Mark variables with segment numbers

[Acentelles]

This PR marks variables with "rounds" or "segments". This will allow us to build randomized circuits made from interactive protocols of the following form:

• The prover sends a vector w1
• The verifier sends a challenge c
• The prover then sends another vector w2.
• Then the verifier performs some checks on the vector (w1, w2, c, 1, x) where x is public IO.

When the witness vector is constructed, we want the code to be able to automatically build w1, w2, etc. so they can be committed separately. We could think of witness to be committed in segments or chunks or rounds. So, when allocating a variable, we want to be able to specify the "segment number".

We want to make sure if within a namespace we call some gadget, all variables allocated inside the gadget fall into the same segment as the caller.

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[srinathsetty]

Nit: bring this back?

[Acentelles]

I can bring it back. The reason is that I get the following error when I compile the code with this default enabled:

Currently feature `asm` can only be enabled on x86_64 arch.

This means that the assembly optimizations feature in halo2curves is only compatible with x86_64 architecture processors, and I am running an ARM processor (Apple M3 in particular).

Maybe I can open an issue with it. What do you think?

[srinathsetty]

For non-x86 platforms, one could use --no-default-features. See for example:

https://github.com/microsoft/Nova?tab=readme-ov-file#tests-and-examples

[srinathsetty]

@Acentelles in the current PR, the witness isn't committed separately i.e., segment by segment, right?

[Acentelles]

Right, the witness isn't committed separately in this PR. Is that something you'd like to add?

[srinathsetty]

Right, the witness isn't committed separately in this PR. Is that something you'd like to add?

Yes, otherwise, we won't have use for marking witness elements with segment numbers. See #338