How do we represent a Gate involving a constant to the left or right of the operator in PLONK?

user93353

8/30/24, 9:10 AM

Let's say I have the following equation to be arithmetised in PLONK

$x^3 + x + 5 = 35$ and the witness is $x = 3$

$3 * 3 = 9$

$9 * 3 = 27$

$27 + 3 = 30$

$30 + 5 = 35$

Now the 4th gate can be expressed as

$Q_l\cdot 30 + Q_r\cdot 5 + Q_m(30\cdot 5) + Q_c + Q_o\cdot(35)$

with $Q_l = 1, Q_r = 1, Q_m = 0, Q_c = 0, Q_o = -1$

Or as

$Q_l\cdot 30 + Q_r\cdot ? + Q_m(30\cdot ?) + Q_c + Q_o\cdot(35)$

with $Q_l = 1, Q_r = 1$ or $0, Q_m = 0, Q_c = 5, Q_o = -1$

In the first case, the element of the $b$ vector would be 5.

But in the second case, the element of the $b$ vector can be anything - it doesn't matter if $Q_r = 0$ but $b$ vector element needs to be $0$ if $Q_r = 1$.

So which of these 2 methods is the right way to represent the 4th gate. If it's the 2nd way, then should $Q_r$ be $0$ or $1$ & what should be the $b$ vector element?

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EN: How do we represent a Gate involving a constant to the left or right of the operator in PLONK?

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