Feat/poly

[Tabaie]

Polynomial module from gkr project. If most things look good I'll clean up into a proper PR.

[Tabaie]

No longer true

[Tabaie]

I've seen the Text method does something similar to this. Should we extract that logic into a ToSignedBigInt that Text also calls?

[gbotrel]

mmhh for the Text one, it actually did give me issues in the past; I think the Text / String should never output "negative" formating. And we should add a "prettyString" or something like that which does.

Current way it's done in Text doesn't allocate a big int I believe ?

[Tabaie]

It does!
https://github.com/ConsenSys/gnark-crypto/blob/master/ecc/bn254/fp/element.go#L824

Also, in this case using a pool makes sense because big.Int.SetBytes does sometimes reuse previously allocated memory correct?

[gbotrel]

yes correct . We should by the way probably mutualize the big int pool declared in fr/ fp/ on each curves. Will create a ticket 👍

[Tabaie]

@gbotrel Is it actually useful to pool string builders? I read somewhere that when a builder is done, the buffer is handed to the generated string and another buffer would be allocated for the next use. If that's the case, it makes no sense to pool these.

[gbotrel]

probably not useful, and fine to do like the go std string builder 👍

[ThomasPiellard]

@Tabaie I don't understand what does computeLagrangeBasis do ? Also, for multilinear polynomials I don't think that it's necessary to express them in another basis than the Lagrange basis on the hypercube. My understanding was that each time we deal with a vector [a0, .. , a_n], the i-th entry would implicitly correspond to the the i-th corner of the cube (i decomposed in base 2)

[Tabaie]

In the GKR project they do a lot of extrapolations on small domains of the form {0, 1, ..., n-1} into an explicit polynomial (I think, I haven't seen all of the repo yet but I assume that's how it is used). So this function is used to precompute the polynomials p_0 = (X-1)...(X-n)/ ((-1)(-2)...(-n)), p_1 = X(X-2)(X-3)...(X-n)/((1-0)(1-2)...(1-n)) etc so that a function X: {0, 1, ..., n-1} can be easily extrapolated as \sum_{i=0}^{n-1} X(i) p_i.

[Tabaie]

The multilinear ones are denoted by their values on the hypercube. This Lagrange thing is for univariate functions.

[ThomasPiellard]

Ah ok. And why the FFT isn't used for this ? Are the degrees in play too small ?

[Tabaie]

Yes the maximum size is 12. I think that's too small for FFT to pay off. What do you think?

[ThomasPiellard]

oh ok yes in that case it's better to do it by hands.