Encryption and Decryption Algorithms in Symmetric Key Cryptography Using Graph Theory

Abstract

The present-day society depends heavily on digital technology where it is used in many applications such as banking and e-commerce transactions, computer passwords, etc. Therefore, it is important to protect information when storing and sharing them. Cryptography is the study of secret writing which applies complex math rules to convert the original message into an incomprehensible form. Graph theory is applied in the field of cryptography as graphs can be simply converted into matrices There are two approaches of cryptography; symmetric cryptography and asymmetric cryptography. This paper proposes a new connection between graph theory and symmetric cryptography to protect the information from the unauthorized parties. This proposed methodology uses a matrix as the secret key which adds more security to the cryptosystem. It converts the plaintext into several graphs and represents these graphs in their matrix form. Also, this generates several ciphertexts. The size of the resulting ciphertexts are larger than the plaintext size.

Full text

PSYCHOLOGY AND EDUCATION (2021) 58(1): 3420-3427 ISSN: 00333077
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Encryption and Decryption Algorithms in Symmetric Key Cryptography
Using Graph Theory
P.A.S.D.Perera
Department of Mathematics, University of Kelaniya, Kelaniya, Sri Lanka. dhanalishalini@gmail.com
G.S.Wijesiri
Department of Mathematics, University of Kelaniya, Kelaniya, Sri Lanka. sujeew@kln.ac.lk
ABSTRACT
The present-day society depends heavily on digital technology where it is used in many applications such as banking and e-commerce
transactions, computer passwords, etc. Therefore, it is important to protect information when storing and sharing them. Cryptography
is the study of secret writing which applies complex math rules to convert the original message into an incomprehensible form. Graph
theory is applied in the field of cryptography as graphs can be simply converted into matrices There are two approaches of
cryptography; symmetric cryptography and asymmetric cryptography. This paper proposes a new connection between graph theory
and symmetric cryptography to protect the information from the unauthorized parties. This proposed methodology uses a matrix as the
secret key which adds more security to the cryptosystem. It converts the plaintext into several graphs and represents these graphs in
their matrix form. Also, this generates several ciphertexts. The size of the resulting ciphertexts are larger than the plaintext size.
Keywords:
Decryption, Encryption, Graph Theory, Symmetric Key Cryptography
Article Received: 18 October 2020, Revised: 3 November 2020, Accepted: 24 December 2020
I. Introduction
The field of cryptology is divided into two parts
namely, cryptography and cryptanalysis [5].
Cryptography is the art of converting the original
message into an incomprehensible form, with the
intention of hiding the meaning, so that only the
parties who are intended can read and process the
information. Cryptanalysis is the science of
breaking cryptosystems.
There are several components in a cryptosystem.
Sender and receiver are the one who sends the
message and the one who receives the message
respectively. Attacker is the one who tries to get the
message in an unauthorized way. The original
message sent by the sender is known as plaintext
and the secret message or the encrypted message is
known as ciphertext. The process of converting the
plaintext into ciphertext is called encryption while
the reverse process is called decryption.
Symmetric key and asymmetric key cryptography
are two main branches of cryptography. In
asymmetric key cryptography, the sender encrypts
the message using a key known as public key and
sends the encrypted message to the receiver. The
receiver then decrypts the ciphertext by using
another key called private key. Yet in symmetric
key cryptography, both the sender and the receiver
use the same key to encrypt as well as to decrypt the
message.
In [2], a new way of applying graph theory in
cryptography is discussed. The original message is
encrypted into a Euler Graph. This graph is then
converted into several matrices and sent to the
receiver. In [4], a modified version of Affine cipher
is proposed. In this algorithm, each character of the
plaintext is converted into a numeric value and
these numeric values are plotted into a graph. The
graph is then sent to the receiver. In [1,3], a new

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encryption algorithm is proposed using graph theory
where the plaintext is divided into several blocks.
These blocks are then converted into graphs and
their graph representations are sent to the receiver.
This paper proposes a new algorithm to encrypt and
decrypt the message using graph theory in
symmetric key cryptography.
The rest of the paper is organized as follows. Some
mathematical preliminaries that are used in the
proposed algorithm are introduced in section II. The
proposed methodology is discussed in section III.
Finally, in section IV, conclusions are stated.
II. Theoretical Preliminaries
2.1 Graph –
A graph G consists of a set of objects V={v1, v2, v3,
…} called vertices and another set E= {e1, e2, e3, …}
called edges. Usually, a graph is denoted as G= (V,
E)[7].
2.2 Complete Graph –
A graph G is said to be complete if every vertex in G
is connected with every other vertex. A complete
graph with n vertices is denoted by Kn. Then Kn has
n(n-1)/2 edges[6].
2.3 Weighted Graph –
A weighted graph is a graph in which each edge is
given a numerical weight[6].
2.4 Matrix Representation –
Adjacency Matrix –
The adjacency matrix of the graph G = (V, E) is a
n×n matrix D = (dij), where n is the number of
vertices in G, V ={v1,v2,v3,…} , E = {e1, e2, e3,…}.
dij is 1 if there is an edge between vi and vj and dij is
0 if there is no edge between vi and vj[7].
III. Proposed Methodology Table -1 Encoding Table
A
B
C
D
E
F
G
H
I
J
K
L
M
0
1
2
3
4
5
6
7
8
9
10
11
12
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
13
14
15
16
17
18
19
20
21
22
23
24
25
3.1 Steps of Encryption –
1) Select a key (K) which is a square invertible
matrix.
2) Define the values for and satisfying the
conditions; and
where m=26.
3) Convert each character of the plaintext into
a numeric value , using the encoding
table(Table -1).
4) Obtain the letters , corresponding to the
value , using the encoding
table.
5) Obtain the numeric values where is the
ASCII value of .
6) Obtain the letters , corresponding to the
value , using the encoding
table. Here is the difference between
maximum and minimum indexes of the
encoding table.
7) Divide these letters into several blocks of
size n-1 If block size <n-1, add padding
characters to complete the block size.
8) Represent each character of the block as
vertices. Connect each vertex with a
weighted edge.

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9) Make the graph complete by adding extra
edges with random weights greater than m.
10)Identify a special character for each block. If
it is the initial block, then the special
character is the letter corresponding to the
summation of elements in K. If it is not the
initial block, then the special character is the
last character of the previous block. Add
this special character to the beginning of the
block.
11)Construct the corresponding adjacency
matrix (M).
12)C = M×K and send C to the receiver.
3.2 Steps of Decryption –
1) Receive several matrices as ciphertext. (C).
2) Calculate the matrix (M) where M =C×K-1
3) Draw the weighted graph whose adjacency
matrix is M.
4) Identify the special character of the initial
block by using K.
5) Convert all the vertices into letters using the
encoding table and the edge weights.
6) Convert all the characters (ignore the special
characters) into numeric values (s), using
the encoding table.
7) Let p = s+(m×q) where
8) Obtain the letters (y) whose ASCII value is
p+r.
9) Obtain the corresponding numeric value (x)
of y using the encoding table.
10) Obtain the letters corresponding to the value
a-1(x-b) which is the plaintext.
3.3 Example –
Suppose the plaintext is “DECEMBER FIRST “.
Let a=17, b=23 and
. Hence exists.
From the encoding table,
From ,
Table -2 Encryption mechanism
Plaintext
D
E
C
E
M
B
E
R
F
I
R
S
T
x
3
4
2
4
12
1
4
17
5
8
17
18
19
(17x+23) mod 26
22
13
5
13
19
14
13
0
4
3
0
17
8
y
W
N
F
N
T
O
N
A
E
D
A
R
I
z
87
78
70
78
84
79
78
65
69
68
65
82
73
(z-25) mod 26
10
1
19
1
7
2
1
14
18
17
14
5
22
d
K
B
T
B
H
C
B
O
S
R
O
F
W
Table- 2 shows the calculations that need to be done in order to obtain the first set of ciphertext.
Convert each character of the block into vertices and connect these vertices with weighted edges. See Figure 1
below.

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Figure 1: Weighted graphs
The weights of the edges are given by using the
encoding table.
For example, the weight of the edge KB is
calculated as follows:
Weight KB = (index of B) - (index of K)
Make these graphs complete by adding extra edges
with random weights greater than m.
Next, identify the special character for each block.
Special characters of the other blocks are assigned
to be the last character of the previous block. Add
these special characters to the beginning of each
block. Figure 2 shows the complete weighted
graphs with the special characters.
Figure 2: Final graph

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The adjacency matrices of the resulted graphs are multiplied with the key matrix . (Here is the index of
the block.)
To decrypt, multiply the received ciphertexts with
Now draw the graphs whose adjacency matrices are s. See Figure 3 below.

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Figure 3:Final graph with extra edges
Ignore the edges having weights greater than . See figure 4.
Figure 4:Simplified graph
We Know,
i.e.
Next, convert all the other vertices into letters using the encoding table.
For example, can be converted into a letter as follows:
1st vertex≡-6+16=10

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Similarly, obtain the letter representation of the other vertices as well. Ignore the letters from special characters.
Table -3 Decryption Mechanism
Ciphertext
K
B
T
B
H
C
B
O
S
R
O
F
W
A
B
C
s
10
1
19
1
7
2
1
14
18
17
14
5
22
0
1
2
p=s+(26×q)
62
53
45
53
59
54
53
40
44
43
40
57
48
52
53
54
p+25
87
78
70
78
84
79
78
65
69
68
65
82
73
77
78
79
y
W
N
F
N
T
O
N
A
E
D
A
R
I
M
N
O
x
22
13
5
13
19
14
13
0
4
3
0
17
8
12
13
14
17-1(x-23) mod 26
3
4
2
4
12
1
3
17
5
8
17
18
19
7
4
1
plaintext
D
E
C
E
M
B
E
R
F
I
R
S
T
A
B
C
The calculations to be done in order to obtain the
plaintext is in Table-3.
Ignore the last three characters as they are the
padding characters.
Therefore,
3.4 Discussion–
The present society depends on digital tools in
almost every activity in their day-to-day life.
Therefore, protecting the information which are
being shared is the most pivotal task today.
There are many algorithms that have been
developed to safeguard the data from unauthorized
parties. An algorithm is claimed to provide a strong
protection if the ciphertext is held unrevealed even
if the attacker has all the information of the
algorithm. Therefore, the security of an algorithm
depends on several factors, such as how hard it is to
guess the secret key and how hard to guess the
plaintext even though the whole ciphertext is
obtained by the attacker.
The proposed methodology generates a ciphertext
where its size is larger than the plaintext size. Also,
this algorithm uses a matrix of order (n×n) as the
secret key, which makes it hard to guess the secret
key. The ciphertexts are obtained by applying
matrix multiplication. This adds more security to
the plaintext. Further, this methodology generates
several matrices as the ciphertext which makes the
algorithm strong against cryptanalysis as the
probability of receiving the whole ciphertext is low.
IV.CONCLUSION
The symmetric key cryptography is used to encrypt
large amounts of data as it is faster than asymmetric
cryptography. In this approach, sharing the shared
key via a secured channel is the most important
thing. This paper proposes a new methodology to
overcome this difficulty using graph theory. The
proposed methodology first converts the original
message into several graphs. And it further
transforms these graphs into matrices. Finally,
obtains the ciphertext by multiplying these matrices
with the secret key. This paper explains the
proposed encryption and decryption processes
further by providing an example as well.
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