Polynomial Interpolation over Finite Domains is a mathematical mechanism in algebraic number theory and coding theory that reconstructs a unique function from its values at distinct points within a finite field, typically represented by Galois Fields (GF(q)). The core principle relies on the Lagrange interpolation formula adapted for modular arithmetic, ensuring that any polynomial of degree k or less can be uniquely determined if it is evaluated at k+1 non-identical elements of the domain. This theoretical construct serves as a foundational subfield within algebraic geometry and cryptography, specifically enabling deterministic information reconstruction without relying on continuous real-number approximations.
Introduction to Lagrange interpolation: constructing the unique polynomial through a set of points via Lagrange basis polynomials and the interpolation formula.