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Efficiently factoring polynomials modulo
Polynomial factoring has famous practical algorithms over fields-- finite,
rational \& -adic. However, modulo prime powers it gets hard as there is
non-unique factorization and a combinatorial blowup ensues. For example, is irreducible, but has exponentially many
factors! We present the first randomized poly(deg ) time algorithm
to factor a given univariate integral modulo , for a prime and
. Thus, we solve the open question of factoring modulo posed in
(Sircana, ISSAC'17).
Our method reduces the general problem of factoring to that
of {\em root finding} in a related polynomial for some irreducible . We could
efficiently solve the latter for , by incrementally transforming .
Moreover, we discover an efficient and strong generalization of Hensel lifting
to lift factors of to those (if possible). This was
previously unknown, as the case of repeated factors of forbids
classical Hensel lifting.Comment: 22 page