Polynomial factoring has famous practical algorithms over fields– finite, rational and p-adic. However, modulo prime powers, factoring gets harder because there is non-unique factorization and a combinatorial blowup ensues. For example, $x^2+p\mathrm{mod}{p}^2$ is irreducible, but $x^2+px\mathrm{mod}{p}^2$ has exponentially many factors in the input size (which here is logarithmic in p)! We present the first randomized poly($\mathrm{deg}f,\log p$) time algorithm to factor a given univariate integral polynomial f modulo $p^k$, for a prime p and $k\le 4$.¹ Thus, we solve the open question of factoring modulo $p^3$ posed in (Sircana, ISSAC'17).

Our method reduces the general problem of factoring $f\mathrm{mod}{p}^k$ to that of root finding of a related polynomial $E(y)\mathrm{mod}〈p^k,\phi {(x)}^{\ell }〉$ for some irreducible $\phi \mathrm{mod}p$. We can efficiently solve the latter for $k\le 4$, by incrementally transforming E. Moreover, we discover an efficient refinement of Hensel lifting to lift factors of $f\mathrm{mod}p$ to those $\mathrm{mod}{p}^4$ (if possible). This was previously unknown, as the case of repeated factors of $f\mathrm{mod}p$ forbids classical Hensel lifting.

多项式因式分解在有限，有理和p -adic领域具有著名的实用算法。但是，由于存在非唯一的因式分解和随之而来的组合爆炸，因此对模素数幂进行因式分解变得更加困难。例如，$X^2+p\mathrm{模}{p}^2$ 是不可约的，但是 $X^2+pX\mathrm{模}{p}^2$输入大小有成倍的因素（这里是p的对数）！我们提出第一个随机poly（$\mathrm{度}F，\mathrm{日志}p$时间算法分解给定的单变量积分多项式f模$p^{ķ}$，对于素数p和$ķ\le 4$。¹因此，我们解决了模因式分解的开放性问题$p^3$ 摆在（Sircana，ISSAC'17）。

我们的方法减少了分解的一般问题 $F\mathrm{模}{p}^{ķ}$到相关多项式求根的结果$Ë（ÿ）\mathrm{模}〈p^{ķ}，\phi {（X）}^{\ell }〉$ 对于一些不可约 $\phi \mathrm{模}p$。我们可以有效地解决后者$ķ\le 4$通过逐步转化ê。此外，我们发现了对Hensel提升的有效改进，以提升以下因素：$F\mathrm{模}p$ 对那些 $\mathrm{模}{p}^4$（如果可能的话）。这是以前未知的，因为重复因素是$F\mathrm{模}p$ 禁止经典的Hensel举升。