Construction of highly non linear component of block cipher based on mclaurin series and mellin transformation with application in image encryption

Abstract

A substitution box (S-box) in data encryption is a non-linear tool that conducts substitution to assure the overall security of the system. S-box is the most important part of the block cipher. The non-linearity trait is critical for the construction of more reliable substitution boxes in data encryption. As a result, new approaches for generating high no n-linear S-boxes are required. Using the Mellin transformation and the McLaurin series, this work suggests an approach for creating Substitution boxes with a high non-linearity value of 112.5. S-box construction consists of three phases. In step 1, we build a sequence from a function's McLaurin series and then apply Mellin transformation to the sequence's terms without substituting limits. In the second phase, we solved all of the coefficients under mod 257, and in the third step, we improved the unpredictability of the initial S-box by using a particular permutation of Symmetric group S256\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${S}_{256}$$\end{document}. Furthermore, the algebraic characteristics of S-box are evaluated using various tests, including non-linearity (NL), Bit Independent Criterion (BIC), Strict Avalanche Criterion (SAC), Linear Approximation Probability (LAP), and Differential Uniformity (DU), all of which certify the algebraic properties of the S-box.

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https://doi.org/10.1007/s11042-023-15965-y
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Construction ofhighly non linear component ofblock
cipher based onmclaurin series andmellin transformation
withapplication inimage encryption
AbidMahboob1· ImranSiddique2 · MuhammadAsif3· MuhammadNadeem4·
AyshaSaleem4
Received: 25 February 2022 / Revised: 13 March 2023 / Accepted: 29 May 2023
© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023
Abstract
A substitution box (S-box) in data encryption is a non-linear tool that conducts substitution
to assure the overall security of the system. S-box is the most important part of the block
cipher. The non-linearity trait is critical for the construction of more reliable substitution
boxes in data encryption. As a result, new approaches for generating high no n-linear S-boxes
are required. Using the Mellin transformation and the McLaurin series, this work suggests an
approach for creating Substitution boxes with a high non-linearity value of 112.5. S-box con-
struction consists of three phases. In step 1, we build a sequence from a function’s McLau-
rin series and then apply Mellin transformation to the sequence’s terms without substituting
limits. In the second phase, we solved all of the coefficients under mod 257, and in the third
step, we improved the unpredictability of the initial S-box by using a particular permuta-
tion of Symmetric group
S256
. Furthermore, the algebraic characteristics of S-box are evalu-
ated using various tests, including non-linearity (NL), Bit Independent Criterion (BIC), Strict
Avalanche Criterion (SAC), Linear Approximation Probability (LAP), and Differential Uni-
formity (DU), all of which certify the algebraic properties of the S-box.
Keywords Block Cipher· Substitution box· McLaurin series· Mellin transformation·
Symmetric group Sn
1 Introduction
It is now well acknowledged that the usage of the internet is rapidly growing in all walks
of life like engineering, medical, military, educational, political, banking, marketing
and vice versa. Everyone wants their data and information to be secured over the inter-
net while transmission third party don’t have unauthorized access to the communica-
tion between two people. In [7], Shannon proposed the concept of block cipher. Block
cyphers such as the Data Encryption Standard (DES)and Advanced Encryption Standard
(AES) are the critical components of multimedia security. DES [32] like cryptosystem
* Imran Siddique
imransmsrazi@gmail.com
Extended author information available on the last page of the article
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break easily due to its short 56 bit key and more complicated due to 16 Feistel rounds.
AES like cryptosystem used 128, 192 and 256-bits keys for 10, 12 and 14 rounds respec-
tively for encryption. In 2001 NIST introduced AES and modern cryptography based on
it [10]. AES is a type of block cypher that encrypts and decrypts data with the same key
using the substitution permutation network (SP-network). The substitution box is still the
most crucial part of block cyphers, including AES (S-box). This is because this ingredi-
ent is the sole source of the nonlinearity effect. Furthermore, according to Shannon’s pio-
neering work [36], S-Box is consistent with the influence of confusion, which obscures
the statistical relationship between the plain text and the private keys.
S-box is a vectorial Boolean function that is defined mathematically as:
𝜑∶
ℤ
u
2
→ℤ
v
2
which map
u
input bit into
v
output bit. The algebraic and statistical features of the S-box,
such as NL, BIC, SAC, DU, LAP, and MLC, are used to assess its validity. Substitution boxes
are essential in symmetric key cryptography because they are used to perform substitutions.
S-boxes are commonly used in block ciphers to conceal the relation among the cipher text and
the key. Because of the usage of algebraically weak S-boxes, cryptosystems are unsecure and
unstandardized. Similarly, constructing an efficient and safe cryptosystem requires the crea-
tion of a robust algebraic S-box. Many methods have been developed as a result of these sig-
nificant applications to create more reliable S-boxes that will be used for robust block encryp-
tion. For these reasons, researchers are currently concentrating on developing new algorithms
for constructing more secure and trustworthy S-boxes. Zhang etal. utilized I-Ching operators
to create an S-box in [43]. The nature of the created S-Box was assessed using various tech-
niques, with great results showing the scheme’s resilience. [21] Introduces a novel approach
for constructing key dependent S-boxes. Firdousi etal. [16] created a modified S-box using
quantum maps. The recommended S-box is strong enough to meet the needed requirement of
secure encryption technique. [26] Presents a powerful image encryption technique based on
a newly developed S-box. Shafique etal. [35] created a secure S-box using Cubic-Logistic
mapping. With the use of integer multiplication, a safe and efficient chaotic map [23] is cre-
ated, which is then used to generate a strong S-box. In [44], authors devised an approach
using the combination of chaotic systems. In [37], the authors have investigated matrix action
on the Galois fieldand created anS-box with good cryptographic features.
The symmetric group
Sn
is utilised to enhance the nonlinearity value of S-box. Rows and
columns of 8 × 8 S-boxes are rearranged using permutations of the S16 group, whereas cells
of S-boxes are rearranged using permutations of the S256 symmetric group. A function’s
Taylor series is used to express a function in the form of the sum of infinite terms of the
function’s derivative at a point. Taylor’s series, which is named after Brook Tayler, extends
a function in terms of Polynomial. The McLaurin series is a special form of the Taylor
series at
x=0
[1]. Mellin transformation is a type of integral transformation that is similar
to Laplace and Fourier transformations. Hjalmar Mellin [25], a mathematician, describes
this transformation. It is employed in number theory, asymptotic expansion theory, and
mathematical statistics. The literature contains various publications on the Mellin trans-
form based cryptography technique [33, 34]. Mellin transformation was utilised by Bhatti
etal. in [8] for encryption and decryption, however the building of S-boxes using Mellin
transformation has never been mentioned before. In this study, we introduce a unique, effi-
cient, and creative method for building robust S-boxes utilizing the Mellin transformation.
The following is the main contribution of our work in this manuscript:
• A new S-box construction process based on the McLaurin and Mellin transformation.
• Permutation of symmetric group
S256
are used to generate the recommended S-box
which have an average non-linearity of 112.5.
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• The recommended S-box is utilized for image encryption, and its validity in image
encryption is assessed using various tests.
• Compare the proposed encrypted image with different algorithm, which shows that our
S-box gives the best encryption scheme for the security of image data.
The body of this article is structured as follows: Basic definitions are provided in Sec-
tion2. In Section3, a mathematical algorithm is described for the construction of S-box
using the McLaurin series and Mellin transformation. Additionally, the nonlinearity of
the initial S-box, which has an average nonlinearity of 112.5 and is higher than the AES
NL score, is increased using a special permutation of the symmetric group
S256
. In Sec-
tion4, the proposed S-box is examined using several tests, including NL, BIC, DP, and
SAC, and it is compared to existing boxes from the literature. In Section5, the recom-
mended S-box is employed for image encryption, and the encryption strength is assessed
using various tests and the proposed algorithm’s conclusion is provided in Section5.
2 Preliminaries
2.1 Mellin Transformation
It is an integral transformation that was named by the mathematician Hjalmar Mellin. It is
defined as,
Hjalmar Mellin also define inverse Mellin transformation as:
which is the line integral in complex plane.
2.2 Taylor series
The formula for Tayler series of a real or complex-valued function is defined as
In sigma notation;
where
n!
denotes factorial of n and
a∈R,C
. McLaurin series is the special case of Taylor
series for which
a=0
3 Algebraic Structure ofS‑box
Step 1: Let S indicates a Set of integers ranging from 0 to 255.
M[
g(x);h
]
=
∫∞
0
g(x)xh−1
dx
M
−1[g(x);h]=1
2𝜆
i
∫
c−i∞
c+i∞
g(x)x(−h)
dx
f(a)+f
�
(a)
1!
(x−a)+ f
�
(a)
2!
(x−a)2+f
���
(a)
3!
(x−a)3+⋯+f
n
(a)
n!+…
f(a)=
∑
∞
n=0
fn(𝛼)
n!
(x−a)
n
S={k|k∈Z∧0≤k≤255}
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A function
f(x)=1∕(1−x)
whose McLaurin series is
Series is the sum of elements of the sequence
Write down all of the components of
S
in ascending order and multiply with the first
256 terms of
{xn}
such that
k.xn
is possible if
k=n
.
The outcomes will be as follows:
Now we will develop a new sequence, as seen below.
This sequence can be extended by multiplying
x2
with each term of the sequence
{nxn}0≤n≤255
as
If we apply the Mellin transformation to the
nth
term of the sequence while neglecting
limitations, we get the following formula:
f(x)=
∞
∑
n=0
x
n
{xn}n=0,1, 2, 3,
0,1x,2x2,3x3,…, 255x255
{nxn}n=0,1, 2, 3, …, 255
{
nx
n+2}
n=0, 1, 2, 3, …
, 255.
M
[
nxn+2,h
]
=n(x)
n+h+2
n+h+2
n=0, 1, 2, 3 …
255
Table 1 Initial S-box1
0 43 37 161 29 129 94 22 139 19 172 81 167 15 136 65
13 176 79 140 155 70 96 10 54 215 34 41 32 84 111 8
112 197 47 33 214 7 228 217 58 40 154 199 153 78 142 164
30 177 71 134 46 156 210 108 220 146 5 21 80 145 116 171
27 56 124 133 184 185 138 99 51 24 66 17 204 236 35 4
137 243 190 109 128 158 241 149 12 206 28 100 250 77 97 168
118 200 233 211 209 144 61 89 166 36 38 196 192 152 68 207
102 234 245 183 173 239 72 202 186 3 90 11 248 169 64 73
57 187 20 86 76 14 25 157 52 191 107 67 63 221 223 93
170 198 115 50 48 26 59 141 91 162 182 9 159 231 53 247
110 18 101 131 150 69 16 122 255 224 23 55 242 193 235 208
160 121 74 75 126 135 203 232 88 143 114 179 238 254 113 39
151 49 103 213 125 188 82 229 95 117 181 106 60 105 219 201
42 31 252 45 226 212 62 147 251 148 175 227 218 225 44 205
249 163 189 104 119 180 83 246 194 123 244 92 178 87 240 120
237 165 130 230 98 222 216 2 195 174 132 6 1 253 127 85
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When we apply the above formula to each term in the sequence
{
nxn+2
}
, we get the new
sequence shown below
Here we will use
h=3
then the sequence will become
{
n(x)n+5
n+5
}
n=0, 1, 2, 3 …255
Step 2: Since an S-box
S∶ {0,1}n
→
{0,1}n
is basically any rearrangements of the
elements of set
S={k|k∈Z∧0≤k≤255}
. We used
mod 257
to solve the coef-
ficients of all sequence terms. As we determined that the
127th
coefficient is equal
to 256, which does not belong to set S, and the
253th
coefficient is equal to
∞
, these
coefficients are replaced by 64 and 1 respectively. Put all of these results in the
16 ×16
table, which is our initial S-box and has an average non-linearity of 102.5.
(See Table1).
A flow chart for the mathematical construction of initial S-box:
{
n(x)
n+h+2
n+h+2
}
n=0, 1, 2, 3 …
255
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Step 3: Since our data is shown in a
16 ×16
tablewith 256 cells, we may improve the
randomization by swapping the placements of these cells. For this reason, a particular
type of permutation from Symmetric group
S256
, as shown in Table2, is applied on
Table1, and the new S-box, which is our suggested S-box with an average non-linearity
of 112.5, is shown in Table3.
4 Security analysis
Within this part, a number of tests have been provided to assess the effectiveness of sug-
gested S-box. Non-linearity test (NL), Bit Independent Criterion (BIC), Strict Avalanche
Criterion (SAC), Linear Approximation Probability (LAP), and differential Uniformity
(DU) of Substitution box are a few examples. Table9 compares the algebraic characteris-
tics of S-boxes to those of many other S-boxes.
4.1 Non‑linearity
In 1988, Pieprzyk and Finkelstein proposed the non-linearity test [31]. It is quite helpful in
determining the effectiveness of the S-box. If the correspondence from plain text to cypher is
linear, S-box is regarded weak, and attackers may easily perform a linear attack on cypher text.
The mathematical formula for determining the non-linearity of S-box is defined as follows:
where
Wf(a)
is the value of Walsh Spectrum
and
a.x
is dot product defined by
(1)
Nf=(2)n−1[1−(2)−nmax
|
|
|
Wf(a)
|
|
|
(2)
W
f(a)=
∑
a∈Fn
2
(−1)f(x)⊕a.
x
Table 2 Permutation of
S256
(1 179 245 93 137 4 20 154 95 130 231 178 170 215 193 140
151 134 6 135 88 149 194 248 143 13 84 225 139 21 187 188
102 3 159 112 31 199 16 177 42 77 104 61 169 105 108 70
97 106 12 91 110 22 55 72 128 148 214 208 60 40 87 183
191 52 161 233 203 66 39 64 163 116 51 152 10 213 101 204
107 221 141 111 241 2 100 99 54 197 226 57 67 90 165 220
160 186 59 240 125 239 176 228 47 46 69 216 28 76 23 37
168 86 126 164 210 181 136 156 123 62 33 195 250 89 212 246
205) (5 247 158 127 254 41 115 118 121 252 19 242 98 36 171
224 129 114 249 219 8 146 65 11 56) (7 94 167 131 211 44
92 172 222 234 184 79 50 82 30 230 74 32 25 253 175 138
83 24 78 133 201 206 109 49 227 166 15 63 218 209 180 117
38 243 255 81 185 68 122 119 113 124 189 251 75 80 145 29
48 26 198 244 71 232 73 27 157 196 217 236 35 9 14 120
174 207 144 53 256 235 238 17 150 192 142 162 153) (18 202 155
173 58 237) (34 45 85 223) (43 182 132 190 229) (96 147 103 200)
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Table4 presents the non-linearity values of eight balanced Boolean functions of con-
structed S-Boxes and Fig.1 compares the non-linearity values of the proposed S-box to
those of several previously developed S-boxes in the literature.
4.2 Bit independent criterion
A Vectorial Boolean
S∶ {0
,
1}n
→
{0
,
1}n
function meets BIC requirements if
∀i,j,k∈ {1,2, 3, …,n}
the inversion of
ith
input modifies output
jth
and
kth
separately [41].
Tables5 and 6 provide the results of the BIC non-linearity values and the BIC SAC values of
the recommended S-box, respectively. Where the relationship is discovered when we modify
the
ith
input and the corresponding change in the
jth
and
kth
output bits. Our S-box has a BIC
nonlinearity score of 103.79, and a BIC-SAC score of 0.4992, which is quite near to the ideal
value of 0.5. As a consequence, our S-box satisfies the BIC’s requirements. Table9 compares
the BIC non-linearity values of the recommended S-boxes to those of numerous other S-boxes.
4.3 Strict avalanche criterion
This is a critical requirement for determining the algebraic characteristics of the S-box
[41]. This S-box feature assures that the output bit changes by 50% or 1/2 probability
(3)
a1.x1⊕a2.x2⊕
⋯
⊕an.xn
Table 3 Proposed S-box
60 237 144 52 108 14 91 175 47 141 27 36 223 139 69 82
87 178 6 161 107 152 17 190 8 164 51 147 170 243 207 24
145 44 92 200 96 173 56 21 253 160 119 252 197 142 104 32
192 35 183 113 93 233 70 172 163 242 143 201 89 90 136 228
198 181 220 88 78 196 230 210 246 180 132 41 40 10 232 66
127 177 191 167 153 122 217 25 174 124 81 199 98 94 162 229
185 165 211 43 226 179 115 204 255 118 106 166 105 28 63 53
72 57 58 101 75 245 3 15 239 133 9 102 120 158 231 99
205 97 16 135 236 59 129 126 250 235 249 151 218 39 2 219
4 22 168 73 149 13 67 71 18 140 117 157 34 216 37 227
134 221 171 169 206 189 77 214 80 121 33 100 182 202 1 240
65 83 0 42 31 154 241 123 137 247 155 114 11 86 203 26
62 48 112 159 156 215 111 61 76 176 194 209 222 95 193 212
148 131 20 12 19 50 224 184 213 116 195 150 38 55 128 23
109 125 30 208 254 84 187 138 110 225 85 251 146 244 248 5
68 79 7 188 74 45 29 49 234 103 238 186 54 64 130 46
Table 4 Non- linearity values Bool Fun
f1
f2
f3
f4
f5
f6
f7
f8
Average
S-box1 100 102 98 104 106 104 104 102 102.5
S-box2 114 112 112 112 114 112 114 110 112.5
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after altering a single input bit. This criteria asserts that if a single input bit is changed,
each output bit will change with a chance of 1/2. [31, 41] provide a strong approach
for calculating SAC of S-boxes using a dependency matrix. The S-Box has an average
SAC value of 0.4995, indicating that the recommended S-Box meets the SAC criteria
adequately. Table7 shows the SAC values of the recommended S-box and Table9 com-
pares them to other S-boxes.
4.4 Differential Uniformity
Biham E and Shamir A [40] proposed this analysis. The differential uniformity of a
Boolean function is calculated by requiring that the XOR values of each output have the
same probability as the XOR values of each input. The following is the mathematical
formula for calculating differential uniformity.
Table8 shows the differential uniformity values of the recommended S-box, and Table9
shows a comparison of the differential probability values of the proposed S-box and many
other existing S-boxes in the literature. Figure2 depicts a graphical comparison between
DU values of the recommended S-box and several existing S-boxes.
(4)
DU =maxΔx≠
0,
Δy(#{x∈X|f(x)
⊕
f(x
⊕
Δx)=Δy})
85
90
95
100
105
110
115
120
Proposed
S-box 2
[14] [15] [16] [18][19][20][21][23][29][27]
Minimum MaximumAverage
Fig. 1 A comparison between the non-linearity values of suggested S-box and various other S-boxes
Table 5 BIC Nonlinearity Values
of S-box2 0 104 108 106 106 98 104 104
104 0 106 106 100 108 104 108
108 106 0 104 104 100 106 100
106 106 104 0 106 100 106 102
106 100 104 106 0 104 104 100
98 108 100 100 104 0 102 102
104 104 106 106 104 102 0 104
104 108 100 102 100 102 104 0
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4.5 Linear approximation probability
The present block cypher’s cryptologist seeks to create enough unpredictability and bit dif-
fusion to protect the data from cryptanalytic efforts. An S-box with a low linear probability
(LP) denotes a mapping that is more nonlinear and resists linear cryptanalysis. This analy-
sis is performed to determine the event’s greatest value of imbalance. Matsui [28] proposed
this approach, and the mathematical formula is as follows,
Table 9illustrates the LP values of the proposed S-boxand compares them to other
S-boxes.
5 Applications ofs‑box inimage encryption
In this phase, we apply the recommended S-box for image encryption scheme using
Advanced Encryption Standard algorithm. MLC specifies a methodology for assessing the
findings of various statistical studies, such as energy, homogeneity, contrast, entropy and cor-
relation. This evaluation determined the validity of S-box for image encryption Figs.3 and 4.
5.1 Contrast
Contrast isthe change in color that sorts an image different from other image within the
similar field of view. The contrast is high means that disorderness in encrypted image
(5)
LP
=maxa1,a2≠0




#{x∈Xx.a
1
=f(x).a
2
}
2n−1
2




Table 6 BIC SAC Values of
S-box2 0 0.5137 0.4688 0.5078 0.5195 0.4961 0.5215 0.4707
0.5137 0 0.4922 0.5117 0.4805 0.4941 0.502 0.5156
0.4688 0.4922 0 0.5078 0.4883 0.498 0.4961 0.5059
0.5078 0.5117 0.5078 0 0.4941 0.4922 0.5234 0.5234
0.5195 0.4805 0.4883 0.4941 0 0.4785 0.5059 0.498
0.4961 0.4941 0.498 0.4922 0.4785 0 0.4805 0.4961
0.5215 0.502 0.4961 0.5234 0.5059 0.4805 0 0.4941
0.4707 0.5156 0.5059 0.5234 0.498 0.4961 0.4941 0
Table 7 SAC values of
recommended S-box 0.5 0.5156 0.4688 0.4688 0.5 0.5312 0.4844 0.5
0.5 0.5312 0.5469 0.4844 0.5 0.5469 0.4531 0.5
0.5312 0.4844 0.4688 0.5156 0.4688 0.4688 0.5625 0.4531
0.5 0.5312 0.4844 0.4844 0.5312 0.5156 0.5 0.4688
0.4844 0.5156 0.5 0.4531 0.4844 0.5 0.5 0.5156
0.4688 0.5156 0.5 0.5156 0.5 0.4688 0.4844 0.4688
0.5 0.5469 0.5 0.4531 0.5 0.4688 0.4688 0.5469
0.5156 0.4844 0.5156 0.4844 0.5625 0.5 0.5 0.5469
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increased. Contrast is related with the amount of confusion which is produced by the s-box
to the plan image. The mathematical form of contrast is,
Here
k,m
denotes the pixels of the image.
Contrast
=
∑|
k−m
|
2p(k,m
)
Table 8 Input / Output XOR
Distribution Table 6866666666666648
6 6 6 6 8 10 6 10 6 8 6 6 6 12 6 6
6466668688666488
6 8 8 6 6 6 6 6 8 6 8 6 8 6 10 6
8 6 10 6 6 6 6 6 6 8 6 6 6 6 8 8
8 8 8 6 6 10 6 6 8 8 6 6 10 12 8 6
8666686866886886
6666666666886886
6 6 6 8 6 6 6 6 6 6 6 6 10 8 6 6
6666668868666866
6 6 6 8 6 6 6 6 8 10 6 6 6 6 4 6
8 6 6 6 6 6 6 10 8 8 8 10 6 6 8 6
6666666664668866
8868866666666686
6866888868868686
6 8 6 6 8 8 6 6 6 6 10 8 6 6 8 0
Table 9 The comparison
between LP, DP, BIL-NL, and
SAC values of S-Boxes
S-boxes LP DP BIC-NL SAC
Recommended
S-box
0.125 0.047 103.79 0.4995
[26] 0.1328 0.039 103.9 0.503
[2] 0.1328 0.031 106.9 0.507
[5] 0.141 0.047 102.3 0.493
[12] 0.137 0.039 102.6 0.497
[42] 0.1406 0.039 103.5 0.498
[18] 0.113 0.031 106.1 0.509
[4] 0.109 0.039 103.9 0.500
[13] 0.139 0.039 103.6 0.501
[22] 0.1328 0.039 104.1 0.501
[17] 0.125 0.039 103.6 0.499
[45] 0.1484 0.047 102.9 0.503
[38] 0.125 0.039 103.5 0.506
[14] 0.141 0.047 102.9 0.499
[24] 0.125 0.047 103.0 0.500
[39] 0.1406 0.039 103.1 0.509
[11] 0.1328 0.055 103.5 0.496
[20] 0.125 0.031 103.0 0.493
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5.2 Correlation
Correlation expands the link among the pixels of the cipher image. This assessment is
divided into three categories.
a. General correlation
b. Diagonal formats
c. Vertical and horizontal
If G, H represents two matrices, the correlation is,
Two distinct images may have similar correlation but dispersal of colors of the pixels
may be totally different as shown in Fig.3.
5.3 Energy
The Grey-level co-occurrence matrix is applied to calculate energy. If energy is nearly
equal to zero then encryption scheme is better. The mathematical form of the energy is
Fig.5,
Correlation
=





mn(Gmn −G)(Hmn −H)

m

n(Gmn −G)
2

m

n(Hmn −H)
2
0
2
4
6
8
10
12
14
16
S-box 2[13][15][17][18][20][21][22][23][24][25][26][28][29]
Fig. 2 A comparison between DU scores of different S-boxes
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5.4 Homogeneity
We put into practice the homogeneity that calculates how closely the scattered components
are together. This is called Grey-tone spatial dependency matrix. The homogeneity close
to zero ensured that encrypted image is good. The GLCM table radiates the frequency of
Grey levels.
The mathematical form of homogeneity is
Here grey level co-occurrence matrices in the GLCM is mentioned by
P(k,l).
5.5 Entropy
Entropy measures the randomness in the picture. Mathematically its denoted as,
where
p(xj)
have the histogram count. Figure6 shows that assessment of lower and higher
entropy. If entropy is close to 8 then encryption quality is better.
Energy
=
∑
P(k,l)
2
Homogeneity
=
∑
kl
p(k,l)
1+|k−l|
Entropy
=−
∑n
j=
1(p
(
xj
)
logbp(xj
))
Fig. 3 Original Lena Image
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Here the Table10 describes the MLC of s-boxes which fulfill all the conditions up to
the mark.
5.6 Complexity analysis
This section explores the computational complexity of the proposed encryption scheme,
which involves dividing the image into Most Significant Bits (MSBs) and Least Signifi-
cant Bits (LSBs). Each pixel in the image is split into two sub-blocks using a constant
time complexity of
O(1).
The initial step requires
O(M×N)
bit operations. Next, the
scheme maps each element of the image onto proposed S-box in constant time since
the image data lies within a fixed range. The preprocessing step requires
O(M×N)
bit
operations to execute. The substitution module is also performed in linear time. As all
Fig. 4 Encrypted Lena Image
Fig. 5 Encrypted image Correlation
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modules of the algorithm operate in linear time, the overall computational complexity
of the scheme is
O(M×N),
which is linear time.
(M×N)
represents the dimension of the
plain image. Comparing the computational time complexities of the proposed encryp-
tion scheme with those of existing algorithms [30], the computational time complexity
for generating the cross-coupled chaotic sequence for one round operation is
O(2×Mx)
,where
Mx
is the maximum value of
M1
and
M2
respectively. The time complexity for the
row-column permutation stage is
O(M1×N1)
, while the computational complexity for
the row-column diffusion operation is
8(M1+N1)
. Therefore, the overall total computa-
tional time complexity of the encryption algorithm [30] is
O(2Mx +9(M1+N1)
which is
almost equal to that of the proposed algorithm.
5.7 Pseudocode ofencoding ofproposed image data
The pseudocode for proposed image encryption scheme is explain in Table10.
6 Analysis andcomparison
In this part, we have applied statistical analysis to proposed encrypted Lena image
Fig. 4. We have computed entropy, contrast, correlation, energy, and homogeneity of
the ciphered image. Table10 shows the comparison of recommended S-box encryption
scheme with other algorithms. This ensures that our highly non-linear S-box is favora-
ble for the image’s encryption reliability (Fig.5).
Table 11 shows that entropy is nearly equal to 8, contrast is higher. Because if
entropy is close to 8 then encryption scheme is good and contrast higher also ensure the
quality of the image encryption scheme. The energy and correlation is close to zero and
homogeneity is also declines. For better encryption quality energy also nearly equal to
zero. From these analysis and comparisons, we can say that proposed S-boxes for image
encryption are good.
Fig. 6 Lower vs higher entropy
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Table 10 Pseudocode of
Proposed Image Encryption
Scheme
1. Start
2. Input Lena Image = Image name. extension
3. Key = []
4.
Output = EncryptedLenaImagedata(.jpg)
5. Compute Numbers from Maclaurin Series
6. Apply Meilin Transformation
7. [m, n] = size(O);
8. Length (m, n)
9. %Sequence generation
10.
fork =1∶Ldo
11.
f(x)
←
S
12.
f(x)
←
xn
13.
f(x)
←
kxn
14. M[g(x);h]
←
[n,
xn+2]
15. End
16. % Binary matrix generation
17.
fork =1∶mdo
18.
forl =1∶ndo
19.
𝛽(k,l)=1
20.
ifSk,l
≥
0
21.
else
22.
𝛽(k,l)=0
23.
FinalS −box =
Abs (S1)
24. % Data conversion
25.
fork =1∶mdo
26.
fork =l∶ndo
27.
ifSk,l
>2
−15
28.
S(k,l)=Sk,l
;
29.
elseSk,l
=2
−15
30.
S(k,l)=Sk,l−1
;
31.
%Dif usionphase
32.
fork =1∶mdo
33.
fork =l∶ndo
34.
S
(k,l)=S
(
M[g(x);h],xn
+2)
35.
endend
36.% Generation of S-box
37.
S1
←
[
n,xn+2
];
38.
S1
←M
[
nx
n+2
,h
]
39.
S2
←
S256[S1]
;
40.% Substitution phase
41.
M
←
S1(S256)
42.
M
←
S1(f(x))
43.
S
←
S2(S1)
44. % Bit-xor operation
45.
I1
←
(f(x)mod257 ⊕S)
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7 Conclusion
In this manuscript, we provided an innovative technique to construct the S-box by
using Maclaurin series of logarithmic function and apply Mellin transformation on it
to erect the initial S-box. After that, specific permutations of
S256
utilized to enhance
the nonlinearity of preliminary S-box and generate the final S-box. The effective-
ness of our suggested S-box is then compared to other famous S-boxes in the litera-
ture. We apply statistical and algebraic tests to evaluate the efficiency of our final
S-box. NL, SAC, DU, LAP and BIC are all part of the algebraic test. Because our
final S-box meet all of the conditions of these analyses, regarded as strong S-box
for secure communication. We use the Majority Logic Criterion in the statistical test
to evaluate the effectiveness of the final S-box in an image encryption application.
In this context, statistical tests such as contrast, energy, homogeneity and entropy
are applied. The output result show that S-box consider more secure and effective
against invaders attacks. In future, we can construct the S-boxes by using other trans-
formations such as Sumudu and Fourier transformations and further those S-boxes
can be utilized in audio, video, and text encryption schemes.
Table 10 (continued) 46.
I2
←
((f(x)mod257 ⊕S)
47.
I3
←(M
[
g(x);h
]
mod257 ⊕S
)
48.
S(i,j)
←
I1⊕I2⊕I3
49.% Reverse conversion
50.
fork =1∶mdo
51.
fork =l∶ndo
52.
if
𝛽
(k,l)
≥
0
53.
S2
←
S(k,l)
54.
else
55.
S2
←
−S(i,j)
56.
end;end;end
57.imagewrite (’encryptedata.jpg’,
S2,F
)
Table 11 Comparison of MLC
S-boxes Entropy Contrast Correlation Energy Homogeneity
Original Lena Image 7.5072 0.4996 0.9175 0.0684 0.7108
Proposed Encrypted Lena Image 7.9872 11.3728 -0.0033 0.0147 0.3742
Skipjack [27] 7.684753 6.814102 0.186839 0.035231 0.486187
APA [15] 7.697393 7.745849 0.235826 0.023841 0.497366
Prime [19] 7.669541 6.477377 0.098544 0.037190 0.48858
AES [9] 7.84018 7.423085 0.088914 0.035476 0.389521
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Data availability The data used to support the findings of this study are available from the corresponding
author upon request.
Declarations
Competing interests The writers affirm that they have no established financial or interpersonal conflicts that
would have seemed to have an impact on the research presented in this study.
Conflict of Interest There is no conflict of interest.
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Authors and Aliations
AbidMahboob1· ImranSiddique2 · MuhammadAsif3· MuhammadNadeem4·
AyshaSaleem4
Abid Mahboob
abid.mahboob@ue.edu.pk
Muhammad Asif
muhammad.asif@math.qau.edu.pk
Muhammad Nadeem
muhammadnadeem4464647@gmail.com
Aysha Saleem
ayeshasaleemch@gmail.com
1 Department ofMathematics, Division ofScience andTechnology, University ofEducation,
Lahore, Pakistan
2 Department ofMathematics, University ofManagement andTechnology, Lahore54770, Pakistan
3 Department ofMathematics, University ofManagement andTechnology, SialkotCampus,
Pakistan
4 Department ofMathematics, University ofEducation, VehariCampus, Pakistan
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1.
2.
3.
4.
5.
6.
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