Mega PR: First iteration of prover and verifier

[luckyyang]

Implemented a rough prover and verifier, includes:

• algebra implementation of Zq, Rq and matrix
• all prover's core computation
• all verifier's core computation

Next stage work:

• choosing proper parameters(now just use fix numbers)
• add recursion

[Junochiu]

perhaps no need to address the comments, we could discuss and if the issue make sense we create issues to fix after the pr is merged

[Junochiu]

thanks for the work 🙏, will approve the pr 🙏 we could still check the discussion here and continue the fix with issues after merging.

[FoodChain1028]

My advice in algebra :)

(1) The coefficient of $R_q$/ PolynomialRing should be a fixed number.

https://github.com/zhenfeizhang/larkworks/blob/10273d10c5dc5efdeb777edaa27bcbeeb57a17b6/algebra/src/polynomial/zz_px.rs#L25-L31C2

impl<C: ConfigZZpX> Default for ZZpX<C> {
    fn default() -> Self {
        Self {
            coeffs: vec![ZZp::<C::BaseConfig>::default(); C::DIM],
        }
    }
}

(2) should check overflow in $Z_q$
(3) We can use a better way to init a PolynomialRing, since we don't want the user can only use just one kind of modulus Q and polynomial (i.e., $X^{64}+1$).
(4) some of the functions in utils.rs can be turned into the arithmetical operations in PolynomialRing

example:

impl Add for PolynomialRing{...}

[Junochiu]

linked the unresolved comments to the issues, we'll merge this pr and continue the fixes from there 🚀