Bounds on the efficiency of black-box commitment schemes

Constructions of cryptographic primitives based on general assumptions (e.g., one-way functions) tend to be less efficient than constructions based on specific (e.g., number-theoretic) assumptions. This has prompted a recent line of research aimed at investigating the best possible efficiency of (black-box) cryptographic constructions based on general assumptions. Here, we present bounds on the efficiency of statistically-binding commitment schemes constructed using black-box access to one-way permutations; our bounds are tight for the case of perfectly-binding schemes. Our bounds hold in an extension of the Impagliazzo–Rudich model: we show that any construction beating our bounds would imply the unconditional existence of a one-way function (from which a statistically-binding commitment scheme could be constructed “from scratch”).