Drinfeld Modules with Complex Multiplication, Hasse Invariants and Factoring Polynomials over Finite Fields

Abstract

We present a novel randomized algorithm to factor polynomials over a finite field $\\F_q$ of odd characteristic using rank $2$ Drinfeld modules with complex multiplication. The main idea is to compute a lift of the Hasse invariant (modulo the polynomial $f \\in \\F_q[x]$ to be factored) with respect to a random Drinfeld module $\\phi$ with complex multiplication. Factors of $f$ supported on prime ideals with supersingular reduction at $\\phi$ have vanishing Hasse invariant and can be separated from the rest. Incorporating a Drinfeld module analogue of Deligne s congruence, we devise an algorithm to compute the Hasse invariant lift, which turns out to be the crux of our algorithm. The resulting expected runtime of $n^{3/2+\\varepsilon} (\\log q)^{1+o(1)}+n^{1+\\varepsilon} (\\log q)^{2+o(1)}$ to factor polynomials of degree $n$ over $\\F_q$ matches the fastest previously known algorithm, the Kedlaya-Umans implementation of the Kaltofen-Shoup algorithm.