A medical image encryption algorithm based on synchronization of time-delay chaotic system

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1 Av. Manuf. (1) 5:15 1 DOI 1.1/s A meical image encryption algorithm base on synchronization of time-elay chaotic system Hua Wang 1 Jian-Min Ye 1 Hang-Feng Liang 1 Zhong-Hua Miao 1 Receive: 9 October 1 / Accepte: 5 April 1 / Publishe online: 1 June 1 Ó Shanghai University an Springer-Verlag Berlin Heielberg 1 Abstract This paper presents a new synchronization metho of the time-elay chaotic system an its application in meical image encryption. Compare with the existing techniques, the error system is greatly simplifie because many couple items can be consiere zero items. An improve image encryption scheme base on a ynamic block is propose. This scheme ivies the image into ynamic blocks, an the number of blocks is etermine by a previous block cipher. Numerical simulations are provie to illustrate the effectiveness of the propose metho. Keywors Time-elay chaotic system Zero items Meical image encryption Chaos synchronization 1 Introuction With the rapily eveloping electronic technology, hospitals are aopting an increasing number of electronic equipment, incluing the picture archiving an communication systems (PACS) [1]. In such systems, the iagnosis results of patients are store in the form of igital images. These images contain the patients personal information. Ranom access to these images may cause serious social problems []. Consequently, storing an safely transferring patient ata are presently becoming hot topics. & Zhong-Hua Miao zhhmiao@shu.eu.cn Hua Wang wanghua91@13.com 1 Shanghai Key Laboratory of Power Station Automation Technology, School of Mechatronic Engineering an Automation, Shanghai University, Shanghai, People s Republic of China Using cryptography technology to protect igital information is an effective metho. In the 19s, conventional encryption algorithms, such as ata encryption stanar (DES), avance encryption stanar (AES) an international ata encryption algorithm (IDEA) [3] processe information as a binary stream. A igital image has its own characteristics, such as high correlation in ajacent pixels, ata reunancy, uneven istribution of energy, etc. []. Hence, the conventional encryption algorithm cannot be successfully applie to a igital image. In recent years, increasing attention has been focuse on the research of an image encryption technique base on chaos synchronization [5]. Many methos were aopte to realize chaos control an synchronization. These inclue the control Lyapunov function [], nonlinear control [], aaptive control [], back stepping control [9], passive control [1], an sliing-moe control [11] methos, etc. In a real physical process, some time-elays are trivial, such that they can be ignore. However, others cannot be isregare, particularly in long-istance communication an with transmission congestion. Therefore, stuying the synchronization in such systems with elays is important an necessary [1 15]. In this stuy, we mainly focuse on the meical image encryption technology base on chaos synchronization. Firstly, we eal with the problem of esigning a linear controller to realize the chaotic synchronization for the time-elay chaotic system. Subsequently, we apply the newly evelope metho in the research of meical image encryption. Thirly, some simulation works are presente to illustrate the feasibility of the propose algorithm. Time-elay chaotic system synchronization Some necessary lemmas are introuce herein before the main results are presente.

2 A meical image encryption algorithm base on synchronization of time-elay chaotic system 159 Suppose C ¼ Cð½ r; Š; R n Þ is the Banach space of continuous functions mapping the interval ½ r; Š into R n with the topology of uniform convergence. Let x t C be efine by x t ðhþ ¼xðt þ hþ; r h. The following autonomous equation must be consiere as _x ¼ f ðx t Þ; ð1þ where f : C! R n is a completely continuous function, an the solutions of Eq. (1) continuously epen on the initial ata. Lemma 1 [1] Let Vðt; x t Þ be a ifferentiable scalar functional for all t > an x t C. M is a positive number. If there is a boune continuous function U 1 ½; 1Þ! ½; 1Þ, which is L 1 ½; 1Þ with _U 1 ðtþ\, U 1 ðtþ! as t!1, such that ( R t W 1 ðkk x Þ Vðt;x t Þ W ðkxðtþkþþw 3 a U 1ðt sþw ðkxðsþkþs ; _Vðt;x t Þj ð5þ W ðkxðtþkþþm; ðþ where W i ði ¼ 1; ; 3; Þ is a wege, an the solutions of Eq. (1) are uniformly boune. Lemma [1] We are particularly intereste in the globally asymptotically stable (GAS) problem of the cascae elay system in the following form _g ¼ f 1 ðg t Þ; ð3þ _w ¼ f ðw t ; g t Þ; where w t ¼ wðt þ hþ; g t ¼ gðt þ hþ; h ½ s; Š; w R n 1 ; g R n. We assume that f 1 an f are completely continuous. They satisfy enough aitional smoothness conitions to ensure the solutions of Eq. (3) continuously epening on the initial ata. We also assume that f 1 ðþ ¼ an f ð; Þ ¼. If (i) the equilibrium g ¼ of system _g ¼ f 1 ðg t Þ is GAS; (ii) the equilibrium w ¼ of w _ ¼ f ðw t ; Þis GAS; an (iii) the solution of the whole system (Eq. (3)) is boune. The equilibrium w ¼ ; g ¼ of the whole system (Eq. (3)) is GAS. Lemma 3 [1] Let a 1 an b 1 be the two n-imensional vectors, an ^g be an arbitrary positive real number. The following inequality then hols a T 1 b 1 ^ga T 1 a 1 þ ^g b T 1 b 1: ðþ This stuy aims to esign a linear controller for the chaotic secure communication system with state elay to realize the synchronization between the transmission an the reception ens. Therefore, the original message can be recovere at the reception en. Consier the following time-elay chaotic system as the transmitter system >< >: _x 1 ¼ ax 1 ðt sþþbx ; _x ¼ ^cx 1 hx ðt sþ x 1 x 3 ; _x 3 ¼ x mx 3 ðt sþþx 1 x ; vðtþ ¼c 1 x 1 þ c x þ sðtþ; ð5þ where x 1 ; x ; an x 3 are state variables; a; b; ^c; ; h an m are the system parameters; s is the time lag; sðtþ is the signal to be transmitte; an vðtþ is the transmitter system output. The system (Eq. (5)) may be in the chaotic state through a suitable selection of the system parameters an time elay. The system (Eq. (5)) is in the chaotic state (see Fig. 1) whena ¼ 3 ; b ¼ 1; ^c ¼ 1, ¼ ; h ¼ 1; m ¼ 1; an s ¼. The receiver system is presente as _y 1 ¼ ay 1 ðt sþþby þ u 1 ; >< _y ¼ ^cy 1 hy ðt sþ y 1 y 3 þ u ; _y 3 ¼ y my 3 ðt sþþy 1 y þ u 3 ; >: w ¼ c 1 y 1 þ c y ; ðþ where w is the output of the receiver system, an u 1 ; u an u 3 are the controllers to be esigne. We assume here that all the states y 1 ; y an y 3 are available for the feeback controller. The synchronization errors are enote as e i ¼ y i x i ði ¼ 1; ; 3Þ. Subtracting Eq. () from Eq. () yiels the following error system < : _e 1 ¼ ae 1 ðt sþþbe þ u 1 ; _e ¼ ^ce 1 he ðt sþ e 1 e 3 e 1 x 3 e 3 x 1 þ u ; _e 3 ¼ e me 3 ðt sþþe 1 e þ e 1 x þ e x 1 þ u 3 : ðþ The slave system (Eq. ()) can successfully synchronize the rive system (Eq. (5)) if the synchronization error system (Eq. ()) is stable. We will aopt three steps in the following to esign the linear controllers that attain this goal. Step 1 Design u 1 ¼ be k 1 e 1, where the controller parameter k 1 can be etermine later by the solution of a Fig. 1 Chaotic attractor of the time-elay system (Eq. (5))

3 1 H. Wang et al. linear matrix inequality (LMI). The first equation of the error system (Eq. ()) will be presente as follows when the controller is applie _e 1 ¼ ae 1 ðt sþ k 1 e 1 ðtþ: ðþ In the following, we will prove that the error system (Eq. ()) is asymptotically stable an boune. The following caniate of the Lyapunov-Krasovskii function is consiere as V 1 ¼ 1 e 1 þ 1 e 1 ðsþs: ð9þ Its erivation along the solution of the system (Eq. ()) is then calculate, an we obtain _V 1 ¼ e 1 ð ae 1 ðt sþ k 1 e 1 ðtþþþe 1 ðtþ e 1ðt sþ ¼ ae 1 e 1 ðt sþ k 1 e 1 þ e 1 e 1ðt sþ k 1 þ 1 a 3 ð1þ ¼ ½e 1 e 1 ðt sþ Š a e 1 5 : e 1 ðt sþ We know from the Lyapunov-Krasovskii stability theory that the error system (Eq. ()) will be asymptotically stable an boune if a scalar k 1 exists, such that the following LMI is solvable X 1 ¼ k 1 þ 1 a 3 a 5\: ð11þ In other wors, a constant q 1 exists, such as je 1 j q 1. Step Consier the secon an thir equations of the error system (Eq. ()) _e ¼ ce 1 he ðt sþ e 1 e 3 e 1 x 3 e 3 x 1 þ u ; _e 3 ¼ e me 3 ðt sþþe 1 e þ e 1 x þ e x 1 þ u 3 : ð1þ The system (Eq. (1)) can be simplifie as follows when e 1 ¼ _e ¼ he ðt sþ e 3 x 1 þ u ; ð13þ _e 3 ¼ e me 3 ðt sþþe x 1 þ u 3 : The linear controllers shoul then be esigne as u ¼ k e ðtþ; u 3 ¼ k 3 e 3 ðtþ, where the controller gains k ; k 3 are to be later etermine by the solution of an LMI. Consier the following caniate of the Lyapunov- Krasovskii function V ¼ 1 e ðtþþ1 e ðsþs þ 1 e 3 ðtþþ1 e 3 ðsþs: ð1þ Its erivation along the solution of the error system (Eq. (13)) is calculate to obtain _V ¼ e ð he ðt sþ k e ðtþþþe ðtþ e ðt sþ þ e 3 ðe me 3 ðt sþ k 3 e 3 ðtþþþe 3 ðtþ e 3ðt sþ ¼ ½e e ðt sþ e 3 e 3 ðt sþ Š k þ 1 h 3 3 h e e ðt sþ k 3 þ 1 m m 5 e 3 e 3 ðt sþ 5 : ð15þ We know from the Lyapunov-Krasovskii stability theory that the simplifie error system (Eq. (13)) is asymptotically stable if two real numbers (i.e., k an k 3 ) exist, such that the following LMI is solvable k þ 1 h 3 h X ¼ k 3 þ 1 m \: ð1þ m 5 We can see that the error system (Eq. (13)) is greatly simplifie than Eq. (1) because we can consier e 1 as zero. Any item couple with e 1 can be treate as zero, which is one avantage of the propose metho. Step 3 In this step, we will prove that the solution of the whole error system (Eq. ()) is boune. Step 1 showe that the error state e 1 was boune by a positive constant q 1. Hence, we only nee to prove that the error states e an e 3 are boune uner the suppression of the linear controller u ; u 3. Consier the following caniate positive function V 3 ¼ 1 e ðtþþ1 e ðsþs þ 1 e 3 ðtþþ1 e 3 ðsþs: ð1þ Its erivation along the error system (Eq. ()) is calculate to obtain _V 3 ¼ e ðce 1 he ðt sþ e 1 e 3 e 1 x 3 e 3 x 1 k e ðtþþþe ðtþ e ðt sþþe 3ðe me 3 ðt sþþe 1 e þ e 1 x þ e x 1 k 3 e 3 ðtþþ þ e 3 ðtþ e 3ðt sþ ¼ ½e e ðt sþ e 3 e 3 ðt sþ Š 1 k h 3 3 h e e ðt sþ 1 k 3 m m 5 þ ce 1 e þ e 1 e 3 x e 1 e x 3 : e 3 e 3 ðt sþ 5 ð1þ

4 A meical image encryption algorithm base on synchronization of time-elay chaotic system 11 We know from Step 1 that je 1 j is boune by je 1 j q 1. The transmitter system (Eq. ()) is a chaotic system; hence its states are boune. Some positive constants, such jx i j k i, are then foun to exist. We can obtain the following equation from Lemma 3 ce 1 e ¼ 1 ce 1 e 1 g 1 c q 1 þ g 1e ; >< e 1 e 3 x ¼ 1 e 1x e 3 1 g k q 1 þ g e 3 ; ð19þ e 1 e x 3 ¼ 1 >: e 1x 3 e 1 g 3 k 3 q 1 þ g 3e ; where g i ði ¼ 1; ; 3Þ are arbitrary positive constants. Subsequently, we can obtain the following inequality _V 3 ½e e ðt sþ e 3 e 3 ðt sþ Š 1 k þ g 1 þ g 3 h 3 h 1 k 3 þ g m m 5 3 e e ðt sþ e 3 5 þ 1 g 1 c q 1 þ 1 g k q 1 þ 1 g e 3 ðt sþ Denote 3 k 3 q 1 : ðþ M ¼ 1 g 1 c q 1 þ 1 g k q 1 þ 1 g 3 k 3 q 1 ; 1 k þ g 1 þ g 3 h 3 h U ¼ 1 k 3 þ g m : m 5 ð1þ We can see that M is a positive constant. We can conclue from Lemma 1 that the solutions of the error system (Eq. ()) will be boune if some positive real numbers g i ði ¼ 1; ; 3Þ an general real numbers k i ði ¼ ; 3Þ, such that the LMI U\, exist. Combining the proof process of Steps 1 3, we know from Lemma that the time-elay chaotic error system (Eq. ()) can be stabilize by the linear controller u i ði ¼ 1; ; 3Þ. 3 Simulation of the chaos synchronization system The simulation work herein was processe in MATLAB R1a. A DDE3 solver was use to solve the elay ifferential equations. The parameter values are as follows: a ¼ 3 ; b ¼ 1; c ¼ 1; ¼ ; h ¼ 1; m ¼ 1; an s ¼ : The simulation results of the transmitter-receiver chaotic system synchronization coul be seen in Figs. an 3. Figure shows that the trajectories of the receiver systems can asymptotically approach the trajectory of the transmitter chaotic systems, while Fig. 3 illustrates that the error states asymptotically approach zero. The controller propose was linear an easy to be applie in a real physical process. Image encryption base on chaos synchronization In the process of secure communication, the signal s to be transmitte is first embee in the chaotic signals. The synthesize signal will then be transmitte through a common channel. A slave chaotic system is use at the receiver terminal. The controller in the receiver terminal can rive the system to synchronize with the transmitter system, an the original signal s can be ecrypte. Figure shows a iagram of the whole process..1 Encryption process In this section, a meical skull image P measuring M N was use for encryption. x 1 ðþ; x ðþ an x 3 ðþ were the initial values of the transmitter system. Although the transmitter (Eq. (5)) an receiver (Eq. ()) systems were continuous systems, the simulation work was conucte using a computer, an iscrete sequences were prouce. Hence, we can irectly utilize the sequence prouce by MATLAB to o the encryption work. An encryption metho was propose in Ref. [19], shown as follows. Step 1 Transform the plain matrix P to a vector ~p of length M N. Step Divie the obtaine vector ~p into S blocks ~p j of length N k. N k is a variable value given by the following function. N kþ1 ¼ N k þðmeanð~p j Þ mo 5Þ; j¼ 1; ; ; N k ; ðþ where N 1 is etermine by the user an P N s k¼1 N k ¼ MN. Step 3 Generate a chaotic sequence ðx k j Þ; j ¼ 1; ; ; N k ; k ¼ 1; of length N k. The sequence is generate by iterating the chaos system given by Eq. (5).

5 1 H. Wang et al. Fig. State trajectories of the transmitter-receiver chaotic systems x i ; y i ði ¼ 1; Þ it into ifferent blocks was har. This metho was not suitable for the meical image, an realizing the encryption took much time. Figure 5 shows the encryption result. We aim to improve an optimize this metho in combination with the inherent characteristics of a meical image. Step Generate the cipher blocks ðc j Þ; j ¼ 1; ; ; N k by applying c j ¼ ~p j z 1 j ; j ¼ 1; ; ; N k: ðþ Fig. 3 States response of the error system Fig. Overall framework StepQuantizetheðx k j Þ; j ¼ 1; ; ; N k coefficients by applying Eq. (3). Here, enote the obtaine key stream as z k j ¼ððx k j 1 1 Þ mo 5Þ; j ¼ 1; ; ; N k ; k¼ 1; : ð3þ Step 5 Sort the chaotic sequences accoring to the ascening orer to get the aress sequence coe ^S MN an R MN. Accoring to the aress sequence coe ^S MN, sort the ^S MN blocks ~p j of length N k. In the same way, we sort the ~p j accoring to the aress sequence coe R MN. After verification, we foun that the metho of Ref. [19] was effective for the general image. The meical image mainly containe the black an white colors. Hence, iviing Step Concatenate. Firstly, let s construct a vector c of length MN from the blocks ðc j Þ; j ¼ 1; ; ; N k. Also, we can enote this vector as cðmnþ. Then two chaotic sequences ðx k j Þ; j ¼ 1; ; ; N k; k ¼ 1; of length M N can be generate by iterating the chaotic system (Eq. (5)). Step Embe the signal s in the output of the rive system by v ¼ c 1 x 1 þ c x þ s. Here, v is a vecotor containing M N elements an we can enote it as vðmnþ. Then the original image can be encrypte by reshaping the vector v to a matrix V of size M N. The whole encryption process can be escribe in Fig... Decryption process The image was encrypte after the abovementione steps have been performe. The image signal can by recovere if Fig. 5 Result of verification

6 A meical image encryption algorithm base on synchronization of time-elay chaotic system 13 5 Performance analyses 5.1 Visual impression Fig. Encryption flow chart the receiver system synchronizes the transmitter system. The ecryption process is the inverse process of the encryption process, an it is omitte here..3 Encryption an ecryption simulation experiment A meical skull image measuring was aopte in this section as the signal to be encrypte (see Fig. ). The encryption key was generate by the transmitter chaotic system. Figure shows the obtaine encryption image. The slave chaotic system (Eq. ()) was use to generate the ecryption key. The ecrypte image can be obtaine after the above ecryption process (see Fig. 9). The peak value signal-to-noise ratio (PSNR) is usually use to evaluate the visual impression of the cipher image. The cipher image was equal to a noise to the plain image. The encryption effect efine as follows will then be better when the PSNR value (R PSN ) is lower: R PSN ¼ 1 lg w max E MS ; ð5þ where w max is the maximum of the pixel color component; E MS ¼ðMNÞ P P i j ðp ij c ij Þ ; p ij an c ij represent the value of the pixel ði; jþ in the plain an cipher images, respectively. R PSN \ B means that the cipher image is impossible to ientify. Table 1 shows that the PSNR values of the encrypte color components are all less than 9. B an inicates that the information is effectively covere. 5. Statistical attack The stanar test meical image of size was selecte herein to test the property of the resisting statistical analysis. We ranomly selecte 1 pairs of two ajacent pixels, incluing vertically, horizontally, an iagonally ajacent pixels, to calculate the correlation of the two ajacent pixels. We then calculate the correlation coefficient of each pair using the two following formulas [] covðx; yþ ¼Eððx EðxÞÞðy EðyÞÞÞ; ðþ covðx; yþ r xy ¼ pffiffiffiffiffiffiffiffiffip ffiffiffiffiffiffiffiffiffi ; ðþ DðxÞ DðyÞ Fig. Original meical skull image Fig. 9 Decrypte skull Table 1 Value of PSNR in the encrypte image PSNR/B Fig. Encrypte skull Encrypte image 5. R 3.9 G.31 9 B

7 1 H. Wang et al. Table Correlation coefficient of ajacent pixels The original image The encrypte image The correlation of horizontal irection The correlation of vertical irection The correlation of iagonal irection where x an y are the gray-scale values of the two ajacent pixels in the image. The following iscrete formulas are employe in the numerical computation EðxÞ ¼ 1 N DðxÞ ¼ 1 N X N i¼1 X N i¼1 covðx; yþ ¼ 1 N x i ; ðx i EðxÞÞ ; X N i¼1 ðþ ð9þ ððx i EðxÞÞðy i EðyÞÞÞ: ð3þ Table shows the results of the horizontal, vertical, an iagonal irections. The correlation coefficients of the cipher images were very small, which implie that no etectable correlation existe between the original an its corresponing cipher images. Therefore, the propose algorithm ha high security against statistical attacks. Conclusions This paper propose a novel linear feeback controller to achieve synchronization an secure communication in time-elay chaotic systems. A secure message can be successfully recovere. The key has a long enough perio an can be use for meical image encryption. Compare with the general image, the meical image has large pieces of black an white areas. This characteristic enables the traitional encryption to have no obvious effect. The propose algorithm mae up for the shortcomings, an the numerical simulations were provie to show the effectiveness of our methos. Acknowlegements This project supporte by the National Natural Science Founation of China (Grant Nos , 31599), an the Science an Technology Commission of Shanghai Municipality (Grant No ). References 1. Zhu Z, Zhang W, Wong K et al (11) A chaos-base symmetric image encryption scheme using a bit-level permutation. Inf Sci 11(): Wang Y, Wong KW, Liao X et al (11) A new chaos-base fast image encryption algorithm. Appl Soft Comput 11(1): Patil P, Narayankar P, Narayan DG et al (1) A comprehensive evaluation of cryptographic algorithms: DES, 3DES, AES, RSA an blowfish. Proc Comput Sci :1. Lei ZK, Sun QY, Ning XX (1) Image scrambling algorithms base on knight-tour transform an its applications. J Chin Comput Syst 31(5): Chen GR, Dong XN (199) From chaos to orer: methoologies, perspectives an applications. Methool Perspect Appl Worl Sci Singap 31(): Wang H, Han ZZ, Zhang W et al (9) Synchronization of unifie chaotic systems with uncertain parameters base on the CLF. Nonlinear Anal Real Worl Appl 1():15. Chen MY, Han ZZ (3) Controlling an synchronizing chaotic genesio system via nonlinear feeback control. Chaos Solitons Fractals 1():9 1. Yassen MT () Aaptive chaos control an synchronization for uncertain new chaotic ynamical system. Phys Lett A 35(1): Li GH () Projective synchronization of chaotic system using back stepping control. Chaos Solitons Fractals 9(): Wang FQ, Liu CX () Synchronization of unifie chaotic system base on passive control. Phys D Nonlinear Phenom 5(1): Tavazoei MS, Haeri M () Determination of active sliing moe controller parameters in synchronizing ifferent chaotic systems. Chaos Solitions Fractals 3(): Mei J, Jiang MH, Wang B et al (13) Finite-time parameter ientification an aaptive synchronization between two chaotic neural networks. J Franklin Instit 35(): Aghababa MP, Khanmohammai S, Alizaeh G (11) Finitetime synchronization of two ifferent chaotic systems with unknown parameters via sliing moe technique. Appl Math Moel 35(): Wang H, Zhang XL, Wang XH et al (1) Finite time chaos control for a class of chaotic systems with input nonlinearities via TSM scheme. Nonlinear Dyn 9(): Jiang GP, Zheng WX, Chen GR () Global chaos synchronization with channel time-elay. Chaos Solitons Fractals (): 5 1. Burton T (195) Stability an perioic solutions of orinary an functional ifferential equations. Acaemic Press, Lonon 1. Wang H, Wu JP, Sheng XS et al (15) A new stability result for nonlinear cascae time-elay system an its application in chaos control. Nonlinear Dyn (1 ):1 1. Cao Y, Sun Y, Cheng C (199) Delay-epenent robust stabilization of uncertain systems with multiple state elays. IEEE Trans Automat Contr 3: Mannai O, Bechikh R, Hermassi H et al (15) A new image encryption scheme base on a simple first-orer time-elay system with appropriate nonlinearity. Nonlinear Dyn (1 ): Zhu CX, Huang DZ, Guo Y (1) A image encryption algorithm with multi chaotic maps an output feeback. Geomat Inf Sci Wuhan Univ 35(5):5 531