Cryptanalysis of log-in authentication based on circle property

A scheme of user authentication had been proposed to authenticate users based on the secret data stored inside a smart card and the property of an n-dimensional circle. We investigate the security of the scheme and show that the scheme is insecure under the dictionary attack, the impersonation attack, and the attack of impersonating the central authority. Due to the insecurity under various attacks, we suggest that a cryptographic scheme should be provably secure.