Trading GRH for algebra: algorithms for factoring polynomials and related structures

TL;DR

This paper develops deterministic algorithms that remove the need for the Generalized Riemann Hypothesis in polynomial factoring over finite fields, leading to new results in algebra and computational number theory.

Contribution

It introduces GRH-free deterministic algorithms for polynomial factoring and algebraic structures, generalizing classical Galois theory constructs for semisimple algebras.

Findings

Deterministic polynomial-time algorithm for factoring polynomials over finite fields.

Subexponential time algorithm for finding zero divisors in noncommutative algebras.

Deterministic polynomial-time algorithm for factoring cyclotomic polynomials with certain properties.

Abstract

In this paper we develop techniques that eliminate the need of the Generalized Riemann Hypothesis (GRH) from various (almost all) known results about deterministic polynomial factoring over finite fields. Our main result shows that given a polynomial f(x) of degree n over a finite field k, we can find in deterministic poly(n^{\log n},\log |k|) time "either" a nontrivial factor of f(x) "or" a nontrivial automorphism of k[x]/(f(x)) of order n. This main tool leads to various new GRH-free results, most striking of which are: (1) Given a noncommutative algebra over a finite field, we can find a zero divisor in deterministic subexponential time. (2) Given a positive integer r such that either 8|r or r has at least two distinct odd prime factors. There is a deterministic polynomial time algorithm to find a nontrivial factor of the r-th cyclotomic polynomial over a finite field. In this paper, following the seminal work of Lenstra (1991) on constructing isomorphisms between finite fields, we further generalize classical Galois theory constructs like cyclotomic extensions, Kummer extensions, Teichmuller subgroups, to the case of commutative semisimple algebras with automorphisms. These generalized constructs help eliminate the dependence on GRH.