Polynomial factorization over finite fields by computing Euler–Poincaré characteristics of Drinfeld modules

Abstract

Abstract.We propose and rigorously analyze two randomized algorithms to factor univariate polynomials over finite fields using rank 2 Drinfeld modules. The first algorithm estimates the degree of an irreducible factor of a polynomial from Euler-Poincare characteristics of random Drinfeld modules. Knowledge of a factor degree allows one to rapidly extract all factors of that degree. As a consequence, the problem of factoring polynomials over finite fields in time nearly linear in the degree is reduced to finding Euler-Poincare characteristics of random Drinfeld modules with high probability. The second algorithm is a random Drinfeld module analogue of Berlekamp's algorithm. During the course of its analysis, we prove a new bound on degree distributions in factorization patterns of polynomials over finite fields in certain short intervals.

Computing the Characteristic Polynomial of a Finite Rank Two Drinfeld Module

“…In [29, Sec. 3.1], Narayanan gives the sketch of a Monte Carlo algorithm to solve Problem 1 for odd q, which applies to those Drinfeld modules (д, ∆) for which the minimal polynomial Γ of Φ x = γ (x)Id + дπ + ∆π 2 has degree n. In this case, it must coincide with the characteristic polynomial of Φ x , which we saw is equal to 1 −A + B (this assumption on Γ holds for more than half of elements of the parameter domain [29,Th. 3.6]).…”

“…Based in part on these similarities, Drinfeld modules have recently started being considered under the algorithmic viewpoint. For instance, they have been proved to be unsuitable for usual forms of public key cryptography [34]; they have also been used to design several polynomial factorization algorithms [7,29,30,38]; recent work by Garai and Papikian discusses the computation of their endomorphism rings [9]. Our goal is to study in detail the complexity of computing the characteristic polynomial of a rank two Drinfeld module over a finite field.…”

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

“…The use of Drinfeld modules for polynomial factorization actually goes back to work of Panchishkin and Potemine [29], whose algorithm was rediscovered by van der Heiden [36]. These algorithms, along with the second author's Drinfeld module black box Berlekamp algorithm [28] are in spirit Drinfeld module analogues of Lenstra's elliptic curve method to factor integers [27]. The Drinfeld module degree estimation algorithm of [28] uses Euler-Poincaré characteristics of Drinfeld modules to estimate the factor degrees in distinct degree factorization.…”

Factoring Polynomials over Finite Fields using Drinfeld Modules with Complex Multiplication

“…The use of Drinfeld modules to factor polynomials over finite fields originated with Panchishkin and Potemine [PP89] whose algorithm was rediscovered by van der Heiden [vdH04]. These algorithms, along with the author's Drinfeld module black box Berlekamp algorithm [Nar15] are in spirit Drinfeld module analogues of Lenstra's elliptic curve method to factor integers [Len87]. The Drinfeld module degree estimation algorithm of [Nar15] uses Euler-Poincare charactersitics of Drinfeld modules to estimate the factor degrees in distinct degree factorization.…”

“…These algorithms, along with the author's Drinfeld module black box Berlekamp algorithm [Nar15] are in spirit Drinfeld module analogues of Lenstra's elliptic curve method to factor integers [Len87]. The Drinfeld module degree estimation algorithm of [Nar15] uses Euler-Poincare charactersitics of Drinfeld modules to estimate the factor degrees in distinct degree factorization. A feature common to the aforementioned algorithms is their use of random Drinfeld modules, which typically don't have complex multiplication.…”