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Password-based key exchange schemes are designed to provide entities communicating over a public network, and sharing a (short) password only, with a session key (e.g, the key is used for data integrity and/or confidentiality). The focus of the present paper is on the analysis of very efficient schemes that have been proposed to the IEEE P1363 Standard working group on password-based authenticated key-exchange methods, but which actual security was an open problem. We analyze the AuthA key exchange scheme and give a complete proof of its security. Our analysis shows that the AuthA protocol and its multiple modes of operations are provably secure under the computational Diffie-Hellman intractability assumption, in both the random-oracle and the ideal-ciphers models.