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DOI: 10.4230/LIPIcs.CCC.2019.15
Counting Basic-Irreducible Factors Mod p^k in Deterministic Poly-Time and p-Adic Applications

Authors: Ashish Dwivedi, Rajat Mittal, and Nitin Saxena

Published in: LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)

Abstract
Finding an irreducible factor, of a polynomial f(x) modulo a prime p, is not known to be in deterministic polynomial time. Though there is such a classical algorithm that counts the number of irreducible factors of f mod p. We can ask the same question modulo prime-powers p^k. The irreducible factors of f mod p^k blow up exponentially in number; making it hard to describe them. Can we count those irreducible factors mod p^k that remain irreducible mod p? These are called basic-irreducible. A simple example is in f=x^2+px mod p^2; it has p many basic-irreducible factors. Also note that, x^2+p mod p^2 is irreducible but not basic-irreducible! We give an algorithm to count the number of basic-irreducible factors of f mod p^k in deterministic poly(deg(f),k log p)-time. This solves the open questions posed in (Cheng et al, ANTS'18 & Kopp et al, Math.Comp.'19). In particular, we are counting roots mod p^k; which gives the first deterministic poly-time algorithm to compute Igusa zeta function of f. Also, our algorithm efficiently partitions the set of all basic-irreducible factors (possibly exponential) into merely deg(f)-many disjoint sets, using a compact tree data structure and split ideals.
Cite as

Ashish Dwivedi, Rajat Mittal, and Nitin Saxena. Counting Basic-Irreducible Factors Mod p^k in Deterministic Poly-Time and p-Adic Applications. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 15:1-15:29, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dwivedi_et_al:LIPIcs.CCC.2019.15,
  author =	{Dwivedi, Ashish and Mittal, Rajat and Saxena, Nitin},
  title =	{{Counting Basic-Irreducible Factors Mod p^k in Deterministic Poly-Time and p-Adic Applications}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{15:1--15:29},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-116-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{137},
  editor =	{Shpilka, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.15},
  URN =		{urn:nbn:de:0030-drops-108373},
  doi =		{10.4230/LIPIcs.CCC.2019.15},
  annote =	{Keywords: deterministic, root, counting, modulo, prime-power, tree, basic irreducible, unramified}
}
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