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I have some issue to understand the verify round of the KZG polynomial commitment scheme. The following diagram is associated to the scheme. I appreciate any help.

To verify, the verifier should compute the pairing of $e(g^{f(\tau)-f(u)}, g)$ and $e(g^{\tau-u}, g^{q(\tau)})$.

However, to compute these pairings, verifier should compute $g^{f(\tau)-f(u)}$ and $g^{\tau-u}$ first. So, we see that $g^{f(\tau)-f(u)}=g^{f(\tau)}/g^{f(u)}$ and this is division of two points of the elliptic curve! However, the division of two elliptic curve points is not defined! We have the same issue with computing $g^{(\tau-u)}$ which is equal to $g^\tau/g^u$.

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  • $\begingroup$ They are using multiplicative notation even though the Elliptic Curve is an additive group - it's a notation thing & not wrong - check this answer of mine - crypto.stackexchange.com/a/105778/3941 $\endgroup$
    – user93353
    May 20, 2023 at 3:38

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In this lecture, they use multiplicative notation for the pairing groups instead of additive notation. Thus, division is well-defined. Division is just the inverse of the group operation.

The choice of additive vs multiplicative notation for a group is purely a semantic choice.

I'll translate some of the items to additive notation. The lowercase letters and symbols will be elements of the scalar field of the pairing group $\mathbb{G}$, while the capital $G$ will denote the generator of the group.

  • The commitment $\mathsf{com}_f = f(\tau) G$.
  • The element $vG$
  • The element $(\tau-u)G = \tau G - uG$
  • The subtraction will be $(f(\tau)-v)G = f(\tau)G - vG$.
  • The proof $\pi=q(\tau)G$
  • The pairing check $e\left(\mathsf{com}_f-vG,G\right)=e\left((\tau-u)G, \pi\right)$
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  • $\begingroup$ Thanks for the explanation. If I want to use additive notation instead of multiplicative notation, would you help me to know how the formulas change? $\endgroup$
    – tesoke
    May 20, 2023 at 4:05
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    $\begingroup$ @tesoke I've added some of the notation changes. Does that help you? $\endgroup$
    – Wilson
    May 20, 2023 at 4:20
  • $\begingroup$ Thanks, yes it helps. But I think that the pairing should be as e(comf−vG,G)=e((τ−u)G,comq). Am I right? $\endgroup$
    – tesoke
    May 21, 2023 at 2:45
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    $\begingroup$ Yep, edited. Minor typo $\endgroup$
    – Wilson
    May 21, 2023 at 3:23

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