Journal of Computational Mathematica

Advancing Post-Cryptographic Schemes Using Number Theoretic Structures: Algorithms, Security, and Real World Implementation

2026-07-03T07:41:32+00:00

Cryptography, a term rooted in Greek that translates to 'secure writing,' encompasses the intricate study of methods employed to encrypt and decrypt information, ensuring secure communication between parties. This fundamental discipline plays a critical role in safeguarding sensitive data from unauthorized access and malicious entities. The process begins with encryption, where plain text, or the original readable information, is transformed into ciphertext through specific algorithms and protocols. This transformation obscures the content, making it unintelligible to anyone who does not possess the appropriate keys or credentials to decode it. Decryption is the complementary process that reverses this transformation, restoring the ciphertext to its original plain text format. This allows authorized users to access the information securely while preventing would-be intruders from deciphering its contents. In addition to these basic principles, modern cryptography incorporates various techniques, such as symmetric and asymmetric encryption, digital signatures, and hash functions, each serving specific purposes in the realm of data protection and secure communication. Overall, cryptography is essential in today's digital landscape, underpinning everything from online banking transactions to secure messaging systems and helping to ensure the integrity and privacy of information exchanged over insecure networks.

Super Mean Graph Labeling: A Novel Cryptographic Framework Using Five-Star Graphs with Applications in Secure Communications and Epidemic Modeling

2026-07-03T07:52:22+00:00

This paper introduces an innovative cryptographic framework based on super mean labeling of five-star graphs K1,V1 ∪ K1,V2 ∪ K1,V3 ∪ K1,V4 ∪ K1,V5, where V1 ≤ V2 ≤ V3 ≤ V4 ≤ V5. We present a comprehensive mathematical foundation for super mean labeling and develop systematic methodologies for encoding messages through carefully constructed graph structures. Three distinct implementation approaches are demonstrated, incorporating computational techniques through C programming for alphabetical mapping, including subtraction-based and division-based numbering schemes. Each approach is thoroughly illustrated with complete message encoding examples, visual cryptography representations, and detailed security analysis. Beyond cryptographic applications, we extend the framework to epidemic modeling, demonstrating how super mean labeling can represent complex disease transmission dynamics in multi-population systems. The epidemic modeling application includes complete mathematical formulations of multi-population SIR models, transmission matrix encoding through graph labeling, and computational implementations for disease surveillance. Our results establish that five-star graphs with super mean labeling provide optimal balance between encoding capacity and structural complexity, offering robust security through multiple layers of mathematical obfuscation. The integration of graph theory with computational algorithms creates a versatile framework applicable to both secure communications and public health informatics.

Detailed Proposed Algorithm: Symmetric Encryption and Decryption Using Eulerian Circuits in Simple Graphs

2026-07-10T08:03:54+00:00

This paper proposes a novel symmetric-key encryption scheme using simple weighted graphs and Eulerian circuits. Unlike NP-complete Hamiltonian cycle methods, it employs polynomial-time Eulerian circuits via Hierholzer‘s algorithm. Plaintext is mapped to a weighted graph using a secret key. An Eulerian circuit generates a dynamic key and Euler Tour Matrix. Encryption uses matrix multiplications with a shared upper triangular key and modular reduction. Decryption reverses the operations using matrix inverses. Statistical tests confirm strong diffusion and randomness. The scheme provides better efficiency and scalability than traditional Hamiltonian-based approaches.

Graph-Based Cryptographic Encoding and Epidemiological Modeling via Even Felicitous Labeling

2026-07-10T08:22:17+00:00

In this work, we have identifified a way for encoding a secret message utilizing the GMJ (Graph Message Jumbled) Code and even felicitous labeling on fifive-star graphs K1,σ1 ∪ K1,σ2 ∪ K1,σ3 ∪ K1,σ4 ∪ K1,σ5 . For every star graph, we give two examples and use a Java program to apply alphabetical techniques to the Corona Number and Triangular Number. A strategy for labeling an even felicitous graph and flflowchart is provided for transforming plaintext into ciphertext (Picture Coding). Furthermore, we demonstrate an innovative application of these graph labeling techniques to epidemiological modeling through a detailed analysis of the SIR (Susceptible-Infected-Recovered) epidemic model on star networks. The paper shows how felicitous labeling can encode varying transmission probabilities and infection states, providing insights for targeted public health interventions. This interdisciplinary approach bridges cryptography, graph theory, and epidemiology, offffering both secure communication methods and practical disease spread modeling frameworks.