad oe cotrol parameter k ad ca be described as follows [22]: y + =cos (k cos - (y )) (7) where y [, ] for =,,2, ad k [2, ). The bifurcatio diagram of the Chepyshev map i Fig. 4 shows that all the (y, k) where y [-, ] ad 2 k < ca be used as secret keys. The Chepyshev map has a positive icreasig Lyapuov expoet at k >= 2, ad thus, it is always chaotic as show i Fig The Permutatio Process We use Chepyshev chaotic map to geerate chaotic sequeces x ad the sort that chaotic umbers i ascedig or descedig order for the geeratio of the permutatio key. For a 256-level gray-scale image, y =.97 ad k= We sort the chaotic sequeces i the idex matrix used i shufflig the origial image to obtai the permuted image. The origial Camerama image ad the permuted image after permutatio are show i Fig. 7 ad Fig. 8. After obtaiig the shuffled image, the correlatio amog the adjacet pixels is completely disturbed ad the image is completely urecogizable. The histogram of the permuted image ad the origial image are the same, sice there is o chage i the itesity of pixels as show i Fig. 9 ad Fig.. Therefore, the permuted image is weak agaist statistical attack, ad kow plai-text attack. As a result, we employ a diffusio process after permutatio to improve the security. Figure 4. Bifurcatio diagram of the Chepyshev chaotic map at x =.2 Figure 7. Origial image k Figure 5. Lyapuov expoet of the Chepyshev chaotic map Figure 8. Permuted image

7 America Joural of Sigal Processig 26, 6(): Figure 9. Histogram of the origial image Figure. Histogram of the permuted image 4.3. Image Diffusio Based o the Proposed Quadratic Chaotic Maps The diffusio step i the proposed ecryptio scheme is performed by the key related to the plai image algorithm which used oly oe roud diffusio operatio ad its key depeds o the iitial key ad the origial image [23]. We will use the proposed Quadratic maps ad also the classical Quadratic map i the diffusio process ad compare betwee them. After compariso, we will get the chaotic map which has the best performace. We will treat the image as a oe dimesioal vector im = {im, im 2 im L } of legth L= M N. The iitial value x =.2. We will discuss the ecryptio process oly, because the decryptio is the reverse process. The details of the ecryptio process ca be summarized as follows: Step : for =, iterate the classical ad proposed Quadratic maps usig Eqs. (), (4), (5) ad (6) for oly oe time to get x. Step 2: modify x accordig to the followig equatio, where im is ay arbitrary image pixel. x = mod (x + (im + )/255,) (8) Step 3: for =+ retur to step util =L to get x L. Let the ew iitial value of the proposed Quadratic map 3 be (x +x L )/2. Step 4: iterate the proposed Quadratic map 3 usig Eq. (8) for L times with the ew iitial value. The, we obtai the sequece. X = x L+, x L+2,, x 2L (9) Step 5: to get the sequece K= {k, k 2,,K L } use k = mod floor x L+ 5, 256 () Where =, 2,.., L Step 6: examie the radomess of the sequece k : H = rustest(k ) () By usig the Matlab fuctio (rustest) for radomess, H returs () if the sequece is radom ad () if ot. I our case, H=. Step 7: compute the first cipher pixel by usig the value of im, the costat c, ad the first key k. c = k mod im + c, 256 (2) Step 8: let =+ Step 9: compute the th pixel of the cipher image by usig the followig equatio i which the cipher output feedback is itroduced. c = k mod im + im, 256 (3) Step : repeat step 8 ad step 9 util reaches L, ad the the cipher image C= {c, c 2,,c } is obtaied. 5. The Proposed Scheme The proposed ecryptio scheme is based o a permutatio-diffusio architecture. The first stage is applyig the permutatio process to the origial image. The, the permuted image is subjected to the diffusio process. The permutatio is achieved usig Chepyshev map. The diffusio process is achieved by the key related to the plai image algorithm based o the proposed Quadratic maps we have just costructed. The decryptio process is simply the reverse of the ecryptio process. See Fig.. 6. Performace Aalysis The quality of the ecryptio algorithm is its ability to resist differet kids of kow attacks such as kow/chose plai-text attack, cipher-text oly attack, statistical attack, differetial attack, ad various brute-force attacks. We will examie the proposed algorithm by measurig security, statistical, ad sesitivity aalysis o differet images. 6.. Security Aalysis 6... Key Space Aalysis The key space is the total umber of differet keys that ca be used i the ecryptio process. The proposed algorithm cosists of two processes; permutatio ad diffusio. I the permutatio process, we use Chepyshev map with two idepedet variables y ad k. I the diffusio process, the Quadratic map has two idepedet variables x ad r. I the key related to the plai text algorithm, we have a costat

8 8 Noha Ramada et al.: Chaos-Based Image Ecryptio Usig a Improved Quadratic Chaotic Map iteger c ad c [, 255]. As a result, the key space is {y, k, x, r, c}. Sice y, k, x ad r are double precisio umbers, the total umber of differet values for y, k, x ad r is more tha 4. So, the key space is larger tha This large key space is eough to resist brute-force attack. O the other had, the key space of RC6, DES ad chaotic Baker map for the same 256 level gray-scale image is 2 28, 2 56, ad 2 63 respectively [24]. See Table (2) Table (2). The key space compariso The ecryptio scheme The key space The proposed hybrid chaotic scheme RC DES 2 56 Chaotic Baker map Statistical Aalysis I order to prove the security of the proposed ecryptio algorithm, the followig statistical tests are performed. (a) Histogram Histogram clarifies that how pixels i a image are distributed by plottig the umber of pixels at each gray-scale level. The histogram of the origial Camerama, Madrill ad Lea images ad their ecrypted versios with the classical Quadratic map, the proposed Quadratic map, 2 ad 3 are show i Figs. 2, 3 ad 4. These figures show that the histograms of the ecrypted images of all Quadratic maps are fairly uiform ad completely differet from those of the origial images. (b) Correlatio Coefficiet For a origial image, adjacet pixels have a large correlatio. For a ecrypted image, the correlatio betwee pixels should be as small as possible. The closer the value of correlatio coefficiet (CC) to zero, the better is the ecryptio. If the correlatio coefficiet equals zero, the the origial image ad its ecrypted versio are totally differet. So, the success of the ecryptio process meas smaller values of the correlatio coefficiet. The CC is measured by the followig equatio [25]: CC N i ( x E( x))( y E( y)) i N 2 N 2 x i E x y i i i E y ( ( )) ) ( ( )) ) N Where ( E( x) x i i ) (4) N where x ad y are the gray-scale pixel values of the origial ad ecrypted images. Table (3) shows the CC of the proposed scheme usig the classical ad proposed Quadratic maps, RC6, DES ad chaotic Baker map for differet test images. The simulatio results show that all the ecryptio algorithms listed have a very small CC, ad the proposed scheme based o the proposed Quadratic map 3 has the best CC. (c) Maximum Deviatio The Maximum Deviatio (MD) measures the quality of the ecryptio i terms of how it maximizes the deviatio betwee the origial ad the ecrypted images [26]. The higher the value of MD, the more the ecrypted image is deviated from the origial image. Table (3) shows that the MD of the proposed scheme based o the proposed Quadratic map 3 is the highest compared to the classical ad the proposed Quadratic maps, 2. The DES has the best MD for Camerama image, oly. The worst result of this test was foud i the chaotic Baker chaotic with a result of zero, because this algorithm depeds oly o permutatio. (d) Irregular Deviatio The Irregular Deviatio (ID) is based o how much the deviatio caused by ecryptio is irregular [27]. The lower the ID value, the better the ecryptio algorithm. Table (3) shows that the ID of the proposed scheme with the proposed Quadratic map 3 has the smallest value for all tested images. i Origial Image Decrypted Image Permutatio (Chepyshev Map) Re-orderig (Chepyshev Map) Diffusio (Proposed Quadratic Map) (Proposed Quadratic Map) Decryptio Ecrypted Image Figure. The proposed scheme

9 America Joural of Sigal Processig 26, 6(): (a) Origial Camerama image. (b) Histogram of origial Camerama image. (c) Ecrypted Camerama image usig the classical Quadratic map. (d) Histogram of a ecrypted Camerama image usig the classical Quadratic map (e) Ecrypted Camerama image usig the proposed Quadratic map. (f) Histogram of a ecrypted Camerama image usig the proposed Quadratic map. (g) Ecrypted Camerama image usig the proposed Quadratic map 2. (h) Histogram of a ecrypted Camerama image usig the proposed Quadratic map (i) Ecrypted Camerama image usig the proposed Quadratic map 3. (j) Histogram of a ecrypted Camerama image usig the proposed Quadratic map 3. Figure 2. Camerama image ecryptio with the classical ad proposed Quadratic maps (a) Origial Madrill image. (b) Histogram of origial Madrill image. (c) Ecrypted Madrill image usig the classical Quadratic map. (d) Histogram of a ecrypted Madrill image usig the classical Quadratic map (e) Ecrypted Madrill image usig the proposed Quadratic map. (f) Histogram of a ecrypted Madrill image usig the proposed Quadratic map. (g) Ecrypted Madrill image usig the proposed Quadratic map 2. (h) Histogram of a ecrypted Madrill image usig the proposed Quadratic map 2.

10 Noha Ramada et al.: Chaos-Based Image Ecryptio Usig a Improved Quadratic Chaotic Map (i) Ecrypted Madrill image usig the proposed Quadratic map 3. (j) Histogram of a ecrypted Madrill image usig the proposed Quadratic map 3. Figure 3. Madrill image ecryptio with the classical ad proposed Quadratic maps (a) Origial Lea image. (b) Histogram of origial Lea image. (c) Ecrypted Lea image usig the classical Quadratic map. (d) Histogram of a ecrypted Lea image usig the classical Quadratic map (e) Ecrypted Lea image usig the proposed Quadratic map. (f) Histogram of a ecrypted Lea image usig the proposed Quadratic map. (g) Ecrypted Lea image usig the proposed Quadratic map 2. (h) Histogram of a ecrypted Lea image usig the proposed Quadratic map (i) Ecrypted Lea image usig the proposed Quadratic map 3. (j) Histogram of a ecrypted Lea image usig the proposed Quadratic map 3. Figure 4. Lea image ecryptio with the classical ad proposed Quadratic maps

11 America Joural of Sigal Processig 26, 6(): -3 Table (3). CC, MD, ID ad processig time of RC6, DES, Baker ad all Quadratic maps for Camerama, Madrill ad Lea images Camerama image The ecryptio scheme CC MD ID Processig time Classical Quadratic map Sec Proposed Quadratic map Sec Proposed Quadratic map Sec Proposed Quadratic map Sec RC Mis DES Sec Chaotic Baker map Sec Madrill image Classical Quadratic map Sec Proposed Quadratic map Sec Proposed Quadratic map Sec Proposed Quadratic map Sec RC Mis DES Sec Chaotic Baker map Sec Lea image Classical Quadratic map Sec Proposed Quadratic map Sec Proposed Quadratic map Sec Proposed Quadratic map Sec RC Mis DES Sec Chaotic Baker map Sec Sesitivity Aalysis I geeral, a ecrypted image must be sesitive to the small chages i the origial image ad secret key. I order to avoid differetial attack, a small chage i the plai image or secret key should cause a sigificat chage i the ecrypted image. Two parameters were used for differetial aalysis; Net Pixel Chage Rate (NPCR) ad Uified Average Chagig Itesity (UACI) [28]. NPCR measures the umber of pixels chage rate of ecrypted image while oe pixel of the origial image is chaged. UACI measures the average itesity of the differeces betwee those two images. The NPCR ad UACI of two ecrypted images are defied i Eqs. (5), (6) respectively. C ad C2 are two ecrypted images, whose correspodig plai images have oly oe pixel chage. NPCR D ( i, j ) i, j % W H C ( i, j) C ( i, j) UACI W H 255 (5) 2 % (6) i, j where D(i,j) represets the differece betwee C(i,j) ad C2(i,j). If C(i,j)= C2(i,j) the D(i,j)=, otherwise D(i,j)=. We ca use these two parameters to measure the key sesitivity as follows. (a) Key Sesitivity We obtaied the NPCR ad UACI of the ecrypted Camerama image uder the chage of the value of x ad r, o which the secret key depeds. We used x =.2 ad r= 2 as the first set of key, ad the chaged it. Table (4) shows the values of NPCR ad UACI betwee ecrypted images with keys (x, r) ad aother slightly differet key (Δx, Δr). Our results show that for all ecryptio algorithms with classical ad proposed Quadratic maps, more tha 99% of the pixels i the ecrypted image chage their gray values, whe the key is just chaged by -4. The ecryptio algorithm with the proposed Quadratic map 3 has the highest chage rate. This meas that the proposed scheme provides high key sesitivity. For a 256 gray-level image, the expected UACI value is 33% ad the proposed scheme has a UACI value equal to %. Furthermore, the proposed scheme has the best results for NPCR ad UACI amog RC6, DES ad chaotic Baker map ecryptio algorithms.

12 2 Noha Ramada et al.: Chaos-Based Image Ecryptio Usig a Improved Quadratic Chaotic Map Table (4). NPCR ad UACI betwee ecrypted Camerama images The ecryptio scheme Classical Quadratic map Proposed Quadratic map Proposed Quadratic map 2 Proposed Quadratic map 3 RC6 DES Chaotic Baker map 6.2. Processig Time Keys Δx = -4, Δr= NPCR UACI Δx =, Δr= Δx = -4, Δr= Δx =, Δr= Δx = -4, Δr= Δx =, Δr= Δx = -4, Δr= Δx =, Δr= The processig time is the time required to ecrypt ad decrypt a image. The smaller the processig time, the better is the ecryptio efficiecy. The proposed scheme uses oly oe roud for diffusio process, ad so this reduces the ecryptio/decryptio time, ad hece the scheme is practicable i real time applicatios. All the calculatios are performed usig MATLAB R27a software uder widows XP operatig system, processor core 2 duo,.6 GHz ad 2G RAM. For the 256 gray level Camerama, Madrill, ad Lea images, the processig time is listed i Table (3). It is clear that the processig time of the proposed scheme is withi 75 secods, which is smaller tha the processig time of RC6, ad DES. Chaotic Baker map has the smallest processig time, because it is just a permutatio map. 7. Coclusios I this paper, ew Quadratic chaotic maps with better chaotic properties have bee proposed. These maps icrease the available chaotic rage of parameter r to ifiity, ad hece are more robust agaist attacks compared to the limited value of parameter r i the classical Quadratic map. The MLE i the proposed Quadratic maps is icreased compared to the classical Quadratic map up to 3.479, while it was.672 i the classical Quadratic map. Therefore, the proposed Quadratic maps are more effective tha the classical Quadratic map i the ecryptio process ad more sesitive to the iitial coditios. A ecryptio scheme of permutatio ad diffusio combiig two chaotic maps has bee preseted ad ivestigated. I the permutatio process, we sort chaotic sequeces of the Chepyshev map to shuffle the image. This procedure avoids the cycle of chaotic umbers i the geeratio of the permutatio key. I the diffusio process, the permuted image is the ecrypted by the key related to the plai image algorithm usig the proposed Quadratic maps. I this ecryptio scheme, the key depeds o both the iitial key ad the origial image. So it ca survive kow plai text attack. This algorithm uses a sigle roud of diffusio, ad hece it has low computatioal complexity. The proposed ecryptio scheme effectively resists the brute-force attack due to its very large key space. Furthermore, security, statistical, ad sesitivity aalysis has bee carried out demostratig high security ad robustess of the proposed scheme. This proposed scheme is suitable for real-time image ecryptio applicatios due to its small processig time compared to the other ecryptio schemes. A comparative study have bee preseted betwee the proposed scheme, RC6, DES ad chaotic Baker map. The results of this compariso show that the proposed scheme usig the proposed Quadratic maps has a superior performace. REFERENCES [] E B William, (2), Data Ecryptio Stadard, i NIST's athology, A Cetury of Excellece i Measuremets Stadards ad Techology: A Chroicle of Selected NBS/NIST Publicatios. [2] FIPS PUB 97, (2), Advaced Ecryptio Stadard (AES), Natioal Istitute of Stadards ad Techology, U.S. Departmet of Commerce. [3] C.E. Shao, Commuicatio Theory of Secrecy Systems, Bell System Techical Joural, vol. 28, No. 4, pp , October 949. [4] Li, T.Y., Yorke, J.A. Period three implies chaos. The America Mathematical Mothly 82, 985{992 (975). [5] Matthews, R. Cryptologia 989, XIII, [6] Habutsu, T.; Nishio, Y.; Sasase, I.; Mori, S., Advacesicryptology-Eurocrypt 9, Lecture otes i computers ciece 547, Pp.27-4, Spiger-Verlag, Berli, 99. [7] Fridrich, J. It. J. Bifurcatio ad Chaos 998, 8, [8] Baptista, M. S. Phys. Lett. A 998, 24, [9] Alexader N. Pisarchik, Massimiliao Zai, Chaotic map cryptography ad security I: Ecryptio: Methods, Software ad Security Editor: Editor Name, pp. -28, 2 Nova Sciece Publishers, Ic. [] Kelber, K.; Schwarz, W. NOLTA 25, Bruges. [] Fial_Report/quadraticmap.pdf [2] S Li, (27), Aalyses ad ew desigs of digital chaotic ciphers, Ph.D. dissertatio, iformatio ad commuicatios egieerig, Xi a Jiaotog Uiveristy, Chia. [3] M. T. Rosestei, J. J. Collis, ad C. J. de Luca, (993), A practical method for calculatig largest Lyapuov expoets from small data sets, Physical D: Noliear Pheomea, vol. 65, o. -2, pp [4] Z. Liu, Q. Guo, L. Xu, M.A. Ahmad, S. Liu, (2) Double image ecryptio by usig iterative radom biary, Optics

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