Noob questions: Plonk Paper

Hi, I really hope this is the right place to post my noob question. Otherwise please let me know where would be better. I am implementing Plonk as a hobby project in Python.

My question is about the linearisation polynomial in the 4th proving round p.28:


The second and last summand I see a plain z. I just wonder if thats a typo? It just feels like that should be the evaluation challenge?

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Hi,
Indeed this is a typo :slight_smile: Thanks for noticing, will be updated on eprint.

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Ah cool! Thanks for the answer! I managed to implement the protocol now. I am really proud now:D.

There was one thing I stumbled over: In round 3 when t is split up, I needed to factor out x^(n+2), not x^n as the paper says. My understanding is that the third summand of t determines the degree as (n+1) + (n+1) + (n+1) + (n+2) - n = 3n+5…

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The splitting of t in the last section indeed seems to ignore the additional factors for zero-knowledge that increase degree a bit

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Could you expand this answer please? Why can we skip those high degree factors?

We can’t :slight_smile: at least if you want zero knowledge. it’s a mistake in the paper that needs to be fixed

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Ah okay I thought I was missing something, thanks a lot!

Finally fixed on eprint

ugg I think I got it wrong, it’s n+5, will upload again

I have some questions about the procedure used to add zero-knowledge to the polynomials the Prover creates.

In Round 1 of the Prover protocol, the wire polynomials a, b, and c each use two random scalars for blinding. In Round 2 of the Prover protocol, the prover uses three random scalars to blind the z polynomial. Why three random scalars here instead of two? Why not some other number of random scalars?

Is there a rule of thumb for knowing how many blinding factors are needed to get zero-knowledge?

Does the number of blinding factors needed change if the polynomials are in Lagrange basis? For instance, can you get the same blinding effect by appending two random values to a vector before interpolating?

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The rule is that if the poly is opened at d points you need d+1 blinding factors; to hide both the commitment (which is an evaluation at a secret point in the exponent, but still to prove zk holds you’ll need this to be totally random, this proof that zk holds is quite similar to existing proofs - e.g. in Marlin, but is a missing hole in the paper right now) and the d evaluation points.
So z is opened at 2 points and hence needs 3 blinding factors.

I think it’s fine to add random multiples of any poly that is zero on the vanishing set you’re actually checking constraints on.
So in Lagrange base you can indeed random coefficients to polys that are zero on the points where the constraints are checked on.