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The Multiplication of z(x) and z(Xw) in the Quotient Polynomial from the PLONK

my flag

From the PLONK paper.

Page 29, Round 3

Quotient Polynomial

Why multiply z(x) and z(Xw) in the quotient polynomial? (why does internal wiring have to multiply input permutation) Why the second term have to "shift by w"?

My observation is:

  1. These 2 lines seem they have to cancel each other out.
  2. z(x) is permutation polynomial and only about input. a(x), b(x), c(x) are about internal wiring.
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de flag

These two lines in the quotient expression make sure $Z$ is accumulating the grand product of the inputs shifted by the identity permutation, divided by the inputs shifted by the permutation $\sigma$ describing the circuit wiring. As described in Section 5 of the paper, this together with $Z$ starting and ending up with the value one, which is checked by the fourth line in the quotient expression, guarantee the inputs $a,b,c$ respect the correct circuit wiring.

Paul Yu avatar
my flag
I presume you're referring to this equation in section5: Z(a)f′(a) = g′(a)Z(a · g). What is your identity permutation?
relG avatar
de flag
that equation covers both identity and sigma - referring to second line of your equation picture above when saying identity - cause beta is multiplied just by X.
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