mapping and double random-phase encoding received on 7th

IET Image Processing

Research Article

Colour image encryption based on logisticmapping and double random-phase encoding

ISSN 1751-9659Received on 7th July 2016Revised 10th November 2016Accepted on 11th December 2016E-First on 12th January 2017doi: 10.1049/iet-ipr.2016.0552www.ietdl.org

Huiqing Huang1,2, Shouzhi Yang1 1Department of Mathematics, Shantou University, Shantou 515063, Guangdong, People's Republic of China2School of Mathematics, Jiaying University, Meizhou 514015, Guangdong, People's Republic of China

E-mail: [email protected]

Abstract: In this study, the authors propose a novel method to encrypt a colour image by use of Logistic mapping and doublerandom-phase encoding. Firstly, use Logistic mapping to diffuse the colour image, then the red, green and blue components ofthe result are scrambled by replacement matrices generated by Logistic mapping. Secondly, by utilising double random-phaseencoding to encrypt the three scrambled images into one encrypted image. Experiment results reveal the fact that the proposedmethod not only can achieve good encryption result, but also that the key space is large enough to resist against commonattack.

1 IntroductionWith the rapid development of networks, people's economic life isincreasingly dependent on the network, so network security hasbecome increasingly important in recent years. Datacommunications have become largely networking in nature withimaging capabilities being embedded in a myriad of portabledevices, such as smart phones and tablet computers.Simultaneously, communication channels, such as internet andwireless networks, these techniques bring great convenience to ourlife, but it also brings new challenges to the privacy. To meet thischallenge, various encryption algorithms have been designed [1–9].Such as image encryption based on pixel permutation, imageencryption based on double random-phase encoding, imageencryption technology based on modern cryptography [10, 11],image encryption technology based on chaotic and so on. Amongthem, due to chaotic have shown some exceptionally goodproperties that are desirable for the cipher system, pseudo-randomness, extremely sensitive for initial value, ergodicity and aperiod and so on, more and more researchers pay attention tochaotic encryption algorithm. In [12], Socek et al. proposed anenhanced one-dimensional (1D) chaotic key-based algorithm forimage encryption. A novel block cryptosystem based on iterating achaotic map has been proposed in [13]. Later in 2007, Li et al. [14]pointed out the encryption scheme presented in [13] is not onlyinsecure against chosen-plaintext attack, but also insecure against adifferential known-plaintext attack. Subsequently, a new blockbased image shuffling is proposed to achieve good shuffling effectusing two chaotic maps and the encryption of the shuffled image isperformed using a third chaotic map to enforce the security of theproposed encryption process [15].

In recent years, plenty of colour image encryption approacheshave been proposed. In some encryption algorithms, colour imageis decomposed into the components of red, green and blue (RGB).The corresponding colour components are encoded by anencryption method of grey-level image. The RGB components ofcolour image are encrypted by chaotic systems. Chaos-based imageencryption algorithm to encrypt colour images by using couplednon-linear chaotic map [16] has been reported. In [17], Liu et al.proposed a colour image encryption by using the rotation of colourvector in Hartley transform domains. Later in 2011, Liu et al. [18]presented a colour image encryption algorithm is designed by useof Arnold transform and discrete cosine transform. Subsequently,Chen et al. have reported a colour image encryption based on theaffine transform and the gyrator transform [19]. Liu et al. [20]

proposed a novel colour image hiding scheme with three channelsof cascaded Fresnel domain phase-only filtering.

In this paper, we propose a colour image encryption by use ofLogistic mapping and phase encoding. Firstly, diffuse the colourimage by using Logistic mapping, then the RGB components of theresult are scrambled and encoded by Logistic mapping and doublerandom-phase encoding. Moreover, finally the numerical resultsare given to illustrate the feasibility and effectiveness of theproposed algorithm.

2 Preliminaries for proposed technique2.1 Logistic chaotic mapping

Logistic mapping is also known as pest model, it is a typicalmathematical model for describing the evolution of species, whichcan be shown with a non-linear repeated equation as follows [21]:

xk + 1 = μxk(1 − xk), (1)

where μ is called branch parameter, and xk ∈ (0, 1). Logisticmapping is in chaotic state when μ ∈ (3.5699456, 4] [22].

Logistic mapping can be generalised to the 2D logisticmapping, which is defined as [23]

xk + 1 = xk + λ(xk − xk2 + yk),

yk + 1 = yk + λ(yk − yk2 + xk) .

(2)

When 0.6 < λ ≤ 0.686, the system of (2) is in chaotic state. Thebifurcation graph shown in Fig. 1 with the initial point (0.4, 0.5)and 0.49 ≤ λ ≤ 0.686. For more detailed analysis of the complexdynamics of the system, please see relative reference [23].

2.2. Double random-phase encoding

In 1995, Refregier and Javidi [24] proposed the double random-phase encoding technique. The encoded image is obtained byrandom-phase encoding in both the input and the Fourier planes. Iftwo random-phase masks are used to encrypt the image in the inputand Fourier planes, respectively, the input image is transformedinto a complex-amplitude stationary white noise.

Let f(x, y) denote the image to be encoded and g(x, y) denote theencoded image. Let ϕ(x, y) and φ(u, v) denote two independentwhite sequences uniformly distributed in [0; 1], (x, y) and (u, v)

IET Image Process., 2017, Vol. 11 Iss. 4, pp. 211-216© The Institution of Engineering and Technology 2017

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denote the spatial plane and the Fourier plane coordinates,respectively. The encoding and decoding procedures are shown asfollows:

g(x, y) = ℱ−1{ℱ{ f (x, y)exp[i2πϕ(x, y)]}exp[i2πφ(u, v)]},f (x, y) = ℱ−1{ℱ{g(x, y)}exp[ − i2πφ(u, v)]}exp[ − i2πϕ(u, v)] .

(3)

where ℱ and ℱ−1 represent the Fourier transform and the inverseFourier transform, respectively.

3 Proposed colour image encryption schemeThis section presents the proposed scheme for colour imageencryption by using Logistic and phase encoding. Assume that thesize of original colour image I is N × N. The schematic diagram ofthe proposed encryption is illustrated in Fig. 2, and the proposedimage encryption algorithm consists of the following steps:

Step 1: Set the values of x0, y0, and λ.Step 2: For the size (N × N) of the colour image, after iterating theLogistic mapping of (2) for L times, we iterate the Logisticmapping continuously. For each iteration, we can get two values xiand yi. These decimal values are preprocessed first to get fourdecimal sequences U, V, W, and Z, as shown in follows:

U = u1, u2, …, u3N2 ,

V = v1, v2, …, v3N2 ,

W = w1, w2, …, w3N2 ,

Z = z1, z2, …, z3N2 ,

(4)

where

ui = 10kxi − floor(10kxi),

vi = 10kyi − floor(10kyi),

wi = floor((10kxi − floor(10kxi))10k) mod 256,

zi = floor((10kyi − floor(10kyi))10k) mod 256.

(5)

Here floor(x) returns the nearest integer less than or equal to x, andmod returns the remainder after division.Step 3: By taking N successive elements of sequence U or V, avector H = (uk + 1, …, uk + N) is obtained, where 0 ≤ k ≤ 3N2 − N.Then we reorder vector H from small to large order to get newvector H′. Via H and H′ we can construct a scrambling matrix A,which satisfy

H = H′A . (6)

So, by this way we can construct six scrambling matrices A1, A2,A3, A4, A5, and A6. Step 4: By taking N × N successive elements of sequence U, andwe convert it into a random matrix Q1 of size N × N. Similarly, wecan obtain another random matrix Q2 by using sequence V.Step 5: Let M1(x, y) = exp(i2πQ1) and M2(x, y) = exp(i2πQ2)present the two random phase masks.Step 6: Reshape the sequences W and Z, respectively, we obtain the3D matrices Y and J with the same size as the original image, thenwe use the matrices Y and J to diffuse the colour image. This ismathematically represented as follows:

F(i, j, k) = ((I(i, j, k) + Y(i, j, k)) mod 256) ⊕ J(i, j, k) . (7)

Here i, j = 1, 2, …, 256 and k = 1, 2, 3. The symbol ⊕ representsthe exclusive OR operation bit-by-bit.Step 7: The matrix F is converted into its RGB components.Afterwards, each colours matrix (R, G or B) is scrambled byscrambling matrices {Ai | i = 1, …, 6}. This is mathematicallyrepresented as follows:

R∗ = A1 × R × A2,G∗ = A3 × G × A4,B∗ = A5 × B × A6 .

(8)

To enhance the security, we can perform more rounds this step, i.e.multiple scramble. In this paper, we take five rounds.Step 8: The normalisation is processed to the matrices R∗, G∗, B∗,and we can obtain R′, G′, B′, respectively.Step 9: Combine R′(x, y) and G′(x, y) to obtain the complexamplitude image

C(x, y) = R′(x, y)exp(i2πG′(x, y)) . (9)Step 10: C(x, y) is first multiplied by the first random phase maskM1(x, y), then transformed by fast Fourier transform (FFT), theamplitude part B1(x, y) and phase part K1(x, y) of the result afterFFT can be represented as, respectively,

B1(x, y) = PT{ℱ[C(x, y)M1(x, y)]},K1(x, y) = AT{ℱ[C(x, y)M1(x, y)]}, (10)

among them, PT{} denotes the extracting amplitude part operator,AT{} denotes the extracting phase part operator, and ℱ[] denotesthe FFT.Step 11: Except replace FFT with IFFT in step 10, repeat steps 9and 10 for B1(x, y), M2(x, y), and B′(x, y), we can obtain B2(x, y) andK2(x, y). Then K1, K2, and B2 are converted into a colour image E.

Fig. 1 Bifurcation graph of system (2) when λ ∈ [0.49, 0.686]

212 IET Image Process., 2017, Vol. 11 Iss. 4, pp. 211-216© The Institution of Engineering and Technology 2017

In the encryption process, E is saved for encrypted image.Moreover, the decryption is the reverse process of the encryption.

4 Numerical simulation and discussionSome numerical simulations are performed to verify the proposedencryption algorithm for one image. In the numerical simulations,colour image of Lena having a size of 256 × 256 pixels is shown inFig. 3a and serves as input original image. For convenience, theencryption key (λ, x0, y0, L) is fixed at (0.65, 1, 0.01, 20,000), and italso is the decryption key. Fig. 3b shows the encrypted output forthe original colour image. It is observed that the encryption imagesare completely unintelligible, and does not reveal any informationabout the original colour image. Fig. 3c shows the decrypted imagewith correct keys.

4.1 Key space and sensitivity analysis

It is well known that high sensitive to initial conditions are inherentto any chaotic system. A good image encryption algorithm should

be sensitive to the cipher keys, and the key space should be largeenough to make any brute-force attacks infeasible.

Fig. 4a shows the decrypted image with an incorrectindependent parameter x0 = 1 + 10−14, while the other keys are allcorrect. Fig. 4b shows the decrypted image with an incorrectindependent parameter y0 = 0.01 + 10−14, while the other keys areall correct. This is observed that even the tiny change of 10−14, thedecrypted image is absolutely different from the plain image.Fig. 4c shows the decrypted image with an incorrect independentparameter λ = 0.65 + 10−16, while the other keys are all correct. Sothe key spaces for x0, y0, and λ are Sx0

= Sy0≃ 1014, Sλ ≃ 1016.

The iteration times L are 16-bit integers, so SL = 216. The totalkey space S = Sx0

Sy0SλSL ≃ 6.5536 × 1048. While advanced

encryption standard (AES) is an acknowledged secure encryptionalgorithm and the key space of the 128-bit AES algorithm is about2128. So it can be seen that the proposed colour image encryptionalgorithm is good at resisting brute-force attack. The key space of

Fig. 2 Schematic of encryption

Fig. 3 Numerical simulations are performed to verify the proposed encryption algorithm(a) Original colour image, (b) Encrypted image, (c) Decrypted image

Fig. 4 Decrypted images with a tiny change of the keys(a) Incorrect independent parameter x0 = 1 + 10−14, (b) Incorrect independent parameter y0 = 0.01 + 10−14, (c) Incorrect independent parameter λ = 0.65 + 10−16


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the proposed algorithm and the algorithms in [4, 7, 12] arecompiled in Table 1, which shows that the proposed algorithm haslarger key space than them.

The mean square error (MSE) between decrypted image andoriginal image is an important factor to evaluate the key sensitivityof an image encryption algorithm

MSE = 1M × N × 3 ∑

i = 1

M

∑j = 1

N

∑k = 1

3(I(i, j, k) − H(i, j, k))2,

where I(i, j, k) and H(i, j, k) mean the pixel values at point (i, y, k)of original colour image and decrypted image, respectively. M × Ndenotes the colour image size. The MSE curves for x0, L arecomputed and shown in Figs. 5a and b. The MSE is very large witha little deviation to the correct keys and the MSE is very small onlywhen the main keys are correct. Thus the decrypted image can berecognised if and only if the keys are correct.

4.2 Statistical analysis

By Marion [25] we know the statistical attack is launched byexploiting the predictable relationship between data segments ofthe original and the encrypted image. Hence to demonstrating theproposed colour image encryption having strongly resistedstatistical attacks, we test on the histograms of the encipheredimages and on the correlations of adjacent pixels in the cipheredimage.

Histograms of RGB colours for the original and the encryptedimages are indicated in Figs. 6a–f and 7a–f, respectively. It isobserved that the histogram of the encrypted image and histogramof the original image significantly different, it illustrates non-existent correlation between the two images.

It is well known that adjacent image pixels are highly correlatedeither in horizontal, vertical or diagonal directions. To test thecorrelations of adjacent pixels in original and encrypted images,randomly select 4000 pairs of two adjacent pixels (in horizontal,vertical, and diagonal directions) from an image. Then, thecorrelation coefficient can be calculated by using the followingformulas [16]:

rxy = cov(x, y)D(x) D(y) (11)

E(x) = 1N ∑

i = 1

Nxi (12)

D(x) = 1N ∑

i = 1

N(xi − E(x))2 (13)

cov(x, y) = 1N ∑

i = 1

N(xi − E(x))(yi − E(y)) (14)

where x and y denote two adjacent pixels and N is the total numberof duplets (x, y) obtained from the image.

The results of correlation analysis are shown in Table 2. Theresult indicates that the correlation of two adjacent pixels of theoriginal image is close to 1 in each direction of each component,while that of the encrypted image is close to 0 in each direction ofeach component, so the encryption effect is rather good.

4.3 Robustness analysis

In this section, we focus on analysing performance of the proposedalgorithm in noisy and lossy. In the first test, we check the

Table 1 Comparison of key spaceAlgorithm Proposed

algorithmPareek et

al. [7]Socek etal. [12]

George andPattathil [4]

Key space 6.5536 × 1048 280 2128 278

Fig. 5 MSE curves(a) x0 and (b) L

Fig. 6 Grey-scale image of the original image of Lena in the(a) Red, (c) Green, (e) Blue components, histogram of the original image of Lena inthe, (b) Red, (d) Green, (f) Blue components


tolerance against the loss of encrypted data. We occlude 1.5625, 25and 50% of the encrypted image pixels. Figs. 8a–c indicate theoccluded encrypted images of Fig. 3b, whereas Figs. 8e–h illustratethe decrypted output for these images. As we can see, in the case ofthe small percentage pixels of Fig. 3b are occluded, in addition tohaving some noise, the decryption image and original image prettymuch the same, while in the case of large percentage pixels ofFig. 3b are occluded, although only some information about theoriginal image can be retrieved from the decrypted output, thebasic outline of the original image can be retrieved.

In the second test, we employ three noises in the simulation,which are random noise, speckle noise and Gaussian noise,respectively. The noise effected encrypted images are thendecrypted, output for which is shown in Fig. 9, Fig. 9a shows therecovered image when a random noise is added to the encryptedimage of Fig. 3b. Fig. 9b shows the recovered image when specklenoise of variation 0.01 is added to the encrypted image of Fig. 3b.Fig. 9c shows the recovered image when a Gaussian noise with azero mean and standard deviation of 0.01 is added to the encryptedimage of Fig. 3b. As shown in Fig. 9, the recovered image in highnoise polluted, but some information about the original image canbe retrieved from the decrypted output.

By above tests, it is shown that the presented algorithm candecrypt the approximate image to the original image even if theencrypted image has been damaged. Therefore, the proposedalgorithm has good robustness.

4.4 Differential analysis

We have also measured the number of pixels change rate (NPCR)to see the influence of changing a single pixel in the original imageon the encrypted image by the proposed algorithm. The NPCRR, G, Bmeasure the number of pixels in difference of a colour componentbetween two images. We take two encrypted images, CR, G, B andC′R, G, B, whose corresponding original images have only one-pixeldifference. We also define a 2D array D, having the same size asthe image CR, G, B(i, j) or C′R, G, B(i, j). The DR, G, B(i, j) is determinedfrom CR, G, B(i, j) and C′R, G, B(i, j). If CR, G, B(i, j) = C′R, G, B(i, j) thenDR, G, B(i, j) = 0 otherwise DR, G, B(i, j) = 1. The NPCRR, G, B isevaluated through the following formula [3]:

NPCR =∑i, j DR, G, B(i, j)

N × M × 100%, (15)

where N × M is the colour image size.

Fig. 7 Grey-scale image of the encrypted image of Lena in the(a) Red, (c) Green, (e) Blue components, histogram of the encrypted image of Lena inthe, (b) Red, (d) Green, (f) Blue components

Table 2 Correlation coefficients of two adjacent pixels intwo imagesScan direction Horizontal Vertical Diagonaloriginal image red 0.9566 0.9812 0.9295

green 0.9432 0.9695 0.9199blue 0.9269 0.9586 0.9020

encrypted image red 0.0027 −0.0013 0.0039green 0.0034 −0.0034 −0.0021blue 0.0046 0.0038 0.0013

Fig. 8 Occluded encrypted images of Fig. 3b and the corresponding recovered images with correct keys(a) Occlude 1.5625% of the encrypted image pixels, (b) Occlude 25% of the encrypted image pixels, (c) Occlude 50% of the encrypted image pixels, (d) Decrypted output for theFig. 8a, (e) Decrypted output for the Fig. 8b, (f) Decrypted output for the Fig. 8c


215

To test our proposed scheme, each colour component isencrypted first. Then, one similar pixel in each component israndomly selected and toggled. The modified component isencrypted again by using the same keys so as to generate a newcipher-component. Moreover, the expected values are presented inTable 3. From Table 3, it can be found that the percentage of pixelschanged in the cipher image is over 99.9985% even with a one-bitdifference in the plain-image. Thus, the proposed encryptionscheme is able to resist the differential attack.

5 ConclusionBy combining Logistic mapping with double random-phaseencoding, a novel colour image encryption algorithm is proposed.To encrypting original colour image involves the XOR, fully phaseencoding and pixel scrambling techniques. We use the Logisticmapping to diffuse the colour image, then the RGB components ofthe result is scrambled and encoded by Logistic mapping anddouble random phase encoding. The performance is analysed withnumerical simulations, it is shown that the key space of the newalgorithm is large enough to make brute-force attacks infeasible,and is sensitive to the parameters of Logistic mapping duringdecryption. Robustness analysis in lossy and noisy demonstratesthe proposed method is robust against the loss of data.

6 AcknowledgmentThis work was supported by the National Natural ScienceFoundation of China (grant nos. 11071152, 11601188, 61403164),the Natural Science Foundation of Guangdong Province (grant no.2015A030313443).

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Fig. 9 Decrypted output for noise attacked images(a) Random noise, (b) Speckle noise, (c) Gaussian noise

Table 3 Sensitivity to plaintextLena Red Green BlueNPCR 0.999985 0.999985 0.999985


mapping and double random-phase encoding received on 7th

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