Round-optimal privacy-preserving protocols with smooth projective hash functions

In 2008, Groth and Sahai proposed a powerful suite of techniques for constructing non-interactive zero-knowledge proofs in bilinear groups. Their proof systems have found numerous applications, including group signature schemes, anonymous voting, and anonymous credentials. In this paper, we demonstrate that the notion of smooth projective hash functions can be useful to design round-optimal privacy-preserving interactive protocols. We show that this approach is suitable for designing schemes that rely on standard security assumptions in the standard model with a common-reference string and are more efficient than those obtained using the Groth-Sahai methodology. As an illustration of our design principle, we construct an efficient oblivious signature-based envelope scheme and a blind signature scheme, both round-optimal.