Noisy polynomial interpolation modulo prime powers

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 $t\in {\mathbb{Z}}_{p^k}^{\ast }$, recovers $f(X)$.