Factoring Polynomials over Finite Fields using Balance Test

We study the problem of factoring univariate polynomials over finite fields. Under the assumption of the Extended Riemann Hypothesis (ERH), (Gao, 2001) designed a polynomial time algorithm that fails to factor only if the input polynomial satisfies a strong symmetry property, namely square balance. In this paper, we propose an extension of Gao's algorithm that fails only under an even stronger symmetry property. We also show that our property can be used to improve the time complexity of best deterministic algorithms on most input polynomials. The property also yields a new randomized polynomial time algorithm

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

Maps between curves and arithmetic obstructions

Let X and Y be curves over a finite field. In this article we explore methods to determine whether there is a rational map from Y to X by considering L-functions of certain covers of X and Y and propose a specific family of covers to address the special case of determining when X and Y are isomorphic. We also discuss an application to factoring polynomials over finite fields.Comment: 8 page