Randomized Polynomial-Time Root Counting in Prime Power Rings

Abstract: Suppose $k,p!\in!\mathbb{N}$ with $p$ prime and $f!\in!\mathbb{Z}[x]$ is a univariate polynomial with degree $d$ and all coefficients having absolute value less than $p<sup>k$. We give a Las Vegas randomized algorithm that computes the number of roots of $f$ in $\mathbb{Z}/!\left(p<sup>k\right)$ within time $d^3(k\log p)<sup>{2+o(1)}$. (We in fact prove a more intricate complexity bound that is slightly better.) The best previous general algorithm had (deterministic) complexity exponential in $k$. We also present some experimental data evincing the potential practicality of our algorithm.