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 multivariate polynomials over finite fields

AbstractThis paper describes an algorithm for the factorization of multivariate polynomials with coefficients in a finite field that is polynomial-time in the degrees of the polynomial to be factored. The algorithm makes use of a new basis reduction algorithm for lattices over Fq[Y]

General Factoring Algorithms for Polynomials over Finite Fields

In this paper, we generate algorithms for factoring polynomials with coefficients in finite fields. In particular, we develop one deterministic algorithm due to Elwyn Berlekamp and one probabilistic algorithm due to David Cantor and Hans Zassenhaus. While some authors present versions of the algorithms that can only factor polynomials of a certain form, the algorithms we give are able to factor any polynomial over any finite field. Hence, the algorithms we give are the most general algorithms available for this factorization problem. After formulating the algorithms, we look at various ways they can be applied to more specialized inquiries. For example, we use the algorithms to develop two tests for irreducibility and a process for finding the roots of a polynomial over a finite field. We conclude our work by considering how the Berlekamp and Cantor-Zassenhaus methods can be combined to develop a more efficient factoring process

Factoring multivariate polynomials over finite fields

This paper describes an algorithm for the factorization of multivariate polynomials with coefficients in a finite field that is polynomial-time in the degrees of the polynomial to be factored. The algorithm makes use of a new basis reduction algorithm for lattices over a finite field

Algebraic Problems Equivalent to Beating Exponent 3/2 for Polynomial Factorization over Finite Fields

The fastest known algorithm for factoring univariate polynomials over finite fields is the Kedlaya-Umans (fast modular composition) implementation of the Kaltofen-Shoup algorithm. It is randomized and takes $\widetilde{O}(n^{3/2}\log q + n \log^2 q)$ time to factor polynomials of degree $n$ over the finite field $\mathbb{F}_q$ with $q$ elements. A significant open problem is if the $3/2$ exponent can be improved. We study a collection of algebraic problems and establish a web of reductions between them. A consequence is that an algorithm for any one of these problems with exponent better than $3/2$ would yield an algorithm for polynomial factorization with exponent better than $3/2$

Splitting full matrix algebras over algebraic number fields

Let K be an algebraic number field of degree d and discriminant D over Q. Let A be an associative algebra over K given by structure constants such that A is isomorphic to the algebra M_n(K) of n by n matrices over K for some positive integer n. Suppose that d, n and D are bounded. Then an isomorphism of A with M_n(K) can be constructed by a polynomial time ff-algorithm. (An ff-algorithm is a deterministic procedure which is allowed to call oracles for factoring integers and factoring univariate polynomials over finite fields.) As a consequence, we obtain a polynomial time ff-algorithm to compute isomorphisms of central simple algebras of bounded degree over K.Comment: 15 pages; Theorem 2 and Lemma 8 correcte

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