LUC-type cryptosystems are asymmetric key cryptosystems based on the Lucas sequence that is extended from RSA. The security challenge is comparable to RSA, which is based on the intractability of factoring a large number. This paper analysed the security of LUC, LUC3, and LUC4, 6 cryptosystems using a common modulus attack. For a common modulus attack to be successful, a message must be transmitted to two distinct receivers with the same modulus. The strengths and limitations of the LUC, LUC3, and LUC4, 6 cryptosystems when subjected to a common modulus attack were discussed as well. The results reveal that the LUC4, 6 cryptosystem provides greater security than the LUC and LUC3.