Implement proof generation

nomos-da/kzgrs/src/kzg.rs

@@ -1,8 +1,12 @@
 use crate::common::KzgRsError;
 use ark_bls12_381::{Bls12_381, Fr};
 use ark_poly::univariate::DensePolynomial;
-use ark_poly_commit::kzg10::{Commitment, Powers, KZG10};
+use ark_poly::{DenseUVPolynomial, EvaluationDomain, GeneralEvaluationDomain, Polynomial};
+use ark_poly_commit::kzg10::{Commitment, Powers, Proof, KZG10};
+use num_traits::One;
 
+/// Commit to a polynomial where each of the evaluations are over `w(i)` for the degree
+/// of the polynomial being omega (`w`) the root of unity (2^x).
 pub fn commit_polynomial(
     polynomial: &DensePolynomial<Fr>,
     roots_of_unity: &Powers<Bls12_381>,
@@ -12,6 +16,26 @@ pub fn commit_polynomial(
         .map(|(commitment, _)| commitment)
 }
 
+/// Compute a witness polynomial in that satisfies `witness(x) = (f(x)-v)/(x-u)`
+fn generate_element_proof(
+    element_index: usize,
+    polynomial: &DensePolynomial<Fr>,
+    roots_of_unity: &Powers<Bls12_381>,
+    domain: &GeneralEvaluationDomain<Fr>,
+) -> Result<Proof<Bls12_381>, KzgRsError> {
+    let u = domain.element(element_index);
+    let v = polynomial.evaluate(&u);
+    let f_x_v = polynomial + &DensePolynomial::<Fr>::from_coefficients_vec(vec![-v]);
+    let x_u = DensePolynomial::<Fr>::from_coefficients_vec(vec![-u, Fr::one()]);
+    let witness_polynomial: DensePolynomial<_> = &f_x_v / &x_u;
+    let proof = commit_polynomial(&witness_polynomial, roots_of_unity)?;
+    let proof = Proof {
+        w: proof.0,
+        random_v: None,
+    };
+    Ok(proof)
+}
+
 #[cfg(test)]
 mod test {
     use crate::kzg::commit_polynomial;