OCML-based colour image encryption

Chaos, Solitons and Fractals 40 (2009) 309–318 www.elsevier.com/locate/chaos

OCML-based colour image encryption Rhouma Rhouma a

a,*

, Soumaya Meherzi

a,b

, Safya Belghith

a

6’tel, Ecole Supe´rieure de Te´le´communications (Sup’com), BP Numero 50, 6014 Mtorech, Gabes, Tunisia b LSS/SUPELEC/CNRS, Plateau de Moulon, 91192 Gif-sur-Yvette, Cedex, France Accepted 23 July 2007

Abstract The chaos-based cryptographic algorithms have suggested some new ways to develop efficient image-encryption schemes. While most of these schemes are based on low-dimensional chaotic maps, it has been proposed recently to use high-dimensional chaos namely spatiotemporal chaos, which is modelled by one-way coupled-map lattices (OCML). Owing to their hyperchaotic behaviour, such systems are assumed to enhance the cryptosystem security. In this paper, we propose an OCML-based colour image encryption scheme with a stream cipher structure. We use a 192-bit-long external key to generate the initial conditions and the parameters of the OCML. We have made several tests to check the security of the proposed cryptosystem namely, statistical tests including histogram analysis, calculus of the correlation coefficients of adjacent pixels, security test against differential attack including calculus of the number of pixel change rate (NPCR) and unified average changing intensity (UACI), and entropy calculus. The cryptosystem speed is analyzed and tested as well.  2007 Elsevier Ltd. All rights reserved.

1. Introduction Digital chaotic ciphers are designed for digital computers. Chaotic systems have several appealing features for secure communications. In particular, such features can be connected to some conventional cryptographic properties of good ciphers [1], namely: (i) Ergodicity in chaos vs confusion in cryptography. (ii) Sensitive dependence on initial conditions and control parameters of chaotic maps vs diffusion property of a good cryptosystem (for a small change in the plaintext or in the secret key). (iii) Random-like behaviour of deterministic chaotic-dynamics which can be used for generating pseudo-random sequences as key sequences in cryptography. Conventional chaos-based cryptographic schemes, such as the well-known Baptista-type [2–9] and the Alvarez-type cryptosystems [10,11], have shown some inherent drawbacks [12–15]. In particular, security weakness even with chaotic dynamics completely hidden and slow performance speed due to analytical floating-point computation and small key *

Corresponding author. Tel.: +216 98967423. E-mail addresses: [email protected] (R. Rhouma), [email protected] (S. Meherzi), [email protected] (S. Belghith).

0960-0779/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.07.083

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space. Therefore it is difficult to promote the chaotic digital encryption into practical service. On the other hand, spatiotemporal chaotic systems [16–18], which are usually modelled by one-way coupled-map lattices (OCML), have been investigated in secure communications as an alternative to low-dimensional ones. Indeed, such systems are hyperchaotic, hence are assumed to enhance the cryptosystem security. Image encryption is somehow different from text encryption due to some inherent features of image, such as bulk data capacity and high correlation among pixels. Therefore, digital chaotic ciphers cited earlier and traditional cryptographic techniques such as DES, IDEA and RSA are no longer suitable for practical image encryption, especially for an on-line communication scenario. We give hereafter a short overview of the main recently proposed chaos-based image encryption schemes [19–25]. In [23], Behnia et al. have proposed an implementation of digital image encryption scheme based on the mixture of chaotic systems. They use high-dimensional chaotic systems such as a coupled map to enhance the cryptosystem security. Further in [24], Pareek et al. have proposed a new approach for image encryption based on chaotic logistic maps. An external secret key of 80-bit length and two chaotic logistic maps are employed. The initial conditions for the logistic maps are derived using the external secret key. Eight different operation-types are used to encrypt the image pixels. More recently, in [25], a fast chaos-based image encryption system with stream cipher structure has been proposed. The major core of the encryption system is a pseudo-random keystream generator based on a cascade of chaotic maps, serving the purpose of sequence generation and random mixing. In this paper, we propose a new OCML-based cryptosystem for colour image encryption operating as a symmetric stream-cipher. An external key of 192-bit length is chosen to generate the initial conditions and the parameters of the OCML by making some algebraic transformations so as to enhance the sensitivity to the change of any bit of the key. The three colour components (Red, Green, and Blue) are encrypted in a coupling fashion in such a way to strengthen the cryptosystem security. The paper is organised as follows. In Section 2, we present the proposed OCML-based image cryptosystem. We explain the session-key generation as well as the encryption and decryption functions. The performance analysis of the cryptosystem is addressed in Section 3. We check in particular the resistance on differential attack, the statistical analysis, the security-key analysis, the information entropy analysis and the speed performance. Finally, we draw some concluding remarks in Section 4.

2. Chaos-based image encryption scheme In this section, we present the proposed scheme for colour image encryption in a stream-cipher fashion. Firstly, the image (A) of size N · M is converted into its RGB components. Afterwards, each colour’s matrix (R, G or B) is converted into a vector of integers within f0; 1; . . . ; 255g. Each vector has a length of L = N · M. The so obtained three vectors represent then the plaintext P(3 · L) which will be encrypted. The OCML used for the plaintext encryption is given by [26]: for n = 1, . . ., L; j = 0, . . ., 3 xnþ1 ðjÞ ¼ ðaj xn ðjÞ þ bj xn ðj  1ÞÞmodð1Þ;

ð1Þ

where aj and bj are the system parameters; x0(j) and xn(0) are the initial conditions and the key sequence of the OCML, respectively. 2.1. Generation of the cryptosystem keys The proposed image encryption process utilizes a 192-bit-long external secret key (K). This key is divided into 32-bitlong blocks (Ki) referred to as session keys. Such session keys are then used for the generation of the initial conditions x0(j) and the parameters aj and bj of the OCML described in Eq. (1). The 192-bit-long external secret-key (K) is given by: K ¼ K 1; K 2; . . . ; K 6: To calculate the initial conditions x0(j) of the OCML, we generate four bytes Kij, i = 1, . . ., 6 as follows: K i1 ¼ ½ðK i  24Þ & 255; K i2 ¼ ½ðK i  16Þ & 255; K i3 ¼ ½ðK i  8Þ & 255; K i4 ¼ K i & 255:

ð2Þ

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311

From these Kij, we generate six numbers (of 32 bits) Qj as follows: Q1 ¼ ½K 01  24 þ ½K 54  16 þ ½K 13  8 þ K 44 ; Q2 ¼ ½K 02  24 þ ½K 34  16 þ ½K 23  8 þ K 62 ; Q3 ¼ ½K 03  24 þ ½K 64  16 þ ½K 33  8 þ K 22 ; Q4 ¼ ½K 04  24 þ ½K 44  16 þ ½K 53  8 þ K 14 ; Q5 ¼ ½K 05  24 þ ½K 14  16 þ ½K 43  8 þ K 52 ; Q6 ¼ ½K 06  24 þ ½K 24  16 þ ½K 63  8 þ K 32 ; where K 01 ¼ K 34  K 63  K 21 ; K 02 ¼ K 44  K 53  K 11 ; K 03 K 04 K 05 K 06

¼ K 14  K 43  K 51 ; ¼ K 24  K 33  K 61 ; ¼ K 64  K 23  K 31 ; ¼ K 54  K 13  K 41 :

The initial conditions are then derived as follows x0 ðjÞ ¼ bin2decðQjþ1 Þ=232 ;

for j ¼ 0; . . . ; 3:

ð3Þ

The parameters aj and bj are generated from Q5 and Q6 as follows: a0 ¼ bin2decðQ5 Þ=232 ak ¼ a0

for k ¼ 1; . . . ; 3:

ð4Þ

32

b0 ¼ 1 þ bin2decðQ6 Þ=2 bk ¼ b0

for k ¼ 1; . . . ; 3:

ð5Þ

The operation bin2dec(x) denotes the decimal equivalent of the binary number x, the operation x  y denotes a right shift of x by y bits; the x  y denotes a left shift of x by y bits; the & operator is bitwise AND. See Fig. 1 for more explanation. 2.2. Design of the encryption and decryption scheme 2.2.1. Encryption First, we generate the OCML dynamics by Eq. (1). Then, the encryption transformation of the proposed scheme is given by: C n ðkÞ ¼ ðP n ðkÞ þ intðxn ðkÞ  LÞ þ C n1 ðkÞÞmod256

for n ¼ 1; . . . ; L; k ¼ 1; . . . ; 3:

ð6Þ

The key sequence of the OCML xn(0) is given by: xn ð0Þ ¼ C n ð3Þ=28 :

ð7Þ

The rows of P being the R, G and B components, respectively, we can rewrite Eq. (6) as follows: C n ð1Þ ¼ ðRn þ intðxn ð1Þ  LÞ þ C n1 ð1ÞÞmod256; C n ð2Þ ¼ ðGn þ intðxn ð2Þ  LÞ þ C n1 ð2ÞÞmod 256; C n ð3Þ ¼ ðBn þ intðxn ð3Þ  LÞ þ C n1 ð3ÞÞmod 256: It is clear that the generation of the keystream (xn(j)) depends on the plaintext P through all the colour components (R, G, B), the length of the plaintext (L) and the cryptosystem keys. Moreover, owing to the coupling structure of the OCML, the stream cipher of the colour components depend on each other. These features strengthen the cryptosystem security.

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K i (32 bits)

a

K i1

K i2

8 bits

b

c

K i3

8 bits

8 bits

Ki4 8 bits

K 34 ⊕

K 63

⊕

K 21

K 01

8 bits

K 44 ⊕

K 53

⊕

K 11

K 02

8 bits

K 14 ⊕

K 43

⊕

K 51

K 03

8 bits

K 24 ⊕

K 33

⊕

K 61

K 04

8 bits

K 64 ⊕

K 23

⊕

K 31

K 05

8 bits

K 54 ⊕

K13

⊕

K 41

K 06

8 bits

K 01

K 54

K 13

K 44

Q1 (32 bits)

K 02

K 34

K 23

K 62

Q2 (32 bits)

K 03

K 64

K 33

K 22

Q3 (32 bits)

K 04

K 44

K 53

K 12

Q4 (32 bits)

α k = α 0 . for k = 1...3.

K 05

K 14

K 43

K 52

Q5 (32 bits)

K 06

K 24

K 63

K 32

Q6 (32 bits)

β 0 = 1 + bin 2dec(Q6 ) / 2 32. β k = β 0 . for k = 1...3.

x0 ( j ) = bin 2dec(Q j +1 ) / 2 32

for

j = 0...3 .

α 0 = bin 2dec(Q5 ) / 2 32.

Fig. 1. (a) Generation of Kij from the session key. (b) Construction of K0j. (c) Initial conditions and parameters generation.

In particular, the dependence on the length L makes the cryptosystem robust against the plaintext-attack presented in [27]. So the proposed scheme does not suffer from the cryptosystem weakness presented in [16–18]. 2.2.2. Decryption In the receiver side, the dynamics of the OCML are regenerated as follows: y nþ1 ðjÞ ¼ ðaj y n ðjÞ þ bj y n ðj  1ÞÞmodð1Þ

for n ¼ 1; . . . ; L; j ¼ 0; . . . ; 3:

ð8Þ

The decryption transformation is then, P n ðkÞ ¼ ðC n ðkÞ  intðy n ðkÞ  LÞ  C n1 ðk  1ÞÞmod 256; y n ð0Þ ¼ C n ð3Þ=28

for n ¼ 1; . . . ; L; k ¼ 1; . . . ; 3:

ð9Þ ð10Þ

We can rewrite Eq. (9) to give the pixels’ values in the RGB components: Rn ¼ ðC n ð1Þ  intðxn ð1Þ  LÞ  C n1 ð1ÞÞmod 256; Gn ¼ ðC n ð2Þ  intðy n ð2Þ  LÞ  C n1 ð2ÞÞmod 256; Bn ¼ ðC n ð3Þ  intðy n ð3Þ  LÞ  C n1 ð3ÞÞmod 256: Note that the receiver must have the same keystream to be able to decrypt the colour image, so it has the same secret key : K = K1, K2, . . ., K6. So that it is possible to generate the same initial conditions and parameters: y0(j) = x0(j),aj and bj for j = 0, . . ., 3. Figs. 2 and 3 show the encryption and the decryption of two 256 · 256 colour images of Lena and Baboon, respectively.

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Fig. 2. (a) Original image of Lena. (b) Encrypted image of Lena. (c) Decrypted image of Lena.

Fig. 3. (a) Original image of baboon. (b) Encrypted image of baboon. (c) Decrypted image of baboon.

3. Performance analysis In this section, the performance of the proposed image encryption scheme is analyzed in detail. We have made several tests to check the security of the proposed cryptosystem: statistical tests including histogram analysis, calculus of the correlation coefficients of adjacent pixels, test security against differential attack including calculus of the number of pixel change rate (NPCR), unified average changing intensity (UACI), and information entropy evaluation. 3.1. Differential attack As a general requirement for all the image encryption schemes, the encrypted image should be greatly different from its original form. Such difference can be measured by means of two criteria namely, the number of pixel change rate (NPCR) and the unified average changing intensity (UACI) [21,25]. The NP CRR,G,B is used to measure the number of pixels in difference of a colour component between two images. Let S(i,j) and S 0 (i,j) be the (i,j)th pixel of two images S and S 0 , respectively. the NPCRR,G,B can be defined as: P NPCRR;G;B ¼

i;j DR;G;Bði;jÞ

L

 100%;

where L is the total number of pixels in the image and DR,G,B(i,j) is defined as  0 if S R;G;B ði; jÞ ¼ S0R;G;B ði; jÞ; DR;G;Bði;jÞ ¼ 1 if S R;G;B ði; jÞ–S0R;G;B ði; jÞ;

ð11Þ

ð12Þ

where SR,G,B(i,j) and S 0R;G;B ði; jÞ are the values of the corresponding colour component red (R), green (G) or blue (B) in the two images, respectively. For instance, for two random images with 512 · 512 pixels and 24-bit true colour: NPCRR = NPCRG = NPCRB = 99.609375%.

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The second criterion, UACIR,G,B, is used to measure the average intensity difference in a colour component and can be defined as ! 1 X jS R;G;B ði; jÞ  S0R;G;B ði; jÞj  100%; ð13Þ UACIR;G;B ¼ L i;j 2BR;G;B  1 where BR,G,B is the number of bits used to represent the colour component of red, green or blue, respectively. In the case of two random images, the expected value of UACIR,G,B is UACIR ¼ UACIG ¼ UACIB ¼ 33:46354%: We have computed the NPCRR,G,B and the UACIR,G,B with the proposed cryptosystem to assess the influence of changing a single pixel in the original image on the encrypted image. We have considered a large number of images by using our encryption scheme. We have found that the NPCR is over 99% and the UACI is over 33%, showing thereby that the encryption scheme is very sensitive with respect to small changes in the plaintext. 3.2. Statistical analysis It is well known that statistical analysis is of crucial importance. Indeed, an ideal cipher should be robust against any statistical attack. In order to prove the robustness of the proposed image encryption scheme, we have performed some statistical tests which are described in the following: (i) Colour histogram: Figs. 4 and 5 show the histograms of RGB colours for the original (Fig. 3a) and the encrypted images (Fig. 3b), respectively. The figures show clearly the random-like appearance of the histogram of the encrypted image. (ii) Correlation of adjacent pixels: It is well known that adjacent image pixels are highly correlated either in horizontal, vertical or diagonal directions. Such high correlation property can be quantified by means of correlation coefficients which are given by: covðp; qÞ r ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi ; DðpÞ DðqÞ

ð14Þ

400

350

400

350

300

350 300

250

200

distribution

250 distribution

distribution

300

200 150

150

200 150

100

100

100

50

50 0 —50

250

0

50

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50

0 —50

300

0

50

100

level

150

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0 —50

300

0

50

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300

level

level

200

200

180

180

180

160

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160

140

140

140

120

120

120

100 80

distribution

200

distribution

distribution

Fig. 4. Histogram of the original image of baboon in the (a) red (b) green (c) blue, components.

100 80

100 80

60

60

60

40

40

40

20

20

20

0 —50

0

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150 level

200

250

300

0 —50

0

50

100

150 level

200

250

300

0 —50

0

50

100

150

200

level

Fig. 5. Histogram of the encrypted image of baboon in the (a) red (b) green (c) blue, components.

250

300

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315

where DðpÞ ¼

S 1X ðp  pÞ2 ; S i¼1 i

covðp; qÞ ¼

S 1X ðp  pÞðqi  qÞ S i¼1 i

qi and pi denote two adjacent pixels (either horizontal or vertical). S is the total number of duplets (pi, qi) obtained from the image; p and q are the mean values of pi and qi, respectively. Table 1 shows the correlation coefficients of Lena image and those of its encrypted image. It can be observed that the encrypted image obtained from the proposed scheme has small correlation coefficients in both horizontal and vertical directions. This result is illustrated in Fig. 6, which presents the distribution of two adjacent pixels in the original and encrypted images of Lena for horizontal (a,b) as well as vertical (c,d) directions.

Table 1 Correlation coefficients of adjacent pixels in the two images Correlation coefficient

Original image

Encrypted image

Horizontal Vertical

0.9006 0.8071

0.0681 0.0845

250

300

250 Pixel value on location (x+1,y)

Pixel value on location (x+1,y)

200

150

100

50

0

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50

0

50

100 150 Pixel value on location (x,y)

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0

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50

100 150 200 Pixel value on location (x,y)

250

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250 Pixel value on location (x,y+1)

Pixel value on location (x,y+1)

200

150

100

50

0

200

150

100

50

0

50

100 150 Pixel value on location (x,y)

200

250

0

Fig. 6. Correlation of two adjacent pixels: Frames (a) and (b), respectively, show the distribution of two horizontally adjacent pixels in the plain and encrypted images of Lena. Frames (c) and (d) respectively, show the distribution of two vertically adjacent pixels in the plain and encrypted images of Lena.

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3.3. Security key analysis Since the secret key is 192-bit long, the key space is about 6.2771 · 1057, which is sufficient enough to resist the bruteforce attack. An encryption scheme has also to be key-sensitive that is a tiny change in the key will cause a significant change in the output. Fig. 7 shows an example of two enciphered images generated from two security keys with only 1-bit difference. Moreover, the average pixel differences of some well-known images over 100 random keys (for each key, all the cases with one-bit difference are computed) are tabulated in Table 2. It can be observed that the values are very close to the expected value of pixel difference on two randomly generated images (NPCR = 99.60937% and UACI = 33.4635%). 3.4. Information entropy analysis It is well known that the entropy H(m) of a message source m can be calculated as   N 2X 1 1 pðmi Þ log H ðmÞ ¼ ; pðmi Þ i¼0

ð15Þ

Fig. 7. (a) Original image of jet; (b) and (c) encrypted images using user keys with 1-bit difference; (d) the difference between (b) and (c). Table 2 Pixel difference between images encrypted by keys with one-bit difference Image

Lena Baboon Jet Peppers

Mean NPCR (%)

Mean UACI (%)

R

G

B

R

G

B

99.5660 99.5469 99.6005 99.6100

99.5860 99.6265 99.6085 99.5790

99.6010 99.5776 99.6080 99.5880

33.2980 33.4600 33.4124 33.4111

33.4137 33.4525 33.4665 33.4236

33.4148 33.3468 33.4633 33.4163

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where N is the number of bits to represent a symbol mi 2 m. p(mi) represents the probability of symbol mi and log represents the base 2 logarithm so that the entropy is expressed in bits. For a purely random source emitting 2N symbols, the entropy is H(m) = N. For encrypted messages, the entropy should ideally be H(m) = N. When a cipher emits symbols with entropy less than N , there exists a certain degree of predictability, which threatens its security. Let us consider the ciphertext of a Lena’s image of size 512 · 512 encrypted using the proposed scheme. The number of occurrence of each ciphertext pixel mi is recorded and the probability of occurrence is computed for the three image colour components (R, G, B). The entropy for the three image colour components is:   8 2X 1 1 H R ðmÞ ¼ pðRi Þ log ¼ 7:9732  8; pðRi Þ i¼0   8 2X 1 1 pðGi Þ log H G ðmÞ ¼ ¼ 7:9750  8; pðGi Þ i¼0   8 2X 1 1 ¼ 7:9715  8: H B ðmÞ ¼ pðBi Þ log pðBi Þ i¼0 With Ri, Gi and Bi are the colour component of the pixel mi. The values obtained are very close to the theoretical value N = 8 for the three image colour entropy. This means that information leakage in the encryption process is negligible and the encryption system is secure against the entropy attack. 3.5. Speed performance Apart from the security considerations, some other issues on image encryption are also important, such as the running speed for real-time image encryption/decryption. The simulator for the proposed cryptosystem is implemented using Matlab 7.0. Performance was measured on a 1.6 GHz Pentium IV with 752 Mbytes of RAM running Windows XP. In addition, to improve the accuracy of our timing measurements, each set of the timing tests was executed 10 times, and we report the average of the times thereby obtained. Simulation results show that the average encryption/decryption speed is 7.47 MB/s for encryption and 7.26 MB/s for decryption.

4. Conclusion In this paper, we have proposed a new OCML-based colour image encryption scheme. We have used a 192-bit-long external key to generate the initial conditions and the parameters of the OCML by making some algebraic transformations to the secret key. Such transformations as well as the coupling structure of the OCML have enhanced the cryptosystem security. Indeed, all performance analysis we have carried out prove the security robustness of the proposed cryptosystem. Furthermore, the proposed scheme shows interesting computation speed making it suitable for real-time encryption and transmission.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

Alvarez G, Li S. Some basic cryptographic requirements for chaos-based cryptosystems. Int J Bifurcat Chaos 2006;16(8):2129–51. Baptista MS. Cryptography with chaos. Phys Lett A 1998;240:50–4. Wong W-K, Lee L-P, Wong K-W. A modified chaotic cryptographic method. Comput Phys Commun 2001;138(3):234–6. Wong K-W. A fast chaotic cryptographic scheme with dynamic look-up table. Phys Lett A 2002;298(4):238–42. Wong K-W. A combined chaotic cryptographic and hashing scheme. Phys Lett A 2003;307(5-6):292–8. Wong K-W, Ho S-W, Yung C-K. A chaotic cryptography scheme for generating short ciphertext. Phys Lett A 2003;310(1):67–73. Li S, Chen G, Wong K-W, Mou X, Cai Y. Baptista-type chaotic cryptosystems: problems and countermeasures. Phys Lett A 2004;332:368–75. Rhouma R, Belghith S. A multidimensional map for a chaotic cryptosystem. In: Proceedings of EUSIPCO’06 Italy, 2006, CDROM, Pisa-Italy. Rhouma R, Belghith S. A piecewice linear chaotic map for baptista-type cryptosystem. In: Proceedings of SSD’07 Tunisia, 2007, CD-ROM, Hammamet-Tunisia. Alvarez E, Fernandez A, Garcia P, Jimnez J. New approach to chaotic encryption. Phys Lett A 1999;263:373–5. Li S, Mou X, Cai Y. Improving security of a chaotic encryption approach. Phys Lett A 2001;290:127–33. Jakimoski G, Kocarev L. Analysis of some recently proposed chaos-based encryption algorithms. Phys Lett A 2001;291(6):381–4.

318

R. Rhouma et al. / Chaos, Solitons and Fractals 40 (2009) 309–318

[13] Alvarez G, Montoya F, Romera M, Pastor G. Cryptanalysis of an ergodic chaotic cipher. Phys Lett A 2003;311(2–3):172–9. [14] Alvarez G, Montoya F, Romera M, Pastor G. Keystream cryptanalysis of a chaotic cryptographic method. Comput Phys Commun 2003;156(2):205–7. [15] Alvarez G, Montoya F, Romera M, Pastor G. Cryptanalysis of dynamic look-up table based chaotic cryptosystems. Phys Lett A 2004;326(3-4):211–8. [16] Wang S, Kuang J, Li J, Luo Y, Lu H, Hu G. Chaos-based secure communications in a large community. Phys Rev Lett E 2002;66(065202):1–4. [17] Tang G, Wang S, Lu¨ H, Hu G. Chaos-based cryptograph incorporated with S-box algebraic operation. Phys Lett A 2003; 318: 388–398. [18] Ye W, Dai Q, Wang S, Lu H, Kuang J, Zhao Z, et al. Experimental realization of a highly secure chaos communication under strong channel noise. Phys Lett A 2004;330:75–84. [19] Yen JC, Guo JI. A new chaotic key based design for image encryption and decryption. In: Proceedings of the IEEE international symposium circuits and systems, vol. 4; 2000. p. 49–52. [20] Li S, Zheng X, Mou X, Cai Y, Chaotic encryption scheme for real time digital video. In: Proceedings of SPIE on electronic imaging, San Jose (CA, USA); 2002. [21] Chen G, Mao Y, Chui CK. A symmetric image encryption based on 3D chaotic maps. Chaos, Solitons & Fractals 2004;21:749–61. [22] Guan ZH, Huang F, Guan W. Chaos-based image encryption algorithm. Phys Lett A 2005;346:153–7. [23] Behnia S, Akhshani A, Mahmodi H, Akhavan A. A novel algorithm for image encryption based on mixture of chaotic maps. Chaos, Solitons & Fractals 2008;35(2):408–19. [24] Pareek NK, Patidar V, Sud KK. Image encryption using chaotic logistic map. Image Vis Comput 2006;24:926–34. [25] Kwok HS, Tang WKS. A fast image encryption system based on chaotic maps with finite precision representation. Chaos, Solitons & Fractals 2007;32:1518–29. [26] Meherzi S, Marcos S, Belghith S. A new spatiotemporal chaotic system with advantageous synchronization and unpredictability features. In: Proceedings of NOLTA’06 Italy, 2006, CD-ROM, Bologna-Italy. [27] Rhouma R, Belghith S. On the security of a spatiotemporal chaotic cryptosystem. Chaos: An Interdisciplinary J Nonlinear Sci 2007;17:1–6.