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

Let $\tilde{f}(X)\in\mathbb{Z}[X]$ be a degree-$n$ polynomial such that $f(X):=\tilde{f}(X)\bmod p$ factorizes into $n$ distinct linear factors over $\mathbb{F}_p$. We study the problem of deterministically factoring $f(X)$ over $\mathbb{F}_p$ given $\tilde{f}(X)$. 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 $\tilde{f}(X)$ is (permutation isomorphic to) a linear group $G\leq \mathrm{GL}(V)$ on the set $S$ of roots of $\tilde{f}(X)$, where $V$ is a finite-dimensional vector space over a finite field $\mathbb{F}$ and $S$ is identified with a subset of $V$. In particular, when $|S|=|V|^{\Omega(1)}$, 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.Comment: To be published in the proceedings of MFCS 202

Deterministic polynomial factoring over finite fields: A uniform approach via P-schemes

We introduce a family of combinatorial objects called P-schemes, where P is a collection of subgroups of a finite group G. A P-scheme is a collection of partitions of right coset spaces H\G, indexed by H ∈ P, that satisfies a list of axioms. These objects generalize the classical notion of association schemes as well as m-schemes (Ivanyos et al., 2009). We apply the theory of P-schemes to deterministic polynomial factoring over finite fields: suppose f(X) ∈ Z[X] and a prime number pare given, such that f(X) :=f(X) modpfactorizes into n =deg(f)distinct linear factors over the finite field F_p. We show that, assuming the generalized Riemann hypothesis (GRH), f(X)can be completely factorized in deterministic polynomial time if the Galois group G of f(X)is an almost simple primitive permutation group on the set of roots of f(X), and the socle of Gis a subgroup of Sym(k)for kup to 2^O(√log n). This is the first deterministic polynomial-time factoring algorithm for primitive Galois groups of superpolynomial order. We prove our result by developing a generic factoring algorithm and analyzing it using P-schemes. We also show that the main results achieved by known GRH-based deterministic polynomial factoring algorithms can be derived from our generic algorithm in a uniform way. Finally, we investigate the schemes conjecturein Ivanyos et al. (2009), and formulate analogous conjectures associated with various families of permutation groups. We show that these conjectures form a hierarchy of relaxations of the original schemes conjecture, and their positive resolutions would imply deterministic polynomial-time factoring algorithms for various families of Galois groups under GRH

Deterministic polynomial factoring over finite fields: A uniform approach via P-schemes

We introduce a family of combinatorial objects called P-schemes, where P is a collection of subgroups of a finite group G. A P-scheme is a collection of partitions of right coset spaces H\G, indexed by H ∈ P, that satisfies a list of axioms. These objects generalize the classical notion of association schemes as well as m-schemes (Ivanyos et al., 2009). We apply the theory of P-schemes to deterministic polynomial factoring over finite fields: suppose f(X) ∈ Z[X] and a prime number pare given, such that f(X) :=f(X) modpfactorizes into n =deg(f)distinct linear factors over the finite field F_p. We show that, assuming the generalized Riemann hypothesis (GRH), f(X)can be completely factorized in deterministic polynomial time if the Galois group G of f(X)is an almost simple primitive permutation group on the set of roots of f(X), and the socle of Gis a subgroup of Sym(k)for kup to 2^O(√log n). This is the first deterministic polynomial-time factoring algorithm for primitive Galois groups of superpolynomial order. We prove our result by developing a generic factoring algorithm and analyzing it using P-schemes. We also show that the main results achieved by known GRH-based deterministic polynomial factoring algorithms can be derived from our generic algorithm in a uniform way. Finally, we investigate the schemes conjecturein Ivanyos et al. (2009), and formulate analogous conjectures associated with various families of permutation groups. We show that these conjectures form a hierarchy of relaxations of the original schemes conjecture, and their positive resolutions would imply deterministic polynomial-time factoring algorithms for various families of Galois groups under GRH

Schemes for deterministic polynomial factoring

In this work we relate the deterministic complexity of factoring polynomials (over finite fields) to certain combinatorial objects we call m-schemes. We extend the known conditional deterministic subexponential time polynomial factoring algorithm for finite fields to get an underlying m-scheme. We demonstrate how the properties of m-schemes relate to improvements in the deterministic complexity of factoring polynomials over finite fields assuming the generalized Riemann Hypothesis (GRH). In particular, we give the first deterministic polynomial time algorithm (assuming GRH) to find a nontrivial factor of a polynomial of prime degree n where (n − 1) is a smooth number

Schemes for Deterministic Polynomial Factoring

In this work we relate the deterministic complexity of factoring polynomials (over finite fields) to certain combinatorial objects we call m-schemes. We extend the known conditional deterministic subexponential time polynomial factoring algorithm for finite fields to get an underlying m-scheme. We demonstrate how the properties of m-schemes relate to improvements in the deterministic complexity of factoring polynomials over finite fields assuming the generalized Riemann Hypothesis (GRH). In particular, we give the first deterministic polynomial time algorithm (assuming GRH) to find a nontrivial factor of a polynomial of prime degree n where (n − 1) is a smooth number