Noisy polynomial interpolation modulo prime powers

TL;DR

This paper introduces a deterministic polynomial-time algorithm for noisy polynomial interpolation over rings modulo prime powers, extending known results from prime moduli to prime power moduli with small primes and large exponents.

Contribution

It provides the first known deterministic algorithm for recovering sparse polynomials over rings modulo prime powers from approximate evaluations.

Findings

Algorithm recovers polynomials with more than half the bits of evaluations.

Works for small primes p and large exponents k.

Applicable to almost all randomly chosen points.

Abstract

We consider the {\it noisy polynomial interpolation problem\/} of recovering an unknown $s$-sparse polynomial $f(X)$ over the ring $\mathbb Z_{p^k}$ of residues modulo $p^k$, where $p$ is a small prime and $k$ is a large integer parameter, from approximate values of the residues of $f(t) \in \mathbb Z_{p^k}$. Similar results are known for residues modulo a large prime $p$, however the case of prime power modulus $p^k$, with small $p$ and large $k$, is new and requires different techniques. We give a deterministic polynomial time algorithm, which for almost given more than a half bits of $f(t)$ for sufficiently many randomly chosen points $t \in \mathbb Z_{p^k}^*$, recovers $f(X)$.