Improved Local Leakage Resilience of Shamir Secret Sharing and Worst-Case Optimal Polynomial Intersection

We study two problems: Local Leakage Resilience (LLR) for Shamir secret sharing, and worst-case Optimal Polynomial Intersection (OPI). Both problems concern polynomials $Q(X)$ of degree less than $k$, over a prime-order finite field $\mathbb{F}_p$. In LLR for Shamir secret sharing, one asks how much