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Received 7 November 2023, accepted 26 November 2023, date of publication 28 November 2023,
date of current version 8 December 2023.
Digital Object Identifier 10.1109/ACCESS.2023.3337443
A Robust S Box Design Using Cyclic Groups and
Image Encryption
RASHAD ALI 1, MUHAMMAD KAMRAN JAMIL 1, AMAL S. ALALI 2,
JAVED ALI1, AND GULRAIZ AFZAL1
1Department of Mathematics, Riphah International University, Lahore 54000, Pakistan
2Department of Mathematical Sciences, College of Science, Princess Nourah Bint Abdulrahman University, Riyadh 11671, Saudi Arabia
Corresponding author: Muhammad Kamran Jamil (m.kamran.sms@gmail.com)
This work was supported by Princess Nourah Bint Abdulrahman University, Riyadh, Saudi Arabia, under Project PNURSP2023R231.
ABSTRACT Modern cryptographic systems use substitution boxes (S-boxes) throughout the encryption
process to enhance the security of the plaintext. The integrity of the communication process is ensured
by these S-boxes, which are vital in converting the ciphertext back into the original plaintext during the
decryption phase. The cryptographic strength of a certain S-box has a substantial impact on the overall
security of a given cipher. As a result, many researchers have used innovative construction techniques to
create robust S-boxes. The method used in this paper is a novel combination of a specific map on the direct
product of cyclic groups of order 16 ×16 and an inversion map of a Galois field with order 256. This
strategy aims to produce dynamic S-boxes. The proposed method can produce a large number of strong
S-boxes by making little changes to the map’s parameters. Four S-boxes were created and their performance
was analyzed using industry standards such as bijectivity, the strict avalanche criterion (SAC), nonlinearity
(NL), the bit independence criterion (BIC), linear probability (LAP), and differential probability (DAP). The
performance of the recommended S-boxes was compared to that of state-of-the-art S-boxes to show their
efficacy. The proposed S-box also exhibits considerable potential as a candidate for modern cryptosystems
aiming at securing multimedia information, as shown by a suggested method for protecting the privacy of
digital images using it. The effectiveness of the encryption method was then assessed using several tests
including contrast, correlation, homogeneity, entropy, energy, Number of Pixel change rate (NPCR), and
Unified Average changing intensity (UACI). We observed the efficacy of the suggested method for image
encryption by comparing our results with different methods.
INDEX TERMS AES, CBC, direct product of cyclic groups, galois field, NPCR, UACI.
I. INTRODUCTION
A major difficulty for cryptographers is to ensure data
security in light of the quick development of communication
technologies. To ensure the security of transmitted data,
a variety of useful encryption methods and approaches
have been created in fascinating literary publications. Block
encryption methods are widely used in modern cryptographic
systems because of their significance in such circum-
stances. Block encryption techniques depend heavily on
the S-box. Numerous cryptographic methods, including the
Advanced Encryption Standard (AES), the International Data
The associate editor coordinating the review of this manuscript and
approving it for publication was Wenming Cao .
Encryption Algorithm (IDEA), and the Data Encryption
Standard (DES) use the S-box. The S-box’s security has an
impact on the overall security of the complete cryptosystem.
To ensure the security of cryptographic systems, the S-box
is thus confirmed to play a significant role as a non-
linear component. After the DES was released in 1977 by
a well-known computer manufacturing company, intensive
research resulted in major improvements to the cryptographic
method. A group of college students eventually managed
to breach the security of DES. After that, it became clear
that a more effective and secure encryption technique needed
to be devised. The AES designed by Daemen and Rijmen
in 2002 is the most extensively used encryption system
nowadays, [1]. The S-box plays an indispensable part in
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2023 The Authors. This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.
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R. Ali et al.: Robust S Box Design Using Cyclic Groups and Image Encryption
determining the reliability of encryption. It is analogous to
undermining the encryption’s security to use a poor S-box
during the encryption procedure. Therefore, before utilizing
an S-box in a cryptosystem, it is imperative to assess its
robustness. The nonlinearity, linear approximation proba-
bility, bit independence criteria, severe avalanche criterion,
and differential approximation probability are some of the
strength assessment techniques used in the evaluation of
S-boxes.
A. LITERATURE REVIEW
There are a lot of methods available in the literature for
building strong S-boxes. The S-box of AES was generated
using an affine map and inversion map of GF(28). The
resulting S-box has very sound cryptographic properties and
it is still used as a standard to compare newly generated
S-boxes using other techniques as developed by researchers.
One of the strong S-boxes is the APA S-box which was
created using the composition of affine, inversion, and affine
maps on a Galois field of order 256. The authors in [49] used
an affine map and inversion map to construct strong S-boxes
using 3 finite fields of order 256. The utilized approach is
simple and very efficient for the generation of strong S-boxes,
but we can generate a limited number of S-boxes using this
approach. Razaq et al. introduced the term of coset graphs
and used symmetric groups to construct robust S-boxes of
good cryptographic properties [41]. The authors further used
S-boxes to encrypt digital images and compared their results
with some available S-boxes. The proposed technique can
be used to generate 462422016 S-boxes. Hussain et al.
used the chaos theory and algebraic theory to design sturdy
S-boxes in [67]. The authors used the chaotic logistic map
and Mobius transformation to design an S-box. They applied
a suitable random permutation of degree 256 to enhance
the cryptographic strength of the designed S-box. Mahboob
et al. [40] proposed a new approach for assembling S-boxes
using a specific quantic fraction transformation. They used
the fractional function over a finite field of order 257 to
generate s-boxes. They used a specific permutation of S256 to
amplify the strength of generated S-boxes. The authors used
these substitution boxes for the image encryption scheme
and the contrast in the encrypted image was noticeable with
a sound score of entropy. Razaq et al. [63] used the coset
graph of the action of the modular group on a finite field to
generate cryptographically robust S-boxes. The authors also
used some specific permutations to enhance the strength of
initial S-boxes.
The main objective of this work is to enhance the security
of the S-box by introducing supplementary measures. There
are a lot of techniques available in literature using symmetric
groups and cyclic groups to construct S-boxes. The usage of
ring and field theory is really common in cryptography for
creating S-boxes. As per our information, there is not a single
S-box developed using direct product of groups. The methods
and strategies discussed in the literature can be categorized
as either being appropriate for producing static S-boxes or
being overly difficult and time-consuming. Static S-boxes
have inherent flaws and restrictions. By crypt-analyzing
the intercepted ciphertext with the help of these S-boxes,
attackers may be able to determine the original plaintext. The
methods described in the literature for developing dynamic
and key-dependent S-boxes are especially complicated and
inefficient. Thus, there is a dire need for a simple and efficient
approach that can generate a large number of S-boxes in a
very short time. In this study, we have presented an efficient
approach to generate a large number of S-boxes using the
composition of a specific map on the direct product of
cyclic groups and inversion of a finite field of order 256.
The S-box created in this way has a high level of security
and closely resembles the ideal values specified by the
conventional S-box. The security strength of the proposed
S-box is thoroughly tested and compared with other S-boxes,
confirming its high level of security.
B. MOTIVATION
The following are the main goals for this study to improve
the strength of S-boxes over algebraic structures and their
applicability in different cryptosystems:
1) There are a few S-boxes in literature based on cyclic
groups with nonlinearity less than 112.
2) There is no usage of the direct product of cyclic
groups in cryptography for designing S-boxes as per
our knowledge.
3) There is a lot of usage of permutations of S256 in
existing schemes as compared to the inversion of the
Galois field.
4) Usage of S-boxes for image encryption to maximize
entropy to 8.
C. CONTRIBUTION
In summary, the important contributions of the proposed
study are:
1) We introduced the concept of Direct Product of Cyclic
groups to generate S-boxes.
2) A new class of bijective functions on the direct product
of cyclic groups is introduced which can be used for the
study of the automorphism group of the direct product
of cyclic groups.
3) We designed 4 S-boxes with each having nonlinearity
112 and we can get 983040 S-boxes of almost optimal
features by this algorithm.
4) Time consumption for generating an S-box by the
proposed algorithm is merely 0.01 sec.
D. STRUCTURE OF THE ARTICLE
The remaining six sections make up the study. In Section II,
we deal with a direct product of cyclic groups and we present
some irreducible polynomials of degree 8. The tables of
multiplicative inverses of the elements of the Galois field
of order 256 were constructed by utilizing the irreducible
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R. Ali et al.: Robust S Box Design Using Cyclic Groups and Image Encryption
polynomials. We presented the proposed algorithm for the
construction of the S-box in section III. The created S-box
was examined using the tests of NL, SAC, BIC, LAP, and
DU in section IV. The application of the created S-box for
picture encryption using AES is covered in Section Vand
the comparison of the outcomes was explored using majority
logic criteria (MLC) and Differential analysis. Section VI
concluded and discussed possible plans.
II. DIRECT PRODUCT OF GROUPS AND GALOIS FIELD
Let G1,G2,G3,...,Gnbe finite groups then the external
direct product of G1,G2,G3,...,Gnis denoted by G1×G2×
G3×. . . ×Gnand is defined as G1×G2×G3×. . . ×Gn=
{(g1,g2,g3,...,gn)|gi∈Gi; ∀i=1,2,3,...,n}where
the operation is component wise. Consider two copies of Z16
then Z16 ×Z16 = {(x,y)|x,y∈Z16}is a group of order 256.
Recall that for any irreducible polynomial r(v) of degree
8 the ring
Z2[v]
<r(v)>= {a7t7+a6t6+. . . a1t+
a0|a0,a1,...,a7∈Z2}is a finite field of order 256 denoted
by GF(28), where tis particular root of r(v). Consider the
four polynomials m1(t)=t8+t6+t5+t+1,m2(t)=
t8+t4+t3+t+1,m3(t)=t8+t7+t6+t5+t4+t+1 and
m4(t)=t8+t4+t3+t2+1, then four finite fields of
order 256 are produced. The Tables 1,2,3,4represent the
multiplicative inverses of the elements of GF(28) with respect
to irreducible polynomials m1,m2,m3and m4.
The algorithm for multiplicative inverse is described as
1) m=8;p=2,irrpolydecimal =283
2) GF =gf (0 :(pm−1),m,irrpolydecimal)
3) GFinv =gf (zeros(1,pm),m,irrpolydecimal)
4) for i=2:pmGFinv(i)=inv(GF(i)) end
5) GFint =double(GF.x)
6) GFinvint =double(GFinv.x)
III. CONSTRUCTION OF S-BOXES
In this section, we will formulate the proposed algorithm for
the construction of new S-boxes. Define a map T:Z16 ×
Z16 →Z16 ×Z16 by T(x,y)=(ay +c( mod 16),bx +d(
mod 16)) for all (x,y)∈Z16 ×Z16, where a,b∈U(16) and
c,d∈Z16. The map Tis an Automorphism of Z16 ×Z16 if
and only if c=0=dbut it does not produce robust S-boxes
as compared to non-zero values of cand d. So, we used non-
zero values of cand dto design S-boxes. For non-zero cand
dthe map T is just a bijection and not a homomorphism.
We choose a=15,b=15,c=7,d=11 for calcula-
tions, then the map T(x,y)=(15y+7( mod 16),15x+11(
mod 16)) is used to generate 256 ordered pairs. We will
use the composition of outputs of Tand inversion map of
Galois field of order 256. Before applying inversion map we
convert x,yin binary and concatenate bits to form 8 bits.
Convert this 8 bit to a decimal and finally we get the result in
{0,1,2, . . . 255}. Let fibe the inversion map of Galois field
of order 256 corresponding to irreducible polynomial mi
fi(t)=(0,if t=0
t−1,if t= 0
FIGURE 1. Nonlinearity Comparison of generated S-boxes with some
8×8 S-boxes.
where i=1,2,3,4. Now, we will formulate all 4 S-boxes
according to the following map
Si(u)=(T◦fi)(u),u∈ {0,1,2,3,255}
The algorithm for generating S-boxes is described as
1) Convert each entry of set {0,1,2,3, , , 255}into binary
and separate each of the 4 bits a part. Form decimal
values from these two binary values and get an ordered
pair (x,y), where 0 ≤x≤15,0≤x≤15.
2) Apply the map Ton these ordered pairs and reverse the
process described in 1st step.
3) Compose the result of step 2 with inverse map of Galois
field to obtain final S-box.
Both compositions (T◦f) and (f◦T) can be used to generate
robust S-boxes of optimal features. In this article, we are only
interested in (T◦f).
IV. ANALYSIS OF PROPOSED S-BOXES
In this section, we evaluate the capability of the provided
algebraic method to generate reliable 8 ×8 S-boxes.
All generated S-boxes are balanced and bijective. The
effectiveness of S-boxes is assessed by accepted testing
standards, such as nonlinearity, strict avalanche criterion,
output bit independence, differential uniformity, and linear
approximation probability. The efficiency of the evaluations
was then assessed by contrasting the results with those of
widely employed S-boxes.
A. NONLINEARITY (NL)
The degree to which an S-box deviates from a linear
relationship between its input and output bits in terms of
magnitude is referred to as its nonlinearity. In simple terms,
it should not be possible to predict the output bits of an S-box
accurately by using a basic linear combination of its input
bits. When considering 8 ×8 S-boxes, the AES S-box (112)
is still thought to be the best option for generating the highest
level of nonlinearity.
B. STRICT AVALANCHE CRITERIA (SAC)
The degree to which a small change in the input bits of
an S-box causes appreciable changes in the output bits
is determined by a feature known as the strict avalanche
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TABLE 1. Multiplicative inverses of GF (28) w.r.t m1.
TABLE 2. Multiplicative inverses of GF (28) w.r.t m2.
TABLE 3. Multiplicative inverses of GF (28) w.r.t m3.
requirements for an S-box. It demands that no matter how
many input bits are left, on average, for any single-bit input
difference, exactly half of the output bits must change. This
characteristic guarantees a high degree of dispersion and
makes it challenging to infer the input from the output even
when a little change in the input to the S-box results in a
complete transformation in the output. The strict avalanche
criteria is a crucial component for symmetric encryption
algorithms to provide robust cryptographic features.
C. BIT INDEPENDENCE CRITERIA (BIC)
A set of requirements called the ‘‘bit independence criteria
for an S-box’’ is used to assess the reliability and security of
an S-box. It assesses the degree of statistical independence
between the input and output bits of an S-box. By assessing
the correlation between input and output bits for various
input differentials (the difference between input pairs), the
bit independence criteria evaluate the behavior of an S-box.
The output bits of a good S-box should be statistically
uncorrelated with the input bits, demonstrating a high degree
of bit independence.
D. LINEAR APPROXIMATION PROBABILITY (LAP)
The linearity of an S-box can be evaluated using a metric
called linear approximation probability. It measures the prob-
ability of a linear relationship between the input and output of
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R. Ali et al.: Robust S Box Design Using Cyclic Groups and Image Encryption
TABLE 4. Multiplicative inverses of GF (28) w.r.t m4.
TABLE 5. Working for S-box S1.
TABLE 6. S-box S1.
TABLE 7. S-box S2.
an S-box. Calculating the likelihood of a linear approximation
involves comparing the number of input-output pairs that
meet a given linear equation to the total number of possible
input-output pairs. For the S-box to have strong cryptographic
qualities, there should be a lower linear approximation
probability, which denotes a higher amount of nonlinearity.
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TABLE 8. S3.
TABLE 9. S4.
TABLE 10. Comparison of SAC of generated S-boxes with some 8 ×8
S-boxes.
FIGURE 2. BIC-Nonlinearity Comparison of generated S-boxes with some
8×8 S-boxes.
TABLE 11. Comparison of BIC-SAC of generated S-boxes with some 8 ×8
S-boxes.
E. DIFFERENTIAL APPROXIMATION PROBABILITY (DAP)
The possibility of a particular input difference resulting in a
particular output difference is quantified by the differential
approximation probability of an S-box. Typically, differential
cryptanalysis or other cryptanalysis methods are used to
FIGURE 3. Comparison of Probability of Linear Approximation of
generated S-boxes with some 8 ×8 S-boxes.
calculate the differential approximation probability. It entails
examining the input-output differences for various inputs
to analyze the behavior of the S-box. A lower probability
suggests more resistance to differential attacks because it
becomes less likely that an attacker will make use of an
S-box’s differential properties.
F. FIXED POINTS
One of the design goals of an S-box is to ensure that it
does not have fixed points, which means that no input value
maps to itself under the S-box transformation. A point x∈
{0,1,2, , , 255}is called a fixed point of an S-box if S(x)=x.
An S-box is considered good if it has no fixed points.
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FIGURE 4. Comparison of Probability of Differential Approximation of
generated S-boxes with some 8 ×8 S-boxes.
TABLE 12. Fixed point analysis of S-boxes.
FIGURE 5. Flowchart of Image Encryption Scheme.
V. IMAGE ENCRYPTION SCHEME
To determine the applicability of the S-box for image security
applications, the suggested S-box presented in Table 7is
employed to encrypt digital photographs. We used different
criteria to measure the strength of the image encryption
scheme and generated an S-box. We employed Barbara,
Baboon, Cameraman, and Pepper of size 512 ×512 as test
images. The key space of the algorithm is 2256, which is a
quite large number and shows the strength of the scheme. The
flowchart of the proposed scheme is depicted in Figure 5.
A. HISTOGRAM ANALYSIS
The distribution of pixel intensities in an image can be exam-
ined using the histogram analysis approach. A consistent and
balanced distribution of the elements in the encrypted data is
ideal for a strong cryptographic technique.
B. ENTROPY
Entropy is a measure used to express the degree of disorder
or randomness in an image’s pixel values. When the entropy
value of scrambled illustrations is close to 8, it indicates
that the pixel values are distributed as uniformly as possible
throughout the image. As a result, it becomes challenging
to anticipate the original image based on the altered or
scrambled version.
Entropy = − X
j
p(vj)log2p(vj).
C. CONTRAST
The difference in brightness and color between the image’s
light and dark portions is referred to as contrast. Any visual
patterns, structures, or statistical dependencies found in the
original image should be disrupted by a powerful encryption
system when it comes to protecting images. Our scheme can
successfully mask the original content and make it difficult
for attackers to decipher important information by achieving
a high level of contrast in the encrypted image. It would be
challenging to extract details, characteristics, or patterns in a
high-contrast encrypted image.
Contrast =X
iX
j
(i−j)2p(i,j).
D. CORRELATION
The statistical link between various elements of an image,
especially between the pixel values, is referred to as
correlation. High degrees of correlation between adjacent
pixels in the encrypted image is a desirable quality in image
encryption. This gives the image a more random appearance
and aids in hindering the recovery of useful information from
local visual patterns. The original image may be recovered
or security flaws in the encryption method may be exposed
if there is a low correlation between neighboring pixels,
which suggests that they are different or independent. Low
cross-correlation and high auto-correlation qualities are the
goals of the correlation analysis used in picture encryption.
By reducing statistical correlations between pixels and
making it more difficult for an attacker to exploit patterns or
retrieve the original image, the encryption approach improves
security in this way.
ruv =
m
P
i=1
(ui− ¯u)(vi− ¯v)
sm
P
i=1
(ui− ¯u)2sm
P
i=1
(vi− ¯v)2
.
E. ENERGY
Energy is a measure that describes the total contrast or
complexity of the texture in an image. It is computed by
summing up the squared values of all the elements in the
Gray-Level Co-occurrence Matrix (GLCM). An image with
a high energy value has fine features and clear edges, which
creates a striking contrast and texture. On the other hand, if an
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FIGURE 6. Plain and Cipher image of Baboon.
FIGURE 7. Histogram of Plain and Cipher image of Baboon.
FIGURE 8. Plain and Cipher image of Barbara.
image has a low energy value, it tends to exhibit a uniform and
less textured appearance.
Energy =
N
X
u=1
N
X
v=1
GLCM(u,v)2.
F. HOMOGENEITY
In an image encryption scheme, homogeneity refers to the
level of uniformity or similarity within an encrypted image.
Homogeneity is a texture attribute that quantifies the level
of similarity between adjacent pixels’ gray or color tones,
measuring how uniform or consistent they are.
Homogeneity =1
1+PM
u=1PM
v=1
(u−v)2
M2
,
G. NUMBER OF PIXEL CHANGE RATE (NPCR)
This metric quantifies the dissimilarity between two images
by measuring the percentage of pixels that are different.
The NPCR (Number of Pixel Change Rate) metric evaluates
how a single pixel alteration affects the entirety of an
FIGURE 9. Histogram of Plain and Cipher image of Barbara.
FIGURE 10. Plain and Cipher image of Cameraman.
FIGURE 11. Histogram of Plain and Cipher image of Cameraman.
FIGURE 12. Plain and Cipher image of Pepper.
image encrypted using the suggested approach. It measures
the frequency of pixel changes in the encrypted image
corresponding to each pixel change in the original image.
Consider two encrypted images C and D with dimensions M
and N, corresponding to two plain images who has one pixel
change. We can measure NPCR by
NPCR =Pi,jE(i,j)
M×N
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FIGURE 13. Histogram of Plain and Cipher image of Pepper.
TABLE 13. Results of majority logic criterion for Plain and Cipher images
of Barbara, Baboon, cameraman and Pepper.
TABLE 14. NPCR and UACI results.
where
E(i,j)=(0,if C(i,j)=D(i,j)
1,if C(i,j)= D(i,j).
Score of our proposed scheme is 99.6060, which is quite
exceptional with very sound score of entropy.
H. UNIFIED AVERAGE CHANGING INTENSITY (UACI)
When a single pixel in the original image is changed, UACI
evaluates the average intensity difference in the encrypted
image compared to the original image. We can measure UACI
by the formula
UACI =1
M×NX
i,j
|C(i,j)−D(i,j)|
255
whereas C,D,M,Nare defined in NPCR. The UACI for
our proposed scheme is 33.4248 which shows quality of
encryption scheme.
VI. CONCLUSION AND FUTURE STUDY
We have demonstrated the utilization of the direct product
of cyclic groups and the Galois field in this manuscript to
obtain a superior S-box for encrypting images. We used a
specific map and then by composing with inversion map of
the Galois field of order 256, we obtained four new S-boxes.
The proposed scheme can generate 983040 robust S-boxes of
almost optimal features. The proposed approach guarantees
good differential and linear probability, as well as the success
of the SAC, nonlinearity, and BIC. We can observe the
strength of generated S-boxes from comparison tables and
bar charts, so our S-boxes can be used to secure plaintext.
We employed one S-box to encrypt digital images using CBC
mode of AES with a key space of 2256. Tables 13 and 14 show
that our proposed S-box and encryption scheme are better as
compared to other image encryption schemes. In the future,
we aim to construct S-boxes by a combination of some other
groups and different algebraic structures.
ACKNOWLEDGMENT
The authors extend their appreciation to Princess Nourah Bint
Abdulrahman University for funding this research under Re-
searchers Supporting Project number (PNURSP2023R231),
Princess Nourah Bint Abdulrahman University, Riyadh,
Saudi Arabia.
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RASHAD ALI received the Graduate degree from
the Government Graduate College of Science,
Lahore, Pakistan, in 2017. He is currently pur-
suing the M.Phil. degree in mathematics with
Riphah International University, Raiwind Campus,
Lahore, Pakistan. He immerses himself fully in the
enigmatic realms of cryptography. He joined HED,
Punjab, in 2018, as a Lecturer in mathematics,
currently posted with the Government Associate
College, Haveli Lakha, Okara, Punjab, Pakistan.
MUHAMMAD KAMRAN JAMIL received the
B.S. degree in mathematics from the University of
Punjab, in 2009, the M.Phil. degree in mathematics
(chemical graph theory) from the Abdus Salam
School of Mathematical Sciences (ASSMS), Gov-
ernment College University Lahore, in 2013,
under the supervision of Prof. Ioan Tomescu, and
the Ph.D. degree from ASSMS, in 2016. He is
currently the Head and an Associate Professor
with the Department of Mathematics, Riphah
International University, Lahore. He was a postdoctoral researcher position
with United Arab Emirates University, United Arab Emirates. During the
Ph.D. degree, he received the Premature-Ph.D. Quality Research Award.
His research interests include topological indices of graphs, extremal graph
theory, graph labeling, coloring in graphs, and distances in graphs. His
major contribution to research is in chemical graph theory. In this area,
he has published more than 100 research articles in international reputed
journals. His research articles are cited more than 1000 times in scientific
papers with H-index 21. He is a Reviewer to various international prestigious
journals, including the Mathematical Reviews under American Mathematical
Society and IEEE. He delivered various scientific lectures at international
and national forums.
AMAL S. ALALI is working at the Department of Mathematical Sciences,
College of Science, Princess Nourah Bint Abdulrahman University, Riyadh,
Saudi Arabia. She has published many articles in well reputed journals.
Her research interests include algebra, coding theory, cryptography, number
theory, semi group theory, graph theory, and fuzzy logic.
JAVED ALI received the Master of Science
degree in mathematics from the esteemed college.
He is currently pursuing the M.Phil. degree
in mathematics with RICAS, Raiwind Campus,
Lahore, Pakistan. He holds the position of an
Assistant Professor with the prestigious Gov-
ernment Graduate College of Science, Lahore.
He is also teaching with the esteemed college.
He has focused his dedication on the algebraic
cryptography, specializing in the intricate field as
he pursued higher education.
GULRAIZ AFZAL received the B.S. degree
(Hons.) in mathematics from the Government
Graduate College of Science, Lahore, Pakistan.
He is currently pursuing the M.Phil. degree in
mathematics with RIU, Raiwind Campus, Lahore.
He is also working under SED Punjab, Pakistan.
He is also working in the domain of cryptogra-
phy where intricate algorithms and cryptographic
protocols interwine forming an enigmatic and
safeguarded world of secure communication.
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