The document proposes an encryption algorithm that uses random graphs and matrices. It generates a random complete graph with 250 vertices and 31,125 edges to represent a computer network. Random start and end nodes are chosen to depict data transfer. Each node is assigned a multi-dimensional weight. A randomly generated path and the node weights are used to build an encryption matrix, increasing randomness. Testing showed encrypting a 16-bit number took around 5 milliseconds. The algorithm could be improved by increasing graph parameters like path length or number of vertices, providing stronger security at the cost of longer runtime.

1. Encryption
Algorithm
Using Random
Graphs
Ameya Meattle
United World College of South East Asia

2. Abstract
In this work, we propose an encryption and decryption algorithm in a computer
network, using ideas from graph theory and matrices. We attempt to simulate a
method in which data can be transferred efficiently post-encryption and study how
such an algorithm performs. Random complete graphs consisting of 250 vertices
and 31125 edges were created, with randomly chosen start and end nodes to depict
the initial source and final destination for data transferred. The algorithm harnesses
the idea of a random graph and assigns a multi-dimensional weight to each node.
These weights, along with a randomly generated path are used to build an
encryption matrix that increases the randomness in the encryption, making it a
more robust encryption algorithm. An analysis of the average time taken for which a
16-bit number was encoded, using an M1 2020 Macbook Air was just over 5
milliseconds, relatively fast for a secure algorithm. The algorithm could be improved
by having additional parameters, such as increasing the random path length, the
number of vertices or even using 3 x 3 matrices, resulting in more data to be
encrypted and a more secure system, though this may lead to a trade-off of a
longer run time.

3. ● Matrices are a set of
elements that are
represented as R rows and C
columns to form an R × C
array.
● Matrix multiplication is
known and defined on a
square matrix, where R = C
● Matrix multiplication is non-
commutative; therefore, the
sequence of multiplication
matters. We use this
property of matrices to
generate unique encrypting
matrices for our algorithm
● Cybersecurity is the practice of
safeguarding systems against
vulnerabilities, threats and
attacks
● Cryptography is a foundational
pillar of cybersecurity. It is the
art of securing communication
through the manipulation of
data and is a process that
involves hiding or coding
information so that only the
recipient (whom the message
was intended for) can read it
Graphs and Networks Linear Algebra Cybersecurity
and
Cryptography
● A graph is a set of nodes and edges
connected to each other to represent
relationships between vertices
● Networks are practical applications of
graphs, where the edges and nodes
represent real-world relationships
between systems

4. Literature Review
● Cryptography is crucial in network security. Recent advanced cryptographic
algorithms have enhanced the fields of cloud computing, wireless sensor
networks, and on-chip networks, according to Sarkar, Chatterjee, and
Chakraborty’s paper, “Role of cryptography in network security”.
● To evaluate the behaviors of networks, concepts from graph theory have been
used previously. The importance of techniques like graph sampling and the
characterization of network features such as node centrality, densification, and
diameter in understanding network vulnerabilities and shifts in network behavior
over time have laid the foundations for using graph theory in the field of
cybersecurity, in Namayanja and Janeja’s paper, “Characterization of evolving
networks for cybersecurity”.
● Various key elements in network security and graph theory can be used to design
public key encryption schemes based on matrices generated from a graph or
graphical codes, according to Sen Sarma’s paper “Application of graphs in
security”. Such schemes can be used to understand and prevent attacks.

5. ● In 2016, Namayanja and
Janeja explored the use of
analytics to model and
evaluate computer network
behavior using graph theory
concepts, focusing on
potential cyber threats, in the
paper, ”Characterization of
Evolving Networks for
Cybersecurity”.
● They highlight the
significance of techniques
like graph sampling and
characterization of network
features for understanding
network vulnerabilities.
● The paper, ”Network Security
and Types of Attacks in
Network” tackles security
issues in mobile ad-hoc
networks (MANETs) due to
node independence and
malicious activities.
● Pawar and Anuradha introduce
network security elements like
confidentiality, integrity, and
availability, classify major
attacks like spoofing, and
emphasize securing individual
computers and network
communication channels.
Cryptography
Graph Theory
Networks Network Security
● Sarkar, Chatterjee, and Chakraborty
talked about the,”Role of Cryptography in
Network Security” in 2020, exploring the
role of cryptography in network security,
focusing on encryption and decryption
techniques.
● It categorizes techniques into symmetric-
key, asymmetric-key, and authentication,
and discusses recent advanced
cryptographic algorithms for cloud
computing, wireless sensor networks, and
on-chip networks.
● The publication provides an
understanding of secure network
communication and the importance of
cryptography in maintaining data
confidentiality.

6. ● The research paper, ”A Survey
on Cryptography Algorithms”,
published in 2018, provides a
comparative study of various
cryptography encryption and
decryption algorithms, such as
AES, DES and RSA.
● Abood and Guirguis explore
these systems by analyzing
the benefits and drawbacks of
factors such as security,
effectiveness, and time.
● We find out that symmetric
algorithms are faster than
asymmetric ones, and AES is
the most reliable algorithm in
terms of speed, decoding,
complexity, flexibility, and
more.
● The paper, ”Network Security
and Types of Attacks in
Network” tackles security
issues in mobile ad-hoc
networks (MANETs) due to
node independence and
malicious activities.
● Pawar and Anuradha
introduce network security
elements like confidentiality,
integrity, and availability,
classify major attacks like
spoofing, and emphasize
securing individual computers
and network communication
channels.
Graphs in Security Cryptography Algorithms Network Security
● In the 2019 research paper,
”Application of Graphs in
Security’, Sensarma and Sarma
explore the concepts of
cryptography and graph theory to
design and propose two public key
encryption schemes:
1. Focuses on matrices generated
from a graph
2. Focuses on properties of
graphical codes.

7. ● The paper by Hogan, Johnson,
and Halappanavar, ”Graph
coarsening for pathfinding in
cybersecurity graphs”,
published in 2013 employs
graph coarsening to detect
hackers using “Pass- the-
Hash Attacks” and assess
network risk by repeatedly
contracting edges to identify
high-risk paths.
● The 2017 paper discussing
the ‘Advanced Encryption
Standard Algorithm To
Encrypt and Decrypt Data’,
Abdullah explains the AES
algorithm as a symmetric
block cipher algorithm and
details the algorithm’s
process of transforming data
through a series of rounds.
● Involves applying
substitutions, permutations,
and bitwise operations to
protect sensitive information.
Graph Theory
Encryption Algorithm
Method for Path Finding in
Cybersecurity Graphs
Encryption and
Decryption of Data
● In the paper written in 2014,
”Encryption Algorithm Using
Graph Theory”, Al Eitawi proposes
a new symmetric encryption
algorithm based on graph theory
properties such as cycle graphs
and complete graphs.
● The algorithm is based on the
foundation of data being
represented as vertices, creating a
cycle graph, with weighted edges
according to an encoding table.
After that, it builds a complete
graph, determines the minimum
spanning tree, and carries out
matrix operations for encryption
and decryption.

8. We define the following variables:
V = Total number of vertices,
L = Path length
How do we know that 1.28 × 1019 paths in a network is secure enough? A 9-character ASCII passcode has
1289 possible combinations. According to the Cyber Security Agency of Singapore, a password of this
nature would take approximately 3 years to break into the system. Since our network exceeds the
number of possible combinations in a 9-character password threshold, we know it would take longer
than 3 years, meeting our security criteria.
To check the number of random
paths, we calculate the number of
permutations, considering the order
of selection and without repetition.
This is done by subtracting 2 from the
total vertices and the path length
since the starting and ending nodes
are already randomly assigned, and
we want to calculate the number of
ways to arrange the remaining
vertices along the path.
Number of Random Paths

9. Procedure: Creating
Random Graph
1. Initially, a network is created using
a complete graph of 250 nodes,
and 31125 edges between them.
2. Next, a random path is created
between two randomly chosen
nodes, with a fixed path length of
10 in the graph.
Fig 1. An example graph with 30 nodes and 100 edges

10. Procedure: Product Matrix
1. 250 2 × 2 predefined matrices, A1, A2, A3 … A249, A250 are put into a list. 2 × 2
matrices are chosen due to their enhanced comprehension efficiency through
the organization of data into categories, therefore being computationally less
intensive while providing an optimal level of security.
2. Based on the random path taken, a product matrix is created by choosing N
numbers of Ai from this list where i is 1 ≤ i ≤ 250, where N represents the number
of elements from the list and the number of vertices in the random path.
3. For each i that represents a node in the random path, starting from the first
node to the Nth node, we multiply the Ai in sequence to obtain the product
matrix. Together, the matrices multiplied to obtain the product matrix are
represented as the sub-keys to the encryption key.

11. Procedure: Encryption and Decryption
1. The user defines a input data to be a 2x2 matrix.
2. To encrypt this data, we multiply the product matrix and the input data. This
encrypted 2x2 matrix is obfuscated and original data cannot be obtained
without the decryption key.
3. The decryption algorithm takes place by multiplying the inverse of the product
matrix by the encrypted data to get back the original input data.
4. To verify that the algorithm works, we subtracted the decrypted data from the
original input data to see if the resulting matrix is a zero matrix. If this works,
the algorithm is a success.

12. Flow Chart

13. Results (Sample Output 1)
● In this sample case, the input data matrix was [-4 5] [6 5].
● A path was randomly generated, with the start and end nodes being
14 and 83 respectively. In the path, the 14th, 47th, 94th, 36th, 25th, 15th
and 83rd vertices were chosen, therefore the corresponding
predefined matrices were multiplied together in that order.
● This final product matrix is multiplied with the input data, to obtain
the encrypted data. Finally, we multiply the inverse of the final
product matrix by encrypted data to return the decrypted data.

14. Results (Case of Final Product Matrix = 0)
● In this output, the inputted data was also kept as [-4 5] [6 5].
● A path was randomly generated, with the start and end nodes being 44 and
18 respectively. In the path, the 44th, 61st, 12th, 51st, 3rd, 69th and 18th vertices
were chosen, therefore the corresponding predefined matrices were
multiplied together in that order.
● Since the determinant of the product matrix, add a randomly predefined
matrix (in this case [1 6] [4 5]).
● This final product matrix is multiplied with the original data, to obtain the
encrypted data. Finally, we multiplying the inverse of the final product matrix
by encrypted data to return the decrypted data.

15. Data Analysis: Performance
An analysis of the average time
taken for the encryption and
decryption algorithm to execute for I
iterations was carried out by
repeating the 50 iteration cycle 10
times, in order to obtain a more
precise result.
The mean time taken is
approximately 5.43 milliseconds,
which is relatively fast for a secure
encryption and decryption
algorithm. In these trials, a 16 bit
number was encoded, using an M1
2020 Macbook Air.

16. Data Analysis:
Error and Special Cases
If during the process of obtaining the product matrix, the product of two of the predefined
matrices results in a zero matrix, then the final product matrix would be a zero matrix itself,
meaning that the encrypted data would also be a zero matrix. For such a matrix, the
inverse would not exist as it’s determinant is 0 and so, the encrypted data cannot be
decrypted.Therefore, if the product matrix is a 0 matrix, we add a random predefined 2x2
matrix to the product matrix, so that it can encrypt and decrypt. If one of the predefined
matrices is a zero matrix and the multiplication of matrices in the random path end up as
zero matrix, the random matrix chosen for addition excludes the zero matrix from the set of
total predefined matrices, so that the zero matrix is not added.

17. Summary and Conclusion
● In this paper, we explored the idea of using random graphs to create an effective
encryption and decryption algorithm, addressing all possible input scenarios.
● The model leverages linear algebra concepts, specifically matrices, to represent
data, bridging theoretical concepts and practical applications in encryption and
decryption.
● The success of the algorithm is proved by an efficient execution time, of just over 5
milliseconds.
● As technology advances and becomes more widely and rapidly utilized, the number
of security and data breaches has been increasing year upon year, hence the
increasing importance of cybersecurity.
● Therefore, an algorithm of this nature enhances data security and serves as a
foundation for advancement in newer encryption algorithms in the future.

18. Summary and Conclusion:
Improvements
● The model could be improved by including additional graph theory parameters such
as the following;
○ increasing the random path length
○ increasing the total number of vertices in the network, resulting in a more
secure system, though this may lead to a trade-off of a longer run time.
● Using 3 x 3 matrices would allow for more data to be encrypted in each run of the
algorithm.
● Exploring whether such an algorithm can have successful implementation of
securely encrypting big data.

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