Factoring Polynomials over Finite Fields with Linear Galois Groups: An Additive Combinatorics Approach.

Abstract

Let f(X) ∈ ℤ[X] be a degree-n polynomial such that f(X): = f(X)od p factorizes into n distinct linear factors over 𝔽_p. We study the problem of deterministically factoring f(X) over 𝔽_p given f(X). Under the generalized Riemann hypothesis (GRH), we give an improved deterministic algorithm that computes the complete factorization of f(X) in the case that the Galois group of f(X) is (permutation isomorphic to) a linear group G ≤ GL(V) on the set S of roots of f(X), where V is a finite-dimensional vector space over a finite field 𝔽 and S is identified with a subset of V. In particular, when |S| = |V|^{Ω(1)}, the algorithm runs in time polynomial in n^{log n/(log log log log n)^{1/3}} and the size of the input, improving Evdokimov’s algorithm. Our result also applies to a general Galois group G when combined with a recent algorithm of the author. To prove our main result, we introduce a family of objects called linear m-schemes and reduce the problem of factoring f(X) to a combinatorial problem about these objects. We then apply techniques from additive combinatorics to obtain an improved bound. Our techniques may be of independent interest.