In some applications of univariate KZG the polynomial is given in evaluation form (as d evaluations p(1),...,p(d) of polynomial p at input points 1,...,d) instead of the conventional coefficient form (as d coefficients of p). Example: KZG as a vector commitment.

Currently we enforce coefficient form at the API level: commit, open and their batch/multi variants all require a Self::Polynomial in coeff form:

This API can be used for polynomials in evaluation form, but it forces the user to do a bunch of unnecessary and expensive FFTs.

It is possible to do all KZG operations natively in evaluation form, with no need for FFT anywhere in the stack: see here. (Skip to "evaluation form" section.)

Suggestions

1. Low-level. Add a mirror API that accepts polynomials in evaluation form. This is a good opportunity to coordinate with Eliminate mem allocation/copy in PCS API #338
2. High-level. Same API. Set Self::Polynomial to a type that abstracts over the underlying representation (coeff vs. eval). Methods such as commit, open, etc select the underlying implementation according to the underlying polynomial representation.

I prefer a low-level solution such as item 1 but that's open to discussion.