Efficient UC-secure authenticated key-exchange for algebraic languages

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., low-entropy) secret keys or credentials. In this paper, we provide a general framework, that encompasses several previous AKE primitives such as (Verifier-based) Password-Authenticated Key Exchange or Secret Handshakes, we call LAKE for Language-Authenticated Key Exchange. We first model this general primitive in the Universal Composability (UC) setting. Thereafter, we show that the Gennaro-Lindell approach can efficiently address this goal. But we need smooth projective hash functions on new languages, whose efficient implementations are of independent interest. We indeed provide such hash functions for languages defined by combinations of linear pairing product equations. Combined with an efficient commitment scheme, that is derived from the highly-efficient UC-secure Lindell's commitment, we obtain a very practical realization of Secret Handshakes, but also Credential-Authenticated Key Exchange protocols. All the protocols are UC-secure, in the standard model with a common reference string, under the classical Decisional Linear assumption.