Add local poseidon2-air crate with extra index columns

[sai-deng]

The local copy will be extended to verify Poseidon2 operations used in challenger/sponge circuits and MMCS verification AIR with CTLs.

[LindaGuiga]

If we're just adding columns, I don't think we need to copy the code. We can make Poseidon2 columns part of our Poseidon2's columns, and those columns can be made into a subair (which applies the same constraints as Plonky3's Poseidon2 (SP1 and OpenVM do that, for Keccak at least)

[Nashtare]

I'd agree with Linda, I don't see why we'd need a blatant copy of the P3 code, unless you'd be modifying the existing internal structure? If yes, please clarify why it's needed and in that case, we'd need a renaming of the copied crate to be able to publish it with our own changes.

[sai-deng]

Could we include a minimal trace-generation helper plus unit tests to harden this PR?

[sai-deng]

I suggest rename it to p3-poseidon2-recursion-air as poseidon2-circuit-air does not emphasize the extra columns you add for the circuit

[hratoanina]

I think that circuit denotes instead that the extra columns are for our circuit in particular, and not for a generic recursion method. But I don't have a strong opinion on this.

[Nashtare]

I think it'd be useful to extract this piece outside of the poseidon2-circuit-air crate to make it widely accessible, as this is fairly valuable even for external users.

[Nashtare]

Suggested change

	/// A submatrix of a matrix. The matrix will contain a subset of the columns of `self.inner`.
	/// A submatrix of a matrix. The matrix will contain a subset of the columns of `self.inner`.

[Nashtare]

also nit but doesn't submatrix imply it's from a "larger" matrix already?

[Nashtare]

Not really blocking but 70% of this file (and a good portion of the air module are due to all these associated constants.

Could be worth diving into some way to refactor this later on to make it a bit more concise / readable.

[4l0n50]

I thought the same. I think something like

pub struct Poseidon2CircuitCols<
    T,
    const WIDTH_EXT: usize,
    const RATE_EXT: usize,
    P: PermutationColumns<T>,
> {
    pub poseidon2: P,

    pub is_sponge: T,
    pub reset: T,
    pub sponge_reset: T,
    pub absorb_flags: [T; RATE_EXT],
    pub input_indices: [T; WIDTH_EXT],
    pub output_indices: [T; RATE_EXT],
}

trait PermutationColumns<T> {}

impl<
    T,
    const WIDTH: usize,
    const SBOX_DEGREE: u64,
    const SBOX_REGISTERS: usize,
    const HALF_FULL_ROUNDS: usize,
    const PARTIAL_ROUNDS: usize,
> PermutationColumns<T>
    for Poseidon2Cols<T, WIDTH, SBOX_DEGREE, SBOX_REGISTERS, HALF_FULL_ROUNDS, PARTIAL_ROUNDS>
{
}

could make the trick.

[4l0n50]

You can also do

const _: () = {
            assert!(CAPACITY_EXT + RATE_EXT == WIDTH_EXT, "capacity+rate mismatch");
            assert!(WIDTH_EXT * D == WIDTH, "width extension mismatch");
        };

[4l0n50]

This is copying the whole poseidon trace. The problem is that generate_trace_rows allocs a new matrix, while we would like to have the matrix at some specific addresses (at every row, before the right_part). But we need to figure out a way of doing so.