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Authenticated Key Exchange (AKE) protocols enable two parties to establish a shared, cryptographically strong key over an insecure network using various authentication means, such as cryptographic keys, short (i.e., lowentropy) secret keys or credentials. In this paper, the authors provide a general framework, that encompasses several previous AKE primitives such as (Verifier-based) Password-Authenticated Key Exchange or Secret Handshakes, they call LAKE for Language-Authenticated Key Exchange. They first model this general primitive in the Universal Composability (UC) setting. Thereafter, they show that the Gennaro-Lindell approach can efficiently address this goal. But they need smooth projective hash functions on new languages, whose efficient implementations are of independent interest. They indeed provide such hash functions for languages defined by combinations of linear pairing product equations.
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