Compact and Efficient UC Commitments under Atomic-Exchanges

We devise a multiple (concurrent) commitment scheme operating on large messages. It uses an ideal global setup functionality in a minimalistic way. The commitment phase is non-interactive. It is presented in a modular way so that the internal building blocks could easily be replaced by others and/or isolated during the process of design and implementation. Our optimal instantiation is based on the decisional Diffie-Hellman (DDH) assumption and the (adversarially selected group) Diffie-Hellman knowledge (DHK) assumption which was proposed at CRYPTO 1991. It achieves UC security against static attacks in an efficient way. Indeed, it is computationally cheaper than Lindell's highly efficient UC commitment based on common reference strings and on DDH from EUROCRYPT 2011.