Trapdoor Smooth Projective Hash Functions

Katz and Vaikuntanathan recently improved smooth projective hash functions in order to build oneround password-authenticated key exchange protocols (PAKE). To achieve security in the UC framework they allowed the simulator to extract the hashing key, which required simulation-sound non-interactive zero-knowledge proofs that are unfortunately ine cient. We improve the way the latter extractability is obtained by introducing the notion of trapdoor smooth projective hash function (TSPHF). A TSPHF is an SPHF with a trapdoor, which may not allow to recover the complete hashing key, but which still allows to compute the hash value, which is enough for an application to PAKE with UC-security against static corruptions. We additionally show that TSPHFs yield zero-knowledge proofs in two ows, with straight-line extractability. Besides those quite interesting applications of TSPHF, we also show how to generically build them on languages of ciphertexts, using any ElGamal-like encryption. Our concrete instantiations lead to e cient one-round UC-secure PAKE, extractable zero-knowledge arguments, and veri able encryption of Waters signatures. In the case of the PAKE, our construction is the most e cient one-round UC-secure PAKE to date.