A Novel Encryption Framework Using Self-Invertible Matrices and Complete Graph Adjacency Matrix Structures

Encryption techniques form the foundation for securing
communication and protecting information in the modern digital
era. With the rapid growth of internet usage and network-based
interactions, the need for robust encryption methods has become
increasingly critical. Transmitting sensitive information over
unsecured or vulnerable networks exposes it to risks such as
cyberattacks, data breaches, and unauthorized access. To address
these challenges, cryptographic techniques have been widely
developed and applied. Classical methods such as the Caesar
Cipher, Atbash Cipher, Hill Cipher, along with various symmetric
encryption schemes, have significantly contributed to secure
communication. In this paper, we propose an encryption approach
that integrates a self-invertible matrix, the adjacency matrix, and
the structure of a complete graph with a Hamiltonian circuit to
generate complex ciphertext from given message units. A key feature
of this method is the use of a self-invertible matrix as the encryption
key, which inherently guarantees the existence of its inverse withoutrequiring explicit computation. This property simplifies the
decryption process and considerably reduces computational
complexity while maintaining the strength of the encryption scheme.