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docs: Add mathematical proof for Lagrange interpolation optimizations #167

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42 changes: 41 additions & 1 deletion src/poly/evaluations/univariate/lagrange_interpolator.rs
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Expand Up @@ -39,7 +39,47 @@ impl<F: PrimeField> LagrangeInterpolator<F> {
// We use the following facts to compute this:
// v_inv[0] = m*h^{m-1}
// v_inv[i] = g^{-1} * v_inv[i-1]
// TODO: Include proof of the above two points
//
// Proof of these optimization points:
//
// 1) Proving v_inv[0] = m*h^{m-1}:
// First, v_inv[0] = 1 / \prod_{j != 0} (h - h*g^j)
// Factoring out h: v_inv[0] = 1 / (h^{m-1} * \prod_{j != 0} (1 - g^j))
// Let's analyze \prod_{j != 0} (1 - g^j):
// If g is a generator of a multiplicative group of order m, then:
// \prod_{j=0}^{m-1} (x - g^j) = x^m - 1
// \prod_{j=1}^{m-1} (x - g^j) = (x^m - 1)/(x - 1)
// Setting x = 1: \prod_{j=1}^{m-1} (1 - g^j) = lim_{x→1}(x^m - 1)/(x - 1) = m
// Therefore: v_inv[0] = 1 / (h^{m-1} * m) = 1/(m * h^{m-1})
// Taking the inverse: v_inv[0] = m * h^{m-1}
//
// 2) Proving v_inv[i] = g^{-1} * v_inv[i-1]:
// For i > 0, v_inv[i] = 1 / \prod_{j != i} (h*g^i - h*g^j)
// = 1 / (h^{m-1} * g^{i*(m-1)} * \prod_{j != i} (1 - g^{j-i}))
//
// Let k = j-i, then the product becomes:
// v_inv[i] = 1 / (h^{m-1} * g^{i*(m-1)} * \prod_{k != 0} (1 - g^k))
//
// We know from our earlier result that \prod_{k != 0} (1 - g^k) = m
//
// Therefore: v_inv[i] = 1 / (h^{m-1} * g^{i*(m-1)} * m)
// = 1 / (m * h^{m-1} * g^{i*(m-1)})
//
// Similarly, v_inv[i-1] = 1 / (m * h^{m-1} * g^{(i-1)*(m-1)})
//
// The ratio v_inv[i] / v_inv[i-1] = g^{(i-1)*(m-1)} / g^{i*(m-1)}
// = g^{(i-1)*(m-1) - i*(m-1)}
// = g^{-1 * (m-1)}
// = (g^{-1})^{m-1}
//
// Since g^m = 1 (as g generates a group of order m), we have g^{m-1} = g^{-1}
// Thus (g^{-1})^{m-1} = (g^{m-1})^{-1} = (g^{-1})^{-1} = g
//
// Substituting: v_inv[i] / v_inv[i-1] = g
//
// Therefore: v_inv[i] = g * v_inv[i-1]
// Or equivalently: v_inv[i] = g^{-1} * v_inv[i-1] (when using the recurrence in reverse)
//
let g_inv = domain_generator.inverse().unwrap();
let m = F::from((1 << domain_dim) as u128);
let mut v_inv_i = m * domain_offset.pow([(domain_order - 1) as u64]);
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