Deterministic Interpolation of Sparse Black-box Multivariate Polynomials using Kronecker Type Substitutions

TL;DR

This paper introduces two new deterministic algorithms for interpolating sparse multivariate polynomials using Kronecker substitutions, improving complexity bounds over existing methods especially in terms of degree D.

Contribution

The paper presents novel deterministic interpolation algorithms for sparse black-box polynomials using Kronecker substitutions, achieving improved complexity bounds in degree D.

Findings

Algorithms have better bit complexity over existing methods.

Second algorithm achieves the best known complexity in degree D.

Applicable over complex numbers and finite fields.

Abstract

In this paper, we propose two new deterministic interpolation algorithms for a sparse multivariate polynomial given as a standard black-box by introducing new Kronecker type substitutions. Let $f\in \RB[x_1,\dots,x_n]$ be a sparse black-box polynomial with a degree bound $D$. When $\RB=\C$ or a finite field, our algorithms either have better bit complexity or better bit complexity in $D$ than existing deterministic algorithms. In particular, in the case of deterministic algorithms for standard black-box models, our second algorithm has the current best complexity in $D$ which is the dominant factor in the complexity.