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Soft Computing (2019) 23:7045–7053https://doi.org/10.1007/s00500-018-3345-0

METHODOLOGIES AND APPL ICAT ION

Image encryption scheme combining a modified Gerchberg–Saxtonalgorithmwith hyper-chaotic system

Huiqing Huang1,2 · Shouzhi Yang1 · Ruisong Ye1

Published online: 29 June 2018© Springer-Verlag GmbH Germany, part of Springer Nature 2018

AbstractWe propose a new image encryption algorithm based on a modified Gerchberg–Saxton algorithm and hyper-chaotic system.First, original image is encoded into a phase function by using the modified Gerchberg–Saxton algorithm, which is controlledby hyper-chaotic system. Then, Josephus traversing is employed to scramble the created phase function. Lastly, the scrambledresult is confused and diffused by using hyper-chaotic system simultaneously. The numerical simulations verify the validityand reliability of the proposed scheme.

Keywords Image encryption · Gerchberg–Saxton algorithm · Hyper-chaotic system · Josephus traversing

1 Introduction

In the past few decades, along with the rapid developmentof computer technology and computer network technology,the unauthorized distribution of data has become a seriousproblem. Aiming at the security and protection of digitalimages, many image encryption algorithms or technolo-gies have been proposed (Matthews 1989; Tang et al. 2017;Refregier and Javidi 1995; Huang and Yang 2017). Amongthem, because chaos has the ability of pseudorandom, sen-sitivity with initial value, unpredictability of path and othergood chaotic characters, attention is paid to it deeply. Asearly as in 1989, Matthews proposed the chaotic encryptionalgorithm in Matthews (1989); afterward, the image encryp-tion based on chaos aroused a number of research effortsin information security. At first, low-dimensional chaos areusually employed in chaos-based image encryption algo-rithms. Those such as logistic map, sine map, and skewtent map have the advantages of simplicity and easy imple-mentation (Socek et al. 2005; Xiang et al. 2006; Alvarezand Li 2006; Pareek et al. 2006). However, with the devel-

Communicated by V. Loia.

B Shouzhi [email protected]

1 Department of Mathematics, Shantou University, Shantou515063, Guangdong, People’s Republic of China

2 School of Mathematics, Jiaying University, Meizhou 514015,Guangdong, People’s Republic of China

opment of computer technology, the encryption algorithmbased on low-dimensional chaos is not secure enough toprotect information safety, because of its small key spaceand weak security (Li and Zheng 2002; Li et al. 2007).To improve the security features of chaos-based encryptionalgorithm, many researchers naturally think through increas-ing the dimension of chaotic system. So the dimension ofchaotic system developed from low-dimensional to high-dimensional, and the performance of chaos is continuallyfashioned or improved (Chen et al. 2004; Wang et al. 2017;Mao et al. 2004; Jin et al. 2017). In 2004, Chen et al. pro-posed a symmetric image encryption algorithm based on 3Dchaotic cat maps. Subsequently, Mao et al. (2004) presenteda new fast image encryption method based on 3D chaoticbaker maps.

Although the high-dimensional chaos brings higher secu-rity, chaos cracking techniques also have been developed,and the image encryption schemes based on chaos are stillunder cracking (Liu and Liu 2014; Zhu et al. 2013; Ye andWong 2013). Thus, to improve the security of the encryptionmethods, the hyper-chaotic systems were introduced to theimage encryption process (Gao and Chen 2008; Wang 2014;Pashakolaee et al. 2017). In 2008, Gao and Chen presented anew image encryption algorithm based on hyper-chaos. Sub-sequently, a new image encryption scheme based on a totalshuffling and parallel encryption algorithm was proposed inMirzaei et al. (2012). The hyper-chaotic systems were com-bined with other encryption methods to enhance the securityof image encryption further. In 2013, Abdo et al. proposed

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7046 H. Huang et al.

a novel image encryption scheme based on elementary cel-lular automata. Afterward, Zhou et al. (2016) presented anefficient image compression–encryption scheme combining2D CS with hyper-chaotic system.

In order to avoid the shortcoming of weak security of tra-ditional Gerchberg–Saxton (G–S) algorithm, we present animproved G–S algorithm based on G–S algorithm that hasbeen brought forward in this paper, which adds an inter-ference phase. Therefore, we will use the improved G–Salgorithm to transform the original image into a phasematrix.Then, the phase matrix is scrambled by the Josephus travers-ing. Lastly, the scrambled result is confused and diffused byusing hyper-chaotic system simultaneously.

The rest of this paper is organized as follows: Sect. 2reviews the hyper-chaotic system, G–S algorithm and Jose-phus traversing. The detailed description of this scheme isprovided in Sect. 3. Simulation outcomes and security anal-ysis are given in Sect. 4. Finally, Sect. 5 concludes the paper.

2 Preliminaries for proposed technique

2.1 Modified Gerchberg–Saxton algorithm

The G–S algorithm (Gerchberg and Saxton 1972; Saxton1974) was originally developed to deal with the problem ofreconstructing phase from two intensity measurements [andfor synthesizing phase codes given intensity constraints ineach of two domains (Hirsch et al. 1971; Gallagher and Liu1973)]. The algorithm consists of the following four simplesteps (Fienup 1982): (1) Fourier-transforming an estimate ofthe object; (2) replacing the modulus of the resulting com-puted Fourier transform with the measured Fourier modulusto form an estimate of the Fourier transform; (3) inverseFourier-transforming the estimate of the Fourier transform;and (4) replacing the modulus of the resulting computedimage with the measured object modulus to form a newestimate of the object. For the kth iteration, the followingequations are shown:

Gk = |Gk |exp[iφk] = F[gk], (2.1)G ′k = |F |exp[iφk], (2.2)g′k = |g′k |exp

[iθ ′k

] = F−1 [G ′k], (2.3)

gk+1 = | f |exp[iθk+1] = | f |exp[iθ ′k

], (2.4)

whereF andF−1 represent the Fourier transformand inverseFourier transform, respectively. With the increasing of itera-tions, the output image Gk+n will gradually converged to theinput image F . When the convergence criteria are satisfied,the iteration finishes, and the output phase θk+n is the desiredphase information.

G–S algorithm has merits of simplicity, feasibility, feweriteration times and rapid convergence, but the algorithm haslow security. In order to enhance the security of G–S algo-rithm, researchers expand the Fourier domain G–S algorithminto Fresnel domain (Situ and Zhang 2004), and the geomet-ric parameters (wavelength and distance) were introduced asauxiliary keys.Amodified algorithmbased onG–Salgorithmhas been brought forward in this paper, and a fixed phase wasintroduced as key. Under the control of the key, the originalimage is transformed into image phase information by iter-ating the Fourier transform and inverse Fourier transform.Therefore, due to the presence of the key, compared to theoriginal algorithm, the security was obviously enhanced.

In the improved algorithm, we let

f (x, y)exp[iθ(x, y)] = F−1{F{ f1(x, y)exp[iα(x, y)]}F{ f2(x, y)exp[iβ(x, y)]}},

= F−1{F{F{exp[iφ(x, y)]}}F{F{exp[iψ(x, y)]}}},

(2.5)

where f (x, y) denotes original image, φ(x, y) denotes thefixed phase as key, ψ(x, y) denotes desired phase, andθ(x, y) is a random phase. For the kth iteration, the mod-ified GS algorithm consists of the following four steps:

(1) f exp[iθk] and exp[iφ] are known, by (2.5), we obtain

f k2 exp[iβk] = F−1{F{ f exp[iθk]}/F{ f1exp[iα]}}.(2.6)

(2) Inverse Fourier-transforming f2exp[iβk], we obtain

gkexp[iψk] = F−1{ f2exp[iβk]}. (2.7)

(3) Let gk = 1, and Fourier-transforming the new complexfunctional exp[iψk], we obtain

f k+12 exp[iβk+1] = F{exp[iψk]}. (2.8)

(4) By (2.5), we obtain

f k+1exp[iθk+1]= F−1

{F { f1exp[iα]}F

{f k+12 exp[iβk+1]

}}

(2.9)

Let f k+1 = f , and the new complex functionalf exp[iθk+1] as input function in the next iteration.The same as G–S algorithm, with the increasing of

iterations, the output image f k+n(x, y)will be gradually con-verged to the input image f (x, y). When the convergence

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Image encryption scheme combining a modified Gerchberg–Saxton algorithm with hyper-chaotic… 7047

Fig. 1 Schematic of the modified Gerchberg–Saxton algorithm

criteria are satisfied, the iteration finishes, and the outputphase ψk+n(x, y) is the desired phase information (Fig. 1).

2.2 Hyper-chaotic system

In the proposed encryption algorithm, a hyper-chaotic sys-tem generated from Chen’s chaotic system is used in keyscheming, which is defined by Gao et al. (2006)

⎧⎪⎪⎨

⎪⎪⎩

ẋ = a(y − x),ẏ = dx − xz + cy − h,ż = xy − bz,ḣ = x + k.

(2.10)

where a, b, c, d, and k are the control parameters of the hyper-chaotic system. When a = 36, b = 3, c = 8, d = − 16 and− 0.7 ≤ k ≤ 0.7, the system is in a hyper-chaotic state. Withparameters a = 36, b = 3, c = 8, d = − 16 and k = 0.2, theLyapunov exponents of the hyper-chaotic system are λ1 =1.552, λ2 = 0.023, λ3 = 0 and λ4 = − 12.573, respectively.Since the hyper-chaotic system has two positive Lyapunovexponents, the prediction time of the hyper-chaotic systemis shorter than that of the original chaotic system (Yanchukand Kapitaniak 2001); as a result, it is safer than chaos insecurity algorithm.

2.3 Josephus traversing

The Josephus Problem: There are n peoples arranged in acircle, and numbered clockwise 1, 2, . . . , n. Each of n peo-ple takes one of the places; beginning with the sth people,we move around the circle and remove every mth people.As each people is removed, the circle closes in. Eventually,all n peoples will have been removed from the circle (Hal-beisen and Hungerbüher 1997). For any given n, s and m,the order of dequeuing for the n peoples is obtained. Letn = 8, s = 1,m = 4, and the order of dequeuing is4, 8, 5, 2, 1, 3, 7, 6. If the order of dequeuing is considered tobe a traversal sequence, we will call them Josephus traversal.

For easy narration in the back rows, Josephus traversalfunction is defined by

fJosephus(a[ ], n, s,m), (2.11)

where a[ ] is a sequence, n is sequence length, s is the startingposition, and m is the number of counted off will dequeue intraversing.

3 The proposed image encryption scheme

This section presents the proposed scheme for image encryp-tion by using a modified GS algorithm and hyper-chaoticsystem. Assume that the size of original image I is N × N .The schematic diagram of the proposed encryption is illus-trated in Fig. 2, and the encryption process is described asfollows:

Step 1 Confirm the values of the initial conditionsx01, y01, z01, h01, k01 and iterate the hyper-chaotic systemfor a suitable times by Runge–Kutta algorithm to avoid theharmful effect of transient procedure.

Step 2The hyper-chaotic system is iterated, and as a result,four hyper-chaotic sequences {x1i }, {y1i }, {z1i } and {h1i }will be generated, respectively. And then, transform thesesequences into sequences {η′i }, and ηi can be replaced byx1i , y1i , z1i and h1i .

η′i = 10kηi − floor(10kηi ), (3.1)

Fig. 2 Schematic of encryption

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where floor(x) returns the nearest integer less than or equalto x .

Step 3 By taking N × N successive elements of sequence{x ′1i }, andwe convert it into a randommatrix Q of size N×N .And let φ(x, y) = exp(2πQ) as the fixed phase.

Step 4 Encrypted original image I via the modified G–S algorithm and the fixed phase, and we could obtain theintermediate encryption result ψ .

Step 5 All pixels of ψ are mapped into an integer rangefrom 0 to 255.

C = round[255 × ψ − minψ

max(ψ − minψ)]

. (3.2)

And then arrange the pixels from row to column, and we canget a sequence of C , as follows:

C = {c1, c2, . . . , cN2}. (3.3)

Step 6 Confirm the values of s and m, we could scramblethe sequenceC by using Josephus traversing, and then a newsequence D is formed.

D = {d1, d2, . . . , dN2}. (3.4)

Step 7 Confirm the values of the initial conditionsx02, y02, z02, h02, k02 and Iterate the hyper-chaotic systemfor a suitable times by Runge–Kutta algorithm to avoid theharmful effect of transient procedure. Four hyper-chaoticsequences {x2i }, {y2i }, {z2i } and {h2i } will be generated,respectively.

Step 8 Transform the four hyper-chaotic sequences {x2i },{y2i }, {z2i } and {h2i } into integer sequences w∗i , w can bereplaced by x2, y2, z2 and h2.

w∗i = floor(10kwi ) mod 256, (3.5)

where mod returns the remainder after division.Step 9Took N×N successive elements of sequences {y∗2i }

and {z∗2i }, and we can get two new sequences M and G.{M = {m1,m2, . . . ,mN2},G = {g1, g2, . . . , gN2}. (3.6)

Step 10 Perform pixel value diffusion according toEq. (4.2), and we can get H .

H = {h1, h2, . . . , hN2}, (3.7)

where

hi = ((di + di−1 + hi−1 + mi ) mod 256) ⊕ gi . (3.8)

Here, i = 1, 2, . . . , N 2 and initial values d0 and h0 are keys.The symbol ⊕ represents the exclusive OR operation bit-by-bit.

Step 11Reshape the sequence H and obtain the encryptedimage E with the same size as the original image.

E = H255

× max(ψ − minψ) + minψ. (3.9)

To enhance the security, we can performmore rounds Step6, that is,multiple scrambling. In this paper,we take 5 rounds.In the decryption process, the encrypted image is first per-formed by the inverse diffusion process, then is performedby the inverse Josephus traversing, and finally is retrievedwith Eq. (2.5).

4 Numerical simulation and discussion

A series of experiment results are performed to demonstratethe performance of the proposed image encryption algo-rithm. In the numerical simulations, the gray image “Lena”with 256 × 256 pixels, shown in Fig. 3a, serves as thetest image of the image encryption scheme combining amodified GS algorithm with hyper-chaotic System. For con-venience, the encryption key1 (x01, y01, z01, h01, k01) andkey2 (x02, y02, z02, h02, k02) are fixed at (1, 0.1, 1.3, 4, 0.2)and (2, 0.5, 0.3, 3, 0.5), respectively. And they also are thedecryption keys. The encrypted “Lena” is shown in Fig. 3b.Thedecrypted imagewith the correct keys is shown inFig. 3c.

4.1 Histogram analysis

Figure 4a is the histogram of “Lena,” while Fig. 4b showsthe histogram of its corresponding encrypted image. It isobserved that the histogram of the encrypted image and his-togram of the original image are significantly different; itillustrates non-existent correlation between the two images.

4.2 Correlation of adjacent pixels

By Marion (1991) we know the statistical attack is launchedby exploiting the predictable relationship between data seg-ments of the original and the encrypted image. So in orderto demonstrate that the proposed image encryption stronglyresists statistical attacks, we test on the correlations of adja-cent pixels in the ciphered image. The correlation coefficientbetween adjacent pixels of an image can be calculatedby using the following formulas (Mazloom and Eftekhari-Moghadam 2009):

Cxy =∑N

i=1(xi − x̄)(yi − ȳ)√(∑Ni=1(xi − x̄)2

) (∑Ni=1(yi − ȳ)2

) , (4.1)

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Fig. 3 a Original image; b the encrypted image; c the decrypted image

Fig. 4 Histograms: a “Lena”; b encrypted “Lena”

where x̄ = 1N∑N

i=1 xi and ȳ = 1N∑N

i=1 yi .To test the correlations of adjacent pixels in original image

and encrypted image, randomly select 4000 pairs of two adja-cent pixels (in horizontal, vertical, and diagonal direction)from an image. And the correlation coefficients of the pro-posed algorithm are shown in Table 1. The result indicatesthat the correlation of two adjacent pixels of the originalimage is close to 1 in each direction; nevertheless, the corre-lation of the encrypted image is close to 0 in each direction,so the encryption effect is rather good. Figure 5. shows thecorrelation distribution of two vertical adjacent pixels in theoriginal image and that in the encrypted image. From Fig. 5,the correlation distributions between adjacent pixels in theoriginal image are similar to a linear-like area in the verti-cal direction, and as might be expected, the pixel statisticalcorrelations of the ciphered image are much weaker. Exper-imental results show that the proposed encryption schemehas the ability to resist statistical attacks. It further demon-strates that the attackers cannot obtain useful information bystatistical analysis.

Table 1 Correlation coefficients of two adjacent pixels in two images

Scan direction Horizontal Vertical Diagonal

Original image 0.9663 0.9491 0.9250

Encrypted image − 0.0019 0.0015 0.000398

4.3 Key space and sensitivity analysis

It is well known that the size of key space reflects thecomplexity and the difficulty in attacking a cryptosystemsuccessfully, so a good cryptosystem should have a largeenough key space S to make brute-force attack invalid. Theparameters x01, y01, z01, h01, k01, x02, y02, z02, h02 and k02are main keys of the proposed image encryption scheme. Thetotal key space S can be expressed as

S = S1S2S3S4S5S6S7S8S9S10, (4.2)

where Si is the key subspace of the i th key, i = 1, 2, . . . , 10.To calculate the key space of x01, two different sequences x

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Fig. 5 Correlation distribution of two horizontally adjacent pixels in a original “Lena” and b encrypted “Lena”

and x ′ are generated with initial values x01 and x01 + ε (εis the deviation of the x01). The mean absolute error (MAE)between two sequences is (Gaurav et al. 2013):

MAN(x, x ′) = 1N

N∑

i=1|xi − x ′i |. (4.3)

The key space of x01 is equal to 1ε , where ε is the right valuesubject toMAN(x, x ′) = 0. The simulation results show thatthe space of x01 is 1015, and so it is with y01, z01, h01, x02,y02, z02 and h02. Similarly, the subspace of k01 and k02 isabout 1016. So the total key space is as large as 10152, whichismuch larger than 2100 (Alvarez and Li 2006); thus, it can beseen that the proposed image encryption algorithm is goodat resisting brute-force attack.

Themean square error (MSE) between original image anddecrypted image is employed here to evaluate the key sen-sitivity of an image encryption scheme. Mathematically, ifI (i, j) and H(i, j) denote the values of the original anddecrypted images of the pixel (i, j), respectively, and then,the MSE can be defined as follows: (Tao et al. 2007):

MSE = 1MN

M∑

i=1

N∑

j=1[I (i, j) − H(i, j)]2, (4.4)

where N and M are the sizes of the images. The MSE curvesfor x01 and x02 are computed and shown in Fig. 6, whichshows the MSE is very large with a little deviation to thecorrect keys and the MSE is very small only when the mainkeys are correct. So the decrypted image can only be recog-

nized when the keys are correct, i.e., the proposed algorithmis very sensitive to the keys.

Figure 7 shows the decrypted image with the incorrectkeys deviated 10−15 from one parameter of (x01, y01, z01,h01, x02, y02, z02, h02), respectively. This is observed thateven in the tiny change of 10−15, the decrypted image isabsolutely different from the plain image.

4.4 Differential analysis

The number of pixels change rate (NPCR) is employed hereto test the influence of changing a single pixel in the originalimage on the encrypted image by the proposed scheme. Forcalculation of NPCR, let us assume two ciphered images, C1andC2, whose corresponding original images have only one-pixel difference, respectively. We define a two-dimensionalarray D, having the same size as the image C1 or C2. Then,D(i, j) is determined by C1(i, j) and C2(i, j); namely ifC1(i, j) = C2(i, j), then D(i, j) = 1; otherwise, D(i, j) =0. The NPCR is evaluated by the following equation (Chenet al. 2004):

NPCR =∑

i, j D(i, j)

N × M × 100%, (4.5)

where N and M are the width and height of C1 or C2.In the proposed scheme, a small difference in the plain

image can affect the whole cipher image. The percentage ofpixel changed in the cipher image is over 99.63% even witha one-bit difference in the plain image [here, we randomlychoose a pixel at position (120,67)]; Fig. 8a shows the cipherimage corresponding to only one-pixel difference in the plain

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Fig. 6 MSE curves: a x01 and b x02

Fig. 7 The decrypted images with a tiny change of the keys

image, and the difference image between Figs. 3b and 8a isshown in Fig. 8b. Thus, the proposed encryption algorithmis able to resist the differential attack.

4.5 Information entropy analysis

Commonly, we take the information entropy as a tool to eval-uate the strength of an encryption algorithm. As is known toall, the entropy H(m) of amessage sourcem can bemeasuredby the following formula:

H(m) = −L−1∑

i=0p(mi ) log p(mi ), (4.6)

where L is the total number of symbolsmi ∈ m, p(mi ) repre-sents the probability of occurrence of mi , and log means thebase 2 logarithm so that the entropy is expressed in bits. Theinformation entropy is 8 bits for any ideal random sequence.Using the proposed algorithm, we can get the entropy for theencrypted image of “Lena” is 7.9972 bits. It indicates that

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Fig. 8 The cipher imagecorresponding to only one-pixeldifference in the plain imageand difference image

the encrypted image is very close to a random source and thecryptosystem can resist the entropy attacks.

5 Conclusion

In this paper, a modified G–S algorithm and hyper-chaoticsystem were used to design an efficient and secure imageencryption algorithm. In this new algorithm, the modifiedGS algorithm is employed to encrypt the original image firstand then uses Josephus traversing to shuffle the positions ofcipher image pixels. Finally, it employs the hyper-chaoticsystem to confuse the shuffled result, thereby significantlyincreasing its resistance to various attacks such as the sta-tistical and differential attacks. These properties are justifiedby the experimental results on statistical, key space and sen-sitivity analyses.

Acknowledgements This work was supported by the National Nat-ural Science Foundation of China (Grant Nos. 11071152, 11601188,61403164), the Natural Science Foundation of Guangdong Province(Grant No. 2015A030313443).

Compliance with ethical standards

Conflict of interest The authors declare that they have no conflict ofinterest.

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Image encryption scheme combining a modified Gerchberg–Saxton algorithm with hyper-chaotic systemAbstract1 Introduction2 Preliminaries for proposed technique2.1 Modified Gerchberg–Saxton algorithm2.2 Hyper-chaotic system2.3 Josephus traversing

3 The proposed image encryption scheme4 Numerical simulation and discussion4.1 Histogram analysis4.2 Correlation of adjacent pixels4.3 Key space and sensitivity analysis4.4 Differential analysis4.5 Information entropy analysis

5 ConclusionAcknowledgementsReferences

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