The security of LUC type cryptosystems was investigated. In this study, chosen message attack was employed to analyze the security of LUC, LUC3 and LUC4,6 cryptosystems. The cryptanalyst invades the system by obtaining a signature without the sender’s consent, and use it to break the system. Finding shows that LUC4,6 cryptosystem is more resilient against chosen message attack compare with LUC and LUC3 cryptosystems.

Elliptic Curve Cryptosystem based on second order Lucas sequenceis a cryptosystem using elliptic curves over finite fields as a mask and incorporate with second order of Lucas sequence. The security of the Elliptic Curve Cryptography cryptosystem depends on the discrete logarithms. In this cryptosystem, Lucas sequence is employed to compute the ciphertext or recover the plaintext. The Elliptic Curve Cryptosystem based on second order Lucas sequence is vulnerable when the bit of the decryption key, $d$ flips by using fault based attack.