Efficiently factoring polynomials modulo p^4
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Polynomial factoring has famous practical algorithms over fields– finite, rational & p-adic. However, modulo prime powers it gets hard as there is non-unique factorization and a combinatorial blowup ensues. For example, x^2+p p^2 is irreducible, but x^2+px p^2 has exponentially many factors! We present the first randomized poly(deg f, log p) time algorithm to factor a given univariate integral f(x) modulo p^k, for a prime p and k ≤ 4. 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(x) p^k to that of root finding in a related polynomial E(y) ⟨ p^k, φ(x)^ℓ⟩ for some irreducible φ p. We could efficiently solve the latter for k≤4, by incrementally transforming E(y). Moreover, we discover an efficient and strong generalization of Hensel lifting to lift factors of f(x) p to those p^4 (if possible). This was previously unknown, as the case of repeated factors of f(x) p forbids classical Hensel lifting.
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