Query-Efficient Algorithms for Polynomial Interpolation over Composites

Parikshit Gopalan

Abstract: The problem of polynomial interpolation is to reconstruct a polynomial based on its valuations on a set of inputs I. We consider the problem over $\mathbb{Z}_m$ when m is composite. We ask the following question: Given $I \subseteq \mathbb{Z}_m$, how many evaluations of a polynomial at points in I are required to compute its value at every point in I? Surprisingly for composite m, this number can vary exponentially between $\log |I|$ and $|I|$, in contrast to the prime case where $|I|$ evaluations are necessary. While we show this minimization problem to be NP-hard, we give an efficient algorithm of query complexity within a factor t of the optimum, where t is the number of prime factors of m. We use our interpolation algorithm to design algorithms for zero testing and distributional learning of polynomials over $\mathbb{Z}_m$. In some cases, we get an exponential improvement over known algorithms in query complexity and running time. Our main technical contribution is the notion of an interpolating set for I which is a subset S of I such that a polynomial which is 0 over S must be 0 at every point in I. Any interpolation algorithm needs to query an interpolating set for I. Our query-efficient algorithms are obtained by constructing interpolating sets whose size is close to optimal.