Summary: Recently, Smith and Lennon proposed a new public key system whose trapdoor is based on the defined Lucas function instead of exponentiation. It is believed that the security of the Lucas function is cryptographically stronger than or at least as strong as the security of exponentiation function, i.e., the discrete logarithm problems. However, it is not really known that the security relationship between the Lucas function and the discrete logarithm problems. We are concerned on the security of the Lucas function and will show that the security of Lucas function is polynomial-time equivalent to the generalized discrete logarithm problems.