Finding Collisions in Interactive Protocols -- Tight Lower Bounds on the Round and Communication Complexities of Statistically Hiding Commitments

We study the round and communication complexities of various cryptographic protocols. We give tight lower bounds on the round and communication complexities of any fully black-box reduction of a statistically hiding commitment scheme from one-way permutations, and from trapdoor permutations. As a corollary, we derive similar tight lower bounds for several other cryptographic protocols, such as single-server private information retrieval, interactive hashing, and oblivious transfer that guarantees statistical security for one of the parties. Our techniques extend the collision-finding oracle due to Simon (EUROCRYPT 98) to the setting of interactive protocols and the reconstruction paradigm of Gennaro and Trevisan (FOCS 00).