Drinfeld modules with complex multiplication, Hasse invariants and factoring polynomials over finite fields

• Published in: Journal of symbolic computation Vol. 105; no. July–August 2021; pp. 199 - 213
• Main Authors: Doliskani, Javad, Narayanan, Anand Kumar, Schost, Éric
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01-07-2021; Elsevier
• Subjects: Complex multiplication; Computer Science; Drinfeld modules; Elliptic modules; Hasse invariant; Polynomial factorization; Symbolic Computation; Drinfeld modules; Polynomial factorization; Hasse invariant; Complex multiplication; Elliptic modules
• ISSN: 0747-7171; 1095-855X
• DOI: 10.1016/j.jsc.2020.06.007

We present a novel randomized algorithm to factor polynomials over a finite field Fq 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∈Fq[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 n3/2+ε(log⁡q)1+o(1)+n1+ε(log⁡q)2+o(1) to factor polynomials of degree n over Fq matches the fastest previously known algorithm, the Kedlaya-Umans implementation of the Kaltofen-Shoup algorithm.