Sparse Polynomial Interpolation and Division in Soft-linear Time

TL;DR

This paper introduces a new soft-linear time Monte Carlo algorithm for sparse polynomial interpolation and division, significantly improving efficiency in terms of bit complexity and handling polynomials with integer coefficients.

Contribution

It presents the first sparse interpolation algorithm with nearly optimal bit complexity, enabling efficient recovery of sparse polynomials and their quotients.

Findings

Algorithm achieves nearly-linear bit complexity in output size.

First sparse interpolation method with soft-linear bit complexity.

Efficient polynomial division for integer coefficient polynomials.

Abstract

Given a way to evaluate an unknown polynomial with integer coefficients, we present new algorithms to recover its nonzero coefficients and corresponding exponents. As an application, we adapt this interpolation algorithm to the problem of computing the exact quotient of two given polynomials. These methods are efficient in terms of the bit-length of the sparse representation, that is, the number of nonzero terms, the size of coefficients, the number of variables, and the logarithm of the degree. At the core of our results is a new Monte Carlo randomized algorithm to recover a polynomial $f(x)$ with integer coefficients given a way to evaluate $f(\theta) \bmod m$ for any chosen integers $\theta$ and $m$. This algorithm has nearly-optimal bit complexity, meaning that the total bit-length of the probes, as well as the computational running time, is softly linear (ignoring logarithmic factors) in the bit-length of the resulting sparse polynomial. To our knowledge, this is the first sparse interpolation algorithm with soft-linear bit complexity in the total output size. For polynomials with integer coefficients, the best previously known results have at least a cubic dependency on the bit-length of the exponents.