Identity Testing and Interpolation from High Powers of Polynomials of Large Degree over Finite Fields

TL;DR

This paper introduces an improved algorithm for identity testing and polynomial interpolation over finite fields using high powers, especially effective for large degree polynomials, with roots in additive combinatorics.

Contribution

It presents a novel algorithm that outperforms naive methods for interpolating high-degree polynomials from power queries over finite fields, leveraging new bounds from additive combinatorics.

Findings

Algorithm reduces the number of queries needed for interpolation.

Provides bounds on rational functions evaluated on subgroups in finite fields.

Demonstrates improved efficiency for large degree polynomials.

Abstract

We consider the problem of identity testing and recovering (that is, interpolating) of a "hidden" monic polynomials $f$, given an oracle access to $f(x)^e$ for $x\in\mathbb F_q$, where $\mathbb F_q$ is the finite field of $q$ elements and an extension fields access is not permitted. The naive interpolation algorithm needs $de+1$ queries, where $d =\max\{{\rm deg}\ f, {\rm deg }\ g\}$ and thus requires $ de<q$. For a prime $q = p$, we design an algorithm that is asymptotically better in certain cases, especially when $d$ is large. The algorithm is based on a result of independent interest in spirit of additive combinatorics. It gives an upper bound on the number of values of a rational function of large degree, evaluated on a short sequence of consecutive integers, that belong to a small subgroup of $\mathbb F_p^*$.