experiment fanin > 2 in product argument

[hero78119]

Context

Currently in ceno, product argument configure with fanin=2, e.g. read set witness. with respective sumcheck arguments $p[\vec{\gamma}] = \sum_{\vec{b}} eq(\vec{\gamma}, \vec{b}) p_{1}[\vec{b}] * p_{2}[\vec{b}]$. The reason is because to simplify, in sumcheck prover_v2 we only support degree <= 3 implementation so far.

We can extend product argument and experiment > 2 fanins on product and benchmark. To eliminate the limitation of degree sumcheck prover, we can use rust macro to generate the boilerplate code of arbitrary degree implementation.

performance gain

Intuitively

1. compiler can do more optimisation https://github.com/scroll-tech/ceno/blob/master/sumcheck/src/prover.rs#L469-L473 if goes from f1 * f2 to f1 * f2 * f3 * f4
2. layer of tower sumcheck product argument can be half, for fanin = 2 -> fanin = 4, number of product argument layers can be divided by 2

Suggested step

• use macro to generate sumcheck protocol boipolate code here https://github.com/scroll-tech/ceno/blob/master/sumcheck/src/prover.rs#L432-L533
• add benchmark follow this to compare sumcheck degree = 2 vs degree = 4
• refactor prover/verifier to support product argument with larger degree