A graph theoretic method for securing key fobs

Abstract

Key fobs are small security hardware devices that are used for controlling access to doors, cars and etc. There are many types of these devices, more secure types of them are rolling code. They often use a light weight cryptographic schemas to protect them against the replay attack. In this paper, by the mean of constructing a Hamiltonian graph, we propose a simple to implement and secure method which overrides some drawbacks of traditional ones. Let $$m,n>1$$ be two integers, and $${\mathbb {Z}}_n$$ be a $${\mathbb {Z}}_m$$ -module. Here, we determine the values of m and n for which the $${\mathbb {Z}}_n$$ -intersection graph of ideals of $${\mathbb {Z}}_m$$ is Hamiltonian. Then a suitable sequence will be produced which by some criteria can be used as the authenticator.