  Oblivious Transfer (OT) is a major primitive for secure multiparty
computation. Indeed, combined with symmetric primitives along with garbled
circuits, it allows any secure function evaluation between two parties. In this
paper, we propose a new approach to build OT protocols. Interestingly, our new
paradigm features a security analysis in the Universal Composability (UC)
framework and may be instantiated from post-quantum primitives. In order to do
so, we define a new primitive named Smooth Projective Hash Function with Grey
Zone (SPHFwGZ) which can be seen as a relaxation of the classical Smooth
Projective Hash Functions, with a subset of the words for which one cannot
claim correctness nor smoothness: the grey zone. As a concrete application, we
provide two instantiations of SPHFwGZ respectively based on the Diffie-Hellman
and the Learning With Errors (LWE) problems. Hence, we propose a
quantum-resistant OT protocol with UC-security in the random oracle model.
