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Key agreement protocols are frequently based on the Diffie-Hellman protocol but require authenticating the protocol messages in two ways. This can be made by a cross-authentication protocol. Such protocols, based on the assumption that a channel which can authenticate short strings is available (SAS-based), have been proposed by Vaudenay. In this paper, we survey existing protocols and we propose a new one. Our proposed protocol requires three moves and a single SAS to be authenticated in two ways. It is provably secure in the random oracle model. We can further achieve security with a generic construction (e.g. in the standard model) at the price of an extra move. We discuss applications such as secure peer-to-peer VoIP