It is well-known that one-way permutations (and even one-to-one one-way functions) imply the existence of non-interactive commitments. Furthermore the construction is black-box (i.e., the underlying one-way function is used as an oracle to implement the commitment scheme, and an adversary attacking the commitment scheme is used as an oracle in the proof of security).
We rule out the possibility of black-box constructions of non-interactive commitments from general (possibly not one-to-one) one-way functions. As far as we know, this is the first result showing a natural cryptographic application that can be achieved in a black-box way from one-way permutations but not from one-way functions.
We next extend our black-box separation to constructions of non-interactive commitments from a stronger notion of a one-way functions, which we refer to as hitting one-way functions. Perhaps surprisingly, Barak, Ong, and Vadhan (Siam JoC '07) showed that there does exist a non-black-box construction of non-interactive commitments from hitting one-way functions. As far as we know, this is the first result to establish a ``separation'' between the power of black-box and non-black-box use of a primitive in a cryptographic construction.