Add FK23 batch opening of univariate polynomials

[alxiong]

Description

Partially completes: #118

Major changes include:

• add trait bound to associated type of the PCS trait: PolynomialCommitmentScheme::SRS: StructuredReferenceString.
• define Toeplitz and Circulant matrix, impl conversion between them.
• Implement fast mul with a vector for circulant matrix
• implement fast mul with a vector for toeplitz by expanding into a 2N circulant matrix and use above subroutine
• add correctness test for matrix multiplications.
• Define a GeneralDensePolynomial that allows its coeffs to be either field or group, and implement batch_evaluate
• FK23 methods using all above, with unit tests.

More important changes when addressing comments:

• hide gen_srs_for_testing under test-srs feature flag (and also available for test) so that downstream and bench code can still use it, but won't be in the default or production-ready binary.

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[alxiong]

dev log (safely skip for reviewer):

I made the wrong decision to make my Toeplitz APIs constant generic rather than just accepting a vector and checking their lengths inside. (the former is good when we know the size at compile-time, the latter is better for dynamic and run-time variables)

I will need to update my code cuz the rigidity of const-generic propagate up to where I try to use my own APIs in batch_open, and it's a hustle.

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Update: fixed in 8e239e1

[chancharles92]

Thanks @alxiong! Generally LGTM but added a few minor comments.

[ggutoski]

Whew that was big. Minor comments only.

[ggutoski]

lgtm only a couple minor tweaks

[ggutoski]

one comment, but we can push that to a future issue if we decide it's worth addressing. lgtm.

[alxiong]

Log for future reference

I shall document briefly on the slight complication of computing the degree/coeffs for FK23 to work for future reference.

Firstly, we are opening a polynomial f(X) = f_0 + f_1 * X + ... + f_d * X^d, (degree: d, num_coeffs: d+1),
the resulting h(X) = h_1 + h_2 * X + ... + h_d * X^d-1 (degree: d-1, num_coeffs: d).

Now we accept f of any degree, and in order for FK23 to work, we need to internally use Toeplitz matrix which works only for degree d power of two, therefore some degree padding is required.
Furthermore, we also need to make sure the roots of unity applied in evaluating h(X) (which are proofs) and in evaluating f(X) (which are evaluations) are the same.

	$$d = 2^k$$	$$d = 2^k - 1$$	$$d = 2^k + 1$$
FFT(n)/rou (i.e. padded h(X).coeffs.len() )	$$2^{k+1}$$	$$2^k$$	$$2^{k+1}$$
Teoplitz padded degree(i.e. next_pow_of_two)	$$2^k$$	$$2^k$$	$$2^{k+1}$$

and for trim_fft_size() API, it should accomondate the FFT size (which is no smaller than the padded degree).

[chancharles92]

Approved. Hooray!