Given a primitive polynomial (gcd of coefficients is $1$) in $\Bbb Z[x]$ we have a polynomial time factoring algorithm for this that runs in time polynomial in degree $d$ and number of bits in coefficients. This is the Lenstra-Lenstra-Lovasz algorithm.
$1$. Arrange members of $\Bbb Z[x]$ with some bound on coefficients in some order. If I ask if the $t$th factor has $j$th bit in $i$th coefficient a $1$ is this algorithm known to give an $ACC^0$ circuit?
$2$. What is the smallest complexity class for the same query as $1$ if I say I only need at least in a randomized sense and for an algorithm that factors only a constant fraction of primitive polynomials?
$3$. Consider the overall Hensel lifting step in Zassenhaus' algorithm. If I ask is the $i$th digit has $j$th bit $1$ when lifting to $p^t$ can we have an $ACC^0$ circuit? Hensel lifting is just reading $p$-adic digits and may be this step is in $ACC^0$? If so Zassenhaus' algorithm would give $ACC^0$ circuit for thus answering $2$. There is a linear system interpretation here in section 8.1.2 https://www.nada.kth.se/~johanh/algnotes.pdf which might put the decision version in here to be in $ACC^0$.
Refer algorithm 10 and theorem 2 here http://www.mmrc.iss.ac.cn/~lzhi/Course/mchapter15.ps.
Please also check corollary $1.79$ here http://compalg.inf.elte.hu/~tony/KedvencKonyvek/InfoKonyvtar/03-Algorithms%20of%20Informatics%201,%202,%203/AlgofInfVol1Oct24.pdf for a log depth implementation of Hensel Lifting. Can it be reduced to constant depth?