No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Factoring Modular Polynomials
Abstract
This paper gives an algorithm to factor a polynomial f (in one variable) over rings like Z=rZ for r 2 Z or F q [y]=rF q [y] for r 2 F q [y]. The Chinese Remainder Theorem reduces our problem to the case where r is a prime power. Then factorization is not unique, but if r does not divide the discriminant of f , our (probabilistic) algorithm produces a description of all (possibly exponentially many) factorizations into irreducible factors in polynomial time. If r divides the discriminant, we only know how to factor by exhaustive search, in exponential time.
... So the hard case is when f mod p is power of an irreducible polynomial. The first resolution in this case was achieved by [28] assuming that k is "large". They assumed k to be larger than the maximum power of p dividing the discriminant of the integral f . ...
... Using this observation they could also describe all the factorizations modulo p k , in a compact data structure. The complexity of [28] was improved by [7]. ...
... The study of [27,28] sheds some light on the behaviour of the factoring problem for integral polynomials modulo prime powers. It shows that for "large" k the problem is similar to the factorization over p-adic fields (already solved efficiently by [5]). ...
Polynomial factoring has famous practical algorithms over fields-- finite, rational and p-adic. However, modulo prime powers, factoring gets harder because 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, łog p) time algorithm to factor a given univariate integral f(x) modulo p^k, for a prime p and k łeq 4. Thus, we solve the open question of factoring modulo p^3 posed in (Sircana, ISSAC'17). Our method reduces the general problem of factoring f(x) mod p^k to that of \em root finding in a related polynomial E(y) \bmodłangle p^k, \varphi(x)^\ell \rangle for some irreducible \varphi \bmod p. We can efficiently solve the latter for kłe4, by incrementally transforming E(y). Moreover, we discover an efficient refinement of Hensel lifting to lift factors of f(x) \bmod p to those \bmod\ p^4 (if possible). This was previously unknown, as the case of repeated factors of f(x) \bmod p forbids classical Hensel lifting.
... So the hard case is when f mod p is power of an irreducible polynomial. The first resolution in this case was achieved by [vzGH98] assuming that k is "large". They assumed k to be larger than the maximum power of p dividing the discriminant of the integral f . ...
... Using this observation they could also describe all the factorizations modulo p k , in a compact data structure. The complexity of [vzGH98] was improved by [CL01]. ...
... The study of [vzGH98,vzGH96] sheds some light on the behaviour of the factoring problem for integral polynomials modulo prime powers. It shows that for "large" k the problem is similar to the factorization over p-adic fields (already solved efficiently by [CG00]). ...
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 of factoring $f(x) \bmod p^k$ to that of {\em root finding} in a related polynomial $E(y) \bmod\langle p^k, \varphi(x)^\ell \rangle$ for some irreducible $\varphi \bmod p$. We could efficiently solve the latter for $k\le4$, by incrementally transforming $E(y)$. Moreover, we discover an efficient and strong generalization of Hensel lifting to lift factors of $f(x) \bmod p$ to those $\bmod\ p^4$ (if possible). This was previously unknown, as the case of repeated factors of $f(x) \bmod p$ forbids classical Hensel lifting.
... Suppose k, p ∈ N with p prime and f ∈ Z[x] is a univariate polynomial with degree d ≥ 1 and all coefficients having absolute value less than p k . Let N p,k (f ) denote the number of roots of f in Z/ p k (see, e.g., [24,23,2,18,14,28] for further background on prime power rings). Computing N p,k (f ) is a fundamental problem occuring in polynomial factoring [21,9,4,25,15], coding theory [3], and cryptography [19]. ...
... Our use of fast factorization (as in [17]) is why we avail to randomness, but this pays off: Gaining access to individual roots in Z/(p) (as suggested in [8]) enables us to give a more streamlined algorithm. [14] a randomized polynomialtime algorithm to compute all factorizations of certain f ∈ (Z/(p k ))[x]. (Examples like ...
... Unfortunately, their algorithm has the restriction that p k not divide the discriminant of f . Their complexity bound, in our notation, is the sum of d 7 k log(p)(k log(p) + log d) 2 and a term involving the complexity of finding the mod p k reduction of a factorization over Z p [x] (see from [14]). The complexity of just counting the number of possible factorizations (or just the number of possible linear factors) of f from their data structure does not appear to be stated. ...
Suppose $k,p\!\in\!\mathbb{N}$ with $p$ prime and $f\!\in\!\mathbb{Z}[x]$ is a univariate polynomial with degree $d$ and all coefficients having absolute value less than $p^k$. We give a Las Vegas randomized algorithm that computes the number of roots of $f$ in $\mathbb{Z}/\!\left(p^k\right)$ within time $d^3(k\log p)^{2+o(1)}$. (We in fact prove a more intricate complexity bound that is slightly better.) The best previous general algorithm had (deterministic) complexity exponential in $k$. We also present some experimental data evincing the potential practicality of our algorithm.
... Even irreducibility testing of a polynomial, with the prime factorization of n given, has no efficient algorithm known. This reduces to prime-power characteristic p k [vzGH98]. Deterministic factoring in such a ring is a much harder question (at least it subsumes deterministic factoring mod p). ...
... Deterministic factoring in such a ring is a much harder question (at least it subsumes deterministic factoring mod p). In fact, even randomized algorithms, or practical solutions, are currently elusive [vzGH96,vzGH98,Kli97,Sȃl05,Sir17,DMS19]. The main obstruction is non-unique factorization. ...
Finding an irreducible factor, of a polynomial $f(x)$ modulo a prime $p$, is not known to be in deterministic polynomial time. Though there is such a classical algorithm that {\em counts} the number of irreducible factors of $f\bmod p$. We can ask the same question modulo prime-powers $p^k$. The irreducible factors of $f\bmod p^k$ blow up exponentially in number; making it hard to describe them. Can we count those irreducible factors $\bmod~p^k$ that remain irreducible mod $p$? These are called {\em basic-irreducible}. A simple example is in $f=x^2+px \bmod p^2$; it has $p$ many basic-irreducible factors. Also note that, $x^2+p \bmod p^2$ is irreducible but not basic-irreducible! We give an algorithm to count the number of basic-irreducible factors of $f\bmod p^k$ in deterministic poly(deg$(f),k\log p$)-time. This solves the open questions posed in (Cheng et al, ANTS'18 \& Kopp et al, Math.Comp.'19). In particular, we are counting roots $\bmod\ p^k$; which gives the first deterministic poly-time algorithm to compute Igusa zeta function of $f$. Also, our algorithm efficiently partitions the set of all basic-irreducible factors (possibly exponential) into merely deg$(f)$-many disjoint sets, using a compact tree data structure and {\em split} ideals.
... As all recurrent sequences are periodic, they are in particular linearly recurrent and satisfy the linear recurrence (of not necessarily minimal degree) defined by x n − 1, with n the period of the sequence. An algorithm for determining all factorisations of a polynomial over a ring of the form Z p a (and some other types of rings) was developed in [13]. One factorisation is derived from the factorisation of the polynomial over the p-adic integers (this can be obtained by the algorithms of Chistov, Ford-Zassenhaus, Buchmann-Lenstra, Cantor-Gordon, Pauli, Ford et. ...
... Factoring over the p-adics and then projecting the factorisation to Z p a [x] does not always result in a factorisation into irreducible factors, as irreducible monic polynomials over the p-adic integers may no longer be irreducible when projected (see Example 4.6 for illustration). The advantage of our results compared to [13] is that they hold for all polynomials, regardless of the value of their discriminant. The disadvantage is that they only hold in Galois rings of characteristic p 2 , with no immediate way of extending them to Galois rings of characteristic p a with a > 2. The paper is organised as follows. ...
This article was published in the journal, Finite fields and their applications [© Elsevier] and is also available at: http://www.sciencedirect.com/science/journal/10715797 It is known that univariate polynomials over finite local rings factor uniquely into primary pairwise coprime factors. Primary polynomials are not necessarily irreducible. Here we describe a factorisation into irreducible factors for primary polynomials over Z4 and more generally over Galois rings of characteristic p2. An algorithm is also given. As an application, we factor xn-1 and xn+1 over such rings.
... On obtient alors, s'il y en a, les racines n-ième modulo p de λ. Un algorithme probabiliste qui fonctionne en temps polynômial pour cette factorisation est donné dans [GH98]. ...
Cette thèse porte sur deux éléments actuellement incontournables de la cryptographie à clé publique, qui sont l’arithmétique modulaire avec de grands entiers et la multiplication scalaire sur les courbes elliptiques (ECSM). Pour le premier, nous nous intéressons au système de représentation modulaire adapté (AMNS), qui fut introduit par Bajard et al. en 2004. C’est un système de représentation de restes modulaires dans lequel les éléments sont des polynômes. Nous montrons d’une part que ce système permet d’effectuer l’arithmétique modulaire de façon efficace et d’autre part comment l’utiliser pour la randomisation de cette arithmétique afin de protéger l’implémentation des protocoles cryptographiques contre certaines attaques par canaux auxiliaires. Pour l’ECSM, nous abordons l’utilisation des chaînes d’additions euclidiennes (EAC) pour tirer parti de la formule d’addition de points efficace proposée par Méloni en 2007. L’objectif est d’une part de généraliser au cas d’un point de base quelconque l’utilisation des EAC pour effectuer la multiplication scalaire ; cela, grâce aux courbes munies d’un endomorphisme efficace. D’autre part, nous proposons un algorithme pour effectuer la multiplication scalaire avec les EAC, qui permet la détection de fautes qui seraient commises par un attaquant que nous détaillons.
... There is a long history and very extensive literature dedicated to algorithms on polynomials in finite fields, see, for example [12]. More recently, there was also increasing interest to algorithms for polynomials over residue rings, especially in residue rings modulo prime powers, see [5,9,10,13,17,19,31] and references therein. Here we continue this directions and consider the noisy polynomial interpolation problem modulo prime powers which is analogue to the same problem in finite fields [28,30], which in turn is an extension of the hidden number problem of Boneh and Venkatesan [2,3]. ...
We consider the {\it noisy polynomial interpolation problem\/} of recovering an unknown $s$-sparse polynomial $f(X)$ over the ring $\mathbb Z_{p^k}$ of residues modulo $p^k$, where $p$ is a small prime and $k$ is a large integer parameter, from approximate values of the residues of $f(t) \in \mathbb Z_{p^k}$. Similar results are known for residues modulo a large prime $p$, however the case of prime power modulus $p^k$, with small $p$ and large $k$, is new and requires different techniques. We give a deterministic polynomials time algorithm, which for almost given more than a half bits of $f(t)$ for sufficiently many randomly chosen points $t \in \mathbb Z_{p^k}^*$, recovers $f(X)$.
... (2) A similar result as in Proposition 4.3 can be found in [vzGH98]. There the authors show that in case R is a discrete valuation ring, the reduced resultant of f , g ∈ R[x] is equal to the largest elementary divisor of S(f, g) (see Lemma 3.8 of op. ...
The resultant of two univariate polynomials is an invariant of great importance in commutative algebra and vastly used in computer algebra systems. Here we present an algorithm to compute it over Artinian principal rings with a modified version of the Euclidean algorithm. Using the same strategy, we show how the reduced resultant and a pair of B\'ezout coefficient can be computed. Particular attention is devoted to the special case of $\mathbf{Z}/n\mathbf{Z}$, where we perform a detailed analysis of the asymptotic cost of the algorithm. Finally, we illustrate how the algorithms can be exploited to improve ideal arithmetic in number fields and polynomial arithmetic over $p$-adic fields.
... Their root will give a nth-root of λ. A probabilistic polynomial time algorithm to achieve this goal is given in [15]. ...
The adapted modular number system (AMNS) is an integer number system which aims to speed up arithmetic operations modulo a prime p. Such a system is defined by a tuple (p,n,γ,ρ,E), where p, n, γ and ρ are integers and E∈Z[X]. In El Mrabet and Gama (in: WAIFI, lecture notes in computer science, Springer, 2012) conditions required to build AMNS with E(X)=Xn+1 are provided. In this paper, we generalise their approach and provide a method to generate multiple AMNS for a given prime p with E(X)=Xn-λ and λ∈Z\{0}. Moreover, we propose a complete set of algorithms without conditional branching to perform arithmetic and conversion operations in the AMNS, using a Montgomery-like method described in Negre and Plantard (in: Information security and privacy, 13th Australasian conference, ACISP 2008, Wollongong, Australia, 2008). We show that our implementation outperforms GNU MP and OpenSSL libraries. Finally, we highlight some properties of the AMNS which state that it could lead to a helpful countermeasure against some side-channel attacks.
... That is, there exist unimodular matrices U and V satisfying U AV = B. Any matrix can be transformed into a diagonal form known as the Smith normal form, which reveals the invariant factors of the matrix [31]. It is also useful for solving systems of linear equations [34,35]. ...
In this thesis we study algorithms for computing normal forms for matrices of Ore polynomials while controlling coe#cient growth. By formulating row reduction as a linear algebra problem, we obtain a fraction-free algorithm for row reduction for matrices of Ore polynomials. The algorithm allows us to compute the rank and a basis of the left nullspace of the input matrix. When the input is restricted to matrices of shift polynomials and ordinary polynomials, we obtain fractionfree algorithms for computing row-reduced forms and weak Popov forms. These algorithms can be used to compute a greatest common right divisor and a least common left multiple of such matrices. Our fraction-free row reduction algorithm can be viewed as a generalization of subresultant algorithms. The linear algebra formulation allows us to obtain bounds on the size of the intermediate results and to analyze the complexity of our algorithms.
... Kaltofen and Lobo use a black box representation for factoring high degree polynomials over finite fields using the Berlekamp algorithm [37]. Algorithms for univariate polynomials over the integers modulo n, which is not a unique factorization domain if n is not prime, are presented by von zur Gathen and Hartlieb [79]. An algorithm for univariate polynomials over Z that relies on factorization over finite fields and uses Hensel lifting to construct integral factors is presented by Zassen- haus [91]. ...
xv Zusammenfassung xvii 1 Introduction 1 1.1 Mathematical preliminaries . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Finite Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Unique Factorization Domains . . . . . . . . . . . . . . . . . 3 1.1.3 Irreducibility . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Factorization Algorithms 9 2.1 Square-free Factorization . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.1 Musser's Algorithm . . . . . . . . . . . . . . . . . . . . . . 12 2.1.2 Yun's Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.3 The Extended Yun Algorithm . . . . . . . . . . . . . . . . . 14 2.1.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2 Univariate Factori...
Text
For any fixed field K∈{Q2,Q3,Q5,…}, we prove that all univariate polynomials f with exactly 3 (resp. 2) monomial terms, degree d, and all coefficients in {±1,…,±H}, can be solved over K within deterministic time log4+o(1)(dH)log3d (resp. log2+o(1)(dH)) in the classical Turing model: Our underlying algorithm correctly counts the number of roots of f in K, and for each such root generates an approximation in Q with logarithmic height O(log2(dH)logd) that converges at a rate of O((1/p)2i) after i steps of Newton iteration. We also prove significant speed-ups in certain settings, a minimal spacing bound of p−O(plogp2(dH)logd) for distinct roots in Cp, and even stronger root repulsion when there are nonzero degenerate roots in Cp: p-adic distance p−O(logp(dH)). On the other hand, we prove that there is an explicit family of tetranomials with distinct nonzero roots in Zp indistinguishable in their first Ω(dlogpH) most significant base-p digits. So speed-ups for t-nomials with t≥4 will require evasion or amortization of such worst-case instances.
Video
For a video summary of this paper, please visit https://youtu.be/npfdxLk04MY.
The applications of solving systems of polynomial equations are legion: The real case permeates all of non-linear optimization as well as numerous problems in engineering. The p -adic case leads to many classical questions in number theory, and is close to many applications in cryptography, coding theory, and computational number theory. As such, it is important to understand the complexity of solving systems of polynomial equations over local fields. Furthermore, the complexity of solving structured systems --- such as those with a fixed number of monomial terms or invariance with respect to a group action --- arises naturally in many computational geometric applications and is closely related to a deeper understanding of circuit complexity (see, e.g., [8]). Clearly, if we are to fully understand the complexity of solving sparse polynomial systems, then we should at least be able to settle the univariate case, e.g., classify when it is possible to separate and approximate roots in deterministic time polynomial in the input size.
We consider the noisy polynomial interpolation problem of recovering an unknown s-sparse polynomial f(X) over the ring Zpk of residues modulo pk, where p is a small prime and k is a large integer parameter, from approximate values of the residues of f(t)∈Zpk. Similar results are known for residues modulo a large prime p, however the case of prime power modulus pk, with small p and large k, is new and requires different techniques. We give a deterministic polynomial time algorithm, which for almost given more than a half bits of f(t) for sufficiently many randomly chosen points t∈Zpk∗, recovers f(X).
Polynomial factoring has famous practical algorithms over fields– finite, rational and p-adic. However, modulo prime powers, factoring gets harder because there is non-unique factorization and a combinatorial blowup ensues. For example, x2+pmodp2 is irreducible, but x2+pxmodp2 has exponentially many factors in the input size (which here is logarithmic in p)! We present the first randomized poly(degf,logp) time algorithm to factor a given univariate integral polynomial f modulo pk, for a prime p and k≤4.¹ Thus, we solve the open question of factoring modulo p3 posed in (Sircana, ISSAC'17).
Our method reduces the general problem of factoring fmodpk to that of root finding of a related polynomial E(y)mod〈pk,φ(x)ℓ〉 for some irreducible φmodp. We can efficiently solve the latter for k≤4, by incrementally transforming E. Moreover, we discover an efficient refinement of Hensel lifting to lift factors of fmodp to those modp4 (if possible). This was previously unknown, as the case of repeated factors of fmodp forbids classical Hensel lifting.
To whom it may concern: You can freely download all my published
papers from my papers page at Texas A&M University:
www.math.tamu.edu/~rojas/list2.html
In this paper, we deal with the problem of finding a factorization of a monic primary polynomial f ∈ Z/(pⁿ)[x] into irreducible factors. This task has been completely solved when pⁿ does not divide the discriminant of f, while there is not an efficient method of determining a factorization when this happens and finding an explicit factorization can be hard for polynomials of high degree. We discuss some techniques to speed up the computation, focusing on the case n=3.
1. Introduction.- 2. Overview.- 3. Technical Prerequisites.- 4. Change of Basis.- 5. Modular Squarefree and Greatest Factorial Factorization.- 6. Modular Hermite Integration.- 7. Computing All Integral Roots of the Resultant.- 8. Modular Algorithms for the Gosper-Petkovsek Form.- 9. Polynomial Solutions of Linear First Order Equations.- 10. Modular Gosper and Almkvist & Zeilberger Algorithms.
We present a new algorithm for determining all factorizations of a polynomial f in the domain Z N [x], a non-unique factorization domain, given in terms of parameters. From the prime factorization of N , the problem is reduced to factorization in Z p k[x] where p is a prime and k 1. If p k does not divide the discriminant of f and one factorization is given, our algorithm determines all factorizations with complexity O(n 3 M(k log p)) where n denotes the degree of the input polynomial and M(t) denotes the complexity of multiplication of two t-bit numbers. Our algorithm improves on the method of von zur Gathen and Hartlieb, which has complexity O(n 7 k(k log p + log n) 2 ). The improvement is achieved by processing all factors at the same time instead of one at a time and by computing the kernels and determinants of matrices over Z p k in an efficient manner. Categories and Subject Descriptors I.1.2 [Symbolic and Algebraic Manipulation]: Algorithms
This paper characterizes all the factorizations of a polynomial with coefficients in the ring Z n where n is a composite number. We give algorithms to compute such factorizations along with algebraic classifications. Contents 1 Introduction 3 1.1 Circuit complexity theory . . . . . . . . . . . . . . . . . . . . . . 3 2 Some Important Tools in Z n [x] 4 2.1 The Z n [x] phenomena . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 The Chinese Remainder Theorem . . . . . . . . . . . . . . . . . . 5 2.3 Irreducibility criteria in Z p k [x] . . . . . . . . . . . . . . . . . . . 7 2.4 Hensel's Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.5 A naive approach to factoring . . . . . . . . . . . . . . . . . . . . 11 3 The Case of Small Discriminants 12 3.1 The p-adic numbers . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Resultants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.3 The correspondence to factoring over the p-adics . . . . ....
A new probabilistic algorithm for factoring univariate polynomials over finite fields is presented. To factor a polynomial of degreen overF
q
, the number of arithmetic operations inF
q
isO((n
2+nlogq). (logn)2 loglogn). The main technical innovation is a new way to compute Frobenius and trace maps in the ring of polynomials modulo the polynomial to be factored.
This chapter discusses algorithms that solve two basic problems in computational number theory—factoring integers into prime factors and finding discrete logarithms. In the factoring problem, one is given an integer n1 and is asked to find the decomposition of n into prime factors. It is common to split this problem into two parts. The first is called primality testing: given n, it is determined whether n is prime or composite. The second is called factorization: if n is composite, a nontrivial divisor of n is to be calculated. In the discrete logarithm problem, one is given a prime number p, and two elements h, y of the multiplicative group F*p of the field of integers modulo p. The algorithms and their analyses depend on many different parts of number theory. Number theory is considered the purest of all sciences, and within number theory the hunt for large primes and for factors of large numbers has always been remote from applications, even to other questions of a number-theoretic nature.
An algorithm is obtained for factoring polynomials in several variables over local fields with complexity which is polynomial in the length of notation of the input data and the characteristic of the residue field of the local field. Here by definition we assume that an infinite series can be calculated in polynomial time if its i-th partial sum can be calculated in time which is polynomial in the length of notation of the input data and i for any i.
We give a computational description of Hensel's method for lifting approximate factorizations of polynomials. The general setting of valuation rings provides the framework for this and the other results of the paper. We describe a Newton method for solving algebraic and differential equations. Finally, we discuss a fast algorithm for factoring polynomials via computing short vectors in modules.
This paper reviews some of the known algorithms for factoring polynomials over finite fields and presents a new deterministic procedure for reducing the problem of factoring an arbitrary polynomial over the Galois field GF(p m) to the problem of finding the roots in GF(p) of certain other polynomials over GF(p). The amount of computation and the storage space required by these algorithms are algebraic in both the degree of the polynomial to be factored and the logarithm of the order of the finite field. Certain observations on the application of these methods to the factorization of polynomials over the rational integers are also included.
We describe a new algorithm for the computation of the Smith normal form of polynomial matrices. This algorithm computes the normal form and pre- and post-multipliers in deterministic polynomial time. Noticing that the computation reduces to a linear algebra problem over the field of the coefficients, we obtain a good worst-case complexity bound.
A p-adic method for the constructive factorization of monic polynomials over a dedekind ring and the ideal theory of [x] are developed.
ac.il Abstract. In this paper we consider the
An O(s5M($2)) algorithm for computing the canonical structure of a finite Abelian group represented by an integer matrix of size (this is the Smith normal form of the matrix) is presented. Moreover, an O(s3M(s2)) algorithm for computing the Hermite normal form of an integer matrix of size is given. The upper bounds derived on the computational complexity of the algorithms above improve the upper bounds given by Kannan and Bachem in (SIAM J. Comput., 8 (1979), pp. 499-507) and Chou and Collins in (SIAM J. Comput., 11 (1982), pp. 687-708).
Let f be a monic separable polynomial over the rational integers ℤ and p be a rational prime. We set up an algorithm which yields the extended p-adic values of a given element a in the semi-simple algebra Af=ℚ[X]/f. Firstly, we determine the decomposition of p in Af and an approximate factorisation of f over the p-adic completion ℚp by means of the ORDMAX algorithm of Ford-Zassenhaus. Then it is an easy matter to calculate the values of a. In case it would be necessary to improve the factorisation, we thirdly give a modification of the HenselZassenhaus factorisation method which works even if the factors are not relatively prime modulo p. Finally, some examples are given at the end of the paper.
Let f be a monic separable polynomial over the rational integers @? and p be a rational prime. We set up an algorithm which yields the extended p-adic values of a given element a in the semi-simple algebra A"f=@?[X]/f. Firstly, we determine the decomposition ...
Bibliogr. na konci kapitol
New probabilistic algorithms are presented for factoring univariate polynomials over finite fields. The algorithms factor a polynomial of degree n over a finite field of constant cardinality in time O(n 1:815 ). Previous algorithms required time Theta(n 2+o(1) ). The new algorithms rely on fast matrix multiplication techniques. More generally, to factor a polynomial of degree n over the finite field F q with q elements, the algorithms use O(n 1:815 log q) arithmetic operations in F q . The new "baby step/giant step" techniques used in our algorithms also yield new fast practical algorithms at superquadratic asymptotic running time, and subquadratic-time methods for manipulating normal bases of finite fields. 1 INTRODUCTION In this paper, we present a new probabilistic approach for factoring univariate polynomials over finite fields. The resulting algorithms factor a polynomial of degree n over a finite field F q whose cardinality q is constant in time O(n 1:815 ). The best ...
Factorization of polynomials modulo small prime powers
- J Zur Gathen
- S Hartlieb
J. von zur Gathen and S. Hartlieb, Factorization of polynomials modulo small
prime powers. Technical report, Universit at-GH Paderborn, 1996a. To appear.
Solving the congruence x 2 a mod n
- M O Vahle
M. O. Vahle, Solving the congruence x 2 a mod n. MapleTech 9 (1993), 69{76.
- P M Cohn
P.M. Cohn, Algebra, vol. 2. John Wiley & Sons, 1977.
Factoring polynomials over large nite elds
- E Bach
E. Bach, Number-theoretic algorithms. Ann. Rev. Comput. Sci. 4 (1990), 119{172.
E. R. Berlekamp, Factoring polynomials over large nite elds. Math. Comp. 24
(1970), 713{735.
E cient factorization of polynomials over local elds
- A L Chistov
A. L. Chistov, E cient factorization of polynomials over local elds. Soviet Math.
- M Cohn
M. Cohn, Algebra, vol. 2. John Wiley & Sons, 1977.