Stwo publics

[ShuangWu121]

Extend stwo to Support Public Input

Description

This PR extends stwo to support public input.

While stwo does not natively implement the public input feature, it is feasible to achieve this by implementing it in the stwo backend.

Implementation Details

The public values are formed using a selector to pick specific values from a witness column, which are then made public to the verifier.

This logic is already implemented in the Plonky3 backend, where public values are passed directly as inputs to the prove and verify functions, simplifying the process.

✅ To-Do List

To achieve similar functionality in stwo, the following steps need to be implemented:

• Add public columns.
• Add constraints linking the public columns, the selector, and the corresponding witness columns.
• Ensure the verifier can verify that the public columns' commitments are correctly used to form the Merkle tree in the proof.