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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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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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7048 H. Huang et al.
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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7050 H. Huang et al.
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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123
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