Replace naive Lagrange coefficient computation with barycentric interpolation

[JesseAbram]

Current Implementation

Currently, the DKG lagrange_at_zero() function computes:

λ_i = ∏_{j ∈ T, j ≠ i} (-j)/(i-j)

This is correct but naive O(t²) per resharing operation. For small committees it’s fine, but it becomes inefficient for larger committees or frequent resharing/refresh rounds.

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Proposed Improvement

Switch to barycentric Lagrange interpolation:

1. Compute barycentric weights once per set of participating nodes:

w_i = 1 / ∏_{j ≠ i} (i - j)

2. Compute Lagrange coefficients at 0:

λ_i = (w_i / i) / Σ_{k ∈ T} (w_k / k)

This reduces repeated computation for multiple resharing operations.

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Benefits

• More efficient for large committees (t > 50) and frequent resharing.
• Coefficients can be cached per participating set.
• Mathematically equivalent to the current computation (same security properties).
• Aligns with practices in FROST, BLS threshold schemes, and other MPC implementations.

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Suggested Implementation

• Add a lagrange_at_zero_barycentric() function.
• Use participating_ids to compute weights once.
• Use the weight for the current node to compute the reshared constant term:

let lambdas = lagrange_at_zero_barycentric(&participating_ids)?;
let index = participating_ids.iter().position(|id| *id == self.id)?;
let constant_term = lambdas[index] * old_share;

• Optional: cache computed weights for reuse.