I need to create a commitment to a value X such that I can provide either proof that the commitment is to X or proof that the commitment is not to some other given value Y != X.

If I use a simple hash with a blinding factor

C = H(BF | X)

I can provide proof by revealing BF,X and "negative proof" that X!=Y by revealing BF, but this would defeat the purpose of the blinding factor, making X guessable as it has a small domain.

Is there some other way to create such commitment?

  • 1
    $\begingroup$ BTW: in your example, just revealing BF wouldn't actually be a proof of commitment to some other value, as you don't give a proof that the value you send isn't just a random value. One could construct such a proof, but it would be a pain... $\endgroup$ – poncho Jan 24 '19 at 15:43
  • $\begingroup$ Yes. I understand that if you want to reveal the BF, you need to include it in the commitment, e.g. C = H(BF | H(BF | X)) $\endgroup$ – Tomas Jan 25 '19 at 10:45

The immediately obvious idea is to use Pedersen commitments. The idea here is that, in some group where the DLog problem is hard, we have two generators of unknown relation $g, h$, and a commitment to $x$ is the value $g^x h^r$ (for some random $r$).

To reveal the commitment is easy; you just give the values $x, r$ (and the verifier can check).

To prove that a value $x'$ is not the committed value to a commitment $g^x h^r$, the prover selects random values $s, t$, and publishes the value $h^t$, along with a zero knowledge proof that he knows the dlog of $h^t$ to the base $h$ (a standard cut-and-choose proof works). Then, both sides compute $g^{-x'} h^t (g^x h^r) = g^{x - x'} h^{r+t}$, which we will call $j$. Now, the prover computes $j^s = g^{s(x-x')} h^{s(r+t)}$, and outputs that, along with the values $s(x-x')$, $s(r+t)$. He also outputs a zero knowledge proof that he knows the discrete log of $j^s$ to the base $j$.

The verifier can check that $s(x-x') \ne 0$, that the values $s(x-x')$, $s(r+t)$ make up the outputted value $j^s$, and that the proofs of knowledge of $s, t$ are valid.

Then, the verifier knows that the prover knows the value $s(x - x')$ and that it's not zero, hence $x - x' \ne 0$ and so whatever $x$ is, it ain't $x'$

There are also standard range techniques that would also work; I believe that this is more straight-forward...

  • $\begingroup$ Thank. That makes sense. As for proof that he knows the dlog of h^t, j^s. What's a cut-and-choose proof? My knowledge is limited, but wouldn't a (DSA) signature work for that? $\endgroup$ – Tomas Jan 25 '19 at 10:50
  • $\begingroup$ @Tomas: to prove that, given $h, h^t$, you know $t$, what the prover does is generate a random $u$ and publish $h^u$. Then, the verifier picks a bit $b$; if $b=0$, then the prover publishes $u$ (and the verifier can check that); if $b=1$, then the prover publishes $u^{-1}t$ (and the verifier can check that $(h^t) = (h^u)^{(u^{-1}t)}$. We do this with a series of random $u, b$ values; if the prover doesn't know $t$, the probability of success after $n$ iterations is at best $2^{-n}$. This is zero knowledge, in the sense that the verifier learns nothing else... $\endgroup$ – poncho Jan 25 '19 at 14:38

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