Uses proof_system crate and implements type serialization for net comms

[gpestana]

The goal of this PR is to refactor current PoC from using marlin to proof_system, which implements serde serialization out of the box.

The steps taken so far have been (split by commits):

Note: Currently using 1.56.0-x8664-apple-darwin

Commit 1 -- Update Cargo.toml to use proof_system and latest zexe

• Remove all dependencies from marlin (submodule) and replace them with impls on proof_system crate.

• Remove 5_wires naming from plonk dlog and circuit dependencies and use proof_system implementation

Result: 80 compilation errors due to different APIs between marlin and proof_system and latest zexe.

Commit 2 -- Fix dependencies names

Result: About 70 compilation errors due to most of proof_system dependencies relying on a different algebra implementation.

Commit 3 -- Replace algebra dependencies for ark and mina curves dependencies

Due to PR refactor zexe -> arkworks renaming imports, replace all algebra instances in bba implementation for ark_* when using crates circuits/plonk, curves, dlog/commitment, dlog/plonk, groupmap, oracle. In addition, use mina::curves instead of algebra curves.

Result: trait CommitmentField is not implemented for ark_ec::AffineCurve, as required by plonk_protocol_dlog::index::Index

Commit 4 -- Add trait bounds to high level types

Commit 5 -- Use algebra_core::FftField instead of ark_ff in fft.rs

Result:

• Multiple trait bounds not implemented;
• ProverCommitments structure changed from :

// plonk_5_wires_protocol_dlog::prover::ProverCommitments
pub struct ProverCommitments<G: AffineCurve>
{
    pub w_comm: [PolyComm<G>; COLUMNS],
    pub z_comm: PolyComm<G>,
    pub t_comm: PolyComm<G>,
}

// plonk_protocol_dlog::prover::ProverCommitments
pub struct ProverCommitments<G: AffineCurve> {
    pub l_comm: PolyComm<G>,
    pub r_comm: PolyComm<G>,
    pub o_comm: PolyComm<G>,
    pub z_comm: PolyComm<G>,
    pub t_comm: PolyComm<G>,
}

However, kimchi prover is similar to used plonk_5_wires

pub struct ProverCommitments<G: AffineCurve> {
    pub w_comm: [PolyComm<G>; COLUMNS],
    pub z_comm: PolyComm<G>,
    pub t_comm: PolyComm<G>,
    pub lookup: Option<LookupCommitments<G>>,
}

• GateType enum changed;

// circuits::plom-5-wires::gate::GateType
pub enum GateType
{
    Zero,       // zero gate
    Generic,    // generic arithmetic gate
    Poseidon,   // Poseidon permutation gate
    Add,        // Gate constraining EC addition in Affine form
    Double,     // Gate constraining EC point doubling in Affine form
    Vbmul1,     // Gate constraining EC variable base scalar multiplication 
    Vbmul2,     // Gate constraining unpacking EC variable base scalar multiplication 
    Endomul,    // Gate constraining EC variable base scalar multiplication with group endomorphim optimization
    Pack,       // Gate constraining packing
}

// plonk_circuits::gate::GateType
pub enum GateType {
    Zero,    // zero gate
    Generic, // generic arithmetic gate

    Poseidon, // Poseidon permutation gate

    Add1, // Gate constraining EC addition in Affine form
    Add2, // Gate constraining EC point abscissa distinctness

    Vbmul1, // Gate constraining EC variable base scalar multiplication
    Vbmul2, // Gate constraining EC variable base scalar multiplication
    Vbmul3, // Gate constraining EC variable base scalar multiplication

    Endomul1, // Gate constraining EC variable base scalar multiplication with group endomorphim optimization
    Endomul2, // Gate constraining EC variable base scalar multiplication with group endomorphim optimization
    Endomul3, // Gate constraining EC variable base scalar multiplication with group endomorphim optimization
    Endomul4, // Gate constraining EC variable base scalar multiplication with group endomorphim optimization
}

However, kimchi circuit gate enum is similar to used plonk_5_wires

CompleteAdd instead of Add, etc

pub enum GateType {
    /// zero gate
    Zero = 0,
    /// generic arithmetic gate
    Generic,
    /// Poseidon permutation gate
    Poseidon,
    /// Complete EC addition in Affine form
    CompleteAdd,
    /// EC variable base scalar multiplication
    Vbmul,
    /// EC variable base scalar multiplication with group endomorphim optimization
    Endomul,
    /// Gate for computing the scalar corresponding to an endoscaling
    EndomulScalar,
    /// ChaCha
    ChaCha0,
    ChaCha1,
    ChaCha2,
    ChaChaFinal,
}

Commit 6 -- Use kimchi PLONK

Current state:

• A bunch of trait bounds are not implemented/respected
• GateType interface still does not match

@imeckler, since you have a much deeper understanding of the crates used, do you have tips/guidance on how to proceed from here? Thanks!