A NEW COLOR IMAGE CRYPTOSYSTEM BASED ON A PIECEWISE LINEAR CHAOTIC MAP Rhouma Rhouma1 , David Arroyo2 and Safya Belghith1 1 2

Syscom Laboratory, Ecole Nationale d’Ing´enieurs de Tunis, 37, Le Belv´ed`ere 1002 Tunis, Tunisia

Instituto de F´ısica Aplicada, Consejo Superior de Investigaciones Cient´ıficas, Serrano 144–28006 Madrid, Spain e-mail: [email protected], [email protected], [email protected]

ABSTRACT

0.6 0.5 Lyapunov exponent

In this paper a piecewise linear chaotic map (PWLCM) is used to build a new digital chaotic cryptosystem. The characteristics of PWLCM are very suitable for the design of encryption schemes. The implicit digital degradation problem of PWLCM has been eluded through the discretization of the phase space. The accuracy, efficiency and security of the proposed encryption scheme is thoroughly analyzed and its adequacy for image encryption is proved.

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Index Terms— Chaos, piecewise linear chaotic map, skew tent map, cryptography, chaotic cryptography, cryptanalysis

Figure 1. Lyapunov exponent of the skew tent map. Although positive for all parameter values, it is small and not uniform.

1. INTRODUCTION

choosing a chaotic map with a high rate of entropy for all the values of the control parameters. In this context, the cryptosystem derived from the chosen chaotic map is associated to a secret key (which corresponds directly or indirectly to the control parameters of the underlying chaotic map) whose valid values can be easily selected. The family of chaotic maps named as piecewise linear chaotic maps or PWLCM [5] is a class of discrete dynamical systems which are always chaotic for all the values of just one control parameter, since they posses a positive Lyapunov exponent for all the values of the control parameter. The skew tent map is a PWLCM defined by fp (u) : [0, 1] → [0, 1] as follows: ½ u/p, if 0 ≤ u < p, fp (u) = (1) (1 − u)/(1 − p), if p ≤ u ≤ 1,

The intrinsic characteristics of images make traditional algorithms such as DES, IDEA and RSA not suitable for practical image encryption. Currently, chaotic maps have been considered as a way to solve this problem [1–4]. In this paper, we propose a new image encryption algorithm based on a piecewise linear chaotic map. The rest of the paper is organized as follows. In Sec. 2, the cryptosystem is presented. Section 3 analyzes the security of the cryptosystem. Finally, some comments and conclusions are given in Sec. 4. 2. DESCRIPTION OF THE CRYPTOSTYSTEM 2.1. Choice and analysis of the used chaotic map The first step when designing a digital chaotic cryptosystem is to choose a chaotic map. The selection criterium must be focused on the capacity of the chaotic map as entropy source. However, the dynamics of chaotic maps are determined by control parameters, so it is necessary to asses that capacity for different values of the control parameters. The best selection, consequently, consists of The work described in this paper was partially supported by Ministerio de Educaci´on y Ciencia of Spain, research grant SEG2004-02418, Ministerio de Ciencia y Tecnolog´ıa of Spain, research grant TSI200762657 and CDTI, Ministerio de Industria, Turismo y Comercio of Spain in collaboration with Telef´onica I+D, Project SEGUR@ with reference CENIT-2007 2004.

where p ∈ (0, 1). The iteration of (1) from a certain initial value for u0 and for a certain value of p determines a sequence of real numbers in [0, 1]. This set of real numbers is generally called an orbit. As mentioned above, the skew tent map has a Lyapunov exponent greater than zero for all the values of the control parameter. The Lyapunov exponent informs about the divergence rate of two orbits generated from very close initial conditions. If the Lyapunov exponent is positive, then the underlying system is chaotic and the derived orbits are called chaotic orbits. A high value of the Lyapunov exponent is very important for the design of a chaos-based cryptosystem. Indeed, a well designed cryptosystem is characterized by the fact that two

very similar plaintexts lead to very different ciphertexts. Moreover, if the same plaintext is encrypted using two very close keys, then the encryption procedure results in two totally different ciphertexts. For that reason, it is very convenient for the design of a chaos-based cryptosystem to use a chaotic map with a large enough Lyapunov exponent. The skew tent map fails in this requirement (see Fig. 1). Nevertheless, it is possible to overcome this problem through the discretization of the phase space based on the following function upon fp (u): fˆp (u) = b2α · fp (u)c mod Θ,

(2)

where α ∈ N and Θ ∈ N. (k) Let u0 be a certain value in (0, 1). Let fˆp (u) be defined in the next fashion: ³ ´ ( ˆp fˆp(k−1) (u) f if k > 1 fˆp(k) (u) = (3) ˆ fp (u) if k = 1, Then, it is possible to generate the next m-length sequence of integer numbers from u0 and using Eqs. (2) and (3): n o fˆp(1) (u0 ), fˆp(2) (u0 ), . . . , fˆp(m) (u0 ) . (4)

Figure 2 informs that the upper bound is around 50 when double precision is considered. Three values of α are used for the cryptographic purpose of this paper: α1 = 33, α2 = 43, and α3 = 50. In the next subsection it is explained how the properties of the discretized skew tent map are used to design a secure cryptosystem. 2.2. Encryption and decryption The encryption procedure includes three different sub-procedures which are interleaved in order to transform the plain-image into the cipher-image. These sub-procedures are: 1. Transformation of the color image into three vectors. The encryption scheme deals with 24-bit color images of size M × N . Any of these color images is a set of three matrixes of size M × N , composed of integer numbers in [0, 255]. Each matrix is transformed into a vector of size m = M × N by scanning the matrix from left to right and from top to bottom. As a result, the original color image leads to three vectors named as PR , PG , and PB for its red, green and blue color components. The i-th element of any of these vectors is denoted as Pc [i] for c ∈ {R, G, B}.

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Figure 2. Shannon entropy of the discretized skew tent map for different values of α and Θ = 256. In our proposed image cryptosystem, this sequence is used to encrypt color images coded with eight bits per pixel. Consequently, Θ is set to 256 so that the generated sequence contains integer numbers inside the interval [0, 255]. On the other hand, α should be selected in such a way that the iteration of (2) leads to random-like sequences. In Fig. 2 it is shown the average entropy of the sequence of integers generated from (2) as a function of α. A large value of α implies a big rate of divergence of the orbits. However, there exists an upper value of α beyond which the entropy of the sequences of integers generated decreases. This upper bound depends on the arithmetic precision involved in the cryptosystem implementation. The entropy of those sequences of integer numbers was measured by applying the Shannon entropy of a source of information S of 8-bit symbols (k) si = fˆp (u) ∈ S given by : ¶ µ m X 1 , (5) H(s) = p(si )log2 p(si ) i=0

2. Mapping the integer values into the phase space of the skew tent map. The phase space of the skew tent map is divided into 256 equal-width subintervals, we consider a function mapping an integer in {0, 1, . . . , 255} to the middle point of one subinterval from the 256 subintervals in (0, 1). The mapping is given by the function g(ν) : {0, 1, . . . , 255} → (0, 1) defined as follows: g(ν) = (2 · ν + 1)/512,

(6)

3. Encryption of a color pixel. The encryption of every single pixel of a plain-image is performed through r rounds and involves the previous encrypted pixel, with the exception of the first pixel at the first encryption round. First of all, the encryption round number j is initialized as 0, and the cipher-image consisting of three vectors is initialized using the values of the three vectors associated to the plain m m m image: {CR [i]}i=1 = {PR [i]}i=1 , {CG [i]}i=1 = m m m {PG [i]}i=1 , and {CB [i]}i=1 = {PB [i]}i=1 , where i is the pixel index (1 ≤ i ≤ m). The encryption of the i-th pixel is given by the updating of the set of values {CR [i], CG [i], CB [i]} as follows: 0 CR = (bxR ·2α1 c+bxG ·2α2 c+bxB ·2α3 c+CR [i])mod 256 0 CG = (bxR ·2α3 c+bxG ·2α1 c+bxB ·2α2 c+CG [i])mod 256 0 CB = (bxR ·2α2 c+bxG ·2α3 c+bxB ·2α1 c+CB [i])mod 256 0 0 0 CR [i] = CR , CG [i] = CG , CB [i] = CB

where

  x0 , fp (g(CR [i − 1])), • xR =  R fpR (g(CR [m])),   y0 , fp (g(CG [i − 1])), • xG =  G fpG (g(CG [m])),   z0 , fp (g(CB [i − 1])), • xB =  B fpB (g(CB [m])),

if j = 1, i = 1, if i > 1, if j > 1, i = 1. if j = 1, i = 1, if i > 1, if j > 1, i = 1. if j = 1, i = 1, if i > 1, if j > 1, i = 1.

This step is repeated until performing j = r rounds of encryption. As a consequence, the secret key of the cryptosystem is given by the vector {x0 , y0 , z0 , pR , pG , pB } ,

(7)

whose components are defined in the interval (0, 1). The encryption and decryption results are illustrated with a sample image (Fig. 3) using the secret key defined by the next set of values: x0 = 0.678541089437521, y0 = 0.128945603563896, z0 = 0.897643190347508, pR = 0.5164321034987619, pG = 0.4951095243167835, pB = 0.5106438925761328. 3. SECURITY ANALYSIS 3.1. Analysis of the key space In order to fulfill the requirements of a complete cryptosystem design, the set of possible values of the secret keys must be carefully detailed [6, Rule 4]. A criterium to specify the key space is based on the concept of key sensitivity [6, Rule 6]. The key sensitivity is the smallest difference between two keys so that the resulting ciphertexts are totally different. In others words, a difference between keys smaller than the key sensitivity implies that the corresponding ciphertexts are identical or closely correlated. As a result, the key space of our cryptosystem is specified through the analysis of the sensitivity with respect to the control parameters and the initial conditions. Nevertheless, the key sensitivity criterium is not the only one to be considered in the specification of the key space. Furthermore, nonchaotic areas should be avoided [6, Rule 5]. The selection of the skew tent map avoids nonchaotic behavior, since it possesses a positive Lyapunov exponent for all values of the control parameter. Moreover, the cryptosystem is based on a discretized version of the skew tent map in order to increase the divergence rate, which further enhances the key sensitivity and thus the key space. Since the divergence rate depends on the values α1 , α2 , α3 , and r, the size of the key space can be determined by these values along with the sensitivity of the cryptosystem with respect to the key. It was experimentally verified that for α1 = 33, α2 = 43, α3 = 50 and r = 2 the sensitivity of the cryptosystem with respect to the control parameters is around 10−16 . On the other hand, the sensitivity with respect to the initial conditions was also experimentally measured as 10−15 . Having in mind that

the encryption of every color component depends on each other and the selection of the keys of every color component is independent from the others, the key space is given by κ ≈ (1016 ×1016 ×1016 )×(1015 ×1015 ×1015 ) = 1093 , which satisfies the security requirement related to the resistance against brute-force attacks [6, Rule 15]. Once a cryptosystem has been designed, the next step is to test it in order to elucidate if it is really secure and efficient. This aim is very difficult to fulfill, since there is no standard framework to examine the quality of a cryptosystem. In other words, it is not possible to be totally sure about the invulnerability of our cryptosystem. Nevertheless, we have to be sure that the cryptosystem is secure against the best-known attacks. This is the goal of this section, along with the evaluation of the cryptosystem’s performance. 3.2. Number of encryption rounds The number of encryption rounds r not only conditions the key sensitivity of the cryptosystem, but also its performance. In this sense, as r increases, the diffusion process is amplified and, in a similar manner, the encryption/decryption speed is decreased. In order to elude a timing-attack based on the relationship between the encryption/decryption speed and r, this parameter has been considered as a design parameter and not being part of the secret key. Its value has to be chosen carefully to fulfill a great level of security and a moderate value of the encryption/decryption time. To do so, a random image and the image resulting from the random modification of one of its pixels are encrypted using a random value for the key and different values for r. As explained in Sec. 2.2, the encryption of every pixel depends on the previous encrypted pixels, with the exception of the first pixel at the first encryption round. In this way, the last pixel only affects the subsequent pixels if r is greater than 1. For that reason, the m-th pixel of the generated random image was chosen as the one to modify along the process of selecting the right value of r. Since the security concerns depending on r are those related to the diffusion property, the proper values of r are selected according to the analysis further described in Sec. 3.3. In this sense, the values of the Number of Pixels Change Rate or NPCR and the Unified Average Changing Intensity or UACI (see the next section for the definition of both concepts) between the resulting cipher-images with respect to r has been measured (see Fig. 4), confirming that a good level of NPCR and UACI is reached for r = 2. Therefore, a number of encryption rounds equal to 2 assures the commitment between the diffusion property and the encryption/decryption speed. For that reason, the remaining considerations and simulations are carried out using r = 2. 3.3. Diffusion analysis Very similar images must lead to very different cipherimages to test and validate the diffusion propriety of the cryptosystem. Therefore, the effect on the ciphertext of a

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Figure 3. (a) Original plain-image; (b) encrypted image; (c) decrypted image. 35

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Figure 4. Analysis of the diffusion property as a function of the number of encryption rounds r: (a) NPCR(%) ; (b) UACI(%). very small modification on the plain-image has to be verified. Such effect can be measured by means of the Number of Pixels Change Rate or NPCR and the Unified Average Changing Intensity or UACI [1, 4]. The used set of keys (pR , pG , pB , x0 , y0 , z0 ) is given in Subsec. (2.2). The NPCRR,G,B is used to measure the number of different pixels of a given color 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 i,j DR,G,B (i, j) NPCRR,G,B = , (8) m where m = M × N is the total number of pixels in the image and DR,G,B (i, j) is defined as : ½

0 0 if SR,G,B (i, j) = SR,G,B (i, j), 0 1 if SR,G,B (i, j) 6= SR,G,B (i, j), (9) 0 where SR,G,B (i, j) and SR,G,B (i, j) are the values of the corresponding color component red (R), green (G) or blue (B) in the two images, respectively. The second criterion, i.e., the UACIR,G,B is used to measure the average intensity difference in a color component and can be defined as :   0 (i, j)| 1 X |SR,G,B (i, j) − SR,G,B , UACIR,G,B = m i,j 2BR,G,B − 1

DR,G,B (i, j) =

(10) where BR,G,B is the number of bits used to represent every color component. As mentioned above, in the context of a well designed cryptosystem the cipher-images resulting

from the encryption of a plain-image and the slightest variation of it must be statistically independent. First of all, it is necessary to establish this statistical independence by means of the NPCR and the UACI. For that reason, a large enough number of pairs of random images were considered to estimate the optimum value for the NPCR and the UACI. Random images with 200 × 200 24-bit true color pixels were considered, resulting NPCRR = NPCRG = NPCRB ≈ 99.6099% and UACIR = UACIG = UACIB ≈ 33.4680%. Once the optimum values to compare with have been determined, the evaluation of the diffusion property can be carried out. In this sense, we have computed the NPCRR,G,B and the UACIR,G,B between pairs of cipherimages for the proposed cryptosystem to assess the influence on the encrypted image of changing a single pixel in the original image. The results of these simulations appear in Table 1. It shows that the NPCR is over 99% and the UACI is over 33%, which are very close to the optimum values previously calculated and thus proves that the encryption scheme is very sensitive with respect to small changes in the plaintext. 3.3.1. Key sensitivity One characteristic of a secure cryptostystem is that two ciphertexts obtained from the same plaintexts but with slightly different keys are totally different [6, Rule 6]. In order to verify this requirement, our cryptosystem is examined using the previous figures NPCR and UACI. This is done by searching the smallest difference between two keys such that the two resulting cipher-images are statistically inde-

Table 1. NPCR and UACI for the ciphertexts corresponding to plain-images with only one pixel in difference. Mean NPCR (%) Mean UACI (%) Image Difference in R G B R G B Lena First pixel 99.6025 99.5625 99.6075 33.6063 33.4807 33.4895 Baboon last pixel 99.6300 99.5825 99.5700 33.4577 33.4641 33.5185 Jet random pixel 99.6350 99.5800 99.5525 33.3019 33.3738 33.5249 Peppers random pixel 99.6325 99.6425 99.5250 33.2427 33.4123 33.3133

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Figure 5. (a) The NPCR(%); (b) the UACI(%) as functions of the precision in the subkey pR ; (c) The NPCR(%); (d) the UACI(%) as functions of the precision in the subkey x0 . pendent. This difference is named key sensitivity. If the difference between two keys is smaller than the key sensitivity, then the cipher-images corresponding to the same plain-image are correlated. This fact has to be avoided forcing the selection of keys so the closest ones are separated by a value greater than the key sensitivity. In order to asses the key sensitivity, the problem must be split into two parts. First of all, the sensitivity with respect to the control parameter is determined. To do so, it is possible to focus the sensitivity evaluation on one of the three control parameters involved in the encryption procedure, since the cryptosystem is equally dependent on any of the three control parameters. In our simulations pR was considered for the sensitivity measure purpose. In Fig. 5.(a)(b) the results of the evaluation are shown. They inform that the sensitivity of the ciphertext with respect to the control parameter is around 10−16 . On the other hand, the sensitivity with respect to the initial condition was also evaluated. It was experimentally verified that the sensitivity of the ciphertext with respect to the initial condition is around 10−15 (see Fig. 5.(c)-(d)). This was further confirmed through the calculation of the NPCR and the UACI for pairs of cipher-images generated from different plainimages using two equal keys but with a difference in x0 equal to 10−15 .

3.4. Confusion assessment Another important aspect for the hardening of our cryptosystem is the elimination from the cipher-image of any possible track or leakage of information related to either the plain-image or the key. It means that the cipher-image must be statistically independent from the key and the plainimage. This is confirmed through the reinterpretation of the proposed cryptosystem by means of a Pseudo Random Number Generator. Indeed, the cipher-image is composed of three sequences (one per color component) of integer numbers each of which must be random and statistically independent from the set {pR , pG , pB , x0 , y0 , z0 }. Therefore, classical randomness tests can be used to test if our cryptosystems satisfies this demand. In this sense, 100 random images of size 64 × 64 were generated and later encrypted with 100 different keys, being the number of encryption rounds 2. The behavior of the resulting cipher-images were analyzed using four randomness tests (the long runs, the mono bit, the poker test, and the Maurer test [7]). The related 300 sequences of integer numbers passed all the randomness tests (only four sequences failed passing the Maurer test, see Table 2). The entropy of those sequences of integer numbers was also measured. The Shannon entropy of a source of information S of L-bit

Table 2. Confusion analysis by means of several randomness tests and the Shannon entropy. 100 random color images were generated and 100 secret keys were also generated randomly. This table presents the result of measuring the randomness of the 300 resulting sequences of integer numbers and the Shannon entropy of the 100 resulting cipher-images. Percentage of sequences passing the tests Entropy per color component (minimum / maximum value) Long runs Monobit Poker Maurer Red Green Blue 100 % 100 % 100 % 98.6667 % 7.9463/7.9635 7.9449/7.9664 7.9442/7.9635 Table 3. Correlation coefficients of adjacent pixels in Baboon plain-image and its corresponding cipher-image. Original image Ciphered image Correlation coefficient R G B R G B horizontal 0.9159 0.8603 0.9231 −0.0035 0.0038 −0.0014 vertical 0.9312 0.8843 0.9333 0.0094 0.0009 −0.0043

symbols si ∈ S is given by H(s) =

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obtained from the proposed scheme has small correlation coefficients in both horizontal and vertical directions.

¶ ,

(11)

where p(si ) represents the probability of occurrence of the symbol si . For a pure random source emitting 2L symbols, the entropy is H(s) = L. For encrypted messages, the entropy should ideally be H(s) = L, where L in the case of dealing with images with 8-bit pixels is equal to 8. In the case of the 300 sequences of integer numbers of our experiment, the mean value of the entropy was 7.9551 and the mean value of its standard deviation 0.0037 (see Table 2). As a result, the statistical behavior of the cipherimage generated by our cryptosystem can be considered as independent from the value of the key and from the plainimage. Furthermore, the cipher-image possesses a high level of entropy very close to the ideal value of a random source of information, which means that the cipher-images are going to be random-like as they must be. This is further confirmed by the study of the correlations of adjacent pixels of the cipher-images. As a result of those analysis, the randomness-like of the cipher-images can be concluded. On the other hand, adjacent pixels of images are usually highly correlated. This feature is a problem if it is present in cipher-images. Therefore, we must quantify this correlation and to assure that it is not a characteristic of the cipher-images generated by our cryptosystems. The correlation property is quantified by means of correlation coefficients given by: cov(p, q) p , D(p) D(q)

r= p

(12)

PS where D(p) = S1 i=1 (pi − p¯)2 , and PS cov(p, q) = S1 i=1 (pi − p¯)(qi − q¯), being qi and pi two adjacent pixels (either in the horizontal or the vertical direction), S the total number of duplets (pi , qi ) obtained from the image and, finally, p¯ and q¯ are the mean values of pi and qi respectively. Table 3 shows the correlation coefficients of Fig. 3 (a) and those of the associated encrypted image. It can be observed that the cipher-image

4. CONCLUSIONS In this paper, a new chaotic cryptosystem for color images has been proposed. A Piecewise Linear Chaotic Map was selected for the cryptosystem’s design based on its capacity as an entropy source and its simple implementation. The proposed cryptosystem has passed some basic common security tests. 5. REFERENCES [1] G. Chen, Y. Mao, and C. K Chui, “A symmetric image encryption based on 3D chaotic maps,” Chaos Soliton Fractals, vol. 21, pp. 749–761, 2004. [2] Z. H. Guan, F. Huang, and W. Guan, “Chaos-based image encryption algorithm,” Physics Letters A, vol. 346, no. 1-3, pp. 153–157, 2005. [3] A.N. Pisarchik, N.J. Flores-Carmona, and M.CarpioValadez, “Encryption and decryption of images with chaotic map lattices,” Chaos, vol. 16, 2006. [4] H. S. Kwok and W. K. S. Tang, “A fast image encryption system based on chaotic maps with finite precision representation,” Chaos Soliton Fractals, vol. 32, pp. 1518–1529, 2007. [5] Shujun Li, Guanrong Chen, and Xuanqin Mou, “On the dynamical degradation of digital piecewise linear chaotic maps,” International Journal of Bifurcation and Chaos, vol. 15, no. 10, pp. 3119–3151, 2005. [6] Gonzalo Alvarez and Shujun Li, “Some basic cryptographic requirements for chaos-based cryptosystems,” International Journal of Bifurcation and Chaos, vol. 16, no. 8, pp. 2129–2151, 2006. [7] “Nist special publication 800-22,” 2001.

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In recent years, the growing number of mobile devices with internet access has ..... 6 The negative extra space is avoided by the zero value in the definition of ...

Color Image Watermarking Based on Fast Discrete Pascal Transform ...
It is much more effective than cryptography as cryptography does not hides the existence of the message. Cryptography only encrypts the message so that the intruder is not able to read it. Steganography [1] needs a carrier to carry the hidden message

A Appendix - Semantic Scholar
buyer during the learning and exploit phase of the LEAP algorithm, respectively. We have. S2. T. X t=T↵+1 γt1 = γT↵. T T↵. 1. X t=0 γt = γT↵. 1 γ. (1. γT T↵ ) . (7). Indeed, this an upper bound on the total surplus any buyer can hope