We study the security requirements for remote authentication with password protected smart card. In recent years, several protocols for password-based authenticated key exchange have been proposed. These protocols are used for the protection of password based authentication between a client and a remote server. In this paper, we will focus on the password based authentication between a smart card owner and smart card via an untrusted card reader. In a typical scenario, a smart card owner inserts the smart card into an untrusted card reader and input the password via the card reader in order for the smart card to carry out the process of authentication with a remote server. In this case, we want to guarantee that the card reader will not be able to impersonate the card owner in future without the smart card itself. Furthermore, the smart card could be stolen. If this happens, we want the assurance that an adversary could not use the smart card to impersonate the card owner even though the sample space of passwords may be small enough to be enumerated by an off-line adversary.