In this paper, we show that the AugPAKE protocol provides the semantic security of session keys under the strong Diffie-Hellman (SDH) assumption in the random oracle model.

At Indocrypt 2005, Viet et al., [22] have proposed an anonymous password-authenticated key exchange (PAKE) protocol and its threshold construction both of which are designed for client's password-based authentication and anonymity against a passive server, who does not deviate the protocol. In this paper, we first point out that their threshold construction is completely insecure against off-line dictionary attacks. For the threshold t > 1, we propose a secure threshold anonymous PAKE (for short, TAP) protocol with the number of clients n upper-bounded, such that n \leq 2 \sqrt{N-1} -1, where N is a dictionary size of passwords. We rigorously prove that the TAP protocol has semantic security of session keys in the random oracle model by showing the reduction to the computational Diffie-Hellman problem. In addition, the TAP protocol provides unconditional anonymity against a passive server. For the threshold t=1, we propose an efficient anonymous PAKE protocol that significantly improves efficiency in terms of computation costs and communication bandwidth compared to the original (not threshold) anonymous PAKE protocol [22].

In this paper, we prove the security of the {\sf RSA-AKE} protocol \cite{SKI05} in the random oracle model. The proof states that the {\sf RSA-AKE} protocol is secure against an adversary who gets the client's stored secret \emph{or} the server's RSA private key.\footnote{The protocol is the same as \cite{SKI05}, but we corrected the security proof partially. The attacks appeared in \cite{TM05} are no longer available in the proof since the adversary has access to either the client's stored secret or the server's private key, not both of them.} To our best knowledge, the {\sf RSA-AKE} protocol is the most efficient among their kinds (i.e., RSA and password based AKE protocols). The other security properties and efficiency measurements of the {\sf RSA-AKE} protocol remain the same as in \cite{SKI05}.