A new four-dimensional hyper-chaotic system for image encryption

International Journal of Electrical and Computer Engineering (IJECE)
Vol. 13, No. 2, April 2023, pp. 1744~1756
ISSN: 2088-8708, DOI: 10.11591/ijece.v13i2.pp1744-1756  1744
Journal homepage: http://ijece.iaescore.com
A new four-dimensional hyper-chaotic system for image
encryption
Huda R. Shakir, Sadiq A. Mehdi, Anwar A. Hattab
Department of Computer Science, Mustansiriyah University, Baghdad, Iraq
Article Info ABSTRACT
Article history:
Received Mar 30, 2022
Revised Sep 16, 2022
Accepted Oct 13, 2022
Currently, images are very important with the rapid growth of
communication networks. Therefore, image encryption is a process to
provide security for private information and prevent unwanted access to
sensitive data by unauthorized individuals. Chaos systems provide an
important role for key generation, with high randomization properties and
accurate performance. In this study, a new four-dimensional hyper-chaotic
system has been suggested that is used in the keys generation, which are
utilized in the image encryption process to achieve permutation and
substitution operations. Firstly, color bands are permuted using the index of
the chaotic sequences to remove the high correlation among neighboring
pixels. Secondly, dynamic S-boxes achieve the principle of substitution,
which are utilized to diffuse the pixel values of the color image. The
efficiency of the proposed method is tested by the key space, histogram, and
so on. Security analysis shows that the proposed method for encrypting
images is secure and resistant to different attacks. It contains a big key space
of (2627) and a high sensitivity to a slight change in the secret key, a fairly
uniform histogram, and entropy values nearby to the best value of 8.
Moreover, it consumes a very short time for encryption and decryption.
Keywords:
Hyper-chaotic system
Image encryption
Permutation
Substitution
This is an open access article under the CC BY-SA license.
Corresponding Author:
Huda R. Shakir
Department of Computer Science, University of Mustansiriyah
Alkadhimiya Street, Baghdad, Iraq
Email: hudarashid@uomustansiriyah.edu.iq
1. INTRODUCTION
With the rapid advancement of digital technologies and communication networks, multimedia
communication such as images, videos, and audio is becoming increasingly significant and indispensable in
today’s society [1]–[3]. However, network openness and sharing introduce several security risks to digital
communication. Digital images, as a significant medium of multimedia communication, play a major part in
medical, biological, military, and social life. Digital images contain a lot of important information, so the
security of image data attracts more and more attention [4], [5].
Due to the strong correlation between image pixels, high redundancy, and large data capacity
[6], [7], therefore, conventional encryption algorithms like advanced encryption standard (AES), data
encryption standard (DES), and Rivest-Shamir-Adleman (RSA) algorithms are not appropriate for image
encryption [8], [9]. As a result, many researchers have proposed new image coding algorithms that satisfy the
criteria of confusion and diffusion [10]. Compared with conventional encryption systems, chaotic systems
have stronger advantages such as sensitivity to initial value, ergodicity, randomness, and other characteristics
[11]–[13], which provide a quick and secure way to protect data transmitted over communication channels
like the internet. In order to ensure the effectiveness of the algorithms, Fredrich was the first to combine the
chaotic system with cryptography theory in 1998 [10], [14].

Int J Elec & Comp Eng ISSN: 2088-8708 
A new four-dimensional hyper-chaotic system for image encryption (Huda R. Shakir)
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Chaotic systems have been used in many studies to encrypt images. Below is a review of some
relevant work. Ali and Ali [15] proposed a new approach for encrypting color images using chaotic maps.
Three stages are involved in the construction of the cipher image. The first phase involves permuting the
digital image using a chaotic map. The second step employs a chaotic substitution box to perform pixel
substitutions, and the third phase employs the Boolean operation exclusive OR (XOR) to blend chaotic
logistic-based random sequences. The red, green and blue (RGB) components of an image that have been
scrambled using permutation, substitution, and XOR operations have good security and resistance to various
types of encryption attacks.
Teng et al. [16] proposed a new image encryption method based on a cross 2D hyper-chaotic map.
The keys are constructed by the hash function (SHA-512) and the plain image. Firstly, the color image is
transformed to a merged bit-level array and scrambled using the column and row cogeneration shift process.
The scrambled matrix is then diffused using a choosing sequence that is dependent on the chaotic sequence.
Lastly, the diffusion matrix is decomposed in order to obtain the color image encryption. Simulations and
security analysis show that the method works well at encrypting the color image and provides adequate
protection against various types of attacks.
Tariq et al. [17], suggested an image encryption scheme based on Lorenz chaotic systems and
analyzed the flaws of the public key encryption method based on logarithm and provided an additional layer
of security to prevent it from cryptographic assaults. According to both security analyses and experimental
findings, the suggested encryption method has proven to be resistant to both linear and differential attacks.
Patro et al. [18], suggested an image encryption approach based on multiple levels of permutations and
diffusions. Bit-level permutations are done with a 4-D hyper-chaotic map. The diffusion step is done with the
piecewise linear chaotic map (PWLCM) system, and block-level permutations are done with Alpar's map.
According to simulation and security analysis, the comparison tests show that the proposed method is secure
and immune to most common threats.
Talhaoui et al. [19], suggested a new image encryption method dependent on the chaotic map of the
Bülban. The new scheme’s security is provided via a substitution and permutation network, which uses a
rotating shift of columns and rows to remove the high correlation among neighboring pixels. Then, the pixels
values are mask the of with XOR and the modulo function to prevent any data from leaking into the system.
Tests and simulations have shown that the system is safe and very fast for real-time images.
Most encryption algorithms have three key problems: insufficient security, limited processing
capacity, and poor encryption efficiency. So, traditional algorithms are not suitable for image encryption, in
addition to the problems of low-chaotic systems such as small key space and low complexity. To overcome
these problems, in this paper, a new color image encryption method built on a novel 4-D chaotic system has
been suggested to offer higher security and randomness. Initially, the position of pixels are scrambled
through the sequences chaotic; then, high confusion is provided by the dynamic S-Boxes constructed using
DNA computing with the new chaotic system. An experimental and security analysis was performed on the
proposed image encryption algorithm to verify the security and efficiency of the encryption.
2. THEORETICAL BACKGROUND
2.1. The new chaotic system
The novel 4D-four-dimensional hyperchaotic system with ten positive parameters is proposed. It has
more complex properties than lower-dimensional chaotic systems. An entirely new 4D autonomous system
can be can be obtained by the following differential equation in system (1).
𝑑𝑥
𝑑𝑡
= −𝑎 𝑥 − 𝑏 𝑤 + 𝑐 𝑦 𝑧 + 𝑧 𝑒𝑦
𝑑𝑦
𝑑𝑡
= 𝑑 𝑦 + 𝑒 𝑥 − 𝑓 𝑥 𝑧 − 𝑥 𝑒𝑧
𝑑𝑧
𝑑𝑡
= −𝑔 𝑧 + ℎ 𝑥 𝑦
𝑑𝑤
𝑑𝑡
= −𝑏𝑤 + 𝑖 𝑥 𝑧 + 𝑗𝑦 𝑧
(1)
The states of system are x, y, z, and w, t  real number and the positive parameters are
𝑏, 𝑑, 𝑎, 𝑐, 𝑔, 𝑒, ℎ, 𝑓, 𝑗, 𝑖 wich displays a chaotic attractors in the new 4-D chaotic system by the initial
conditions of: 𝑥0 = 0.2, 𝑦0 = 0.4, 𝑧0 = 1.5, and 𝑤0 = 0.8 and parameter values of: 𝑎 = 3.1, 𝑏 = 2.1, 𝑐 =
15.8, 𝑑 = 1.1, 𝑒 = 16.5, 𝑓 = 1.5, 𝑔 = 2.4, 𝑖 = 5.1, ℎ = 26.6, and 𝑗 = 12.9. Figure 1(a) shows the chaotic
attractors in (y-x-w), Figure 1(b) in (x-y-z), Figure 1(c) in (x-z-w), and Figure 1(d) in (y-x-z). The four
Lyapnov Exponents of system (1), by parameters are obtained: LE1=4.05761, LE2=0.347562, LE3=-3.94257
and LE4=-6.61896. Due to LE1 and LE2 are positive Lyaponov exponents. Therefore, the new system is a
hyper chaotic.

 ISSN: 2088-8708
Int J Elec & Comp Eng, Vol. 13, No. 2, April 2023: 1744-1756
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(a) (b) (c) (d)
Figure 1. Attractors of chaotic (a) (y-x-w), (b) (x-y-z), (c) (x-z- w), and (d) (y-x-z)
2.2. Waveform analysis of the novel chaotic system
The waveform of a chaotic system should be aperiodic in order to show that the suggested system is
a chaotic system, as shown in Figure 2. Figure 2(a) depicts the x(t) waveform, Figure 2(b) the y(t) waveform,
Figure 2(c) the z(t) waveform, and Figure 2(d) the w(t) waveform in the time domain. By plotting the results
of a MATHEMATICA simulation, it is clear that the waveform of the system has complex behavior, chaotic
motion, and non-periodic characteristics. So, the time-domain waveform has non-cyclical characteristics so
that it can tell the difference between several periodic motions with complicated behaviors and chaotic
motion.
(a) (b)
(c) (d)
Figure 2. Waveforms of the novel chaotic system (a) x(t), (b) y(t), (c) z(t), and (d) w(t)
2.3. The sensitivity to initial conditions of the novel chaotic system
Long-term unpredictability is perhaps the most key characteristic of a chaotic system. This is due to
the sensitivity of solution dependency on initial variables. Two distinct initial states, despite how close they
begin, will eventually separate widely. As a result, for every initial condition with a finite number of digits of
accuracy, a future time will come when no reliable predictions can be formed about the state of a system.
Figures 3(a) to 3(d) demonstrate the sensitivity of the chaotic trajectories’ evolution to the initial conditions
in x(t), y(t), z(t), and w(t). For the solid line, the initial values are 𝑥0=0.2, 𝑦0=0.4, 𝑧0=1.5, and 𝑤0=0.8; for the
dashed line, they are x(0)=0.2, y(0)=0.400000000000001, z(0)=1.5, and w(0)=0.8.

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(a) (b)
(c) (d)
Figure 3. Sensitivity tests of the new system (a) x(t), (b) y(t), (c) z(t), and (d) w(t)
3. RESEARCH METHOD
In order to increase the level of security, we proposed a new algorithm for color image encryption
based on a new four-dimensional chaotic system and dynamic S-boxes. The suggested method consists of
two main phases: the encryption phase and the decryption phase. These steps will be explained in detail in
the following sections.
3.1. Encryption phase
The encryption phase uses the chaotic sequences to transform red, green and blue (RGB) images
into encoded images with unexpected features to withstand statistical attacks. It includes four stages:
generating four chaotic sequences from the new 4D-hyperchaotic system, permutation, S-box generation, and
substitution, in order to produce the final encrypted image. Figure 4 explains the total steps of the encryption
phase.
3.1.1. Generate chaotic sequences from the new chaotic system
The generation of chaotic sequences begins by entering the parameters with the initial values that
belong to the 4D chaotic system. In this step, we generate four chaotic sequences: xn, yn, zn, and wn; the
length of each sequence is the same as the size of the image dimensions (Row×Column). Then, sort
sequences in ascending order for xn, yn, and zn; and take the locations for each element of these sequences
and store them in the index vectors Xi, Yi, and Zi, which represent the keys.
3.1.2. Permutation
There are strong correlations between neighboring pixels in most of the images, so we utilize chaotic
sequences to permute the RGB image pixels in order to eliminate the correlations. The first step in the
proposed algorithm is image permutation. The plain color image RGB is split into three channels and
converted into three vectors: VR, VG, and VB and then is permuted depending on three sequences: xi, yi, and zi
that were generated in the previous step, so that the correlation between them is lessened. In order to
scramble the position of every value in the vectors, the sequence {xi} scrambles the red vector (VR), the
sequence {yi} scrambles the green vector (VG), and the sequence {zi} scrambles the blue vector (VB). Hence,
these vectors represent the permutation image.
3.1.3. S-boxes generation
At this stage, we proposed new S-boxes to achieve the substitution process in image encryption,
built on a hybrid method combining a chaotic system and DNA encoding. Firstly, depending on the chaotic

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sequences (xi, yi, zi, and wi), which have been transformed to vectors k1, k2, k3, and k4 from chaotic
sequences, also, using algebraic DNA processes (exclusive-or, subtraction, and addition), which are used to
make a set of S-boxes (1616), each chaotic sequence is turned into a binary sequence and then changed to a
DNA sequence so that each pair of bits is replaced by a DNA code. For example, (00) is replaced by (A),
(01) by (C), (10) by (G), and (11) by (T), giving the four sequences (k1, k2, k3, and k4). The addition
operation is applied to the first and second sequences (k1, k2), while the subtraction operation is applied to
the third and fourth sequences (k3, k4). Finally, the XOR process is performed on the results of the previous
subtraction and addition stages. The results of these operations are used for generating a set of S-boxes by
getting each digit as one cell in the S-box, and this cell should not be repeated before. The process continues
until unique 256 values are placed in the S-box. Table 1 denotes the results of applying the suggested method
for creating S-box.
Figure 4. Block diagram of encryption algorithm
3.1.4. Substitution
The S-box achieves the principle of substitution. In this stage, the permuted pixels in the previous
step will be substituted based on new S-boxes have been generated, where three permuted vectors merge into
one vector and are split into blocks, each block having a size of 8×8 bytes. Each block is converted into a
hexadecimal string and substituted by one S-box. Based on a (wi) chaotic sequence for the selection No. of
S-box, (wi mod n), where wi represents the No. of S-box. After the S-box selection, each two-digit
hexadecimal is substituted in the S-box. To change the value of each pixel, the first hexadecimal digit is
given to the rows and the second hexadecimal digit is given to the columns. The value resulting from the
intersection of the row and column in the s-box is put in the new matrix. The previous steps are repeated until
all the blocks are completed and the get the image confusion. Algorithm 1 shows the total steps of the
encryption method.
Sequence wi
Dynamic S-boxes
Substitution Stage
Permutation Stage
Image (RGB)
Divide color channels
Red
Reshape to
vector (Vr)
Green Blue
Reshape to
vector (Vb)
Reshape to
vector (Vg)
Input parameters & iInitial value
Apply 4-D Chaotic to generate
sequences xi, yi, zi, wi
Sort ascending xi, yi, zi
sequences
Index xi
(k1)
Index yi
(k2)
Index zi
(k3)
Split into three vectors
Encrypted Image
Reshape
to matrix
Concatinating into 3D matric
VsR Vs
G
VsB
Transform every value
in to (integer value)
Wi mod n
to select the
No. of S-box
Reshape
to matrix

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Table 1. S-box values results
0 1 2 3 4 5 6 7 8 9 A B C D E F
0 50 41 33 36 91 177 51 21 118 25 159 237 62 1 169 103
1 87 200 39 4 40 246 229 213 139 110 134 69 190 119 235 142
2 17 70 170 97 194 147 3 211 135 2 143 243 116 5 35 67
3 78 167 121 145 204 37 42 132 140 188 141 90 57 14 201 18
4 89 133 20 196 205 185 191 117 210 228 131 56 197 179 249 32
5 52 181 250 66 47 0 82 166 83 199 58 175 100 216 182 113
6 176 45 202 55 48 157 178 184 12 88 239 232 248 44 6 99
7 102 122 238 163 22 128 168 92 43 187 98 11 72 74 24 153
8 165 207 215 34 148 26 150 149 227 127 192 64 107 209 95 120
9 171 96 231 241 198 212 164 254 46 236 253 81 230 125 138 245
A 161 7 195 31 38 75 233 77 30 71 65 189 158 162 106 19
B 146 203 61 60 244 86 208 123 222 224 180 151 206 156 79 152
C 193 252 94 54 85 242 8 240 255 68 124 214 247 15 111 109
D 126 154 53 218 49 217 155 59 225 105 226 173 221 16 104 130
E 234 219 115 136 76 13 28 174 9 108 10 29 172 23 186 137
F 63 114 220 144 80 27 73 101 93 129 160 251 112 84 223 183
Algorithm 1. Encryption
Input: plain color image (PI), initial values: Y0, X0, Z0, W0, parameters: b, d, a, c, g, e, h, f, j, i
Output: Encrypted image (EI)
Begin
Step 1: Read color image (PI)
Step 2: Mhight of PI
Nwidth of PI
S M×N
Step 3: Iterate proposed chaotic system with parameter and initial conditions to
Create four series xn, yn, zn, and wn, size of sequences ≥S
Step 4: split color image (im) into R, G, B bands
Step 5: Each color channel (Red, Green, and Blue) are reshape to (VR, VG, and VB ) vector
Step 6: Sort (xi, yi, zi) sequences in ascending order, get the index, and find the
position of each sequence in the index (k1, k2, k3)
Step 7: Using k1 to permutate the vector (VR) // (Red scrambling)
using k2 to permutate the vector (VG) // (Green scrambling)
using k3 to permutate the vector (VB) // (Blue scrambling)
Step 8: Merging the three (VR, VG and VB) vectors to one (VRGB) vector
Step 9: divide the (VRGB) vector to blocks of size 8×8
Step 10: using the 4th
sequence wi as index (wi mod n) of dynamic S-Boxes, to select the
No. of S-box // n the number of S-boxes produced
Step 11: After selecting the No. of S-box, for each block we substitute each value that
denotes the address of the row and column in the S-box and get the new value
by the intersection of the row and column.
Step 12: Continue until all the blocks are complete to get the confused image (Vsbox).
Step 13: split Vsbox into three vectors VsR, VsG, and VsB
Step 14: reshape VsR, VsG and VsB into 2D (mR, mG, and mB) matrix
Step 15: combine (mR, mG, and mB) to create 3-D of matrix represented the enciphered image
End
3.2. Decryption stage
Decryption is the inverse operation of encryption. The inputs to this stage are the encrypted image,
initial condition values, and parameters, while the output is the extracted image. At first, the chaotic
sequences are generated by iterating the proposed new chaotic system to create sequences: xi, yi, zi, and wi.
These sequences are based on the same values of parameters and primary values as the encryption phase.
Second, in the inverse substitution, the encrypted image is split into three bands and reshaped into vectors.
The vectors are divided into blocks of size 8×8. Then each block is substituted by one from an inverse S-box.
The chaotic {wi} sequence is used to select the inverse S-box (wi mod n), which is the number (NO. of
inverse S-boxes). Thirdly, the inverse permutations are performed by the index of the chaotic sequences (k1,
k2, and k3) and the image vectors (ViR, ViB, and ViB) to get the re-permuted image. Lastly, the image's
vectors are combined and turned into a matrix that shows the extracted image. Figure 5 shows the total steps
in the decryption stage, and algorithm 2 displays the main steps of decryption:

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Figure 5. Block diagram of decryption algorithm
Algorithm 2. Decryption
Input: Encrypted image (imx), initial values: Y0, X0, Z0, W0, parameters: b, d, a, c, g, e,
h, f, j, i
Output: Original image (Ex) of size (h×w×3)
Begin
Step 1: read encrypted image (imx)
Step 2: mhight of imx
nwidth of imx
S m×n
Step 3: Iterate proposed chaotic system with parameter and initial conditions to
Create four series xn, yn, zn, and wn, size of sequences ≥S
Step 4: split image (imx) into three R, G, B bands
Step 5: Each color channel (Red, Green, and Blue) are reshape to (IVR, IVG, and IVB)vector
Step 6: Merge IVR, IVG, and IVB to one (IVRGB) vector
Step 7: divide the (IVRGB) vector to blocks of size 8×8
Step 8: using the 4th
sequence wi as index (wi mod n) of inverse dynamic S-Boxes, to
select the No. of Inverse S-box // n the number of inverse S-boxes produced
Step 9: After selecting the No. of inverse S-box, for each block we substitute each
value that denotes the address of the row and column in the S-box and get
the new value by the intersection of the row and column.
Step 10: Continue until all the blocks are complete to get the Re-confused image(IVsbox)
Step 11: split (IVsbox) into three vectors ViR, VsG, and VsB
Step 12: Sort (xi, yi, zi) sequences in ascending order, get the index, and find the
position of each sequence in the index (k1, k2, k3).
Step 13: Using (k1) to re-permutate the vector(ViR)
using (k2) to re-permutate the vector(ViG)
using (k3) to re-permutate the vector(ViB)
Step 14: reshape ViR, ViG and ViB into 2D matrices MRi, MGi, and MBi
Step 15: combine MRi, MGi, and MBi to create 3D matrix represented the ciphered image(Ex)
End
Inverse Substitution
Merging into one vector (VRGB)
& divide into blocks
Ciphered Image
Split color bands
Red
Reshape
to vector
(VR)
Green Blue
Reshape
to vector
(VB)
Reshape to
vector
(VG)
Input parameters& initial
value
Apply 4-D Chaotic to generate
sequences xi, yi, zi, wi
Sort ascending xi, yi, zi
sequences
Sequence
wi
Index xi
(k1)
Index yi
(k2)
Index zi
(k3)
Inverse Permutation
Split into three vectors
Original Image
Reshape into 3-D array
ViR ViG ViB
Wi mod n
select No.of S-box
Convert each
value to
integer
Inverse Dynamic
S-Boxes

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4. RESULTS AND ANALYSIS
In order to illustrate our algorithm’s security and efficiency, some security analysis has been
implemented on the suggested enciphering algorithm. In this paper, several standard images (256×256×3 and
512×512×3 pixels) have been tested. Furthermore, the proposed algorithm is simulated in MATLAB R2020a,
and our testing results were compared with those of other encryption methods of the same type. These
analyses are discussed.
4.1. Key space analysis
A cryptosystem’s key space should be at least 2100
to withstand brute force attacks [20], [21]. If the
precision is in the range of 10–14
, then the key space is 10196
≈2627
. Therefore, the key space is large enough to
withstand brute force attack. The key space comparative results between the proposed method and the related
works are shown in Table 2.
Table 2. Comparison the key space with the other methods
Algorithm key space
Proposed method 2627
Ref. [15] 2299
Ref. [16] 2425
Ref. [17] ≅ 4×1012
4.2. Key sensitivity analysis
A secure encryption system should be extremely sensitive to small changes in the key. In other
words, the encryption technique should be able to withstand brute-force attacks and must be sensitive to any
changes in the keys [22]. As explained in Figure 6, the key sensitivity for images is Figure 6(a) show the
original image, Figure 6(b) show an encrypted image, Figure 6(c) show a decrypted image with the correct
key, and Figure 6(d) show a decrypted image with the incorrect key.
(a) (b) (c) (d)
Figure 6. The key sensitivity for images (a) original image, (b) encrypted image, (c) decrypted image with the
correct key, and (d) decrypted image with the incorrect key
4.3. Histogram analysis
To stop the attacker from extracting any data contained in the plain image, any statistical
relationship or similarity between the plain and decrypted images should be avoided. From a histogram
analysis of an image, the statistical properties of that image can be derived. The plain and ciphered images
must have completely different statistical properties [23]. Figure 7 displays the histogram of images tested
before and after encryption.
4.4. Entropy analysis
Entropy is the significant characteristic that reflects information’s randomness and unpredictability
[24], [25]. The entropy of the cipher image should ideally be 8 [11]. The entropy of H (s) can be calculated:
𝐻(𝑠) = − ∑ 𝑝(𝑠𝑖) log2 𝑝(𝑠𝑖)
𝑁−1
𝑖=0 (2)
where N denotes the number needed to represent the symbol si. s denotes the source, and P (si) is the
symbol’s probability. Table 3 shows the entropy values of encryption images.

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Figure 7. The histogram of image tested before and after encryption

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Table 3. The entropy test of images
Name image Entropy of original images Entropy of encryption
Lina 7.7516 7.99857
palace 7.09016 7.99896
Yaght 7.6035 7.99802
sailboat on lake 7.7632 7.99897
4.5. Correlation coefficients analysis
Correlation coefficient analysis is used to determine the degree of similarity between plaintext and
ciphertext images [1], [26]. Table 4 shows the correlation coefficients analysis. The test was applied to red,
green, and blue colors, respectively, between an original image and an encrypted image. As shown in
Figure 8, the correlation is applied in horizontal, which tests the pixel with a neighbor in a row, vertical,
which tests the pixel with a neighbor in a column, and diagonal, which tests the pixel with a neighbor pixel in
the diagonal.
Figure 8. Correlation coefficients of plain and encrypted image
Table 4. Correlation coefficients analysis
Direction image Lena palace Yaght Sailboat Direction
Horizontal original 0.96438 0.89285 0.97313 0.96793 original
encryption -0.00020 0.00036 0.00068 -0.00121 encryption
vertical original 0.98031 0.92422 0.96652 0.96448 original
encryption -0.00045 0.00130 0.00356 -0.00126 encryption
diagonal original 0.94832 0.84829 0.93995 0.95048 original
encryption -0.00474 -0.00612 -0.00141 -0.00090 encryption
4.6. Number of pixels change rate (NPCR) and unified average changing intensity (UACI) analysis
Two differential assault metrics are used to evaluate how vulnerable the original data is to slight
alterations: NPCR as well as UACI. Suppose the enciphered image is (C and C’) before and after changing
one pixel in the original image [27]–[29]. The results of applying the proposed method on tested images are
represented in Table 5. The (3) and (4) [30], express the NPCR and UACI formulas, respectively:
𝑁𝑃𝐶𝑅 =
∑ 𝐷(𝑖,𝑗)
𝑖,𝑗
𝑊×𝐻
× 100% (3)
𝑈𝐴𝐶𝐼 = 1
𝑊×𝐻
[∑
|𝑐(𝑖,𝑗)−𝑐′(𝑖,𝑗)|
255
𝑖,𝑗 ] × 100% (4)

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4.7. Peak signal-to-noise ratio (PSNR) and mean square error (MSE) analysis
The PSNR and MSE are two of the more commonly used tests for image encryption techniques; the
PSNR may be used to evaluate an encrypting method. It is a metric that indicates the difference in pixel
values between the plain and cipher images. A lower PSNR value indicates a higher encoding quality. The
PSNR and MSE formulas are [31]:
𝑃𝑆𝑁𝑅 = 20 log10( 255
√𝑀𝑆𝐸
)𝑑𝐵 (5)
𝑀𝑆𝐸 = 1
𝑀×𝑁
∑𝑖.𝑗 (𝑝0(𝑖. 𝑗) − 𝑝1(𝑖. 𝑗))2
(6)
where 𝑀 and 𝑁 represent the height and width of images. For the original and ciphered images, the intensity
values of the pixels are P0 (i, j) and P1 (i, j). According to the results, a high MSE value and a low PSNR
value between the original and encrypted images indicate desirable encryption quality, as shown in Table 6.
Table 5. UCAI and NPCR result
Image NPCR UCAI
Lena 99.6367 33.0305
Palace 99.6329 29.4177
Yaght 99.6183 31.0267
Sailboat 99.6152 34.0265
Table 6. The MSE and PSNR test
Image MSE PSNR
Lena 8914.693 3.8508
Palace 8327.430 3.4910
Yaght 9262.862 3.2850
Sailboat 10171.02 2.9726
4.8. Comparison results
The proposed method is compared to other algorithms in order to see if it satisfies general
requirements and how well it performs. Lena is used as the test image. Entropy, key space, NPCR, UACI,
and correlation coefficient are utilized for the comparison. By looking at the algorithms, the suggested
method is more secure against differential, statistical, and brute-force attacks than the methods that are
currently being used. The results of the comparison are listed in Table 7.
Table 7. Comparison results of the proposed method with other methods
Image method Key space Entropy NPCR UACI Correlation.H Correlation.V Correlation.D
Encrypted Lena proposed 2627
7.9985 99.6367 33.0305 -0.00020 -0.00045 -0.00474
Ref. [15] 2299
7.9984 99.6094 33.4635 -0.00216 0.00103 0.0004
Ref. [16] 2425
7.9912 99.6235 33.4620 0.000617 -0.00053 -0.00041
Ref. [17] 4×1012
7.9974 99.62 31.03 -0.0026 0.0031 -0.0043
Ref. [20] 2260
7.929 99.65 32.4966 -0.0921 -0.0372 -0.1013
Ref. [28] - 7.9976 99.64 28.66 - - -
4.9. Time consuming
The execution time is also an important factor in evaluating the performance of an encryption
algorithm. The encryption algorithm is more efficient when the execution time is less. Table 8 shows the
average consumption time for two-stage encryption and decryption.
Table 8. Time-consuming for encryption/decryption technique
Image size Encryption Time (sec) Decryption time (sec)
256×256 0.58233 0.79361
512×512 1.29370 1.36520
5. CONCLUSION
In this study, a robust color image encryption algorithm built on a novel chaotic four-dimensional
system has been suggested to achieve a high level of security, fast speed, and high performance. A variety of
security analyses, including entropy analysis and histogram analysis, have been performed on the proposed
scheme. The results of the simulations and performance measurements have shown that the encryption effect
on security and reliability of our suggested method is good and it is well suited for image encryption. The
proposed method has a large key space of (2627
) and a high sensitivity to a slight change in the secret key of
decrypted images, a fairly uniform histogram, and entropy values close to the ideal value of 8, mean values

1755
for NPCR and UACI values of (99.63) and (33.03), respectively, and a very short time for encryption and
decryption. In future work, the proposed method could be applied to other types of data, such as video and
audio data, also by merging the proposed system with other techniques like watermarking or steganography
techniques.
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BIOGRAPHIES OF AUTHORS
Huda R. Shakir received a B.Sc. in Computer Sciences from Mustansiriyah
University, Baghdad, Iraq. Currently, she is studying M. Sc. in Computer Science, College of
Education, Department of Computer Science, Mustansiriya University, as well as working in
the Iraqi Ministry of Education. She can be contacted at hudarashid@uomustansiriyah.edu.iq.
Sadiq A. Mehdi received the B.Sc. in Mathematical Sciences from Mustansiriyah
University, Baghdad, Iraq in 1996, M.Sc. in Applied Mathematics Sciences-Modeling and
Simulation from Al al-Bayt University, Jordan in 2002, and Ph.D. in Applied
Mathematics/Data Cryptography from University of Mustansiriyah, Baghdad, Iraq in 2011.
Current position and Functions: computer science from Mustansiriyah University. His research
interest is in the fields of Dynamical system, Chaotic system, Chaotic Encryption and
Modeling and Simulation. He can be contacted at sadiqmehdi71@uomustansiriyah.edu.iq.
Anwar A. Hattab received the B.Sc in Computer Science from Baghdad
University and her MSc degree in Network Management in 2003 from the Iraq Commission
for Computer and Informatics, Institute for Post Graduate Studies in Informatics. Currently she
is a lecturer in computer science. Anwar has more than 18 years of experience and has
supervised Msc and BSc final year project. Her research interests include cryptography, image
processing, data security, network security, and databases. She can be contacted at email:
anwarabbas76@uomustansiriyah.edu.iq.

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