Analysis of Rainbow Vertex Antimagic Coloring and its Application to Cryptographic Secret Sharing with Affine Cipher Technique

Abstract

Rainbow vertex antimagic coloring is a novel concept in graph theory that combines rainbow vertex connection with antimagic labeling. Rainbow vertex connection is a vertex coloring where each vertex in a simple connected graph G=(V,E) is connected by a path such that all interior vertices have distinct colors. The antimagic labeling assigns a bijective function f:E(G)→ {1,2,3,...,|E(G)|} to the edges, and the vertex weight w_f(v) = ∑_(e∈ E(v))▒〖f(e)〗, where E(v) is the set of edges adjacent to vertex 𝑣. A graph 𝐺 achieves rainbow vertex antimagic coloring if all its internal vertices have unique vertex weights. This research investigates the application of rainbow vertex antimagic coloring to Shadow D_2 (S_n) graphs and Amal(V_n,v,m) graphs in cryptographic secret sharing and encryption using the affine cipher technique. The study employs mathematical modeling, graph visualization tools, and cryptographic software to ensure methodological rigor. The encryption and decryption processes are evaluated based on effectiveness, including brute force test resistance, encryption time, and encryption size. The results demonstrate that rainbow vertex antimagic coloring is an effective method for dividing cryptographic keys into segments during the secret sharing stage and serves as a robust key in the affine cipher technique. The method offers significant advantages, including faster encryption times for Shadow D_2 (S_n) graphs compared to Amal(V_n,v,m) graphs and reduced encryption size for Amal(V_n,v,m) graphs. Both graphs exhibited strong resistance to brute force attacks. In conclusion, this study highlights the relevance of rainbow vertex antimagic coloring in advancing graph theory applications and its utility in developing secure and efficient cryptographic systems. These findings contribute to bridging theoretical graph concepts with practical cryptographic implementations.