Recently, Smith and Lennon proposed a new public key cryptosystem, called LUC, which uses the Lucas function as the one-way function in their cryptographic mechanisms instead of using the exponentiation function. They conjectured that LUC is cryptographically stronger than RSA in 1993. Since then, many weaknesses of LUC have been discoverd, e. g., similar to RSA, LUC also suffers from the chosen-message attacks and the evaluation in LUC is slightly less efficient than that in RSA. In this paper, we analyze another possible weakness of LUC that was not pointed out before. We show that the number of messages which cannot be concealed in LUC is at least as the same as that in RSA regardless of the choice of public keys. In particular, in many cases, the number of messages which cannot be concealed in LUC is greater than that in RSA. This implies that the choice of public keys in LUC needs more limitations than that used in RSA. Our results are useful to designers who consider to use LUC type systems.