In this paper, we investigate and review public-key cryptosystems which are derived from higher order linear recurrence relation
which are based on the Lucas function. The RSA scheme is based on hard mathematical problem, the intractability of factoring large integers. This application of a hard mathematical problem to cryptography revitalized efforts to find more efficient methods to factor. The first motivation to develop a new
cryptosystem analogous to RSA is the possibility that the higher order
analogues are more secure than the RSA. The explicit formulation involves a generalization of the Euler Totient function, which underlie the algebra of the RSA cryptosystem.