New PCS trait and implementation

[mrain]

Description

closes: #62

• Introduct new PolynomialCommitmentScheme trait and basic implementations
• Now PlonkKzgSnark use our own KZG10 implementation

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[alxiong]

Can add more trait bounds such as:
type Proof: Clone + Serialize + CanonicalSerialize + Debug + PartialEq...

can take a look at PairingEngine's associated type, they inherented a lot of basic traits.

[alxiong]

@mrain, @chancharles92 maybe use this?

    /// Prover parameters
    type ProverParam: Clone;
    /// Verifier parameters
    type VerifierParam: Clone + CanonicalSerialize + CanonicalDeserialize;
    /// Structured reference string
    type SRS: Clone + Debug;
    /// Polynomial and its associated types
    type Polynomial: Sized + Clone + Debug + Hash + PartialEq + Eq + Add + Neg + Zero + CanonicalSerialize + CanonicalDeserialize + for<'a> AddAssign<&'a Self> + for<'a> AddAssign<(F, &'a Self)> + for<'a> SubAssign<&'a Self>;
    /// Polynomial input domain
    type Point:: Clone + Ord + Debug + Sync + Hash + PartialEq + Eq;
    /// Polynomial Evaluation
    type Evaluation: Field;
    /// Commitments
    type Commitment: Clone + CanonicalSerialize + CanonicalDeserialize + Debug + PartialEq + Eq;
    /// Batch commitments
    type BatchCommitment: Clone + CanonicalSerialize + CanonicalDeserialize + Debug + PartialEq + Eq;
    /// Proofs
    type Proof: Clone + CanonicalSerialize + CanonicalDeserialize + Debug + PartialEq + Eq;
    /// Batch proofs
    type BatchProof: Clone + CanonicalSerialize + CanonicalDeserialize + Debug + PartialEq + Eq;

[alxiong]

• type Polynomial should be greatest common subset of traits bound from Polynomial and MultilinearExtension.
• type Point maybe exclude Copy

(I've updated my comment above)

[mrain]

Thanks @alxiong . But for Polynomial, Rc used in multilinear setting could not fit in this trait bound.
I think it make sense to put polynomial in heap because it's usually quite large.

[alxiong]

I agree, but similar to the argument for impl Borrow<>, how do you know what type of smart pointer one wants?

I don't think we should fix smart pointer type at the associate type level, (namely MultilinearKzgPCS::Polynomial = Rc<DenseMultilinearExtension<E::Fr>> is not as flexible as just DenseMultilinearExtension<E::Fr>.

and you can always wrap this MLE into some smart pointer, such as Rc, RefCell, or Arc etc.

@chancharles92 what's your opinion?

[chancharles92]

I don't have a strong opinion. Actually we were using Polynomial = DenseMultilinearExtension, but later Zhenfei changed it to Polynomial = Rc<Dense...> which has a few advantages:

1. It significantly simplifies the code, o/w we need to add Rc everywhere.
2. It make sure that the developer never forget to use Rc, otherwise the code performance can be accidentally effected.
3. When we want to use other smart pointers, we can e.g. use Polynomial = Arc<...> in associated type rather than change it everywhere.

[alxiong]

I give a provisional approval on my comments #116 (comment) being settled.