We consider the noisy polynomial interpolation problem of recovering an unknown $s$-sparse polynomial $f\left(X\right)$ 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 $f\left(t\right)\in {\mathbb{Z}}_{p^k}$. 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 $f\left(t\right)$ for sufficiently many randomly chosen points $t\in {\mathbb{Z}}_{p^k}^{\ast }$, recovers $f\left(X\right)$.

中文翻译：

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噪声多项式内插模幂

我们考虑恢复未知数的噪声多项式插值问题$s$-稀疏多项式 $F（X）$ 在环上 ${\mathbb{ž}}_{p^{ķ}}$ 模余数 $p^{ķ}$， 在哪里 $p$ 是一个小的素数 $ķ$ 是一个大的整数参数，来自残差的近似值 $F（Ť）\in {\mathbb{ž}}_{p^{ķ}}$。对于以大素数为模的残基，已知相似的结果$p$但是，素数功率模量的情况 $p^{ķ}$，小 $p$ 和大 $ķ$，是新的，需要使用不同的技术。我们给出了确定性的多项式时间算法，该算法几乎给出了一半以上的$F（Ť）$ 对于足够多的随机选择的点 $Ť\in {\mathbb{ž}}_{p^{ķ}}^{\ast }$，恢复 $F（X）$。