Verified-zkEVM / Verified-zkEVM/CompPoly

Add fixed-domain barycentric interpolation for repeated-query univariate evaluation

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Dominant language
Lean
Stars
47
Forks
43
Avg merge
2d 6h
Merged PRs (30d)
53

Description

Summary

Add a fixed-domain barycentric interpolation surface to CompPoly.CPolynomial.CLagrange for repeated-query univariate evaluation over a field.

For pairwise distinct nodes x : Fin n → R, store the barycentric weights
w_i = ∏_{j ≠ i} (x_i - x_j)⁻¹
once and evaluate by the classical second barycentric formula:

  • if z = x_i, return y_i;
  • otherwise return
    (∑ i, w_i * y_i * (z - x_i)⁻¹) / (∑ i, w_i * (z - x_i)⁻¹).

Correctness target

Prove that for every y : Fin n → R and z : R,
BarycentricDomain.eval y z = (Lagrange.interpolate Finset.univ x y).eval z.

Also derive the CompPoly-facing corollary
BarycentricDomain.eval y z = (CLagrange.interpolate Finset.univ x y).eval z,
together with the roots-of-unity specialization agreeing with CLagrange.interpolatePow.

Motivation

This supports the Phase 2 evaluation/interpolation track by separating one-time nodal preprocessing from per-query evaluation on a fixed domain.

Contributor guide

Open the contributing guide

First steps

  1. Read the whole issue, then the project's contributing guide.
  2. Comment on the issue to say you are picking it up — it saves two people doing the same work.
  3. Fork the repository and make your change on a branch.
  4. Open a pull request that references the issue number.

Research direction

Start with the CompPoly.CPolynomial.CLagrange surface and the proposed BarycentricDomain.eval definition. Read the existing Lagrange.interpolate and CLagrange.interpolatePow results, then establish the stated equality for all y and z and the roots-of-unity specialization. Done means the fixed-domain evaluation, CompPoly corollary, and specialization are proved.

Written by the indexing model from the issue text.

Assessment

Domain
tooling
Issue type
Feature
Difficulty
5/5
Estimated time
Over a week
Activity status
Quiet
Clarity
Mostly clear
Newbie friendliness
45/100

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