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Soft Computing
https://doi.org/10.1007/s00500-021-06423-8
FOUNDATION, ALGEBRAIC, AND ANALYTICAL METHODS IN SOFT
COMPUTING
Improved seven-dimensional (i7D) hyperchaotic map-based image
encryption technique
Manjit Kaur1·Dilbag Singh2·Vijay Kumar3
Accepted: 17 September 2021
© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021
Abstract
With the advancements in Internet technologies, a huge amount of images are transferred over the public networks. Therefore,
during the transmission process, the images are prone to various security threats. To protect images against any unauthorized
access, many researchers have designed various approaches such as encryption, stenography, and visual cryptography. Each
approach has its own benefits over the others. The main objective of this paper is to design an efficient image encryption
approach. Initially, various image encryption approaches have been studied. It is found that the majority of the existing image
encryption approaches suffer from the hyperparameters tuning problem. Also, it is desirable to develop secret keys which
are highly sensitive to plain images. Therefore, in this paper, an improved seven-dimensional hyperchaotic system (i7DHS)
is adopted to generate secret keys. But, i7DHS is sensitive to its initial parameters. Therefore, a dual local search-based
evolutionary algorithm (DEA) is utilized to tune the initial parameters of i7DHS. The computed optimal keys are utilized
to encrypt the input images. Performance analysis shows that DEA-based i7DHS outperforms the competitive approaches.
DEA-based i7DHS is highly sensitive to the input images and has large secret key space. It is found that DEA-based i7DHS
can resist various security attacks.
Keywords Encryption ·Hyperchaotic system ·Hyperparameters ·Remote sensing images
1 Introduction
In the recent years, researchers are attracted toward the
protection of multimedia data during the transmission over
Internet. Due to the advancements in Internet and artificial
intelligence techniques, multimedia data can be easily tem-
pered (Wang et al. 2015). It can be seen from the literature
that images are mostly transmitted over Internet. In modern
society, the color images are preferred over the grayscale
images. Because the color images have more information
BDilbag Singh
dggill2@gmail.com; dilbag.sng@yahoo.com
1School of Electrical Engineering and Computer Science,
Gwangju Institute of Science and Technology, Gwangju,
South Korea
2School of Electrical Engineering and Computer Science,
Gwangju Institute of Science and Technology, Gwangju,
South Korea
3Computer Science and Engineering Department, NIT
Hamirpur, Hamirpur, India
than the grayscale images (Xiong et al. 2019), there is a need
to protect the color images from attackers during the trans-
mission, especially for medical imaging and defense (Kaur
et al. 2021). For this, a large number of techniques are avail-
able in the literature. Among them, encryption techniques are
widely used to protect the contents of color images. Nowa-
days, color image encryption has become a recent research
topic for young researchers and scientists. Researchers have
tried to encrypt the color image by using the characteristics of
image such as correlation among pixels, high volume of data,
and redundancy (Pak et al. 2019). The classical techniques
namely Data Encryption Standard (DES) and International
Data Encryption Algorithm (IDEA) are used to encrypt the
images (Xuejing and Zihui 2020). For color images, these
techniques are unable to provide the best security. Nowa-
days, chaos-based image encryption techniques have been
developed. These techniques have better security than the
classical techniques. The main characteristics of chaotic sys-
tem such as highly sensitive toward the initial values of
parameters, ergodicity, and unpredictability are most suitable
for color image encryption (Yang et al. 2019). The chaos-
123
M. Kaur et al.
based encryption technique generates chaotic sequences,
which are used to modify the position or value of pixels.
The well-known chaotic systems are Logistic map, Cat map,
Tent map, etc. These are low-dimensional chaotic systems,
which were utilized in the development of image encryption
techniques. However, these chaotic systems suffer from small
key size and simple structure (Wang and He 2011). Therefore,
the attackers easily decrypt these systems. To overcome the
problems associated with low-dimensional chaotic system,
the complex chaotic system is required. Recently, researchers
suggested some modifications in chaotic system to enhance
the performance of image encryption technique. The chaotic
system can be enhanced through Lyapunov exponent (LE)
(Sayed et al. 2018). If the given system has one positive LE,
then it is considered as a chaotic system, while it is con-
sidered hyperchaotic if it has more than two positive LEs
(Nu nez-Perez et al. 2021). The hyperchaotic-based image
encryption is more secure than the chaotic systems (Kaur
et al. 2020). The well-known hyperchaotic system is Lorenz
system (Rodriguez et al. 2017). Hyperchaotic-based image
encryption techniques are widely used for secure commu-
nication. Hyperchaotic system can be combined with other
chaotic system to further enhance the performance of encryp-
tion system (Liu and Wang 2011). However, these systems
require the optimal initial parameters for better security.
The estimation of initial parameter is still a challenging
problem. The above facts motivate us to develop the novel
hyperchaotic-based image encryption technique.
From the existing techniques, it is found that the major-
ity of the existing image encryption approaches suffer
from the hyperparameters tuning problem. Also, it is desir-
able to develop secret keys which are highly sensitive to
plain images. Therefore, in this paper, an improved seven-
dimensional hyperchaotic system (i7DHS) is adopted to
generate secret keys. Since i7DHS is sensitive to its initial
parameters, a dual local search-based evolutionary algorithm
(DEA) is utilized to tune the initial parameters of i7DHS.
The remaining structure of this paper is as follows.
Section 2discusses the literature review. The proposed
methodology is discussed in Sect. 3. Performance analysis
of DEA-based i7DHS is presented in Sect. 4. Section 5con-
cludes the paper.
2 Literature review
Yuan et al. (2017) presented a parallel image cryptographic
system. The pixels of plain images were decomposed into dif-
ferent levels. Logistic map and 5D hyperchaotic system were
combined to generate the chaotic sequences. This method
attained the better sensitivity and computational speed than
the other algorithms. Hua et al. (2018) developed a 2D logis-
tic sine coupling map (2D-LSCM)-based image encryption
technique. The chaotic sequences were generated through
2D-LSCM. These sequences were used to perform the both
diffusion and permutation operations on the plain image. The
developed method attained better security and fast encryp-
tion speed than the other algorithms. This method can be
extended to video encryption. Cheng et al. (2019) developed
a color image encryption technique. They used block per-
mutation to reduce the dependency among the color planes
of an image. The key sequences were generated through
hyperchaotic system. This algorithm withstands against the
differential and statistical attacks. Wu et al. (2016) developed
an image encryption algorithm based on 6D hyperchaotic
system and 2D discrete wavelet transform (DWT). The image
was decomposed into four sub-images using 2D-DWT. The
key stream was generated from 6D hyperchaotic system and
used to permute the sub-images. This algorithm achieved
the fast computational speed and withstands against various
attacks. Yang et al. (2021) designed an encryption technique
using DNA and fractional hyperchaotic system (DFHS).
DCT was used to process the two images. These processed
images are combined together and formed a new image.
The new image was encrypted through DNA encoding. The
encoding process was controlled through chaotic sequences,
which were produced from fractional hyperchaotic system.
This technique was able to withstand against various attacks.
Yang et al. (2020) studied the characteristic of Galois field
(GF). An image encryption algorithm was proposed by using
fractional-order hyperchaotic system and GF (FHGF). This
algorithm provided better encrypted image and security facil-
ities. Wang and Yang (2021) investigated the characteristics
of fractional-order cellular neural network (FO-CNN) hyper-
chaotic system. FO-CNN hyperchaotic system was used to
design the novel image encryption algorithm. This algo-
rithm was able to encrypt image in an efficient manner. Wu
et al. (2018) designed an encryption technique using DNA
encoding and 2D Henon-Sine map (2D-HSM). The pixels
of an image were diffused through DNA encoding and per-
mutated through 2D-HSM. This algorithm attained better
computational complexity and withstands against differen-
tial attacks. Sun et al. (2019) investigated a hash function
to generate parameters for 7D hyperchaotic system (7DHS).
The chaotic sequences were generated through 7D hyper-
chaotic system. These sequences were used to scramble the
plain image. Thereafter, the diffusion process was applied on
the scrambled image. This technique provides better perfor-
mance than the other algorithms in terms of key sensitivity
and security. Wang and Gao wang2020image designed an
image encryption technique using semi-tensor product. The
plain image was decomposed into four sub-images. These
sub-images were scrambled through Arnold transformation
and combined these scrambled sub-images to produce an
image. They used a Boolean network to produce a secret
key. The secret key was used in mixed linear–nonlinear
123
Improved seven-dimensional (i7D) hyperchaotic map...
coupled map lattice to produce a chaotic sequence. These
chaotic sequences were applied in semi-tensor product to
generate the encrypted image. This technique was able to
produce the more secure encrypted image. Wang et al. (2019)
designed an image encryption algorithm in parallel fashion.
They combined both cyclic shift and sorting to attain the
high efficiency. The parallel diffusion process was used to
improve the robustness of algorithm. The proposed algorithm
was able to withstand against the various attacks. However,
the parallel implementation of permutation process has to be
investigated. Wang and Gao (2020) studied a chaotic image
encryption algorithm based upon the concepts of semi-tensor
product and Boolean network. The proposed algorithm was
evaluated on three images and attained the entropy of 7.9974.
Teng et al. (2021) developed an image encryption algo-
rithm that based on cross 2D hyperchaotic system. The color
image was transformed to bit-level matrix. This matrix was
permuted through chaos-based column and row cycle shift
scrambling technique. Thereafter, this scrambled matrix was
diffused according to the keys generated from the hash func-
tion. The proposed method provided better security than the
other algorithms. Ding and Ding (2020) designed an image
encryption algorithm by utilizing the concepts of 2D-DWT
and 4D hyperchaotic system. The plain image was scram-
bled through 2D-DWT. The scrambled image was shuffled
by using fractional-order Henon chaotic system. The shuf-
fled image was diffused using 4D hyperchaotic system. The
proposed approach was able to withstand against the various
attacks. Li and Zang li2021hyperchaotic proposed a hyper-
chaotic image encryption through diffusion and permutation
process. 4D hyperchaotic system (4DS) was used to generate
the chaotic sequences. These sequences were used to perform
the encryption process. This approach performed better than
the existing algorithms. Benlashram et al. (2020) designed
an encryption technique using 3D chaotic system and pixel
shuffling. The proposed algorithm attained the NPCR of
99.66% and UACI of 33.68%. Fang and Sun (2020)devel-
oped an image encryption based on 5D hyperchaotic system.
5D hyperchaotic system was used to produce the chaotic
sequences. These sequences were used to encrypt the given
image. The proposed approach was also able to withstand
against various statistical and differential attacks. Ul Haq
and Shah (2021) utilized the concepts of 1D Sine and 2D
Thinker bell maps to design the image encryption algorithm.
They use 4D mixed chaotic system to generate the chaotic
sequences for encryption process. The proposed approach
provided better performance than the existing techniques in
terms of performance measures. Zhou et al. (2020)devel-
oped a DNA-based image encryption algorithm. SHA-512
was used to perform the permutation and diffusion process.
Four-wing chaotic and Lorentz systems were used to gener-
ate the sequences. These sequences were used to encrypt the
given image. This method attained the better performance
than the other algorithms.
It has been found that the majority of competitive tech-
niques achieve either significant convergence speed or signif-
icant diversity but fail to achieve both at a time. To overcome
this issue, Hong et al. (2018) designed a dual local search-
based evolutionary algorithm (DEA). It achieves a balance
between convergence and diversity to obtain efficient solu-
tions.
3 Proposed approach
The hyperparameters of i7DHS are tuned using DEA. To
obtain encrypted image, the computed optimal secret keys are
utilized to implement diffusion and permutation operations
on input image.
3.1 Seven-dimensional hyperchaotic system
A seven-dimensional hyperchaotic system (7DHS) was pro-
posed by Yang et al. (2018). 7DHS was obtained from the
combination of 6D hyperchaotic and 1D linear systems. It has
complex structure that provides better hyperchaotic behav-
ior as compared to the existing chaotic systems (Kaur et al.
2020; Haspolat and Yildiz 2021). Due to its complex dynamic
behavior, it can be utilized in the secure communications. It
can be mathematically defined as:
c
1=k(c2−c1)+c4+mc6
c
2=lc1−c2−c1c3+c5
c
3=−nc3+c1c2
c
4=oc4−c1c3
c
5=−wc2+c6
c
6=t1c1+t2c2
c
7=hc7+uc4.
(1)
Here, initial states are denoted as c1,c2,c3,c4,c5,c6,
and c7. Control parameters are defined using m,o,w,t1,t2,
and h. Constant parameters of System (1)arek,n, and l.u
denotes the coupling parameter. c1,c2,c3,c4,c5,c6, and
c7are the obtained random sequences. Figure 1a, b shows the
hyperchaotic attractor of 7DHS with k=10, m=1, l=28,
n=8/3, o=2, w=9.9, t1=1, t2=2, h=1, and u=1
in c2−c5−c6space and c1−c2−c6, respectively.
System (1) states c
5and c
6are not dependent on their
initial states but depend on other states’ initial values. The
chaotic behavior of System (1) can be improved by modifying
the equations of the random sequences c
5and c
6. Because the
dependency on initial states plays an important role in secure
123
M. Kaur et al.
Fig. 1 Hyperchaotic attractor of 7DHS in ac2−c5−c6space and bc1−c2−c6space
communication, an improved 7DHS (i7DHS) is defined as
c
1=k(c2−c1)+c4+mc6
c
2=lc1−c2−c1c3+c5
c
3=−nc3+c1c2
c
4=oc4−c1c3
c
5=−wc2+c5
c
6=t1c1+t2c6
c
7=hc7+uc4.
(2)
Figure 2a, b shows the hyperchaotic attractor of i7DHS
with k=10, m=1, l=28, n=8/3, o=2, w=9.9, t1=1,
t2=2, h=1, and u=1inc2−c5space and c1−c6,
respectively.
3.2 Dual local search-based evolutionary algorithm
Since i7DHS is sensitive to its initial parameters, DEA
(Hong et al. 2018) is utilized to tune the parameters of
i7DHS. In DEA, initially, random population is obtained.
External archive (EA) is then formed with the help of devel-
oped population. EA is utilized to generate new solutions
in every iteration. New solutions are then appended to the
developed population. Hypervolume-based selection is then
implemented to select solutions. To improve the diversity,
various rules are implemented on the updated EA (refer Hong
et al. 2018).
3.3 Optimization using DEA
Algorithm 1shows various steps to obtain the optimal secret
keys from i7DHS. Initially, various parameters of DEA are
defined such as number of decision variables (D), popula-
tion size (PS), and maximum number of iterations (MI). D
contains c1,c2,c3,c4,c5,c6,c7,k,m,l,n,o,w,t1,t2,h,
and uof i7DHS and encryption parameters (i.e., βand γ).
Initial population is then obtained using normal distribution
(see line (1)). Evolution counter (c) is set to 0. New offspring
are computed by considering the traditional solution gener-
ator in every evolution step (see lines (6)–(8)). EA is then
initialized using non-dominated sorting (see line (13)). Prob-
ability vectors (lpM×Dand rpM×D) are then defined (see
line (14)) to form new solution generator. Lines (18) - (21):
An updated offspring is then obtained using self-evaluation
evolution (refer Yang et al. 2017). A single member (ρi)of
archive κis modified and forms an archive κ(i)with E
i.κ(i)
is then compared with κusing hypervolume indicator ()
as:
κ=κ, if (κ) ≥(κ(i))
κ(i),Otherwise.(3)
Line (20): Probability vectors (lpM×Dand rpM×D)are
then updated by utilizing 1/5 rule (refer Hansen et al. 2015).
Newly developed offsprings are then appended to ωby
considering (see line 21). The final solution composes non-
dominated offsprings as an output. Hypervolume by slicing
objectives (refer While et al. 2006) is used to compute .
Entropy, peak signal-to-noise ratio (PSNR), and number
of pixels change rate (NPCR) are used to form the multi-
objective fitness function.
Algorithm 2shows step-by-step image encryption pro-
cess. It accepts input image and returns an encrypted image.
If image is color, then we will call same algorithm for red,
green, and blue channels one by one. Finally, we will con-
catenate encrypted red, green, and blue channels. Algorithm
3shows step-by-step image decryption process. It requires
encrypted image along with various encryption parameters.
These parameters are ci,i=1,2, ...7, k,m,l,n,o,w,t1,
123
Improved seven-dimensional (i7D) hyperchaotic map...
Fig. 2 Hyperchaotic attractor of i7DHS in ac2−c5space and bc1−c6space
Input:PS,D,andMI
Output: Non-dominated solutions (ω)
1Develop initial population (ω);
2while Termination Criterion not satisfied do
3Define required parameter;
4set c=0;
5while c<MIdo
6for i=1:PSdo
7Select q1and q2from ωon random basis;
8Compute new solution pusing mutation and
crossover operators by considering q1and q2;
9Append ωwith pusing ;
10 end
11 c=c+1;
12 end
13 M=PS/2;
14 Archive κ={ρ1,...,ρ
M}is defined as a partner of ωby
using non-dominated sorting;
15 Set lpM×Dand rpM×Dto 1.0;
16 Set c=0;
17 while c<MIdo
18 for i=1:Mdo
19 Obtain ρ
ifrom ρiby utilizing SEE with lpM×Dand
rpM×D;
20 Update κby κ(i)={ρ1,...,ρ
i,...,...,ρ
M}using
Eq. (3);
21 Update lpiand rp
iusing 1/5−rule;
22 Update ωwith ρ
iusing ;
23 end
24 end
25 end
26 return ω
Algorithm 1: Dual local search-based evolutionary
algorithm
t2,h, and uof Eq. (2). It also needs encryption parameters
(i.e., βand γ).
Input: Input image (II)with size m×n
Output: Encrypted image CI
1Decompose IIto 1Dvector;
2Call Algo. 1to obtain optimal parameters such as
ci,i=1,2, ...7, k,m,l,n,o,w,t1,t2,h,anduof Eq. (2)and
encryption parameters (i.e., βand γ);
3Obtain secret keys c
ii=1,2,...,7 using i7DHS (Eq. (2));
4Change the pixel values of IIusing c
1and c
2as:
S=mod(II+c
1,256)and χ=mod(S+c
2,256);
5Obtain c
3by sorting c
3. Evaluate location values of c
3in c
3and
store the updated locations in uc.;
6Permute χby using ucas:
χ(j)=χ(uc(j)), j=1,2,...,m×n;
7χpermuted again by considering c
4;
8Obtain c
4by sorting c
4. Evaluate location values of c
4in c
4and
store the updated locations in u
c.;
9Permute χby using u
cas:
χ(j)=χ(u
c(j)), j=1,2,...,m×n;
10 Diffuse χ by considering c
5and γto compute encrypted vector
C
Ias: C
I(j)=mod(c
5×χ(j)+(1−γ) ×c
5,256)];
11 Compute key (K) using all ci,i=1,2, ...5as
K=mod(c
1⊕c
2⊕c
3⊕c
4⊕c
5,⊕c
6⊕c
7,256);
12 Use Kand βto diffuse C
Ias
CI(j)=mod(K×C
I(j)+(1−β) ×K,256);
13 Decompose CIinto 2D;
14 return CI
Algorithm 2: Image encryption
4 Performance analysis
To evaluate the performance of DEA-based i7DHS, experi-
ments are performed on MATLAB2021a. Comparisons of
DEA-based i7DHS are performed with five competitive
image encryption techniques by considering four remote
sensing images. Initial parameters of DEA are selected same
as mentioned in Hong et al. (2018).
123
M. Kaur et al.
Input:CI,cii=1,2,...,7, k,m,l,n,o,w,t1,t2,h,u,βand γ
Output: Decrypted image CI
1Obtain secret keys c
ii=1,2,...,7 using i7DHS (Eq. (2));
2K=mod(c
1⊕c
2⊕c
3⊕c
4⊕c
5⊕c
6⊕c
7,256); Decompose
CIto 1D vector.;
3C
I=CI−(1−β) ×K/β;
4χ =C
I−(1−γ)×c
5/γ ;
5Sort c
4and store into c
4;
6Evaluate elements positions of c
4in c
4and store them into u
c;
7χ(j)=χ(C
p(j));
8Sort c
3and store into c
3;
9Evaluate elements positions of c
3in c
3and store them into uc;
10 χ(j)=χ(uc(j));
11 S=mod(χ −c
2,256);
12 DN=mod(S−c
1,256);
13 Decompose DNto 2D matrix;
14 Return DN
15 return CI
Algorithm 3: Image decryption
4.1 Visual analysis
Figure 3demonstrates the visual analysis of DEA-based
i7DHS on four remote sensing images. It is found that the
encrypted images are completely random in nature and do not
show any kind of information to attackers. Also, decrypted
images are identical to the input images; therefore, visually
there is no much loss in the visual quality of images.
4.2 Histogram analysis
Figure 4shows the histogram analysis of DEA-based
i7DHS on four remote sensing images. Evenly balanced
histogram states that the given image encryption technique
achieves efficient encrypted images (Hussain et al. 2012). We
have considered only red color channel analysis only. Figure
4a–d shows histograms of input images. These images clearly
show that the histograms of input images provide statistical
information about the respective images. Figure 4e–h shows
histograms of encrypted images. It is found that histograms
are evenly balanced and it does not provide any statistical
information regarding the input images.
4.3 Entropy analysis
For efficient image encryption technique, entropy of encrypted
8−bit images should approach toward 8 (Kaur and Singh
2021). Table 1shows the entropy analysis of DEA-based
i7DHS. It is found that DEA-based i7DHS achieves better
entropy values as compared to the existing techniques.
4.4 Correlation analysis
Generally, pixels of input images are highly correlated hori-
zontally (H), vertically (V), and diagonally (D). Attacker can
Table 1 Entropy analysis of DEA-based i7DHS
Method Canal Quickbird Sea Blue sea
7DHS 7.9981 7.9895 7.9898 7.9981
4DS 7.9937 7.9918 7.9939 7.9898
FHGF 7.9912 7.9926 7.9918 7.9895
DFHS 7.9913 7.99291 7.9909 7.9911
i7DS 7.9898 7.9912 7.9915 7.9921
DEA-based i7DHS 7.9985 7.9991 7.9994 7.9987
utilize correlation values (Pak and Huang 2017) to decrypt
the images. Therefore, it should be minimized (Kaur and
Kumar 2018).
Tables 2and 3depict the correlation analysis of input
and encrypted images obtained from DEA-based i7DHS,
respectively. It clearly shows that the pixels of input images
are highly correlated. The correlation analysis of encrypted
images obtained from DEA-based i7DHS shows signifi-
cantly lesser correlation values. Thus, DEA-based i7DHS
can resist statistical attacks.
4.5 Analysis of key space
Ahmad et al. have shown that if key space is more than
2100 ≈1030, then the given image encryption approach
can efficiently handle brute-force attack (Ahmad and Hwang
2016).
DEA-based i7DHS generates seven secret keys such as
c
1,c
2,c
3,c
4,c
5,c
6, and c
7(using Eq. (2)) from initial states
such as c1,c2,c3,c4,c5,c6, and c7, control parameters such
as m,o,w,t1,t2, and h, constant parameters, i.e., k,n, and
l, and coupling parameter u. Here, all the required initial
parameters are considered double precision numbers (i.e.,
10−16). The possible values of each initial parameter can
be more than 1016. Therefore, the key space of proposed
technique becomes approximately 10272. The key space of
DEA-based i7DHS is quite large and can easily defend the
brute-force attack.
4.6 Differential attack analysis
Sensitivity of encryption technique against tiny modifica-
tions in input images can be evaluated by considering NPCR
(Bechikh et al. 2015; Benrhouma et al. 2015) and UACI (Ye
2010; El Assad and Farajallah 2016). If computed NPCR
>99.5693% and UACI >33.6447, then the given image
encryption can resist the differential attacks.
The sensitivity of proposed DEA-based i7DHS is also
tested against small changes in the input images. Tables 4
and 5show the NPCR and UACI values of existing and pro-
posed techniques, respectively. It can be seen that DEA-based
123
Improved seven-dimensional (i7D) hyperchaotic map...
Fig. 3 Visual analysis of DEA-based i7DHS: aCanal image, bQuickbird image, cSea image, dBlue sea image, e–hencrypted images of (a–d),
respectively, and i–ldecrypted images of (a–d), respectively
Fig. 4 Histogram analysis of DEA-based i7DHS: a–dhistograms of input images and e–hhistograms of encrypted images
123
M. Kaur et al.
Table 2 Correlation analysis of
input remote sensing images Images Horizontal Vertical Diagonal
RGBRGBRGB
Canal 0.9752 0.9754 0.9726 0.8177 0.9943 0.9940 0.6306 0.9912 0.8820
Quickbird 0.9727 0.9524 0.9642 0.8333 0.9888 0.9775 0.6306 0.9868 0.9748
Sea 0.9732 0.9656 0.9646 0.0004 0.6840 0.9835 0.9300 0.8542 0.9858
Blue sea 0.9762 0.9743 0.9826 0.8157 0.9854 0.9667 0.8338 0.6641 0.9632
Table 3 Correlation analysis of DEA-based i7DHS
Images Horizontal Vertical Diagonal
RGBRGBRGB
Canal 0.0252 −0.0048 0.0026 −0.0022 −0.0011 0.0024 −0.0008 −0.0122 0.0023
Quickbird −0.0231 −0.0094 0.0073 0.0020 −0.0024 0.0006 0.0401 −0.0202 0.0013
Sea 0.0051 −0.0228 −0.0066 −0.0004 0.0021 −0.0042 0.0142 −0.0011 0.0042
Blue sea −0.0080 −0.0033 −0.0059 0.0029 −0.0009 −0.0027 −0.0076 −0.0032 0.0084
Table 4 Comparison based on NPCR
Method Canal Quickbird Sea Blue sea
7DHS 0.9949 0.9945 0.9941 0.9947
4DS 0.9940 0.9640 0.9640 0.9643
FHGF 0.9944 0.9942 0.9946 0.9941
DFHS 0.9957 0.9957 0.9954 0.9959
i7DS 0.9956 0.9952 0.9958 0.9954
DEA-based i7DHS 0.9965 0.9968 0.9971 0.9962
Table 5 Comparison based on UACI
Method Canal Quickbird Sea Blue sea
7DHS 0.3348 0.3323 0.3312 0.3322
4DS 0.3329 0.3315 0.3341 0.3319
FHGF 0.3340 0.3333 0.3334 0.3332
DFHS 0.3349 0.3317 0.3331 0.3343
i7DS 0.3332 0.3331 0.3330 0.3334
DEA-based i7DHS 0.3352 0.3338 0.3339 0.3335
i7DHS generates almost identical encrypted images when a
tiny change is made in plain images.
4.7 Peak signal-to-noise ratio
The quality of decrypted image can be computed with the
help of peak signal-to-noise ratio (PSNR) (Rawat et al. 2016).
PSNR between decrypted and actual image should be max-
imum. Also, PSNR between encrypted and actual image is
desirable to be minimum (Chen et al. 2018). Table 6demon-
strates PSNR analyses of DEA-based i7DHS. It is found that
Table 6 PSNR analysis
Method Canal Quickbird Sea Blue sea
7DHS 70.180 71.909 70.670 73.121
4DS 64.572 62.843 68.868 61.982
FHGF 59.992 74.259 75.842 70.613
DFHS 79.848 69.579 63.347 77.817
i7DS 66.011 62.129 69.986 67.553
DEA-based i7DHS 81.750 88.545 89.415 83.983
Table 7 Key sensitivity analysis
Canal Quickbird Sea Blue sea
Difference 99.943 99.969 99.945 99.962
DEA-based i7DHS achieves better PSNR values than the
competitive techniques.
4.8 Key sensitivity
An encryption technique should be sensitive toward itsinitial
parameters. It means even a tiny change in initial parameters
should generate almost identical encrypted image. Figure 5
shows the key sensitivity analysis of DEA-based i7DHS by
considering the Canal image and a secret key (Bk). Also
another secret key (Bk) is also considered with single-bit
difference with Bk.Bkand Bkare used to obtain encrypted
Canal images as Emg and Emg , respectively. The difference
between Emg and E
mg is computed using NPCR. Table 7
demonstrates the difference between Emg and E
mg. The table
clearly shows DEA-based i7DHS is highly sensitive to the
secret key.
123
Improved seven-dimensional (i7D) hyperchaotic map...
Fig. 5 Analysis of key sensitivity: aCanal image, bBk-based encrypted
image, cBk-based encrypted image, ddifference between (b)and(c)
5 Conclusion
The existing image encryption approaches suffer from the
hyperparameters tuning problem. Therefore, an DEA-based
i7DHS was proposed to generate secret keys. The computed
optimal keys were then utilized to implement permutation
and diffusion on input image to develop the encrypted image.
Extensive experiments were carried out by considering the
well-known remote sensing images. Performance analysis
has shown that DEA-based i7DHS outperforms the compet-
itive approaches. DEA-based i7DHS has been found to be
highly sensitive to the input images and has large secret key
space. It was also found that DEA-based i7DHS can resist
various security attacks.
Declarations
Conflict of interest The authors declare no competitive interest regard-
ing the publication of this paper.
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