Sparse Polynomial Interpolation Based on Derivative

TL;DR

This paper introduces two new algorithms for sparse multivariate polynomial interpolation over finite fields, one randomized and one deterministic, both with improved complexity bounds compared to existing methods.

Contribution

The paper presents novel randomized and deterministic algorithms for sparse polynomial interpolation with better complexity over large characteristic finite fields.

Findings

Randomized algorithm has linear complexity in non-zero terms and variables.

Deterministic algorithm has quadratic complexity in terms of variables, terms, and degree.

Both algorithms outperform existing methods in their respective categories.

Abstract

In this paper, we propose two new interpolation algorithms for sparse multivariate polynomials represented by a straight-line program(SLP). Both of our algorithms work over any finite fields $F_q$ with large characteristic. The first one is a Monte Carlo randomized algorithm. Its arithmetic complexity is linear in the number $T$ of non-zero terms of $f$, in the number $n$ of variables. If $q$ is $O((nTD)^{(1)})$, where $D$ is the partial degree bound, then our algorithm has better complexity than other existing algorithms. The second one is a deterministic algorithm. It has better complexity than existing deterministic algorithms over a field with large characteristic. Its arithmetic complexity is quadratic in $n,T,\log D$, i.e., quadratic in the size of the sparse representation. And we also show that the complexity of our deterministic algorithm is the same as the one of deterministic zero-testing of Bl\"{a}ser et al. for the polynomial given by an SLP over finite field (for large characteristic).