perf: non-native multilinear polynomial evaluation

[ivokub]

Description

This PR optimizes evaluation of multilinear polynomials.

Previously, we folded the multilinear polynomial one variable at a time, but this is sub-optimal approach when using non-native arithmetic as we have to perform many modular multiplications.

Now, instead, we split the variables x1, ..., xn into two similar size partitions x1, .., x_{n/2} and x_{n/2+1}, ..., xn. We then compute recursively all possible combinations prod({x1, (1-x1)}, ..., {x_{n/2}, 1-x_{n/2}}) and prod({x_{n/2+1}, (1-x_{n/2+1})}, ..., {x_n, 1-x_n}) first and then compute the corresponding evaluated coefficients (1-x1)*...*(1-xn) f0 ... x1*...*xn f_{2^n} one by one with the combinations using non-reducing multiplication and addition chains.

With doing the addition chains and non-reducing multiplications, we can save significantly on modular reductions. Additionally, we can reuse the computations when we're computing several ML evaluations which is useful later for the sumcheck verifier.

Type of change

• New feature (non-breaking change which adds functionality)

How has this been tested?

• TestEvalMultiLin2 -- checks consistency with the previous implementation (which is kept for testing purposes)
• TestPartialMultilinearEval

How has this been benchmarked?

Ran the tests from the local scalar multiplication with sumcheck -- for MLE of size 2^17 the reduction is from 33039039 constraints to 4502862 constraints.

Checklist:

• I have performed a self-review of my code
• I have commented my code, particularly in hard-to-understand areas
• I have made corresponding changes to the documentation
• I have added tests that prove my fix is effective or that my feature works
• I did not modify files generated from templates
• golangci-lint does not output errors locally
• New and existing unit tests pass locally with my changes
• Any dependent changes have been merged and published in downstream modules