Graph isomorphism (GI) has turned out to be an attractive basis of the construction of secure cryptographic protocols because it has a special set of computational hardness properties, not in the classical NP-complete classes. This article introduces a new cryptographic system that utilizes the graph isomorphism hardness where pairs of graphs of the same structure are used to conduct secure communication through the use of permutation-based mappings. The protocol proposed presents a strong public- private key architecture in which isomorphic graph transformations are encryption primitives. The security properties and resistance to structural inference attacks are modeled by creating a mathematical formulation. The encryption algorithm uses randomized permutations to hide the correspondences between the vertex with confidentiality and unpredictability. The efficiency and the possible quantum resistance of the proposed approach is compared to the NP-complete and post-quantum cryptographic assumptions, which helps to draw the comparative analysis.