We present a novel randomized algorithm to factor polynomials over a finite field ${\mathbb{F}}_q$ 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 $f\in {\mathbb{F}}_q[x]$ to be factored) with respect to a random Drinfeld module ϕ with complex multiplication. Factors of f supported on prime ideals with supersingular reduction at ϕ 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 $n^{3/2+\epsilon }{(\log q)}^{1+o(1)}+n^{1+\epsilon }{(\log q)}^{2+o(1)}$ to factor polynomials of degree n over ${\mathbb{F}}_q$ matches the fastest previously known algorithm, the Kedlaya-Umans implementation of the Kaltofen-Shoup algorithm.

我们提出了一种新颖的随机算法来分解有限域上的多项式 ${\mathbb{F}}_q$使用具有复杂乘法的2级Drinfeld模块求奇数特性。主要思想是计算Hasse不变量的提升（对多项式取模）$F\in {\mathbb{F}}_q[X]$关于带复数乘法的随机Drinfeld模块ϕ。的因素˚F支持与超奇异还原素理想φ已经消失哈瑟不变，并且可以从其他部分隔开。结合了Deligne的一致性的Drinfeld模块模拟，我们设计了一种算法来计算Hasse不变升力，这正是我们算法的症结所在。产生的预期运行时间${ñ}^{3/2+\epsilon }{（\mathrm{日志}q）}^{\mathrm{1个}+Ø（\mathrm{1个}）}+{ñ}^{\mathrm{1个}+\epsilon }{（\mathrm{日志}q）}^{2+Ø（\mathrm{1个}）}$以度系数多项式ñ过${\mathbb{F}}_q$ 匹配最快的已知算法，即Kaltofen-Shoup算法的Kedlaya-Umans实现。