Revisit Sparse Polynomial Interpolation based on Randomized Kronecker Substitution

TL;DR

This paper introduces a new Monte Carlo-based reduction method for multivariate polynomial interpolation over finite fields, improving efficiency by combining it with a modified univariate algorithm.

Contribution

It presents a novel reduction technique and a modified univariate interpolation algorithm, achieving better or comparable complexity for sparse polynomial interpolation.

Findings

Enhanced multivariate interpolation efficiency over finite fields

Reduced complexity compared to existing algorithms

Effective for black-box polynomial sparse interpolation

Abstract

In this paper, a new reduction based interpolation algorithm for black-box multivariate polynomials over finite fields is given. The method is based on two main ingredients. A new Monte Carlo method is given to reduce black-box multivariate polynomial interpolation to black-box univariate polynomial interpolation over any ring. The reduction algorithm leads to multivariate interpolation algorithms with better or the same complexities most cases when combining with various univariate interpolation algorithms. We also propose a modified univariate Ben-or and Tiwarri algorithm over the finite field, which has better total complexity than the Lagrange interpolation algorithm. Combining our reduction method and the modified univariate Ben-or and Tiwarri algorithm, we give a Monte Carlo multivariate interpolation algorithm, which has better total complexity in most cases for sparse interpolation of black-box polynomial over finite fields.