Add Logup-GKR support

[ShuangWu121]

Logup-GKR Integration in STARK Provers

Context

I have been researching how different implementations integrate Logup-GKR with STARK proofs. My focus is on two implementations:

• Miden VM, which uses Winterfell (Facebook’s STARK prover).
• Stwo Logup-GKR implementation.

Current Progress

The main challenge is understanding how these implementations use a univariate polynomial IOP (STARK) to integrate GKR, which requires a multivariate polynomial IOP.

• Miden VM provides a clear approach to this integration (detailed in [this doc] and [this doc]). However, its GKR and related APIs are from Winterfell (logup-gkr branch), making direct application to the Stwo backend difficult. - This may not be a blocker, as Stwo already implements GKR and sumcheck, meaning integration could be possible with modifications.

• Stwo's GKR implementation is clear for proving a logup lookup alone, but its integration with STARK is less clear.

  • One key aspect is understanding the "univariate IOP for multilinear evaluation at a point" in this example, which seems central to the integration.
  • My current research is focused on understanding this component.

Next Steps

I will continue investigating this and update the issue as I gain a clearer picture. Any insights or related references are welcome!