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Consider a pairing $\mathbb{e}: \mathbb{G}_1\times \mathbb{G}_2\longrightarrow \mathbb{G}_T$ with generators $g_1$, $g_2$ for $\mathbb{G}_1$, $\mathbb{G}_2$ respectively. The groups $\mathbb{G}_1$, $\mathbb{G}_2$, $\mathbb{G}_T$ are of some prime order $p$.

For a trapdoor $s$, let $[g_1,g_1^s,\cdots,g_1^{s^N}], [g_2,g_2^s,\cdots,g_2^{s^N}]$ be the common reference string (although for some Snarks and polynomial commitment schemes, the public parameter does not contain $g_2^{s^i}$ for $i\geq 2$).

Given elements $a,b\in \mathbb{G}_1$, I would like to prove in ZK that I know a constant (as opposed to a larger degree polynomial) $\alpha$ such that $a^{\alpha} = b$. What's the most efficient way to do this?

A couple of ideas I had:

Idea 1:

  1. For a randomly generated element $a_2\in \mathbb{G}_2$ (the challenge), the Prover sends the element $b_2:= a_2^{\alpha}$.

  2. The Verifier performs the pairing check $\mathbb{e}(a,b_2) = \mathbb{e}(a_2,b)$

Idea 2

  1. The Prover proves in zero knowledge that he knows some polynomial $f(X)$ such that $a^{f(s)} = b$ (there are straightforward ways to do this, not too different from Schnorr's protocol for PoKs of discrete logs)

  2. The Prover sends the element $b_2:= g_2^{s^N\cdot \alpha}$ (which is not possible if $\alpha = f(s)$ for some non-constant polynomial $f(X)$).

  3. The Verifier verifies the proof sent in Step 1.

  4. The Verifier performs the pairing check $\mathbb{e}(a,b_2) = \mathbb{e}(b,g_2^{s^N}) $

Are there more efficient protocols that would do the job? I am not particularly fond of the idea of relying on a hashing algorithm that generates random elements of the group $\mathbb{G}_2$ as challenges.

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I don't think that either of these ideas is zero knowledge, as I don't see how the verifier would be able to produce valid transcripts without the co-operation of the prover.

On the other hand if we just do the Schnorr protocol for $\mathbb G_1$ without using the bilinear structure, that would seem to do the job. Is there a reason why this would not be acceptable?

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  • $\begingroup$ Thanks. You are right, Schnorr's protocol should work without any modifications. I thought there would be a gap in the protocol because the blinding factor chosen by the Prover could be a non-constant polynomial. But if the polynomial $\gamma\cdot f(X) + c(X)$ is a constant for a committed blinding polynomial $c(X)$ and a randomly generated challenge $\gamma$, it would follow that the polynomial $f(X)$ is constant. No need for pairings or common reference strings. $\endgroup$ Feb 3 at 8:13

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