Factoring polynomials over global fields

Factoring polynomials in the ring of formal power series over Z

We consider polynomials with integer coefficients and discuss their factorization properties in Z[[x]], the ring of formal power series over Z. We treat polynomials of arbitrary degree and give sufficient conditions for their reducibility as power series. Moreover, if a polynomial is reducible over Z[[x]], we provide an explicit factorization algorithm. For polynomials whose constant term is a prime power, our study leads to the discussion of p-adic integers.

$\mathcal{P}$-schemes and Deterministic Polynomial Factoring over Finite Fields

We introduce a family of mathematical objects called $\mathcal{P}$-schemes, where $\mathcal{P}$ is a poset of subgroups of a finite group $G$. A $\mathcal{P}$-scheme is a collection of partitions of the right coset spaces $H\backslash G$, indexed by $H\in\mathcal{P}$, that satisfies a list of axioms. These objects generalize the classical notion of association schemes as well as the notion of $m$-schemes (Ivanyos et al. 2009). Based on $\mathcal{P}$-schemes, we develop a unifying framework for the problem of deterministic factoring of univariate polynomials over finite fields under the general…

Note on Integer Factoring Methods IV

This note continues the theoretical development of deterministic integer factorization algorithms based on systems of polynomials equations. The main result establishes a new deterministic time complexity bench mark in integer factorization.

Efficiently factoring polynomials modulo $p^4$

Polynomial factoring has famous practical algorithms over fields-- finite, rational \& $p$-adic. However, modulo prime powers it gets hard as there is non-unique factorization and a combinatorial blowup ensues. For example, $x^2+p \bmod p^2$ is irreducible, but $x^2+px \bmod p^2$ has exponentially many factors! We present the first randomized poly(deg $f, \log p$) time algorithm to factor a given univariate integral $f(x)$ modulo $p^k$, for a prime $p$ and $k \leq 4$. Thus, we solve the open question of factoring modulo $p^3$ posed in (Sircana, ISSAC'17). Our method reduces the general problem…

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-dimen…

Deterministic factoring with oracles

We revisit the problem of integer factorization with number-theoretic oracles, including a well-known problem: can we factor an integer $N$ unconditionally, in deterministic polynomial time, given the value of the Euler totient $$Φ$(N)$? We show that this can be done, under certain size conditions on the prime factors of N. The key technique is lattice basis reduction using the LLL algorithm. Among our results, we show for example that if $N$ is a squarefree integer with a prime factor $p > $\sqrt$ N$ , then we can recover p in deterministic polynomial time given $$Φ$(N)$. We also shed some li…

Factoring Polynomials over Finite Fields using Drinfeld Modules with Complex Multiplication

We present novel algorithms to factor polynomials over a finite field $\F_q$ of odd characteristic using rank $2$ Drinfeld modules with complex multiplication. The main idea is to compute a lift of the Hasse invariant (modulo the polynomial $f(x) \in \F_q[x]$ to be factored) with respect to a Drinfeld module $ϕ$ with complex multiplication. Factors of $f(x)$ supported on prime ideals with supersingular reduction at $ϕ$ have vanishing Hasse invariant and can be separated from the rest. A Drinfeld module analogue of Deligne's congruence plays a key role in computing the Hasse invariant lift. We …

Drinfeld Modules with Complex Multiplication, Hasse Invariants and Factoring Polynomials over Finite Fields

We present a novel randomized algorithm to factor polynomials over a finite field $\F_q$ of odd characteristic using rank $2$ Drinfeld modules with complex multiplication. The main idea is to compute a lift of the Hasse invariant (modulo the polynomial $f \in \F_q[x]$ to be factored) with respect to a random Drinfeld module $ϕ$ with complex multiplication. Factors of $f$ supported on prime ideals with supersingular reduction at $ϕ$ have vanishing Hasse invariant and can be separated from the rest. Incorporating a Drinfeld module analogue of Deligne's congruence, we devise an algorithm to compu…

Digit Polynomials and their application to integer factorization

This paper presents the concept of digit polynomials, which leads to a deterministic and unconditional integer factorization algorithm with the runtime complexity $\mathcal{O}(N^{1/4+ε})$. Strassen's well known factoring approach is a special case of our method. We will also consider a possibility to improve upon the complexity bound.

A deterministic version of Pollard's p-1 algorithm

In this article we present applications of smooth numbers to the unconditional derandomization of some well-known integer factoring algorithms. We begin with Pollard's $p-1$ algorithm, which finds in random polynomial time the prime divisors $p$ of an integer $n$ such that $p-1$ is smooth. We show that these prime factors can be recovered in deterministic polynomial time. We further generalize this result to give a partial derandomization of the $k$-th cyclotomic method of factoring ($k\ge 2$) devised by Bach and Shallit. We also investigate reductions of factoring to computing Euler's totient…

因式分解行列式及其在数论中的应用

Factoring determinants and applications to number theory

本文利用Toeplitz/Hankel算子和相关恒等式，将经典紧致群上的移位特征多项式乘积平均值因式分解为辛群与正交群平均值，并给出数论平均值公式的新证明。

Non-unique factorization of polynomials over residue class rings of the integers

We investigate non-unique factorization of polynomials in Z_{p^n}[x] into irreducibles. As a Noetherian ring whose zero-divisors are contained in the Jacobson radical, Z_{p^n}[x] is atomic. We reduce the question of factoring arbitrary non-zero polynomials into irreducibles to the problem of factoring monic polynomials into monic irreducibles. The multiplicative monoid of monic polynomials of Z_{p^n}[x] is a direct sum of monoids corresponding to irreducible polynomials in Z_p[x], and we show that each of these monoids has infinite elasticity. Moreover, for every positive integer m, there exis…

模多个素数的确定性多项式分解

Deterministic polynomial factorisation modulo many primes

针对有限域上单变量多项式分解的开放问题，本文提出一种无条件确定性算法，计算不可约多项式模所有不超过给定界限的素数的分解，并在Galois多项式情况下实现摊销确定性多项式时间分解。

通过求解范数方程来因式分解线性微分算子

Factoring Linear Differential Operators in Positive Characteristic by means of Solving a Norm Equation

本文提出了一种在正特征下求解线性微分算子因式分解的算法，通过求解范数方程和Riemann-Roch空间计算来高效获得解。

Degrees of irreducible polynomials over binary field

An algorithm for factoring polynomials over finite fields is given by Berlekamp in 1967. The main tool was the matrix Q corresponding to each polynomial. This paper studies the degrees of polynomials over binary field that associated with their corresponding matrices Q and irreducibility.

Montgomery's method of polynomial selection for the number field sieve

The number field sieve is the most efficient known algorithm for factoring large integers that are free of small prime factors. For the polynomial selection stage of the algorithm, Montgomery proposed a method of generating polynomials which relies on the construction of small modular geometric progressions. Montgomery's method is analysed in this paper and the existence of suitable geometric progressions is considered.

Root optimization of polynomials in the number field sieve

The general number field sieve (GNFS) is the most efficient algorithm known for factoring large integers. It consists of several stages, the first one being polynomial selection. The quality of the chosen polynomials in polynomial selection can be modelled in terms of size and root properties. In this paper, we describe some algorithms for selecting polynomials with very good root properties.

Complexity of strong approximation on the sphere

By assuming some widely-believed arithmetic conjectures, we show that the task of accepting a number that is representable as a sum of $d\geq2$ squares subjected to given congruence conditions is NP-complete. On the other hand, we develop and implement a deterministic polynomial-time algorithm that represents a number as a sum of 4 squares with some restricted congruence conditions, by assuming a polynomial-time algorithm for factoring integers and Conjecture~\ref{cc}. As an application, we develop and implement a deterministic polynomial-time algorithm for navigating LPS Ramanujan graphs, und…

最近用于因数分解和计算离散对数的量子算法的无条件正确性

Unconditional correctness of recent quantum algorithms for factoring and computing discrete logarithms

本文通过解析数论工具证明了Regev改进的量子算法在因数分解和计算离散对数问题上的无条件正确性。

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 compu…

有限域上不可约多项式的伪确定性构造

Pseudo-Deterministic Construction of Irreducible Polynomials over Finite Fields

本研究扩展Shoup的确定性算法，通过结合快速随机因式分解算法，提出了一种运行时间为$\tilde{O}(d^4 \u2009\text{log}^4 q)$的多项式时间伪确定性算法，用于构造有限域上的不可约多项式。

The diophantine exponent of the $\mathbb{Z}/q\mathbb{Z}$ points of $S^{d-2}\subset S^d$

Assume a polynomial-time algorithm for factoring integers, Conjecture~\ref{conj}, $d\geq 3,$ and $q$ and $p$ are prime numbers, where $p\leq q^A$ for some $A>0$. We develop a polynomial-time algorithm in $\log(q)$ that lifts every $\mathbb{Z}/q\mathbb{Z}$ point of $S^{d-2}\subset S^{d}$ to a $\mathbb{Z}[1/p]$ point of $S^d$ with the minimum height. We implement our algorithm for $d=3 \text{ and }4$. Based on our numerical results, we formulate a conjecture which can be checked in polynomial-time and gives the optimal bound on the diophantine exponent of the $\mathbb{Z}/q\mathbb{Z}$ points of $…

使用 Amitsur 上同调的中心单代数的高效计算

Efficient computations in central simple algebras using Amitsur cohomology

本文提出基于Brauer因子集的中心单代数高效计算表示，结合数论多项式量子算法，给出了数域上显式同构问题的多项式量子算法，并提供了无需启发式假设但有界次数下的替代版本。

一个蕴含多项式因式分解与整数因式分解更快算法的数论猜想

A number-theoretic conjecture implying faster algorithms for polynomial factorization and integer factorization

本文针对有限域一元多项式因式分解的3/2指数壁垒，提出基于数论猜想的新策略，可优化多项式因式分解指数，还能降低确定性整数因式分解算法的指数。

Splitting quaternion algebras over quadratic number fields

We propose an algorithm for finding zero divisors in quaternion algebras over quadratic number fields, or equivalently, solving homogeneous quadratic equations in three variables over $\mathbb{Q}(\sqrt{d})$ where $d$ is a square-free integer. The algorithm is randomized and runs in polynomial time if one is allowed to call oracles for factoring integers.

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.

Polynomial Factorization over Finite Fields By Computing Euler-Poincare Characteristics of Drinfeld Modules

We propose and rigorously analyze two randomized algorithms to factor univariate polynomials over finite fields using rank $2$ Drinfeld modules. The first algorithm estimates the degree of an irreducible factor of a polynomial from Euler-Poincare characteristics of random Drinfeld modules. Knowledge of a factor degree allows one to rapidly extract all factors of that degree. As a consequence, the problem of factoring polynomials over finite fields in time nearly linear in the degree is reduced to finding Euler-Poincare characteristics of random Drinfeld modules with high probability. Notably, …

Computing explicit isomorphisms with full matrix algebras over $\mathbb{F}_q(x)$

We propose a polynomial time $f$-algorithm (a deterministic algorithm which uses an oracle for factoring univariate polynomials over $\mathbb{F}_q$) for computing an isomorphism (if there is any) of a finite dimensional $\mathbb{F}_q(x)$-algebra $A$ given by structure constants with the algebra of $n$ by $n$ matrices with entries from $\mathbb{F}_q(x)$. The method is based on computing a finite $\mathbb{F}_q$-subalgebra of $A$ which is the intersection of a maximal $\mathbb{F}_q[x]$-order and a maximal $R$-order, where $R$ is the subring of $\mathbb{F}_q(x)$ consisting of fractions of polynomi…

Polynomials over Ring of Integers of Global Fields that have Roots Modulo Every Finite Indexed Subgroup

A polynomial with coefficients in the ring of integers $\mathcal{O}_{K}$ of a global field $K$ is called intersective if it has a root modulo every finite-indexed subgroup of $\mathcal{O}_{K}$. We prove two criteria for a polynomial $f(x)\in\mathcal{O}_{K}[x]$ to be intersective. One of these criteria is in terms of the Galois group of the splitting field of the polynomial, whereas the second criterion is verifiable entirely in terms of constants which depend upon $K$ and the polynomial $f$. The proofs use the theory of global field extensions and upper bound on the least prime ideal in the Ch…

Preperiodic points of polynomials over global fields

Given a global field K and a polynomial f defined over K of degree at least two, Morton and Silverman conjectured in 1994 that the number of K-rational preperiodic points of f is bounded in terms of only the degree of K and the degree of f. In 1997, for quadratic polynomials over K=Q, Call and Goldstine proved a bound which was exponential in s, the number of primes of bad reduction of f. By careful analysis of the filled Julia sets at each prime, we present an improved bound on the order of slog(s). Our bound applies to polynomials of any degree (at least two) over any global field K.