The Curious Case of Non-Interactive Commitments – On the Power of Black-Box vs. Non-Black-Box Use of Primitives

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 task 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 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 to implement a natural cryptographic task.

We finally show that unless the complexity class

$\ensuremath{\ensuremath{\mathsf{NP}} }$

has program checkers, the above separations extend also to non-interactive instance-based commitments, and 3-message public-coin honest-verifier zero-knowledge protocols with

O

(log

n

)-bit verifier messages. The well-known classical zero-knowledge proof for

$\ensuremath{\ensuremath{\mathsf{NP}} }$

fall into this category.