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 polynomials 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 $f(X)$.