Synthesis of Index Difference Graph Structures for Cryptographic Implementation

Abstract

Cryptography stands out as a scientific methodology for safeguarding communication against unauthorized access. This article proposes a newly formulated graph termed the Index Difference Graph (IDG). The proposed graph model serves as the secret key in the encryption process. Furthermore, we present a new graph-based algorithm, the Index Difference Modular Cryptographic (IDMC) Algorithm, and analyze it using centipede and path graphs. The goal of this graph-based approach is to increase the encryption rate while maintaining computational efficiency. This research investigates different types of index difference graphs and analyzes the time and space complexity of the algorithm. IDMC exhibits a lower collision probability, thereby enhancing encryption security. When employing a graph that admits an Index Difference Graph structure in the cryptographic algorithm, both the sender and receiver must be aware of the graph’s precise structure, as this strengthens the robustness of the cryptographic key. The application of the index difference centipede graph $P_n\odot 2k_1$ in cryptography, examined through the IDMC algorithm, demonstrates exceptionally high brute-force resistance estimated at approximately $2.6\times {10}^{39}$ for smaller instances with $n\le 7$ and escalating to $6.93\times {10}^{163}$ for larger graphs with $n\ge 20$. This resistance underscores the algorithm’s efficiency and cryptographic resilience.

1. Introduction

Cryptographic security is essential for data integrity and secure communication. Graph theory has recently gained importance in cryptography, where researchers apply various graph-theoretic concepts to security operations. The aim of labeling techniques is to address theoretical challenges, identify structural patterns, and leverage graph properties in real-world applications such as data management, network design, coding theory, etc. J.A. Bondy and U.S.R. Murty [1] present fundamental concepts of of graph theory along with a wide variety of applications. Jingxiang Jin and Zhuojie Tu [2] surveyed antimagic labelings, while Julien Bensmail [3] studied edge labeling, ensuring that adjacent vertices have distinct incident sums. Mirka Miller et al. [4] proposed 1-vertex-magic vertex labeling. Griggs et al. [5] introduced $L_d(2,1)$ labeling, which requires adjacent vertices to differ by at least $2d$ and distance-2 vertices by at least d. Adarsh Kumar Handa et al. [6] investigated local distance antimagic labeling on paths, cycles, wheels, friendship graphs, multipartite graphs, and special caterpillars. Arumugam S. et al. [7] examined key results on the local antimagic chromatic number, a parameter arising from local antimagic labeling. Stefko Miklavic et al. [8] established a sufficient condition for Hamming graphs to be distance magic graphs, while T. Singh et al. [9] determined the distance magic index for trees and complete bipartite graphs. Joseph A. Gallian [10] compiled all published studies on graph labeling into a single comprehensive volume. Syed Ahtsham Ul Haq Bokhar et al. [11] proposed three graph-based encryption schemes using bipartite and Cartesian graphs, offering enhanced security and efficiency, along with formal analyses of complexity and implementation. Motivated by these observations, we introduce the new graph, the Index Difference Graph (IDG). We elucidate the structural framework of graphs that admit such a technique and propose a new graph-based algorithm, the Index Difference Modular Cryptography (IDMC) algorithm. In this framework, the Index Difference graph serves as a powerful secret key within the cryptographic process. Graph labeling methodologies are generally characterized by three fundamental attributes: a set of numbers for assigning vertex labels, a rule for assigning labels to edges, and specific conditions that these labels must satisfy. The topology Index Difference Graph structure incorporates vertex positions; vertices are labeled after being placed according to specific conditions and their edge connectivity. Unlike other graphs, the distinct positioning of the vertices enhances brute-force resistance and strengthens the encryption key.

In this study, we examine graph classes that admit the Index Difference Graph technique with time and space complexity of $O\left(n\right)$, and we incorporate them into the algorithm to improve its efficiency.

1.1. Related Works

Cryptography is the discipline of transmuting confidential information into an enciphered form to guarantee its secure transmission to the intended recipient without the risk of exfiltration. It was basically utilized for war time plans. Classical cryptography goes back over two thousand years. Modern cryptography was established by Shannon in 1949 [12]. Rosen K. H. et al. [13] defined encryption as the process of converting an original message into a coded format, with decryption being its reverse operation. A key is used to transform raw data into a ciphered form during encryption and to reconstruct the plaintext during decryption. Through a provided key, the authorized recipient can open up the hidden message with full ease, but this is not possible for an interceptor. With the provided key, the authorized recipient can easily decrypt and access the hidden message, whereas it remains inaccessible to any interceptor. Stinson D. R. et al. [14] outlined three principal schemes in modern cryptography: symmetric key cryptography, public key cryptography, and hash functions. Priyadarsini P. et al. [15] emphasized the integration of graph theory with cryptography as a significant advancement in cybersecurity research. Cusack and Chapman [16] presented a study on graphical methods for evaluating and challenging cryptographic performance. West D. B. [17] commenced by presenting several fundamental concepts of graph theory as a foundation for subsequent developments. Graphs can be used for designing different encryption algorithms. The interaction between graph theory and cryptography is quite interesting. A. Netto Mertia and M. Sudha [18] introduced a newly formulated Index Difference Graph and explored its structural properties as a foundation for cryptographic systems. Beaula C. [19] proposed a cryptosystem based on the Turan graph, leveraging its dense multipartite structure to enhance robustness. The study introduced a novel approach by decomposing the Turan graph into paths and stars, applying edge labeling for encrypting and decrypting sentences, along with corresponding algorithms. Meenakshi [20] investigated a combinatorial approach to graph networks using efficient domination and total magic labeling. The study introduced a novel method for encrypting and decrypting confidential numbers by employing efficient domination concepts together with labeled graphs. For applications of graph theory in cryptography, refer to [21,22,23]. In [24], Selvakumar and Gupta proposed an innovative algorithm for encryption and decryption using connected graphs. In [25], Yamuna and Karthika describe a unique method of transferring data by using bipartite graph. Hu Liang and Dong [26] proposed a bipartite graph-based propagation approach to address fraud detection in large-scale advertising systems. In recent years, there has been increasing interest in utilizing graphs as a framework for developing novel methodologies across various domains of cryptography (refer [27,28,29,30,31,32,33]). A few more applications of graph theory in coding theory can be referred to in [34,35,36]. Shahnawaz Ahmad et al. [37] proposed the Hybrid Cryptographic Approach (HCA), which was introduced to improve the Key Management System (KMS) in cloud environments. Corthis et al. [38] proposed a fog computing-based framework for secure healthcare IoT, integrating Elliptic Curve Cryptography and Proxy Re-encryption with an Enhanced Salp Swarm Algorithm (ESSA). The hybrid model, combining WOA with SSA, optimizes key parameters to strengthen real-time data sharing and authentication of electronic health records. Hongyu Zhu [39] addressed the computational challenges of the SM2 public key cryptographic algorithm in IoT devices, where large integer operations hinder efficiency. The study proposed an optimized SM2 implementation tailored to enhance performance in IoT contexts. Singh. P et al. [40] explored the use of Generative Adversarial Networks (GANs) in cryptography, modeling encryption and decryption through adversarial neural networks. The study conducted a comparative analysis of four activation functions to evaluate their impact on the efficacy of the cryptographic process. Tsmots et al. [41] presented a neural network-based approach for real-time cryptographic data protection with symmetric keys, tailored for embedded systems. The study emphasized its applicability to UAV onboard communication and evaluated feasibility through FPGA hardware implementation. Zahrah Asri Nur Fauzyah et al. [42] proposed a hybrid image encryption scheme that integrates Quantum Key Distribution (QKD) using the BB84 protocol with seven-dimensional ($7D$) and two-dimensional ($2D$) hyperchaotic systems. The scheme enhances key exchange security, achieves high randomness through hyperchaotic dynamics, and demonstrates strong resistance to statistical and differential attacks, making it suitable for post-quantum secure communications.

1.2. Preliminaries

Graph labeling is a technique that assigns values, labels, or numbers to the vertices, edges, or both, according to specific rules. Each labeling method follows distinct criteria and serves different applications, with the choice of labels depending on the problem under investigation. This technique facilitates the study of symmetry, balance, and other combinatorial aspects of a graph, and the process of studying them is made easier with the aid of the graph-labeling technique. Throughout this manuscript, all graphs are assumed to be simple, finite, connected, and undirected.

Design of the secret key in graph-based index difference modular cryptography

Let G be a graph of order n, and let S be a set of positive integers with $\left|S\right|=n$. Assign each vertex $v\in V\left(G\right)$ a unique positional index i, which is selected uniformly at random from the discrete ordered set S. Then G is said to be an Index Difference Graph if there exists a bijection $\mathrm{\Omega }:V\left(G\right)\to \{0,1,2,\dots ,n\}$ that maps each vertex to an element of the aforementioned codomain, according to its positional index i, such that $\mathrm{\Omega }\left(v_i\right)={\displaystyle \frac{i}{2}}$ when $i\equiv 0\left(mod2\right)$ and ${\displaystyle \frac{i-1}{2}}$ when $i\equiv 1\left(mod2\right)$. The edge labeling exhibits a bounded imbalance, with the edge discrepancy lying within the closed interval $[0,1]$. Specifically, this means that $0\le |{\mathrm{\Delta }}_{L_1}-{\mathrm{\Delta }}_{L_1^c}|\le 1$, where ${\mathrm{\Delta }}_{L_1}$ represents the number of edges labeled with 1, and ${\mathrm{\Delta }}_{L_1^c}$ denotes the number of edges not labeled with 1. If a graph G allows for a specific labeling technique, it is classified as an index difference graph.

Perception of the graph model key structure

The proposed graph model acts as the secret key, aiming to bolster the security of the encryption mechanism. The key space of this encryption technique is determined by the number of possible permutations of the connections in the scrambled graph. In IDMC, the key length is determined by the structural complexity of the graph used as the encryption key. This complexity is influenced by several factors, including the positioning of vertices, the number of vertices, the number of edges, the labeling schemes, and the type and structure of the graph.

In the next section, we have investigated its applicability across different graph structures, such as the wheel graph, centipede graph, double triangular snake graph, bipartite graph and subdivision graph, centipede graph, and path graph. A new graph-based algorithm is implemented on graphs adhering to the index difference graph structure, ensuring that the encryption complexity remains linear.

2. Main Results

The Index Difference Modular Cryptography Algorithm is implemented on graphs adhering to the index difference graph structure, ensuring that both time and space complexities remain linear. As in the graph-based cryptography, time complexity depends on the number of vertices and edges, the type of labeling method, and the graph structure. In the next section, we have identified the graph structures that conform to the criteria of the Index Difference Graph.

Proposition 1.

Let $n\ge 1$ be a positive integer. Then the centipede graph $P_n\odot 2k_1$ is an index difference graph.

Proof. 

The centipede graph $P_n\odot 2k_1$ is a graph constructed by starting with a path graph $P_n$ on n vertices $v_1,v_2,\dots ,v_n$ and then two pendant vertices, $u_i$ and $w_i$, are attached to each vertex $v_i$ in $P_n$. Edges are formally added to each vertex $P_n$ to connect it to $u_i$ and $w_i$, where $u_i{w}_i\notin P_n$.

Thus, the vertex set is $V\left(G\right)=\{v_1,v_2,\dots ,v_n\}\cup \{u_1,u_2,\dots ,u_n\}\cup \{w_1,w_2,\dots ,w_n\}$. And the edge set is $E\left(G\right)=\{(v_i,v_{i+1})/1\le i\le n-1\}\cup \{(v_i,u_i)/1\le i\le n\}\cup \{(v_i,w_i)/1\le i\le n\}$

$$\left|V\right(G\left)\right|=3n\mathrm{and}\left|E\right(G\left)\right|=3n-1.$$

This structure resembles a centipede, with the central spine formed by the path $P_n$ and each vertex having two legs as pendant edges.

Initially, the positional indices are designated to each vertex in the centipede graph $P_n\odot 2k_1$. The vertices should be systematically assigned positions pursuant to the following scheme:

$$\mathrm{Position}\left(\phi \right)=\left\{\begin{array}{cc}3i-2,\hfill & \mathrm{if}\phi =u_i\left(\mathrm{upper}\mathrm{pendent}\mathrm{vertex}\mathrm{of}{v}_i\right)\hfill \\ 3i-1,\hfill & \mathrm{if}\phi =v_i\left(\mathrm{path}\mathrm{vertex}\right)\hfill \\ 3i,\hfill & \mathrm{if}\phi =w_i\left(\mathrm{lower}\mathrm{pendent}\mathrm{vertex}\mathrm{of}{v}_i\right)\hfill \end{array}\right.$$

where, $1\le i\le n$.

Now, define a function $\mathrm{\Omega }:V\left(G\right)\to \{0,1,2,3,\dots ,n\}$ that allocates numerical values to the vertices of $P_n\odot 2k_1$, according to their designated positions, as governed by the index difference graph conditions. Subsequently, the edges labeled with the value 1 in the centipede graph are determined as follows.

The edges $e(u_1,v_1),e(v_1,v_2)$, and $e(v_2,w_2)$ consistently retain the label 1 for all permissible n in $P_n\odot 2k_1$, provided they are included in the graph structure. Beyond this invariant subset, the labeling schema can be encapsulated as follows:

For all $i\in N$,

$\chi \left(e(u_i,v_i)\right)=1$ if $i\equiv 1\left(mod2\right)$

$\chi \left(e(v_i,w_i)\right)=1$ if $i\equiv 0\left(mod2\right)$

$\chi \left(e(v_{2i-1},v_{2i})\right)=1$$\forall i\in N$

In this context, $\chi :E(P_n\odot 2k_1)\to Z$ represents the characteristic labeling function assigning value 1 to the specified edge set in accordance with the parity and indexing structure of the centipede graph. All remaining edges in $E(P_n\odot 2k_1)\setminus E_1$ are assigned values strictly different from 1, where $E_1$ denotes the set of edges assigned the value 1.

Upon a comprehensive examination of the edge label assignments throughout the graph $P_n\odot 2k_1$ for arbitrary n, it can be deduced that the total number of edges labeled with 1 denoted as ${\mathrm{\Delta }}_{L_1}$, asymptotically conforms to $⌊{\displaystyle \frac{3n}{2}}⌋$.

Owing to the fact that the total number of edges is $3n-1$, the number of edges labeled 1, denoted as ${\mathrm{\Delta }}_{L_1^c}$ is given by ${\mathrm{\Delta }}_{L_1^c}=(3n-1)-⌊{\displaystyle \frac{3n}{2}}⌋$.

It is necessary to prove that $|{\mathrm{\Delta }}_{L_1}-{\mathrm{\Delta }}_{L_1^c}|\le 1$

Substituting the expressions

$$\left|{\mathrm{\Delta }}_{L_1}-{\mathrm{\Delta }}_{L_1^c}\right|=\left|⌊{\displaystyle \frac{3n}{2}}⌋-\left((3n-1)-⌊{\displaystyle \frac{3n}{2}}⌋\right)\right|.$$

$$=\left|2⌊{\displaystyle \frac{3n}{2}}⌋-(3n-1)\right|$$

(1)

Since $⌊{\displaystyle \frac{3n}{2}}⌋$ represents the greatest integer less than or equal to ${\displaystyle \frac{3n}{2}}$.

That is,

$${\displaystyle \frac{3n}{2}}-1<⌊{\displaystyle \frac{3n}{2}}⌋\le {\displaystyle \frac{3n}{2}}$$

So,

$$3n-2<2⌊{\displaystyle \frac{3n}{2}}⌋\le 3n$$

Owing to the fact that $(3n-1)$ lies between the two values $3n-2$ and $3n$.

From Equation (1), it must be shown that

$$\left|2⌊{\displaystyle \frac{3n}{2}}⌋-(3n-1)\right|\le 1$$

When n is even. Let $n=2k$, an even number.

Then,

$${\displaystyle \frac{3n}{2}}=3k\left(\mathit{aninteger}\right)\Rightarrow 2⌊{\displaystyle \frac{3n}{2}}⌋=3n.$$

Thus, $\left|2⌊{\displaystyle \frac{3n}{2}}⌋-(3n-1)\right|=1$. So, the difference is exactly 1.

When n is odd. Let $n=2k+1$, where $k\in N$.

Then,

$${\displaystyle \frac{3n}{2}}={\displaystyle \frac{(6k+3)}{2}}=3k+1.5\Rightarrow 2⌊{\displaystyle \frac{3n}{2}}⌋=3n-1.$$

Therefore, $\left|2⌊{\displaystyle \frac{3n}{2}}⌋-(3n-1)\right|=0$. So, the difference is exactly 0.

Thus, in both cases, the difference between $2⌊{\displaystyle \frac{3n}{2}}⌋$ and $(3n-1)$ is at most 1, there by proving that $\left|2⌊{\displaystyle \frac{3n}{2}}⌋-(3n-1)\right|\le 1$.

From the Equation (1) ⇒

$$|{\mathrm{\Delta }}_{L_1}-{\mathrm{\Delta }}_{L_1^c}|=\left|2⌊{\displaystyle \frac{3n}{2}}⌋-(3n-1)\right|\le 1.$$

Hence, the centipede graph $P_n\odot 2k_1$ is an index difference graph. □

3. Foundational Graph Concepts in Modular Cryptographic Design

Cryptography can be used to conceal the content of the message so that it is visible solely to the sender and the intended receiver. Graph-based cryptography is a relatively new area of study, and these methods are constantly being developed, and the specific labeling techniques can offer new avenues for security. The secret key plays a crucial role in the cryptographic framework. This secret key impedes brute-force attacks and renders unauthorized decryption attempts considerably more challenging. We have applied the index difference graph model, integrated with modular arithmetic, to perform the encryption and decryption of plaintext in the cryptographic process. This new process is known as the graph-based Index Difference Modular Cryptographic algorithm.

In the subsequent section, we have discussed the Index Difference Modular Cryptography algorithm, which utilizes both the Centipede graph $P_n\odot 2k_1$ and path graph $P_n$ structures along with the modular algorithm and the formally defined secret key used in the cryptographic process.

4. Graph-Theoretic Construction of an Index Difference Modular Algorithm Using a Centipede Graph ${\mathit{P}}_{\mathit{n}}\odot \mathbf{2}{\mathit{k}}_{\mathbf{1}}$

The structural model of the index difference centipede graph $P_n\odot 2k_1$, which includes vertex positioning, vertex labeling schemes, and edge connectivity, plays a pivotal role in the encryption key structure. This study elucidates the process of efficiently encoding and decoding the plaintext message ‘THE EAGLE HAS LANDED’ by employing the index difference graph technique over a centipede graph topology, integrated with a modular algorithm.

In graph-based cryptography, plaintext letters are transformed into the vertices of a graph, and thus, the plaintext letters become the vertices of the graph.

Mapping Characters to Numbers: Each character in the plaintext is converted into a corresponding integer value using the following scheme: Letters A–Z are mapped to values 1–26, and the space character is mapped to 0. This mapping ensures that all characters fall within the range [0, 26], which will be essential for applying modular arithmetic with a modulus of 27. The plaintext consists of 20 characters, including spaces, so it is essential to choose a centipede graph that includes 20 or more vertices. Accordingly, the graph $P_7\odot 2k_1$, which consists of 21 vertices, is deemed appropriate for this purpose.

Formulation of the Secret Key for Index Difference Modular Cryptography: In the graph-based Index Difference Modular Cryptographic Algorithm, the encryption process entails vertex positioning and labeling, computation of edge values, and edge/vertex values are used to generate the encrypted formula. These serve as the cryptographic key in the proposed algorithm.

Secret Key Generation: Assign the first n plaintext characters, including spaces, to the path vertices $v_1,v_2,\dots ,v_n$ of the centipede graph $P_n\odot 2k_1$. Subsequently, allocate the remaining n characters sequentially to all upper pendant vertices $u_i$. Upon completion of assignments to all $u_i$ vertices, continue the allocation to the lower pendant vertices $w_i$, maintaining the original order. In instances where the total number of characters is insufficient to populate all vertices, the surplus lower pendant vertices shall remain unassigned. Only those vertices to which characters have been explicitly mapped are to be considered in the encoding procedure.

Once the characters of the plaintext are mapped to the vertices, assign a position systematically to each character (vertex) as delineated in Proposition 1. Subsequently, assign values to each vertex according to the rules of the Index Difference Graph, followed by computing the corresponding values for the edges.

Encryption Algorithm: Each plaintext value is encrypted using the formula

[Plaintext − vertex label] mod27

For encryption, a secret key vector of equal length is used. Each element of the plaintext is subtracted by the corresponding secret value (vertex label).

Table 1. Encrypted value of the plaintext.

Plaintext	Plaintext	Secret Keys
Character	Value	Position of the Character	Assigned Vertex Label	Encrypted Value = (Plaintext Value − Vertex Label) mod27
T	20	2	1	19
H	8	5	2	6
E	5	8	4	1
-	0	11	5	22
E	5	14	7	25
A	1	17	8	20
G	7	20	10	24
L	12	1	0	12
E	5	4	2	3
-	0	7	3	24
H	8	10	5	3
A	1	13	6	22
S	19	16	8	11
-	0	19	9	18
L	12	3	1	11
A	1	6	3	25
N	14	9	4	10
D	4	12	6	25
E	5	15	7	25
D	4	18	9	22

presents the integer values of the plaintext, their positions after being placed on the graph, the corresponding vertex labels based on their positions, and the final encrypted values of the plaintext.

Encrypted message: SFAVYTXCXCVKRKYJYYV

Decryption Algorithm: To retrieve the original plaintext, each encrypted value is decrypted using the formula: $Plaintext=(Cipher+secretcode)mod27$, where the secret code refers to the vertex label of the character.

This technique proficiently reverses the encryption process and enables the reconstruction of the original message.

Recovered Plaintext: THE EAGLE HAS LANDED

This study unveils how to efficiently encode and decode the plaintext message ‘THE EAGLE HAS LANDED’ using the graph topology, which admits an index difference graph.

Low Collision Probability in IDMC

In this example, we have placed the plaintext letters starting with the vertices of the path, following the order from $v_1$ to $v_n$. Once all the path vertices are filled, we then assign the characters to the upper pendent vertices, followed by the lower pendent vertices, in the same order. The selection of vertices to place the plaintext characters is also part of the secret key in IDMC. We either shuffle the plaintext characters on a given graph or select a graph containing more vertices than the number of plaintext characters. This increases the strength of the encryption. Even when using two different large keys, the input can produce the same ciphertext only with extremely low probability. When using a complex index-difference graph with the same plaintext, the encryption and decryption consistently yield the original message, while maintaining a minimal collision probability. A lower collision probability occurs when the graph key is large, complex, uniquely structured, and labeled in a highly variable manner, making it extremely unlikely for two different graphs to produce the same encrypted output. By virtue of this, IDMC exhibits a lower collision probability, enhancing the security of the encryption.

Remark 1.

The selection of vertices and their order can also be changed. Shuffling the letters across the graph vertices enhances the cryptographic strength of the secret key, thereby increasing its resistance to unauthorized decryption. In this study, a 21-character plaintext is utilized; the proposed method is inherently scalable and can accommodate any desired number of characters, demonstrating its adaptability for broader cryptographic applications. In the current work, the encryption results are demonstrated for the alphabet A–Z with space to illustrate the feasibility of the proposed method in a simplified setting. Extending the scheme to ASCII/Unicode to expand the key space and improve practical applicability is among our plans for future future work. The key factors in selecting the index difference modular cryptography algorithm were as follows:

• Structural novelty—the use of graph-theoretic properties, particularly the index difference graph, provides a new mathematical key foundation for encryption.
• Low collision probability—a modular encryption technique based on IDG lowers the probability of message collisions.
• Computational efficiency—the algorithm maintains linear time complexity.
• Flexibility across graph classes—the framework can be applied to various graph structures (e.g., paths, cycles, bipartite graph ($K_{1,n}$), Double Triangular Snake Graph $\left(D{T}_n\right)$, and the subdivision graph $S\left(L_n\right)$), allowing adaptability.
• Cryptographic strength—the inherent complexity of the key structure enhances the algorithm’s robustness against brute-force and structural attacks.

Time Complexity Analysis of Graph-Based Cryptography Using an Index Difference Centipede Graph $P_n\odot 2k_1$

In graph-based cryptography, time complexity depends on the number of vertices and edges, the type of labeling method, and the graph structure. The centipede graph $P_n\odot 2k_1$, consists of $3n$ vertices and $3n-1$ edges. In the proposed method, the positions of the plaintext, as well as the vertex and edge labels, are predetermined. We are counting the number of edges labeled by 1, and the number of edges other than 1 depends on the total number of edges. As a result, the time complexity of the graph-based cryptography algorithm when implemented on an index difference centipede graph $P_n\odot 2k_1$ is $O\left(n\right)$.

Proposition 2.

For every integer $n\ge 2$, the path $P_n$ is an index difference graph.

Proof. 

A path graph $P_n$ with n vertices, where each vertex is connected linearly to the next, has exactly $n-1$ edges. For a path $P_n$, assign indices to all its n vertices. Index the first vertex as the first position, the second vertex as the second position, and continue this pattern sequentially until the last vertex, which is indexed as the $n^{th}$ position.

Now define a function $\mathrm{\Omega }:V\left(G\right)\to \{0,1,2,3,\dots ,n\}$ that assigns labels to the vertices of the graph $P_n$ according to their designated positions. Hence, for each vertex $v_i$, the label is determined by the expression $\mathrm{\Omega }\left(v_i\right)=⌊{\displaystyle \frac{i}{2}}⌋$, where i denotes the index or position of the vertex. As the edges in the graph $P_n$ connect consecutive vertices $v_i$ and $v_{i+1}$, where $i=1,2,3,\dots ,n-1,$ the weight assigned to an edge $e_i=(v_i,v_{i+1})$ is given by $w\left(e_i\right)=\left|\mathrm{\Omega }\left(v_{i+1}\right)-\mathrm{\Omega }\left(v_i\right)\right|$.

• If i is even, then $\mathrm{\Omega }\left(v_{i+1}\right)=\mathrm{\Omega }\left(v_i\right),$ so $w\left(e_i\right)=0$.
• If i is odd, then $\mathrm{\Omega }\left(v_{i+1}\right)=\mathrm{\Omega }\left(v_i\right)+1$, so $w\left(e_i\right)=1$.

Since the edge weights alternate between 0 and 1, the absolute difference between any two consecutive edge weights satisfies $\left|w\left(e_{i+1}\right)-w\left(e_i\right)\right|\le 1$. Hence, the difference between edges labeled 0 and 1 is always, at most, 1.

Therefore, $P_n$ is an index difference graph for all $n\ge 2$. □

Result 1.

There are n vertices and $n-1$ edges in the path graph. As a result, the encryption algorithm’s time complexity is $O\left(n\right)$.

5. Graph-Theoretic Construction of an Index Difference Modular Algorithm Using Path Graph ${\mathit{P}}_{\mathit{n}}$

We now use a path graph that satisfies the index difference condition for secure encoding and decoding in graph-based cryptography.

Consider a path graph $P_n$ if the plaintext contains n characters, including spaces. Character-to-number mapping is applied in the same manner as demonstrated in the preceding example.

In this example, three cryptographic keys are utilized: Secret Keys 1 and 2 dictate the spatial allocation of positional indices to the vertices (characters) and the values of the edges as specified by Proposition 2, and Secret Key 3 is the encrypted formula. $Encryptedformula=[Plaintext-Edgelabel]mod27$. For encryption, a secret key vector of equal length is used. Each element of the plaintext is subtracted by the corresponding secret value (Edge label).

Here, we retain the plaintext value of the final character of the plaintext, unaltered, since it does not have an outgoing edge on the path graph $P_n$, where the last letter is placed.

Then, compute the cipher text values and the encrypted message. Finally, use the formula below to decode the original message: $Decrypted=(Ciphertextvalue+Edgelabel)mod27$.

This study unveiled how to effectively use a path graph to encode and decode a plaintext message.

Observation 1.

The proposed algorithm facilitates secure message transmission regardless of word length by leveraging the structural properties and mathematical characteristics of these graphs.

Proposition 3.

Show that the wheel graph $W_5$ admits an index difference graph.

Proof. 

A wheel graph $W_n$ is created by adding a central vertex (called the ‘hub’) to a cycle graph $C_n$, and then connecting this hub to every vertex on the cycle.

$V\left(W_n\right)=V\left(C_n\right)\cup \left\{u\right\}$ that is, $\left|V\left(W_n\right)\right|=n+1$.

$E\left(W_n\right)=E\left(C_n\right)\cup \{(u,u_i):1\le i\le n\}$. That is, $\left|E\left(W_n\right)\right|=2n$.

The wheel graph $W_5$ includes the five vertices of the cycle $C_5$, namely, $u_1,u_2,u_3,u_4,u_5$ and one central hub vertex, say u. It contains a total of 10 edges: 5 rim edges forming the cycle $C_5$, and 5 spoke edges connecting the central vertex u to each vertex of the cycle. Initially, assign a position to each vertex in the wheel graph. Thereafter, assign values to the vertices pursuant to Definition 1. Based on their positions, the following values are to be assigned to the vertices: $3,1,0,2,1,$ and 2. The vertices should be positioned and labeled in such a way that all the conditions of the index difference graph are satisfied. The vertex labeling is examined under the following cases:

Case(i) Assume that the value assigned to the central vertex is 0.

If the value assigned to the central vertex is zero, then the remaining values $3,1,2,1$, and 2 can be assigned to the other vertices according to their positions.

Now, determine the values of all edges by calculating the absolute differences between the values of the vertices that each edge connects. Among the five spoke edges, two have a value of one.

To prove the theorem, it must be shown that exactly three of the five rim edges are assigned the value one.

There are six distinct circular permutations that are used to assign the values $1,1,2,2,$ and 3 to the five vertices. Among all six permutations, the number of edges labeled one appears two times in three of the six different circular permutations and four times in three of them. This contravenes the requirement that the absolute difference between the number of edges labeled with 1 and those labeled with values other than 1 must not exceed 1. This violates the condition for an index difference graph.

Hence, in this case (i), $W_5$ does not admit an index difference graph.

Case(ii) Assume that the value assigned to the central vertex is three.

If the central vertex has a value of 3, the remaining values $2,1,1,0,$ and 2 can be allocated among the five vertices based on how they are positioned.

In this instance, two of the five spoke edges have magnitudes of precisely one.

To prove the theorem, it must be proven that exactly three of the five rim edges have a magnitude of one. Consequently, the values $1,1,0,2,$ and 2 can be assigned to the five vertices in six distinct ways. The number of edges labeled one appears two times in three of the six different circular permutations and four times in three of them. This breaches the norm that the number of edges with the label 1 and those without the label 1 cannot differ by more than one.

This, in turn, indicates that the criteria for the index difference graph have been violated. As an outcome, in this case (ii), $W_5$ does not accept an index difference graph.

Case(iii) Assume that the value assigned to the central vertex is one.

Let the central vertex be assigned the value one, and let the remaining values from the set $\{3,1,0,2,2\}$ be assigned to the five vertices. So, the total number of spoke edges labeled one is three. There are 12 distinct ways to assign the values $3,1,2,2,$ and 0 to the five vertices. As a result, the total number of edges labeled with a value of 1 is given by $3+x$, where x represents the number of such edges among the rim edges. Accordingly, we define ${\mathrm{\Delta }}_{L_1}=3+x$.

Now, the condition $|{\mathrm{\Delta }}_{L_1}-{\mathrm{\Delta }}_{L_1^c}|=|2x-4|\le 1$ must hold. This equation is satisfied only when $x=2$. Out of the 12 possible arrangements of the vertex values, 5 specific arrangements fulfil this condition.

Therefore, in this scenario (Case iv), the wheel graph $W_5$ admits an index difference graph.

Case(iv) Assume the value assigned to the central vertex is two.

In the current case, the five vertices can be assigned the values $3,1,2,1,$ and 0 in 12 different circular ways. There are three spoke edges in all that are labeled one. Thus, the total number of edges labeled with the value 1 can be expressed as $3+y$, where y denotes the count of such labels among the rim edges. This leads to the expression ${\mathrm{\Delta }}_{L_1}=3+y$.

The condition $|{\mathrm{\Delta }}_{L_1}-{\mathrm{\Delta }}_{L_1^c}|=|2y-4|\le 1$ must be satisfied for an index difference graph technique to hold. This equation is valid only when $y=2$. This condition is met by exactly 6 arrangements out of the 12 total circular permutations of vertex values.

Thus, under this configuration (Case iv), the wheel graph $W_5$ supports an index difference graph.

Hence, based on cases (iii) and (iv), it is conclusively proven that the graph $W_5$ admits an index difference graph. □

Observation 2.

Exhaustive (brute-force) enumeration of labeling is commonly used as a baseline in graph-labeling studies. It involves checking up to $k^n$ assignments (where n is the graph size and k label options) to find a satisfying labeling. In the above proposition, cases (i) and (ii) do not satisfy the criteria of the Index Difference Graph. If these cases are considered in the cryptographic process, they may lead to confusion during encoding and decoding, as vertex values in this graph are assigned based on the positions of the vertices. Cases (iii) and (iv) meet the desired criteria. Employing a structured, symmetric, and balanced Index Difference Graph can improve the robustness of the cryptographic key.

Corollary 1.

The wheel graphs $W_2,W_3,W_4,W_5$ and $W_8$ are index difference graphs.

Proof. 

Initially, assign every vertex of the wheel graphs $W_2,W_3,W_4,W_5$ and $W_8$ a position, as specified in the previous theorem.

Let $\mathrm{\Omega }:V\left(W_n\right)\to \{0,1,2,\dots ,n\}$ be a vertex labeling function that assigns numerical values to the index vertices.

For each $uv\in E\left(W_n\right)$, define the edge difference as $\left|\mathrm{\Omega }\left(u\right)-\mathrm{\Omega }\left(v\right)\right|,$ and define

$$\chi \left(\left|\mathrm{\Omega }\left(u\right)-\mathrm{\Omega }\left(v\right)\right|\right)=\left\{\begin{array}{cc}1,\hfill & \mathrm{i}\mathrm{f}\left|\mathrm{\Omega }\left(u\right)-\mathrm{\Omega }\left(v\right)\right|=1\hfill \\ 0,\hfill & \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\hfill \end{array}\right.$$

Then define ${\mathrm{\Delta }}_{L_1}={\sum }_{uv\in E\left(W_n\right)}\chi \left(\left|\mathrm{\Omega }\left(u\right)-\mathrm{\Omega }\left(v\right)\right|\right)$ and ${\mathrm{\Delta }}_{L_1^c}={\sum }_{uv\in E\left(W_n\right)}\left(1-\chi \left(\left|\mathrm{\Omega }\left(u\right)-\mathrm{\Omega }\left(v\right)\right|\right)\right)$

To prove that a graph shows an index difference, we must show that $\left|{\mathrm{\Delta }}_{L_1}-{\mathrm{\Delta }}_{L_1^c}\right|\le 1.$ Case Verification:

Specifically…

For $W_2:{\mathrm{\Delta }}_{L_1}=2$, ${\mathrm{\Delta }}_{L_1^c}=2$; $W_3:{\mathrm{\Delta }}_{L_1}=3$, ${\mathrm{\Delta }}_{L_1^c}=3$; $W_4:{\mathrm{\Delta }}_{L_1}=4$, ${\mathrm{\Delta }}_{L_1^c}=4$; $W_5:{\mathrm{\Delta }}_{L_1}=5$, ${\mathrm{\Delta }}_{L_1^c}=5$; $W_8:{\mathrm{\Delta }}_{L_1}=8$, ${\mathrm{\Delta }}_{L_1^c}=8$

In each case, using a previously established technique, we have

${\mathrm{\Delta }}_{L_1}={\mathrm{\Delta }}_{L_1^c}={\displaystyle \frac{2n}{2}}=n$ where $n=2,3,4,5,8$

Therefore,

$\left|{\mathrm{\Delta }}_{L_1}-{\mathrm{\Delta }}_{L_1^c}\right|=0$.

The requirement for the index difference graph technique is met, since $\left|{\mathrm{\Delta }}_{L_1}-{\mathrm{\Delta }}_{L_1^c}\right|\le 1$ in all scenarios.

Therefore, the wheel graphs $W_2,W_3,W_4,W_5$ and $W_8$ admit the index difference graph. □

Observation 3.

In graph-based cryptography, time complexity depends on the number of vertices and edges, the type of labeling method, and the graph structure. The wheel graph, being a dense structure, is inherently more complex than sparse graphs like paths or centipedes. To minimize time complexity, the index difference encoding algorithm selects either edge-based or vertex-based processing, depending on the structural characteristics of the graph. When a vertex-based encryption algorithm is applied to a structured index difference wheel graph, the time complexity is $O\left(n\right)$.

Proposition 4.

The bipartite graph $K_{1,n},2\le n\le 9$ is an index difference graph.

Proof. 

Consider the bipartite graph $K_{1,n}$ where there are two sets of vertices. Set A contains a single vertex. Set B contains n vertices. Each vertex in Set B is connected to a single vertex in Set A, forming edges, for a total of n edges. First, allocate positions to each vertex in the graph. Since the graph $K_{1,n}$ has $n+1$ vertices, the possibilities for positions when allocating the indices for its vertices are $1,2,3,\dots ,n+1$. To meet the conditions set forth by the theorem, each vertex must be positioned in a specific manner. The methods for assigning these positions are detailed below.

For the graph $K_{1,n}$, when $2\le n\le 5$, the vertex in set A is assigned to the $n^{th}$ position, while the vertices in set B are randomly distributed among positions $1,2,3,\dots ,n-1,n+1$. When $n=6,9,$ the vertex in A is placed at the ${(n-3)}^{rd}$ position, and the remaining vertices in B are assigned randomly to positions $1,2,3,\dots ,n-4,n-2,n-1,n,n+1$. Similarly, for $n=7,8,$ the vertex in A is located at the 4^th position, with the vertices in B being randomly allocated to positions $1,2,3,5,\dots ,n-4,n-3,n-2,n-1,n+1$.

Next, define a function $\mathrm{\Omega }:V\left(G\right)\to \{0,1,2,3,\dots ,n\}$ that allocates numerical values to the vertices of the graph $K_{1,n}$ for $2\le n\le 9$, pursuant to their assigned positional order, as articulated in the definition. As a result, the vertex in Set A was assigned the value $⌊{\displaystyle \frac{n}{2}}⌋$ when $2\le n\le 5$, and also maintained the value $⌊{\displaystyle \frac{n-3}{2}}⌋$ for $6\le n\le 9$.

Now compute the absolute difference between the values assigned to the edges.

When n is even, for the graph $K_{1,2n},$ with $n\ge 1$, the graph consists of $2n$ edges. Among these, n edges are assigned a value of 1, while n edges are assigned values different from 1.

When n is odd—that is, for the graph $K_{1,2n+1}$ with $n\ge 1$—the graph contains $2n+1$ edges. In this case, $n+1$ edges are assigned a value of 1, whereas the remaining n edges are assigned values other than 1, according to the predefined labeling scheme.

To evaluate the absolute disparity between the number of edges labeled with weight 1 and those assigned a weight other than 1, consider the following cases:

When n is even; that is, $K_{1,2n},n\ge 1$. $\left|{\mathrm{\Delta }}_{L_1}-{\mathrm{\Delta }}_{L_1^c}\right|=|n-n|=0$

When n is odd; that is, $K_{1,2n+1},n\ge 1$. $\left|{\mathrm{\Delta }}_{L_1}-{\mathrm{\Delta }}_{L_1^c}\right|=|n+1-n|=1$

In both scenarios, the magnitude of the difference does not exceed 1. Hence, it can be affirmed that the condition $\left|{\mathrm{\Delta }}_{L_1}-{\mathrm{\Delta }}_{L_1^c}\right|\le 1$ is consistently upheld, thereby satisfying the criterion.

Therefore, it is proven that $K_{1,n}$ for $2\le n\le 9$ is an index difference graph. □

Remark 2.

When employing a graph that admits an Index Difference Graph structure, both the sender and receiver must be aware of the graph’s precise structure, as this enhances the robustness of the cryptographic key.

Proposition 5.

The Double Triangular Snake Graph $\left(D{T}_n\right)$ is an index difference graph for all n.

Proof. 

The Double Triangular Snake Graph $\left(D{T}_n\right)$ is a graph constructed from the path graph $P_{n+1}$, which consists of $n+1$ vertices, denoted as $v_1,v_2,\dots ,v_n,v_{n+1}$, connected in a linear sequence. Each edge of $P_{n+1}$ serves as the base for two parallel triangles, forming a zigzag chain of $2n$ triangles. Every successive pair of triangles shares a common edge, ensuring connectivity throughout the graph. The ancillary vertices constituting the upper and lower triangular substructures are designated as $u_1,u_2,\dots ,u_n$ and $w_1,w_2,\dots ,w_n$ respectively.

Thus, $V\left(D{T}_n\right)=3n+1$ and $\left|E\left(D{T}_n\right)\right|=5n$. This structure ensures that $D{T}_n$ is a connected, planar graph with a well-defined arrangement of triangular subgraphs.

Begin by delineating the positions of all vertices constituting the Double Triangular Snake Graph $D{T}_n$.

Let us define $\chi \left(v_i\right)$ as the position of vertex $v_i$ in graph $D{T}_n$.

$$\chi (v_i)=\left\{\begin{array}{cc}1,\hfill & i=1\hfill \\ 2,\hfill & i=2\hfill \\ 6,\hfill & i=3\hfill \\ \chi \left(v_{i-1}\right)+3,\hfill & i\ge 4\hfill \end{array}\right.$$

Let us define $\psi \left(u_i\right)$ as the position of vertex $u_i$ in graph $D{T}_n$.

$$\psi (u_i)=\left\{\begin{array}{cc}3,\hfill & i=1\hfill \\ 5,\hfill & i=2\hfill \\ \psi \left(u_{i-1}\right)+3,\hfill & i\ge 3\hfill \end{array}\right.$$

Let us define $\vartheta \left(w_i\right)$ as the position of vertex $u_i$ in graph $D{T}_n$.

$$\vartheta (w_i)=\left\{\begin{array}{cc}4,\hfill & i=1\hfill \\ \vartheta \left(w_{i-1}\right)+3,\hfill & i\ge 2\hfill \end{array}\right.$$

Subsequently, define a function $\mathrm{\Omega }:V\left(G\right)\to \{0,1,2,3,\dots ,n\}$ that allocates numerical values to the vertices of $D{T}_n,$ contingent upon their designated spatial positions and the stipulated criteria. The weight of an edge is ascertained by computing the absolute difference between the numerical values allocated to its incident vertices.

For any $n\ge 1$, in the graph $D{T}_n$, the edges that are invariably assigned the value 1 are $e\left(v_1{v}_2\right)=1,$$e\left(v_1{u}_1\right)=1$ and $e\left(v_2{w}_1\right)=1$. From $n=2$ onward, additional edges are assigned value 1, contingent upon the parity of n.

Case 1 When n is even

For even values of n, the graph $D{T}_n$ incorporates two supplementary edges labeled 1 relative to $D{T}_{n-1}$: $e\left(v_n{u}_n\right)=1$ and $e\left(u_n{v}_{n+1}\right)=1$.

Case 2 When n is odd

For odd values of n, the graph $D{T}_n$ encompasses three additional edges consistently labeled 1 when compared to the preceding graph: $e\left(v_n{u}_n\right)=1$, $e\left(w_n{v}_{n+1}\right)=1$, and $e\left(v_n{v}_{n+1}\right)=1$.

Let ${\mathrm{\Delta }}_{L_1}$ denote the number of edges in $D{T}_n$ that are endowed with the label 1, and let ${\mathrm{\Delta }}_{L_1^c}$ denote the number of edges assigned labels distinct from 1. The resultant quantities, derived through the aforementioned procedure, are expressed as follows: ${\mathrm{\Delta }}_{L_1}\left(n\right)=⌈{\displaystyle \frac{5n}{2}}⌉$ and ${\mathrm{\Delta }}_{L_1^c}\left(n\right)=⌊{\displaystyle \frac{5n}{2}}⌋$.

We consider two exhaustive cases based on the nature of ${\displaystyle \frac{5n}{2}:}$

If ${\displaystyle \frac{5n}{2}}$ is an integer, then ${\mathrm{\Delta }}_{L_1}={\mathrm{\Delta }}_{L_1^c}\Rightarrow {\mathrm{\Delta }}_{L_1}-{\mathrm{\Delta }}_{L_1^c}=0$ (since in this case, the ceiling and floor functions coincide).

If ${\displaystyle \frac{5n}{2}}$ is not an integer, then ${\mathrm{\Delta }}_{L_1}={\mathrm{\Delta }}_{L_1^c}+1\Rightarrow {\mathrm{\Delta }}_{L_1}-{\mathrm{\Delta }}_{L_1^c}=1.$

In both cases, the absolute difference satisfies $\left|{\mathrm{\Delta }}_{L_1}-{\mathrm{\Delta }}_{L_1^c}\right|\le 1$.

This proves that for all n, the absolute difference between the number of edges labeled by 1 and those labeled by other than 1 in graph $D{T}_n$ is always at most 1.

Therefore, the Double Triangular Snake Graph $D{T}_n$ is an index difference graph for all n. □

Proposition 6.

For all integers $n>1$, the subdivision graph $S\left(L_n\right)$ of the line graph $L_n$ is an index difference graph.

Proof. 

The subdivision graph $S\left(L_n\right)$ denotes the graph obtained by performing a subdivision operation on $L_n$, wherein a new vertex is interposed at the midpoint of each edge. Consequently, every edge of $L_n$ is transformed into a path of length 2. Let $u_1,u_2,\dots ,u_n$ and $v_1,v_2,\dots ,v_n$ be the vertices of the first (lower) and second (upper) rows of $L_n$, respectively. Let $v_i^{\prime }$, $u_i^{\prime }$ and $w_i$ denote the newly added vertices inserted between the vertex pairs $(v_i,v_{i+1}),(u_i,u_{i+1}),$ and $(u_i,v_i)$, respectively. Then, $S\left(L_n\right)$ is of order $5n-2$ and size $6n-4$.

The position of the vertices of the graph $S\left(L_n\right)$, for $n>1$ is delineated as follows:

$$P(u_i)=\left\{\begin{array}{cc}i,\hfill & i=1\hfill \\ i+5,\hfill & i=2\hfill \\ P\left(u_{i-1}\right)+\chi (i\equiv 1\left(mod2\right)).3+\chi (i\equiv 0\left(mod2\right)).7\hfill & i>2\hfill \end{array}\right.$$

where $\chi (i\equiv 1(mod2\left)\right)$ = 1 if the condition is true; otherwise, it is 0.

     $\chi (i\equiv 0(mod2\left)\right)$ = 1 if the condition is true; otherwise, it is 0.

$$P(v_i)=\left\{\begin{array}{cc}i+2,\hfill & i=1\hfill \\ i+3,\hfill & i=2\hfill \\ P\left(v_{i-1}\right)+\chi (i\equiv 1\left(mod2\right)).7+\chi (i\equiv 0\left(mod2\right)).3\hfill & i>2\hfill \end{array}\right.$$

where $\chi (i\equiv 1(mod2\left)\right)$ = 1 if the condition is true; otherwise, it is 0.

     $\chi (i\equiv 0(mod2\left)\right)$ = 1 if the condition is true; otherwise, it is 0.

$$P(w_i)=\left\{\begin{array}{cc}i+1,\hfill & i=1\hfill \\ i+4,\hfill & i=2\hfill \\ P\left(w_{i-1}\right)+5,\hfill & i\ge 3\hfill \end{array}\right.$$

$$P(u_i^{\prime })=\left\{\begin{array}{cc}i+7,\hfill & i=1\hfill \\ P\left(u_{i-1}^{\prime }\right)+\chi (i\equiv 1\left(mod2\right)).9+\chi (i\equiv 0\left(mod2\right)).1\hfill & i>1\hfill \end{array}\right.$$

where $\chi (i\equiv 1(mod2\left)\right)$ = 1 if the condition is true; otherwise, it is 0.

     $\chi (i\equiv 0(mod2\left)\right)$ = 1 if the condition is true; otherwise, it is 0.

$$P(v_i^{\prime })=\left\{\begin{array}{cc}i+3,\hfill & i=1\hfill \\ P\left(v_{i-1}^{\prime }\right)+\chi (i\equiv 1\left(mod2\right)).9+\chi (i\equiv 0\left(mod2\right)).1\hfill & i>1\hfill \end{array}\right.$$

where, $\chi (i\equiv 1(mod2\left)\right)$ = 1 if the condition is true; otherwise, it is 0.

     $\chi (i\equiv 0(mod2\left)\right)$ = 1 if the condition is true; otherwise, it is 0.

Thereafter, define a function $\mathrm{\Omega }:V\left(G\right)\to \{0,1,2,3,\dots ,n\}$ that allocates numerical values to the vertices of $S\left(L_n\right)$, contingent upon their designated positions and the stipulated criteria.

In light of the preceding observation, it is evident that in the graph $S\left(L_n\right),$ for all $n>1$, the cardinalities of the edge sets labeled 1 and those labeled distinctly from 1 are both identically equal to $3n-2$.

Therefore, $\left|{\mathrm{\Delta }}_{L_1}-{\mathrm{\Delta }}_{L_1^c}\right|\le 1$. Hence, the graph $S\left(L_n\right)$ meets the criteria for an index difference graph.

Thus, the graph $S\left(L_n\right)$ is an index difference graph for all $n>1$. □

Remark 3.

Depending on the length of the plaintext or the desired strength of the secret key, one may use only the ladder line, the lower and upper rows of the line graph, or randomly select vertices from the graph and assign letters to them accordingly.

6. Key Management Strategies

For the reinforced key system, the suggested encryption algorithm makes use of three secret keys, which are quite different from the traditional ones because of the differences in the properties of the new methods. A highly resilient key is used.

7. Algorithmic Complexities of the Index Difference Graph Technique

The complexity of the Index Difference Graph Technique is as follows:

Encryption Complexity: The time required to position the vertices, assign values, and compute the encrypted outputs.

Decryption Complexity: Time to reverse the process (usually simpler once labeling is known).

Computational Complexity Analysis of the Index Difference Modular Cryptography Algorithm

The significant strength of our proposed algorithm is its implementation within a structured graph framework that adheres to the criteria of index difference graph structures. In the proposed method, the positions of the plaintext, as well as the vertex and edge labels, are predetermined. As a result, it takes constant time to fix the vertex position.

In Section 4, we have employed the centipede graph for IDMC. In the graph-based cryptography framework, plaintext characters are mapped to the vertices. During the encryption process, each character is mapped onto the graph. Assigning a character to a vertex requires constant time. Therefore, to assign at most $3n$ characters to the centipede graph, the overall complexity is $O\left(3n\right)\approx O\left(n\right)$. In our proposed algorithm, we have employed the centipede graph, which admits IDG. Consequently, the positions of the vertices are predetermined, as established in Proposition 2. Since assigning a position to a vertex requires constant time, i.e., $O\left(1\right)$, fixing the positions of all characters in a graph of size $3n$ requires at most $O\left(3n\right)\approx O\left(n\right)$ time.

The next step in the encryption process is to fix values to the vertices according to the condition defined in the design of the secret key. The condition is as follows: let $\mathrm{\Omega }:V\left(G\right)\to \{0,1,2,\dots ,n\}$, that maps each vertex to an element of the aforementioned codomain, according to its positional index i, such that $\mathrm{\Omega }\left(v_i\right)={\displaystyle \frac{i}{2}}$ when $i\equiv 0\left(mod2\right)$ and and $\mathrm{\Omega }\left(v_i\right)={\displaystyle \frac{i-1}{2}}$, when $i\equiv 1\left(mod2\right)$. Fixing the values to the vertices using the above formula requires constant time for each vertex. Therefore, for a graph with $3n$ vertices, the overall average time complexity is $O\left(n\right)$. In the next step of the encryption process, the ciphertext value is obtained by subtracting the vertex value from the plaintext value. Alternatively, the edge value can be used instead of the vertex value, depending on the graph employed and the placement of the plaintext. However, in that case, the corresponding decryption formula must be adjusted accordingly. The value of each edge is defined as the difference between its adjacent vertices. For a graph with $3n-1$ edges, computing the values of all edges requires $3n-1$ operations. Hence, the overall average time complexity is $O(3n-1)\approx O\left(n\right)$. In turn, the overall computational complexity of IDMC using the centipede graph is $O\left(n\right)$.

The path graph $P_n$, double triangular snake $D{T}_n$, subdivision of ladder graph $S\left(L_n\right)$, bipartite graph $K_{1,n}$, and wheel graph $W_n$ have vertices of n, $3n+1$, $5n-2$, $n+1$, and $n+1$, respectively, with edges of $n-1$, $5n$, $6n-4$, n and $2n$. Consequently, all of the graphs have an $O\left(n\right)$ time complexity. We are counting the number of edges labeled by 1, and the number of edges other than 1 depends on the total number of edges. As a result, when employing the proposed algorithm, the time complexity for each of these graphs is $O\left(n\right)$.

Concerning space complexity, since all vertices are stored, the overall space complexity across all graphs is $O\left(n\right)$ with respect to the vertices. Additionally, the space complexity for storing the number of edges labelled by 1 and those labelled by other values is constant. Therefore, the final space complexity of the graphs is $O\left(n\right)$. In conclusion, the aforementioned graphs have an $O\left(n\right)$ computational complexity.

As is well known, an algorithm with a complexity of $O\left(n^k\right)$ indicates a slower process that is feasible only for small values of n due to high computational overhead. In contrast, the proposed technique achieves a complexity of $O\left(n\right)$ across the entire graph, making it linearly efficient even for large graphs. This method enables the secure transmission of large messages using a strong key, as it allows characters to be shuffled across the graph’s vertices, followed by a structured encryption process that is particularly reliant on the positions of the vertices.

8. Discussion

In our approach, the index difference graph structure is mathematically formulated, and modular arithmetic is then integrated to define the encryption process. We use a combination of theoretical analysis and empirical testing to evaluate the security and effectiveness of the proposed algorithm.

Performance Analysis

The proposed encryption techniques provide new approaches to improve data security within confidential communications by utilising path graphs as well as particular centipede graphs. The suggested encryption approach will make use of the graphs that satisfy the index difference graph, which will be addressed in this paper. When an index difference graph is used in graph-based cryptography, labeling optimization ensures that the encoding is secure and efficient. The labeling is unique and reversible. The computational cost (time/space) of encryption and decryption is optimized. Here, we analyse the performance of these schemes, while taking into account their affirmative outcomes and limitations:

Affirmative Outcomes of Index Difference Graph-Based Modular Cryptography:

• Enhanced security: The encryption algorithms provide increased security by combining modular algebra with the index difference graph structure, making it difficult for unauthorized parties to decode messages. The proposed cryptographic framework, predicated on IDG, employs a tri-key architecture, whereas conventional systems typically rely solely on a public and private key. This enhanced multi-key paradigm substantially strengthens security.
• Resistance to traditional attacks: The distinct structure and functionality of these nonstandard graph-based encryption algorithms offer a defense mechanism against conventional cryptographic attacks such as frequency analysis and brute-force attacks. IDMC renders brute-force attacks computationally infeasible, as the key-search complexity escalates rapidly with graph size, yielding exceptionally high resistance for larger graphs. This will be demonstrated in the next section.
• Scalability: The technique remains effective even when the communication system is complex or when there is a massive quantity of data. For large plaintext messages, brute-force guessing becomes exceedingly difficult; however, the encryption process remains tractable, as all values are predetermined and the proposed algorithm utilizes graphs that admit IDG.
• Algorithm flexibility: Using different types of graphs makes this algorithm highly flexible and adaptable to various communication scenarios.

Limitations of Index Difference Graph -Based Modular Cryptography

A high level of expertise may be required to implement and comprehend the mathematical key concepts of the proposed encryption techniques, possibly hindering broad acceptance.

9. Brute-Force Resistance Analysis of Index Difference Modular Cryptography

Incorporating practical experiments with real data or simulations would further strengthen the validation of the proposed algorithm against specific attack models, such as brute-force and side-channel attacks. This section is primarily concerned with brute-force attacks.

In IDMC encryption, security arises from the graph structure and its index difference properties. IDMC encryption is robust because it makes brute-force guessing extremely difficult due to complex graph-based relationships. As a result, if an attacker attempts to guess the correct keys, the problem quickly becomes infeasible for large graphs, as the difficulty escalates rapidly with the graph size.

In Section 4, for the encryption process, we used a plaintext of 20 characters and accordingly selected a centipede graph of order 21. According to the proposed IDMC algorithm, 20 plaintext characters are assigned to the 21 vertices of the centipede graph. Therefore, the maximum allocation is achieved by selecting 20 vertices out of 21 to occupy the 20 positions. The number of such selections is given by $\left({\displaystyle \genfrac{}{}{0pt}{}{21}{20}}\right)=21$. Once the 20 vertices are chosen, the 20 positions can be assigned to them in all possible permutations, e.g., $20!$. Hence, the total number of distinct ways to assign the positions is $\left({\displaystyle \genfrac{}{}{0pt}{}{21}{20}}\right)\times 20!=21!$.

Thus, the total number of possible assignments is $21!\approx 5.11\times {10}^{19}$.

In general, when $3n-1$ plaintext characters are assigned to the $3n$ vertices in the centipede graph, the total number of possible assignments is $\left({\displaystyle \genfrac{}{}{0pt}{}{3n}{3n-1}}\right)·(3n-1)!=\left(3n\right)!$. If a centipede graph of order $3n$ is employed to enhance the strength of the secret key for a plaintext of 20 characters, then the total number of possible assignments is given by $\left({\displaystyle \genfrac{}{}{0pt}{}{3n}{20}}\right)\times \left(20\right)!={\displaystyle \frac{\left(3n\right)!}{(3n-20)!}}=\left(3n\right)(3n-1),\dots ,(3n-19)$, is well approximated by ${\left(3n\right)}^{20}$, illustrating the extremely rapid growth as n increases.This dramatic growth in brute-force resistance enhances the cryptographic strength as the order of the centipede graph increases.

In the proposed cryptography process, the next step is to assign values to the vertices (plaintext characters) as explained in the design of the secret key, once their positions are fixed. The number of possible assignments of 20 values in 21 vertices becomes $21!$.

Hence, if the 20 distinct positions $(1--20)$ are assigned to the 21 vertices in the centipede graph, and the vertex values are determined after assigning the positions, then the total number of possibilities is ${(21!)}^2\approx 2.6\times {10}^{39}$.

Considering a centipede graph with $3n$ vertices and $3n-1$ plaintext characters, the total number of possibilities to assign the both the values of the vertex position and values of the vertices is ${\left(\left({\displaystyle \genfrac{}{}{0pt}{}{3n}{3n-1}}\right).(3n-1)!\right)}^2={(\left(3n\right)!)}^2\approx {10}^{2lo{g}_{10}\left(3n\right)!}$.

• For small graphs ($n\le 7$, i.e., 21 vertices), brute-force resistance ≈$2.6\times {10}^{39}$.
• For larger graphs ($n\ge 20$, i.e., 60 vertices), brute-force resistance ≈$6.93\times {10}^{163}$.

Hence, the utilization of sufficiently large graphs can substantially enhance brute-force resistance, even for relatively short plaintexts or small datasets, thereby reinforcing the robustness of the cryptosystem.

10. Weaknesses of the Index Difference Graph Structure

In graph-based IDMC, certain weaknesses may arise depending on the choice of graph structure. For instance, index difference modular cryptography may become weaker, or the computational complexity may no longer remain linear, when using graphs that do not admit index difference properties. Even in cases where the graph satisfies the requirements of an index difference graph, structural weaknesses can still exist. Specifically, graphs with highly regular or predictable structures (such as cycle graphs or wheel graphs) may allow an attacker to more easily infer the mapping or significantly reduce the brute-force search space. Such regularity and predictability represent important vulnerabilities in graph-based IDMC cryptographic systems.

11. Comparative Performance Analysis of Existing Cryptographic Models

It is significant to compare our proposed graph-based encryption methods with other encryption techniques in order to highlight their benefits and distinctive characteristics. Prathipa Murugan et al. [43] proposed a graph-based geometric mean labeling approach to enhance encryption and key management, and compared various graph labeling methods in cryptography.

Table 2. Graph-based cryptography complexity comparison table.

Labeling	Security	Computational	Encryption	Key
Method	Strength	Complexity	Efficiency	Management
Antimagic	High	$O\left(n^3\right)$	Slow	High Complexity
Labeling		(Vertex-Edge		in Large
		Weights		Graphs
Arithmetic	Moderate	$O\left(n^2\right)$	Moderate	Limited
Mean		(Arithmetic		Scalability
Labeling		constraints
Cordial	Moderate	$O\left(n^2\right)$	Moderate	Simple,
Labeling		(Balance		but Not
		constraints)		Cryptographically
				Strong
Prime-	Moderate	$O\left(n^3\right)$	Slow	High Storage
Based		(Multiplicative		Requirement
Labeling		Inversions)
Graph	Moderate	$O\left(n^2\right)$	Moderate	Key
Coloring		(Graph		Distribution
Labeling		Traversal)		Challenges
Edge	High	$O\left(n^3\right)$	Slow	Requires
Magic		(Algebraic		Complex
Labeling		Transformations)		Computation
Graceful	Moderate	$O\left(n^2\right)$	Moderate	Efficient
Labeling		(Vertex Edge		but Limited
		Mapping)
Harmonious	Moderate	$O\left(n^2\right)$	Slow	Limited
Labeling		(Edge sum		Applications
		constraints)
Radio	High	$O\left(n^3\right)$	Moderate	Useful
Mean		(Frequency		for Wireless
Labeling		Constraints)		Security
Geometric	Very	$O\left(V^3\right)$	Fast	Scalable and
Mean	High	(Matrix		Efficient
Labeling		Operations)
Index	Exceptionally	$O\left(n\right)$	Fast	Robust
Difference	High	(Vertex Positions, as		Cryptographic
Graph		well as Vertex and		Resilience
(Proposed)		Edge Labels,		IoT and
		are Predetermined)		Standard
				Cryptography

presents a comparative analysis of various graph labeling methods utilized in cryptography.

Therefore, relative to other graph-theoretic cryptography methods, the proposed IDMC transcends in scalability and exhibits strengthened security.

In conventional cryptographic paradigms such as AES, RSA, and QKD, the framework generally relies on a dual-key mechanism, wherein a public key facilitates encryption while a private key enables decryption. However, within military-grade and high-security domains, augmenting the number of cryptographic keys can substantially fortify the resilience of the system against sophisticated adversarial attacks. In the proposed methodology, a tri-key architecture is employed, exclusively integrated into the encryption phase. The first key assigns values based on the positions of the plaintext characters. The second key reassigns these values through a transformation governed by a formula embedded in the structural design of the secret key within the IDMC framework. Subsequently, a third key is applied to produce the definitive ciphertext representation. This multi-layered key mechanism amplifies cryptographic robustness, thereby rendering the scheme exceptionally well suited for defense and security-critical applications.

12. Conclusions

The proposed graph-based Index Difference Modular Cryptography (IDMC) algorithm stands out due to its innovative use of complex graph structures, detailed key management protocols, maintenance of low complexity, enhanced security features, thorough analysis of computational efficiency, and low collision probability, making it a significant advancement in the field of cryptographic techniques. Future research should focus on further improving computational efficiency by leveraging more sophisticated graph structures, aiming to achieve higher security while preserving linear complexity. In this paper, we have presented the theoretical implementation of the proposed IDMC method. Moreover, the strengths, limitations, and unique position of the proposed method have been discussed through comparisons with other graph-based cryptographic techniques. Future research will also aim to conduct a comprehensive comparative analysis of IDMC with state-of-the-art cryptographic technologies such as AES, RSA, and QKD, alongside the implementation and validation of the Index Difference Modular Cryptography algorithm through rigorous computational experiments.