Boolean algebra and lattice of graphs with their applications in cryptography

This paper introduces a new conceptual framework of the intersection of graph theory and abstract algebra, creating a new avenue for the applications of Boolean algebra and graph theory in cryptography. The integration of Boolean graphs with cryptographic principles offers a novel approach in the construction of the Camouflaged-Graph Cryptosystem[Formula: see text]CGC[Formula: see text] using symbolic representations of the graphs. We explore the graphs satisfying Boolean identities, and introduce Boolean structures on the set of all simple undirected graphs with a common vertex set. We introduce the concepts of Boolean graphs, atoms, and coatoms of Boolean graphs, and explore properties of the lattices of graphs, presenting the intricate relationships within graph structures. The introduction of auxiliary vertices representing Boolean OR/AND operations and defining the Boolean graph complements is a novel approach in the construction of the CGC, building on the symbolic Boolean to graph representation that may be used during the decryption process to restore the graph’s canonical form after a cryptographic obfuscation or a camouflaged Boolean graph has been created during encryption.