chore: Add more description for the kzg verify algorithm #3703
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| This is done by checking if the following equation holds: | ||
| Q(x) Z(x) = f(X) - r(X) | ||
| Where: | ||
| f(X) is the polynomial that we want to show opens at `k` points to `k` values |
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"we want to show" seems like the wrong PoV here since we are documenting a verification function
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Changed it to "we want to verify" let me know if that also sounds off
| Where: | ||
| f(X) is the polynomial that we want to show opens at `k` points to `k` values | ||
| Q(X) is the quotient polynomial computed by the prover | ||
| r(X) is the degree `k-1` polynomial that agrees with f(x) at all `k` points |
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"agrees with f(x)" seems like the wrong PoV here, since the verifier does not know if f(x) actually evaluates to the right values at the right points.
Pehraps "r(x) is the degree k-1 polynomial that evaluates to ys at all zs points"? Or something similar that indicates that r(x) is something that the verifier actually "trusts".
It seems a bit weird to not use the ys and zs inputs anywhere in the doc description.
Finally, what about renaming r(x) to I(x) standing for interpolation polynomial? No strong opinion about this one. I know Dankrad used this notation in his post and we also used it in ours.
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Will take your suggestion for the description and rename r(X) to I(X) -- I think r(X) describes what it is, while I(X) describes how it was made (by interpolating the values). In this case, to avoid confusion from folks reading multiple documents, using I(X) is the sound approach
This fleshes out the description for the KZG verify algorithm being used, similar to how the proving algorithm has been expanded.