Explore sumcheck optimizations for small fields

[tcoratger]

Description

We have made significant progress on univariate skip to optimize the sumcheck protocol. This work is ongoing, but it already impacts a large portion of the codebase, since the verifier changes ripple throughout WHIR.

While this continues, it would be valuable to explore additional sumcheck optimizations from recent research.

Relevant references

• The Sum-Check Protocol over Fields of Small Characteristic
  Paper: https://eprint.iacr.org/2024/1046
  Implementation: https://github.com/ingonyama-zk/smallfield-super-sumcheck/tree/main/src/algorithms

• Speeding Up Sum-Check Proving (appears to be an optimization over the above)
  Paper: https://eprint.iacr.org/2025/1117
  Blog post: https://blog.lambdaclass.com/optimizing-sumcheck/

• Packed Sumcheck over Fields of Small Characteristic with Application to Verifiable FHE
  Paper: https://eprint.iacr.org/2025/719
  (Unclear yet if this approach is directly applicable to WHIR)

Motivation

Exploring these directions could lead to further improvements in WHIR’s prover performance and verifier efficiency, especially in small characteristic settings. This may complement or extend the ongoing univariate skip work.

Next steps

• Review the referenced papers and implementations.
• Evaluate which techniques are applicable to WHIR.
• Prototype promising approaches in parallel to univariate skip.

[tcoratger]

@diegokingston and his team are working on this :)

[MuhtasimTanmoy]

I will be looking into it as well.

Do these seem relevant?

• Using system-native types for efficient arithmetic
• Ark-ff Small Field Support