We presented a simple and efficient password-based encrypted key exchange protocol that allows a user to establish secure session keys with remote servers from client terminals in low resource environments. He does not need to carry smart card storing his private information but just needs to know his identity and password. For this purpose, the scheme was implemented over elliptic curves because of their well-known advantages with regard to processing and size constraints. Furthermore, the scheme is provably secure under the assumptions that the hash function closely behaves like a random oracle and that the elliptic curve computational Diffie-Hellman problem is difficult.

In the last couple of years, ID-based cryptography has got fruitful achievements. Proxy multi-signature allows a designated person, called a proxy signer, to sign on behalf of two or more original signers. In this paper, we present a general security model for ID-based proxy multi-signature (ID-PMS) schemes. Then, we show how to construct a secure ID-PMS scheme from a secure ID-based signature scheme, and prove that the security of the construction can be reduced to the security of the original ID-based signature scheme.

Provable security has been widely used for analyzing the security of cryptosystems. Its main idea is to reduce the security to some well-defined computational assumption. The reduction process is called the security proof. In this paper, we find a flaw in the security proof of BDOP-PEKS and PEKS-STAT, present a new conclusion for the security of BDOP-PEKS, and give a security proof. The flaw in the security proof of PEKS-STAT can be fixed in the same way. Finally we conclude some steps of security proof, and emphasize that the probability is as important as the construction.