Confusion and Diffusion of Grayscale Images Using Multiple Chaotic Maps

Sukalyan Som Department of Computer Science

Barrackpore Rastraguru Surendranath College Kolkata, West Bengal, India

sukalyan.s@gmail.com

Atanu Kotal Department of Computer Science

Techno India College of Technology Kolkata, West Bengal, India

atanukotal@gmail.com

Abstract—In present age chaos based cryptosystem has gained attention in research of information security and number of chaos based image encryption algorithms has been proposed. In this paper, a new symmetric image encryption algorithm based on multiple chaotic maps is proposed. In the proposed algorithm, the plain image is first scrambled using generalized Arnold Cat Map. Further, the scrambled image at a particular iteration is encrypted using chaotic sequences generated by one-dimensional Logistic Map after preprocessing them to integers. The experimental results depicts that the proposed algorithm can successfully encrypt and decrypt grayscale images with secret keys. The simulation analysis also exhibit that the proposed method is loss-less, secure and efficient measured in terms of statistical tests, key sensitivity test and some well known tests like NPCR, UACI and entropy.

Keywords-Arnold Cat Map, Logistic Map, Scrambling, encryption, Correlation Coefficient, Histogram, NPCR, UACI.

I. INTRODUCTION In present age chaotic maps are being considered to be

extremely efficient for practical use because of their inherent properties like they high sensitivity to initial conditions and system parameters, pseudo-random behavior and non-periodicity etc. The combination of chaos theory and cryptography forms an important field of information security. Chaos based encryption systems are robust against statistical attacks.

Due to some inherent features of images like huge data storage and high data redundancy, the encryption of images is different from that of texts; therefore it is difficult to be handled by traditional encryption methods.

Chaos based cryptosystems are gaining the attention of the researchers now-a-days to develop secure image encryption techniques to meet the demand for real-time image transmission over the communication channels. A number of image encryption algorithms based on chaotic systems have been proposed. There have been many image encryption algorithms based on chaotic maps like the Logistic map [1-3], the Standard map[4], the Baker map [5, 6], the PWNLCM [7] the Cat map [8], the Chen map [9] etc. In order to improve the security performance of the image encryption algorithm, the concept of shuffling the positions of pixels in the plain-image and then changing the gray values of the shuffled image pixels is used. In this paper, image shuffling is proposed using generalized Arnold Cat Map to achieve good shuffling effect and the encryption of the shuffled image is performed using 1D Logistic map to enforce the security of the proposed encryption process.

In section II a brief overview on Arnold Cat Map and 1D Logistic Map is presented. The encryption and Decryption algorithms are presented in section III with the security analysis and tests being stated in section IV. Conclusions are drawn in section V.

II. BACKGROUND

A. Generalized Arnold Cat Map An Arnold Cat Map is a discrete system that stretches

and folds its trajectories in phase space. Vladimir Arnold proposed the map in 1960 and he used the image of a cat while working on it. [10]

Let X = , where X is a vector, then the transformation would be : 1 1 Mod N, where ( , are the pixel position of N N image, p, q are the parameters which are positive integers. ( , ) is the new position of the original pixel position ( , ) when Arnold Cat Map is performed once. To better understand the mechanism of the transformation , let us decompose it into its elemental pieces.

• Shear in the direction of x by a factor of 1. • Shear in the direction of y by a factor of 1. • Evaluate the modulo.

Figure 1 provides a visual aid illustrating these steps. The first step shows the shearing in the x and y directions, followed by evaluation of modulo and reassembly of the image.

Figure 1. Arnold Cat Map

2012 National Conference on Computing and Communication Systems (NCCCS)

978-1-4673-1953-9/12/$31.00 ©2012 IEEE

Some elementary properties of Arnold csummarized as follows:

• Experimentally, no elegant modeveloped to establish the relationsperiod of scrambling and numbcolumns. In general, it may be clasize of an image increases, the increases. However, this is not alway

• The period T of the Map depends op, q and the size N of the original imperiod be T then T (n) = 3n if n = 2.5k k =1, 2,…. T (n) = 2n if n = 5k or n = 6.5k k=T (n) 12 /7 for all other choices o

• The determinant of the transforma[12], thus the map is such a map warea i.e. no attractor.

• The map is a one-to-one mapping [1• Plain image or original image can b

secret values of p and q after iteratiand thus the correlation among thecan be disturbed completely.

B. 1D Logistic Map The one-dimensional Logistic map is pr

May [14]. It is one of the simplest nonlineasystems that exhibit chaotic behavior, equation: (1 , where is initithe system parameter and n is the number omap is chaotic for (3.56994,4 and sequence generated by 1D Logistic map isthe shuffled image.

III. PROPOSED ALGORITH

The entire algorithm has two componstrong correlation between adjacent pixelsimage has to be disturbed. To achieve Arnold Cat Map is used. After scrambling thdiffuse it 1D Logistic map is used.The entthe encryption scheme is as follows:

A. Method of Encryption Step 1: Consider the plain image to be INXN where x=0, 1, 2, … , N-1 and y = 0, 1, Step 2: Scramble the plain image using Arnold Cat Map with given values of p, qthe scrambled image at nth iteration to beI Step 3: Keys for diffusing the gray scale pixusing 1D Logistic Map with given vconditions x and µ as follows: (1 10 , 256 1, Step 4: Each scrambled image pixel pwith the key k generated in step 3 to get the

cat map can be

odel could be ship between the ber of rows or aimed that as the

period tends to ys true.

on the parameters mage [11]. Let the

=1, 2,…. of n. ation matrix is 1 which is keeping

13]. be scrambled via ing m (>1) times e adjacent pixels

roposed by R. M. ar chaotic discrete

defined by the

ial condition, is of iterations. The (0, 1) n. The s used to encrypt

HM nents. Firstly, the s in the original this, generalized he plain image to tire procedure of

(x, y of size , 2, …, N-1.

the generalized q and n. Consider (x, y .

xels are generated values of initial

2, 3, … , ( is encrypted

e encrypted pixel

p at the coordi, 0, 1, 2, … , ( 1 as mod ( denotes the exclusive-OR operatio

B. Method of Decryption The plain image can be recovered the encryption algorithm in reverse q and n for generalized Cat Map image or descramble the cipher conditions for the 1D generation which would be used dudecryption has to be transmitted thro

IV. SECURITY ANALYSIS A

A. Statistical analysis • Visual test and Histogram anAn image histogram illustrates

are distributed by graphing the nucolor intensity level. The histogramwell as encrypted images have beenof such histogram analysis is presen

(a) (b)

(d) (e) Figure 2. (a) Input image, (b) shuffled imPlain-image histogram (e) Shuffled imagehistogram

The histogram of shuffled imhistogram of plain-image. This meastatistical information in shuffled iplain-image. The encrypted image by a uniform distribution, is quite dhistogram.

• Correlation Coefficient analIn most of the plain images, the

among adjacent pixels whereas pooneighbouring pixels of corresponobserved. Karl Pearson’s Producoefficient stated as follows is usedcorrelation of horizontally and vertic

nate (x,y) where , 256 where on successfully by applying order. The parameters p, for scrambling the plain

image and the initial Logistic Map for key

uring both encryption and ough a secure channel.

AND TEST RESULTS

nalysis how pixels in an image

umber of pixels at each ms of several original as n analyzed. One example ted in Figure 2.

(c)

(f)

mage and (c) cipher-image (d) e histogram (f) Cipher-image

mage is similar to the ans that the corresponding image is as equal as the histogram, approximated ifferent from plain-image

lysis ere exists high correlation or correlation between the nding cipher image is

uct Moment correlation d as a measure to find the cally adjacent pixels:

( ,

Where ( , ∑ ( ( 1 ( 1 (With 0 0

An extensive study of the correlatiooriginal image and encrypted image has using the USC-SIPI [15] image databacollection of digitized images available anUniversity of Southern California primaresearch in image processing, image analyvision. The database is divided into fourdifbased on the basic character of the picturevolumes available atUSC-SIPI site are—miscellaneous and sequences. We miscellaneousvolume tomeasure the correlofUSC-SIPI image database. The miscelconsists of 44 imagesout of which 16 aremonochrome. In table 1 correlation coevertically, horizontally and diagonally adjacsample original images and corresponding eare presented from which it can be conclunegligible correlation between the two adencrypted image but high correlation in orig

TABLE I. CORRELATION COEFFICIENT

Image name

Horizontal pixels

Vertical pixels

Plain Image

Cipher Image

Plain Image

Cipher Image

Boat 0.9400 0.0045 0.9704 0.0015 Bridge 0.9404 0.0016 0.9275 0.0013 Clock 0.9894 -0.0080 0.9938 -0.0114 Elaine 0.9757 0.0037 0.9730 -0.0026

• Measures of Central Tendency and DIn the following table mean, median of

and standard deviation of them is presentedimages. The comparative analysis as presedepicts that the measures are different in orencrypted image. The correlation between thand corresponding cipher image reveals correlation between them.

TABLE II. MEASURES OF CENTRAL TENDENCY

Image Name

Mean Median StandaDeviat

Orig

inal

Im

age

Encr

ypte

d Im

age

Orig

inal

Im

age

Encr

ypte

d Im

age

Orig

inal

Im

age

Boat 129.3 127.2 142 125 46.6 Clock 186 124.7 214 123 36.7 Elaine 136.4 127.7 135.0 128.0 46.1 Bridge 113.8 128.3 105.0 129.0 54.7

on between the been performed

ase which is a nd maintained by arily to support ysis and machine fferent categories es.Currently, four —textures,aerials,

have chosen lation coefficient llaneous volume e colored and 28 efficient of two

cent pixels of four encrypted images uded that there is djacent pixels in inal image.

T ANALYSIS

Diagonal Pixels

Plain Image

Cipher Image

0.9223 0.0021 0.8975 -0.0007 0.9840 -0.0116 0.9692 -0.0013

Dispersion gray pixel values d on four sample ented in Table II riginal image and he original image that there poor

Y AND DISPERSION

ard tion

Correlation between original

and encrypted

image Encr

ypte

d Im

age

73.8 0.0102 74.1 0.0066 73.3 -0.0057 73.6 -0.0020

B. Key Sensitivity analysis A good cryptosystem should be sensecret keys i.e. a small changdecryptingprocess results into adecrypted image. Ourproposed esensitive to a tiny change inthe secwe have taken the secret keys for p=1,q=1,x0=0.7199, 3.57 to emade a small change in the

0.7199000001 only.The resultsa sample image clock are presented

(a) (b)

(d) Figure 3. Key-sensitivit

(a) Original Image Clock (b) Scrambled Imagp=1,q=1,x0=0.7199, 3.57 (d)Decrypted 3.57 (e) Decrypted image with p=1,q=1,

C. Some other analysis • Information Entropy Information theory is the math

communication and storage foundeModern information theory is correction, data compression, cryptosystems, and related topics. It is weH(s) of a message source s can be ca( ∑ ( ( ,the probability of symbol s and thebits. Let us suppose that the sourcequal probability, i.e., sevaluating the above equation, we o8, corresponding to a truly randomthat a practical information source smessages, in general its entropy videal one. However, when the messentropy should ideally be 8. If theemits symbols with entropy less thdegree of predictability, which threa

Information entropy for a few imIII which is H(s)=7.99 and very clomeans a high permutation and substproposedalgorithm and has a robustentropy attack.

sitive to a small changein ge in secret keys in a completely different encryption algorithm is cret keys. As an example a round of encryption as encrypt the images and e initial condition as s of key sensitivity test on in Figure 3.

(c)

(e) ty analysis ge (c) Encrypted Image with image with p=1,q=1,x0=0.7199, , 0.7199000001

hematical theory of data ed in 1949 by Shannon. concerned with error-

ography, communications ll known that the entropy alculated as: , where P(s ) represents e entropy is expressed in ce emits 28symbols with s , s , … . . , s .After obtain the entropy H(s) =

m source. Actually, given seldom generates random value is smaller than the sages are encrypted, their output of such a cipher

han 8, there exists certain atens its security. mages is shown in Table se tothe ideal value. This titution is achieved by the t performance against the

• Measurement of Encryption Quality Image encryption quality measure is a figure of merit

used for the evaluation of image encryption techniques. With the application of encryption to an image a change takes place in pixels values as compared to those values before encryption. Such change may be irregular. This means that the higher the change in pixels values, the more effective will be the image encryption and hence the encryption quality. So the encryption quality may be expressed in terms of the total changes in pixels values between the original image and the encrypted one. A measure for encryption quality may be expressed as the deviation between the original and encrypted image [16] [17]. The quality of image encryption may be determined as follows:

Let P and C denote the original image and the encrypted image respectively, each of size H×W pixels with L grey levels. P(x,y),C(x,y) {0,...,L-1}are the grey levels of the images P and C at position (x, y), 0 < x < H-1, 0 < y < H-1. We will define HL(P as the number of occurrence for each grey level L in the original image (plain-image), and HL(C as the number of occurrence for each grey level L in the encrypted image (cipher-image). The encryption quality represents the average number of changes to each grey level L and it can be expressed mathematically as:

Encryption Quality ∑ | HL(P HL(C | L 256 The encryption quality test was performed using the

input image of size 5212X512 shown in Figure 2. The encryption quality of the proposed scheme is 68.8787 while the encryption quality of AES is 69.5250.

A very useful measure of the performance of the decryption procedure is the Peak Signal-to-Noise Ratio (PSNR). The PSNR of a given image is the ratio of the mean squaredifference of the component for the two images to the maximum mean square difference thatcan exist between any two images. It is expressed as a decibel value. The greater PSNR value(>30dB), the better the image quality recovered. For encrypted image, smaller value of PSNR isexpected.

Let C(i, j) and P(i, j)be the gray level of the pixels at the ith row and jth column of a H×W cipher and plain-image, respectively. The MSE between these two images is defined in MSE 1H X W |C(i, j P(i, j |WH

20 ( 255( PNSR for the cipher images with respect to their

plainimages for several images have been calculated. In all test cases, it was found to be small(<10dB). The computed results of peak-signal-to-noise ratio for four sample images are presented in table III. The low PSNR values reflect the difficulty inretrieving the plain image from the cipher image, without the knowledge of secret key.

• NPCR, UACI and MSE In general, a desirable property for an encrypted image is

being sensitive to the small changes in plain-image (e.g., modifying only one pixel). Opponent can create a small change in the input image to observe changes in the result. By this method, the meaningful relationship between original image and encrypted image can be found. If one small change in the plain-image can cause a significant change in the cipher-image, with respect to diffusion and confusion, then the differential attack actually loses its efficiency and becomes practically useless.

To test the influence of one-pixel change on the whole image encrypted by the proposed algorithm, three common measures were used NPCR and UACI [18][19]. NPCR means the number of pixels change rate of ciphered image while one pixel of plain-image is changed. UACI which is the unified average changing intensity, measures the average intensity of the differences between the plain-image and ciphered image.

Consider two cipher-images, C1 and C2, whose corresponding plain-images have only one pixel difference. The NPCR of these two images is defined in

NPCR ∑ ( ,,HXW , where D(i, j)is defined as D(i, j 0 , if C (i, j C (i, j1 , if C (i, j C (i, j

Another measure, UACI, is defined by the following formula 1H X W | C (i, j C (i, j |255 X 100%,

Tests have been performed on the proposed scheme, about the one-pixel change influence on 256 gray-scale image of size 512X512 and results on four sample images is presented in table III which c

TABLE III. DIFFERENTIAL ATTACK ANALYSIS

Image Name NPCR UACI PSNR Entropy Boat 0.999996 0.0018 9.3378 7.9978 Clock 0.999996 0.0027 7.2264 7.9953 Elaine 0.999992 0.0022 9.3315 7.9953 Bridge 0.999996 0.0019 8.7891 7.9988

• Encryption and Decryption Time Comparison For images of different size we have presented the time

taken by the proposed algorithm to perform encryption, decryption, scrambling and descrambling of images.

TABLE IV. TIME ANALYSIS OF THE ALGORITHM

Image Size

Key Generation

Scram bling

Encryp tion

Decryp tion

Descram bling

256X256 0.009 0.12 0.42 1.08 0.17 512X512 0.01 0.23 0.97 1.96 0.26

V. CONCLUSION In this communication, a new algorithm for confusing

and diffusing grayscale images using multiple chaotic maps have been presented which is based on shuffling the pixel

positions and changing the grayscale intensities of plain image. To perform shuffling of pixel positions generalized Arnold Cat map is used and to change the pixel values of the shuffled image 1D Logistic map is used. All the simulation and experimental analysis show that the proposed image encryption system has (1) a very large key space, (2) high sensitivity to secret keys, (3) has information entropy close to the ideal value 8 and (4) has low correlation coefficients close to the ideal value 0. Hence, we can say that all the analysis prove the security, effectiveness and robustness of the proposed image encryption algorithm. Block based shuffling of plain images by using pseudorandom sequences generated chaotic maps may be a future scope to work.

ACKNOWLEDGMENT The authors express a deep sense of gratitude to the

Department of Computer Science, Barrackpore Rastraguru Surendranath College, Kolkata, West Bengal, India as well as for Techno India College of Technology, Kolkata, West Bengal, India for providing necessary support for the work and their family members for being a source of constant inspiration and motivation for pursuing research works.

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