In 2010, Grigoriev and Shpilrain, introduced some graph-based authentication schemes. We present a cryptanalysis of some of these protocols, and introduce some new schemes to fix the problems.

Let  be any connected graph with vertices set  and edges set .  For any two distinct vertices  and , the detour distance between  and  which is denoted by  is a longest path between  and  in a graph . The detour polynomial of a connected graph  is denoted by ;  and is defined by . In this paper, the detour polynomial of the theta graph and the uniform theta graph will be computed.