A fast image encryption scheme based on chaotic standard map

Physics Letters A 372 (2008) 2645–2652

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A fast image encryption scheme based on chaotic standard map

Kwok-Wo Wong ∗, Bernie Sin-Hung Kwok, Wing-Shing Law

Department of Electronic Engineering, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon Tong, Hong Kong

Received 4 December 2006; received in revised form 11 December 2007; accepted 12 December 2007

Available online 23 December 2007

Communicated by A.P. Fordy

Abstract

In recent years, a variety of effective chaos-based image cryptosystems have been proposed. One of the architectures of this kind of cryptosys-tems is composed of multiple rounds of substitution and diffusion. As the confusion and diffusion effects are solely contributed by the substitutionand the diffusion stages, respectively, the required overall rounds of operations in achieving a certain level of security is found more than necessary.In this Letter, we suggest to introduce a certain diffusion effect in the substitution stage by simple sequential add-and-shift operations. Althoughthis leads to a longer processing time in a single round, the overall encryption time is reduced as fewer rounds are required. Simulation resultsshow that at a similar performance level, the proposed cryptosystem needs less than one-third the encryption time of an existing fast cryptosystem.The effective acceleration of chaos-based image cryptosystems is thus achieved.© 2008 Elsevier B.V. All rights reserved.

1. Introduction

Nowadays, communication infrastructure such as mobilenetworks and the Internet are well developed. However, they arepublic networks and are not suitable for the direct transmissionof confidential messages. To make use of the communicationinfrastructure already developed and to maintain the secrecy,cryptographic techniques need to be applied. Traditional sym-metric ciphers such as data encryption standard (DES) andadvanced encryption standard (AES) are designed with goodconfusion and diffusion properties [1]. These two properties arealso found in chaotic systems which are usually ergodic and aresensitive to system parameters and initial conditions. In recentyears, a number of chaos-based cryptographic schemes havebeen proposed. Some of them utilize one-dimensional chaoticmaps for data sequence or document encryption [2–4]. Forimage encryption, two-dimensional (2D) or high-dimensionalchaotic maps are naturally employed as the image can be con-sidered as a 2D array of pixels [5–9].

In [5], Fridrich suggested that a chaos-based image encryp-tion scheme should compose the iterations of two processes:

* Corresponding author.E-mail address: [email protected] (K.-W. Wong).

0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2007.12.026

substitution and diffusion. The former is achieved by permut-ing all the pixels as a whole using a 2D chaotic map. Thenew pixel moved to the current position is taken as a substi-tution of the original pixel. In the diffusion process, the pixelvalues are altered sequentially and the change made to a partic-ular pixel depends on the accumulated effect of all the previouspixel values. This architecture forms the basis of a number ofchaos-based image ciphers proposed subsequently. For exam-ple, Chen and his research group employed a three-dimensional(3D) cat map [6] and a 3D baker map [7] in the substitutionstage. Guan et al. used a 2D cat map for pixel position permu-tation and the discretized Chen’s chaotic system for pixel valuemasking [8]. Lian et al. [9] used a chaotic standard map in thesubstitution stage and a quantized logistic map in the diffusionstage. The parameters of these two chaotic maps are determinedby a keystream generated in each round.

Besides this substitution–diffusion architecture, there areother chaos-based image cryptosystems with their own struc-ture. For example, Pisarchik et al. suggested to convert im-age pixels to chaotic maps coupled unidirectionally to form achaotic map lattice (CML) and the cipher image is obtainedby iterating the CML with secret system parameters and num-ber of cycles [10]. Pareek et al. extended the concept of theirdocument cipher [4] to image encryption by using two logis-tic maps and an external key [11]. Kwok et al. [12] proposed

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2646 K.-W. Wong et al. / Physics Letters A 372 (2008) 2645–2652

a fast chaos-based image cryptosystem with the architecture ofa stream cipher. In particular, the plain-image pixels are maskedby a pseudo-random keystream generated by a cascade of theskewed tent map and a high-dimensional cat map.

Lian et al. [9] pointed out that there exist some weak keysfor ciphers employing the cat and the baker maps. Moreover,the key space of these two maps is not as large as that of thestandard map. Therefore they suggested using a standard mapfor permutation while keeping the logistic map for diffusion.To achieve a satisfactory level of security, they recommendedperforming at least four overall rounds of substitution and diffu-sion [9]. In each substitution stage, 4 permutation rounds shouldbe performed. These lead to a total of 16 permutation and 4 dif-fusion rounds. Indeed, measures such as pre-computation of thepermutation mode and the sine table were suggested to reducethe computational complexity caused by iterating the standardmap. Nevertheless, the overall encryption speed is still not fastenough.

To further accelerate the encryption speed of substitution–diffusion type chaos-based image cryptosystems, we suggestthat the diffusion effect is not necessarily contributed solely bythe diffusion process. Instead, it can also be introduced in thesubstitution stage. As a result, satisfactory security performanceis achieved in fewer overall rounds and the total encryptiontime is shorter. Simulation results confirm that although theprocessing time required in a single round is increased slightly,a considerable amount of overall encryption time is saved.

The Letter is organized as follows. In the next section, thearchitecture of cryptosystems based on iterative substitution–diffusion processes is introduced with Lian et al.’s scheme [9]as an example. The design concept of our approach is describedin Section 3. Simulation results and performance analyses arereported in Section 4. In Section 5, a conclusion is drawn.

2. Architecture of substitution–diffusion type chaos-basedimage cryptosystems

The architecture of substitution–diffusion type chaos-basedimage cryptosystems is shown in Fig. 1.

There are two iterative stages in this type of chaos-basedimage cryptosystems. The substitution stage permutes all thepixels as a whole, without changing their values. In the diffu-sion stage, the pixel values are modified sequentially so that atiny change in one pixel is spread out to as many pixels as possi-ble, hopefully the whole image. To decorrelate the relationshipbetween adjacent pixels, there are n permutation rounds in the

Fig. 1. Architecture of substitution–diffusion type chaos-based image cryp-tosystems.

substitution stage with n � 1. The whole substitution–diffusionround repeats for a number of times to achieve a satisfactorylevel of security. To further enhance the security, the parametersof the chaotic maps governing the permutation and the diffusionshould better be distinct in different rounds. This is achieved bya round key generator with a seed secret key as input.

In Lian et al.’s cryptosystem [9], the substitution process isrealized solely by permuting all the pixels by an invertible dis-cretized 2D standard map, without mixing their values. As thecorner pixel (x = 0, y = 0) is not permuted at all under the stan-dard map, a random scan couple (rx, ry ) is included to permutethis corner pixel together with other pixels. The modified stan-dard map equations are given by Eq. (1).

xk+1 = (xk + yk + rx + ry)modN,

(1)yk+1 =(

yk + ry + KC sin2πxk+1

N

)modN,

where (xk, yk) and (xk+1, yk+1) is the original and the permutedpixel position of an N × N image, respectively. The standardmap parameter KC is a positive integer.

In the diffusion stage, each pixel of the 2D permuted imageis scanned sequentially, usually start from the upper left corner.The diffusion effect is achieved by Eq. (2).

(2)

{c−1 = Kd,

ci = vi ⊕ q[f (ci−1),L],where vi is the value of the ith pixel of the permuted image,ci−1 and ci are the values of the (i − 1)th and ith pixels of thediffused image, respectively. The seed of the diffusion functionis c−1 which is obtained from the diffusion key Kd . The non-linear function f (·) is the logistic map given by Eq. (3).

(3)f (ci−1) = 4ci−1(1 − ci−1).

The bit extraction function q(·) extracts some bits just afterthe decimal point, as defined by Eq. (4).

(4)q(X,L) = ⌊2L · X⌋

,

where X = 0 . b1b2b3 · · ·bL · · · is the binary representation ofX and bi is either 0 or 1.

The new pixel value is obtained by exclusive-OR (XOR)the current pixel value vi of the permuted image with an L-bit sequence obtained from the logistic map taking the previousdiffused pixel value ci−1 as input. As the previous diffused pixelwill affect the current one, a tiny change in the plain image isreflected in more than one pixel in the cipher image and so thediffusion effect is introduced in this stage. To generate the dis-tinct substitution and diffusion keys used in different rounds,a key generator composed of skewed tent maps is suggestedin [9].

3. The proposed scheme

As illustrated in the previous sections, the confusion anddiffusion effects are solely contributed by the substitution anddiffusion stages, respectively. To achieve a satisfactory levelof security, a total of 16 permutation and 4 diffusion rounds

K.-W. Wong et al. / Physics Letters A 372 (2008) 2645–2652 2647

are recommended in Lian et al.’s cryptosystem [9]. Each per-mutation step of the cryptosystem involves a pre-computationof standard map with multiple times of pixel moving. To en-crypt a 512 × 512 gray-scale image, the computational timerequired in different parts of their cryptosystem is summarizedin Table 1. All simulations in this work are performed using Mi-crosoft Visual Studio 6.0 on a personal computer (PC) with a3 GHz Pentium D processor, 512MB memory and 80GB hard-disk capacity. For the recommendation of m = 4 and n = 4,the total encryption time is approximately equal to 95.44 ms(0.04 + 4 × 3.39 + 16 × 4.54 + 4 × 2.30 ms), of which 90%(86.2 ms) is dedicated to the 16 permutation rounds. On theother hand, the diffusion process only requires 2.30 ms. Thisis because the length of ci−1 in Eq. (2) is L bits. There are atmost 2L distinct values and so all the possible outputs of thebit extraction function q[f (ci−1),L] can be pre-computed andstored in a lookup table. As a result, the diffusion stage only in-volves a table lookup and a logical XOR operation, and can becompleted in a very short time.

Table 1Time required in different parts of Lian et al.’s cryptosystem for encryptinga gray-scale image of size 512 × 512 under the recommended cipher roundsm = 4 and n = 4

Task Keygeneration

Permutation(pre-computationof standard map)

Permutation(Pixel moving)

Diffusion

Average timeRequired (ms)

0.04 3.39 4.54 2.30

Fig. 2. Architecture of the proposed chaos-based image cryptosystem.

The total encryption time of Lian et al.’s cryptosystem isquite long as too many rounds of permutation and diffusion arerequired to achieve a satisfactory level of security. The reasonis that the confusion effect is contributed solely by the substi-tution stage while the diffusion effect is only introduced in thediffusion stage. A better way is to introduce certain diffusioneffect in the substitution stage as well. The architecture of theproposed cryptosystem is shown in Fig. 2. In the substitutionstage, both the shuffling of pixels and the change of their valuesare carried out at the same time while the diffusion process re-mains unchanged. Consequently, the pixel value mixing effectis contributed by two levels of diffusing operations: the modi-fied substitution process and the original diffusion function. Asthe diffusion effect is not solely contributed by the diffusionfunction, the same level of security is achieved in fewer cipherrounds. The overall encryption speed is thus accelerated.

In our modified substitution stage, the new position of a pixelis calculated according to Eq. (1). However, before relocatingthe pixels, diffusion effect is injected by adding the current pixelvalue of the plain image to the previous permuted pixel and thenperforms a cyclic shift on their sum. Of course, this combina-tion is not the only choice. For example, other simple logicaloperations such as XOR can be used instead of the additionoperation, or the shift operation can be performed before addi-tion. However, simulation results found that the “add-and-shift”combination leads to the best performance and so it becomesthe choice in our cryptosystem. The new pixel value is thengiven by Eq. (5),

(5)vi = Cyc[(pi + vi−1)mod 2L, LSB3(vi−1)

],

where pi is the current pixel value of the plain image, 2L isthe number of possible gray levels in each pixel, vi−1 is thevalue of the (i − 1)th pixel after permutation, Cyc[s, q] per-forms the q-bit right cyclic shift on the binary sequence s,LSB3(s) refers to the value of the least three significant bits of s,vi is the resultant pixel value in the permuted image. The seedv−1 ∈ (0,2L − 1] is a subkey obtained from the key generator.

Similar to the effect of using higher-dimensional chaoticmaps for image encryption [6,7], our modification makes thehistogram of the confused image uniform in only a few rounds.

(a) (b)

Fig. 3. A white homogeneous image processed by 3 rounds of our modified confusion stage. (a) The resultant “noise-like” cipher image; (b) the correspondinghistogram.


A 512 × 512 image composed of all white pixels (pixel value255) is taken as an example of homogeneous image. Our cryp-tosystem gives a “noise-like” cipher image and a uniform his-togram in only three confusion rounds, as shown in Figs. 3(a)and 3(b), respectively. By the property of pixel value mixing,the value of every single pixel is diffused to nearly the wholeimage. It is regarded as the first level diffusion of the proposedcryptosystem.

As the pixel value mixing depends on the value of thepreviously processed pixel, the operation sequence cannot bechanged. This may lead to a problem in the reversed substitu-tion process required in decryption. A solution is to make thefirst decipher round perform the reverse position permutationonly. Then both reverse position permutation and pixel valuechange are performed from the second decipher round. In thismanner, an additional decipher round is required for the reverseof pixel value modification. It composes of the simple add-and-shift operation and adds only little cost to the overall decryptionprocedures.

As mentioned in Section 2, an explicit chaotic diffusionfunction based on the logistic map is employed in Lian et al.’scryptosystem [9]. It is preserved in ours as the second leveldiffusion for achieving a higher diffusion rate. Similar to theircryptosystem for higher security requirements, the plain-imageis firstly processed by the modified substitution for n rounds,followed by a diffusion round. Then the whole process is re-peated for m rounds.

4. Experimental results

Simulation results and performance analyses of the proposedimage encryption scheme are provided in this section. In par-ticular, the performance of our scheme is compared with Lianet al.’s. This is because their scheme is the fastest substitution–diffusion cryptosystem, due to the use of a lookup table to avoidthe slow real number arithmetic in the diffusion stage. The com-parisons show that at a similar performance level, the proposedcryptosystem leads to a higher encryption speed than Lian etal.’s.

Firstly, the performance of the substitution stage in ourscheme is compared with Lian et al.’s. The standard Lena im-age of size 512 × 512 and 256 gray levels is employed in thistest. It is shown in Fig. 4(a) while its histogram is given inFig. 4(b). After three rounds of the substitution process, theintermediate cipher image obtained by Lian et al.’s scheme isshown in Fig. 4(c). As only pixel permutation is performed,the corresponding statistical information depicted in Fig. 4(d)is exactly the same as that of the plain image. Fig. 4(e) is the ci-pher image obtained after the substitution stage of our scheme.The corresponding histogram found in Fig. 4(f) indicates apromising degree of pixel value mixing in only 3 substitutionrounds.

In addition to the above test, an experiment dedicated to thesubstitution process has been carried out. The results shown inFig. 5 demonstrate that our proposed substitution method is sen-sitive to two plain-images different by as little as only one bit.Fig. 5(a) is the original Cameraman image of size 256×256

in 256 gray levels. Figs. 5(b) and 5(c) are the cipher imagesobtained after 3 rounds of the proposed substitution process,whose corresponding plain images have only a 1 bit differenceat the lower right corner. The two cipher images have 99.36%pixels different from each other. The difference image betweenthe two cipher images can be found in Fig. 5(d). Such resultsare benefited from the pixel value modification introduced inthe substitution process. During the pixel position permutation,the gray level values of all pixels are changed pixel-by-pixel,depending on their neighborhood. This acts as the first leveldiffusion in our scheme. As a result, the one-bit difference inthe plain-image diffuses substantially over many pixels in thecipher image.

Indeed, the diffusion effect introduced in the substitutionprocess supplements that contributed by the explicit diffusionfunction. Therefore our cryptosystem achieves a similar perfor-mance in fewer cipher rounds. This is supported by the simula-tion results of the proposed and Lian et al.’s schemes at differentcombinations of permutation (n) and overall (m) rounds usingthe 512×512 Lena image in 256 gray scales. Two performanceindices, namely, number of pixels change rate (NPCR) and uni-fied average changing intensity (UACI) as adopted in [6,7] arelisted in Table 2. Their trend at different overall rounds withn fixed to 4 is plotted in Figs. 6(a) and 6(b), respectively. Thegraphs show that both performance indices rise rapidly in ourscheme, which indicate good confusion and diffusion effect. Ta-ble 2 also lists the encryption and decryption time required inboth schemes.

The simulation results listed in Table 2 show that to achievea similar performance of Lian et al.’s recommended cryptosys-tem (m = n = 4) [9], the proposed scheme only requires oneoverall round with three permutation rounds in the confusionstage, i.e., m = 1 and n = 3. The corresponding encryptiontime is 20.79 milliseconds (ms) which is slightly higher thanone-fifth of Lian et al.’s (95.81 ms). To achieve a higher perfor-mance such as NPCR > 0.996 and UACI > 0.334, Lian et al.’srequirement is m = 6 and n = 3 while the proposed schemeonly needs m = 2 and n = 2. Our encryption time (31.86 ms)is less than one-third of Lian et al.’s (116.33 ms). The substan-tial acceleration in encryption speed is due to the reduction inthe number of overall rounds m and permutation rounds n. Theadditional computation complexity of the simple add-and-shiftoperation in the modified substitution stage is insignificant. Itleads to an extra encryption time of only 0.43 ms per permu-tation, as given by the difference between the two encryptiontime data in the first row (m = 1 and n = 2) of Table 2. Be-cause of the programming arrangement, the decryption time forboth the proposed and Lian et al.’s schemes are longer thanthe corresponding encryption time. As found from the sametable, the increase in decryption time is not substantial, only2.15 ± 1.57% in Lian et al.’s scheme and 7.24 ± 1.89% inour scheme (with consideration of the proposed decryption ap-proach explained in Section 3). Similar results are obtained onvarious sample gray scale images [13], namely, Elaine (Fig. 7)with size 512 × 512, Peppers with size 512 × 512 (Fig. 8) andCameraman (Fig. 5(a)) with size 256 × 256, as listed in Ta-bles 3–5, respectively.


(a) (b)

(c) (d)

(e) (f)

Fig. 4. (a) Plain Lena image; (b) Histogram of the plain image; (c) Intermediate cipher image using Lian et al.’s substitution; (d) Histogram of the intermediatecipher image shown in (c); (e) Intermediate cipher image using the proposed substitution; (f) Histogram of the intermediate cipher image given in (e).

In general, adjacent pixels of most natural images are highlycorrelated. However, one of the requirements of an effectiveimage cryptosystem is the generation of a cipher image withsufficiently low correlation of adjacent pixels. To analyze the ef-fectiveness of our cryptosystem in this aspect, the correlationsbetween two adjacent pixels in horizontal, vertical and diago-nal directions are calculated. In the experiment, four images,namely, a 256 gray scale plain Lena image of size 512 × 512,the cipher images obtained using the proposed scheme (m = 2and n = 2) and Lian et al.’s scheme (m = 6 and n = 3), and

a random image with each pixel value selected randomly areemployed. The correlation coefficients are calculated by the for-mula stated in [6,7] and are listed in Table 6. The data for thetwo cipher images are in the same order of magnitude as thosefor the random image. This implies that both cryptosystems caneffectively decorrelate adjacent pixels in the plain image. As anexample, the correlation distribution of two horizontally adja-cent pixels of the plain image and the cipher image obtainedusing the proposed scheme is shown in Figs. 9(a) and 9(b), re-spectively.


(a) (b)

(c) (d)

Fig. 5. (a) Plain Cameraman image; (b) and (c) cipher images whose corresponding plain images have one-bit difference only; (d) difference between cipher imagesshown in (b) and (c).

Table 2Execution time and performance indices NPCR and UACI of the proposed and Lian et al.’s schemes, for some selected combinations of m and n (test image: Lena)

(m,n) Encryption time (ms) Decryption time (ms) NPCR UACI

Proposed Lian et al. Proposed Lian et al. Proposed Lian et al. Proposed Lian et al.

(1,2) 15.65 14.80 16.95 15.35 0.686642 0.000179 0.208793 0.000040(1,3) 20.79 19.39 22.37 19.78 0.994370 0.000252 0.328191 0.000061(1,4) 25.63 24.13 27.14 24.27 0.996040 0.000423 0.334905 0.000093(2,2) 31.86 30.12 34.77 30.77 0.996086 0.009903 0.334273 0.002623(2,3) 41.26 38.69 44.09 39.77 0.996014 0.019802 0.334172 0.005082(2,4) 50.97 48.04 54.17 48.75 0.996014 0.031902 0.334896 0.008362(3,2) 47.00 44.64 51.15 46.14 0.996353 0.446320 0.334267 0.121025(3,3) 61.85 58.15 66.47 59.84 0.996063 0.647816 0.335562 0.176647(3,4) 76.77 72.02 81.59 73.17 0.995911 0.748901 0.335196 0.205962(4,2) 62.74 59.29 68.10 61.43 0.996025 0.984406 0.334703 0.300348(4,3) 82.56 77.58 88.43 79.96 0.996178 0.99091 0.333724 0.311426(4,4) 102.85 95.81 108.65 97.90 0.996143 0.992676 0.334972 0.317068(4,5) 122.22 113.69 129.03 116.16 0.996239 0.960281 0.334382 0.280655(5,3) 103.35 96.88 110.77 99.56 0.996243 0.995861 0.334087 0.330970(5,4) 127.94 119.69 136.15 122.39 0.995956 0.995892 0.334679 0.333018(5,5) 152.79 142.25 160.95 144.78 0.99604 0.995693 0.333184 0.327371(6,2) 95.06 89.50 103.10 92.45 0.996304 0.996109 0.334044 0.333748(6,3) 124.40 116.33 134.18 119.53 0.996181 0.995865 0.334865 0.334197


(a) NPCR vs. overall rounds m (n = 4) (b) UACI vs. overall rounds m (n = 4)

Fig. 6. Performance of the proposed and Lian et al.’s cryptosystems in terms of (a) number of pixels change rate (NPCR); and (b) unified average changing intensity(UACI) at different overall rounds (m) with 4 permutation rounds in each substitution stage (n = 4).

Table 3Execution time and performance indices NPCR and UACI of the proposed and Lian et al.’s schemes, for some necessary combinations of m and n (test image:Elaine)



(1,3) 20.73 19.65 22.33 20.09 0.995483 0.000233 0.332477 0.000044(2,2) 31.71 29.88 34.73 31.13 0.996071 0.024055 0.334804 0.005726(4,4) 102.81 96.11 108.36 98.35 0.996082 0.985256 0.334849 0.302132(6,3) 124.42 116.66 134.14 119.94 0.995922 0.995953 0.334185 0.333885

Table 4Execution time and performance indices NPCR and UACI of the proposed and Lian et al.’s schemes, for some necessary combinations of m and n (test image:Peppers)



(1,3) 20.71 19.41 22.48 19.90 0.99596 0.000053 0.333139 0.000012(2,2) 31.60 29.92 34.54 31.14 0.996052 0.00211 0.334019 0.000605(4,4) 102.91 95.86 108.94 98.20 0.995953 0.986649 0.334727 0.302256(6,3) 124.47 116.49 134.49 120.34 0.996178 0.996086 0.334878 0.333595

Table 5Encryption time and performance indices NPCR and UACI of the proposed and Lian et al.’s schemes, for some necessary combinations of m and n (test image:Cameraman)



(1,3) 4.91 4.70 5.30 4.79 0.992142 0.002289 0.321162 0.000442(2,3) 9.85 9.40 10.63 9.50 0.99614 0.143021 0.334562 0.035714(4,4) 24.16 22.82 25.64 23.24 0.996094 0.989059 0.335884 0.306595(6,3) 29.44 28.16 31.82 28.99 0.99646 0.995743 0.334223 0.333586

Table 6Correlation coefficients of adjacent pixels of different images

Plain Lenaimage

Cipher image bythe proposed scheme(m = n = 2)

Cipher image by Lianet al.’s scheme(m = 6, n = 3)

Random image

Horizontal 0.975103 0.006816 0.005343 0.001562Vertical 0.988925 0.007827 0.008460 0.005922Diagonal 0.96704 0.003233 0.003557 0.004006


Fig. 7. Test image “Elaine”.

Fig. 8. Test image “Peppers”.

5. Conclusions

The substitution–diffusion type chaos-based image cryp-tosystem has been studied. It consists of the alternate operationsof substitution and diffusion to produce the required confusionand diffusion effect, respectively. In this Letter, we have pro-posed to introduce certain diffusion effect in the substitutionstage by simple sequential add-and-shift operations. Althoughthis leads to a longer processing time in a single round, theoverall encryption time is reduced as fewer rounds are required.Simulation results show that at a similar performance level, theproposed cryptosystem needs less than one-third the encryp-tion time required by Lian et al.’s. The effective acceleration ofsubstitution–diffusion type chaos-based image cryptosystem isthus achieved.

Acknowledgement

The work described in this paper was fully supported by agrant from CityU (Project No. 7002216).

References

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(a)

(b)

Fig. 9. Correlation analysis of two horizontally adjacent pixels in (a) the plainLena image; (b) the cipher image obtained using the proposed scheme.

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A fast image encryption scheme based on chaotic standard map