Sparse Polynomial Interpolation with Finitely Many Values for the Coefficients

TL;DR

This paper introduces new algorithms for sparse polynomial interpolation with coefficients from a finite set, utilizing a modified Kronecker substitution to efficiently handle multivariate cases with polynomial complexity.

Contribution

The paper presents novel sparse interpolation algorithms for polynomials with coefficients from finite sets, including a univariate method and a multivariate approach using modified Kronecker substitution.

Findings

Univariate interpolation from a single evaluation at a large point a.

Multivariate interpolation reduced to univariate case via modified Kronecker substitution.

Algorithms achieve polynomial bit-size complexity.

Abstract

In this paper, we give new sparse interpolation algorithms for black box polynomial f whose coefficients are from a finite set. In the univariate case, we recover f from one evaluation of f(a) for a sufficiently large number a. In the multivariate case, we introduce the modified Kronecker substitution to reduce the interpolation of a multivariate polynomial to the univariate case. Both algorithms have polynomial bit-size complexity.