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 pk. 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(2n)⋅poly(m,n) to O(20.8765n)⋅poly(m,n); which currently is improved to O(20.6943n)⋅poly(m,n) in Dinur (SODA'21). For large p but constant n, Huang and Wong (FOCS'96) gave a randomized 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 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 pk; 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 pk); when k>1 is constant. The challenge here is to find a lift of singularFp-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∈Z[x] and a prime-power pk (k≥2), finding factors of fmodpk 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≥5 (Dwivedi, Mittal, Saxena, ISSAC'19). One issue is that degree-δ factors of f(x)modpk could be exponentially many, as soon as k≥2. We give the first randomized poly(deg⁡(f),log⁡p)-time algorithm to find a degree-δ factor of f(x)modpk, when k+δ 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.