Factoring Polynomials over Finite Fields using Drinfeld Modules with Complex Multiplication

TL;DR

This paper introduces new algorithms leveraging Drinfeld modules with complex multiplication to efficiently factor polynomials over finite fields, improving computational methods in algebraic number theory.

Contribution

It develops novel algorithms using Drinfeld modules with complex multiplication for polynomial factorization over finite fields, including a randomized quadratic-time and a deterministic sublinear-time method.

Findings

Randomized algorithm has quadratic expected runtime.

Deterministic algorithm runs in O(√p) time.

Effective separation of factors with supersingular reduction.

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 $\F_q$.