Factoring Polynomials over Finite Fields using Drinfeld Modules with Complex Multiplication

Abstract: We present novel algorithms 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 $f(x) \in \F_q[x]$ to be factored) with respect to a Drinfeld module $\phi$ with complex multiplication. Factors of $f(x)$ supported on prime ideals with supersingular reduction at $\phi$ have vanishing Hasse invariant and can be separated from the rest. A Drinfeld module analogue of Deligne's congruence plays a key role in computing the Hasse invariant lift. We present two algorithms based on this idea. The first algorithm chooses Drinfeld modules with complex multiplication at random and has a quadratic expected run time. The second is a deterministic algorithm with $O(\sqrt{p})$ run time dependence on the characteristic $p$ of $\F_q$.