Key Generation in Cryptography Using Radio Path Coloring

Abstract

An L(p 1 , p 2 , p 3 , . . . , p m )labeling of a graph G is an assignment of positive integers to the vertices of G such that the difference in the labels assigned to the vertices at distance i should be at least p i . The particular case of p 1 = d, p 2 = d − 1, p 3 = d − 2, . . . , p d = 1 where d is the diameter of the graph, was known as the radio labeling of G. The minimum value of the maximum integer used in any feasible radio labeling of G was called as radio number of G denoted by rn(G). The idea of radio path coloring was conceived to ensure secure communication in networks. If there exists a path between each pair of vertices, such that, the labeling in that path is an L(p 1 , p 1 − 1, p 1 − 2, . . . , 1)− labeling, then such a labeling was called an L(p 1 , p 1 −1, p 1 −2, . . . , 1)− path coloring or a radio path coloring of G. The minimum value of the largest label used in such a coloring was called as the radio path connection number. Earlier researchers have studied the case of p 1 = 2 for different classes of graphs. We focus on the more general case of p 1 ≥ 3 and obtain an upper bound on the radio connection number k p1c (G), of any semi-Hamiltonian graph G. An algorithm to obtain the radio path coloring of a Semi-Hamiltonian graph G is also discussed here and the same is used to generate keys for secure communication in cryptography. INDEX TERMS Cryptography, radio labeling, path coloring I. INTRODUCTION