Pattern Recognition Letters 31 (2010) 347–354

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Pattern Recognition Letters

journal homepage: www.elsevier .com/locate /patrec

Image scrambling encryption algorithm of pixel bit based on chaos map

Guodong Ye *

College of Science, Guangdong Ocean University, Zhanjiang 524088, China

a r t i c l e i n f o

Article history:Received 5 August 2008Received in revised form 31 October 2009Available online 16 November 2009Communicated by F. Hsu

Keywords:Image scrambling encryptionChaos mapGray scramblingPosition permutationPixel bit

0167-8655/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.patrec.2009.11.008

* Fax: +86 759 2383636.E-mail address: guodongye@gmail.com

a b s t r a c t

This paper presents an image scrambling encryption algorithm of pixel bit based on chaos map. The algo-rithm takes advantage of the best features of chaos maps, such as their pseudorandom property, systemparameters, sensitive dependence on initial conditions and un-periodicity, combined with the pixel valuebits. The new algorithm uses a single chaos map only once to implement the gray scrambling encryptionof an image, in which the pixel values ranging from 0 to 255 are distributed evenly, the positions of allpixels are also permutated. In this way, the proposed method transforms drastically the statistical char-acteristic of original image information, so, it increases the difficulty of an unauthorized individual tobreak the encryption. Finally, the numerical experimental results show that the image encryption algo-rithm suggested has perfect hiding ability including large key space, sensitive key to initial conditions,high gray scrambling degree, and is suitable for practical use to protect the security of digital image infor-mation over the Internet.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

It is a mainstream task to transmit information using the Inter-net, main reasons for this include: independence of geographicalposition, no temporal limit, and low cost. However, while weare taking advantage of the Internet’s capabilities, unauthorizedindividuals have the chance to intercept our information and thenvisit, copy, and destroy it. So, the security and protection ofinformation becomes a hot problem studied by experts andresearchers. Image scrambling encryption algorithms realign allthe pixels in an image to different positions to permutate an ori-ginal image into a new image that is non-recognizable in appear-ance, disorderly, and unsystematic. In the past, many classicalencryption algorithms were proposed such as Arnold transform,Josephus traversing, chaos map (Guan et al., 2005; Zhang et al.,2005; Fang et al., 2007; Fang and Tong, 2007; Ye et al., 2007; Liet al., 2006; Pareek et al., 2006; Liu and Liu, 2007; Gao and Chen,2008). All these algorithms could encrypt images better than tra-ditional cipher theory, but, many of them were to consider the po-sition scrambling encryption, for example, the algorithm used in(Liu et al., 2005), and the pixel values did not change. So theyare not the ideal algorithms considering statistical characteristic,which is the same in the original image and the cipher image Grayencryption (Zhang et al., 2005; Pareek et al., 2006; Liu and Liu,2007; Gao and Chen, 2008) was also adopted to enhance the secu-rity further.

ll rights reserved.

Chaos-based cryptographic algorithm (Matthew, 1989) is anefficient encryption first proposed in 1989. It has many uniquecharacteristics different from other algorithms such as the sensi-tive dependence on initial conditions, non-periodicity, non-conver-gence and control parameters (Baptista, 1999; Chung and Chang,1998). The one-dimensional chaos system has the advantages ofsimplicity and high security (Gao and Chen, 2008; Sun et al.,2008). And many studies (Zhang et al., 2005; Fang et al., 2007; Fangand Tong, 2007; Ye et al., 2007; Li et al., 2006; Pareek et al., 2006;Liu and Liu, 2007; Gao and Chen, 2008) were proposed to adoptand improve it. Some of them use high-dimensional dynamicalsystems (Sun et al., 2008), other studies use couple maps (Behniaet al., 2008); however, all research is mainly to increase parametersto enhance the security. As we know, a good encryption methodshould be sensitive to keys and space should be large enough(Schneier, 1995), moreover, it must have some ability to resist out-er attack from statistic analysis, i.e., gray distribution. However, toomany parameters will influence the implementation and increasecomputation, for example, there are eight parameters in (Gaoand Chen, 2008). The efficiency will also be low using more thanone encryption algorithm to an image.

An image scrambling encryption algorithm of pixel bit based onchaos map is reported in this paper. On one hand, the algorithmuses the unique significant features such as sensitivity to initialcondition, control parameters to permutate efficiently the pixelpositions of original image. On the other hand, the gray scramblingencryption is carried out at the same time according to pixel bit.The gray distribution in our algorithm is changed in differentiteration. The rest of this paper is organized as follows. Section 2introduces chaos and its unique characteristics. In Section 3, the

348 G. Ye / Pattern Recognition Letters 31 (2010) 347–354

image encryption based on chaos is proposed including new algo-rithm steps. The simulated experiments are tested in Section 4.And in Section 5, the security of new algorithm is analyzed. Finally,the conclusions will be discussed in Section 6.

2. Chaos map and its unique characteristics

The classical chaos system in one-dimension is a logistic map,which can be defined by following:

xkþ1 ¼ lxkð1� xkÞ: ð1Þ

Here 0 6 l 6 4 is called parameter, and xk 2 ð0;1Þ.

0 10 20 30 40 50.3

0.4

0.5

0.6

0.7

0.8

0.9

1a

x va

lue

Number

0 10 20 30 40 50.3

0.4

0.5

0.6

0.7

0.8

0.9

1b

y va

lue

Number o

Fig. 1. Chaos phenomena: (a) x0

From chaos dynamical system, the logistic map becomes confu-sion if 3:569945 � � � 6 l 6 4, that is to say, the chaos sequencesfxkg1k¼0 and fykg

1k¼0 are non-periodic, non-convergent and pseudo-

random given by two different initial conditions x0 and y0, forexample, with l ¼ 3:5786; x00:3333 and y0 ¼ 0:5656, the chaosbehavior are shown in Fig. 1. Moreover, the propositions (Fangand Tong, 2007) of chaos are listed as follows.

Proposition 1. The probability density function of sequence general-ized by chaos system is

q ¼1

pffiffiffiffiffiffiffiffiffiffixð1�xÞp ; 0 < x < 1

0 others:

(ð2Þ

0 60 70 80 90 100

of iteration

0 60 70 80 90 100f iteration

=0.3333 and (b) y0=0.5656.

G. Ye / Pattern Recognition Letters 31 (2010) 347–354 349

Proposition 2. Average value of contrail points in chaos sequence is

x ¼ limN!1

1N

XN�1

k¼0

xk ¼Z 1

0xpðxÞdx ¼ 0:5: ð3Þ

Proposition 3. Self-correlation of chaos sequence is

TðsÞ ¼ limN!1

1N

XN�1

k¼0

ðxk � xÞðxkþs � xÞ ¼Z 1

0xf sðxÞpðxÞdx ¼ 0: ð4Þ

Proposition 4. Self-correlation function between two chaossequences is

CðsÞ ¼ limN!1

1N

XN�1

k¼0

ðxk � xÞðykþs � yÞ ¼ 0; ð5Þ

where x0 and y0 are the different initial values of chaos map.

3. Proposed image encryption algorithm based on chaos map

3.1. Image encryption algorithm based on chaos map

For a gray image of size M � N, it is a digital matrix of M rows Ncolumns, in which the values range from 0 to 255. In the process ofencryption using chaos map, fist, the N + 1 shuffling addressesTM ¼ ft1; t2; . . . ; tMg and TN1 ¼ fp11; p12; . . . ; p1Ng; . . . ; TNN ¼ fpN1;

pN2; . . . ; pNNg are generated by initial value x0 after doing iterationsin chaos map, the details can be seen from next section. The next

Fig. 2. Image encryption and decryption test: (a) original image of size 256� 256, (b) enwrong keys.

step is to exchange the rows by vector TM, and exchange columnposition of all values from first row to last row according toTN1; . . . ; TNN . So the position of pixels is completely shuffled, i.e.,position scrambling encryption. Practically, set eh ¼ feðt1Þ;eðt2Þ; . . . ; eðtMÞg; el ¼ feðp1Þ; eðp2Þ; . . . ; eðpNÞg, here eðtiÞ; eðpjÞði ¼1;2; . . . ;M; j ¼ 1;2; . . . ;NÞ denote unit column vectors, the cipherimage will be obtained when the original image is multiplied leftby matrix eh and right by matrix el, by which it will implementthe scrambling encryption for row and column. In order todecrease the computation complexity and save time, we only usetwo transform vectors to scramble row and column separately,For example, Fig. 2b is the encrypted image of Fig. 2a withx0 ¼ 0:364 and l ¼ 3:92 (m ¼ 13 and n ¼ 27). The original imageFig. 2c is restored having correct key, else, Fig. 2d is got by wrongkeys. The histogram pictures of original and encrypted images arealso given in Fig. 3, so, we can see that there is no change in thehistogram.

3.2. Generation of chaos sequence

Given an initial value l; x0 and a positive integer m, a chaossequence fxkg1k¼0 will be obtained by formula (1). Then for thesequence generated, we take out M values fxmþ1; xmþ2; . . . ; xmþMgfrom the mth number. And ordering for this M values and gettingfxmþ1; xmþ2; . . . ; xmþMg. Following, we find the position of valuesfxmþ1; xmþ2; . . . ; xmþMg in fxmþ1; xmþ2; . . . ; xmþMg and mark downthe transform positions TM ¼ ft1; t2; . . . ; tMg. So the TM is the

crypted image, (c) decrypted image with correct keys, and (d) decrypted image with

350 G. Ye / Pattern Recognition Letters 31 (2010) 347–354

sequence vector we want. In this case the numbers l and x0 aremade up of keys. The same will be done for TN with n.

3.3. Proposed encryption algorithm and encryption steps

In the new algorithm, the chaos-based image encryption iscombined with pixel bit. Considering the number of parametersand computational complexity, we use only one single chaosmap which is applied directly to the position scrambling encryp-tion; fortunately, it carries out the gray encryption at the sametime, of which is the significant idea of our paper. Before applyingthe new algorithm, we decompose the pixel values into bits. LetPði; jÞ denote the pixel in the ith row and jth column of original

0 50 100

0

100

200

300

400

500

600

700

800

900

1000a

0 50 100

0

100

200

300

400

500

600

700

800

900

1000b

Fig. 3. Histogram: (a) original im

image, and let ptði; jÞ denote the tðt ¼ 0;1; . . . ;7Þ digit after trans-forming Pði; jÞ. Then

ptði; jÞ ¼ 1 if ðPði; jÞ=2tÞmod2 ¼ 10 others:

(ð6Þ

So, we can transform a gray image into bit matrix composed of zeroand one by formula (6). Contrarily, we also can transform theptði; jÞðt ¼ 0;2; . . . ;7 into gray pixels Pði; jÞ if using the followingformula:

Pði; jÞ ¼X7

t¼0

2t � ptði; jÞ: ð7Þ

150 200 250

150 200 250

age and (b) encrypted image.

G. Ye / Pattern Recognition Letters 31 (2010) 347–354 351

For an original image of size of M � N, a matrix of M � 8N will be gotaccording to formula (6), the pixels of it are zeros and ones. Forexample, a matrix A3�2 are transformed into B3�16. Then we encryptevery row and column of this bit matrix of M � 8N according to thetwo chaos vector sequences generated by formula (1). Theencrypted bit matrix is also of size M � 8N. An encrypted decimalimage will finally be obtained by computing formula (7). For exam-ple, the former matrix B3�16 can be encrypted in matrix C3�16, thencipher matrix D3�2 is eventually obtained.

The encryption process for a gray image using the new algo-rithm is performed according to the following steps:

Step 1. Reading a gray image and expressing it with a decimalmatrix A of size M � N.

Step 2. Transforming all pixels in A into bit pixels by formula (6)and noting it as matrix B of size M � 8N.

Step 3. Applying chaos encryption to rows and columns ofmatrix B, then matrix C is got.

Step 4. Transforming contrarily the encrypted matrix C by for-mula (7) and getting decimal matrix D of size M � N.

Step 5. The encryption process is complete and a gray image isgenerated.

4. Simulated experiments

In this section, we do experiments for validating the securityand practicability of proposed algorithm. All the work was done

Fig. 4. Image encryption and decryption test using new algorithm: (a) original image, (bwith wrong keys.

by Matlab in a computer of Core 2, CPU 2.0 GHz and EMS memory1.99 GB. Lena image of size 256� 256 is shown in Fig. 4a, the initialvalues x0 ¼ 0:2009;l ¼ 3:98 and integers m ¼ 20; n ¼ 51 are takento generate the chaos sequences. The encrypted image of Fig. 4b isobtained from Fig. 4a, we cannot find out any useful informationfrom Fig. 4b related to original image. The cost of whole encryptionis approximately 2.3 s which is acceptable for security. In next sec-tion, the classical Arnold method is taken to compare with the pro-posed method.

5. Security analysis

(1) Key space and sensitivity analysis.Key space refers to the total number of different keys that canbe used in an algorithm. In proposed algorithm, the keysinclude l; x0 (not including m;n) compared with a; b in Arnoldmethod. So, the key space is no less than Arnold. With thegiven keys, we encrypt the image Fig. 4a into Fig. 4b. Usingthe wrong keys, even with little movement, for example0.20090000000001, will result in an incorrectly decryptedimage, as displayed in Fig. 4d. Of course, the original imagewill be restored with correct secret keys seen Fig. 4c. There-fore, we can conclude that encryption algorithm is sensitiveto initial conditions.

(2) Gray difference degree.The gray difference of a pixel with a neighbor pixel in imageis computed as follows:

) encrypted image, (c) decrypted image with correct keys, and (d) decrypted image

352 G. Ye / Pattern Recognition Letters 31 (2010) 347–354

GN ¼P½Gðx; yÞ � Gðx0; y0Þ�2

4; hereðx0; y0Þ ¼

ðx� 1; yÞðxþ 1; yÞðx; y� 1Þðx; yþ 1Þ;

8>>><>>>:

ð8Þ

where Gðx; yÞ denotes the gray value in position ðx; yÞ. The aver-age neighborhood gray difference of the whole image can becomputed by formula (9).

ANðGNðx; yÞÞ ¼PM�1

x¼2

PN�1y¼2 GNðx; yÞ

ðM � 2Þ � ðN � 2Þ : ð9Þ

0 10 20 30 40 500.83

0.84

0.85

0.86

0.87

0.88

0.89

0.9

0.91

0.92

Number of

Gra

y va

lue

degr

ee

a

0 10 20 30 40 500.952

0.9525

0.953

0.9535

0.954

0.9545

0.955

0.9555

Number of

Gra

y va

lue

degr

ee

b

Fig. 5. Gray value degree: (a) Arnold meth

And the gray value degree is defined by

GVD ¼ AN0ðGNðx; yÞÞ � ANðGNðx; yÞÞAN0ðGNðx; yÞÞ þ ANðGNðx; yÞÞ

; ð10Þ

where AN and AN0 denote the average neighborhoodgraydifference of image before and after being encrypted sepa-rately.Fig. 5a and b shows the gray value degree generated by Arnoldmethod and the proposed method, we can see that the valueof gray degree in new algorithm is nearer to 1.

60 70 80 90 100

Iteration

60 70 80 90 100 Iteration

od, and (b) proposed method.

0 50 100 150 200 250

0

100

200

300

400

500

600

700

a

0 50 100 150 200 250

0

100

200

300

400

500

600b

0 50 100 150 200 250

0

100

200

300

400

500

600

700

c

Fig. 6. Histogram (a) original image, (b) encrypted image by proposed method, and(c) encrypted image by Arnold method.

Table 1Correlation coefficients of two adjacent pixels in Rice image and its encrypted image.

Model Original image By proposed method By Arnold method

Horizontally 0.9690 �0.0134 0.0787Vertically 0.9637 0.0012 �0.0793Diagonally 0.9492 0.0398 �0.0633

G. Ye / Pattern Recognition Letters 31 (2010) 347–354 353

(3) Histogram analysis.A histogram, also called a gray distribution picture, is a chartof the pixel distribution order for an image. It reflects thedistribution of pixels in an image after being disposed.

Fig. 6a is the histogram of Lena image, Fig. 6b and c denoteseparately the histogram of two encrypted images by theproposed method and Arnold method. So, we can hardlydo any statistical analysis to decrypt an encrypted imagebecause the histogram is changed greatly by the proposedmethod.

(4) Correlation analysis of two adjacent pixels.The following formula (11) is commonly applied to test thecorrection between two adjacent pixels in original imageand encrypted image. We select randomly 2500 pairs oftwo adjacent pixels including horizontal, vertical and diago-nal directions from another in Rice image. The results can beseen in Table 1, from which we find that the correlation isalmost zero.

rxy ¼covðx; yÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDðxÞDðyÞ

p ; ð11Þ

here covðx; yÞ ¼ 1N

PNi¼1ðxi � EðxÞÞðyi � EðyÞÞ; DðxÞ ¼ 1

N

PNi¼1

ðxi � EðxÞÞ2, and EðxÞ ¼ 1N

PNi¼1xi.

6. Conclusions

A novel image scrambling encryption algorithm based on chaosmap together with pixel bit is presented in this paper. The newalgorithm can implement the position encryption and gray valueencryption simultaneously. The algorithm proposed in this papercan also be extended to high-dimensional chaos map and can beapplied to 3D images. It is an efficient encryption algorithm thatcan be used directly to transmit securely all kinds of image infor-mation over the Internet. Additionally, the gray distribution canalso be discussed as (Guan et al., 2005; Xiao et al., 2009) further.We just do the scrambling encryption and do not consider it here.

Acknowledgements

The author would like to thank the Editor and the anonymousReferees for their valuable comments and suggestions to improvethis paper. This research was partially supported by the NaturalScience Foundation of Guangdong Ocean University of PR China(No. 0812109).

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