Solving polynomial systems over non-fields and applications to modular polynomial factoring

Abstract

We study the problem of solving a system of m polynomials in n variables over the ring of integers modulo a prime-power $p^k$. The problem over finite fields is well studied in varied parameter settings. For small characteristic $p=2$, Lokshtanov et al. (SODA'17) initiated the study, for degree $d=2$ systems, to improve the exhaustive search complexity of $O\left(2^n\right)\cdot \text{poly}(m,n)$ to $O\left(2^{0.8765n}\right)\cdot \text{poly}(m,n)$; which currently is improved to $O\left(2^{0.6943n}\right)\cdot \text{poly}(m,n)$ in Dinur (SODA'21). For large p but constant n, Huang and Wong (FOCS'96) gave a randomized $\text{poly}(d,m,\log p)$ time algorithm. Note that for growing n, system-solving is known to be intractable even with $p=2$ and degree $d=2$.

We devise a randomized $\text{poly}(d,m,\log p)$-time algorithm to find a root of a given system of m integral polynomials of degrees bounded by d, in n variables, modulo a prime power $p^k$; when $n+k$ is constant. In a way, we extend the efficient algorithm of Huang and Wong (FOCS'96) for system-solving over Galois fields (i.e., characteristic p) to system-solving over Galois rings (i.e., characteristic $p^k$); when $k>1$ is constant. The challenge here is to find a lift of singular $F_p$-roots (exponentially many); as there is no efficient general way known in algebraic-geometry for resolving singularities.

Our algorithm has applications to factoring univariate polynomials over Galois rings. Given $f\in Z\left[x\right]$ and a prime-power $p^k$ ($k\ge 2$), finding factors of $f\mathrm{mod}{p}^k$ has a curious state-of-the-art. It is solved for large k by p-adic factoring algorithms (von zur Gathen, Hartlieb, ISSAC'96); but unsolved for small k. In particular, no nontrivial factoring method is known for $k\ge 5$ (Dwivedi, Mittal, Saxena, ISSAC'19). One issue is that degree-δ factors of $f\left(x\right)\mathrm{mod}{p}^k$ could be exponentially many, as soon as $k\ge 2$. We give the first randomized poly$(\mathrm{deg}(f),\log p)$-time algorithm to find a degree-δ factor of $f\left(x\right)\mathrm{mod}{p}^k$, when $k+\delta$ is constant. Our method has potential application in algebraic coding theory. In particular, extending algebraic geometric and Reed-Solomon codes to Galois rings could enable new and improved bounds on their underlying efficiency parameters.