Low Complexity High Security Image EncryptionBased on Nested PWLCM Chaotic Map

Hadi H AbdulredhaComputer Sceince Department

Petra Private University, Faculty of Information TechnologyAmman, Jordan

halsaadi@uop.edu.jo

Qassim NasirDepartment of Electrical and Computer Engineering

University of SharjahSharjah, UAE

nasir@sharjah.ac.ae

Abstract— Chaotic systems produce pseudo-random sequenceswith good randomness; therefore, these systems are suitable toefficient image encryption. In this paper, a low complexity imageencryption based on Nested Piece Wise Linear Chaotic Map(NPWLCM) is proposed. Bit planes of the grey or color levelsare shuffled to increase the encryption complexity. A securityanalysis of the proposed system is performed and presented.The proposed method combine pixel shuffling, bit shuffling, anddiffusion, which is highly disorder the original image. The initialvalues and the chaos control parameters of NPWLCM maps arederived from external secret key. The cipher image generatedby this method is the same size as the original image and issuitable for practical use in the secure transmission of confidentialinformation over the Internet. The experimental results of theproposed method show advantages of low complexity, and high-level security.

I. INTRODUCTION

In recent years, and due to the development of multimediaand information technology, digital images are shared overthe public communication networks. Therefore, there is poten-tial risk of vulnerable access to sensitive documents. Imageencryption and robust cryptographic become an essential toprotect these multimedia files from leakage. Conventionalcipher algorithms such as DES, IDEA ..etc are not suitablefor multimedia files due to public data capacity, strong pixelcorrelation and high redundancy which reduces the encryptionperformance[1]. The chaos based multimedia files encryptionis not new idea. Matthew[2] was the fist step in this lineof research. Large amount of work using digital chaotictechniques to construct cryptosystems has been studied and hasattracted more and more attention in the last years [3] - [19].Researchers are especially interested in enhancing the chaoticgenerators and the diffusion stage of the cryptosystems. Ac-cording to the classification of chaotic systems, the securityapplication of chaos can be divided into analog chaotic securecommunications utilizing continuous dynamical systems [3]and digital chaotic cryptosystems utilizing discrete dynamicalsystems.

Discrete chaotic system is easy to implement and has goodstatistical properties, which can be used as random numbergenerator. Chaotic system is extremely sensitive to parametersand initial conditions. If the system parameters or initial valueis seen as the key, chaotic systems have become a good pass-

word system [4]. Recently some researchers such as [5], [6],they used two chaotic maps to encrypt the image to enhancethe security. Similarly, Ashtiyani et al. [7] also employedchaotic maps and other method to encrypt the images. Ahmad[8] introduced a new method using two logistic chaotic mapsand a large enough external secret keys for image encryption.This method exhibits a high security, but they did not proofthis method is robust or not to common signal processingattacks. The scheme proposed in this paper is based on twochaotic maps which can overcome the periodicity of Arnoldmap and is more security; besides, it is robust to the commonsignal attacks. The researchers in [9], [10], proved that logisticmap, that was widely used in the encryption domain, is notenough random and uniform. Then, they propose to use otherchaotic maps like Piece Wise Linear Chaotic Map (PWLCM).In [11] and [12] presents a chaos-based cryptosystem forsecure transmitted images abased on a 2D chaotic map whichis used to shuffle the image pixel positions, accompanied withsubstitution (confusion) and permutation (diffusion) operationson every block. A multiple rounds, are combined using twoperturbed chaotic PWLCM maps. Indeed, to obtain betterdynamical statistical properties and to avoid the dynamicaldegradation caused by the digital chaotic system working in afinite state, a perturbation technique is used.The objective of this research work is specially orientedtowards using Nested PWLCM map based image encryptionschemes. Two enhancement measures in the system efficiencyhave been proposed to the main components of typical chaos-based image cryptosystems: chaotic confusion and pixel diffu-sion processes. In the first block confusion and shuffle of bitpositions is performed, while the other block is diffused by addand shift algorithm. The resultant image is of high encryptionsecurity as diffusion is performed twice on the bit and bytelevels. Bit shuffling has been introduced by other researcherssuch in Pixel Chaotic Shuffle (PCS) [13], Pixel Shuffle (PS)[14] but the proposed method if a low complexity one as ituse a simple NPWLCM.The paper is organized as follows: Section II briefly intro-duces the Nested PWLCM. Section III describes the proposedencryption algorithm; The proposed system performance mea-sures and security analysis are given in section IV. And finally,we summarize our conclusions in section V.

II. NESTED PWLCM CHAOTIC MAP

The general description of chaos is an unpredictable andrandom-like long-term evolution that results from determinis-tic nonlinear systems. The simplest class of chaotic dynamicsystems is one-dimensional chaotic map of the form

xn+1 = f(xn, λ), n = 0, 1, 2, ... (1)

where the state variable x and the system parameter λ arescalars, and f is a mapping function defined in the real domain.

A. The Piecewise Linear Map [15]:

This map is given by:

x(n+1) =

{Bx(n) +A x(n) < 0Bx(n)−A x(n) ≥ 0

n = 0, 1, 2, ... (2)

where parameters A and B are chosen to be 1 and 1.998respectively to generate chaos. This map is extensively usedfor chaos generation due to its perfect properties such asuniform invariant density function; exactness, mixing andergodicity; exponentially decaying correlation function andsimple realization in both hardware and software [16]. Sincethe parameter A represents just a scaling factor, the stochasticproperties of the generated bit sequence are dependent onlyon the parameter B, which must assume values greater than 1for the system to be chaotic and not greater than 2 to avoidthe state x(n) being attracted to either +∞ or -∞ [17].

B. The Nested Piecewise Linear Chaotic Map (NPWLCM):

The proposed Nested Piecewise Linear Chaotic Maps areexpressed as [18], [19]:

x1(n) =

{x4(n− 1)−Ax4(n− 1) +A

x4(n− 1) > 0x4(n− 1) < 0

(3)

x2(n) =

{Bx1(n)−ABx1(n) +A

x1(n) > 0x1(n) < 0

(4)

x3(n) =

{Bx2(n)−ABx2(n) +A

x2(n) > 0x2(n) < 0

(5)

x4(n) =

{Bx3(n)−ABx3(n) +A

x3(n) > 0x3(n) < 0

(6)

where parameters A and B are chaos generation parameters.The bifurcation diagram [19] of the NPWLCM shows thatwhen the control parameter B is 1.998, chaos is still generated.

III. ENCRYPTION SYSTEM DESCRIPTION

For image encryption, 2D or higher-dimensional chaoticmaps are naturally employed for a reason that the image canbe considered as a 2D array of pixels. Let’s assume that thesize of the plain image is N x M. If the image is of graylevels then each pixel can be represented by 2G, where G(=8)is the number of bits per pixel. If the image is of a colorformat, then it can represented by 3D dimension array of Red,Green, and Blue (RGB) levels. Since images are composedof finite lattice called pixels, the domain of the map f(.) ischanged to the discretized form Image permutation can beachieved through shuffling the position of image bits and then

pixels. It is necessary to introduce a diffusion mechanismafter the bit permutation stage. The idea is to spread theinfluence of every single pixel over the entire image. Thecomplexity evaluation is important to image encryption aswell since it always indicates the feasibility of encryptionschemes. Some special attentions should be given in termsof computational speed, size and quality of the encryptedimages. The block diagram is composed of two processes:chaotic confusion and pixel diffusion as shown in Figure (1) .In confusion module, each bit position at each column in bit-

Confusion Diffusion Stage

Bits Level Permutation

Plain

Image Diffusion Stage

(Add-Shift Value

Transformation)

Ciphered

Image

Extractor PWLCM PWLCM

Password

Secret Key

A,B

x(0)PWLCM PWLCM

m x n Rounds

8 x m x n Rounds

Nested Piece Wise Linear Chaotic Map

Fig. 1. Standard Chaos based Image Encryption

plane is shuffled by using the pseudorandom sequence whichis generated by NPWLCM chaotic map. The change in thebit position at each pixel will modify the pixel value. Thusthe security of confusion module is significantly improveddue to introducing of diffusion effect throughout the bit-levelpermutation operation. Moreover, the overall security -level isfurther enhanced by introducing another diffusion operation atthe pixel -level by using a simple Add and shift operations. Toachieve a satisfactory level of security the confusion- diffusionoperations are repeated ( 8xmxn where m and n are the imagewidth and high respectively) rounds iteration.

The proposed image encryption is summarized by thefollowing steps

1) The proposed image encryption process utilizes a 96bit external secret key. This key is partitioned intothree 32 blocks. The chaotic sequence generation controlparameters (A, B), and the initial condition x(0) arederived from this secret key blocks.

2) Generate four chaotic bit sequences by NPWLCM usingformulas (3-6). XOR post processing is used to generatethese random bit sequences.

3) Sort four chaotic binary sequences and generatefour new integer sequences by ordinal numbers

corresponding to the original sequences, such as[sqk,i(i, j)]

N−1,4k=0,i=1 = sort([xk,i]

N−1,4k=0,i=1.

4) Each bit-plane is shuffled separately by using the sort-ing sequences generated in previous step as shown inFig. 2 . Thus the pixel value (Grey Levels bits, R-level bits, G-level bits, Blue level bits) matrices arecolumn shuffled by using the indexing generated chaoticsequences. So each level will be shuffled using thefollowing formula by assuming the level matrix definedas [B(i, j)]M−1,8N−1j=0,j=0 :

([B′(i, j, k)]M−1,N−1,8j=0,j=0,k=1 =

B(:, sq1, k)k = 1, 2B(:, sq2, k)k = 3, 4B(:, sq3, k)k = 5, 6B(:, sq4, k)k = 7, 8

The shuffled bit-planes are shown in Figure 2.

5) the permuted image is achieved by combining all the 8shuffled and confused bit-planes together

6) The pixels (Grey , or RGB) value will be diffused usingthe following Add-Shift formula

mask = Ti−1&7

XPi = (XPi + Ti−1)mod(256)

Ti = (XPi >> mask)|(XPi << (8−mask))

Where XPi be the value of the (i)th value in the imageafter confusion and permutation operations, Ti−1 and Tibe temporary values , ’mask’ be the number of bits tobe used in right and left shifting.

bit1 bit2 bit8 bit7 bit4 bit3 bit5 bit6

Bit 8n-2 Bit 8n Bit 8n-1 bit1 bit2 bit4 bit3

Bit Shuffled to a different position according to chaotic sequences

x1, x2, x3, x4

x1 x2 x3 x4

Bit Position in each Row of the RGB Bit plans

Each Byte of the

RGB bit planes

Sequence

Fig. 2. Bit indexing and shuffling within a row

IV. SYSTEM PERFORMANCE

A. System Performance for Grey Image Pictures

Three test images are chosen (Lena , Baboon and cameraMen) and encrypted by the proposed method, and visual testis performed as shown in Figure 3 (a,c,e) and Figure 3 (b,d,f),where each image is in 8 bit gray color with 256x256 pixels.

By comparing the original and the encrypted images in Figure(3) , there is no visual information observed in the cipheredimage. In order to further demonstrate the effectiveness ofproposed encryption scheme, some more tests suggested byother researchers have been carried out and the results areto be explained below [12]. An image-histogram illustrateshow pixels in an image are distributed. The histograms forthe original and ciphered test images are shown in Figure 4.As we can see, the histogram of the ciphered image is fairlyuniform and is significantly different from that of the originalimages. Hence the ciphered image does not provide any clue toemploy any statistical attack on the proposed image encryptionprocedure, which makes statistical attacks difficult.

a b

c d

e f

Fig. 3. Images before and after Encryption

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Fig. 4. Histogram Analysis of test Images before and after Encryption

Correlation coefficient measures the dependence of two adja-cent variables at a certain direction. The more closely relatedthese two variables are, the closer the correlation coefficientapproaches 1. Conversely, if they are less closely related,the value of correlation coefficient approaches 0. The twovariables are not related and unpredictable when the coefficientis close to 0. For an ordinary image, each pixel is highlycorrelated with its adjacent pixels either in horizontal, verticalor diagonal direction. However, an efficient image cryptosys-tem should show sufficiently low correlation in the adjacent

pixels and in all directions. The correlation distribution of twoadjacent pixels of the plain images and the cipher images onthe horizontally, vertical, and diagonal directions are shownin Figures 5-7. All adjacent pixels are selected and then thepixel value on the location (x+1,y) over the value on (x,y)in the case of horizontal direction. Similar tests are donefor pixel vale on (x,y+1) over (x,y) for vertical, and pixelvalue on (x+1,y+1) over (x,y) in case of diagonal as shownin the Figures 5-7 for the sampled images (Lena, baboon, andcamera man). To quantify and compare the correlations ofadjacent pixels, the correlation coefficient rxy of each adjacentpair is calculated for the plain and ciphered images using thefollowing formulas:

rxy =cov(x, y)√Dx

√Dy

(7)

cov(x, y) = E(x− E(x))(y − E(y)) (8)

E(x) =1

L

L∑i=1

xi (9)

Dx =1

L

L∑i=1

(xi − E(x))2 (10)

where x and y are gray scale values of two adjacent pixels inthe image, and L denotes the total number of samples (numberof pixels in the image).The results of the correlation coefficients for horizontal, ver-tical and diagonal adjacent pixels for the plain image and itscipher image are given in Table I. It is clear from Figures 5- 7 and Table I that the strong correlation between adjacentpixels in plain image is greatly reduced in the cipher imageproduced by the proposed chaos based encryption scheme.

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I(x,y

+1)

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I(x,y)

I(x+

1,y

+1)

Fig. 5. Correlation of two Horizontal, vertical and diagonal adjacent pixels- Baboon Test Image

Two common measures can be used to measure encryptionperfroamnce such as number of pixels change rate (NPCR) andthe unified average changing intensity (UACI)which measuresthe normalized mean difference between the plane and ci-phered images. Let the plane, and the ciphered image denoted

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+1)

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I(x,y)

I(x+

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Fig. 6. Correlation of two Horizontal, vertical and diagonal adjacent pixels-Lena Baboon Image

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I(x+

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+1)

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I(x,y)

I(x+

1,y

+1)

Fig. 7. Correlation of two Horizontal, vertical and diagonal adjacent pixels-Camera Test Image

by P and C [12] .Define a bipolar array, D, with the samesize as the images P and C. D(i,j) is determined as follows :

D(i, j) =

{1 if P (i, j) 6= C(i, j)0 elsewhere

(11)

The NPCR is defined as

NPCR =

∑i,j D(i, j)

MxNx100 (12)

Where M and N are the width and height of the P or C.While The UACI measures the average intensity of differencesbetween the plain image and the ciphered image. UACI isdefined as follows [12]:

UACI =1

MxN

M−1∑i=0

N−1∑j=0

|C1(i, j)− C2(i, j)|255

x100 (13)

Table (II) summarizes the mean value of NPCR and UACIbetween the original image and the ciphered image for theGrey Images. The NPCR and UACI are high enough to saythat the two images are very different.Information Entropy will be used to measure the encryption

TABLE ICORRELATION COEFFICIENTS FOR TWO ADJACENT PIXELS IN THE

ORIGINAL AND ENCRYPTED IMAGES.

Correlation Baboon Lena Camera

Orig.l Encry. Orig.l Encry. Orig.l Encry.

Horizonal 0.8464 -0.0203 0.9471 -0.0159 0.9201 0.0202Vertical 0.8108 -0.0435 0.9665 -0.0195 0.9443 -0.0771

Diagonal 0.7170 -0.0334 0.8985 0.0135 0.9074 -0.0496

TABLE IINPCR AND UACI BETWEEN ORIGINAL AND CIPHERED IMAGES

Image NPCR UACI

Baboon 99.6003 27.3953Lena 99.6292, 28.505

Camera Man 99.5682 31.0874

performance as it is defined to express the degree of uncer-tainties in the system. It is well known that the entropy Hm

of a message source m can be calculated as :

Hm =

2G−1∑i=0

p(mi) log2(1

p(mi)) (14)

TABLE IIIENTROPY OF THE CIPHERED IMAGES

Image Entropy

Baboon 7.9975Lena 7.9975

Camera Man 7.9966

Where p(mi) represents the probability of symbols (mi) ,G= number of gray level( =8). If a source emits 28 symbolswith equal probabilities, then source entropy Hm is equal to8. Actually, given that a practical information source seldomgenerates random messages, in general its entropy is smallerthan the ideal one. If the entropy of encrypted images isless than, but approaching eight, it will reduce the probabilityof successful restoration of images by interceptors. On thecontrary, if the entropy is more than eight, then interceptorseasily decode the encrypted images. Table(III) shows that theentropy of the encrypted images using the proposed encryptionscheme is close to theoretical value 8. This means thatinformation leakage in the encryption process is negligibleand the encryption system is secure upon the entropy attack.

B. System Performance for Color Image Pictures

Figures (8) and (9) demonstrate that the proposed encryptionalgorithm change the RGB histogram to be uniform which isrequired by any encryption method. Table (IV) summarizesthe mean value of NPCR and UACI for Lena color imagecompared with PCS and PS. The proposed method has NPCRhigher than 99.6

V. CONCLUSIONS

In this paper, an image encryption scheme based on NestedPiece Wise Linear Chaotic Map (PWLCM) is proposed. The

TABLE IVMEAN VALUE OF NPCR AND UACI TEST FOR COLOR LENA IMAGE WITH

DIFFERENT ENCRYPTION METHODS

Encryption NPCR% UACI%

NPWLCM 99.6323 99.6307 99.5941 33.1035 30.6367 27.6208PCS[13] 99.42 99.6 99.54 27.78 27.66 24.94PS[14] 99.26 99.45 99.13 21.41 23.42 15.08

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Fig. 8. Lena color image and RGB intensity Histogram Analysis

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Fig. 9. Encrypted Lena color image and RGB intensity Histogram Analysis

system is stream cipher architecture. The NPWLCM chaoscontrol parameters (A,B) and initial values x(0) of the mapsare derived from an external password (secret key). Thediffusion and confusion are conducted on the bit as well ason the pixel level and the method is applicable for both grayand color images. Detailed security analysis of the proposedscheme is presented which show a high security performancemeasures. The correlation coefficients of the encrypted imagesare below 0.07 for all test images and in all directions.In addition, high values (more than 99%) of NPCR andUACI of (27-33)% prove its ability to resist against pixelchanges attacks. Based on the results, we conclude that theproposed simple NPWLCM is suitable for real time secureimage transmission over public networks. Implementation ofthe proposed encryption scheme on DSP chip is one of our

future directions.

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